diff --git "a/YIG/all_papers.json" "b/YIG/all_papers.json" new file mode 100644--- /dev/null +++ "b/YIG/all_papers.json" @@ -0,0 +1 @@ +[ { "title": "2105.11057v1.Phase_resolved_electrical_detection_of_coherently_coupled_magnonic_devices.pdf", "content": "Phase-resolved electrical detection of coherently coupled magnonic devices\nYi Li\u0003,1Chenbo Zhao\u0003,1Vivek P. Amin,2, 3Zhizhi Zhang,1Michael Vogel,1, 4Yuzan Xiong,5, 1Joseph Sklenar,6\nRalu Divan,7John Pearson,1Mark D. Stiles,3Wei Zhang,5, 1Axel Ho\u000bmann,8and Valentyn Novosad1,\u0003\n1Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USAy\n2Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA\n3Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA\n4Institute of Physics and Center for Interdisciplinary Nanostructure Science and Technology (CINSaT),\nUniversity of Kassel, Heinrich-Plett-Strasse 40, Kassel 34132, Germany\n5Department of Physics, Oakland University, Rochester, MI 48309, USA\n6Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202, USA\n7Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439, USA\n8Department of Materials Science and Engineering,\nUniversity of Illinois at Urbana-Champaign Urbana, IL 61801, USA\n(Dated: May 25, 2021)\nWe demonstrate the electrical detection of magnon-magnon hybrid dynamics in yttrium iron\ngarnet/permalloy (YIG/Py) thin \flm bilayer devices. Direct microwave current injection through\nthe conductive Py layer excites the hybrid dynamics consisting of the uniform mode of Py and the\n\frst standing spin wave ( n= 1) mode of YIG, which are coupled via interfacial exchange. Both the\ntwo hybrid modes, with Py or YIG dominated excitations, can be detected via the spin recti\fcation\nsignals from the conductive Py layer, providing phase resolution of the coupled dynamics. The phase\ncharacterization is also applied to a nonlocally excited Py device, revealing the additional phase\nshift due to the perpendicular Oersted \feld. Our results provide a device platform for exploring\nhybrid magnonic dynamics and probing their phases, which are crucial for implementing coherent\ninformation processing with magnon excitations.\nHybrid magnonic systems have recently attracted wide\nattention due to their rich physics and application in\ncoherent information processing [1{15]. The introduc-\ntion of magnons has greatly enhanced the tunability\nin hybrid dynamics, the capability of coupling to di-\nverse excitations for coherent transduction [16{20], as\nwell as the potential for on-chip integration [12{14]. Re-\ncently, thin-\flm-based magnon-magnon hybrid systems\nhave provided a unique platform for implementing hy-\nbrid magnonic systems [21{30]. Coupling between mate-\nrials in the hybrid structure can arise through the inter-\nfacial exchange interaction. Because magnon excitations\nare con\fned within the magnetic media, it is convenient\nto build up more compact micron-scale hybrid platforms\ncompared with millimeter-scale microwave circuits. Fur-\nthermore, abundant spintronic phenomena, such as spin-\ntorque manipulation and spin pumping, can be used to\ncontrol and engineer the hybrid dynamics especially for\nmagnetic thin-\flm devices.\nOne important aspect of hybrid magnonic systems\nis controlling and engineering the phase relation be-\ntween di\u000berent dynamic components, leading to phenom-\nena such as exceptional points [31, 32], level attraction\n[33, 34] and nonreciprocity [35, 36] in cavity spintron-\nics. Phase resolved detection of individual magnetiza-\ntion dynamics has been extensively explored electrically,\noptically, and with advanced light sources. In particu-\nlar, electrical measurements of magnetic thin-\flm devices\nvia spin recti\fcation e\u000bects [37{42] can directly trans-\nform microwave magnetic excitations into sizable dc volt-\nages. This technique has been used to sensitively mea-sure nanoscale magnetic devices and, more importantly,\nthe phase of magnetization dynamics in order to quan-\ntify the spin torque generated from charge current \row\n[43{54].\nIn this work, we establish the usefulness of electrical ex-\ncitation and detection for the study of coherently coupled\nmagnon-magnon hybrid modes in Y 3Fe5O12/Ni80Fe20\n(YIG/Py) thin-\flm bilayer devices. This approach dif-\nfers from the previous work on inductive microwave mea-\nsurements [21, 23, 24] in its applicability to nanoscale\ndevices and its phase sensitivity. The coupled YIG/Py\nmagnetization dynamics are excited by directly apply-\ning microwave current through the conductive Py layer.\nOnly the Py layer contributes to the spin recti\fcation sig-\nnal because the YIG is insulating, enabling clear phase-\nresolved detection of the Py component of the YIG-Py\nhybrid modes. We measure a constant phase for the Py-\ndominated hybrid modes and a \u0019phase o\u000bset across the\navoided crossing for the YIG-dominated modes. From\nthe slope of the \u0019phase shift we can determine the inter-\nlayer coupling strength, agreeing with the measurement\nfrom the avoided crossing. We have also characterized a\nnonlocally excited YIG/Py sample, in which a phase o\u000b-\nset compared with the single bilayer device suggests the\nexistence of a large perpendicular Oersted \feld driving\nthe dynamics. Our results open an avenue of building up,\nreading out and designing circuits of on-chip magnonic\nhybrid devices for the application of coherent magnonic\ninformation processing.\nYIG thin \flms (50 nm, 70 nm and 85 nm) were sput-\ntered on Gd 3Ga5O12(111) substrate with lithographi-arXiv:2105.11057v1 [cond-mat.mes-hall] 24 May 20212\n(a)\nYIGPyAu electrode\nAu electrodeBias-T\nVIrf\nVdc \n(b) Irf\n(d)\n(c)45 o\nHB\n50 μV(f) (e) \nPy (n=0) \nYIG (n=0) \nYIG (n=1) \n8 GHz 11 GHz \nYIG\n(n=1)Py \n(n=0)\n+anti- \ncrossing \nFIG. 1. (a) Illustration of electrical excitation and detection of YIG/Py bilayer devices. The in-plane external biasing \feld is\nkept as 45\u000efrom theIrfdirection along the Py devices. (b-c) Optical microscope images of the device and Au coplanar waveguide\nantenna for (b) single devices and (c) nonlocally excited device. (d-f) Spin recti\fcation signals for the YIG(70 nm)/Py(9 nm)\nsingle device, with the mode anti-crossing between YIG ( n= 1) and Py ( n= 0) modes marked by the dashed box. (e) Zoom-in\nlineshape of (d) at 11 GHz where the extrapolated YIG ( n= 1) and Py ( n= 0) peaks are well separated. (f) Lineshape at 8\nGHz where the YIG ( n= 1) and Py ( n= 0) modes are degenerate in \feld. The red and green dashed curves in (f) denote the\n\ft of the two YIG-Py hybrid modes.\ncally de\fned device patterns, followed by lifto\u000b and an-\nnealing in air at 850\u000eC for 3 h [24, 55]. Then a second-\nlayer Py device (8 nm or 9 nm) was de\fned on the YIG\ndevice with lithography and sputtering, with 1 min ion\nmilling of YIG surface in vacuum right before deposition.\nLastly a 200 nm thick Au coplanar waveguide (CPW)\nwas fabricated which was in contact with the Py device\nfor electrical excitations and measurements. Fig. 1(a)\nshows the schematic of the spin recti\fcation measure-\nment. The top-view optical microscope images of the\ndevices are shown in Fig. 1(b) for the single devices and\n(c) for the nonlocally excited devices. The dimensions\nof the Py devices are 10 \u0016m\u000240\u0016m in Fig. 1(b) and\n6\u0016m\u000220\u0016m in Fig. 1(c). The two Py devices in Figs.\n1(c) are separated by 2 \u0016m, one for applying nonlocal\nexcitation signals and the other for the spin recti\fcation\nmeasurements. Throughout the measurements, the ex-\nternal biasing \feld is applied in the sample plane and\ntilted 45\u000eaway from the microwave current direction,\nwhich is the commonly used con\fguration in spin recti\f-\ncation measurements for maximizing the output signals\n[51].\nFig. 1(d) shows the \feld-swept spin recti\fcation sig-\nnals of the YIG(70 nm)/Py(9 nm) single device at dif-\nferent frequencies. We observe both the nominal Py and\nYIG uniform mode resonances, as reported previously\n[56, 57], even though the signals come from only the Py\nlayer. For the Py uniform mode, the excitation is mainly\ndue to a \fnite Oersted \feld projection to the dynamic\nmode, which has also been observed in a single CoFe\nlayer in our prior work [58]. For the YIG excitation,\nthe interfacial exchange coupling creates coupled modes\nwith \fnite amplitude on the Py, leading to a modulation\nof the Py resistance even for mode that is nominally the\nYIG uniform mode. In addition, the YIG ( n= 1) PSSW\nYIG(a) (b)\n(c) (d)\n85 nm70 nm50 nmFIG. 2. (a-c) Extracted resonance peak positions of (a)\nYIG(50 nm)/Py(8 nm), (b) YIG(70 nm)/Py(9 nm) and (c)\nYIG(85 nm)/Py(9 nm) single devices. The mode degeneracy\nbetween the Py uniform mode and YIG ( n= 1) mode happens\nat!c=2\u0019= 19:5 GHz for (a), 7.9 GHz for (b) and 4.7 GHz\nfor (c), denoted by vertical dashed lines. (d) Exchange \feld\nfor di\u000berent tYIG.\nmode is also excited when it intersects with the Py uni-\nform mode, forming an avoided crossing between the two\nmodes at!c=2\u0019= 7:9 GHz (Fig. 1f). The separation of\nthe two hybrid peaks is 8.5 mT, which is larger than the\nextrapolated individual linewidths of the Py ( n= 0) and\nYIG(n= 1) modes ( \u00160\u0001HPy= 5:5 mT,\u00160\u0001HYIG= 3:0\nmT). Far away from the avoided crossing, the excitations\nof the YIG ( n= 1) mode are almost unnoticeable (Fig.\n1e), which is due to the weak coupling of the uniform3\nOersted \feld to the odd PSSW modes. Thus the drive of\nthe YIG (n= 1) mode is dominated by excitation of the\nadmixture of the Py mode due to the interfacial exchange\n[21{24].\nTo analyze the spin recti\fcation signals, the measured\nlineshape for each peak can be \ftted to the following\nfunction:\nVdc= Im\u0014V0ei\u001e\u0001H\n(HB\u0000Hr)\u0000i\u0001H\u0015\n(1)\nwhereHBis the biasing \feld, Hris the resonance \feld\nas a function of frequency, \u0001 His the half-width-half-\nmaximum linewidth, V0is the peak amplitude, and \u001e\nrepresents the mixing of the symmetric and antisymmet-\nric Lorentzian lineshapes and re\rects the phase evolu-\ntion of the Py component in the YIG-Py hybrid dynam-\nics. The operator Im[] takes the imaginary part. This\ntechnique has been used to probe the dampinglike and\n\feldlike torque components [43{46, 48{53] as well as in\nrecent optical recti\fcation experiments [59{64]. More-\nover, the single source of spin recti\fcation signal from\nPy allows convenient theoretical analysis for studying the\nphase evolution of the hybrid dynamics, as will be shown\nbelow.\nFigs. 2(a-c) show the extracted Hras a function of\nfrequency!fortYIG= 50 nm, 70 nm and 85 nm, re-\nspectively. The two hybrid modes, marked as blue and\ngreen circles, are formed between Py uniform and YIG\n(n= 1) modes. With di\u000berent tYIG, the mode intersec-\ntion happens at di\u000berent frequencies due to the e\u000bective\nexchange \feld \u00160Hex= (2Aex=Ms)k2withk=\u0019=tYIG,\nwhich shifts the YIG ( n= 1) mode towards higher fre-\nquencies. Fig. 2(d) plots the extracted \u00160Hexas a func-\ntion of (\u0019=tYIG)2; good linear dependence con\frms the\nrole of the exchange \feld. By \ftting the data to the Kit-\ntel equation plus the exchange \feld, we obtain similar\nvalues of magnetization in all \flms as \u00160MPy= 0:81 T,\n\u00160MYIG= 0:19 T. From the linear \fts in Fig. 2(d) we\nobtainAex= 4:7 pJ/m for the YIG \flm.\nThe mode anticrossing behaviors in Figs. 2(a-c) can\nbe \ftted to the equation developed by the two coupled\nmagnon resonances [24]:\n\u00160H\u0006=\u00160HYIG+HPy\n2\u0006s\u0012\n\u00160HYIG\u0000HPy\n2\u00132\n+g2\nH\n(2)\nwhere\u00160HYIG=p\n\u00162\n0M2\nYIG=4 +!2=\r2\u0000\u00160MYIG=2 +\n\u00160Hexand\u00160HPy=q\n\u00162\n0M2\nPy=4 +!2=\r2\u0000\u00160MPy=2 are\nthe solutions of the Kittel equation for the YIG ( n= 1)\nand Py modes, \r=2\u0019= (geff=2)\u000227:99 GHz/T and\ngHis the interfacial exchange coupling strength in the\nmagnetic \feld domain. In our previous work [24], we de-\nrived thatgH=f(!)p\nJ=M PytPy\u0001J=M YIGtYIGwhereJ\nis the interfacial exchange coupling strength. The fac-\ntorf(!)\u00190:9 accounts for the nonlinearity due to thedemagnetizing \feld. Data \fttings to Eq. (2) yield an\naveragedgH= 8:7 mT andJ= 0:066 mJ/m2; the lat-\nter is consistent with the reported value of 0.06 mJ/m2\nfor continuous thin \flms [24]. We also double-check the\nvalue ofJby measuring the inductive ferromagnetic res-\nonance of a 200 \u0016m\u000240\u0016m YIG stripe from the same\nfabrication as the YIG(50 nm)/Py(8 nm) device, with\nthe peak dispersion shown as stars in Fig. 2(a). From\nthe Kittel \ftting, we obtain a constant resonance \feld\no\u000bset of\u00160Hk= 13:8 mT between the YIG stripe and\nthe YIG/Py device. From this static o\u000bset, we extract\nJ=\u00160HkMYIGtYIG= 0:110 mJ/m2[24], in good agree-\nment with the value of 0.112 mJ/m2obtained above from\nthe avoided crossing for tYIG= 50 nm. The YIG/Py de-\nvice shows a higher resonance \feld than YIG, con\frming\nthe antiferromagnetic exchange coupling between YIG\nand Py.\nNext, we show the evolution of \u001efor the hybrid modes,\nwhich are the main results of this work. Fig. 3(a)\nshows the extracted phases for the three modes in the\nYIG(70 nm)/Py(9 nm) single device, with the color cor-\nresponding to the resonance \feld plot in Fig. 2(b). The\ntwo hybrid modes exhibit a clear phase crossing where\ntheir resonance \felds intersect at !c=2\u0019= 7:9 GHz (ver-\ntical dashed line). For the Py-dominated hybrid modes\nwhich are represented by the blue circles lower than !c\nand the green circles higher than !c, the phase stays at a\nconstant level ( \u001ePy=\u00000:23\u0019). This is expected in spin\nrecti\fcation measurements, where a consistent phase re-\nlation between the Py dynamics and the microwave cur-\nrent is maintained in a broad frequency domain. For\nideal \feldlike excitations as illustrated in Fig. 3(c) in the\nsingle device, we expect \u001ePy=\u0000\u0019=2. Experimentally,\nthe deviation of \u001ePymay be due to the self spin torque\nproviding a \fnite dampinglike drive component [65]. Al-\nternatively, the phase o\u000bset may be also a re\rection of\nthe inhomogeneous mode pro\fle of Py in the presence of\nthe YIG/Py interfacial exchange boundary as well as the\nnonuniform current distribution across the thickness of\nPy.\nThe phase of the YIG-dominated hybrid modes, on the\nother hand, evolve from below \u001ePyto above\u001ePywith an\nincrement of nearly \u0019. As a rough explanation, by pass-\ning through the avoided crossing, the frequency of the\nYIG-dominated mode evolves from below the Py reso-\nnance frequency to above it. This leads to a phase shift of\n\u0019for the Py susceptibility. Because the YIG dynamics is\ndriven by the interfacial exchange from the Py excitation,\na phase shift of \u0019is also expected in the YIG-dominated\nmode. Furthermore, due to the strong magnon-magnon\ncoupling, the \u0019phase shift does not take a sharp transi-\ntion at!c, but takes a gradual transition with the transi-\ntion bandwidth determined by the coupling strength gH.\nTo quantitatively understand the phase evolution of\nthe hybrid mode, we follow the susceptibility tensor\nwhich has been derived in our prior work see Eq. (S-4\nhrf z Irf \nPy \nYIG Py \nYIG hrf \nIrf VVrad \nrad (a) (b) (e) gH=1.0 mT \ngH=4.0 mT \ngH=7.0 mT Theory \n(c) (d) rad \nFIG. 3. Phase evolution of the spin recti\fcation signals for (a) YIG(70 nm)/Py(9 nm) single device and (b)\nYIG(85 nm)/Py(9 nm) nonlocally excited device, with their microwave current \row and \feld distribution illustrated in (c)\nand (d), respectively. The blue and green curves show the theoretical prediction from Eq. (5) with (b) gH= 4:0 mT for (a)\nand 5.3 mT for (b). The error bars indicate single standard deviation uncertainties that arise primarily from the \ftting of the\nresonances. (e) Theoretical plots of phase evolution from Eq. (5) using the Hrin (a) and\u001ePy= 0 for di\u000berent gH.\n4) in the Supplemental Materials of Ref. [24]. In the\nlimit of weak damping and ignoring the precession ellip-\nticity, the dynamics of the Py uniform and YIG ( n= 1)\nmodes can be expressed as:\n~mPy=~hx\nPy\nHB\u0000HPy\u0000i\u0001HPy\u0000g2\nH\nHB\u0000HYIG\u0000i\u0001HYIG(3a)\n~mYIG=gH~mPy\nHB\u0000HYIG\u0000i\u0001HYIG(3b)\nwhere ~mPyand ~mYIGdenote the unitless transverse com-\nponents for Py and YIG, \u0001 HPyand \u0001HYIGdenote their\nlinewidths. For the Py layer, the e\u000bective \feld ~hx\nPyis ex-\nerted from the microwave current \rowing through. For\nthe YIG layer, the e\u000bective \feld gH~mPyis provided by\nthe interfacial exchange when the Py magnetization pre-\ncesses. Note that because YIG is an insulator, the spin\nrecti\fcation signal is only contributed by ~ mPy, which sig-\nni\fcantly simpli\fes the theoretical analysis. Eq. (3a) can\nbe rewritten as:\n~mPy=~hx\nPy(HB\u0000HYIG\u0000i\u0001HYIG)\n(HB\u0000H+\u0000i\u0001H+)(HB\u0000H\u0000\u0000i\u0001H\u0000)(4)\nwhere the values of H\u0006are de\fned in Eq. (2) and \u0001 H\u0006\nare the linewidths for the two hybrid modes. Compared\nwith Eq. (1), the phase for the H\u0006resonance can be\n\fnally expressed as:\n\u001e\u0006=\u001ePy+ tan\u00001\u0012\u0000\u0001HYIG\nH\u0006\u0000HYIG\u0013\n\u0000tan\u00001\u0012\u0000\u0001H\u0007\nH\u0006\u0000H\u0007\u0013\n(5)\nIn Eq. (5) the \frst term comes from a \fnite phase o\u000bset\nbetween ~hx\nPyand the microwave current, the second termcomes from the numerator and provides the \u0019phase shift,\nand the last term is usually close to zero in the strong\ncoupling regime as the linewidth is much smaller than\nthe resonance detuning. The calculation results of Eq.\n(5) are plotted in Fig. 3(a), which nicely reproduce the\nexperimental data and the positive increment of phase\nfor the YIG-dominated hybrid mode. We also plot the\ncalculated phase evolution for di\u000berent values of gHin\nFig. 3(e). For small gH, the YIG-dominated mode shows\na rapid phase shift near the mode crossing frequency.\nAsgHincreases, the phase transition regime broadens\nbecausegHde\fnes how quickly the hybrid mode evolves\nto uncoupled individual modes.\nThe phase-resolved spin recti\fcation measurement of\nthe hybrid modes are also repeated on a nonlocally ex-\ncited device. With the excitation and detection schemat-\nics shown in Fig. 3(d), the microwave current \rows\nthrough a nonlocal Py electrode, which provides an Oer-\nsted \feld that is perpendicular to the Py device be-\ning measured. For the detection, due to the induc-\ntive coupling between the two adjacent Py devices, a\n\fnite microwave current \rows through the second Py\ndevice which leads to a measurable spin recti\fcation\nvoltage when the Py magnetization dynamics is excited.\nFig. 3(b) shows the measured \u001efor the three modes.\nAbove!c=2\u0019= 4:7 GHz, the YIG-dominated mode ex-\nhibits a phase advance close to \u0019=2 compared with the\nPy-dominated mode, which agrees with the theoretical\nprediction. For the Py-dominated mode, the extracted\nvalue of\u001ePy=\u00000:99\u0019also agrees with theoretical pre-\ndiction of\u001ePy=\u0000\u0019due to the additional \u0000\u0019=2 phase\no\u000bset from the perpendicular Oersted \feld from the non-\nlocal antenna. Below 4.7 GHz, the anomalous phase drift5\nis accompanied with the linewidth drift and is due to\nthe weak signals. Thus we consider this low-frequency\nphase drift as to be an artifact due to weak signals rather\nthan a signi\fcant e\u000bect. Note that the nonlocal excita-\ntion schematic should eliminate the spurious phase o\u000bset\ndue to the complex excitation pro\fle, because the out-\nof-plane Oersted \feld is rather uniform.\nThe YIG uniform modes exhibit a consistent phase of\n\u001ePy=\u0019=2 in both Figs. 3(a) and (b). Note that we\nstill use\u001ePyto represent the phase because the spin rec-\nti\fcation signals come from the motion of the Py layer\ninduced by the resonance of YIG via the interfacial ex-\nchange [51, 66]. The value of \u001ePysuggests a dominat-\ning in-plane Oersted \feld on the YIG layer from the mi-\ncrowave current \rowing through the adjacent Py layer.\nFor the YIG(70 nm)/Py(9 nm) single device, the only\nPy layer acts as an antenna which is highly e\u000ecient\nin exciting the YIG uniform mode [Fig. 3(c)]. For the\nYIG(85 nm)/Py(9 nm) nonlocally excited device, the un-\nchanged\u001ePy=\u0019=2 shows that the perpendicular \feld\nfrom the nonlocal Py antenna is still insigni\fcant com-\npared with the induced microwave current in the Py de-\nvice being electrically measured, with the latter much\nmore e\u000ecient in producing an in-plane Oersted \feld on\nthe YIG layer underneath. Note that the sign change\nof\u001ePyfrom the Py-dominated uniform mode is caused\nby the negative value of gHfrom antiferromagnetic cou-\npling, adding an additional \u0019phase to the YIG uniform\nmode. A similar observation has also been reported in\nRef. [57].\nIn conclusion, we have demonstrated phase-resolved\nelectrical measurements of YIG/Py bilayer devices with\nstrong magnon-magnon coupling. The micron-wide and\nnanometer-thick devices serve as an on-chip miniatur-\nized two-cavity hybrid system, where the two microwave\ncavities are composed of two exchange-coupled thin lay-\ners of magnon resonators. Furthermore, the unique cou-\npling mechanism and the con\fned magnon resonance al-\nlow versatile geometric con\fguration, such as the non-\nlocal device, as well as convenient electrical excitation\nand detection. In the recent rapid development of cav-\nity spintronics and magnon hybrid systems [67{71], lots\nof emerging physics and device engineering including ex-\nceptional points [31, 32], level attraction [11, 33, 34] and\nnonreciprocity [35, 36] have utilized coherent interaction\nof di\u000berent microwave ingredients. Our results provide a\nplatform for implementing and realizing these \fndings in\ngeometrically con\fned, thin-\flm based dynamic systems\nand for studying the driving and coupling interactions,\nwhich are critical for applications in coherent information\nprocessing.\nWork at Argonne on sample preparation and char-\nacterization was supported by the U.S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences, Mate-\nrials Sciences and Engineering Division, while work at\nArgonne and National Institute of Standards and Tech-nology (NIST) on data analysis and theoretical modeling\nwas supported as part of Quantum Materials for Energy\nE\u000ecient Neuromorphic Computing, an Energy Frontier\nResearch Center funded by the U.S. DOE, O\u000ece of Sci-\nence, Basic Energy Sciences (BES) under Award #DE-\nSC0019273. Use of the Center for Nanoscale Materials,\nan O\u000ece of Science user facility, was supported by the\nU.S. Department of Energy, O\u000ece of Science, O\u000ece of\nBasic Energy Sciences, under Contract No. DE-AC02-\n06CH11357. W. Z. acknowledges support from AFOSR\nunder grant no. FA9550-19-1-0254.\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\n\u0003novosad@anl.gov\nyYi Li and Chenbo Zhao contributed equally to this paper\n[1] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 111, 127003 (2013).\n[2] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603\n(2014).\n[3] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys.\nRev. Lett. 113, 156401 (2014).\n[4] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan,\nM. Kostylev, and M. E. Tobar, Phys. Rev. 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Hehn2 and J.-E. \nWegrowe*1 \n \n1Ecole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palaiseau , France . \n2Institut Jean Lamour UMR 7198 CNRS, Université de Lorraine, Vandoeuvre les Nancy France . \n3Unité Mixte de Physique CNRS/Thales and Universit é Paris Sud, Palaiseau , France . \n*Correspondence to :jean-eric.wegrowe@polytechnique. edu. \n \nAbstract: \nThe Righi -Leduc effect refers to the thermal analogue of the Hall effect, for which the electric \ncurrent is replaced by the heat current and the electric field by the temperature gradient. In both \ncases, the magnetic field generates a transverse force that deviates the carriers (electron, phonon, \nmagnon) in the direction perpendicular to the current. In a ferromagnet, the magnetization plays \nthe role of the magn etic field, and the corresponding effect is called anomalous Hall effect . \nFurthermore, a second transverse contribution due to the anisotropy, the planar Hall effect, is \nsuperimposed to the anomalous Hall effect. We report experimental evidence of the ther mal \ncounterpa rt of the Hall effect s in ferromagnet s, namely the magnon Hall effect (or equivalently \nthe anomalous Righi -Leduc effect) and the planar Righi -Leduc effect , measured on ferromagnets \nthat are either electrical conductor (NiFe) or insulator (YIG) . The study shows the universal \ncharacter of these new thermokinetic effects, related to the intrinsic chirality of the anisotropic \nferromagnetic degrees of freedom. \n \n \n 2 \nIntro duction \nWe report a new effect that could be added to the large family of the rmokinetic transport \nphenomena . It consists in the observation of both anomalous Righi -Leduc effect - or magnon \nHall effect [1,2]- and planar Righi -Leduc effect , measured on YIG and NiFe ferromagnets . The \nconventional Righi -Leduc effect is the thermal counterpart of the well -known Hall effect , and it \naccounts for the temperature gradient developed transversally to a heat current under a magnetic \nfield. The adjectives anomalous and planar – that characterize the effect reported here - refer to \nthe action of the magnetization axial vector (instead of a magnetic field) and the corresponding \nvector potential . \nThe application of a magnetic axial vector result s in the partial break ing of two different \nsymmetries . These symmetries are , on the one hand , the invariance under time reversal of the \ndynamical equations at the microscopic scale [3], and on the other hand, the rotational \ninvariance (for an initially isotropic system ). However, the symmetry breaking is partial . Indeed, \nin the first case, the time reversal invariance is recovered by the application of a rotation to the \nmagnetization , and i n the second case, the symmetry breaking is partial because the system is \nstill invariant under any rotation around the magnetization . The consequence of the se reduced \nsymmetries is to impose a specific form to the heat transport coefficients [4] (see \nSupplementary), so that the temperature gradient becomes a very specific function of the \nmagnetization states (as shown in Eq. (1) below) . \nSince t he addition of a thin electrode in thermal contact with both edges of the \nferromagnetic layer (see the set -up of Fig1) plays the role of a Seebeck thermometer (or \nthermocouple), the temperature difference T is converted to a voltage difference V, allowing \n 3 the measurement of a magneto -voltaic signal. Such a device defines the principle of a magneto -\nthermal sensor. The studies of magneto -voltaic signals measured in response to thermal \nexcitations on ferromagnetic layers has attracted considerable a ttention in the last years, with the \nobservation of similar signals in conductor (NiFe), semiconductor (GaMnAs), and insulator \n(YIG) [5-13]. \n \nFig.1: Schematic views of typical device s including a ferromagnet (either a conductor or an \ninsulator ) and two transversal non-ferromagnet ic electrode s (noted heater and probe) . In \nour study, the electrodes have a width of 200µm and are spaced 5mm apart. The direction \nof the magnetization is defin ed by the angles θ and φ. \n \nThe anomalous Righi -Leduc effect has been predicted in various magnetic systems [14-\n17] and it has been measured recently in peculiar insulating ferromagnetic materials that possess \na chiral crystalline structure [1,2]. The study of the anomalous Righi -Leduc effect in usual \nferromagnetic layers ( e.g. NiFe and YIG) has however been overlooked. On the other hand, the \nplanar Righi -Leduc effect refers to the contribution of the anisotropy in the thermal conductivity. \nThis anisotropy originates from the difference r between the t hermal resistivity measured along \n 4 the magnetization axis and the thermal resistivity perpendicular to the magnetization axis [18]. \nThe comparative study between anomalous and planar Righi -Leduc effects, in NiFe and YIG \nferromagnets , allows us to make a call in favor of a unifying interpretation in terms of \nanisotropic thermal transport (AThT), and to point out the universality of the phenomenon. \n \nExperimental angular dependences \nThe sample s contained electrodes that have been fabricated using the same se t of shadow \nmask s in a sputtering deposition system . They are fixed on top of two different magnetic \nmaterial s. The first sample include s a 20nm thick Ni80Fe20 conductor stripe while the second \nsample contains a 20nm thick ferromagnetic YIG insulator [19]. Electrode s composed of \nPlatinum (Pt) are deposited on top of each magnetic layer (see fig . 1). An ac electric current I(t) \n= I0 cos(t) is injected into the heater electrode (the power is of the order of a fraction of Watt \nand the frequency is a fraction of Hz ). It produces a heat current \n2/)1)2 (cos( )(2\n0 t cRItJQ \nhaving twice the frequency of the electric current (c is a constant th at takes into account the \npower dissipation (see Supplementary ). The voltaic response ΔV y to the thermal excitation is \nmeasured using a lock-in method via a probe electrode placed 5mm away from the heater \nelectrode . All the measurements are done under a magnetic field H of 1 Tesla that serves to \nrotate the magnetization . The voltage \n),(H H yV has been recorded for the two \naforementioned devices either by varying the azimuthal angle φH while keeping the polar angle \nθH fixed to 90° or by varying the polar angle keeping φH fixed to 90° (see fig. 2) . \nFirst we observe that in both cases (conductor and insulator ferro magnetic material ), π-\nperiodic signals are measured in the magnetization in-plane (IP) configuration , while 2π -periodic \nsignals are observed in the out-of-plane (OOP) configuration . Second w e find that the angular \n 5 voltage variations display opposite phase on NiFe/Pt and on YIG/Pt . Finally , a triangular rather \nthan a sinusoidal feature is observed in the NiFe sample for the measurements under an out -of-\nplane field . \n \nFig.2 Transverse voltages vs. direction of the 1T magnetic field H. The results collected \nin two different measu rement geometries are presented: In-plane configuration for which \nthe polar angle θH is fixed and equal to 90° , the azimuthal angle φH is varied ( ); Out-of-\nplane configuration for which the azimuthal angle φH is fixed to 90° and the polar angle is \nvaried ( ). The data corresponding to the ferromagnetic conductor case (Ni 80Fe20) and the \nferromagnetic insulator case (YIG) are represented in the top graphs (purple color) and in \nthe bottom graphs ( red colors) respectively. \n \nAnisotropic Thermal Transport (AThT) \nAccording to the AThT phenomenology (see Supplementary ), a heat current \nQJ injected \ninside the ferromagnet generates a thermal gradient \n),(T related to the orientation of the \nmagnetization (θ,φ). Its description , based on t he anisotropic Fourier equation, is valid both for \n 6 the electric conductor and for the insulator . The probe electrode serves as a thermocouple that \nconvert s a local transverse temperature difference into a voltage \nrJSQ\nx. . The parameter \nstands for the difference between the Seebeck coefficients of the materials that compose the \ndevice . Considering that the heat current is along the x direction, the transverse voltage \nyV is \ngiven by the expression [4]: \n\n\n cos )2sin().(sin2.2\nRLQ\nx y rrJS V\n Eq.(1), \nwhere is the planar Righi -Leduc coefficient and is the an omalous Righi -Leduc coefficient. \nFrom Eq.(1), the period s observed in Fig.2 can be easily understood . The term allows \nto explain the π -periodic ity in the IP configuration while the 2π -periodic signals are linked the \n term that occur only in the OP configuration . Moreover we can also predict from Eq.( 1), \nthat the magnitude of the oscillations are equal to\nrJSQ\nx. for the IP configurations and equal to \nARLQ\nxrJS.\n for the OOP configuration . Using an independent measurement setup, we have \ndetermined S for the NiFe and YIG based devices to be respectively -16.2µV.K-1 and \n0.69µV.K-1 (see Supplementary ). The opposite signs of S provide a straightforward explanation \nfor the aforementioned \"antiphase\" feature observed in Fig. 2 comparing the results on NiFe/Pt \nand YIG/Pt . Finally taking into account the magnetic propert ies of the ferromagnetic layers (see \nSupplementary ), we were able to fit al l measurements the only free parameters were either \nrJSQ\nx.\n (in the IP configuration ) or \nARLQ\nxrJS. (in the OP configuration ). It can be seen on \nFig.2 (grey lines) that all the experimental results are in excellent agreements with our \ninterpr etation based on anisotropic thermal transport in the ferromagnet . The triangular profile \n(rather than sinusoidal ) exhibited by the Ni80Fe20 device in the OP configuration is simply due to \n 7 the fact that a 1 Tesla magnetic field i s not large enough to fully saturate the magnetization \nperpendicular to the plane of the film (see Supplementary ). \nTo test the robustness of the AThT explanation , we have first varied the thickness of the \nPt probe from 5nm to 100nm. Defining the maximum amplitude of the magneto -voltaic signal \n 0 180y y y V V V\n. A decrease of \ndVy as a function of the probe thickness is \nobserved ( Fig.3a). Such a decrease is often interpreted as the effect of spin injection at the \ninterface [20-22]. Here we demonstrate that the thermal shunt effect suffice s to explain the data . \nNot all the injected heat current \nQ\nxJ* contributes to the AThT effect since a part of this current is \nalso flowing into the neutral Pt electrode. In order to evaluate the active part of the heat current, \nwe can rewrite : \nQ\nx NiFe\nTh NiFePt\nThNiFePt\nTh Q\nx Jd ddJ*\n. ..\n \n\n Eq.(2), \nassuming a simple scheme of two thermal conductors in parallel , as for anisotropic Hall \nmeasurements [23] (see Supplementary ). The dependence of the signal on the thickness d of the \nPt probe is calculate d using the tabulated value s \nPy\nTh/1 =72W.m-1.K-1 for Ni 80Fe20 and \nPt\nTh/1 = \n46W.m-1.K-1 for Pt, and the value of \nARLQ\nxrJS ..* presented in Fig. 2 (see Supplementary ). From \nthe good agreement between the experimental curve and the prediction of Eq. (2) in Fig.3a) , we \nconclude that the sole thermal shunt effect suffices to reprodu ce the observed decrease (the \nelectrical counter part of the shunt effect is also reproduced without adjustable parameter , as \nshown in of Supplementa ry Fig. S9. \n \n \n 8 \n \nFig 3 . a) Voltages difference \nyV vs. thickness of the Pt electrode d (). The gray line \npresents the expected dependence taking into account only a thermal shunt effect. b) Transverse \nvoltage \nyV vs. θH ( ) for a device composed of a C u electrode (No PE effect, no ISHE). \n \nMoreover, the AThT interpretation does not require to invoke the hypothesis on inverse \nspin Hall effect (ISHE) or of a proximity effect arising from induced magnetic moments [11, 13]. \nIn order to verify this stat ement , we have replaced the Pt electrodes by ultra -pure copper ones \n(99.9999% purity target). Indeed, the use of Cu electrodes allows to test at the same time the \nISHE and PE hypotheses since both effects are absent in Cu [24,25 ,11]. We observed the \nmagneto -voltaic signal even wi th pure copper electrode (Fig .3 b), as expected for AthT . \n \nDiscussion and Conclusion . \nWe have observed the coexistence of both anomalous and planar Righi -Leduc \ncontributions in NiFe and YIG, of comparable amplitude s (leading to a transverse temperature \ndifference of the order of 10 mK ). \n 9 Although t he anomalous Righi -Leduc effect can simply be understood on the basis of the \nOnsager reciprocity relations [3] (Supplementary Equations S2), a microscopic description can \nalso been performed with a dedicated vector potential – or the corresponding local gauge and \nBerry phase – associated to the ferromagnetic system under consideration [26-28,14 -17]. This \nproblem generalizes sixty years of intensive theoretical development related to the anomalous \nHall effect (starting with the work of Karplus and L uttinger in 1954 [29], and summarized e.g. in \nthe review by Nagaosa et al [30]). Like the Lorentz force in the case of the conventional Hall \neffect, and like the spin -orbit scattering force in the case of anomalous Hall effect, the transversal \nforce measured in this study can be derived from a vector potential . This force is thus neither \nconservative (it cannot be derived from a scalar potential) nor dissipative (no power can be \nextracted). \nA second transverse force is observed , which is generated by the anisotropy of the \nferromagnetic excitations r ≠0. T he measurements show that the two forces are not \nindependent: the anomalous Righi -Leduc coefficient is associated to the planar Righi -Leduc \ncoefficient . The same ferromagnetic axial vector is indeed responsible for both the anisotropy of \nthe heat resistance (r ≠ 0) and the breaking of the time invariance symmetry . \n \nIn conclusion, our results show that the anomalous Righi -Leduc effect, which has already \nbeen observed in specific ferroma gnetic structures, is universal . This effect is observed in \nparallel to the planar Righi -Leduc effect. Both planar and anomalous Righi -Leduc effects should \nbe present in any ferromagnetic materials in the same manner as anomalous and planar Hall \neffects can be expected a priori in any ferromagnetic conductors. \n \n 10 References and Notes: \n[1] Onose, Y. et al. Observa tion of the M agnon Hall effect. Science 329, 297-299 (2010). \n[2] Ideue , T. et al. Effect of lattice geometry on magnon Hall effect in ferromagnetic \ninsulators. Phys. Rev. B 85, 134411 (2012). \n[3] Onsager , L. RECIPROCAL RELATION IN IRREVERSIBLE PROCESS II. Phys. Rev. \n38, 2265 -2279 (1931). \n[4] Wegrowe, J.-E., Lacour, D., & Drouhin , H.-J. Anisotropic magnetothermal transport and \nspin Seebeck effect . Phys. Rev. B 89, 094409 (2014). \n[5] Uchida , K. et al. Observation of the spin Seebeck effect . Nature 455, 778-781 (2008). \n[6] Uchida , K. et al . Spin Seebeck insulator. Nature Mater. 9, 894 -897 (2010) . \n[7] Jaworsky , C. M. et al. Observation of the spin -Seebeck effect in a ferromagnetic \nsemiconductor . Nature. Mater. 9, 898-903 (2010). \n[8] Huang, S. Y. , Wang, W. G. , Lee, S. F., Kwo, J. & Chien, C. L. Intrinsic Spin-Dependen t \nThermal Transport. Phys. Rev. Lett. 107, 216604 (2011). \n[9] Avery, A. D. , Pufall, M. R. & Zink , B. L. Observation of the Planar Nernst effect in \npermalloy and Nickel thin films with in -plane thermal gradient. Phys. Rev. Lett. 109, 196602 \n(2012). \n[10] Schultheiss, H., Pearson , J. E., Bader, S. D. , & Hoffmann, A. Thermoelectric Detection \nof Spin Waves. Phys. Rev. Lett. 109, 237204 (2012). \n[11] Huang , S. Y. et al. Transport M agnetic Proximity Effects in P latinium. Phys. Rev. \nLett. 109, 107204 (2012). \n[12] Schmid , M. et al. Transverse Spin Seebeck Effect versus Anomalous and Planar Nernst \nEffects in Permalloy Thin F ilms. Phys. Rev. Lett. 111, 187201 (2013). \n 11 [13] Kikkawa , T. et al. Longitudinal Spin Seebeck Effect Free from the Proximity Nernst \nEffect . Phys. Rev. Lett. 110, 067207 (2013). \n[14] Fujimoto , S. Hall Effect of Spin Waves in Frustrated M agnets. Phys. Rev. Lett. 103, \n047203 (2009). \n[15] Katsura, H., Nagaosa, N., & Lee, P. A. Theory of the M agnon Hall E ffect in Quantum \nMagnets. Phys. Rev. Lett. 104, 066403 (2010). \n[16] Matsumoto, R. & Murakami , S. Theor etical Prediction of a Rotatin Magnon Wave \nPacket in F erromagn ets. Phys. Rev. Lett. 106, 197202 (2011). \n[17] Qin, T., Niu, Q. & Shi, J. Energy Magnetization and the Thermal Hall effect. Phys. Rev. \nLett. 107, 236601 (2011). \n[18] Kimling, J., Gooth, J., & Nielsch, K. Anisotropic Magnetothermal Resistance in Ni \nNanowires. Phys. Rev. B 87, 094409 (2013). \n [19] O. d’Allivy Kelly et al. Inverse spin Hall effect in nanometer -thick yttrium iron \ngarnet/Pt system. Appl. Phys. Lett. 103, 082408 (2013). \n [20] Nakayama, H. et al. Geometry dependence on inverse spin Hall effect induced by spin \npumping in Ni 81Fe 19/Pt films. Phys. Rev. B 85, 144408 (2012). \n[21] Althammer , M. et al. Quantitative study of the spin Hall magnetoresistance in \nferromagnetic insulator/normal metal hybrids . Phys. Rev. B 87, 224401 (2013). \n[22] Castel, V., Vlietstra, N., Ben Youssef , J. & van Wees , B. J. Platinum thickness \ndependence of the inverse spin -Hall voltage from spin pumping in a hybrid yttrium iron \ngarnet/platinum system. Appl. Phys. Lett. 101, 132 414 (2012). \n[23] Xu, W. J. et al. Scaling law of anomalous Hall effect in Fe/Cu bilayers. Eur. Jour. Phys. \n 12 B 65, 233-237 (2008). \n[24] Niimi, Y. et al. Extrinsic Spin Hall Effect Induced by Iridium Impurities in Copper. \nPhys. Rev. Lett. 106, 126601 (2011). \n[25] Harii , K., Ando , K., Inoue , H. Y., Sasage , K. & Saito h, E. Inverse spin -Hall effect and \nspin pumping in metallic films. J. App l. Phys. 103, 07F311 (2008). \n[26] Holstein , B. R. The adiabatic theorem and Berry’s phase . Am. J. Phys. 57, 1079 -1084 \n(1989 ). \n[27] Bruno , P. Nonquantized Dirac monopoles and strings in the Berry ph ase of anisotropic \nspin systems. Phys. Rev. Lett. 93, 247202 (2004 ). \n[28] Matsumoto, R., Shindou, R. & Murakami , S., Thermal Hall effect of magnon in \nmagnets with dipolar interaction. Phys. Rev. B 89, 054420 (2014 ). \n[29] R. Karplus and J. M. Luttinger, Hall Effect in Ferromagnetics . Phys. Rev. 95, 1154 -1160 \n(1954 ). \n[30] Nagaosa, N., Sinova, J., S. Onoda, MacDonald, A. H. & Ong, N. P. Anomalous Hall \nEffect. Rev. Mod. Phys. 82, 1539 -1592 (2010). \n \nAcknowledgements \nThe Authors acknowledge H. Molpeceres and A. Jacquet for their assistance and O. d’Allivy \nKelly for fruitful discussion. Financial funding RTRA `Triangle de la physique’ Projects DEFIT \nn° 2009 -075T and DECELER 2011 -085T , the FEDER, France, La région Lorraine, Le grand \nNancy, ICEEL and the ANR -12-ASTR -0023 Trinidad is greatly acknowleged . \n \n 13 Suppl ementary M aterials \n \nI Magnetic and electric charact erization of the 20 nm thick permalloy (Ni 80Fe20) samples . \n \nI -1. Ferrom agnetic quasi -static states. \n \nDue to the thin layer structure, the magnetization of t he Permalloy (Ni80Fe20 or Py) layer is \nsingle domain. As a consequence, the magnetization \nmM Ms\n is a vector of constant modulus \nMs (magnetization at saturation) oriented along the unit vector\nm . The quasi -static magnetizati on \nstates are given by the minimum of the ferromagnetic free energy. This energy depends on three \nparameters, namely the magnetization at saturation M s, the demagnetizing field H d, and the \nmagnetocrystalline anisotropy field Han, confined in the plane of t he layer. The corresponding \nenergy is the s um of the three terms: \n 2 2cos21sin21.s d a s an MH MH MH F \n Eq.(S1) \nwhere\n),( mHan a\n is the angle between the magnetocrystalline anisotropy axis and the \nmagnetization, and is the angle between t he vector\nn normal to the plane of the layer and the \nmagnetization. \nThe minimum of the energy F (Eq.(S1)) sets the position of the magnetization, i.e. the radial \nangle and the azimuthal angle as a f unction of the amplitude H and direction H andH of \nthe applied field. The minimum is calculated through numerical methods (Mathematica® \nprogram). \nThe magnetization states were characterized using anisotropic electric transport properties, with \nthe use of three different experimental configurations , which correspond to anisot ropic \nmagnetoresistance (AMR) (31), planar Hall effect (PHE), and anomalous Hall effect (AHE )(30). \n \nI-2. Electric properties \n \nThe electric transport is described by the Ohm’s law that relates the electric field \n to the \nelectric current\neJ with the use of the conductivity tensor: \neJ.ˆ (note that for convenience, \nthe experiments are usual ly performed in a galvanostatic mode, i.e. with constant current \ndistribution \neJ). For a polycrystalline conducting ferromagnet, the conductivity tensor is \ndefined by three parameters . If the reference frame is such that the unit vector is aligned along \nOz, the parameters are the resistivity measured perpendicular to magnetization, the resistivity \nz measured parallel t o magnetization, and the Hall cross -coefficient H. According to Onsager \nreciprocity relation H xy = -yx and we have in the reference frame {x,y,z} : \n\n\n\n\n\nzHH\n\n\n0 000\nˆ\n \nAccordingly, t he Ohm’s law can be expressed in an arbitrary reference frame, as (31): \n 14 \ne\nHe\nzeJm mmJ J . . \nor, explicitly: \n \n \n \n\n\n\n \n\ne\nz ze\ny x H z ye\nx y H z xe\nz x H z ye\ny xe\nx z H y xe\nz y H z xe\ny z H y xe\nx x\nJm Jm mm Jm mmJm mm Jm Jm mmJm mm Jm mm Jm\n222\n,\n \n\n Eq. (S2) \nwhere =z \ncos sinxm , \nsin sinym , \ncoszm . The angle θ is the same radial \nangle as the one introdu ced in the magnetic free energy, is the azimuthal angle between the \ndirection Ox and the projection of the magnetization in the film plane . After integration, Eq. ( S2) \ngives the magneto -voltaic signals that corresponds to the Anisotropic magnetoresistance \n(diagonal terms) , the anomalous magn etoresis tance (second term of the non -diagonal matrix \nelements) , and the planar magnetoresi stance (first term of the non -diagonal matrix elements) . \nThe same line of reasoning is applied in section II-1 below for the transport of heat . \n \nI-2-1. Anisotropic magnetoresistance (AMR) \n \nFor AMR measurements , the voltage is measured along the same axis as the current flow (see \nFig.1). The voltage is given by the integration over x of the first line in Eq. (2) with \n0e\nye\nzJ J . \n \n \nFig.S1: Resistance as a function of the amplitude of the external perpendicular field at = 0° \nfor (a) =0 and (b) zoom for =5°, =23° and =50°. The points are the measured data and the \nline is the fit calculated from the minimization of the en ergy Eq.(S1) and Eq.(S2). \n \nFigure S1 shows the resistance as a function of the external perpendicu lar field at = 0. The \nfitted parameters are Hd = 1T and the AMR ratio is found to be R/R= 1.83% . Note that the \nsaturation is not reached for H=1T. Consequently, the direction of the magnetization ( ,) does \nnot exactly coincide with that of the external field (H,H): Indeed we have exploited this \nbehavior in order to show that the magneto -voltaic signal is not a response to the external \nmagnetic field (i.e. it is not the usual Nernst or Righi -Leduc effect) , but a response to the \nmagnetization (i.e. it is either the anisotropic Nernst or the anisotropic Righi -Leduc effect) . \n \nOn the other hand, t he in-plane magnetocrystalline anisotropy field Han is very weak , about 5.10-\n4 T, but its effect is rather dramatic as shown in Fig. 2. In the vicinity of H = 0° (modulo 180°) , \nthe magnetization suddenly switche s from its initial position imposed by the applied field from \n \n 15 H = 0° or = 90° to = 30° which is the direction of in plane anisotropy . This jump is well \nreproduced by the numerical simulation shown in Fig.S2. \n \nFig.S2: Magnetoresistance ratio (AMR) as a function of the out -of-plane angle for an \nexternal field of H=0.2T at (black upper curve), and at (lower \ncurve ). If is close to zero modulo the magnetization swi tches to the direction \n (which corresponds to the plane defined by the external field and the anisotropy \nfield). \n \nI-2-2. Anomalous Hall effect (AHE) and Planar Hall effect (PHE) \n \nFig.S3: Configuration for AHE and PHE measurements. \n \nFor planar Hall e ffect (PHE) and anomalous Hall effect (AHE) , the electric current is injected \nalong 0x axis, but the voltage is now measured on the transverse electrode , along 0y (see F ig.S3). \nThe voltage is given by the integration along the electrode of the second line of equation (2) with \n0e\nye\nzJ J\n : \n cos 2sin sin2' 2\nH x y BRRAI V\n Eq. S3 \nThe first term is due to PHE while the second term is due to AHE. The coefficients A’ and B are \nfitting parameters of the order of L/A where L is the distance between the two co ntacts and A is \n \n 16 the section of the electrode (A’ and B also include the contact resistance , so that th ey differ \nslightly from one sample to the other ). The two contributions co -exist for an arbitrary direction \nof the magnetization , except if the configurat ions are fixed for the external magnetic field = \n90° (in plane measurements as a function of for pure planar Hall effect) or at =0° or =90° \n(out-of plane measurements as a function of for pure anomalous Hall effect) . \nFigure S4 show s out-of-plane measurements (AHE) as a function of the angle , performed at \n(A) H=0.2T and H=1T. The calculated curve (continuous line s) follows closely the experimental \ndata for H=0.2T. The jump of the magnetization for close to zero [ resp. 180°] is that \ndescribed on the AMR measurements presented in Fig.S2. The deviation between calculation and \nexperimental data in Fig. S4(B ) is explained by the metastable states due to the irreversible jump \n(the hysteresis loop is time dependent) , that are not taken into account in the calculation of the \nquasi -static states. \n \nFig S4 : (A) Out-of-plane (AHE) voltage as a function the angle H for H=0.2T at H=0°. \n(B) Same c onfiguration for H =1T. The symbols are th e experimental data and the l ine is \ncalculated based on Eq.(S3) and on minimization of Eq.(S1). \n \nFigure S5 shows the in plane measurement s with a saturation field of H=1T. The cur ve follows \nexactly the expected with a single adjustable parameter RH. \n \n \nFig S5 : Planar Hall voltage as a function of the angle for an in-plane field (=H=90° of \nH=1T for the Cu and Pt electrodes. (a) Py(20nm)/Cu(5nm)/ Pt(10nm) and (b) \nPy(20nm) /Pt(10nm). The presence of Cu does not change the magnetization states. \n \n 17 \nFig.S6: (a) Measurements of the Hall voltage as a function of the out -of-plane external \nmagnetic field (=0) for different angle . (b) Calculation based on Eqn.S1 and Eqn.S3 \nPlanar Hall effect dominates. Note the brutal reversal from H=180° to H=180.5° . It is \nthe same as the one shown in Fig .S2 and Fig .S4. \n \nThe m easurements presented in Fig .S6 show that the magnetization states are well \ncharacterized by the simulation based on Eqn.S1 and Eqn.S3, and using the parameters fitted \nas described previously (with in-plane and out -of-plane angular dependence ). \n \nI-2-3. AHE and PHE as a function of the thickness of the electrodes \n \nFigure S7 show s the dependence of both AHE (a) and PHE (b) as a function of electrodes \nthicknesses ranging from 5nm to 100nm under an applied field of H=1T. The profile of the \ncurve is not changed by the variation of the thickness, which means that the magn etization \nstates are not impacted by the electrode thickness. Fig .S7 shows that the a mplitude of the \nsignal changes dramatically between 5 and 50 nm. \n \n \nFig.S7: Measurement of the voltage for different thicknesses of the Pt electrode as a \nfunction of the a ngles at H =1T for (a) planar Hall effect ( H = ) and (b) anomalous Hall \neffect (H ≠ ). The signal V is defined as the voltage difference between the maxima and \nminima. \n \nIn order to justify the thickness dependence of the AHE and PHE signals , we first take the \nassumption that the non -ferromagne tic electrode is passive . The effective current that flows \n \n 18 inside the ferromagnetic layer is not t he initial current but it is divided into t wo branches \n(Fig. S8). A first branch is defined by the resistance of the ferromagnetic layer (shunt \neffect )(23). \n \nFig. S8: Illustration of the shunt effect that takes place at the level of the electrode. The \neffect is well described by a two resistor model R Pt and R Py. \n \nThe thickness dependence is given by the coefficient such that I eff = I. We have: \nPt Py Py PtPy Pt\nd dd\n\n Eq.S4 \nThe Py thickness is dPy and that of the Pt electrode is dPt. The corre sponding resistivities are Py \nand Pt that have been determined by independent resistance measurements. \n \n \nTable.S1 : Parameters used for the calculation of Fig.S9 \n \nThe typical profile s of the thickness dependence of both the AHE and ANE are presented in \nFig.S9. The measured data follows perfectly the profile predicted taking into account the shunt \neffect. There is no adjustable parameter in the calculation . We took the mean value s of \nand obtained by averag ing the parameters (Table.S1 ) over all samples. \n \n \n 19 \n \nFig.S9: (A) Anomalous and (B) planar Hall signal s V as a function of the thickness d of the \nPy electrode. The points are the measured data and the line is the correction (coefficient ) \ndue to the shunt ing effect Eq.(4 ). \n \nThe excellent agreement between experiments and the predictions show n in Fig. S9 bring as a \nclear conclusion that the typical thickness dependence is only due to the shunt ing effect. \n \nII) Anisotropic Thermal Thransport (AThT) \n \nII-1 The anisotropic Fourier equation \n \nIn order to describe the transport of heat in a ferromagnetic system, we follow an equivalent \napproach of the one used in section I -2. Indeed, b oth electric and t hermal transport phenomena \nobey the same symmetry properties, namely the rotational invariance of the system through any \nrotation around the magnetization axis and the time reversal invariance associated to the rotation \n. The Fourier law takes th us the same form as the Ohm’s law (the electric current is replaced by \na heat current and the electric field by a gradient of temperature). \nFourier law relates the gradient of the temperature \nT to the electric current \nQJrT\nˆ . The \nconductivity tensor \nrˆ of a polycrystalline conducting ferromagnet (this is the case of the NiFe \nsamples) is defined by three parameters . If th e reference frame is such that the unit vector \nm is \nalong Oz, we define the thermal resistance r measured perpendicular to the magnetization, the \nthermal resistance measured parallel to the magnetization, and the Righi -Leduc cross -\ncoefficient rARL. According to Onsager reciprocity relation, we have in the reference frame \n{x,y,z} : \n\n\n\n\n\nzARLARL\nrr rr r\nr\n0 000\nˆ\n \nThe Fourier ’s law can then be expressed in an arbitrary reference frame, as (4): \nQ\nARLQ QJmrmmJrr JrT \n . .//\n \nwhere r = rz – r. Explicitly: \n 20 \n \n \n \n\n\n\n \n\nQ\nz zQ\ny x ARL zyQ\nx y ARL zxQ\nz x ARL zyQ\ny xQ\nx z ARL yxQ\nz y ARL zxQ\ny z ARL yxQ\nx x\nJrm Jmr mrm Jmr mrmJmr mrm Jrm r Jmr mrmJmr mrm Jmr mrm Jrm r\nT\n222\n,\n Eq.(S5) \nwhere \ncos sinxm , \nsin sinym , \ncoszm , θ is the same as the one introduced in the \nmagnetic free energy and is the angle between the direction Ox and the projection of the \nmagnetization in the plane of the sample. \nThe temperature difference Ty can be measured between the two edg es of the ferromagnetic \nlayer along 0y, thanks to the thermocouple effect. The voltage is given by the Seebeck \ncoefficient S, such that Vy = S Ty (see II -3 below) . Since the heat current is mainly along \n0x, we obtain the main equation used in this stud y: \n cos 2sin sin22\nARLQ\nx y rrSJ V\n Eq.(S6) \nThe second term in the r ight hand side of Eq.( S6) – proportional to cos - defines the anomalous \nRighi-Leduc coefficient rARL , that can be measured directly with setting =0 or =90° (out-of-\nplane measurements) . On the other hand, the first term in the right hand side of Eqn.S6 – \nproportional to sin(2) (in-plane angle) – defines the planar Righi -Leduc coefficient r, that can \nbe measured directly with se tting (in-plane measurements) . \n \nII-2 AThT on NiFe sample \n \nIn complement to the measurements on NiFe ferromagnet presented in the main text, \ncomplementary results obtained with a Cu(5nm)/Pt(10nm) electrode are shown in Fig. S10 and \nFig.S11. We observe that the results are identical to that corresponding to the Pt(5 0nm) presented \nin Fig.2 of the main text (after correction due to the shunting effect ). We can conclude that the \nCu(5nm) electrode deposited between the ferromagnet and the Pt does not modify the signals \nsignificantly , in agreement with Eq.( S6). The angular dependence s (radial and a zimuthal) for \nH=1T are plotted in Fig.S10, with the numerical simulation , according to equation ( S6). Fig.S10 \nand Fig.S11 display supplementary measurements with Cu electrodes. \n \n 21 \n \nFig.S10 :(A-C) Transverse voltages vs. dire ction of the 1T magnetic field . Cu(5nm)/Pt(10nm) \nelectrode: (A) In-plane configuration at θH = 90°and (B) out-of-plane configuration for φH = \n90°. (C) Cu(20nm) electro de: in -plane configuration (see Fig3B for the out -of-plane \nconfiguration). The line s correspond to the calculation of Eq.(6) with minimization of the energy \nEq.(1). \n \nThe Anosotropic Thermal Transport (AThT) signals have been measured a function of the \nmagn etic field (see Fig.11(a) ) for three values of the direction of the applied field ( ). The \nnumerical simulations (continuous lines) are in excellent agreement with the experimental \nresults . The magnetization reversal at small field is shown in the inset . The out -of plane angular \nvariation at H = 0° for a medium magnetic field (H = 0.18T) is plotted in Fig. S11(B) . The \nirreversible jump of the magnetization (presented in Fig. S2 and Fig. S4) is clearly observed , and \ndescri bed by the numerical simulations . \n \n \n \n 22 \nFig.S11. Transverse voltages on Py/Cu/Pt electrode as a function of (A) amplitude of the magnetic field H \nfor three out -of-plane angles, (B) radial angle H for an applied field of 0.18T. The line correspond to the \ncalculat ion of Eq.( S6) with minimization of the energy Eq.( S1). (C). Transverse voltage on \nPy(40nm)/Cu(40nm) as a function of the amplitude of the magnetic field for out -of-plane config uration at \n180° . \n \nII-3. Heat power and magneto -voltaic signal \n \nIn our experimen t, Joule heating is generated using AC current of pulsation , injected into a \nresistance through a second electrode deposited on the ferromagnetic layer (see Fig.1 of the main \ntext). The h eat power flowing through the sample is proportional to the square of the current . As \na consequence, the magneto -voltaic response to th e heat excitation is measured at the double \nfrequency 2. \nWe checked that the signal is proportional to the injected power as shown in Fig. 12. The \nextrapolation to zero shows that the hea t current JQ measured at the level of the electrode is \nsimply proportional to the heat power injected by Joule effect: JQ\nx = c P Joul , where the constant c \n(such that 0 3,000) [17, 18, \n19, 20, 21, 22, 23, 24, 25]. V[TCNE] x thin films are deposited via chemical vapor deposition (CVD) at \nrelatively low temperature and high pressure (50 ◦C and 35 mTorr, respectively) and is largely insensitive \nto substrate lattice constant or surface termination [17, 20]. Further, V[TCNE] x can be patterned via e-beam \nlithography techniques without increase in its damping [19]. The highly coherent and ultra-low loss \nmagnonic properties of V[TCNE] x have driven interest in applications in microwave electronics [26, 29] \nand magnon-based quantum information science and engineering (QISE) [21, 30, 31]. These benign \ndeposition conditions, combined with patterning that does not degrade performance, highlight the versatility \nof V[TCNE] x for facile on-chip integration with pre-patterned microwave circuits and devices [26, 27, 29, \n32, 33, 34]. These excellent magnetic properties are even more surprising given that V[TCNE] x lacks long-\nrange structural order [25]. Early studies indicated that V[TCNE] x films do not exhibit magnetic anisotropy \nbeyond shape effects due to this lack of long-range ordering [19, 20]. However, recent FMR studies on \nV[TCNE] x nanowires, microstructures, and thin films [19, 20, 21], coupled with combined DFT and \nelectron energy loss spectroscopy (EELS) of the crystal structure [25], suggest there is a residual nematic 4 ordering of the c-axis of the V[TCNE] x unit cell, giving rise to an averaged crystal field anisotropy that is \nsensitive to structural and thermally induced strain. However, dynamic measurements of these crystal fields \nand their dependence on strain are currently lacking, preventing a quantitative analysis of the \nmagnetostrictive properties of this material. \nFor this work, PMN-PT/epoxy/V[TCNE] x/glass heterostructure devices are fabricated such that upon \nelectrically biasing the PMN-PT[001] substrate, the piezoelectric effect produces a lateral in-plane strain in \nthe V[TCNE] x thin film, schematically shown in Fig. 1a. PMN-PT is selected for its strong piezoelectric \neffects and high strain coefficients ( d31 ∼ −(500 to 1 ,000) pm/V) to maximize the strain in the devices [3], \nand the epoxy encapsulation layer is selected to allow for device operation under ambient conditions [22]. \nThis device structure allows investigation of the magnetoelastic properties of V[TCNE] x via standard FMR \ncharacterization and analysis. In the main text of this work, measurements on three devices denoted Samples \n1 - 3 are presented. Sample 1 is measured via broadband FMR (BFMR) techniques. Sample 2 is studied via \nX-band (∼9.8 GHz) cavity FMR techniques. Sample 3 is used to directly measure and calibrate strain in the \ndevices via optical techniques. Additional devices characterized via BFMR techniques are presented in the \nSupplemental Information and their characteristics summarized in Table 1. \nThe BFMR response of Sample 1 is described in Fig. 1(b), where individual scans (inset) are fit to a \nLorentzian lineshape to extract the resonance frequency as a function of applied field, HR vs fR. This data \ncan be modeled by considering the V[TCNE] x thin film as an infinite sheet with attendant shape anisotropy \nand with a uniaxial crystal field anisotropy oriented in the out-of-plane direction (as described above). \nAccordingly, the Kittel equation for ferromagnetic resonance reduces to [19, 20, 21] \n (1) \nwhere fR = ω/2π is the FMR resonance frequency, γ is the gyromagnetic ratio, HR is the applied magnetic \nfield at resonance, Heff = 4πMeff =4πMs−H⊥ is the effective magnetization of the V[TCNE] x film with \nsaturation magnetization 4 πMs and uniaxial strain-dependent anisotropy field H⊥, and θ describes the \norientation of the external field as defined in Fig. 1(a). This equation is valid for films where HR ≫ 4πMeff, \nand is appropriate here as the effective magnetization for V[TCNE] x is typically ∼100 G [19] while the \nresonance field is typically between 3500 – 3650 G at X-band frequencies (9.86 GHz) for all magnetic field \norientations. As-grown films exhibit no in-plane anisotropy, consistent with the literature [19, 20, 21], and \nso ϕ-dependences are neglected. When the external magnetic field is held out-of-plane ( θ = 0◦), Eq. (1) \nreduces to \n (2) \n5 Further information about the magnetic damping in thin films can be revealed by comparing the FMR \nfull-width-half-max (FWHM) frequency linewidth, Γ, to the FMR resonance frequency, fR, via (for \n4𝜋𝑀≪𝐻ோ and Γ ≪𝑓ோ [19]) \n Γ = 2αfR + Γ0 (3) \nwhere α is the (dimensionless) Gilbert damping constant and Γ 0 is the inhomogeneous broadening. It \nshould be noted this form for the Gilbert damping utilizing the frequency-swept linewidth is appropriate \ndirectly for out-of-plane magnetized thin films due to symmetry conditions resulting in the linear \nrelationship between 𝐻ோ and 𝑓ோ [19]. Accordingly, Eqs. (1 - 3) show that performing FMR at various \nfrequencies, fields, and magnetization orientations with and without applied strains in V[TCNE] x thin films \nshould provide information regarding the magnetoelastic properties of V[TCNE] x. \nFitting the data from Sample 1 to Eq. (2) reveals an effective magnetization 4 πMeff = 106.2 G and \ngyromagnetic ratio | γ|/2π = 2.756 MHz/Oe, consistent with literature [17, 18, 19, 20, 21, 22]. The Lorentzian \nfits of the FMR response also reveal the linewidth Γ as a function of the resonant frequency, seen in Fig. \n1(c), where fitting to Eq. (3) yields α = 1.02 ± 0.52 × 10−4 and Γ0 = 8.48 ± 1.22 MHz in Sample 1. These \ndamping characteristics are also consistent with literature values for V[TCNE] x [19, 27], and show that the \ndevices incorporate high-quality magnetic films exhibiting superb low-damping properties [32, 33, 34]. \nMoving beyond measurements of the as-grown strain-free sample, the FMR response of the device is \nnow measured while straining the V[TCNE] x film (Fig. 1(d)). Comparing the FMR response of Sample 1 \nwith no applied strain ( EB = 0 kV/cm) and maximum-applied strain ( EB = 13.3 kV/cm) yields a shift in the \nresonance frequency of 45.5 MHz at a resonance frequency fR = 9.8 GHz, corresponding to a CME \ncoefficient A = 3.38 MHz cm/kV (1.23 Oe cm/kV). It is worth noting that while this absolute shift in \nfrequency, and consequent value for CME, is modest when compared to other magnetostrictive materials \n[4], it represents a shift of over 6 magnetic resonance linewidths due to the ultra-low damping and narrow \nFMR linewidths of the V[TCNE] x thin film. This ability to shift cleanly on and off resonance with an applied \nelectric field is central to the functionality of many dynamically tuned MMIC devices, motivating a more \nin depth and systematic investigation of this phenomenon. \nThe magnetostriction in this composite device is explored by biasing the piezoelectric transducer \nbetween 0 kV/cm and 13.3 kV/cm, and the shift in the resonance frequency (for 𝜃= 0∘) tracks the linear \nstrain produced by the transducer [35], as seen in Fig. 2(a). For maximally strained films, fitting to Eq. (2) \nnow reveals 4 πMeff = 122.9 G, a difference of +16 .7 G between EB = 0 kV/cm and EB = 13.3 kV/cm (14% \nchange). Panels (b-d) of Fig. 2 show the FMR linewidth Γ, inhomogeneous broadening Γ 0, and Gilbert \ndamping α, for Sample 1 as a function of applied electric field (strain). These parameters do not vary over \nthe entire tuning range and are robust to repeated cycling ( >300 cycles - see Supplemental Information). 6 This stability indicates that the shift in resonance frequency is due to a true magnetoelastic effect under \nlinear deformation rather than some fatigue induced structure or morphology change in the film, and further \ndemonstrates the potential for device applications. Finally, it is noteworthy that the linewidths and damping \ncoefficients observed in these proof of principle devices are much narrower than typical magnetostrictive \nmaterials, but are roughly twice the value observed in optimized bare V[TCNE] x films (Γ is typically ~3 \nMHz [17, 19, 25]). This suggests that the tuning ratio of 6 times the linewidth may be further extended to \nmore than 10 times the linewidth in fully optimized devices [19, 25]. \nTo confirm that these shifts in the resonance position are due to strain-dependent crystal-field anisotropy \nin V[TCNE] x as prior studies suggest [20, 21], angular-dependent measurements on unstrained and \nmaximally strained films are performed. Sample 2 is mounted in an X-band (∼9.8 GHz) microwave cavity \nso that the structure can be rotated to vary the polar angle, θ. In-plane ( 𝜃= 90∘) and out-of-plane ( 𝜃= 0∘) \nFMR spectra are shown in Supplemental Fig. S1, with FWHM linewidths of 2.17 Oe (5.97 MHz) and 2.70 \nOe (7.45 MHz), respectively. By tracking the resonance field as a function of rotation and fitting to Eq. (1) \nthe effective magnetization Heff = 4πMeff = 74.0 G is extracted for Sample 2, as seen in Fig. 3. The difference \nin 4πMeff between Samples 1 and 2 can be attributed to sample-to-sample variation and remains consistent \nwith literature values [19]. Repeating the measurement with an applied bias of 13.3 kV/cm to the PMN-PT \nreveals an increase of 4 πMeff to 79.4 G, an increase of 5.4 Oe (8% change), which is like the change observed \nin Sample 1. This confirms that strain is modulating the magnetic anisotropy in V[TCNE] x through the \ncrystal field term H⊥ where 4πMeff = 4πMs−H⊥. This strain-dependent crystal field H⊥ is consistent with and \nsupports previous measurements of V[TCNE] x with both thermally and structurally induced strain [19, 20, \n21]. \nAn approximate upper bound to the strain in these devices can be simply calculated through the relation \nε = d31EB = (d31VB)/t ∼ −(6 − 12) × 10−4 for typical PMN-PT d31 piezo coefficients [4, 35]. However, the \naddition of epoxy, V[TCNE] x, and the glass substrate affect the overall stiffness of the device, thereby the \npiezo coefficient changes from d31 of the bare piezo to an effective coefficient deff of the entire stack. This \ndeff is directly measured by exploiting the color change of V[TCNE] x upon laser heating [23] to pattern \nfiducial marks on the samples and monitor their positions under strain using optical microscopy (see \nSupplemental Information). This approach yields an effective piezoelectric coefficient of deff ∼ −180 pm/V \nor strain of ε ∼ −2.4×10−4, reasonable for the PMN-PT heterostructures used here [4, 35]. \nDensity functional theory calculations on the relaxed and strained V[TCNE] x unit cell provide further \ninsight into the elastic and magnetoelastic properties of V[TCNE] x. These properties are calculated using \nthe Vienna ab initio Simulation Package (VASP) (version 5.4.4) with a plane-wave basis, projector-\naugmented-wave pseudopotentials [36, 37, 38, 39], and hybrid functional treatment of Heyd-Scuseria-\nErnzerhof (HSE06) [40, 41]. The experimentally verified [25] local structure of the V[TCNE] x unit cell is 7 found by arranging the central V atom and octahedrally-coordinated TCNE ligands according to \nexperimental indications [42, 43, 44, 45, 46], and subsequently allowing the structure to relax by \nminimizing the energy. These DFT results previously produced detailed predictions of the structural \nordering of V[TCNE] x, along with the optoelectronic and inter-atomic vibrational properties of V[TCNE] x \nverified directly by EELS [25] and Raman spectroscopy [23], respectively. This robust and verified model \ntherefore promises reliable insight into the elastic and magnetoelastic properties of V[TCNE] x. \nThe magnetoelastic energy density for a cubic lattice f = fel + fme = E/V is a combination of the elastic \nenergy density \n (4) \nwhere Cij are the elements of the elasticity tensor and εij are the strains applied to the cubic lattice, and \nthe magnetoelastic coupling energy density \n \nwhere Bi are the magnetoelastic coupling constants and αi where i ∈ {x,y,z} represent the cosines of the \nmagnetization vector [47]. \nThe elastic tensor C = Cij for V[TCNE] x is found by applying various strains to the unit cell and observing \nthe change in the energy. The calculated Cij tensor results in a predicted Young’s modulus for V[TCNE] x YV \n= 59.92 GPa. By directly applying compressive and tensile in-plane strains to the DFT unit cell (i.e. in the \nequatorial TCNE ligand plane [25]), one may calculate the overall change in the total energy density, both \nparallel and perpendicular to the easy axis. The difference between these two, Δ𝐸, is the magnetic energy \ndensity change, which is proportional to the magnetoelastic coupling constant B1 [47] \n ∆E/V = −(ν2D + 1)B1ε|| (6) \nwhere ν2D is the 2-dimension in-plane Poisson ratio and ε|| is the applied in-plane (equatorial TCNE \nplane) epitaxial strain while allowing out-of-plane (apical TCNE direction) relaxation. The elastic and \nmagnetoelastic coefficients are related via the magnetostriction constant λ100 = λs via \n . (7) \nAs a result, the calculated changes of the magnetoelastic energy density with strain provide direct \npredictions of the elasticity tensor ( Cij) and magnetoelastic coefficients ( Bi) for V[TCNE] x. For \npolycrystalline samples of cubic materials, the overall (averaged) magnetoelastic coefficient λs also \nconsiders the off-axis contribution from λ111 such that λs = (2/5)λ100 + (3/5)λ111. However, the off-axis \ncomponent is not considered here for two reasons: (i) the apical TCNE ligands are assumed to align along \n8 the out-of-plane direction ( z-axis, θ = 0◦), and (ii) difficulties in calculating the magnetoelastic energy \ndensity changes upon applying a shear strain that provides the estimate of B2 needed to calculate λ111. The \nformer argument is reasonable as previous experimental results indicate the magnetocrystalline anisotropy \nfrom strain is out-of-plane [21], consistent with the ligand crystal field splitting between the equatorial and \napical TCNE ligands [25]. Further, the lack of in-plane ( ϕ-dependent) anisotropy suggests the distribution \nin the plane averages out to zero. Therefore, the magnetoelastic coefficient calculated here considers an \naverage of the in-plane Cii components in determining λ100. That is, the DFT predicts a magnetoelastic \ncoefficient for V[TCNE] x of \n (8) \nwhere 𝐶𝐼𝑃 = (1/2)(𝐶11 + 𝐶22) = 60.56 GPa and C12 = 37.84 GPa. Accordingly, utilizing the \ncalculated value of B1 = 85.85 kPa (see Supplemental Information) predicts a theoretically calculated 𝜆ሚଵ = \n−2.52 ppm for V[TCNE] x magnetized along the apical TCNE ligand (i.e. θ = 0◦). \nCombining these ferromagnetic resonance, direct strain measurements, and DFT calculations provides \nthe information necessary to determine the magnetoelastic properties of V[TCNE] x. Here, we follow the \nconvention in the literature using the magnetoelastic free energy form from the applied stress σ = Y ε to the \nmagnetostrictive material, Fme = (3/2)λsσ [2, 3, 4]. Accordingly, this free energy yields an expression for the \nstrain-dependent perpendicular (out-of-plane) crystal field [3] \n (9) \nwhere λS is the magnetoelastic coefficient, Y is the Young’s modulus of the magnetic material, d31 is the \npiezoelectric coefficient of the (multiferroic) crystal, and EB is the electric field bias. Here, ε = d31EB is the \nstrain in the magnetic layer obtained based on the assumption that the electrically induced strain is perfectly \ntransferred to the magnetic film. For this study, the direct optical measurement of the strain in the V[TCNE] x \nfilms allows the modification of Eq. 9 by replacing d31EB by the measured ε = deffEB = −2.4 × 10−4 to account \nfor the mechanical complexity of the multilayered device. The magnetoelastic coefficient of V[TCNE] x can \nthen be calculated from Eq. 9 using the values of 4 πMS and H⊥ from FMR characterization, the direct \nmeasurement of ε from optical measurements, and the calculated value of YV = 59.92 GPa from DFT. \nAccordingly, inserting the corresponding values into Eq. 9 yields a magnetoelastic constant for V[TCNE] x \nof λs ∼ −1 ppm to λs ∼ − 4 ppm for the devices measured here. This range shows excellent agreement with \nthe DFT calculations of the magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm from Eq. 8. This agreement \nprovides additional support for the robustness of the DFT model developed in previous work [23, 25]. \n9 Further, comparison with past studies of the temperature dependence of Heff [21] allows for the extraction \nof the thermal expansion coefficient of V[TCNE] x, 𝛼௧ = 11 ppm/K, at room temperature. \n \nDiscussion \nThe results above compare V[TCNE] x thin films to other candidate magnetostrictive materials using the \nestablished metrics of CME and λs. However, while these parameters are effective in capturing the impact \nof magnetoelastic tuning on the DC magnetic properties of magnetic thin films and magnetoelectric devices, \nthey fail to capture the critical functionality for dynamic (AC) magnetoelectric applications: the ability to \ncleanly tune on and off magnetic resonance with an applied electric field. For example, Terfenol-D is \nconsidered a gold standard magnetostrictive material due to its record large magnetoelastic coefficient λs up \nto 2,000 and CME coefficients 𝐴 as large as 590 Oe cm/kV [3]. However, due to its broad ∼1 GHz FMR \nlinewidths, large 4 πMeff > 9,000 G, high Gilbert damping α = 6×10ିଶ, and brittle mechanical nature, it is \nnot practical for many applications in MMIC. As a result, we propose a new metric that appropriately \nquantifies the capability of magnetostrictive materials for applications in microwave magnonic systems [28, \n33, 34] that takes into account both the magnetostrictive characteristics and the linewidth (loss) under \nmagnetic resonance of a magnetically-ordered material. Accordingly, a magnetostrictive agility ζ is \nproposed here, which is the ratio of the magnetoelastic coefficient λs to the FMR linewidth (in MHz) ζ(fR) \n= |λs|/Γ. For the V[TCNE] x films studied here the magnetostrictive agility at X-band frequencies (9.8 GHz) \nis in the range ζ = {0.164 – 0.660}, comparable to YIG ζ = {0.139 − 0.455} and Terfenol-D ζ = {0.301 – \n0.662} as shown in Table 1. Further, we note that the growth conditions under which high-quality V[TCNE] x \nfilms can be obtained make on-chip integration with microwave devices significantly more practical than \nfor YIG, and that the narrow linewidth (low loss) is more attractive for applications such as filters and \nmicrowave multiplexers than Terfenol-D. \nConclusion \nWe have systematically explored indirect electric-field control of ferromagnetic resonance in the low-\nloss organic-based ferrimagnet V[TCNE] x in V[TCNE] x/PMN-PT heterostructures. These devices \ndemonstrate the ability to shift the magnetic resonance frequency of V[TCNE] x by more than 6 linewidths \nupon application of compressive in-plane strains 𝜀 ~ 10ିସ . Further, we find there is no change in the \nmagnetic damping of the films with strain and that the samples are robust to repeated cycling (> 300 cycles), \ndemonstrating the potential for applications in MMIC without sacrificing the ultra-low damping of \nmagnetic resonance in V[TCNE] x. The changes in the FMR characteristics along with direct optical 10 measurements of strain provide an experimentally determined range for the magnetoelastic coefficient, λS \n= −(1 − 4 .3) ppm, showing excellent agreement to DFT calculations of the elastic and magnetoelastic \nproperties of V[TCNE] x. Finally, we present a discussion on the metrics used in the magnetostriction \ncommunity wherein we point out the shortcomings on the commonly used metrics of the magnetostriction \nand CME coefficients. In this context, we propose a new metric, the magnetostrictive agility, ζ, for use of \nmagnetoelastic materials for coherent magnonics applications. \nThese results develop the framework necessary for extended studies into strain-modulated magnonics \nin V[TCNE] x. Additionally, these magnetoelastic properties in V[TCNE] x suggest that large phonon-\nmagnon coupling in V[TCNE] x might be achieved, necessary and useful for applications in acoustically-\ndriven FMR (ADFMR) or, in conjunction with high-Q phonons, for quantum information applications [28]. \nOther recent work has identified V[TCNE] x as a promising candidate for QISE applications utilizing \nsuperconducting resonators [31] and NV centers in diamond ranging from enhanced electric-field sensing \n[5] to coupling NV centers over micron length scales [30]. These findings lay a potential framework for \ninvestigating the utilization of V[TCNE] x in quantum systems based on magnons and phonons. \n \nAcknowledgments \nS. W. K. developed the project idea, and S. W. K., E. J.-H., P. S., and M. P. developed the project plan for \nexperimental analysis. S. W. K. fabricated the V[TCNE] x heterostructure devices, performed FMR \ncharacterization and analysis, and wrote the manuscript. A. F. developed the analysis software used for \nfitting FMR linewidths and extracting parameter fits. P. S., G. S., and M. P. provided PMN-PT substrates. \nY. S. performed and analyzed DFT calculations of the elasticity tensor and the magnetoelastic coefficients \nfor V[TCNE] x. H. F. H. C. performed and analyzed optical measurements of strain in the devices. K. E. \nN. and M. S. performed BLS measurements on V[TCNE] x/Epoxy devices to extract elastic properties. All \nauthors discussed the results and revised the manuscript. S. W. K., A. F., and E. J.-H. were supported by \nNSF DMR-1808704. P. S., G. S, and M. P. were supported by the Air Force Office of Scientific Research \n(AFOSR) Award No. FA955023RXCOR001. 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Appl. \nPhys. 99, 093909 (2006). \nMethods \nSynthesis of V[TCNE] x and Device Fabrication. \nV[TCNE] x films are deposited via ambient-condition chemical vapor deposition (CVD) in a custom CVD reactor \ninside an argon glovebox (O 2 < 1 ppm, H 2O < 1 ppm) in accordance with literature [17, 18, 19, 20, 21, 22, 23, 24, \n25, 26, 27, 29, 31]. Argon gas flows over TCNE and V(CO) 6 precursors that react to form a V[TCNE] x thin film on \nthe substrates. The pressure inside the CVD reactor for all growths was 35 mmHg, and TCNE, V(CO) 6, and the \nsubstrates are held at 65◦C, 10◦C, and 50◦C, respectively. All substrates were cleaned via solvent chain (acetone, \nmethanol, isopropanol, and deionized (DI) water (×2)) and dried with N 2, followed by a 10 minute UV/Ozone clean \nin a UVOCS T10x 10/OES to remove any residual organic contaminants. \nNominally 400 nm V[TCNE] x films are deposited onto microscope cover glass substrates ( t = 100µm). These \nV[TCNE] x/glass substrates are then mechanically fixed to a PMN-PT transducer (4 mm×10 mm ×0.15 mm) with an \nOLED epoxy (Ossila E130) to create a PMN-PT/Epoxy/V[TCNE] x/Glass heterostructure. The epoxy here not only \nprotects V[TCNE] x from oxidation [22], but also propagates lateral strain into V[TCNE] x film from the piezo \ntransducer upon biasing. While the primary deformation in the piezo transducer is along the poling direction of the 16 PMN-PT ( z), the distortion of the PMN-PT in the thickness direction also produces a lateral in-plane strain in the \nPMN-PT through the Poisson effect (i.e. one must consider here the d31 piezo coefficient of PMN-PT). Therefore, the \nprimary strain experienced by the V[TCNE] x film is in-plane. The PMN-PT electrodes are connected to a Keithley \n2400 voltage source so that electric fields up to EB = VB/tPMN−PT = 13.3 kV/cm can be applied across the PMN-PT layer. \nFerromagnetic Resonance Characterization \nBroadband FMR (BFMR) measurements on Sample 1 and Supplemental Devices A-C were taken using a \ncommercial microstrip (Southwest Microwave B4003-8M-50) and Agilent N5222A vector network analyzer \n(VNA). The devices are mounted so that the magnetic field is normal to the V[TCNE] x film (𝜃 = 0∘ ). S21 \nmeasurements (P = −20 dBm) show the FMR peak upon matched magnetic field and frequency conditions \nin accordance with Eq. 2. A Keithley 2400 Sourcemeter is used to apply up to 200 V to the piezoelectric \ntransducers – accordingly, the maximum-applied strain in the 150 μm PMN-PT corresponds to an electric \nfield 𝐸 = 13.3 kV/cm as mentioned in the main text. \nAll angular-dependent FMR measurements (Sample 2) were performed in a Bruker X-band (~9.6 GHz) \nEPR (Elexsys 500) spectrometer. The frequency of the microwave source is tuned to match the resonant \nfrequency of the cavity before each scan to ensure optimal cavity tuning. All scans had a 0.03 G modulation \nfield at 100 kHz modulation frequency and were performed at the lowest possible microwave power (0.2 \nμW) to prevent sample heating and non-linear effects distorting the FMR lineshape. The V[TCNE] x/PMN-PT \ndevices are mounted on a sapphire wafer and loaded into glass \ntubes for FMR measurements such that the samples can be rotated in-plane (IP: 𝜃 = 90∘ ) to out-of-plane \n(OOP: 𝜃 = 0∘ ) for FMR measurements in 10 degree increments, where resonance occurs upon matched field \nand frequency conditions according to Eq. 1. \nDensity Functional Theory Calculations \nThe pseudopotentials used are default options from VASP’s official PAW potential set, with five valence electrons \nper vanadium, four per carbon and five per nitrogen [36, 37, 38, 39]. For the rest of the calculation we used 400 eV \nfor the energy cutoff and a Γ centered 5x5x3 k-mesh sampling. From these results, the elastic tensor Cij for V[TCNE] x \nis calculated. Using the elastic tensor, the Young’s modulus for V[TCNE] x is averaged over the C11, C22, and C33 \ncomponents to yield YV = 59.92 GPa. From the DFT calculations, the full elastic matrix from the Cij is given by (in \nunits of GPa) \n𝐶=\n⎣⎢⎢⎢⎢⎡66.4437.847.96\n37.8454.683.79\n7.96 3.79 58.641.38−0.200.53\n0.09−1.55−0.37\n−0.69 0.76 0.31\n1.38 0.09 −0.69\n−0.20 −1.55 0.76\n0.53 −0.37 0.3135.16 0.25 −0.95\n0.25 6.65 −0.17\n−0.95 −0.17 9.94 ⎦⎥⎥⎥⎥⎤\n \n \n 17 Optical Measurements of Strain in V[TCNE] x \nV[TCNE] x films can be patterned via laser heating techniques, whereupon the material changes color when heated \nabove its thermal degradation temperature ( ∼ 370 K) [16, 23]. To more appropriately calibrate strain in the \nV[TCNE] x films versus applied bias, we directly measure the deformation in the films by exploiting the color change \nof V[TCNE] x upon laser heating [23] and optical microscopy techniques. Fresh V[TCNE] x/PMN-PT devices are \nexposed to a focused laser spot to create ad hoc fiducial marks on the film in Sample 3 (Supplementary Fig. S5) in \na 50 µm × 50 µm square. By measuring the distance between these laser-written structures with and without applied \nstrain, we can precisely and directly measure the strain in the V[TCNE] x films upon electric bias thus allowing a \nmore precise calculation of magnetoelastic coefficients. Using these methods, we apply a bias of 13.3 kV/cm on \nSample 3 and find a strain ε ∼ 2.4×10−4 which is in reasonable agreement with estimated values of strain using the \nthickness of the PMN-PT (150 µm) and typical piezo coefficient d31 ∼ 500 − 1000 pm/V). 18 \n \n \n \nFigure 1: (a) Effective device schematic, coordinate system, and wiring diagram for V[TCNE] x/PMN-PT \nheterostructures. (b) Ferromagnetic resonance frequency fR vs external field Hext with the field held OOP ( θ \n= 0◦) measured via BFMR. The external field is held constant as the microwave frequency is swept. (Inset) \nRepresentative BFMR scan at f0 = 9.8 GHz, Hext = 3,660 Oe. (c) FMR linewidth Γ versus FMR frequency \nfor OOP field ( θ = 0◦). A linear fit (red line) extracts the dimensionless Gilbert damping parameter α = 1.02 \n± 0.52 × 10−4 and the inhomogeneous broadening Γ 0 = 8.48 ± 1.22 MHz. (d) BFMR scans for unstrained (0 \nkV/cm – black) and maximally strained (13.3 kV/cm – red). The shift in the FMR frequency is ∼45 MHz, \na shift ∼4 linewidths. \n 19 \nFigure 2: V[TCNE] x damping analysis with applied strain: (a) Plot showing differential (shifted) \nFMR resonance position f R−f0 where f 0 is the resonance at 9.8 GHz, (b) FWHM linewidth Γ, (c) \ninhomogeneous broadening Γ 0, and (d) Gilbert damping α versus applied electric field bias E B. \nWhile the resonance position shifts by multiple linewidths, there is negligible effect in the \nlinewidth or damping of the material. 20 \n \n \n \n \n \n \nFigure 3: Cavity X-band FMR measurements on V[TCNE] x/PMN-PT devices. Angular dependence of \nFMR resonant field H R at 𝑓ோ ∼ 9.6 GHz is measured without (black squares) and with (red circles) strain. \nIn-plane and out-of-plane peak-to-peak (FWHM) linewidths are 1.25 (2.16) Oe and 1.56 (2.70) Oe, \nrespectively. \n 21 \n \n \n \n \n \n \n \n \nTable 1: Extracted parameters from V[TCNE] x strain devices compared to YIG, Terfenol-D, and \nother magnetostrictive materials. Asterisk indicates the frequency-equivalent linewidth calculated \nfrom the field-swept FMR linewidth and accounts for the ellipticity of FMR precession for in-plane \nmagnetized materials following the method in Ref. [56]. \n22 Supplemental Information: In situ electric-field control of \nferromagnetic resonance in the low-loss organic-based ferrimagnet \nV[TCNE] x∼2 \n \n \nCavity X-band FMR of Sample 2 \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S1: X-band (9.8 GHz) cavity FMR scans of Sample 2 for DC magnetic field in-plane ( 𝜃=90°) and out-\nof-plane ( 𝜃 = 0° ). Peak-to-peak (p2p) linewidths and resonance positions determined from a fit to a \nLorentzian derivative, from which the full-width-half-max (FWHM) linewidth Γ is found by multiplying by \n√3. 23 V[TCNE] x Resonance Frequency with Piezo Switching Strain \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S2: V[TCNE] x out-of-plane magnetized differential resonance frequency 𝛿𝑓ோ=9.8 𝐺𝐻𝑧−𝑓ோ \nas a function of applied bias voltage to 150 µm PMN-PT . The “butterfly” hysteresis arises from the \nhysteretic behavior of the piezo strain upon the polarization direction switching. The inset shows \nfrequency-swept FMR spectra and fits at maximum and minimum frequency shift. 24 \n \n \n \n \nFigure S3: Differential resonance frequency ( 𝑓ோ,=9.8 GHz) in the same device from Fig. S2 switching \nbetween 𝑉= ±25 V (𝐸= ±1.67 kV/cm) as a function of the number of the number of switches \nbetween positive and negative applied strain. The FWHM linewidth of the V[TCNE] x remains \neffectively constant for over 300 positive/negative (tensile/compressive) strain applications, and the \nresonant frequency for the respective compressive and tensile additionally remains effectively constant. \nThese conditions were selected based on the linewidth and resonance frequency tuning such that the \nresonance features do not overlap, thereby demonstrating a means to electrically-bias a device on and \noff resonance. \n25 V[TCNE] x Density Functional Theory Calculations of Strain-Dependent \nMagnetoelastic Energy \n \nV[TCNE] x Optical Strain Characterization \nFigure S5: Positions of fiducial marks “burned” onto the V[TCNE] x film measured via optical techniques \nupon biasing a 150 µm PMN-PT piezo transducer. The extracted strain in x and y is averaged to 𝜀 =\n2.4×10ିସ and is used to calculate the magnetoelastic coefficients 𝜆�� presented in the text. \nFigure S4: DFT-calculated magnetic energy difference of the V[TCNE] x unit cell upon manipulating \nthe applied strain. The orange line is a tangential linear fit at 𝜀 = 0 to solve for 𝛥𝐸/𝑉 in the main \ntext that provides the magnetoelastic coupling 𝐵ଵ. 26 \nAdditional V[TCNE] x Device Strain Characterizations \n \n \n \nFigure S6: Supplemental devices measured via BFMR techniques ( 𝜃=0°). Supplemental device C varies from the \nothers only by the thickness of the PMN-PT piezo ( 𝑡= 500 𝜇𝑚 ), so that the electric field across the device \n(hence the strain) is adjusted accordingly. 27 Additional Linewidth and Damping Analysis: Supplemental Device C ( 𝒕𝑷=\n𝟓𝟎𝟎 𝝁𝒎) \n \n \n \n \nFigure S7: Gilbert analysis of Supplemental Device C as a function of applied electric-field bias. (a) Resonance \nfrequency for an out-of-plane magnetization orientation and applied external field 𝐻ோ= 3,658.8 G (𝑓ோ,=\n9.83 GHz). (b) FWHM linewidth corresponding to the resonance frequencies in panel (a). (c) \nInhomogeneous broadening and (d) Gilbert damping parameters. Note there is negligible change in \nlinewidth, inhomogeneous broadening, and Gilbert damping for positive and negative bias up to the piezo \nswitching fields at ±3.2 kV/cm. The error bars in (a) and (b) are smaller than the markers used for the data \npoints. " }, { "title": "2402.03734v1.Magnon_mediated_spin_pumping_by_coupled_ferrimagnetic_garnets_heterostructure.pdf", "content": "Magnon mediated spin pumping by coupled ferrimagnetic garnets\nheterostructure\nAnupama Swain*,1Kshitij Singh Rathore*,1Pushpendra Gupta,1Abhisek Mishra,1Gary Lee,2Jinho Lim,3Axel\nHoffmann,3Ramanathan Mahendiran,2and Subhankar Bedanta1, 4,a)\n1)Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences,\nNational Institute of Science Education and Research (NISER), An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n2)Department of Physics, 2 Science Drive 3, National University of Singapore, 117551,\nRepublic of Singapore\n3)Department of Materials Science and Engineering and Materials Research Laboratory,\nUniversity of Illinois Urbana-Champaign, Urbana, Illinois 61801, USA\n4)Center for Interdisciplinary Sciences (CIS), NISER, An OCC of Homi Bhabha National Institute (HBNI),\nJatni-752050, Odisha, India\n*equal contribution\nABSTRACT\nSpin pumping has significant implications for spintronics, providing a mechanism to manipulate and trans-\nport spins for information processing. Understanding and harnessing spin currents through spin pumping is\ncritical for the development of efficient spintronic devices. The use of a magnetic insulator with low damping,\nenhances the signal-to-noise ratio in crucial experiments such as spin-torque ferromagnetic resonance (FMR)\nand spin pumping. A magnetic insulator coupled with a heavy metal or quantum material offers a more\nstraight forward model system, especially when investigating spin-charge interconversion processes to greater\naccuracy. This simplicity arises from the absence of unwanted effects caused by conduction electrons unlike\nin ferromagnetic metals. Here, we investigate the spin pumping in coupled ferrimagnetic (FiM) Y 3Fe5O12\n(YIG)/Tm 3Fe5O12(TmIG)bilayerscombinedwithheavy-metal(Pt)usingtheinversespinHalleffect(ISHE).\nIt is observed that magnon transmission occurs at both of the FiMs FMR positions. The enhancement of\nspin pumping voltage ( Vsp) in the FiM garnet heterostructures is attributed to the strong interfacial exchange\ncoupling between FiMs. The modulation of Vspis achieved by tuning the bilayer structure. Further, the spin\nmixing conductance for these coupled systems is found to be ≈1018m−2. Our findings describe a novel\ncoupled FiM system for the investigation of magnon coupling providing new prospects for magnonic devices.\nKeywords: Ferrimagnet, Spin Pumping, Magnons, Coupling, Thin Films.\nThe need for ultra-low power consumption devices has\ngiven rise to the field of spintronics. This field uses\nthe spin degree of freedom of electrons1,2. Spintronic-\nbased applications rely on the generation of spin cur-\nrents and their conversion into charge currents in mag-\nnetic heterostructures3–6. However, in this process, the\nscattering of conduction electrons in the magnetic layer\nresults in Joule heating. In this context, moving to-\nwards the magnon-based information processing in in-\nsulators will solve this issue as there is no physical move-\nment of the electrons7. Magnons are the quanta of spin\nwaves, defined as the collective excitations of magnetic\nmoments in magnetically ordered materials. Magnons\ncan propagate over distances ranging from micrometers\nin low damping metallic thin films to around cm in high\nquality magnetic insulators8,9. The magnonic spin cur-\nrent can be employed to carry, transport, and process\ninformation10,11, as well as generate a spin torque act-\ning on the local magnetic moment that can be exploited\nto drive magnetization dynamics12,13and magnetic do-\nmain walls14–16. A promising technique for detecting\nmagnonic spin currents is the spin pumping induced in-\nverse spin Hall effect (ISHE)10,17. Spin pumping refers\nto the transfer of spin angular momentum via magne-tization precession from the ferromagnetic material to\nthe adjacent spin sink layer18. These pure spin currents\nare transformed into conventional charge currents by the\nISHE, which allows for a convenient electrical detection\nof spin-wave based spin currents11. After the discovery\nof the spin-pumping effect and the ways for enhancement\nof spin current in ferrimagnetic insulator (yttrium iron\ngarnet, Y 3Fe5O12, YIG)/non-magnetic metal (platinum,\nPt), there was rapidly emerging interest in the investi-\ngation of these structures19,20. In these magnetic insu-\nlators movement of individual electron gets restricted,\nwhich helps to avoid Joule heating dissipation and there-\nfore benefits modern upcoming spintronic devices21.\nYIG is often considered as the best medium for spin\nwavepropagationbecauseofitsverysmallGilbertdamp-\ning coefficient ( 2×10−5for bulk YIG)22. Being an elec-\ntrical insulator, electron-mediated angular momentum\ntransfer can only occur at the interface between YIG and\na metallic layer. In that context, metals with large spin\norbit coupling (SOC) like Pt where a pure spin current\ncan be generated through spin Hall effect (SHE) have\nbeen used to excite or amplify propagating spin waves\nthrough loss compensation in YIG23–25. Recently, other\ninsulator garnets like Tm 3Fe5O12(TmIG), Gd 3Fe5O12arXiv:2402.03734v1 [cond-mat.mtrl-sci] 6 Feb 20242\n(GdIG) etc. have been investigated with a focus on low\nGilbert damping, interlayer coupling and spintronics ap-\nplications. Here we investigate the manipulation, genera-\ntionanddetectionofmagnon-basedspincurrentsinferri-\nmagnetic insulators capped with heavy metal Pt. These\ninsulators with Pt have been explored a lot for magnon\ntransport physics in effects like ISHE26, Spin Hall mag-\nnetoresistance (SMR)27and spin-Seebeck effect (SSE)28.\nThere are few results published on spin waves in coupled\nferrimagnetic layers29,30, however, the effect of coupling\non spin pumping in such FiM system is scarce in liter-\nature. In order to study the interface and the growth\neffects on spin pumping, this work presents a systematic\nstudy by considering YIG/Pt, TmIG/Pt and bilayers of\nYIG/TmIG with Pt. This study reveals the increase in\nthe spin pumping voltage in the bilayers which can be\nattributed to the interfacial exchange coupling between\nYIG and TmIG.\nHigh-quality garnet films were prepared on (111)\noriented GGG (GdGa 5O12)single crystal substrate by\npulsed laser deposition (PLD) technique using an ex-\ncimer laser (λ= 248 nm ). The garnet targets were com-\nmercially purchased from M/s. Testbourne, UK. The\nbase pressure of the chamber was 8×10−7mbar. The\nsubstrate temperature was maintained at 560◦Cin a\n9×10−4mbar of oxygen partial pressure during YIG\nlayer deposition. The laser fluence and repetition rate\nwere 1.8 J/cm2and8 Hz, respectively. After deposition,\nthe sample was annealed for 2 hat 800◦Cin 300 mbar\nof ambient oxygen environment and cooled at 10◦C/min\nrate. TheTmIGlayerwasgrownbymaintainingthesub-\nstrate temperature at 750◦C, laser fluence at 1 J/cm2,\nwith a repetition rate of 6 Hzin 0.26 mbar of oxygen par-\ntial pressure. Post deposition, the prepared TmIG film\nwas in-situ annealed at same growth temperature for 25\nmins at 100 mbar oxygen pressure followed by cooling\nat 5◦C/min. During deposition the substrate to target\ndistance was kept 5 cmfor both the films. The bilayer\nsamples were prepared by following the same growth con-\ndition as the individual single layers. For the bilayer\nsamples, each FiM layer was first deposited and then an-\nnealed in similar conditions as of its corresponding sin-\ngle reference layer samples and then the subsequent layer\nwas deposited followed by its post-annealing. The details\nof prepared sample structure are mentioned in Table I.\nThe thickness of the corresponding layer is mentioned\nin the brackets. The Pt layer has been prepared via dc\nmagnetron sputtering in a high vacuum multi-deposition\nchamber manufactured by Mantis Deposition Ltd., UK.\nThe growth quality and thickness of the prepared\nfilmswereinvestigatedby X-raydiffraction(XRD).High-\nresolution transmission electron microscopy (HR-TEM)\nhas been performed in this study to verify the epitaxy of\ndeposited films. The saturation magnetization value has\nbeen taken from the superconducting quantum interfer-\nencedevice(SQUID)magnetometrydata. Theferromag-\nnetic resonance (FMR) and spin pumping induced ISHE\nmeasurements were performed using a coplanar waveg-TABLE I. Details of the sample structure studied in this work\nSample Sample\nname structure\nS1 GGG(111)/YIG(100 nm)/Pt( (5 nm)\nS2 GGG(111)/TmIG(30 nm)/Pt(5 nm)\nS3GGG(111)/YIG(100 nm)/TmIG(30 nm)/Pt(5 nm)\nS4GGG(111)/TmIG(30nm)/YIG(100 nm)/Pt(5 nm)\nuide (CPW) based setup. The sample was placed up-\nside down on the CPW. FMR measurements were car-\nried out in the 3-12 GHz of frequency range with 25 mW\nmicrowave power. The ISHE measurements were carried\noutat7GHz rffrequencybyconnectingananovoltmeter\nat the opposite edges of the sample. The measurement\ndetails are mentioned in our previous work31.\nFIG. 1. XRD patterns of (a) S3 and (b) S4 samples (the\ncorresponding insets show the XRD pattern for the 2 θrange\n20◦-80◦). HRTEM image and SAED pattern for sample S3\nare shown in (c) and (d), respectively. The inset shows the\nmagnified part of YIG and TmIG interface.\nThe prepared samples were structurally characterized\nby XRD to confirm the phase and growth quality. The\nXRDpatternofS3andS4samplesareshowninFig. 1(a-\nb). The XRD pattern of S1 and S2 is given in the supple-\nmentary file. The YIG and TmIG lattice parameters are\nveryclosetothatofthesubstrateGGGwhichensuresthe\nepitaxial growth of the structure. The observed diffrac-\ntion peaks were indexed with the corresponding crystal\nplane (h k l) values, and it is evident from the analysis\nthatthegrowthofpreparedsamplesisalongthe(111)di-\nrection. The absence of any additional peaks other than\nthe peaks corresponding to YIG, TmIG, and GGG in\nthe patterns ensures the phase purity of the grown struc-\ntures (shown in the corresponding insets). In Fig. 1(a),\nthe peak corresponding to YIG layer is dominating over\nthe TmIG peak as the YIG thickness is high. By ana-\nlyzing the XRD peak at (444) reflection yields a cubic\nlattice parameter of YIG in S1 is 12.46 Å, which is com-3\nparable to 12.38 Åfor the bulk YIG32. Moreover, the\nlattice parameters of YIG and TmIG are estimated for\nall the samples. The YIG lattice parameter in S3 and\nS4 is 12.43 Åand 12.48 Å, respectively. Likewise, the lat-\ntice parameters of TmIG in S2, S3, and S4 are 12.45 Å,\n12.54 Å, and 12.60 Å, respectively. It is to be noted that,\nthe lattice parameter has changed in S3 and S4 samples\nfor both YIG and TmIG with respective to their single\nlayer i.e., S1 and S2.This indicates the presence of strain\nin the films which may have the impact on the physical\nproperties of the samples.\nThe interface is further explored by cross-sectional\nHRTEM for the sample S3. The zoomed-in image [shown\nin the inset of Fig. 1(c)] reveals the epitaxial growth\nof YIG and TmIG. These high-quality images provide\nclearevidenceofthewell-definedinterfaceandcrystalline\nstructure of the film showing in-plane lattice matching\nwith the substrate. The thicknesses of YIG and TmIG\nwere found to be 100 nm and 30 nm respectively. No-\ntably, no defects or misalignment in lattice planes were\nobserved in the HRTEM image shown in Fig. 1(c). Fig.\n1 (d) shows the selected area electron diffraction (SAED)\npattern of sample S3, which confirms the single crys-\ntalline nature. The validation provided by these HRTEM\nimages and SAED pattern is crucial evidence, confirm-\ning the successful fabrication of the thin films with sharp\ninterface.\nThe spin dynamics properties were investigated by\nFMR measurements at room temperature. Fig. 2 (a-\nd) show the FMR spectra of the prepared samples at\ndifferent frequencies ( f) ranging from 3- 12 GHz. Dis-\ntinct FMR peaks were observed for YIG and TmIG in\nall samples. This confirms the room temperature mag-\nnetic phase of the prepared samples. A shift in Hres\ncorresponding to YIG and TmIG in the S3 and S4 sam-\nples compared to S1 and S2 samples is observed. This\ncould be attributed to the interfacial exchange coupling\nbetween YIG and TmIG, similar to other multilayered\nsystems33.\nFurther, the FMR signals were fitted by Lorentzian\nfunction to obtain line width (∆H)and resonance field\n(Hres) at each frequency. Later, the damping analy-\nsis is carried out for all the samples by plotting fvs\nHresand∆Hvsf(shown in Fig. 3 ). In case of S1,\nthe damping value is estimated considering the uniform\n(n=0) mode. Here, for the bilayer samples, the damping\nanalysis is done for the respective resonance fields of YIG\nand TmIG.\nThe plotted data in Fig 3 (a) were fitted by the Kittel\nequation34,\nf=γ\n2πq\n(Hres+HK)(Hres+ 4πMeff+HK)(1)\nwhere, γ\u0000\n=gµB\nℏ) is gyromagnetic ratio ( gis Lande g-\nfactor, µBis Bohr magneton), HKis in-plane anisotropic\nfield), 4πMeff\u0010\n= 4πMs+2KS\nMstFM\u0011\nis the effective de-\nmagnetizing field (KS, Ms, and tFMare perpendicu-\nFIG. 2. FMR spectra of (a) S1 (b) S2 (c) S3 and (d) S4\nsamples at different frequencies.\nlarsurfaceanisotropyconstant, saturationmagnetization\nand thickness of the magnetic layer, respectively). Af-\nterwards, the Gilbert damping constant ( α) values were\nestimated by fitting the plot in Fig 3(b) and (c) by the\nequation\n∆H= ∆H0+4παf\nγ. (2)\nTheobtainedvaluesof g,Msandα(tabulatedinTable\nII)areingoodagreementwiththeexistingliterature26,35.\nAn increase in αvalue is observed for both YIG and\nTmIG in S3 and S4 samples compared to the S1andS2\nsamples.\nThe ISHE measurements were carried out by a FMR\nbased setup. The observed voltage signal corresponding\nto the FMR resonance is shown in Fig 4 (a-d) for the\nprepared samples. In general case, for ISHE two layers\nare required, one is the source for spin current i.e., the\nmagnetic layer and the other one is the spin sink i.e., the\nhigh spin orbit coupling (HS) material. In this study, the\nHS layer is the Pt layer where the spin current source is\nthe YIG/TmIG bilayer in the samples S3 and S4. In-\nterestingly, here we have observed ISHE voltage at two\ndistinct FMR resonance field positions corresponding to\nthe resonances and concomitant spin currents mainly ex-\ncited from different layers, i.e., YIG and TmIG layers in\nboth S3 and S4 samples as shown in Fig. 4.\nIn order to extract the spin pumping voltage from the\nrectifications such as anomalous Hall effect (AHE) and\nanisotropic magneto resistance (AMR), the voltage was\nmeasured at different value of the angle ( ϕ) at a step of\n5◦in the range of 0◦to360◦at a constant frequency of 7\nGHz. We deliberately used higher frequency to avoid the\n3 magnon splitting phenomenon which in general takes4\nFIG. 3. (a) fvsHresplot for all the samples. ∆Hvsfplot for (b) YIG and (c) TmIG for respective resonance in FMR signals\nwith corresponding fittings.\nplace at lower frequencies36. Here ϕis defined as the\nangle between the direction of Hand the perpendicular\ndirection to contacts for voltage measurement. The mea-\nsured voltage is fitted by the Lorentzian equation. The\nsymmetric ( Vsym) and antisymmetric ( Vasym) contribu-\ntions of the voltage were extracted. The obtained Vsym\nandVasymvalues were plotted as a function of angle ϕ\n(shown in Fig. 5). It is to be noted that the antisym-\nmetric component is almost negligible compared to the\nsymmetric component. This clearly indicates the domi-\nnance of the spin pumping induced ISHE in the samples.\nMoreover, to quantify the spin pumping voltage and\nother rectification effects the estimated VsymandVasym\nvalues in Fig. 5 were fitted by the given equations (3)\nand (4), respectively37.\nVsym=Vspcos3(ϕ) +VAHE cos(θ)cos(ϕ)\n+VAMR⊥\nsym cos(2ϕ)cos(ϕ)\n+VAMR||\nsym sin(2ϕ)cos(ϕ)(3)\nVasym =VAHE sin(θ)cos(ϕ)+\nVAMR⊥\nasym cos(2ϕ)cos(ϕ)+\nVAMR||\nasym sin(2ϕ)cos(ϕ)(4)\nwhere VspandVAHEare voltages due to spin pumping\nand the anomalous Hall effect. Furthermore, VAMR∥\nasym,sym\nandVAMR⊥\nasym,sym are the parallel and perpendicular com-\nponents of the AMR voltage, respectively. θis the angle\nbetween the electric and magnetic fields of the microwave\nwhich is 90◦.\nThe extracted value from fitting is tabulated in Table\nII. The V∥,⊥\nAMRcomponent is calculated by the following\nformula38\nV∥,⊥\nAMR =r\u0010\nVAMR∥,⊥\nsym\u00112\n+\u0010\nVAMR∥,⊥\nasym\u00112(5)\nFrom Table II, it is further confirmed quantitatively\nthat the spin pumping voltage Vspis the dominating\nFIG. 4. (a)FMR and ISHE spectra for (a) S1, (b) S2, (c) S3\nand S4 samples at 7 GHz.\ncontribution to the measured ISHE voltage. The power-\ndependent data (shown in supplementary file Fig. 2) fur-\nther ensure the spin pumping dominance in all the sam-\nples. It is observed that the Vsphas decreased by one\norder for YIG in S3 sample as compared to the S1 sam-\nple. This is expected as in S3 sample, the spin current\ngenerated at YIG layer has to pass through the TmIG\nlayer before converting to charge current at Pt layer. In\nthis process, there may be dissipation of the spin angular\nmomentumwhichledtothedecreaseinthespinpumping\nvoltage. The Vspisalmost20timeslargerforTmIGinS3\nas compared to the S2 sample. The observed enhanced\nVspfor TmIG in S3 and S4 as compared to the S2 sam-\nple can be ascribed to the interfacial exchange coupling.\nThe presence of interlayer exchange coupling in magnetic\nbilayersystemshasalreadybeenshowntochangetheam-\nplitudes of different ferromagnetic resonance modes due\nto the dynamic exchange field at the interface39,40. As\nobservedfromtheFMRdata, thecouplingofthemagnon\ncurrent at the interface of YIG and TmIG led to the en-\nhancement of the Vspfor TmIG. Interestingly, the Vspis\nmaximum for YIG in S4 sample. This may be ascribed\nto the interfacial exchange coupling between YIG and\nTmIG.5\nTABLE II. Fitted parameters\nSample S1(GGG/YIG/Pt) S2(GGG/TmIG/Pt) S3(GGG/YIG/TmIG/Pt) S4(GGG/TmIG/YIG/Pt)\nYIG TmIG YIG TmIG\nα×10−30.51±0.07 17.00±0.03 1.10±0.0317.00±0.04 2.60±0.01 30.00±0.38\nVsp(µV) 180.0±7.8 0.06±0.01 18.0±0.71.00±0.01220.00±0.52 0.16±0.12\nVasym\nAHE (µV) 17.0±1.5 0.010±0.004 0.94±0.070.16±0.07 3.20±0.82(0.40±0.02)×10−2\nV⊥\nAMR(µV) 12.0±1.0 0.04±0.01 11.0±0.80.58±0.01 1.10±0.16 0.17±0.15\nV∥\nAMR(µV) 0.62±0.04 (0.40±0.04)×10−21.40±0.380.110±0.002 1.40±0.14 0.050±0.006\ng↑↓\neff 4.78×10182.39×10171.56×10171.12×10171.24×10181.76×1018\nFIG. 5. Angular dependence of VsymandVasymwith corre-\nsponding fits for (a) S1, (b) YIG in S3, (c) TmIG in S3, and\n(d) YIG in S4.\nIn this context, in order to quantify the spin current\npropagation, effective spin mixing conductance (g↑↓\neff) is\nevaluated by the following expression using the obtained\ndamping constant value6.\ng↑↓\neff=4π∆αMstFM\ngµB(6)\nwhere MS,tFM, and ∆αare the saturation magneti-\nzation, thickness of magnetic layer and change in Gilbert\ndamping from bilayer to single layer films, respectively.\nThese g↑↓\neffvalues (of the order of 1018m−2) are well\nmatched with the existing literature26. Hence, the inter-\nfacial exchange coupling led to the enhancement of spin\npumping for YIG and TmIG layer41.\nWehavestudiedspinpumpingandISHEforgarnet/Pt\nsystems. The damping analysis exhibits an enhancement\nofαin the bilayer garnet sample. The measured ISHE\nvoltage for YIG/Pt layer is around 180 µV where the ma-\njor contribution is from spin pumping. A decrease in the\nISHE voltage is observed by one order i.e., 17 µV in S3\nwhich is attributed to the presence of TmIG layer play-\ning as a hindrance to the transfer of angular momentumfrom the YIG layer. Whereas, an increase in spin pump-\ning voltage for YIG in S4 and TmIG in S3 is observed as\ncompared to their respective single layers. This may be\nattributed to the interfacial exchange coupling between\nYIG and TmIG. Moreover, further study can be carried\nout to have an insight to interface exchange coupling and\nthe effects of spin pumping on magnon-magnon interac-\ntions.\nACKNOWLEDGEMENT\nS.B., A.S., K. S. R., P.G., and A.M. thank the De-\npartment of Atomic Energy, Department of Science and\nTechnology, Science and Engineering Research Board\n(Grant No. CRG/2021/001245), Government of In-\ndia, Chanakya Post-doctoral fellowship, i-Hub quan-\ntum technology foundation (Sanction Order No. I-\nHUB/PDF/2022-23/04) for providing financial support.\nWork from J. L. and A. 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Maier-Flaig,\nM. Althammer, H. Huebl, R. Gross, R. D. McMichael, M. D.\nStiles, et al., Physical review letters 120, 127201 (2018).\n41J. Liu, Y. Xiong, J. Liang, X. Wu, C. Liu, S. K. Cheung,\nZ. Ren, R. Liu, A. Christy, Z. Chen, et al., arXiv preprint\narXiv:2309.03116 (2023)." }, { "title": "1601.05605v1.Detection_of_spin_pumping_from_YIG_by_spin_charge_conversion_in_a_Au_Ni___80__Fe___20___spin_valve_structure.pdf", "content": "Detection of spin pumping from YIG by spin-charge conversion in a Au jNi80Fe20\nspin-valve structure\nN. Vlietstra and B. J. van Wees\nPhysics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Groningen, The Netherlands\nF. K. Dejene\nMax Planck Institute for Microstructure Physics, Weinberg 2, 06120 Halle(Saale), Germany\n(Dated: June 20, 2021)\nMany experiments have shown the detection of spin-currents driven by radio-frequency spin pump-\ning from yttrium iron garnet (YIG), by making use of the inverse spin-Hall e\u000bect, which is present\nin materials with strong spin-orbit coupling, such as Pt. Here we show that it is also possible to\ndirectly detect the resonance-driven spin-current using Au jpermalloy (Py, Ni 80Fe20) devices, where\nPy is used as a detector for the spins pumped across the YIG jAu interface. This detection mech-\nanism is equivalent to the spin-current detection in metallic non-local spin-valve devices. By \fnite\nelement modeling we compare the pumped spin-current from a reference Pt strip with the detected\nsignals from the Au jPy devices. We \fnd that for one series of Au jPy devices the calculated spin\npumping signals mostly match the measurements, within 20%, whereas for a second series of devices\nadditional signals are present which are up to a factor 10 higher than the calculated signals from\nspin pumping. We also identify contributions from thermoelectric e\u000bects caused by the resonant\n(spin-related) and non-resonant heating of the YIG. Thermocouples are used to investigate the pres-\nence of these thermal e\u000bects and to quantify the magnitude of the Spin-(dependent-)Seebeck e\u000bect.\nSeveral additional features are observed, which are also discussed.\nPACS numbers: 72.25.-b, 75.78.-n, 76.50.+g, 85.75.-d\nI. INTRODUCTION\nEmploying a ferro/ferrimagnetic insulating material\n(FMI) for spintronics research has attracted a lot of in-\nterest in the past years owing to the possibility of gen-\nerating pure spin-currents, without accompanying spu-\nrious charge-currents. Besides, in these materials, it is\nshown that spin information can be transported over\nlarge distances on the \u0016m-scale1or even mm-scale2,3,\nopening up new possibilities for spin-based data stor-\nage and transport. In these devices, Yttrium iron garnet\n(YIG), which is a room-temperature FMI with very low\nmagnetic damping, is most often employed. Together\nwith the (inverse) spin-Hall e\u000bect ((I)SHE) in Pt, it of-\nfers a platform for studying pure spin-current generation,\ntransport and detection. An example of such an exper-\niment is the electrical detection of spin pumping in a\nYIGjPt system, where the resonance of the YIG mag-\nnetization leads to a spin-current pumped into the adja-\ncent Pt layer, which can electrically be detected via the\nISHE.2,4{7\nPure spin-currents can also be generated and detected\nby making use of metallic magnetic jnon-magnetic nanos-\ntructures such as permalloy (Ni 80Fe20, Py) or cobalt.8\nThis method is mostly used in spin-valve structures,\nwhere a spin-current is generated by sending a charge-\ncurrent through one magnet, which can be detected (ei-\nther locally or non-locally) by a second magnetic strip,\nas a change in electric potential when switching between\nrelative parallel and anti-parallel magnetic states of both\nmagnets.9,10In this manuscript, we show that by combining a FMI\n(YIG) with a conducting magnetic material (Py) it is\npossible to electrically detect the magnetic resonance of\nthe FMI, without the need of a high spin-orbit coupling\nmaterial like Pt. Here the magnetic-resonance-induced\ndc spin-current pumped into an adjacent Au layer is de-\ntected as an electrical voltage by a Py detector connected\nto a Au spacer.\nThis alternative method for detection of spin-currents\nfrom FMI-materials opens up new ways of investigating\nthe origin of the spin-Seebeck e\u000bect11without the possi-\nble presence of non-equilibrium proximity magnetization\nin the heavy metal Pt.12{14Besides, because of its anal-\nogy to measuring a conventional spin-valve structure, this\nmethod also helps to determine the sign of the pumped\nspin-current from YIG into Pt,15and expands the pos-\nsibilities for designing devices, including spin transport\nthrough FMI materials.\nIn the experiments we \frst induce magnetic resonance\nin the YIG by sending RF currents through a microwave\nstripline, which is placed near the Au jPy devices that are\nconnected in series to maximize the total signal.\nPart of the build-up potential we attribute to the spin-\ncurrent generation by spin pumping from the YIG into\nthe adjacent structure [schematically shown in Fig. 1(a)].\nHereby we compare spin pumping signals from a standard\nYIGjPt device structure with the signal from YIG jAujPy\ndevices placed in series. Furthermore, we also identify\nsignals that are related to heating and induction e\u000bects,\nwhich are rather small to explain the observed signals.\nIt is found that we not only detect the resonance spin\npumping from the magnetic YIG layer, but also observearXiv:1601.05605v1 [cond-mat.mes-hall] 21 Jan 20162\nthe Py resonance state. This self-detection of FMR by\na Py strip has been observed before,16however, here we\ndiscuss that the mechanism is possibly di\u000berent and re-\nlated to the interaction of spins at the YIG interface.\nII. SAMPLE CHARACTERISTICS\nThe studied devices are fabricated on a 4 \u00024 mm2\nsized sample, which is cut from a wafer consisting of\na 500- \u0016m-thick single crystal (111)Gd 3Ga5O12(GGG)\nsubstrate and a 210 nm thick layer of YIG, grown by\nliquid phase epitaxy (from the company Matesy GmbH).\nThe YIG magnetization shows isotropic behavior of the\nmagnetization in the \flm plane, with a low coercive \feld\nof less than 1 mT (measured by SQUID).\nFig. 1(b) shows a schematic of one device from the\nstudied series, fabricated by several steps of electron\nbeam lithography. It consists of an 8-nm-thick Au layer\ndeposited on YIG by dc sputtering, followed by a 20-nm-\nthick Py layer (30 \u00022:5\u0016m2for area 1, and 60 \u000210\u0016m2\nfor area 2), contacted with a top Ti jAu layer of 5j100 nm,\nboth deposited by e-beam evaporation. To prevent short-\ning between the top and bottom Au layers, when placing\nseveral devices in series, a 60-nm-thick Al 2O3layer was\ndeposited over the edges of the Au jPy stack before the\ndeposition of the top Ti jAu layer. Ar-ion milling has\nbeen used to clean the surfaces and etch the native oxide\nlayer of Py before deposition of the Py and Ti jAu layers,\nrespectively. Fig. 1(c) shows a microscope image of a\nfull series of devices. With the employed meander struc-\nture for the series of devices possible signals generated\nby the ISHE cancel out. In Fig. 1(c) also the reference\nPt-strip (400\u000230\u0016m2, 7-nm-thick, dc sputtered) can be\nseen, placed below and perpendicular to the 60 \u0016m wide\nTijAu microwave stripline (5 j100 nm thick) used to ex-\ncite the magnetization resonance in the magnetic layers.\nBetween the Pt-strip and the microwave stripline a 60-\nnm-thick Al 2O3layer (dark brown) is added, to prevent\nelectrical shorts. Fig. 1(d) and 1(e) show a close-up of\nthe devices in area 1 and 2, respectively.\nFor the spin pumping experiment using the YIG jPt\nsystem, the obtained signal scales linearly with the length\nof the Pt detection strip. For the YIG jAujPy devices\nthe detected signal is not directly scalable by the size\nof the device, rather by the number of YIG jAujPy de-\nvices connected in series. As the expected signal for one\nYIGjAujPy device is below the signal-to-noise ratio of\nour measurement setup, we fabricated a structure where\nwe increase the detected signal by placing many separate\ndevices in series. Two sets of devices were investigated,\nhaving di\u000berent surface area, as is shown in \fg. 1(c).\nFor the presented experiments 96 (area 1) and 62 (area\n2) YIGjAujPy devices in series were used.\n(c) \n(e) BVISHEVISHE\nYIG Ti/Au \nPy Al2O3\nAu Ti/Au Au \nYIG MYIG Au V\nPy MPy \n(d) Stripline\n+-\nVPy -VPy+Pt VPy -\nVPy+\nArea 1Area 2\n(b) (a) \nTi/Au \nAl 2O3 Py Au \nArea 1\n Area 2Ti/Au \nAl 2O3 Py Au 100 µmFIG. 1. (a) Schematic representation of the spin pumping\nprocess and voltage detection in a YIG jAujPy device. (b)\nSchematic drawing of one YIG jAujPy device. Each device\nconsists of Au (8 nm), Py (20 nm), Al 2O3(60 nm), and TijAu\n(5j100 nm) layers. (c) Microscope image of the \fnal device\nstructures. In area 1 (area 2) 96 (62) YIG jAujPy devices are\nplaced in series. The Pt strip is used as a reference for the\nmeasurements on the YIG jAujPy devices, and this strip is\nelectrically insulated from the stripline by an Al 2O3layer (60\nnm thick). During the measurements, an external magnetic\n\feld is applied and three separate sets of voltage probes are\nconnected as pointed out in the \fgure. (d) and (e) show close-\nups of the devices placed in area 1 and 2, respectively. The\ndi\u000berent material layers are marked.\nIII. MEASUREMENT METHODS\nThe microwave signal is generated by a Rohde-Schwarz\nvector network analyzer (ZVA-40), connected to the\nwaveguide on the sample via a picoprobe GS microwave\nprobe. To be able to use the low-noise lock-in detection\nmethod, the power of the applied microwave signal is\nmodulated between `on' and `o\u000b' using a triggering sig-\nnal which is synchronized with the trigger of the lock-in\nampli\fers. Typically used `on' and `o\u000b' powers are 10\ndBm and -30 dBm, respectively. The RF frequency is\n\fxed for each measurement (ranging from 1 GHz to 10\nGHz), while sweeping the static in-plane magnetic \feld.\nDuring this magnetic \feld sweep, the voltage from the\nPt strip and both series of Py devices are separately\nrecorded by connecting them to three di\u000berent lock-in\nampli\fers, using the connections as is shown in Fig. 1(c).\nThe ISHE voltage detected from the Pt strip is used as\na reference for the data obtained from the Py devices.\nAs the power is modulated during all measurements, the\nabsorbed power cannot be measured simultaneously, and\ntherefore no detailed information is obtained about the\ndependence of power absorption on applied RF frequency.\nThe frequency dependence of the absorbed power was ob-3\n(a)\n(b)-120-80-4004080120\n-101234567\n-300 -200 -100 0 100 200 300-0.4-0.3-0.2-0.10.00.10.2 VISHE [µV]Frequency [GHz]\n 1 5 9 \n 2 6 10\n 3 7 \n 4 8\nP = 10mW\nArea 1VPy [µV]\nArea 2VPy [µV]\nMagnetic Field [mT]-0.50.00.51.01.5\n0 20 40 60 80-0.2-0.10.00.1-120-80-400VPy [µV]\nMagnetic Field [mT] VISHE [µV] VPy [µV]\n(f)(e)(d)\n(c)\nFIG. 2. Magnetic \feld sweeps for di\u000berent applied frequen-\ncies, with an applied RF power of 10 mW for (a) the Pt strip,\ndetecting the ISHE voltage and (b), (c) voltage generated by\nthe series of Py-devices in area 1 and 2, respectively. (d)-(f)\nclose-up of marked areas in (a)-(c).\ntained by separate measurements of the S 11parameter as\nin Ref.6. All measurements are performed at room tem-\nperature.\nIV. RESULTS AND DISCUSSION\nResults of the magnetic \feld sweeps for di\u000berent RF\nfrequencies are shown in Fig. 2 where the ISHE voltage\nsignal detected by the Pt strip [Fig. 2(a)] and the corre-\nsponding signals from the series of Py devices in area 1\n[Fig. 2(b)] and area 2 [Fig. 2(c)] are shown. The detected\nvoltage of the Pt strip shows the expected peaks for YIG\nresonance, changing sign when changing the magnetic\n\feld direction, as also observed in for example Refs.4,6,17.\nThe magnitude of the peaks is in the order of 100 \u0016V.\nComparing these results to the data shown in Figs. 2(b)\nand 2(c); peaks at exactly the same position are observed\nfor the Py devices. Here the detected peaks do not change\nsign by changing the magnetic \feld direction, as the sign\nof the signal in these devices is determined by the relative\norientation of the YIG and Py magnetization. Because\nof the low coercive \felds ( <10 mT) of both the YIG and\nPy layers, their magnetizations always align parallel to\neach other when applying a small magnetic \feld.\nAt \frst glance, it appears that when the YIG is excited\ninto resonance conditions, the pumped spin-current into\nthe Au layer is detected as an electrical voltage gener-\nated by a conducting ferromagnet placed on top of the\nAu layer. Fig. 3 shows the frequency dependence of the\nmagnitude of the observed peaks at YIG resonance for all\nthree types of devices (Pt strip and Py devices area 1 and2). A few points can be made regarding these dependen-\ncies. First, where the dependence of the Pt shows some\nanalogy with the dependence observed in area 2, the fre-\nquency dependence of the signals from area 1 (narrow\nPy strips) and area 2 (wider Py strips) largely di\u000bers.\nBesides, the signals from area 1 and 2 show about one\norder di\u000berence in magnitude. These observations show\nthat by changing the surface area of the YIG jAujPy de-\nvices, di\u000berent physics phenomena can be present. In\nsection IV B we calculate the contribution of spin pump-\ning to the observed signals in both Py device areas and\ndiscuss these results.\nSecondly, from Fig. 3, we observe that the signals\nfor positive and negative applied magnetic \felds consis-\ntently di\u000ber. Interestingly, for area 1 the higher signals\nare observed for positive applied \felds, whereas for area\n2 the higher signals appear for negative applied \felds.\nAt \frst glance, the device is fully symmetric and there-\nfore one would not expect any dependence on the sign of\nthe applied magnetic \feld. Further investigation leads to\nthe existence of nonreciprocal magnetostatic surface spin-\nwave modes (MSSW), whose traveling direction (perpen-\ndicular to the stripline) is determined by the applied mag-\nnetic \feld direction.18,19As the spin pumping process in\ninsensitive to the spin-wave mode, and the device areas\n1 and 2 are placed on opposite sides of the stripline, the\npresence of these unidirectional MSSWs can lead to the\nobserved di\u000berence in peak-height for positive and nega-\ntive applied \felds.\nBesides the peaks at YIG resonance, the Py devices\nshow more peaks at lower applied magnetic \felds (most\nclearly visible in area 1, Fig. 2(b)), these peaks are at-\ntributed to the ferromagnetic resonance of the magnetic\n(a)\n0 2 4 6 8 100.00.10.20.30.40.5Peak Height Py [ µV]Area 2\n -B\n +B\nFrequency [GHz](c) (b)0 2 4 6 8 10406080100120140\nFrequency [GHz]Pt\n -B\n +B Peak height Pt [ µV]\n0 2 4 6 8 10012345Peak Height Py [ µV]\n Area 1\n -B\n +B\nFrequency [GHz]\nFIG. 3. Magnitude of the peaks at YIG resonance obtained\nfrom the measurements shown in Fig. 2, for positive and\nnegative applied magnetic \felds as a function of applied RF\nfrequency. (a) For the Pt strip, (b) and (c) for the Py devices\nof area 1 and 2, respectively.4\nPy layers, as will be shown in section IV A and further\ndiscussed in section IV E.\nA. Position of the resonance peaks\nTo check the position of the observed peaks with\nrespect to the predicted resonance conditions of the\nmagnetic layers (YIG and Py), the Kittel equation is\nused:20,21\nf=\r\n2\u0019q\n(B+Nk\u00160Ms)(B+N?\u00160Ms); (1)\nwhere\ris the gyromagnetic ratio ( \r= 176 GHz/T), B\nis the applied magnetic \feld, NkandN?are in-plane\nand out-of-plane demagnetization factors and \u00160Msis\nthe saturation magnetization. Taking Nk\u00160Ms= 10 mT\nandN?\u00160Ms= 1:1 T (consistent with previously re-\nported values for Py)16,22, we obtain the red solid curve\nshown in Fig. 4, and \fnd that the position of the inner\npeaks, detected only for the series of Py devices, matches\nthe Py resonance conditions (the shown data is obtained\nfrom area 1). Therefore, the origin of the inner peaks is\nrelated to the Py layers being in ferromagnetic resonance.\nThe position of the outer peaks, detected for both se-\nries of Py devices, corresponds to the voltage peaks ap-\npearing in the Pt strip, which are ascribed to the YIG\nmagnetization being in resonance. To check the expected\nYIG resonance conditions, a more simpli\fed form of the\nKittel equation can be used, because of its isotropic in-\nplane magnetization behavior ( Nk= 0 andN?= 1):\nf=\r\n2\u0019p\n(B(B+\u00160Ms): (2)\nFor the curve corresponding to the YIG resonance\npeaks as shown in Fig. 4, \u00160Ms= 176 mT is used,\n0 2 4 6 8 10 12050100150200250300350\n Magnetic Field [mT]\nFrequency [GHz]Kittel Equation:\n YIG\n Py \nMeasured:\nYIG/Pt peak\nPy outer peak\nPy inner peak\nFIG. 4. The measured peak positions for both the Py devices\narea 1 (red symbols) as well as the Pt strip (black symbols),\nplotted together with the calculated resonance conditions by\nthe Kittel equation [Eqs. (1) and (2)].which is the reported bulk saturation magnetization of\nYIG.2,6,23The close match of the calculated curve and\nthe measured data proves that the outer peaks from the\nPy devices and the peaks from the Pt strip indeed origi-\nnate from the YIG magnetization being in resonance.\nB. Estimation of the spin pumping signal in\nYIG jAu jPy devices\nIn this section we will only focus on the origin of the\npeaks at YIG resonance. The possible origin of the de-\ntected inner peaks for the Py devices will be discussed in\nsection IV E. Using the data from the Pt strip as a ref-\nerence, we calculate the expected signal caused by spin\npumping in the YIG jAujPy devices. Di\u000berent steps in\nthis calculation are: 1) calculate the pumped spin-current\ndensity from the ISHE voltage detected by the Pt strip, 2)\nobtain an estimate for the spin-mixing conductance of the\nYIGjAu interface, and 3) use a \fnite element spin trans-\nport model to \fnd the expected spin pumping signal in\nthe YIGjAujPy devices, setting the results of 1) and 2) as\nboundary conditions and input parameters, respectively.\nFor the calculations we assume that the spin accumula-\ntion in the layer adjacent to the YIG, de\fned as the ratio\nbetween the injected spin-current Jsand the real part of\nthe spin-mixing conductance Gr, is constant when con-\nsidering di\u000berent types of interfaces and devices.24\nAs the magnitude of the excited spin-waves decays\nwith distance from the stripline, we fabricated another\nsample, where a 6-nm-thick Pt strip and a Au jPt strip\n(8j6 nm) were placed exactly on the location of the series\nof Py devices in the previous batch, as is shown in Fig. 5\n(The dimensions of these strips are 334 \u000230\u0016m). Using\nthis sample, the average injected spin-current on the loca-\ntion of the Py devices can directly be calculated. Signals\nobtained from these new devices show similar behavior\nas the data shown in Fig. 2(a) and Fig. 3(a), only the\nmagnitude of the signals di\u000bers: The Pt (Au jPt) strip on\nthe new sample results in ISHE-voltages around 30 \u0016V (3\n\u0016V), compared to 120 \u0016V for the Pt strip directly below\nthe RF line. These signi\fcantly lower signals prove the\ndecay of spin-waves with distance from the stripline. The\nsignal for the AujPt strip is even more suppressed com-\npared to the Pt strip, as the Au layer short-circuits the\nstructure because of its lower resistance compared to Pt,\nwhile the inverse spin-Hall voltage is mainly generated in\nthe Pt layer. In the following calculation, the Au jPt strip\nis modeled using the 3D \fnite-element modeling software\nComsol Multiphysics, taking into account these losses.\nStep 1: From the measured ISHE-voltage VISHE in the\nPt strip, the injected spin-current density can be calcu-\nlated by\nJs=VISHEtPt\nl\u001a\u0015\u0012 SH1\n\u00111\ntanh(tPt\n2\u0015); (3)\nwheretPt,l,\u001a,\u0015, and\u0012SHare the thickness (6 nm),\nlength (330 \u0016m), resistivity (3 :5\u000210\u00007\nm), spin relax-5\nB\nV+V-\nV+V-100 µm\nPt Au/Pt Pt \nFIG. 5. Microscope image of the Pt and Au jPt strips used\nto estimate the injected spin-currentby spin pumping at the\nlocation of the Py-devices.\nation length (1.2 nm) and the spin-Hall angle (0.08) of the\nPt strip, respectively, and \u0011= [1+Gr,Pt\u001a\u0015coth(tPt\n2\u0015)]\u00001is\nthe back\row term,25whereGr,Ptis the spin-mixing con-\nductance of the YIG jPt interface (4 :4\u00021014\n\u00001m\u00002).\nThe given material properties are taken from previously\nreported spin-Hall magnetoresistance measurements.26\nUsing Eq. (3), an injected spin-current density of 2 :0\u0002\n107A/m2is found for a detected VISHE of 30 \u0016V.\nStep 2: The spin-mixing conductance of the YIG jPt\ninterface di\u000bers from the YIG jAu interface, as present\nin the measured YIG jAujPy devices. Therefore we use\nthe obtained signals from the combined Au jPt strip to\n\fnd an estimate of the spin-mixing conductance of the\nYIGjAu interface Gr,Au. Spin pumping into the Au jPt\nstrip results in ISHE signals in the order of 3 \u0016V. Using\nComsol Multiphysics, we model the YIG jAujPt device,\nincluding spin-di\u000busion by the two-channel model and\nthe spin-Hall e\u000bect as explained in Ref.27. To include\nthe contribution of the spin-mixing conductance in this\nmodel, such that back\row is accounted for, a thin inter-\nface layer (t= 1 nm) is de\fned between the YIG and Au\nlayers (resulting in a stack: YIG jinterfacejAujPt). The\ninterface layer acts as an extra resistive channel for the in-\njected spin-current, parallel to the spin-resistance of the\ndevice on top (in this case the Au jPt strip), such that\nthere e\u000bectively are two spin-channels: one for back\row\nand one for injection into the Au layer. The conductivity\nof this interface layer is de\fned as \u001bint=Gr,Au\u0001t. The\ninput parameter at the interface jAu boundary is the dc\nspin-current Jsobtained in step 1 for the YIG jPt device.\nBy scaling Jswith the ratio of Gr,AuandGr,Ptwe take\ninto account that the injected spin-current is lower when\nthe spin-mixing conductance is lower. We now tune the\nvalue ofGr,Auin the model such that the modeling re-\nsult matches the measured VISHE of the AujPt strip. By\ndoing so we \fnd Gr,Au= (2:2\u00060:2)\u00021014\n\u00001m\u00002, in\norder to match the measured VISHE of the AujPt strip.\nThis value is similar as reported by Heinrich et al. ,28who\nobtainedGr,Au-values up to 1 :9\u00021014\n\u00001m\u00002. Used\nmodeling parameters for the Au layer are \u001bAu= 6:8\u0002106\nS/m and\u0015Au= 80 nm.29The modeling parameters for\nthe Pt layer are as mentioned above.\nBy replacing the Pt layer by the Au jPt strip, we \fnd\nthat the back\row spin-current almost doubles (around75% increase). This increased back\row is mainly caused\nby the larger spin-di\u000busion length in Au as compared to\nPt, resulting in a higher spin-resistance for the injection\nof spins into the thin Au layer, as compared to the back-\n\row spin-channel (In other words: Pt is a better spin-\nsink). Furthermore, the initially injected spin-current is\na factor 2 lower for the YIG jAujPt strip, compared to\nYIGjPt, caused by the lower spin-mixing conductance.\nThese two cases result in intrinsically lower signals when\nplacing Au on top of YIG as compared to Pt.\nStep 3: After having calculated the injected spin-\ncurrent and the spin-mixing conductance of the YIG jAu\ninterface, these parameters are used as input for the Com-\nsol Multiphysics model of one YIG jAujPy device. This\nmodel is again based on the two-channel model for spin\ntransport, including an additional interface layer to add\nback\row to the model, as described above. Detailed in-\nformation about the modeling and the used equations can\nbe found in Ref.27. The di\u000berent sizes of devices present\nin area 1 and area 2 are both separately modeled.\nThe properties of the Py layer added to the model are\nP= 0:3 (de\fning the spin polarization), \u001bPy= 2:9\u0002\n106S/m and\u0015Py= 5 nm.29The spin-current density\nobtained from Eq. (3) is used as input, and is set as a\nboundary \rux/source term at the bottom interface of the\nAu layer. As explained in step 2, also here a thin interface\nlayer is placed below the Au layer, such that it acts as\na spin-current channel parallel to the injection of spin-\ncurrent into the device. The above estimated value for\nGr,Auis used to de\fne the interface layer conductivity.\nAlso from this model we \fnd a large back\row from the\ninitially injected Js, which is mainly caused by the spins\naccumulating at the Au jPy interface, increasing the spin-\nresistance in the injection-channel with respect to the\nback\row spin-channel.\nTo obtain the dependence on RF frequency of the ex-\npected voltage signal, Jswas calculated from Eq. (3) for\neach frequency, using the measurements on the YIG jPt\nstrip, and the YIG jAujPy model was run for each ob-\ntainedJs. The calculated voltage signal caused by spin\npumping for one Py device was multiplied by 96 (62), the\nnumber of devices placed in series in area 1 (area 2), and\nthe \fnal results of these calculations are shown as the red\ncurve in Fig. 6(a) and 6(b), together with the absolute\nvalues of the measured peaks, shown in black, for area 1\nand 2, respectively.\nFor the Py-devices in area 1, the calculated spin pump-\ning voltages are one order of magnitude smaller than the\nmeasured signals, from which we conclude that the major\ncontribution of the measured signal is caused by another\nsource than spin accumulation created by spin pump-\ning. Additionally, the calculated signals generated by\nspin pumping clearly show a di\u000berent dependence on fre-\nquency, as compared to the measured signals from the\nYIGjAujPy devices. This also indicates that besides spin\npumping there are other phenomena present which in-\nduce voltage signals in our devices.\nInterestingly the peaks obtained from the devices in6\n012345\n0 100 200 3000.00.20.40.6\n0 50 100 150 200 250 3000.00.10.20.3\n Vpeak [µV] Spin Pumping \n from Model \n Vpeak [µV]\nMagnetic Field [mT]Area 1\nArea 2(a)\n(b)\nFIG. 6. Calculated spin pumping voltage (red squares) ver-\nsus measured signal (black peaks) as a function of applied RF\nfrequency and magnetic \feld. The peaks from left to right\ncorrespond to applied RF frequencies from 1 to 10 GHz, re-\nspectively, directly copied from the measurements shown in\nFig 2. (a) Results for the Py-devices in area 1. The inset\nshows a close-up of the calculated spin pumping voltage. (b)\nResults for the Py-devices in area 2.\narea 2 show a very good agreement with the calculated\nspin pumping signals. From these results we conclude\nthat in this case the major contribution of the measured\nsignals is caused by spin pumping.\nComparing the results from area 1 and area 2, it is clear\nthat in the experiments the exact device geometry largely\nin\ruences the signal and even results in a totally di\u000berent\ndependence on applied RF frequency. From the calcula-\ntions of the spin pumping signals, such a big change in\nbehavior cannot be reproduced and therefore additional\nphenomena must be present and becoming more promi-\nnent for more narrow devices, as present in area 1.\nAs thermoelectric e\u000bects might also play a role in\nthe performed experiments, next section discusses some\nfurther investigation of possible signals related to RF-\nheating.\nC. Thermal e\u000bects\nWhile an RF current \rows through the stripline, heat is\nabsorbed by the YIG layer causing the YIG temperature\nto rise. Together with Eddy currents that are induced\nin the Py devices, which can result in Joule heating, this\npower absorption leads to local heating. Besides heat-\ning at the non-resonance conditions, especially at mag-\nnetic resonance additional heat will be dissipated into\nthe YIG due to the continuous damping of the YIG mag-\nnetization precession.30The generation of temperature-\n(a) (b)\n100 µmV+ V-\nB\n0 2 4 6 8 100.00.20.40.60.8\n VBackground [µV]\nFrequency [GHz](c)-300 -200 -100 0 100 200 3000.40.60.81.01.2\n VThermocouple [µV]\nMagnetic Field [mT]F = 7 GHz\nP = 10 mW\n(d)\n0 2 4 6 8 100.00.10.20.30.40.50.6 VPeak [µV]\nFrequency [GHz]05101520\n∆T [mK] +B\n -BFIG. 7. (a) Microscope image of the thermo-couples placed\nnear the microstripline. Below all contact-leads a 70-nm-thick\nAl2O3layer is present (visible as the dark areas), to avoid\nspurious signals generated in these leads. Only the Au pad\nin the center of each thermo-couple is in direct contact to the\nYIG substrate. Each Au pad is contacted with a 40-nm-thick\nPt lead on the left side and a 40-nm-thick NiCu lead on the\nright side. (b) Detected voltage from the thermocouple most\nclose to the stripline for F= 7 GHz and P= 10 mW. (c)\nFrequency dependence of the detected peaks for positive and\nnegative applied magnetic \felds. The right axis shows the\ncorresponding temperature change \u0001 T= \u0001V=(SPt\u0000SNiCu).\n(d) Dependence of the thermo-couple background voltage on\napplied frequency. (c) and (d) are both for P= 10 mW and\nfor the thermo-couple most close to the stripline.\ngradients caused by local heating gives rise to thermo-\nelectric e\u000bects such as the Seebeck e\u000bect (caused by the\ndi\u000berence in Seebeck coe\u000ecient of Au and Py), the spin-\ndependent-Seebeck e\u000bect (SdSE) (due to the spin depen-\ndency of the Seebeck coe\u000ecient in Py, resulting in ther-\nmal spin injection at the Au jPy interface), and the spin-\nSeebeck e\u000bect (SSE) (spin pumping caused by thermally\nexcited magnons in the YIG, leading to spin accumula-\ntion at the YIGjAu interface).\nIn order to probe the RF induced heating, NiCu jPt\nthermocouples were placed near the stripline as is shown\nin Fig. 7(a). In this way, the temperature of the sub-\nstrate can locally be measured by making use of the See-\nbeck e\u000bect. From the measured thermo-voltage signals\nthe increase in temperature at the NiCu jPt junction with\nrespect to the reference temperature of the contact pads\ncan be obtained using \u0001 V=\u0000(SPt\u0000SNiCu)\u0001T, where\nSPt=\u00005\u0016V/K andSNiCu =\u000032\u0016V/K are the See-\nbeck coe\u000ecient of Pt and NiCu, respectively.30Besides\na constant background voltage signal, indicating heating\nwhen the YIG magnetization is not in resonance, clear\npeaks are observed at the YIG resonance conditions, as\nis presented in Fig. 7(b) for an applied RF frequency of\n7 GHz. The magnitude of the peaks at resonance is in\nthe order of 40 nV ( F= 1 GHz,P= 10 mW, distance\nfrom microstrip 195 \u0016m) up to 0.6 \u0016V (F= 10 GHz,\nP= 10 mW, distance from microstrip 50 \u0016m); Higher\nsignals were measured for higher frequencies and for de-\nvices closer to the RF line.7\nFig. 7(c) shows the extracted peak-height of the mea-\nsurements for the thermocouple most close to the mi-\ncrostrip (50 \u0016m), and the corresponding temperature in-\ncrease is added on the right vertical axis. Additionally,\nFig. 7(d) gives the evolution of the background voltage\nas a function of applied RF frequency. A maximum tem-\nperature increase at resonance conditions of 22 mK is ob-\nserved in this thermocouple. Interestingly, all measure-\nments show di\u000berent behavior at the YIG resonance con-\nditions for positive and negative applied magnetic \felds;\nConsistently, the peak at negative applied magnetic \felds\nis larger than the one for positive \felds, increasing to a\nfactor 2 in magnitude at F= 10 GHz.\nThese observations are in agreement with the dif-\nference in peak-height between positive and negative\napplied magnetic \felds as observed for the Py-devices\nin area 2, which are placed on the same side of the\nstripline as the thermocouples. Also here the existence of\nnonreciprocal magnetostatic surface spin-waves (MSSW)\nmight explain the observed behavior, as they will in\ru-\nence the heating and lead to unidirectional heating of\nthe substrate, as observed by An et al. ,31who used a\nmeasurement con\fguration very similar to the one we\ndescribe in this paper.\nThe thermocouple measurements show non-negligible\nheating of the YIG surface, and therefore thermal ef-\nfects are likely to play a role in the observed voltage\ngeneration in the YIG jAujPy devices. To obtain the\nquantitative contribution of the Seebeck e\u000bect, SdSE,\nand SSE in the studied device geometry, it is needed\nto model the YIG jAujPy device, including its thermal\nproperties. However, from the observed dependence on\napplied RF frequency of the thermocouple signals, even\nwithout knowing the expected quantitative contribution\nof the thermal e\u000bects, it can be concluded that these ef-\nfects cannot explain the large signals in the YIG jAujPy\ndevices of area 1. As observed for spin pumping, the\nthermocouple signals saturate at higher RF frequencies,\nwhereas the series of permalloy devices in area 1 shows\na continuously increasing signal. This indicates that ad-\nditional e\u000bects are present, which scale linearly with fre-\nquency, such as for example inductive coupling, where\nthe RF current in the microstrip induces a current in the\nPy, causing Joule heating.\nD. Finite element simulation of the SdSE and SSE\nTo determine the contribution of the SdSE (at the\nAujPy interface) and the SSE (at the YIG jAu interface)\nwe performed a three-dimensional \fnite element (3D-\nFEM) simulation of our devices27where the charge- ( ~J)\nand heat- ( ~Q) current densities are related to the cor-\nresponding voltage- ( V) and temperature- ( T) gradients\nusing a 3D thermoelectric model. Details of this model-\ning can be found in Refs.27,29,32{34. The input material\nparameters, such as the electrical conductivity, thermal\nconductivity, Seebeck coe\u000ecients and Peltier coe\u000ecientare adopted from Table I of Refs.32,33.\n1. SdSE\nIn the SdSE, the heat current \rowing across the Au jPy\ninterface causes the injection of spins which are anti-\naligned to the magnetization of the Py layer. In our\nmodeling, we use the temperature values measured by\nthe NiCujPt thermocouples, shown in Fig. 7(c), as a\nDirichlet boundary condition in the 3D-FEM.\nSpeci\fcally, for a given microwave power and fre-\nquency, by \fxing the temperature of the Au jYIG in-\nterface to the measured values and anchoring the leads\n(TijAu contacts) to the reference temperature, we can\ncalculate the resulting temperature gradient rTFin the\nPy and hence the SdSE voltage. From this model, for a\nsingle AujPy interface, a total spin-coupled voltage drop\nof approximately \u00001 nV is obtained, which corresponds\nto\u000096 nV for the series of Py devices in area 1.\nWe can also compare this result with one obtained from\na simple one-dimensional spin-di\u000busion model using the\nfollowing equation35\n\u0001Vs=\u00002\u0015sSSrTPyP\u001bRm; (4)\nwhere\u0015s= 5 nm is the spin-di\u000busion length, SS=\n\u00005\u0016V/K is the spin-dependent Seebeck coe\u000ecient, P\u001b=\n0:3 is the bulk spin-polarization, rTPy= 105K/m is the\ntemperature gradient in the Py, obtained from a simple\n1D heat di\u000busion model across the interfaces, and Rm\nis a resistance mismatch term which is a value close to\nunity for such metallic interfaces considered here. The es-\ntimated signal from this 1D-model is a factor two larger\nthan that obtained from the 3D-FEM. Note that, in\nboth modeling schemes, the distance dependence from\nthe strip-line has not been taken into account, which\nwould lead to an even lower signal. Therefore, we con-\nclude that the SdSE does not contribute signi\fcantly to\nthe measured signal.\n2. SSE\nTo estimate the maximum contribution of the SSE,\ncaused by spin pumping due to thermal magnons, we\nneed to obtain the temperature di\u000berence between the\nmagnons and electrons \u0001 Tmeat the YIGjAu interface.\nWe again use the 3D-FEM, but this time, extended to\ninclude the coupled heat transport by phonons, elec-\ntrons and magnons with the corresponding heat exchange\nlengths between each subsystem. The detailed descrip-\ntion of this model, which was used earlier to describe the\ninterfacial spin-heat exchange at a Pt jYIG interface, can\nbe found in Ref.33along with the used modeling param-\neters.\nIn our model, we set the bottom of the GGG substrate\nto the surrounding (phonon) temperature T0, the AujPy8\ninterface at the equilibrium temperatures of both elec-\ntrons and phonons, i.e., Te=Tph=T0+ 20 mK while\nusing a magnon heat conductivity \u0014m= 0:01 Wm\u00001K\u00001\nand phonon-magnon heat exchange length \u0015m-ph = 1 nm.\nFrom our 3D-FEM model, we \fnd an interface temper-\nature di\u000berence of \u0001 Tme= 25 \u0016K between the magnon\nand electron subsystems. While \u0001 Tmeat the YIGjAu\ninterface seem a rather small value, the comparison with\nearlier reports indicate equivalence between the ratio of\n\u0001Tmeto the temperature increase of the YIG \u0001 Tph.\nIn the SSE, the spin-current density Jspumped across\nthe YIGjAu interface can be obtained using Js=\nLS\u0001Tme, whereLS=Gr\r~kB=2\u0019MsVa= 7:24\u0002109\nAm\u00002K\u00001is the interface spin Seebeck coe\u000ecient11,33\nwhereGr,Ms, andVaare the real part of the spin-mixing\nconductance per unit area, the YIG saturation magneti-\nzation, and the magnetic coherence volume (3pVa= 1:3\nnm)25, respectively. Because the thickness t of the Au\nis much smaller than the spin di\u000busion length \u0015in\nAu, we can assume a homogeneous spin accumulation\n\u0001\u0016=Jst\u001a. The voltage drop at the Au jPy interface is\nthus \u0001VSSE=P\u001b\u0001\u0016=2, which gives the maximum SSE\nvoltageVSSEdetected by the Py. Using \u0001 Tme= 25 \u0016K,\nwe obtainVSSE= 54 pV for a single Py device and a total\nof 5 nV for 96 Py serially connected devices. From this\ndiscussion we conclude that the combined voltage contri-\nbutions from the SSE and the SdSE cannot explain the\nobserved enhancement at higher frequencies, suggesting\nthat there are additional e\u000bects that need to be consid-\nered here.\nE. Discussion\nBesides the measured signals at YIG resonance, a few\nother present features need attention. First, at low ap-\nplied frequency and magnetic \feld, the voltages gener-\nated by the Py-devices in series in area 1 show resonance\nbehavior, as is clearly visible in Fig. 2(e). These res-\nonating signals decrease and \fnally disappear for higher\nfrequencies. In area 2 also some small resonances are ob-\nserved [see Fig. 2(f)], but these resonances are far less\nprominent. The origin of the resonances might be related\nto the fact that this system, like the YIG jPt system, is\nnot only sensitive to the ferromagnetic resonance (FMR)\nmode, but to any spin-wave mode. This means that addi-\ntional signals can appear when multiple spin-wave modes\nexist, which might be more strongly present at lower fre-\nquencies, and for some reason more sensitively detected\nby the narrow strips of Py-devices in series in area 1 as\ncompared to the wider Py-devices and the YIG jPt sys-\ntem. Furthermore, for the lower frequencies the YIG\nresonance conditions are very similar to those of the Py\nlayer, which could lead to coupling between those states,\nresulting in a broader range of possible resonance mag-\nnetic \felds for a certain applied frequency.\nA second feature that needs attention is the back-\nground signal for the YIG jAujPy devices. The magni-tude of the background signal increases with the applied\nRF power, similar in magnitude as the resonance-peaks,\nonly having opposite sign. From the evolution of the\nbackground heating, as depicted in Fig. 7(d), it is ob-\nserved that the background heating decreases by increas-\ning the RF frequency. Therefore it is not possible to di-\nrectly attribute the measured background signals of the\nYIGjAujPy devices to heating of the substrate, and the\norigin of these signals is still unclear.\nThird are the peaks observed at Py resonance con-\nditions. As a control experiment the same sequence of\nYIGjAujPy devices was fabricated on a Si jSiO2substrate,\nincluding the waveguide and stripline. In these samples\nno resonance peaks were present, neither at the YIG res-\nonance conditions nor at the Py resonance conditions.\nThis experiment proves the need of the YIG substrate in\norder to detect Py resonance.\nA possible explanation of the origin of the detected\npeaks is as follows: By having the Py magnetization in\nresonance, a pure spin-current is pumped into the adja-\ncent layers. The polarization of this spin-current consists\nof both an ac- and a dc-component. The spin-current\npumped into the upper contact will relax and does not\ngive rise to any signal. The spin-current pumped into\nthe thin Au layer below the Py strip will not relax before\narriving at the YIG jAu interface. At this interface the\ncomponent of the spin angular momentum perpendicular\nto the YIG-magnetization (here the ac-component of the\npumped spins) will be absorbed and the parallel com-\nponent (the dc-component of the pumped spins) will be\nre\rected (as is the case for the spin-Hall magnetoresis-\ntance). This interaction with the YIG jAu interface re-\nsults in only the dc-component of the initially pumped\nspin-current being re\rected. The re\rected spins will dif-\nfuse back to the Py strip, where they accumulate, as\ntheir polarization direction is changed as compared to\nthe spins being pumped. This spin-accumulation results\nin a build-up potential, which is measured. In the case\nYIG is replaced by SiO 2this mechanism does not work,\nas the absorption and re\rection of spins at the SiO 2jAu\ninterface is not spin-dependent.\nFinally, to prove the obtained signals are caused by\nspin pumping, and detected by the di\u000busion of the gen-\nerated spin accumulation to the Py layers, as in typical\nnon-local spin-valve devices, the observed peaks should\nchange sign when the magnetization directions of the\nYIG and Py are placed anti-parallel, as is shown in Fig.\n8(a). In the experiments, this situation turned out to be\nhard to accomplish, as the switching \felds of both YIG\nand Py are relatively low: for YIG smaller than 1 mT,\nand for the 20-nm-thick Py strips on YIG a maximum\nswitching \feld of 10 mT was obtained for Py dimen-\nsions of 0:3\u00026\u0016m2. So when sweeping the magnetic\n\feld, only \felds between 1 and 10 mT will result in anti-\nparallel alignment of the magnetic layers. Besides this\n\feld window being rather narrow, the resonance peaks\nin this regime are no clear single peaks [see Fig. 2(e)].\nNevetheless, one batch of samples was fabricated where9\n-15 -10 -5 0 5 10 15-0.3-0.2-0.10.00.10.20.3\nF = 1 GHz \nP = 10 mW\nPy [0.3 x 6 µm2]VPy [µV]\nMagnetic Field [mT]Py \nYIG\n-10 0 10\nMagnetic Field [mT](b) (a) \nFIG. 8. (a) Theoretically expected resonance peaks, when in-\ndividual switching of the YIG and Py layers is obtained. The\ninsets show the magnetization orientation of the YIG and Py\nlayers for each resonance peak. (b) Measurement result for\na batch of samples having smaller Py-strips (0 :3\u00026\u0016m2) as\ncompared to the afore described devices, in order to observe\nthe anti-parallel state of the YIG and Py magnetization direc-\ntions. The trace and retrace of the measurement are marked\nby the black and red data, respectively.\nthe Py dimensions were set to 0 :3\u00026\u0016m2, and one result-\ning measurement is shown in Fig. 8(b). While sweeping\nthe magnetic \feld from negative to positive values (black\nline), the resonance peak clearly changes sign. However,\nfor the reverse \feld sweep (red line) the expected switch-\ning of the peak is not observed: There is a narrow positive\npeak, but less than halfway the expected peakwidth, it\nreverses sign. From this measurement it is not possible\nto unambiguously state the presence of the sign reversal\nfor the anti-parallel state. To do so, a device is needed\nhaving a switching \feld of the second magnetic layer in\nthe order of 100 mT or higher (possibly accomplished by\nreplacing Py for cobalt, which has a larger coercive \feld),\nsuch that the anti-parallel magnetization state can also\nbe obtained for slightly higher magnetic \felds.\nV. SUMMARY\nIn summary, we have observed the generation of volt-\nage signals in YIG jAujPy devices placed in series, caused\nby YIG magnetization resonance. Furthermore, the reso-\nnance of the magnetic Py layers, caused by direct excita-\ntion of the magnetization, or indirect dynamic coupling,\nis detected. By modeling our device structure, we \fnd\nthat the signals of the wider Py structures (area 2) can\nvery well be reproduced by the calculated spin pumping\nsignals. For the narrow structures (area 1) additional\nsignals are detected. The origin of these additionally\nobserved signals and some other features, such as thedependence of the resonance peaks on applied RF fre-\nquency, the resonating peaks at low applied frequencies,\nand the increasing background voltage as a function of\nRF frequency, remain to be explained.\nSpin-dependent thermal e\u000bect are also quanti\fed; The\nheating caused by the applied RF current was studied by\nplacing thermocouples in close proximity to the stripline.\nDue to the temperature increase at the surface of the\nYIG substrate, especially at YIG resonance conditions,\ncontributions of thermal e\u000bects to the generated voltage\nin the YIGjAujPy devices cannot be excluded. Never-\ntheless, from \fnite element simulations we \fnd that the\ncontribution of the SdSE and SSE are rather small and\ncannot explain the observed features. Additionally, the\nthermocouples showed that the heating of the substrate\nis dependent on the applied \feld direction, indicating the\npossible presence of nonreciprocal magnetostatic surface\nspin-wave modes.\nConcluding, we have shown the possibility to electri-\ncally detect magnetization resonance from an electrical\ninsulating material by a spin-valve-like structure, with-\nout making use of the ISHE. For the presented work, 96\nand 62 AujPy devices were placed in series to increase\nthe magnitude of the generated signal. By comparing\nthe obtained data with signals from a reference Pt strip\nand using a \fnite element model of the devices, we \fnd\nthat part of the detected signals can be ascribed to spin-\ncurrent generation by spin pumping (for the 62 devices in\narea 2 an agreement between measurements and calcu-\nlated signals within 20% is found), however, especially for\nthe 96 devices in area 1, additional signals are present, of\nwhich the origin remains to be explained. Once a better\nunderstanding of the origin of the full signals is obtained,\nthe device geometry and injection e\u000eciency can be im-\nproved, such that the number of needed devices can be\ndecreased, which opens up possibilities for new types of\nspintronic devices, where magnetic insulators can be in-\ntegrated.\nACKNOWLEDGEMENTS\nWe would like to acknowledge J. Flipse for sharing\nhis ideas and M. de Roosz, H. Adema and J. G. Hol-\nstein for technical assistance. 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Hieberta,b, M.R.\nFreemana,b\naDepartment ofPhysics, University ofAlberta, Edmonton, Al berta, Canada T6G 2G7\nbNational Institute for Nanotechnology, Edmonton, Alberta , Canada T6G 2M9\nAbstract\nA novel method for simultaneous detection of both DC and time -dependent magnetic signatures in individual\nmesoscopic structures has emerged from early studies in spi n mechanics. Multifrequencynanomechanicaldetection\nof AC susceptibility and its harmonics highlights reversib le nonlinearities in the magnetization response of a single\nyttriumirongarnet(YIG)element,separatingthemfromhys tereticjumpsintheDC magnetization.\nThismanuscriptwasacceptedforpublicationin SolidState Communications(SpecialIssue: SpinMechanics).\nKeywords:\nA.magneticallyorderedmaterials,A.nanostructures,B. n anofabrications,D.spin dynamics\nNanomechanical torque magnetometry of quasi-\nstatic magnetization processes has sparked recent in-\nterest due to its exceptional sensitivity (with room-\ntemperature magnetic moment resolution approaching\n106µB) and ability to non-invasively measure single,\nmesoscopicelements1,2,3. Thestudyofindividualstruc-\ntures is a necessity to probe local e ffects from varia-\ntionin themagneticmicrostructuredueto grainbound-\naries, vacancies, and other intrinsic or extrinsic inho-\nmogeneities. These are masked in measurementsof ar-\nrays of magnetic structures but can have a tremendous\nimpact on the magnetization response of single ele-\nments. Inthistechniqueamagnetostatictorque, τ,isex-\nerted on the magnetization, M, by an external field, H:\nτ=MV×µ0H,inwhich Visthesamplevolume. Ifthe\nmagneticmaterialisa ffixedto atorsionalresonator,the\ninduced magnetic torque is converted to a mechanical\ndeflection proportional to its magnetization. For struc-\ntures with strong in-plane magnetic anisotropy, a small\ntorquingAC (dither)field is appliedout-of-planeto the\nsurfacewhile a DC field biases the in-planemagnetiza-\ntion. By ramping the DC field, the quasi-static magne-\ntizationevolutionwithappliedfieldcanberecorded.\nTechniques to probe time-dependent magnetic phe-\nnomena yield information complementary to the DC\nmagnetization. In particular,the AC magneticresponse\nEmailaddress: mark.freeman@ualberta.ca (M.R.Freeman)aids in discriminatingreversible fromirreversiblemag-\nnetization changes. In the small signal regime, an AC\nmagnetization Mac=χHacisinducedbyanalternating\nfield, whereχis a susceptibility tensor. In bulk mag-\nnetic systems, low frequency susceptibility (including\nthe use of higher order susceptibility4,5) measurements\nare used extensively to monitor myriad phenomena\nfromthe onsetof ferromagnetismat theCurie tempera-\nture6,7,magnetizationreversalprocessesinthinfilms8,9\nand patternednanostructures10, exchangeanisotropyin\nexchangebiasedsystems11, anddynamicsin spin-glass\nsystems12. The advantages o ffered by nanomechanical\ntransductionhaveyet to befullyexploitedformagnetic\nsusceptibility measurements. It is highly desirable to\ndevelop an experimental technique for measuring both\nDCandACcomponentsofmagnetization,complemen-\ntarytoSQUIDmeasurements13,14byvirtueofoperating\nin a wider range of sample environments. Micro Hall\nmagnetometerscan also be verysensitive15,16, but their\nlow bandwidth will limit applicability to susceptibility\nmeasurements. Synchrotron-based XMCD-PEEM has\nalsobeenutilizedtorecordlow-frequencydynamicsus-\nceptibilityin magneticthin-films17,18.\nHerewereportananomechanicalplatformforsimul-\ntaneous DC and AC magnetic measurements through\nthe introduction of an AC field component along the\nbias field direction, to probe the quasi-static longitudi-\nnal susceptibility. Two orthogonalAC fields, if applied\nPreprintsubmitted to Solid State Communications August14,2018at different frequenciessimultaneously, give rise to AC\nmagnetictorquesatthesumanddi fferencefrequencies.\nThese can be tuned to match the natural resonance fre-\nquency of the resonator, allowing it to function e ffec-\ntively as a signal mixer. Di fferent frequency compo-\nnentscanbeextractedfromthemechanicalsignalasthe\nDC field sweeps the magnetization texture to measure\nsimultaneously the DC magnetization, AC susceptibil-\nity,andhigherorderACsusceptibilityterms.\nFigure 1: Instrument schematic for frequency-mixed nanome chani-\ncal detection of AC susceptibility. The lock-in amplifier pr ovides the\nreference/drive frequencies, f1andf2, which induce orthogonal AC\nfields through aHelmholtz coil assembly ( Hac\nx) and single coil ( Hac\nz),\nboth illustrated in the bottom-right inset). The lock-in ou tput signal\nis amplified by audio (AF) and radio (RF) frequency power ampl i-\nfiers, respectively. A HeNe laser is used to interferometric ally detect\nthe mechanical motion of the torsional resonator (BS =beam splitter,\nPD=photodetector). The di fference and/or sum frequencies are de-\nmodulated simultaneously at the lock-in. Top left inset: fa lse-colour\nscanning electron micrograph of thedevice (scale bar =1µm)\nDetermination of the susceptibility requires some\nlevel of description of the spin-mechanical coupling,\ntaking in to account the conversion of magnetic torque\ninto mechanical torque. The full tensor analysis will\ninclude Einstein-de Haas terms, which we will neglect\nhere to concentrate on the net mechanical torque aris-\ning throughmagneticanisotropy. Conceptually,we can\nviewthe systemas two torsionspringsconnectedin se-\nries. The geometrically-confinedmagnetizationtexture\nof the disk is the first spring, as is the one acted on di-\nrectly by the torquing fields. In turn, the net magnetic\ntorquewill twist the torsionrod until an equivalentme-\nchanicalcounter-torquedevelops. Forthepresentanaly-\nsis,wedefinethee ffectivemagnetictorsionspringcon-\nstantsbyconvertingmagneticsusceptibilitiesintoangu-\nlar changesas if the magnetizationof the element werea single macrospin. Although far from the real case,\ntheparallel axistheoremallowsforthe samenet torque\nfrom the same net magnetization independent of how\nthemagnetizationisdistributed.\nThe net magnetic anisotropy (shape plus magne-\ntocrystalline) of the mesoscale magnetization in this\nsimplepicturemanifestsitselfthroughdi fferencesindi-\nagonal components of the magnetic susceptibility ten-\nsor. If no anisotropy exists, then the equilibrium mag-\nnetization will always be parallel to the applied field,\nwith a resultant torque of zero. In general, one will\nencounterstructureswith significantlydi fferentsuscep-\ntibilities parallel and perpendicular to the plane of the\ntorsionpaddle,but whereneither componentis negligi-\nble. This is the case for the specific example we con-\nsider below, a 3D vortex state in a short cylinder of\nYIG, where the low-field linear in-plane susceptibility\nisapproximatelytwicetheout-of-planesusceptibility19.\nThen,theresultanttorqueinsimultaneoussmall(forlin-\nearmagnetizationresponse)fields HxandHzis:\n−τy=µ0(MxHz−MzHx)V=µ0(χx−χz)HxHzV(1)\nThis net torque will transfer to the lattice and twist the\nresonator. For a low frequency AC field, an AC torque\nensuesinphasewiththefield. Ifthefrequencyisnottoo\nhigh, the magnetization remains in quasi-static equilib-\nrium with the field at all times. For the vortex structure\nin the absence of pinning (no slow thermal dynamics),\nthis criterion is satisfied at frequenciesin the low MHz\nregime,convenientforresonantenhancementoftheme-\nchanical response in nanoscale torsional structures. In\nthe presentexperiment,correctionsforthe appliedfield\nangles changing in the frame of reference of the pad-\ndlearenegligible: themechanicalspringismuchsti ffer\n(∼106×) than the magnetic spring, and the mechanical\nQ isonly2600.\nWhentwoorthogonalACfieldsactsimultaneouslyat\ndifferentfrequencies, f1andf2,ACmagnetictorquesat\nthesumanddifferencefrequenciesalsoarise,according\nto:\n−τy=µ0(χx−χz)Hac\nxcos(2πf1t)Hac\nzcos(2πf2t)V\n=µ0(χx−χz)Hac\nxHac\nz\n2(cos(2π(f2−f1)t)+cos(2π(f2+f1)t))V\n(2)\nWith the simultaneous application of a DC field to\nsweep the magnetization texture through a hysteresis\ncycle, this becomes the basis of the AC susceptome-\ntrymeasurement,to complementtheDC magnetization\ndata.\nFor demonstration of the technique, a single-crystal,\nmesoscale YIG disk (radius =600nm,thickness =500\n2Figure 2: DC torque magnetometry of an individual micromagn etic\nYIGdisk, (a) and the corresponding numerical derivative, ( b).\nnm)wasfocusedionbeam(FIB)-milledfromanepitax-\nialthickfilmandnanomanipulated in-situontoaprefab-\nricated torsional resonator19. A scanning electron mi-\ncrograph of the completed device is shown in the inset\nof Fig 1. The FIB milling and ‘pick and place’ proce-\ndureswereemployedtoovercomethedi fficultyofform-\ning monocrystalline mesoscale objects which are to be\nmeasured singularly. The pristine magnetic nature of\nthe YIG disk is exhibited, within experimental resolu-\ntion,bythelackofBarkhausensignaturesinthechange\nof the magnetization with applied field, which are as-\nsociated with nanoscale imperfections such as grain\nboundaries in polycrystalline materials. In the present\ncontext YIG simplifies interpretation of the demonstra-\ntion susceptibility signals. Hysteretic minor loops as-\nsociated with Barkhausen transitions in the magnetiza-\ntion evolution2, can strongly influence and complicate\ntheAC susceptibility. The YIG disk,on the otherhand,\nwasshowntohavenoobservableminorhysteresis.\nThe resonatorwas drivenby the magnetictorque ex-\nerted on the YIG disk. The mechanical deflection was\noptically detected through the interferometric modula-\ntion of the light reflected from the device. The mag-\nnetization was biased by a variable external field, ( Hdc\nx,\nCartesiancoordinatesindicatedinFig. 1)intheplaneof\nthe resonator and perpendicularto the torsion rod. The\nAC torquing field was supplied by a small copper wire\ncoilwhichproducedanout-of-planeditherfield, Hac\nz,to\ncomplete the field orientationsnecessary for the torque\nmagnetometry measurement. The AC field was driven\natthefundamentaltorsionfrequencytomaximizedetec-\ntionsensitivity. ThemagnetizationoftheYIGdiskwith\nappliedbias field is shown in Fig. 2, characterizedby a\nunipolarhysteresissimilartowhathasbeenobservedin\ntwo-dimensional disks where the sharp transitions areattributed to the nucleation and annihilation of a mag-\nnetic vortex. However, the thickness of the YIG disk\npromotes additional three dimensional structure in the\nmagnetizationtexture(detailedin Ref. 19).\nAresultfromthesimultaneousacquisitionoftheDC\nmagnetization and AC susceptibility is shown in Fig 3.\nFeaturesagreeingcloselyto the numericalderivativeof\nthe DC magnetization curve, Fig. 2b, are evident with\nthe differencesarisingfromthe susceptibilityscreening\noutirreversibleeventsinthefieldsweep. Here,thelock-\nin amplifier providedboth orthogonalac drive frequen-\ncies,f1(Hac\nx)andf2(Hac\nz),whilerecordingtheresponse\nof the resonator through simultaneous demodulation20\natf2andatf2−f1(seeFig. 1formeasurementscheme).\nIt is interesting to note that a softening of the magne-\ntization texture (peak in the susceptibility) occurs just\nafter the irreversible annihilation transition on the field\nsweep-up. Thehighestfieldsusceptibilityisat opposite\nphase in this measurementconsistent with the possibil-\nityinherentinEqn. 2for χzcontributestoexceedthe χx\ncontributionasthemagnetizationapproachessaturation\ninx. Thisfeatureisreplicatedagainonthesweepdown\nataslightlylowerfield–ahystereticfeaturenotreadily\napparentfromtheDCmagnetization.\nFigure 3: Simultaneous acquisition of the DC magnetometry t hrough\nthe driven signal at f2=fres+f1, a), and low frequency AC suscepti-\nbility detected at the mixing (resonance) frequency, fres, b). Here, f1\nwas set to 500 Hz.\nThe technique can easily be extended to detect the\nharmonics of AC susceptibility arising when the mag-\nnetization response is nonlinear in field. For example,\nfor a magnetization describable by a polynomial series\ninfield,\nMx=a1Hx+a2H2\nx+a3H3\nx+a4H4\nx+a5H5\nx+a6H6\nx+...,\n(3)\nwhereHx=Hdc\nx+Hac\nx,multipleharmonicsof f1arise\nand also mix with f2to generate unique torque terms.\n3For this representation of nonlinear magnetization, Ta-\nble 1 shows the amplitude coe fficients for the first six\nentriesin the Fouriersum forthe full nonlinearsuscep-\ntibility,\nχx(Hx)=∞/summationdisplay\nn=−∞χx\nn(Hx)e−i(2πnf1)t. (4)\nTable1: Harmonics of the AC susceptibility\nf2±nf1Amplitudecoefficients relatedto\nf2+0a1Hdc\nx+a2[(Hdc\nx)2+1\n2]+\na3[(Hdc\nx)3+3\n2Hdc\nx]\n+a4[(Hdc\nx)4+3Hdc\nx)2+3\n8]+\na5[(Hdc\nx)5+5(Hdc\nx)3+15\n8Hdc\nx]\n+a6[(Hdc\nx)6+15\n2(Hdc\nx)4+\n45\n8(Hdc\nx)2+5\n16]χx\n0(Hx)\nf2±f1a1+2a2Hdc\nx+a3[3(Hdc\nx)2+\n3\n4]+a4[4(Hdc\nx)3+3Hdc\nx]\n+a5[5(Hdc\nx)4+15\n2(Hdc\nx)2+5\n8]\n+a6[6(Hdc\nx)5+15(Hdc\nx)3+\n15\n4Hdc\nx]−dMz\ndHz|Hdcxχx\n1(Hx)\nf2±2f11\n2a2+3\n2a3Hdc\nx+a4[3(Hdc\nx)2+\n1\n2]+a5[5(Hdc\nx)3+5\n2Hdc\nx]\n+a6[15\n2(Hdc\nx)4+15\n2(Hdc\nx)2+\n15\n32]χx\n2(Hx)\nf2±3f11\n4a3+a4Hdc\nx+a5[5\n2(Hdc\nx)2+\n5\n16]+a6[5(Hdc\nx)3+15\n8Hdc\nx]χx\n3(Hx)\nf2±4f11\n8a4+5\n8a5Hdc\nx+a6[15\n8(Hdc\nx)2+\n3\n16]χx\n4(Hx)\nf2±5f11\n16a5+3\n8a6Hdc\nx χx\n5(Hx)\nf2±6f11\n32a6 χx\n6(Hx)\nA spectroscopic frequency-versus-field mapping of\nthe higher order AC susceptibilities for the YIG disk\nis shown in Fig. 4 for the sweep-down portion of\nthe hysteresis cycle (31 to 4 kA /m). Here the fre-\nquency response was recorded at each magnetic field\nstep while keeping both f1(Hac\nx) andf2(Hac\nz) constant\n(500 Hz and 1.8095 MHz, respectively). The f2drive\nwasshifted fromthe resonancefrequency(1.808MHz)\ninordertoallowformoresusceptibilityharmonicstobe\nrecorded within the mechanical resonance line-width.\nThe brightest band is the demodulation of the f2drive,\nwhile the subsequent bands represent the harmonics of\nthemagneticsusceptibility. Thefeaturesobservedinthe\nsecondhalfofthefieldsweepinFig. 3barereproduced,\nincludingthestrongsusceptibilitypeakjustbefore(very\nclosetotheonsetof)thenucleationtransition. Through\nsuch strong nonlinearities in the magnetization curve,\nmixingfrequencies f2±nf1areobservedupto n=7inFig. 4.\nFigure 4: Spectroscopic mapping of higher harmonics of the A C sus-\nceptibility. f1(Hac\nx) andf2(Hac\nz) were each driven simultaneously\nat constant frequency (500 Hz and 1.8095 MHz respectively) w hile\nthe frequency response (100 Hzbandwidth) was measured with mag-\nnetic field. The image shows the sweep-down portion of the fiel d\nsweep. The band of highest intensity is the demodulation of f2, with\nthe successive bands representing theharmonics ofAC susce ptibility,\nf2±nf1. A line-scan thorough the high susceptibility peaks is show n\nin the bottom left, showing up to the n =7 harmonic. The original in-\ntensity scale was cropped to show the highest order signals.\nThe multifrequency responses are all harmonics\nof the low AC frequency dithering the magnetiza-\ntion, mixed to fall within the mechanical resonance\nlinewidth. High harmonics of susceptibility have been\nusedpreviouslytocharacterizehysteresisforbulkferro-\nmagnetismandsuperconductivity. Inbulksystems, and\nin arrays of nanostructures, the strongest nonlinearities\ninmagnetizationcomefromhystereticfeatures. Typical\nACfieldamplitudescanprobeminorhysteresisloops21,\noreventhefullhysteresiscloseenoughtotheCurietem-\nperature4. By contrast, the single mesoscale structure\nwe examine here exhibits no minor hysteresis, and its\nmagnetization nonlinearities are observed without any\ndiminution from array averaging. The harmonics char-\nacterizethenon-hysteretic,nonlinearresponse.\nIn summary, nanomechanical detection of the low-\nfrequency AC susceptibility (along with higher order\nharmonics)ofanindividualmicromagneticdiskwasac-\ncomplishedthroughfrequencymixingoforthogonalAC\ndrivingfieldsmanifestingasamechanicaltorqueonthe\nresonator. In addition to providing further insight into\nquasi-static magnetization processes in geometrically-\nconfined magnetic elements, this signal-mixing ap-\nproach, in principle, will enable very broadband mea-\nsurementsoffrequency-dependentsusceptibility.\n4Acknowledgements\nThe authors would like to thank NSERC, CIFAR,\nAlberta Innovates and NINT for support. Partial de-\nvice fabrication was done at the University of Alberta\nNanoFab. FIBmillingandmanipulationwascarriedout\nbyD.VickandS.R.ComptonattheNINTelectronmi-\ncroscopyfacility.\nReferences\n[1] J. Moreland, A. Jander, J.A. Beall, P. Kabos and S.E. Russ ek,\nIEEET.Magn., 37,2770 (2001).\n[2] J.A.J. Burgess, A.E. Fraser, F. Fani Sani, D. Vick, B.D. H auer,\nJ.P.Davis and M.R.Freeman, Science 339,1051 (2013).\n[3] J.E. Losby, J.A.J. Burgess, Z. Diao, D.C. Fortin, W.K. Hi ebert\nand M.R.Freeman, J.Appl. Phys. 111, 07D305 (2012).\n[4] C. R¨ udt, P J. Jensen, A. Scherz, J. Lindner, P. Poulopoul os and\nK.Baberschke, Phys.Rev. 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Phys. 36, 31 (1964).\n5" }, { "title": "2002.12550v1.Spin_Hall_magnetoresistance_in_Pt_YIG_bilayers_via_varying_magnon_excitation.pdf", "content": "1 \n Spin Hall magnetoresistance in Pt/YIG bilayers via varying magnon \nexcitation \nQ. B. Liu1,2, K. K. Meng1*, S. Q. Zheng1, Y . C. Wu1, J. Miao1, X. G. Xu1 and Y . \nJiang1** \n1Beijing Advanced Innovation Center for Materials Genome Engineering, School of Materials \nScience and Engineering, University of Science and Technology Beijing, Beijing 100083, China \n2Applied and Engineering Physics , Cornell University, Ithaca, NY 14853, USA \n \nAbstract: Spin Hall magnetoresistance (SMR) and magnon excitation \nmagnetoresistance (M MR) that all generate via the spin Hall effect and inverse spin \nHall effect in a nonmagnetic material are always related to each other. However, the \ninfluence of magnon excitation for SMR is often overlooked due to the negligible \nMMR. Here, we investigate the SMR in Pt/Y 3Fe5O12 (YIG) bilayers from 5 to 300K, \nin which the YIG are treated after Ar+-ion milling. The SMR in the treated device is \nsmaller than in the non -treated. According to theoretical simulation, we attribute this \nphenomenon to the reduction o f the interfacial spin -mixing conductance at the treated \nPt/YIG interface induced by the magnon suppression. Our experimental results point \nout that the SMR and the MMR are inter -connected, and the former could be \nmodulated via magnon excitation. Our fin dings provide a new approach for separating \nand clarifying the underlying mechanisms. \nKey words: Spin-orbit coupling; Spin transport in metals ; Spin transport through \ninterfaces \n \n*Authors to whom correspondence should be addressed: \n*kkmeng@ustb.edu.cn \n**yjiang@ustb.edu.cn 2 \n Ferromagnetic insulators (FMIs) has emerged as a promising and novel \ntechnology platform to generate, modulate and detect spin information over long \ndista nces [1,2]. The advantage of using FMIs against metallic ones is avoiding the \nflow of charge current, preventing ohmic losses and the emergence of undesired \nspurious effects [3]. Yttrium Iron Garnet (YIG) is one of the most prominent FMIs for \nthe investiga tion of spintronics, magnonics [4,5], and spin caloritronics [6,7], due to \nits extremely low damping, soft ferrimagnetism and negligible in -plane magnetic \nanisotropy even in limit thickness. One can efficiently investigate and control the \nmagnetization direction and spin waves propagation in FMIs through transport \nmeasurements based on the spin Hall effect (SHE) and the inverse spin Hall effect \n(ISHE) [2,8,9], by which the mutual conversion between magnon or spin current and \ncharge current can be realized in a heavy metal (HM) with strong spin-orbit coupling \n(SOC) . Therefore, t he combin ation of S HE and ISHE can give rise to the spin Hall \nmagnetoresistance (SMR) and magnon excitation magnetoresistance (MMR) [10,11], \nwhile the former can be ascribed to the spi n transfer torque at the HM/FMIs interface. \nOwing to the SHE, the spin current would accumulate at the HM/FMIs interface with \nspin polarization σ parallel to the surface. If σ is not collinear with the magnetization \nM of the FMIs, the accumulated spin current can exert a torque proportional to \nM×(M×σ) on the M of FMIs. It suggests that a finite spin current will be absorbed by \nFMIs if M and σ have a finite angle. The spin current absorption by FMIs represents \nthe spin current reflection that is suppressed, and its resistance therefore will change \nwith the magnetization orientation which is expected to be maximized (minimized) \nwhen M is perpendicula r (parallel) to σ [8,9]. On the other hand, the MMR is firstly \nobserved in Pt/YIG heterostructures in which the two parallel Pt strips were separated \nby a distance. When the two Pt electrodes are closed enough, the spin accumulation at 3 \n one Pt strip will transmit into anot her strip by magnon, and the spin current will be \nconverted into charge current via ISHE. Therefore, a large non -local charge current is \nsupposed for M//σ, since the magnons beneath the first Pt strip can diffuse across the \ngap to the second Pt strip. In c ontrast, for M⊥σ, the non -local signals should be \nsignificantly reduced, since the spin transfer torque absorbs the magnon propagation. \nTherefore, the measured resistance or voltage signal is expected to be maximized \n(minimized) when M is parallel (perpend icular) to σ [12]. The MMR can thus be seen \nas a magnon -mediated, non -local counterpart of the S MR. In short , both the SMR and \nthe MMR stem from spin accumulation and spin transport at the HM/FIMs interface s. \nTheoretically, the efficiency and strength of t he spin transfer across the interface \ndepend on the magnitude of interfacial spin -mixing conductance (SMC) G↑↓=(G \nr+iGi), where Gr (Gi) denotes the real (imaginary) part [1 3,14]. Therefore, if the \nmagnon transport can be suppressed or enhanc ed, it will cha nge the magnitude of Gr \nand the SMR is expected to change correspondingly . However, the influence of \nmagnon excitation for SMR is often overlooked due to the negligible MMR, and few \nmethods have been reported to separate and clarif y the underlying mechanis ms. \nIn this work, we have employed the etching treatment to suppress the in -plane \nmagnon transport and found that the SMR in Pt/YIG -based devices was dramatically \naltered when the part of YIG film out of the hall devices is treated with Ar+-ion \nmilling etc hing. At room temperature, the SMR in the treated device was much \nsmaller than that in the non -treated, while at low temperature the SMR for the two \ndevices was similar. The anomalous SMR reduction has similar temperature \ndependence with magnon excitation. According to the theoretical simulation, we have \nattribute d this phenomenon to the reduction of the G↑↓ at the Pt/YIG interface induced \nby magnon suppression after etching . Our experimental results point ed out that the 4 \n SMR and the MMR were interconnected, while the former can be modulated via \nmagnon excitation even though the value due to magnon excitatio ns was much smaller \nthan the SMR . \nThe epitaxial YIG film was grown on a [111] -oriented GGG substrate (lattice \nparameter a = 1.237 nm) by pulsed laser deposition technique with the substrate \ntemperature TS = 780℃ and the oxygen pressure 10 Pa. Then the sam ples were \nannealed at 780℃ for 30 min at the oxygen pressure of 200 Pa . The base pressure of \nthe PLD cavity was better than 2 ×10-6 Pa. Then, the Pt layer was deposited on YIG at \nroom temperature by magnetron sputtering. In order to avoid the run -to-run err or, \neach large Pt/YIG sample was then cut into two small pieces. After the deposition, the \nelectron beam lithography and Ar ion milling were used to pattern Hall bars, and a \nlift-off process was used to form contact electrodes. The size of all the Hall bar s is 20 \nμm×120 μm. For comparison , a part of YIG film out of the Hall bars was etched \naway by Ar+-ion milling which was defined as YIG+ as shown in Fig. 1(a). Fig. 1 (b) \nshows the XRD ω-2θ scan spectra of the 40-nm-thick YIG thin film, which was taken \nfrom represe ntative thin film of each type, and it shows predominant (444) diffraction \npeaks with no diffraction peaks occurring from impurity phases or other \ncrystallographic orientation, indicating the single phase nature. According to the (444) \ndiffraction peak pos ition and the reciprocal space maps of the (642) reflection of \n30-nm-thick YIG films grown on GGG as shown in Fig. 1(c), we have found that the \nlattice constant of YIG layer is similar with the value of GGG substrate, indicating the \nhigh quality epitaxial growth without mismatch. Moreover, the saturated \nmagnetization of the YIG layer measured by a vibrating sample magnetometer was \ndetermined to be 140 emu/cm3 as shown in Fig. 1(d), which is similar with the value \nof the bulk YIG. All the magnetotransport me asurements performed in the multilayers 5 \n were carried out using a Keithley 6221 sourcemeter and a Keithley 2182A \nnanovoltmeter. These measurements were performed at different temperatures from 5 \nand 300 K in a liquid -He cryostat that allows applying magneti c fields H up to 3 T \nand rotating the samples by 360 º. \n Using small and non -perturbative current densities (~ 106 A/cm2), we have \ninvestigated the angular -dependent magnetoresistance (ADMR) in Pt (5 nm)/YIG (40 \nnm) and Pt (5 nm)/YIG+ (40 nm) devices at roo m temperature. The measurement \nconfiguration, the definition of the axes, and the rotation angles (α, β, γ) are defined in \nthe sketches as shown in Fig. 2(a). Figs. 2(b) and (c) show the longitudinal ADMR \ncurves with applying magnetic field of 3T, and the ADMR was defined as \nADMR =[ρ−ρ(0 deg)]/ρ(0 deg) [15]. The ADMR of the two devices show the \nexpected behavior of the SMR, in agreement with the earlier report in Pt/YIG bilayers, \nand the values in the treated and non -treated devices are about 5.723 × 10−4 and 7.814 × \n10−4, respectively. One can find that the SMR of Pt/YIG is 1.4 times larger than that in \nPt/YIG+ device. In order to further investigate the anomalous SMR reduction in \nPt/YIG+ device, we carried out the ADMR measurements at the temperature range \nfrom 5 to 300 K. The temperature dependent ADMR of Pt/YIG and Pt/YIG+ bilayers \nin the β scan with 3 T field are shown in Figs. 3 (a) and (b), which all can be well \nfitted by the SMR mechanism. The Fig. 3(c) displays the temperature dependent SMR \nof the two devices. It is obvious that the SMR changes non -monotonically with \ndecreasing temperature , which is in debate since it might stem from the competition \nof two physical mechanisms: the spin Hall effect -induced magnetoresistance \n(SHE -MR) and the magnetic proximity effect -induced magnetoresistance (MPE -MR) \n[16]. The MPE -MR becomes evident a t relatively lower temperature due to the \nmagnetization induced by the MPE , while at higher temperature, the thermal 6 \n fluctuations will dominate, disrupting the spontaneous Pt magnetization and \neliminating the MPE -MR. More interestingly, the temperature dependence of SMR in \nPt/YIG bilayers in the previous reports was weak, while the sharp drop of SMR below \n100 K in our Pt/YIG bilayers would be discussed latter via systematic ADMR \nmeasurements with varying the thickness of Pt. Furthermore, we have defined the \nratio ΔSMR as ΔSMR =[SMR (YIG)-SMR (YIG+)]/SMR (YIG), and the temperature \ndependence is shown in Fig. 3(d). It seems to be constant below 100 K and linearly \nincreases from 100K to 300K. The anomalous temperature dependen ce behavior may \nbe related to the magnon excit ations, which should also be suppressed by the MPE \nand increase with temperature. To further verify our speculation about the magnon \nexcitation modulat ed SMR, we have carried out the Pt thickness (t) dependent \nmeasurements. The unusual ΔSMR fluctuation cur ve with increasing t was shown in \nFig. 4 (a). We can find that the ΔSMR is irrelevant with the bulk spin Hall angle \n(SHA) and spin diffusion length (SDL) of Pt that should exhibit a peak value at t ~ 3 \nnm [1 7]. Therefore, the only possibility of the unusua l ΔSMR should originate from \nthe change of the interface SMC due to the suppress ion of magnon transport after \netching. \nTo qualitatively analyze the experimental results, we employ a simulation within \nthe spin drift -diffusion theoretical framework. Accordi ng to the SMR theory, the \nlongitudinal resistivities of the Pt layer are given by\n）（2\ny 1 0 0 m-1 , where \nm(mx,my,mz)=M/Ms are the normalized projections of the magnetization of the YIG \nfilm to the three main axes, Ms the saturated magnetization of the YIG , ρ0 is the Drude \nresistivity , Δρ0 accounts for a number of corrections due to the SHE and Δρ1 is the \nmain SMR term . The SMR is quantified by [10,1 8]: \n 7 \n \n)/(coth) 2/(1)2/( tanh/2 2\nsd r sdsd sd SH\nλt Gρλλt\ntλθρΔρ (1) \nWhere λsd, θSH, t, and Gr are the spin Hall angle , the spin diffusion length , the Pt layer \nthickness, and the real part of the SMC at the YIG/Pt interface, respectively. The \nthickness dependence of the longitudinal resistance is shown in Fig. 4(b), and the \nproduct of the longitudinal resistivity ρ and th e Pt film thickness t is found to change \nlinearly with the film thickness as shown in the inset of Fig. 4(b). Considering the \nproduct is found to well obey the equation ρt = ρbt + ρs with the bulk resistivity ρb and \nthe interfacial resistivity ρs, which are determined to be 8.0 μΩ.cm and 32.0 μΩ.cm2 \nbased on the fitted lines, respectively. The bulk resistivity is close to the value of 10.0 \nμΩ.cm of the bulk Pt [ 19]. Here, we have used the SHA, the SDL and the Gr as 0.07, \n1.5 nm and 5 ×1015 Ω-1 m-1 [16,18,20]. We can find a large discrepancy between the \nfitted and the measured results as shown in Fig. 4(c). However, the variation trend of \nthe SMR could be fitted with giving SHA = 0.131 and SDL = 0.864 nm as shown in \nFig. 4(d), and the large SHA should stem fro m the interfacial contribution [2 1]. \nTherefore, the sharp drop of SMR ratio below 100 K in our Pt/YIG bilayers could be \nderived from the large interfacial SHA. \nNotably, the Pt/YIG+ device should have similar SHA and SDL with Pt/YIG \ndevice. We can fit the Gr in Pt/YIG+ bilayers through giving SHA 0.131 and SDL \n0.864 nm with Eq. (1). According to the fitted results as shown in Fig. 5(a), we can \ndetermine the Gr at the Pt/YIG+ interface is one order of magnitude smaller than \nPt/YIG interface. Furthermore, the Hanle magnetoresistance (HMR) cloud modulate \nthe resistance of the HM layer with H instead of M, exhibiting the similar angular \ndependent behavior with SMR: no resistance correction is observed for H parallel to σ, \nwhereas a resistance increase is obtained for H perpendicular to σ [22,23]. Therefore, 8 \n the anomalous SMR reduction in our devices may also stem from the different HMR. \nNotably, the HMR is due to the spin precession around the external magnetic field H, \nleading to the spin relaxation , so we could distinguish SMR and HMR from the field \ndependent MR measurement. As shown in Fig. 5(b), we can find that the distinction \nof HMR is negligible as compared with SMR in Pt/YIG and Pt/YIG+ devices at H =3 \nT. Recently , Y . Dai, et al. have found that the simulation method by Eq. (1) exhibits \ndiscrepancy because the SHA is fluctuation with varying the thickness of Pt layer. \nThey put forward the electron diffusion coefficient (EDC) and SDL that could be \nprecisely estimate d through the ratio of HMR and SMR [2 1]. Therefore, we would \nuse it to determine the EDC and SDL in the Pt/YIG devices. Then, the SMC at \nPt/YIG+ interface could be calculated with the similar EDC and SDL in Pt/YIG. \nNotably, the HMR/SMR ratio is independent of the SHA and thus the HMR/SMR \nratio reads [21]: \n\n\n\n\n\n\n\n\n\n\n\nDBigμλDBigμ\nλt Gρt/λ Gλρ\nλt//DBigμ\nλt\nDBgμiλSMR HMR\nB\nsdB\nsdr xxsd r sdxx\nsdB\nsd\nB sd\n22222\n2\n1)1(coth 2\n1) (coth 21\n)2( tanh)21( tanh\n11Re /\n(2) \nwhere g, μB, D, and B are the Landé factor, the Bohr magneton, the EDC, and the \nmagnetic induction intensity, respectively . The SDL and the EDC are determined to be \n2.02 nm and 4 × 10−6 m2s-1, respectively, where the longitudinal resistivity ρ = 25.5 \nμΩ.cm, Gr = 5 × 1015 Ω-1m-1, and the Landé factor g ≈ 2.0. As shown in Fig. 6, if one \nassigns the sd and the D with other values that deviate from 2.02 nm and 4 × 10−6 \nm2s-1, the fitted results cannot reproduce for all the samples. Based on the determined \nsd and D, the Gr for Pt/YIG+ device is five times smaller than Pt/YIG device. 9 \n However, there is still a problem which needs to be further clarified. From Fig. 6 (c), \nwe can find that the fitted curves are insensitivity to the SMC which is used as the \nonly free paramet er. It is difficult to point out which of the two fitted methods is more \nprecise. Here, we put forward a simple model to further explain this result. According \nto the report by X. P. Zhang in Ref. [2 3], the Gr should be read as Gr = e2vF(1/τP-1/τT) \n= e2vF/τP + Gs, where e, vF, τP and τT are the elementary charge ( e > 0), the density of \nstates per spin species at the Fermi level, the longitudinal and transverse spin \nrelaxation times per unit length for the itinerant electron, respectively. We note that Gr \nrepresents the difference between the longitudinal spin relaxation with transverse spin \nrelaxation and it does not have a physical meaning on its own. The Gs=-e2vF/τT \noriginates entirely from spin -flip processes and associates with magnon emi ssion and \nabsorption [2 4,25]. Therefore, magnon transport could affect the Gr. In the Pt/YIG+ \ndevice, the part of YIG film around the Hall bar is etched, which produces infinite \nhigh barriers at two sides of the Hall bar and suppresses the in -plane magnon transport. \nObviously, it will increase the magnon accumulation and suppress the spin absorption \nat Pt/YIG+ interface, which will reduce the Gr and the corresponding SMR [2 6]. Q. \nShao et al. also found that the measured SOT efficiency was significantly enha nced \nwith increasing the FMIs (TmIG) thickness, which is completely different from the \nFMs based devices. Similarly, we also found that the SMR is significantly enhanced \nwith increasing the YIG thickness as shown in Fig. 6(d). Therefore, we also ascribe \nthe modulated SMR to the magnon excitations because the thicker YIG is benefit for \nmagnon diffusion, reducing magnon accumulation [2 7]. \nIn conclusion, we have found the SMR in Pt/YIG -based devices was dramatically \naltered when the part of YIG film out of the hall devices was etched by the Ar+ -ion \nmilling. At room temperature, the SMR effect in the treated device was smaller than 10 \n in the non -treated one. According to theoretical simulation, we attributed this \nphenomenon to the reduction of the G↑↓ at the Pt/YIG interface induced by the \nsuppressed magnon transport. Our experimental results pointed out that the SMR and \nthe MMR that were inter -connected, and the SMR could be modulated via magnon \nexcitation/suppression even though the magnitude of MMR from magnon ex citations \nwas much smaller than the SMR . Our findings provide a new approach for modulating \nSMR for spintronic applications. \n \nAcknowledgements: This work was partially supported by the National Science \nFoundation of China (Grant Nos. 51971027 , 51731003, 51 671019, 51602022, \n61674013, 51602025), and the Fundamental Research Funds for the Central \nUniversities ( FRF-TP-19-001A3). 11 \n References \n[1] M. I. Dyakonov, and V . I. Perel, Phys. Lett. A 35, 459 (1971). \n[2] Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. \nUmezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh. Nature \n464, 262 (2010). \n[3] M.-Z.Wu and A. Hoffmann, Academic, New York, 2013. \n[4] Demokritov, S. O., Hillebrands, B. and Slavin, A. N. Phys. Rep. 348, 441 (2001). \n[5] A. A. Serga, A. V . Chumak, and B. Hillebrands, J. Phys. D 43, 264002 (2010). \n[6] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, \n054436 (2015). \n[7] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. 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Schreier, S.Altmannshofer, M. \nWeiler, H. Huebl, S. Geprä gs, M. Opel,R. Gross, D. Meier, C. Klewe, T. Kuschel, \nJ.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y . -T. Chen, G. E. W. Bauer, E. \nSaitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). \n[15] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. \nSaitoh, and G. E.W. Bauer, Phys. Rev. B 87, 144411 (2013). \n[16] Q. Shao, A. Grutter, Y . Liu, G. Yu, C. Y . Yang, D. A. Gilbert, E. Arenholz, P. \nShafer, X. Che, C. Tang, M. Aldosary, A. Navabi, Q. L. He, B. J. Kirby, J. Shi, \nand K. L. Wang, Phys. Rev. B 99, 104401 (2019). \n[17] M. H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 116, 126601 \n(2016). \n[18] X. Jia, K. Liu, K. X ia, and G. E. W. Bauer, Europhys. Lett. 96, 17005 (2011). \n[19] S. Dutta, K. Sankaran, K. Moors, G. Pourtois, S. Van Elshocht, J. Bö mmels, W. \nVandervorst, Z. Tӧei, and C. Adelmann, J. Appl. Phys. 122, 025107 (2017). \n[20] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ral ph, and R. A. Buhrman, Phys. Rev. \nLett. 109, 096602 (2012). \n[21] Y . Dai, S. J. Xu, S. W. Chen, X. L. Fan, D. Z. Yang, D. S. Xue, D. S. Song, J. Zhu, \nS. M. Zhou, and X. P. Qiu , Phys. Rev. B 100, 064404 (2019). \n[22] S. Vé lez, V . N. Golovach, A. Bedoya -Pinto, M. Isasa, E. Sagasta, M. Abadia, C. \nRogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett. 116, \n016603 (2016). \n[23] H. Wu, X. Zhang, C. H. Wan, B. S. Tao, L. Huang, W. J. Kong, and X. F. Han, \nPhys. Rev. B 94, 174407 (2016). \n[24] X.-P. Zhang, F. S. Bergeret, and V. N. Golovach , Nano Lett. 19, 9, 6330 (2019). \n[25] F. K. Dejene, N. Vlietstra, D. Luc, X . Waintal, J. B en Youssef, and B. J. Van Wees, 13 \n Phys. Rev. B 91, 100404 (R) (2015) . \n[26] S. Vélez, A. Bedoya -Pinto, W. Yan, L. E. Hueso, and F. Casanova Phys. Rev. B \n94, 174405 (2016). \n[27] Q. Shao, C . Tang, G . Yu, A . Navabi, H . Wu, C . He, J . Li, P. Upadhyaya, P . Zhang, \nS. A. Razavi, Q . L. He, Y . Liu, P . Yang , S. K. Kim, C . Zheng, Y . Liu, L . Pan, R . K. \nLake, X . Han, Y . Tserkovnyak , J. Shi and K. L. Wang , Nat. Commun. 9, 3612 \n(2018). \nFigure captions \nFigure 1 (a) Sample and measurement configurations with the definition of the axes. \n(b) The XRD ω-2θ scans of the YIG film grown on GGG substrate. (c) \nHigh -resolution XRD reciprocal space maps of the YIG film grown on GGG substrate. \n(d) Fi eld dependence of out -of-plane magnetization (black) and in -plane \nmagnetization (red) of YIG film. \n \nFigure 2 (a) Notations of different rotations of the angular α, β, and γ. (b) and (c) \nADMR in Pt/YIG and Pt/YIG+ devices at 300 K with H = 3 T in the three rotation \nplanes ( α, β and γ ). \n \nFigure 3 (a) and (b) ADMR curves measured in Pt/YIG and Pt/YIG+ devices with \nvarying β at different temperature, and the applied magnetic field is 3 T. (c) \nTemperature dependence of the ADMR of Pt/YIG (black) and Pt/YIG+ (red) devices. \n(d) Temperature dependence of the ΔSMR . \n \nFigure 4 The ΔSMR (a) and longitudinal resistance in the two devices (b) at room \ntemperature with varying Pt layer thickness (t). The inset of (b) shows the Pt layer 14 \n thickness dependence of the produc t of the longitudinal resistivity and thickness in the \ntwo devices . (c) and (d) The Pt layer thickness dependence of the experimental and \nfitted SMR in Pt/YIG with different parameters. \n \nFigure 5 (a) The experimental and fitted SMR in Pt/YIG+ with varying the Pt layer \nthickness, and Gr is parameter for fitting. (b) The MR versus the external magnetic \nfield H along Y axis and Z axis. \n \nFigure 6 (a) and (b) The measured and fitted of HMR/SMR ( H) curves in Pt/YIG \ndevice . In panel (a), the red lines refer to the fitted results where the SDL and the \nEDC are free parameters. The blue and pink lines refer to the fitted results with fixed \nbut different EDC. In panel (b), the red lines refer to the fitted results where the SDL \nand the EDC are free parameters. The blue and pink lines refer to the fitted results \nwith fixed but different EDL. (c) The measured and fitted of HMR/SMR ( H) curves in \nPt/YIG+ device. T he red lines refer to the fitted results where the SMC is free \nparameter. The blue and pink lines refer to the f itted results with fixed but different \nSMC. (d) Temperature dependence of the ADMR of Pt/YIG (20 nm) and Pt/YIG (60 \nnm) bilayers. 15 \n \n16 \n \n17 \n \n " }, { "title": "1509.04440v1.Magnetic_Nernst_effect.pdf", "content": "Magnetic Nernst e\u000bect\nSylvain D. Brechet\u0003and Jean-Philippe Ansermet\nInstitute of Condensed Matter Physics, Station 3,\nEcole Polytechnique F\u0013 ed\u0013 erale de Lausanne - EPFL, CH-1015 Lausanne, Switzerland\nThe thermodynamics of irreversible processes in continuous media predicts the existence of a\nMagnetic Nernst e\u000bect that results from a magnetic analog to the Seebeck e\u000bect in a ferromagnet\nand magnetophoresis occurring in a paramagnetic electrode in contact with the ferromagnet. Thus,\na voltage that has DC and AC components is expected across a Pt electrode as a response to the\ninhomogeneous magnetic induction \feld generated by magnetostatic waves of an adjacent YIG slab\nsubject to a temperature gradient. The voltage frequency and dependence on the orientation of the\napplied magnetic induction \feld are quite distinct from that of spin pumping.\nI. INTRODUCTION\nIn spincaloritronics, there has been recently quite some\ninterest in the study of the propagation of spin waves\nacross a ferromagnetic \flm in the presence of a tempera-\nture gradient1,2. For a con\fguration where the external\nmagnetic induction \feld is parallel to the temperature\ngradient, the propagation of magnetization waves induces\na magnetic induction \feld of magnitude proportional to\nthe temperature gradient. Since the Seebeck e\u000bect refers\nto an electric \feld induced by a temperature gradient,\nthis e\u000bect demonstrated in a YIG slab3can be called the\nMagnetic Seebeck e\u000bect.\nThe Magnetic Seebeck e\u000bect is a dynamical e\u000bect re-\nsulting from the precession of the magnetization. Thus,\nit should not be confused with the Spin Seebeck e\u000bect4{8\nwhere the magnetization is at equilibrium. A theoreti-\ncal model of the Spin Seebeck e\u000bect was established by\nAdachi et al.9in a quantum framework, while Schreier et\nal.10attribute the e\u000bect to a di\u000berence in the tempera-\ntures of the lattice and the magnetization.\nHere, we point out that the thermodynamics of irre-\nversible processes in a magnetic continuous medium11\npredicts that the electrostatic potential depends on the\ngradient of the magnetic induction \feld applied to it.\nThis is a consequence of the magnetophoretic force ex-\nerted on a magnetized charge carrier. Magnetophoresis is\ncommonly used in electrochemistry12and biophysics13.\nThe Magnetic Seebeck e\u000bect implies the existence of a\nmagnetic induction \feld normal to the interface between\nthe ferromagnet and the electrode. Thus, the magne-\ntophoretically induced electric \feld, the thermally in-\nduced magnetic induction \feld and the temperature gra-\ndient are orthogonal to one another as in the Nernst ef-\nfect. Hence, we call it the Magnetic Nernst e\u000bect14. Here,\nthe YIG ferromagnet is an insulator and the Pt electrode\nis a paramagnetic conductor. The Magnetic Nernst ef-\nfect is thus a combination of the Magnetic Seebeck e\u000bect\npresented in reference3and magnetophoresis15. It is to\nbe distinguished from the anomalous Nernst16e\u000bect and\nfrom the planar Nernst e\u000bect17.\nAs shown below, the composition of these two e\u000bects\nresults in a voltage that has a DC component and an AC\ncomponent oscillating at twice the frequency of the mag-netization. The maximum amplitude occurs when the\nmagnetic induction \feld applied to carry the ferromag-\nnetic resonance is oriented with a 45\u000eangle with respect\nto the orientation of the electrode and of the temperature\ngradient.\nII. MAGNETIC SEEBECK EFFECT\nAs shown below, the composition of these two e\u000bects\nresults in a voltage that has a DC component and an AC\ncomponent oscillating at twice the frequency of the mag-\nnetization. The maximum amplitude occurs when the\nmagnetic induction \feld applied to carry the ferromag-\nnetic resonance is oriented with a 45\u000eangle with respect\nto the orientation of the electrode and of the temperature\ngradient.\nFor the sake of clarity, we recall here in what sense\na magnetic induction \feld is induced by an out-of-\nequilibrium magnetization in a temperature gradient.\nThe formalism presented in reference11implies the exis-\ntence of a magnetic counter-part to the well-known See-\nbeck e\u000bect, where a magnetic induction \feld BTis in-\nduced by a temperature gradient ryTin a YIG slab in\nthe presence of an oscillating magnetic excitation \feld\nand a constant external magnetic induction \feld Bext\nin the slab plane (see axes ^x,^y,^zon Fig. 1). The\nexistence of a magnetic induction \feld BTcan be un-\nderstood as follows. In an insulator like YIG, there is\nno drift current. Thus, an induced magnetization force\ndensity MYryBTbalances the thermal force density\n\u0000nYkBryT, i.e.\nMYryBT=\u0015YnYkBryT; (1)\nwhere the index Yrefers to YIG, \u0015Y>0 is a phenomeno-\nlogical dimensionless parameter, MYis the magnetiza-\ntion of YIG and nYis the Bohr magneton number density\nof YIG. The magnetization MYis the sum of the satu-\nration magnetization and the magnetic linear response,\ni.e.\nMY=MSY+mYwhere mY\u0001Bext= 0:(2)\nAs detailed in reference11, the magnetic induction \feld\nBTinduced by the temperature gradient ryTcan bearXiv:1509.04440v1 [cond-mat.mes-hall] 15 Sep 20152\nwritten as,\nBT=\"MY\u0002ryT; (3)\nwhere the phenomenological vector \"MYis given by,\n\"MY=\u0000\u0015YnYkB\nM2\nSY\u0000\nr\u00001\ny\u0002mY\u0001\n: (4)\nHence, the thermally induced magnetic induction \feld\nBTis oscillating at the frequency of mY.\nIII. MAGNETOPHORESIS\nThe continuity of the orthogonal component of the\nthermally induced magnetic induction \feld BTacross\nthe junction between the YIG and the Pt is ensured by\nThomson's equation, i.e.\nr\u0001BT= 0: (5)\nTherefore, the normal component of the magnetic induc-\ntion \feld BTis acting also on the Pt electrode. Mag-\nnetophoresis occurs in a Pt electrode that is su\u000eciently\nthick to be treated thermodynamically and su\u000eciently\nnarrow for the temperature gradient to be neglected.\nThe interaction between the magnetization of the con-\nduction electrons of the paramagnetic Pt electrode and\nthe thermally induced magnetic induction \feld in the fer-\nromagnetic YIG slab results in a magnetization force that\nleads to the di\u000busion of the conduction electrons along\nthe electrode, i.e. magnetophoresis. This generates in\nturn an electrostatic potential gradient rxVacross the\nPt electrode (see Fig. 1) which can be thought of as a\nmagnetophoretic electrochemical voltage. The existence\nof an the electrostatic potential gradient orthogonal to\nthe temperature gradient depends on the orientation of\nthe external magnetic induction \feld, as we shall show.\nConcretely, in the Pt electrode, the thermodynamic\nformalism11yields linear phenomenological relations be-\ntween the electric current density and the magnetization\nand electrostatic forces densities respectively. By identi-\nfying the electric current in these relations, the electro-\nstatic force density \u0000qPrxVresulting from the drift of\nthe conduction electrons is found to be proportional to\nthe magnetization force density MPrxBTgenerating\nthe drift, i.e.\n\u0000qPrxV=\u0015PMPrxBT; (6)\nwhere the index Prefers to Pt, qP<0 is the charge\ndensity of conduction electrons, MPis the magnetiza-\ntion of the conduction electrons in the Pt electrode and\n\u0015P>0 is a phenomenological dimensionless parameter.\nThe magnetization MPis the sum of the paramagnetic\ncontribution due to the constant external \feld Bextand\nthe linear response mPto the stray magnetic induction\feld generated by the propagating magnetization waves\nin the ferromagnet, i.e.\nMP=\u001fP\n\u00160Bext+mPwhere mP\u0001Bext= 0;(7)\n\u00160is the magnetic permeability of vacuum, \u001fPis the\nPauli susceptibility of conduction electrons in Pt. As\nshown in reference18, the magnetization force density can\nbe expressed in terms of the magnetization current den-\nsity, i.e.\nMPrxBT= (rx\u0002mP)\u0002BT: (8)\nThus, the relations (6) and (8) imply that the electro-\nstatic potential gradient generated by transport of the\nconduction electrons is given by,\nrxV=\u0000\u0015P\nqP\u0010\n(rx\u0002mP)\u0002BT\u0011\n: (9)\nThis e\u000bect is shown on Fig. 1 for a YIG slab with a Pt\nelectrode.\nIV. MAGNETIC NERNST EFFECT\nThe voltage di\u000berence derived from rxVin equa-\ntion (9) is a Magnetic Seebeck e\u000bect detected electrically\nthrough the magnetophoresis of the conduction electrons\nin the Pt electrode. We show now explicitly this e\u000bect in\nthe form of a Nernst e\u000bect. In a sense the Magnetic\nNernst e\u000bect is a thermally induced magnetophoresis.\nIn the Magnetic Seebeck e\u000bect, the temperature gradi-\nentryTimposed on the YIG slab induces a magnetic\ninduction \feld BTthat is oscillating in an orthogonal\nplane. Through magnetophoresis, this \feld generates an\nelectrostatic potential gradient rxVacross the Pt elec-\ntrode. Using the Magnetic Seebeck e\u000bect (3) and the\nde\fnition (4), the electrostatic potential gradient gener-\nated by magnetophoresis (9) is recast explicitly as,\nrxV=\rPY(rx\u0002mP)\u0002\u0010\u0000\nr\u00001\ny\u0002mY\u0001\n\u0002ryT\u0011\n;\n(10)\nwhere\n\rPY=\u0015P\u0015YnYkB\nqPM2\nSY: (11)\nUsing the Jacobi identity for the cross product, the linear\nrelation (10) yields the Magnetic Nernst e\u000bect, i.e.\nrxV=Nz\nm\u0002ryT; (12)\nwhere the phenomenological vector Nz\nmis given by,\nNz\nm=\rPY(rx\u0002mPz)\u0002\u0000\nr\u00001\ny\u0002mY z\u0001\n(13)\nwithmPz= (^z\u0001mP)^zandmY z= (^z\u0001mY)^zin order\nto satisfy the vectorial symmetries and contribute to the3\ne\u000bect. The structure of equation (12) relating the gradi-\nentsryTandrxVis that of a Nernst e\u000bect. In place\nof a magnetic induction \feld, there is a phenomenologi-\ncal vector Nz\nm, which depends on the out of equilibrium\nmagnetization in the YIG slab. This e\u000bect is illustrated\non Fig. 1 for a YIG slab with a Pt electrode, and a surface\ncoil is presumed to excite the ferromagnetic resonance.\nBext\nˆyˆxˆzBT\n∇yTPtYIGθ\n∆Vω\nFIG. 1: YIG slab with a Pt electrode connected to a voltmeter\nand excited by a local probe.\nIn order to determine the structure of the Magnetic\nNernst vector Nz\nm, we perform a Fourier series expan-\nsion of the linear response \felds mPzandmY z. In a\nstationary regime, the Fourier transform of the response\n\felds are expressed in terms of real parameters as,\nmPz=X\nkPmkPzsin (kP\u0001r\u0000!kPt+\u001ekP)^z;(14)\nmY z=X\nkYmkYzsin (kY\u0001r\u0000!kYt+\u001ekY)^z;(15)\nwhere\u001ekand'kare the dephasing angles and !kis\nthe angular frequency of the eigenmodes k. Finally, the\nFourier decompositions (14) and (15) imply that the re-\nlation (13) is recast in terms of the eigenmodes as,\nNz\nm=\rPYX\nkP;kYkPxk\u00001\nYymkPzmkYz (16)\n\u0001cos (kP\u0001r\u0000!kPt+\u001ekP) cos (kY\u0001r\u0000!kYt+\u001ekY)^z;\nwherekPx=^x\u0001kPandk\u00001\nYy=^y\u0001k\u00001\nY. The magnetic\nwaves vectors in YIG and Pt are collinear to the external\nmagnetic induction \feld, i.e. kP\u0002Bext=0andk\u00001\nY\u0002\nBext=0, which implies that kPx=kPsin\u0012andk\u00001\nYy=\nk\u00001\nYcos\u0012where\u0012is the orientation angle between the\ntemperature gradient and the external magnetic \feld in\nthe plane of the YIG slab as shown on Fig. 1. Moreover,\nthe speci\fc mode k=kY=kPcorresponding to the\nexcitation frequency !\u0011!kof the magnetization waves\nin YIG and Pt is determined by the quadratic dispersion\nrelation of the magnetization waves , i.e. !k=Ak2whereAis the sti\u000bness. Thus, choosing the initial time\nto cancel the dephasing of the magnetization in YIG and\nusing the trigonometric identity,\ncos (k\u0001r\u0000!t+\u001e) cos (k\u0001r\u0000!t) =\n1\n2\u0010\ncos\u001e+ cos (2 k\u0001r\u00002!t+\u001e)\u0011\n;(17)\nthe Magnetic Nernst vector (16) is recast as,\nNz\nm=\rPY\n4mkPzmkYzsin (2\u0012)\n\u0001\u0010\ncos\u001e+ cos (2 k\u0001r\u00002!t+\u001e)\u0011\n^z;(18)\nwhere\u001e\u0011\u001ekPand\u001ekY= 0 . In the homogeneous elec-\ntrode, the electrostatic potential varies linearly along the\n^x-axis, i.e. ^x\u0001rV= \u0001V=`xwhere`xis the length of\nthe electrode. Thus, the voltage across the electrode is\ngiven by,\n\u0001V=`x^x\u0001(Nz\nm\u0002ryT): (19)\nThe expressions (18) and (19) imply that the voltage \u0001 V\nalong the electrode consists of DC and AC contributions.\nThe DC contribution is proportional to cos \u001e, which im-\nplies that it is maximal in the absence dephasing between\nthe magnetization in Pt and YIG. The AC contribution\nis oscillating with an angular frequency 2 !that corre-\nsponds to the double of the excitation angular frequency\n!. The Magnetic Nernst e\u000bect vanishes if the external\nmagnetic \feld Bextis collinear ( \u0012= 0) or orthogonal\n(\u0012=\u0019=2) to the temperature gradient ryTand it is\nmaximal if there is a \u0019=4 angle between these vectors in\nthe YIG slab plane.\nIt is important to mention that the Magnetic Nernst ef-\nfect is not equivalent to thermal spin pumping19. The an-\ngular dependence of these two e\u000bects are di\u000berent. Ther-\nmal spin pumping is maximal for \u0012= 0 and minimal for\n\u0012=\u0019=2 whereas the Magnetic Nernst e\u000bect is maximal\nfor\u0012=\u0019=4 and minimal for \u0012= 0 and\u0012=\u0019=2 .\nV. CONCLUSION\nIn summary, a Nernst e\u000bect is predicted, which results\nfrom the interplay between the magnetization dynamics\nof a ferromagnet driven in a temperature gradient, and\nthe linear response of the paramagnetic electrode to the\ninhomogeneous \feld produced by the magnetization.\nFirst, in a magnetic insulating YIG slab, there is the\nMagnetic Seebeck e\u000bect, i.e. a magnetic induction \feld\ninduced by a temperature gradient. Second, the electrical\ndetection of this e\u000bect in a paramagnetic Pt electrode\ncontacted to the slab relies on the voltage induced by the\ninhomogeneous \feld acting on the conductive electrode,\ncausing a drift of charges that carry a magnetic dipole.\nThe voltage has a DC component and an AC compo-\nnent that oscillates at twice the frequency of the \feld\ndriving the magnetization. The e\u000bect is zero when the4\napplied \feld is parallel or perpendicular to the tempera-\nture gradient, and maximum at a 45\u000eangle in between.\nHence, this e\u000bect is quite distinct from the angular de-\npendence of spin pumping or the Spin Seebeck e\u000bect.\nAcknowledgments\nWe would like to thank Fran\u0018 cois Reuse and Klaus\nMaschke for useful comments and acknowledge the fol-lowing funding agencies : Polish-Swiss Research Program\nNANOSPIN PSRP-045 =2010; Deutsche Forschungsge-\nmeinschaft SS1538 SPINCAT, no. AN762 =1.\n\u0003Electronic address: sylvain.brechet@ep\r.ch\n1R. O. Cunha, E. Padr\u0013 on-Hern\u0013 andez, A. Azevedo, and S. M.\nRezende, Phys. Rev. B 87, 184401 (2013).\n2B. Obry, V. I. Vasyuchka, A. V. Chumak, A. A. Serga,\nand B. Hillebrands, Applied Physics Letters 101, 192406\n(2012).\n3S. D. Brechet, S. D., F. A. Vetro, F. A., E. Papa, S.-E.\nBarnes, and J.-P. Ansermet, Phys. Rev. Lett. 111, 087205\n(2013).\n4K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n5S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota,\nE. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401\n(2011).\n6C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P.\nHeremans, and R. C. Myers, Nat Mater 9, 898 (2010).\n7K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi,\nJ. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai,\net al., Nat Mater 9, 894 (2010).\n8K.-I. Uchida, T. Nonaka, T. Ota, and E. Saitoh, Appl.\nPhys. Lett. 97, 262504 (2010).\n9H. Adachi, K.-i. Uchida, E. Saitoh, and S. Maekawa, ArXiv\n1209.6407 (2012).\n10M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W.Bauer, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B\n88, 094410 (2013).\n11S. D. Brechet and J.-P. Ansermet, The European Physical\nJournal B 86, 1 (2013).\n12N. Leventis and X. Gao, Analytical Chemistry 73, 3981\n(2001).\n13E. P. Furlani, Journal of Physics D: Applied Physics 40,\n1313 (2007).\n14H. B. Callen, Thermodynamics and an Introduction to\nThermostatistics, 2nd Edition (Wiley & Sons: New York,\n1960).\n15J. Lim, C. Lanni, E. R. Evarts, F. Lanni, R. D. Tilton,\nand S. A. Majetich, ACS Nano 5, 217 (2011).\n16M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas,\nR. Gross, et al., Phys. Rev. Lett. 108, 106602 (2012).\n17M. Schmid, S. Srichandan, D. Meier, T. Kuschel, J.-M.\nSchmalhorst, M. Vogel, G. Reiss, C. Strunk, and C. H.\nBack, Phys. Rev. Lett. 111, 187201 (2013).\n18D. J. Gri\u000eths, Introduction to Electrodynamics (Prentice-\nHall, Upper Saddle River, 1999), 3rd ed.\n19K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kaji-\nwara, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Journal\nof Applied Physics 111, 103903 (2012)." }, { "title": "1712.04304v2.Microwave_to_optical_photon_conversion_by_means_of_travelling_wave_magnons_in_YIG_films.pdf", "content": "1 \n Microwave to optical photon conversion by means of travelling-wave magnons in YIG \nfilms \nM.Kostylev1 and A.A. Stashkevich2 \n1School of Physics, the University of Western Australia, Crawley, 6009 WA, Australia \n2LSPM (CNRS-UPR 3407), Université Paris 13, Sorbonne Paris Cité, 93430 Villetaneuse, \nFrance \n \nAbstract: In this work we study theoretically the efficiency of a travelling magnon based \nmicrowave to optical photon converter for applications in Quantum Information (QI). The \nconverter employs an epitaxially grown yttrium iron garnet (YIG) film as the medium for \npropagation of travelling magnons (spin waves). The conversion is achieved through coupling \nof magnons to guided optical modes of the film. The total microwave to optical photon \nconversion efficiency is found to be larger than in a similar process employing a YIG sphere \nby at least 4 orders of magnitude. By creating an optical resonator of a large length from the \nfilm (such that the traveling magnon decays before forming a standing wave over the resonator \nlength) one will be able to further increase the efficiency by several orders of magnitude, \npotentially reaching a value similar to achieved with opto-mechanical resonators. \nAlso, as a spin-off result, it is shown that isolation of more that 20 dB with direct insertion \nlosses about 5 dBm can be achieved with YIG film based microwave isolators for applications \nin Quantum Information. \n An important advantage of the suggested concept of the QI devices based on travelling spin \nwaves is a perfectly planar geometry and a possibility of implementing the device as a hybrid \nopto-microwave chip. \n \n \nI. Introduction \nThe magnon-based microwave-light converter is very attractive from the viewpoint of \nenlarging the potential of the superconducting qubits [1-5]. Such device whose first conversion \nstage is based on coherent coupling between a ferromagnetic magnon and a superconducting \nqubit is expected to have a bandwidth of around 1 MHz and thus operates faster than the \nlifetime of a superconducting qubit currently available (around 100 μs [1]). Moreover, the \nferromagnetic magnon coherent coupling to a superconducting qubit has recently been \nconfirmed experimentally [2,3]. The most natural and direct way of up-converting, at the \nsecond stage, thus excited microwave magnons (typical frequencies lying in the lower part of \nthe GHz range) to the optical domain is via magneto-optical interactions, thus realizing \ncoherent connection between distant superconducting qubits. In the basic configuration studied \nso far the microwave magnons, known as magnetostatic Kittel mode are excited in an uniformly 2 \n magnetized YIG sphere with a diameter of about 1 mm. This mode is spatially homogenuous \nand can be considered as the lowest ferromagnetic resonance (FMR) localized within a \nspherical ferromagnetic resonator. The FMR line-width Δ H is its major parameter determining \nthe quality factor. To improve the efficiency of the magnon-photon magneto-optical (MO) \ncoupling, the latter should be kept as large as possible. This implies a very narrow FMR line \nwhich in its turn corresponds to a narrow frequency bandwidth of effective MO interaction \nΔfFMR. Thus, the value of FWHM 2Δ H=0.3 Oe typical for high-quality YIG specimen \ncorresponds to a very modest bandwidth of 1 MHz. \nOne has to note that the currently achieved quantum efficiency of the microwave to optical \nphoton coupling via magnons in a YIG sphere is also quite modest 1010 [4]. This is much \nsmaller than already achieved with other methods [6-10]. In particular, a 10% coupling was \ndemonstrates through an opto-mechanical route [6], and electro-optics allows one to reach \n0.1% [7]. \nTherefore, in this paper we investigate theoretically an alternative solution to the problem of \nmagnon-to-light conversion. More specifically, we consider travelling magnons in the form of \nmagnetostatic spin waves as a candidate for this role. A spin wave is a wave of electromagnetic \nnature which is propagating in a saturated ferrimagnetic film and which is strongly retarded \n(1000 - 10000 times with respect to the velocity of a normal electromagnetic wave) due to a \nstrong resonance interaction with the matter. As a consequence of the slow propagation, the \nmagnetic component of the wave is extremely strong. This is why magnetostatic spin waves \ncan be exceptionally effective in MO modulating devices based on the Faraday Effect. \nFurther, such a microwave to optical photon converter has an intrinsically planar geometry. \nTherefore, contrary to the YIG sphere in a microwave cavity concept [4], it potentially allows \nintegration into a hybrid opto-microwave chip of reasonable sizes and fabrication of the \nmicrowave-photon to magnon transducers using optical lithography. \nFirst experiments on MO diffraction of light by magnetostatic surface spin waves (MSSW) \ndate back to the early 1970s [11-12]. However, the interaction efficiency was low due to the \ninadequately chosen configuration; the light was impinging on a YIG film normally. Thus, the \ninteraction length was equal to the film thickness. A major breakthrough was achieved a decade \nlater through introduction of the so-called waveguide MO interaction [13] in which case both \nthe MSSW and the optical wave propagate simultaneously in the YIG film as in a waveguiding \nstructure. In this geometry the interaction length is determined by the MSSW free propagation \npath, which typically is on the order of several millimeters. As a result, the interaction intensity \nwas increased several orders of magnitude. In the course of investigations that followed in the \nlate 1980s, several basic SW-optical device geometries have been studied, pure YIG films \ntypically having been used as a medium for magneto-optical interaction [14-16]. These \nexperiments have stressed the significance of the improvement of such an important parameter \nas the interaction efficiency. Not surprisingly, theoretical effort was focused on the specificity \nof strong MSSW-Light interactions involving multiple-scattering mechanisms depicted \nmathematically by Feynman-type multiple-scattering diagrams [17]. 3 \n Further improvement of the interaction intensity (20 – 50 times) was due to the progress in the \ntechnology of highly bismuth-doped YIG films with higher values of the Faraday constant (5 \n– 10 times) [18-20]. Also, a more sophisticated optical geometries were suggested by \ntheoreticians – the focus has been on using bi-layer ferromagnetic films and confining the \noptical modes in the film plane in order to improve the structure of optical modes and the \nlight/spin wave interaction efficiency [21,22]. All these experiments have been performed at \nroom temperatures and microwave and optical power levels well above the single \nphoton/magnon one. No experiments have been carried out so far at cryogenic temperatures \nand at the single optical-photon / magnon power level. On the other hand, the first experiments \nprobing microwave properties of magnons in YIG films at millikelvin temperatures have been \nalready carried out [23,24]. Therefore, it is very timely to look at this problem from the \ntheoretical prospective. \nThe latest theoretical effort [21,22] in this field was focused on telecom applications, therefore \nits goal was to achieve a 100% conversion of incident light into scattered light by employing a \nhigh-power spin wave signal. The incident spin wave power was of no importance in that case. \nFurthermore, the efficiency of the conversion of the microwave signal into spin waves by \nmicrowave stripline transducers was not included in the theory. \nIn the present work we focus on a qualitatively different situation. We assume that the \nmicrowave signal incident onto the device input port is at a single-photon level. Therefore, a \n100% conversion of the incident light into an output optical signal is impossible. And we are \nnot interested in achieving the 100% light-to light conversion. Instead, we are interested in a \n100% conversion of the weak input microwave signal into the output optical one, expressed in \na number of generated optical photons per one input microwave photon. The incident optical \npower can be arbitrary in this situation. We investigate this configuration theoretically and \nmake predictions about the rate of scattering of the guided light from single traveling magnons. \nIn the process of the calculation, the efficiency of microwave photon to magnon conversion is \ntaken into account in full. \nIn contrast to the case of an YIG sphere in a microwave cavity, theoretical description of \ncoupling of traveling spin waves to microwave photons in microwave stripline transmission \nlines is much more involved (see e.g. [25]). Furthermore, it is difficult to obtain efficient \nmicrowave photon to magnon coupling experimentally, unless the coupling geometry is \ncarefully optimized (which is feasible even in a mass-production setting). Therefore, an \nextended part of the present paper (Section IIA) is devoted to theoretical treatment of spin wave \nexcitation by stripline antennas. Section IIB presents theoretical details of coupling of \ntravelling-wave magnons to guided optical photons in the YIG film. Section IIIA reports on \nthe results of numerical calculations of the microwave to optical photon conversion efficiency \nemploying the formalism presented in Section II. Ways to further improve the conversion \nefficiency are discussed in Section IIIB. Conclusions are contained in Section IV. \nAs a side result of this calculation, we also evaluate the efficiency of a microwave isolator \nbased on travelling Damon-Eshbach magnetostatic surface spin waves [26] in YIG films. It has \nbeen recently shown that these devices are very important for isolation of qubits from the noise 4 \n in the microwave circuits to which they are connected [27]. Previous calculations and \nexperiments showed that losses inserted by a travelling spin wave device in the forward \ndirection can be small – about 5 dB theoretically and 10 dB experimentally (see Figs. 2-4 in \nRef. [28]). In our calculation, we evaluate transmission of the device in both directions and \nfind that a YIG-film based isolator employing asymmetric coplanar transducers has very good \nperformance characteristics. \n \nII. Theory \nA. Excitation and propagation of spin waves in the YIG film waveguide. \nIn this work we will be dealing with rather thick YIG films, that is why the Damon-Eshbach \nexchange-free approximation [26] is appropriate. However, if needed the exchange interaction \ncan be easily included in the theory by using the approach from [26]. \nThe geometry of the problem is shown in Fig. 1. It represents a YIG film of thickness L in the \ndirection y. In this section, we consider the film being continuous in both in-plane directions x \nand z. The film is placed on top of a microwave stripline. The stripline runs along the z-axis \nand consists of a number of parallel conductors (lines). We will be interested in a general case \nof a backed coplanar line shown in Fig. 1. The line consists of three parallel conductors of \nwidths w1, w, and w2 in the direction x. The central conductor of a width w is called the signal \nline. The two lines of widths w1 and w2 are ground lines. They are separated from the signal \nline by gaps of widths 1 and 2 respectively. If one of the ground lines is absent ( w2=2=0), \none deals with a microwave transmission line called the asymmetric coplanar line. If both \nground lines are absent ( w1=1=w2=2=0), but the ground plane at y=d is still present one \ndeals with a microstrip line (We will use an asymmetric coplanar line in examples which will \nbe considered in the discussion sections of the paper.) \n \nThe stripline length in the direction z is assumed to be infinite at the first stage of the problem \nsolution. Also, the YIG film will be considered as infinite in both in-plane directions x and z at \nthe first stage. At the second stage, we will assign specific values to the stripline length and the \nfilm width in the direction z, once an expression for stripline’s linear impedance has been \nobtained. The thickness of the microstrip in the direction y is assumed to be zero. All these \nassumptions significantly simplify the calculations without any significant loss of generality \n[25,30,31]. \nThe stripline is supported by a dielectric substrate of thickness d whose other surface (located \nat y=d) is metalized thus forming a backed coplanar line. The dielectric permittivity of the \nsubstrate is . A microwave current I flows through the line. The linear density of this current \nis j(x) may be non-uniform across the widths of both signal and ground lines and obeys the \nrelation as follows: \n 5 \n \nFig. 1. Vertical cross-section of the geometry of the problem. 1: Stripline ground plane. 2. \nStripline substrate. 3 Stripline (ground, signal and ground lines of widths w1, w and w2 \nrespectively). 4. Yttrium iron garnet (YIG) film. 5: Film substrate made of gadolinium gallium \ngarnet. The static magnetic field H is applied along the z-axis. \n \n/2\n/2( )w\nwj x dx I\n (1). \nA microwave Oersted field of the current drives magnetisation precession in the film. The total \nwidth of the stipline w1+1+w+2+w2 is assumed to be much smaller that the free propagation \npath of the magnetostatic surface spin waves in epitaxial YIG films (typically several mm for \n4+ micron thick YIG films). Therefore the excited magnetization oscillation propagates as a \nplane wave from the stripline transducer in both directions along the x-axis. We will term this \nstripline “the input stripline transducer”. \nWe also assume that a magnetic field zHH uis applied along the axis z thus forming \nconditions for propagation of a Damon-Eshbach surface wave (MSSW) [26] along the x-\ndirection (here zuis the unit vector along z). Also, an optical guided mode can propagate in \nthe YIG film along the x-direction with its evanescent field penetrating into the gadolinium \ngallium garnet (GGG) substrate of the film. We will deal with optical properties of this system \nin Section II B, but now let us focus on the formalism of excitation of the spin waves in the \nfilm by the input transducer and propagation of the excited spin waves in the film. \nThe goal of this section is to express the efficiency of spin wave excitation in terms of a number \nof excited magnons per one microwave photon incident onto the input port of the input \ntransducer. We will use a classical formalism for spin waves for the calculation; the quantum \ntheoretical notions of the numbers of photons and magnons will appear at the very last step of \nthis derivation (Eq.(21)). The reader not interested in the details of this calculation can skip \ndirectly to this equation. \n6 \n The most comprehensive theory of the efficiency of spin wave excitation in this geometry was \ndeveloped by Vugalter and Gilinski [31]. Our formalism will be similar to the one suggested \nby them, but because we are interested in maximising the efficiency of all involved interactions, \nwe will be paying greater attention to detail. In Ref.[31] the focus was on obtaining analytical \nsolutions. This was achieved by introducing a number of approximations valid in particular \nlimiting cases, for instance, large values of d or small values of w with respect to spin wave \nwavelength. We will keep the theory as general as possible, keeping in mind that derived \nequations will be solved numerically at the last stage of the calculation, in order to obtain results \nwhich are as rigorous as possible. This is because the focus of the present paper is not on \nderivation of equations but on getting numbers which reflect efficiency of microwave to optical \nphoton conversion via the travelling-magnon route. \nOur analysis will be similar to one we used in Refs. [31,26]. The translational invariance of the \ngeometry in Fig. 1 in the z-direction enables calculation of a quantity called the complex \nimpedance of stripline. This is achieved by using a quasi-static approach for the description of \nmicrowave transmission lines whose central point is an assumption that all the fields can be \nconsidered as uniform in the direction z. This allows decomposing the problem to a two \ndimensional one, where all dynamic variables depend only on x and y [25,30,31]. \nThe magnetization dynamics in the films are described by the linearized Landau-Lifshitz \nequation \n( )t mm H M h , (2) \nwhere the static magnetization within the film is given by zMM u where M is the saturation \nmagnetization for the film. The dynamic magnetization vector m has only two components \n(perpendicular to M) and can be represented as mxex+myey. The dynamic dipole field of \nprecessing magnetization h is attained from solving Maxwell’s equations in the magnetostatic \napproximation \n0 h, (3) \n h m , (4) \n 0( )i e h m , (5) \nwhere e is the microwave electric field generated by the dynamic magnetization due to Faraday \ninduction. In the framework of the quasi-static approach for the stripline description, from the \nsymmetry of the problem it follows that e has only a z-component: e=ezuz. A solution for all \ndynamic variables entering the equation – m, h and e – is obtained representing it as a set of \nplane waves propagating along x: \n, , , , exp( )k k k i t ikx dk\n m h e m h e , (6) 7 \n where is the angular frequency of the microwave current flowing through the input \ntransducer and k is the Fourier wave number. \nThe analytical expressions for km, and kh are given in Appendix A. Once these expressions \nhave been derived, one can calculate ( )xm, ( )xh and ( )xe by carrying out the inverse Fourier \ntransformation (6). It this work, the transformation is carried out numerically. In order to enable \nthis, magnetic losses H in the material are introduced into the expressions. We do this by \nreplacing H with H+iH in the final expressions [32]. Also, a simple analytical solution exists \nfor the far zone of the “spin wave antenna” which the input transducer actually represents. \nWe will return to the far-zone solution below. Let us now turn to the expression for the complex \nlinear impedance of the transducer rZ (often called the “Radiation impedance of a spin wave \nantenna” [31]). It may be obtained in the framework of the “Induced Electromotive force” \nmethod, as suggested in [30]. Following this method, \n/2\n2\n/21( ) ( , 0)w\nr z\nwZ j x e x y dxI , (7) \nwhere I is the microwave current through the antenna, j(x) is its linear density and ( , 0)ze x y \nis the z-component of the microwave electric field at the level of the antenna ( y=0) and the dash \nabove j denotes complex conjugation. This electric field has two components – the self-field \nof the microstrip line and the electric field of the precessing magnetization in the film. Outside \nthe spin wave frequency band the latter field vanishes, and ( , 0)ze x y reduces to the self-field \nof the stripline [31,33]. This property will be used in the following in order to calculate the \ncharacteristic impedance and the complex propagation constant for the transducer. \n Starting with this expression, a solution for rZ is obtained. We do not derive this solution here \nbecause it is a simplified version of a more general result from [29] and also because very \nsimilar solutions exist in the literature [25,30,31,33]. The final formula for the complex \nradiation impedance has the form as follows \n22( 0)r k zkZ j e y dk\nI\n , (8) \nwhere the Fourier transform of the electric field at y=0 reads \n 0( 0) cosh( ) tanh( ) ( 0)zk k xie y k d j k d h yk . (9) \nIn order to find the Fourier transform kj of the microwave current density in the transducer, \nwe may use the fact that in the quasistatic approximation, the continuity equation for j(x) [31] \nreduces to 8 \n ( ) ( / ) ( )c j x i x , (10) \nwhere x is the linear charge density distribution across the transducer width for the current \nand c is the complex propagation constant for the stripline. This implies that kj scales as the \nFourier transform of the charge density k ( / )k c kj i , where the concrete value of the \nconstant ( / )ci K is of no importance for the following. The charge density can be obtained \nby solving the electrostatic problem for the transducer [29]. Because in the following we will \nbe dealing with transducers of an unusual shape, in the present work we find x self-\nconsistently. \nTo formulate the self-consistent problem, we use the fact that in the electrostatic \napproximation, the electric potential u(x) should be uniform across the widths (in the direction \nx) of the transducer electrodes (strips). Requiring this and using Eq.(B11) from [29] which \nrelates the Fourier transform of the potential uk to the Fourier transform of the charge density \nk one arrives at an integral equation for x This equation can be easily solved numerically \nto yield x for a stripline geometry of interest. x is then Fourier transformed numerically \nto yield k. Then k kj K. This method takes into account the dielectric properties of the \nferromagnetic film and its substrate while calculating kj and k, but it does not take into \naccount the magnetic properties of the film. However, in [28] it has been shown that the \ninfluence of the excited spin waves on the x-dependence of j is negligible for most of the spin \nwave vector range, except the upper edge of the range. In the present work, we will be interested \nin smaller spin wave wavenumbers, therefore this approximation is appropriate. \nOnce we have obtained kj, the complex linear impedance rZ is easily computed numerically. \n(K cancels out in the course of this derivation, because of the static nature of the involved \nfields.) rZ is then transformed into the input impedance for the stripline by utilizing established \nformulas [33,29]. To maximize the efficiency of spin wave excitation, one has to maximize the \nmicrowave current through the stripline. This is obtained by shorting the end of the antenna. In \norder to incorporate this requirement in our model, now we assume that the stripline has a finite \nlength al (in the direction z) and the width of the film in the direction z is also finite and equals \nthe same al. \nThe input impedance for a stripline obeys Telegrapher Equations [34]. For a stripline of a length \nal whose other end is shorted one has \ntanh( )in c c aZ Z l , (11) \nwhere cZ is the characteristic impedance for the stripline. Both cZ and c relate to the linear \nparallel capacitive conductance iC and in-series inductive ZL reactance of the stripline. The \nlinear capacitance C is obtained as a by-product of the solution of the electrostatic problem for 9 \n x (see Eq.(B9) in [29]), and rZ plays the role of ZL in our case, as it reduces to ZL outside \nthe spin wave band (see the comment after Eq.(7) above). Hence, \n/( )c rZ Z i C , (12) \nc ri CZ . (13) \nThen the complex reflection coefficient from the input transducer input port reads: \n0 0( )/( )in inZ Z Z Z , (14) \nwhere 0Z=50 is the characteristic impedance of the microwave feeding line. \nLet us now assume that the microwave voltage inV incident onto the input port of the transducer \nhas an amplitude of 1 Volt. The theory of transmission lines then allows us to express the \ncurrent in the transducer as a function of the position 0 z la (z=0 coincides with the transducer \ninput port and z=al with the shorted antenna end): \n 1( ) exp( ) exp( )1 exp( 2 )in\ns\nc sVI z z l zZ l . (15) \nAs follows from Eq.(15), the amplitude of the driven precession of magnetization below the \ninput transducer will be a function of the co-ordinate z along the transducer. So, each point z \nwill be a point source of a travelling spin wave whose complex amplitude scales as I(z), and \nthe wave front in the far field of antenna will be a result of interferences of these partial waves. \nHence, strictly speaking, the problem of formation of the wave front of a travelling spin wave \nin the far zone of the antenna is 2-dimensional (see e.g. [31]). Solving the two-dimensional \nproblem is beyond the scope of the current work, as it involves consideration of caustic beams \nwhich spin waves in in-plane magnetized films are prone to form (see e.g. [35]) if the films are \nnot confined in the z direction. However, if they are confined, a guided spin wave mode will \nbe formed. The m(z) distribution for the guided spin wave is not perfectly uniform [36] across \nthe width of the waveguide, however we may neglect this non-uniformity as it is not of central \nimportance for the present paper. Accordingly, below we will assume that the m(z) is uniform \nacross the film width (the width coincides with the antenna width la) everywhere in the far zone \nof the antenna (the latter also implies that we assume that the guided mode is formed straight \nafter the wave escapes the near antenna zone.) \nThe energy conservation law tells us that the spin wave energy adjusted for energy losses due \nto the intrinsic magnetic damping in the film should be the same for any waveguide cross-\nsection. Therefore, we may introduce an effective z-uniform current through the input antenna \nwhich produces the same spin wave energy carried through a waveguide cross-section Lxla. \nThe magnitude of the effective current reads 10 \n 2\n01( )al\neff\naI I z dzl. (16) \nNote that this definition leaves the phase of the effective current undefined. \nThe dynamic magnetisation of the travelling spin wave at any position x is given by Eq.(6), \nwhere mk is given by Eqs. (A1-A2) from Appendix A. In the closed form, the Fourier \ncomponent of the dynamic magnetisation (for the input microwave signal of 1 Volt, see above) \nreads: \n ( ) ( )\n( )( )exp( ) ( )exp( ) 2( , )ˆdet( ( ))mx y mx y\nkx y k\neffAC k k y BC k k ym x y jI W k , (17a) \nwhere Cmx=1 and Cmy is given by Eq.(A3) in Appendix A. \nThe inverse Fourier transformation of (Eq.(17a)) can be carried out analytically. The analytical \nsolution of the integral in (6) is expressed in terms of exponential integral functions and two \ncomplex exponential functions [37]. The exponential integral term describes the near field of \nthe transducer [38] and the complex exponential ones two travelling spin waves propagating in \nthe two opposite directions from the transducer. A similar analytical solution can be obtained \nfor the integral in Eq.(8). The near-field term of that solution yields the antenna linear reactance \nand the real-valued term represents the spin wave radiation resistance of the antenna. We will \nnot evaluate analytically Zr in the present work. Instead we will use numerical integration in \norder to calculate Zr, which is just easier and more accurate. \nThe near field of an MSSW transducer is localised in the closest vicinity of the transducer, as \nwe previously demonstrated experimentally in Ref.[38]. Outside this area, only the travelling \nwave contributions to the integral exist. Evaluation of the integral in (6) in this far zone of the \ntransducer using the Residues Theory is straightforward and results in a simple expression as \nfollows \n \n0( )\n3\n( ) 0 0 ( ) 0 0\n0\n0( , )\n( )exp( ) ( )exp( ) 2exp( )exp( )\nˆ(det( ( ))x y\nmx y mx y\nk\neffm x y\nAC k k y BC k k yj ik x vxd IW kdk\n (17b) \nwhere ˆdet( ( ))W k denotes the determinant of the matrix ˆ( )W k shown in Appendix A and A, \nB, mxC and myC are coefficients also given in Appendix A. k0 is the value of spin-wave wave \nnumber which satisfies the implicit dispersion relation for spin waves ˆdet( ( )) 0W k. In the \nlimiting case d=∞ it reduces to the Damon-Eschbach dispersion relation [26] for spin waves in \na film not sitting on top of a backed stripline: \n2\n2 2( )/ ( ) [1 exp( 2 )]4Mk H H M kL . 11 \n In the following we will refer to the case d=∞ as “a film in vacuum”. \nThe quantity\n0kj is obtained as the Fourier transform of j(x)=const(x), where the value of the \nconstant is given by the condition 2\n2/2\n/2( )w\neff\nwj x dx I\n. \n There are two solutions to the dispersion relation ˆdet( ( )) 0W k, one for a positive wave \nnumber k and one for a negative one. The magnitudes of the two wave numbers are different, \nas the presence of the metal ground plane of a stripline at y= d makes the spin wave spectrum \nnon-reciprocal. \nThe spatial decay factor for the wave is given by the expression as follows: \n\n 0\n0ˆdet ( ) /\nˆdet ( ) /d W k dk\nH\nd W k dH \n , \nWhere H is the loss parameter for the film, and, as in Eq.(17b), the argument k0 means that \nthe derivatives are evaluated for k= k0 . \nThe analytical expression (17b) is in excellent agreement with the direct numerical evaluation \nof the integral in (17a) in the far zone of the transducer. This excellent agreement allows us to \nuse the same analytical approach in order to evaluate the Poynting vector for spin waves [31]. \nAn expression for the Poynting vector for a somewhat simpler geometry of a film in vacuum \nis shown in Appendix B. The derivation of an expression for the Poynting vector for the \ngeometry of a film on top of a backed coplanar line is also straightforward (not shown here). \nNumerical evaluation of this expression demonstrates excellent agreement with the expression \nfor the power irradiated by the transducer in the form of spin waves. \nThe microwave power incident onto the input port of the transducer reads \n2\n02in\ninVPZ. (19) \nThen from the telegraphist equations it follows that the incident microwave power converted \ninto the magnon power is given by \n \n2\n01Re (1 )(1 )2inVPZ . \nThis power is redistributed between the two waves excited by the transducer in the two opposite \npropagation directions + k and –k. Excitation of the Damon-Eshbach spin waves is highly non-12 \n reciprocal (see e.g. [39]). The portion of the power nr carried by the spin wave propagating in \nthe positive direction of the axis x (i.e. in the + k direction) is given by [30] \n/Re( )nr rR Z , \nwhere R+ is the component of the total radiation resistance associated with excitation of spin \nwaves travelling in the + k direction: \n2\n02Re ( 0)k zk R j e y dk\nI\n \n . \nHence, the power P+ carried by spin waves in the + k direction is given by \n2\n01( ) Re (1 )(1 ) exp( 2 )2in\nnrVP x xZ . (20) \nA result of numerical evaluation of this expression in the far zone of an asymmetric coplanar \ntransducer ( x>w2+2+w3) is in excellent agreement with the value of the Poynting vector for \nspin waves, as has already been stated above. \nEq.(20) also yields an expression for the efficiency of the microwave photon to magnon \nconversion \n( 0)\nm\ninP xNP . (21) \nSince the incident microwave photons and the generated magnons have the same frequency \n(energy), the efficiency of the photon-magnon transduction Nm actually represents the number \nof generated magnons per one incident photon. \n \nB. Scattering of optical photons from magnons \nIn this section we consider two guided optical modes propagating in the YIG film. One is \nincident, the other is generated due to a photon-magnon interaction. An incident optical mode \nscatters from a spin wave whose wave vector is collinear to light. This creates the second \noptical mode which we will call “scattered light”. The physics behind the process of the \ninteraction of the guided light with the spin wave is light scattering from a diffraction grid \nformed by a spatial harmonic modulation of the material’s refractive index along the light \npropagation path. The modulation originates from the dynamic magnetisation of the spin wave \n– the spatial variation of the magnetisation vector leads to spatially periodic modulation of the \nstrength of the Faraday and Cotton-Mouton Effects. The “diffraction grid” moves with the \nphase velocity of the spin wave which leads to important peculiarities of light scattering from \nit. They will be discussed in the very end of the Discussion section. 13 \n From the point of view of wave interactions, the magnon-optical photon interaction is a three-\nwave process. Classically, this process is described by a theory of coupled modes (see Eqs. \n(16-17) in [21]). The mode coupling coefficient scales as the overlap intergral for the three \nwaves (which reflects spatial correlation of the interacting waves) \n𝐼௩=∭𝐄(௦)(x,y,z)∙𝐦(x,y,z)∙𝐄()(x,y,z)dV. (22) \nHere functions E(i) (x,y,z), E(s) (x,y,z), m(x,y,z) describe the spatial distribution of the electric \nfield in the incident and scattered optical waves and that of the magnetization in the scattering \nspin wave. \nSecondly, the efficiency of the MO interactions depends on the “correlation” of their \npolarizations (the vector factor). Mathematically, the symmetry of MO coupling in an optically \nisotropic media is described by the totally antisymmetric Levi-Civita tensor [40]. As a result, \nthe vector factor is expressed via the mixed product of the interacting waves \n𝐼௩=ቀ𝑒⃗(௦)∙൫𝐦×𝑒⃗()൯ቁ=∑ ∑ ∑ 𝛿𝑒(௦)𝑚 ୀ௫,௬,௭ ୀ௫,௬,௭ ୀ௫,௬,௭ 𝑒(). (23) \nCorrespondingly, their polarizations are given by unit vectors 𝑒⃗(),𝑒⃗(௦),𝐦. An interested reader \ncan find all details in an exhaustive theoretical analysis presented in Ref . [41]. \nA YIG film can be regarded as a planar dielectric waveguide. Due to its specific symmetry \nsuch a waveguide supports two types of electromagnetic waves, namely TE-modes \n𝐄()்ா(𝑥,𝑦)=𝐸௭()்ா(𝑦)exp (−𝑖𝛽()்ா𝑥)𝑒⃗௭ and TM-modes 𝐄()்ெ(𝑥,𝑦)=ቀ𝐸௫()்ெ(𝑦)𝑒⃗௫+\n𝐸௬()்ெ(𝑦)𝑒⃗௬ቁ∙exp (−𝑖𝛽()்ெ𝑥) with 10, the \nscattering MSSW propagates in the same direction as the optical modes. Consequently, the \nscattered TM-mode is frequency up-shifted 𝛺்ா=𝛺்ெ+𝜔, here 𝜔 is the frequency of the \nscattering MSW (anti-Stokes process). On the other hand, if <0, the scattering MSW 14 \n propagates in the opposite direction with respect to the optical modes, as a result the scattered \nTM-mode is frequency down-shifted 𝛺்ா=𝛺்ெ−𝜔 (Stokes process). \n \nFig. 2. Geometry of the magnon-light interaction. (1,2,3) is the stripline used to excite magnons \nin the fiim. (1) stripline ground plane. (2) its dielectric substrate. (3) stripline itself (asymmetric \ncoplanar). (4) YIG film. (5) film substrate. “TE” denotes the non-diffracted light beam and \n“TM” the diffracted beam (light scattered from magnons). Magnons excited by a microwave \ncurrent in the stripline propagate in the film collinear with the optical photons. This results in \nlight scattering from the magnons. \n \nIn the TM →TE configuration , the Bragg condition reads \n𝛥𝛽=𝛽()்ா−𝛽()்ெ=𝑘, (25) \nand the previous paragraph can be directly applied to the analysis of the scattering mechanism, \nexcept that the superscript TM should be replaced with TE and vice versa. \nNow consider the MO coupling coefficient, describing the efficiency of the MO interaction. \nAs mentioned above, in garnets the magnetically induced perturbation of the dielectric \npermittivity at optical frequencies is comprised of two major contributions due to the Faraday \n(F) and Cotton-Mouton (CM) effects \n442F CM\nij ij ij F ijl l i j if M g M M , \nwhere Ff is the Faraday constant, g44 is a Cotton-Mouton constant, Mi is the component of the \ntotal vector zM M e m along the axis i, and summation over the dummy indices is assumed \n(Einstein convention). Here we have neglected the anisotropic part of the Cotton-Mouton \ncontribution proportional to 11 12 442 g g g which is not essential. \nIn our case, the total magnetization vector M is comprised of the following components \n( ) ( ) exp( )exp( )z x x y yM m y m y i t ikx x M e e e \n(see (17)). \nThe amplitude of the scattered optical mode, either TE or TM, i.e. the efficiency of the MO \ninteraction follows from Eq.(22). One finds that it scales as a parameter vMO referred to as the \nindex of phase modulation. Its magnitude is given by \n15 \n 0\n, , 44 , , 2 ( ) 2 ( )2S F xz y k yz x kk xz x k y\nez y\nffk i f I m I m g M I m I mN \n (26) \nfor the Stokes process and \n 0\n, , 44 , , 2 ( ) 2 ( )2AS F xz y k yz x k x\neffz x k yz y k f I m iI m g M I m iI mN \n (27) \nfor the anti-Stokes one. In Eqs.(26,27), 02 / is a light wavenumber in vacuum, and Neff \nis an effective refraction index of the “incident” waveguide mode; ( )\n0/TE n TEN N for \nthe TE→TM optical mode conversion and ( )\n0/TM TM\neffnN N for TM →TE. The \nquantities ,x kkm and ,y km are thickness averages of the respective vector components of \nthe amplitude mk (Eq.(17a)), calculated for k=. Stripline transducers excite spin waves with \nkL<1, therefore the y-dependencies of mkx(y) and mky(y) are close to uniform on the scale of the \nfilm thickness. For the purpose of our calculation, it is appropriate to assume that they are \nperfectly uniform and equal to the thickness averages of these quantities. (The expressions for \nthe latter are given by (A16-A17) in Appendix B.) Under this assumption, Eq.(23) yields \n0xzI and 1zyI. \nThe latter relations illustrate symmetry of the optical waveguide modes. More specifically, the \ntransverse components ( )( )n TE\nzE y and ( )( )n TM\nyE y have practically identical spatial distribution \nacross the waveguide and 1zyI, while the longitudinal component ( )( )n TM\nxE y is characterized \nby a symmetry that is opposite to that of the transverse components, hence 0xzI. Thus, both \neffects, Faraday and Cotton-Mouton, couple only the transverse components of the optical \nfields, i.e. ( )( )n TE\nzE y and ( )( )n TM\nyE y . \nTaking all this into account, we finally arrive at \n \n 0\n442S F kx ky\neffi f m g MmN , (28) \n 0\n442 )AS F kx\neffky i f m g MmN . (29) \nHere we denoted ,x k kxm m and ,y k kym m , in order to simplify notations . \nThe amplitude of the scattered optical mode is proportional to that of the incident one and the \ncoupling coefficient /2MO. Hence the power of the scattered light Is is given by \n2\n4MO\ns LP P, (30) 16 \n where PL is the power of the incident light and ,MO S AS , depending on whether one deals \nwith a Stokes or an anti-Stokes scattering process. The microwave-photon to optical-photon \n(microwave-to-light) conversion efficiency is given by Eq.(6) in [7]. \n \ns\ninP\nP\n, (31) \nwhere Pin is given by Eq.(19) and is the frequency of the light. \n \nIII. Discussion \nA. Calculation results \nIn this section, we show results of calculations by employing the theory above. For these \ncalculations, we use the parameters as follows. The thickness of the YIG film L=20 micron \n(Fig. 1). The input transducer is an asymmetric backed coplanar line, having a 400 micron wide \nsignal line and a 200 micron wide ground line ( w2=400 micron, w3=200 micron). The gap \nbetween the two 2=25 micron. The coplanar line is supported by a 0.5mm-thick dielectric \nsubstrate ( d=0.5mm) with =11 whose other surface is metalized and grounded. The length of \nthe transducer la= 5 mm; it coincides with the width of the film. The applied field H=1000 Oe. \nWe use magnetic parameters for the film extracted from a recent microwave experiment at 16 \nmK [23]: saturation magnetization 4 M=2360 G and gyromagnetic ratio =2.83 MHz/Oe. As \nthe resonance linewidth could not be measured in that experiment (because traveling waves \nwere excited in [23]) we use the fact that no difference between the resonance linewidths at the \nsame temperature and at room temperature was found in [42]. Therefore we use the same \nmagnetic loss parameter as typical for bismuth-substituted YIG films at room temperature: \nFWHM 2 H=0.8 Oe. \nBismuth-dopped YIG is the best candidate for the magneto-optical interaction. Unfortunately, \nits optical constants at 16 mK are not known. However, in [4] it was found that the Verdet \nconstant for a YIG sphere at 16 mK does not differ significantly from the literature value for \nthe room temperature. Therefore, in our calculation we use the literature value for the bismuth-\nsubstituted YIG: fF=103/(4M) (measured in G1) obtained for room temperature [43]. The \nconstant g44 can then be estimated based on fF and the Stokes/anti-Stokes peak asymmetry from \n[18]; one obtains 2 g44=0.58 103/(4M)2 (measured in G2). Optical wavelength is 1.15 µm \nwhich corresponds to β0 = 5.46∙106 rad/m and optical frequency =22.6 1014 Hz. N =2, and \nLP=15 mW as in the recent experiment with a YIG sphere [4]. \nLet us first calculate the efficiency of the YIG film with asymmetric coplanar transducer as a \nnon-reciprocal device called a microwave isolator. The power transmitted by the device is \nobtained by calculating the electric field of the spin waves at the position of the output \ntransducer. Then Telegraphist Equations which include a distributed source of a microwave 17 \n electric voltage are employed, in order to convert the electric field into the output voltage of \nthe output transducer [31]. A result of this calculation is shown in Fig. 3. The distance between \nthe transducers is 4 mm. One sees that transmission characteristics for the + k (from Port 1 to \nPort 2) and – k (from Port 2 to Port 1) directions differ by at least 20 dB. Also, the transmission \nlosses in the + k directin are just 5 dBm. This characterizes the spin wave device as a very \nefficient microwave isolator. Note that the magnitude of losses in the + k direction is consistent \nwith previous calculations and experiments [28,44]. \n \nFig. 3. Non-reciprocity of the spin wave device. Solid line: loss of power transmitted from Port \n1 to Port 2 (+ k direction), dotted line: loss microwave power for transmission from Port 2 to \nPort 1. Parameters of calculation: L=20 micron, la=5 mm, w2=400 micron, w3=200 micron, \n=25 micron and the distance between the Port 1 and Port 2 transducers is 4 mm. Applied \nfield is 1000 Oe. \n \nLet us now proceed to discussion of the microwave photon to optical photon conversion \nefficiency . Central to the discussion is Eq.(17a), as it enters the expression for (31). One \nsees that this expression has a resonant denominator – for some k=k0 the real part of the \ndenominator vanishes and the denominator value becomes equal to ˆIm[det( ( ))]i W k . Hence, \nˆRe[det( ( ))] 0W corresponds to resonant interaction of light with a spin wave with \neigenfrequency given by the condition ˆRe[det( ( ))] 0W k . In the vicinity of \n0 k k and for small magnetic losses ( H H ), the denominator can be expanded into \nTaylor series \n18 \n \n0 0 0 0ˆdet( ( , ))\nˆ ˆ ˆRe[det( ( , )] ( ) Re[det( ( , ))]/ Re[det( ( , ))]/W k H i H\nW k H k k W k H k i H W k H H \n \nThis expression can be re-written spin-wave frequency resolved: \n0\n0 0 0\n0ˆdet( ( , ))\n( )ˆ ˆ ˆRe[det( ( , )] Re[det( ( , ))]/ Re[det( ( , ))]/( )gW k H H\nW k H W k H k i H W k H HV k \n \nwhere 0 0( , )k H and ( , )k H is the dispersion law for spin waves given by \nˆdet ( ) 0W k. The expression for the spin wave group velocity gV reads \n\n ˆ Re det /\nˆ Re det /gd W dk\nV\nd W d \n . Substituting this expression into the expression above one finds that \nthe bandwidth of the optical guided mode conversion is given by ˆRe[det( )]/\nˆRe[det( )]/WH\nW H \n \nevaluated at 0( )k,H. The first term of this expression is close to the gyromagnetic ratio \n(Eq.(2)). Therefore, the bandwidth of the optical-photon/magnon interaction is given by \n2H, which is the same as in the case of a YIG sphere. \nHowever, for the sphere, the microwave photon to magnon conversion bandwidth is given by \nthe same H, but for the travelling spin waves the latter bandwidth is much larger – several \nhundreds of MHz, as one sees from Fig. 3. This implies that multi-channel regime of magnon-\noptical photon conversion can be implemented, with each individual channel corresponding to \na specific pair of optical modes “n” with a specific value of TE n-TMn birefringence n. \n(Recall that the k= condition should be satisfied, therefore the central microwave frequency \n𝜔௧(𝑘=𝛥𝛽) for each independent MO channel will correspond to a particular point in \nthe Damon-Eshbach MSW dispersion curve.) The bandwith of each channel, scaling as the \ninverse of the effective length of the MO interaction leff, typically varies from 10-1 to 10 MHz. \nThus, it is considerably smaller than the above-mentioned frequency band of MSSW excitation \nby an MSSW transducer. \nThe important point of the frequency characteristics of a MO channel steming from the Bragg \ncondition based phase synchronism for the optical guided modes and MSSW will be addressed \nin the last section “B. Ways to further improve the efficiency”. In the following, we assume \nthat the phase synchronism Bragg condition k=has been satisfied at any microwave \nfrequency by properly choosing a respective pair of guided optical modes. Therefore, in all \nfigures below we show the efficiencies of the microwave to optical photon conversion \ncalculated for the maximum of the conversion bandwidth k= In other words, we assume \nthat the condition k= is fulfilled for each frequency shown in the graphs. Mathematically 19 \n this means that the real part of the denominator of Eq.(17a) vanishes for each frequency and k \nfor each frequency is chosen such that it makes ˆRe[det( ( , )]W k H H vanish. \nFigure 4 shows the result of our calculation for the asymmetric coplanar geometry. One sees \nthat the efficiency of the anti-Stokes process is larger than that of the Stokes one in our \ngeometry. Therefore, below we will focus on the anti-Stokes process. \n \n \n \nFig. 4. MOP conversion efficiency for the parameters of Fig. 3 (except the second coplanar \ntransducer is absent, because not needed). Solid line: anti-Stokes process; dotted line: Stokes \nprocess. The length of the area where magnons interact with guided optical photons is 3 lf=20 \nmm. \nIn this figure, we actually demonstrate the efficiency of the magnon to optical photon (MOP) \nconversion. To carry out this calculation, we use the same Eq.(31) but equate Pin in it to the \npower P+ of spin waves radiated by the stripline transducer in the direction of optical mode \npropagation. The latter is given by Eq.(21). \nFrom the figure, one sees that the light scattering process is characterised by a pronounced \nmaximum. One also sees that the efficiency of MOP conversion is quite high – on the order of \n105, or tens of parts per million (PPM). In order to understand formation of the maximum, in \nFig. 5 we compare this result to the MOP efficiency in the assumption that a traveling spin \nwave only exists in a medium. In this case, Eq.(29) can be cast in the form given by Eq.(A18) \nin Appendix B. This form of the equation for \nMOis convenient for calculation of MOP \nefficiency of eigen-waves. Let us use the case of a film in vacuum ( d=∞) as a reference. In \nAppendix B, we show a simple analytic expression (A20) for the Poynting vector eig for the \n20 \n eigen-waves in this geometry. By using (A19) and equating Pin in (31) to | eig| one obtains the \ndotted line in Fig. 5. \n \nFig. 5. MOP conversion efficiency for the anti-Stokes process. Dotted line: eigen-waves in a \nfilm in vacuum ( d=∞). Solid line: Film on top of the backed stripline, but only the traveling \nspin wave (i.e. far-field) contribution is accounted for. Dashed line: total MOP conversion \nefficiency for the backed stripline (contributions of both far and near spin wave field of the \nasymmetric coplanar transducer are taken into account). \nThe solid line in this figure is obtained by calculating the amplitude of the traveling wave \ncomponent of the total spin wave field excited by the transducer (in real space, Eq.(17b)). This \ncomponent and P+ (Eq.(20)) are calculated for x=0 and then Eq.(A18) is applied to compute \nthe MOP conversion efficiency. The latter is obtained by equating Pin to P+(x=0) in Eq.(31). \nOne sees that the two curves agree perfectly for frequencies larger than 5.4 GHz. Below this \nfrequency, the efficiency of MOP conversion for the real coplanar device goes down. This is \nexplained by the effect of the metal of the stripline ground plane located at y=d. The effect of \nthe ground plane is present for kd<1. The latter wave number range corresponds to the \nfrequency range below 5.4 GHz. When d is increased in the numerical simulation, the \nmaximum in the solid line shifts to lower frequencies. This confirms that the maximum is \nformed due to the effect of the ground plane. \nOne also sees that the total MOP conversion efficiency (dashed line) is two times bigger than \nthe traveling spin wave contribution. This evidences that the contribution of the near field of \nthe transducer to the total MOP conversion efficiency may be very significant. Again, the \ndecrease in the efficiency below 5.4 GHz is due to the effect of the metal ground plane. \nThe respective total microwave photon to optical photon (MPOP) efficiency (Eq.(31)) and \nthe efficiency of microwave photon to magnon conversion Nm (Eq.(21)) is shown in Fig. 6. \nOne sees that the MPOP efficiency is of the same order of magnitude as the MOP one. This is \n21 \n because the efficiency of the microwave photon to magnon conversion Nm by the asymmetric \ncoplanar transducer is close to 1. \n \n \nFig. 6. Total (MPOP) conversion efficiency (left-hand axis) for the asymmetric coplanar line \ngeometry from Fig. 5. Solid line: anti-Stokes process; dotted line (Stokes process). Dashed \nline: microwave photon to magnon conversion efficiency by the asymmetric coplanar \ntransducer (right-hand axis). \n \nB. Ways to further improve the efficiency \nEven though MPOP efficiency achievable with travelling magnons is significantly larger than \none achievable with a standing wave oscillation [4], it is still much smaller than for \noptomechanical converters (10%). Therefore, below we discuss ways to further improve the \nconversion efficiency. However, one has to keep in mind that all these measures will decrease \nthe available frequency bandwidth. \nThe first obvious way to increase efficiency is by decreasing the area of cross-section for light \nscattering from magnons. Figure 7 demonstrates the results of calculations of MOP conversion \nefficiency for three different film cross-sections. The first one is the same as in Fig. 4 – L=20 \nmicron, la=5 mm. The second one is for the film thickness L=4 micron, which is the same \nthickness as in the experiments [18,19], but keeping the film width the same la=5 mm. The \nthird calculation is for a microscopic film cross-section L=4 micron, la=50 micron. One sees \n22 \n that the decrease in the film thickness alone does not change the efficiency; this is because, \nultimately, the efficiency scales as kL, as does the frequency of the Damon-Eshbach waves for \nkL<<1. In other words, k-vector resolved, one gains in efficiency by decreasing L. However, \nfrequency-resolved, the net gain is zero, because the spin wave frequency for a given k \ndecreases proportionally, such that for the same frequency the efficiency remains the same. \nThis is illustrated in Fig. 7 by showing k() dependences for both film thicknesses. \n On the other hand, the decrease in the film width has an impact on the efficiency, as the same \nfigure shows. Here we have to note that for the 4x50 square micron cross-section, L and la are \ncomparable, therefore, strictly speaking, our theory developed under assumption L<< la is not \nfully applicable in this case. Notwithstanding this, it demonstrates that the reduction of the film \nwidth is a valid way to significantly boost the efficiency of the MOP conversion. However, the \ndecrease in la decreases coupling of the microwave field of the transducer to the precessing \nspins. This happens because the number of spins in the volume where the microwave field is \npresent shrinks with the decrease in la. In our case of a stripline shorted at its end this is seen \nas a decrease in the input antenna impedance for la =50 micron. It is possible to compensate \nthe drop in the impedance to a significant extent by playing around with the geometry of the \ncoplanar line, as shown in Fig. 8. To this end, one has to use a coplanar line with a microscopic \ncross-section. As this calculation shows, in this way it is possible to achieve an impressive total \nconversion efficiency of 0.3 percent. Importantly, the difference between the MPOP and MOP \nefficiencies is about 3 times in this case, which is not a huge number, and the input impedance \nof the coplanar transducer is large enough – about 8 Ohm. Hence, it should be technically easy \nto increase MPOP to 0.9% by inserting an impedance matching circuit between the feeding \nline and the coplanar transducer. \nThe conversion efficiency of almost 1% is a very good result. However, the possibility of its \nfurther improvement through additional reduction of the YIG film width should be taken with \ncaution, as our theory is not fully applicable for la comparable to L. \nOne more way to improve the efficiency is by further increasing the time of spin wave \ninteraction with the optical modes. This can be done by confining the optical field inside a \nlength equal to lf by forming an optical resonator of this length. Then the time of spin wave \ninteraction with the optical modes will increase by Q times, where Q is the quality factor for \nthe optical resonator. \nTo be specific, let us begin with a modest and easily realizable quality factor of Q = 100. \nNaturally, this will increase the length of magnon-light interaction in a 4 µm thick YIG film \nfrom 0.6 cm to 0.6 m which will increase the efficiency accordingly from 102 to 1 (or 100%), \nas the efficiency scales as lf (see Eq.(A18)). At the same time, this increase in the interaction \nlength will inevitably lead to restrictions imposed by the Bragg selectivity in the reciprocal \nspace. As a result, even a slight deviation from the phase-synchronism Bragg condition will \nlead to a drop in the efficiency of the MO interaction according to 2sinc ( ) where is the \nphase mismatch due to the above-mentioned deviation. To be specific, let us consider the \nTE→TM configuration with Δβ > 0 (see Eq.(24)), in which case the magnon propagating in \nthe same direction as the “incident” TE optical mode will contribute to the anti-Stokes process 23 \n (the up-shifted scattered TM mode). There are two mechanisms, both dispersion related, \ncontributing to this mismatch: magnetic (MSSW mode) and optical (optical waveguide mode). \n \nFig. 7. Thick solid line: conversion efficiency for L=20 micron, la=5 mm; dotted line: the same, \nbut L=4 micron, la=5 mm; dashed line: the same, but L=4 micron, la=50 micron. Thin solid \nline: spin wave wave number for L=4 micron (right-hand axis); dash-dotted line: the same, but \nfor L=20 micron. \n \nFig. 8. Solid line: MOP conversion efficiency or the microscopic device (left-hand axis). \nDotted line: total (MPOP) conversion efficiency (left-hand axis). Dashed line (right-hand axis): \nmicrowave photon to magnon conversion efficiency by the asymmetric coplanar transducer \n(w1==0). Film thickness: L=4 micron, film width: 50 micron. Width of the signal line of the \nasymmetric coplanar transducer: w2=4 micron, width of the ground line: w3=20 micron, gap \nbetween the two: =250 micron. Distance to the ground plane d=500 micron. Dielectrip \npermittivity of the substrate of the coplanar line: 11. \n \n24 \n First, numerical estimation of the value of the wave number of the scattered optical mode \nshould take into account the Doppler shift imposed by the moving magnon, i.e. \n( ) ( ) /TM TM opt\ngV , where opt\ngV is the group velocity of the optical mode. The \nlatter approximation is perfectly justified since ω is really very small with respect to Ω and \ntypically ω/Ω ~10-5. Thus, the Doppler related additional phase shift reads \n/2 /(2 )opt\neff eff gl l V , here leff is the effective interaction length, while ( ) \nis the perturbation of the wavenumber of the scattered optical TM mode due to the Doppler \nfrequency shift and, consequently, (0) 0. The optical wave number frequency dependence \n( ) makes the total phase mismatch also frequency dependent ( ) . Suppose that \nin the absence of the Doppler shift the Bragg condition is fully satisfied and the phase \nsynchronism is perfect, i.e. (0) 0 . Now let us estimate the consequences of the Doppler \neffect, namely the deviation from the phase synchronism ( ) ( ) (0) for a MSSW \nfrequency of 4 GHz and two characteristic values of the effective interaction length mentioned \nabove leff = 0.6 m (with the optical resonator) and leff = 6 mm (without the resonator). In the \nconventional no-resonance case one obtains (4GHz,6mm) 0.5rad which is not enough to \ninfluence appreciably the mechanism of the MO interaction that is why this effect was \nneglected in the earlier papers of the 1980s – 1990s. It follows from the same analysis that the \neffective bandwidth of the MO interaction defined as full width at half maximum (FWHM) \nand adapted for the Sinc function is equal to (6mm) 30GHzMSSWf . In the resonator case one \nobtains an impressive figure of 2( ) 0.5 10 rad and, as a result, (0.6m) 300 HzMSSWf M . \nSecond, the MSSW dispersion also appears in the Bragg condition. Moreover, the phase \nsynchronism is even more sensitive to MSSW frequency variations through this mechanism \nwhich is due to the direct presence of the wavenumber of extremely slow MSSW modes in the \nBragg condition. As in the previous case, the coefficient weighing this contribution is the \ninverse group velocity, but this time that of the MSSW that is about 104 times slower. Let us \nsuppose that the 4µm thick YIG film is magnetized to saturation by a 800 Oe magnetic field. \nIn this case a MSSW with a frequency of 4 GHz will propagate with a group velocity gV of \napproximately 4∙104 m/s. Correspondingly, in the “without-resonator” and with-resonator \nconfigurations one obtains (6mm) 8MHzMSSWf and (0.6m) 80KHzMSSWf respectively. \nThe former figure was confirmed in the experiments in the 1980s-1990s. In any case, it is the \nsecond mechanism that bottlenecks the frequency properties, its bandwidth in the first place, \nof the MO interaction. \nThus, in this paragraph we provide useful quantitative data on the « bandwidth – efficiency » \nlimitations, classical in photonics, which stem from the general laws of three-wave interactions \nrequiring phase synchronism between the interacting waves. This information is indispensable \nfor the design of specific devices specialized in coherently connecting distant superconducting \nqubits via light. \nOn the other hand, our analysis emphasizes the importance of the role played by the Doppler \nshift in the scattering of light by MSSW in the case where the effective interaction length leff 25 \n exceeds a critical value of several centermeters in the lower GHz band addressed in this paper. \nWhile it is of less importance in the case of Brillouin light scattering by thermally excited \nincoherent magnons [5], it should not be overlooked when one considers coherent magnon-\nqubit up conversion to optical frequencies [45]. It is especially important for MO interactions \ninvolving the homogeneous Kittel mode in which case Δβ→0, especially if the optical resonator \nquality factor is as high as 105 [46]. In other words, the actual pertinent criterion of the \nsmallness of the Doppler shift [45] cannot be formulated solely in terms of the ratio of the \nfrequencies of the interacting waves ఠ\nఆ≪ 1, even if it is as small as 10-5. It must follow directly \nfrom the Bragg phase synchronism and be expressed in terms of the maximum tolerable phase \nmismatch, thus reading \n∆𝜑=ఠ\n௩\nଶ< 1. \nIt is should emphasized that boosting the efficiency of the MO interaction through a radical \nincrease of the interaction length cannot be implemented in the absence of a reliable mechanism \nof fine-tuning to the Bragg condition. In this regard, the configuration relying on travelling spin \nwave has an advantage of an additional flexible degree of freedom, namely the tunable spatial \nperiodicity in the form of the spin wave wavenumber k. \n Another important aspect of the Doppler frequency shift is its asymmetry with respect to the \ninversion of the direction of the incident optical mode which can be exploited in order to create \nnon-reciprocal MO devices. Thus, if both interacting waves, the spin wave and the incident \noptical mode, propagate collinearly this shift will be positive ( +ω), whereas anti-collinear \npropagation will produce a negative shift ( ω). This, in its turn, means that if the Bragg \ncondition is perfectly satisfied in the collinear geometry ∆𝜑(+𝜔) = 0, reversal of propagation \ndirection of the optical incident wave will lead to a double phase asynchronism ∆𝜑(−𝜔) =\n2ఠ\n௩\nଶ . As a result, the amplitude of the “new-born” scattered optical wave will reduce \naccordingly. In other words, such a MO element can be regarded as an optical isolator with a \nratio of nonreciprocity (which is defined as the ratio of amplitudes of counter propagating \noptical modes) equal to 𝑆𝑖𝑛𝑐(∆𝜑(−𝜔)). \n \nIV. Conclusions \nIn this work we evaluated theoretically the efficiencies of a travelling magnon based \nmicrowave to optical photon converter for applications in Quantum Information. The \nmicrowave to optical photon conversion efficiency was found to be larger than in a similar \nprocess employing a YIG sphere by at least 4 orders of magnitude. By employing an optical \nresonator of a large length (such that the traveling magnon decays before forming a standing \nwave over the resonator length) it will be possible to further increase the efficiency by several \norders of magnitude, potentially reaching a magnitude similar to one achieved with opto-\nmechanical resonators. However, this measure will decrease the frequency bandwidth of \nconversion. 26 \n Also, as a spin-off result, it has been shown that microwave isolation of more that 20 dB with \ndirect insertion loss of about 5 dBm can be achieved with YIG film based isolators. These \ndevices are needed to isolate qubits from noise in a microwave circuit to which they are \nconnected. \nAn important advantage of the concept of the travelling spin wave based Quantum Information \ndevices is a perfectly planar geometry and a possibility of implementing a device as a hybrid \nopto-microwave chip. \n \nAcknowledgement \nResearch Colaboration Award from the University of Western Australia is acknowledged. The \nauthors also thank M. Goryachev, M. Tobar and V.N. Malyshev for fruitful discussions. \n \n \nAppendix A: Solution of the system of equations (2-5). \nThe solution is obtained in the Fourier space (Eq.(6)). \nexp( ) exp( )kxm A k y B k y , (A1) \n( )exp( ) ( )exp( )ky my mym AC k k y BC k k y (A2) \nwhere \n2 2\n2 2 2( )( )( )M\nmy\nH M Hqk q kC q iq q k \n . (A3) \nSimilarly, \n \n( )exp( ) ( ( ))exp( )kx hx hxh AC k k y B C k k y , (A4) \n( )exp( ) ( ( ))exp( )ky hy hyh AC k k y B C k k y , (A5) \nwith \n2 2 2( )( )( )H\nhx\nH M Hk q kC qq q k \n , (A6) \nand \n2 2 2( )( )( )H\nhy\nH M Hq q kC q iq q k \n , (A7) 27 \n where HH and MM (or 4MM in Gaussian units ). \nApplication of the electro-dynamic boundary conditions results in a vector-matrix equation \nsign( )ˆ\n0k A i k jWB , (A8) \nwhere \n(| |) ( | |)ˆ\n(| |) ( | |)k kWk k \n , (A9) \n \n( ) ( ) ( ) coth(| | ) sign( ) ( )my hy hxq C q C q q d i q C q , (A10) \nand \n( ) ( ) ( ) sign( ) ( ) exp( )my hy hxq C q C q i q C q qL . (A11) \nSolving (A8) with respect to the vector on its left-hand side yields \n1sign( )ˆ\n0k A i k jWB , (A12) \nwhere \n1 ( | |) ( | |)1ˆ\nˆ(| |) (| |) det( )k kWk k W \n , (A13) \nand ˆdet( )W denotes the determinant of the matrix ˆW. \nAccordingly, \nˆ sign( ) (| |) /det( )k A i k k j W , (A14) \nˆ sign( ) ( | |) /det( )k B i k k j W . (A15) \n \nThis concludes the solution of the problem of calculation of the spin wave amplitude mk. The \nclosed form of the expression for mk is given by Eq.(17(a)). \n \nAppendix B: Calculation of quantities entering expressions for magnon to optical photon \nconversion efficiency \nThe thickness-averaged spin wave amplitude is obtained from Eqs. (A1) and (A2) and reads: \n 28 \n ( ) ( )kxm AF k BF k , (A16) \n( ) ( ) ( ) ( )ky my mym AC k F k BC k F k , (A17) \nwhere ( ) sign( )(exp( ) 1)/( )F k k kL kL . \nFor eigen-waves we may set A=1. Then (A14) and A(15) yield ( | |)/ (| |)B k k . Substituting \nthese A and B into (A16) and (A17) and the result into (28) and (29) we obtain the MOP \nconversion efficiency. In doing this we need to specify the length of the MO interaction area. \nIt is found introducing the spin wave propagation path as /f gl H V , where His the \nmagnetic loss parameter and /gV k is the group velocity of spin waves. Then the wave \ndecays exponentially during its propagation 0exp( / )exp( )k k f x l ikx m m , where 0\nkm is its \ninitial amplitude (at x=0). Substituting this expression into (17b) and it into the overlap integral \n(22) we arrive at the expression for the light-magnon coupling coefficient for the eigen-waves. \n 0 0 0 0 0\n44 ( ) 2 ( )2f\nAS F xz y yz x xz x yz ylf I m iI m g M I m iI mN . (A18) \nIn order to calculate the MOP conversion efficiency for the eigenwaves we also need the \nPoynting vector for them. It is obtained from (A2), (A4), (A5) and (5). 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Lett . 116, 223601 \n(2016). \n \n " }, { "title": "2308.13586v3.Macroscopic_distant_magnon_modes_entanglement_via_a_squeezed_reservoir.pdf", "content": "Macroscopic distant magnon modes entanglement via a squeezed reservoir\nKamran Ullah,1,∗M. Tahir Naseem,2,†and ¨Ozg¨ur E. M ¨ustecaplıo ˘glu1, 3,‡\n1Department of Physics, Ko c ¸University, 34450 Sarıyer, Istanbul, T ¨urkiye\n2Faculty of Engineering Science, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology,\nTopi 23640, Khyber Pakhtunkhwa, Pakistan\n3T¨UB˙ITAK Research Institute for Fundamental Sciences, 41470 Gebze, T ¨urkiye\n(Dated: December 19, 2023)\nThe generation of robust entanglement in quantum system arrays is a crucial aspect of the realization of efficient\nquantum information processing. Recently, the field of quantum magnonics has garnered significant attention as\na promising platform for advancing in this direction. In our proposed scheme, we utilize a one-dimensional array\nof coupled cavities, with each cavity housing a single yttrium iron garnet (YIG) sphere coupled to the cavity\nmode through magnetic dipole interaction. To induce entanglement between YIGs, we employ a local squeezed\nreservoir, which provides the necessary nonlinearity for entanglement generation. Our results demonstrate the\nsuccessful generation of bipartite and tripartite entanglement between distant magnon modes, all achieved through\na single quantum reservoir. Furthermore, the steady-state entanglement between magnon modes is robust against\nmagnon dissipation rates and environment temperature. Our results may lead to applications of cavity-magnon\narrays in quantum information processing and quantum communication systems.\nI. INTRODUCTION\nQuantum entanglement plays a pivotal role in various ap-\nplications of quantum information processing [ 1], encompass-\ning quantum cryptography [ 2], quantum teleportation [ 3], and\nquantum metrology [ 4]. It is a crucial resource for enhanc-\ning the performance of quantum devices and technologies [5].\nHowever, preparing long-lived entangled states, especially at\nthe macroscopic scale, becomes challenging due to inevitable\ninteractions between quantum systems and their environments.\nConsequently, the quest for generating macroscopic entangled\nstates using various physical setups has gained significant at-\ntention. In this context, steady-state entanglement between\natomic ensembles has been successfully demonstrated [ 6–8].\nAdditionally, entangled states involving macroscopic systems\nhave been reported in different setups, such as entanglement\nbetween a single cavity mode and a mechanical resonator in\nan optomechanical configuration [ 9]. Recently, experimental\nachievements include entanglement between two macroscopic\nresonators coupled to a common cavity mode through optome-\nchanical interaction [10].\nAnother central challenge in harnessing entanglement to\nimprove various quantum tasks lies in the ability to create and\ndistribute entanglement across large arrays of quantum systems\n[11–17]. An intriguing approach to address this challenge is\nreservoir engineering, where external control drives are used\nto engineer desirable dissipative dynamics, leading the quan-\ntum system to relax in the desired quantum state [ 14,18–25].\nHowever, most reservoir engineering schemes require multiple\nexternal control fields to be applied over different elements of\nthe quantum system array [26, 27].\nIn recent years, hybrid quantum systems, combining differ-\nent physical subsystems, have received significant attention\n∗kullah19@ku.edu.tr\n†mnaseem16@ku.edu.tr\n‡omustecap@ku.edu.tr[28–30]. Among these systems, cavity magnon setups offer a\nunique platform to investigate light-matter interactions [ 31,32].\nIn cavity quantum electrodynamics, the strong and ultrastrong\ncoupling between magnons and photons has been increasingly\nstudied [ 31,33–37], with experimental demonstrations of these\nregimes [ 38–40]. This strong coupling opens possibilities for\nexploring novel phenomena, such as the magnon Kerr effect\n[41], cavity-magnon polaritons [ 42], magnon-induced trans-\nparency [ 43], magnomechanically induced slow-light [ 44],\nbistability [ 45], exceptional points [ 42,46], magnon block-\nade [47], and nonreciprocity [48].\nRecent proposals aim to realize genuine quantum effects\nin macroscopic magnon-cavity systems, including magnon\nblockade [ 49], magnon squeezed states [ 50], Schr ¨odinger cat\nstates [ 51,52], Bell states [ 53], and nontrivial bipartite and\nmultipartite entangled states [ 54,55]. Of particular interest\nis the generation of entanglement between distant magnon\nmodes [ 56], which holds promise for quantum information\nprocessing applications. However, to the best of our knowledge,\nexisting studies have primarily focused on entangling only\ntwo magnon modes, such as entangling two YIG spheres in\na single cavity via Kerr nonlinearity [ 57], magnetostrictive\ninteraction [ 58], or applying a vacuum-squeezed drive [ 59,\n60]. Further, stationary quantum entanglement between two\nmassive magnetic spheres can be induced by subjecting each\nsphere to two-tone Floquet fields, which effectively generates\nparametric interaction between magnon modes [ 61]. In the case\nof two cavities, each containing a single YIG, entanglement\nbetween the YIGs has been generated through optomechanical-\nlike coupling [ 62], vacuum squeezed drive [ 63], or reservoir\nengineering [ 56,64,65]. Recently, the bipartite entanglement\nbetween magnon modes is investigated in a one-dimensional\narray of cavities. The entanglement is generated by exploiting\nthe coupling of the magnon modes with a central giant atom\nvia virtual photons [66].\nHybrid quantum magnonic systems hold great promise for\nefficient quantum networks [ 67]. Therefore, a crucial task is to\nestablish and control entanglement between multiple magnon\nmodes in arrays based on quantum magnonic systems. In thisarXiv:2308.13586v3 [physics.optics] 17 Dec 20232\nFigure 1. (a) Schematic description of magnon-magnon entanglement generation scheme, consisting of a one-dimensional 2N+ 1array of\nidentical lossy cavities coupled via hopping interaction J. Each cavity contains a yttrium iron garnet (YIG) sphere with volume V. The cavities’\ndecay and dissipation rates are κaandκj, respectively. The central YIG is coupled to a quantum-squeezed reservoir, enabling our entanglement\ngeneration scheme. (b) Describes the effective model, which consists of only magnon modes obtained after removing the cavities via employing\nSchrieffer-Wolf transformation. (c) shows the behavior of the dispersion relation of the cavity mode in momentum space.\nregard, we propose a scheme to generate entanglement between\nmultiple yttrium iron garnet (YIG) spheres using a reservoir\nengineering approach with a single squeezed bath. Specifically,\nour setup consists of an array of cavities, each housing a macro-\nscopic YIG coupled to the cavity through beam splitter-like\ninteraction. However, entanglement cannot be generated with\nthis interaction alone; thus, we introduce the necessary nonlin-\nearity by driving one of the magnon modes with a squeezed\nreservoir. Previous studies have demonstrated that a single and\ntwo-mode squeezed drive can create entanglement between\nmagnon modes [ 59,60,62,63]. In our work, we show that\nwhen one of the magnon modes is subjected to a squeezed\nreservoir, it is sufficient to generate steady-state bipartite and\ntripartite entanglement between distant magnon modes.\nWe would like to highlight that, in principle, the local\nsqueezed reservoir on the central magnon mode ( m2) can be\nrealized by coupling an auxiliary quantum system. Genera-\ntion of the squeezed input field can be accomplished using a\nmethod similar to the proposal discussed in [ 68]. For example,\nthe input squeezed reservoir can be engineered by coupling\nan auxiliary microwave cavity to the central magnon mode\nthrough beam-splitter-type interaction. The weak squeezed\nvacuum field, generated via a flux-driven Josephson parametric\namplifier, acts as the driving force for the auxiliary cavity. Con-\nsequently, the auxiliary cavity serves as a quantum squeezed\nreservoir for the central magnon mode [ 50,69]. Similarly, the\nsqueezed reservoir can also be achieved by coupling the central\nmagnon mode to a superconducting qubit [ 70]. More specif-\nically, the approach proposed in [ 71] can be implemented\nby replacing the coplanar waveguide resonator with a YIG\nplaced inside a microwave cavity. This scheme enables the\ncoupling of a superconducting qubit with an adjustable en-\nergy gap to a magnon mode characterized by the frequency\nω2. By modulating the energy gap of the qubit with a bichro-\nmatic field, the squeezed reservoir for the magnon mode can\nbe engineered [ 50]. Alternatively, the input squeezed radia-\ntion can be generated by applying a two-tone microwave field\ndrive to the central YIG, where the magnon mode is coupledto its mechanical vibrational mode through magnetostrictive\ninteraction [72, 73].\nThe paper is structured into the following sections: In Sec. II,\nwe introduce the model; and in Sec. III derivation of effective\nHamiltonian is presented. The results for bipartite and tripartite\nentanglement between the magnon modes are discussed in\nSecs. IV A and IV B, respectively. Finally, in Sec. V, we offer\nconcluding remarks and summarize the key findings.\nII. THE MODEL\nWe investigate a cavity-magnon system comprising a one-\ndimensional array of 2N+ 1cavities, as illustrated in Fig. 1.\nThese cavities are interconnected through photon-hopping in-\nteractions. Inside each cavity, a single YIG sphere is present,\ncoupled to the cavity mode via magnetic dipole interaction. In\nthis study, we consider a single magnon mode, which repre-\nsents quasi-particles with collective spin excitations, associated\nwith each YIG in the cavities. The Hamiltonian governing the\nfield for the cavity array is as follows:\nHa/ℏ=ωc/summationdisplay\njˆa†\njˆaj−J/summationdisplay\nj(ˆajˆa†\nj+1+ ˆa†\njˆaj+1).(1)\nThe first term represents the free energy of the cavities, while\nthe second term accounts for the exchange energy of the field.\nˆajandˆa†\njare the bosonic annihilation and creation operators of\nthej-th cavity and the corresponding resonance frequency ωc.\nHerej(j=−N,.., 0,...,N ) describes the index of the cavity\nin a one-dimensional array, with each cavity interconnected\nthrough a hopping coupling called J. We consider only a single\nphoton process in each cavity.\nThe ferromagnetic sample (YIG) holds within it scattering\nspin waves, with the assumption that only the spatially uniform\nKittel mode [ 74] exhibits a pronounced interaction with pho-\ntons within the cavity. The free Hamiltonian for these magnon3\nmodes is given by\nHm/ℏ=/summationdisplay\nnωnˆm†\nnˆmn, (2)\nhere ˆmn(ˆm†\nn) is the annihilation (creation) operator of the\nmagnon mode, and ωnis the associated frequency of the mode.\nIn addition, the index nin the summation is an integer number\nsuch thatn∈[−N,N ]. In general, some cavities can be\nempty while some can contain YIG spheres. For example, in\nSec. IV, we assume that three of the cavities are occupied\nwhile the rest remain empty. In this scenario, Eq. (2) includes\nonly three terms for magnon modes, however, 2N+ 1cavities\nare present. The magnon frequencies can be calculated based\non the applied magnetic fields Hn, and given by ωn=γHn,\nwhereγ/2π= 28 GHz/T represents the gyromagnetic ratio.\nThe interaction between the magnon and cavity modes is given\nby\nHI/ℏ=/summationdisplay\nngn(ˆanˆm†\nn+ ˆa†\nnˆmn), (3)\nhere,gnis the coupling strength between the magnon\nand associated cavity modes and is given by gn=\nζγ/2/radicalbig\n5ℏωnµ0N/V . The volume of the cavity is given by V,\nNrepresents the total number of spins in the YIG, µ0is the\npermeability of free space, and ζdescribes the spatial over-\nlap between the magnon and photon modes. We note that the\ninteraction term in Eq. (3) is obtained after performing the\nHolstein-Primakoff transformation in which collective spin\noperators are written in the form of bosonic magnon operators\nˆm(ˆm†) [57]. In addition, the counter-rotating terms ˆmˆa,ˆm†ˆa†\nare ignored assuming the validity of rotating wave approxima-\ntion [59]\nIII. THE SCHRIEFFER–WOLFF APPROXIMATION AND\nEFFECTIVE HAMILTONIAN\nTo generate entanglement between two magnon modes, a\nnonlinear interaction such as effective parametric-type nonlin-\near coupling ( ˆm†\n1ˆm†\n2+ ˆm1ˆm2)between the modes is required.\nThis interaction can be achieved by introducing strong Kerr\nnonlinearity [ 57] or nonlinear magnomechanical interaction\n[58]. It is typically easier to engineer an effective beamsplitter-\ntype coupling ( ˆm†\n1ˆm2+ ˆm1ˆm†\n2)between the magnon modes.\nFor instance, in a system with two YIGs spheres placed in-\nside a microwave cavity, it is possible to generate the de-\nsired coupling ( ˆm†\n1ˆm2+ ˆm1ˆm†\n2)by carefully selecting the\nsystem parameters that allow adiabatic elimination of the cav-\nity mode [ 75]. Another approach involves an array of three\ncavities, where the central cavity contains a qubit, and each end\ncavity houses a YIG sample [ 65]. By employing the dispersive\nregime and using the Schrieffer-Wolff (Frohlich-Nakajima)\napproximation [ 76–78], the cavity modes can be eliminated,\nresulting in an effective beamsplitter-type coupling between\nthe magnon modes. In our case, we adopt the latter method\nto create an array of YIGs with an effective coherent coupling( ˆm†\nnˆmn+1+ ˆmnˆm†\nn+1). Our strategy is to utilize these easier-\nto-engineer beam-splitter-type magnon-magnon interactions\nbut introduce the required nonlinearity for generating entan-\nglement between distant magnon modes through a squeezed\nthermal bath [68, 79–82].\nThe Hamiltonian Ha(Eq. (1)) associated with a one-\ndimensional array of coupled cavities represents the tight-\nbinding bosonic model. To diagonalize it, we introduce new\noperators in the momentum space ( k-space). The resulting\ndiagonal Hamiltonian is given by\nHdiag/ℏ=/summationdisplay\nkωkˆa†\nkˆak, (4)\nhere, we have introduced\nˆak=1√\n2N+ 1/summationdisplay\njˆajeikj, (5)\nˆa†\nk=1√\n2N+ 1/summationdisplay\njˆa†\nje−ikj. (6)\nWe have assumed the periodic boundary conditions such that\nk:=km= 2πm/(2N+ 1) withm∈[−N,N ], and con-\nsidering a large cavity array ( N≫1) results ink∈[−π,π].\nMoreover,ωk=ωc−2Jcoskis the dispersion relation of\nthe cavity mode. The interaction Hamiltonian, as presented in\nEq. (3), modifies to\nHI/ℏ=1√\n2N+ 1/summationdisplay\nk,n/bracketleftig\ngn(ˆakˆm†\nneikln+ ˆa†\nkˆmne−ikln)/bracketrightig\n.\n(7)\nWe note that ln=n, and it is associated with the position\nof the YIG in the array (see Fig. 1). To derive an effective\nHamiltonian involving only magnon modes, the elimination\nof cavity modes is necessary. This can be accomplished by\ninvoking the Schrieffer-Wolf (Frohlich-Nakajima) transforma-\ntion [76, 77, 83, 84].\n[S,H 0] =−HI. (8)\nWhereH0is the free Hamiltonian consisting of cavity ( Hdiag),\nand magnon ( Hm) modes. In addition, HIis the interaction\nHamiltonian between the cavity and magnon modes, as given\nin Eq. (7).Sis called a generator and it is anti-Hermitian in\nnature i.e.,S†=−S, and it is given by\nS=/summationdisplay\nk,n′/bracketleftig\nαn′\nkˆakˆm†\nn′−αn′\nk⋆ˆa†\nkˆmn′/bracketrightig\n, (9)\nhereαn′\nk=gn′/(√\n2N+ 1(ωn′−ωk))e−ikln′. The effec-\ntive Hamiltonian of the system can be determined by uni-\ntary transformation eSHe−S. In the dispersive regime, such\nthatωn′−ωk≫gn′/√\n2N+ 1, we can ignore higher or-\nder terms in the expansion of the unitary transformation, i.e.,\nH0+1/2[HI,S], and keep terms up to the second order in αn′\nk.\nConsequently, the approximate effective Hamiltonian, compris-\ning solely of magnon modes, can be explicitly expressed as4\nfollows [65, 66, 85, 86]:\nHeff/ℏ=/summationdisplay\nnω′\nnˆm†\nnˆmn+/summationdisplay\nn,n′Gnn′( ˆm†\nnˆmn′+ ˆmnˆm†\nn′).\n(10)\nWhere we have replaced discrete modes with a continuous\ndistribution\n1\n2N+ 1/summationdisplay\nk=1\n2π/integraldisplayπ\n−πdk, (11)\nin addition, the following integral identity is employed in the\nderivation of Eq. (10)\n/integraldisplayπ\n−πdk\n2πe−ilx\nU+Vcosx= (−1)|l|/radicalbigg\n1\nU2−V2e−|l|arccosh (U/V).\n(12)\nThe effective frequency ω′\nnof then-th magnon mode is given\nby\nω′\nn=ωn+g2\nn/radicalbig\n∆2n+ 4J∆n,and (13)\nGnn′=gngn′(−1)|lnn ′|\n2/parenleftbigge−|lnn ′|arccosh (1+∆n′/2J)\n/radicalbig\n∆2\nn′+ 4J∆n′+\ne−|lnn ′|arccosh (1+∆ n/2J)\n/radicalbig\n∆2n+ 4J∆n/parenrightbigg\n, (14)\nis spatially dependent effective coupling strength between the\nmagnon modes. Further, ∆n′= (ωn′−δa)2+ 4J(ωn′−δa),\n∆n= (ωn−δa)2+ 4J(ωn−δa), andδa=ωa−2Jis\nthe lower bound frequency of the cavity mode. In addition,\nlnn′=ln−ln′is the distance between n-th andn′-th YIG\nplaced inside the cavity array. In our numerical simulations, we\nconsider identical magnon modes: Since all the YIGs oscillate\ninside the cavity with the same frequency, therefore, resonance\ncondition ∆n′=∆n= ∆ , andδn′=ωn′−ωaandδn=ωn−\nωa=δ. The effective Hamiltonian in Eq. (10) takes on the\nform of a beam splitter for magnon modes, which can be used\nfor state transfer between distant magnon modes, as discussed\nin the previous studies [ 65,66]. We note that, in our case,\nthe interaction between magnon modes is mediated by virtual\nphotons; and when contrasted with actual photons, virtual\nphotons possess the advantages of being non-propagating and\nnon-radiative. Consequently, energy exchange facilitated by\nvirtual photons results in entirely coherent dynamics devoid of\ndissipation [65].\nIV . RESULTS\nTo illustrate the working principles of our scheme, we ini-\ntially focus on a simple scenario where only three cavities are\noccupied, each containing one YIG. The remaining cavities\nin the array are unoccupied. It is worth noting that the place-\nment of YIGs is not restricted to neighboring cavities; they\ncan be situated in any of the three cavities within the array. Inaddition, we consider a squeezed thermal reservoir coupled\nto the central YIG. It is worth noting that this entanglement\ngeneration scheme remains applicable for arrays with arbitrary\ncavities lengths [68].\nWe employ the quantum Langevin equations to describe\nthe system’s dynamics. By working in the interaction\npicture through the unitary evolution operator ˆU(t) =\nexp[−it(/summationtext\nnω′\nnˆm†\nnˆmn)], the equations of motion take the\nfollowing form (for convenience, we will now omit the hat\nsymbol from operators)\n˙m1=−κ1m1−iG12m2−iG13m3+/radicalbig\n2κ1min\n1,\n˙m2=−κ2m2−iG12m1−iG23m3+√2κ2min\n2,\n˙m3=−κ3m3−iG13m1−iG23m2+/radicalbig\n2κ3min\n3.(15)\nHere,κnis the dissipation rate of the n-th magnon modes, and\nmin\nnrepresents the corresponding input noise operator. The in-\nput noise operator min\n2accounts for driving the central magnon\nmode through a squeezed vacuum field. It is characterized by\nzero mean and following correlation functions [87]\n⟨min\n2(t)min†\n2(t′)⟩=(N+ 1)δ(t−t′),\n⟨min†\n2(t)min\n2(t′)⟩=Nδ(t−t′),\n⟨min\n2(t)min\n2(t′)⟩=Mδ(t−t′),\n⟨min†\n2(t)min†\n2(t′)⟩=M∗δ(t−t′). (16)\nHereN= sinh2r+ ¯n2(sinh2r+ cosh2r)andM=\neiθsinhrcoshr(1 + 2¯n2)withθandrbeing the phase and\nsqueezing parameter of the input squeezed reservoir, respec-\ntively. In addition, NandMaccount for the number of exci-\ntations and correlations, respectively. We observe that due to\nthe Schrieffer-Wolf transformation as described in Eq. (9), ad-\nditional terms are introduced into the bath correlation function.\nNonetheless, the alterations induced by this transformation in\nthe correlation function are deemed negligible. Among these\nparameters,Mplays the most influential role in entanglement\ngeneration. The equilibrium mean thermal magnon number\ncan be determined by ¯n2(ω2) = [exp( ℏω2/KBT2)−1]−1.\nThe input noise operators for the other two magnon modes\nare characterized by the following correlation functions\n⟨min\nα(t)min†\nα(t′)⟩=(¯nα+ 1)δ(t−t′),\n⟨min†\nα(t)min\nα(t′)⟩=¯nαδ(t−t′), (17)\nhereα= 1,3, and ¯nα= [exp( ℏωα/KBTα)−1]−1.\nIn our scheme, we showcase the possibility of generating\nentanglement between all potential bipartitions of the magnon\nmodes by driving the central magnon mode with an input\nquantum-squeezed field. The quadratures for both the magnon\nmodes and the input noise operators are defined as follows\nxn= (mn+m†\nn)/√\n2,yn= (mn−m†\nn)/√\n2i, andxin\nn=\n(min\nn+min†\nn)/√\n2,yn= (min\nn−min†\nn)/√\n2i, respectively.\nIn terms of quadrature fluctuations, the quantum Langevin\nequation (15) can be rewritten as\n˙F(t) =F(t)A+N(t), (18)5\n0 0.1 0.2 0.300.050.10.15\nT(K)ℰ\nFigure 2. Magnon-magnon bipartite entanglement as a function of\nthe environment temperature T. The blue solid line represents the\nlogarithmic negativity E1,2, betweenm1andm2modes, while the\nblack dashed line indicates the logarithmic negativity E2,3between\nm2andm3modes. Further, the red solid line shows the logarithmic\nnegativity E1,3betweenm1andm3modes. Parameters: ωn/2π= 10\nGHz,gj/2π= 10 MHz,J/2π= 12 MHz,κn/2π=5MHz, and\nr= 1.\nwhereF(t)=[x1(t),y1(t),x2(t),y2(t),x3(t),y3(t)]T, and\nN(t)=[xin\n1(t),yin\n1(t),xin\n2(t),yin\n2(t),xin\n3(t),yin\n3(t)]Tdenote\nthe quadrature fluctuation vectors of magnon and input noise\noperators, respectively. The drift matrix Fis given by\nF=\n−κ10 0G12 0G13\n0−κ1−G12 0−G13 0\n0G12−κ20 0G23\n−G12 0 0−κ2−G23 0\n0G13 0G23−κ30\n−G13 0−G23 0 0−κ3\n. (19)\nAs the effective Hamiltonian of the magnon modes, as given\nin Eq. (10), is quadratic, and the input quantum noise is Gaus-\nsian, as a result, the state of the system also remains Gaussian.\nThe reduced state, consisting of three magnon modes, forms\na continuous variable three-mode Gaussian state. This state\ncan be entirely characterized by a 6×6covariance matrix V,\nwhich is expressed as\nVij=1\n2⟨Fi(t)Fj(t) +Fj(t)Fi(t)⟩i,j= 1,2,..., 6.\n(20)\nThe steady-state solution can be obtained by solving the Lya-\npunov equation\nFV+VFT=−D (21)\nwhereDis the diffusion matrix and can be derived from the\nnoise correlation matrix; ⟨Ni(t)Nj(t) +Nj(t)Ni(t)⟩/2 =\nDijδ((t−t′). It can be written as a direct sum of D=D1⊕\nD2⊕D3, withDα=diag[κα(2¯nα+ 1),κα(2¯nα+ 1)] , and\nD2=/parenleftbigg\nκ2U1iκ2(M∗−M )\niκ2(M∗−M )κ2U2/parenrightbigg\n. (22)\n0 0.2 0.4 0.6 0.8 1 1.200.050.10.150.2\nrℰFigure 3. Magnon-magnon bipartite entanglement Eas a function of\nsqueezing parameter r. The solid blue line illustrates the logarithmic\nnegativity E1,2characterizing the entanglement between m1andm2\nmagnon modes. The red dashed curve showcases the logarithmic\nnegativity E2,3representing the entanglement between the m2andm3\nmodes. Furthermore, the blue dot-dashed line shows the logarithmic\nnegativity E1,3betweenm1andm3considering the same parameters\nas those specified in Fig. 2.\nwhereU1= (2N+ 1 +M+M∗), andU2= (2N+ 1−\nM−M∗).\nA. Bipartite entanglement\nTo investigate the entanglement between the magnon modes,\nwe compute the logarithmic negativity EN. This quantity has\npreviously been proposed as a measure of entanglement and\nhelps establish the conditions under which the two modes are\nentangled [ 88]. In the continuous variable case, the logarithmic\nnegativity is defined as [89]\nE=max[0,−ln 2V−], (23)\nwhereV−=/radicalbig\n1/2(Σ−(Σ2−4 detV′)1/2with Σ =\ndetA+ detB−2 detC.V′is a4×4matrix obtained from\nsteady-state covariance matrix Vby removing the two rows\nand associated columns related to the traced-out magnon mode.\nThe reduced covariance matrix V′is given by\nV′=/parenleftbiggA C\nCTB/parenrightbigg\n.\nWe compute the bipartite entanglement among all possible\npairs of magnon modes by employing the logarithmic neg-\nativityEN. The analytical solution of equation (21) is too\ncumbersome and we don’t report it here. Instead, we exten-\nsively examine the logarithmic negativity across various system\nparameters. For numerical evaluations, we employ experimen-\ntally attainable parameters reported in recent studies [ 55,90–\n93]. We consider the magnon density at low temperature with\nground state spin s= 5/2of theFe+\n3ion in the YIG sphere.\nThe total number of spins N=ρVwithρ= 4.22×1027/m36\ncharacterizing the density of each YIG and Vis the volume of\neach sphere with diameter 250 µ-meter; this results in the total\nnumber of spins N= 3.5×106in each YIG.\nIn Fig. 2, we present a plot illustrating bipartite entan-\nglement, quantified using the logarithmic negativity, among\nall possible pairs of magnon modes within an effective sys-\ntem comprising three magnon modes. These results are de-\nrived using a set of experimentally feasible parameters [ 43]:\nωn/2π= 10 GHz,gj/2π= 10 MHz,J/2π= 12 MHz,\nωc/2π= 10 GHz,κ1/2π=κ3/2π=5MHz,r= 1, and the\nphase angle θ= 0. The entanglement is robust against tem-\nperature and survives up to about T= 0.2K (solid blue line\nand dashed black line), and T= 0.15K (red solid line). The\ncouplingG13between YIG1 and YIG3 is relatively weak as\ncompared toG12orG23due to their farthest location and as a re-\nsult, YIG1 and YIG3 are relatively weakly entangled described\nby red line in Fig. 2. However, the more pronounced squeezing\nparameterrwould result in a more robust entanglement against\nthe environment temperature. Here, we consider a reasonable\nvalue of the squeezing parameter r= 1. The logarithmic neg-\nativityE1,2between YIG1 and YIG2 and E2,3between YIG2\nand YIG3 are the same due to the same couplings; G12=G23\nrepresented by solid blue and black dashed lines. The quan-\ntum squeezed reservoir coupled to the central magnon mode is\nresponsible for entanglement generation between the magnon\nmodes. The degree of entanglement decreases by reducing\nthe squeezing parameter rand dies out in the absence of the\nsqueezed reservoir. We observe that the difference in the order\nof magnon decays at resonance frequencies can produce sig-\nnificant changes in the amount of steady-state entanglement\nbetween the two magnon modes. Each magnon-pair has a\nstate-swap interaction and the squeezing can be transferred\nfrom a squeezed magnon state to another magnon state. As a\nresult, each magnon pair is to be entangled due to swap-state\ntype interaction via squeezing. This implies that squeezing can\naffect the bipartite entanglement generated in between a pair\nof the magnon modes.\nTo observe the impact of squeezing on entanglement gen-\neration, we plot the bipartite entanglement, quantified using\nlogarithmic negativity ( E), as a function of the squeezing pa-\nrameter (r). This can be observed in Fig. 3. The results indi-\ncate that without the presence of the squeezed thermal bath,\nentanglement generation within our scheme is unattainable.\nTherefore, it is evident that squeezing plays a pivotal role in\nthe generation of entanglement, with logarithmic negativity\nexhibiting a notable increase as the squeezing parameter ( r) is\nraised. The entanglement between the magnon modes m1-m2\nandm2-m3is identical, owing to the equal coupling strength\nbetween these magnon pairs. However, the magnon modes\nm1-m3are relatively weakly coupled, primarily because of the\ngreater distance between the associated YIGs. Consequently,\nthe entanglementEm1,m3between modes m1-m3is lower in\ncomparison to the other two pairs.\nFigure 4. Two-dimensional plot for tripartite entanglement, quantified\nby minimal residual contangle Ras a function of bath temperature T,\nand squeezing parameter r. The rest of the system parameters are the\nsame as given in Fig. 2.\nB. Tripartite entanglement\nWe further investigate the possibility of the generation of the\ndistant magnon-magnon tripartite entanglement in the array as\nshown in Fig. 1. To this end, we employ the minimal residual\ncontangle given by [94, 95]\nRi|jk=Ci|jk−Ci|j−Ci|k. (24)\nHere, the contangle of subsystems xandyis denoted by Cx|y,\nandymay represent more than one mode. Cx|yis a proper\nentanglement monotone, and it is determined by the square\nof the logarithmic negativity between the respective modes,\nwhich is given by\nEi|jk=max[0,−ln 2Vi|jk], (25)\nwhereVi|jk=min/vextendsingle/vextendsingleeigiΩ3˜V/vextendsingle/vextendsingleis the smallest symplectic\neigenvalues. Ω3is the symplectic matrix Ω3=⊕3\nj=1iσy,\nwithσybeing they-Pauli matrix and ⊕symbol describes\nthe direct sum of the σymatrices. The 6×6covariance\nmatrix ˜Vis obtained by inverting the momentum quadra-\nture of one of the magnon modes. The transformed covari-\nance matrix ˜Vis determined by ˜V=Pi|jkVPi|jkwhere\nP1|23=diag[1,−1,1,1,1,1],P2|13=diag[1,1,1,−1,1,1],\nandP3|12=diag[1,1,1,1,1,−1]are partial transposition di-\nagonal matrices. The steady state of the magnon modes can\nbe fully characterized by the 6×6covariance matrix because\nof its Gaussian nature. The tripartite entanglement for Gaus-\nsian states can be determined by minimum residual cotangle\n[94, 95]\nRmin=min[R1|23,R2|13,R3|12]. (26)\nThis ensures that the tripartite entanglement remains un-\nchanged regardless of how the modes are permuted. The7\ndensity plot of magnon-magnon tripartite entanglement de-\ntermined via minimum residual cotangle Rminis shown in Fig.\n4. It is evident from the results that considerable tripartite\nentanglement can be generated between magnon modes when\nthe central magnon mode is coupled to a squeezed reservoir.\nFig. 4 shows that the degree of entanglement is significantly\nenhanced with the increase in the squeezing parameter r.\nV . CONCLUSION\nIn summary, we have shown that steady-state bipartite\nand tripartite entanglement can be generated between distant\nmagnon modes when only one of these magnon modes is cou-\npled to a quantum-squeezed reservoir. In particular, we con-\nsidered an array of Ncavities each of which houses a single\nYIG sphere, and the central YIG is coupled to a single-mode\nsqueezed vacuum bath. We have demonstrated that steady-\nstate bipartite and tripartite entanglement can be generated\namong magnon modes hosted by YIGs when some of the cav-\nities, up to five, are occupied. In contrast to prior proposals\nfor magnon-magnon entanglement generation [ 56–60,62–65],\nboth bipartite and tripartite entanglement are possible in our\nscheme. Further, in principle, our scheme can be extended to\nan arbitrary number of YIGs within the cavity array [ 68,81].\nHowever, achieving entanglement across the entire array may\nnecessitate negligible or extremely small magnon decay rates.\nTo address this challenge, we propose the use of an external\nsqueezed drive on each cavity in the array, enabling strong\nlong-range interactions between YIGs positioned at greater dis-\ntances within the array [ 86]. In our scheme, the entanglement\ngeneration does not require nonlinearity, instead, entanglement\nis generated via a squeezed reservoir and its strength dependson the squeezing parameter r. The generated entanglement\nis robust against the environment temperature provided the\nmagnons dissipation rates are sufficiently low. It may be in-\nteresting to look for multipartite entangled states of magnons,\nwhich is left for future works. Our work may be useful in\ndesigning quantum networks based on cavity-magnon systems\n[67].\nAppendix A: Bipartite entanglement with five YIGs\nHere, we extend our results for five YIGs placed inside the\ncavities array, and the squeezed reservoir is coupled to the\ncentral YIG. The Langevin equations of motion in this case are\ngiven by\n˙m1=−κ1m1−i5/summationdisplay\nn′=2G1n′mn′+√2κ1m1in,\n˙m2=−κ2m2−iG12m1−i5/summationdisplay\nn′=3G2n′mn′+√2κ2m2in,\n˙m3=−κ3m3−i2/summationdisplay\nn=1Gn3m3−i5/summationdisplay\nn′=4G3n′mn′+√2κ3m3in,\n˙m4=−κ4m4−i3/summationdisplay\nn=1Gn4mn−iG45m5+√2κ4m4in,\n˙m5=−κ5m5−i4/summationdisplay\nn=1Gn5m5+√2κ5m5in. (A1)\nAgain we write the Langevin equations in matrix form as\nequation (18). The A matrix is now a 10×10matrix which\ncan be described as\nA=\n−κ10 0G12 0G13 0G14 0G15\n0−κ1−G12 0−G13 0−G14 0−G15 0\n0G12−κ20 0G23 0G24 0G25\n−G12 0 0−κ2−G23 0−G24 0−G25 0\n0G13 0G23−κ30 0G34 0G35\n−G13 0−G23 0 0−κ3−G34 0−G35 0\n0G14 0G24 0G34−κ40 0G45\n−G14 0−G24 0−G34 0 0−κ4−G45 0\n0G15 0G25 0G35 0G45−κ50\n−G15 0−G25 0−G35 0−G45 0 0−κ5\n. (A2)\nWithF(t)=[(xi(t),yi(t)]TandN(t)=[xiin(t),yiin(t)]T\ni=1,2,3,. . . , 10. Since the squeezed reservoir for a 5 YIG sam-\nple is taken on YIG3, this shifts the frequency of each magnon\nmode by the amount of the frequency i.e., ∆′\nj=ω′\nj−ω0after\napplying the rotating wave approximation corresponding to\neach magnon mode. We can write equation (A2) for N magnon\nmodes in general m×nmatrix form in the interaction picture\ncan be expressed asQnn′=/parenleftbigg\nKnGnn′\nGnn′Kn/parenrightbigg\nwhere,Kn=/parenleftbigg\n−κn0\n0−κn/parenrightbigg\n, andGij=/parenleftbigg\n0Gnn′\n−Gnn′0/parenrightbigg\nfor\nn=1 . . . N, and n′= 2,. . . are the sub-block 2×2matrices of8\n0 0.1 0.20.000.050.100.15\nT(K)ℰ\n(a)\n0 0.1 0.20.0000.0050.0100.0150.0200.025\nT(K)ℰ (b)\nFigure 5. (Color online). In the given panel, the pairs of bipartite entanglement for a 5-YIG sample are shown. (a) The black solid, blue\ndashed, and red dotdashed lines indicate the entanglement between m1andm4modes,m2andm4, andm3andm4, respectively. (b) Describes\nthe bipartite entanglement between m1andm5(solid black curve), m2andm5(red dashed curve), m3andm5blue dotdashed Curve. The\nparameters remain the same as used in Fig. 3\nn-th YIG decays with corresponding zero detuning frequency\nin the interaction picture, and the effective spatially dependent\ncouplingGnn′as discussed earlier.Our model can be extended to 2N+ 1cavities with a squeezed\nreservoir coupled to the central YIG and can be manifested as\na quantum network.\n[1]F. Flamini, N. Spagnolo, and F. 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Theor. 40, 7821 (2007)." }, { "title": "1912.13407v2.Probing_magnon_magnon_coupling_in_exchange_coupled_Y__3_Fe__5_O___12___Permalloy_bilayers_with_magneto_optical_effects.pdf", "content": "Probing magnon-magnon coupling in exchange coupled\nY3Fe5O12/Permalloy bilayers with magneto-optical e\u000bects\nYuzan Xiong,1, 2Yi Li,3,a)Mouhamad Hammami,1Rao Bidthanapally,1Joseph Sklenar,4\nXufeng Zhang,5Hongwei Qu,2Gopalan Srinivasan,1John Pearson,3Axel Ho\u000bmann,6, 3\nValentine Novosad,3and Wei Zhang1, 3,b)\n1)Department of Physics, Oakland University, Rochester, MI 48309,\nUSA\n2)Department of Electronic and Computer Engineering, Oakland University,\nRochester, MI 48309, USA\n3)Materials Science Division, Argonne National Laboratory, Argonne, IL 60439,\nUSA\n4)Department of Physics and Astronomy, Wayne State University, Detroit,\nMI 48201, USA\n5)Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439,\nUSA\n6)Department of Materials Science and Engineering, University of Illinois at\nUrbana-Champaign, Urbana, IL 61801, USA\n(Dated: 13 July 2020)\nWe demonstrate the magnetically-induced transparency (MIT) e\u000bect in\nY3Fe5O12(YIG)/Permalloy(Py) coupled bilayers. The measurement is achieved\nvia a heterodyne detection of the coupled magnetization dynamics using a single\nwavelength that probes the magneto-optical Kerr and Faraday e\u000bects of Py and\nYIG, respectively. Clear features of the MIT e\u000bect are evident from the deeply\nmodulated ferromagnetic resonance of Py due to the perpendicular-standing-spin-\nwave of YIG. We develop a phenomenological model that nicely reproduces the\nexperimental results including the induced amplitude and phase evolution caused\nby the magnon-magnon coupling. Our work o\u000bers a new route towards studying\nphase-resolved spin dynamics and hybrid magnonic systems.\na)Electronic mail: yili@anl.gov\nb)Electronic mail: weizhang@oakland.edu\n1arXiv:1912.13407v2 [cond-mat.mtrl-sci] 10 Jul 2020I. INTRODUCTION\nHybrid magnonic systems are becoming rising contenders for coherent information\nprocessing1{3, owing to their capability of connecting distinct physical platforms in quantum\nsystems as well as the rich emerging physics for new functionalities4{21. Magnons have been\ndemonstrated to e\u000eciently couple to cavity quantum electrodynamics systems including\nsuperconducting resonators and qubits4{8; magnonic systems are therefore well-positioned\nfor the next advances in quantum information. In addition, recent studies also revealed the\npotential of magnonic systems for microwave-optical transduction22{28, which are promising\nfor combining quantum information, sensing, and transduction.\nTo fully leverage the hybrid coupling phenomena with magnons, strong and tunable cou-\nplings between two magnonic systems have attracted considerable interests recently29{32.\nThey can be considered as hosting hybrid magnonic modes in a \\magnonic cavity\" as op-\nposed to microwave photonic cavity in cavity-magnon polaritons (CMPs)1{3, which allows\nexcitations of forbidden modes and high group velocity of spin waves owing to the state-\nof-the-art magnon bandgap engineering capabilities30,33. The detuning of the two magnonic\nsystems can be easily engineered by the thickness of the thin \flms, which set the wavenum-\nbers and the corresponding exchange \feld. Furthermore, in such strongly coupled magnetic\nheterostructures, both magneto-optical Kerr and Faraday e\u000bects can be utilized for light\nmodulation, in terms of light re\rection by metals and/or transmission in insulators, re-\nspectively. In this architecture, the freedom of lateral dimensions is maintained for device\nfabrication and large-scale, on-chip integration.\nTo date, both magnon-photon and magnon-magnon couplings are predominantly investi-\ngated by the cavity ferromagnetic resonance (FMR) spectroscopy, i.e. microwave transmis-\nsion and/or re\rection measurements, typically involving a vector-network analyzer (VNA)\nor a microwave diode4,6{12,29{32,34. Strong magnon-magnon couplings have been observed\nin yttrium iron garnet (Y 3Fe5O12, YIG) coupled with ferromagnetic (FM) metals, where\nexchange spin waves were excited by a combined action of exchange, dampinglike, and/or\n\feldlike torques that are localized at the interfaces29{32.\nIn this work, we investigate the magnon-magnon coupling in YIG/Permalloy(Py) bilayers\nby a phase-resolved, heterodyne optical detection method. We reveal the coupled magnon\nmodes in the regime exhibiting the magnetically-induced transparency (MIT) e\u000bect, i.e. the\n2GGS\nHYIG Faraday effect\nPy Kerr effect(1)(2)\n(3)(4)\nhrf(a)\n(b)VO(µV)Py/YIGPy/SiO2/YIG\n40\n(c)Re[VO]\nIm[VO]\n|VO|FIG. 1. (a) Schematic illustration of the experimental setup. Modulated and linearly-polarized\n1550-nm light enter the sample at a polarization angle (1); dynamic Faraday e\u000bect of the YIG\ncauses the polarization to rotate (2); dynamic Kerr e\u000bect of the Py causes polarization to further\nrotate (3); the re\rected light, upon the returning path, picks up again the Faraday e\u000bect and causes\nthe polarization to further rotate (4), before entering light detection and analysis. The applied\ndc magnetic \feld is parallel to the ground-signal-ground (G-S-G) lines of the CPW. (b) Example\nsignal trace for YIG/Py (solid) and YIG/SiO 2/Py (dashed) measured at 5.85 GHz, showing the\nin-phase X(top) and quadrature Y(middle), and the total amplitude,p\nX2+Y2(bottom). (c)\nPlotting and the \ftting of the observed PSSW modes versus the resonance \felds.\nmagnetic analogy of electromagnetically induced transparency (EIT)10,35{39, akin to a spin-\nwave induced suppression of FMR. In the hybrid magnon-photon systems, the MIT e\u000bect\narises when the coupling strength, g=2\u0019is larger than the photon dissipation rate \u0014p=2\u0019\nbut smaller than the magnon dissipation rate \u0014m=2\u001910. Under such a condition, the mode\nhybridization leads to an abrupt suppression of the microwave transmission at a certain\nfrequency range. A transparency window, whose linewidth is determined by the low-loss\nmode, can be observed in the broad resonance of the other lossy mode. Such resonant\ntransparency is controlled by an external magnetic \feld. Our measurement is achieved via\ndetecting the coupled magnetization dynamics of the insulating and metallic FMs using a\nsingle 1550-nm telecommunication wavelength. Unlike the ultrafast optical pump-probes26,\nthe method herein is a continue-wave (cw), heterodyne technique in which the 1550-nm\n3laser light is modulated at the FMR frequencies (in GHz range) simultaneously with the\nsample's excitation. This feature makes the method e\u000bectively an optical \\lock-in\" type\nmeasurement, akin to the electrical lock-in detection40{42. The phase information between\nthe Py and YIG FMRs, as well as the YIG perpendicular standing spin waves (PSSWs) can\nbe obtained by simultaneously analyzing both the Kerr and Faraday responses.\nII. SAMPLES AND MEASUREMENTS\nThe commercial YIG \flms (from MTI Corporation) used in this work are 3- \u0016m thick,\nsingle-sided grown on double-side-polished Gd 3Ga5O12(GGG) substrates via liquid phase\nepitaxy (LPE). The Py \flms (t Py= 10 nm and 30 nm) were subsequently deposited on\nthe YIG \flms using magnetron sputtering following earlier recipes32. To ensure the strong\ncoupling, we used in situ Ar gas rf-bias cleaning for 3 minutes, to clean the YIG surface\nbefore depositing the Py layer. Reference samples of GGG/YIG/SiO 2(3-nm)/Py(10-nm)\nGGG/YIG/Cu(3-nm)/Py(10-nm) were also prepared at the same growth condition.\nFigure 1(a) illustrates the measurement con\fguration. The modulated and linearly-\npolarized 1550-nm light passes through the transparent GGG substrates and detects the\ndynamic Faraday and Kerr signals upon their FMR excitation. As the light travels through\nthe YIG bulk, the dynamic Faraday rotation due to the YIG FMR is picked up. Similarly,\nthe dynamic Kerr rotation caused by the Py FMR is then picked up, when the light reaches\nthe Py layer. The Py layer also serves as a mirror and re\rects the laser light. Upon re\rec-\ntion, the dynamic Faraday e\u000bect from the YIG is picked up again, making the e\u000bective YIG\nthickness 6- \u0016m, i.e. twice the \flm thickness. It should be noted that the Faraday rotations\nfor the incoming and returning light add up as opposed to cancel, due to the inversion of\nboth the chirality of the Faraday rotation and the projection of the perpendicular magne-\ntization of YIG along the wavenumber direction, whose mechanism is akin to a commercial\n\\Faraday rotator\" often encountered in \fber optics.\nThe YIG/Py samples are chip-\ripped on a coplanar waveguide (CPW) for microwave\nexcitation and optical detection, as depicted in Fig. 1(a). An in-plane magnetic \feld, H,\nalong they-direction saturates both the YIG and Py magnetizations. We scan the frequency\n(from 4 to 8 GHz) and the magnetic \feld, and then measure the optical responses using a\nlock-in ampli\fer's in-phase X(Re[VO]) and quadrature Y(Im[VO]) channels as well as the\n4microwave transmission using a microwave diode. A detailed description of the measurement\nsetup is in the Supplemental Materials (SM), Figure S1.\nIII. RESULTS AND DISCUSSIONS\nFigure 1(b) compares the optical recti\fcation signals between the 10-nm-Py sample and\nthe YIG/SiO 2/Py reference sample measured at 5.85 GHz. The 10-nm-Py sample (solid\nline) shows the representative features of the detected FMR and hybridized PSSW modes.\nThe complete \fne-scan including the FMR diode dataset are summarized in the SM, Fig.\nS2. The optical signals with the phase information are obtained by the lock-in's in-phase\nX(Re[VO], top panel) and quadrature Y(Im[VO], middle panel), which are further used\nto calculate the total amplitude,p\nX2+Y2, (bottom panel). The technical details of the\nmeasured signal versus the optical and electrical phases are summarized in the SM.\nThe YIG FMR signal at \u00181.3 kOe is accumulated from the Faraday e\u000bect corresponding\nto the spatially uniform precession of the YIG magnetization. The FMR dispersion is de-\nscribed by the Kittel formula: !2=\r2=HFMR(HFMR+Ms), where!is the mode frequency,\n\r=2\u0019= (ge\u000b=2)\u000228 GHz/T is the gyromagnetic ratio, ge\u000bis the g-factor, HFMRis the reso-\nnance \feld, and Msis the magnetization. The excitation of the YIG PSSW modes introduces\nan additional exchange \feld Hexto the Kittel equation, as \u00160Hex= (2Aex=Ms)(n\u0019=d YIG)2,\nwhich de\fnes the mode splitting between the PSSW modes and the uniform mode. Here\nAexis the exchange sti\u000bness, and dYIGis the YIG \flm thickness. A total of more than\n30 PSSW modes can be identi\fed for the 10-nm-Py sample. In Fig. 1(c), the quadratic\nincrease of Hexwith the mode number ncon\frms the observation of the PSSWs. Fittings\nto the Kittel equation and the exchange \feld expression yield Ms= 1.97 kOe and Aex=\n3.76 pJ/m, which are in good agreement with the previously reported values29{32.\nIn Fig. 1(b), the Py FMR at \u00180.6 kOe is strongly modulated by the YIG PSSWs,\nexhibiting the MIT e\u000bect, due to the formation of hybrid magnon modes. Besides, the YIG\nPSSW signals near the Py FMR regime ( n>25) are much stronger than the o\u000b-resonance\nregime (n < 25), which indicates the important role of the Py/YIG coupling in exciting\nthe relevant PSSW modes and resonantly enhancing the magnetization dynamics. As a\ncomparison, no apparent PSSW modes are observed for the Py/SiO 2/YIG reference sample,\nin Fig.1(b) (dashed line), indicating that only the Py but not the YIG PSSWs couples to\n5the microwave drive in the MIT regime. The Py resonance linewidth also is much narrower.\n(e)\n(a)(b)(c)\nPyYIG\n(d)\n(f)PyYIG\nFIG. 2. (a) Theoretical signal trace of the MIT e\u000bect of the YIG/Py bilayer. 7 hybrid PSSW\nmodes are shown as an example. (b) Example \ftting of the complex optical singal at 6 GHz for\nthe 30-nm-Py sample. (c) Full scan of the signal as a function of the magnetic \feld and frequency.\n(d) Theoretical calculated dispersion using the \ftting parameters, reproducing the experimental\ndata in (c). (e) and (f) are the \fne-scans at smaller \feld and frequency steps corresponding to the\nboxes in (c) and (d) (5.7 - 6.3 GHz).\nOur experimental con\fguration, similar to previously reported29{32, is relevant to the\nSchl omann excitation mechanism of spin wave43with a dynamic pinning at the interface44.\nThe interfaces of two distinct, coupled magnetic layers have been recently recognized as\nan interesting source of spin dynamics generation and manipulation45,46. In particular, the\n6critical role of the microwave susceptibilities of the distinct magnetic layers has been theoret-\nically laid out47,48that are directly relevant to the magnon-magnon coupled experiments29{32.\nHere, we introduce a phenomenological model by considering a series of YIG harmonic os-\ncillators coupled with the Py oscillator, and using practical experimental \ftting parameters,\nin which the measured complex optical signal, VO, can be expressed as:\nVO=Aei(\u001eL\u0000\u001em)\ni(HPy\nFMR\u0000H)\u0000\u0001HPy+ \u0006jg2\ni(HYIG\nPSSW ;j\u0000H)\u0000\u0001HYIG;j(1)\nwhereAis the total signal amplitude, HPy\nFMR andHYIG\nPSSW is the resonance \feld of Py and\nYIG-PSSWs, respectively, \u0001 HYIG(Py) is the half-width-half-maximum linewidth, and gis the\n\feldlike coupling strength from the interfacial exchange. This model disregards the linear\nfrequency-dependent phase (exists in the Re[ VO] and Im[VO] signals) but directly analyzes\nthe total optical signal.\nFigure 2(a) plots the theoretically predicated MIT e\u000bect according to Eq.1, showing the\nlineshape of the Py FMR mode that is coupled to the YIG PSSW modes. The center curve\nwith a zero resonance detuning denotes the MIT e\u000bect. The magnon-magnon coupling\ninduces a set of sharp dips in the spectra. Such dips in the optical re\rection means a\npeak in their transmission, which is referred to as a \\transparency window\" in quantum\noptics, resembling the EIT phenomenon in photonics37and optomechanics38,39. Using a\nsinglegvalue, the amplitude and phase of the hybrid modes display a clearly evolution with\nrespect to the di\u000berent Py-YIG resonance detuning. Away from the Py FMR, the lineshape\nappears to be more antisymmetric, whilst around the Py FMR, the hybrid mode appears\nto be more symmetric. Such a phase evolution is contained in the Eq.1 and is not a \ft\nparameter. Fig.2(b) is an example \ftting result of a signal trace at 6 GHz. The \fttings\nnicely reproduce the complex lineshapes arising from the coupled YIG PSSW modes and the\nPy FMR, including the phase evolution across the involved PSSW modes ( n). Our model\nallows extracting the YIG and Py magnon dissipation rates, \u0001 HYIG= 1.8 Oe, \u0001 HPy=\n43.3 Oe, and the coupling strength: g= 18.7 Oe. In the frequency domain, these values\ncorrespond to g=2\u0019= 90 MHz, \u0014YIG=2\u0019= 6 MHz, and \u0014Py=2\u0019= 308 MHz. The numbers\nsatisfy the condition for the MIT e\u000bect: \u0001 HYIG1012A/m2) into a pristine 7 nm thick Pt nanostrip evaporated on top of yttrium iron garnet (YIG), can\nimprove the spin transmission up to a factor 3: a result of particular interest for interfacing ultra thin garnet\n\flms where strong chemical etching of the surface has to be avoided. The e\u000bect is con\frmed by di\u000berent\nmethods: spin Hall magnetoresistance, spin pumping and non-local spin transport. We use it to study the\nin\ruence of the YIG jPt coupling on the non-linear spin transport properties. We \fnd that the cross-over\ncurrent from a linear to a non-linear spin transport regime is independent of this coupling, suggesting that\nthe behavior of pure spin currents circulating in the dielectric are mostly governed by the physical properties\nof the bare YIG \flm beside the Pt nanostrip.\nThe transport of pure spin information through local-\nized magnetic moments is at the heart of a new research\ntopic called insulatronic (forinsula -tor spin- tronic )1{4.\nInterest stems here from the recognition that magnetic\ninsulators are superior spin conductors than metals or\nsemi-conductors. Among magnetic insulators, garnets,\nand in particular yttrium iron garnet (YIG), have the\nlowest known magnetic damping5. One can exploit here\nthe spin Hall e\u000bect (SHE) to interconvert pure spin cur-\nrents circulating inside the dielectric into charge currents,\nwhich can then be probed electrically. This is usually\nachieved by depositing a heavy metal electrode, advan-\ntageously in Pt6,7, on top of the YIG surface. The en-\nsuing \row of spin escaping through the metal-oxide in-\nterface can be measured through spin pumping (SP)8,\nspin Hall magnetoresistance (SMR)9{11, spin Seebeck ef-\nfect (SSE)12or spin orbit torque (SOT) using non-local\ntransport devices1,13.\nThe e\u000eciency of the process is determined by the spin\ntransparency of the interface and parametrized by the\nso-called spin mixing conductance, G\"#. To optimize its\nstrength, the YIG surface is usually treated by strong\nprocess such as O+/Ar+plasma14, by annealing15and\npiranha etching16,17prior to the metal deposition in\norder to achieve good chemical and structural YIG jPt\ninterface. However these treatments are performed on\n\u0016m thick YIG samples and are di\u000ecult to implement\nonce the thickness of the \flm is in the nanometer\nrange as a modulation of surface roughness signi\fcantly\ndisturbs its magnetic properties. Additionally an\nenhancement of SMR by global annealing at very high\ntemperature18has been reported, yet this process is not\nnecessarily compatible at the device level. In addition\na)Corresponding author: oklein@cea.frto the cleaning of the YIG surface, the use of sputtering\ntechnique to deposit the metal is known to lead to\nbetterG\"#than the evaporation technique19. The later\nhowever usually leads to lower resistivity, which should\nbe favored when one wants to inject large current densi-\nties. Considering that evaporation is also advantageous\nto have better lift-o\u000b during nanofabrication, certainly\na process allowing to improve interfacial quality of\nevaporated Pt is required.\nIn this paper, we investigate the impact of local Joule\nheating at 550 K on the spin transport between thin YIG\n\flm and evaporated Pt. We use SMR and spin pumping\nmeasurements to show a clear 3 times post-deposition\nenhancement of the interfacial spin transmission, which\nis irreversible. We have exploited this feature to study\nthe in\ruence of the interfacial spin conductivity on non-\nlinear spin transport properties in lateral devices. We\n\fnd that the later are mostly governed by the physical\nproperties of the bare YIG \flm not covered by Pt. Any\nenhancement of the coupling to the electrodes seems to\nplay a negligible role in determining the non-linear char-\nacteristics of pure spin transport.\nWe use a tYIG = 56 nm thick YIG \flm grown by\nliquid phase epitaxy on a 500 \u0016m Gd 3Ga5O12(GGG)\nsubstrate9,20. Ferromagnetic resonance experiments have\nshown a damping parameter of \u000bYIG= 2:2\u000210\u00004re-\nvealing an excellent crystal quality of the YIG \flm21.\nTwo similar Pt nanostrips, respectively Pt 1and Pt 2,\nare patterned by e-beam lithography to have a width\nof 300 nm and length of 30 \u0016m. A 7 nm thick Pt layer is\nthen deposited by e-beam evaporation on the YIG \flm.\nThe nanostrips are connected to Ti jAu (5 nmj50 nm)\nelectrical contacts. The sample is mounted on a rota-\ntional stage and exposed to an in-plane magnetic \feld of\n\u00160H0= 200 mT to fully magnetize the YIG \flm. All the\nmagneto-transport experiments are performed at roomarXiv:2009.02785v1 [cond-mat.mtrl-sci] 6 Sep 20202\nFIG. 1. a) Schematic of the YIG jPt structure with two nanos-\ntrips oriented along the y-direction. The \frst one (Pt 1) is\nconnected to a current source and it is used to measure the\nSMR ratio. The second electrode (Pt 2) is used to measure the\nnon-local spin transport properties. b) Resistance of the Pt 1\nnanostrip,RI=V1=I, as a function of the injected current\nIinside. c) Angular dependence of the SMR ratio when an\napplied magnetic \u00160H0= 200 mT is rotated in the xyplane.\nThe data shows the result before (blue dots) and after (red\ndots) local Joule annealing at 550 K. Solid lines are \ft with\ncos2'.\ntemperature, T0. The schematic of the sample geometry\nis shown in Fig.1a). Transport measurements are per-\nformed by injecting 10 ms current pulses with a duty cycle\nof 10% into the Pt 1nanostrip via a 6221 Keithley current\nsource which is synchronized to a 2182A Keithley nano-\nvoltmeter. We \frst investigate the con\fguration where\nthe nanovoltmeter is connected to the same nanostrip\nto extract the magnetoresistive response through V122.\nFig.1b) represents the evolution of the Pt 1resistance,\nRI, as a function of the electrical current, I, showing a\nquadratic dependence due to Joule heating. The Pt elec-\ntrode can also be used as a temperature sensor. The tem-\nperature rise of the Pt is simply inferred from the change\nof Pt resistance with T\u0000T0=\u0010Pt(RI-R0)/R0, where\nR0= 2:6 k\n is the Pt resistance at I= 0 and\u0010Pt= 478 K\nis a thermal coe\u000ecient speci\fc to our Pt nanostrip. In\nour structure, the local temperature reaches about 550 K\nwhen the current density is Jmax= 1:2\u00021012A/m2.\nThe spin transparency of the interface can be evalu-\nated from the SMR ratio de\fned as ( R'\u0000Rk)=R0, which\nmeasures the relative change of the Pt resistance as a\nfunction of ', the azimuthal angle between H0(and thus\nthe magnetization) and the x-axis. In this notation Rkis\nthe resistance when H0is applied parallel to the nanos-\ntrip direction ( '=\u000690\u000eory-axis). Fig.1c) presents the\nSMR signal measured with a bias current of I= 100\u0016A.\nThe data in blue dots show the values obtained directly\nafter the nanofabrication process. The angular depen-dence follows a cos2'behavior (see solid line \ft), and\nthe maximum SMR deviation is observed when H0is ap-\nplied perpendicular to the nanostrip direction ( '= 0\u000e\norx-axis) with a value of ( R?\u0000Rk)=R0=\u00002:9\u000210\u00005\nextracted from the \ft (blue line). From the theory of\nSMR11,23, the amplitude of the SMR ratio is expressed\nas :\nR?\u0000Rk\nR0=\u0000\u00122\nSHE2(\u00152\nsf=tPt)\u001ag\"#tanh2\u0012tPt\n2\u0015sf\u0013\n1 + 2\u0015sf\u001ag\"#coth\u0012tPt\n\u0015sf\u0013;(1)\nwhere\u001a= 19:5\u0016\ncm is the Pt resistivity with tPtits\nthickness,\u0015sfis the spin di\u000busion length, and \u0012SHEis the\nSHE angle inside the Pt layer. In our notation, g\"#is an\ne\u000bective spin mixing conductance of the YIG jPt interface\n(see discussion below).\nUsing SMR measurement, we then investigate the\ne\u000bect of 550 K local Joule annealing on the e\u000bective\nspin mixing conductance of YIG jPt. The electrical\nannealing is provided by applying current density pulses\nofJmax= 1:2\u00021012A/m2for about 60 minutes. The\nred dots shows the SMR ratio measured after. The\namplitude is increased to ( R?\u0000Rk)=R0=\u00008:9\u000210\u00005\n(red line \ft in Fig.1c), which is 3 times larger than the\nvalue before annealing (blue line). This enhanced SMR\nis irreversible and increasing the annealing time above\nan hour leads to negligible gain in the SMR value. We\nalso observe that the Pt 1resistance is changed by less\nthan 1%, indicating no major structural changes in the\nPt, which suggests that \u0012SHEand\u0015sfremain the same\nthroughout this treatment24,25. Knowing the product\n\u0012SHE\u0015sfto be 0.18 nm26, we conclude that the spin mix-\ning conductance is improved from g\"#= 0:64\u00021018m\u00002\nto 1.90\u00021018m\u00002. The enhanced value is comparable\nto the ones obtained after Ar+-ion milling process14\nand \"piranha\" etch16,17, which means that the observed\nincrease of g\"#is more a catching up of the de\fcit of\nspin conductance probably associated with the use of\nevaporation rather than an overall improvement of the\nresult from what is obtained by sputtering.\nNext, we focus on spin pumping measurements using\nthe same sample batch. The experiment is performed at\na \fxed frequency of 9.65 GHz in a X-band cavity while\napplying a static in-plane magnetic \feld perpendicularly\nto the Pt nanostrip ( x-axis). Conversely the rf magnetic\n\feldhrfis applied along the Pt nanostrip ( y-axis). The\nrf power is \fxed at 5 mW ( \u00160hrf= 2\u0016T) to maximize\nthe SP signal while minimizing non-linear e\u000bects such\nas distorsion of the lineshape (foldover e\u000bect)27,28. At\nthe ferromagnetic resonance condition, a \row of angular\nmomentum from the YIG relaxes into the Pt29.\nThe normalized spin-pumping spectra ispare obtained\nby dividing the generated inverse spin Hall e\u000bect (ISHE)\nvoltage by both h2\nrfand the Pt resistance R0,i.e.isp=\nVISHE=(R0h2\nrf). The shape of the spectral line does not\nfollow a simple Lorentzian, which is attributed to inho-3\nFIG. 2. Spin pumping spectra at di\u000berent annealing step\nby injecting a current density Jmax= 1:2\u00021012A/m2into\nthe Pt nanostrip for three di\u000berent annealing time. The inset\nshows the normalized spin pumping spectra at each annealing\ntime.\nmogeneous broadening. The main peak of the spectrum\nis identi\fed as the Kittel mode (uniform precession) and\nwe have measured its amplitude to estimate the e\u000eciency\nof spin transmission through the interface. After char-\nacterizing the pristine YIG jPt interface, we performed\na local annealing by applying pulse current density of\nJmax=1.2\u00021012A/m2for various durations (10, 30 and\n60 minutes) following the same procedure as before. The\ne\u000bect of annealing on the spectra are shown in Fig.2.\nSimilar to the SMR measurement, Joule annealing in-\ncreases the spin pumping signal. From the amplitude\nof the main peak of the spectra and using the model\nin ref.29, one can estimate the enhancement of g\"#from\n0.60\u00021018m\u00002to 1.23\u00021018m\u00002, compatible with our\nprevious SMR estimation.\nThe interesting feature of this experiment lies in\nthe measurement of the full width at half maximum\n(FWHM) of the main peak. As the amplitude of the\npeak become larger with the annealing time, FWHM re-\nmains constant over the whole annealing process (see the\ninset of Fig.2). The modulation of FWHM can be in\ngeneral attributed to the extra damping induced by the\ncoupling between YIG and Pt30,31. The constant FWHM\nreveals that the linewidth is mostly controled by the bare\nYIG \flm beside the nanometric Pt nanostrip, rather than\nthe sole relaxation of precession dynamics at the YIG jPt\ninterface. This suggests that the additional relaxation\nchannel provided by the adjacent metallic nanostrip is\na weak perturbation of the overall relaxation of the ex-\ntended YIG thin \flm underneath. Since this \fnding dif-\nfers from what is observed when the adjacent Pt covers\nthe whole YIG \flm21, we attribute the di\u000berence to \f-\nnite size e\u000bects. For nanostructured Pt electrodes, the\nadditional relaxation channel of the magnons in the YIG\nprovided by the adjacent metal becomes weak when the\nlateral size of the Pt is smaller than the magnon wave-\nlength. We emphasize that such scenario would then\nmostly concern the long wavelength magnons, such as\nFIG. 3. Non-local spin signals \u0006 I(left) and \u0001 I(right) as a\nfunction of injected current for di\u000berent annealing steps. The\ninset of the left panel displays the current variation of \u001b0=\u0006I,\nwhere\u001b0is the initial slope of the spin current increase (see\ngray dashed line). The intercept with the 0.75 level is a land-\nmark that de\fnes Jc13. The (a) panel represents non-local\nspin signals measured directly after the nanofabrication. The\n(b) panel shows the result when the injector is annealed. The\n(c) panel shows the result when both injector and detector\nnanostrip are annealed.\nthose excited by the cavity.\nFinally we study the impact of local Joule annealing\non the non-local spin transport properties. In these sam-\nples, the second Pt nanostrip (Pt 2), placed 2 \u0016m away\nfrom the \frst nanostrip (Pt 1), is used as a detector of\nthe spin current (see Fig.1a). When a charge current\nis sent to the Pt 1nanostrip (injector), a spin accumu-\nlation is generated at the YIG jPt interface due to the\nSHE and its angular momentum is transferred from Pt to\nYIG32,33. The angular momentum is then carried in YIG\nby magnons , which are then detected via the ISHE volt-\nage at the Pt 2nanostrip (detector). Again, the non-local\nspin transport is probed by applying pulses of electrical\ncurrent in the injector while simultaneously monitoring\nthe non-local voltage V'on the detector as a function\nof the angle '. Similar to the SMR measurement, we4\nuse the background-subtracted voltage \u000eV'=V'\u0000Vk\nto extract the spin contribution13. The background is\nagain measured by applying the external \feld H0par-\nallel to the nanostrip direction ( y-axis). We distinguish\nSSE from SOT by de\fning two quantities based on the\nyzmirror symmetry13: \u0006';I;\u0001';I\u0011(\u000eV';I\u0006\u000eV';I)=2,\nwhere'=\u0019\u0000'.\nPreviously we have reported13that in 18 nm thick\nYIG \flm, injecting electrical current above a cross-over\nthreshold of the order of Jc=6.0\u00021011A/m2is su\u000ecient\nto excite low energy magnetostatic magnons (in the GHz\nrange). This phenomenon is characterized by the emer-\ngence a non-linear spin conduction as a function of the\napplied current. In the following, we want to use this\nincident change of the interface transmission to investi-\ngate the in\ruence of non-equilibrium spin accumulation\nat the YIGjPt interface on the non-linear properties.\nLet us \frst consider in Fig.3a) the pristine state where\nneither interface of YIG jPt at the injector nor the de-\ntector has been treated by Joule annealing. In the top\npanel a) of Fig.3, we display the non-local \u0006 '=0;Iand\n\u0001'=0;Ias a function of the applied current in the injec-\ntor. In this con\fguration Pt 1is connected to the current\nsource while we record V2the voltage drop across Pt 2(cf.\nFig.1a). It can be seen that the \u0001 Ishows a quadratic\nrise due to its thermal origin while the \u0006 Iseems to evolve\nquasi-linearly with current Ion the range explore. The\nnon-linear properties are analyzed by plotting in the in-\nset of Fig.3a) the current dependence of \u001b0=\u0006I, where\n\u001b0= (@\u0006I=@I)jI=0is the slope of a linear regression\nthrough the \u0006 Idata measured at I <0:5 mA (see gray\ndashed lines). We de\fne Jcas the intercept with the 75%\ndecrease1334. We report on the main graph the estimate\nofIc, which is here not precise due to the low spin con-\nductivity of the device. Next, we perform Joule annealing\nof the injector (Pt 1) for 60 minutes with the same proce-\ndure as before. The impact on the non-local signals can\nbe seen in the second panel b). We observe that the \u0006 I\nis now 3 times larger, indicating that the excited magnon\ndensity in YIG is enhanced due to the higher spin trans-\nmission at the injector. Remarkably, now we are able to\ndistinguish clearly on the \fgure at Jc= 7\u00021011A/m2,\nthe cross-over threshold from a linear to a non-linear spin\ntransport regime, which is the signature of the participa-\ntion of low energy magnetostatic magnons to spin trans-\nport. Such magnons are in principle solely excited by\nSOT exerted at the interface between YIG and the injec-\ntor. We also note that the \u0001 I-signal produced by ther-\nmally generated magnons remains unchanged. This is\nexpected because the spin conversion of SSE signal oc-\ncurs only at the detector Pt 2nanostrip (which is not an-\nnealed for this moment) whereas the injector nanostrip\nonly plays the role of a heater. The last step, shown in\npanel c), both injector and detector are annealed. The\n\u0001Iis enhanced by a factor of 3 due to the higher spin\nconversion of the probing interface. As can be seen there\ntoo is that the \u0006 Iis now 9 times larger than the ref-\nerence (panel a). It follows the fact that injection anddetection of magnons at each YIG jPt interface are now 3\ntimes more e\u000bective, leading to a factor of 9 increase of\nthe SOT signals. Nonetheless, the crossover current den-\nsityJc13is not a\u000bected by the annealing and it occurs\nat the same value for both cases (b) and (c). It supports\nthe observation in SP experiments, where local annealing\nhasn't induced additional broadening the linewidth (in-\nset of Fig.2). It also highlights the striking di\u000berence of\nout-of-equilibrium behavior between closed (nano-pillar)\nand open (extended \flms) magnetic geometries.\nA possible explanation compatible with the observed\nbehavior could be an improved wettability of the Pt on\nYIG after local Joule annealing35. The e\u000bect could be\nparametrized by introducing an additional transmission\ncoe\u000ecient 0 1\n3andK≥1nHz). The result illustrates that\nthe counterintuitive asymmetric responds of the steady-\nstate mean values are affected by the nonlinearity term\nin equations ( 2). Meanwhile, it also plays an important\nrole in evolutions of the perturbation terms.\nExpandingequations( 2a)-(2c) with the ansatzofoper-\nators and removingthe steady-state terms, the perturba-\ntion terms canbe expressedby the linearizedHeisenberg-\nLangevin equations\nδ˙a=−(i∆c+κa\n2)δa−iJδb+√ηaκasae−i∆pt,(7a)\nδ˙b=−(i∆c+κb\n2)δb−iJδa−igδm\n+√ηbκbsbe−i∆pt, (7b)\nδ˙m=−[i(∆m+4K|̟|2+K)+κm\n2]δm−igδb\n−i2K(̟2δm∗+̟∗δm2+2̟δm∗δm)\n−i2Kδm∗δm2, (7c)4\nthe quadratic terms −i2K̟∗δm2,−i4K̟δm∗δmin\nequation ( 7c) denote the system nonlinearity and can\nnot be sneezed at. They inevitably cause asymmetric re-\nsponses of the perturbation terms, just as the responses\nof the steady-state mean values in Fig. 2. However, the\ncubicterm −i2Kδm∗δm2inequations( 7)issosmallthat\nit will be ignored in the following calculation.\nTo study asymmetric responses of the perturbations\nfrom a new perspective, we explore the nonreciprocal\ntransmission of the output fields, mainly including the\nfirst-, second- and third-order sidebands, in case 1 and\ncase 2. To get analytical solutions of these high-order\nsidebands, we define the perturbation terms have the fol-\nlowing forms\nδa=A−\n1e−i∆pt+A+\n1ei∆pt+A−\n2e−2i∆pt+A+\n2e2i∆pt\n+A−\n3e−3i∆pt+A+\n3e3i∆pt, (8a)\nδb=B−\n1e−i∆pt+B+\n1ei∆pt+B−\n2e−2i∆pt+B+\n2e2i∆pt\n+B−\n3e−3i∆pt+B+\n3e3i∆pt, (8b)\nδm=M−\n1e−i∆pt+M+\n1ei∆pt+M−\n2e−2i∆pt+M+\n2e2i∆pt\n+M−\n3e−3i∆pt+M+\n3e3i∆pt, (8c)\nthis ansatz is rooted from the internal mechanism of\nthe system, where a series of high-order sidebands [ 51]\nwith frequency ωd+n∆p(n= 1,2,3...) are generated. n\nrepresents the sideband order as shown in Fig. 1(b). The\nnegativeandpositiveexponentsofthepowersareinpairs\nand respectively denote the upper and lower sidebands.\nFor examples, the term A−\n1e−i∆ptdenotes the first-order\nupper sideband with coefficient A−\n1, andA+\n1ei∆ptcor-\nresponds to the first-order lower sideband (the Stokes\nfield [52] in Fig. 1(b)) with coefficient A+\n1. For the sake\nof simplicity, in what follows we only take the high-order\nupper sidebands into consideration, because of similar\ncharacteristics in the lower sidebands. Of course the cor-\nresponding lower sidebands can also be studied by utiliz-\ning the same method.\nWith the ansatz we can analytically solve equa-\ntions (7), the coefficients of the high-order sidebands can\nbe derived, respectively. Firstly, the coefficients of the\nfirst-order upper sideband are derived as\nB−\n1=i√ηbκbsb(∆p−∆c+iκa\n2)(iC −∆p+2̟2KL)\ng2(∆p−∆c+iκa\n2)+(iC −∆p+2̟2KL)S(−)\n+iJ√ηaκasa(iC −∆p+2̟2KL)\ng2(∆p−∆c+iκa\n2)+(iC −∆p+2̟2KL)S(−),\n(9a)\nA−\n1=JB−\n1+i√ηaκasa\n∆p−∆c+iκa\n2, (9b)\nM−\n1=−gB−\n1\niC −∆p+2K̟2L, (9c)\nM+\n1=L∗M−∗\n1, (9d)with\nC=−i(∆m+K+4K|̟|2)−κm\n2, (10)\nS(±)= (∆p±∆c+iκa\n2)(∆p±∆c+iκb\n2)−J2,(11)\nL=2KS(+)̟2∗\ng2(∆p+∆c+iκa\n2)−(∆p−iC)S(+).(12)\nFor the second-order upper sideband, the coefficients are\ngiven as\nB−\n2=g(2∆p−∆c+iκa\n2)M−\n2\nG, (13a)\nA−\n2=JB−\n2\n2∆p−∆c+iκa\n2, (13b)\nM−\n2=−2KG(̟2E∗+̟∗M−2\n1+2̟M−\n1M+∗\n1)\n(iC −2∆p+2K̟2D∗)G+g2(2∆p−∆c+iκa\n2),\n(13c)\nM+\n2=DM−∗\n2+E, (13d)\nwhere\nD=2̟2KF\ng2(2∆p+∆c−iκa\n2)−F(2∆p+iC),(14)\nE=2̟∗KFM+2\n1+4̟KFM+\n1M−∗\n1\ng2(2∆p+∆c−iκa\n2)−F(2∆p+iC),(15)\nF= (2∆ p+∆c−iκb\n2)(2∆p+∆c−iκa\n2)−J2,(16)\nG= (2∆ p−∆c+iκb\n2)(2∆p−∆c+iκa\n2)−J2.(17)\nTo the third-order upper sideband, the coefficients have\nthe following forms\nB−\n3=g(3∆p−∆c+iκa\n2)M−\n3\n(3∆p−∆c+iκa\n2)(3∆p−∆c+iκb\n2)−J2,(18a)\nA−\n3=JB−\n3\n3∆p−∆c+iκa\n2, (18b)\nM−\n3=2K̟2H∗\n++H−(3∆c+gT∗−iC∗)\n(3i∆p+C −igQ)(−3∆p−gT∗+iC∗)−4iK2|̟|4,\n(18c)\nM+\n3=2̟2KM−∗\n3+iH+\n−3∆p−gB3−iC, (18d)\nin which\nQ=g(3∆p−∆c+iκa\n2)\n(3∆p−∆c+iκa\n2)(3∆p−∆c+iκb\n2)−J2,(19)\nT=−g(3∆p+∆c−iκa\n2)\n(3∆p+∆c−iκa\n2)(3∆p+∆c−iκb\n2)−J2,(20)\nH+=−4i̟∗KM+\n1M+\n2−4i̟K(M+\n1M−∗\n2+M+\n2M−∗\n1),\n(21)\nH−=−4i̟∗KM−\n1M−\n2−4i̟K(M−\n2M+∗\n1+M−\n1M+∗\n2).\n(22)5\nIt can be seen that the coefficients A−\n1,B−\n1,A−\n2,B−\n2,\nA−\n3andB−\n3of the high-order sidebands are closely de-\npendent on the magnon mode including the steady-state\namplitude and the perturbation terms.\nWith above analytical solutions, we calculate the out-\nput fields of each sideband in two cases. In case 1, the\ncontrol and probe fields are injected from cavity aand\nthe output fields from cavity bare explored. With input-\noutput relation, the output fields of high-order sidebands\nfrom cavity bcan be expressed as\nsaout=√ηbκbB−\n1e−i(ωd+∆p)t+√ηbκbB−\n2e−i(ωd+2∆p)t\n+√ηbκbB−\n3e−i(ωd+3∆p)t. (23)\nIn the direction from cavity atob, the transmission co-\nefficients of the first-, second- and third-order sidebands\nare respectively given as\ntba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB−\n1\nsa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24a)\nτba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB��\n2\nsa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24b)\nℓba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB−\n3\nsa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (24c)\nIn case 2, the same control and probe fields are ap-\nplied on cavity band the output fields from cavity aare\nresearched. In order to avoid confusion, we use A−\nj,B−\nj\nandM−\nj(j= 1,2,3) to distinguish from the coefficients\nin case1. Combiningthis condition andthe input-output\nrelation, output fields of the high-order upper sidebands\nare given as follows\nsbout=√ηaκaA−\n1e−i(ωd+∆p)t+√ηaκaA−\n2e−i(ωd+2∆p)t\n+√ηaκaA−\n3e−i(ωd+3∆p)t. (25)\nSimilarly, along the direction from cavity btoa, the\ntransmission coefficients of the first-, second- and third-\norder sidebands are defined as\ntab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA−\n1\nsb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (26a)\nτab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA−\n2\nsb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (26b)\nℓab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA−\n3\nsb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (26c)\nIn order to more clearly describe nonreciprocal trans-\nmissions of the high-order sidebands in two cases, we\nintroduce definition of the nonreciprocal isolation ratio,\nwhich is expressed as\nIo=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglelog10|√ηbκbB−\nj|2\n|√ηaκaA−\nj|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (27)hereostands for either t,τorℓcorresponding to j=\n1,2,3.It,IτandIℓrespectively denote isolation ratio\nof the first-, second- and third-order sidebands. Then\nwe use isolation ratio Ioto quantitatively describe de-\ngree of the nonreciprocal transmissions in two cases. It\nis decided by the value of Io: ifIo= 0, equivalently,\n|√ηbκbB−\nj|2/|√ηaκaA−\nj|2= 1, it means transmissions of\nthehigh-ordersidebandsarereciprocal(symmetry)along\ntwo opposite directions. For other situations, the higher\nthe value of Io, the stronger is the nonreciprocity of the\ntransmissions in two cases. For example, Io≥1, which\nillustrates transmission coefficient in one direction is at\nleast 10 times that of the other direction. It also shows\nthat transmittances along two opposite directions have a\nstrong asymmetry. This situation can be approximated\nas one-way transmission of high-order sidebands.\nIII. NONRECIPROCAL TRANSMISSIONS OF\nTHE HIGH-ORDER SIDEBANDS IN A CAVITY\nMAGNONICS SYSTEM\nSince the transmission coefficients in equations ( 24)\nand (26) and the isolation ratio in equation ( 27) are\nmainly dominated by coefficients of each sideband, which\nare closely associated with the steady-state amplitude\nand the perturbation terms of magnon mode. Under this\nsituation, we analyze influence of the driving power, ap-\nplied on the magnon mode, and the frequency detuning\nbetween the driving field and the magnon mode on the\nnonreciprocal transmission of each sideband.\n0 1 2 3 4051015\nFIG. 3: (Color online) The logarithmic amplitudes lo g10S(ω)\nof the transmitted high-order sidebands vs frequency ω/∆p\nin case 1 and case 2. Here P= 15mW, ∆ p/∆c= 0.7, other\nparameters are same as in Fig. 2.\nFor the same purpose, but utilizing a different method,\nwe first exhibit an overview comparison of the high-order\nsideband spectrums between case 1 and case 2. Here\nthe magnon mode is resonantly driven by a 15 mW driv-\ning field. As shown in Fig. 3, spectrums of the adjacent\nsidebands are equally spaced apart from each other at6\nfrequency ∆ pin the rotating frame. Where ω/∆p= 0\ndenotes the control field, ω/∆p=n(n= 1,2,3) cor-\nresponds to the nth-order sideband. It can be seen\nthat there exists amplitude gap between the sidebands\nwith the same order in case 1 and case 2. Scilicet,\nthe transmissions of the high-order sidebands are non-\nreciprocal in two cases. What’s more, amplitudes of\nthe high-order sidebands decrease rapidly as the side-\nband order increases. Specifically, S(∆p)/S(0)<10−3,\nS(2∆p)/S(0)<10−7andS(3∆p)/S(0)<10−12in both\ncases. This proves that it is valid to regard high-order\nsidebands as perturbation comparedwith the strongcon-\ntrol field. Then, we analyze in detail the transmission\nnonreciprocity of each sideband in the following content.\nA. Dependence of nonreciprocal transmissions for\nhigh-order sidebands on the resonant driving power\nP\nFigure4shows transmission coefficients |tba|2and\n|tab|2vary with frequency detuning δ/∆cunder differ-\nent resonant driving powers P. In Fig. 4(a), we choose\ndriving power P= 0. Under this situation, |tba|2and\n|tab|2aresmallerthan0 .2andevolutionsofthem aresyn-\nchronous. It means the transmissions of the first-order\nFIG. 4: (Color online) Transmission coefficients |tba|2and\n|tab|2of the first-order sideband vs frequency detuning δ/∆c.\nTo observe transmission coefficients clearly, |tab|2in (b) and\n|tba|2in (c) have been increased by one order of magnitude.\nThe driving powers P= 0, 20 mW, 30mW in (a), (b) and\n(c), respectively. Other parameters are same as that in Fig. 2.sidebandintwocasesarereciprocalwhenthedrivingfield\non magnon mode is absent. Increasing driving power to\n20mW in Fig. 4(b), intuitively, only a peak with value\n150 of|tba|2is seen at δ/∆c=−0.3; and a peak value\n5.6 of|tab|2is shown at δ/∆c=−0.25. However, if one\nzooms in locally, a local peak of |tba|2and a tiny one\nof|tab|2can be found at the position of the blue-dotted\nline. This illustrates that the first-order sideband (probe\nfield) is magnified hundreds of times compared with the\noriginal input probe field and mainly transmitted at\nδ/∆c=−0.3 in case 1. Whereas, in case 2, magnifica-\ntion and transmission of the first-order sideband mainly\noccurs at δ/∆c=−0.25. From another point of view,\nthe transmissions of the first-order sideband in two cases\nare robustly nonreciprocal near δ/∆c=−0.3. Continue\nto enhance the driving power to 30 mW in Fig. 4(c), we\nfind|tba|2with two peaks is always smaller than 2; while\n|tab|2reaches its maximum value 15 .5 atδ/∆c=−0.31,\nwith the other invisible peak at the blue-dotted line. The\nresults illustrate that the first-order sideband is magni-\nfied less than twice in case 1, but more than ten times\nin case 2; and the strong transmission nonreciprocity ex-\nists near δ/∆c=−0.3. From above numerical analysis,\nwe find the transmission nonreciprocity of the first-order\nsideband is effectively modulated by the driving power\nacted on the magnon mode. This stems from the fact\nthatthe increased driving power excites more magnon\npolarons and simultaneously enhances the effective Kerr\nFIG. 5: (Color online) Transmission coefficients τba,τaband\nℓba,ℓabof the second- and third-order sidebands vs frequency\ndetuning δ/∆c. In order to clearly observe the transmission\ncoefficients, the values of τabin (b) and ℓabin (e) have been\nrespectively magnified by three and five orders of magnitude.\nHereP= 0 in panels (a) and (d), P= 20mW in panels\n(b) and (e), and P= 30mW in panels (c) and (f). Other\nparameters are same as that in Fig. 2.7\nnonlinearity of the magnon ,which eventually strengthens\nthe nonlinear asymmetric responses of the system in two\ncases. In addition, it needs to be emphasized that peaks\non each curve should have been resonant with the effec-\ntivesupermodesmadeupoftwomicrowavecavitymodes.\nFrequencies of the supermodes shift ±/radicalBig\nJ2−(κb−κa\n2)2\n(equal to ±0.3∆c) from the original cavity frequency ωc.\nThis is originated from the strong cavity-cavity coupling\nrate, i.e., J > κ a,κbin the system. Practically, the ac-\ntural resonant frequency exhibit shifts due to the joint\ninterference of the strong cavity-magnon coherent inter-\naction (g > J) and the magnon Kerr nonlinearity [ 20].\nFigure5shows transmission coefficients τba,τaband\nℓba,ℓabof the second- and the third-order sidebands\nversus frequency detuning δ/∆cwith different resonant\ndriving powers. When the driving field is absent in\nFigs.5(a) and 5(d), although the transmission coeffi-\ncients are very tiny, the transmission nonreciprocity is\nvery obvious in the regime δ/∆c∈[−0.5−0.2]. Once\nthedrivingfieldisconsidered,boththetransmissioncoef-\nficients and nonreciprocity will be further enhanced. For\ninstance, P= 20mW in Figs. 5(b) and 5(e),τbaand\nℓbareach respective unique maximum 0 .7 and 1.5×10−3\natδ/∆c=−0.3. While τabandℓabkeep much smaller\nmaximums. Continue to increase the driving power to\n30mW in Fig. 5(c) and 5(f). Both τbaandℓbahave\nvery tiny maximums. While τabandℓabhave soared\nto their peaks 4 .8×10−3and 4.6×10−7at the point\nδ/∆c=−0.31. This illustrates that the resonant driving\nfield can not only effectively manipulate the transmis-\nsion nonreciprocity of the first-order sideband, but also\nof the second- and the third-order sidebands. What’s\nmore, the transmission nonreciprocity of the high-order\nsidebands have a cooperative and consistent characteris-\ntic when facing different driving powers. Besides being\napplied in high performance nonreciprocal directional-\nswitching isolator[ 33,53] and diode [ 54–56], such a series\n-0.4 -0.2 0 0.2 0.401020304050\n012345\nFIG. 6: (Color online) The isolation ratio Itof the first-order\nsideband vs the resonant driving power Pand frequency de-\ntuningδ/∆c. Parameters are the same as that in Fig. 2.of equally spaced high-order sidebands also have poten-\ntial applications in frequency comb-like precision mea-\nsurement through simply operating the driving power.\nThen, we quantitatively describe the degree of trans-\nmission nonreciprocity for each sideband by isolation ra-\ntio. Figure 6intuitively displays isolation ratio Itvaries\nwith the resonant driving power Pand frequency detun-\ningδ/∆c. As the driving power increases, one bright re-\ngion, composed of an upper and a lowerparts, appearsat\nthe center of δ/∆c=−0.3. In this region the optimal It\nis 5.6 dB obtained with P= 21.8mW. This can also be\nexplained by the result in Fig. 4, i.e., the prominent large\ndifferencesoftransmissioncoefficients |tba|2and|tab|2be-\ntween the left resonant peaks near δ/∆c=−0.3.\nFor the second- and third-order sidebands, the cor-\nresponding isolation ratios IτandIℓversus the driv-\ning power Pand frequency detuning δ/∆care given in\nFig.7(a) and7(b). In Fig. 7(a), the value of Iτin most\nareas is lower than 3 .5 dB, except two bright areas cen-\ntered at δ/∆c=−0.3, where Iτreaches its maximum\n9.9 dB. While the distribution of Iℓis quite different.\nAs presented in Fig. 7(b), except a small dark area, the\nvalues of Iℓin other areas are higher than 10 dB. Es-\npecially in the yellow area centered at δ/∆c=−0.4\nwithP∈[0 10]mW and the other one centered at\nδ/∆c=−0.3 withP∈[22 28]mW,Iℓcan be higher\nthan 30 dB. By an overallobserving, what IℓandIτhave\nin common is that the optimal value is distributed near\nδ/∆c=−0.3, this is consistent with the result in Fig. 5.\nTakentogetherFig. 6andFig. 7, wefind that the isola-\ntion ratios It,IτandIℓshow an overall increasing trend\nwith the increase of the sideband order. Specifically, op-\ntimalIt,IτandIℓare enhanced in turn from 5 .6 dB,\nto 9.9 dB and lastly 30 dB near detuning δ/∆c=−0.3.\nThe reasonis rooted fromthe strengthened effective Kerr\nnonlinearity in each sideband. Besides the steady-state\namplitude of the magnon mode, the first-order sideband\nis also associated with the first-order perturbation terms\n(in equations ( 9)) of the magnon mode; the second-order\nsideband is simultaneously related to the first- and the\nsecond-orderperturbation terms of the magnon mode (in\n-0.5 -0.4 -0.3 -0.2 -0.101020304050\n02468\n-0.5 -0.4 -0.3 -0.2 -0.101020304050\n51015202530\nFIG. 7: (Color online) isolation ratios (a) Iτof the second-\norder sideband and (b) Iℓof the third-order sideband vs the\nresonant driving power Pand frequency detuning δ/∆c. Pa-\nrameters are same as that in Fig. 2.8\nequations. ( 13)); and the third-order sideband is totally\nrelated to the first-, second-and third-orderperturbation\nterms (in equations ( 18)) of the magnon mode. Except\nfor the steady-state amplitude of the magnon, the strong\ntransmission nonreciprocity of each sideband originates\nfrom the aggregatenonlinear effects in each perturbation\nterm. Therefor, the higher the order of the sideband,\nthe stronger nonlinear asymmetry of the transmission, it\nlastly leads to much higher isolation ratio.\nFIG. 8: (Color online) The isolation ratios (a) It, (b)Iτ\nand (c) Iℓvsδ/∆cwith different detunings ∆ mbetween\nthe magnon mode and the correspong driving field. Here\nP= 25mW, and the other parameters are the same as in\nFig.2.\nB. Dependence of the nonreciprocal transmissions\nfor high-order sidebands on frequency detuning ∆m\nIn this section, we research influence of frequency de-\ntuning ∆ monthe nonreciprocaltransmissionofthe high-\nordersidebands. InFig. 8,Isolationratios It, IτandIℓvs\nδ/∆cwith different detunings ∆ mare intuitively given.\nWhenmagnonmodeisresonantwiththedrivingfieldi.e.,\n∆m= 0, isolation ratios It, IτandIℓincrease in turn.\nEspecially in the optimal detuning regime (centered at\nthe resonant frequency of one effective supermode of the\nmicrowave cavities [ 20]),It,IτandIℓreach their max-\nimum values. The same evolution trend is also suitable\nfor non-resonant situations, i.e., ∆ m/2π=−19.2 MHz,\n20 MHz, except for the reduced optimal isolation ratios\nand the changed optimal detuning regimes. It is because\nexcitation of the magnetic polarons is suppressed in thenon-resonant situations and results in much weaker effec-\ntive Kerr nonlinearity . This lastly induces much weaker\ntransmission nonreciprocity and shifts the optimal de-\ntuning regime.\nFor actual operation to obtain strong nonreciprocity\nof the high-order sidebands, the bandwidth of the op-\ntimal detuning regime is often more concerned. In the\nresonant situation, we centralize It,IηandIℓin Fig.9.\nIt can be obviously seen It,IηandIℓincrease in turn\nFIG. 9: (Color online) The isolation ratios It,Iτand (c)Iℓ\nvsδ/∆cwhen the magnon mode is resonant with the driving\nfield, i.e., ∆ m= 0. Here P= 25mW and the other parame-\nters are the same as in Fig.2.\nand reach their optimal values in the gray shaded area.\nThis means an overall controlling of the strong nonre-\nciprocity for high-order sidebands can be realized in the\noptimal detuning regime ranging from δ/∆c=−0.301 to\nδ/∆c=−0.273. The bandwidth of the optimal detun-\ning regime is seven megahertz with ∆ c= 2π×40 MHz in\nthe system. It is achievablefor experimentaloperationto\ncapture strong transmission nonreciprocity of the high-\norder sidebands simultaneously in such a bandwidth.\nIV. CONCLUSION\nIn summary, we have theoretically provided a method\ntorealizeanoverallcontrollingofthe strongtransmission\nnonreciprocity of the first-, second- and third-order side-\nbands in a cavity magnonics system. It is consist of two\ncoupledmicrowavecavitiesandoneYIG (oranothersuit-\nable magnonic material) sphere. Our approach utilizes\nthe self-Kerr nonlinearity of magnon in the YIG sphere.\nWe have shown that strong transmission nonreciprocity\nof the high-order sidebands can be achieved by respec-\ntively regulating the driving power on the magnon mode\nand the frequency detuning between the magnon mode\nand the driving field. We also showed that the higher the\norder of the sideband, the stronger is the nonreciprocity\nmarkedby the enhancedisolationratioin the optimalde-\ntuning regime. We have illustrated that the bandwidth9\nof the optimal detuning regime can reach several mega-\nhertz when magnon mode is resonant drived. This im-\nplies that it is experimentally feasible to capture robust\ntransmission nonreciprocity of the high-order sidebands\nsimultaneously. 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Stamps 1,4 \n \n1School of Physics M013, University of Western Austr alia, Crawley 6009, Australia \n2Department of Physics, Indian Institute of Technolo gy Bombay, Powai, Mumbai 400076, India \n3Department of Metallurgical Engineering and Materia ls Science, Indian Institute of Technology \nBombay, Mumbai 400076, India \n4SUPA, University of Glasgow, Glasgow, G12 8QQ, Unit ed Kingdom \n \nAbstract: By using the stripline Microwave Vector–Network Ana lyser Ferromagnetic Resonance \nand Pulsed Inductive Microwave Magnetometry spectro scopy techniques, we study a strong \ncoupling regime of magnons to microwave photons in the planar geometry of a lithographically \nformed split–ring resonator (SRR) loaded by a singl e–crystal epitaxial yttrium–iron–garnet (YIG) \nfilm. Strong anti-crossing of the photon modes of S RR and of the magnon modes of the YIG film is \nobserved in the applied-magnetic-field resolved mea surements. The coupling strength extracted \nfrom the experimental data reaches 9% at 3 GHz. \n Theoretically, we propose an equivalent circuit mo del of the SRR loaded by a magnetic \nfilm. This model follows from the results of our nu merical simulations of the microwave field \nstructure of the SRR and of the magnetisation dynam ics in the YIG film driven by the microwave \ncurrents in the SRR. The equivalent-circuit model i s in good agreement with the experiment. It \nprovides simple physical explanation of the process of mode anti-crossing. \nOur findings are important for future applications in microwave quantum photonic devices \nas well as in nonlinear and magnetically tuneable m etamaterials exploiting the strong coupling of \nmagnons to microwave photons. \n \n \n1. Introduction \n \n In order to be useful for quantum application, a p roposed technology has to be able to \nexchange information with preserved coherence [ 1-3]. To this end, a system which consists of two \nsub–systems has to operate in a regime called 'stro ng coupling'. The strong coupling regime is \ncharacterised by strength of coupling between the s ubsystems which is larger than the mean energy \nloss in both of them. A straightforward way to decr ease losses is to make use of resonant systems \n[4–6]. Systems that exploit plasmonic resonances make u se of strong coupling between electric \ndipoles and optical fields localised at the sub-wav elength scale (i.e. on a spatial scale smaller than \nthe wavelength of light). This makes possible the c reation of solid–state sources of quantum states \nof light (single photons, indistinguishable or enta ngled photons) such as those based on \nsemiconductor quantum dots (QDs) or nitrogen–vacanc y centres in diamonds (NVs), embedded in \noptical microcavities [ 7–10 ] or precisely placed in close proximity to an opti cal nanoantenna [ 11–\n13 ]. \n \n* Both authors contributed equally to this paper. \n† Corresponding author On the other hand, strong light–matter coupling is also achievable by using magnetic dipoles \n[14 ]. These schemes make possible the processing of qu antum information in various microwave \nresonator systems comprising ultra–cold atomic clou ds [ 15 ], molecules [ 16 ], NVs [ 17, 18 ] and ion–\ndoped crystals [ 19 ]. \n Recently, strong coupling of microwave photons to magnons was demonstrated for a system \nrepresenting a microwave cavity loaded by a single– crystal yttrium–iron–garnet (YIG) sphere [ 1, \n20–22 ]. Furthermore, a more planar geometry of a microwa ve stripline resonator loaded by a \nsingle–crystal YIG film was shown to produce streng th of coupling of 150 MHz [ 23 ]. \n Nowadays, a microstrip line has largely replaced a traditional microwave cavity as a source \nof driving microwave magnetic field in the ferromag netic resonance (FMR) experiment [ 24, 25 ]. \nOn the other hand, the stripline arrangement is use d to investigate planar microwave metamaterials \n[26 ]. Consequently, it makes this technique very suita ble for the investigation of coupling of \nmicrowave photons to magnons. Additionally, the us e of microstrip lines opens up avenues for the \ndevelopment of magnetically tuneable metamaterials based on arrays of highly–conductive meta–\nmolecules (resonant elements that comprise metamate rials) coupled to magnetically–active \nmaterials of different sizes and shapes [ 27, 28 ]. \n In a recent work – Ref [ 23 ], the interaction between a non–magnetic split–rin g resonator \n(SRR) and a thin film of YIG was investigated with the Microwave Vector–Network Analyser \n(VNA) Ferromagnetic Resonance (FMR) method. Strongl y hybridised resonances were observed. \nThe YIG film was grown on a single–crystal yttrium aluminium garnet substrate by pulsed laser \ndeposition using an excimer laser. Although the SRR supported an anti–symmetric (low–frequency) \nand a symmetric (high–frequency) mode, only strong coupling of the YIG static magnetic field–\ndependent mode to the symmetric mode was studied, a nd a coupling strength of 150 MHz was \nobserved. \n In the present work, we study the interaction of m agnetic resonances in a YIG film with \nmicrowave photon resonances in SRRs. We use both th e VNA–FMR (frequency domain) and \nPulsed Inductive Microwave Magnetometry ('PIMM', ti me domain) spectroscopy techniques to \ninvestigate the coupling between photons and magnon s. We observe a strong coupling between the \nYIG mode and both low–frequency anti–symmetric and high–frequency symmetric modes of the \nSRR. In the VNA–FMR experiment, this interaction ma nifests itself as a strong anti–crossing \nbetween the photon and magnon mode. Naturally, the same result is confirmed by Fourier–\ntransforming time–domain PIMM traces into the frequ ency domain. Additionally, the inspection of \ntime–domain traces reveals the presence of a strong beat effect. The presence of the beat signal in \ntime–domain traces is a signature of the entangleme nt of qubits [ 29, 30 ]. \n We conduct numerical simulations of the microwave field structure of the split rings and of \nthe magnetisation dynamics driven by the microwave currents in the ring. We also suggest an \nequivalent circuit of an SRR loaded by a magnetic f ilm. These calculations are in very good \nagreement with the experiment and deliver a clear p hysical picture of the process of anti–crossing \nof the SRR and magnon modes. \n \n \n2. Experimental arrangement \n \n The investigated SRR (Fig. 1) represents a small m etallic loop with a slit in it. It is a kind of \nLC resonant contour in which the loop acts as an indu ctance and the slit represents a lumped \ncapacitance. Due to the lumped–element approach an SRR can support resonance wavelengths noticeably larger than the linear size of the SRR. An epitaxial single–crystal YIG film also \nrepresents a microwave resonant system which suppor ts FMR. FMR is the collective precession of \nspins about the equilibrium spin direction in the m aterial. In a ferro– or ferrimagnetic materials the \nspins are coupled by the exchange and dipole–dipole interactions. The energies of the \n(inhomogeneous) exchange and dipole–dipole interact ions determine the FMR frequency. For the \npresent work it is important that the FMR frequency also depends on the static magnetic field [ 31 ] \nin which the material is placed. The dependence on the applied field is through the Zeeman energy \ncontribution to the energy of magnons which represe nt quanta of FMR [ 32 ]. \n \n \n \nFig. 1. Sketch of the split ring resonator structur e. The split ring is inductively coupled to a \nmicrostrip feeding line. In experiment, the input a nd output of the microstrip line are connected to a \nVNA and the static applied magnetic field H is created by an electromagnet (not shown). For th e \nmeasurements, a YIG film (not shown) with the dimen sions 10 mm × 15 mm × 25 µm is placed on \ntop of the split ring. The measured dimensions of t he split ring and the microstrip line are: a = \n8.5 mm, b = 7.5 mm, g = 0.06 mm (the distance between the microstrip lin e and the SRR is also g), \nw = 3 mm. Also not shown: the total thickness of the SRR plus the dielectric substrate (grey area, \nε = 2.3) and the back–side metallisation is h = 0.84 mm. The thickness of the copper lines of th e \nSRR and the microstrip is 0.01 mm [ 33 ]. \n \n \nUsually, FMR in a sample is excited by placing the sample in a uniform (or quasi–uniform) \nmicrowave magnetic field. An onset of FMR is easily seen as applied–field dependent resonance \nabsorption of the microwave power by the magnetic m aterial. A convenient way to excite FMR in \nferro– and ferrimagnetic films is by placing them on top of a microstrip or copla nar microwave \nstripline transmission line [ 24, 25, 34 ] or forming a microstrip line directly on top of t he film [ 35 ]. \nA microwave current flowing through the stripline i nduces a microwave Oersted field in the space \nabove the stripline. This field drives spin precess ion in the material. \n A split–ring resonator can be formed lithographica lly as a loop of a microstrip line on top of \na microwave substrate. One can use the Oersted fiel d of a microwave current flowing through such \nan SRR to drive FMR in a ferrimagnetic film sitting on top of the SRR. In this way, we realise a simple and effective means of implementing addition al functionality into the split ring design. This \nis the central point of our paper. \n The geometry of the device under test is shown in Fig. 1. It represents an SRR inductively \ncoupled to a microwave transmission line. A single– crystal yttrium–iron–garnet film grown on a \n1 mm–thick gadolinium–gallium garnet (GGG) substrat e sits on top of the resonator with the YIG \nlayer facing the SRR. The film is 25 µm thick and was grown with liquid–phase epitaxy (LP E). A \nstatic magnetic field H is applied in the plane of the film in the directi on perpendicular to the \nmicrostrip line (Fig. 1). \n The measurements have been taken at room temperatu re. In order to take the measurements \nin the frequency domain, the input and the output o f the microstrip line have been connected to the \nports of a VNA and the transmission characteristic of the microstrip line (S21=Re(S21)+ iIm(S21)) \nhas been measured as a function of microwave freque ncy f and the strength H of the applied field. \n In the alternative time–domain (PIMM) arrangement, the input port of the microstrip line is \nconnected to the output port of a generator of shor t pulses. We excite our system with a pulse of a \nrectangular shape which is 1 V in amplitude and 10 ns in duration. The nominal pulse rise time at \nthe output port of the pulse generator is 55 ps. \n \n \n3. Experimental results \n \n A set of representative |S21| vs. microwave freque ncy f dependencies taken with VNA–\nFMR for a number of values of the applied field is shown in Fig. 2. In each of the panels (except \none for 300 Oe) one observes two peaks. One peak is very strongly dependent on the applied field. \nEssentially it moves across the displayed frequency range with an increase in the frequency. For \nH=300 Oe this peak has an almost vanishing amplitude and is located at about 2.25 GHz. Its \nrelative amplitude becomes much larger when it move s closer to second – higher–frequency – peak \n(the peaks at 2.7 GHz and 3.4 GHz respectively in t he panel or H=430Oe). For H=450 Oe the \nlower–frequency peak ceases moving across, but the higher–frequency peak starts to move quickly \nwith H. Its amplitude drops with an increase in the frequ ency separation from the lower–frequency \npeak. The respective frequency vs. field dependence of the peak positions are shown in Fig. 3(a). \nOne clearly sees strong anti–crossing of the two li nes which suggests strong coupling of the modes. \nIn order to identify the resonances these measureme nts have been repeated when the YIG \nfilm covered not only the SRR but also a section of the feeding microstrip line. These \nmeasurements have been taken on a different SRR str ucture whose geometry was able to \naccommodate the film in this position. The shape of the SRR for this structure was the same as in \nFig. 1, but its sizes were slightly different, ther efore the frequencies of the resonances in Fig. 3(b ) \ndo not coincide with the ones in Fig. 3(a) (and in Fig. 4 which we will discuss later on). From \nFig. 3(b) one sees that for this YIG film placement one observes a third mode located in between \ntwo peaks of the type shown in Fig. 2. The frequenc y vs. applied field dependence for this extra \nmode is well fitted by the Kittel formula [ 36 ] \n \n0 s ( ) 2f H H M γµπ= + (1) \n for the FMR frequency of an in–plane magnetised fil m with a saturation magnetisation value \nµ0Ms=0.2 T=2 kOe. (Here γ/(2 π) is gyromagnetic ratio; typically 2.8 MHz/Oe for Y IG.) This value \nis very close to the standard value of µ0Ms for YIG: 0.175 T. \n \n300 Oe |S21| (dB) \n-4 -2 02\n400 Oe |S21| (dB) \n-4 -2 02\n430 Oe |S21| (dB) \n-4 -2 02\n470 Oe |S21| (dB) \n-4 -2 0\n500 Oe \nFrequency (GHz) 2.0 2.5 3.0 3.5 4.0 |S21| (dB) \n-4 -2 02450 Oe |S21| (dB) \n-4 -2 0\n \nFig. 2. Representative |S21| vs. microwave frequenc y traces taken with the vector network analyser. \nThe numbers in the panels are the strengths of the applied static magnetic fields applied to the YIG \nfilm to take the measurements. \n \n \nThis suggests that this extra mode is the ferromagn etic resonance mode ('magnon mode') of \nthe YIG film excited directly by the Oersted field of the feeding microstrip line. Because of this \norigin, it is decoupled from the other two modes. F or this reason its frequency vs. applied field \ndependence is a smooth monotonic function. From Fig . 3(b) one also sees that the four other modes \nconverge with this extra line either at higher or l ower frequencies. These four modes clearly separate into two pairs. Each of the pairs contains a mode located on the lower–field side from the \nmagnon mode and a mode located at the higher–field one. Importantly, far away from the “anti–\ncrossing” with the dispersion line the frequencies are almost the same within each pair and the line \nslope is practically vanishing. This identifies the horizontal sections of these four lines as uncoupl ed \nSRR resonances ('photon modes'). The sections of th ese lines with significant slopes (close to the \nanti–crossing area) are SRR resonances coupled to m agnon modes of the YIG film. \n(a) \nApplied field (Oe) 0 200 400 600 800 Frequency (GHz) \n2.0 2.5 3.0 3.5 4.0 \n \n(b) \nApplied field (kOe) 1 2 3 4 5Frequency (GHz) \n46810 12 14 16 \n \nFig. 3. (a) Frequencies of the peaks from Fig. 1 as functions of the applied field. Dots – experiment , \ndashed lines – fits with Eq. (2). (b) Results of a measurement when a YIG film covers both an SRR \nand a section of the microstrip line. Dots – experi ment, dashed line – fit with the Kittel formula (1) \nfor FMR frequency of an in–plane magnetised film. T he solid lines in both panels are the guides for \nthe eye. \n \nThe strong anti–crossing between the photon and mag non modes seen in Fig. 3 suggests a \nstrong coupling between them. Figure 4 displays the results we obtained on the sample from Fig. 1 \nin a broad range of frequencies and applied fields. Two SRR modes are visible in it: at 3.2 GHz and \n6.6 GHz. [the lower one is the same as in Fig. 3(a) ]. Both modes strongly interact with the YIG \nmagnon mode. \n The dashed lines in Fig. 3(a) are the fits of the experimental data for the lower–frequency \nsection of Fig. 4 with the model of two coupled res onators (see, e.g., Ref. [ 22 ]) \n 20 0 0 0 \n2 1 2 1 2 \n1(2) 2 2 f f f f f + − = ± + ∆ , (2) \nwhere 1f and 2fare the frequencies of the coupled resonances, 0\n1f and 0\n2f are the respective \nresonance frequencies in the absence of coupling an d ∆ is the coupling strength (measured in \nfrequency units). By assuming that the frequency 0\n2f is given by the Kittel formula Eq. (1) \n(depends on the applied field) and 0\n1fis independent on the frequency (uncoupled SRR mode ), from \nthe fit we obtain ∆ = 270 MHz or 0\n1/ 9% f∆ = . Similarly, for the higher–frequency anti–crossing \n(located between 5 and 7 GHz) the best fit with Eq. (1) is obtained for ∆ = 450MHz or 6.8%. \n This latter value is significantly larger than the one previously observed for a pulse–laser \ndeposited YIG film (1.3%) but weaker than the one f or a bulk YIG crystal (approximately 20%) \n[23 ]. The SRR geometries and sizes in Fig. 1 and in Re f. [23 ] are essentially different, therefore it is \ndifficult to estimate the contribution of the SRR d esign to the performance of our device–prototype. \nHowever, from the comparison of the three results i t becomes certain that the usage of a much \nthicker YIG film in the present work (25 µm) than in Ref.[ 23 ] (2.4 µm), combined with potentially \nsmaller intrinsic magnetic losses for the film in o ur case delivers a significant contribution to the \nimprovement of the device performance observed in t he present work. Indeed, it is known that \ncoupling of magnon dynamics in ferromagnetic films to microwave fields of striplines scales with \nthe film thickness [ 37 ] and that the amplitude of any resonance driven by an external source is \ninversely proportional to the intrinsic losses in t he resonating medium. Although the magnetic loss \nparameter for the YIG film from [ 23 ] is not specified in the paper, it is natural to s uppose that the \nfilm from [ 23 ] is characterised by an at least half an order of magnitude larger loss parameter than \nour YIG film, because the former was grown with pul se laser deposition and the latter with liquid–\nphase epitaxy (LPE). It is known that LPE is the on ly method able to produce extremely low–loss \nfilms [ 38 ]. The strength of coupling of 20% observed in [ 23 ] for the case of the bulk YIG crystal is \nexplained in a similar way. It is due to the very l arge thickness of this material potentially combine d \nwith very low magnetic losses, if the used YIG slab represents a single–crystal material. \n In Ref. [ 23 ] a strong positive peak of |S21| was observed betw een the two usual negative \npeaks of resonance absorption for the maximum of an ti–crossing of the two resonances (middle line \nin Fig. 3 in Ref. [ 23 ]). It was claimed that this peak could be explaine d as due to a negative \nrefraction index for the YIG+SRR meta–material in t hese conditions. Interestingly, we do not \nobserve this behaviour in our Fig. 2. Instead, for H=450 Oe we see strong reduction in the peak \namplitudes. It reaches 3dB with respect to the peak shown in the panel for 300 Oe. (Note that our \nsimulations in Sect. 3 do not reproduce this extra peak either.) \nThe result of the time–domain measurements is shown in Fig. 5(a). The main observation in \nthis figure is the damped high–frequency oscillatio ns on top of the rectangular pulse. The shape and \nthe amplitude of this oscillatory pattern strongly depend on the applied field. The results of the \nFourier analysis of these time sequences are consis tent with the frequency–resolved measurements \nin Fig. 4 and Fig. 2. In particular, for 430 Oe one observes strong reduction in the amplitude of the \nhigh–frequency oscillation and noticeable change in the regularity of oscillations. Also, for 900 Oe \nand 1300 Oe the dominant period of the temporal seq uence is practically the same and practically \nequal to the one for 0 Oe. This fact can be explain ed as the dominant contribution to the total \noscillation amplitude originating from the lowest S RR resonance (at 3 GHz for this field). This is \nexpected since the Fourier component of the frequen cy spectrum of the rectangular excitation pulse of duration τ is given by the simple expression F(f)=sin( πfτ)/( πfτ). For τ= 10 ns, \nF(3.0 GHz)/ F(6.6 GHz) = 15. Hence, the oscillations at 6.6 GHz may be excited by a rectangular \npulse less efficiently than at 3.0 GHz by an order of magnitude or so. For all these reasons, the \nchange in the oscillation pattern between 0 Oe and 430 Oe is very noticeable and may be explained \nas the beat of two damped resonance modes from the respective panel in Fig. 2. On the contrary, the \nrespective beat pattern is practically invisible in the time trace for 1400 Oe. This is because the \ndominating contribution to this pattern is delivere d by the SRR resonance at 3 GHz. However, the \nFourier analysis in Fig. 5(b) reveals the presence of this beat pattern. \n \nFig. 4. Grey–scale plot of the linear magnitude of the real part of S21 (Re(S21)) as a function of the \napplied field and microwave frequency obtained usin g VNA–FMR. \n \n \n \n3. Numerical results and discussion \n \n We have conducted rigorous numerical three–dimensi onal finite–difference time–domain \n(FDTD) simulations in order to understand electrody namic properties of the SRR. The FDTD \nmethod is a very well–known time–domain Maxwell's e quations solver, which allows simulating of \nopen–space problems using so–called absorbing bound ary conditions [ 39 ]. Conceptually, this \nnumerical method is a counterpart of the time–domai n PIMM technique. In brief, we simulate the \npropagation of a short pulse of microwave current t hrough the microstrip line coupled to the SRR. \nFor the sake of clarity, in the first approximation we consider the SRR without the YIG film. This \nsimulation is repeated for an isolated microstrip l ine (the SRR is absent). Similar to PIMM, the \nsimulated time–domain traces are Fourier–transforme d to obtain transmission characteristics. The \ntransmission characteristic of the isolated microst rip line is used for the normalisation, which removes a standing wave pattern originating from ar tificial reflections at the ends of the microstrip \nline, which are unavoidable in our numerical model. \n The numerical results are shown in Fig. 6. We see that the FDTD simulation qualitatively \nreproduces the experimental picture. The low– and t he high–frequency resonances of the SRR are \neasily identified [Fig. 6(a)]. The inspection of th e magnetic field profiles of these resonances \n[Fig. 6(b–e)] reveals that the microwave magnetic f ield of SRR is strongly localised in close \nproximity to the split ring; it drops sharply with distance from the ring edges. From this point of \nview the double–SRR structure from Ref. [ 23 ] may be more advantageous since it ensures more \ncomplete filling of the area inside the SRR with th e microwave magnetic field. Figure 6(f) also \nconfirms the result in Ref. [ 23 ] that the profile of the microwave magnetic field for the low–\nfrequency mode is anti–symmetric but the one for th e high–frequency mode is symmetric with \nrespect to an imaginary horizontal axis running thr ough the middle of the ring gap in Fig. 1. In the \nfollowing we will call these SRR modes as 'symmetri c' and 'anti–symmetric', respectively. \n \n \n \nFig. 5 (a) Time–domain traces for the selected valu es of the applied field. For the sake of clarity, \neach trace has been offset vertically by –0.2 dB. ( b) Grey–scale plot of the linear magnitude of \nFourier–transformed PIMM traces as a function of th e applied field and microwave frequency. Note \na different grey–scale bar as compared with Fig. 4. We used the 'Window filtering procedure' to \nremove noise from the time traces and thus improve the quality of the Fourier–space data. \n \n \n In our VNA–FMR experiment we noticed that the reso nant frequencies shift when we \nremove the YIG film from the SRR. The YIG film is k ept on top of a significantly thicker GGG \nsubstrate. Due to computation constraints, in our s imulations the GGG substrate cannot be as thick \nas it is in the experiment. Nevertheless, simulatio ns with a significantly thinner GGG substrate (not \nshown) qualitatively reproduce the experimental res ults: the resonance frequencies of the SRR are \nshifted and an additional very strong peak arises a t ~6 GHz [ 40 ]. This peak is probably due to the \nmicrowave resonance of the GGG substrate, and it se ems to be realistic as confirmed by the data in \nFigs. (4) and 5(a). In these figures one sees a wea k resonance at about 5.2 GHz which does not \ninteract with the magnon mode at all. The unrealist ically large amplitude of this peak in our \nsimulation seems to be an artefact, possibly due to constraints of the numerical model (which \nassumes much smaller GGG thickness, etc.). \n A separate numerical simulation was run to qualita tively explain the origin of the mode \nanti–crossing. It was based on a completely differe nt approach. As has been mentioned in Ref. [ 23 ], \nthe magnetic dynamics excited by microwave magnetic field of the SRR is actually not a genuine \nferromagnetic resonance, i.e. the dynamics does not take the form of magnetisation precession \nwhose amplitude is (quasi) uniform and the phase is constant over the film area covered by the \nSRR. The excitations driven by the microwave field of the SRR are actually travelling spin waves. \nTwo types of spin waves are relevant for our geomet ry. Both exist in in–plane magnetised \nferromagnetic films [ 41 ]. The Damon–Eshbach (DE) wave propagates in the fi lm plane at a right \nangle to the applied magnetic field H. It is characterised by a positive dispersion (gro up velocity) – \ndω/d k > 0, where k is the wave number. The backward volume magnetostat ic spin wave (BVMSW) \nis characterised by a negative dispersion d ω/d k < 0. Its wave vector is parallel to the applied field . \nThe frequency (energy) gaps ω(k=0) for both types of waves are the same and are gi ven by the \nKittel formula Eq. (1). \n Microwave Oersted fields of microstrip lines are a ble to efficiently excite spin waves in the \nwave number range \n \n−2π/s> (γ ∆ Η)/(2π ), where \n∆Η= 0.5 Oe, is the standard magnetic loss parameter for LPE YI G films. The latter parameter \ndetermines the intrinsic resonance linewidth for FM R. (Basically, (γ ∆ Η)/(2π ) is the frequency–\nresolved linewidth and ∆Η is the field–resolved one). The fulfilment of the condition \nδf >> (γ ∆ Η)/(2π ) formally implies that the travelling–spin–wave con tribution to the FMR linewidth \nis significant [ 45 ]. This happens because s is significantly smaller that the free propagation path for \nspin–waves in YIG films. (For our 25 µm–thick LPE-grown film the latter amounts to severa l \nmillimetres.) \n Indeed, in Fig. 8(a) one clearly sees travelling s pin waves propagating away from the two \nmicrostrip lines, placed at a distance a from each other. The existence of a travelling spi n wave is \nevidenced by a linear spatial dependence of the amp litude of dynamic magnetisation seen in this \nfigure for | z| > 5 mm or so, i.e. outside the SRR perimeter. Thi s linear dependence on the \nlogarithmic scale corresponds to an exponential dec ay of the dynamic magnetisation amplitude my \non the linear scale. The exponential decay of ampli tude is characteristic to a travelling wave in any \nmedium with non–negligible losses. \nOne notices a large difference in the wave amplitud es propagating in the positive and the \nnegative direction of the z–axis: my(z<−5mm) > my(z>+ 5mm). This is due to the strong non–\nreciprocity of the DE wave excitation by microstrip transducers (see, e.g., Ref. [ 42 ]). \nBecause the free propagation path of spin waves in YIG is very large, the partial wave \nexcited by the right–hand–side microstrip in Fig. 8 (a) reaches the left–hand–side microstrip. This \nleads to interference of the two partial waves exci ted by the two parallel sides of the SRR. The \ninterference forms an oscillatory amplitude pattern characteristic to a partial standing wave in the \nspace between the two microstrips (e.g., inside the SRR). This standing–wave pattern is very pronounced in Fig. 8(a) for −4.25 mm = −a/2 < z < +a/2. A similar oscillatory pattern is seen in \nFig. 8(a) for the radiation impedance and is formed because the DE wave number varies with \nfrequency according to the dispersion law for the D E wave [ 41 ]. As a result, the phase φ= ka \naccumulated by the wave after crossing the distance a between the two sides of the SRR is a \nfunction of frequency. The maxima and the minima of the 'interference pattern' in Fig. 7(a) \ncorrespond to φ equal to either an even or odd multiple of 2 π. \n Figure 8(b) demonstrates an equivalent circuit for the fundamental mode of SRR resonance \nloaded by the magnon mode in YIG. Because of the mu ch smaller length of the SRR perimeter with \nthe respect to the wavelength of the electromagneti c wave in the SRR substrate, one can consider \nthe SRR as a series LC contour with lumped L and C. Physically, the capacitance C is concentrated \nin the gap. Although the inductance L is actually distributed along the perimeter of the SRR we can \nalso consider it as a lumped one for the same reaso n of the smallness of the perimeter with respect \nto the microwave wavelength at 3 GHz. The losses in this contour are due to ohmic losses in the \nSRR’s metal; in our model they are accounted for by an equivalent resistance R0. Zr is an equivalent \nlumped element whose complex impedance is given by the plots in Fig. 7(a). \nThe dashed line in Fig. 7(b) shows the reactance X0 of this contour for H = 0. As stated \nabove, for this field Zr = 0, hence \n \nX0 = −1/( ωC)+ ωL, (4) \n \nwhere ω= 2πf. The resonance frequency f10 of the unloaded SRR follows from the condition X0 = 0. \nIt is given by the well–known formula \n \n0\n11/ f LC = . (5). \n \nX0 = 0 corresponds to a maximum of the real part of t he complex conductance Y 0 \n \n0 0 0 1/ ( ) Y R iX = + . (6). \n \nIn Fig. 7(b) one clearly sees a sharp resonance pea k of Re( Y0) at f10 = 3.06GHz. \n For H = 485 Oe one has to add Zr in series to 0 0 R iX + . From Fig. 7(a) one sees that there are \ntwo frequencies for which Im( Zr) = − X 0. The existence of the two compensation points is p ossible \nbecause the SRR resonance represents an 'anti–reson ance' (given by a maximum of the complex \nconductance ) [ 46 ], but the spin wave excitation represents a 'reson ance' (given by a maximum of \nthe complex impedance ). \nThe resonance condition for the whole sequence of t he equivalent elements connected in \nseries follows from X0+Im( Zr) = 0 or, equivalently, it is given by the frequenc y position of the \nmaximum of the real part of the total complex condu ctance \n \n0 0 1/ ( ) r Y R iX Z = + + . (7) \n \nAccordingly, the presence of the two frequencies fo r which Im( Zr) = −X0 results in two resonances \nfor the coupled system, seen as the anti–crossing o f f1 and f2 [in the notations of Eq. (1)] in the field \nand frequency resolved data in Figs. 3 and 4. The two peaks for H=485 Oe in Fig. 7(b) are the result of calculation using Eq. (7). In this \ncalculation we assume that the value of C is given. Then the value of L is obtained from Eq. (5) by \nsetting f10 to the frequency corresponding to the experimental SRR frequency for H = 0 (Step 1). \nThe extracted value of L allows us to determine R0 as \n \nR0 = ωL/Q, (8) \n \nwhere Q is the experimental quality factor for the SRR res onance for H = 0 (Step 2). The last step \nof the calculation (Step 3) is to determine the two frequencies for which X0+Im( Zr) = 0. An \nadditional step of the calculation is converting th e results for the 'internal' dynamics of the SRR in to \nthe signal from the output of the feeding microstri p line. To include the coupling of the loaded SRR \nto the feeding microstrip line into our model, we i ntroduce an equivalent lumped mutual inductance \nM into the equivalent circuit in Fig. 8(b). It can b e then shown that \n \n S21 = 1 −KY, (9) \n \nwhere K is a coefficient – 'SRR to microstrip coupling coe fficient' – which depends on M. \n To obtain the traces of Re( Y) shown in Fig. 7(b) we fit the experimental data w ith Eqs. (4–8) \nusing C as a single fit parameter. To this end, we iterati vely perform Steps 1–3 for different values \nof C until the frequency difference between the positio ns of the two peaks, shown by the solid line \nin Fig. 7(b), becomes equal to the experimental one from the panel for 470 Oe in Fig. 2: 592 MHz. \nThe Zr(f) profile is kept the same throughout the fitting p rocedure; we use the one shown in \nFig. 7(a). We choose the particular value of the ex perimental field – 470 Oe – because from Fig 3(a) \none sees that for this field value the frequency se paration of the two coupled resonances is a \nminimum. According to Eq. (1) this implies that for this field f20(H) = f10 = 3.106 GHz and hence \nthe results of the calculation in Fig. 7(a) corresp ond to this particular experimental field value. \nAs an initial (guess) value for C we employ a value which we extracted from the FDTD \nsimulation of the microwave electric field inside t he gap: 4.9 pF. Once the value of C has been \nextracted from the fit, the value of K may be found by equating Re(S21) in Eq. (9) to the maximum \nof the negative peak of the experimentally measured Re(S21) for H = 0 [Fig. 7(d)]. \nThe best fit is obtained for C=2.0 pF which is of the same order of magnitude as the above \nmentioned guess value. The corresponding values of L and R0 are 1.01 nH and 0.71 Ohm \nrespectively. We believe that both quantities are q uite reasonable. The respective plots of Re(S21) \nare shown in Fig. 7(c). These data correspond to K = 0.23. \nThe main observation from Fig. 7(c) is in good qual itative agreement with the experimental \ndata in Fig. 7(d). Furthermore, in both Fig. 7(c) a nd 7(d) one sees a fine structure (small amplitude \noscillations) on top of the higher–frequency peak. Comparison of Figs. 7(a) and 7(c) reveals that \nthis fine structure is due to the spin–wave interfe rence (see above). \nOne more observation is that the linewidth of the l ower–frequency coupled resonance is \nsmaller than that of the uncoupled one, both in exp eriment and theory. The decrease in the peak \nlinewidths, due to coupling to the magnon system, m ay be explained based on the observation that \nthe slope of the Re( Zr) vs. f dependence, near the lower–frequency compensation point, is of the \nsame sign as of X0(f). Therefore Im{ Zr(f)}+ X0(f) is a steeper function of f than X0(f). This leads to \nquicker detuning from the resonance with f than for the uncoupled resonance and hence a quick er \ndrop in the amplitude with the frequency. This idea is in agreement with the much larger expe rimental and theoretical resonance \nlinewidths for the higher–frequency coupled resonan ce [Figs. 7(c) and 7(d)]. Indeed, if one neglects \nthe fast oscillations of Im( Zr) [Fig. 7(a)] and follows the local mean value of I m{ Zr(f)}, then one \nfinds that at the frequencies slightly below this r esonance Im{ Zr(f)} first becomes flatter and then \nthe slope changes from positive to negative. This c hange in the slope results in a long tail of the \nhigher–frequency coupled resonance peak, spanning o ver the range of 400 MHz towards smaller \nfrequencies. This tail is very visible in Figs. 7(c ) and 7(d); it significantly broadens the upper–\nfrequency peak with respect to the lower–frequency one. \nTwo small discrepancies between the theory and the experiment become clear from the \ncomparison of Fig. 7(c) and 7(d). The first one is a noticeable shift upwards in frequency of the \ntheoretical resonance pair for H = 485 Oe with respect to H = 0. Indeed, the experimental pair is \nlocated more symmetrically with respect to the H = 0 data than the theoretical one. We believe that \nthis may be due to the fact that in our calculation we neglected the BVMSW contribution to Zr. \nInclusion of the excitation of BVMSW would modify t he lower frequency slope of the peak of \nRe( Zr) by making it less steep. Accordingly, the lower–f requency slope of Im( Zr) would become \nless steep and non–vanishing values of Im( Zr) would extend to smaller frequencies. This would \npotentially move the lower–frequency peak for H = 485 Oe in Fig. 7(c) to lower frequencies, thus \nmaking the location of the two peaks more symmetric with respect to the H = 0 peak. \nThe other noticeable discrepancy is the peak amplit udes. The experimental data in Fig. 7(d) \ndemonstrate a larger reduction in the peak amplitud es for H = 470 Oe with respect to the H = 0 case \nthan the theoretical ones. This difference may sugg est that K is dependent on H. Indeed, the YIG \nfilm covers the SRR but does not cover the feeding microstrip line. Therefore one may expect that \nthe mutual inductance M is a function of H. With the approach of the magnon–mode resonance, t he \nmicrowave magnetic permeability of the YIG film [ 47 ] grows. This affects the microwave fields of \nthe SRR. However, the microwave magnetic permeabili ty in the vicinity of the feeding microstrip \nremains the same, since this part is not covered by the film. This applied–field dependent jump in \nthe spatial profile of the permeability may make th e strength inductive coupling of the microstrip to \nthe SRR magnetic–field dependent. \n \n \n4. Conclusions \n \nBy using both the frequency–domain VNA–FMR and time –domain PIMM spectroscopy \ntechniques, we have demonstrated a strong coupling regime of magnons to microwave photons in \nthe planar geometry of a lithographically formed sp lit–ring resonator loaded by a single–crystal \nepitaxial YIG film. Whereas in the VNA–FMR experime nt this interaction manifests itself as a \nstrong anti–crossing between the photon and magnon mode, the time–domain PIMM traces exhibit \na signature of a strong beat effect. \n We have conducted numerical simulations of the mic rowave field structure of the SRR and \nof the magnetisation dynamics driven by the microwa ve currents in the SRR and suggested an \nequivalent circuit of an SRR loaded by a magnetic f ilm. These calculations are in very good \nagreement with the experiment and reveal the physic al origins of the effect of anti–crossing. \nOur results are important for the progress of micro wave quantum photonic devices such as, \ne.g., generators of entangled microwave radiation [ 48] required for quantum teleportation, quantum \ncommunication, or quantum radar with continuous var iables at microwave frequencies [49]. \nMoreover, our findings are of immediate relevance t o the development of nonlinear metamaterials [50] and magnetically tuneable metamaterials [23] e xploiting the strong coupling of magnons to \nmicrowave photons. \n \nFig. 6 Results of 3D FDTD simulations. The YIG film and its GGG substrate are not taken into \naccount. (a) Spectrum of the SRR excited by the mic rostrip line. (b–e) Intensity profiles of the out–\nof–plane ( hout = |hy|) and in–plane [ hin = (| hx|2 + |hz|2)1/2 ] fields. The false colour is slightly \noversaturated for the sake of illustration. (f, g) Re( hz)–field and Re( hx)–field profiles across the z–\ndirection and x–direction, respectively, for the resonance frequen cies 3.7 GHz (solid line) and \n7.2 GHz (dashed line). These results, which are com plementary to those in panels (b–e), show the \nphase relationship for the in–plane field component s. \n \nFrequency (GHz) Reactance (Ohm) \n-50 -40 -30 -20 -10 010 20 30 \nResistance (Ohm) \n0246810 12 14 16 18 20 \n(b) \nFrequency (GHz) Conductance Re( Y) (Sm) \n0.0 0.2 0.4 0.6 0.8 1.0 1.2 (a) Re(S21) (dB) \n-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 \n(c) \n(d) \nFrequency (GHz) 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 Re(S21) (dB) \n-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 \n \nFig. 7. (a) Complex radiation impedance Zr of two 0.5 mm–wide and 8.5 mm long parallel \nmicrostrip lines (solid lines). Left–hand axis: Im( Zr); right–hand axis: Re( Zr). The distance between \nthe longitudinal axes of the microstrips is 8 mm. T he static magnetic field is applied along the \nmicrostrips and is 485 Oe. Film saturation magnetis ation 4 πMs = 2000 G. This geometry mimics \nexcitation of the Damon–Eshbach spin waves by the t wo sides of SRR which are parallel to the x–\naxis in Fig. 1. Dashed line: reactance for the unco upled SRR (Im( X0)). (b) Calculated real part of \nconductance Re( Y). Solid line: H = 485 Oe; dashed line: H = 0. (In the latter case it is assumed that \nZr(H=0) = 0). (c) Simulated signal from the output of t he feeding microstrip line Re(S21). Solid \nline: H=485 Oe; dashed line: H = 0. (d) Experimental data for Re(S21). Solid line: H = 470 Oe; \ndashed line: H = 0. \n(a) \nCo-ordinate z across SRR (mm) -15 -10 -5 0 5 10 15 Magnetization amplitude (dB) -14 -12 -10 -8 -6 -4 -2 0\n \nFig. 8. (a) Calculated profile of the dynamic magne tisation amplitude in the z direction. The SRR \nsides are located at z = ±4 mm. Frequency is 3.242 GHz. 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Zverev(a)\naKazan Federal University, 420008 Kazan, Russia\nbInstitute of Applied Research of Tatarstan Academy of Scien ces, 420111 Kazan, Russia\ncKotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, 125009 Moscow, Russia\ndRussian Quantum Center, Skolkovo, 143025 Moscow, Russia\n(Dated: November 2, 2018)\nThe conventional magnonBose-Einstein condensation (BECo f magnons with /vectork= 0) characterized\nby a macroscopic occupation of the lowest-energy state of ex cited magnons. It was observed first\nin antiferromagnetic superfluid states of3He. Here we report on the discovery of a very similar\nmagnon BEC in ferrimagnetic film at room temperature. The exp eriments were performed in\nYttrium Iron Garnet (YIG) films at a magnetic field oriented pe rpendicular to the film. The\nhigh-density quasi-equilibrium state of excited magnon wa s created by methods of pulse and/or\nContinuous Waves (CW) magnetic resonance. We have observed a Long Lived Induction Decay\nSignals (LLIDS),well knownas asignature ofspin superfluid ity. Wedemonstrate thattheBEC state\nmay maintain permanently by continuous replenishment of ma gnons with a small radiofrequency\n(RF) field. Our finding opens the way for development of potent ial supermagnonic applications at\nambient conditions.\nPACS numbers: 67.57.Fg, 05.30.Jp, 11.00.Lm\nKeywords: Supermagnonics, spin supercurrent, magnon BEC, YIG, time crystal\nThe superfluid current of spins – spin supercurrent – is\none more representative of superfluid currents in the sys-\ntems, where the symmetry under spin rotations is spon-\ntaneously broken. It carries spin without dissipation in\nthe ordered magnets, such as solid ferro and antiferro-\nmagnets, and spin-triplet superfluid3He. It governed\nby spatial gradients of angles of rotation of spin system.\nThis type of spin transport in magnetically ordered sys-\ntems have been discussed for a long time [1, 2]. It man-\nifests itself by a formation of spin waves oscillations and\ntopological defects [3].\nIn 1984 the new state of matter has been discov-\nered in antiferromagnetic superfluid3He – the sponta-\nneously self-organized phase-coherent precession of spins\n[4, 5]. This state is radically different from the con-\nventional ordered states in magnets. It is the quasi-\nequilibrium state, which emerges on the background of\ntheorderedmagneticstate, andwhichcanbe represented\nin terms of the Bose condensation of magnetic excita-\ntions – magnons [6]. It corresponds to a macroscopic\noccupation of non-equilibrium magnons on the lowest-\nenergy state for given density of magnons. These phe-\nnomena for non-equilibrium magnons was first discussed\nin [7]. Owing the coherence the magnon BEC radiates a\nvery long living induction decay signal. It may be con-\nsidered as time crystals [8] with a very long, but finite\nlifetime. It may reach minutes in antiferromagnetic su-\nperfluid3He-B. Furthermore, the Goldstone modes - the\ntime-space excitations of the time crystal (the analog of\nsecond sound in superfluid4He) have been observed in\nmagnonBECs[9,10]. ThelifetimeofmagnonBECstates\nmay be infinite in the case, when the losses (evaporation)\nof quasiparticles are replenished by an excitation of newquasiparticles. The very interesting unconventional BEC\nstate of propagating spin waves was reliably observed in\n[11–13]. Owing the interference of spin waves with the\nopposite wave vectorsit shows the properties of quantum\ncrystal [14].\nThe magnonBECopened the new classofthe systems,\nthe Bose-Einstein condensates of quasiparticles, whose\nnumber is not conserved. Representatives of this class\nin addition to BEC of magnons are the BEC of phonons\n[15], excitons [16], exciton-polaritons [17], photons [18],\nrotons [19] and other bosonic quasiparticles.\nThe similar phenomenon for stable particles - the\natomic BEC in diluted gases - was found in 1995 [20].\nAccording Einstein [21] the critical temperature of BEC\nformationTBECat given density of atoms NCreads:\nTBEC≃3.31/planckover2pi12\nkBm(NC)2/3, (1)\nBelow this temperature the BEC state should forms.\nSince magnonsobey the Bose statistics, they should form\nthe magnons BEC state at about the same ratio between\nthe temperature and density.\nThe magnon BEC is a quantum phenomenon, which\ncannot be described by a mean field approximation. For\na proper description of magnon BEC we should use the\nquantum presentation, described by Holstein-Primakoff\ntransformation, which expresses the spin operators in\nterms of the operators of creation and annihilation of\nmagnons [22]. The density of magnons ˆNrelates to\nthe deviation of spin ˆSzfrom its equilibrium value S=\nχH0V/γ.\nˆN= ˆa†\n0ˆa0=S −ˆSz\n/planckover2pi1. (2)2\nMagnonsarebosonicquasiparticleswithspin −/planckover2pi1. Thatis\nwhy after pumping of ˆNmagnons into the system the to-\ntal spin projection is reduced by the number of magnons,\nˆSz=S −/planckover2pi1ˆN.\nThedensityofthermalmagnonsdecreaseswithcooling\n(and reach zero at T= 0) and it is always below the crit-\nical density of magnons BEC formation. However, the\ndensity of excited non-equilibrium magnons NMcan be\ndrastically increased up to about Avogadro density by\nmagnetic resonance methods. Owing the two magnons\nscattering the gas of excited magnon constitute a quasi-\nequilibrium state for a short time after excitation, with\na time scale of about few quasiparticles scattering time.\nThe density matrix of this states exhibiting off-diagonal\nlong-range order (ODLRO) and spin superfluidity [23].\nThe critical density of magnons for a BEC formation\nNBECcan be estimated from equation (1). The tem-\nperature of magnons corresponds to phonon subsystem\ntemperature, which determines the value of magnetiza-\ntionManddensityofthermalmagnons. Magnonsshould\nform a BEC state when NM> NBEC, under certain\nconditions, which will be discussed below. The critical\nmagnons concentration NBECfor different magnetic sys-\ntems was calculated in [24, 25]. Particularly for easy\nplane antiferromagnets with wave spectrum\nεk=/radicalBig\nε2\n0+ε2ex(ak)2 (3)\nit reads:\nNBEC≃(kBT)2\n2π2ε0\na3ε3ex, (4)\nwherekBis a Boltzmann constant.\nThe critical magnon density can be reached at dynam-\nical deflection of magnetization on the angle about few\ndegrees. For antiferromagnetic superfluid3He this angle\nis very small, below 1◦[23]. In the case of ferromag-\nnetic resonance in YIG film, we will discuss in this ar-\nticle, the estimation of critical magnon density is more\ncomplicated. Indeed, as it was shown in [25], the critical\ndensity of magnons can be obtained at a magnetization\ndeflection on the angle about only 3◦.\nThe formation of a magnon BEC state was first ob-\nserved in antiferromagnetic superfluid3He-B as a for-\nmation of extremely Long Lived Induction Decay Signal\n(LLIDS) [4, 26]. The LLIDS manifests the condensa-\ntion of magnons in a common wave function in the whole\nsample with a common phase and frequency of preces-\nsion. The LLIDS obeys the entire requirement for BEC\nof quasiparticles, which later was postulated as an re-\nquirementofmagnonBECinwell-knownarticlebySnoke\n[27]. Magnon BEC has one to one analogy with the ex-\nperiments of atomic BEC [28]. Owing the slow magnons\nrelaxation, the number of magnons decreases, but the\nmagnons remains in a coherent state. It is important tonote that the BEC state is the eigen state for given den-\nsity of excited magnons. It was shown experimentally,\nthat the small RF pumping on a frequency of magnon\nBECωBECcan compensate the magnons relaxation. In\nthis case the magnons BEC may maintains permanently\nfor an infinite time [29]. The physics of excited magnons\nBEC states and phenomena of spin superfluidity are well\nestablished during the 30 years of investigations. The re-\nview of this investigations one can found, for example in\n[30, 31] and in the book [32].\nFIG.1: Thefrequencyshiftofmagnetic resonanceatanormal\nandtangential orientation ofmagnetic field(left) andthe s hift\nof resonance magnetic field for 9.26 GHz (right) as function\nof magnetization deflection angle β.\nIn this article we describe the similar experiments, we\nhave performed in a normally magnetized thin film of\nYttrium Iron Garnet (YIG). We have observed a similar\nLLID signal, as well the permanent magnon BEC state,\nstabilized by a small RF pumping. YIG film is character-\nized by a very small Gilbert [33] damping factor αabout\n10−5, the one of the best value for solid magnetic ma-\nterials. That is why the formation of magnon BEC in\nYIG and observation of spin supercurrent, like in3He,\nshould leads to a development of new branch of physics\n- Supermagnonics. However, the observation of a con-\nventional BEC in YIG has some fundamental difficulties\nassociated with a sufficiently large value of ferrimagnetic\nmagnetization. The magnetization of the YIG at room\ntemperature 4 πMSis about 1750 Oe, which is several\norders of magnitude greater than in all magnetic me-\ndia where BEC magnons were previously observed. High\nmagnetization leads to significant inhomogeneities in the\neffective magnetic field within the sample, which requires\na substantially larger magnon super current and spatial\nvariation of magnons density to equalize the phase of\nthe wave function. For this reason, the first experiments\non the observation of BEC in YIG were carried out in\nYIG magnetic films with a plane magnetization and with\n/vectork/negationslash= 0 [14, 34, 35].\nIn this article we describe the first observation of\nmagnon BEC and spin supercurrent in a normally mag-\nnetized YIG film. At these conditions the minimum of3\nenergy corresponds to /vectork= 0 like in atomic BEC and in\nsuperfluid3He. It was shown [36] that at these condi-\ntions the magnetization precession is stable in the field\nabove 2 kOe. The spectrum of the ground mode of mag-\nnetic resonance for a normally magnetized film for a first\napproximation, reads [37]:\nω=γ(H0−4πMScosβ), (5)\nwhereβis an angle of magnetization deflection. The\nfrequency of precession increases with the density of\nmagnons, which corresponds to repulsive interaction be-\ntween magnons.\nIn Fig.(1) is shown the frequency shift of magnetic\nresonance for /vectork= 0 for normal and tangential orienta-\ntion of magnetic field is shown as function of magnetiza-\ntion deflection angle βfor resonance frequency 9.26 GHz\n(left) as well the shift of resonance magnetic field for\n9.26 GHz (right). For the case of normal magnetization\nwe have a classical potential energy trap for magnons\nwith/vectork= 0. Furthermore, the magnetization precession\nis stable against the splitting on the spin waveswith non-\nzero/vectork. There are very nice conditions for magnons BEC\nformation, very similar for one in antiferromagnetic su-\nperfluid3He-A [38].\nPulsed FMR\nWe have used single-crystal YIG films of thickness\nabout 6µm in the shape of a disk with diameter of 0.5\nand 0.3 mm, which have been grown in the (111) crystal-\nlographic plane on a Gadolinium Gallium Garnet (GGG,\nGd3Ga5O12) substrate by liquid-phase epitaxy [39]. In\norderto overcomethe influence ofthe inhomogeneousde-\nmagnetization field, we modified the shape of the edges\nof the sample by chemical etching (see Methods).\nThe external magnetic field was oriented perpendicu-\nlar to the film. The RF field was oriented in plane of\nthe film. Following the procedure, developed for BEC\nformation in3He-A in aerogel [40], we have excited the\novercritical density of magnons by a relatively long RF\npulse atthe frequencyhigherthe resonanceone. We have\nobservedthe typicalLLIDS, whichringsonthe frequency\nof RF pumping and not on the resonance frequency. In\nFig.(2) are shown the signals at room temperature after\nthe RF pulse of about 20 Oe and duration of 400 ns. At\nresonance excitation (Fig.(2) a) the magnon BEC does\nnot form. The length of signal is about 200 ns and has\nthe same length as in the case of a short pulse (with a\ndecay time constant of 50 ns. It corresponds to the in-\nhomogeneity of the resonance line. In the case of a non-\nresonance excitation (Fig.(2) b-d) the signals are drasti-\ncally changes. The signals are characterized by a time\ndecay on an order of magnitude longer (see Fig.(3)). We\nhaveobservedtheLLIDS with time decayconstantabout\nFIG. 2: The spectroscopic records of LLID signal in normally\nmagnetized YIG film at different frequency shift of RF pulse\nfrom the resonance frequency. The lines corresponds to a sig -\nnals, measured with 0◦,90◦,180◦and 240◦phase shift. Point\n(a) corresponds to the end of RF pulse of 400 ns duration and\n(b) - the end of spectrometer dead time.\n0.8 - 1.6µs for different samples. The amplitude of LLID\nsignalsis comparablewith the amplitude ofinitial partof\nsignals. Its field dependence does not correspond to the\nshapeofCWsignal. We haveobservedthe LLIDS signals\nin different samples and at different temperatures from\nroom to 100 K. The formation of LLIDS is very robust.\nAll its properties well correspond to a BEC signals from\nother systems and particularly antiferromagnetic super-\nfluid3He-A. It is well known from previous investigations\nthat the LLIDS signals radiates by a coherent system of\nexcited magnons, which forms during a non-resonance\nexcitation. Its observation confirms the magnon BEC\nformation in the YIG film.4\nFIG. 3: The LLID signals at 11.2 (above) and 22.4 (below)\nMHz frequency shift. The decay time constant for this sample\nis 0.84µs.\nFIG. 4: The amplitude of CW signal of absorption at different\npower of RF pumping at sweep down of magnetic field. The\nenlarged scale is shown in the inset. Here and in the next\nfigures signals marked a - h corresponds to an RF power 80,\n40, 20, 10, 1, 0.4, 0.1 and 0.05 mWt.\nContinuous wave FMR\nWe have performed the detailed investigations of CW\nsignals in normally magnetized YIG film. The RF field\nwas applied in plain of the film and the magnetic field\nsweep down. At a small power we can see the linear\nresponse of the magnetic system (Fig.(4)). It is impor-\ntant to point out, that between the power of excitation\n0.4 mW and 1 mW the behavior of the signal drastically\nchanges. The frequency of FMR signal begins to follow\nto the frequency, corresponds to descended field. This\nchange appears at the region of field at a shift about 2Oe. At further increase of RF power we have observed\nthe region of fields, where FMR frequency follows to the\nfield as large as 90 Oe, which correspond to a frequency\nshift about 240 MHz.\nThe usual explanation of this phenomenon is well\nknown as a ferromagnetic resonance or be-stable reso-\nnance[41]. ThetheoryofFMRbasedonthe propertiesof\nnon-linear oscillator, which frequency increases with ex-\ncitation. There aresupposing that the higher RF power-\nthe biggerangleofdeflection and consequentlythe bigger\nfrequency shift. This model of FMR was tested exper-\nimentally [42]. There was found that the FMR theory\nworks well only in the limit of a small RF power. At rel-\natively big power the amplitude and frequency shift does\nnot follow to this model. Based on our previous investi-\ngations of magnon BEC in other system we may suggest\nthat the observed phenomena are the result of magnons\nBEC formation.\nFIG. 5: The energy dissipated by a magnon spin system at\ndifferent level of exciting power. The energy was calculated as\na product of absorption signal on the amplitude of magnetic\nfield. The enlarged scale is shown in the inset.\nTo test this hypothesis we have calculated the energy,\ndissipated in the sample at different level of excitation.\nFor this purpose we have multiplied the amplitude of ab-\nsorption signal on the RF field of excitation. The results\nare shown in Fig. (5). It is clearly seen that the dis-\nsipated energy does not depend on the RF power and\ndetermines only by a frequency shift and corresponding\nmagnetic field shift! But this is the property of magnon\nBEC, the eigen state of non-equilibrium magnons, which\ndensity correspond to a resonance at a given frequency\nshift. This effect for antiferromagnetic superfluid3He-\nA was investigated in [43]. The analogy is even more\nclear if we will consider the energy loses as function of an\nangle of magnetization deflection, recalculated from the\nfrequency shift. The results are shown in Fig. (6). There\nis the squaredependence ofenergy losses, similar to BEC5\nin antiferromagnetic3He-A [44], which was explained by\na relaxation due to a spin diffusion of a transverse mag-\nnetization. The signals at different excitations fall on a\nuniversal curve independent of the amplitude of the RF-\nfield. Thisdemonstratesthatatlarge βthemagnonBEC\nis self-consistent and is not sensitive to the amplitude of\nRF field; the latter is only needed for compensation of\nlosses.\nFIG. 6: The dissipated energy as a function of angle of deflec-\ntion. The fitting lines correspond to square dependence form\ndeflected magnetization.\nThe properties of magnon BEC\nLet us consider the basic principles of magnon BEC.\nThe atomic BEC state described by a wave function\nψ=/angb∇acketleftˆa0/angb∇acket∇ight=N1/2\n0eiµt+iα. (6)\nwhereN0is the density of particles and µis a chemi-\ncal potential of Bose condensate. The coherent magnons\nsystem described by a very similar impression:\nψ=/angb∇acketleftˆa0/angb∇acket∇ight=N1/2eiωt+iα=/radicalbigg\n2S\n/planckover2pi1sinβ\n2eiωt+iα.(7)\nwhereβis the angle of deflection in the mean field ap-\nproximation. The role of the global chemical potential\nof magnons µis played by the global frequency of the\ncoherent precession ω, i.e.µ≡ω. The frequency of\nlocal precession in a linear approximation (at a small\nexcitation) is determined by the sum of local external\nH0and demagnetization 4 πMSfields. The interaction\nbetween the excited magnons leads to dynamical fre-\nquency shift due to decrease of local demagnetization\nfield, which for the normally magnetized YIG film reads:\nωN= 4πMS(1−cosβ). It corresponds to a repulsive in-\nteraction and, consequently the magnons BEC formationis possible. In the opposite case of attractive interaction\nthe spatial profile of magnons density has a tendency to\nsplits on a spatially inhomogeneous distribution [45].\nThere are two approaches to study the magnons non-\nequilibrium systems: at fixed particle number NMor at\nfixed chemical potential µ. These two approaches cor-\nrespond to two different experimental conditions: the\npulsed and continuous wave (CW) resonance, respec-\ntively. In the first case the RF pulse excite a number\nof non-equilibrium magnons. If the density of magnons\nis higher than NBEC, the magnon BEC state should be\ncreated with the frequency ω0+ωN. But the local fre-\nquency of magnon BEC may be different for different\nparts of the sample due to the inhomogeneity of local\nfrequencyω0r. Owing to its inhomogeneity ∆ ωthe in-\nduction decay signal decreases at the time scale about\n1/∆ω. It was shown in the experiments with magnon\nBEC in3He-B that the spatial inhomogeneity leads to\nformation of a gradient of magnon BEC wave function\nand to a spin supercurrent of magnons [4, 26]. The spin\nsupercurrents redistribute the density of magnons. As\na result the spatial magnetic inhomogeneity is compen-\nsated by a spatial distribution of magnons density. Fi-\nnally the global BEC state appears with the frequency of\nprecessionωBEC=ω0r+ωNr=constthrows the entire\nsample and radiates a LLID signal. This phenomenon is\na magnon analogy of a global coherent states in other\ncoherent liquids, like superfluids and superconductors,\nwhere supercurrents leads to an inhomogeneities of the\nground state.\nThe signals of CW FMR well correspond to a prop-\nerties of magnon BEC. The critical density for magnon\nBEC in YIG film at room temperature was calculated in\n[25].NBEC≃M−Mz=M(1−cosβ) whereβ= 3◦.\nThis angle of deflection leads to a frequency shift of FMR\nof 7.1 MHz and field shift of about 2.5 Oe. These param-\neters are very close to the point of signals transforma-\ntion we have observed in our experiments. The collective\nquantum state is formed at higher density of magnons.\nIt is an eigen state of excited magnons. It does not de-\npendonthe powerofexcited RFfield, startingfromsome\ncritical value. Particularly this effect was investigated in\n3He-A [43, 46].\nThe BEC state formation is usually analyzed by min-\nimum of Gross-Pitaevskii (GP) equations for Ginzburg-\nLandau (GL) free energy. For magnons in normally mag-\nnetized YIG film it reads:\nF=/integraldisplay\nd3r/parenleftbigg|∇Ψ|2\n2mM+(ω0−ω)|Ψ|2+1\n2b|Ψ|4/parenrightbigg\n,(8)\nwhere parameter bis a repulsive magnon interaction,\nb=4πMS\n2S(9)\nAtω>ωo, magnon BEC must be formed with density\n|Ψ|2=ω−ω0\nb. (10)6\nThis corresponds to the following dependence of the fre-\nquency shift on tipping angle βof coherence precession:\nω−ω0=γ4πMS(1−cosβ) (11)\nIf the precession is induced by continuous wave FMR,\none should also add the interaction with the RF field,\nHRF, which is transverse to the applied constant field\nH0. In continuous wave FMR experiments the RF field\nprescribes the frequency of precession, ω=ωRF, and\nthus fixes the chemical potential µ=ω. In the preces-\nsion frame, where both the RF field and the spin Sare\nconstant, the interaction term is\nFRF=−γHRF·S=−γHRFS⊥cos(α−αRF),(12)\nwhereHRFandαRFare the amplitude and the phase of\nthe RF field. In the language of magnon BEC, this term\nsoftly breaks the U(1)-symmetry and serves as a source\nof magnons [47]:\nFRF(ψ) =−1\n2η(ψ+ψ∗), (13)\nThe phasedifference α−αRFis determined bythe energy\nlosses due to magnetic relaxation, which is compensated\nby the pumping of power from the CW RF field:\nW+=ωSHRFsinβsin(α−αRF),(14)\nthe phase difference between the condensate and the\nRF field is automatically adjusted to compensate the\nlosses. If dissipation is small, the phase shift is small,\nα−αRF≪1, and can be neglected. The neglected\n(α−αRF)2term leads to the nonzero mass of the Gold-\nstone boson – quantum of second sound waves (phonon)\nin the magnon superfluid [47]. The signal of magnon\nBEC collapses at the moment, when the RF power is not\nenough for compensating the magnons dissipation. Since\nthe pumping (14) is proportional to sin βsin(α−αRF),\na critical tipping angle βc, at which the pumping can-\nnot compensatethe losses, increaseswith increasing HRF\n(see Fig.(5)).\nIn a perfect homogeneous sample, the collapse occurs\nwhen the phase shifts α−αRFreaches about 90◦. How-\never, in real systems the phase shift α−αRFat the\ncollapse is smaller owing the inhomogeneity of magnons\ndessipation. This indicates that YIG sample is not ho-\nmogeneous but contains some regions with higher dissi-\npation. In this case collapse may start when the local\nvalueα(r)−αRFreaches 90◦within one of such regions.\nCandidates for regions with high dissipation could be the\nregions with high impurities density, or topological de-\nfects.\nDiscussion\nIn the case of a non-coherent precession, like in the\nmodeldescribedin[41]theabsorption-dispersionrelationshould have a form of a circle. Moreover, if local oscilla-\ntors are independent, then after the precession in regions\nwith high dissipation collapses, the precession will con-\ntinue in regions with smaller dissipation. This, however,\ndoes not occur. The collapse of precession in our exper-\niments is very sharp. The sharp feature of collapse and\ntheshapeoftheabsorption-dispersionhistogramindicate\nthe coherence between different parts of the sample: the\nspin supercurrents transfer the deflected magnetization\nbetween the parts of the cell.\nThe rough estimates of the magnitude of the absorp-\ntion and dispersion signals indicate that almost all the\nmagnetization of the film is deflected and precess at the\nfrequency of the RF pumping. It is clearly visible that\nthe signal amplitude is very large and practically has no\nnoise. This behavior is usual for magnon BEC signals\nbecause of their coherence. We have investigated the\ndependence of the magnitude of the transverse magneti-\nzation as a function of the frequency shift (shift of mag-\nnetic field). To do this, we plot in Fig.(7) the magnitude\nof absorbtion signals at the moments of it collapse. At\nthis condition the phase difference α(r)−αRFshould be\nabout 90◦. Thus, the absorption signal supposed to be\nproportional to the magnitude of the transverse magne-\ntization. The theoretical line in Fig.(7) is calculated for\nassumption that BEC state fills up the entire volume of\nthe samplewhereinthemagneticfield islessthan ωRF/γ.\nIn this case the signal should be proportionalto sin βand\nthis volume. In Fig.(8) is shown the distribution of an\neffective field in the sample as it was calculated by a mi-\ncromagnetic simulation. In result the calculated depen-\ndence shows the good agreement with the experimental\npoints. The contributions of the volume and angle of de-\nflection to the amplitude of the signals in shown in insert\nin Fig.(7).\nThe amplitude of the LLID signals turned out to be\nan order of magnitude smaller than should be in the case\nwhen it is formed by the complete magnetization of the\nsample. Their properties correspond to the signals of\nthe so-called Q-ball that was discovered in antiferromag-\nnetic superfluid3He-B [48] and later explained as a sta-\nble droplet of coherently precessed magnetization [49]. It\nforms as a result of BEC instability near the surface of\nthe sample [50, 51] which pushed the BEC to the central\npart of the sample. The Q-balls excited on the frequency\nof RF pulse. The time dependence of Q-ball frequency is\ndetermining by a dynamical frequency shift and the pres-\nsureofthe orderparametertexture. It is likelythat these\ntwo forces well compensate each other in our case. We\nhave registered a very week time dependence of Q-ball\nfrequency. The stable objects, which radiates a signal\non the frequency of RF excitation and not on the reso-\nnance frequency was first observed in a coupling nuclear-\nelectron precession in MnCO 3and CsMnF 3and named\n“Capturedecho signals”[52]. The observationsofsimilar\neffect in YIG films requires a more detailed study of this7\nFIG. 7: The amplitude of absorption signals at the moment of\ncollapse ofprecession (points)andtheoreticalcurvecalc ulated\nfrom the angle of magnetization deflection and the volume of\nthe sample with local frequency below the frequency of RF\npumping. The inset shows the contributions of the angle of\nthe magnetization deflection and the volume of the sample\nwith BEC from the magnetic field shift.\nFIG. 8: The distribution of a local magnetic field from the\ncenter of the sample and the frequency of BEC pumping\n(dashed line). The BEC state filled up all the region, where\nlocal field is below ω/γ.\nphenomenon take place in spin systems with a dynamical\nfrequency shift.\nConclusion\nIn conclusion, we have observed the CW and pulse\nFMR signals, which properties corresponds to a magnon\nBEC states early observed in other non-linear spin sys-\ntemswithrepulsiveinteractionbetweenmagnons. InCWFMR the signals correspond to a formation BEC in the\nvolume of the entire sample. In pulsed FMR the insta-\nbility of homogeneous precession near the edges leads to\nformation of the BEC droplet in the central part of the\nsample, like in superfluid3He-B. The magnon BEC in\nYIGfilmaddstotheothercoherentstatesofmagnonsob-\nserved in antiferromagnetic superfluid states of3He and\nin antiferromagnets with coupled nuclear-electron pre-\ncession [53, 54]. We expect to conduct new experiments\nto observe the spin supercurrent, Josephson phenomena\nand spin vortex in YIG. It would be interesting to search\nsimilar dynamic coherent states of excitations in other\ncondensed matter systems. Owing the exceptionally long\nlifetime ofmagnetic excitationsYIG is used in microwave\nand spintronic devices that can operate at room temper-\nature. It makes YIG as an ideal platform for the devel-\nopment of microwave magnetic technologies, which have\nalready resulted in the creation of the magnon transis-\ntor and the first magnon logic gate [55, 56]. There is\na significant interest to investigate quantum aspects of\nmagnon dynamics. The YIG can be used as the basis for\nnew solid-state quantum measurement and information\nprocessing technologies including cavity-based QED, op-\ntomagnonics, and optomechanics [57]. That is why the\nformationofmagnonBECinYIGandobservationofspin\nsupercurrent, like in3He, should leads to a development\nof new branch of physics - Supermagnonics.\nMETHODS\nThe sample\nAll samples for experiments were prepared from gar-\nnet ferrite films grown by liquid phase epitaxy on GGG\nsubstrates with the (111) orientation. To reduce the ef-\nfect of cubic anisotropy, we used scandium substituted\nLu1.5Y1.5Fe4.4Sc0.6O12ferrite garnet films; the introduc-\ntion of lutetium ions was necessary to match the param-\neters of the substrate and film gratings. It is known that\nthe introduction of scandium ions in such an amount re-\nduces the field of cubic anisotropy by more than an or-\nder ofmagnitude [58]. In addition, the used lutetium and\nscandium ions practically do not contribute to additional\nrelaxation in the YIG, therefore the width of the FMR\nline of the obtained samples did not exceed 1 Oe at a\nfrequency of 5 GHz. Samples were prepared in the form\nof a disk with a diameter of 500 and 300 µm and a thick-\nness of 6 microns. The disk was located on the front side\nof the substrate with a thickness of 500 µm. The disk\nwas made by photolithography to avoid pinning on the\nsurface of the sample was etched in hot phosphoric acid\n[59]. As a result, the edges of the disk had a slope of 45\ndegrees and had a smooth surface.8\nSpectrometry\nThe CW FMR experiments was performed on Varian\nE-12 X-band EPR spectrometer at the room tempera-\nture and the frequency 9.26 GHz. The amplitude and\nthe frequency of magnetic field modulation were 0.05 Oe\nand 100 kHz, respectively. This frequency is much lower\nthe estimated frequency of second sound of magnon BEC\n(Goldstoun mode). That is why we may consider these\nconditions as stationary.\nThe pulsed FMR experiments were performed on\nBruker ELEXSYS E-580 X-band spectrometer at a fre-\nquency about 9.76 GHz. We were able to use the tem-\nperature from a room to 100 K. The results were prac-\ntically the same since the relaxation processes changes\nvery a little in this region of temperature. Indeed at the\ncondition of cooling by a gas stream the stability of tem-\nperature and consequently the resonance frequency was\nmuch better.\nAcknowledgments\nTheauthorswishtothankG.E.Volovik,V.P.Mineev,\nV. Lvov and O. A. Serga for helpful comments. This\nwork was financially supported by the Russian Science\nFoundation (grant RSF 16-12-10359).\n∗Electronic address: Yury.bunkov@neel.cnrs.fr\n[1] Andreev, A. F. & Marchenko, V. I. “Macroscopic theory\nof spin waves”. JETP43,794-803 (1976).\n[2] Dzyaloshinskii, I. E. & Kukharenko V. G. “The phe-\nnomenological theory of magnetic resonance and of\nspin waves in antiferromagnetics”. JETP43,1232-1239\n(1976).\n[3] Mermin, N. D. “The topological theory of defects in or-\ndered media”. Reviews of Modern Physics 51, 591-648\n(1979).\n[4] Borovik-Romanov, A. S., Bunkov, Yu. M., Dmitriev, V.\nV. & Mukharskii, Yu. M. “Long-lived induction signal in\nsuperfluid3He-B”.JETP Lett. 40,1033-1037 (1984).\n[5] Fomin, I. 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Yuan1,2and Yaroslav Blanter1\n1Department of Quantum Nanoscience, Kavli Institute of Nanoscience,\nDelft University of Technology, 2628 CJ Delft, The Netherlands and\n2Institute for Advanced Study in Physics, Zhejiang University, 310027 Hangzhou, China\n(Dated: February 8, 2024)\nSurface plasmons in two-dimensional (2D) electron systems have attracted great attention for their\npromising light-matter applications. However, the excitation of a surface plasmon, in particular,\ntransverse-electric (TE) surface plasmon, remains an outstanding challenge due to the difficulty to\nconserve energy and momentum simultaneously in the normal 2D materials. Here we show that\nthe TE surface plasmons ranging from gigahertz to terahertz regime can be effectively excited and\nmanipulated in a hybrid dielectric, 2D material and magnet structure. The essential physics is that\nthe surface spin wave supplements an additional freedom of surface plasmon excitation and thus\ngreatly enhances the electric field in the 2D medium. Based on widely-used magnetic materials like\nyttrium iron garnet (YIG) and manganese difluoride (MnF 2), we further show that the plasmon\nexcitation manifests itself as a measurable dip in the reflection spectrum of the hybrid system while\nthe dip position and the dip depth can be well controlled by the electric gating on the 2D layer\nand an external magnetic field. Our findings should bridge the fields of low-dimensional physics,\nplasmonics and spintronics and open a novel route to integrate plasmonic and spintronic devices.\nIntroduction.— Plasmons are collective excitations of\nelectronic charge density in metallic structures. In three-\ndimensional (3D) systems, one has to overcome a gap\nof several electronvolts to excite the bulk plasma oscilla-\ntions, which makes it challenging to be manipulated. The\nsituation in two-dimensional (2D) systems is very differ-\nent since the plasmon frequency is usually proportional to√qwith qbeing the propagating wavevector of plasmons\n[1], implying that the excitation energy can be desirably\ntuned far below the optical regime. Another benefit of\nthe a 2D configuration is the electrical tunability of the\nFermi energy and thus of the charge carrier density [2, 3].\nAs a result, surface plasmons in 2D materials, for exam-\nple, graphene, have attracted significant attention with\nthe well-developed fabrication technology of 2D materials\n[2–7]. In particular, transverse magnetic (TM) plasmons\nare broadly studied while transverse electric (TE) plas-\nmons are seldom studied for its restrictive excitation con-\ndition. For usual 2D systems with the parabolic electron\ndispersion, it is widely believed that the TE plasmons\nare not present for their positive imaginary component\nof conductivity, which is well described by the Drude\nmodel [8]. For graphene, it was theoretically proposed\nthat the sign of the imaginary part of the conductivity\nmay reverse near the spectral onset of intraband scatter-\ning to unlock the TE modes [9]. However, the resulting\nTE plasmons locating in infra and terahertz regime are\nyet to be verified.\nOn the other hand, spin waves – collective excitations\nof spins in ordered magnets – can carry information even\nin magnetic insulators, which largely reduces the Joule\nheating problem during information processing [10, 11].\nSpintronic systems are also easily integrated with other\nphysical systems, for example, photonic platforms, qubits\nand phonons, to form hybrid systems for multifunctional\ninformation processing [12, 13]. The frequency of spin\nFIG. 1. Schematic of the modified Otto configuration com-\nposing of a prism, a dielectric layer, a 2DEG layer and a fer-\nromagnetic layer. An incident electromagnetic wave induces\nan evanescent wave in the dielectric layer above a critical in-\ncident angle. The evanescent wave propagates toward + ez\ndirection and excites a surface plasmon in the 2DEG layer as\nwell as surface spin waves in the magnetic layer.\nwaves ranges from gigahertz (GHz) in ferromagnets to\nterahertz (THz) in antiferromagnets [14]. This makes it\npossible to couple them to surface plasmons in 2D mate-\nrials that have a continuous spectrum [15–17]. Hybrid 2D\nmaterials which include magnetic films, which attracted\na lot of interest recently, [18–22] also provide an accessi-\nble platform to investigate the hybrid magnon-plasmon\nexcitation.\nIn this work, we show how the conventional constraint\non a TE surface plasmon can be overcome by the inter-\nplay of surface plasmons and spin waves. In particular,\nwe investigate the wave propagation in a hybrid dielectric\n(DE), 2D electron gas (2DEG) and magnetic insulator\nstructure as shown in Fig. 1. An incident electromag-\nnetic wave first induces a surface wave at the interfacearXiv:2402.04626v1 [cond-mat.mes-hall] 7 Feb 20242\nof media 4-3 and an evanescent wave inside medium 3.\nThe evanescent wave propagates towards the 2DEG layer\nand excites a TE surface plasmon and surface spin waves\nsimultaneously free from the constraint of 2DEG conduc-\ntivity. The frequency of this surface plasmon-magnon po-\nlariton can be well tuned by an external magnetic field\nand falls into the GHz regime for ferromagnets (FM) and\nTHz for antiferromagnets (AFM). For comparison, plas-\nmons are barely generated when a magnet is replaced by\na (non-magnetic) dielectric material. Furthermore, the\nexcitation of the surface plasmon manifests as a sharp\ndip in the reflection spectrum of the layered structure.\nThe depth and position of the dip are tunable by elec-\ntric gating and by external magnetic field. These find-\nings give a state-of-art demonstration of surface-plasmon\nexcitations in hybrid 2D material-magnet structures and\nthey should provide a feasible platform to study the inter-\nplay of magnon spintronics and plasmonics. The reported\nGHz and THz plasmons may find promising applications\nin designing novel plasmonic devices.\nPhysical model and the excitation spectrum. —Let us\nfirst look at the excitation spectrum of the hybrid system\nDE/2DEG/FM(DE) shown in Fig. 1, where the DE and\nFM layers are semi-infinite. The electromagnetic proper-\nties of the hybrid structure should satisfy the Maxwell’s\nequations\n∇ ×E=−∂tB,∇ ×H=∂tD, (1)\nwhere EandHare respectively electric and magnetic\nfields, while D=ϵ0ϵEandB=µ0(H+M) are respec-\ntively the electric displacement and the magnetic induc-\ntance with ϵ0,µ0andϵbeing the vacuum permittivity,\nvacuum permeability and material permittivity, respec-\ntively. After eliminating the electric components, Eqs.\n(1) can be combined to\n(∇2+k2)H− ∇(∇ ·H) +k2M= 0, (2)\nwhere k2=ϵµ0ω2,M=Msmwith Msbeing the satura-\ntion magnetization and mthe normalized magnetization\nvector of the FM layer.\nOn the other hand, the magnetization dynamics in\nthe FM layer is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation [23–25]\n∂tm=−γm×Heff+αm×∂tm. (3)\nThe first and second terms on the right-hand side of Eq.\n(3) describe the precessional and damped motion of the\nmagnetization toward the effective field Heffwith γand\nαbeing the gyromagnetic ratio and the Gilbert damp-\ning parameter, respectively. In general, Heffis a sum\nof the external field He, the dipolar field H, the crys-\ntalline anisotropy field, and the exchange field. It is as-\nsumed that the external field is applied along the x-axis\nHe=Hexand is strong enough to generate a uniformequilibrium state M0=Msex. Then the spin-wave ex-\ncitation above this ground state can be represented as\nM=M0+Myey+Mzezwith My,z≪Msand the\ndynamics of ( My, Mz) is derived by linearizing the LLG\nequation (3) around M0as\n\u0012\nMy\nMz\u0013\n=\u0012\nκ−iν\niν κ\u0013\u0012\nHy\nHz\u0013\n, (4)\nwhere κ= (ωh−iαω)ωm/((ωh−iαω)2−ω2),ν=\nωmω/((ωh−iαω)2−ω2) with ωh=γH,ωm=γMs.\nWithout loss of generality, we have neglected the ex-\nchange field, because it does not contribute significantly\nto the low-energy excitation of spin-waves in the soft\nmagnets like yttrium iron garnet (YIG).\nBy substituting Eq. (4) into the Maxwell’s equations\n(2), we can derive self-contained equations of Hyand\nHz. We consider an incident wave with momentum\nk(i)= (0, k4cosθ, k4sinθ). Then the spins mainly os-\ncillate in the yandzdirections, and the combined LLG\nand Maxwell equations in medium 2 read\n\u0012\n∂zz+k2\n2(1 +κ)−∂yz−ik2\n2ν\n−∂yz+ik2\n2ν ∂ yy+k2\n2(1 +κ)\u0013\u0012\nHy\nHz\u0013\n= 0.(5)\nBy solving Eqs (5), we derive the surface spin-wave\nmode with H2= (0 , H(−)\n2,y, H(−)\n2,z)eik2,yy−κ2z,E2=\n(E(−)\n2,x,0,0)eik2,yy−κ2z,H(−)\n2,y=iκ2E(−)\n2,x/(µ0ω) and\nk2,y, κ2, ωare related to each other by the determinantal\nequation. Unless stated otherwise, we always label the\nwavevector and decay exponent in medium ibyki,yand\nκi, and they satisfy k2\ni,y−κ2\ni=ω2/c2ϵi(i= 3,4).\nIn the dielectric medium 3, M= 0, the TE\nwave solution to the Maxwell’s equations reads H3=\n(0, H(+)\n3,y, H(+)\n3,z)eik3,yy+κ3z,E3= (E(−)\n3,x,0,0)eik3,yy+κ3z\nwith the relation H(+)\n3,y=−iκ3E(+)\n3,x/(µ0ω). At the in-\nterface of media 3 and 2, the tangential components of\nelectric field should be continuous while the tangential\ncomponents of magnetic field are connected by the sur-\nface electric currents jx=σE2,xcorresponding to surface\nplasmons excitations in the 2DEG layer, i.e.\nH(+)\n3,y−H(−)\n2,y=σE(−)\n2,x, E(+)\n3,x=E(−)\n2,x, (6)\nwhere we shifted the z= 0 plane to the interface of 2DEG\nfor simplicity and imposed the requirement of in-plane\nmomentum conservation k3,y=k2,y≡q. Nontrivial so-\nlutions of Eqs. (6) appear provided\nr\nq2−ω2ϵ3\nc2+ϵ2ω2\nc2δp−iµ0ωσ= 0, (7)\nwhere δp(p∈ {FM,DE}) depends on the nature of\nmedium 2 such that\nδFM=−i \nk2,yk2\n2,z−k2\n2(1 +κ)\nk2,yk2,z−ivk2\n2−k2,z!\n, (8a)\nδDE=k2\n2/κ2. (8b)3\n(a) (b)\nLight coneH=0.3T\nH=0.4T\n0 2 4 6 805101520\nq(mm-1)ωr/2π(GHz )\nωh+ωm/2\nDE/GRA /DEEF=0.1eV\nEF=0.2eV\nEF=0.8eV\n0.0 0.1 0.2 0.3 0.4 0.505101520\nField (T)ωr/2π(GHz )\nFIG. 2. (a) Dispersion relation of the surface plasmon-\nmagnon polariton. The light cone is bounded by ω=cq/√ϵ4.\nϵ4= 14 , ϵ3= 2, EF= 0.8 eV. The parameters of YIG\nare used with ϵ3= 10.8,Ms= 0.175 T [28]. (b) Resonant\nfrequency of the surface plasmon-magnon polariton as a func-\ntion of external field at different values of the Fermi energy\nin the hybrid structure shown in Fig. 1. θ= 1.1θc. The\nblack line at ω= 0 is the solution to resonant condition (7)\nin DE/GRA/DE structure.\nThis is the first key result of our work. When medium\n2 is a dielectric, the resonance condition is reduced to\nthe familiar form in literature by inserting δDEinto Eq.\n(7) [9]. This condition only has real solutions when σis\npurely imaginary and otherwise has a negative imaginary\ncomponent. Therefore, it cannot be fulfilled for conven-\ntional 2DEG whose conductivity is well described by the\nDrude model [9, 26, 27] i.e. σ=σ0EF/(πΓ−iπℏω) with\nσ0=e2/4ℏ,EFbeing the Fermi energy, and Γ being\nthe relaxation rate of carriers. This implies that the TE\nsurface plasmons cannot be resonantly excited using the\nconventional Otto setup.\nThe situation changes dramatically when medium 2 is\na ferromagnet. By plugging δFMinto Eq. (7), we find\nthat the resonant frequency should satisfy the equation\nr\nq2−ω2ϵ3\nc2+q2−k2\n2ν\nq(1 +κ+ν)+µ0σ0EF\nπℏ= 0.(9)\nFirstly, we recover the frequency of surface magnon mode\nin the magnetostatic limit ( ω≪cq) when EF= 0 as\nωr=ωh+ωm/2 (blue and red dashed lines in Fig. 2(a))\n[29, 30]. As we go beyond this limit, the spectrum can\nbe obtained by numerically solving Eq. (9), with the\nresult shown in Fig. 2(a). Clearly, there is a crossing be-\ntween the light cone and surface plasmon-magnon disper-\nsion, suggesting the possibility to match both momentum\nand frequency between the incident photons and hybrid\nplasmon-magnon modes and thus enabling the plasmon\nexcitations. It is noteworthy that the resonant frequency\ncan be well tuned in the GHz regime by the external field,\nas shown in Fig. 2(b).\nReflection rate. –Now we proceed to demonstrate that\nthe surface plasmons and magnons can be simultane-\nously excited by shining a proper wave on the hy-\nbrid system. The excited surface plasmon will carry\naway electromagnetic energy and reduce the reflec-tion rate of the system, which provides a feasible way\nto detect the excitation of surface plasmons in our\nsetup. Here we consider a s-polarized incident wave\nwith electric field perpendicular to the incident plane\nk(i)= (0 , ky, kz) in medium 4, where ky=k4sinθ\nand kz=k4cosθ, as shown in Fig. 1. To sat-\nisfy the Maxwell’s equations, the magnetic and electric\nfields should read H(i/r)\n4 = (0, H(i/r)\n4,y, H(i/r)\n4,z)ei(kyy±kzz)\nandE(i/r)\n4 = (E(i/r)\n4,x,0,0)ei(kyy±kzz)with H(i/r)\n4,y =\n±E(i/r)\n4,xkz/(µ0ω), H(i/r)\n4,z=−E(i/r)\n4,xky/(µ0ω), where i, r\nlabel the incident and reflected waves, respectively. The\nfinite thickness of medium 3 allows for the coexistence of\nexponential increase and decay modes, i.e.\nH3= (0, H(−)\n3,y, H(−)\n3,z)eik3,yy−κ3z\n+ (0, H(+)\n3,y, H(+)\n3,z)eik3,yy+κ3z,\nE3= (E(−)\n3,x,0,0)eik3,yy−κ3z+ (E(+)\n3,x,0,0)eik3,yy+κ3z,\n(10)\nwhere H(±)\n3,y =∓iE(±)\n3,xκ3/(µ0ω) and H(±)\n3,x =\n−iE(±)\n3,xk3,y/(µ0ω).\nNow the boundary conditions require the continuity of\nthe tangential components of both electric and magnetic\nfields at interfaces of media 4-3 and 3-2, i.e.\nH(i)\n4,y+H(r)\n4,y=H(+)\n3,y+H(−)\n3,y, (11a)\nE(i)\n4,x+E(r)\n4,x=E(+)\n3,x+E(−)\n3,x, (11b)\nH(+)\n3,ye(κ2+κ3)d+H(−)\n3,ye(κ2−κ3)d−σE(−)\n2,x=H(−)\n2,y,(11c)\nE(+)\n3,xe(κ2+κ3)d+E(−)\n3,xe(κ2−κ3)d=E(−)\n2,x. (11d)\nBy expressing all the magnetic fields by their elec-\ntric fields counterparts and solving the resulting lin-\near equations, we can derive the reflection coefficient\nR≡E(r)\n4,x/E(i)\n4,xas [31]\nR=k2\n2(kzsinh(κ3d)−iκ3cosh( κ3d)) +δpc+\nk2\n2(kzsinh(κ3d) +iκ3cosh( κ3d)) +δpc−,(12)\nwhere c±= (∓µ0σω+kz)κ3cosh( κ3d)∓i(κ2\n3±\nkzµ0σω) sinh( κ3d). For very thin dielectric medium 3\n(κ3d→0), the reflection coefficient is simplified as\nR=−ik2\n2+δp(kz−µ0σω)\nik2\n2+δp(kz+µ0σω). (13)\nThis is the second key result of our work. Figure 3(a)\nshows the reflection rate |R|2as a function of the fre-\nquency of incident wave when θ= 1.1θcwith the critical\nangle θc= arcsinp\nϵ2/ϵ4(ϵ4> ϵ3,2). A sharp dip in the\nreflection rate appears at the resonant frequency (vertical\ndashed line), implying a resonant excitation of the sur-\nface plasmon-magnon polariton. As a comparison, the\nreflection rate is approximately one when the magnetic\nlayer is replaced by a normal dielectric with the same4\n(b)\n(c) (d)\nFano -likeLorentz -like\n0.20.30.40.50.60.70.80.050.100.150.20\nEF(eV)Γ(meV )ρ\n0246810(a)\nH=0.3T\nH=0.4T\nH=0.5T\n0.05 0.10 0.15 0.200.0.10.20.3\nEF(eV)|Rm2\nΓ=0.01 meV\nFano -like\nΓ=0.2meV\nLorentz -like\n6 810 12 14 160.50.60.70.80.91.0\nω/2π(GHz )|R2\nDE/GRA /FM\nDE/GRA /DE\nRe(E/E(i))FM\nRe(E/E(i))DE\n3 my2+mz2\n8 10 12 14 160.00.20.40.60.81.0\nω/2π(GHz )|R2\nFIG. 3. (a) Reflection rate of the hybrid system, electric field\nstrength at the FM surface, spin-wave excitation amplitude as\na function of the incident wave frequency. d= 2.5µm, H=\n0.3 T, α= 10−4,Γ = 0 .01 meV , θ= 1.1θc, EF= 0.3 eV. (b)\nFano-like and Lorentz-like reflection spectrum at small relax-\nation rate and large relaxation rate of carriers, respectively.\nThe dashed lines are the results of analytical formula Eq.\n(14). (c) Density plot of the lineshape index ρin the EF−Γ\nplane. The lineshape is Fano-like for ρ≪1 and Lorentz-like\nforρ≫1. The black dashed line is ρ= 1. (d) The minimum\nreflection rate as a function of the Fermi energy at different\nexternal fields. Γ = 0 .01 meV.\npermittivity ϵ2(blue line), indicating very weak plasmon\nexcitations. This comparison explicitly confirms that the\nmagnetic layer releases the constraint to excite the TE\nsurface plasmon. To understand the essential physics, we\nfurther plot the electric field in the 2DEG layer as well\nas the spin-wave amplitude as a function of the wave\nfrequency in Fig. 3(a). When medium 2 is a magnetic\nlayer, the spin-wave is maximally excited at the resonant\nfrequency, which also significantly enhances the electric\nfield in the 2DEG layer and thus strongly excites the sur-\nface plasmon mode. However, there is no enhancement\nof electric fields when medium 2 is a dielectric. Now it\nseems safe to conclude that the surface spin waves boost\nthe surface plasmon excitations, which carry away signifi-\ncant amount of electromagnetic energy and thus generate\na considerable dip in the reflection spectrum.\nLineshape of the reflection spectrum. — We further no-\ntice that the lineshape of the reflection spectrum near the\nresonance is asymmetric. Physically, this may be inter-\npreted as an interference effect between the background\ncontinuum spectrum and a discrete mode. Here the con-\ntinuous mode is the flat reflection spectrum without con-\nsidering the magnetic properties of medium 2 (blue line\nin Fig. 3(a)) while the discrete mode is the hybrid sur-\nface plasmon-magnon mode. Specifically, we can expand\nthe reflection rate (13) around the resonance frequencyand derive that [31]\n|R|2=A0(ω−ω0+λβ)2+η2\n(ω−ω0)2+β2, (14)\nwhere λis the well-known Fano parameter, ω0=ωr−∆ω\nis the modified resonance frequency, βis the effective\nlinewidth, and ηis the strength of the Lorentz contri-\nbution. In general, near the resonance position, we may\ncharacterize the relative weight of the Fano and Lorentz\nlineshapes by defining a lineshape index ρ≡η/(∆ω+λβ)\nas [31]\nρ=\f\f\f\fπkzΓ2−µ0σ0ωEFΓ +πkz(ℏω)2\nµ0σ0EFℏω2\f\f\f\f. (15)\nWhen the relaxation rate of carriers Γ in 2DEG is very\nsmall, the ratio ρ≈πkzℏ/µ0σ0EFis much smaller than\none for higher Fermi energy, then the reflection spectrum\nis Fano-like [32], as shown in Fig. 3(b) (red line). When\nthe relaxation rate Γ is very high, the ratio becomes\nρ≈πkzΓ2/(µ0σ0EFℏω2). In this regime, the Lorentz\ncontribution can be comparable and even dominate the\nFano contribution (orange line in Fig. 3(b)). The com-\nplete phase diagram of ρin the EF−Γ plane is shown in\nFig. 3(c). It is noteworthy that the Fermi energy of the\n2DEG can be tuned by electric gating [33], which makes\nit possible to tune the lineshape and thus the minimum\nreflection rate of the hybrid system. Figure 3(d) shows\nthat the minimum reflection rate |Rm|2can reach zero\nif the Fermi energy and external fields are appropriately\ntuned. In this situation, all the incident wave energy is\nconverted to excite surface plasmons.\nExtension to antiferromagnet. — The essential physics\npresented above can be extended to AFMs. As an exam-\nple, we consider a two-sublattice AFM insulator with the\neasy axis and external field both aligned along the x-axis.\nFollowing the theoretical approach presented above, we\nderive a similar form of reflection coefficient (13) with κ\nandνreplaced by their AFM counterparts [31]. Figure\n4 shows the reflection rate as a function of the incident\nwave frequency. Unlike the ferromagnetic case, two dis-\ntinguished dips appear in the sub-THz regime (red and\nblue lines) depending on the direction of in-plane mo-\nmentum ( q) of incident wave. This difference is because\nthere are two surface spin-wave modes in an AFM prop-\nagating in ±eydirections respectively [36]. The incident\nwave with q > 0 (q < 0) only excites the surface spin\nwave and plasmon propagating in the + ey(−ey) direc-\ntions. Therefore one may generate nonreciprocal surface\nplasmons by properly choosing the wave frequency.\nDiscussions and conclusions. —In conclusion, we have\nshown that surface spin waves in both ferromagnet and\nantiferromagnet can boost the excitation of TE surface\nplasmons ranging from GHz to THz regime in 2D materi-\nals. The excitation condition does not require the purely\nnegative imaginary conductivity and is thus applicable5\nωh/ωspq>0\nq<0ωr/ωspDE/GRA/AFM(q<0)100 my2+mz2DE/GRA/AFM(q>0)\n265 270 275 280 285 2900.00.20.40.60.8\nω/2π(GHz )|R20 0.602\nFIG. 4. (a) Reflection rate and spin-wave excitation am-\nplitude of an AFM as a function of incident frequency in\nan DE/GRA/AFM structure. Here, mrefers to the mag-\nnetization order of an AFM. The inset shows the external\nfield dependence of the frequency of plasmon-magnon mode.\nωsp=γp\nHan(2Hex+Han). Parameters of MnF 2are used\nwith the exchange field Hex= 55 .6 T, anisotropy Han=\n0.88 T, Ms= 0.059 T , ϵ2= 7.645 [34, 35], Γ = 0 .8 meV.\nOther parameters are the same as Fig. 3(a).\nto a wide class of 2D systems. The excitation of surface\nplasmons carries away electromagnetic energy and gen-\nerates a local minimum in the reflection spectrum of the\nsystem. 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Seehra and R. E. Helmick, J. Appl. Phys. 55,\n2330 (1984), ISSN 0021-8979, URL https://doi.org/\n10.1063/1.333652 .\n[36] R. E. Camley, Phys. Rev. Lett. 45, 283 (1980), URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.45.\n283." }, { "title": "2309.05982v3.Anisotropy_assisted_magnon_condensation_in_ferromagnetic_thin_films.pdf", "content": "Anisotropy-assisted magnon condensation in ferromagnetic thin films\nTherese Frostad,1Philipp Pirro,2Alexander A. Serga,2\nBurkard Hillebrands,2Arne Brataas,1and Alireza Qaiumzadeh1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nRheinland-Pf¨ alzische Technische Universit¨ at Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\nWe theoretically demonstrate that adding an easy-axis magnetic anisotropy facilitates magnon\ncondensation in thin yttrium iron garnet (YIG) films. Dipolar interactions in a quasi-equilibrium\nstate stabilize room-temperature magnon condensation in YIG. Even though the out-of-plane easy-\naxis anisotropy generally competes with the dipolar interactions, we show that adding such magnetic\nanisotropy may even assist the generation of the magnon condensate electrically via the spin transfer\ntorque mechanism. We use analytical calculations and micromagnetic simulations to illustrate this\neffect. Our results may explain the recent experiment on Bi-doped YIG and open a pathway toward\napplying current-driven magnon condensation in quantum spintronics.\nIntroduction—. Magnon condensate with nonzero\nmomentum at room temperature is a fascinating phe-\nnomenon first observed in 2006 [1]. Condensed magnons\nwere observed at two degenerate magnon band minima\nof high-quality yttrium iron garnet (YIG), an easy-plane\nferrimagnetic insulator with a very low magnetic dissi-\npation [2–4], as the spontaneous formation of a quasi-\nequilibrium and coherent magnetization dynamics in mo-\nmentum space [5–8]. To generate magnon condensate,\nnonequilibrium magnons must be pumped into the sys-\ntem by an incoherent stimulus such as parametric pump-\ning [1, 9–18], rapid cooling of thermal magnons [19–\n21], and spin-transfer torque [22–29]. Above a critical\nnonequilibrium magnon density, magnons may finally\n(quasi)thermalize to form a quasi-equilibrium magnon\ncondensate at the bottom of magnon bands.\nThe study of magnon condensation is not only inter-\nesting from an academic point of view but also of great\nimportance in various areas of emerging quantum tech-\nnology and applied spintronics [15, 30–33]. Therefore, it\nis crucial to clarify the intricate microscopic mechanisms\nat play and to present theoretical proposals to electrically\ncontrol the generation of magnon condensate.\nAt low magnon densities, the interaction between\nmagnons is weak, and they behave as free quasiparti-\ncles. But when the magnon population increases, as\nin magnon condensation experiments, the interactions\nbetween magnons become stronger and more crucial.\nMoreover, nonlinear magnon interactions facilitate the\nquasi-thermalization process of injected nonequilibrium\nmagnons. A stable and steady quasi-equilibrium magnon\ncondensation requires an effective repulsive interaction\nbetween injected magnon quasiparticles. It is known that\nin a system mainly influenced by Heisenberg exchange\ninteractions, interaction between magnons is attractive.\nHowever, it was shown that dipolar interactions may as-\nsist in the generation of a metastable double-degenerate\nmagnon condensate in YIG [12–14, 34–42].\nRecently, it was theoretically shown that the quasi-\nthermalization time of magnon condensation is reducedin confined nanoscopic systems [43]. It was also demon-\nstrated that the lateral confinement in YIG thin films\nenhances the dipolar interaction along the propagation\ndirection of magnons and causes a deeper band depth,\ni.e., the difference between the ferromagnetic resonance\n(FMR) and magnon band minima. Increasing the life-\ntime of the magnon condensate was attributed to this\nenhancement of the band depth [43].\nIn another recent achievement in magnon condensa-\ntion experiments, Divinsky et al. [22] found evidence of\ncondensation of magnons by spin-transfer torque mech-\nanism. They introduced a small perpendicular magne-\ntocrystalline anisotropy (PMA) through bismuth doping\nin the thin film of YIG, while the magnetic ground state\nstill resides within the plane. This discovery opens a\nroute toward electrical control of magnon condensation.\nHowever, the interplay between the dipolar interac-\ntions, which was previously shown to be essential for\nthe stability and thermalization of magnon condensation,\nand the counteracting out-of-plane easy-axis magnetic\nanisotropy is so far uncharted. In this Letter, we ana-\nlyze the stability of condensate magnons in the presence\nof a PMA in YIG. We present simulations within the\nLandau-Lifshitz-Gilbert framework [44–46] that support\nour analytical calculations.\nModel—. We consider a thin ferromagnetic film in the\ny−zplane to model YIG. The magnetic moments are di-\nrected along the zdirection by an in-plane external mag-\nnetic field of strength H0. The magnetic potential energy\nof the film contains contributions from the isotropic ex-\nchange interaction Hex, Zeeman interaction HZ, dipolar\ninteraction Hdip, and additionally a PMA energy Hanin\nthexdirection, normal to the film plane. YIG has a\nweak in-plane easy-axis that can be neglected compared\nto the other energy scales in the system. The total spin\nHamiltonian of the system reads,\nH=Hex+HZ+Hdip+Han. (1)arXiv:2309.05982v3 [cond-mat.mes-hall] 10 Feb 20242\nThe PMA energy is given by,\nHan=−KanX\nj(Sj·ˆx)2, (2)\nwhere Kan>0 is the easy-axis energy, ℏSjis the vector\nof spin operator at site j, and ℏis the reduced Planck\nconstant. Details of the Hamiltonian can be found in the\nSupplemental Material (SM) [47].\nThe Holstein-Primakoff spin-boson transformation [48]\nallows us to express the spin Hamiltonian in terms of\nmagnon creation and annihilation operators. The ampli-\ntude of the effective spin per unit cell in YIG at room\ntemperature is large S≈14.3≫1, [39, 49, 50], and thus\nwe can expand the spin Hamiltonian in the inverse pow-\ners of the spin S, which is equivalent to the semiclassical\nregime. Up to the lowest order in nonlinear terms, the\nmagnon Hamiltonian Hof a YIG thin film can be ex-\npressed as the sum of two components: H2andH4. The\nformer represents a noninteracting magnon gas compris-\ning quadratic magnon operators. The latter, on the other\nhand, constitutes nonlinear magnon interactions charac-\nterized by quartic magnon operators; see SM for details\n[47]. Note that three-magnon interactions are forbidden\nin our geometry by the conservation laws [51]\nMagnon dispersion of YIG with a finite PMA—. The\nmagnon dispersion in YIG is well known and has been\nstudied extensively in both experimental and theoreti-\ncal works [2, 52, 53]. Magnons traveling in the direc-\ntion of the external magnetic field have the lowest en-\nergy. These so-called backward volume magnetostatic\n(BVM) magnons have a dispersion with double degener-\nate minima at finite wavevectors qz=±Q. When pump-\ning magnons into the thin film, the magnons may ther-\nmalize and eventually form a condensate state in these\ntwo degenerate minima with opposite wavevectors.\nThe noninteracting magnon Hamiltonian and the dis-\npersion of BVM magnons, along the zdirection, in the\npresence of a finite PMA reads,\nH2=X\nqzℏωqzˆc†\nqzˆcqz, (3a)\nℏωqz=q\nA2qz−B2qz, (3b)\nwhere ˆ c†\nqz(ˆcqz) are magnon creation (annihilation) oper-\nators, which are Bogoliubov bosons [47], and\nAqz=Dexq2\nz+γ(H0+ 2πMSfq)−KanS, (4a)\nBqz= 2πMSfq−KanS. (4b)\nHere, Dex=JexSa2is the exchange constant, Jexis the\nHeisenberg exchange coupling and MS=γℏS/a3is the\nsaturation magnetization, where γ= 1.2×10−5eV Oe−1\nis the gyromagnetic ratio, and a= 12 .376˚A is the\nlattice constant of YIG. The form factor fq= (1 −\ne−|qz|Lx)/(|qz|Lx) stems from dipolar interactions in a\nthin magnetic film with thickness Lx[54, 55].\nFIG. 1. The analytical dispersion of noninteracting BVM\nmagnons in a YIG thin film for various PMA strengths, Eq.\n(3b). The inset shows the depth of the magnon band minima\nas a function of the PMA strength. We set the thickness Lx=\n50 nm and the magnetic field in the z direction H0= 1 kOe.\nFigure 1 shows the effect of PMA on the magnon\ndispersion of YIG. PMA decreases the FMR frequency\nωqz=0, in addition to a more significant decrease in the\nmagnon band minima ωqz=±Q. Therefore the band depth\n∆ω=ωqz=0−ωqz=±Qis increased. The position of\nthe band minima at qz=±Qis also shifted to larger\nmomenta. In addition, the curvature of the minima in-\ncreases as a function of the anisotropy strength. Above\na critical PMA, Kc2an, the magnetic ground state is desta-\nbilized and the in-plane magnetic state becomes out-\nof-plane. We are interested in the regime in which\nthe magnetic ground state remains in the plane, and\nthus the effective saturation magnetization is positive\nMeff=MS−2Kan/(µ0MS)>0.\nThe effect of PMA on magnon dispersion resembles the\neffect of confinement in the magnon spectra of YIG. In\nRef. 43, it was shown that transverse confinement in a\nYIG thin film leads to an increase of the FMR frequency,\nthe band depth, as well as shifting the band minima to\nhigher momenta while the magnon band gap at the band\nminima is also increased. It was shown that this change of\nthe spectrum in confined systems increases the magnon\ncondensate lifetime. Therefore, we expect PMA to in-\ncrease the magnon condensate lifetime and assist in the\ngeneration of magnon condensation.\nNonlinear magnon interactions in the presence of\nPMA—. To check the stability of condensate magnons at\nlow energy, we should show that the interactions do not\ndestabilize and destroy magnon condensation. To achieve\nthis goal, we turn on the magnon interaction between\nthe condensate magnons at qz=±Q. This magnon in-\nteraction consists of intra- and inter-band contributions,3\nH4=Hintra\n4+Hinter\n4, where\nHintra\n4=A(ˆc†\nQˆc†\nQˆcQˆcQ+ ˆc†\n−Qˆc†\n−Qˆc−Qˆc−Q), (5a)\nHinter\n4= 2B(ˆc†\nQˆc†\n−QˆcQˆc−Q) +C(ˆc†\nQˆc−QˆcQˆc−Q\nˆc†\n−Qˆc−Qˆc−QˆcQ+ H.c.) + D(ˆc†\nQˆc†\nQˆc†\n−Qˆc†\n−Q+ H.c.) .(5b)\nThe intraband magnon interaction, parametrized by A,\npreserves magnon number. However, the interband\nmagnon interaction includes both a magnon conserving\ncontribution, parametrized by B, and nonconserving con-\ntributions, parametrized by CandD, see SM [47]. The\ninteraction amplitudes are given by\nA=−γπM S\nSN\u0002\n(α1+α3)fQ−2α2(1−f2Q)\u0003\n−DexQ2\n2SN(α1−4α2) +Kan\n2N(α1+α3), (6a)\nB=γ2πMS\nSN\u0002\n(α1−α2)(1−f2Q)−(α1−α3)fQ)\u0003\n+DexQ2\n2SN(α1−2α2) +Kan\nN(α1+α3), (6b)\nC=γπM S\n2SN\u0002\n(3α1+ 3α2+ 4α3)fQ−8\n3α3(1−f2Q)\u0003\n+DexQ2\n3SNα3+Kan\n4N(3α1+ 3α2+ 4α3), (6c)\nD=γπM S\n2SN\u0002\n(3α1+ 3α2+ 4α3)fQ−2α2(1−f2Q)\u0003\n+DexQ2\n2SNα2+Kan\n2N(3α2+α3). (6d)\nHere, Nis the total number of spin sites. The dimen-\nsionless parameters α1,α2, and α3are related to the\nBogoliubov transformation coefficients, listed in SM [47].\nAn off-diagonal long-rage order characterizes the con-\ndensation state. The condensate state is a macroscopic\noccupation of the ground state and can be represented by\na classical complex field. Therefore, to analyze the sta-\nbility of the magnon condensate, we perform Madelung’s\ntransform ˆ c±Q→p\nN±Qeiϕ±Q, in which the macroscopic\ncondensate magnon state is described with a coherent\nphase ϕ±Qand a population number N±Q[39, 40]. The\ntotal number of condensed magnons is Nc=N+Q+N−Q,\nwhile the distribution difference is δ=N+Q−N−Q.Ncis\nset in the system by an external magnon pumping mech-\nanism and is a constant. We also define the total phase\nas Φ = ϕ+Q+ϕ−Q.\nFinally, the macroscopic four-magnon interaction energy\nof condensed magnons is expressed as ,\nV4(δ,Φ) =N2\nc\n2\u0002\nA+B+ 2Ccos Φs\n1−δ2\nN2c\n+Dcos 2Φ −\u0000\nB−A+Dcos 2Φ\u0001δ2\nN2c\u0003\n.(7)\nThis expression is similar to the one recently obtained\nwithout PMA [56], but the interaction amplitudes, Eq.\nFIG. 2. The analytical nonlinear interaction energy of\nmagnon condensate state, Eq. (7), as a function of the PMA\nstrength. NandNcare the total spins and condensate\nmagnons, respectively. Kc1anrepresents the critical value of\nthe PMA at which the sign of nonlinear interaction energy is\nchanged. On the other hand, Kc2ancorresponds to the critical\nvalue of PMA at which the in-plane magnetic ground state\nbecomes unstable. We set Lx= 50 nm and H0= 1 kOe.\nKsim\nan= 0.5µeV denotes the PMA used in our micromagnetic\nsimulations.\n(6), depend on the PMA through the Bogoliubov coeffi-\ncients, see SM [47] .\nWe now look at the total interaction energy and am-\nplitudes of condensate magnons in more detail. Figure\n2 shows the effective interaction potential of condensate\nmagnons as a function of the PMA. In a critical PMA\nstrength, Kc1an, the sign of the interaction changes. This\nmeans that below Kc1an, the interaction reduces the total\nenergy of condensate magnons while above Kc1anthe inter-\naction increases its energy. This critical anisotropy is well\nbelow the critical magnetic anisotropy strength Kc2anthat\ndestabilizes the in-plane magnetic ground state. In the\nfollowing, we consider a PMA strength below the critical\nanisotropy Kan< Kc1an.\nThe interacting potential energy of the condensate\nmagnons, Eq. (7), has five extrema, ∂V4(δi,Φi) = 0,\nat,\nδ1= 0,Φ1= 0; (8a)\nδ2= 0,Φ2=π; (8b)\nδ3= 0,Φ3= cos−1(−C\nD); (8c)\nδ4=Nc\u0002\n1−(C\nB−A+D)2\u00031\n2,Φ4= 0; (8d)\nδ5=δ4,Φ5=π. (8e)\nδi= 0 indicates condensate states with symmetric\nmagnon populations in the two magnon band minima\nwhile δi̸= 0 represents states with nonsymmetrical4\nTABLE I: The material parameters used in the\nmicromagnetic simulations.\nParameter Symbol Value\nSaturation magnetization 4 πMS1.75 kOe\nEffective spin S 14.3\nExchange constant Dex 0.64 eV ˚A2\nGilbert damping parameter α 10−3\n(a)\n(b)\nFIG. 3. The theoretical phase diagram of the condensate\nmagnons in the absence (a) and presence (b) of PMA. We plot\nthe magnon interaction energy V4/N2\nc, Eq. (7), as a function\nof the film thickness Lxand the magnetic field strength H0,\napplied along the zdirection. Different states are labeled\nbased on different extrema listed in Eq. (8). The dashed black\nlines indicate the boundaries between the different condensate\nphases, Eq. (8). We set Kan= 0.5µeV in (b).\nmagnon populations. Whether any of these extrema rep-\nresents the actual minimum of the interacting potential\nenergy, i.e., ∂2V4(δi,Φi)>0, depends on the system pa-\nrameters. Finding these minima allows us to construct\nthe phase diagram for magnon condensate.\nPhase diagram for magnon condensate—. Now, we\ntheoretically explore the (meta)stability of the magnoncondensate as a function of the film thickness Lxand the\nmagnetic field strength H0, applied along the zdirection\nin the plane, using the YIG spin parameters, see Ta-\nble I. We characterize a (meta)stable state of a magnon\ncondensate in the phase diagram as one that minimizes\nthe interaction potential V4while satisfying the condi-\ntionV4<0, ensuring a reduction in the total magnon\ncondensate energy at qz=±Qthrough interactions.\nFirst, we present the phase diagram for magnon con-\ndensation in YIG, in the absence of PMA, in Fig. 3a.\nThe thinner films are expected to have a symmetric dis-\ntribution of magnons between the two magnon band min-\nima, the state with δ1= 0, and only thicker films with\nlarger applied magnetic fields tend to have nonsymmetric\nmagnon populations, the state with δ4̸= 0. This phase\ndiagram is in agreement with previous studies [39, 56].\nNext, we add a PMA, with strength Kan= 0.5µeV,\nand plot the phase diagram of the magnon condensate in\nFig. 3b for different thicknesses. Compared to the case\nwithout PMA, we see that the condensate magnons can\nonly be stabilized for thinner films, and within our mate-\nrial parameters, we do not have any metastable conden-\nsation above 90nm since the sign of the total interaction\nenergy becomes positive. In addition, we have a richer\nphase diagram in the presence of PMA. PMA tends to\npush the magnon condensate within our material param-\neters toward a more nonsymmetric population distribu-\ntion between the two magnon band minima, states with\nδ4̸= 0 and δ5̸= 0. Since both minima are degener-\nate, there is an oscillation of magnon population between\nthese two minima. In very thin films, less than 30 nm, we\nmay have a symmetric condensate magnon state, δ1= 0,\nin our system.\nThis phase diagram shows that in the presence of a\nPMA, magnon condensate can be still survived as a\nmetastable state. In addition, as we discussed earlier,\na PMA increases the band depth and reduces the cur-\nvature of noninteracting magnon dispersion, see Fig. 1,\nleading to an enhancement of the condensate magnon\nlifetime. Thus, we expect that introducing a small PMA\ninto a thin film of YIG facilitates the magnon conden-\nsation process. It is worth mentioning that the stabi-\nlizing condensated magnons in thinner films with finite\nPMA, is not a real problematic issue since the injection\nof magnons into the system by electrical means is more\nefficient in thin films.\nMicromagnetic simulation of magnon condensate—.\nTo validate our theoretical predictions and illustrate the\nfacilitation of magnon condensate formation by incor-\nporating a PMA, we conducted a series of micromag-\nnetic simulations. Simulations were performed using\nMuMax3[57], which solves the semiclassical Landau-\nLifshitz-Gilbert (LLG) equation that describes magneti-\nzation’s precessional motion; see SM [47]. In the limit\nof large S, it is important to note that we enter the\nsemiclassical regime and hence the LLG equation may\neffectively capture and accurately describe the spin dy-\nnamics of the system. In our ferromagnetic thin film5\n(a)\n (b)\n(c)\n (d)\nFIG. 4. Micromagnetic simulation of nonequilibrium\nmagnon distribution injected by spin torque mechanism for\na YIG thin film with a thickness of Lx= 50 nm and lat-\neral sizes of Ly=Lz= 5µm,in the presence of an external\nmagnetic field along the zdirection H0= 1 kOe. (a) and\n(b) show magnon distributions of the initial nonequilibrium\ninjected magnons and final quasi-equilibrium magnon con-\ndensate steady state, respectively, when Kan= 0. (c) and\n(d) show magnon distributions of initial nonequilibrium ex-\ncited magnons and final quasi-equilibrium magnon conden-\nsate steady state, respectively, when Kan= 0.5µeV. The\ndotted line indicates the analytical dispersion relation of non-\ninteracting magnons, Eq. 3b. Because of magnon-magnon\ninteractions, the simulated magnon dispersion has a nonlin-\near spectral shift compared to the analytical noninteracting\nmagnon dispersion. Although the duration of magnon pump-\ning by spin-transfer torque is the same in the absence or pres-\nence of the PMA, the critical torque amplitude is lower in the\npresence of PMA.\nsimulation, magnons are excited via spin-transfer torque\nat zero temperature, eliminating thermal magnons. The\nnonequilibrium magnons in the film are the result of in-\njection of spin current across the sample surface [22].\nOptimal spin torque strength ensures that the magnon\npopulation reaches the critical density required for form-\ning condensed magnons; see SM for simulation details\n[47]. By introducing spin torque into the system, we\nexcite magnons with varying wavevectors and frequen-\ncies, as illustrated in Figs. 4(a) and 4(c). A portion of\nthese nonequilibrium magnons undergoes thermalization\nthrough nonlinear magnon-magnon interactions, leading\nto the establishment of a stable and quasi-equilibriumstate of condensed magnons located at the minima of the\nmagnon band spectra, ±Q, as depicted in Figs. 4(b) and\n4(d). This sharp peak in the number of magnons at the\nband minima is a signature of magnon condensate.\nThe numerical simulations confirm the supportive role\nof PMA in the condensation process. First, there is a\nreduction in the threshold of spin-transfer torque neces-\nsary to inject the critical magnon density into the sys-\ntem, enabling the system to attain said critical magnon\ndensity even at lower torque amplitudes. Second, the fi-\nnal condensate magnons in the presence of the PMA are\nmore localized around the band minima than in the ab-\nsence of PMA. Simulations also indicate that PMA shifts\nthe population of condensate magnons from a symmetric\ndistribution between two band minima to a nonsymmet-\nric distribution, Fig. 4. This agrees with the analytical\nphase diagram in Fig. 3(b).\nSummary and concluding remarks—. Dipolar inter-\nactions are assumed to be relevant to stabilizing the\nmagnon condensate within YIG. The presence of a PMA\nis expected to counteract dipolar interactions. In this\nLetter, we show that even at intermediate strengths of\nthe PMA field, a magnon condensate state can exist as a\nmetastable state. We note that the anisotropy increases\nthe band depth and curvature of the magnon disper-\nsion. These adjustments to the magnon spectrum are ex-\npected to facilitate magnon condensate formation. From\nthe calculations of effective magnon-magnon interactions\nand minimizing the interaction potential at the band\nminima, we find the magnon condensate phase diagram.\nWe demonstrate that the inclusion of PMA results in a\nmagnon condensate with a more intricate phase diagram\ncompared to when PMA is absent. A finite PMA has\nthe tendency to drive the magnon condensate towards\na nonsymmetric magnon population at band minima in\nthinner films and lower magnetic fields, as compared to\nthe absence of PMA. Micromagnetic simulations within\nthe LLG framework confirm our analytical results and\nanalyses.\nACKNOWLEDGEMENTS\nThe authors thank Anne Louise Kristoffersen for help-\nful discussions. We acknowledge financial support from\nthe Research Council of Norway through its Centers\nof Excellence funding scheme, project number 262633,\n”QuSpin”. A.Q. was supported by the Norwegian Fi-\nnancial Mechanism Project No. 2019/34/H/ST/00515,\n”2Dtronics”. 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B 106,\n024423 (2022).8\nAppendix A: Diagonalization of Magnon Hamiltonian\nThe total spin Hamiltonian of a thin film, in the y−zplane with a small perpendicular anisotropy along the ˆ x\ndirection reads,\nH=Hex+HZ+Hdip+Han. (A1)\nThe exchange energy between neighboring spins reads\nHex=−1\n2JexX\ni,jSi·Sj, (A2)\nwhere Jex>0 is the ferromagnetic exchange constant and ais the lattice constant. The Zeeman energy due to an\ninplane external magnetic field of strength H0along the ˆ zdirection reads,\nHZ=−gµBH0X\njSz\nj, (A3)\nwhere µBis the Bohr magneton and gis the the effective Land´ e g-factor. The dipolar field is expressed as [58],\nHdip=−1\n2X\ni,jX\nα,βDα,β\ni,jSα\niSβ\nj, (A4a)\nDα,β\ni,j= (gµB)2(1−δi,j)∂2\n∂rα\nij∂rβ\nij1\n|rij|, (A4b)\nwhere α, βdenote the spatial components x, yandz; and rijis the distance vector between the spin sites iandj.\nFinally, the PMA anisotropy is given by,\nHan=−KanX\nj(Sj·ˆx)2. (A5)\nThe Holstein-Primakoff transformation allows us to express the spin operators in terms of bosonic creation and\nannihilation operators ˆ a†and ˆarespectively. Using the large- Sapproximation, we have, S+≈ℏ√\n2S(ˆa−ˆa†ˆaˆa/(4S))\n,S−≈ℏ√\n2S(ˆa†−ˆa†ˆa†ˆa/(4S)), and Sz=ℏ(S−ˆa†ˆa) .\nThe corresponding noniteracting boson Hamiltonian in the Fourier space reads,\nH2=X\nqAqˆa†\nqˆaq+1\n2Bqˆaqˆa−q+1\n2B∗\nqˆa†\nqˆa†\n−q, (A6)\nwhere ˆ a†\nj=1\nNP\nqeik·rjˆa†\nqandAq(Bq) is presented in Eq. (4) in the main text. We utilize the following Bo-\ngoliubov transformation to diagonalize this bosonic Hamiltonian and find the corresponding noninteracting magnon\nHamiltonian,\nˆaq=uqˆcq+vqˆc†\n−q, (A7a)\nˆa†\n−q=v†\nqˆcq+uqˆc†\n−q, (A7b)\nwhere uq=u−qandvq=v−qare the Bogoliubov coefficients, given by,\nuq=\u0000Aq+ 2ℏωq\n2ℏωq\u00011\n2, (A8a)\nvq= sgn( Bq)\u0000Aq−2ℏωq\n2ℏωq\u00011\n2. (A8b)\nThe Bogoliubov coefficients depend on the easy-axis magnetic anisotropy Kanvia both magnon dispersion ωqand\nAq. It can be shown that the off-diagonal terms ( α̸=β) of the dipolar interaction vanish in the uniform mode\napproximation for a thin film of infinite lateral lengths [41, 58]. In this case, the dipolar interaction contains no\nthree-magnon operator terms. We omit any renormalization correction to the noninteracting magnon Hamiltonian, as9\nFIG. S1. The interaction parameters, Eq. (6) in the main text, as a function of PMA. AandBare, respectively, the\nmagnon-conserving intraband and interband interaction parameters while CandDare the magnon-non-conserving interband\ninteraction parameters. We set Lx= 50 nm and H0= 1 kOe.\nthey are of the order of 1 /Sand small. We define the following parameters in the 4-magnon interaction amplitudes,\nintroduced in the main text Eq. (6), [39, 40].\nα1=u4\nQ+v4\nQ+ 4u2\nQv2\nQ, (A9a)\nα2= 2u2\nQv2\nQ, (A9b)\nα3= 3uQvQ(u2\nQ+v2\nQ). (A9c)\nThese parameters depend on Kanvia Bogoliubov coefficients.\nAppendix B: The nonlinear interaction amplitudes\nIn Fig. S1, we plot the different intraband and interband interaction apmlitudes, see Eq. (6) in the main text, as\na function of PMA.10\nTABLE II: Simulation parameters\nParameter Value\nExcitation time, first interval 100 ns\nExcitation time, second interval 200 ns\nExcitation strength, first interval −1.2×1010A m−2\nExcitation strength, second interval −0.35×1010A m−2\n(Kan= 0)\nExcitation strength, second interval −0.2×1010A m−2\n(Kan= 0.5µeV)\nAppendix C: Micromagnetic Simulations\nWe solve the following LLG equation in the continuum model for the magnetization direction m(r, t) =S/S, using\nmicromagnetic simulation code MuMax3 [57], to study spin dynamics in the system,\n∂m\n∂t=γ\n1 +α2\u0000\nm×Beff+αm×(m×Beff)\u0001\n+τSTT, (C1)\nwhere γis the gyromagnetic ration, αis the Gilbert damping parameter, Beff=−M−1\nSdH/dmis the effective magnetic\nfield,τSTT=βm×(mp×m) is the Slonczewski spin-transfer torque, with βis depends on material parameters and\napplied charge current density and mpis the polarization of the spin current. In the continuum model, the exchange\nstiffness in the SI unit is related to the Heisenberg exchange interaction via Aex= 104MSDex/(2γ).\nWe perform simulations of magnon creation by spin-transfer torque with and without out-of-plane anisotropy. We\ndefine an initial state in which the spins, on average, point along the ˆ zdirection. We introduce a random noise in the\nspin direction to mimic the thermal noise as the initial condition of our simulation. Next, we excite magnons in the\nferromagnetic thin film by applying a spin torque, with mp∥ˆ z, to the entire film surface. The film is discretized in the\nlateral directions and uniform in the ˆ xdirection. The lateral dimensions of the film are large compared to the film\nthickness. In this way, the film is effectively a 2D magnetic system. The LLG equation is solved for each successive\ntime step, using the open-source software MuMax3 [57]. We refer to Ref. [59] for more numerical details.\nThe spin accumulation determines the strength of the spin torque, see Ref. [59]. We start the simulations by\nexciting magnons with a strong spin torque (interval I1), before lowering the torque strength to keep the magnetization\ndynamics in a semi-stable state where the total number of magnons does not change dramatically over time (interval\nI2). In this semistable state, the two magnon populations may interact with each other, and the density of magnons\nin both minima may oscillate in time. However, the total number of magnons remains relatively unchanged. The\ncurrent strength and time duration of the two intervals are listed in Table I in the SM. The magnetization data in\nFig. 4 in the main text are from the last 50 ns of the intervals.\nThe nonequilibrium magnon density in the film is proportional to the deviation of total magnetization for the\nground state, η= 1− ⟨mz⟩[59].\nThe torque strength determines the number of excited magnons. A finite PMA lowers the magnon spectrum minima,\nmeaning that one can use a weaker spin torque to generate the critical magnon density needed for the generation\nof condensate magnons. In Fig. 4 in the main text, we choose a spin-transfer torque strength for the generation of\ncondensate magnons, resulting in a magnon density of approximately η≈0.01. This density is small enough to only\ntake into account two-magnon scattering possesses and not higher order processes.\nThe distribution of magnons in the magnon spectrum ξcan be found by performing a Fourier transform of the\ntransverse magnetization components, ξ(qy, qz;ω) =|F[mx(y, z;t)]|2+|F[my(y, z;t)]|2[59].\nTo reduce the consequences of the finite-size effect in our results, we analyze the magnetization data in the middle\nregion of the thin film, ( y, z)∈[L/8,7L/8], where L=Ly=Lz= 5µm are the lateral dimensions of the square thin\nfilm." }, { "title": "2308.13144v1.Giant_orbit_to_charge_conversion_induced_via_the_inverse_orbital_Hall_effect.pdf", "content": "Giant orbit-to-charge conversion induced via the inverse orbital Hall effect Renyou Xu,1,2 Hui Zhang,1* Yuhao Jiang,1 Houyi Cheng,1,2 Yunfei Xie,3 Yuxuan Yao,1 Danrong Xiong,1 Zhaozhao Zhu,4,5 Xiaobai Ning,1 Runze Chen,1 Yan Huang,1 Shijie Xu,1,2 Jianwang Cai,4 Yong Xu,1,2 Tao Liu,3 Weisheng Zhao1,2* 1Fert Beijing Institute, School of Integrated Circuit Science and Engineering, Beihang University, Beijing 100191, China 2Hefei Innovation Research Institute, Beihang University, Hefei 230013, China 3National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, Chengdu 610054, China 4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China *Corresponding authors: weisheng.zhao@buaa.edu.cn; huizh@buaa.edu.cn; These authors contributed equally: Renyou Xu, Hui Zhang, Yuhao Jiang, Houyi Cheng Abstract: We investigate the orbit-to-charge conversion in YIG/Pt/nonmagnetic material (NM) trilayer heterostructures. With the additional Ru layer on the top of YIG/Pt stacks, the charge current signal increases nearly an order of magnitude in both longitudinal spin Seebeck effect (SSE) and spin pumping (SP) measurements. Through thickness dependence studies of the Ru metal layer and theoretical model, we quantitatively clarify different contributions of the increased SSE signal that mainly comes from the inverse orbital Hall effect (IOHE) of Ru, and partially comes from the orbital sink effect in the Ru layer. A similar enhancement of SSE(SP) signals is also observed when Ru is replaced by other materials (Ta, W, and Cu), implying the universality of the IOHE in transition metals. Our findings not only suggest a more efficient generation of the charge current via the orbital angular moment channel but also provides crucial insights into the interplay among charge, spin, and orbit. The observation of the orbital torque generated by the orbital Hall effect (OHE) and orbital Rashba-Edelstein effect (OREE), has recently drawn a great deal of attention due to the utilization of orbital angular moment (OAM) as a new information carrier for future information technology [1-3]. In the scenario of OHE, a flow of OAM or nonequilibrium OAM accumulation, whose orbital polarization is transverse to the charge current, can be generated in transition metals independent of strong spin-orbit coupling (SOC), which is a bulk effect [4,5]. The confirmation of charge-to-orbit conversion, as evidenced by the orbital torque switching of magnetization and theoretical calculations, provides a strong indication of the possibility of orbit-to-charge conversion, considering the Onsager reciprocity [6,7]. It has been suggested that OHE is significantly stronger than the spin Hall effect (SHE) by an order of magnitude in most transition metal materials [1]. Consequently, we anticipate that the inverse orbital Hall effect (IOHE) is considerably stronger than the inverse spin Hall effect (ISHE), which could fundamentally contribute to the development of memory storage devices and enhance the efficiency of ISHE-based technologies, such as magnetoelectric spin-orbit (MESO) logic [8,9]. Over the past years, plenty of works have been devoted to exploring the ISHE [10-13]. However, thus far, only a few studies have explored the IOHE through the spin terahertz measurements [14,15]. The film thickness-dependent absorption of the terahertz signal, coupled with spin/orbit accumulation induced by ultrafast demagnetization, makes it difficult to disentangle the contributions of ISHE and IOHE [16-19]. To unlock the underlying physics, it is essential to employ alternative spin/OAM injection methods and perform theoretical analyses encompassing the interactions between charge, spin, and orbital degrees of freedom. In this work, we demonstrate the unambiguous detection of IOHE in YIG/Pt/NM heterostructures through longitudinal spin Seebeck effect (SSE) and spin pumping (SP), where NM represents nonmagnetic material. Remarkably, a substantial enhancement in SSE and SP signals was observed upon depositing a light metal, Ru, on top of the YIG/Pt sample. Furthermore, the signal reached saturation with increasing thickness of the Ru layer, as confirmed by the measurements of the thickness dependence of orbit-to-charge conversion. Theoretical analysis based on a spin-orbit diffusion model exhibited good agreement with the experimental results, indicating that the IOHE of Ru primarily contributes to the increased SSE signal. Notably, our findings demonstrate the existence of an orbital sink effect in the Ru layer, which partially promotes the SSE signal. A similar enhancement of SSE(SP) signals is also observed when Ru is replaced by Ta, W, and Cu, suggesting that the IOHE is a universal phenomenon in these materials. These results shed light on the mystery of the IOHE and offer a promising avenue for the development of spin-orbitronics devices based on the IOHE. YIG/Pt/NM trilayer heterostructures were prepared using DC/RF magnetron sputtering with the basic pressure of the chamber below 10-8 mbar [20]. A 40-nm-thick epitaxial Y3Fe5O12 (YIG) film was deposited on a (111)-oriented gadolinium gallium garnet (GGG) single crystal substrate. A slight roughness of 0.078 nanometers of the YIG layer was obtained through Atomic Force Microscopy (AFM) (see S1 in the Supplemental Material for details [21]). We further confirmed the high structural quality of the YIG layer by X-ray diffraction (XRD). Fig. 1(a) shows the representative θ-2θ scan of the GGG/YIG heterostructure, demonstrating the purity of the YIG phase. The appearance of Laue oscillations near the YIG (444) diffraction peak provides evidence of the film's exceptional uniformity. Fig. 1(b) displays the hysteresis loop of the GGG/YIG sample under the in-plane magnetic field obtained by a Lake Shore vibrating sample magnetometer (VSM) at room temperature. The GGG/YIG sample demonstrates a saturation magnetization of 95.1 emu/cc, with a coercive field below 2 Oe, which agrees well with the previous work [22,23]. The thin film multilayers consist of Pt(1.5)/NM(tNM)/MgO(3)/Ta(2) (tNM denotes the thickness of the NM layer, and numbers indicate the thickness in nm) deposited on top of the GGG/YIG sample (see S1 in the Supplemental Material for details [21]). MgO(3)/Ta(2) serves as the capping layer to prevent further oxidation of the NM layer, which will be omitted in electric transport experiments for convenience. Since the thin Ta layer exhibits high resistivity when exposed to the atmosphere, the current-shunting effect induced by the capping layer was negligible in our experiments. Fig. 1(c) presents the cross-section of the YIG(40)/Pt(1.5)/Ru(6)/MgO(3)/Ta(2) stack, as captured by high-resolution transmission electron microscopy (HR-TEM). The YIG layer, MgO layer, and Ru layer exhibit well-crystallized structures. Fig. 1(d) showcases the corresponding energy dispersive spectroscopy (EDS) images, revealing the distribution of Y , Pt, Ru, and Mg elements, respectively. The interdiffusion of elements between different layers is minimal, as no additional annealing process was applied. The HR-TEM and EDS results validate the high quality of the sample and the sharp interfaces between the different layers. \n FIG. 1. Structure and magnetism characterization. (a) The XRD pattern of the GGG/YIG sample. (b) The magnetic loop of the GGG/YIG sample under the in-plane magnetic field. (c) The cross-sectional HR-TEM image of the YIG(40)/Pt(1.5)/Ru(6)/MgO(3)/Ta(2) multilayer. (d) The EDS mapping of corresponding elements. We carry out longitudinal SSE measurements in a Quantum-designed physical property measurement system (PPMS), and a sketch for the experiment setup is shown in Fig. 2(a). The sample was thermally connected to a heater, with its temperature measured via a thermocouple. At the same time, a copper block served as a heat sink at the top of the sample surface, which was thermally contacted to the chamber of PPMS, maintaining a constant temperature of 300 K. A temperature difference, denoted as ∆T, was established along the z-axis of the film, with most of the ∆T occurring in the thicker YIG layer due to its lower thermal conductivity compared to the metal layers. The temperature difference ∆T within the YIG layer induces a pure spin current 𝑆!\"#$%&', which is then injected into the Pt layer. The spin index σ aligns parallel to the magnetization of the YIG layer, and its orientation can be controlled by applying an in-plane magnetic field. As a result of the ISHE, the charge current 𝐼($ generated in the Pt layer induces an electric field in the direction of σ×𝑆!\"#$%&', giving rise to a voltage denoted as VSSE along the x-axis [24]. Fig. 2(b) displays the SSE signals ISSE of YIG(40)/Pt(1.5) (black) and YIG(40)/Pt(1.5)/Ru(4) (red) as a function of the magnetic field 𝐻)\talong the y-axis. The charge current ISSE was defined as VSSE/R, where the voltage VSSE was measured by a Keithley 2182 nanovoltmeter, and R corresponds to the resistance of the electrical contacts between the sample and the nanovoltmeter. The magnetic field swept between -20 Oe and 20 Oe, and the heating power was fixed at 300 mW to maintain a constant \n2nmYIGPtRuMgOTaOX\nY\nPt\nRu\nMg2nm505152YIG(444)Intensity (a.u.)2q (degree)GGG(444)(a)(b)\n(c)(d)-20-1001020-2-1012M (10-4 emu)Hin-plane (Oe)temperature difference (∆T =13 K) for both samples during the test. By adding a 4-nm-thick Ru layer on the top of the YIG(40)/Pt(1.5) bilayer, the ISSE dramatically increased nearly an order of magnitude[Fig. 2(b)], rising from 0.17 nA to 1.58 nA. We modulated the temperature difference by adjusting the heating power. With the increased heating power, a larger temperature difference leads to a larger injected spin current, and a significant enhancement of the ISSE was observed (see S2 in the Supplemental Material [21]). Considering the weak SOC of Ru, ISHE in the Ru layer is much smaller compared to ISHE in the Pt layer [25-27]. As a result, the traditional ISHE theory cannot explain the significant SSE signal enhancement. Considering the negligible ISHE of Ru, we argued that the plausible mechanism to explain this phenomenon is the occurrence of IOHE in the Ru layer. Despite the quenching of OAM in transition metal, OHE leads to the local accumulation of OAM due to the applied transverse electric field, which has been proven by the Kerr effect and orbital torque effect experiments recently [28-30]. Analogous to the ISHE that can convert spin current into charge current, the IOHE enables the conversion between OAM and the charge current, which means the injection of an orbital current will leads to induced charge current in a direction that is perpendicular to the orbital current and orbital polarization. The orbital current can be understood as a wave packet carrying OAM, which can be induced through OHE and OREE [31,32]. \n FIG. 2. SSE measurements. (a) Longitudinal SSE in YIG/Pt bilayer. (b) A comparison of the SSE signals between the YIG(40)/Pt(1.5)/Ru(6) (red) and YIG(40)/Pt(1.5) (black) is presented. These signals were obtained using a heating power of 300 mW. (c) Longitudinal SSE in YIG/Pt/Ru trilayer heterostructures considering spin-to-orbit conversion (L-S conversion), ISHE, and IOHE. (d) SSE signal of YIG(40)/Pt(1.5)/Ru(tRu) samples as a function of the thickness of the Ru layer (red circle). Here the purple dotted line, green dashed line, and blue dashed line indicate fitting curves of the SSE signal with contributions from charge current induced by the ISHE of Pt (𝐼&*+,($), the IOHE of Pt (𝐼&-+,($), and the IOHE of Ru (𝐼&-+,./), respectively. The ISHE of Ru (𝐼&*+,./) is much weaker compared to the other contributions, thus not shown in this figure. The solid red line represents the fitting curves 𝐼0-012 with all four contributions mentioned above. In consideration of the large orbital Hall conductivity of Ru (𝜎-+./ = 9100 (ħ/e) (Ω cm)-1) that was \n(a)(b)\n(c)(d)-20-1001020-2-1012ISSE (nA)Hy (Oe)YIG/Pt(1.5)/Ru(4)YIG/Pt(1.5)\nspincurrentISHE\nGGGYIGPtRuorbitalcurrentIOHE with L-Sconversion\nxzHy>0V\nxzHyGGGYIGPtobserved in recent works [33-35], we propose a physical mechanism as shown in Fig. 2 (c). The temperature difference drives spin current 𝑆($ diffuses into the Pt layer, a transverse charge current 𝐽3 ( 𝐽3=4\"ħ6!\"#$6#$𝑆($, where 𝜎*+($ and 𝜎($ is the spin Hall conductivity and conductivity of Pt) is generated as a result of ISHE. At the same time, an orbital current 𝐿($ (𝐿($=𝜂2*($𝑆($, where 𝜂2*($ is the conversion coefficient of spin-to-orbit for Pt) generates in the Pt layer due to the strong SOC of Pt and then diffuses into the Ru layer. Based on the scenario of IOHE, the orbital current 𝐿./can also be converted to a transverse charge current 𝐽3 (𝐽3=4\"ħ6%\"&'6&'𝐿./, where 𝜎./\tis the conductivity of Ru) in the Ru layer. Since 𝜎-+./, 𝜂2*($, and 𝜎*+($ have the same positive sign, the additional Ru layer shall enlarge the SSE signal of the YIG/Pt bilayer. To further understand the properties of the observed SSE signal enhancement, we conducted the Ru layer thickness tRu-dependent SSE measurements. As shown in Fig. 2(d), the SSE signal of YIG(40)/Pt(1.5)/Ru(tRu) samples increases with tRu increases and then reached saturation when tRu = 4 nm (see S3 in the Supplemental Material for details [21]). To better understand the contributions to the SSE signal, we have developed a one-dimension spin-orbit diffusion model (refer to Supplemental Material S4 for detailed calculations [21]), which contains the spin-to-orbit conversion in Pt due to its strong SOC [2,36,37]. As illustrated in Fig. 2(d), the fitting curves 𝐼0-012 (red solid line) exhibit a good agreement with our experimental results (red circle), where 𝐼0-012 encompasses four distinct SSE contributions involving ISHE and IOHE in both Pt and Ru layers, expressed as 𝐼0-012=𝐼&-+,./+𝐼&*+,./+𝐼&*+,($+𝐼&-+,($. As the thickness of the Ru layer increases, the 𝐼&-+,./ (blue dashed line) experiences a rapid increase, ultimately becoming the predominant factor among all the contributions, which indicates the increased SSE signal mainly comes from the IOHE of Ru. The 𝐼&*+,./ is nearly constant at zero due to its weak SOC, thus is negligible in our experiment. As the spin current permeates through the Pt layer, a portion of this spin current reflects at the edge of the Pt layer, compensating 𝐼&*+,($\t. Consequently, introducing an extra Ru layer restrains the spin backflow, resulting in an increase of 𝐼&*+,($[38,39]. This phenomenon is evident in the slight elevation of the 𝐼&*+,($ (purple dotted line) as the Ru layer's thickness increases, denoting a spin sink influence (see S5 in the Supplemental Material for detail). However, the rise in 𝐼&*+,($ is notably minuscule in comparison to 𝐼&-+,./, suggesting that the influence of the spin sink effect can also be disregarded in our experiment. Furthered, 𝐼&-+,($ (green dashed line) exhibit an obvious increase and even exceed the 𝐼&*+,($ at the thicker Ru layer. We note that this phenomenon can be explained by an orbital sink effect. Since the considerable orbital current is generated and diffusion in the Pt layer, partial orbital current shall be reflected at the edge of the Pt layer and compensate for the IOHE that occurred in the Pt layer (see S5 in the Supplemental Material for detail). However, the additional Ru layer restains the orbital backflow and boosts the orbit-to-charge conversion occurring in the Pt, thereby promoting the SSE signal[5,21,40,41]. By discerning between different contributions, we demonstrate that the impact of the spin sink of Ru appears to be negligible in our model, but the IOHE of Ru is primarily responsible for the increased SSE signal, while the orbital sink effect partially enhances the SSE signal. We also performed spin injection measurements using SP driven by ferromagnetic resonance (FMR) to validate our findings from the longitudinal SSE measurements. Fig. 3(a) illustrates the experimental setup for the SP measurements. Upon RF excitation, a spin current is initially injected into the Pt layer, which then converts into an orbital current that diffuses into the Ru layer. A similar process is expected to occur as in the longitudinal SSE experiment, where both ISHE and IOHE contribute to the charge current signal in the YIG/Pt/Ru heterostructure. The detailed experiment setup was similar to our previous work [42]. Fig. 3(b) shows the measured SP signal of YIG(40)/Pt(1.5) sample as a function of the magnetic field under different frequencies, exhibiting a symmetrical Lorentzian shape. For a clear comparison, the charge current ISP is defined as USP/R, where USP is directly measured by the lock-in amplifier, and R is the electric resistance of the sample contacts. Upon reversing the external magnetic field (The angle between the external DC magnetic field and the x-axis (θH) changes from 90° to 270°), the measured signal shows an opposite sign with an almost unchanged magnitude, indicating that the spin rectification effects are negligible in our experiment [43]. Fig. 3(c) displays the SP signals of YIG(40)/Pt(1.5) (red) and YIG(40)/Pt(1.5)/Ru(4) (black), both measured at 8 GHz. It can be observed that the ISP of YIG(40)/Pt(1.5)/Ru(4) is over 10 times higher than that of YIG(40)/Pt(1.5). Further information about the IOHE was obtained through a Ru thickness dependence experiment, as shown in Fig. 3(d). The ISP first increases as the Ru layer's thickness increases, then saturates at about 4 nm (see S5 in the Supplemental Material for details [21]). These results support the conclusion obtained by SSE measurements that the additional light metal Ru layer strongly enlarges the charge signal, which can be well explained by IOHE. \n FIG. 3. SP measurements. (a) An illustration of FMR-driven SP. H is the in-plane external DC magnetic field, and HRF is the out-of-plane radiofrequency magnetic field. V oltage is measured along the x direction. (b) Representative SP signal of YIG(40)/Pt(1.5). θH is 90° or 270° which denotes the angle between the external DC magnetic field and the x-axis. (c) SP charge current of YIG(40)/Pt(1.5) (black) and YIG(40)/Pt(1.5)/Ru(4) (red) samples measured at 8 GHz. (d) SP charge current as a function of Ru thickness in YIG(40)/Pt(1.5)/Ru(tRu) samples measured at 8 GHz. The Ru element exhibits a large 𝜎-+ compared to other transition metals, as evidenced by recent studies [5,44]. To corroborate the hypothesis that the dominant factor responsible for signal enhancement is the IOHE, additional experiments were conducted involving other transition metals possessing large 𝜎-+ [1,15]. We investigate the IOHE in the YIG/Pt(1.5)/NM(2) heterostructures (NM = Ta, W, Cu, and Ru) via both SSE and SP experiments, as shown in Figs. 4(a) and 4(b). The results are ordered from weak to strong signals, and it is evident that the sample with the Ru layer exhibits the largest signal. In addition, Figs. 4(a) and 4(b) also demonstrate the consistent agreement between SSE and SP measurements. Surprisingly, the signals of samples with NM=Ta and W also Pt (1.5 nm)YIG (40 nm)Ru (tRunm)(a)(b)\n(c)(d)1.21.62.02.4-40-2002040ISP (μV)H (Oe) qH = 90°qH = 270°6 GHz7 GHz8 GHz\n-500500306090120150ISP (nA)H-HR (Oe) YIG/Pt(1.5)/Ru(4) YIG/Pt(1.5)024120306090120150ISP(nA)tRu (nm)V\n𝑆\"#$%&'(GGGYIGPt𝑆)%𝐿)%𝑆+,𝐿+,RuxyHshow a significant increase, which contradicts the theory of ISHE. According to ISHE, the SSE and SP signals generated in Ta and W should cancel out the Pt signal due to the opposite signs of the spin Hall conductivity (𝜎*+) compared to Pt. However, the 𝜎-+ of Ta, W, and Pt have the same positive sign. Consequently, the IOHE overcomes the cancellation effect of ISHE in Ta and W, resulting in an increased signal. Furthermore, the sample of Cu, which is a light metal and a poor spin sink material (spin diffusion length over 200 nm measured at room temperature [45]), exhibits a substantial increase, supporting the notion that the spin sink effect is not prominent in our experiments. \n FIG. 4. (a) SP measurements and (b) SSE measurements of YIG/Pt/NM samples with different NM (NM = Ta, W, Cu, and Ru). TABLE I. The sign of the 𝐼&*+, and 𝐼&-+, from the NM layer in YIG/Pt(Gd)/NM heterostructures. NM1 NM2 NM3 𝜎*+78/𝜎-+78 +/+ -/+ +/- \t𝜂2*($>0 𝐼&*+,/𝐼&-+, +/+ -/+ +/- \t𝜂2*'9\t<0 𝐼&*+,/𝐼&-+, +/- -/- +/+ In YIG/Pt/NM trilayer heterostructures, Pt serves as an efficient spin-to-orbit converter which enables the orbital current to inject into the NM layer, and promotes the orbit-to-charge conversion in the NM layer. We note that the SSE(SP) signal of a single NM layer in YIG/Pt/NM has a strong correlation with its 𝜎*+, 𝜎-+, and 𝜂2*($('9). While recent work introduces the rare earth element Gd have stronger SOC enables the conversion between spin and orbital current. We compare ISHE and IOHE of different NM layers in YIG/Pt(Gd)/NM heterostructures SSE(SP) measurements, which is summarized in TABLE I. By assuming a positive spin current initially diffuses into the Pt layer, the utilization of Pt as a positive spin-to-orbit converter (𝜂2*($>0) facilitates the injection of positive spin and orbital currents into the NM layer. Consequently, NM1 with positive 𝜎-+ and 𝜎*+, such as Ru, exhibit stronger signals, while NM2(NM3) with opposite signs of 𝜎*+ and 𝜎-+ offset the signals. On the other hand, employing Gd as a negative spin-to-orbit converter (𝜂2*'9<0) leads to the injection of positive spin and negative orbital currents [4]. In such cases, the stronger signals are expected to obtain by using NM2(NM3) with large 𝜎*+ and 𝜎-+ but oppositive signs, such as transition metals (NM2 = Ta and W), two-dimensional electron gas materials (NM2 = 2DEG at the KTaO3 or SrTiO3 interface [46]), as well as transition metal disulfides (NM3 = MoTe2 and WTe2 [47,48]). (a)\n(b)-500500306090120-50050-50050-50050-50050-20020-2-1012-20020-20020-20020-20020 Pt Pt-Ta Pt-W Pt-Cu Pt-RuISSE (nA) ISP (nA) \nH (Oe)In conclusion, this study presents a novel efficient approach to generating charge current utilizing the OAM channel. It is found that the Ru layer with negligible SOC can significantly boost the inverse charge signal through the orbit-to-charge conversion, as evidenced by the SSE and SP measurements. In addition to the previous studies of IOHE, our work combines thickness-dependent measurements with theoretical analysis, disentangles IOHE from ISHE, and surprisingly observes an orbital sink effect. These results shed light on the interaction between charge, spin, and orbit, offering potential benefits for advancing emerging spin-orbitronics applications such as MESO and spin terahertz emitters. The authors thank Albert Fert for his useful discussion. The authors gratefully acknowledge the National Key Research and Development Program of China (No. 2022YFB4400200), National Natural Science Foundation of China (No. 92164206, 52261145694, 52072060, and 52121001), Beihang Hefei Innovation Research Institute Project (BHKX-19-01, BHKX-19-02). All the authors sincerely thank Hefei Truth Equipment Co., Ltd for the help on film deposition. This work was supported by the Tencent Foundation through the XPLORER PRIZE. REFERENCES [1] D. Lee et al., Nat. Commun. 12, 6710 (2021). [2] G. Sala and P. Gambardella, Phys. Rev. Research 4, 033037 (2022). [3] S. Ding et al., Phys. Rev. Lett. 128, 067201 (2022). [4] S. Lee et al., Commun. Phys. 4, 234 (2021). [5] L. Salemi and P. M. Oppeneer, Phys. 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Varotto et al., Nat. Commun. 13, 6165 (2022). [47] S. Bhowal and S. Satpathy, Phys. Rev. B 102, 035409 (2020). [48] X. Wang et al., Cell Rep. Phys. Sci. (2023). " }, { "title": "1808.05785v2.Temperature_Dependence_of_Magnetic_Properties_of_an_18_nm_thick_YIG_Film_Grown_by_Liquid_Phase_Epitaxy__Effect_of_a_Pt_Overlayer.pdf", "content": "Temperature Dependence of Magnetic Properties of\nan 18-nm-thick YIG Film Grown by Liquid Phase\nEpitaxy: Effect of a Pt Overlayer\nNathan Beaulieu\u0003;y, Nelly Kervarecz, Nicolas Thieryx, Olivier Kleinx, Vladimir Naletovy;x;{,\nHerv ´e Hurdequinty, Gr´egoire de Loubensy, Jamal Ben Youssef\u0003and Nicolas Vukadinovick\n\u0003LabSTICC, CNRS, Universit ´e de Bretagne Occidentale, 29238 Brest, France\nySPEC, CEA, CNRS, Universit ´e Paris-Saclay, 91191 Gif-sur-Yvette, France\nzPlateforme technologique RMN-RPE, Universit ´e de Bretagne Occidentale, 29238 Brest, France\nxUniv. Grenoble Alpes, CEA, CNRS, Grenoble INP, INAC-Spintec, 38000 Grenoble, France\n{Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation\nkDassault Aviation, 92552 Saint-Cloud, France\nAbstract —Liquid phase epitaxy of an 18 nm thick Yttrium\nIron garnet (YIG) film is achieved. Its magnetic properties are\ninvestigated in the 100 – 400 K temperature range, as well as the\ninfluence of a 3 nm thick Pt overlayer on them. The saturation\nmagnetization and the magnetocrystalline cubic anisotropy of\nthe bare YIG film behave similarly to bulk YIG. A damping\nparameter of only a few 10\u00004is measured, together with a\nlow inhomogeneous contribution to the ferromagnetic resonance\nlinewidth. The magnetic relaxation increases upon decreasing\ntemperature, which can be partly ascribed to impurity relaxation\nmechanisms. While it does not change its cubic anisotropy, the Pt\ncapping strongly affects the uniaxial perpendicular anisotropy of\nthe YIG film, in particular at low temperatures. The interfacial\ncoupling in the YIG/Pt heterostructure is also revealed by an\nincrease of the linewidth, which substantially grows by lowering\nthe temperature.\nI. I NTRODUCTION\nYttrium Iron garnet (Y 3Fe5O12, YIG) is the marvel material\nfor ferromagnetic resonance (FMR) [1], with the lowest known\nGilbert damping parameter, \u000b= 3\u000110\u00005for bulk [2],\n[3]. Since the 1970s, liquid phase epitaxy (LPE) has been\nthe reference method to grow micrometer-thick YIG films\nwith bulk-like dynamical properties [4], [5], and numerous\nmicrowave devices based on the propagation of spin-waves in\nsuch films have been developed [6]. In the past years, thin\nfilms of YIG have become highly desirable in the context of\nmagnonics [7], [8] and its coupling to spintronics [9] for three\nmain reasons. Firstly, in magnonics, one wishes magnetic films\nwith low damping to ensure large propagation length and with\nthickness limited to a few tens of nanometers so that fun-\ndamental spin-wave modes do not interact with higher order\nthickness modes, two conditions met in nanometer-thick YIG\nfilms [10]. Secondly, coupling YIG magnonics to spintronics\nwas made possible by the discovery that pure spin currents\ncan be transmitted at the interface between YIG (a magnetic\ninsulator) and an adjacent normal metal [11], [12]. In these\nhybrid bilayers, ultrathin YIG layers are required to enhance\nthe interfacial effects. For instance, the damping of nanometer-thick YIG can be controlled by the spin-orbit torque produced\nby an electrical current flowing in an adjacent Pt layer [13],\nleading to the generation of coherent spin-waves above a\nthreshold instability [14]. Thirdly, due to the high resilience\nof YIG, only nanometer-thick films can be patterned through\nstandard nanofabrication techniques, which is an important\nasset to design magnonic crystals [15] and engineer the spin-\nwave spectrum of individual YIG nanostructures [16].\nDue to this renew of interest for YIG thin films, lots of\nefforts have been put in the last few years to produce ultrathin\nfilms with high epitaxial and dynamical qualities. Pulsed laser\ndeposition (PLD) is a technique of choice to deposit ultrathin\nYIG layers ablated from stoichiometric targets on Gd 3Ga5O12\n(GGG) substrates [17], [18], [19], [20], [21], [22]. Magnetic\nproperties close to bulk ones [19] and Gilbert damping pa-\nrameters below 10\u00004have been reported for films thinner\nthan 50 nm [22], even though such low intrinsic damping\noften comes at the detriment of the full linewidth due to in-\nhomogeneous broadening [20], [21]. Another method to grow\nepitaxial nanometer-thick YIG films is off-axis sputtering [23],\nwhich can be used to control the strain-induced anisotropy on\nlattice-mismatched substrates [24]. Finally, even though LPE\nis not well suited to grow sub-micron thick films, it has been\nsuccessfully used to produce YIG films with thicknesses in\nthe 100 – 200 nm range and damping parameters approaching\n10\u00004[25], [26].\nHere, we demonstrate that the growth of a YIG film as\nthin as 18 nm and of high quality can be achieved by LPE.\nWe present a detailed investigation of its magnetic properties,\nincluding relaxation, as a function of temperature, as well as\nthe effect of a 3 nm thick Pt overlayer on them.\nII. S AMPLE PREPARATION AND PRELIMINARY\nCHARACTERIZATIONS AT ROOM TEMPERATURE\nThe ultrathin YIG film under consideration was grown by\nLPE from PbO and B 2O3flux on a 1 inch (111)-oriented\nGGG substrate. High-purity oxides (6N) were used. The keyarXiv:1808.05785v2 [cond-mat.mtrl-sci] 24 Aug 2018Fig. 1. (a) X-ray reflectometry of the 18 nm thick LPE grown YIG film.\nThe inset shows the AFM surface topography. (b) Resonance linewidth vs.\nfrequency of the bare YIG layer measured by broadband FMR at room\ntemperature. The inset displays the FMR line at 10 GHz.\nparameter is the growth rate that depends on the growth\ntemperature. To get an ultrathin YIG film, a very low growth\nrate is required which means that the growth temperature has\nto be close to the saturation temperature (small supercooling).\nFor the selected sample, the growth rate was around 1 nm/s.\nAnother important parameter is the speed of the substrate\nrotation during the growth process that must be controlled.\nThe quality of the utrathin YIG film was checked by atomic\nforce microscopy (AFM) measurements from which a surface\nroughness of about 0.25 nm was determined (see inset in\nFig.1(a)). The film thickness of 18 nm was confirmed by X-ray\nreflectivity measurements (Fig.1(a)). From the experimental\nin-plane magnetization curves recorded along the [110]and\n[112]axes (not shown here), several conclusions can be drawn:\n(i) the saturation magnetization 4\u0019MSat room temperature is\nequal to 1700 G, close to the bulk value, (ii) the magnetization\ncurves are isotropic in the film plane, and (iii) the film has a\nvery weak coercive field ( Hc'0:3Oe).\nBroadband FMR on a millimeter-size slab of the grown\nfilm was used to investigate its magnetic relaxation at room\ntemperature. The full width at half maximum (FWHM) of the\nresonance line measured in in-plane magnetized configuration\nas a function of the excitation frequency is shown in Fig.1(b).\nFig. 2. (a) Dependence of the magnetization of the 18 nm thick YIG film on\ntemperature. (b) Resonance field vs.polar angle measured at X-band at three\ndifferent temperatures. (c) Cubic and (d) uniaxial perpendicular anisotropies\nextracted as a function of temperature. The red dashed lines in (a) and (c)\nare the dependences for bulk YIG from literature, the dotted line in (d) is a\nguide to the eye.\nThe linear dependence of the linewidth on frequency allows to\nfit a Gilbert damping parameter \u000b= (3:4\u00060:2)\u000110\u00004and an\ninhomogeneous contribution \u0001H0= 2:44\u00060:24Oe. These\nvalues, which are close to those reported on PLD grown YIG\nfilms of similar thickness [18], highlight the good dynamical\nquality of our film.\nIII. T EMPERATURE DEPENDENCE OF MAGNETIC\nPROPERTIES FOR THE BARE YIG FILM\nThe temperature dependence of the saturation magnetization\nfor the bare YIG film measured by means of a supercon-\nducting quantum interference device (SQUID) magnetometer\nand a vibrating sample magnetometer (VSM) is displayed inFig.2(a). The 4\u0019MS(T)profiles are consistent between the\ntwo methods and are in very good agreement with the one\nreported for a bulk YIG sample [27] for T\u0015200 K. For\nlower temperatures, the saturation magnetization for the bulk\nsample exceeds the one found for our ultrathin YIG film.\nNext, the anisotropy constants were extracted from FMR\nmeasurements performed in a X-band cavity (magnetic field\nswept at fixed frequency f= 9:3GHz). We note that the slab\nused in this experiment has a slightly larger inhomogeneous\ncontribution to the linewidth than the one measured by broad-\nband FMR. Using the Makino’s procedure [28], the first-order\ncubic anisotropy constant K1, the first-order uniaxial perpen-\ndicular anisotropy constant KUand the gyromagnetic ratio \r\nwere determined as a function of temperature by exploiting\nthe polar angle variation of the polarizing magnetic field, the\n4\u0019MS(T)profile being known for that very same slab of\nfilm (red stars in Fig.2(a)). Examples of such variations are\nreported in Fig.2(b) for three temperatures: T= 140 K, 300 K,\nand 405 K, where \u0012His the polar angle of the polarizing\nmagnetic field. \u0012H= 0\u000ecorresponds to the film normal ( [111]\ncrystallographic axis) and \u0012H= 90\u000eis the in-plane [112]axis.\nTwo remarks can be made. First, the amplitude of the angular\nvariation increases for decreasing temperature. Second, this\nvariation becomes more and more asymmetrical with respect\nto the film plane as the temperature is lowered. This last point\nmeans that the absolute value of K1is enhanced for decreasing\ntemperature. The experimental temperature dependence of K1\nis displayed in Fig.2(c). The increase of the absolute value of\nK1for decreasing Tsatisfactorily matches with the variation\nfound for a bulk YIG sample [29]. On the other hand, the\ntemperature dependence of KUis reported in Fig.2(d). It\nappears that the sign of KUis positive (easy axis along the\nfilm normal) and KUincreases with T.\nThe origin of KUin micrometer-thick LPE grown YIG\nfilms has been discussed in detail in the eighties. Two main\ncontributions were identified, namely, the growth and the stress\nanisotropies. For pure YIG films, the growth anisotropy is\nmainly induced by the Pb impurities from solvent. It has\nbeen established that it raises with the Pb content [30].\nIn addition, the Pb content increases with the supercooling\n[31]. In our case, the growth of the ultrathin YIG film with\na small supercooling prevents a large contribution of the\ngrowth anisotropy. On the other hand, the uniaxial stress\nanisotropy arises from the lattice parameter mismatch between\nthe GGG substrate ( as= 12:383 ˚A) and the YIG film\n(af= 12:376 ˚A). Introducing \u0001a?= (as\u0000a?\nf)=aswherea?\nf\nis the lattice parameter of the YIG film in the growth direction,\nthe uniaxial stress anisotropy constant Ks\nUis expressed by:\nKs\nU= (\u00003=2)\u0015111\u001bwhere\u001b=E=(1\u0000\u0017)\u0001a?, whereE\nis the Young’s modulus and \u0017the Poisson’s ratio. For our\nfilm grown under tension, \u001bis positive,\u0015111is negative and\na positive value of Ks\nUis expected in agreement with the\nexperimental data. An estimate of Ks\nUbased on the values of\nE,\u0017and\u0015111for a bulk YIG sample leads to about 40% of the\nKUvalue deduced from FMR measurements. The remaining\ndifference can be ascribed to a surface induced contribution to\nFig. 3. (a) FMR lines measured at X-band on the bare YIG layer for three\ndifferent temperatures. (b) FMR linewidth vs.temperature. The inset shows\nthe comparison of the data to the slowly relaxing impurity model (Eq.1).\nthe uniaxial perpendicular anisotropy, which is also present in\nultrathin films. In fact, we have observed at room temperature\nthatKUvaries with the YIG thickness, and becomes negative\n(easy plane) above about 50 nm.\nMoreover, a value of the gyromagnetic ratio nearly inde-\npendent of temperature was extracted from the angular fit of\nthe resonance field, j\rj= 1:7685\u00060:002s\u00001Oe\u00001. This value\nis consistent with the one deduced from the broadband FMR\nmeasurements,j\rj= 1:771\u00060:0007 s\u00001Oe\u00001.\nAnother interesting feature is the temperature dependence of\nthe magnetic relaxation. Fig.3(a) shows the absorption spectra\n(derivative form) versus magnetic field recorded at 9.3 GHz\nin the parallel configuration for three temperatures. These\nspectra reveal that the FMR linewidth increases for decreasing\ntemperature. The value of \u0001HFWHM at 140 K is nearly four\ntimes greater than the one at 405 K. Such a behavior has been\nrecently reported on ultrathin YIG films grown either by a\nspin coating method [32] or by off-axis sputtering [33], and\non YIG spheres [34]. The broader temperature range probed\nin these studies allows to evidence a peak-like maximum\nin the temperature dependence of the FMR linewidth. The\nposition of this peak is frequency dependent and is located\naroundT= 25 K at X-band. Based on these observations,\nthe origin of the resonant temperature dependent linewidth\nwas ascribed to the slowly relaxing impurity mechanism\nextensively investigated on bulk YIG samples in the sixties\n[2], [3]. The existence of rare earth or Fe2+impurities induced\nduring the growth process was put forward. In both cases, the\ncontribution of the slowly relaxing impurity mechanism to theFig. 4. Dependences on temperature of (a) cubic anisotropy, (b) uniaxial\nperpendicular anisotropy and (c) resonance linewidth for both the bare YIG\nlayer (black dots) and the YIG layer capped with 3 nm Pt (red squares). The\ninset shows the FMR lines measured at 300 K on both samples. All dashed\nlines are guides to the eye.\nlinewidth can be described by the following expression [33]:\n\u0001HSR=A(T)!\u001c\n1 + (!\u001c)2; (1)\nwhereA(T)is a frequency-independent prefactor and \u001ca\ntemperature-dependent time constant. The total FMR linewidth\n\u0001HFWHM = \u0001H0+4\u0019\u000bf\nj\rj+ \u0001HSRis displayed in the\ninset in Fig.3(b) (red dashed curve). It can be seen that the\nexperimental temperature dependence of the FMR linewidth\ncannot be fully reproduced by the model, where only \u0001HSRis\na function of temperature. Other relaxation mechanisms seem\nto contribute to the FMR linewidth. As highlighted in Ref.[26],\nthe existence of a transition layer between LPE grown YIG\nfilms and the GGG substrate, whose thickness is estimated to\nbe around 5 nm (nearly a third of our film thickness), could\nproduce additional relaxation channels. For our ultrathin film,\na sizeable surface induced contribution to the linewidth can\nalso be expected.\nIV. T EMPERATURE DEPENDENCE OF MAGNETIC\nPROPERTIES FOR THE YIG/P T HETEROSTRUCTURE\nNext, we are interested in the influence of a thin overlayer\nof Pt on the magnetic properties of our ultrathin YIG film.\nUsing electron beam evaporation, a 3 nm thick Pt layer was\nevaporated on top of the YIG film after a very soft in situdry etching of its surface. Similarly to the bare YIG film,\nthe obtained YIG/Pt heterostructure was then studied at X-\nband as a function of temperature. The main results of this\ncharacterization are compared to those obtained on the bare\nYIG film in Fig.4. In order to extract the anisotropy constants\nof the YIG/Pt bilayer, we used the same 4\u0019MS(T)profile\nas in Fig.2(a), since it is known that no magnetic moment\nis induced in Pt by proximity effects at ferrites/Pt interfaces\n[35]. As can be seen in Fig.4(a), the Pt overlayer does not\nchange the cubic anisotropy of the bare YIG film, which is\nexpected from its bulk magnetocrystalline origin. In contrast,\nit does strongly affect the uniaxial perpendicular anisotropy,\nwhich becomes negative (easy plane) at low temperature, as\nshown in Fig.4(b). It means that an interfacial mechanism [36]\nof spin-orbit origin changes the surface anisotropy component\nofKUdiscussed above for the bare YIG film. The gyromag-\nnetic ratio remains unaffected by the Pt overlayer within our\ndetermination uncertainty.\nAs expected, the interfacial coupling between YIG and Pt\nalso leads to the increase of the FMR linewidth [37], [38],\nwhich is obvious from the comparison presented in Fig.4(c).\nAt 300 K, the measured linewidth is nearly doubled by the\ndeposition of Pt on top of YIG. To prove its interfacial\norigin, we have checked that this enhancement is inversely\nproportional on the YIG thickness by varying the latter in\nthe 15 – 200 nm range (not shown here). It is interesting\nto note that the increase of the linewidth due to the Pt\noverlayer is not constant as a function of temperature. The\nincrement in the damping depends both on the strength of\nthe interfacial coupling (whose microscopic origin is rarely\ndiscussed, most often, it is parametrized by the so-called spin\nmixing conductance of the hybrid interface) and on the spin\ntransport parameters in the Pt layer (the thickness of the\nlatter being here comparable to the spin diffusion length in\nPt at room temperature [39]). These two contributions can be\ntemperature dependent, leading to a non trivial behavior of the\nincrement in the damping versus temperature. As a matter of\nfact, the latter seems to be minimum around 250 K, and its\nrapid increase below this temperature should be confirmed by\nmeasurements at even lower temperature [32].\nV. C ONCLUSION\nIn sum, this study demonstrates that the LPE growth method\ncan be used to produce YIG films with thickness below 20 nm\nand good structural parameters. 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Klein, “Electrical properties of epitaxial yttrium iron garnet ultrathin\nfilms at high temperatures,” Phys. Rev. B , vol. 97, p. 064422, 2018." }, { "title": "1810.00380v1.Magnon_Valves_Based_on_YIG_NiO_YIG_All_Insulating_Magnon_Junctions.pdf", "content": "arXiv:1810.00380v1 [physics.app-ph] 30 Sep 2018Magnon Valves Based on YIG/NiO/YIG All-Insulating Magnon J unctions\nC. Y. Guo,∗C. H. Wan,∗X. Wang, C. Fang, P. Tang, W. J. Kong,\nM. K. Zhao, L. N. Jiang, B. S. Tao, G. Q. Yu, and X. F. Han†\nBeijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, University of Chinese Academy of Scie nces,\nChinese Academy of Sciences, Beijing 100190, China.\n(Dated: October 2, 2018)\nAs an alternative angular momentum carrier, magnons or spin waves can be utilized to encode\ninformation and breed magnon-based circuits with ultralow power consumption and non-Boolean\ndata processing capability. In order to construct such a cir cuit, it is indispensable to design some\nelectronic components with both long magnon decay and coher ence length and effective control\nover magnon transport. Here we show that an all-insulating m agnon junctions composed by a\nmagnetic insulator (MI 1)/antiferromagnetic insulator (AFI)/magnetic insulator (MI2) sandwich\n(Y3Fe5O12/NiO/Y 3Fe5O12) can completely turn a thermogradient-induced magnon curr ent on or\noff as the two Y 3Fe5O12layers are aligned parallel or anti-parallel. The magnon de cay length in NiO\nis about 3.5 ∼4.5 nm between 100 K and 200 K for thermally activated magnons . The insulating\nmagnon valve (magnon junction), as a basic building block, p ossibly shed light on the naissance of\nefficient magnon-based circuits, including non-Boolean log ic, memory, diode, transistors, magnon\nwaveguide and switches with sizable on-off ratios.\nPACS numbers: 72.25.Rb, 72.25.Ba, 73.50.Bk, 73.40.Rw\nINTRODUCTION\nData processing and transmission in sophisticated mi-\ncroelectronics rely strongly on electric current, which in-\nevitably wastes a large amount of energy due to Joule\nheating. Magnons represent the collective excitations in\nmagnetic systems. Though charge neutral, they possess\nangular momenta and can also transfer the momenta as\ninformation carrierfree from Joule heating [1]. The main\ndissipation channel of magnons in magnetic insulators is\nspin-lattice coupling which is much weaker than Joule\nheating. Moreover, the wave nature of magnons pro-\nvides additional merits, (1) long propagation distance\nup to millimeters [2, 3] and (2) a new degree of free-\ndom (magnon phase) owing to which non-Boolean logic\nprocessing [1, 4–6] are anticipated. These characteristics\nmake magnons the ideal information carriers based on\nwhich some electronic components for future magnonic\ncircuits are being developed recently [7–11].\nBenderet al.[7] theoretically proposed a spin valve\nwith magnetic insulator (MI)/nonmagnetic metal/MI\nstructure where magnetization switching induced by\nthermally driven spin torques was expected. Wu et al.[8]\nproposed a concept of magnon valve with heterostruc-\nture of MI/spacer/MI and experimentally realized the\nheterostructure in a YIG/Au/YIG sandwich whose spin\nSeebeck effect (SSE) depends on the relative orientation\nof the top and bottom YIG layers. In this structure,\nmagnon transport occurs in two YIG layers while spin\ntransport in Au remains limited to the electrons. Angu-\nlar momentum transfer through the structure thus relies\non mutual conversion between magnon spin current and\nelectron spin current. Cramer et al.[9] prepared another\ntype ofhybridspin valvesin YIG/CoO/Co. In this valve,\ninverse spin Hall voltage (ISHE) in Co induced by spinpumping effect is determined by spin configurations be-\ntween YIG and Co. The ferromagnetic Co electrode can\nsupportboth electronand magnoncurrents, andthus the\noutput signalhaveboth electronspin and magnoncontri-\nbutions [3, 9]. In the above cases, additional conversions\nbetween magnon current and spin current or vice visa\ncould first reduce effective decay length of angular mo-\nmentum. Second, more noticeably, magnon phase would\nbe lost in the above conversions since phase information\ncannot be encoded by ordinary spin current.\nThus, it is very desirable to construct pure magnon\nvalves in an all-insulating structure such that the spin\ninformation propagation is uniquely limited to magnons.\nVery recently, a sandwichconsisting oftwo ferromagnetic\ninsulators and an antiferromagnetic spacer was proposed\nby Cheng et al.[11] where both giant spin Seebeck effect\nand magnon transfer torques were predicted.\nHere we design and further experimentally realize a\ntypical MI/antiferromagneticinsulator (AFMI)/ MI het-\nerostructure using YIG/NiO/YIG sandwiches. We en-\ntitle such a MI/AFMI/MI heterostructure as insulating\nmagnon junction (IMJ) for short. Output magnon cur-\nrent of an IMJ generated by SSE can be regulated by its\nparallel (P) or antiparallel (AP) states. Especially, the\noutput magnon current can be totally shut down in the\nAP state while superposed in the P state near room tem-\nperature,contributingtoalargeon-offratio. Demonstra-\ntion of the pure magnon junction based on all-insulators\nmay further help to develop magnon-based circuits with\nfast speed and ultralow energy dissipation in the coming\nfuture.2\nFIG. 1 Microstructure of the GGG//YIG(100)/NiO(15)/YIG(60 n m) IMJ. (a) The cross-sectional TEM im-\nage of the sample. HRTEM images of (b) the GGG//YIG(100 nm) and ( c) YIG(100)/NiO(15)/YIG(60 nm)\ninterfaces, respectively. Fourier transformation of HRTEM for ( d) the bottom YIG and (e) the top YIG\nonly. (f) Larger area HRTEM for the YIG/NiO(15 nm)/YIG IMJ and ( g) inverse Fourier Transformation\nof the yellow ring in the inset diffraction pattern which is obtained by Fo urier transformation of Fig.1(f).\nEXPERIMENTS\nIMJs stacks YIG(100)/NiO( t)/YIG(60 nm) ( t= 4, 6,\n8, 10, 15, 20, 30, 60, all thickness number in nanometers)were deposited on Gd 3Ga5O12(GGG) (111) substrates3\nin a sputtering system (ULVAC-MPS-4000-HC7 model)\nwith base vacuum of 1 ×10−6Pa. After deposition, high\ntemperature annealing in an oxygenatmosphere was car-\nried out to further improve the crystalline quality of the\nYIG layers. The stacks with t= 6, 8, 15, 20, 30 and 60\nnm were fabricated in the same round. Then a 10 nm Pt\nstripe with 100 µm×1000µm lateral dimensions for SSE\nmeasurement were fabricated by standard photolithog-\nraphy combined with an argon-ion dry etching process.\nFinally, an insulating SiO 2layer of 100 nm and a Pt/Au\nstripe were successively deposited on top of the Pt stripe\nfor on-chip heating. Before platinum deposition, vibrat-\ning sample magnetometer (VSM, EZ-9 from MicroSense)\nwas used to characterize magnetic properties of the YIG\nlayers. After microfabrication process, SSE of the IMJs\nwere measured in a physical property measurement sys-\ntem (PPMS-9T from Quantum Design). Keithley 2400\nprovided a heating current ( I) to the Pt/Au stripe while\nKeithley 2182 picked up a voltage along the Pt stripe\ninduced by SSE and ISHE. Magnetic field was applied\nalong the transverse direction of the Pt stripe. Control\nsamples YIG(100)/Pt(10nm), YIG(100)/NiO(15)/Pt(10\nnm) and NiO(15)/YIG(60)/Pt(10 nm) were also pre-\npared on GGG substrates and measured for comparison.\nWe have also confirmed insulating nature of oxide parts\nin the YIG/NiO/YIG and the control stacks by electrical\ntransport measurements.\nRESULTS AND DISCUSSION\n(1) STRUCTURE CHARACTERIZATION\nFig.1 shows crystalline structure of an IMJ stack\nGGG//YIG(100)/NiO(15)/YIG(60 nm). The NiO\nspacerhas uniform thickness without pinholes (Fig.1(a)).\nThe bottom YIG (B-YIG) is epitaxially grown on GGG\nsubstrate with atomically sharp interface (Fig.1(b)).\nFourier transformation of the high-resolution transmis-\nsion electron microscope (HRTEM) in the inset only\nshows one diffraction pattern, further confirming the\nepitaxial relation. However, the interfaces of the NiO\nspacer with adjacent YIG layers become rougher than\nthe GGG//YIG interface, especially for the interface\nwith the top YIG (T-YIG) (Fig.1(c)). It is worth not-\ning the two YIG layers are both single-crystalline but\nwith different orientations (Fig.1(d) and (e)). The B-\nYIG grows along [111] direction of the substrate while\ntheT-YIGdoesnot, probablybecausethepolycrystalline\nNiO spacer broke epitaxial relation (Fig.1(c)). Fig.1(f)\nshows a HRTEM image acquired across the bottom-\nYIG/NiO/top-YIG interfaces. Two sets of diffraction\npatterns corresponding to Fig.1(d) and Fig.1(e) can be\nidentified. From the patterns and lattice parameters of\nYIG, we can accurately calibrate camera length of the\nTEM. Then, besides of the already-known patterns ow-\ningtoYIGs, diffractionpatternsowingtoNiOcanbealsoidentified as highlighted by the yellow ring and the two\nred circles in Fig.1(g) inset. The yellow ring corresponds\nto (200) plane of NiO while the red circles correspond to\n(111) plane of NiO. After inversely Fourier transforming\nthe yellow ring pattern, NiO polycrystals can be clearly\nobserved as shown in Fig.1(g).\n(2) SPIN SEEBECK EFFECT MEASUREMENT\nFig.2(a) schematically shows setup to measure SSE of\nan IMJ. A Pt/Au electrode on top of a 100 nm SiO 2\ninsulating layer is heated by a current and then a tem-\nperature gradient ∇Talong the stack normal (+ zaxis)\nis built. ∇Tintroduces inhomogeneous distribution of\nmagnons inside a MI and produce a magnon current\nalong∇T[12, 13]. The magnon current can be fur-\nther transformed as a spin current penetrating into an\nadjacent heavy metal and then generate a sizable volt-\nage by ISHE [14–17], which is so-called longitudinal SSE.\nHere we use a Pt stripe to measure the voltage ( VSSE)\ninduced by ISHE and monitor the magnitude and direc-\ntion of spin current exuded from the top YIG. Fig.2(b)\nshows angle scanning of VSSEof an IMJ with tNiO=8\nnm. The Pt stripe is along the yaxis. Therefore SSE\ncan be observed only if magnetization has component in\nthexaxis. This requirement gives the angle dependence\nshowninFig.2(b). Fig.2(c)showsspinSeebeckvoltageas\na function of applied field measured at different heating\ncurrents. Saturated VSSEparabolicly depends on current\n(Fig.2(d)). This parabolic dependence confirms the ther-\nmopower essence of the measured voltage signals while\nthe angle dependences confirm the voltage signals are in-\nduced by spin Seebeck effect. We have also estimated\ntemperature rise (∆ T) of spin detector (Pt stripe) as el-\nevating heating current by calibrating resistivity of the\nPt stripe. Heating current of 10 mA, 15 mA and 20 mA\nwould lead to ∆ Tof 2.7 K, 6.0 K and 10 K, respec-\ntively, as background temperature within (50 K, 325 K).\nIn order to enhance signal-to-noise ratio and minimize\ninfluence of heating current on IMJs, we have selected\n15 mA as heating current to conduct the following SSE\nmeasurement in Fig.4 and Fig.5.\n(3) FIELD DEPENDENCE OF VSSEOF AN\nINSULATING MAGNON JUNCTION\nFig.3(a) shows a typical hysteresis of an IMJ\nYIG(100)/NiO(8)/YIG(60 nm). Two magnetization re-\nversals have been identified. Epitaxial growth and larger\nthickness endow the bottom YIG with lower coercivity\n(HC) and larger magnetization than the top YIG on the\nNiO spacer. Thus, the reversal with HC≈3 Oe and\n∆M≈1.3(MS,B+MS,T) is attributed to the switching\nofthe bottom YIG while the other reversalwith HC≈16\nOe and ∆ M≈0.7(MS,B+MS,T) is attributed to the top\nYIG.MS,BandMS,Tare saturated magnetization of the4\nFIG. 2 (a) Schematic diagram of spin Seebeck ef-\nfect and its measurement setup for an IMJ. Dur-\ning measurement, a field in the x-axis is applied.\n(b) Angle scanning of VSSEof an IMJ with tNiO=8\nnm. The Pt stripe is along the yaxis. (c) Spin\nSeebeck voltage as a function of applied field mea-\nsured at elevated heating currents. (d) Parabolic\nfitting of the field dependence of saturated VSSE.\nbottom and top YIG layers, respectively. The obtained\nMS,B/MS,Tis 13/7, close to the ratio 5/3 in the nomi-\nnal thickness. Due to the difference in coercivity, parallel\nand antiparallel spin configurations can be formed, as il-\nlustrated in the figure.\nFig.3(b) shows field-dependence of VSSE. Besides of a\nlarge ∆VSSE≈1.6VSSEmaxoccurring at 13 Oe, VSSEalso\nsharply changes by about 0.4 VSSEmaxat 2 Oe. VSSEmax\nis the saturation value (2.6 µV) in Fig.3(b). d VSSE/dH\nand dM/dH(Fig.3(c)) are used to show correspondence\nbetween SSE and VSM results. A peak in Fig.3(c) rep-\nresents a sharp reversal of a YIG layer. There are four\npeaks in both field dependences. The middle two la-\nbeled as (P1-) and (P1+) originate from the reversal of\nthe bottom YIG while the outer ones marked as (P2-)\nand (P2+) are caused by the reversal of the top YIG.\nSSE of the control samples YIG/NiO/Pt, NiO/YIG/Pt\nand YIG/Pt have also been measured (Fig.3(d)). Ex-\ncept YIG/NiO/Pt, the other samples show comparable\nVSSEat the same heating current, which may be owing\nto higher spin mixing conductance of YIG/Pt interface\nthan that of NiO/Pt interface in our case. YIG/NiO/Pt\nand YIG/Pt are indeed much softer than NiO/YIG/Pt.\nFurthermore, only onemagnetizationreversalis observed\nfor the control samples. If the NiO spacer is replaced by\nan MgO spacer, VSSEsignal due to the reversal of the\nbottom layer disappears as shown in Ref [8]. The above\nobservation indicates that the magnon current from the\nbottom layer can flow through the NiO spacer and thetop YIG and finally penetrate into Pt. Magnon decay\nlength in epitaxial and polycrystalline YIG is about 10\nµm [2] and several tens of nanometers [18, 19], respec-\ntively. Wang et al. [20] reported spin relaxation length\nof 9.8 nm in NiO. Our IMJs have comparable dimensions\nwith those reported values, indicating magnon current\nfrom the bottom YIG layer capable of flowing into Pt.\nRemarkably, though solidly confirmed in experiments,\nlongitudinal spin Seebeck effect is regarded to be po-\ntentially caused by (1) difference in electron tempera-\nture and magnon temperature across an interface be-\ntween heavy metal and magnetic insulator [12, 13] or by\n(2) inhomogeneous magnon distribution inside bulk re-\ngionof amagnetic insulatorand as-inducedpure magnon\nflow [21, 22]. These two mechanisms are hard to tell in\nclassic magnetic insulator/heavy metal bilayer systems.\nThough not ruled out possibility of the 1stmechanism,\nnevertheless, our experiment strongly proved rationality\nof the 2ndmechanism since the bottom YIG layer could\nonly deliver magnon current into Pt via the bulk effect.\nFIG. 3 (a) Hysteresis loop and (b) field depen-\ndence of VSSEfor an IMJ with t= 8 nm at 300\nK. (c) Corresponding field dependences of d VSSE/dH\nand dM/dH. (d) VSSEof the control samples.\n(4)TANDtNiODEPENDENCE OF SSE OF\nINSULATING MAGNON JUNCTIONS\nWe have measured field dependence of VSSEat 15 mA\nwith elevating Tfor different IMJs as shown in Fig.4. In\norder to distinguish switching fields from different YIG\nlayers, their d VSSE/dHare shown in Fig.5. First, all\nIMJs show a significant exchange bias below blocking\ntemperature of about 100 K (Fig.4, Fig.5 and Fig.6(b)),\nwhich evidences the appearance of NiO antiferromag-\nnetism. Similarblockingtemperaturesforallthe samples\nindicate interfacial nature of exchange bias effect whose5\nmagnitude is dominantly determined by exchange cou-\npling strength between interfaciallayersofNiO and YIG.\nSecond, only two peaks are unambiguously identifiable\nfor the IMJ with 30 nm and 60 nm NiO at all temper-\natures (Fig.5(g,h)). These peaks belong to (P2+) and\n(P2-) because their positions are identical with those\nof YIG(100)/NiO(15)/YIG(60 nm) determined by VSM\nandHCof NiO/YIG/Pt as shown in Fig.6(a). Due to\ntoo thick NiO spacer and its blocking effect on magnon\ncurrent, the Pt stripe can only detect magnon current\nfrom the top YIG. Thus it is nature that the peaks in\nFig.5(g,h) share the same positions with P2+ and P2-.\nFort=6∼20 nm, temperature evolution of d VSSE/dH\nvs.Hcurves seem nontrivial. Within a certain tempera-\nture region, 4 peaks corresponding to 4 switching events\nof two YIG layers can be clearly resolved. For the t=\n15 nm IMJ, for example, four peaks can be clearly re-\nsolved between 50 K and 275 K. The relative intensity\nof (P1+/P2+) or (P1-/P2-) decreases gradually with in-\ncreasing T(Fig.5(e)). The trend has also been repro-\nduced in the IMJ with t= 8 nm (Fig.5(c)). At T <175\nK, the pair of peaks (P1+) and (P1-) are dominant. At\nintermedium temperature from175K to325K, the other\npair of peaks (P2+) and (P2-) emerge and are enhanced\nwith increasing T. AtT >325 K, (P1+) and (P1-) even-\ntually fade away but (P2+) and (P2-) remain (Fig.5(c)).\nThis trend indicates the layer dominating SSE of an IMJ\ncan be changed between the two YIG layers though spin\ndetector is only directly connected to the top layer.\nThe IMJ with 4 nm NiO is unique, in which only P1+\nand P1- are observed at all temperatures, which is also\nindicated in Fig.6(a). Though P2+ and P2- are absent\nhere, two broad shoulders outsides of P1+ and P1- can\nbe identified above 275 K, which probably still originate\nfrom switching of the top YIG. For thin enough NiO\nspacer, magnon current generated in the bottom layer\ncan still survive after a weak decay in the NiO spacer at\nhigh temperatures. Thus VSSEdue to the bottom layer\nis still observable or even dominated in this case.\n(5) MAGNON DECAY LENGTH OF NIO\nIn order to further check the correlation between the\npeaks at different temperatures in Fig.5 and switch-\ning fields of the YIG layers, we have plotted them\n(open symbols) together with HCof YIG/NiO/Pt and\nNiO/YIG/Pt (solid triangles) determined by SSE and\nHCof YIG/NiO(15)/YIG determined by VSM (solid\nhexagons) in Fig.6(a). Remarkably, nearly all the peaks\nlie on 4 branches defined by HCof the control samples\nandHCfrom the VSM results, which unambiguously\ndemonstrates origin of the four peaks in the entire tem-\nperature range, i.e., the outer peaks from the top and the\ninnerpeaksfromthe bottom YIG. Thus, wecanconclude\nthe magnon current that overwhelmingly contributes to\nSSE comes from the bottom YIG at low temperatureswhile the magnon current from the top YIG becomes\nsignificant at high temperatures. A plausible explana-\ntion of the relative contributions of the magnon current\nfrom the two YIG layers is as follows. The bottom YIG\ngrownon GGG has much better quality and thus a larger\nSSE coefficient. At low temperature and for a small NiO\nthickness, the magnon current from the bottom layer is\nable to propagate through both NiO and the top YIG\nwithout noticeable decay. When NiO becomes thicker,\nmagnon current from the bottom YIG decays significant.\nOn the other hand, the magnon current of the top YIG\ndoes not suffer such decay since it is in direct contact\nwith Pt. Thus the bottom and top YIG contribute more\ndominantly at low and high temperatures, respectively.\nFig.6(c) summarizes magnon valve ratio ηmvof the\nIMJs as a function of T. Here ηmvis defined as\nVSSE,AP/VSSE,P.VSSEinduced by ISHE is propor-\ntional to the injected magnon current Jmflowing to-\nward Pt from the top YIG layer. According to the\nSSE theory [12, 13, 21, 22], Jm,T/Bis proportional to\nST/B(∇T)T/B. Here Jm,Tis the spin Seebeck coeffi-\ncient of the top/bottom YIG and ( ∇T)T/Bis the in-\nduced temperature gradient across the top/bottom YIG\nand proportional to I2. ThusVSSE∝I2, as confirmed in\nFig.2(d). The Jm,P/AP=Jm,T±aNiOaYIGJm,Bin the P\nand AP states. Here, Jm,TandJm,Bare the generated\nmagnon currents by SSE in the top and bottom YIG,\nrespectively, while aNiOandaYIGare the decay ratio of\nthe magnon current from the bottom YIG in NiO and\nthe top YIG, respectively. The magnon valve ratio ηmv\ncan be thus rewritten as\nηmv=Jm,AP\nJm,P=Jm,T−aNiOaYIGJm,B\nJm,T+aNiOaYIGJm,B(1)\nηmvsign reflects the relative magnitude of the magnon\ncurrents from the top and the bottom layers. The nega-\ntive values in Fig.6(c) indicates the magnon current from\nthe bottom YIG is larger at low temperature. The posi-\ntiveηmvseen in the 6 and 8 nm NiO IMJs of Fig.6(c) at\nhigh temperatures means a larger magnon current from\nthe top YIG. The most interesting case is ηmv= 0 where\nthe net magnon current at the YIG/Pt interface becomes\nzero, i.e., the exact cancellation of the two magnon cur-\nrents generated by two YIG layers (inset of Fig.6(c)).\nSuch cancellation only occurs at the AP state. The fig-\nurealsoshowsatrendthatthecriticaltemperaturewhere\nηmv=0 increases with decreasing tNiO.\nNext, weestimatethemagnondecaylength λNiOinthe\nNiO spacers. Magnon decay ratio in NiO is aNiO. From\nEq.(1), one can easily see aNiOis proportional to δ= (1-\nηmv)/(1+ηmv). We then plot ln δas a function of tNiObe-\ntween 100 K to 200 K (Fig.6(d)). In this temperature re-\ngion, the switching fields of both YIGs are well separated\nfor the IMJs. Worth noting, data from the IMJs with 4\nnm and 10 nm NiO spacer are not used here. For the\nIMJ with 4 nm NiO, only two peaks are clearly observed\n(Fig.5(a)). Thus it is hard to obtain reliable ηmvfor the6\nFIG. 4 Field dependence of VSSEat elevated temperatures for IMJs with (a-h)\nt=4 nm, 6 nm, 8 nm, 10 nm, 15 nm, 20 nm, 30 nm and 60 nm, respectively.\ndevice. For the IMJs with 10 nm and 4 nm NiO, the\nstacks were deposited and annealed in different rounds\nwith the others. The other stacks were fabricated in the\nsameroundandtheirdataweresystematicandthusmore\ncomparable. Fig.6(d) shows a good linear dependence\nof lnδontNiO, which suggests that magnons generated\nin the bottom YIG can pass through NiO in a diffusive\nway [11]. The inset in Fig.6(d) shows T-dependence of\nthe derived λNiOwhich increases slightly with T.λNiO\nis about 3.5 nm ∼4.5 nm between 100 K and 200 K.\nThough λNiOslightly increases with T. However, VSSE\nfrom the top YIG gradually dominates at higher temper-\natures. Itindicatesnotonlymagnontransportproperties\nof an IMJ but also T-dependent spin Seebeck coefficients\nof YIG layers and thermoconductivity of YIG and NiO\nlayers will eventually affect the magnon valve effect. The\ninfluence of these parameters is already out of scope of\nthis article. Therefore, we only use magnon valve ratio\nand its thickness dependence at a fixed temperature to\nmeasuremagnondecaylengthasshowninthemainpanel\nof Fig.6(d).\nSpin decay length in antiferromagnetic materials such\nas NiO, CoO, Cr 2O3and IrMn have been measured\nby spin pumping or spin Hall magnetoresistance tech-\nniques [20, 23–29]. For NiO thin films, λNiOis reported\nabout 10 nm [20, 23]. Our value has the same orderof magnitude with theirs. Qiu et al[27, 28] reported\nan enhanced spin pumping effect near N´ eel temperature\nTNof antiferromagnetic materials. For 6 nm CoO and\n1.5 nm NiO, they observed the most significant enhance-\nment at about 200 K and 285 K, respectively [27]. In\nour case, NiO spacers have thickness of 4 nm ∼60 nm.\nThey probably have even higher TN. Then the increase\ninλNiOwithin (100 K, 200 K), we think, is also due to\nsimilar enhancement in magnon transport efficiency as T\napproaching TN.\n(6) EXCHANGE BIAS DEPENDENCE OF SSE OF\nIMJS\nWe have also changed exchange bias direction of the\nsame IMJ with 8 nm NiO by field cooling technique. In\nthis case, we first elevated temperature to 400 K (the\nhighest temperature of our PPMS system) and then ap-\nplied 5 T field along different directions (along the Pt\nstrip or in-plane vertical to the Pt stripe or normal to\nstacks) and then cooled the device down to 10 K with\nthe field of 5 T maintained. Finally, the high field was\nreduced to 0 in oscillating mode to minimize remanence\nfield of magnet. After all the above procedures, we be-\ngan SSE measurement with the Pt stripe along the yaxis7\nFIG. 5 Field dependence of d VSSE/dHat elevated temperatures for IMJs with (a-\nh)t=4 nm, 6 nm, 8 nm, 10 nm, 15 nm, 20 nm, 30 nm and 60 nm, respectively.\nand field applied along the xaxis. At 10 K, only the case\nwith cooling field in-plane vertical to the stripe shows re-\nmarkable exchange bias effect while the other two cases\nshow small or even negligible exchange bias effect. This\nmeans we have successfully changed the exchange bias\ndirections by the above procedures. However, as shown\nin Fig.7, all the three cases show nearly the same sat-\nurationVSSEat all the temperatures. Furthermore, we\ncan see from the data between 150 K and 300 K that\nmagnon valve ratio was also independent on exchange\nbias directions. It indicates different exchange coupling\ndirections at the interfaces would not deterioratetransfer\nefficiency of magnon current across the NiO/YIG inter-\nfaces, which is luckily benefit for applications. Indepen-\ndenceofspintorquetransferondirectionofexchangebias\nwas very recently reported in metallic system [29]. Our\ndata show this independence was also solidly reproduced\nfor magnon transfer.\nCONCLUSION\nIn conclusion, fully electric-insulating and merely\nmagnon-conductive magnon valves have been demon-\nstrated by magnetic insulator YIG/antiferromagnetic in-sulator NiO/magnetic insulator YIG IMJs in which out-\nput spin current in Pt detector can be regulated with\nan high on-off ratio between P and AP states of the\ntwo YIG layers near room temperatures. The magnon\ncurrent is dominated by the bottom (top) YIG layer at\nlow (high) temperature regions. The transition temper-\nature depends on tNiO. The magnon decay length in\nNiO is about several nanometers. Magnon transfer effi-\nciency is independent on exchange bias directions. Most\nimportantly, similar to the fundamental role played by\nmagnetic tunnel junction (MTJ) in spintronics, the IMJ\ncan also provide a basic building block for magnonics\nand oxide spintronics. Pure magnonic devices/circuits\nbased on IMJs can be constructed in an ideally insu-\nlating system with magnon being the only angular mo-\nmentum carrier. In these devices/circuits, information\nprocessing and transport can be accomplished only by\nmagnons without mobile electrons and ultralow energy\nconsumption and more versatile functions such as non-\nBoolean logics utilizing magnon phase coherence, mag-\nnetic memory based on insulators, magnon diode, tran-\nsistors, waveguide and switches with large on-off ratios\ncan be expected.8\nFIG. 6 The dependence of (a) switching fields and (b)\nexchange bias of all the IMJs and the control samples on\ntemperatures. (c) VSSE,AP/VSSE,Pratios for the IMJs.\nInset shows the field dependence of VSSEfor the IMJ\nwith 6 nm NiO spacer and sizeable on-off ratio at 260\nK. (d) Thickness dependence of ln δat medium tem-\nperatures. Insets show derived λNiOas function of T.\nACKNOWLEDGEMENTS\nWe gratefully thank Prof. S. Zhang in Univer-\nsity of Arizona for enlightening discussions. This\nwork was supported by the National Key Research\nand Development Program of China [MOST, Grants\nNo. 2017YFA0206200], the National Natural Sci-\nence Foundation of China [NSFC, Grants No.11434014,\nNo.51620105004, and No.11674373], and partially sup-\nported by the Strategic Priority Research Program (B)\n[Grant No. XDB07030200], the International Partner-\nship Program (Grant No.112111KYSB20170090), and\nthe Key Research Program of Frontier Sciences (Grant\nNo. QYZDJ-SSWSLH016) of the Chinese Academy of\nSciences (CAS).\n∗These two authors contributed equally to this work.\nFIG. 7 Independence of magnon transfer on di-\nrections of exchange bias. Three field cooling ge-\nometries are also shown as insets. Colors of the\nVSSEvs.Hcurves are the same with the cooling\nfields in the left insets. FC, IP and OOP denote\ncooling field, in-plane and out-of-plane, respectively.\n†Corresponding Author: xfhan@iphy.ac.cn\n[1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. 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Using these ultralow damping YIG film s, we demonstrate for the first time strong \ncoupling between magnons in patterned YIG thin films and microwave photons in a \nsuperconducting Nb resonator. This result pave s the road towards scalable hybrid quantum \nsystems that integrate superconducting microw ave resonators, YIG film magnon conduits, and \nsuperconducting qubits into on-chip QIS devices. 2\nY3Fe5O12 (YIG) is a well-known ferrimagnetic in sulator with extremely low magnetic \ndamping, which makes it one of the best ma terials for fundamental studies and potential \napplications in magnonics, spintronics, and QIS.1-8 To date, mm-scale YIG single-crystal \nspheres have been the material of choice fo r coherently coupling magnons to superconducting \nqubits4 as well as for quantum sensing.4, 7, 9, 10 For QIS applications that require scalable on-\nchip integration, ultralow-dam ping magnetic films, which can be patterned and integrated in \nhybrid quantum systems, are highl y desired. However, there is a major barrier for using YIG \nthin films in QIS because epitaxial YIG films are mostly grown on Gd 3Ga5O12 (GGG) or other \ngarnet substrates containing ra re-earth elements, which induce excessive damping loss in YIG \nfilms across the interface at low temperatures.11 If this substrate-induced damping enhancement \ncan be eliminated while maintaining high YIG film quality, it will enable scalable on-chip QIS \ndevices based on ultralow-d amping YIG films operati ng at mK temperatures. \nPreviously, we have shown that a diam agnetic epitaxial buffer layer of Y 3Sc2.5Al2.5O12 \ncan effectively separate the magnetic coupli ng between the YIG film and GGG substrate, \nresulting in much improved damping at low temperatures.12 Here, we demons trate the growth \nof YIG epitaxial films on a new diamagnetic Y 3Sc2Ga3O12 (YSGG) single-crystal substrate \ncontaining no rare-earth elemen ts, which exhibit extremely low damping that decreases rapidly \nwith temperature below 5 K. By integrating patterned YIG epitaxial films on YSGG with \nsuperconducting Nb coplanar waveguide (CPW) microwave resonators, we observe strong \ncoupling between microwave phot ons and magnons at 2 K. \nYIG thin films are epitaxially grown on YSGG (111) substrates using off-axis \nsputtering13 at a substrate temperature of 675 ℃. The bulk lattice constant of YSGG is a = \n12.466 Å, which causes a 0.72% tensil e strain in the YIG film (bulk a = 12.376 Å). The \ncrystalline quality of the YIG/ YSGG films is characterized by X-ray diffraction (XRD) and X-\nray reflectivity (XRR), as shown in Figs. 1a and 1b, respectivel y. The clear Laue oscillations 3\nof the YIG (444) peak indicate highly coherent crystalline orde ring. The XRR peaks reveal the \nfilm thickness (30 nm) and low interfacial and surface roughness. \nBroadband ferromagnetic resonance (FMR) measurement is performed on a YIG(30 \nnm)/YSGG(111) film at va rious temperatures ( T) between 2 and 300 K in a Physical Property \nMeasurement System (PPMS) from Quantu m Design, as described previously.12 We obtain \nFMR absorption spectra as a function of an in -plane magnetic field at various microwave \nfrequencies ( f) and temperatures. Figure 2a shows a re presentative derivative FMR spectrum \nat T = 2 K and f = 4.3 GHz. The resonance field 𝐻୰ and linewidth Δ𝐻 are extracted from fitting \nthe FMR spectrum using 𝑦ൌሾ𝐴െ𝐵ሺ𝐻െ𝐻୰ሻሿ ሾሺ𝐻െ𝐻୰ሻଶሺΔ𝐻/2ሻଶሿଶ⁄ , where 𝐴and 𝐵 are \nthe symmetric and antisymmetric amplitudes of th e lineshape, respectively. From Fig. 2a, we \nobtain 𝐻୰ = 710 Oe and Δ𝐻 = 13.2 Oe ( √3ൈ of peak-to-peak linewidth of 7.6 Oe). \nWe also measure the frequency-dependent FMR absorption at 2 K with a fixed in-plane \nfield of 710 Oe using a vector network analyzer (VNA), as shown in Fig. 2b. The resonance \nfrequency 𝑓୰ and linewidth ∆𝑓 are extracted from fitting the spectrum with a Lorentzian \nfunction 𝑦ൌ𝐴ሾሺ𝑓െ𝑓୰ሻଶሺΔ𝑓/2ሻଶሿ ⁄ . The obtained 𝑓୰ = 4.307 GHz and ∆𝑓 = 36.3 MHz \nagree well with the field linewidth Δ𝐻 = 13.2 Oe as related by th e gyromagnetic ratio of free \nelectron, 𝛾2π⁄ 2.8 MHz/G, where 36.3 MHz corresponds to 13.0 Oe. \nFigure 2c shows the extracted field linewidths of the YIG(30 nm)/YSGG film as a \nfunction of frequency at T = 2 to 300 K. As temperature decreases from 300 K, the linewidth \nand the slope of their frequency dependence in crease and peak at 20-30 K, after which both \nΔ𝐻 and the slope decrease quickly down to 2 K ( limited by our PPMS). This behavior agrees \nwith the slow-relaxation theory in YIG,14-16 where the existence of rare-earth impurities in YIG \ninduces a significant enhancement of relaxation at temperatures of 10’s K. At temperatures \nbelow this regime, the linewidth an d the slope decrease significantly. \nAccording to the Landau-Lifs hitz-Gilbert (LLG) equation,17 the FMR linewidth is 4\nlinearly dependent on the microwave fre quency with the slope determined by the \nphenomenological Gilbert damping coefficient ( 𝛼): Δ𝐻 ൌ 4π𝛼𝑓 𝛾⁄Δ𝐻, where Δ𝐻 is the \ninhomogeneous broadening whic h is generally attributed to magnetic nonuniformity18 and \nsurface defects.19 We note that the LLG equation is a simplified theory and does not explain \nthe nonlinearity observed in the frequency dependence of Δ𝐻, particularly at low temperatures, \nwhich requires additional contri butions such as the slow-relax ation mechanism. Nonetheless, \nthe phenomenological damping coefficient α from the LLG equation can serve as a figure of \nmerit for a magnetic material. \nWe fit the linewidth vs. frequency data in Fig. 2c using the LLG equation and extract \ndamping 𝛼and inhomogeneous broadening Δ𝐻 for each temperature, as shown in Fig. 2d. The \ndamping starts at a very low value of 𝛼 = 3.3 10-4 at room temperature and increases as \ntemperature drops. After reaching a peak of 𝛼 = 1.3 10-3 at 45 K, the damping decreases \nquickly at lower temperatures. Remarkably, the FMR linewidths ar e essentially frequency \nindependent at 5 and 2 K, indi cating an essentially zero dampin g according to the LLG equation. \nThe rapid decrease of linewidth below 10 K implies that Δ𝐻 will likely decrease even further \nat sub-Kelvin temperatures. \nThe extremely low damping of the YI G epitaxial films on diamagnetic YSGG \nsubstrates at very low temperatures is highly pr omising for QIS studies. As an initial step in \nthis regard, we integrate the YIG/YSGG film s with superconducting resonators for the study \nof coupling between magnons in a YIG film and microwave photons emitted by a \nsuperconducting CPW resonator. First, we pattern a YIG(30 nm)/YSGG film into 10- μm wide \nstrips using photolithography a nd Ar ion milling. Then, 300-nm thick Nb resonators are \ndeposited by sputtering on the YS GG substrate with photolithogra phy patterns followed by lift-\noff such that the YIG strips lie within the gaps of the CPW resonators. \nThe microwave resonator design includes a main bus channel which is capacitively 5\ncoupled to two resonators, as shown in Fig. 3a . The two resonators are 13- and 13.5-mm long \nwith both ends open-circuit, so they resona te with half-wavelengt h standing waves, as \ndetermined by 𝑓ൌ𝑐2𝑙ඥ𝜖ୣ ⁄ , where 𝑐 is the speed of light, 𝑙 is the length of the resonator, \nand 𝜖ୣ is the average dielectric constant of vacuum and YSGG substrate. The center conductor \nof the CPW resonators is 20- μm wide with a spacing gap of 15- μm from the group conductor \non either side. A YIG strip is located within a gap of each resonator near the middle of the \nresonator length, as shown in the insets of Fig. 3a, which enables the ma gnetic field component \nof the resonating microwave to drive the FMR of the YIG strip. We fabricate two such devices \nwith a total of four Nb resonators coupled to four YIG strips with 10- μm width and lengths of \n300, 600, 900, and 1200 μm. In addition, we fabricate an other such device with two Nb \nresonators on YSGG without YI G strips as a reference sample . The devices are mounted on a \nhome-made sample holder and wire -bonded as shown in the insets of Fig. 3a, which is loaded \ninto the PPMS and cooled down to 2 K. \nWe measure microwave transmission spectr a of these devices using a VNA with an \noutput power of -40 dBm, where each spectrum is averaged over 40 scans. Figure 3b shows \nthe transmission spectrum (S21) of a device with two Nb resonato rs and no YIG, which exhibits \ntwo sharp dips at f = 4.203 and 4.364 GHz, corresponding to the resonances of the 13.5- and \n13-mm resonators, with high quality factors (Q) of 56,800 and 49,000, respectively. It is noted \nthat the measured resonance frequency ratio of (4.364 GHz)/(4.203 GHz) = 1.0383 is almost \nidentical to the length ratio of (13.5 mm)/( 13 mm) = 1.0385, validating the resonator design \nand fabrication. \nFigure 3c compares the transmission spectra of a bare Nb resonator (no YIG) and a Nb \nresonator coupled to a 10 1200 μm2 YIG strip at zero field or in the presence of a magnetic \nfield of 550 Oe, while Fig. 3d shows similar sp ectra for a Nb resonator coupled to a 10 600 \nμm2 YIG strip. All Nb resonators with or wit hout coupling to YIG strips exhibit high quality 6\nfactors between 44,000 and 72,700 at zero magnetic field. To evaluate the performance of the \nNb resonators in a magnetic field needed for strong microwave phot on-magnon coupling, we \napply an in-plane field of 550 Oe, wh ich lowers Q by a factor of ~10 , although the quality \nfactor remains quite high (>4,000). This behavi or together with shift of resonance frequency \ndue to the applied field and coupling to YIG are commonly seen in other microwave \nphotonmagnon coupling reports.20, 21 \nSuch resonator-ferromagnet hybr id systems can be modeled as a macrospin coupled to \nan LC resonator by the radio-frequenc y (rf) magnetic field component brf.20 The \neigenfrequencies of this syst em can be calculated as, \n𝜔േൌ൫𝜔𝜔ሺ𝐻ሻ൯2⁄േඥሺ𝜔ሺ𝐻ሻെ𝜔ሻଶ4𝑔ଶ2⁄, (1) \nwhere 𝜔୰ is the resonance frequency of a standalone microwave resonator, 𝜔ሺ𝐻ሻ is the \nstandalone ferromagnet’s FMR frequency (depende nt on the applied field 𝐻), and 𝑔 is the total \nphoton-magnon coupling strength. Th e total coupling strength scales with the number of spins \n𝑁 in the ferromagnet as 𝑔ൌ𝑔௦√𝑁, where 𝑔௦ is the coupling strength between microwave \nphotons and individual spins in the ferromagne t, which depends on the device design and \nmicrowave magnetic field strength brf at the ferromagnet. Note that the number of spins here \nis a net value due to YIG’s ferrimagnetism. \nWe perform field-dependent transmission measurement on the 13- and 13.5-mm Nb \nresonators to quantify their coupling strength to the 10 300 μm2, 10 600 μm2, 10 900 μm2, \nand 10 1200 μm2 YIG strips. Figures 4a-4d show the microwave transmission of the four \nresonator-YIG hybrid devi ces as a function of field and frequency measured at 2 K. The \nanticrossing features in the plots are the signatures of mi crowave photon-magnon coupling, \nwhere YIG’s FMR frequency 𝜔ሺ𝐻ሻ meets the microwave resonator’s resonance frequency \n𝜔 and the degeneracy is lifted. By comb ining Eq. (1) and the Kittel equation, 𝜔୫ሺHሻൌ7\n𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻ, where 𝑀ୣ is the effective saturation magnetization of the YIG film, we \nobtain the angular frequencies of the hybrid systems’ modes, \n𝜔േൌ൫𝜔୰𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻ൯2⁄േට൫𝛾ඥ𝐻ሺ𝐻4π𝑀ୣሻെ𝜔୰൯ଶ\n4𝑔ଶ2ൗ (2) \nWe point out that different aspect ratios of the YIG strips can give rise to different \ndemagnetizing factors, which slightly shift 𝜔୫ሺ𝐻ሻ. However, this shift is smaller than 1 Oe \nover the field and frequency range of interest and thus is ignored here. We fit the anticrossing \nfeatures with Eq. (2) and extract the coupling strength 𝑔 which is dependent on the magnetic \nvolume and the number of spins 𝑁. For the fitting, a background linearly dependent on the field \nis added to Eq. (2) to account for the fiel d-induced microwave resonator frequency shift. \nTo determine the number of spins in a YIG st rip, we measure magnet ic hysteresis loops \nfor the YIG(30 nm)/YSGG film using a superc onducting quantum interference device (SQUID) \nmagnetometer at 2 and 300 K, as shown in Fig. 4e. The saturation magne tization is determined \nto be 𝑀ௌ = 180 emu/cm3 at 2 K and 131 emu/cm3 at 300 K. The hysteresis loops remain largely \nsquare with a small coercivity of 1.1 and 3.1 Oe at 300 and 2 K, respectively, corroborating the \nhigh magnetic uniformity at both ro om and cryogenic temperatures. \nFigure 4f shows the linear de pendence of coupling strength 𝑔2𝜋⁄ on the square root of \nboth the YIG volume and the number of sp ins. From Fig. 2b, the decay rate 𝜅୫2π⁄ of YIG’s \nresonance mode near 4.3 GHz at 2 K and H = 710 Oe is determined to be 36.3 MHz. Next, the \ndecay rate 𝜅୰2π⁄of the Nb resonator is estimated to be 1.037 MHz based on the transmission \nlinewidth of the resonator coupled to the 10 600 μm2 YIG strip at H = 550 Oe as shown in \nFig. 3d. Thus, the cooperativity for the 10 600 μm2 YIG is calculated to be 𝑔ଶሺ𝜅𝜅ሻ ⁄ \n57.9, showing that the coupling strength is much stronger than the decay rates of individual \ncomponents, which is required for coherent coupling . Using 𝑀ௌ = 180 emu/cm3 at 2 K for the \nYIG/YSGG film, we determine the average couplin g strength to an individual spin to be 𝑔௦/2𝜋 8\n= 25.1 Hz. This coupling strength 𝑔௦/2𝜋is comparable to other reports on microwave photon-\nmagnon coupling using ferromagnetic metals20-22 and similar CPW re sonator-ferromagnet \nstructures. Considering that the microwave magne tic field in the gap of a CPW (where our YIG \nstrip is located) is weaker than that for a ferromagnet directly on top (or underneath) the \nsuperconducting center conductor channel, on e can achieve stronger microwave photon-\nmagnon coupling by positioning the YIG strip near a stronger microwave magnetic field or \nutilizing resonator designs such as lumped-element LC resonator, which can provide far \nstronger coupling strength.20 \nThis work provides the first demonstration that ultralow-damping YIG epitaxial films \non YSGG can be integrated with superconducto r resonators to achieve strong microwave \nphoton-magnon coupling at few Kelvin temperatur es. Such ultralow-damping YIG films offer \nclear advantages (in terms of decay rate) over metallic ferromagnets for on-chip hybrid \nquantum systems that incorporate magnonic c onduits, microwave superconductor resonators, \nand superconductor qubits for QIS applica tions that operate in the mK regime. \nIn summary, we demonstrate the growth of high-quality epitaxial YIG thin films on \ndiamagnetic YSGG substrate, which exhibi t narrow FMR linewidt h and extremely low \ndamping at 2 K. We couple these YIG films to superconducting Nb resonators to create hybrid \nstructures that achieve strong microwave photon-magnon coupling. This work demonstrates \nthe potential power of ultralow-damping YIG films for scalable, integrated QIS applications at \nlow temperatures. \nThis work was primarily supported by the Center for Emergent Materials: an NSF \nMRSEC under award number DMR-2011876. D.R. acknowledges partial support from the \nNational Science Foundation under award numbe r DMR-2225646 (YIG film growth and X-\nray characterizations). 9\nFigure Captions: \nFigure 1 . X-ray diffraction results of YIG films. (a) 2𝜃/𝜔XRD scan and (b) XRR scan of \na YIG(30 nm) epitaxial film grown on YSGG (111) substrate, indicati ng the high crystalline \nquality of the YIG film. \nFigure 2 . FMR measurements of a YIG(30 nm)/YSGG film. (a) Field-dependent derivative \nFMR spectrum at 2 K driven by a microwave frequency of 4.3 GHz. (b) Frequency-dependent \nFMR spectrum at 2 K with an in-plane fiel d of 710 Oe. (c) Frequency dependence of FMR \nlinewidth at different temperatures, from wh ich the damping constant and inhomogeneous \nbroadening are obtained by linear fitting. (d ) Temperature dependence of damping (red) and \ninhomogeneous broadening ∆𝐻 (blue). The error bars are from the linear fitting in (c). \nFigure 3 . Microwave transmission of Nb CPW resona tors with or without YIG strips and \nmagnetic field at 2 K. (a) Schematic of Nb resonator devi ce with YIG strips placed within \ntheir gaps. The size of the whole device is 3.5 4.4 mm2. The lengths of the two Nb resonators \nare 13 mm and 13.5 mm. Insets: optical microscope images of selected areas with the same \nmagnification. The YIG strips shown (color contrast augmented) are 10 900 μm2 (top) and \n10 300 μm2 (bottom). (b) Microwave transmission (S21) spectrum of two Nb resonators \nwithout a YIG strip in their gaps. The two sharp dips (resonances) at 4.364 and 4.203 GHz \ncorrespond to the resonance frequencies of th e 13 mm and 13.5 mm resonators, respectively. \n(c) Microwave transmission spectra of the 13.5 mm resonators without YIG at zero field (blue), \nwith a 10 1200 μm2 YIG strip at zero field (orange), and with a 10 1200 μm2 YIG strip in \nthe presence of a 550 Oe field (green). (d ) Microwave transmission spectra of the 13 mm \nresonators without YIG at zero field (blue), with a 10 600 μm2 YIG strip at zero field (orange), \nand with a 10 600 μm2 YIG strip at 550 Oe (green). 10\nFigure 4. Microwave photon-magnon coup ling in Nb resonator-YIG(30 nm) hybrid \nstructures on YSGG. Microwave transmission in Nb resona tors coupled to YIG strips of (a) \n10 300 μm2, (b) 10 600 μm2, (c) 10 900 μm2, and (d) 10 1200 μm2, as a function of \nmicrowave frequency and in-plane magnetic field at 2 K. (e) Ma gnetic hysteresis loops of a \nYIG(30 nm)/YSGG film at 2 and 300 K. (f ) Coupling strength between microwave photons \nand magnons in the YIG films as a function of th e square root of the magnetic volume as well \nas the number of spins in YIG. 11\nReferences: \n1. A. A. Serga, A. V. Chumak and B. Hillebrands, \"YIG magnonics,\" J. Phys. D-Appl. \nPhys. 43, 264002 (2010). \n2. A. V. Chumak, V. I. Vasyuchka, A. A. Serga and B. Hillebrands, \"Magnon \nspintronics,\" Nat. 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Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Hoffmann and V. Novosad, \n\"Hybrid magnonics: Physics, ci rcuits, and applications for coherent information \nprocessing,\" J. Appl. Phys. 128, 130902 (2020). \n9. S. P. Wolski, D. Lachance-Quirion, Y. Ta buchi, S. Kono, A. Noguchi, K. Usami and Y. \nNakamura, \"Dissipation-Based Quantum Se nsing of Magnons with a Superconducting \nQubit,\" Phys. Rev. Lett. 125, 117701 (2020). \n10. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami and Y. Nakamura, \n\"Hybridizing Ferromagnetic Magnons and Mi crowave Photons in the Quantum Limit,\" \nPhys. Rev. Lett. 113, 083603 (2014). \n11. C. L. Jermain, S. V. Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brangham, M. R. \nPage, P. C. Hammel, F. Y. Yang and D. C. Ralph, \"Increased low-temperature damping \nin yttrium iron garnet thin films,\" Phys. Rev. B 95, 174411 (2017). \n12. S. D. Guo, B. McCullian, P. C. Hamm el and F. Y. Yang, \"L ow damping at few-K \ntemperatures in Y 3Fe5O12 epitaxial films isolated from Gd 3Ga5O12 substrate using a \ndiamagnetic Y 3Sc2.5Al2.5O12 spacer,\" J. Magn. Magn. Mater. 562, 169795 (2022). \n13. F. Y. Yang and P. C. Hammel, \"Topica l review: FMR-Driven Spin Pumping in \nY3Fe5O12-Based Structures,\" J. Phys. D: Appl. Phys. 51, 253001 (2018). \n14. H. Maier-Flaig, S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. Huebl and \nS. T. B. Goennenwein, \"Temperature-dep endent magnetic damping of yttrium iron \ngarnet spheres,\" Phys. Rev. B 95, 214423 (2017). \n15. P. E. Seiden, \"Ferrimagnetic Resonanc e Relaxation in Rare-Earth Iron Garnets,\" Phys. \nRev. 133, A728-A736 (1964). \n16. E. G. Spencer, R. C. LeCraw and A. M. Clogston, \"Low-Temperature Line-Width \nMaximum in Yttrium Iron Garnet,\" Phys. Rev. Lett. 3, 32-33 (1959). \n17. S. V. Vonsovskii, Ferromagnetic Resonance: The Phen omenon of Resonant Absorption \nof a High-Frequency Magnetic Fi eld in Ferromagnetic Substances . (Pergamon, 1966). \n18. E. Schlömann, \"Inhomogeneous Broade ning of Ferromagnetic Resonance Lines,\" Phys. \nRev. 182, 632-645 (1969). \n19. S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko, R. Gross, H. Huebl, S. T. B. \nGoennenwein and M. Weiler, \"Gilbert damp ing of magnetostatic modes in a yttrium \niron garnet sphere,\" Appl. Phys. Lett. 110, 092409 (2017). 12\n20. J. T. Hou and L. Q. Liu, \"Strong Coupling between Microwave Photons and \nNanomagnet Magnons,\" Phys. Rev. Lett. 123, 107702 (2019). \n21. Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Z. Zhang, J. J. Ding, T. Khaire, \nH. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. L. Xiao, V. Novosad, A. Hoffmann \nand W. Zhang, \"Strong Coupling between Magnons and Microwave Photons in On-\nChip Ferromagnet-Superconduc tor Thin-Film Devices,\" Phys. Rev. Lett. 123, 107701 \n(2019). \n22. I. W. Haygood, M. R. Pufall, E. R. J. Edwards, J. M. Shaw and W. H. Rippard, \"Strong \nCoupling of an FeCo All oy with Ultralow Damping to Superconducting Co-planar \nWaveguide Resonators,\" Phys. Rev. Appl. 15, 054021 (2021). \n \n 13\n \n \nFigure 1. \n 101103105\n1234\n2 (deg)XRR Intensity (c/s)YIG(30 nm)/YSGG(b)100102104106\n47 49 51 53\n2 (deg)XRD Intensity (c/s)(a) YIG(30 nm)/YSGG14\n \n \nFigure 2. 680 700 720 740FMR Int. (arb. unit)H (Oe)(a)\nT = 2 K\nf = 4.3 GHzH = 13.2 Oe\n4.2 4.3 4.4VNA Int. (arb. unit)f (GHz)T = 2 K\nH = 710 Oef = 36.3 MHz (b)\n0102030\n0 5 10 15 20FMR Linewidth (Oe)\nf (GHz)T = 300 K150 K45 K 75 K30 K\n20 K\n10 K\n5 K\n2 K(c)\n00.00050.001\n01020\n0 100 200 300Damping H0 (Oe)\nT (K)YIG(30 nm)/YSGG\n(d)15\n \n \nFigure 3. \n-40-30-20-10\n4.2 4.21 4.22 4.23S21 (dB)\nf (GHz)Resonator 1\nH = 0\nNo YIG\nQ = 56,800H = 550 Oe\n10x1200 m YIG\nQ = 5,054\nH = 0 Oe\n10x1200 m YIG\nQ = 44,900(c)\n-40-30-20-10\n4.36 4.37 4.38 4.39S21 (dB)\nf (GHz)Resonator 2\nH = 0\nNo YIG\nQ = 49,000H = 550 Oe\n10x600 m YIG\nQ = 4,230\nH = 0\n10x600 m YIG\nQ = 72,700(d)-30-20-10\n4.2 4.25 4.3 4.35S21 (dB)\nf (GHz)(b)\nResonator 1\n(13.5 mm)\nResonator 2\n(13 mm)\nfRes1 = 4.203 GHz fRes2 = 4.364 GHzT = 2 K16\n \n \n \nFigure 4. \n020406080\n0 5 10 15 20Coupling g/2 (MHz)\nYIG Volume1/2 (m3/2)Number of spins (1012)\n0 14 9\n(f)\n-200-1000100200\n-20 -10 0 10 20Ms (emu/cm3)\nH (Oe)T = 2 K300 KYIG\nYSGG\n(e)" }, { "title": "2211.05514v2.High_wave_vector_non_reciprocal_spin_wave_beams.pdf", "content": "NRSWB\nHigh wave vector non-reciprocal spin wave beams\nL. Temdie,1,a)V. Castel,1,a)C. Dubs,2G. Pradhan,3J. Solano,3H. Majjad,3R. Bernard,3Y. Henry,3M.\nBailleul,3and V. Vlaminck1,a)\n1)IMT- Atlantique, Dpt. MO, Technopole Brest-Iroise CS83818, 29238 Brest Cedex 03 France\n2)INNOVENT e.V. Technologieentwicklung, Pruessingstrasse 27B, 07745 Jena, Germany\n3)IPCMS - UMR 7504 CNRS Institut de Physique et Chimie des Matériaux de Strasbourg France\n(*Electronic mail: vincent.vlaminck@imt-atlantique.fr)\n(Dated: 13 January 2023)\nWe report unidirectional transmission of micron-wide spin waves beams in a 55 nm thin YIG. We downscaled a chiral\ncoupling technique implementing Ni 80Fe20nanowires arrays with different widths and lattice spacing to study the\nnon-reciprocal transmission of exchange spin waves down to l\u001980 nm. A full spin wave spectroscopy analysis of\nthese high wavevector coupled-modes shows some difficulties to characterize their propagation properties, due to both\nthe non-monotonous field dependence of the coupling efficiency, and also the inhomogeneous stray field from the\nnanowires.\nI. INTRODUCTION\nNon-reciprocity is an essential properties of today’s\ninformation systems1. The ability to inhibit signal flow in one\ndirection while allowing it in the reverse direction is crucial\nto either protect devices from reflection, isolate transmitters\nand receivers in radar architecture, or even shield qubits from\nits environment in quantum computers. Most of the current\nnon-reciprocal functionalities rely on the gyrotropic nature\nof the magnetization dynamics in field-based ferrimagnetic\nsystems (namely Yttrium Iron Garnet - YIG), which tend\nto be large, and costly to assemble2. Future progress in\ncommunication systems, and most critically in quantum\ntechnologies, rely heavily on the possibility to miniaturize\nand integrate these non-reciprocal devices3.\nThe field of magnonics, which implements magnetic excita-\ntion called spin waves -or their quanta magnons-, is actively\ninvolved in the search of non-reciprocal scalable solutions4.\nExtensive research in the last decade revealed the possibility\nto engineer both amplitude and frequency non-reciprocity of\nspin waves in many different ways5. Firstly, the well-known\nDamon-Eshbach (DE) configuration, where the equilibrium\nmagnetization of a thin film lies in the plane of the film and\nperpendicular to the wavevector6, displays non-reciprocal\ndynamic amplitude across the thickness for oppositely travel-\ning waves, which couple chirally to an excitation antenna7.\nMoreover, this non-reciprocity was recently proven to be\nstrongly enhanced in the presence of magnetic nanostructures,\nwhere unidirectional transmission of spin waves was achieved\nusing the chiral coupling between the FMR of Co nanowires\nand exchange spin waves in a thin YIG film8,9. Secondly,\nsmall frequency non-reciprocity (namely f(k) 6=f(-k)) was\ndemonstrated more recently in various systems, either with\nasymmetrical surface anisotropies between top and bottom\nsurfaces10, or with the coupled dynamics of ferromagnetic\nmultilayers11, or also with the interfacial Dzyaloshinskii-\nMoriya interaction (DMI) of a ferromagnetic layer coupled\na)Also at Lab-STICC - UMR 6285 CNRS, Technopole Brest-Iroise CS83818,\n29238 Brest Cedex 03 Franceto a high spin-orbit material12,14 ?. Additionally, asymmetric\nspin-wave dispersion was recently predicted in non-planar\ngeometries due to a topographically induced dynamic dipolar\neffect15. Nevertheless, all these frequency non-reciprocity\neffects only becomes significant when the magnons wave-\nlength reaches sub micrometric sizes. Lastly, non-reciprocal\nfunctionalities have also been predicted very recently using\nan innovative inverse design approach16,17.\nIn this communication, we further miniaturized the method\nof Chen et al.8and demonstrate the possibility of shaping\nnon-reciprocal spin wave beams in a continuous thin YIG\nfilm. This paper is organized as follows: in Sec. II, we\npresent the sample design and the experimental protocol used\nto measure broadband multi-modes spin waves transmission.\nIn Sec. III, we present the non-reciprocal spin wave beams\nspectra, and address through a spin wave spectroscopy study\nthe peculiarities encountered in shaping narrow spin wave\nbeams via this method.\nII. EXPERIMENTAL SET-UP\nA. Sample design and fabrication\nThe chiral excitation of propagating spin waves is achieved\nby coupling the ferromagnetic resonnance (FMR) of mag-\nnetic nanowires array (NWA) to a low damping continuous\nfilm such as YIG. As illustrated in Fig.1-(a), the phase pro-\nfile of a propagating spin wave excited by the dynamic dipo-\nlar field of nanowires, all precessing in phase, only matches\nfor a single propagation direction. The degree of chirality de-\npends strongly on the ellipticity of the dynamic dipolar field\nof the NWA18. Typically, the elliptical polarization of the Kit-\ntel mode in flat rectangular nanowires breaks the perfect chi-\nrality. For this reason we made the nanowires 60 nm thick to\ncome as close as possible to an aspect ratio t=w=1 which pro-\nduces circularly polarized dipolar field. Moreover, the Kittel\nmode frequency of the nanowires can be tuned up with de-\ncreasing their width w, therefore the NWA can excite prop-\nagating modes ( kNW) with much higher wavevector than thearXiv:2211.05514v2 [physics.app-ph] 12 Jan 2023NRSWB 2\n(a)\n𝒎\nYIG \nfilm𝒉𝒅𝑘𝑁𝑊\n𝑴𝒆𝒒𝒕𝒘𝒂\nn=4n=6\n𝑘𝐶𝑃𝑊\n𝑘𝑁𝑊\n𝑴𝒆𝒒\n(b)\nFIG. 1. (a) Sketch of the NWA-FMR mediated chiral cou-\npling mechanism. (b) SEM image of device I, w=300 nm width and\na=500 nm lattice constant of Ni 80Fe20NWA grown on top of 55 nm\nthickYIG film, 80 nm Au-antennas grown on top of YIG|Ni 80Fe20.\nRed and Yellow color represent respectively antenna ( kCPW) and\nNi80Fe20NWA( kNW) excite mode .\none directly coupled by the microwave field of the antenna\n(kCPW). In the case of an extended array of NWA, these high\nvector ( kNW) correspond simply to integer values of2p\naac-\ncordingly with the periodic boundary conditions of the phase\nprecession19,20, minus some possible extinction related to the\nratio of w=a. In order to study the dependence of the NWA ar-\nray density on the excitation efficiency of the spin wave beam,\nwe designed two devices with different width wand lattice\nconstant a. Namely, for device I, w=300 nm and a=500 nm;\nand for device II, w=200 nm and a=400 nm. The NWA length\nof 10 mm was kept the same for both devices. Such a localized\ndistribution of excitation field produces a focused emission of\nspin waves in a similar fashion to the spin wave beam excited\nfrom a constricted CPW21?. In the present geometry, the dis-\ntance of propagation is well within the near-field region de-\nfined as the ratio L2=lNW, with Lbeing the antenna length and\nlNWthe magnon wavelength, so that the near-field diffraction\npattern from such a localized excitation follows very closely\nits shape. For this reason, the emission of kNWmagnons re-\nmains confined within the length of the NW, thus forming a\nspin wave beam of width closely equal to 10 µm as sketched\nin Fig. 1-(b).\nFig.1-(b) shows a SEM image of device I. The sample fabri-\ncation required two steps of e-beam lithography followed bye-beam evaporation lift-off process on a 55 nm thick liquid\nphase epitaxy YIG film23. To circumvent the insulating na-\nture of the substrate, we resorted to an extra layer of conduc-\ntive resist AR Electra 9224on top of PMMA. In the first step,\nwe structured the 60 nm thick Ni 80Fe20(Permalloy) using a\n3 nm thin Ti adhesion layer, and a 8 nm Au capping layer via\ne-beam evaporation. In the second step, we aligned for both\ndevices the same 20 mm long coplanar wave guides (CPW)\nwith the following lateral dimensions: 2 mm wide central line\nand 1 mm wide ground lines each spaced by 1 mm with the\ncentral line. These pairs of spin wave antenna made of 4 nm Ti\n+ 80 nm Au also via ebeam evaporation. The center-to-center\ndistance D between the two CPW are respectively 20 mm for\ndevice I, and 15 mm for device II.\nB. Broadband Spin Wave Spectroscopy\nWe incorporated an home-made confined electromagnet\nonto a PM8 probe station to perform VNA spin wave\nspectroscopy using a Rohde & Schwarz ZNA43GHz Vector\nNetwork Analyzer, calibrated via a SOLT procedure with\n150mm pitch picoprobes. The sample is carefully placed\nwithin the 11 mm gap of the electromagnet, whose poles are\n15 mm in diameter, and produce an homogeneous in-plane\nfield along the poles axis (namely less than 0.3% variation\nat most) up to 500 mT at 3 A. Due to the small hysteresis\nof the electromagnet, we always initialize a measurement\nwith a high current of 5A in order to be consistent with the\ncalibration of our magnet.\nWe performed broadband frequency sweep in the\n[0.5;20] GHz range at constant applied field acquiring\n3201 points with a resolution bandwidth of 100 Hz in order to\nresolve widely spread-out multi-modes transmission spectra.\nWe measure 2-ports S-parameters for several field values,\nwhich we translate into the corresponding Zi jimpedance\nmatrices. Furthermore, due to the small area of NWA (namely\n10*5 mm2), the typical range of Si jsignal amplitudes are of\nthe order of 10\u00004in linear scale (or -80 dB in log scale), while\nthe noise floor lays at 10\u00006. We retrieve a flat base line by\nsubtracting the measurement at given field with a reference\nmeasurement at another field value Hre ffar enough that there\nare no dynamic feature in the frequency span. Finally, as the\ncoupling of the spin wave to an antenna is of inductive nature,\nwe chose to represent our relative measurements in units of\ninductance defined as21,25,26:\nDLi j(f;H) =1\ni2pf(Zi j(f;H)\u0000Zi j(f;Hre f)) (1)\nwhere the subscripts (i;j)=1 or 2 denote either a transmission\nmeasurement between two antennas ( i6=j), or a reflection\nmeasurement done on the same port ( i=j).NRSWB 3\n(c)\n(d)\n𝒌𝑪𝑷𝑾\n𝒌𝑵𝑾𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟓𝒎𝑻\n𝜇0𝐻𝑒𝑥𝑡(𝑚𝑇)\n15\n269Δ𝐿11\n(a)Im(∆𝐿11) Im(∆𝐿21) Im(∆𝐿12)(b)\n(e) (f) (g) 𝒌𝑪𝑷𝑾 𝒌𝑵𝑾0.00.20.40.60.81.0\n0.00.20.40.60.81.0\n𝑘𝑎𝐼𝑘𝑏𝐼\n𝑘𝑎𝐼𝑘𝑏𝐼∆𝑘\nFIG. 2. (f,H) mapping of the spin wave spectra obtained for device I for (a) DL11, (b)DL21, and (c) DL12. (d) Spectra obtained at\nm0Hext=+15 mT from (b) and (d). (e) Zoom of (d) on the kCPW region, and (f) on the kNWregion. (g) Dispersion relation for field rang-\ning from 50 mT to 269 mT.\nIII. EXPERIMENTAL RESULTS AND DISCUSSION\nA. Perfect non-reciprocal transmission\nWe present in Fig.2 a mapping in the (f,H) plane of the spec-\ntraDL11,DL21, andDL12acquired with device I for field grad-\nually changing from +46 mT to -46 mT. We observe a myriad\nof modes that we can separated into two main regions. The\nfirst region at lower frequencies corresponds to all the kCPW\nmodes coupled directly by the microwave field of the antenna,\nand which range typically from k \u00190 (namely the section of\nthe CPW around the picroprobes) to k \u001915 rad. mm\u00001(the 8th\nsatellites peak of the antenna). As shown in Fig.2-(e) with\nthe spectra measured at m0Hext=+15 mT, some partial non-\nreciprocity occurs between DL21andDL12due to the elliptic-\nity of the microwave field of the antenna as mentioned above.\nThe second region above 8 GHz corresponds to the higher\nkNWmodes mediated by the FMR of the permalloy NWA.\nIn this frequency region, Fig.2-(b) and 2-(c) show a perfect\n100% non-reciprocal transmission of spin waves. Namely, at\npositive field values, DL21shows large oscillations, while no\ntransmission occurs from port 2 to port 1. Conversely at neg-\native fields, the non-reciprocity is reversed, and no transmis-\nsion occurs from port 1 to port 2. We emphasize that the dy-\nnamic range employed for our measurements (iBW=100Hz,\nP=-10dBm) gives a noise floor of about 5 fH, while typical\ntransmission amplitude are of the order of 100 fH. Besides,\none notices between 0 and -15 mT that the transmission for\nDL12is somewhat dispersed, indicating that the magnetiza-\ntion of the permalloy NWA do not conserve an anti-parallel\norientation with the YIG magnetization, and that a gradual\nswitching of the Py NWA is likely occurring. Furthermore, as\nshown with the spectra at +15 mT of Fig.2-(f), a closer lookin this region shows an additional perfectly non-reciprocal\nmode around 8.4 GHz on top of the more pronounced mode at\n9.4 GHz. Nevertheless, the frequency spacing between these\ntwo modes is much smaller than expected, as the wavevec-\ntor difference between two adjacent modes, kn+1\u0000kn=2p\nawould result in a frequency difference of about 3 GHz, as can\nbe deduced from Fig.2-(g). We investigate further the nature\nof these kNWmultimodes in the next section.\nB. High-k spin wave beam characterization\nWe make use of several distinct peaks to characterize the\npropagation of high k-vector magnons on both devices I and\nII. To begin with, we use the Kittel mode to fit the lower\nbranch of the reflection spectra in Fig.2-(a), which corre-\nsponds to the quasi k=0 FMR resonance of the YIG happen-\ning in the larger portion of the CPW close by the picoprobe\nand far from the NWA. From this analysis, we extract a value\nof the gyromagnetic ratio of g=2p=27.7\u00060:2 GHz.T\u00001and an\neffective magnetization m0Me f f=185\u00061 mT, which is identi-\ncal to the saturation magnetization M s23, suggesting no in-\nplane anisotropy. Then, we track the small reflection peak\nvisible in between the two regions (e.g. the peak at 7.4 GHz in\nFig.2-(d)), which corresponds to the first perpendicular stand-\ning spin wave (PSSW) mode, and we fit the difference of the\nsquare of the frequencies between the PSSW and the Kittel\nmodes27:\nf2\nPSSW\u0000f2\nFMR= (g\n2pm0)2[2MsL2p2\nt2Hext\n+M2\nsL2p2\nt2(1+L2p2\nt2)](2)NRSWB 4\n𝑘(𝑟𝑎𝑑.µ𝑚−1)\n 𝑘(𝑟𝑎𝑑.µ𝑚−1)\nDevice I\n𝝁𝟎𝑯𝒆𝒙𝒕=𝟑𝟕𝒎𝑻(a)\n𝑘𝑏𝐼\n𝑘𝑐𝐼\n𝑘𝑎𝐼\n𝑘𝑏𝐼𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟔𝟑𝒎𝑻\nDevice II(c)\n𝑘𝑎𝐼𝐼\n𝑘𝑎𝐼𝐼\n𝑘𝑎𝐼𝐼𝝁𝟎𝑯𝒆𝒙𝒕=𝟑𝟕𝒎𝑻\n𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟔𝟑𝒎𝑻\n(b)(d)\n𝜇0𝐻𝑒𝑥𝑡(𝑚𝑇)\n23\n106\n269\n𝑘𝑎𝐼,𝐶𝑃𝑊=8\n𝑘𝑐𝐼,𝐶𝑃𝑊=15.2𝑘𝑏𝐼,𝐶𝑃𝑊=11.2(e)\n(f)\n(g)𝑘(𝑟𝑎𝑑.µ𝑚−1)\n𝑘𝑎𝐼\n𝑘𝑎𝐼𝐼𝑓𝑜𝑠𝑐𝑘𝑐𝐼\nFIG. 3. Transmission spectra at 37 mT and 163 mT for (a) device I and (c) device II, blue and red line represent respectively transmission\nspectra DL21andDL12. Field dependence of the frequency for the several mode of (b) device I and (d) device II. (e) Field dependence of all\nthe mode amplitudes. (f) and (g) Comparison of the measured group velocity with theoretical expression.\nwhere L=q\n2Aex\nm0M2sis the exchange length. In doing so, we ob-\ntain the following exchange constant A exch=3.85\u00060:1 pJ.m\u00001.\nAt last, we study the field dependence of the several kNW\nmodes features. For this purpose, we use a simple Gaus-\nsian function multiplied by a cosine to fit the oscillations of\neach transmission peak (see colored fit in Fig.3-(a) and 3-(c)),\nwhich gives us the peak position, its amplitude, and the pe-\nriod of oscillation. This leaves us fitting the field dependence\nof the frequency of each mode with only the wavevector as\nfitting parameter.\nWe show in the Fig.3 the results of this methodology.\nFor device I with a lattice constant a=500 nm, we fol-\nlowed three modes which correspond to kI\na=61.3 rad. mm\u00001,\nkI\nb=66.7 rad. mm\u00001, and kI\nc=74 rad. mm\u00001; and for device II\nwith a=400 nm, we tracked two modes kII\na=72 rad. mm\u00001, and\nkII\nb=77 rad. mm\u00001.\nIt may seem curious at first that none of these wavevec-\ntors corresponds to an integer value of2p\na. However, the\nFourier transform of the NWA dynamic dipolar field distri-\nbution, which gives the spectral efficiency of the excitation25,\nis rather complex to depict as it consists of the modulation of\nthe NWA periodicity with the field distribution of the CPW.\nWe therefore perceive these kNWmodes to be neighboring rip-\nples of a convoluted spectral distribution that depends on the\nlattice periodicity, the width of the nanowire, and the antenna\nfield distribution.\nWe also reported the field dependence of the mode amplitudes\nin Fig.3-(e), which shows non-monotonous behavior. As onecan see in the spectra of Fig.3-(a) and Fig.3-(c), the efficiency\nof the coupling to a particular mode will be all the stronger\nthat its frequency matches with the one of the NWA FMR.\nAs the gyromagnetic ratio of YIG is smaller than the one\nof permalloy ( gPy=2p=29.5 GHz.T\u00001), the NWA dispersion\nwill gradually cross the kNWmultimodes dispersion across the\nfield range. Unfortunately, this non-monotonous field depen-\ndence of the excitation efficiency makes it arduous to charac-\nterize the attenuation of these high k-vector with just one kind\nof device. A series of devices with different distance D would\nbe more suited for determining the attenuation length of these\nhigh-k SWB.\nFinally, from the period of oscillation f oscof the transmis-\nsion spectra, we estimate the group velocity according to\nvg=fosc*D21. We compare our measurement of v gwith the\ntheoretical expression in Fig.3-(f) and Fig.3-(g). Although the\nagreement at lower wavevector is fair, we observe some sig-\nnificant discrepancies in the group velocity among the kNW\nmodes. We foresee that an additional phase delay must oc-\ncur through the propagation path. Namely, considering that\nthe length of NWA is comparable to the propagation distance\nD, one can expect the static stray field of the wires to cause\nsome inhomogeneities in the static field distribution between\nthe two antennas.NRSWB 5\nIV. CONCLUSION\nWe carried out a spin wave spectroscopy study on two dif-\nferent 50 mm2Ni80Fe20nanowire arrays resonantly coupled\nto a continuous 55 nm thin YIG film. We demonstrated uni-\ndirectional transmission of 10 mm wide spin wave beams up\nto 77 rad. mm\u00001in the [8;20] GHz frequency range. An at-\ntempt to characterize the propagation properties of these high\nk-vector spin wave beams reveals several peculiarities regard-\ning the modes selection, their coupling efficiency, and pos-\nsible additional phase lag due to inhomogeneous stray field\nfrom the nanowires. These findings serve as a guideline for\nfuture miniaturization of nonreciprocal magnonic devices.\nACKNOWLEDGMENTS\nThe authors acknowledge the financial support from the\nFrench National research agency (ANR) under the project\nMagFunc , the Region Bretagne with the CPER-Hypermag\nproject, and the Département du Finistère through the project\nSOSMAG . We also want to thanks Bernard Abiven for the\nimplementation of the electromagnet and Guillaume Bourcin\nfor fruitful discussions. CD acknowledges funding by the\nDeutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) - 271741898.\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\n1W. Palmer, et al., IEEE Micr. Magazine 20, 36 (2019).\n2V . G. Harris, Modern microwave ferrites, IEEE Trans. Magn. 48, 3 (2012).\n3M. Devoret, et al., Superconducting circuits for quantum information, Sci-\nence 339, 1169 (2013).\n4A. V . 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C: Solid State Phys. 19, 7013\n(1986)." }, { "title": "1709.08997v1.Magnetic_field_induced_suppression_of_spin_Peltier_effect_in_Pt____rm_Y__3_Fe__5_O__12____system_at_room_temperature.pdf", "content": "Magnetic-\feld-induced suppression of spin Peltier e\u000bect in\nPt/Y3Fe5O12system at room temperature\nRyuichi Itoh,1Ryo Iguchi,1, 2,\u0003Shunsuke Daimon,1, 3Koichi\nOyanagi,1Ken-ichi Uchida,2, 4, 5and Eiji Saitoh1, 3, 5, 6\n1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n2National Institute for Materials Science, Tsukuba 305-0047, Japan\n3WPI Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n4PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan\n5Center for Spintronics Research Network,\nTohoku University, Sendai 980-8577, Japan\n6Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\nAbstract\nWe report the observation of magnetic-\feld-induced suppression of the spin Peltier e\u000bect (SPE)\nin a junction of a paramagnetic metal Pt and a ferrimagnetic insulator Y 3Fe5O12(YIG) at room\ntemperature. For driving the SPE, spin currents are generated via the spin Hall e\u000bect from applied\ncharge currents in the Pt layer, and injected into the adjacent thick YIG \flm. The resultant\ntemperature modulation is detected by a commonly-used thermocouple attached to the Pt/YIG\njunction. The output of the thermocouple shows sign reversal when the magnetization is reversed\nand linearly increases with the applied current, demonstrating the detection of the SPE signal.\nWe found that the SPE signal decreases with the magnetic \feld. The observed suppression rate\nwas found to be comparable to that of the spin Seebeck e\u000bect (SSE), suggesting the dominant and\nsimilar contribution of the low-energy magnons in the SPE as in the SSE.\n\u0003IGUCHI.Ryo@nims.go.jp\n1arXiv:1709.08997v1 [cond-mat.mtrl-sci] 26 Sep 2017I. INTRODUCTION\nThermoelectric conversion is one of the promising technologies for smart energy utilization\n[1]. Owing to the progress of spintronics in this decade, the spin-based thermoelectric\nconversion is now added to the scope of the thermoelectric technology [2{7]. In particular,\nthe thermoelectric generation mediated by \row of spins, or spin current, has attracted much\nattention because of the advantageous scalability, simple fabrication processes, and \rexible\ndesign of the \fgure of merit [4, 8{12]. This is realized by combining the spin Seebeck e\u000bect\n(SSE) [13] and spin-to-charge conversion e\u000bects [14{16], where a spin current is generated by\nan applied thermal gradient and is converted into electricity owing to spin{orbit coupling.\nThe SSE has a reciprocal e\u000bect called the spin Peltier e\u000bect (SPE), discovered by Flipse\net al. in 2014 in a Pt/yttrium iron garnet (Y 3Fe5O12: YIG) junction [5, 17]. In the SPE, a\nspin current across a normal conductor (N)/ferromagnet (F) junction induces a heat current,\nwhich can change the temperature distribution around the junction system.\nTo reveal the mechanism of the SPE, systematic experiments have been conducted [17{\n19]. Since the SPE is driven by magnetic \ructuations (magnons) in the F layer, detailed\nstudy on the magnetic-\feld and temperature dependence is indispensable for clarifying the\nmicroscopic relation between the SPE and magnon excitation and the reciprocity between\nthe SPE and SSE [20{29]. A high magnetic \feld is expected to a\u000bect the magnitude of\nthe SPE signal via the modulation of spectral properties of magnons. In fact, the SSE\nthermopower in a Pt/YIG system was shown to be suppressed by high magnetic \felds\neven at room temperature against the conventional theoretical expectation based on the\nequal contribution over the magnon spectrum [22]. This anomalously-large suppression\nhighlights the dominant contribution of sub-thermal magnons, which possess lower energy\nand longer propagation length than thermal magnons [23, 25, 28, 30]. Thus, the experimental\nexamination of the \feld dependence of the SPE is an important task for understanding the\nSPE. Although the SPE has recently been measured in various systems by using the lock-in\nthermography (LIT) [17], it is di\u000ecult to be used at high \felds and/or low temperatures. For\ninvestigating the high-magnetic-\feld response of the SPE, an alternative method is required.\nIn this paper, we investigate the magnetic \feld dependence of the SPE up to 9 T at 300\nK by using a commonly-used thermocouple (TC) wire. As revealed by the LIT experiments\n[17, 19], the temperature modulation induced by the SPE is localized in the vicinity of N/F\n2interfaces. This is the reason why the magnitude of the SPE signals is very small in the \frst\nexperiment by Flipse et al. [5], where a thermopile sensor is put on the bare YIG surface,\nnot on the Pt/YIG junction. Here, we show that the SPE can be detected with better\nsensitivity simply by attaching a common TC wire on a N/F junction. This simple SPE\ndetection method enables systematic measurements of the magnetic \feld dependence of the\nSPE, since it is easily integrated to conventional measurement systems. In the following,\nwe describe the details of the electric detection of the SPE signal using a TC, the results\nof the magnetic \feld dependence of the SPE signal in a high-magnetic-\feld range, and its\ncomparison to that of the SSE thermopower.\nII. EXPERIMENTAL\nThe spin current for driving the SPE is generated via the spin Hall e\u000bect (SHE) from\na charge current applied to N [15, 16]. The SHE-induced spin current then forms spin\naccumulation at the N/F interface, whose spin vector representation is given by\n\u0016s/\u0012SHEjc\u0002n; (1)\nwhere\u0012SHEis the spin Hall angle of N, jcthe charge-current-density vector, and nthe unit\nvector normal to the interface directing from F to N. \u0016sat the interface exerts spin-transfer\ntorque to magnons in F via the interfacial exchange coupling at \fnite temperatures, when\n\u0016sis parallel or anti-parallel to the equilibrium magnetization ( m) [31, 32]. The torque\nincreases or decreases the number of the magnons depending on the polarization of the\ntorque ( \u0016skmor\u0016sk\u0000m), and eventually changes the system temperature by energy\ntransfer, concomitant with the spin-current injection [5, 17, 29]. The energy transfer induces\nobservable temperature modulation in isolated systems, which satis\fes the following relation\n\u0001TSPE/\u0016s\u0001m/(jc\u0002n)\u0001m: (2)\nA schematic of the sample system and measurement geometry is shown in Fig. 1(a). The\nsample system is a Pt strip on a single-crystal YIG. The YIG layer is 112- \u0016m-thick and\ngrown by a liquid phase epitaxy method on a 500- \u0016m-thick Gd 3Ga5O12substrate with the\nlateral dimension 10 ×10 mm2, where small amount of Bi is substituted for the Y-site of\nthe YIG to compensate the lattice mismatching to the substrate. The Pt strip, connected\n3(a)\nVres Current Source Voltmeter \nsyncVTC \nJc\nH\n(b)\nThermocouple Varnish \nAl 2O3\nYIG \nSubstrate Pt JcμsThermo \nCouple \n0Jc\ntime0VTC ∆Jc\n-∆Jc\nJoule heating ( ∝Jc2)spin Peltier effect\n(∝Jc)T\nsignal acquisition\ntdelay ∆T\n-∆TTYIG Pt \nVTC +\nVTC -convert\nJq\n2∆VTC mThermal\nanchoringFIG. 1. (a) Schematic of the Pt/YIG sample and measurement system with a thermocouple\nwire. Jc,\u0016s,m,H, and Jqdenote the applied current, the spin accumulation, the unit vector\nof the equilibrium magnetization, the magnetic \feld, and the heat current concomitant with the\nspin-current injection. In an isolated system, Jqinduces a temperature gradient by accumulating\nheat, which can be detected by the thermocouple. The resistance of the Pt strip is obtained by\nmeasuringVres. The measurements were carried out by using the physical property measurement\nsystem, Quantum Design. (b) Expected responses due to the SPE and Joule heating when Jcis\nperiodically changed from \u0001 Jcto\u0000\u0001Jcwith the zero o\u000bset current J0\nc= 0. Corresponding voltage\nsignalsV+\nTCandV\u0000\nTCare obtained after the time delay tdelay. The SPE signal is extracted from the\ndi\u000berence \u0001 VTC=\u0000\nV+\nTC\u0000V\u0000\nTC\u0001\n=2 .\nto four electrodes, is 5-nm-thick, 0.5-mm-wide, fabricated by a sputtering method, and\npatterned with a metal mask. Then, the whole surface of the sample except the electrodes\nis covered by a highly-resistive Al 2O3\flm with a thickness of \u0018100 nm by means of an\natomic layer deposition method. We attached a TC wire to the top of the Pt/YIG junction,\nwhere the wire is electrically insulated from but thermally connected to the Pt layer owing\nto the Al 2O3layer. We used a type-E TC with a diameter of 0.013 mm (Omega engineering\nCHCO-0005), and \fxed its junction part on the middle of the Pt strip using varnish. The\nrest of the TC wires were \fxed on the sample surface for thermal-anchoring and avoiding\nthermal leakage from the top of the Pt/YIG junction [see the cross sectional view in Fig.\n1(a)]. The expected thickness of the varnish between the TC and the sample surface is in\n4the order of 10 \u0016m [33]. Note that the thickness of the varnish layer does not a\u000bect the\nmagnitude of the signal while it a\u000bects the temporal response of the TC [34]. The ends\nof the Pt strip are connected to a current source and the other two electrodes are used for\nmeasuring resistance based on the four-terminal method. The magnetic \feld His applied in\nthe \flm plane and perpendicular to the strip, thus is along \u0016s, satisfying the symmetry of the\nSPE [Eq. (2)]. The TC is connected to electrodes on a heat bath, which acts as a thermal\nanchor, and further connected to a voltmeter via conductive wires. The measurements were\ncarried out at 300 K and \u001810\u00003Pa.\nFor the electric detection of the SPE, we measured the amplitude di\u000berence (\u0001 VTC) of\nthe TC voltage VTCin response to a change (\u0001 Jc) in the current Jc(so-called Delta-mode\nof the nanovoltmeter Keithley 2182A). After the current is set to Jc=J0\nc\u0006\u0001Jc, time delay\n(tdelay) is inserted before measuring the corresponding voltages V\u0006\nTC. Then, \u0001VTCis obtained\nas \u0001VTC=\u0000\nV+\nTC\u0000V\u0000\nTC\u0001\n=2. The time delay tdelay is necessary because the temperature\nmodulation occurred at the Pt/YIG junction takes certain time to reach and stabilize the\nTC. The appropriate value of tdelaycan be determined from the delay-dependence of the SPE\nsignal, which will be shown in Sec. III. For the SPE measurements, we set no o\u000bset current\n(J0\nc= 0) so that \u0001 VTCis free from Joule heating ( /J2\nc), and the SPE (/Jc) is expected\nto dominate the \u0001 VTCsignal [see Fig. 1(a)]. By using the measured values of \u0001 VTC, the\ntemperature modulation (\u0001 T) is estimated via the relation \u0001 T=STC\u0001VTC, whereSTC\nis the Seebeck coe\u000ecient of the TC. For the low-\feld measurements ( \u00160H < 0:1 T), the\nreference value of STC= 61\u0016V=K at 300 K is used, while, for the high-\feld measurements,\nthe \feld dependence of STC, determined by the method shown in Appendix A, is used.\nIII. RESULTS AND DISCUSSION\nFirst, we demonstrated the electric detection of the SPE at low \felds. Figure 2(a) shows\n\u0001VTCas a function of the \feld magnitude Hat \u0001Jc= 10 mA and tdelay= 50 ms. \u0001 VTC\nclearly changes its sign when the \feld direction is reversed and the appearance of the hystere-\nsis demonstrates that it re\rects the magnetization curve of the YIG, showing the symmetry\nexpected from Eq. (2) [5, 17]. The small o\u000bset of \u0001 VTCmay be attributed to the tempera-\nture modulation by the Peltier e\u000bect appearing around the current electrodes, Joule heating\ndue to small uncanceled current o\u000bsets, and possible electrical leakage of the applied current\n5(a)\n(b) (c)\n1.0\n0.9\n0.8\n0.7normalized ∆ TSPE \n1 100 tdelay (ms)10 1000\n∆J2\n1\n0∆TSPE (mK) \n15 10 5 0\nc (mA)310\n305\n300TPt (K) -1 01∆T (mK) \n-20 -10 0 10 20 \nµ0H (mT)0.1\n0.0\n-0.1∆VTC (uV) 2∆ TSPEFIG. 2. (a) Field magnitude Hdependence of \u0001 VTC(the right axis) and \u0001 Twithout o\u000bset\n(the left axis), measured at \u0001 Jc= 10 mA and tdelay = 50 ms. (b) Current Jcdependence of\n\u0001TSPEat\u00160jHj= 0:1 T, where \u00160represents the permeability of vacuum. (c) Delay-time tdelay\ndependence of \u0001 TSPE. Plotted data are estimated from the results measured at \u0001 Jc= 10 mA and\n5 mT< \u0016 0jHj<20 mT and normalized based on the \ftting with B[1\u0000exp (\u0000tdelay=\u001c)], where\nBdenotes the proportional constant and \u001c= 0:4 ms denotes the characteristic time scale. The\nacquisition time of 20 ms is used for measurements.\nfrom the sample to the TC. Since the Peltier and resistance e\u000bects are of even functions\nof the magnetization cosines though the SPE is of an odd function, the SPE-induced tem-\nperature modulation \u0001 TSPEcan be extracted by subtracting the symmetric response to the\nmagnetization: \u0001 TSPE= [\u0001T(+H)\u0000\u0001T(\u0000H)]=2 [5]. Figure 2(b) shows the \u0001 Jcdepen-\ndence of \u0001TSPEand the temperature ( TPt) of the Pt strip, estimated from the resistance of\nthe strip. While TPtincreases parabolically with the magnitude of \u0001 Jcfor Joule heating,\n\u0001TSPEincreases linearly as is expected from the characteristic of the SPE. This distinct de-\npendencies show negligibly small contribution to \u0001 TSPEfrom the Joule heating in this study\n[35]. The magnitude of the SPE signal is estimated to be \u0001 TSPE=\u0001jc= 3:4\u000210\u000013Km2=A,\nwhere \u0001jcis the di\u000berence in jc. This value is almost same as the value obtained in the\nthermographic experiments [17, 19]; since in Ref. [19] the sine-wave amplitude Aof \u0001TSPEis\ndivided by the rectangular-wave amplitude of \u0001 jc, a correction factor of \u0019=4 is necessary, i.e.\n6(a) (b) (d)\n2\n1\n0∆R/R0 (x10 -4 )\n-10 -5 0 510 \nµ0H (T)2\n1\n0∆TSPE (mK) \n15 10 5 0\nJc (mA) 0.1 T\n 8.0 T\n-2 -1 012∆TSPE (mK) \n-10 -5 0 5 10 \nµ0H (T)-1.0-0.50.00.51.0normalized SSSE \n-10 -5 0 5 10 \nµ0H (T) 1 mA\n 5 mA\n 10 mA\n 15 mA(c)\nH\nJqV+\n-FIG. 3. (a) Field dependence of \u0001 TSPEincluding the high-magnetic-\feld range measured at various\n\u0001Jcvalues. (b) Jcdependence of \u0001 TSPEat\u00160jHj= 0:1 T and 8:0 T. The solid lines represent the\nlinear \ftting. (c) Field dependence of the resistance change \u0001 R=R(H)\u0000R0, whereRdenotes\nthe resistance and R0that at\u00160H= 0:1 T. (d) Field dependence of the spin Seebeck voltage VSSE\nnormalized at \u00160H= 0:1 T, where VSSEis obtained by measuring the voltage under heater power\nof 100 mW and subtracting the component symmetric to H.\n\u0001TSPE=\u0001jc=\u0019A=(4\u0001jc) = 3:7\u000210\u000013Km2=A in the previous study. We note that, in the\nabove and following measurements, tdelay= 50 ms is chosen based on the tdelaydependence\nof \u0001TSPE[Fig. 2(c)], where \u0001 TSPEis almost saturated at tdelay>10 ms. Such \fnite but\nsmall thermal-stabilization time can be explained by the thermal di\u000busion from the junction\nto the TC and rapid thermal stabilization of the SPE-induced temperature modulation [17].\nNext, we measured the \feld dependence of the SPE at higher \felds up to \u00160H= 9:0 T.\nFigure 3(a) shows \u0001 TSPEas a function of H, where \u0001Jcis changed from 1 to 15 mA. The\nsuppression of the \u0001 TSPEsignal at higher \felds is clearly observed for all the \u0001 Jcvalues.\nAs shown in Fig. 3(b), the signal shows a linear variation with Jcboth at\u00160H= 0:1 T\nand 8.0 T, demonstrating a constant suppression rate. Since the resistance of the sample\nvaries only 1 % at most [Fig. 3(c)], the junction temperature keeps constant during the\n\feld scan. The \feld dependence of the thermal conductivity of YIG is also irrelevant to the\n\u0001TSPEsuppression as it is known to be negligibly small at room temperature [36]. Thus we\ncan conclude that the suppression is attributed to the nature of the SPE. By calculating the\nsuppression magnitude as\n\u000eSPE= 1\u0000\u0001TSPE(\u00160H= 8:0 T)\n\u0001TSPE(\u00160H= 0:1 T);\n7we obtained \u000eSPE= 0:26.\nTo compare \u000eSPEto the \feld-induced suppression of the SSE, we performed SSE mea-\nsurements in a longitudinal con\fguration using a Pt/YIG junction system fabricated at the\nsame time as the SPE sample. The SSE sample has the lateral dimension 2 :0\u00026:0 mm2and\nthe same vertical con\fguration as the SPE sample except for the absence of the Al 2O3layer.\nThe detailed method of the SSE measurement is available elsewhere [27, 37, 38]. Figure\n3(d) shows the \feld dependence of the SSE thermopower in the Pt/YIG junction. The clear\nsuppression of the SSE thermopower is observed. Importantly, the high-\feld response of the\nSSE is quite similar to that of the SPE in the Pt/YIG system. The suppression magnitude of\nthe SSE\u000eSSE, de\fned in the same manner as the SPE, is estimated to be \u00180:22, consistent\nwith the previously reported values [22, 23, 25].\nThe observed remarkable \feld-induced suppression of the SPE at room temperature shows\nthat the SPE is likely dominated by low-energy magnons because the energy scale of the ap-\nplied \feld is less than 10 K and thus much lower than the thermal energy of 300 K [22]. The\norigin of the strong contribution of the low-energy magnons in the SPE can be (i) stronger\ncoupling of the spin torque to the low-energy (sub-thermal) magnons and (ii) greater propa-\ngation length of the low-energy magnons than those of high-energy (thermal) magnons [30].\nWhile (i) is not well experimentally investigated, the existence of the \u0016m-range length scale\nin the SPE [19] and the similarity between \u000eSPEand\u000eSSEsuggest the dominant contribution\nfrom (ii) as in the case of the SSE [25, 30]. In fact, recently, it has been demonstrated that\nthe high magnetic \felds reduce the propagation length of magnons contributing to the SSE\n[30]. This length-scale scenario can qualitatively explain the suppression in the SPE. Recall-\ning that a heat current density ( jq) existing over a distance ( l) generates the temperature\ndi\u000berence \u0001 T=\u0014\u00001jqlin an isolated system, \u0001 Tshould decrease when ldecreases, where\n\u0014is the thermal conductivity of the system. In the SPE, lcorresponds to the magnon prop-\nagation length [39], and a \row of magnons accompanies a heat current [36]. Consequently,\nwhen the high magnetic \feld is applied and the magnons with longer propagation length are\nsuppressed by the Zeeman gap, the averaged magnon propagation length decreases and thus\nresults in the reduced \u0001 T. To further investigate the microscopic mechanism of the SPE,\nconsideration of the spectral non-uniformity may be vital both in experiments and theories.\n8IV. SUMMARY\nIn this study, we showed the magnetic \feld dependence of the spin Peltier e\u000bect (SPE)\nup to 9.0 T at 300 K in a Pt/YIG junction system. We established a simple but sensitive\ndetection method of the SPE using a commonly-available thermocouple wire. The SPE\nsignals were observed to be suppressed at high magnetic \felds, highlighting the stronger\ncontribution of low-energy magnons in the SPE. The similar suppression rate of the SPE-\ninduced temperature modulation to that of the SSE-induced thermopower suggests that\nthe suppression originates the decrease in the magnon propagation length as in the case of\nthe SSE. We anticipate that the experimental results and the method reported here will be\nuseful for systematic investigation of the SPE.\nACKNOWLEDGMENTS\nThe authors thank T. Kikkawa for the aid in measuring the SSE and G. E. W. Bauer\nand Y. Ohnuma for the valuable discussion. This work was supported by PRESTO \\Phase\nInterfaces for Highly E\u000ecient Energy Utilization\" (Grant No. JPMJPR12C1) and ERATO\n\\Spin Quantum Recti\fcation Project\" (Grant No. JPMJER1402) from JST, Japan, Grant-\nin-Aid for Scienti\fc Research (A) (Grant No. JP15H02012), and Grant-in-Aid for Scienti\fc\nResearch on Innovative Area \\Nano Spin Conversion Science\" (Grant No. JP26103005) from\nJSPS KAKENHI, Japan, NEC Corporation, the Noguchi Institute, and E-IMR, Tohoku\nUniversity. S.D. was supported by JSPS through a research fellowship for young scientists\n(Grant No. JP16J02422). K.O. acknowledges support from GP-Spin at Tohoku University.\n9(a)\n(b)0Jc\ntime0VTC Jc+∆Jc\nJoule heating \nVTC +\nVTC -\n2∆V TC Jc-∆Jc00\n2.26\n2.24\n2.22∆VJoule (µV) \n-10 -5 0 5 10 \nµ0H (T)FIG. 4. (a) Expected responses due to the Joule heating at the \fnite o\u000bset current J0\nc6= 0. (b)\nField dependence of \u0001 VJoule at \u0001Jc= 0:5 mA and J0\nc= 5 mA. The solid curve represents the\ncalibration line determined by the \ftting.\nAppendix A: Calibration of Thermo Couple at High Magnetic Fields\nTo measure the \feld dependence of STC, we used the Joule-heating-induced signal as\na reference. By adding a non-zero o\u000bset ( J0\nc) to the applied current, we obtained the\ntemperature modulation induced by the Joule heating, of which the power Pchanges from\nP(H) =R(H) (J0\nc\u0000\u0001Jc)2toP(H) =R(H) (J0\nc+ \u0001Jc)2, whereRdenotes the resistance\nof the strip [Fig. 4(a)]. Figure 4(b) shows the magnetic \feld dependence of the component\nof \u0001VTCsymmetric to the \feld (\u0001 VJoule= [\u0001TTC(+H) + \u0001TTC(\u0000H)]=2). As the change in\nR, due to the ordinary, spin Hall, and Hanle magnetoresistance e\u000bects [40, 41], is in the order\nof 0.02 % [Fig.3(c)], its contribution to Pcan be neglected. 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Heremans, Phys. Rev. B 90, 064421 (2014).\n12[37] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa,\nand E. Saitoh, J. Appl. Phys. 111, 103903 (2012).\n[38] A. Sola, P. Bougiatioti, M. Kuepferling, D. Meier, G. Reiss, M. Pasquale, T. Kuschel, and\nV. Basso, Sci. Rep. 7, 46752 (2017).\n[39] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys.\nRev. B 94, 014412 (2016).\n[40] H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani,\nS. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and\nE. Saitoh, Phys. Rev. Lett. 110, 206601 (2013).\n[41] S. V\u0013 elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E.\nHueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett. 116, 016603 (2016).\n13" }, { "title": "1912.11634v1.Hybrid_nanophotonic_nanomagnonic_SiC_YiG_quantum_sensor__I__theoretical_design_and_properties.pdf", "content": "arXiv:1912.11634v1 [quant-ph] 25 Dec 2019Hybrid nanophotonic-nanomagnonic\nSiC-YiG quantum sensor:\nI/ theoretical design and properties.\nJ´ erˆ ome TRIBOLLET∗\nInstitut de Chimie de Strasbourg, Strasbourg University, U MR 7177 (CNRS-UDS),\n4 rue Blaise Pascal, CS 90032, F-67081 Strasbourg Cedex, Fra nce\nE-mail: tribollet@unistra.fr\nAbstract\nHere I present the theory of a new hybrid paramagnetic-ferri magnetic SiC-YiG\nquantum sensor. It is designed to allow sub-nanoscale singl e external spin sensitivity\noptically detected pulsed electron electron double resona nce spectroscopy, using an X\nband pulsed EPR spectrometer and an optical fiber. The sensor contains one single V2\nnegatively charged silicon vacancy color center in 4H-SiC, whose photoluminescence\nis waveguided by a 4H-SiC nanophotonic structure towards an optical fiber. This V2\nspin probe is created by ion implantation at a depth of few nan ometers below the\nsurface, determined by optically detected paramagnetic re sonance under the strong\nmagnetic field gradient of a YiG ferrimagnetic nanostripe lo cated on the back-side\nof the nanophotonic structure. This gradient also allow the study, slice by slice at\nnanoscale, of the target paramagnetic sample. The fabricat ion process of this quantum\nsensor, its magnetic and optical properties, its external s pins sensing properties in a\nstructural biology context, and its integration to a standa rd commercially available\npulsed EPR spectrometer are all presented here.\n1Introduction\nElectron paramagnetic resonance1(EPR) investigation of electron spins localized inside, at\nsurfaces, or at interfaces of ultrathin films is highly relevant to man y fields.2–6For example,\nin the fields of photovoltaic7and photochemistry8, EPR is useful to study the spins of\nphoto-created electron-hole pairs, their dissociation, and their e ventual transport or chemi-\ncal reaction occurring at some relevant interface. Also, in struct ural biology, it is relevant to\nstudy by EPR spin labeled proteins9,10introduced in model lipid bilayers membranes.11,12\nIn the context of the development of new theranostic agents for nanomedicine, it is also\nrelevant to study ligand-protein molecular recognition events occu rring on surfaces by EPR,\nusing for example, bifunctional spin labels.13As various nanotechnologies now allow to pro-\nduce biological, chemical or physical samples with nanoscale thicknes s, one needs to perform\nsensitive surface EPR. However, commercial EPR spectrometers have generally not enough\nsensitivity1415forstudying thosethinfilms when targetspins arediluted, and, f orsure, they\ncannot detect a single external spin. While home-made EPR experime ntal setups have been\ndeveloped recently allowing to reach the single spin sensitivity, by opt ically15–20(ODMR),\nelectrically21ormechanically22detected EPR,apowerfull upgradestrategyallowing toreach\nboth the sub-nanoscale resolution and the single external spin sen sitivity, while still using\na commercially available pulsed EPR X band spectrometer, is clearly lack ing and is highly\nrelevant for most EPR users worldwide.\nHere I present a new methodology for state of art optically detect ed pulsed double electron\nelectron spin resonance spectroscopy (OD-PELDOR), allowing SiC b ased sub-nanoscale sin-\ngle external spin sensing. This new methodology requires only a stan dard X band pulsed\nEPR spectrometer, as well as an optical fiber and a new SiC-YiG quan tum device, which\ncan be both conveniently introduced in a standard EPR tube. The 4H SiC part of the device\ncontains a nanophotonic structure with a single V2 negatively charg ed silicon vacancy color\ncenters (spin S=3/2), introduced below the surface by low energy ion implantation and used\n2as a quantum coherent ODMR spin probe19,20,27,28,42for sensing14–16external spins through\nmagnetic dipolar couplings. The V2 depth below the surface can be de termined by optical\nfiber based Optically Detected paraMagnetic Resonance (ODMR) un der the strong mag-\nnetic field gradient produced by a YiG ferrimagnetic nanostripe, app ropriately designed,26\nlocated nearby on the back-side of the 4H-SiC nano-photonic stru cture. This field gradient\nis also designed to allow the ODPELDOR investigation, slice by slice at sub -nanoscale, of\nthe target paramagnetic sample located on the 4H-SiC surface. Th e fabrication process and\nthe properties of this SiC-YIG quantum sensor are all presented h ere.\nPrinciples for sub-nanoscale single external spin sensing\nwith a V2 color center in a SiC-YiG quantum sensor.\nA single color center spin probe can sense an external spin located n earby at a nanoscale\ndistance through its dipolar magnetic coupling with this spin (fig.1a). S ince decades, the\npulsed EPR experiments called DEER (Double Electron Electron Reson ance) or PELDOR\n(Pulsed ELectron electron DOuble Resonance)1,9,10,17,30allow to measure this dipolar mag-\nneticcoupling between pairsofspins A-B using pump-probemagnetic resonance experiments\nwith two different microwave frequencies, experiments performed over macroscopic ensem-\nble of identical pairs of spins A-B. As this dipolar coupling scales with1\nR3\nAB,RABbeing\nthe distance between the two spins A and B, measuring the dipolar co upling gives acces\nto the nanoscale distance RAB. It was thus particularly used in structural biology for the\ndetermination of constraints on the 3D conformation of some spin la belled proteins. The\nsitutation becomes however much more complicated for some studie s like study of trans-\nmembrane proteins which are extremely difficult to obtain in large quan tities and even more\nin 2D model lipid bio-membranes. Thus, despite the fact that 50 perc ent of all known small\nmolecule drugs bind to transmembrane proteins, those proteins re mains difficult to study by\n3PELDOR spectroscopy by lack of sensitivity.\nFigure 1: Basic principles for external spins sensitive and nanoscale Quantum Sensing.a/\n4H-SiC-YIG sensor for ODPELDOR with a single V2 spin probe (red dot ) and a plane of\nexternal target spins, like spins labels (arrows); on the left of das hed-dot line, a 2D spin\nlabeled lipid membrane 5 nm thick, on the right, the model in green. Boz and B1y: external\nstatic and microwave magnetic fields. Bdip,z (x) is the spatially depend ent dipolar magnetic\nfield produced by the YIG ferromagnetic nanostripe located on the opposite 4H SiC surface.\nb/ OD-PELDOR pulse sequence for quantum sensing including synchr onized optical and\nmicrowave pulses. c/ Simplified scheme showing a top view of the SiC-YI G quantum sensor:\nSiC is in gray,the red dot is the V2 single color center, the dashed red r ectangle is the\nYIG nanostripe present on the opposite surface of the SiC membra ne, through which a 1D\nnanophotonic waveguide has been designed in a 2D photonic cristal m ade of air holes in SiC.\nThe 1D optical waveguide is coupled to the optical fiber by an approp riate coupler. Ky,ex,\nKy,det : excitation and PL light wavevectors.\nAs it was previously demonstrated with a single NV color center in diamo nd and a home\nmade setup,17it is now possible to detect one single or several external electron s pin by\noptically detected PELDOR spectroscopy (fig.1a). The idea is to tak e advantage of the spin\ndependent photoluminescence rate and/or of spin dependent non -radiative relaxation rates\nof some optically excited color centers found in wide gap semiconduct ors like diamond17\n(NV- center) or 4H-SiC27,28,42(V2 negatively charged silicon vacancy), firstly to perform\noptical pumping43of the ground spin state of the color center, and secondly to perf orm an\n4ultra-sensitive photoluminscence measurement19,20ofthe groundspin statepopulationof the\ncolor center by mapping this spin state population difference onto a g iven photoluminescence\namount. Single photon detectors in the visible or near infrared rang e combined with efficient\noptical pumping thus allow altogether to reach the single spin detect ion level. This is orders\nof magnitude better than the sensitivity of standard PELDOR spec troscopy performed using\na standard pulsed EPR spectrometer operating at X band (around 10 GHz). The ODPEL-\nDOR pulse sequence (fig.1b) is thus similar to the older PELDOR pulse se quence, the main\ndifference being the need of an initial optical pumping pulse and the ne ed of a second optical\nreadout pulse after the PELDOR microwave pulse sequence at two m icrowave frequencies,\nwhich ends by a last microwaveπ\n2pulse at probe frequency to convert the spin state coher-\nence into a spin state population difference. The time domain ODPELDO R curve (versus\ntd, see fig.1b for definition of td) recorded by the single V2 photolum inescence detection can\nbe either a simple oscillation at the dipolar magnetic coupling frequency if the V2 spin is\ncoupled to a single external spin, or a decaying curve if the single V2 s pin is coupled to a\nspins bath (a spin ensemble), as it was previously demonstrated with a single NV- center\nspin probe in diamond.17As I also suggested it in a previous work on quantum computing\nwith spins,26a strong dipolar magnetic field gradient produced by a ferromagnet ic nanos-\ntripe (see Bdip,z (x) on fig.1a) located nearby the V2 spin probe color center and its nearby\nexternal target spins, allows to encode the spatial position of eac h of those spins onto its\nparamagnetic resonance field or frequency, allowing to reach nano scale spatial resolution\nwith paramagnetic resonance.\nThe idea of the SiC-YIG quantum sensor presented here (fig.1c) is t o provide a convenient\nexperimental acces to this ultra-sensitive nanoscale resolution OD PELDOR spectroscopy to\nthe many worlwide pulsed EPR users, by interfacing a standard pulse d EPR spectrometer\nwith the SiC-YIG quantum sensor described here, using an optical fi ber (fig.1c and fig.2).\nThis 4H-SiC-YIG quantum sensor contains one single sub surface V2 negatively charged\nsilicon vacancy color center, whose infrared photoluminescence is w aveguided by a 4H-SiC\n5Figure 2: A laser (9) at 785 nm can conveniently excite the photolumin escence (PL) of\nthe single V2 spin probe through a dichroic miror (5), the optical fibe r (1) , the optical\ncoupler (2) and the 1D photonic waveguide of the SiC-YIG quantum s ensor (3). The PL\nis then collected by the same path in the reverse direction till the dich roic miror, and then\nit is filtered (7) before being send to the time gated sensitive photod etector (8), either a\ngated photomultiplier tube or a gated avalanche photodiode, having single photon counting\ncapabilities. The photodetector signal, which is proportionnal to th e photoluminescence\nsignal, has then to be sent in a voltage form to the pulsed EPR spectr ometer input channel\nof the transient recorder. Microwave pulses at two different freq uencies are fabricated by the\npulsed EPR/ELDOR spectrometer (10) and synchronized with a trig er voltage pulse used to\npulse the exciting laser, when necessary. For the much simpler ODMR experiments requiring\na single microwave frequency, a Si photodiode can be used for PL de tection and its voltage\nsignal send to a lock in amplifier. (4) and (6) are short focal lenses o r objectives. Static field\nB0 and microwave magnetic field B1 are perpendicular. The pulsed EPR resonator is inside\na standard pulsed EPR cryostat (4K-300K).\nnanophotonic structure (fig.1c) coupled to the optical fiber by an appropriate optical cou-\npler.47The SiC nanophotonic structure can be either a simple SiC nanobeam48,49or a 1D\noptical waveguide created in a 2D photonic crytsal,50,51itself made of air holes in a thin\n4H-SiC membrane.50–52A YIG nanostripe53on the backside of the photonic waveguide pro-\nvide a strong permanent magnetic field gradient.26Outside the pulsed EPR spectrometer, a\nsimple photoluminescence setup using a dichroic miror and few other o ptics (fig.2) is used\nto excite and collect the photoluminescence from the V2 spin probe ( at around λ= 915nm\nat low temperature), the light being waveguided by the optical fiber in both directions. A\n6commercially available pulsed EPR spectrometer operating at X band is by this way up-\ngraded into a state of art ODPELDOR spectrometer allowing single ex ternal spin sensing\nwith nanoscale resolution.\nFabrication process of the SiC-YiG quantum sensor\nThe SiC-YIG quantum sensor proposed here is an hybrid nanophoto nic-nanomagnonics de-\nvice containing a 4H-SiC1D nanophotonic waveguide and aferrimagne tic nanostripeof YIG.\nHereIpresent onepossible fabricationmethodofthequantum sen sor, but othersarepossible\nwith the advanced SiC technology.\nIt requires firstly the fabrication of a 2D photonic cristal in a thin 4H -SiC membrane, as\nshown on fig.3a. For this one can start with anappropriate 4H-SiC HP SI substrate with very\nlow n type residual doping and very few residual V2 silicon vacancy an d also very few other\nkinds of paramagnetic defects, eventually also containing a top epila yer of isotopically puri-\nfied 4H-SiC to eliminate all nuclear spins from the SiC membrane to fabr icate. If necessary\na high temperature pre-processing annealing can be used to remov e all residual V2 silicon\nvacancies in the starting 4H-SiC material. Then, using for example a fi rst RIE etching step\nor a SICOI approach, a thin 4H-SiC membrane can be produced (aro und 100 nm thickness),\nfollowing previous demonstrated methods.50–52If one wish to have the full SIC-YIG quan-\ntum sensor, then the SiC membrane need to be thin enough, typically between 100 nm and\n300nm in order that the subsurface V2 silicon vacancy feels the opt imal dipolar magnetic\nfield gradient (see next sections). But if one wishes to produced a m uch simpler SiC sensor,\nwithout the YIG nanostripe, then a thicker membrane, convenient ly produced by RIE can\nbe fabricated. Then, electron beam lithography is used to fabricat e an appropriate mask for\nthe subsequent second RIE etching step allowing to produce the air holes for the photonic\n7Figure 3: Six main steps in the fabrication process of the 4H-SiC-YIG quantum sensor\nproposed. a/ SiC membrane and then 2D photonic cristal fabricatio n; b/ oxydation of SiC;\nc/ YIG nanostripe fabrication on the backside of the SiC membrane; d/ annealing under\noxygen of YIG nanostripe; e/ Fabrication of a single V2 spin probe be low SiC surface by ion\nimplantation through a mask; f/ Removal of mask and final quantum device structure. See\ntext for details.\nnanostructure. This mask has thus to reproduce the 2D photonic crystal structure wished,\nincluding the 1D photonic waveguide. This mask for RIE can be create d either on the top\nside of SiC membrane if the membrane is thick, or on the back side of th e SiC membrane if\nthis one is very thin, using through membrane ebeam lithography54of a photoresist (fig.3a).\nThen, after this RIE step and after removing the mask used, a the rmal oxydation of the 4H\nSiC is performed (fig.3b), creating a SiO2 layer of around 10 nm. As sh own on fig.3c, the\nsingle YIG nanostripe has then to be fabricated on the backside of t he 4H-SiC 1D photonic\nwaveguide. The width Wnanobeamof this 4H-SiC 1D waveguide in the 2D photonic crystal\n(or the width of the alternative SiC nanobeam surrounded by air) ha s to be larger than\nthe width WYIGof the YIG nanostripe. As the YIG nanostripe is on the backside of t he\n8device, one has to use through membrane ebeam lithography54of a photoresist deposited\non the backside(fig.3c) of the SiC membrane. Then, room temperat ure sputtering of YIG53\non SiO2 through this photoresist mask can be used to deposit YIG on the backside of the\nmembrane. Removing the photoresist will let an amorphous YIG nano stripe deposited on\nthebackside of the4H-SiC1Dwaveguide. Asubsequent thermal an nealing inoxygen (fig.3d)\nhas then to be performed to obtain a poly-cristalline YIG nanostripe with improved spin\nwave resonance properties. Alternatively, direct Pulsed Laser De position (PLD) of YIG at\nhigh substrate temperature, followed by an appropriate backside etching of the YIG film\nthrough a mask can be used to produce such a YIG nanostripe, but in that case one should\ncreate the YIG nanostripe on the backside of the membrane befor e creating the air holes\nof the photonic nanostructure by RIE. The mask can again in this ca se be defined on the\nbackside by through membrane ebeam lithography.54\nThen, a single V2 color center has to be introduced just few nanome ters below the 4H SiC\nsurface on the front side of the SiC membrane, at a position located vertically above the\ncenter of the YIG nanostripe itself located on the backside of the S iC membrane (fig.3e).\nThe detailed ion implantation methods proposed to create such a sing le V2 spin probe in\nthe quantum sensor are described in the appendix 1 and are inspired from a recent work\nreporting the fabrication of a silicon vacancies array in 4H-SiC.55A common key point to\nthe various proposed statistical fabrication processes by ions imp lantation through a single\naperture mask is the need to evaluate each quantum sensor after fabrication, because over\naround 100 devices fabricated in parallel, around just 5 contains an appropriate single V2\nspin probe located few nm below the SiC surface, at a given depth with a precision of around\n+/- 1 nm. Firtsly, the second order photoluminescence auto-corr elation function has to be\nmeasured to test whether agiven sensor hasa singleV2 spinprobeo rnot.55Secondly, thanks\nto the strong dipolar magnetic field gradient produced by the YIG na nostripe, one can de-\ntermine by ODMR spectroscopy under such strong gradient, the d epth of this single V2 spin\nprobe in all sensors having a single V2. By time resolved ODMR, with an o ptically detected\n9hahn echo decay measurement for example, one can also determine the T2 spin coherence\ntime of this single V2 spin probe. Thus, the various post fabrication t ests combined with\nan appropriate YIG nanostripe design allow an efficient selection of th e quantum sensors\nfabricated having the desired properties (depth below the surfac e and T2 spin coherence\ntime).\nAs a last remark about the whole fabrication process, one should no te that in 4H-SiC, just\nlike for diamond, surface defects exist and can induce some magnet ic noise on the V2 sub-\nsurface single spin probe. To limit this possible spin decoherence effec t, a low temperature\npassivation treatment can be applied to the top 4H-SiC surface, like for example, a H+N\nplasma treatment40at 400 degree C, which is known to reduce its surface density of sta te to\n61010cm−2. Thermal cooling of the quantum sensor is an alternative method to reduce or\nsuppress the thermal fluctuations of those surface defects an d thus to increase in this case\nthe T2 of sub surface V2 spin probes.\nDesign of the YIG ferrimagnetic nanostripe and mag-\nnetic properties of the SiC-YiG quantum sensor\nThe quantum sensor proposed here contains a YIG nanostripe, as already said above, whose\ndesign should allow to perform ODMR test of the sub surface V2 spin p robe under a strong\ndipolar magnetic field gradient, as well as ODPELDOR under this stron g dipolar magnetic\nfield gradient to improve the spatial resolution of ODPELDOR down to nanoscale (see fig.1\nand fig.2).\nThe figure 4 summarizes the static and dynamic magnetic properties of the YIG nanos-\ntripe located on the backside of the 1D nanophotonic waveguide. Th e fig.4d shows that the\nmaximum magnetic field gradient in the x direction, perpendicular to 4H -SiC surfaces, is\n10Figure 4: YIG nanostripes: width W=500 nm , thickness T=100 nm, len gth L=100 µ m,\nBsat=1700 G; x,y,z as in fig.1. a/ and b/ EPR spectrum at X (9.7 Hz) and Q (34 GHz)\nband resp., showing, in blue, YIG SWR and, in red, the shifted EPR of g =2.00 electron\nspins placed at xopt=150 nm above the YIG nanostripe center (x=0 ) (1 G linewidth for all\nhere; SWR oscillator strength not calculated). c/ 1D eigenenergies of the spin waves along\nz axis; here z*=300+z; d/ z component of the dipolar magnetic field o f the YIG nanostripe\nas a function of x (black), as well as its gradient along x (red) multiplie d here by 100 for\nclarity. e/ and f/ Total effective Zeeman splitting at X band (dot line) , expressed in Gauss\n(thus divided by ( g µB), assuming g=2.00), as well as its two contributions: Bdz (to first\norder in blue), and Bdx (in red to second order), at xopt(e/) and a t xopt +/-10 nm (f/).\nof around 0.5 G/nm and is obtained at a distance xopt=150 nm from th e center of a given\nYIG nanostripe for the dimension chosen here (see legend of fig.4). That is why the SiC\nmembrane has to have a thickness of xopt - T/2 =100 nm here, such that the V2 spin probe\non the opposite face of the 4H-SiC nanophotonic waveguide feel th e maximum magnetic\nfield gradient. Increasing the width W of the YIG nanostripe will incre ase xopt (xopt=230\nnm for W=800 nm, corresponding to a 180 nm thick 4H-SiC membrane) , but it will also\nreduce the maximum gradient available at xopt, thus a compromise ha s to be found. The\nmagnetic field gradient produced by the YIG nanostripe considered here (W=500 nm) is\n11not rigorously one dimensional along x. However, as I previously exp lained in the context\nof quantum computing,26locally, around xopt = 150 nm here, and laterally at z=0 +/- 30\nnm along z, detailed calculations clearly show (fig. 4e) that in this port ion of plane above\nthe YIG nanostripe, the dipolar magnetic field can be considered as la terally homogeneous\nwith a precision better than 0.1 G, thus for external spins on the se nsor surface. Even in the\nportion of plane located at around xopt - 10 nm, where the V2 spin pr obe could be placed,\nand laterally at z=0 +/- 30 nm along z, the dipolar magnetic field can be c onsidered as lat-\nerally homogeneous with a precision better than 0.3 G (fig. 4f). In th e proposed fabrication\nprocess of the V2 color center, its position is well defined at +/- 10 n m along z, the diameter\nof the hole in the mask used for low energy ion implantation being 20 nm. I will also assume\nhere that it is possible to fabricate a quantum sensor device having a single isolated sub\nsurface V2 with a spin coherence time comprised between 10 µsand 50µs, in principle at\nroom temperature, and at least at a sufficiently low temperature (s ee appendix 2 for more\ndetails). Those spin coherence times thus correspond to a homoge neous V2 linewidth com-\nprised between 7 mG and 36 mG, smaller than the 50 mG variation of the dipolar magnetic\nfield which is expected with the variation of the lateral position of the V2 spin, as shown on\nfig.4f. This further implies, with a 0.5 G/nm gradient along x, that the V 2 coordinate along\nthe x axis, relatively to the center of the YIG nanostripe, or equiva lently its depth below the\n4H-SiC surface, could be determined in principle with a precision of aro und 1 Angstrom.\nAs ODPELDOR spectroscopy also allow the indirect detection of the p aramagnetic reso-\nnance of an external spin through the V2 photoluminescence signa l, and as this external spin\nfeels also the field gradient of the YIG nanostripe, its paramagnetic resonance frequency (or\nresonant magneticfield) necessary also encodeits coordinatealon gthe xaxis. Thus ODPEL-\nDOR spectroscopy performed under the strong magnetic field gra dient produced by the YIG\nnanostripe could allow to determine the coordinate along the x axis of the external spin,\nrelatively to the center of the YIG nanostripe, or equivalently its ve rtical distance above the\n124H-SiC surface. At X band (around 10 GHz), at T=100K and in water -glycerol, a trityl\nspin label56has a long longitudinal relaxation time T1= 1 ms, and also a rather long s pin\ncoherence time T2, limited by solvant proton spin diffusion and equal t o around 4 µs. For a\nsingle trityl spin labelled transmembrane protein that would be locate d just few nm above\nthe 4H-SiC surface of the SiC YIG quantum sensor, this T2 corresp onds to a homogeneous\nlinewidth of the single trityl of around 90 mG, which is slightly larger tha n the 0.05 G = 50\nmG variation of the dipolar magnetic field which is expected with the var iation of the lateral\nposition of the spin label located betwen xopt and xopt + 10 nm, as sh own on fig.4f. Thus in\nthis case, the spatial resolution along x would be limited by the 90 mG line width of the trityl\nspin label and thus to a precision of0.09\n0.5= 1.8 Angstrom. This opens exciting possibilities\nin the context of structural biology to study the structure of sp in labelled transmembrane\nprotein reconstituted in a model lipid bilayer or lipid nanodisk, itself anc hored on the 4H\nSiC-YIG sensor surface. This important application of this SiC-YIG O DPELDOR quantum\nsensor is further discussed in one of the next sections.\nAs a last remark, one can note that the YIG nanostripe design also a llows to avoid the\nspectral overlap between the many spin wave resonance (SWR) line s of YIG and the dipo-\nlar magnetic field shifted electron paramagnetic resonance lines of t he single V2 and of the\nsingle external spin label, as shown on fig.4a (at X band) and fig.4b (at Q band). As pre-\nviously explained,26this is necessary to avoid the microwave driving of some spins in the\nYIG ferromagnetic nanostripe because this could produce some un wanted decoherence ef-\nfects. Without spectral overlap between SWR and EPR lines, the re sidual V2 decoherence\nrelated to YIG is due to thermal fluctuations of the spins in YiG, which are expected to\nbe negligigle here compared to other decoherence processes for t he isolated sub surface V2\nspin probe, following my previous estimates made for a permalloy nano stripe with a higher\nsaturation magnetization in the context of spin based quantum com puting.26Note also that\nthe SWR mode which has the highest resonant magnetic field corresp onds to a SWR mode\nconfined on the lateral edges of the YIG nanostripe and is thus exp ected to have a much\n13smaller oscillator strength compared to the quasi uniform SWR mode (FMR mode) having\nthe largest oscillator strength and a much smaller resonant magnet ic field.\nDesign of the nanophotonic structure and optical prop-\nerties of the SiC-YiG quantum sensor\nThe quantum sensor proposed here contains a 4H-SiC 1D nanophot onic waveguide, as al-\nready said above, whose design should allow the efficient optical excit ation of the V2 spin\nprobe and the efficient collection of its photoluminescence, which has to be directed towards\nthe optical fiber integrated by a coupler47to this nanophotonic structure (see fig.1 and fig.2).\nThis is necessary to further integrate this sensor to a widely availab le pulsed EPR spectrom-\neter. For the coupling of the fiber to the nanophotonic waveguide,47one can use either\na tapered waveguide coupler, a diffraction grating based coupler, a n adiabatic coupler (or\nevanescent coupler), or a cylindrical GRIN microlens integrated or placed at the end of the\nfiber. Recently, a high optical coupling efficiency above 90 percent h as been demonstrated\nwith an optical fiber coupled to a diamond nanophotonic structure.57An optical coupling\nefficiency above 25 percent has also been recently obtained betwee n an optical fiber and a\nSilicon nanophotonic structure, with another approach allowing cry ogenic cooling down to\nvery low temperatures.58\nAs explained in the previous section, the optimisation of the in plane ho mogeneity and ver-\ntical strength of the dipolar magnetic field gradient produced by th e YIG nanostripe close to\nthetop4H-SiCsurfacewhere theV2spinandexternal spinsareloc atedprovides aconstraint\non the thickness of the SiC membrane that has to be fabricated (SiC membrane thickness =\nxopt - T/2, which is 100 nm for a W=500 nm YIG nanostripe width). The YIG nanostripe\nshould also be fabricated on the backside of the 1D SiC photonic wave guide, requiring that\nthe width of the 1D photonic waveguide be larger than the width of th e YIG nanostripe.\nTypically, the width of the 1D photonic waveguide has to be close to an integer number of\n14times half the wavelength of the low temperature zero phonon line of the V2 spin probe,\nwhen the 1D waveguide is a simple SiC nanobeam surrounded by air, in or der to optimize\nphoton confinement in the waveguide. With a wavelength of around 9 15 nm for the ZPL of\nV2 in 4H-SiC, this gives the following possible values of the width of such SiC nanobeam:\n457 nm, 915 nm, 1372 nm,... As we also wish to have a maximum optical elec tric field in\ninteraction with the sub surface V2 electric dipole, aligned along the c axis of 4H-SiC,20,27,28\nwe further chose a nano-beam width either of 457 nm or 1372 nm, bo th having an anti-node\nat half such a width. For a YIG nanostripe having a width of 500 nm, on e has thus to chose\na SiC nanobeam width of around 1372 nm.\nThis value can be considered as a good starting value for designing als o the width of another\nkind of 1D photonic waveguide fabricated in a 4H-SiC membrane and ma de of defects in a\n2D photonic cristal of SiC, itself made of an array of cylindrical air ho les in the SiC mem-\nbrane.50–52The main optical property of this 2D photonic crystal is its optical b andgap.50,51\nIt should be centered at the zero phonon line (ZPL) wavelength of t he V2 spin probe in\norder that the ensemble of defects (one or several lines of defec ts), that are here missing air\nholes in such a 2D photonic crystal, create a volume for photon confi nement and provide\nefficient 1D waveguiding properties to the V2 spin probe photolumines cence in this photonic\nnanostructure. The photonic band structure for TM modes prop agating in the plane of a\n2D photonic crystal with a triangular array of air holes in 4H-SiC can b e simulated numeri-\ncally.59Here I focus on TM modes, that means on electromagnetic modes ha ving an optical\nelectric field aligned along the c axis of 4H SiC and thus having the maxima l coupling with\nthe V2 electric dipole also aligned along the c axis.\nThe results of such a numerical simulation (see Appendix 3), show th at an isotorpic optical\nbandgap exist for TM mode if one fabricate such a nanophotnic stru cture with a center to\ncenter inter-hole distance equal to a= 622nmin the triangular array of air holes in SiC,\nthe air holes having a diameter of 2 ∗r=360nm. This is quite feasible with available elec-\ntron beam lithography and SiC RIE etching methods refsPCSiC. Othe r alternative designs\n15based on larger but partial optical bandgap for TM modes are also p ossible (see discussion\nin Appendix 3).\nSignal to noise ratio of the photoluminescence ODPEL-\nDOR signal obtained with the SiC-YiG quantum sensor\nUsing the normalized DEER signal expression,1,10,30Vdeer, whose value is comprised be-\ntween 0 and 1, and which is directly related to the ODPELDOR experime nt shown on\nfig.1b, considering the effect of the last additional −π\n2microwave pulse, which converts\nthe V2 spin quantum coherence into a V2 spin state populations differ ence, and assum-\ning a photon shot noise limited signal to noise ratio,60one obtains the following expres-\nsion for the photoluminescence signal to noise ratio R, in the case where ODPELDOR\nis obtained by off resonant optical excitation of the V2 spin probe, a t 785 nm for exam-\nple:R=Ropt(1−Vdeer(td, dx, C 2D,target))X, and in the case where ODPELDOR is\nobtained by a spin state selective resonant optical excitation of th e V2 spin probe, one\nfindsR=Ropt(1−Vdeer(td, dx, C 2D,target)), with Roptgiven by the formula: Ropt=\nexp/parenleftbig\n−2t0\nT2/parenrightbig/radicalBig\npcollpdetσ\nAP0T\nhν/angbracketleftΦ/angbracketright. I used above the following notations, /angbracketleftΦ/angbracketright=ΦH+ΦL\n2, and\n2X=ΦH−ΦL\n/angbracketleftΦ/angbracketright, taking into account the two possible different photoluminescence q uantum\nyield for the V2 spin probe, which depend on the spin state associate d to those optical tran-\nsisitons (see Appendix 4). The delay td is defined onfig.1b, C2D,targetis the 2D concentration\nof the target spin plane, and dxis the distance between the V2 spin probe and the target\nspin plane. σis the absorption cross section of the V2 spin probe, Ais the area on which the\noptical power P0is sent,hνis the photon energy, and Tis the integration time of the photo-\nluminescence by the photodetector over one single cycle. Xis given above, and has a value\nclose to 0.02 at room temperature according to previous ODMR expe riments on V2 with\noff resonant excitation of photoluminescence.19,20,27,28,42Vdeercan be numerically computed\nusing the linear approximation and shell factorization model.10This model was previously\n16introduced for calculating the standard DEER time domain signal in th e case of a three-\ndimensional distributions of spins. Here, this model has been adapt ed to take into account\nthebidimensional randomdistribution ofthetargetexternal spins intheir well-defined plane,\nparallel to the SiC sensor surface (see also next section). Assumin g realistic parameters esti-\nmated in the previous sections (see Appendix 4), one finds in the cas e of off resonant optical\nexcitation of the V2 spin probe, R= 90 for a single one shot one point ODPELDOR experi-\nment with off resonant optical excitation and a photoluminescence in tegration time T= 1µs.\nR can be off course increased in several ways. Firstly, R becomes 50 times larger when\nresonant optical excitation of V2 at low temperature is used, but t his is more complicated\nin practice (see appendix 4). Secondly, using averaging over 5000 c ycles of ODPELDOR\nexperiments, R is multiplied by 70, thus reaching R= 6300 for off reson ant excitation, which\ntakes in practice around 1 second (see appendix 4). Assuming Ncyc les=5000 per point and\na 100 points ODPELDOR spectrum as a function of fpump (1 point eac h 2 MHz, 200 MHz\nscanned), one could obtain such a 200 MHz ODPELDOR spectrum by o ff resonant opti-\ncal excitation (see next experimental section for an exemple of PE LDOR spectrum obtained\nwith a standard detection of EPR) in 100 s with a large signal to noise r atio R=6300, assum-\ning negligible hardware and software delays for changing the pumping microwave frequency\n(otherwise, the experimental time is determined by those delays). Thus the sensitive opti-\ncal, but indirect, detection of the paramagnetic resonance spect rum of Dark external spins\n(not photoluminescent paramagnetic centers or molecules) is poss ible by ODPELDOR spec-\ntroscopy. Note also that if the quantum sensor has many identical but isolated spin probes\nV2,NV2≥≥1, located at the same depth below the SiC sensor surface, then th e signal to\nnoise ratio is in principle also enhanced by a factor√NV2, but this last option for further\nimprovement seems difficult in practice (see appendix 4).\n17ODPELDOR quantum sensing with a SiC-YiG quantum\nsensor applied to structural biology\nThere are several ways to perform ODPELDOR quantum sensing, d epending whether the\nquantum sensor device presented here has a YIG nanostripe or no t. Also, in order to make\nthe discussion more explicit and as structural biology studies of tra nsmembranes proteins\nis expected to be an important application of this new quantum senso r, I will present the\nquantum sensing properties in the context of structural biology. The external spin consid-\nered here are thus spin labels, like nitroxide radicals,1gadolinium spin labels61and trityl\nradicals,56whose spin hamiltonians are well known.\nFigure 5: X band (9.369 GHz) EPR/ODMR spectrum of V2 in 4H SiC (in blac k) with\nan external magnetic field applied along the c axis of 4H-SiC, as well as a powder EPR\nspectrum of a/ nitroxide spin labels (red), b/ gadolinium spin labels (gr een), and c/ trityl\nspin labels (pink). Such powder spectrum is expected for spin labels in a frozen solution\nfor example. Blue squarres for probe microwave pulse and dashed d ark squarre for pump\nmicrowave pulse(s). Short pulses are spectrally broad by Fourier t ransform. Long pulse are\nspectrally selective. Multifrequency simultaneous microwave pumpin g is also now possible\nwith arbitrary waveform generators (AWG).\nSo let consider firstly the quantum sensor having one single V2 spin pr obe and no YIG\n18nanostripe. Asalreadyexplainedinthefirstsection, theODPELDOR pulsesequence(fig.1b)\nis similar to the older PELDOR pulse sequence implemented on a standar d pulsed EPR\nspectrometer. Before performing a two microwave frequencies O DPELDOR experiment in\ntime domain, one needs to record experimentally, or at least to nume rically simulate, the\nEPR spectrum of both the V2 spin probe (or several ones to get all the EPR lines) and of\nthe external spin labels. Such EPR spectrum in the first situation co nsidered here without\nany YIG nanostrip is numerically simulated at X band on fig.5, using Easy spin.62In this\nsimulation I considered many V2 all having the same uniaxial magnetic a nisotropy axis\noriented along the C axis of 4H-SiC, also assumed here to be the direc tion of the externally\napplied magnetic field. For the spin labels, a powder spectrum was simu lated with all\npossible orientations, providing broad spin labels EPR lines for nitroxid e radicals (fig.5a)\nand gadolinium radicals (fig.5b), which have anisotropic g tensor and/ or hyperfine coupling\ntensor. This is not the case for trityl spin labels, which have rather well defined magnetic\nparameters providing a narrow EPR line, typically of around 1 G at 70K in frozen water-\nglycerol.56It can be seen on fig.5a and 5b, that with nitroxyde radicals and gado linium\nspin labels, it is possible to excite a large part of their broad powder sp ectrum (dashed\ndark squarre), as it could be observed at low temperature, withou t exciting the V2 spin\nprobe. Inversely, it is possible to excite the V2 spin probe (blue squa rre) without exciting\nthe gadolinium spin labels. In the other important case of nitroxide ra idcals, which are the\nmost common radicals used in EPR based structural biology, one can excite the V2 spin\nprobe with a long selective resonant microwave pulse, thus minimally ex citing the nitroxide\nspin labels. The small amount of nitroxide radicals excited by this selec tive excitation\ncorrespond to a very diluted external spin bath of nitroxide radica ls (considering the very\nbroad EPR spectrum associated to all the nitroxide radicals), which is thus not expected to\nimpact significatively the spin coherence time of the V2 spin probe. On e can thus perform\nODPELDOR with such a SiC quantum sensor with nitroxide and gadolinium spin labels. In\nthe case of trityl radicals however, one can see on fig.5c that the n arrow EPR line of trityl\n19raidcals will spectrally overlap one of the EPR possible transition of th e V2 spin probe, the\ncentral one. Thus ODPELDOR is not possible with trityl radicals in this configuration with\na 4H-SiC quantum sensor which do not have a YIG nanostripe. For th e two favorable cases\nof nitroxide and gadolinium spin labels, the quantum sensing propertie s of a plane of spin\nlabels by a single V2 spin probe are numerically simulated on fig.6 for vario us experimental\nparameters.\n0 5 10 15 20 25 30 35\nd (n m)00.51\n0 5 10 15 20 25 30 35\nd (n m)00.51\n0 5 10 15 20 25 30 35\n 00.511 - V\n1 - V\n1 - V\nd (n m)1/ dx = 5 nm\n3/ dx = 15 nm2/ dx = 10 nm\nFigure 6: Dependence of the ODPELDOR normalized net signal to nois e ratio (see text),\n1-V, on the relative distance dx between the V2 spin probe and the p lane of external spin\nlabels (dx= 5 nm (1/) , 10 nm (2/), or 15 nm (3/), from top to bottom ), and on the target\nspin plane concentration ( C2D,target=1\nd2, with d in nm). Dark trace is for td=5µs, red trace\nis fortd=3µs, blue trace is for td=1µs. See fig.1 for the standard definition of td in the\n(OD)PELDOR sequence.\nIt is assumed on fig.6 that the spins labels are either anchored on lipids or on transmem-\nbrane proteins and located few nm above the SiC sensor surface, a ll in the same plane. If\nthe ODPELDOR normalized net signal to noise ratio, 1-V, is comprised between 0.5 and 1,\n20ODPELDOR is clearly feasible as explained in the previous section, wher eas if it is between\n0 and 0.5, it is much more difficult. A small inter-spin labels distance d valu e on fig.6 means\na large 2D concentration of spin labels in the target plane to sense. O ne can thus see on\nfig.6 that ODPELDOR is much easier to perform at large 2D concentra tion of spin labels,\nat smaller dx distance between the V2 and the plane of spin labels, and it is also easier if\nthe V2 has a long spin coherence time T2 allwoing to use a large value of t he time delay td\nin the ODPELDOR sequence (typically T2≥2td).\nIn practice, one would perform the quantum sensing ODPELDOR exp eriment described\nhere, firstly as a function of the microwave pump frequency, at fix edtdvalue, in order to\ndetect indirectly by the V2 photoluminescence, the EPR spectrum o f the external target\nspin labels. Then, one would perform a time domain ODPELDOR experime nt, varying td\nand using a microwave pump frequency appropriate to resonantly fl ip the external target\nspin labels. Then, the time domain decay curve obtained can be fitted with only dx and\nC2D ,target=1\nd2as fitting parameters. Note that this 2D concentration is in practic e often\nnot the total 2D concentration of spin labels, but the effective con centration if one excites\nonly one part of the target external spins spectrum, like in the cas e of nitroxide spin labels\nand gadolinium spin labels. Thus the factor associated to the propor tion of the spin labels\nEPR spectrum effectively pumped has to be taken into account in pra ctice.\nOne can note also that state of art commercial pulsed EPR spectro meters offer the pos-\nsibility to excite a given EPR spectrum over several parts by multifre quencies excitations,\nwhich is possible using an arbitrary waveform generator and which ca n provide a 2D con-\ncentration of spin labels effectivelly pumped close to the total 2D spin label concentration.\nPerforming such kind of time domain ODPELDOR experiments at sever al spin labels con-\ncentration C2D,target=1\nd2, it is thus possible to extract the parameter dx, directly related to\nthe important average insertion depth of the spin labelled proteins in the model 2D biomem-\nbrane. In order to really extract this average insertion depth of t he spin labelled proteins in\nsuch kind of SiC quantum sensor without a YIG nanostripe on the bac kside of the photonic\n210 500 1000 1500 2000 2500 3000 3500 4000 4500 5000\ntd (ns)00.20.40.60.81VDeer\nFigure 7: Time domain signal Vdeer(td), expected at dx=6nm (cont inuous line), and at\ndx=8nm (dashed dot line), each time for three different 2D externa l spins concentration:\nC2D,target=1\n(5nm)2,C2D,target=1\n(7nm)2,C2D,target=1\n(9nm)2. The lower C2D,targetis, the\nslower is the decay.\nwaveguide, a first solution consist in taking advantage of the residu al paramagnetic surface\ndefects at the SiC surface as a mean to determine the exact depth of the V2 spin probe below\nthe surface of SiC. Or, alternatively, one can fabricate on top of t he SiC surface a very thin\nSiO2 sacrificial layer on top of which a sacrificial YIG nanostripe can b e fabricated, whose\nmagnetic field gradient can encode the depth of the V2 on its resona nt frequency. Once\nthe depth of the V2 has been determined by ODMR of the V2 spin prob e under the strong\nmagnetic field gradient of the YIG nanostripe, this YIG nanostripe a nd the SiO2 ultrathin\nlayer can be removed by HF etching of the SiO2 layer. Thus simple 4H-S iC quantum sensors\nwithout a YIG nanostripe could be provided with a datasheet indicatin g the V2 depth below\nthe SiC surface and its T2 spin coherence time with a free SiC surface . Note that the case\nof a SiC quantum sensor having an ensemble of V2 spin probe and no YI G nanostripe may\nhave a larger signal to noise ratio but will probably have a lower spatia l resolution due to\nthe depth distribution expected by the ion implantation process (se e appendix 4).\nAsecond, very importantkindofquantumsensor proposedhere, istheSiC-YIG quantum\n22sensor containing a single V2 spin probe and a permanent YIG nanost ripe fabricated on the\nbackside ofthe1Dphotonicwaveguide, corresponding tothecase described inthefabrication\nprocess represented on fig.3. The experimental configuration he re corresponds to an external\nmagnetic field applied perpendicular to the c axis of 4H-SiC and here all spins are submitted\nto the strong gradient of dipolar magnetic field produced by the nea rby YIG nanostripe. The\nsimulated X band EPR/ODMR spectrum of the V2 and of the possible ta rget spin labels,\nwithout (left) and with(right) this strong gradient is described on fi g.8. This figure clearly\nshows that in this configuration and with the YIG nanostripe, it is now possible to perform\nODPELDOR spectroscopy with a single V2 and a plane of trityl spin labe ls because the\nresonant magnetic fields of the V2 and of the trityl can be spectra lly distinguished due to\nthe different YIG induced dipolar magnetic field experienced by those spins.\nThe detailed magnetic properties of this SiC YIG quantum sensor hav e been already\ndiscussed inaprevious section. Briefly, itwas shown, thatdue toth estrongdipolarmagnetic\nfield gradient produced by the YIG nanostripe, of 0.5 G per nm, and a ssumingT2,V2≥10µs\nfor a single isolated V2, and T2,trityl≥4µsfor a single isolated trityl spin label at a\ntemperature equal or below 100 K, then one can determine the x co ordinate of the V2 spin\nprobewithaspatial resolutionofaround1 A◦byODMRoftheV2under thisstronggradient,\nand the x coordinate of the trityl with a spatial resolution of aroun d 2A◦by ODPELDOR\nspectroscopy under this strong gradient, assuming a single trityl is present in the biolgical\nsample above the V2 spin probe. Thus ODPELDOR under the strong g radient of the YIG\nnanostripe of the quantum sensor allows to indirectly detect and loc ate with a high precision\nalong x a single trityl, through the measurement at a given pump freq uency of an oscillatory\nODPELDOR time domain signal (like the one obtained in standard DEER, but here with\njust two spins involved).\nOf course, it could be interesting to peform ODPELDOR spectrosco py with such an hybrid\nSiC-YIG quantum sensor having a doubly spin labelled single protein loca ted just above\nthe top SiC surface, inside a model 2D lipid bilayer or lipid nanodisc. Using the simple\n23Figure 8: X band (9.369 GHz) EPR/ODMR spectrum of V2 in 4H SiC (black ) with an\nexternal magnetic field applied perpendicular to the c axis of 4H-SiC, as well as a powder\nEPR spectrum of a/ nitroxide spin labels (red), b/ gadolinium spin labels (green), and c/\ntrityl spin labels (pink); left side: without the YIG nanostripe gradie nt, right: with the YIG\nnanostripe gradient. Such powder spectrum is expected for spin la bels in a frozen solution\nfor example. Blue squarres for probe microwave pulse and dashed d ark squarre for pump\nmicrowave pulse(s). Short pulses are spectrally broad by Fourier t ransform. Long pulse are\nspectrally selective. Multifrequency simultaneous microwave pumpin g is also now possible\nwith arbitrary waveform generators (AWG).\nODPELDORsequence describedonfig.1butunderthestrongdipolar magneticfieldgradient\nof the YIG nanostripe, then one can access, as explained just abo ve, to the X1cooordinate\nof trityl 1 and also here to the X2coordinate of trityl 2, through the value of the resonant\npump microwave frequencies at which one observe an oscillating ODPE LDOR signal. But\ntwo other bio-structural informations can be obtained through t he fourier transformation of\nthose two oscillating time domain ODPELDOR signal, because the oscillat ion period gives\naccess to the dipolar spin-spin couplings and thus to the two relative distances between the\nV2 spin probe and the two trityl spin labels anchored on the target p rotein under study,\ncalledRV2,Tri1andRV2,Tri2. Thus, ODPELDOR spectroscopy alone applied on such doubly\nspin labeled protein in a model lipid membrane provides already four con straints ( X1,X2,\n24RV2,Tri1,RV2,Tri2) on the 3D model of insertion or interaction of the protein with the m odel\nlipid biomembrane.\nConclusion\nIn this article, I have presented the theory of a new SiC-YiG hybrid q uantum sensor. I have\ndescribed a complete fabrication process, taking advantages of io n implantation methods\nand available SiC technologies, allowing to produce such a quantum sen sor. The sensor has\na single V2 negatively charged silicon vacancy color center introduce d inside a 4H-SiC 1D\nnanophotonic waveguide based on defects in a 2D photonic crytsal. The waveguide is ap-\npropriately designed to waveguide efficiently the V2 spin probe photo luminescence towards\nan optical fiber, interfacing the pulsed EPR spectrometer with an o utside standard photo-\nluminescence setup. I showed that adding a YIG ferrimagnetic nano stripe on the backside\nof the SiC photonic structure is important to determine precisely th e depth of the V2 spin\nprobe below the SiC surface with angstrom resolution. I also showed that under optimal\nconditions, and even under off resonant optical excitation of the V 2 photoluminescence,\nthis SiC-YIG quantum sensor should allow the sub-nanoscale investig ation of a single trityl\nspin labelled transmembrane protein by optically detected pulsed elec tron electron double\nresonance spectroscopy, with a large signal to noise ratio obtaine d in just one second of\nmeasurement per point, and using a simple standard X band pulsed EP R spectrometer up-\ngraded by this quantum sensor device and a simple optical fiber. This SiC-YiG quantum\nsensor should thus be of great interest for all the biophysicists, c hemists and physicists which\nare already worldwide pulsed EPR user and who wants to reach the sin gle spin sensitivity\nand sub nanoscale spatial resolution offered by quantum sensing me thodologies, just slightly\nmodifying their experimental EPR setup. The next challenges are th e fabrication of the\nproposed 4H-SiC-YIG hybrid spin-photonic-magnonic structure w ith state of art available\nSiC technology and ion implantation methods, as well as the experimen tal demonstration of\n25fiber based preparatory experiments combining an external optic al setup and a standard X\nband pulsed EPR spectrometer, both interfaced by an optical fibe r.\nAcknowledgments\nThe author thanks the University of Strasbourg and the french C NRS for the reccurent\nresearch fundings.\nAppendices\nAppendice 1: Ion implantation fabrication process.\nIt was recently demonstrated that low energy C+ ions ensemble impla ntation in 4H-SiC\nthrough an array of 65 nm diameter apertures patterned on a 300 nm thick PMMA layer\nusing electron beam lithography allows to produce, statistically, an a rray of isolated single\nsub surface V2 silicon vacancies (see main text), having, in statistic al average, a depth of 42\nnm and a longitudinal straggling of about 35 nm.\nThe idea here is thus to produce a mask having a single narrow apertu re and thus a single\naccess to the SiC substrate for implantation, to chose the energy of C+ ions implanted, the\nthickness of a sacrificial barrier layer for implantation, and also to c hose the dose of C+\nions implanted, such that, in statistical average, one single V2 color center can be produced\nper aperture and with a control over the statistical average dep th of the V2 color center. If\none also targets a lateral precision better than+/- 10 nm, this cor responds to a cylindrical\naperture in the mask used for ion implantation having a diameter equa l to 20 nm, which is\naccessible by state of art electron beam lithography. The implanted carbon ions profile and\nthe silicon vacancy concentration profile, as predicted by the SRIM software and assuming\n30 keV C+ ions implantation (with 7 degree of tilt) in a trilayer made of Zn O (60 nm), SiO2\n260\r 50\r 100\r 150\r 200\r 250\r0,0\r2,0x10\r4\r4,0x10\r4\r6,0x10\r4\r8,0x10\r4\r1,0x10\r5\r1,2x10\r5\rnormalized C+ concentration (cm\r-1\r)\r\ndepth (nm)\r120\r 140\r 160\r0,00\r0,01\r0,02\r0,03\r0,04\r0,05\rVSi number / (A˚ - ion)\r\ndepth (nm)\r\nFigure 9: Implanted carbon ions profile (left) as predicted by the SR IM software and\nassuming 30 keV C+ ions implantation (with 7 degree of tilt) in a trilayer m ade of ZnO (60\nnm), SiO2 (60 nm) and 4H-SiC (infinity); silicon vacancy concentratio n profile in 4H-SiC\n(right).\n(60 nm) and SiC (infinity), are presented on fig.9. Further assuming a dose of 30 keV C+\nions implanted equal to 4 .41012cm−2, as well as a decimation factor equal to 0.01 (to take\ninto account the more realistic amount of silicon vacancies created in practice compared to\nthe one calculted by SRIM, possibly due to interstitial-vacancy reco mbination processes or\ndefects complex formation), then I find that in statistical averag e, one single V2 spin probe\nis created through an aperture of 20 nm of diameter, the other C+ ions being stopped by\nthe 300 nm PMMA film. This also implies, in view of the Poisson Statistic exp ected, that if\none produces in parallel 100 similar quantum sensors on the same 4H- SiC wafer, 37 quantum\nsensors fabricated will have exactly 1 single V2 in the aperture, 37 w ill have 0 V2, 18 will\nhave 2 V2, 6 will have 3 V2,... Also, taking into account the V2 depth dist ribution profile\nof fig.9 (right), one can also calculate the probability that the V2 spin probe be contained\nin a thin layer of SiC of 2.5 nm thickness, as obtained by SRIM. Taking th e beginning of\nthe 4H-SiC material (absolute position 122.5 nm) as the origin of dept h, I find here P(0;\n272.5 nm)=0.21, P(2.5; 5 nm)=0.17,P(5; 7.5 nm)=0.14,P(7.5; 10 nm)=0.12, ... Thus one can\nconclude that with the fabrication process proposed here, which is statistical in essence, over\n100 quantum sensor fabricated in the same way, one will obtain arou nd 37∗0.14 = 5\nquantum sensors having a single V2 in their aperture located above t he YIG nanostripe, and\nwith a depth comprised between 5nm and 7.5 nm. Note also that at the end of this method\nof implantation through a sacrificial layer, this sacrificial layer has t o be removed. Here,\nHCl can be used to remove the ZnO sacrificial layer, while not etching the SiO2. Then HF\netching can be used to remove the SiO2 sacrificial layer. Note that t he SiO2 sacrificial layer\nbelow the dense ZnO layer is necessary to stop the Zn ions impacted b y the C+ ions im-\nplanted, avoiding Zn contamination of the SiC top surface. As an alte rnative to 30 keV C+\nions implantation through a single aperture in PMMA over such trilayer (ZnO/SiO2/SiC),\none could also perform a similar implantation but through a bilayer made of a thicker SiO2\nsacrificial layer on SiC, targeting a maximum carbon ions and V2 conce ntration close to the\nSiO2/SiC interface and then etching the SiO2 by HF. One could also per form direct 5 keV\nC+ ions implantation in 4H-SiC through a single PMMA aperture of diamet er 20 nm over\nSiC at a dose of 11011cm−2. SRIM simulation then predict in average 105 silicon vacancies\nper aperture, but taking into account once again the decimation fa ctor equal to 0.01, one\nexpect in practice in this case a single V2 spin probe per aperture. SR Im simulation (not\nshown here) also predict that the single silicon vacancy will be create d with a maximum\nprobability at zmax=6nm.\nAppendice 2: homogeneous linewidth and spin coherence time for the isolated V2 in\n4H-SiC.\nAn ensemble of V2 spins in 4H-SiC can already have a narrow inhomogen eous linewidth\nof few Gauss or even less in some previous reports. A thin epitaxially g rown isotopically\npurified layer of SiC at the top surface of the 4H-SiC device propose d could probably further\n28reduce this inhomogeneous linewidth. But, a single isolated V2 spin pro be in 4H-SiC is\nexpected to have an homogeneous linewidth of less than 1 G. In princ iple, the homogeneous\nlinewidth should be inversely proportional to the T2 spin coherence t ime of the isolated V2\nsingle spin probe. A room temperature spin coherence time for bulk ( isolated or diluted) V2\nof around 50 µsare often measured in bulk 4H-SiC HPSI, corresponding to a homoge neous\nlinewidth of around 7 mG, and previous studies reported or estimate d the T2 of such bulk\nisolated V2 to more than 300 µ sat room temperature. A much longer T2 is expected at\nlow temperature at few Kelvin, because it is ultimately limited by T1 relat ed longitudinal\nrelaxation processes, and the T1 value of isolated V2 has been show n to exceed few minutes\nat few Kelvins (see main text). Indeed, using dynamical decoupling p ulses sequences at low\ntemperature, it was recently experimentally demonstrated that t he effective T2 obtained us-\ning this methodology can reach 10 msat a temperature of few K. In practice however, for sub\nsurface V2 created by such a fabrication process, the T2 is expec ted to be much smaller due\nto the residual surface defects density, but this remained unexp lored to date and is largely\ndependent on the material and fabrication methodology used. A re sidual surface density of\nstate of 41010cm−2, correspond to an average lattice of surface defects, separat ed in plane\nby 50 nm. In the proposed quantum sensor, the V2 spin probe could be placed around 6 nm\nbelow the 4H-SiC surface. That means that over the many SiC-YiG de vices identically fabri-\ncated in parallel, there should be some of them having all the required properties, including\nsome with a long spin coherence time due to their large distance to the nearest fluctuating\nsurface defects. Note also that in all cases, it is in principle possible t o reduce the magnetic\nnoise due to surface defects by cooling down the sample to sufficient ly low temperature, thus\nincreasing their T1 relaxation time.\nAppendice 3: Optical bandgap of TM modes of the 2D photonic cryst al\nHere I present a numerical simulation of the optical bandgap prope rties of a 2D photonic\n29crystal, assumingthefollowingparameters: therefractiveindexu sedare,nair= 1,nSiC= 2.5\nandr= 0.29∗a, withrtheradiusofairholesandatheinter-center distancebetwe en neighbor\nair holes in the triangular lattice.\n00.10.20.30.40.50.60.70.80.91\nGammaGammaK M\nFigure 10: Numerical simulation of the TM modes dispersion relation an d optical bandgaps\nfound in a triangular lattice of cylindrical air holes in 4H-SiC. The norma lized frequency\nω=a\nλis plotted as a function of the in plane photon wave vector considere d. The horizontal\naxis indicates the in plane wavevector of the photon, along the direc tion ΓK,K M, and\nMΓ. Parameters: refractive index nair= 1,nSiC= 2.5 andr= 0.29∗a, with r the radius of\nair holes and a the inter-center distance between neighbor air holes in the triangular lattice.\nOne can see on fig.10 that a single isotropic very narrow optical band gap for TM modes\nexist for a normalized frequency ω=a\nλ= 0.680. Thus a good design for this 2D photonic\ncrystal consist in fabricating a triangular array of air holes in SiC, wit h a center to center\ninter-hole distance equal to a= 915∗0.680 = 622 nm, the air holes having a diameter of\n2∗r= 2∗0.29∗622=360 nm. This is quite feasible with available electron beam lithography\nand SiC RIE etching methods (see main text). One can also see on fig.1 0 that two larger but\npartial optical bandgap exist for TM modes propagating along the K Mdirection in such a\n2D photonic crystal, at normalized frequencies ω=a\nλ= 0.467 andω=a\nλ= 0.345. Thus if\nfor example, one define a line of defects in this 2D photonic crystal a long the direction Γ M,\nwhich is perpendicular to the K Mdirection, then one expects some photon confinement in\nthe direction perpendicular to this 1D photonic waveguide, and thus a still interesting col-\nlection efficiency for the V2 spin probe photoluminescence. More pre cisely, considering that\n30a photon is emitted by the V2 with an in plane wave vector forming an an gleθwith the Γ M\ndirection of this second kind of 1D photonic waveguide, then the pro bability of collecting\nthis photon in the direction of the waveguide is equal to ( cos(θ))2. The probability that the\nphoton is emitted in the orthogonal KM direction is ( sin(θ))2, and in this case it could be\nreabsorbed by the V2 spin probe and then reemitted in another dire ction more appropriate\nfor collection by the 2D waveguide. Thus, integrating over the θrange, one finds a collection\nefficiency of at least 0.5 for this 1D photonic waveguide build in a 2D phot onic crystal hav-\ning an anisotropic optical bandgap for TM modes. This is of course les s than the complete\ncollection (collection efficiency equal to 1) which is possible with the tru e isotropic optical\nbandgap case, but it is still interesting and offer an alternative optic al nanophotonic design.\nOne possible advantage here is the larger optical partial bandgap, maybe more appropriate\nfor room temperature operation where the V2 photoluminescence is broadened by phonon\nreplica. In the case of a partial optical bandgap case for TM modes withω=a\nλ= 0.467,\none thus has to fabricate a triangular array of air holes in SiC, with a c enter to center inter-\nhole distance equal to a= 915∗0.467 = 427 nm, the air holes havind a diameter of\n2∗r= 2∗0.29∗427=248 nm. This is also feasible with available electron beam lithography\nand SiC RIE etching methods. In the third case of a partial optical b andgap case for TM\nmodes with ω=a\nλ= 0.345, one thus has to fabricate a triangular array of air holes in SiC,\nwith a center to center inter-hole distance equal to a= 915∗0.345 = 316 nm, the air\nholes havind a diameter of 2 ∗r= 2∗0.29∗316=184 nm. This is still feasible with available\nelectron beam lithography and SiC RIE etching methods, but maybe it is mechanically less\nrobust.\nAppendice 4: Signal to noise ratio of the ODPLEDOR experiment\nAs already said above, coupling strategies between such a 1D photo nic waveguide and an\noptical fiber exist, with demonstrated coupling efficiencies comprise d between 0.25 and 0.90.\n31In the following, I will thus assume a V2 photoluminescence collection e fficiency pcollcom-\nprised between 0.125=0.25*0.5 (case of partial optical bandgap) a nd 0.90=0.90*1 (isotropic\noptical bandgap), thus pcoll= 0.51 in average, further assuming no loss of photons be-\ntween the SiC-YIG quantum sensor and the photo-detector. The photo-detector is assumed\nhere to have a quantum efficiency for near infrared photon detect ion at 915 nm of around\npdet= 0.4. Also, one has to note that the excitation efficiency of the photolu minescence of\nthe V2 spin probe is proportionnal to a wavelength dependent term (related to the absorp-\ntion cross section) and to the photoluminescence quantum yield, giv en quite generally by\nΦ =krad\nkrad+knot−rad, withkradthe rate of radiative recombination of the V2 spin probe from a\ngiven excited state and knot−radthe rate of non radiative recombination from a given excited\nstate, which includes the inter-system crossing rate. The V2 spin p robe color center has a\nspin3\n2, andthegroundandexcited electronicstatearebothsplittedby t heso calledzerofield\nsplitting in the ground and excited states. Thus, optical transition s noted L (like low value\nof ms) between GS (ms=+/-1/2) and ES (ms=+/-1/2) can be disting uished in energy from\nthe optical transitions between GS (ms=+/-3/2) and Es (ms=+/-3 /2), noted H (like high\nvalue of ms), even without application of a magnetic field. Thus, the e xcitation efficiency\nof the V2 spin probe can become spin state dependent by two effect s: either by spin state\nselective optical excitation of the V2 spin probe at the end of the DE ER sequence (after the\nthird microwave pulse at fprobe), which is possible by energy selective resonant excitation of a\ngiven narrow optical transition by a narrow linewidth laser, or by the fact that the quantum\nyield is spin state dependent for the V2 spin probe, due to the spin de pendent intersystem\ncrossing rate of the V2 spin probe. I now introduce the following not ations,/angbracketleftΦ/angbracketright=ΦH+ΦL\n2,\nand 2X=ΦH−ΦL\n/angbracketleftΦ/angbracketright, taking into account the two possible different photoluminescence q uan-\ntum yield. Note also that it was previously shown (see main text) that the optical power\nnecessary to obtain saturation values of optical V2 spins pumping is inversely proportional\nto their longitudinal spin-lattice relaxation time T1. AsT1increases up to several tens of\nsecond at 5K, then less than 1 mW at 780 nm spread over a 1mm*1mm s quare sample is\n32sufficient at 5K for obtaining optical pumping saturation, but more a t room temperature.\nWith the above remarks, one can estimate the signal to noise ratio R of an ODPELDOR ex-\nperiment, asdescribedbythesequence onfig.1b, assuming tdisfixedandthepumpfrequency\nfpumpvary. The photoluminescence signal Splin ODPELDOR is expected to vary depending\non the value of the microwave pump frequency chosen, because wh en the target spins are\nmicrowave manipulated on resonance, the V2 spin probe feels an acc elerated spin echo decay\nat a given tdparameter value. In optimal experimental conditions, the Noise is d ominated by\nthe optical shot noise, and is given by Npl=/radicalbig\nSpl(pB= 0), where pB= 0 means that the\nmicrowave pump frequency is off resonant with the resonant frequ ency of the target external\nspin(s).pB= 1 means on the contrary, that the microwave pump frequency is r esonant with\nthe resonant frequency of the target external spin(s). Thus, the net signal to noise ratio R\nin ODPELDOR spectroscopy is given by the formula R=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(Spl(pB=1)−Spl(pB=0))√\nSpl(pB=0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nUsing the normalized DEER signal expression (see main text), Vdeer, whose value is com-\nprised between 0 and 1, directly related to the ODPELDOR experimen t shown on fig.1b,\nand considering the effect of the last additional −π\n2microwave pulse, which converts the\nV2 spin quantum coherence into a V2 spin state populations differenc e, one obtains the\nfollowing expression for the signal to noise ratio Rin the case where ODPELDOR is ob-\ntained by off resonant optical excitation of the V2 spin probe, at 78 5 nm for example:\nR=Ropt(1−Vdeer(td,dx,C2D,target))X, and in the case where ODPELDOR is obtained\nby a spin state selective resonant optical excitation of the V2 spin p robe (assumed on the\nHH optical transition), one finds R=Ropt(1−Vdeer(td,dx,C2D,target)), withRoptgiven by\nthe formula: Ropt=exp/parenleftbig\n−2t0\nT2/parenrightbig/radicalBig\npcollpdetσ\nAP0T\nhν/angbracketleftΦ/angbracketright.\nσis the absorption cross section of the V2 spin probe, Ais the area on which the optical\npowerP0is sent,hνis the photon energy, and Tis the integration time of the photolumines-\ncence by the photodetector over one single cycle. Xis given above, and has a value close to\n0.02 at room temperature according to previous ODMR experiments on V2 with off resonant\nexcitation of photoluminescence. Thus one see immediatly that R can be greatly improved\n33by resonant spin state selective optical excitation of the photolum inescence of the V2 spin\nprobe, on the well defined Zero phonon line (ZPL) at low temperatur e, but off resonant exci-\ntation is much more convenient in practice, particularly at high tempe rature or even at room\ntemperature. Vdeercan be numerically computed using the linear approximation and shell\nfactorization model (see main text). This model was previously intr oduced for calculating\nthe standard DEER time domain signal in the case of a three-dimensio nal distributions of\nspins. Here, this model has been adapted to take into account the bidimensional random\ndistribution of the target external spins in their well-defined plane, parallel to the SiC sensor\nsurface.\nNow, assuming a sensor operating with t0= 6.25µ sand 2t0=T2= 12.5µ s, as-\nsumingC2D,targetis sufficiently large for a given td, such that Vdeer(td,dx,C2D,target) = 0 ie\n1−Vdeer= 1 (see main text for examples),further assuming, /angbracketleftΦ/angbracketright ≈1,pcoll= 0.5,\npdet= 0.4,σ\nA≈1, and choosing a photoluminescence integration time per ODPELDOR\nsequence T= 1µsfor example and an optical power at 780 nm of 20 µ Wspread over\nA≈3λ\n2λ\n10, according to the previous nanophotonic structure design, one fi nds approx-\nimatelyRopt= 5000, and thus, in the case of off resonant optical excitation of t he V2\nspin probe, R= 5000 (1 −0) 0.02≈90 for a single one shot one point ODPELDOR\nexperiment with off resonant optical excitation. R can be off course 50 times larger for\nresonant optical excitation at low temperature. The optical re-p umping time of V2 spins\nis short in practice with an appropriate laser, typicaly of around 150 µs, and much less at\nlow temperature. The ODPELDOR microwave pulses sequence after optical initialization\nof V2 spins last around 20 µs, such that the shot repetition time of the full ODPELDOR\nsequence can be taken equal here to Texp= 200µs, meaning one can perform 5000 cycles\nof averaging in one second, which further improve the signal to nois e ratio by a factor of\n√\n5000≈70. As a consequence, for a SiC quantum sensor having a single V2 sp in probe,\none finds, assuming averaging over 5000 cycles ie over 1s of experim ent in practice, a signal\nto noise ratio equal to R1sec,off optex = 6300 for off resonant excitation of the photolumines-\n34cence, while for resonant optical spin state selective excitation of photoluminescence at low\ntemperature, one finds R1sec,onoptex = 315000. In both case, ODPELDOR is clearly feasible\nand should have a high signal to noise ratio. Even if one consider some pessimistic values\nlikepcoll= 0.01 with some photon loss betwen the quantum sensor and the photo detector,\nand a low efficiency detector with pdet= 0.04, then one still finds for one second of averaging\n(5000 cycles) a signal to noise ratio of R1sec,off optex = 286. Assuming Nshot=5000 per\npoint and a 100 points ODPELDOR spectrum as a function of fpump (1 point each 2 MHz,\n200 MHz scanned), one could obtain such a 200 MHz ODPELDOR spect rum by off resonant\noptical excitation in 100 s with a signal to noise ratio R comprised betw een 286 and 6300,\nassuming negligible hardware and software delays for changing the p umping microwave fre-\nquency (otherwise, the experimental time is determined by those d elays). Thus the optical,\nbut indirect, detection of Darkexternal spins (not photoluminesc ent paramagnetic centers or\nmolecules) is possible by ODPELDOR spectroscopy. Those estimates also assume a photon\nshot noise limited noise. If some photoluminescence background is pr esent in the SiC-YIG\nquantum sensor device, of course the signal to noise ratio will be low er. But according to\nthe above estimates, it seems clearly possible to tolerate some amou nt of photoluminescence\nbackground for performing ODPELDOR spectroscopy as long as on e has sufficiently good\nphotodetection and photon collection efficiencies, and further usin g averaging. Note also\nthat the large signal to noise ratio estimated above also assume exp erimental parameters\nsuch that Vdeer(td,dx,C2D,target) = 0, but of course, values of Vdeermuch larger, between 0.5\nand 1, meaning a smaller decoherence effect of the spin bath on the V 2 spin probe, could\nalso be measured in the optimal situation of an optical shot noise limite d measurement.\nAs a last remark, it has to be also noted that if the quantum sensor h as many identical\nbut isolated spin probes V2, NV2≥≥1, located at the same depth below the SiC sen-\nsor surface, then the signal to noise ratio is in principle enhanced by a factor√NV2if it\nremains shot noise limited. 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Bailleul,4and V.Vlaminck1\n1Colegio de Ciencias e Ingeniera, Universidad San Francisco de Quito, Quito, Ecuador\n2Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA\n3Department of Physics and Astronomy,\nUniversity of Delaware, Newark, Delaware 19716, USA\n4Institut de Physique et Chimie des Mat\u0013 eriaux de Strasbourg,\nUMR 7504 CNRS, Universit\u0013 e de Strasbourg,\n23 rue du Loess, BP 43, 67034 Strasbourg Cedex 2, France\n(Dated: November 8, 2021)\nAbstract\nThe propagation of magnetostatic forward volume waves excited by a constricted coplanar waveg-\nuide is studied via inductive spectroscopy techniques. A series of devices consisting of pairs of\nsub-micrometer size antennae is used to perform a discrete mapping of the spin wave amplitude\nin the plane of a 30-nm thin YIG \flm. We found that the spin wave propagation remains well\nfocused in a beam shape of width comparable to the constriction length and that the amplitude\nwithin the constriction displays oscillations, two features which are explained in terms of near-\feld\nFresnel di\u000braction theory.\n1arXiv:1807.11754v1 [cond-mat.mes-hall] 31 Jul 2018The emerging \feld of magnonics [1, 2] has sparked a renewed interest in non-uniform\nmagnetization dynamics. Spin-waves are now considered as a very promising information\ncarrier for performing basic logic operations [3{7], or implementing novel computation archi-\ntectures [8, 9]. A key advantage of spin-waves is that their dispersion can be easily tailored\nin a wide band of the microwave spectrum, particularly in the so-called magnetostatic wave\nregime for which the magnetic dipolar interaction plays the dominant role. Recently, it has\nbeen demonstrated that the propagation of spin waves in ferromagnetic thin \flm could be\nshaped using several concepts borrowed from optics [10]. A special attention has been set\non understanding their refraction and re\rection e\u000bects [11{14], and also on generating and\nmanipulating spin-wave beams. The latter is of particular importance in order to exploit\nthe potential of multi-beam interference.\nSo far, three di\u000berent mechanisms have been investigated to shape spin-wave beams: (i)\nthe so-called caustic e\u000bect [15{17, 21] associated with the very strong anisotropy of mag-\nnetostatic wave dispersions; (ii) the con\fnement by the strongly inhomogeneous internal\nmagnetic \felds existing at strip edges [10], in magnetic domain-walls [18], or close to\nnano-contact spin torque nanoscillators [19, 20]; (iii) the coupling to specially designed\nconstricted microwave antennae providing a suitable non-uniform magnetic pumping \feld\npro\fle [22, 23]. The last method appears the most versatile, being able to produce a co-\nherent spin wave beam in a homogeneous magnetic layer without any special requirement\non its magnetic con\fguration. It was \frst proposed theoretically by Gruszecki et al. via\nmicromagnetic simulation [22], and was recently veri\fed experimentally by K orner et al. via\ntime resolved magneto-optical imaging of magnetostatic surface wave beams generated in\na relatively thick NiFe \flm [23]. In this letter, we show experimentally that the spin wave\nbeam generated by a constricted coplanar waveguide (CPW) follows closely a near-\feld\ndi\u000braction pattern. To this objective, we resort to all-electrical measurements performed in\na con\fguration providing isotropic spin-wave propagation (thin Yttrium Iron Garnet \flm\nmagnetized out-of-plane) and analyze them using elementary Fresnel di\u000braction modeling.\nThe spin wave antennae are designed in such a way that the constricted region of the\nCPW reproduces as closely as possible the case of an isolated rectangular slit. We \frst focus\non the geometry-A of spin-wave antennae shown in Fig. 1-(a),(b). It consists of a pair of\nidentical shorted CPWs, whose constriction is shaped symmetrically with a gradual bend\n2in order to have two narrow sections of CPW facing each other. The constricted region\nof the CPW consists of a central track of width wS= 400nmand two ground tracks of\nwidthwG= 200nmwith a gap of 200 nm. The generated spinwaves have a wavelength of\nthe order of the distance between the center of the ground tracks, i.e. \u0015'1\u0016m, which\nremains much smaller than the constriction length. We adopted a much sharper constric-\ntion than in the geometries study by Gruszecki et al. [22], and K orner et al. [23], with\na factor of ten between the widths of the constricted section I and the non-constricted\nsection II. This allows us to fully separate the peaks associated with the excitation of spin\nwaves in the two sections, as illustrated in Fig. 1-(c) which shows the corresponding Fourier\ntransforms of the current density (assumed to be uniform in each CPW track). For section\nI, one distinguishes a main peak centered at kI= 5:92rad:\u0016m\u00001with a full width at half\nmaximum \u0001 kI= 4:15rad:\u0016m\u00001, and for section II a main peak at kII= 0:59rad:\u0016m\u00001with\n\u0001kII= 0:41rad:\u0016m\u00001.\nWe fabricated \fve spin-wave transducers of geometry\u0000Awith separation distance D=f4;6;8;10;12g\u0016m\nand constriction length lexc= 5\u0016m, which we used for preliminary characterization and val-\nidation of the spin-wave transduction in continuous layer. The antennae were fabricated\nby e-beam lithography and lift-o\u000b of 5 nmTi= 80nmAu directly on top of 30 nmthin\nsputtered YIG (Y 3Fe5O12) \flms deposited on gadolinium gallium garnet by magnetron\nsputtering and post-annealed [24{26]. The fabricated device is then placed in the center of\nthe lower pole of an electromagnet \ftting in a home-made probe station, and we proceed\nto the propagative spin-wave spectroscopy measurement [27] while applying an external\nmagnetic \feld Hlarge enough to magnetize the \flm out of the plane. This corresponds to\nthe so-called magnetostatic forward volume wave (MSFVW) con\fguration, for which the\nisotropic dispersion relation does not favor any propagation direction. This is in strong\ncontrast with the situation of in-plane magnetized \flms, for which the spin-wave dispersion\nis strongly anisotropic with a maximal group velocity in the so-called magnetostatic surface\nwave con\fguration. For practical reasons, most studies of nanomagnonics, including recent\nones in YIG have been done in this last con\fguration [28, 29]. In the present case, we\nsimplify the analogy with optics by employing the isotropic MSFVW con\fguration, and\ntake directly advantage of the low damping and low magnetization of the YIG \flms.\nThe microwave spectra were acquired using a vector network analyzer ( AgilentE 8342B,\n10MHz - 50GHz ) at low input power ( \u000020dBm ), 100Hzbandwidth, and in a single sweep\n3kIkII k=0D = 4 /uni03BCm\nD = 8 /uni03BCm\nD = 12 /uni03BCmfosc\nkIkII(b)\n5/uni03BCm\nkIIkI(a)\nlexc\n2/uni03BCmD\nλIλII\n(c)(d)FIG. 1. (a) Geometry-A spin-wave antennae with a separation distance D= 12\u0016m. (b) Scanning\nelectron microscope image of the same device zoomed in the region of the constriction. (c) Fourier\ntransform of the current distribution for the two sections I and II, and MSFVW dispersion relation\nfor\u00160Hext= 308mT. (c) Self- and mutual inductance spectra obtained at \u00160Hext= 308mTfor\nthree devices of geometry-A with separation distance D= 4\u0016m,D= 8\u0016m, andD= 12\u0016m.\nmode in order to limit the possible temperature drift of the electromagnet. We always per-\nform two measurements: a \frst one at a resonant \feld ( Hres), followed by a second one at\na reference \feld ( Href) for which no resonance occurs within the frequency range swept. In\nthis manner, we retrieve the variation of inductance \u0001 Lij=Lij(Hres)\u0000Lij(Href) due to\nspin wave excitation [29]; \u0001 L11the self-inductance measured on antenna 1, and \u0001 L21the\n4mutual inductance characterizing the transduction of spin wave excited by antennae 1 and\ndetected by antennae 2. Figure 1-(d) shows typical spectra obtained with identical antennae\nofgeometry-A atHext= 308mT, for three di\u000berent separation distances (4, 8, and 12 \u0016m).\nWe can identify from the re\rection spectra three main peaks which are attributed to the\ndi\u000berent parts of the CPW. Namely, the lowest frequency peak corresponds to the quasi-\nuniform resonance ( k= 0) of the wide section of the CPW where the 150 \u0016mpitch coplanar\nprobe are contacted. The second peak corresponds to the non-constricted region II of the\nCPW, and the last peak to the constricted region I. For the mutual inductance spectra,\nwe observe oscillations only underneath the last peak con\frming that only the constricted\nregion of the CPW contributes to the spin wave transduction between antennae. These\noscillations are attributed to the phase delay kDaccumulated by the spin-waves during its\npropagation between the two antennas, and therefore are more numerous the longer the\nseparation distance between antennae. The level of amplitude of the mutual inductance\nspectra is comparable to the one found when performing simulation of MSFVW transduc-\ntion [27, 30] on a stripe of width equal to the length of the constricted region. This suggests\nalready that the excitation of the spin wave from this type of constriction should remain\nfairly focused.\nTo validate our spin wave transduction technique when applied to a continuous magnetic\nlayer, we \frst analyze the microwave spectra measured for di\u000berent \felds and di\u000berent dis-\ntances between the antennas. In particular, we can take advantage of the three section of our\nwaveguides to perform k-resolved ferromagnetic resonance (FMR). As shown in Fig. 2-(a), we\ntrack the peak position in function of applied \feld respectively for k= 0,kII, andkI. Fitting\nit to the MSFVW dispersion relation [31] for \fve di\u000berent devices, we obtain an average value\nfor the gyromagnetic ration \r=2\u0019= 28:26\u00060:07GHz:T\u00001and the e\u000bective magnetization\n\u00160Meff= 136\u00062mT. Furthermore, we can estimate independently the saturation magneti-\nzationMsby plotting the \feld-dependence of the di\u000berence f2\nres(kI)-f2\nres(kII) =\r2\u00162\n0(Hext-\nMeff)Ms(kI\u0000kII)t=2 (wheretis the YIG \flm thickness) as shown in Fig. 2-(b), from which\nwe \fnd a nice linear dependence and the average value \u00160Ms= 196\u00068mT. Next, we use the\nobserved decay of the amplitude of the mutual-inductance as a function of the distance as\nshown in Fig. 2-(c) to extract the characteristic attenuation length of the spin wave. For each\napplied \feld, we observe a clear linear dependence of ln(j\u0001L21j) on the antenna separation\nD, which is consistent with an exponential decay jL21j/e\u0000D=L att). This constitutes another\n5(a)\n(b)\nγμ0MS(kI-kII)t/4π(c)\n1/vg-1/L att\n(d)2πα eff\n(g)(e)\n(f)FIG. 2. (a) Field dependence of the resonance peaks k= 0,kII, andkI. (b) Di\u000berence of the square\nof the resonance frequencies f2(kI) -f2(kII). Separation distance dependence of (c) ln(\u0001L21), and\n(d) the inverse of the oscillation period of \u0001 L21. Frequency dependence of: (e) the ratio of the\ngroup velocity to the attenuation length, (f) the attenuation length, and (g) the group velocity.\nevidence for a proper focusing of the spin wave excitation. Obviously, a di\u000bused emission\nof opening angle \u0012would reduce the amplitude by an additional factor ldet=(D\u0012), which\nis not observed here. Then, from the period of oscillation foscof the mutual inductance\nspectra, we can estimate the group velocity vgaccording to vg=foscD[27]. Fig. 2-(d) shows\nclear linear dependence of 1 =foscwithDfor the di\u000berent applied \felds. Finally, we perform\na linear \ft of the frequency dependence of the ratio vg=Latt[32],and identify the slope to\n2\u0019\u000beff[cf. Fig. 2-(e)], which gives us a value of the e\u000bective damping \u000beff= 7:5\u00060:2 10\u00004in\ngood agreement with previous measurements on similar \flms [33, 34]. We obtain fairly good\nagreements with the theoretical group velocity and attenuation length estimated from the\nMSFVW dispersion relation [dotted lines in Fig. 2(f,g)], which validate the implementation\nof the spin wave transduction technique to continuous layers for this geometry of CPW.\nWe now turn to the main result of this work, which is the evolution of the ampli-\ntude \u0001L21between several pairs of antennae at various separation distances D, and with\nvarious shift swith respect to their axis in order to map in a discrete manner the spin-\n6wave emission from a constriction. We fabricated two series of pairs of non-identical\nspin-wave antennae with a long excitation antenna ( lexc= 10\u0016m), and a shorter detection\nantenna of ( ldet= 2\u0016m) in order to re\fne the spatial resolution of the mapping. The \frst\nseries consists of the symmetrical geometry-A as shown in Fig. 3-(a), for which we fabri-\ncated six devices having the same separation distance D= 5\u0016m, and only one-sided shift\ns=f0;1;2;3;4;6g\u0016m. For the second series, geometry-B shown in Fig. 3-(b), which con-\nsists of an asymmetrical constriction short-circuited right at its end and also with a steeper\nbend, we fabricated eighteen devices covering two separation distances D=f8;12g\u0016m, and\nnine shifts=f\u00008;\u00006;\u00004;\u00002;0;2;4;6;8g\u0016m. Fig. 3-(c) shows the shift dependence of the\npeak amplitudej\u0001L21jmaxforgeometry-A at various applied \feld (see typical examples of\nmutual-inductance spectra in the supplementary materials [35]). We observe an oscillation\nof the amplitude within the width of the constriction and a clear drop of amplitude for the\ndevices= 6\u0016m, which lays just entirely outside of the constriction. Similar observations\nare made with geometry-B shown in Fig. 3-(d) although the drop of amplitude outside the\nconstriction is slower for negative shifts due to the non-symmetrical shape of the antennae.\nIndeed, the shorted ends of the constrictions, which come close to each other for positive s\n[see Fig. 3(b)], radiate much less spin-wave power out of the constriction than the broader\nconvex CPW access, which come close to each other for negative s.\nTo describe these features of spin wave emission from a constricted CPW, we propose\nto implement the common equations of optics used in the case of the Fresnel di\u000braction\nfrom a rectangular slit [36]. This choice is particularly relevant for the range of wavelength\nconsidered, for which the Fresnel radius RFremains much smaller than the length of the\nconstriction: RF=p\n\u0015D << l exc. We simplify the problem by considering that each track\nof the CPW [ j=fG\u0000;S;G +g, see sketch in Fig. 3-(e)] acts a single rectangular source of\ncoherent, circular, and monochromatic waves, of wavelength \u0015=2\u0019\nkI. We also account for\nthe spin wave attenuation with an exponential decay factor ( e\u0000r=L att), whereLattis \feld- (or\nfrequency-) dependent with a value given in Fig. 2-(f). The normalized spin-wave amplitude\n~mFresnel (D;s) at a distance Dand a shift semitted by the track jof the CPW is written\nas:\n~mj(D;s) =Rl=2\n\u0000l=2dyRwj=2\n\u0000wj=2dx1prje\u0000rj=Latte\u0000ikrj (1)\n7(c) (d)\n 5 10 15 | | | G- G+ S\nλ+(x,y)\n+\n+1015\n1510\nD [μm]s [μm] mFresnel(e)\n2 μmD\ns\n2 μms\nD(a) (b)\n(x,y)0FIG. 3. (a) Geometry-A device for the mapping of spin wave with a separation distance D= 5\u0016m\nand a shifts=\u00005\u0016m. (b) geometry-B with a separation distance D= 12\u0016mand a shifts= +8\u0016m.\nEvolution of the measured mutual inductance amplitude with the antennae shift sfor: (c) geometry-\nAmapping devices, and (d) geometry-B antennae separated by D= 8\u0016m, andD= 12\u0016m. The\ndotted lines are the calculated spin-wave amplitude from the Fresnel di\u000braction model with the\ncorresponding Latt. The symbols are the measured ampitude for the di\u000berent devices. (e) Color\nmapping of the wave amplitude for Latt= 10\u0016mtaking into account the probe size ldet= 2\u0016m.\nThe vertical dotted lines indicate sections of amplitude <~mCPW (s;D)>atD= 5;8;12\u0016m,\n8Whererj=p\n(D\u0000xj)2+ (s\u0000y)2is the distance between an element of surface dxdy of\nthe source centered at ( xj;y) and a detection point of coordinates ( D;s);lis the antennae\nlength and wjthe width of the CPW track. Now, the complete amplitude of the Fresnel\ndi\u000bracted spin-wave ~ mCPW results from the linear combination of the three branches of the\nCPW:\n~mCPW (D;s) =\u0000~mG\u0000(D+\u0015\n2;s) + ~mS(D;s)\u0000~mG+(D\u0000\u0015\n2;s) (2)\nWhere the negative signs accounts for the opposite phase of the excitation in the ground\nlines with respect to the central line. Fig. 3-(e) shows the color mapping of the spin-wave am-\nplitude in the ( D;s) plane calculated from Eq. (2) with an attenuation length Latt=10\u0016m.\nIn order to compare our measurement with this Fresnel di\u000braction model, we took into\naccount the non-punctual aspect of the detection antenna by averaging the amplitude over\nthe probe antenna extension ( ldet=2\u0016m). This near-\feld di\u000braction patterns reproduces\nthe main features of our measurement, which are on one hand an emission that remains\nfocused in a beam shape of width similar to the CPW length, and on the other hand, some\noscillations of the amplitude within the beam width that depend mostly on the distance\nD. Finally, we compare the measured amplitudes with the calculated ones <~mCPW(s;D)>\n[dotted lines in Fig. 3-(c),(d)] for the speci\fc distances D, and with the corresponding values\nof attenuation length obtained in Fig. 2-(f). We \fnd a remarkable agreement between this\nFresnel di\u000braction model of spin waves emitted from an antenna of \fnite extension and\nour measurements in the two di\u000berent geometries of waveguide, which constitutes a direct\ndemonstration of the focused nature of spin wave beams in constricted CPW.\nIn summary, we \frst demonstrated the possibility of performing spin-wave spectroscopy\nin thin magnetic \flms without the need to structure a spin-wave guide, only by using su\u000e-\nciently sharp constrictions in CPWs. We \frstly showed that the signal amplitudes measured\nfor pairs of identical antennae shifted gradually along the beam direction follow precisely\nan exponential decay, which suggests that the emission remains well-focused. Secondly, via\na series of devices consisting of pairs of non-identical antennae covering di\u000berent location of\nthe 2D-plane, we performed a discrete mapping of the spin-wave amplitude for two di\u000berent\ngeometries conceived in such a way to reproduce the case of an optical rectangular slit. We\nfound that the spin wave amplitude oscillates within the constriction zone, while it decays\n9rapidly outside of it, which is notably well-explained with a Fresnel di\u000braction model of\ncircular waves. These \fndings draw a deeper parallel between the excitation of spin-waves\nfrom sub-micrometric antennae and the basic concepts of optics, and therefore pave the\nway for future studies of spin wave beam interference, which could \fnd applications for spin\nwave logic devices.\nWe thank Olga Gladii, Hicham Majjad, Romain Bernard, and Alain Carvalho for sup-\nport with the nanofabrication in the STnano platform, and Guy Schmerber for X-ray\nmeasurements. This work was supported by the French Agence National de la Recherche\ngrant ANR-11-LABX-0058 NIE, and the USFQs PolyGrant] 431 program. The synthesis\nof the YIG \flms at Argonne was supported by the U.S. Department of Energy, O\u000ece of\nScience, Materials Science and Engineering Division.\n[1] S. O. Demokritov, A. N. Slavin, Magnonics, From Fundamentals to Applications, Springer\n(2012).\n[2] A.V. Chumak, el al., Nature Physics 11, 453-461 (2015).\n[3] T. Schneider, et al., Appl. Phys. Lett. 92, 022505 (2008).\n[4] A. Chumak, et al., Nat. Commun. 5, 4700 (2014).\n[5] K. Vogt, et al., Nat. Commun. 5, 3727 (2014).\n[6] S. Klingler, et al., Appl. Phys. Lett. 106, 212406 (2015).\n[7] S. Louis, et al., AIP Advances 6, 065103 (2016).\n[8] A. Kozhevnikov, et al., Appl. Phys. Lett. 106, 142409 (2015).\n[9] A. Papp, et al., Sci. Rep. 7, 9245 (2017).\n[10] V. Demidov et al., Appl. Phys. Lett. 92, 232503 (2008).\n[11] P. Gruszecki, et al., Appl. Phys. Lett. 105, 242406 (2014).\n[12] J. Stigloher, et al., Phys. Rev. Lett. 117, 037204 (2016).\n[13] P. Gruszecki, et al., Phys. Rev. B 95, 014421 (2017).\n[14] J. Gr afe et al., arXiv:1707.03664.\n[15] V. Demidov, et al., Phys. Rev. B 80, 014429 (2009).\n10[16] T. Sebastian, et. al, Phys. Rev. Lett. 110, 067201 (2013).\n[17] J-V. Kim, et. al, Phys. Rev. Lett. 117, 197204 (2016).\n[18] K. Wagner, et al., Nature Nanotechnology 11, 432 (2016).\n[19] A. Houshang, et al., Nature Nanotechnology 11, 280 (2016).\n[20] V. Demidov et al., Nature Comm. 7, 10446 (2016).\n[21] T. Schneider, et al., Phys. Rev. Lett. 104, 197203 (2010).\n[22] P. Gruszecki, et al., Sci. Rep. 6, 22367 (2016).\n[23] H. S. K orner, et al., Phys. Rev. B 96, 100401(R) (2017).\n[24] S. Li et al, Nanoscale 8, 388 (2016).\n[25] M. B. Joung\reisch, et al. Nano Lett. 17, pp 814 (2017).\n[26] Supplementary online materials (S-1).\n[27] V. Vlaminck, et al., Phys. Rev. B 81, 014425 (2010).\n[28] S. Maendl, et. al., Appl. Phys. Lett. 111, 012403 (2017).\n[29] M. Collet, et al., Appl. Phys. Lett. 110, 092408 (2017).\n[30] Supplementary online materials (S-2).\n[31] A. G. Gurevich, G. A. Melkov, Magnetization Oscillations and Waves, CRC (1996).\n[32] O. Gladii et al. Appl. Phys. Lett. 108, 202407 (2016)\n[33] T. Liu, et al., J. Appl. Phys. 115, 17A501 (2014).\n[34] M. B. Jung\reisch, et. al, J. Appl. Phys. 117, 17D128 (2015).\n[35] Supplementary online materials (S-3).\n[36] E. Hecht, Optics, Addison Wesley (2002).\n11" }, { "title": "2311.07293v1.Microwave_to_Optical_Quantum_Transduction_Utilizing_the_Topological_Faraday_Effect_of_Topological_Insulator_Heterostructures.pdf", "content": "Microwave-to-Optical Quantum Transduction Utilizing the Topological Faraday Effect\nof Topological Insulator Heterostructures\nAkihiko Sekine,1,∗Mari Ohfuchi,1and Yoshiyasu Doi1\n1Fujitsu Research, Fujitsu Limited, Kawasaki 211-8588, Japan\n(Dated: November 14, 2023)\nThe quantum transduction between microwave and optical photons is essential for realizing\nscalable quantum computers with superconducting qubits. Due to the large frequency difference\nbetween microwave and optical ranges, the transduction needs to be done via intermediate bosonic\nmodes or nonlinear processes. So far, the transduction efficiency ηvia the magneto-optic Faraday\neffect (i.e., the light-magnon interaction) in the ferromagnet YIG has been demonstrated to be small\nasη∼10−8−10−15due to the sample size limitation inside the cavity. Here, we take advantage\nof the fact that three-dimensional topological insulator thin films exhibit a topological Faraday\neffect that is independent of the sample thickness. This leads to a large Faraday rotation angle\nand therefore enhanced light-magnon interaction in the thin film limit. We show theoretically that\nthe transduction efficiency can be greatly improved to η∼10−4by utilizing the heterostructures\nconsisting of topological insulator thin films such as Bi 2Se3and ferromagnetic insulator thin films\nsuch as YIG.\nIntroduction.— The quantum transduction, or quan-\ntum frequency conversion, is an important quantum tech-\nnology which enables the interconnects between quan-\ntum devices such as quantum processors and quantum\nmemories. In particular, the quantum transduction be-\ntween microwave and optical photons has so far gath-\nered attention in pursuit of large-scale quantum com-\nputers with superconducting qubits [1–3]. Due to the\nlarge frequency difference between microwave and opti-\ncal ranges, the transduction needs to be done via inter-\nmediate interaction processes with bosonic modes or via\nnonlinear processes, such as the optomechanical effect [4–\n13] electro-optic effect [14–21], and magneto-optic effect\n[22–26]. To date, the transduction efficiency η, whose\nmaximum value is 1 by definition, has recorded the high-\nest value η∼10−1with a bandwidth ∼10−2MHz [6]\n(η∼10−2with∼1 MHz [15]) among the transductions\nutilizing the optomechanical effect (electro-optic effect).\nThe focus of this Letter is the microwave-to-optical\nquantum transduction via the magneto-optic Faraday ef-\nfect, i.e., the light-magnon interaction. Such a quan-\ntum transduction mediated by ferromagnetic magnons\ncan have a wide bandwidth ∼1 MHz and can be oper-\nated even at room temperature [1–3]. Also, the coherent\ncoupling between a ferromagnetic magnon and a super-\nconducting qubit has been realized [27, 28]. However, the\ncurrent major bottleneck when using the ferromagnetic\ninsulators (FIs) such as YIG is the low transduction ef-\nficiency η∼10−8−10−15[22–26, 29] due to the sample\nsize limitation inside the microwave cavity, as can be un-\nderstood from the relation η∝dFIwith dFIthe sample\nthickness [22]. The purpose of this study is to challenge\nthis issue by utilizing topological materials, which are a\nnew class of materials that are expected to exhibit un-\nusual materials properties due to their topological nature.\nIn this Letter, we take advantage of the fact that three-dimensional (3D) topological insulator (TI) thin films ex-\nhibit a topological Faraday effect that is independent of\nthe sample thickness, thus leading to a large Faraday ro-\ntation angle in the thin film limit. To this end, we partic-\nularly consider the heterostructures consisting of TI thin\nfilms such as Bi 2Se3and FI thin films such as YIG. We\nfind that the transduction efficiency ηis inversely pro-\nportional to the thickness of the FI layers ( η∝1/dFI),\nwhich is in sharp contrast to the above case of conven-\ntional FIs. We show theoretically that the transduction\nefficiency can be greatly improved to η∼10−4in a het-\nerostructure of a few dozen of layers of nanometer-thick\nTI and FI thin films.\nQuantum transduction.— Let us start with a generic\ndescription of our setup depicted in Fig. 1. We consider\nthe interaction Hamiltonian Hint=Hκ+Hg+Hζ[22],\nwhere\nHκ=−iℏ√κcZ∞\n−∞dω\n2πh\nˆa†ˆain(ω)−a†\nin(ω)ˆai\n(1)\ndescribes the coupling between the microwave cavity pho-\nton ˆaand an itinerant microwave photon ˆ ain(ω),\nHg=ℏg\u0000\nˆa†ˆm+ ˆm†ˆa\u0001\n(2)\ndescribes the coupling between the microwave cavity pho-\nton and the magnon (in the ferromagnetic resonance\nFIG. 1. Schematic illustration of our setup in terms of the\noperators, coupling strengths, and losses.arXiv:2311.07293v1 [cond-mat.mes-hall] 13 Nov 20232\nstate) ˆ m, and\nHζ=−iℏp\nζ\n×Z∞\n−∞dΩ\n2π\u0000\nˆm+ ˆm†\u0001h\nˆbin(Ω)eiΩ0t−ˆb†\nin(Ω)e−iΩ0ti\n(3)\ndescribes the coupling between the magnon and an itiner-\nant optical photon ˆbin(Ω), which is indeed the sum of the\nbeam-splitter-type and parametric-amplification-type in-\nteractions [30]. Ω 0is the input light frequency. In order\nto relate the incoming and outgoing itinerant photons,\nwe can employ the standard input-output formalism, to\nobtain ˆ aout= ˆain+√κcˆaandˆbout=ˆbin+√ζˆm. We\nsolve the equations of motion for the cavity and magnon\nmodes in the presence of intrinsic losses κandγ:\n˙ˆa=i\nℏ[Htotal,ˆa]−κc+κ\n2ˆa−√κcˆain, (4)\n˙ˆm=i\nℏ[Htotal,ˆm]−γ\n2ˆm−p\nζˆbin, (5)\nwhere Htotal=H0+Hintis the total Hamiltonian of the\nsystem with H0being the noninteracting Hamiltonian for\nthe cavity and magnon modes.\nThe microwave-to-optical quantum transduction effi-\nciency, which is defined by the ratio between the outgo-\ning and incoming photon numbers η(ω) =\f\f\f⟨ˆaout(ω)⟩\n⟨ˆbin(Ω)⟩\f\f\f2\n=\n\f\f\f⟨ˆbout(Ω)⟩\n⟨ˆain(ω)⟩\f\f\f2\n, is obtained as [22]\nη(ω) =4Cκc\nκc+κζ\nγ\u0010\nC+ 1−4∆c\nκc+κ∆m\nγ\u00112\n+ 4\u0010\n∆c\nκc+κ+∆m\nγ\u00112,(6)\nwhere C=4g2\n(κc+κ)γis the cooperativity, ∆ c=ω−ωc\nis the detuning from the microwave cavity frequency ωc,\nand ∆ m=ω−ωmis the detuning from the ferromagnetic\nresonance frequency ωm.\nTopological insulator heterostructures.— It has been\nshown that the magnitude of the magnon-mediated\ntransduction efficiency (6) is essentially determined by\nthe light-magnon coupling strength ζ, i.e., η∝ζ∝\nϕ2\nF/Ns[22], where ϕFandNsare respectively the Fara-\nday rotation angle and total number of spins of the ferro-\nmagnet. From this relation we see that the transduction\nefficiency can be improved in materials which exhibit a\nlarge Faraday rotation angle even with a small sample\nsize.\nWe take advantage of the fact that 3D TI thin films\nexhibit a topological Faraday effect arising from the sur-\nface anomalous Hall effect, whose rotation angle ϕF,TI\nis independent of the material thickness [31–34]. Here,\nthe bandgap 2∆ of the surface Dirac bands, generated\nby the exchange coupling between the surface electrons\nand the proximitized magnetic moments having the com-\nponents perpendicular to the surface, is essential for the\nFIG. 2. (a) Heterostructure consisting of a magnetically\ndoped TI and a nonmagnetic insulator. (b) Heterostructure\nconsisting of a (nonmagnetic) TI and a FI.\noccurrence of the surface anomalous Hall effect. In other\nwords, ϕF,TI= 0 in the absence of proximitized mag-\nnetic moments. The applicable range of the input light\nfrequency Ω 0is limited by the cutoff energy εcof the\nsurface Dirac bands (given typically by the half of the TI\nbulk bandgap), such that ℏΩ0< εc[31]. In particular,\nϕF,TItakes a universal value in the low-frequency limit\nℏΩ0≪εcand when the Fermi level µFis in the bandgap\n2∆ [31–33]\nϕF,TI= tan−1α≈α, (7)\nwhere α=e2/ℏc≈1/137 is the fine-structure con-\nstant. This universal behavior has been experimentally\nobserved [34–37].\nWe propose to utilize two types of TI heterostructures\n[38–40], as shown in Fig. 2. One is the heterostructures\nconsisting of magnetically doped TIs and nonmagnetic\ninsulators [38, 40]. The other is the heterostructures\nconsisting of (nonmagnetic) TIs and FIs [39–43]. In what\nfollows, we focus on the latter because the surface anoma-\nlous Hall effect (and thereby the quantum transduction)\ncan occur at a higher temperature ∼100 K than the for-\nmer [41, 42].\nLight-magnon interaction in the TI heterostructure.—\nSuppose that a linearly polarized light is propagating\nalong the zdirection. Microscopically, the Hamiltonian\nfor the Faraday effect in a ferromagnet is described by\nthe coupling between the z-component of the magnetiza-\ntion density and the z-component of the Stokes operator\nof the light [22, 30]. We extend this Hamiltonian to the\nheterostructure of NLTI layers and NLFI layers [see\nFigs. 2(b) and 3] as\nHF=ℏANLX\ni=1Zti+τ\ntidt G i(t)mi,z(t)Sz(t), (8)\nwhere idenotes the i-th FI layer, Gi(t) is the coupling\nconstant, Ais the cross section of the light beam, and\nτ=dFI/c(with dFIthe thickness of each FI layer and\ncthe speed of light in the material) is the interaction\ntime. We assume that the coupling constant describing\nthe topological Faraday effect [Eq. (7)] takes a δ-function\nform, since it is a surface effect. Taking also into account3\nFIG. 3. Enlarged view of a heterostructure consisting of TIs\nand FIs. δm⊥(t) is the small precessing component around\nthe direction of the effective field Beff. The applied magnetic\nfield needs to be tilted from the zaxis in order to induce a\nfinite angle θbetween the zaxis and Beff.\nthe conventional contribution to the Faraday effect, cG0,\nwhich takes a constant value across the sample [22], we\nobtain\nGi(t) =cG0+1\n2GTIδ(t−ti) +1\n2GTIδ(t−ti−τ).(9)\nThezcomponent of the magnetization density ˆ mi,z(t) is\ngiven by [44]\nmi,z(t) =δm⊥(t) sinθ=√Ns\n2Vsinθh\nˆmi(t) + ˆm†\ni(t)i\n,\n(10)\nwhere V(Ns) is the volume (total number of spins) of\neach FI layer, and ˆ mi(t) is the magnon annihilation op-\nerator satisfying [ ˆ mi(t),ˆm†\nj(t)] = δij. Note that a fi-\nnite angle θbetween the zaxis and the effective field\nBeff=−∂F/∂mi(with Fthe free energy of each FI\nlayer), which can be realized by a tilt of the applied mag-\nnetic field from the zaxis, is required. Here, let us define\na collective magnon operator ˆ m(t)≡1√NLPNL\ni=1ˆmi(t)\nsatisfying [ ˆ m(t),ˆm†(t)] = 1, in a similar way as spin en-\nsembles [45, 46]. Then, Eq. (8) is simplified to be\nHF=ℏAp\nNLZτ\n0dt\u0002\ncG0+1\n2GTIδ(t) +1\n2GTIδ(t−τ)\u0003\n×mz(t)Sz(t), (11)\nwhere mz(t) =√Ns\n2Vsinθ[ ˆm(t)+ ˆm†(t)]. The zcomponent\nof the Stokes operator for the polarization of light Sz(t)\nis given by [30]\nSz(t) =1\n2Ah\nˆb†\nR(t)ˆbR(t)−ˆb†\nL(t)ˆbL(t)i\n, (12)\nwhere ˆbR(t) [ˆbL(t)] is the annihilation operator of the\nmode of the right-circular (left-circular) polarized light\npropagating in the zdirection. For a strong x-polarized\nlight we have ˆbR,L(t) =1√\n2(ˆbx±iˆby)≃1√\n2(⟨ˆbx⟩ ±iˆby)[30], where ⟨ˆbx⟩=q\nP0\nℏΩ0e−iΩ0twith P0(Ω0) the power\n(angular frequency) of the input light.\nBecause the interaction time τ=dFI/c∼10−8m/(3×\n108m/s)∼10−17s is much shorter than the time scale of\nthe magnon dynamics (i.e., the ferromagnetic resonance\nfrequency) 1 /ωm∼10−10s, the operators in Eq. (11) can\nbe regarded as constant during the interaction. Then,\nsetting ˆby≡ˆbin, we arrive at Eq. (3). The light-magnon\ncoupling strength ζis obtained as\nζ=ϕ2\nFNL\nNssin2θP0\nℏΩ0, (13)\nwhere the Faraday rotation angle ϕF= (G0dFI+\nGTI)ns/4 with ns=Ns/Vthe spin density. As expected,\nthis expression for ϕFproperly describes the physical sit-\nuation: the conventional contribution is proportional to\nthe thickness dFI, while the topological contribution is\nindependent of the thickness, i.e., is a surface contribu-\ntion.\nCoupling between magnon and microwave-cavity\nphoton.— Next, we calculate the coupling strength gin\nthe heterostructure. The total Hamiltonian describing\nthe coupling between magnon and microwave-cavity pho-\nton is given by the sum of the contribution from each FI\nlayer, Hg=PNL\ni=1ℏgi(ˆa†ˆmi+ ˆm†\niˆa), where gi=g0√Ns\nwith g0the single-spin coupling strength [47, 48]. As\nwe have done in the case of the light-magnon coupling,\nwe may introduce a collective magnon operator ˆ m(t)≡\n1√NLPNL\ni=1ˆmi(t) satisfying [ ˆ m(t),ˆm†(t)] = 1. Then, we\nobtain Eq. (2) with the coupling strength\ng=g0p\nNLNs. (14)\nMagnetization dynamics in the FI layer.— So far, we\nhave treated the electronic response of the TI surface\nstate, i.e., the topological Faraday effect [Eq. (7)] as the\nmagnonic response of the FI layer. Indeed, these two pic-\ntures are equivalent because the effective spin model is\nderived by integrating out the electronic degree of free-\ndom in the surface Dirac Hamiltonian coupled to the FI\nlayer via the exchange interaction [49, 50]. The derived\nspin model takes the form of the exchange interaction and\nthe easy-axis anisotropy when the chemical potential µF\nlies in the mass gap of the surface Dirac fermions, i.e.,\nµF<|∆|, while it takes the form of the Dzyaloshinskii–\nMoriya interaction when µF>|∆|[49].\nIt has been shown that the rotation angle of the topo-\nlogical Faraday effect decreases as the carrier density\n(i.e., the value of µF) becomes larger [31, 50]. We point\nout that such a behavior is consistent with the above-\nmentioned effective spin model analysis. Generally, the\nFaraday rotation angle is proportional to the spin density\nasϕF∝ns. On the other hand, as represented by the\nskyrmion lattice, a canted spin structure is favored due\nto the Dzyaloshinskii–Moriya interaction, which leads to4\nFIG. 4. The input light frequency Ω 0dependence of the\ntransduction efficiency η. We set θ= 30◦,εc= 150 meV,\nand ∆ = 15 meV. The input photon number is fixed here\ntoP0\nℏΩ0= 1.5×1019Hz, which can be obtained, for example,\nwith P0= 10 mW and Ω 0/2π= 1 THz.\nthe decrease of the magnetization density, i.e., the spin\ndensity ns. Therefore, the decrease of the value of ϕF,TI\nfrom the case of µF<|∆|to the case of µF>|∆|can\nalso be explained from the effective spin model analysis.\nTransduction efficiency.— We are now in a position\nto obtain the transduction efficiency ηin the TI het-\nerostructures. We use the typical (possible) values of\nYIG for the FI layer and those of microwave cavity: ns=\n2.1×1019µBmm−3,g0/2π= 40 mHz, κ/2π= 1 MHz,\nκc/2π= 3 MHz, γ/2π= 1 MHz [22, 47]. We assume\nthe sample area size of 0 .5 mm×0.5 mm, and treat NL\nanddFIas key variables. We also assume the topological\ntransport regime of µF<|∆|. In the following, we focus\non the thin film regime for the FI layer where the conven-\ntional contribution to the Faraday rotation angle can be\nneglected as ϕF,0=G0nsdFI/4≪1/137, as well as the\nthin film regime for the TI layer where the thickness of\nthe TI thin film can be neglected as dTI≪λ= 2πc/Ω0.\nAs can be seen from Eq. (6), the transduction effi-\nciency ηis proportional to the light-magnon interaction\nstrength ζ. Thus, we firstly consider the dimensionless\nprefactor η/(ζ/γ). The cooperativity Cis obtained as\nC= 1.6×10−15×NLNs. Assuming NL∼101and\nd∼1 nm, we find that C ∼10−1. Accordingly, it turns\nout that the prefactor η/(ζ/γ) takes a maximum value\n≈0.5 when ∆ c= ∆ m= 0, i.e., ω=ωc=ωm[51]. From\nthis result we see that the magnitude of ηin our case is\nalso essentially determined by ζ.\nNext, we show the dependence of the transduction ef-\nficiency on the input light frequency Ω 0in Fig. 4. Here,\nnote that the input photon numberP0\nℏΩ0is fixed in Fig. 4.\nIn other words, the Ω 0dependence in Fig. 4 originates\nfrom the topological Faraday rotation angle [31, 50]. Im-\nportantly, the Faraday rotation angle needs not be the\nuniversal value ϕF,TI≈1/137 (in the low frequency limit)\nand the transduction efficiency can be enhanced more\nthan an order of magnitude near the sharp peak at the\ninterband absorption threshold 2∆ [31] by tuning Ω 0. As-\nFIG. 5. Schematic Illustration of the microwave-to-optical\nquantum transduction utilizing a TI heterostructure.\nsuming the cutoff energy εc= 150 meV and the surface\nmass gap ∆ = 15 meV [38–40], we find that the frequency\nat which ηtakes the maximum value is Ω 0/2π= 7.3 THz.\nNote that the maximum value of ηis mainly determined\nbyNLanddFI, not by εcor ∆.\nThe transduction efficiency η∼10−3−10−4obtained\nin Fig. 4 is greatly improved [52] compared to that of a\nspherical YIG of 0 .75 mm diameter, η∼10−10, obtained\nwithP0= 15 mW and Ω 0/2π= 200 THz [22, 53]. Here, it\nshould be noted that the Verdet constant V(=G0ns/4)\nin YIG in the terahertz range ( ∼1 THz) is about an\norder of magnitude smaller than that in the telecom\nfrequency range ( ≈200 THz) [54]. This means that\nthe light-magnon interaction strength ζdoes not change\nlargely even in the terahertz range due to the relation\nζ∝ϕ2\nF,0/Ω0where ϕF,0=VdFI. Then, it follows that\nthe transduction efficiency using YIG would be as small\nasO(10−10) even in the terahertz range.\nThere is a fundamental difference in the sample size\ndependence of the transduction efficiency ηbetween pre-\nvious studies and our study, while the expression η∝\nϕ2\nF/Nsis the same. In conventional ferromagnets such\nas YIG, one obtains η∝dFIsince both ϕFandNsare\nproportional to dFI. This indicates that the value of η\nbecomes very small in the thin film limit. On the other\nhand, in TI heterostructures, one obtains η∝1/dFIsince\nϕFis constant whereas Nsis proportional to dFI. This is\nthe mechanism for the enhancement of ηin the thin film\nlimit.\nFinally, we show in Fig. 5 a schematic illustration of\nthe microwave cavity setup in our microwave-to-optical\nquantum transduction. Here, a linearly polarized light\nin the terahertz range is applied perpendicular to the\nheterostructure plane. A finite tilt angle between the\nlight propagation direction and the ground state direc-\ntion of the ferromagnetic moments is required by apply-\ning a static magnetic field.\nDiscussion.— We briefly discuss a possible application\nof our finding. While optical fibers in the telecom fre-5\nquency range are currently used widely, we would like to\nstress that optical fibers in the terahertz range are also\nin principle able to interconnect quantum devices. Actu-\nally, terahertz optical fibers are under active research and\ndevelopment [55]. Thus, we expect that in the future TI\nheterostructures might be used as a quantum transducer\nfor superconducting quantum computers interconnected\nvia terahertz optical fibers.\nOne possible way for bringing the light frequency closer\nto the telecom frequency range is to find TIs with a large\nbulk bandgap ( ≈2εc), as well as combinations of FIs\nand such TI surface states that allow a strong exchange\ninteraction between them and therefore enable a large\nsurface bandgap 2∆. If a TI heterostructure with ∆ ≈\n100 meV is discovered, then the maximum transduction\nefficiency will be obtained at Ω 0/2π= 2∆ ≈48 THz. In\nthis case, infrared optical fibers can be used.\nSummary.— To summarize, we have shown that the\ntransduction efficiency of microwave-to-optical quantum\ntransduction mediated by ferromagnetic magnon can be\ngreatly improved by utilizing the topological Faraday ef-\nfect in 3D TI thin films. 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Rep. 5, 13638 (2015).\n[50] See Supplemental Material for details.\n[51] Note that even when the cooperativity Cis as large as\nO(102), the maximum value of the prefactor η/(ζ/γ) is\n≈0.7 at the nonzero detunings ∆ c̸= 0 and ∆ m̸= 0.\n[52] If a heterostructure of NL= 500 can be realized (where\nthe total thickness of the heterostructure 2 NL(dTI+\ndFI)≈10µm is still shorter than the wavelength λ=\n2πc/Ω0), the maximum value of ηin Fig. 4 can be fur-\nther improved to η∼10−2.\n[53] To make a more precise comparison, the input photon\nnumber should be the same as in Fig. 4. The value in\nRef. [22] (P0\nℏΩ0= 1.1×1017Hz obtained with P0= 15 mW\nand Ω 0/2π= 200 THz) is two orders of magnitude\nsmaller than that in Fig. 4. IfP0\nℏΩ0= 1.5×1019Hz is7\nemployed in the setup in Ref. [22], then the transduction\nefficiency would be η∼10−8.\n[54] Y. Li, T. Li, Q. Wen, F. Fan, Q. Yang, and S. Chang,\nTerahertz magneto-optical effect of wafer-scale La: yt-trium iron garnet single-crystal film with low loss and\nhigh permittivity, Opt. Express 28, 21062-21071 (2020).\n[55] M. S. Islam, C. M. B. Cordeiro, M. A. R. Franco, J.\nSultana, A. L. S. Cruz, and D. Abbott, Terahertz Optical\nFibers [Invited], Opt. Express 28, 16089 (2020).\nSupplemental Material for\n“Microwave-to-Optical Quantum Transduction Utilizing the Universal Faraday Effect\nof Topological Insulator Heterostructures”\nMAGNETIZATION DYNAMICS OF THE FERROMAGNETIC INSULATOR LAYER\nWe consider the surface Hamiltonian of the topological insulator (TI) layer exchange-coupled to the ferromagnetic\ninsulator (FI) layer, which is given by H=P\nkψ†\nkH(k)ψkwith\nH(k) =ℏvF(kxσx+kyσy) +Jm·σ\n≡ℏvF(kxσx+kyσy) + ∆ σz+X\nαJαδmασα, (S1)\nwhere vFis the Fermi velocity, σiare the Pauli matrices describing electrons’ spin on the TI surface, Jis the strength\nof the exchange coupling, and m=m0+δm(with m0the ground state direction) is the magnetization density\nvector of the FI layer. The effective action for the FI layer can be derived by integrating out the electronic degree of\nfreedom:\nZ=Z\nD[ψ,¯ψ]eiS≡eiSeff[δm]= exp\"\nTr\u0000\nlnG−1\n0\u0001\n−∞X\nn=11\nnTr (−G0V)n#\n, (S2)\nwhere G0(k, iωn) = [iωn−H(k)−µF]−1is the unperturbed Green’s function and V=P\nαJαδmασαis a perturbation.\nAt the one-loop level and in the low-frequency limit [49], the effective action for δmis written in terms of the static\nsusceptibility χαβ(q,0) as\nSeff[δm] =1\n2iTr (G0V)2=1\n2X\nqX\nα,βJαJβδmα(q)χαβ(q,0)δmβ(−q), (S3)\nwhich takes the form of the exchange interaction and the easy-axis anisotropy when the chemical potential µFlies\nin the mass gap of the surface Dirac fermions, i.e., µF<|∆|, while it takes the form of the Dzyaloshinskii–Moriya\ninteraction when µF>|∆|[49]. This indicates that the magnetization dynamics of the FI layer is modified by the\npresence of the TI surface state, depending on the value of the chemical potential.\nTOPOLOGICAL FARADAY EFFECT IN TOPOLOGICAL INSULATOR THIN FILMS\nGeneral expression for the Faraday rotation angle\nWe consider the electronic response of the TI surface state which is described by the effective Hamiltonian of the\nform,\nH(k) =ℏvF(kxσx+kyσy) + ∆ σz, (S4)\nwhere vFis the Fermi velocity, σiare the Pauli matrices describing electrons’ spin on the TI surface, and ∆ is\nthe mass gap induced by the exchange coupling between the proximitized ferromagnetic moments. The optical\nconductivity of the system described by Eq. (S4) can be obtained by solving a quantum kinetic equation [31]. In\nthe following, the conductivities are given in units of e2/ℏ=cα, where we set c= 1. The longitudinal conductivity8\nσxx(Ω0) =σR\nxx(Ω0) +iσI\nxx(Ω0) and the transverse conductivity σxy(Ω0) =σR\nxy(Ω0) +iσI\nxy(Ω0) are written explicitly\nas [31]\nσR\nxx(Ω0) =\"\u0012∆\n2Ω0\u00132\n+1\n16#\nθ[Ω0−2 max( µF,∆)] +µ2\nF−∆2\n4µFδ(Ω0)θ(µF−∆), (S5)\nσI\nxx(Ω0) =1\n8π(\n1\n2\"\n4\u0012∆\nΩ0\u00132\n+ 1#\nF(Ω0)−2∆2\nΩ0\u00121\nεc−1\nmax( µF,∆)\u0013)\n+1\n4πµ2\nF−∆2\nµFΩ0θ(µF−∆), (S6)\nand\nσR\nxy(Ω0) =−∆\n4πΩ0F(Ω0), (S7)\nσI\nxy(Ω0) =∆\n4Ω0θ[Ω0−2 max( µF,∆)], (S8)\nwhere\nF(Ω0) = ln\f\f\f\fΩ0+ 2εc\nΩ0−2εc\f\f\f\f−ln\f\f\f\fΩ0+ 2 max( µF,∆)\nΩ0−2 max( µF,∆)\f\f\f\f, (S9)\nεcis the cutoff energy of the surface Dirac bands ±q\nℏ2v2\nF(k2x+k2y) + ∆2which is given typically by the half of the\nTI bulk bandgap, and we have considered the case of µF>0 and ∆ >0 without loss of generality.\nIn what follows, we derive a general expression for the Faraday rotation angle at a single interface. Suppose that\nan electromagnetic wave is propagating along the zdirection from medium ito medium jwith dielectric constant\nϵiandϵjand magnetic permeability µiandµj, respectively. The boundary conditions for the electric and magnetic\nfields are [33]\nEi=Ejand−iτy(Bj−Bi) =µ0Js, (S10)\nwhere the first equation follows from the continuity of the electric field by Faraday’s law and the second equation\nfollows from Amp` ere’s law integrated over the zdirectionR\ndz∇ ×B=R\ndz µ 0j.τyis the y-component of the Pauli\nmatrices. Here, we have assumed that the bulk of the media is insulating, i.e., the electric current Js=σEiflows\nonly at the boundary. Note also that Ez=Bz= 0 because we are considering an transverse wave.\nLetE0,Er, andEtbe incident, reflected, and transmitted electric fields, respectively. Then, the electric fields in\nmedia iandjare given by\nEi=eikizEti+e−ikizEri,\nEj=eikjzEtj. (S11)\nAt the boundary, the incoming fileds\u0002EtiErj\u0003Tand the outgoing fields\u0002EriEtj\u0003Tare related by the scattering\nmatrix S=\u0002r t′\nt r′\u0003\nwith r=\u0002rxxrxy\n−rxyryy\u0003\nandt=h\ntxxtxy\n−txytyyi\n(and similarly for r′andt′) [33]. We therefore have\n\u0014\nEri\nEtj\u0015\n=\u0014\nr t′\nt r′\u0015\u0014E0\n0\u0015\n=\u0014rE0\ntE0\u0015\n. (S12)\nThe explicit forms of randtcan be obtained by solving the boundary condition equations (S10) with the use of\nFaraday’s law in media iandj,∇ ×E=−(1/c)∂B/∂t.\nThe Faraday rotation angle is calculated from the arguments of the transmitted electric field (in medium j):\nϕF(Ω0) =\u0002\narg(Et\n+)−arg(Et\n−)\u0003\n/2, (S13)\nwhere Et\n±=Et\nx±iEt\nyare the left-handed (+) and right-handed ( −) circularly polarized components of the transmitted\nelectric field Et. The explicit forms are given by\narg(Et\n±) = tan−1\"\n4πασI\nxx(Ω0)±4πασR\nxy(Ω0)p\nϵi/µi+p\nϵj/µj+ 4πασRxx(Ω0)−4πασIxy(Ω0)#\n. (S14)9\nFIG. S1. (a) The input light frequency Ω 0dependence and (b) the chemical potential µFdependence of the Faraday rotation\nangle ϕFin the thin film regime. We set εc= 150 meV and ∆ = 15 meV. In (a) and (b), we also set µ/∆ = 0 .5 and Ω 0/εc= 0.05,\nrespectively.\nFaraday rotation angle in the thin film regime\nHere, we calculate the Faraday rotation angle in a TI thin film. In the thin film regime where dTI≪λ= 2πc/Ω0,\nthe thickness of the TI thin film can be neglected. In other words, we may regard the system as a 2D system consisting\nonly of the top and bottom surfaces. Then, we can simply setp\nϵi/µi=p\nϵj/µj→1, as well as σxx(Ω0)→2σxx(Ω0)\nandσxy(Ω0)→2σxy(Ω0), which accounts for the contributions from both the top and bottom surfaces.\nFigure S1(a) shows the input light frequency Ω 0dependence of the Faraday rotation angle ϕF, which reproduces\nthe result in Ref. [31]. The value of ϕFin the low-frequency limit Ω 0/εc≪1 is almost universal such that ϕF=\ntan−1[α(1−∆/εc)]≃tan−1α. Figure S1(b) shows the chemical potential µFdependence of ϕF. The value of ϕFis\nconstant as long as the chemical potential is in the surface gap (i.e., µ/∆<1), reflecting the constant anomalous Hall\nconductivity σR\nxy= (α/4π)(1−∆/εc) in the topological transport regime. On the other hand, we can see that the\nvalue of ϕFbegins to decrease as the carrier density, i,e., the value of µ/∆(>1), becomes larger." }, { "title": "1710.05995v1.Positive_Negative_Birefringence_in_Multiferroic_Layered_Metasurfaces.pdf", "content": "Positive-Negative Birefringence in Multiferroic Layered Metasurfaces\nR. Khomeriki\nInstitut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, D-06099 Halle/Saale, Germany\nPhysics Department, Tbilisi State University, 3 Chavchavadze, 0128 Tbilisi, Georgia\nL. Chotorlishvili\nInstitut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, D-06099 Halle/Saale, Germany\nI. Tralle\nFaculty of Mathematis and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland\nJ. Berakdar\nInstitut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, 06099 Halle/Saale, Germany\u0003\nWe uncover and identify the regime for a magnetically and ferroelectrically controllable nega-\ntive refraction of light traversing multiferroic, oxide-based metastructure consisting of alternating\nnanoscopic ferroelectric (SrTiO 2) and ferromagnetic (Y 3Fe2(FeO 4)3, YIG) layers. We perform an-\nalytical and numerical simulations based on discretized, coupled equations for the self-consistent\nMaxwell/ferroelectric/ferromagnetic dynamics and obtain a biquadratic relation for the refractive\nindex. Various scenarios of ordinary and negative refraction in di\u000berent frequency ranges are ana-\nlyzed and quanti\fed by simple analytical formula that are con\frmed by full-\redge numerical simula-\ntions. Electromagnetic-waves injected at the edges of the sample are propagated exactly numerically.\nWe discovered that for particular GHz frequencies, waves with di\u000berent polarizations are charac-\nterized by di\u000berent signs of the refractive index giving rise to novel types of phenomena such as a\npositive-negative birefringence e\u000bect, and magnetically controlled light trapping and accelerations.\nINTRODUCTION\nAccording to Veselago's predictions [1, 2] that were\nlater con\frmed experimentally [3{5], negative refraction\nphenomena occur in metamaterials where both the elec-\ntric permittivity \u000fand the magnetic permeability \u0016are\nnegative. The Poynting and the wave vectors are then\nantiparallel resulting in a phase decrease during the prop-\nagation process. A clear example of this phenomena is\nbased on metallic heterostructures [6] which are inher-\nently absorptive at relevant frequencies [7]. To avoid\nlosses, insulating multiferroics may o\u000ber a solution [8, 9],\nbut also new possibilities for external control and ex-\nploitations of functional materials. Multiferroics are one-\nphase or composite, synthesized structures exhibiting si-\nmultaneously multiple orderings such as ferromagnetic,\nferro and/or piezoelectric order and respond thus to a\nmultitude of conjugate \felds. This class of materials\nplays a key role for addressing fundamental issues regard-\ning the interplay between electronic correlation, symme-\ntry, magnetism, and polarization. Potential applications\nare diverse, ranging from sensorics and magnetoelectric\nspintronics to environmentally friendly devices with ul-\ntralow energy consumption [10, 11].\nHere, we demonstrate how multiferroics properties lead\nto exotic electromagnetic wave propagation features. In\nparticular, we demonstrate the existence of a negative\nrefraction in ferroelectric (FE)/ferromagnetic(FM) mul-\ntilayers. The large (FE and FM) resonance frequency\nmismatch between ferroelectric and ferromagnetic mediais usually an obstacle. Indeed, the paradigm ferroelectric\nBaTiO 3has a resonance frequency in THz range [12{15],\nwhile the insulating ferromagnet, rhodium-substituted \"-\nRhxFe2\u0000xO3with largest known coercivity has charac-\nteristic frequencies in 200GHz range [16]. On the other\nhand, ferroelectric SrTiO 2(STO) [17] does possess over-\nlapping resonances with the well-investigated insulating\nferromagnet Y 3Fe2(FeO 4)3(also called YIG) [18]. A\nnumber of further insulating FE, and FM insulating ox-\nides are also possible, for concreteness we present and dis-\ncuss here the results for STO/YIG/STO/ \u0001\u0001\u0001structures.\nIn the pilot numerical simulations below we choose the\nFE layer to be 10 nmand the FM layer to be 1 \u0016m.\nAs we will be working in reduced units, meaning the ef-\nfects are scalable; for materials composites responding at\nhigher frequencies, smaller structures are appropriate.\nIn earlier studies the negative refraction e\u000bect was ob-\nserved in speci\fc systems embracing two di\u000berent sub-\nparts with \u000f <0,\u0016 > 0, and\u000f >0,\u0016 < 0 respectively\n[8, 9]. We note that in the same medium propagating pos-\nitively refracting mode can exist as well [9]. In the present\npaper, we study the FE/FM composite (cf. the schemat-\nics setup in Fig. 1a) with an eye to \fnd the frequency\ndomain where both negatively and positively refracting\nwaves with di\u000berent polarizations simultaneously coex-\nist for the same excitation frequency. By this we predict\nthat in the suggested system unpolarized electromagnetic\nwave undergoes both positive and negative refractions,\nmanifesting a novel positive-negative birefringence e\u000bect.arXiv:1710.05995v1 [physics.optics] 16 Oct 20172\n𝑷𝟐 \n𝑷𝟒 \n𝑷𝟔 \n𝑺𝟓 \nx \ny \nz \n𝑺𝟑 \n𝑺𝟏 \n(a) \n(c) (b) \nFIG. 1: (a) Schematics for the pho-\ntonic/ferroelectric/ferromagnetic heterostructure. The\nelectric polarization Pjin the layer jis aligned along xaxis.\nThe incident light wave propagating along zaxis.EandH\ndenote the electric and magnetic \feld components. The mag-\nnetization Siin the layer ipoints also along the zaxis. (b)\nand (c) graphs show real (dashed blue curve) and imaginary\n(dashed-dotted green curve) parts of the refraction index\nnversus the mode frequency according to the biquadratic\nequation (5). Red solid curve is the zcomponent of the\ntime averaged dimensionless Poynting vector hWzi=P0S0as\ncalculated by means of the formula (6). Graphs (b) and\n(c) correspond to the negatively and positively refracting\nwaves, respectively for the same excitation frequency 30GHz\nindicated by vertical dotted lines in the insets. Insets display\nenlarged views of the frequencies in interest. The FE/FM\nspeci\fc materials are detailed in the text.\nMODEL AND PARAMETERS\nLet both the thickness of the YIG and the STO layers\nbe equala, for clarity. We denote the positions of STO\nand YIG layers by even (2 m) and odd (2 m\u00001) inte-\nger numbers respectively, where m= 1;2;:::; N= 2. For\nthe description of light-induced FE/FM dynamics and its\nbackaction on the light propagation properties we will\nutilized a discretized Maxwell materials equation self-\nconsistently coupled to the FE dynamics as described by\nthe Ginzburg-Landau-Devonshire (GLD) method, and a\nclassical Heisenberg model for the magnetization preces-\nsion. This low-energy e\u000bective treatment is well-justi\fed\ndue to the choice of the appropriate frequency and the\n(low to moderate) intensity of the incident light wave.\nThe discretized FE polarization P2m(initially along x\naxis) and FM magnetization ~S2m\u00001(initially along zaxis) (cf. Fig. 1a) are thus described by the energy func-\ntional\nH=HP+HS;HP=N=2X\nm=1\u0014\u000b0\n2\u0012dP2m\ndt\u00132\n\u0000\n\u0000\u000b1\n2(P2m)2+\u000b2\n4(P2m)4\u0000P2mEx\n2m\u0015\n; (1)\nHS=\u0000N=2X\nm=1h\nH0Sz+D\u0000\nSz\n2m\u00001\u00012+~H2m\u00001~S2m\u00001i\n;\nwhere\u000b0stands for the kinetic constant, \u000b1and\u000b2are\npotential coe\u000ecients of the FE part, and Dis a uniaxial\nanisotropy constant in FM layers. H0is a static external\nmagnetic \feld applied along the zaxis which will prove\nuseful for tuning the functional properties of the setup,\ne.g. for switching between the ordinary and the negative\nrefraction regimes (see below).\nWe assume that the multilayer structure is \frst driven\nto saturation by appropriately strong \felds. The rem-\nnant FE and FM polarizations are then denoted by P0\nandS0. Let us introduce dimensionless photonic \feld\n~h\u0011~H=S 0, and~E\u0011~E=P 0and denote small deviations\naround the ordering directions by p2m\u0011P0\u0000P2m, and\n~ s2m\u00001\u0011~S0\u0000~S2m\u00001. The thickness of the layers (along\nthe propagation direction) should be small enough such\nthat no domains are formed along the zaxis (the gen-\neral case of large ais captured also with this model by\nadding pinning sites and appropriate energy contribu-\ntions to eq.(1), but this is expected to be subsidiary to\nthe e\u000bects discussed here). Our propagating electromag-\nnetic wave cannot create domains since the wavelength\nfar exceeds a. The discretized form of Maxwell's equa-\ntions for the electromagnetic \feld vectors read\n1\ncd\ndt\u0000\nhx\n2m+1+ 4\u0019sx\n2m+1\u0001\n=1\n2a\u0000\nEy\n2m+2\u0000Ey\n2m\u0001\n\u00001\ncd\ndt\u0000\nhy\n2m+1+ 4\u0019sy\n2m+1\u0001\n=1\n2a\u0000\nEx\n2m+2\u0000Ex\n2m\u0001\n\u00001\ncd\ndt(Ex\n2m+ 4\u0019p2m) =1\n2a\u0000\nhy\n2m+1\u0000hy\n2m\u00001\u0001\n1\ncd\ndtEy\n2m=1\n2a\u0000\nhx\n2m+1\u0000hx\n2m\u00001\u0001\n:(2)\nThese equations need to be propagated simultaneously\nwith the dynamics governed by eqs.(1). Nonlinear cor-\nrections are irrelevant when the relative values of the po-\nlarization and the magnetization of eigenmodes are much\nsmaller than unity. Thus, the validity of the linear ap-\nproximation can be checked directly by monitoring the\nrelative eigenmodes. After these simpli\fcations, from (1)\nwe infer the linearized, coupled photonic-matter evolu-3\ntion equations\nd2p2m\ndt2=\u0000!2\nPp2m+!2\nP\n4\u0019\u000bEx\n2m (3)\n@sx\n2m+1\n@t=\u0000!0sy\n2m+1+!M\n4\u0019hy\n2m+1\n@sy\n2m+1\n@t=!0sx\n2m+1\u0000!M\n4\u0019hx\n2m+1:\nThe FE resonance frequency !Pof STO is around\n!P= 10 GHz. The GDL potential curvature at equi-\nlibrium\u000b= 2\u000b1is related to the electric susceptibil-\nity at the zero mode frequency as \u001f(0) = 1=4\u0019\u000b. For\nthe large permittivity observed in Ref. [17] the potential\ncurvature is of the order of \u000b\u001810\u00004. For YIG [18] the\nLarmor frequency is !0=\rH0+2D\rS 0(in zero external\n\feld!0= 20GHz) and !M= 4\u0019\rS 0= 30GHz (\ris the\ngyromagnetic ratio for electrons).\nFor a linearly polarized electromagnetic waves in the\nFE/FM multilayers the refractive index follows from the\nmatter equations (3). The expressions for the permittiv-\nity\u000f= 1+!2\nP=\u000b\n!2\nP\u0000!2and the permeability \u0016= 1+!0!M\n!2\n0\u0000!2for\nthe linearly polarized wave components Ex,hyindicate\nthat\nn2\n1=\u000f\u0016=\u0012\n1 +!2\nP=\u000b\n!2\nP\u0000!2\u0013\u0012\n1 +!0!M\n!2\n0\u0000!2\u0013\n:(4)\nIn spite of the fact that a linearly polarized wave is not\nan eigenmode of the FE-FM system, the eigenmode (4)\nhas a certain merit. The asymptotic solution corresponds\nto the large susceptibility limit (see below). In order to\nprecisely calculate the refractive index, one needs to solve\na complete set of coupled Maxwell (2) and matter (3)\nequations. Thus looking for a general solution we proceed\nfurther and adopt an ansatz presenting \feld and matter\nwave components as follows: Ex\nm=Exei(!t\u0000kam). Herea\nis a FE/FM lattice constant !andkare the eigenmode\nfrequency and wavenumber, respectively. Analyzing the\nlinear algebraic equations (see Supporting Information\nfor the details) we arrive at the following biquadratic\nequation for the refractive index n\u0011ck=! .\n\u000b(!2\n0\u0000!2)(!2\nP\u0000!2)n4\u0000 (5)\n\u0000(!2\n0+!M!0\u0000!2)\u0002\n2\u000b(!2\nP\u0000!2) +!2\nP\u0003\nn2+\n+\u0002\n\u000b(!2\nP\u0000!2) +!2\nP\u0003\u0002\n(!0+!M)2\u0000!2\u0003\n= 0;\nand a set of amplitudes for the \feld and matter wave com-\nponents. Then it is straightforward to calculate the time\naveraged dimensionless Poynting vector W=hWzi=P0S0\nas\nW=hExHy\u0000EyHxi=P0S0=\u0002\nExh\u0003\ny\u0000Eyh\u0003\nx\u0003\n+c:c:\n=n(\n(!2\u0000!2\nP)\u0002\n(!2\u0000!2\n0)(n2\u00001) +!0!M\u00032\n(!2\u0000!2\nP\u0000!2\nP=\u000b)!2!2\nM+ 1)\n:(6)\nFIG. 2: Graphs (a) and (c) represent negative refraction in-\njecting the signal with a polarization Ex;y= 0:19 + 1:81i.\nwhile Graphs (b) and (d) display positive refraction of the\nwave with polarization Ex;y= 0:015+1:985icorresponding to\na positive refraction. In graphs (c) and (d) blue (solid), green\n(dashed-dotted) and red (dashed) curves correspond to time\noscillations of the FE/FM layers with increasing site numbers.\nIt is straightforward to derive from (5) the two obvious\nlimiting cases of pure ferromagnet or ferroelectrics. For\nzero magnetization, we set !M= 0 and from (5) obtain\ntwo linearly polarized eigenmodes with the refractive in-\ndexesn1= 1 andn2=r\n1 +!2\nP=\u000b\n!2\nP\u0000!2corresponding to the\npolarizations along yandxaxis, respectively. In the case\nof a vanishing polarization in the system we set \u000b!1\nand \fnd two circularly polarized eigenmodes character-\nized by di\u000berent refractive indexes n1=q\n1 +!M\n!0\u0000!and\nn2=q\n1 +!M\n!0+!.\nObviously all of these modes in both limiting cases\nare characterized by ordinary refraction properties for\nlow excitation frequencies !, while for large !one of re-\nfractive indexes becomes purely imaginary corresponding\nto the non-propagating regime, while other remains real\nand corresponds to an ordinary refraction. It should be\nemphasized that in the above mentioned cases with the\nlinear polarized wave at the input always obtain linearly\npolarized wave at the output: In purely ferromagnetic\ncase we \fnd a Faraday rotation and for pure ferroelectrics\njust a phase shift (that however can be retrieved by in-\nterference measurements).\nThe situation changes drastically in the presence of both\nferroelectric and ferromagnetic layers. An injected linear\npolarized electromagnetic wave emerges after traversing\nthe FE/FM heterostructure with an elliptical polariza-\ntion, as detailed below.\nWe can further simplify the analytic solutions (5) and4\n(6) in the general case (both ferroelectric and ferromag-\nnetic layers are present in the system) assuming \u000b!0\nwhich is just the case of a large susceptibility in FE [17].\nThen one infers two roots of (5). The \frst one matches\nexactly the asymptotic result (4), and the second one\nreads\nn2\n2=(!0+!M)2\u0000!2\n!0(!0+!M)\u0000!2: (7)\nNow it is evident that in the same limit of \u000b!0,\nfor the \frst mode (4) the Poynting vector has the val-\nuesW1\u0018\u0000n, meaning that a negative refraction takes\nplace. For the second mode (7) the Poynting vector is\nW2\u0018ncorresponding to a positive refraction case. To\nidentify the positive-negative birefringence regime both\npropagating modes should be present, i.e. n2\n1>0 and\nn2\n2>0. From these relations we deduce the restrictions\non the mode frequency\nMaxf!P;!0g0 of the refractive in-\ndexn=n0+n00are displayed as to pinpoint the wavevec-\ntor direction and compare it with the sign of the Poynting\nvector as calculated according to (6). Apparently in Fig.\n1 (b), the sign of the Poynting vector is negative in the\nfrequency range close to != 30GHz, i.e. the Poynting\nvector is antiparallel to the wave vector direction and\ntherefore we are in the negative refraction regime, while\nin graph (c) another root with a positive n0is presented.\nThe Poynting vector in this case is positive, that means\nwe have a second mode with an ordinary (positive) re-\nfraction for the same wave frequency != 30GHz. The\nclear evidence of the coexistence of a positive and nega-\ntive refractions for an unpolarized electromagnetic wave\nis fully compatible with the approximate conditions (8).\nTo substantiate the analytical estimation we per-\nformed full numerical experiment considering two wave\nmodes with di\u000berent polarizations being injected into the\nFE/FM composite metastructure. An oscillating electric\n\feld with the characteristic frequency != 30GHz oper-\nating on the left edge of the dipolar/spin chain is due to\nthe action of the light source on the sample. The wave\npropagation proceeds self-consistently as governed by the\nset of equations (2) and (3). The results shown in Fig. 2\ncon\frm evidently the existence of the birefringence e\u000bect.\nIn numerical simulations we act on the left end of\nthe system with an electric \feld having the polarizationEx;y= 0:19 + 1:81iand corresponding to the negative\nrefraction regime. Obviously a Gaussian pulse prop-\nagates with a positive group velocity see Fig. 2 (a),\nwhile according to Fig. 2 (c) the phase velocity is neg-\native (blue, green and red curves correspond to time os-\ncillations of subsequent sites with increasing site num-\nber). Fig. 2 (b) displays the wave propagation process\n(again with positive group velocity) with the polarization\nEx;y= 0:015 + 1:985i. However, the corresponding phase\nvelocity is now positive and ordinary (positive) refraction\nscenario holds, see Fig. 2 (d).\nIn the above considerations the damping e\u000bects were\nneglected which allows obtaining analytical expressions.\nIn practice, ferromagnetic layers can be engineered such\nthat the damping [20] is very small (in \u0016s range) and\nhence can neglected on our relevant ns time scale. In fer-\nroelectrics damping is much stronger and its e\u000bect should\nbe considered. FE damping impacts the \frst mode only\n(4). In the limit \u000b!0 (i.e. for large susceptibility) the\nmodi\fed expression of the refraction index of the \frst\nmode reads\nn2\n1=\u0012\n1 +!2\nP=\u000b\n!2\nP\u0000!2+i!\u0000\u0013\u0012\n1 +!0!M\n!2\n0\u0000!2\u0013\n;(9)\n(here \u0000 is the damping parameter) while the second\nmode (7) is left unchanged. The experimentally observed\npeaks in the susceptibility correspond to the frequency\n!p= 10GHz see Ref [17]. In our case != 30GHz,\n\u0000\u001c!P\u001c!. Hence, we conclude that also FE damping\nhas no signi\fcant e\u000bect on the \frst mode (9) clarifying\nso the role of losses for the predicted phenomena. Until\nnow we considered FM and FE layers of equal thickness.\nAbsorption e\u000bects are rather sensitive to the thickness of\nFE layers and much less sensitive to the thickness of FM\nlayers. A recipe for minimizing losses is thus straightfor-\nward by fabricating thinner FE layers. In the numerical\nsimulations the thickness enters through the values of \u000b\nin (3). For example, taking \u000b= 10\u00002instead of\u000b= 10\u00004\nin (3), one can make the average polarization 100 times\nsmaller (which a\u000bects the electromagnetic waves). If we\ntake the FE layer 100 times thinner than the FM layer,\nthen the wave spends much less time in FE layer and the\nabsorption e\u000bects are reduced drastically. On the other\nhand, tuning \u000bleaves the main qualitative characteristics\nof the considered e\u000bect unchanged.\nThe frequency range for which the negative refraction\ntakes place depends on the external magnetic \feld (see\nFig. 3). The external magnetic \feld induces a shift in the\nLarmor frequency !0=!a+\rH0, while the anisotropy\nfrequency is \fxed to !a= 20GHz. By means of an exter-\nnal magnetic \feld the frequency !0is tunable within an\ninterval 10GHz to 50GHz. The results for the negative\nrefraction are shown in Fig. 3 (left graph). These results\ncon\frm that the frequency range for which a negative\nrefraction takes place can be controlled by an external\nmagnetic \feld. In the experiment one may switch so neg-5\n0 50 100−15−10−505\nω [GHz]Wz / P0S0\n−0.100.1−0.0200.020.040.060.08\nH0 [ T ]\nvg [units of c]\nω0=50GHzω0=40GHzω0=20GHzω0=10GHz\nbomgr\nω0=30GHz\nFIG. 3: Left graph: The Poynting vector magnitude ver-\nsus the mode frequency for di\u000berent external magnetic \felds:\n!0= 10;20;30;40;50 GHz correspond to blue (b), orange\n(o), magenta (m), green (g), and red (r) lines. Solid curves in-\ndicate the modes with a negative refraction regime and dashed\nlines describe the positively refractive ones. The frequency\nranges with a vanishing Poynting vector correspond to non-\npropagating evanescent modes. Right graph: The mode group\nvelocity dependence on the static magnetic \feld for a \fxed ex-\ncitation frequency != 24GHz. Blue (dashed) and red (solid)\ncurves describe the group velocities (in units of the speed of\nlight) of positively and negatively refracting modes, respec-\ntively.\native refraction media to ordinary media and vice versa\nsimply by turning the magnetic \feld o\u000b and on.\nIn the right graph of Fig. 3 for negatively and pos-\nitively refracting waves we plot the dependence of the\ngroup velocity on a static magnetic \feld. It is evident\nthat by a ramped static magnetic \feld we can achieve ac-\nceleration or even trapping of the electromagnetic waves.\nFinally we note that e\u000bects related to photonic-magneto-\nelastic and/or piezoelectric couplings are straightfor-\nwardly incorporated in the above formulism by including\nthe respective energy term in eqs.(1), along the lines as\ndone in Refs.[21, 22]. Furthermore, to access a higher\nfrequency regime (cf. eq.7)) it would be advantages\nto utilize FE/aniferromagnetic/FE/.... layer structures.\nFor example, recently an antiferromagnetic resonance fre-\nquency of 22 THz were observed for KNiF 3[23].\nSUMMARY\nSummarizing, in the present work we illustrated\ntheoretically an insulating multiferroic metamate-\nrial featuring simultaneous positive and negative re-\nfraction (positive-negative birefringes) and provided\nconcrete predictions for a realization on the ba-\nsis ofSrTiO 2=YIG multilayers. In addition to\nfull-\redge numerical simulations for the coupled\nMaxwell/ferroelectric/ferromagnetic dynamics, we wereable to derive credible analytical solutions and concrete\nfrequency regimes in which the predicted e\u000bects are to\nbe expected. The theory and predictions are of a general\nnature and are applicable to a wide range of material\nclasses. The \fndings point to new exciting applications\nof insulating nanostructured oxides in photonics.\nThis research is funded by the German Science founda-\ntion under SFB 762 \"Functionality of Oxide Interfaces\".\nWe thank J. Schilling for comments on the experimen-\ntal aspects. R. Kh. acknowledges \fnancial support from\nGeorgian SRNSF (grant No FR/25/6-100/14) and travel\ngrants from Georgian SRNSF and CNR, Italy (grant No\n04/24) and CNRS, France (grant No 04/01).\nAPPENDIX\nTo obtain wave solutions of the set Eqs.(2,3) in\nthe main text we express the \feld and the polariza-\ntion/magnetization components as follows:\nEx\nm=Exei(!t\u0000kam)+c:c;Ey\nm=Eyei(!t\u0000kam)+c:c;\nhx\nm=hxei(!t\u0000kam)+c:c; hy\nm=hyei(!t\u0000kam)+c:c;\nsx\nm=Sxei(!t\u0000kam)+c:c; sy\nm=Syei(!t\u0000kam)+c:c;\npm=Pei(!t\u0000kam)+c:c: (10)\nSubstituting this into Eqs.(2,3) of the main text and con-\nsidering the large wavelength limit k!0 we \fnd\ni!(hx+ 4\u0019Sx) =\u0000ikEy;\u0000i!(hy+ 4\u0019Sy) =\u0000ikEx\n\u0000i!(Ex+ 4\u0019P) =\u0000ikhy; i!Ey=\u0000ikhx\ni!Sx+!0Sy\u0000!M\n4\u0019hy= 0i!Sy\u0000!0Sx+!M\n4\u0019hx= 0\n\u000b\u0000\n!2\u0000!2\nP\u0001\nP+!2\nP\n4\u0019Ex= 0: (11)\nAfter some algebra one can reduce this equation to the\nthree coupled equations\ni!(n2\u00001)hx+\u0002\n!0(n2\u00001)\u0000!M\u0003\nhy\u0000!0nP= 0;\n\u0000\u0002\n!0(n2\u00001)\u0000!M\u0003\nhx+i!(n2\u00001)hy\u0000i!nP= 0;\n!2\nPnhy+\u0002\n\u000b(!2\u0000!2\nP)\u0000!2\nP\u0003\nP= 0; (12)\nwheren\u0011ck=! is a refractive index. Thus \fnally we\narrive at the matrix\n0\n@i!(n2\u00001)!0(n2\u00001)\u0000!M!0n\n!M\u0000!0(n2\u00001)i!(n2\u00001)\u0000i!n\n0 !2\nPn \u000b (!2\u0000!2\nP)\u0000!2\nP1\nA\nthe determinant of which should be equal to zero which\nleads to the relation\n\u0002\n(!2\u0000!2\nP)\u000b\u0000!2\nP\u0003n\u0002\n!0(n2\u00001)\u0000!M\u00032\u0000!2(n2\u00001)2o\n+!2\nPn2\u0002\n(!2\n0\u0000!2)(n2\u00001)\u0000!M!0\u0003\n= 06\nThis could be straightforwardly simpli\fed to a bi-\nquadratic equation for the refractive index n\n\u000b(!2\n0\u0000!2)(!2\nP\u0000!2)n4\u0000\n\u0000(!2\n0+!M!0\u0000!2)\u0002\n2\u000b(!2\nP\u0000!2) +!2\nP\u0003\nn2+\n+\u0002\n\u000b(!2\nP\u0000!2) +!2\nP\u0003\u0002\n(!0+!M)2\u0000!2\u0003\n= 0\nwhich is exactly the same relation (5) as in the main text.\nThe above matrix together with the relations (11) of\nthis supplementary materials gives the eigenmodes of the\nsystem and de\fnes the time averaged Poynting vector\nvalue as\nhWzi=hExHy\u0000EyHxi=P0S0[Ex(hy)\u0003\u0000Ey(hx)\u0003]+c:c:\nDe\fning now the dimensionless version of the Poynting\nvector asW=hWzi=P0S0we obtain eq.(6) of the main\ntext\nW=n(\n(!2\u0000!2\nP)\u0002\n(!2\u0000!2\n0)(n2\u00001) +!0!M\u00032\n(!2\u0000!2\nP\u0000!2\nP=\u000b)!2!2\nM+ 1)\nwhich in the limit \u000b!0 gives the following expressions\nfor the negatively refracting n=n1and positively re-\nfractingn=n2modes\nW1=\u0000n(!2\n0+!0!M\u0000!2)2!2\nP\n\u000b(!2\u0000!2\nP)!2!2\nM;\nW2=n\u001a\n1\u0000\u000b(!2\u0000!2\nP)!2!2\nM\n(!2\n0+!0!M\u0000!2)2!2\nP\u001b\nand as it could be easily seen W1\u0018\u0000nfor! >!Pand\nW2\u0018nfor\u000b!0.\n\u0003Electronic address: Jamal.Berakdar@physik.uni-halle.de\n[1] V.G. 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Secchi, R.\nV. Pisarev, Th. Rasing, G. Cerullo, and A. V. Kimel\nMacrospin dynamics in antiferromagnets triggered by\nsub-20 femtosecond injection of nanomagnons. Nat. Com-\nmun. 7:10645 doi: 10.1038/ncomms10645 (2016)." }, { "title": "2212.02257v3.Propagating_spin_wave_spectroscopy_in_nanometer_thick_YIG_films_at_millikelvin_temperatures.pdf", "content": "PSWS at millikelvin temperatures\nPropagating spin-wave spectroscopy in nanometer-thick YIG films at\nmillikelvin temperatures\nSebastian Knauer,1Kristýna Davídková,2David Schmoll,1, 3Rostyslav O. Serha,1, 3Andrey Voronov,1, 3Qi\nWang,1Roman Verba,4Oleksandr V. Dobrovolskiy,1Morris Lindner,5Timmy Reimann,5Carsten Dubs,5Michal\nUrbánek,2and Andrii V. Chumak1\n1)University of Vienna, Faculty of Physics, A-1090 Vienna, Austria\n2)CEITEC BUT, Brno University of Technology, 612 00 Brno, Czech Republic\n3)University of Vienna, Vienna Doctoral School in Physics, A-1090 Vienna, Austria\n4)Institute of Magnetism, Kyiv 03142, Ukraine\n5)INNOVENT e.V. Technologieentwicklung, Prüssingstraße 27B, Jena, Germany\n(*Electronic mail: knauer.seb@gmail.com)\n(Dated: 24 January 2023)\nPerforming propagating spin-wave spectroscopy of thin films at millikelvin temperatures is the next step towards the\nrealisation of large-scale integrated magnonic circuits for quantum applications. Here we demonstrate spin-wave prop-\nagation in a 100nm-thick yttrium-iron-garnet film at the temperatures down to 45mK, using stripline nanoantennas\ndeposited on YIG surface for the electrical excitation and detection. The clear transmission characteristics over the\ndistance of 10 mm are measured and the subtracted spin-wave group velocity and the YIG saturation magnetisation\nagree well with the theoretical values. We show that the gadolinium-gallium-garnet substrate influences the spin-wave\npropagation characteristics only for the applied magnetic fields beyond 75mT, originating from a GGG magnetisation\nup to 47kA =m at 45mK. Our results show that the developed fabrication and measurement methodologies enable the\nrealisation of integrated magnonic quantum nanotechnologies at millikelvin temperatures.\nI. INTRODUCTION\nYttrium-Iron-Garnet (YIG, Y 3Fe5O12) is the ideal choice\nof material to build and develop classical and novel quan-\ntum technologies1,2by coupling spin waves, and their single\nquanta magnons, to phonons3, fluxons4, or to microwave and\noptical photons5–8. These technologies may be realised by\nthe coupling to bulk or spherical YIG samples (e.g. Ref.9,10),\nor by the fabrication of integrated structures in thin YIG\nfilms11,12. Such nanometer-thick films can be grown using\nliquid phase epitaxy (LPE)13,14, exhibiting long spin wave\npropagation lengths, narrow linewidths and low damping con-\nstants13–16. Significant progress was made in realising YIG\nnano-waveguides with lateral dimensions down to 50nm11,\nin understanding the spin-wave properties in these waveg-\nuides17, and in using them for room-temperature data process-\ning12.\nTo create, propagate and read out spin waves at single\nmagnon level, millikelvin temperatures are required to sup-\npress thermal magnons according to the Bose-Einstein statis-\ntics2. The established technique of ferromagnetic resonance\n(FMR) spectroscopy was used to characterise YIG films of\nmicrometer18and nanometer thicknesses19–21at kelvin tem-\nperatures. At millikelvin temperatures FMR measurements\nwere performed on micrometer22and nanometer-thick19–21\nYIG films. Another method, propagating spin-wave spec-\ntroscopy (PSWS), is often used to characterise magnon trans-\nport between spatially-separated sources and detectors. This\ntechnique was successfully used in thin films at room23,24and\nnear room temperature25, and at millikelvin temperatures for\nmicrometer-thick YIG slabs26,27and micrometer-scaled hy-\nbrid magnon-superconducting systems28.\nThe ability to process information in sub-100nm sized\nmagnonic structures is one of the key advantages of magnon-ics, which translates also to the fields of hybrid opto-magnonic\nquantum systems and quantum magnonics. To couple PSW\nto these nanostructures efficiently at millikelvin temperatures,\nintegrated nanoantennas24,29are required. Here we demon-\nstrate PSWS at millikelvin temperatures, with base tempera-\ntures reaching 45mK in a 100nm-thick YIG film, using inte-\ngrated nanoantennas separated by 10 micrometers for excita-\ntion and detection. The analysis is focused on magnetostatic\nsurface spin waves (MSSWs, also called “Damon-Eshbach”\nmode) that propagate perpendicular to an in-plane magnetic\nfield k?B. We find that magnon transport at the nanome-\nter structure scale can be measured also down to millikelvin\ntemperatures. Although the propagation signal is measurable\nacross a wide field and temperature range, we observe that\nthe transmitted signal is distorted for applied magnetic fields\nabove 75mT. This effect is largely caused by the magneti-\nsation of the gadolinium-gallium-garnet (GGG) substrate, on\nwhich the YIG film is grown. It reaches 47kA =m for 75mT\nof applied external magnetic field at 45mK temperature. In\ngeneral, our findings agree with the increase in the damping\nof YIG grown on GGG at low-temperatures reported in the\nliterature19–22.\nFirst, we explain the sample preparation and experimental\ntechniques, before we continue to pre-characterise the sam-\nple at room and base temperature, using standard FMR tech-\nniques. Then we discuss the first PSWS experiment, in which\na fixed external magnetic field is applied and the temperature\nis swept from base to room temperature. We continue to anal-\nyse the propagation characteristics in more detail by compar-\ning the low-temperature measurements to room-temperature\nresults and extract the spin-wave group velocities. Finally, we\nperform PSWS at higher external magnetic fields, to inves-\ntigate the propagation characteristics between the room and\nbase temperature. The magnetisation of GGG is measured byarXiv:2212.02257v3 [physics.app-ph] 22 Jan 2023PSWS at millikelvin temperatures 2\n(a)\n(b) (c)\n1μm380μm\ne-\n(I) (II) (III)\n(IV) (V) (VI)\nYIG GGG PMMA Ti/Au\nAntenna 1\nAntenna 2S21,21Stripline nanoantennas with CPWReference CPW\nGGGYIGPort 2Port 1Antenna 1\nAntenna 2\nFIG. 1. Overview of the electron-beam lithographed stripline nanoantennas on the yttrium-iron-garnet film. (a) Sketch of the\nsample used in these measurements. Stripline nanoantennas coupled to coplanar waveguide (CPW) and the reference CPW are fabricated\natop a 100nm-thick yttrium-iron-garnet film on a 500 mm-thick gadolinium-gallium-garnet substrate. (b) The coplanar-waveguide coupled\nnanoantennas are fabricated with electron-beam lithography. These nanoantennas are made of Ti(5nm)/Au(55nm) (more details in main text).\n(c) Optical and secondary-electron images of the CWP nanoantennas used in the manuscript. These nanoantennas are 10 mm spaced apart and\nhave a width of 330nm and length of 120 mm. The propagating spin waves (PSW) are excited and detected by the stripline nanoantennas 1 and\n2 respectively. The transmission is measured through the S-parameters, acquired by a vector network analyser.\nvibrating sample magnetometry (VSM) of a GGG-only sub-\nstrate at low-temperatures.\nII. SAMPLE AND EXPERIMENTAL SETUPS\nIn our experiments we use an LPE-grown 100nm-thick\n(111)-orientated YIG film on a 500 mm-thick GGG substrate,\nas sketched in Fig. 1 (a). Atop the YIG film we fabricate\nnanoantennas connected to CPWs, using an electron-beam\nlithography process Fig. 1 (b). First, a single layer of PMMA\nis spin-coated and baked. After, we use electron-beam lithog-\nraphy to write the antenna structures, develop the sample and\ndeposit a layer of Ti(5nm)/Au(55nm), using electron beam\nphysical vapour deposition, followed by lift-off. Figure 1(c)\nshows an optical (top) and a secondary-electron image (bot-\ntom) with the coplanar waveguides and stripline nanoanten-\nnas used in this work. Here the nanoantennas have a spac-\ning of 10 mm. The stripline nanoantennas possess a width of\n330nm and a length of 120 mm. Additionally, we fabricate\na reference coplanar waveguide, to measure the FMR signals\nonly (see Fig. 1 (a)). After fabrication, the sample is glued\nand then wire-bonded, with a 75 mm diameter gold wire, to\na high-frequency printed circuit board and mounted into the\ndilution refrigerator.\nOur setup is based on a cryogenic-free dilution refrigerator\nsystem (BlueFors-LD250), which reaches base temperatures\nbelow 10mK at the mixing chamber stage. The sample space\npossesses a base temperature of about 16mK. During oper-\nation, the sample space heats up to about 45mK. At these\ntemperatures, the thermal excitations of gigahertz-frequency\nmagnons and phonons are still suppressed. The input signal\nis transmitted and collected from the sample (ports 1 and 2\nFig. 1(a)), using high-frequency copper and superconducting\nwiring each attenuated by 7dB to reduce thermal noise. The\nsignals are collected with a 70GHz vector network analyser\n(Anritsu VectorStar MS4647B).The room-temperature measurements are carried out on a\nhome-built setup. The setup consists of a VNA (Anritsu\nMS4642B) connected to an H-frame electromagnet GMW\n3473-70 with an 8cm air gap for various measurement config-\nurations and magnet poles of 15cm diameter to induce a suf-\nficiently uniform biasing magnetic field. The electromagnet\nis powered by a bipolar power supply BPS-85-70EC (ICEO),\nallowing it to generate up to 0 :9T at 8cm air gap. The in-\nput powers are adjusted, to obtain the same power levels at\nthe sample as in the cryogenic measurements, to account for\ncable losses and the previously mentioned attenuators. The\nprecise microwave powers for each individual experiment are\nstated later.\nIII. RESULTS AND DISCUSSION\nFirst, we use the reference CPW to pre-characterise the\nsample and to estimate the Gilbert damping as. We plot our\nFMR and PSWS data according to\nS0\n21(f) =S21;sig(f)\u0000S21;ref(f)\nS21;ref(f); (1)\nwhere fdenotes the set of frequency points of the com-\nplex transmission signal S21;sig(f)and reference S21;ref(f)\nvalues30. The reference signal is obtained by detuning the\nexternal magnetic field by +50mT. For the FMR refer-\nence measurements, we find the Gilbert damping parameter\nas= (5:98\u00060:3)\u000210\u00004for room temperature, and as=\n(3\u00061:5)\u000210\u00003at 45mK respectively. The large error in\nthe low-temperature case originates from the fit uncertainty\nin the slope of FMR linewidth versus FMR frequency. The\nmethodology developed in Ref.30, which accounts for asym-\nmetry and phase offset in the distorted FMR signal, was used.\nThe order of magnitude in the Gilbert damping at 45mK is in\ngood agreement with previously reported values for thin YIG\nfilms at Kelvin temperatures19.PSWS at millikelvin temperatures 3\n0.5K\n0.045K\n1.0K\n1.5K2.0K2.5K\nFIG. 2. Linear magnitude, real and imaginary part of the S0\n21\nparameters for propagating spin waves (PSW) in the Damon-\nEshbach mode, using 50mT of external magnetic field and dif-\nferent temperatures. The applied microwave power was set to\n\u000028dBm (at the sample) with an average sampling of 50 for 45mK-\n1K and 100 for 1 :5K-2 :5K. The FMR point ( k=0) is constant at\n3:36GHz (189kA =m) for all measured PSW.\nWe perform the first PSWS experiment at a fixed exter-\nnal magnetic field of 50mT, using the stripline nanoanten-\nnas shown in Fig. 1. Figure 2 displays the linear magni-\ntude (black), real (blue) and imaginary (red) part of the trans-\nmission data (cw-mode), together with a temperature sweep\nfrom the base temperature of 45mK up to 2 :5K, i.e. about\nthe Curie-Weiss temperature of GGG31,32. The spin waves\nare excited with a power of \u000028dBm at the sample, with the\nexternal magnetic field applied perpendicular to the propaga-\ntion direction. In Fig. 2 we verify the ability to measure the\ntransmission across the entire temperature range and observe\na propagation signal with a fixed FMR point ( k=0) of about\n3:36GHz, corresponding to an effective saturation magnetisa-\ntion of about 189kA =m. The signal amplitude increases by\nabout 30% from 45mK to 2 :5K.\nWe continue to investigate the spin-wave propagation in\nmore detail and compare the results to room-temperature mea-\nsurements. Figure 3 (first column) depicts the imaginary part\nof the S0\n21parameters for PSWs between the two nanoanten-\nnas at three different selected temperatures: (a) 297K, (b)\n500mK, and (c) 45mK, at a fixed external magnetic field of\n50mT. The second column in Fig. 3 shows the correspond-\ning calculated dispersion relations for MSSWs (black), us-\ning the Kalinikos-Salvin model33. The maximum excitation\nefficiency J(green line Fig. 3) is governed by the 330nm\nstripline nanoantennas23. The third column in Fig. 3 shows\n(a)\n(b)\n(c)297K\n0.5K\n0.045K\nFIG. 3. Imaginary part of the S0\n21parameter, calculated disper-\nsion relation, antenna excitation efficiency and group velocity for\nPSW (Damon-Eshbach mode), using 50mT external field at dif-\nferent temperatures. The theoretical group velocity is calculated as\nthe derivation of the dispersion relation and measured as vg=df\u0001D,\nwith the periodicity of the transmission in the Im(S0\n21) parameters\ndfand the gap between the nanoantennas D(see Ref.23). The pa-\nrameters measured and used for the calculation are the following:\n(a) 297K, Ms=142kA =m, (b) 500mK, Ms=189kA =m, (c) 45mK,\nMs=189kA =m. The effective saturation magnetisation increases\nand thus group velocity increases by about 50% at millikelvin tem-\nperatures.\nthe theoretical group velocities as the derivation of the dis-\npersion relation (black curve) and the measured values given\nbyvg=df\u0001D(red dots), where dfis the periodicity of\nthe oscillations in the Im (S0\n21)parameters and Dthe gap be-\ntween the nanoantennas23. The errors in the calculated group\nvelocities are estimated from the error propagation of the\nfrequency reading. We observe a reduction in propagation\namplitude by about 50% between the room and both cryo-\ngenic temperatures caused by the increase in Gilbert damping.\nWe find values for the effective saturation magnetisation of\nMs=142kA =m at room temperature and Ms=189kA =m for\n45mK and 500mK. The constant effective saturation mag-\nnetisation at millikelvin temperatures is in good agreement\nwith literature34,35, with a value close to the observed ones\nin micrometer-thick YIG samples26. In accordance with the\nincrease in effective saturation magnetisation, we observe an\nincrease of the group velocity by about 50%. The measured\nvalues are in good agreement with the theoretically calculated\ngroup velocities.\nWe continue our investigations by comparing the spin-\nwave propagation for higher external magnetic fields than inPSWS at millikelvin temperatures 4\n(a)\n(b)50mT297K175mT0.045K25mT 200mT 75mT\nFIG. 4. Linear magnitude, real and imaginary part of the S0\n21parameters for PSWS in the Damon-Eshbach mode at different external\nfields. The applied microwave power was set to \u000028dBm (at the sample) with an averaging of 10 (for 297K) and 25 (for 0 :045K).\n(a) Room temperature (297K): The spin-wave propagation can be measured over a wide magnetic field range. (b) Base temperature (45mK):\nThe spin-wave propagation for magnetic fields in the range from about 25mT to 75mT is trackable, while above 75mT the magnitude and its\npropagation characteristics start to be distorted. This effect is a result of the increased magnetisation of the GGG substrate (see Fig. 5).\nthe previous measurements, at 297K (Fig. 4 (a)) and 45mK\n(Fig. 4 (b)). Figure 4 shows the linear magnitude (black),\nreal (blue) and imaginary (red) part for PSW in the Damon-\nEshbach mode at selected magnetic fields. At room temper-\nature, we measure the spin-wave signal over a wide external\nmagnetic field range up to about 900mT. Examples for low\nfields are given in Fig. 4 (a). However, at 45mK the propa-\ngation characteristics are changing (Fig. 4 (b)). After about\n75mT the magnitude of the spin-wave signal is reduced sig-\nnificantly and only a signature in the oscillation behaviour\ncan be observed. Moreover, the fixed phase relation between\nthe imaginary and real parts disappears, causing challenges in\nplotting the linear magnitude of the propagation signal. Exam-\nples for the reduced spin-waves signals are given for 175mT\nand 200mT. This opposing behaviour between the room and\nbase temperature is a clear indication, that beyond an external\nfield of about 75mT the GGG substrate magnetises enough\nto influence the propagation characteristics of the spin waves.\nThus, future millikelvin measurements at high magnetic fields\nmay rely on suspended YIG membranes or triangular nanos-\ntructures, which have already been demonstrated in other ma-\nterial systems (e.g. Ref.36).\nTo estimate the influence of the paramagnetic GGG sub-\nstrate on the spin-wave propagation in YIG, we conclude our\ninvestigations by measuring the GGG magnetisation MGGG of\na 4\u00024\u00020:5mm GGG-only substrate, using a vibrating sam-\nple magnetometer (VSM) in the temperature range from 2K\nto 300K in the presence of fields up to 9T. The results at 2K\nfor our magnetic fields of interest are shown in Fig. 5 (dark-\nblue dots). As the VSM is limited to kelvin temperatures,\nwe extrapolate magnetisation values for GGG at 45mK (blue\ndashed line), using the 2K data. For example at 75mT (Fig. 5\nblack dots) we find, that GGG possesses a magnetisation value\nof 28 :5kA =m at 2K, which increases to about 47kA =m at\n45mK. Thus, the temperature and magnetic field dependant\nGGG magnetisation may explain the observed reduction in the\n47kA/m\n28.5kA/mFIG. 5. Magnetisation of the GGG substrate versus the applied\nmagnetic field. A GGG-only sample is measured using a vibrating\nsample magnetometer (VSM) at 2K (dark-blue dots), leading for ex-\nample to an effective magnetisation of 28 :5kA =m. From the data the\nmagnetisation values for 45mK are extrapolated (blue dashed line).\nAt 75mT the magnetisation increases to about 47kA =m.\nPSW amplitudes and the propagation distortions above exter-\nnal magnetic fields of 75mT.\nHowever, the role of the paramagnetic GGG substrate on\nspin waves in YIG is the subject of separate systematic stud-\nies. Our PSWS measurements, supported by the FMR and\nVSM studies, suggest that the magnetic moment induced in\nGGG at millikelvin temperatures by the application of rela-\ntively large magnetic fields is at least partly responsible for the\nincrease in spin-wave damping. The increase in the Gilbert\ndamping constant acan only be approximately quantified,\nas this requires plotting the FMR linewidth DBagainst the\nFMR frequency fFMR over a wide range of applied fields.PSWS at millikelvin temperatures 5\nHowever, since the FMR linewidth depends on the degree\nof the magnetisation of the GGG (given by the temperature\nand the applied field - see Fig. 5), the dependence DB(fFMR)\nbecomes nonlinear and the parameter aloses its original\nphysical meaning. Moreover, the measurement of FMR on\nnanometer-thick samples requires the careful subtraction of\nthe reference microwave transmission signal (see Eq. 1) at\na 50mT detuned magnetic field. Since this reference signal\nalso depends significantly on the GGG magnetisation at low-\ntemperatures, the measurement uncertainties increase. Nev-\nertheless, we can qualitatively conclude that the increase in\nspin-wave damping in the nanometer-thick YIG films on GGG\ncorresponds to the previously reported increase in damping in\nthe micrometer-thick films on GGG18,19,22. Other phenomena\nthat could contribute to the distortion of the PSWS experi-\nments at the nanoscale at fields above 75mT are the possible\ndependence of the magnetocrystalline anisotropy caused by\nthe dependence of the YIG/GGG lattice mismatch on temper-\nature and the absorption/distortion of the microwave signal in\nthe CPW transmission lines (see Fig. 1(a)) by the magnetised\nGGG substrate.\nIV. CONCLUSIONS\nIn conclusion, we have shown for the first time that propa-\ngating spin-wave spectroscopy in 100nm-thin YIG films can\nbe performed in a wide temperature range, from millikelvin to\nroom temperature, without changing the propagation charac-\nteristics. At a fixed external magnetic field of 50mT we con-\nfirm that the propagating spin waves maintain a constant fer-\nromagnetic resonance frequency below temperatures of about\n2:5K. However, the signal amplitude increases by 30% be-\ntween 45mK and 2 :5K, and further by about 50% when\nthe temperature is raised to room temperature. In contrast\nto previous work we demonstrate, that only beyond an ex-\nternal field of about 75mT the GGG substrate magnetises\nup to 47kA =m influence the spin-wave propagation at low-\ntemperatures. With our experiments, we illustrate that al-\nthough the GGG substrate influences the spin-wave propaga-\ntion characteristics at millikelvin temperatures, future large-\nscale integrated YIG nanocircuits can be realised and mea-\nsured.\nACKNOWLEDGMENTS\nThe authors thank Vincent Vlaminck for useful discussions\nand feedback. SK acknowledges the support by the H2020-\nMSCA-IF under the grant number 101025758 (OMNI). KD\nwas supported by the Erasmus+ program of the European\nUnion. The authors acknowledge CzechNanoLab Research\nInfrastructure supported by MEYS CR (LM2018110). The\nwork of CD was supported by the Deutsche Forschungsge-\nmeinschaft (DFG, German Research Foundation) under grant\n271741898. The work of ML was supported by the German\nBundesministerium für Wirtschaft und Energie (BMWi) under\ngrant 49MF180119. CD thanks O. Surzhenko and R. Meyer(INNOVENT) for their support. The authors thank Oleksandr\nDobrovolskiy for his support in the initial configuration of the\ndilution refrigerator.\nAUTHOR DECLARATIONS\nConflict of Interest\nThe authors have no conflicts to disclose.\nAuthors Contributions\nSK and MU conceived the experiment in discussion with\nAC. SK and KD performed the experiments under the guid-\nance of MU and AC. SK and KD analysed and interpreted\nthe data with support from AC. RS and OD performed the\nVSM measurements at kelvin temperatures, and A V interpo-\nlated the data for millikelvin temperatures. ML and TR pre-\npared the LPE sample. CD conceived and supervised the LPE\nfilm growth. QW and RV supported the measurements with\ntheoretical expertise. DS and SK set up the cryogenic system.\nRS supported the measurements and analysis of the measure-\nments. SK wrote the manuscript with support from all co-\nauthors.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nREFERENCES\n1A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Grafe,\nC. Adelmann, S. Cotofana, A. Naeemi, V . I. Vasyuchka, B. Hillebrands,\nS. A. Nikitov, H. Yu, D. Grundler, A. V . Sadovnikov, A. A. Grachev,\nS. E. Sheshukova, J. Y . Duquesne, M. Marangolo, G. Csaba, W. Porod,\nV . E. Demidov, S. Urazhdin, S. O. Demokritov, E. Albisetti, D. Petti,\nR. Bertacco, H. Schultheiss, V . V . 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Zeiske, and M. Yethiraj, “Mag-\nnetic frustration and order in gadolinium gallium garnet,” Physica B: Con-\ndensed Matter 266, 41–48 (1999).\n32M. Sabbaghi, G. W. Hanson, M. Weinert, F. Shi, and C. Cen, “Terahertz\nresponse of gadolinium gallium garnet (GGG) and gadolinium scandium\ngallium garnet (SGGG),” Journal of Applied Physics 127(2020).\n33B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave\nspectrum for ferromagnetic films with mixed exchange boundary condi-\ntions,” Journal of Physics C: Solid State Physics 19, 7013–7033 (1986).\n34P. Hansen, P. Röschmann, and W. Tolksdorf, “Saturation magnetization\nof gallium-substituted yttrium iron garnet,” Journal of Applied Physics 45,\n2728–2732 (1974).\n35I. V . Zavislyak and M. A. Popov, “Microwave Properties and Applications\nof Yttrium Iron Garnet (YIG) Films: Current State of Art and Perspectives.\nIn Yttrium: Compounds, Production and Applications,” (Nova, 2009) pp.\n87–125.\n36M. J. Burek, Y . Chu, M. S. Liddy, P. Patel, J. Rochman, S. Meesala,\nW. Hong, Q. Quan, M. D. Lukin, and M. Loncar, “High quality-factor op-\ntical nanocavities in bulk single-crystal diamond,” Nature Communications\n5, 1–7 (2014)." }, { "title": "2308.00412v2.Crystallization_Dynamics_of_Amorphous_Yttrium_Iron_Garnet_Thin_Films.pdf", "content": "Crystallization dynamics of amorphous yttrium iron garnet thin films\nSebastian Sailler,1Gregor Skobjin,1Heike Schlörb,2Benny Boehm,3Olav Hellwig,3, 4Andy Thomas,2, 5Sebastian\nT. B. Goennenwein,1and Michaela Lammel1\n1)Department of Physics, University of Konstanz, 78457 Konstanz, Germany\n2)Leibniz Institute of Solid State and Materials Science, 01069 Dresden, Germany\n3)Institute of Physics, Technische Universität Chemnitz, 09126 Chemnitz, Germany\n4)Center for Materials Architectures and Integration of Nanomembranes (MAIN), Technische Universität Chemnitz,\n09107 Chemnitz, Germany\n5)Institut für Festkörper- und Materialphysik (IFMP), TUD Dresden University of Technology, 01069 Dresden,\nGermany\n(*Electronic mail: michaela.lammel@uni-konstanz.de)\n(*Electronic mail: sebastian.sailler@uni-konstanz.de)\n(Dated: 19th April 2024)\nYttrium iron garnet (YIG) is a prototypical material in spintronics due to its exceptional magnetic properties. To\nexploit these properties high quality thin films need to be manufactured. Deposition techniques like sputter deposition\nor pulsed laser deposition at ambient temperature produce amorphous films, which need a post annealing step to induce\ncrystallization. However, not much is known about the exact dynamics of the formation of crystalline YIG out of the\namorphous phase. Here, we conduct extensive time and temperature series to study the crystallization behavior of\nYIG on various substrates and extract the crystallization velocities as well as the activation energies needed to promote\ncrystallization. We find that the type of crystallization as well as the crystallization velocity depend on the lattice\nmismatch to the substrate. We compare the crystallization parameters found in literature with our results and find an\nexcellent agreement with our model. Our results allow us to determine the time needed for the formation of a fully\ncrystalline film of arbitrary thickness for any temperature.\nI. INTRODUCTION\nYttrium iron garnet (Y 3Fe5O12, YIG) is an electrically in-\nsulating ferrimagnet, crystallizing in a cubic crystal lattice\nwith Ia ¯3d symmetry.1,2Its electric and magnetic properties\ninclude a long spin diffusion length, which makes YIG an\nideal material for spin transport experiments with pure spin\ncurrents.3–5Additionally, YIG shows an exceptionally low\nGilbert damping and a low coercive field, which allows in-\nvestigations of magnon dynamics via e.g. ferromagnetic res-\nonance experiments.6–10These exceptional properties caused\nYIG to be intensively studied and made it the prototypical ma-\nterial in the field of spintronics, which almost exclusively re-\nlies on devices in thin film geometry.\nSeveral deposition techniques are known to produce high\nquality YIG thin films, including pulsed laser deposition\n(PLD),11–18liquid phase epitaxy (LPE)10,19–23and radio-\nfrequency (RF) magnetron sputtering.24–41Some deposition\ntechniques like magnetron sputtering give the opportunity to\ndeposit both, amorphous and crystalline thin films, depend-\ning on the process temperatures during deposition.25,42Here,\nroom temperature magnetron sputtering processes yield amor-\nphous films.24–33,35–42For the deposition of YIG onto gadolin-\nium gallium garnet (Gd 3Ga5O12, GGG) substrates, which fea-\nture a lattice constant very similar to the one of YIG, direct\nepitaxial growth was observed for process temperatures of\n700◦C.25,42On quartz a post annealing step is needed to en-\nable the formation of polycrystalline YIG.43\nThe annealing process is usually performed in air26,36or\nreduced oxygen atmosphere28,40,44,45to counteract potential\noxygen vacancies in the YIG lattice. For amorphous PLD\nfilms annealing in inert argon atmosphere has been reported tohave no deteriorating influence.15Annealing crystalline, sput-\ntered YIG films in vacuum, however, showed a reduction in\ntypical characteristic properties like the spin Hall magnetore-\nsistance in YIG/Pt.44\nFurthermore, the annealing process itself can lead to an in-\nterdiffusion at the substrate interface,36,46often leading to the\nformation of a magnetic dead layer,23,36,46as well as an in-\ncrease of the ferromagnetic resonance linewidth, especially at\nlow temperatures.13,47On the one hand, YIG grown on GGG\nby LPE requires no post annealing, which allows for the sup-\npression of the gadolinium interdiffusion, leading to an ex-\ntremely sharp interface.23On the other hand, scaling the LPE\nprocess is not straightforward. Sputter deposition31or solu-\ntion based methods45,48allow for wafer scale processes, but\nthe mandatory post annealing step should be optimized to al-\nlow fast processing, which then could simultaneously reduce\nthe interdiffusion of yttrium and gadolinium. To achieve this,\nthe annealing time required to yield fully crystalline YIG films\nneeds to be kept as low as possible.\nHowever, the dynamics describing the crystallization of\nYIG thin films during the post annealing step are only selec-\ntively reported in the literature. Typically, only the tempera-\nture and a time proven to yield a completely crystalline thin\nfilm with the desired properties are reported.\nHere, we present an extended picture of the crystalliza-\ntion dynamics of YIG at different temperatures and annealing\ntimes, which allows us to define different crystallization win-\ndows depending on the substrate material. Our results pro-\nvide a general crystallographic description of the crystalliza-\ntion process for YIG thin films and summarize the crystalliza-\ntion parameters found in the literature.arXiv:2308.00412v2 [cond-mat.mtrl-sci] 18 Apr 20242\nII. METHODS\nAhead of the deposition, all substrates were cleaned for\nfive minutes in aceton and isopropanol, and one minute in\nde-ionized water in an ultrasonic bath. YIG thin films were\nthen deposited at room temperature onto different substrate\nmaterials using RF sputtering from a YIG sinter target at\n2.7·10−3mbar argon pressure and 80 W power, at a rate of\n0.0135 nm /s. The nominal thickness upon deposition was\n33 nm. The post-annealing steps were carried out in a tube\nzone furnace under air.\nAs substrates yttrium aluminum garnet (Y 3Al5O12, YAG,\nCrysTec ) and gadolinium gallium garnet (Gd 3Ga5O12, GGG,\nSurfaceNet ) with a <111> crystal orientation along the sur-\nface normal were used. Additionally, silicon wafers cut along\nthe <100> crystal direction with a 500 nm thick thermal ox-\nide layer (Si/SiO x,MicroChemicals ) were used. Since GGG\nand YAG crystallize in the same space group Ia ¯3d as YIG and\ntheir lattice parameters are 1 .2376 nm49and 1 .2009 nm,50re-\nspectively, they are considered lattice matched in regards to\nthe 1 .2380 nm for YIG.51The lattice mismatch εcan be cal-\nculated with Eq. (1)\nε=aYIG−asubstrate\nasubstrate·100% (1)\nand translates to 0 .03 % for GGG and 3 .09 % for YAG.52Due\nto the amorphous SiO xlayer the Si/SiO xsubstrates do not pro-\nvide any preferential direction for crystallization. But even\nconsidering the underlying Si layer, we do not expect it to\ninfluence the crystallization direction in any way, as it fea-\ntures a fundamentally different space group (Fd ¯3m) and lat-\ntice constant.53Therefore, Si/SiO xis considered non lattice\nmatched and fulfills the function as an arbitrary substrate.\nFor the structural characterization X-ray diffraction mea-\nsurements (XRD) were performed using a Rigaku Smart Lab\nDiffractometer with Cu Kαradiation. Scanning electron mi-\ncroscopy as well as electron backscatter diffraction (EBSD)\nmeasurements were conducted using a Zeiss Gemini Scan-\nning Electron Microscope (SEM). The magnetic properties\nwere characterized via magneto-optical Kerr effect measure-\nments in longitudinal geometry (L-MOKE) in a commercial\nKerr microscope from Evico Magnetics.\nIII. RESULTS AND DISCUSSION\nThe crystallization mechanism of a thin film crucially de-\npends on the substrate: for substrates where the lattices of\nfilm and substrate are sufficiently similar, the thin film layer\ncrystallizes epitaxially, whereas for a substrate where the two\nlattices do not match, nucleation is needed.\nFigure 1 shows the different crystallization mechanisms\nand the resulting YIG micro structure depending on the cho-\nsen substrate. As depicted in Figure 1(a), a lattice matched\nsubstrate acts as a seed on which the film can grow epitaxi-\nally. Therefore, a single crystalline front is expected to move\nfrom the substrate towards the upper boundary of the film,54,55which is commonly referred to as solid phase epitaxy (SPE) in\nthe literature. For a substrate with a sufficiently large lattice\nmismatch or no crystalline order, no such interface is given,\nsee Fig. 1(b). Here, a nucleus needs to be formed first from\nwhich further crystallization takes place. The formation of the\ninitial seeds by nucleation is expected to yield random crys-\ntal orientations. The polycrystalline seeds grow until reaching\nanother grain or one of the sample’s boundaries. For any of\nthese processes, SPE or nucleation, to take place, the system\nneeds to be at a temperature characteristic for this specific thin\nfilm/substrate system.56\nTo distinguish between amorphous, partly and fully crys-\ntalline films we apply several characterization methods, prob-\ning the structural and magnetic properties of the YIG thin\nfilms. The typical fingerprints of amorphous versus crystalline\nYIG on different substrates as determined by X-ray diffrac-\ntion (XRD), the longitudinal magneto-optical Kerr effect (L-\nMOKE) and electron back scatter diffraction (EBSD) are de-\npicted in Figure 2. From top to bottom we gain an increased\nspacial resolution, probing increasingly smaller areas of the\nsample.\nWith XRD, the structural properties of YIG on YAG and\nGGG can be evaluated. For the amorphous films, the XRD\nmeasurements in Fig. 2 (a-c) show a signal stemming only\nfrom the substrate (cp. gray dashed lines). Upon annealing,\nYIG is visible in the form of Laue-oscillations on GGG (pur-\nple) and as a peak on YAG (red). In stark contrast to that\nno signal, which could be attributed to YIG, can be found\non SiO x, even when annealing at 800◦C for 48 h. The sharp\npeak in Fig. 2(c) at 32 .96◦can be attributed to a detour ex-\ncitation of the substrate, as it is visible in the as deposited\nstate and fits the forbidden Si (200) peak.57In the literature,\nYIG on SiO xhas been reported to be polycrystalline at lower\nannealing temperatures than in the exemplary data shown in\nFig. 2(c).26,28,43These films show peaks in the XRD, however\nthey were at least one order of magnitude thicker. We there-\nfore do not expect the YIG layer on Si/SiO xto be amorphous,\nwhich will be confirmed in the following.\nBy probing the magnetic properties of the thin films with\nFig. 1. Expected crystallization of an amorphous, as-deposited\n(a.d.) YIG thin film on lattice matched substrates (a) and non lat-\ntice matched substrates (b). In the first case of solid phase epitaxy,\na homogeneous crystal front forms at the substrate and propagates\ntowards the upper thin film border. For the latter, nucleation is nec-\nessary and crystallites form in various orientations. This results in a\nsingle crystalline (sc) film for the epitaxy and a polycrystalline (pc)\nfilm when nucleation occurs.3\nFig. 2. (a)-(c) XRD analysis of YIG thin films pre and post annealing on different substrates as given above in the respective column. The\nnominal positions of the substrate and the thin film are marked by the grey and black dashed lines, respectively. The additional peak marked\nwith Si(200) in (c) is a detour reflex from the substrate. (d)-(f) Background corrected Kerr microscopy data in L-MOKE configuration for the\nsame samples before and after the annealing procedure. The change in the measured gray value corresponds to a change in the magnetization\nof the sample. The data was acquired from a central spot on the sample. (g)-(i) crystal orientation of the post annealed YIG thin films normal\nto the surface normal as extracted from the Kikuchi-patterns determined by EBSD. The as-deposited films showed no Kikuchi-Patterns and\nare therefore not shown here.\nL-MOKE (cp. Fig. 2(d-f)), a clear distinction between amor-\nphous and crystalline YIG can be made. While the film\nshows a linear L-MOKE signal in the as-deposited state, it\nchanges to a hysteresis for all three samples upon anneal-\ning. In general, the sharpest hysteresis is visible for YIG\non GGG, which becomes broader for an increasing structural\nmisfit. Naïvely polycrystalline samples are expected to con-\nsist of multiple domains pointing towards different directions,\nwhich lead to an increase of the coercive field. This is con-\nsistent with our results and also with the magnetic properties\nfound in literature.14,28,35,58These coercive fields are below0.1 mT for YIG on GGG14,35and between 2.2-3 mT for YIG\non Si/SiO x.28,58The L-MOKE measurements therefore indi-\ncate the spontaneous formation of a phase with magnetic or-\ndering on all three substrates.\nFor additional characterization of the magnetic properties\nof the films via ferromagnetic resonance and SQUID magne-\ntometry please refer to the supplemental information.59The\ncorresponding data show the same dependence on the type of\nsubstrate, that is also apparent in the L-MOKE measurements.\nOnce the YIG is fully crystallized, however, we do not find a\ndependence of the magnetic parameters of our thin films on4\nthe annealing parameters.\nWhile L-MOKE correlates the magnetic properties with\namorphous and crystalline films, it lacks the ability to quan-\ntify the amount of crystalline YIG. The hysteretic response\nfor the annealed YIG on SiO xstrongly supports the forma-\ntion of crystalline YIG, however, we cannot correlate this to\na percentage of crystalline material. Therefore, a structural\ncharacterization with higher spacial resolution than XRD is\nneeded.\nTo that end electron back scatter diffraction (EBSD) mea-\nsurements were performed. With this technique Kikuchi pat-\nterns, which are correlated to the crystal structure, are detected\nand later evaluated. The results are shown for crystalline sam-\nples only, as the amorphous film showed no Kikuchi patterns.\nThis confirms, that the detected patterns stem from the YIG\nthin film itself and not from the crystallographically simi-\nlar substrates of YAG or GGG. This is consistent with the\nEBSD signal depth given in the literature of 10 to 40 nm.60\nThe extracted crystal orientations along the surface normal\ncan be seen in Fig. 2 (g-i). On YAG and GGG a single\ncolor corresponding to the <111> direction is visible in the\nmapping, which is consistent with the XRD data and corrob-\norates SPE from the substrate in the <111> direction. On\nSiO x, however, various crystal directions are present, con-\nfirming the polycrystalline nature of the YIG. The crystallo-\ngraphic data from our EBSD measurements show random nu-\ncleation. The cross shape of the individual crystalline areas\npoint towards an anisotropic crystallization with a preferen-\ntial direction along <110> or higher indexed directions like\n<112>, which is consistent with earlier studies on YIG and\nother rare earth garnets24,61–63as well as PLD grown bismuth\niron garnet.11\nThe use of EBSD enables the quantification of the amount\nof crystalline material in a YIG thin film on SiO xor any ar-\nbitrary substrate. Combining the magnetic and structural data\nfrom L-MOKE and EBSD, respectively, allows for an unam-\nbiguous identification of the formation of polycrystalline YIG\non SiO x.\nWe presume that the absence of any XRD peaks in the sym-\nmetric θ−2θscan results from the small volume of the in-\ndividual crystallites of YIG on SiO x.34,37,48We approximate\nthe volume of a single polycrystalline grain, i.e. one cross\nfrom the EBSD data (cp. Fig. 2(i)) to be 0 .5µm3, stemming\nfrom an area of about 15 µm and a film thickness of 32 nm.\nThis is also the size of individually contributing grains to the\ndiffraction within the XRD. Assuming a single crystalline thin\nfilm, where the whole irradiated area contributes additively,\nthe contributing area amounts to 7 ·105µm3, which is six\norders of magnitude larger than that of an individual grain.\nTherefore, the contributions of the individual grains of the\nYIG layer on SiO xto the XRD intensity are too small to result\nin a finite peak for a 30 nm thick film.\nThese results provide the basis for the investigation of the\ncrystallization behavior and reveal how different techniques\nenable us to distinguish between amorphous, partly and fully\ncrystalline films. We utilize the structural information to ana-\nlyze the crystallization dynamics on the different substrates.\nThe percentage of crystalline YIG was quantified differ-\nFig. 3. Evolution of the crystallinity as a function of the annealing\ntemperature for a constant annealing time of 4 h (a,c,e) and for dif-\nferent times at a constant temperature of 600◦C on GGG, YAG (b,d)\nor 800◦C on SiOx(f). The dotted lines act as a guide to the eye.\nently for the three different substrates. For YIG on YAG the\namount of crystalline YIG correlates to the intensity of the\nBragg peak. A certain film thickness corresponds to a max-\nimum area under the peak, to which the intensity is normal-\nized. For YIG on GGG, the percentage of crystalline YIG\nis extracted from the Laue oscillations (cp. Fig 2(a)). The\nfrequency of the oscillation corresponds to the number of in-\nterfering lattice planes, enabling the calculation of the thick-\nness of the crystalline layer. Using X-ray reflectivity, the ab-\nsolute film thickness was measured for each film. For these\nmeasurements and evaluation, please refer to the supplemen-\ntal information.59Comparing the thickness of the crystalline\nlayer with the film thickness then enables to monitor the crys-\ntallization of YIG on GGG. For the non lattice matched sub-\nstrates, EBSD mappings were taken to extract the amount of\ncrystalline YIG. Further evaluation of partly crystalline YIG\non SiO xcan be found in the supplemental information.59For\neach of the YIG thin films, a percentage of crystalline YIG\nat a given time and temperature is extracted, which allows an\nevaluation of the crystallization process for this specific tem-\nperature.\nFirst, we find the onset temperature for the crystalliza-\ntion of YIG on each substrate. As crystallization is ther-\nmally activated, it depends exponentially on the annealing\ntemperature,64which leads to a very narrow temperature win-\ndow of incomplete crystallization. To extract this window,\nmultiple samples were annealed for four hours at different\ntemperatures. Figure 3 shows the results for YIG on GGG\n(a), YIG on YAG (c) and YIG on SiO x(e). At substrate de-5\npendent temperatures of 550◦C, 575◦C and 700◦C for YIG\non GGG (a), YAG (c) and SiO x(e), respectively, a steep in-\ncrease in the crystallinity can be seen. Towards higher tem-\nperatures the extracted value stays the same or is only slightly\nreduced, which suggests, that the YIG film is fully crystal-\nlized and no further changes are expected. A crystalline YIG\nfilm on YAG and GGG can therefore be obtained at a temper-\nature range around 600◦C, whereas on SiO x, temperatures of\napproximately 700◦C are necessary.\nFor our samples, the heating up and cooling down is in-\ncluded in the annealing time. An in-situ study on a represen-\ntative sample with dYIG=100nm yielded data in good agree-\nment with the crystallization behavior in the one zone furnace.\nIt should be noted that the use of different equipment led to a\nsmall variation in the absolute temperature, see supplemental\ninformation.59\nThe lower extracted crystallinities for YIG on YAG and\nGGG at 800◦C and above (cp. Fig. 3(a)+(c)) hint towards\nthe occurrence of competing crystallization processes. We at-\ntribute the reduction in crystallinity at annealing temperatures\nabove 800◦C to additional formation of polycrystals enabled\nby the elevated temperatures, which competes with the solid\nphase epitaxy and by that reduces the crystal quality of the\nthin film. Analyzing the rocking curves of these samples (see\nsupplemental information) confirms an increased full width\nat half maximum value at higher temperatures.59This can be\ncorrelated with a lower crystal quality, which supports an ad-\nditional crystallization process.\nTo study the crystallization dynamics, the time evolution of\nthe normalized crystallinity for a given temperature is eval-\nuated, shown in Fig. 3 for YIG on GGG(b), YAG(d) and\nSiO x(f). Here, a sample was subjected to the same temper-\nature for multiple repeats until the extracted value and there-\nwith the crystalline amount did saturate. This saturation can\nbe seen on all substrates and represents a fully crystallized\nthin film, where no further changes are expected.\nTo describe the crystallization at an arbitrary temperature,\nwe find a general crystallographic description for each of the\nsubstrates. A phase transition in a solid like crystallization can\ngenerally be described by the Avrami equation:64–67\nθc=1−e−k·tn(2)\nwhere θcis the crystallinity normalized to one, with respect to\na complete crystallization, kthe rate constant and tthe anneal-\ning time. The exponent nis often referred to as the Avrami ex-\nponent and describes how the crystallization takes place.66It\ncan take values between 1 and 4, where one contribution stems\nfrom the nucleation and takes values of 0 for controlled and\n1 for random nucleation, while the other contributions origi-\nnate from the type of crystallization in the three spacial direc-\ntions. For the rate constant k we use an exponential Arrhenius\ndependency:54,68\nk=k0·e−EA\nkB·T (3)\nwhere both the pre-factor k0and the activation energy EAare\nunique for each combination of film and substrate material.The Avrami equation (cp. Eq.(2)) lets us describe the crys-\ntallization on all three substrates. To that end we fit the nor-\nmalized crystallinity values of YIG with the Avrami equation\n(cp. Eq.(2)), where we fix the Avrami exponent n between 1\nand 4 (cp. Fig. 4(a)+(b)). The rate constants kthen describe\nthe crystallization velocities on the respective substrate in h−1.\nThe crystallization behavior of YIG on GGG and YAG at an\nannealing temperature of 600◦C is shown in Fig. 4(a).\nOn GGG at 600◦C (cp. Fig. 4(a)), YIG immediately starts\nto crystallize with a rate constant of 1 .96 h−1and an Avrami\nexponent of 1. This means, that the crystallization takes place\nwithout nucleation and in one spacial direction, which is con-\nsistent with the monotonously moving crystallization front ex-\npected for SPE. The rate constant translates to a initial veloc-\nity of 0 .98 nm /min for the 30 nm films. Towards longer an-\nnealing times, the curve flattens, meaning that the crystalline\nmaterial reaches the sample’s surface.\nThe crystallization of YIG on YAG shows an initial time de-\nlay, despite the comparably small lattice mismatch of 3 .09 %\nFig. 4. (a) and (b) show the time evolution of the YIG crystallization\non the three substrates after normalizing the data with the maximum\nvalue to 1. The dots represent the crystallinity values from XRD\n(YAG/GGG) and EBSD (SiOx), while the solid lines show the fit of\nthe data using Eq.(2). Because of the inherently different crystalliza-\ntion processes, the time scales and the temperatures differ. Conduct-\ning these time evolutions at different temperatures for each substrate\nresult in a rate constant k(T)for this temperature. A logarithmic rep-\nresentation of the k(T)values over the inverse temperature is given\nby the symbols in (c). For each substrate a linear expression was\nfitted, where the slope represents the activation Energy EAand the\nintercept of the y-axis the pre-factor k0for YIG on each substrate.6\nFig. 5. (a) Annealing parameters to obtain a fully crystalline YIG film on the respective substrates. We expect every point in the colored area\nto yield a fully crystalline sample. We use Eq.(4) with the values obtained in Fig. 4(c) to determine the boundary separating crystalline YIG\n(shaded areas, sc = single crystalline, pc = poly crystalline) from amorphous YIG (white areas). The open circles represent the samples from\nFig. 4(c) which are used for the fit. Further studied, fully crystalline samples are marked by the full circles. There are different regions where\nthe YIG is fully crystalline depending on the substrate. Panel (b) gives a comparison of our crystallization diagram with other studies.24–33,35–41\nNote that, while we here consider only the crystallization of sputtered thin films by post annealing, the crystallization diagram also fits for\ncomparable samples obtained by PLD (not shown here).11–17\n(cp. Fig. 4(a)). The fitting of the data at 600◦C leads to a rate\nconstant of 0 .10 h−1with n=3.8. This means, that the crys-\ntallization does not follow a typical SPE behavior and nucle-\nation processes in the thin film cannot be excluded. However,\nalso for the crystallization on YAG, single crystalline YIG is\nobtained (cp. Fig. 2(b)+(h)). This deviation from YIG on\nGGG is most likely due to the larger lattice mismatch which\ncauses an energetically costly strain in the film.69The crys-\ntallization velocity along the surface normal direction is ob-\ntained by the n-th root out of the rate constant and translates\nto 0.27 nm /min.\nThe crystallization of YIG on SiO xis fundamentally differ-\nent (cp. Fig. 4(b)). Here, polycrystalline grains were found\nat temperatures of 675◦C and above. The time evolution of\nthe crystallinity is depicted in Figure 4(b), where fitting the\ndata by the Avrami equation (Eq. (2)) yields n=4 and a rate\nconstant of 9 .9·10−5h−1. This confirms our initial hypothesis\nof nucleation and subsequent crystallization in three dimen-\nsions. Higher temperatures compared to the garnet substrates\nare needed to provide enough energy for nucleation, which\ncauses the crystallization process to be visible at 675◦C and\nabove.\nAn approximation of the crystallization velocity can be ex-\ntracted from the EBSD data. Here, we assume that the crys-tallization starts in the middle of a cross shape structure (cp.\nFig. 2(i)) and stops when reaching a boundary given by neigh-\nboring crystallites. The distance covered depends on the num-\nber of nuclei formed and is highly dependent on the crystal-\nlographic direction. To ensure comparability with the two lat-\ntice matched substrates, we consider grains growing in plane\nalong the <111> direction. At 700◦C, the YIG crystallites\non SiO xmeasured up to 10 µm in length after at least 12 h\nof annealing. This translates into a propagation velocity of\n16.7 nm /min at 700◦C on an arbitrary substrate along the\n<111> direction.\nTo compare the three crystallization velocities, the temper-\nature dependence of the rate constants kneeds to be taken into\nconsideration. Using the Arrhenius equation (Eq. (3)) we are\nable to extrapolate the crystallization rate at any temperature.\nTo that end, the logarithm of each rate constant is plotted over\nthe inverse temperature. The linear dependency of Eq. (3) in\nthe logarithmic plot allows us to extract the activation energy\nand the pre-factor k0for YIG on each substrate. The resulting\nvalues are plotted in Tab. I. While at first glance the crys-\ntallization velocity for YIG on SiO xseems faster, the differ-\nent annealing temperatures of 600◦C for the garnet substrates\nand 700◦C for SiO xneed to be taken into account (cp. Fig.4).\nExtrapolating the crystallization velocity for YIG on GGG at7\n700◦C reveals that here YIG would crystallize approximately\n30 times faster than on SiO x.\nOur activation energy of 3 .98 eV for YIG on GGG is in\ngood agreement with the literature. For the formation of bulk\nYIG from oxide powders, a value of 5 .08 eV was reported.70\nFurther, for the crystallization of bulk polycrystalline YAG,\nwhich is expected to behave similarly as it has the same crys-\ntal structure, an activation energy of 4 .5 eV was found.71The\nlower value of 3 .98 eV for YIG on GGG highlights the re-\nduced energy needed, due to the SPE from the lattice matched\nGGG.\nThe activation energies for YIG on YAG as well as on SiO x\nare much higher than the value on GGG. As the general crys-\ntallization windows and times needed for a fully crystalline\nfilm stay the same, we ascribe this behavior to a kinetic block-\ning, originating from the lattice mismatch and the nucleation.\nUnderstanding the exact mechanism however, would need fur-\nther study.\nThese results allow to establish a diagram to underline\nwhich annealing parameters will lead to a fully crystalline\nYIG thin film on the three substrates (cp. Fig. 5(a)). For a\nmathematical description, we combine the Avrami equation\nEq. (2) with the Arrhenius equation Eq. (3) to be able to\nexpress the crystallinity in terms of annealing time and tem-\nperature.\nt=\u0012\u0014\n−ln(1−θc)\nk0\u0015\n·eEA\nkBT\u00131\nn\n(4)\nWe use a crystallinity θcof 0.999 to avoid the divergence of\nthe logarithm and the respective n,k0andEAfound in Tab. I.\nFigure 5(a) outlines the temperature and time combination\nwhere crystalline YIG (shaded areas) can be obtained. Re-\ngions where the YIG thin film remains amorphous are left in\nwhite. The boundary between non crystalline and crystalline\nfor each substrate is given by Eq. (4). Each of the circles seen\nin Fig. 5(a) represents one fully crystalline sample obtained as\ndescribed for Fig. 3(b). The filled circles represent fully crys-\ntalline samples, where no time dependence of the crystallinity\nwas measured. As already anticipated, YIG exhibits different\ncrystallization behavior depending on the substrate. Note, that\nthe formation of polycrystalline YIG on SiO xor any arbitrary\nsubstrate needs notably higher temperatures than SPE, where\nan annealing at 660◦C for 100 h would be necessary to result\nin a fully crystalline film.\nThe different temperatures and times necessary to induce\ncrystallization stem from the different types of substrates. For\nTab. I. Extracted activation energies EAand pre-factors k0for YIG\non each substrate\nEA(eV) k0(1/h) n\nYIG on GGG 3 .98±0.32 2 .0·10231\nYIG on YAG 15 .70±1.59 2 .6·10893.8\nYIG on SiOx 16.37±0.85 8 .4·10804YIG on GGG and YAG the seed for the crystallization is given\nby the lattice of the substrate. Therefore, we ascribe the dis-\ncrepancy between YAG and GGG to the different lattice mis-\nmatch compared to YIG. In the YIG thin films on YAG a\nhigher strain is expected to exist in the film, which leads to\nthe formation of energetically costly dislocations. This in turn\nresults in the slightly higher temperature needed for YIG to\ncrystallize on YAG. On SiO x, however, a significantly higher\ntemperature than for the lattice matched substrates is needed\nfor crystalline YIG to form. Here, as no initial seed is given\nby the substrate, nucleation is required, which is a thermally\nactivated process that needs additional energy, i.e. higher tem-\nperatures. This random formation of seeds leads to a polycrys-\ntalline YIG thin film on SiO x\nA comparison with the literature shows, that parame-\nters which have been previously reported to result in a\nfully crystalline YIG layer, fit into our extracted area, (cp.\nFig. 5(b)).24–33,35–41Additionally to the sputtered films, also\namorphous films obtained from PLD with subsequent anneal-\ning fit in the observed regions.11–17The extracted diagram in\nFig. 5 therefore acts as a general description for the crystal-\nlization of YIG thin films out of the amorphous phase.\nIV. CONCLUSION\nExtensive time and temperature series were used to ana-\nlyze the crystallization kinetics of sputtered amorphous YIG\nthin films on different substrates. We find the formation of\nsingle crystalline YIG thin films on garnet substrates where\nthe crystallization on gadolinium gallium garnet can be co-\nherently described in a solid phase epitaxy picture, whereas a\nmore complicated crystallization scheme is found on yttrium\naluminum garnet. On SiO xa polycrystalline YIG thin film\ndevelops, with slower crystallization dynamics than for the\ngarnet substrates.\nA fully crystalline YIG film on GGG was found for tem-\nperatures as low as 537◦C and annealing times of 110 h. On\nsilicon oxide (representing any type of amorphous or non lat-\ntice matched substrate), the nucleation of the YIG crystals is\nnot expected for reasonable time scales below 660◦C. The\nresults summarized in Tab. 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Res. 16, 1795 (2001)." }, { "title": "2309.03116v1.Strong_magnon_magnon_coupling_in_an_ultralow_damping_all_magnetic_insulator_heterostructure.pdf", "content": "1 \n Strong magnon -magnon coupling in an ultralow damping all-magnetic -insulator heterostructure \nJiacheng Liu*1,5, Yuzan Xiong*2, Jingming Liang*3, Xuezhao Wu1,5, Chen Liu4, Shun Kong Cheung1,5 \nZheyu Ren1,5, Ruizi Liu1,5, Andrew Christy2, Zehan Chen1,5,6, Ferris Prima Nugraha1,5, Xi-Xiang Zhang4, \nChi Wah Leung3, Wei Zhang#2, Qiming Shao#1,5,6 \n1Department of Electronic and Computer Engineering, The Hong Kong University of Science and \nTechnology, Hong Kong SAR \n2Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC \n27599, USA \n3Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong SAR \n4 Physical Science and Engineering Division (PSE), King Abdullah Universi ty of Science and Technology \n(KAUST), Thuwal 23955 –6900, Saudi Arabia \n5IAS Center for Quantum Technologies, The Hong Kong University of Science and Technology, Hong \nKong SAR \n6Department of Physics, The Hong Kong University of Science and Technology, Hong K ong SAR \n* Equal contributions # Corresponding emails: zhwei@unc.edu ; eeqshao@ust.hk \n \nMagnetic insulators such as y ttrium iron garnets (YIGs) are of paramount importance for spin-wave \nor magnonic devices as their ultralow damping enables ultra low power dissipation that is free of \nJoule heating , exotic magnon quantum state, and coherent coupling to other wave excitations. \nMagne tic insulator heterostructures bestow superior structural and magnetic properties and house \nimmense design space thanks to the strong and engineerable exchange interaction between individual \nlayers. To full y unleash their potential, realizing low damping and strong exchange coupling \nsimultaneously is critical , which often requires high quality interface . Here, we show that such a \ndemand is realized in an all-insulator thulium iron garnet (TmIG) /YIG bilayer system. The ultralow \ndissipation rates in both YIG and TmIG , along with their significant spin-spin interaction at the \ninterface, enable strong and coherent magnon -magnon coupling with a benchmarking cooperativity \nvalue larger than the conventional ferromagnetic metal -based heterostructures . The coupling \nstrength can be tuned by varying the magnetic insulator layer thickness and magnon mode s, which \nis consistent with analytical calculations and micromagnetic simulatio ns. Our results demonstrate \nTmIG/YIG as a novel platform for investigating hybrid magnonic phenomena and open \nopportunities in magnon devices comprising all-insulator heterostructures. \n \nSpin-wave (or magnonic ) devices utilize magnon spin degree of freedom to process information, which can \noccur in magnetic insulators free from any charge current , and therefore, are promising contenders for \nultralow -power functional circuits 1–4. Magnetic garnets such as yttrium iron garnet ( Y3Fe5O12, YIG) have \nan ultralow damping factor , and they have enabled long magnon spin transmission 5, efficient magnon spin \ncurrent generation 6, and magnon logic circuits 2,3. Another type of magnetic garnet, thulium iron garnet \n(Tm 3Fe5O12, TmIG), has been engineered to a binary memory with a robust perpendicular magnetic \nanisotropy 7,8. Besides, TmIG thin films can exhibit topological magnetic skyrmion phase 9,10, promising \nfor future magnetic insulator -based racetrack memory devices. In addition to these promising practical 2 \n applications, magnetic insulators are well-known for hosting novel quantum phases such as Bose –Einstein \ncondensate11, spin superfluidity12, and topological magnonic insulators13. \nMagnetic heterostructures can provi de more functionalities and richer properties because exchange \ninteractions between different layers provide another control knot 14,15. While ferromagnetic metal -based \nheterostructures have been extensively studied and applied in commercial devices such as magneto -resistive \nrandom -access memor y 15, magnetic insulator -based heterostructures are still on the horizon yet already \nshowcased a few promises, including strong interfacial coupling s 16–20, magnon valve effects21–23, control \nof magnon transport in the magnetic insulator layer using another magnetic layer24,25, magnonic crystal 26, \ncoherent magnon -magnon coupling 27–30, and topological spin textures 10. Magnetic insulator \nheterostructures are also theoretically predicted to host exotic quantum phase such as magnon flat band 31. \nHowever, to date, coherent magnon -magnon coupling has only been studied in hybrid systems consisting \nof a low damping YIG and another ferromagnetic metal 27–30. The d emonstration of low damping and strong \ncoherent coupling in purely magneti c insulator bilayers is lacking. \nIn this work, we demonstrate ultralow damping and strong magnon -magnon coupling in a TmIG/YIG \nheterostructure. We characterize the structural and magnetic properties of our TmIG/YIG heterostructures \non gadolinium gallium garnet (Gd 3Ga5O12, GGG) using high-angle annular dark -field scanning \ntransmission electron micro scopy (HAADF -STEM ), X-ray diffraction (XRD), and vibrating sample \nmagnetometry (VSM) . Then, we investigate the magnetic dynamics in these bilayers by using a broadband \nferromagnetic resonance (FMR) technique . We observe a strong coupling between the Kittel mode of YIG \nand perpendicular standing spin wave (PSSW) mode of TmIG. By matching t he experimental FMR spectra \nwith analytical calculations and micromagnetic simulations, we obtain the exchange coupling strength at \nthe interface, which is dependent on the magnetic insulator layer thickness and coupling mode. Finally , we \nbenchmark the di ssipation rates and cooperativity in our samples against these in ferromagnetic metal -based \nheterostructures. \nWe prepare our TmIG/YIG on GGG substrates using pulsed laser deposition (see Methods). Atomic images \nfrom HAADF -STEM show a single crystallinity and perfect interfaces at the YIG/GGG and TmIG/YIG \nboundaries (Fig. 1a). Elemental mapping (Fig. 1b) proves there is no inter diffusion between different layers. \nFig. 1c presents the high -resolution XRD spectra of TmIG/GGG, YIG/GGG, and TmIG/YIG/GGG bilayer \nfilms measured with the scattering vector normal to the <001> oriented cubic substrate. Along the sharp \n<004> peaks from the GGG substrate, the XRD spectra shows Laue oscillations, indicating a smooth \nsurface and interface. We also measured the magnetic hysteresis loops for YIG, TmIG, and TmIG/YIG \nsamples to quantify their saturation magnetizations (see Supplementary Note 1). In pr inciple, e xchange \ncoupling strength (J) between different layers can be estimated from major and minor hysteresis loops 15. \nWe can estimate the interfa cial J at the CoFeB(50 nm)/ TmIG(350 nm) interface is −0.031 mJ/m2, \nindicating an antiferromagnetic exchange coupling (see Supplementary Note 1). However, YIG and TmIG \nhave very similar coercive fields, preventing us from obtaining the coupling strength directly from the \nhysteresis loop measurements. \nWe measure the magnetization dynamics in TmIG(200 nm)/ YIG(200 nm) bilayers using a field modulated \nFMR technique (see Methods). We mount the sample on a coplanar waveguide and apply a microwave \ncurre nt that generates radiofrequency magnetic fields (Fig. 2a) . The absorption coefficient exhibits a peak \nwhen the FMR conditions for YIG and TmIG are met (Fig. 2b) . We experimentally extract the resonance \nfrequency at a specific field by fitting the frequenc y scan at the field using Lorentz functions (Fig. 3a). In \naddition to regular FMR peaks, we also observe anti -crossing at specific field-frequency points, which are \nsignatures of exchange interaction -driven coupling of Kittel mode in YIG and PSSW modes in TmIG. To 3 \n identify the underlying magnon modes responsible for the coupling, we list the formula of generalized \nexcited spin wave modes in two layers (𝜔𝑖\n2𝜋 𝑜𝑟 𝑓𝑖): \n𝜔𝑖\n2𝜋=𝑓𝑖=𝛾𝑖\n2𝜋√(𝜇0𝐻ext+2𝐴ex,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2)(𝜇0𝐻ext+2𝐴ex,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝜇0𝑀𝑠,𝑖), (1) \nwhere i=YIG or TmIG, 𝛾𝑖\n2𝜋=(𝑔eff,i/2)×28 GHz /T is the gyromagnetic ratio, 𝜇0 is the permeability, 𝐻ext \nis the external field, 𝑀𝑠 is the effective magnetization, 𝐴ex is the exchange stiffness, and 𝑘 is the \nwavevector of the excited spin wave. Note that if there is no exchange interaction between YIG and TmIG, \n𝑘=𝑛𝜋\n𝑑, where n is an integer and 𝑑 is the thickness of the magnetic insulator . By fitting the Kittel mode \nwith n=0, we get 𝑔eff,YIG=2 (𝜇0𝑀𝑠,𝑌𝐼𝐺=0.25 T) and 𝑔eff,TmIG =1.56 (𝜇0𝑀𝑠,𝑇𝑚𝐼𝐺 =0.24 T) for YIG \nand TmIG , respectively, which are consistent with the previous report 34. Then, by assuming zero exchange \ninteraction and matching 𝜔YIG=𝜔TmIG from Eq. (1), we can understand the first (second) anti -crossing \nshown in Fig. 2b is from the coupling between n=0 mode in YIG and n=1 (n=2) mode in TmIG. A schematic \nof n=0 mode in YIG and n=1 mode in TmIG is shown in Fig. 2a. In addition, we determine t he exchange \nstiffness of the TmIG to be 2.69 pJ/m, which is consistent with the previous report 35. When there is an \nexchange interaction between YIG and TmIG, we expect an anti -crossing gap , whic h can be described by \nthe minimum frequency separation of 2g. However, with only Eq. (1) the relation between the exchange \ninteraction and the g value cannot be uniquely determined . \nTo fully understand the exchange coupling -driven magnon -magnon coupling, we perform the \ncomprehensive numerical analysis and micromagnetic simulations (see Method s). We consider the \nboundary conditions at the interface and two surface s of the TmIG/YIG bilayers and arrive at the formula \n(see Supplementary Note 2) : \n2𝐴ex,YIG\n𝑀s,YIG𝑘YIGtan(𝑘YIG𝑑YIG)∙2𝐴ex,TmIG\n𝑀s,TmIG𝑘TmIG tan(𝑘TmIG 𝑑TmIG )=\n2𝐽\n𝜇0(𝑀𝑠,YIG+𝑀𝑠,TmIG )[2𝐴ex,YIG\n𝑀s,YIG𝑘YIGtan(𝑘YIG𝑑YIG)+2𝐴ex,TmIG\n𝑀s,TmIG𝑘TmIG tan(𝑘TmIG 𝑑TmIG )], (2) \nwhere J is the interfacial exchange coupling strength. By solving 𝜔YIG=𝜔TmIG from Eq. (1) and Eq. (2) \ntogether , we can get a set of ( 𝑘YIG, 𝑘TmIG ) values that correspond to different modes. In the presence of \nexchange interaction, 𝑘 will not be precisely equal to 𝑛𝜋\n𝑑 anymore . As a result, the degeneracy is lifted at \nthe crossing point and we have two frequencies corresponding to two ( 𝑘YIG, 𝑘TmIG ) values . By employing \n𝐽=−0.057 mJ/m2 (see Supplementary Table 1) , we have obtained high consistency between the \nexperimental and calculated spectra of field-frequency points in the entire range (Fig. 3a). The negative \nsign suggests an antiferromagnetic exchange coupling between TmIG and YIG. The strength is also \ncomparable with the ferromagnetic metal/YIG bilayers 27. We have also carried the FMR measurement on \nthe reference TmIG(350 nm)/CoFeB(50 nm) sample (see Supplementary Note 4). We get 𝐽=\n−0.032 𝑚J/m2, which is close to the result from the VSM loop measurements. This consistency suggests \nthat we can reliably extract 𝐽 values of TmIG/YIG samples from the FMR measurement. \nWe further study the thickness and mode dependence of anti -crossing gap (2 g). We extract the g value from \nthe frequency scan , for example, g = 85 MHz for the TmIG(200 nm)/YIG(200 nm) bilayer ( Fig. 3b). We \nfind the gap reduces as the layer thickness increases (Fig. 3c). To understand this, we derive the approximate \nsolution (see Supplementary Note 3) : 4 \n 𝑔≈𝛾𝑌𝐼𝐺𝛾𝑇𝑚𝐼𝐺\n4𝜋2𝐽\n(𝑀𝑠,𝑌𝐼𝐺+𝑀𝑠,𝑇𝑚𝐼𝐺 )∙√(2𝜇0𝐻res+𝜇0𝑀𝑠,𝑌𝐼𝐺)(2𝜇0𝐻res+𝜇0𝑀𝑠,𝑇𝑚𝐼𝐺 )\n𝑓𝑟𝑒𝑠∙1\n√𝑑𝑌𝐼𝐺𝑑𝑇𝑚𝐼𝐺, (3) \nwhere 𝜔res and 𝐻res are the resonance frequency and field in the gap center, respectively. The calculated \nresults show the same trend as in the experiment s (Fig. 3c). Also, Eq. (3) allows us to analyze the g valu e \nfor coupling of the YIG Kittel mode to different TmIG PSSW modes. We compare the experimental and \ncalculated g values for the coupling of n=0 mode in YIG and n=2 mode in TmIG in Fig. 3c, where we \nconfirm that the higher mode coupling results in a lower g in our case. \nFinally, t o evaluate the coupling cooperativity in TmIG/YIG bilayers , we have determine d the individual \ndissipation rates. We first get Gilbert damping factors for YIG and TmIG from field scans at different \nfrequencies when they are not coupled (see Supplementary Note 4). The extracted damping factors are \nplotted in Fig. 3d, where we find a damping factor as low as 4.91 (± 0.79) × 10-4 in the 350 nm -thick TmIG. \nWe also extract the dissipation rates for YIG and TmIG from f requency scans at different fields when they \nare not coupled (see Supplementary Note 4). As an example, 𝜅𝑌𝐼𝐺=10 𝑀𝐻𝑧 and 𝜅𝑇𝑚𝐼𝐺 =29.5 𝑀𝐻𝑧 for \nthe TmIG(200 nm)/YIG(200 nm) bilayer. Therefore, 𝑔>𝜅𝑌𝐼𝐺,𝜅𝑇𝑚𝐼𝐺 , and 𝐶=𝑔2\n𝜅𝑌𝐼𝐺𝜅𝑇𝑚𝐼𝐺=24.5, \nconcluding a strong coupling in the bilayer. In Fig. 4, we summarize the dissipation rates and cooperativity \nfor TmIG - and ferromagnetic metal -based heterostructures that show magnon -magnon coupling. The TmIG \nhas a very low dissipation rate compared to ferromagnetic metals, which is consistent with the ultralow \nGilbert damping . \nIn summary, we demonstrate ultralow damping and dissipation rates in the TmIG and achieve strong \nmagnon -magnon coupling and high cooperativity in the TmIG/YIG bilayers. The combined experimental \nand theoretical analyses allow us to determine the interfacial exchange coupling strength s in our all-\ninsulator bilayers . The all-magnetic -insulator bilayers allow us to achieve ultralow damping insulati ng \nsynthetic antiferromagnets, magnonic crystals, and other artificial structures to realize energy -efficient spin \nwave devices. Besides, the strong coupling between two distinct magnetic insulators with ultralow damping \nallows to explore the novel quantum phases, such as topological magnon insulators and magnon flatband . \n 5 \n Figures and Captions \n \nFigure 1. Structural characterizations of YIG /TmIG heterostructures on GGG substrates. a, High-\nangle annular dark -field scanning t ransmission electron micro scopy (HAADF -STEM ) image for the \nYIG/TmIG heterostructure on GGG substrates . The YIG/GGG and YIG/TmIG interfaces are denoted by \nthe green and red lines in partial enlarged figure, respectively. Pt is used as a capping layer to pre vent \ndamage when preparing TEM samples. The inset of a demonstrates that two 1/8 of unit cells corresponding \nto YIG and TmIG at the interface, which have a garnet -type structures (𝐶)3[𝐴]2(𝐷)3𝑂12. b, Energy \ndispersive x -ray (EDX) spectra of different elements in the TmIG/YIG/GGG heterostructure. The images \nwere taken along the <100> direction of the GGG substrate, and the distribution of elements is marked in \nthe figure with color, respectively. c, High-resolution of X-ray di ffraction spectr a for the YIG(100 \nnm)/GGG, TmIG(100 nm)/GGG, and TmIG(100 nm)/YIG(100 nm)/GGG samples . \n \n26 27 28 29 30 31\n2q (degrees) TmIG(100nm)XRD Intensity (arb.units) YIG(100 nm) TmIG(100 nm)/YIG(100 nm)\n¨\n·\n§¨ GGG\n· TmIG\n§ YIG\nTmIG\nYIG\nGGG\nGGGYIGYIGTmIG\nPt(a) (c)\n(b)\nYIGTmIG\nTm\nFe\nY\nMd\nMd\nMaMa Mc\nO2-\nC-site Y3+\nA-site Fe3+D-site Fe3+C-siteTm3+\nHAADF Pt Tm Y O Fe Ga Gd\n50 nm 50 nm 50 nm 50 nm 50 nm 50 nm 50 nm 50 nm6 \n \nFigure 2. Schematic diagram of the spin waves in the heterostructure and the measured resonance \nspectra . a, Schematic illustration of the measurement set -up, where ℎ𝑟𝑓 and 𝐻𝑒𝑥𝑡 stand for the microwave \nmagnetic field and external static magnetic field, respectively. Spin -wave spectra are obtained by placing \nthe sample face -down on a coplanar waveguide (CPW ). The inset depicts the Kittel uniform spin wave \nmode in the YIG and the perpendicular standing spin wave (PSSW) mode in the TmIG . b, Experimentally \ncolor -coded spin -wave absorption spectra of the YI G(200 nm)/TmI G(200 nm) for the first three resonance \nmodes of TmIG (n=0, 1, 2) and the uniform mode of YIG (n=0). \n \ny\nxzHext\n(a) (b)\n0 200 400 6001.02.03.04.0\nm0Hr (Oe)w/2p (GHz)-3E-4 -2E-4 -3E-5 9E-5 2E-4\n2g=\n0.06GHz\n2g=\n0.16GHzAmplitude (V)\n0 20 40 601.02.03.04.0\nm0Hext (mT)w/2p (GHz)-3E-4 -2E-4 -3E-5 9E-5 2E-47 \n \nFigure 3. Observation of s trong magnon -magnon coupling and ultralow damping in YIG/TmIG \nbilayers . a, Resonant absorption peaks of the two hybrid modes as a function of external magnetic field \nwith the YIG(200 nm)/TmI G(200 nm) bilayer. Solid curves show the numerical theory method fitting as \nhybrid modes. Data points are extracted from experimental data by reading out the minimum of each \nresonant peak from Fig. 2(b). b, Spin wave spectra at minimum resonance separation ( 𝜇0𝐻𝑒𝑥𝑡=10 mT) \nin magnetic insulator bilayers with the YIG(200 nm)/TmI G(200 nm) bilayer . c, Coupling strength g \nbetween TmIG (n=1,2) mode and YIG (n=0) mode as a function of the YIG thickness. Red circles are \nexperimental results and black squares are from the oretical calculations . Red dots: Experiments. d, \nThickness dependence of Gilbert damping factors of YIG and TmIG in the YIG/TmIG bilayers . \n \n0 20 40 601.02.03.04.0\n1.0 1.2 1.4 1.6 1.8-15-10-505\n012345\n0.040.080.120.160.20w/2p (GHz)\nm0Hext (mT)dots: Experiment\ndash line: Fitting\nAmplitude (V)\n0.030.060.090.120.150.18\ndTmIG/dYIG (nm)100/100 140/140 200/200TmIG (n=2) couple YIG (n=0)TmIG (n=1) couple YIG (n=0)Frequency (GHz)2g m0Hext:\n10 mT200nm/200nm YIG/TmIG\nsignal ´105\n1.56-1.40=2g\nk1=0.032/2\nk2=0.04821 /2\nC=16.59 Experiment\n Lorentz Fitsignal ´10-5\n0.17 GHzaYIG\nThickness (nm) TmIG\n YIG´10-3\n012345\naTmIG´10-3\n100 140 200 350g (GHz) Experiment\n Calculation(a) (b)\n(c) (d)8 \n \nFigure 4. Summary of dissipation rates in TmIG and ferromagnetic metals versus cooperativities in \nTmIG - and ferromagnetic metal -based heterostructures. Star and square symbols are dissipation rates \nfor TmIG and ferromagnetic metals, respectively. 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Tunable Magnon -Magnon 11 \n Coupling Mediated by Dynamic Dipolar Interaction in Synthetic Antiferromagnets. Phys. R ev. \nLett. 125, 017203 (2020). \n39. Wang, H. et al. Hybridized propagating spin waves in a CoFeB/IrMn bilayer. Phys. Rev. B 106, \n064410 (2022). \n40. Hayashi, D. et al. Observation of mode splitting by magnon –magnon coupling in synthetic \nantiferromagnets. Appl . Phys. Express 16, 053004 (2023). \n \n 12 \n Methods \nMaterial growth \nHigh -quality epitaxial YIG (lattice constant 12.376 Å), TmIG (lattice constant 12.324 Å), and YIG/TmIG \nthin films were deposited on <100> -oriented GGG (lattice constant 12.383 Å) single -crystal substrates \nusing pulsed laser deposition (PLD). Prior to deposition, the substrates were cleaned with acetone, alcohol, \nand deionized water, followed by annealing in air at 1000° C for 6 hours. The films were deposited at 710 °C \nin an oxygen atmosphere of 100 mTorr, with a base pressure of better than 2×10−6 𝑚𝑇𝑜𝑟𝑟 , using a 248 \nnm KrF excimer laser with a repetition rate of 10 Hz. In -situ post -anneali ng was carried out at the deposition \ntemperature for 10 minutes in 10 Torr oxygen ambient, followed by natural cooling to room temperature. \n \nMaterial characterizations \nThe cross -sectional (TmIG/YIG/GGG) TEM lamella with a thickness of ~70 nm was fabricated by the \nfocused ion beam (FIB) technique in a Thermofisher Helios G4 UX dual beam system. The atomic \nresolution high -angle annular dark -field scanning transmission electron microscopy (HAADF -STEM) \nimages viewed along <100> orientation and the EDS data were obtained using FEI Titan Themis G2 TEM \nwith a probe corrector at an acceleration voltage of 300 kV. The C2 aperture is 70 µ m and the camera length \nis 115 mm, corresponding to a probe convergence semi -angle of 23.9 mrad. The image filtering and EDS \nmapping analysis were processed using the Velox software. \nFilm thickness was measured using Surface Profiler (Bruker DektakXT). The microstructure of the samples \nwas analyzed using X -ray diffractometry (SmartLab, Rigaku Co.) with Cu Ka1 radiation (λ = 1.5406 Å). \nMagnetic properties were assessed with vibrating sample magnetometer s (Physical Property Measurement \nSystem , Quantum Design , or Lakeshore ). \n \nResonance studies \nThe magnetization dynamics measurement was performed using the field -modulation FMR technique at \nroom temperature. During the measurement, the GGG/YIG/TmIG sample was mounted in the flip -chip \nconfiguration (TmIG side facing down) on top of the signal -line of a coplanar waveguide for broadband \nmicrowave excitation. An external bias field, H, was applied in -plane perpendicular to the rf field of the \nCPW. We used a modulation frequency of Ω/2π = 81.57 Hz (supplied by a lock -in amplifier and provided \nby a pair of modulation coil) and a modulation field of about 1.1 Oe. The microwave signal was delivered \nfrom a signal generator (0 - 5 dBm) to one port of the board. The field -modulated FMR signal was measured \nfrom the other port by the lock -in amplifier in the form of a dc voltage, V, by using a sensitive rf diode. We \nswept the bias field, H, and at each incremental frequency f, to construct the V[f, H] dispersion contour \nplots. \n \nMicromagnetic simulations \nOur finite -element model implements coupled LLG equation with antiferromagnetic interfacial exchange \ninteraction in magnetic insulator heterost ructure in order to simulate the strong magnon -magnon coupling \nin frequency -domain. The technical details are provided in Supplementary Note 2. 13 \n \nAcknowledgements \nThe authors appreciate insightful discussions with S. S. Kim, Y. Tserkovnyak , Z. Zhang , A. Com stock, D. \nSun, Y. Li . The sample fabrication, structural characterization, and data analysis at HKUST were supported \nby National Key R&D Program of China (Grants No.2021YFA1401500). The magnetic dynamics \nmeasurement and analysis were supported by the U.S. National Science Foundation under Grant No. ECCS -\n2246254. The thin film deposition work in PolyU through the GRF grant 15302320 . The authors also \nacknowledge support from RGC General Research Fund (Grant No. 16303322), State Key Laboratory of \nAdvanced Displays and Optoelectronics Technologies (HKUST), and Guangdong -Hong Kong -Macao Joint \nLaboratory for Intelligent Micro -Nano Optoelectronic Technology (Grant No. 2020B1212030010). \n \nData availability \nThe data that support the plots within this paper and other findings of this study are available at XXX. \n(Authors’ note: the data will be uploaded after the acceptance of this manuscript.) \n \nAuthor contributions \nW. Z. and Q. S. conceived the idea. J. L. did the VSM, XRD, partial FMR measurements, and data analysis \nwith help from X. W., S. K. C., Z. R., R. L., Z. C., F. P. N. , and S. K. Kim . Y. X. and W.Z. did FMR \nmeasurements with help from A. C. J. L. and D. C.W. L. grew the samples. C. L. and X.X. Z. did the TEM. \nJ. L. and Q. S. and W. Z. wrote the manuscript with help from other co-authors. \n \nCompeting interests \nThe authors de clare that they have no competing financial interests. \n \n \n \n \n1 \n Supplementary Information \nJiacheng Liu, et al. \n2 \n Supplementary Note 1. Hysteresis loops by vibrating sample magnetometer (VSM) \nmeasurement for Tm IG, YIG , and TmIG/Y IG samples and TmIG, CoFeB, and \nTmIG /CoFeB samples \nIn principle , due to the existence of antiferromagnetic coupling at the heterostructure interface , \nwe could also measure the major and minor hysteresis loop s through VSM to estimate the \ninterfac ial coupling energy (J). We prepared 2 series of samples to do a comparison , [TmIG( 100 \nnm), YIG(100 nm), TmIG(100 nm)/YIG (100 nm)) and TmIG (350 nm), CoFeB(50 nm), \nTmIG(350 nm)/CoFeB(50 nm)]. Due to the extremely similar coercive fields of YIG and TmIG , \nwe cannot get the J directly from hysteresis loop data in Supplementary Fig. 1. Then , we made \nanother series of samples ( TmIG, CoFeB, and TmIG/CoFeB ) for comparison. Measuring the \nhysteresis loops of TmIG(350 nm) /CoFeB(50 nm) and individual layer s in Sup plementary Fig. \n2a, we could clearly observe that the process of magnetization flipping (black solid line) from \nTmIG(350 nm)/CoFeB(50 nm), which is caused by the different coercivity between TmIG \n(light blue solid line) and Co FeB (red solid line). We noticed that no sharp switching (like \narrow points A to B to C) of the CoFeB layer is visible but a smooth increase (like arrow points \nA to C) of the measured magnetic moment until the bilayer magnetization is saturated. This \ncould be explained by a direct interfacial exchange coupling between TmIG and CoFeB \nmagnetizations 321-3. Process (1 -5) in Supplementary Figs. 2b -c show a possible magnetization \nflipping process in an exchange coupled heterostructure at an exte rnal magnetic field . \nSubsequently, to quantify the interfacial exchange coupling field and energy , we measured the \nminor hysteresis loops of TmIG(350 nm)/CoFeB(50 nm) in Supplementary Fig. 3a . In an ideal \nstate, (2) and (3) state is similar, but the difference is that the moment of (3) is about to flip the \nspin of the Tm IG layer, and the moment of (2) has not yet reached the flip condition. The -\nminor loop measured from the experiment is a C -A-B-C loop (Supplementary Fig. 3b) , \nindicating the existence of antiferromagnetic interfacial coupling . If there is no interfac ial \ncoupling, the loop will be like E -A-D-E, because it only depends on its coercivity of TmIG , \nthat is, the forward and reverse min or loops should basically coincide (just like a single TmIG \n3 \n hysteresis loop). The shift from the minor hysteresis loops is precisely because of the interfac ial \nexchange coupling that the two layers are in an antiferromagnetic relationship such that the \nextra 𝐻𝑒𝑥 want s to make spins parallel, which is why the flip ping (3) will be performed faster \nby (𝐻𝑒𝑥−𝐻𝑒𝑥𝑡). So, we can estimate the interfacial exchange coupling J value from the minor \nloops: 𝐽=𝜇0𝐻𝑒𝑥×∆𝑀𝑠\n𝑉×𝑡=−0.394×10−3(𝑇)∙1.975×105(𝐴/𝑚)∙400×10−9(𝑚)=\n−0.03113 𝑚𝐽/𝑚2, where 𝐻𝑒𝑥 is exchange field induced by the interfacial coupling, ∆𝑀𝑠 is \nthe value of magnetization reduction in the shad ow region (C-B-D-E loop) for additional \nexchange filed , 𝑉=(350𝑛𝑚+50𝑛𝑚)∙5𝑚𝑚∙5𝑚𝑚 is volume of the heterostructure, t is \nthickness of the sample. After the analysis of the interfacial exchange coupling by the m inor \nloops, we also extract the interfacial coupling energy of the TmIG (350nm)/CoFeB (50nm) \nsample by ferromagnetic resonance ( FMR ) measurement , which is described in detail in \nSupplementary Note 3. \n \nSupplementary Note 2. Numerical analysis and micromagnetic simulation for the strong \nmagnon -magnon coupling induced by the interface exchange interaction \nNumerical analysis \nTo solve the eigenfrequency of the anti -crossing curve under the strong magnon -magnon \ncoupling (Fig. 3a) , we conduct numerical analysis and micromagnetic simulations on the \nbilayer heterostructure following the method in ref erence s 4,5. We assume that magnetic \ninsulator layer 1 (MI1) describes index 1, and magnetic insulator layer 2 (MI2) describes index \n2. The interface of the MI1 and MI2 is at z=0 (Supplementary Fig. 4a). \nDispersion relation : for the magnetic system, Landau -Lifshitz -Gilbert ( LLG) equation [Eq. \n(S1)] describe s the intrinsic precess ion of the spin in the materials : \n𝜕𝑴𝑖̇⃗⃗⃗⃗⃗ \n𝜕𝑡=−𝜇0𝛾𝑖𝑴𝑖̇⃗⃗⃗⃗⃗ ×𝑯𝑒𝑓𝑓,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ +𝛼𝑖\n𝑀𝑠,𝑖(𝑴𝑖̇⃗⃗⃗⃗⃗ ×𝜕𝑴𝒊̇⃗⃗⃗⃗⃗ \n𝜕𝑡), (S1) \n4 \n where whole magnetic insulator bilayers subjected to the static magnetic field 𝑯𝒆𝒙𝒕⃗⃗⃗⃗⃗⃗⃗⃗ in the x \ndirection and the dynamic magnetic field 𝒉𝒓𝒇⃗⃗⃗⃗⃗⃗ generated by the coplanar waveguide (CPW) in \nthe y direction. Since the dynamic magnetic field induced by the microwave is so small \n|𝒉𝒓𝒇⃗⃗⃗⃗⃗⃗ |≪|𝑯𝒆𝒙𝒕⃗⃗⃗⃗⃗⃗⃗⃗ | that precessing amplitude of spin moment can be divided into static and \ndynamic parts: \n𝑴𝑖̇⃗⃗⃗⃗⃗ =𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ),𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ =[1\n0\n0],𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =𝛿𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ 𝑒𝑗𝜔𝑡=[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖],(S2) \nwhere 𝒎0⃗⃗⃗⃗⃗⃗ is normalization constant modulo 1, which is the static expression of spin moment . \n𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ is micro -perturbations induced by microwave fields, and its amplitude is much smaller \nthan |𝒎0⃗⃗⃗⃗⃗⃗ |. 𝑯𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ include s an external magnetic statistic magnetic field 𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ [Eq. (S3)] in x \ndirection, dynamic magnetic field 𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ , |𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ |≪|𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ |,𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ ⊥𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ , and demagnetization \nfield 𝑯𝑑𝑒,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ [Eq. (S4)]. In particular, the sample is considered as a thin film, so 𝑁𝑥=𝑁𝑦=\n0,𝑁𝑧=1. \n𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ =[𝐻𝑥\n0\n0],𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ∥𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ , (S3) \n𝑯𝑑𝑒,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ =−𝒩∙𝑴𝑖̇⃗⃗⃗⃗⃗ =−𝑀𝑠,𝑖[𝑁𝑥00\n0𝑁𝑦0\n00𝑁𝑧](𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )=−𝜇0𝑀𝑠,𝑖[0\n0\n𝛿𝑚𝑧,𝑖],(S4) \nSubstitute 𝑴𝑖̇⃗⃗⃗⃗⃗ and 𝑯𝑒𝑓𝑓,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ into Eq. (S1): \n𝑀𝑠,𝑖𝜕(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )\n𝑑𝑡=−𝜇0𝛾𝑖𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )×(𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ +𝑯𝑑𝑒⃗⃗⃗⃗⃗⃗⃗ +𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ ) \n+𝛼𝑖\n𝑀𝑠,𝑖𝑀𝑠,𝑖(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )×𝑀𝑠,𝑖𝜕(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ )\n𝜕𝑡, (S5) \nwhere 𝜕𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗⃗ \n𝜕𝑡=0, and 𝜕𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ \n𝑑𝑡=𝑗𝜔𝛿𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ 𝑒𝑗𝜔𝑡=𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ . In Eq. (S5) , 𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ =0 is due to \n5 \n the magnetization and the static component of the external magnetic fiel d are parallel to each \nother , 𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ =0 is due to the second -order epsilon, and 𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝜕𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ \n𝜕𝑡=𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×\n𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =0. So, Eq. (S5) could be simplified to: \n𝑗𝜔𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ =−𝜇0𝛾(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝑯𝑑𝑒⃗⃗⃗⃗⃗⃗⃗ +𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝒉𝑟𝑓⃗⃗⃗⃗⃗⃗ +𝛿𝒎𝑖̇⃗⃗⃗⃗⃗ ×𝑯𝑒𝑥𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ )+𝑗𝜔𝛼(𝒎0,𝑖̇⃗⃗⃗⃗⃗⃗⃗⃗ ×𝛿𝒎𝑖̇⃗��⃗⃗⃗ ),(S6) \nThe matrix form of Eq. (S6) is: \n𝑗𝜔[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]=−𝜇0𝛾𝑖[[1\n0\n0]×[0\n0\n−𝑀𝑠,𝑖𝛿𝑚𝑧,𝑖]+[1\n0\n0]×[ℎ𝑥\nℎ𝑦\nℎ𝑧]+[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]×[𝐻𝑥\n0\n0]]\n+𝑗𝜔𝛼𝑖([1\n0\n0]×[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]), (S7) \n𝑗𝜔\n𝜇0𝛾𝑖[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖]=−[[0\n𝑀𝑠,𝑖𝛿𝑚𝑧,𝑖\n0]+[0∙ℎ𝑥 \n−ℎ𝑧\nℎ𝑦]+[0\n𝐻𝑥𝛿𝑚𝑧\n−𝐻𝑥𝛿𝑚𝑦]]+𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖[0\n−𝛿𝑚𝑧\n𝛿𝑚𝑦],(S8) \nSimplify Eq. (S8) into 𝒉⃗⃗ =𝜒−1𝛿𝒎⃗⃗⃗ : \n[0∙ℎ𝑥\nℎ𝑦\nℎ𝑧]=\n[ 𝑗𝜔\n𝜇0𝛾0 0\n0(𝐻𝑥+𝑗𝜔𝛼\n𝜇0𝛾) −𝑗𝜔\n𝜇0𝛾\n0 𝑗𝜔\n𝜇0𝛾(𝑀𝑠+𝐻𝑥−𝑗𝜔𝛼\n𝜇0𝛾)] \n[𝛿𝑚𝑥,𝑖\n𝛿𝑚𝑦,𝑖\n𝛿𝑚𝑧,𝑖], (S9) \nWhen the FMR condition is met , the system will reach maximum resonance, that means the \nsystem, ℎ⃗ =0,𝜒−1𝑚⃗⃗ =0 has a non -zero solution [Eq. (S10)]: \n|𝜒−1|=0, (S10) \n(𝑀𝑠,𝑖+𝐻𝑥−𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖)(𝐻𝑥+𝑗𝜔𝛼𝑖\n𝜇0𝛾𝑖)−𝜔2\n𝜇02𝛾𝑖2=0, (S11) \n(1−𝛼𝑖2)𝜔2−𝑗𝜇0𝛾𝑀𝑠,𝑖𝛼𝑖𝜔−𝜇02𝛾𝑖2𝐻𝑥(𝐻𝑥+𝑀𝑠,𝑖)=0, (S12) \n6 \n 𝜔𝑐𝑜𝑚𝑝𝑙𝑒𝑥=𝑗𝜇0𝛾𝑖𝑀𝑠,𝑖𝛼𝑖\n2(1−𝛼𝑖2)+\n1\n2(1−𝛼𝑖2)√𝜇02𝛾𝑖2𝑀𝑠,𝑖2𝛼𝑖2+4(1−𝛼𝑖2)[𝜇02𝛾𝑖2𝐻𝑥,𝑖(𝐻𝑥,𝑖+𝑀𝑠,𝑖)], (S13) \nWhen 𝛼≪1, the Eq. (S13) will be simplified to : \n𝜔𝑖=𝛾𝑖√𝜇0𝐻𝑥(𝜇0𝐻𝑥+𝜇0𝑀𝑠,𝑖), (S14) \nEq. (S14) is Kittel mode by solving the linearized LLG equation with wave vector 𝑘=0. If \nconsider the exchange field 𝑯𝑒𝑥⃗⃗⃗⃗⃗⃗⃗ =2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖𝑘𝑖2, the Eq. (S14) will be revised to: \n𝜔𝑐𝑜𝑚𝑝𝑙𝑒𝑥=𝑗𝜇0𝛾𝑖𝑀𝑠,𝑖𝛼𝑖\n2(1−𝛼𝑖2)+ \n1\n2(1−𝛼𝑖2)√𝜇02𝛾𝑖2𝑀𝑠,𝑖2𝛼𝑖2+4(1−𝛼𝑖2)𝛾𝑖2[ (𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2)(𝐻𝑥,𝑖+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝑀𝑠,𝑖)] (S15) \nWhen 𝛼≪1, the Eq. (S15) will be simplified to : \n𝜔𝑖=𝛾𝑖√(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2)(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 𝑘𝑖2+𝜇0𝑀𝑠,𝑖), (S16) \nwhere 𝑘𝑖 is the wave vector of the perpendicular standing spin wave (PSSW) , and the \npropagation of the spin wave is in the z direction . \n𝜔1−𝜔2=0, (S17) \nFor the magnetic insulator bilayer, Eq. (S15) with 𝛼 and Eq. (S16) without 𝛼 describe the \ndispersion relation within each layer. Eq. (S17) means that the necessary condition for the \nbilayer spin waves to exist simultane ously in the same external condition ( same ℎ𝑟𝑓 and \n𝐻𝑒𝑥𝑡) in the two layers . \nBoundary condition s—free boundary condition and interface exchang e boundary \ncondition : Now, we can focus on the boundary conditions in the magnetic bilayers. For the \nmagnetic insulator bilayer system, free boundary condition should be applied to the top (𝑧=\n𝑑1) [Eq. (S18)] and the bottom (𝑧=−𝑑2) [Eq. (S19)] of the bilayer system, which al so \nmean no pining at the surfaces: \n7 \n 𝜕𝛿𝑚𝑧,1\n𝜕𝑧|\n𝑧=𝑑1=0, (S18) \n𝜕𝛿𝑚𝑧,2\n𝜕𝑧|\n𝑧=−𝑑2=0, (S19) \nConsidering that the dynamic magnetization in x and y directions is uniform, there is a wave \nvector (𝑘⊥) along the thickness direction. The spatial distribution of the PSSW can be expressed \nas: \n𝛿𝑚𝑧,𝑖=𝛿𝑚0,𝑖cos(𝑘𝑖𝑧+𝜙𝑖), (S20) \nwhere 𝜙𝑖 is the phase of the spin wave . Substitute Eq . (20) into Eq . (S18, S19), we can obtain: \n−𝑘1𝑑1+𝑛1𝜋=𝜙1, (S21) \n 𝑘2𝑑2+𝑛2𝜋=𝜙2, (S22) \nInterface exchange energy J describes the number and strength of exchange bonds between \nmagnetic insulator layer 1 and layer 2. The effect of exchange coupling energy 𝐽 (𝑚𝐽/𝑚2) at \nthe interface (z = 0) will be shown by the interface boundary conditions generated by the \ncombination of the Huffman boundary conditions 6-8. The conservation of magnetic energy flow \nat the interface leads to: \n2𝐴𝑒𝑥,1\n𝑀𝑠,1𝜕𝛿𝑚𝑧,1\n𝜕𝑧+2𝐽\n(𝑀𝑠,1+𝑀𝑠,2) (𝛿𝑚𝑧,2−𝛿𝑚𝑧,1)|\n𝑧=0=0, (S23) \n−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝜕𝛿𝑚𝑧,2\n𝜕𝑧+2𝐽\n(𝑀𝑠,1+𝑀𝑠,2) (𝛿𝑚𝑧,1−𝛿𝑚𝑧,2)|\n𝑧=0=0, (S24) \nThe matrix form of Eqs. (S23, S24) is as follows : \n[0\n0]=[−2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙1−2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑠𝑖𝑛𝜙12𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙2\n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙1 −2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙2+2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑠𝑖𝑛𝜙2][𝛿𝑚0,1\n𝛿𝑚0,2] (S25) \nThe resonant condition requires that the determinant of the coeffic ient matrix of Eq. S25 \nvanishes: \n8 \n [2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙1+2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑠𝑖𝑛𝜙1][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)cos𝜙2−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑠𝑖𝑛𝜙2]\n=[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙2][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑐𝑜𝑠𝜙1], (S26) \nEq. (S26) is the relationship under the interface -exchanged boundary conditions, combined \nwith the free boundary conditions of Eqs. (S21, S22) , we can obtain: \n[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)−2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑡𝑎𝑛(𝑘1𝑑1)][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)−2𝐴𝑒𝑥,2\n𝑀𝑠,2𝑘2𝑡𝑎𝑛(𝑘2𝑑2)]\n=[2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)][2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)], (S27) \n2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1)∙2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2)\n=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)[2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1)+2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2)], (S28) \nTo get the eigenfrequency, we can s olve the transcendental equations of the di spersion \ncondition [Eq. (S17)] and boundary condition [Eq. (S28)] for each external magnetic field 𝐻𝑒𝑥𝑡 \nby traversal . \nAlthough 𝑓1(𝑘1,𝑘2,𝐻𝑒𝑥𝑡)=0 [Eq. (S17)] and 𝑓2(𝑘1,𝑘2,𝐻𝑒𝑥𝑡)=0 [Eq. (S28)] cannot be \nexplicitly functionalized , we can solve them numeri cally by handling them as implicit functions. \nSupplementary Fig. 4b shows the frame work of the numerical analysis using MATLAB \nsoftware. Next, we will solve the eigenfrequency in two cases (𝐽=0 𝑚𝐽/𝑚2,𝐽≠0 𝑚𝐽/𝑚2): \nWhen 𝐽=0 𝑚𝐽/𝑚2, transcendental Eq. (S28) will degenerate to: \n𝑘1𝑘2tan(𝑘1𝑑1)tan(𝑘2𝑑2)=0, (S29) \n𝑘1=𝑛𝜋/𝑑1 or 𝑘2=𝑛𝜋/𝑑2 is the solution of Eq. (S29), which means spin wave in magnetic \ninsulator layer 1 and layer 2 is quantized as 𝑛𝜋/𝑑 and not influenced with each othe r. \nSubstitute Eq. (S29) into Eq. (S16) (assuming that 𝛼≪1): \n9 \n 𝜔𝑖=𝛾𝑖√(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 (𝑛𝜋\n𝑑)2\n)(𝜇0𝐻𝑥+2𝐴𝑒𝑥,𝑖\n𝑀𝑠,𝑖 (𝑛𝜋\n𝑑)2\n+𝜇0𝑀𝑠,𝑖), (S30) \nThe resonant peaks in Supplementary Fig. 5a shows the FMR and PSSW modes of TmIG and \nYIG FMR mode as a function of magnetic field when 𝐽=0 𝑚𝐽/𝑚2. We also show the process \nof solving 𝑘1,𝑘2 by graphical method in Supplementary Figs. 5b-c. Without inter facial \nexchange energy, the hybrid modes will be degenerate to the uniform mode and PSSW modes, \nwhich also means that spin -wave vector is quantized with 𝑛𝜋/𝑑. \nWhen 𝐽≠0=−0.0573𝑚𝐽/𝑚2 , we can clearly see the anti -crossing curve in the \nSupplementary Fig. 6a. A set of degenerate solutions of the shaded region in the Supplementary \nFig. 6a are composed of a set of intersection points of Figs. 6 b-c. It also needs to be noted that \nthe evanescent wave emerges in the YIG layer when the frequency of the PSSW in TmIG layer \nis lower than the uniform mode in YIG layer ( 𝑘1 is pure imaginary part). \nMicromagnetic simulation s based on finite -element model \nIn addition to proposing a relatively complete theory, we show the strong magnon -magnon \ncoupling regime by solving the LLG equation in the frequency domain base d on COMSOL's \nmicromagnetic simulation module 9. We apply the frequency -domain LLG equations of Eq. \n(S31 , S32) to the two magnetic thin films respectively : \n−𝑖𝜔𝛿𝒎𝒊=−𝛾𝑖𝒎𝒊×(𝑯𝒆𝒇𝒇+𝑯𝒆𝒙)−𝑖𝜔𝛼𝑖𝒎𝒊×𝛿𝒎𝒊 (S31) \n−𝑖𝜔𝛿𝒎𝒋=−𝛾𝑗𝒎𝒋×(𝑯𝒆𝒇𝒇+𝑯𝒆𝒙)−𝑖𝜔𝛼𝑗𝒎𝒋×𝛿𝒎𝒋 (S32) \nwhere all parameters are consistent with the theory calc ulations. 𝐻𝑒𝑥 is the effective filed-like \ntorque induced by the interfacial exchange energy, which is defined as: \n𝐻𝑒𝑥=2𝐽\n𝑀𝑖+𝑀𝑗𝛿(𝑧)𝒎𝒋,∫𝛿(𝑧)=1+∞\n−∞ (S33) \nwhere 2𝐽/(𝑀𝑖+𝑀𝑗) means that t he average value of the interfac ial exchange energy for \ndifferent magnetization materials on both sides of the interface . 𝛿(𝑧) is impulse function, \n10 \n which shows that the interaction only occur s at the interface of the TmIG/YIG heterostructure. \nThe simulation parameters to be set are shown in the following Supplementary Table S1. \nTo demonstrate the coupling strength between YIG and TmIG, we perform full -frequency full -\nmagnetic field simulations . We characterize the strength of resonance through the response of \n𝛿𝑚𝑧 to magnetic field and mi crowave field . Through the simulation results shown in the \nSupplementary Fig. 7a, we can clearly see that TmIG (n=1,2) is coupled with YIG (n=0) to \nform anti -cross ing at 105 Oe and at 470 Oe. Supplementary Fig. 7b shows the spectral \nresonance curve of the spin wave at 105 Oe. Then, w e can see the internal dynamics of spin \nwaves through simulation . To perform the resonance direction of the spin wave on the interface \nof the bilayers, we show the 𝛿𝑚𝑧 distribution o f the normalized intensity in the z direction for \nthe two hybrid modes A and B (marked) in Supplementary Fig. 8. \n \nSupplementary Note 3. Thickness dependence of interfacial exchange coupling energy \nand coupling strength for TmIG/YIG samples \nThe sign judgment of interfacial exchange coupling energy \nJudgment 1: we obtained the interfacial coupling energy (𝐽<0 𝑚𝐽/𝑚2) by fitting the \nexperimental data with the theory of Supplementary Note 2. The values of the J are summarized \nin the Supplementary Table S2. \nJudgment 2: We plot the spin -wave absorption spectra for TmIG(100 nm)/YIG(100 nm) in \ncomparison with YIG(100 nm)/GGG (grey dashed line) in Supplementary Fig. 9. The lower \nresonant magnetic field (𝐻𝑟𝑒𝑠) or higher resonant frequency (𝜔𝑟𝑒𝑠) means that for the \nTmIG/YIG sample, the interfacial exchange field applied on YIG is opposite to the external \nmagnetic field, which supports the antiferromagnetic coupling nature concluded in the main \ntext. In the meantime, Eq. (S38) shows that 𝛿𝑘1(𝑌𝐼𝐺)2 has the same sign as J, which means if \n𝐽<0 𝑚𝐽/𝑚2 , 𝛿𝑘1,𝑌𝐼𝐺2<0 . According to the Eq. (S16), we can know the u nder the same \n11 \n external magnetic field, the resonant frequency required by TmIG/YIG sample is smaller than \nthat of YIG/GGG, which is consisten t with our results (Supplementary Fig.9). We co uld also \nroughly estimate the coupling strength (𝐽≈𝐻𝑠ℎ𝑖𝑓𝑡×𝑀𝑠1+𝑀𝑠2\n2×𝑑1=−0.0042 𝑇×\n133.3 𝑘𝐴/𝑚×100𝑛𝑚=−0.0598 𝑚𝐽/𝑚2) by observing the shift [Eq. (S38)] in the FMR \ncurve between the TmIG/YIG/GGG heterostructure and YIG/GGG single layer . \nThickness dependence of interfacial coupling energy and coupling strength for \nTmIG/YIG samples \nInterfacial coupling energy: Supplementary Figs. 10a-c show the colormap of the spin-wave \nabsorption spectra for the PSSW modes of TmIG and the uniform mode of YIG measured for \nTmIG(100 nm)/YIG(100 nm), TmIG(140 nm)/YIG(140 nm), and TmIG(200 nm)/YIG(200 nm) \nheterostructures . We fit the experimental data (Supplementary Figs. 10e -f) through numerical \nanalysis in Supplementary Note 2 . The results are summarized in the Supplementary Table 2. \nCoupling strength g : We could also extract the magnon -magnon coupling strength, which is \ndefined as the half of the minimal peak to peak frequency spacing in the anti -crossing induced \nby the interface exchange energy10,11. When the coupling strength is 0 (𝐽=0 𝑚𝐽/𝑚2), the \nhybrid modes will be decoupled. The resonant magnetic field (𝐻𝑒𝑥𝑡) and resonant frequency \n(𝑓) where the minimum resonant separation is located should intersect , which means that \n𝑘1=0,𝑘2=𝑛𝜋/𝑑2 . At the existence of the interfacial exchange coupling energy (𝐽≠\n0 𝑚𝐽/𝑚2), we can obtain the perturbative solution. we make a difference to Eq. (S16) to get \nan expression of the coupling strength g (𝛿𝑓\n2): \n2fres𝛿𝑓=(𝛾\n2𝜋)2\n∙[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠𝑘𝛿𝑘+(2𝐴𝑒𝑥\n𝑀𝑠)24𝑘3𝑑𝑘],(S34) \nFor Eq. (S28), we can let 2𝐴𝑒𝑥,1\n𝑀𝑠,1k1tan(𝑘1𝑑1) be A and 2𝐴𝑒𝑥,2\n𝑀𝑠,2k2tan(𝑘2𝑑2) be B. So, the Eq. \n(S28) can be simplified: \n1−2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)(1\n𝐴+1\n𝐵)=0, (S35) \n12 \n We now consider 𝑘1=0+𝛿𝑘1,𝑘2=𝑛𝜋/𝑑2+𝛿𝑘2,|𝛿𝑘1|≪1 ,|𝛿𝑘2|≪1 as the \nperturbation solution corresponding to the minimum resonance separation of YIG's FMR \nmode and TmIG PSSW mode (n) . Eq. (S35) will yield layer 1 -dominated and layer 2 -\ndominated reson ance. For layer 1 -dominated resonance, we have: \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐴≈1 and 2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐵≪1, (𝑆36) \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)≈2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝑡𝑎𝑛(𝑘1𝑑1)=2𝐴𝑒𝑥,1\n𝑀𝑠,1𝑘1𝛿𝑘1𝑑1=2𝐴𝑒𝑥,1\n𝑀𝑠,1𝛿𝑘12𝑑1,(S37) \n𝛿𝑘12=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑀𝑠,1\n2𝐴𝑒𝑥,11\n𝑑1, (S38) \nFor 𝑘1=𝛿𝑘1, (2𝜇0𝐻+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠2𝛿k1≫(2𝐴𝑒𝑥\n𝑀𝑠)2\n4𝛿𝑘13 \n𝛿𝑓1≈(𝛾1\n2𝜋)2\n[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝑑1]1\n2𝑓𝑟𝑒𝑠, (S39) \nFor layer 2 -dominated resonance, we have: \n2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐵≈1 and 2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝐴≪1, (S40) \n𝛿𝑘2=2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)𝑀𝑠,2\n2𝐴𝑒𝑥,21\n𝑛𝜋/𝑑21\n𝑑2, (S41) \nFor 𝑘2=𝑛𝜋\n𝑑2+𝛿𝑘2 , (2𝜇0𝐻+𝜇0𝑀𝑠)~10−1,2𝐴𝑒𝑥\n��𝑠~10−18,𝑘2~𝑛𝜋\n𝑑2~107 , so we can get that \n(2𝜇0𝐻+𝜇0𝑀𝑠)2𝐴𝑒𝑥\n𝑀𝑠2𝑘2~10−12≫(2𝐴𝑒𝑥\n𝑀𝑠)2\n4𝑘3~10−15. \n𝛿𝑓2≈(𝛾2\n2𝜋)2\n[2(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)2𝐽\n(𝑀𝑠,1+𝑀𝑠,2)1\n𝑑2]1\n2𝑓𝑟𝑒𝑠, (S42) \n𝛿𝑓2=(𝛾1\n2𝜋)2\n(𝛾2\n2𝜋)2\n(2𝐽\n(𝑀𝑠,1+𝑀𝑠,2))2[(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)]\n𝑓𝑟𝑒𝑠21\n𝑑1𝑑2,(S43) \n𝑔=𝛿𝑓\n2=𝛾1\n2𝜋𝛾2\n2𝜋𝐽\n(𝑀𝑠,1+𝑀𝑠,2)√(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,1)(2𝜇0𝐻𝑟𝑒𝑠+𝜇0𝑀𝑠,2)\n𝑓𝑟𝑒𝑠1\n√𝑑1𝑑2,(S44) \n13 \n Where 𝑔 is the coupling strength, 𝐻𝑟𝑒𝑠,𝑓𝑟𝑒𝑠 are the resonant magnetic field and frequency at \nthe minimum resonance separation , 𝑑1,𝑑2 is the thickness of the YIG layer (FMR mode) and \nTmIG layer (PSSW mode). The g value extracted in the experiment (Supplementary Figs. 11a -\nc) and calcu lation from Eq. (S4 4) are shown as Fig. 3c. \n \nSupplementary Note 4. The extraction of Gilbert d amping , dissipation rate, and \ncooperativity from the FMR measurements \nGilbert damping 𝜶: High coupling strength 𝑔 and extremely low Gilbert damping 𝛼 are \nkey factors to realize long -distance information transmission that is free of Joule heating. \nGilbert damping refers to an intrinsic feature of magnetic substances which dictates the speed \nat which angular momentum is transferred to the crystal lattice , which is the key parameter that \ndetermines the spin -wave relaxation 13,14 . To demonstrate the superiority of low damping \nMI/MI bilayer system of this work , we focus on the line widths variation with the frequency of \nYIG/ TmIG heterostructure samples. The linewidth versus frequency experiment data with \nerror bar for different thicknesses were extracted by fitting Lorentz equations in Supplementary \nFig. 12. In strong coupling region, we can clearly observe that the line width value of the pink \ncircle is significantly higher than that of the light blue circle, which is suggested that a coherent \ndamping -like torque which acts along or against the intrinsic damping torque depending on the \nphase difference of the coupled dynamics of YIG and TmIG 4; Away from the strong coupling \nregion, we use ∆𝐻=∆𝐻0+4𝜋𝛼\n𝛾𝑓 for linear characterization, and the result is indicated by \nthe dashed line in the Supplementa ry Fig. 12 . The fitting effective damping constants are \nsummarized in the Supplementary Table 2. \nDissipation rate: We extract the dissipation rate (half width at half maximum ) from frequency \nscans at different fields when they are away from the coupling region (Supplementary Fig. 13) . \nWe fit the FMR mode of YIG and TmIG in Supplementary Fig. 13 through the Lorentz peak \nto get the dissipation rate. \n14 \n Cooperativity 𝑪: The strong c oupling implies coherent dynamics between the magnon and \nthe magnon. In our case, all samples’ 𝑔>𝜅1,𝑔>𝜅2 mean that strong coupling is formed \n11,12,15. Through Lorentz peak fitting, we found that the dissipation rate will decrease as the \nthickness increases, which is consistent with the damping trend we calculated. Due to the \nmagnetic insulator heterostructure, the dissipation rate s in TmIG/YIG bilayers are particularly \nlow compared w ith ferromagnetic metal -based heterostructures so that we can get a largest \ncooperativity 𝐶=(𝑔/𝜅1)(𝑔/𝜅2)=24.5 in the TmIG(200 nm)/YIG(200 nm) case. The \nresults are summarized in the Supplementary Table 2 . As the thickness increases, the \ncooperativity increases significantly, mainly because the decrease rate of the dissipation rates \nis greater than that of coupling strength. The cooperativity is summarized in the Supplementary \nTable S2. \nAntiferromagnetic interfacial exchange coupling energy for the TmIG(350 \nnm)/CoFeB(50 nm) sample \nWe also verify antiferromagnetic interfacial exchange coupling energy of the MI/FM \nTmIG(350 nm)/CoFeB(50 nm) by FMR modulation . We use theory ( Supplementary Note 2) \nto analyze and fit the color -coded experiment data (Supplementary Figs. 14a -b) and get the \nTmIG/CoFeB interfacial coupling energy 𝐽=−0.0321 𝑚𝐽/𝑚2 , showing that the FMR \nmethod is consistent with the minor hysteresis loop method (𝐽=−0.0311 𝑚𝐽/𝑚2). \n \n15 \n Parameters Symbol Value \nMesh 𝑁 20×20×50 \nMicrowave field ℎ𝑟𝑓 0.001𝑇 \nInterfacial exchange energy 𝐽 −0.0537 𝑚𝐽/𝑚2 \nYIG/TmIG gyromagnetic ratio 𝛾𝑌𝐼𝐺/𝛾𝑇𝑚𝐼𝐺 28/21.81 𝐺𝐻𝑧/𝑇 \nYIG/TmIG stiffness constant 𝐴𝑒𝑥,𝑌𝐼𝐺/𝐴𝑒𝑥,𝑇𝑚𝐼𝐺 3.08/2.67 𝑝𝐽/𝑚 \nYIG/TmIG Gilbert damping 𝛼𝑌𝐼𝐺/𝛼𝑇𝑚𝐼𝐺 6.66×10−4/1.17×10−3 \nFrequency range (𝑓𝑠𝑡𝑎𝑟𝑡,∆𝑓,𝑓𝑒𝑛𝑑) (1,0.01,4) 𝐺𝐻𝑧 \nMagnetic field range (𝐻𝑠𝑡𝑎𝑟����,∆𝐻,𝐻𝑒𝑛𝑑) (30,5,700) 𝑂𝑒 \nSupplementary Table S1. | Parameters used to do the micromagnetic simulation with \nTmIG 200nm/ YIG 200nm bilayer in Supplementary Figs. 7-8. \n16 \n Materials 𝜅𝑌𝐼𝐺 (𝛼𝑌𝐼𝐺) 𝜅𝑇𝑚𝐼𝐺 (𝛼𝑇𝑚𝐼𝐺) \nor 𝜅𝐹𝑀 (𝛼𝐹𝑀) g C J (mJ/m2) \nYIG(100 nm)/TmIG \n(100 nm) (this work) 0.058 GHz \n(2.98e -3 ± 1.4e-4) 0.046 GHz \n(4.4e -3 ± 3e -4) 0.155 GHz 8.76 -0.0618 \nYIG(140 nm)/TmIG \n(140 nm) (this work) 0.0185 GHz \n(7.78e -4 ± 1.7e -4) 0.054 GHz \n(1.69e -3 ± 2e -4) 0.105 GHz 11.04 -0.072 0 \nYIG(200 nm)/TmIG \n(200 nm) (this work) 0.01 GHz \n(6.64e-4 ± 4.2e-5) 0.0295 GHz \n(1.07e -3 ± 1e -4) 0.085 GHz 24.49 -0.0573 \nTmIG(350 nm)/CoFeB \n(50 nm) (this work) -- 𝜅𝑇𝑚𝐼𝐺 : 0.0255 GHz \n(4.91e -4±8e-5) \nor \nκFM: 0.232 GHz \n(3e-3±3.9e-5) 0.263GHz 11.69 -0.0321 \nYIG(20 nm)/Ni(20 nm)29 * 0.06GHz 0.63 GHz 0.12 GHz 0.38 -- \nYIG(20 nm)/Co(30 nm)29 * 0.06GHz 0.5 GHz 0.79 GHz 21 -- \nYIG(100 nm)/ Ni80Fe20 \n(9 nm)27 0.106GHz \n(2.3e -4) 0.192 GHz \n(1.75e -3) 0.35 GHz 6 -0.060 ± \n0.011 \nYIG(1 µm)/Co(50 nm) 28 (7.2e -4 ± 3e -5) (7.7e -3 ± 1e -4) -- -- -- \nNi80Fe20(20 nm)34 * -- 0.66 GHz -- 0.6 -- \nNi80Fe20(20 nm)35 * -- 0.31 GHz -- 2.25 -- \nIrMn(10 nm)/CoFeB \n(80 nm)36 -- 0.26 GHz -- 2 -- \nCoFeB(15 nm)/Ru(0.6 nm) \n/CoFeB(15 nm )37 * -- 0.31 GHz -- 5.39 -- \nCoFeB(15 nm)/Ru(0.6 nm) \n/CoFeB(15 nm)38 * -- 0.23 GHz -- 8.4±1.3 -- \nSupplementary Table 2. Summary of dissipation rates, damping factors, coupling strengths, \ncooperativities, and exchange interaction strengths in the studied bilayers and some reported \nYIG/ferromagnetic metal (FM) bilayers that show magnon -magnon coupling. All reference \nnumbers are consistent with the main text . * In -plane confinement introduces extra dynamical \ndipolar coupling. \n17 \n \nSupplementary Figure 1 | Hysteresis loop s for YIG (100 nm), TmIG(100 nm) and Y IG \n(100nm) /TmIG (100nm) samples. Since YIG and TmIG have approximately equal coercive \nfields (~1.5Oe), we could not obtain the interfacial coupling energy through the dynamic \nprocess of the switching of the two layers in the hysteresis loop. \n-50 -40 -30 -20 -10 0 10 20 30 40 50-0.6-0.4-0.20.00.20.40.6M (memu)\nField (Oe)In-plane\n YIG (100nm)\n TmIG (100nm)\n YIG (100nm)/TmIG (100nm) \n18 \n \nSupplementary Figure 2 | Hysteresis loop s for TmIG (350 nm), CoFeB(50 nm), and TmIG \n(350 nm)/CoFeB(50 nm) samples . a, The individual magnetic thin film layers TmIG (light \nblue solid line) and Co FeB (red solid line) exhibit different coercivity, which could be directly \nobserved the process of magnetization flipping between the two coercivity (black solid line). \nFurthermore, due to the existence of interfacial exchange coupling (J), no sharp switching (like \nthe arrow point A to B to C) of the CoFeB layer is visible but a smooth increase (the arrow \npoint A to C) of the measured magnetic moment until the bilayer magnetization is saturated. \nb, Enlarged detail of the TmIG(350 nm)/CoFeB(50 nm) heterostructure hysteresis loop. c, \nProcess (1 -5) shows a possible magnetization flipping in an exchange coupled heterostructure \nat an external magnetic field in b. \nHext(3) Hext(1)TmIG CoFeBHext(4) Hext(5) Hext(2)\n-40 0 40 80-1.00.01.0\n-44 -22 0 22 44M/Ms\nField (Oe) CoFeB (50nm)\n TmIG (350nm)\n TmIG (350nm)/CoFeB (50nm)\nIn-plane\nTmIG (350nm) /CoFeB (50nm)\n-40 0 40 80-1.00.01.0\n-44 -22 0 22 44M/Ms\nField (Oe) CoFeB (50nm)\n TmIG (350nm)\n TmIG (350nm)/CoFeB (50nm)\nIn-plane\nTmIG (350nm) /CoFeB (50nm)(1)\n(3)\n(4)\n(5)AB\nC(2)\n(a) \n(c) \n(b) \n19 \n \nSupplementary Figure 3 | Hysteresis loop and minor loops for TmIG (350 nm)/CoFeB (50 \nnm) sample . a, Minor loops for TmIG(350 nm)/CoFeB(50 nm) sample . The non -coincidence \nof +minor loop and -minor loop indicates the existence of interface coupling. b, The process \nof analyzing interfacial exchange coupling using minor loop s. The result of the experiment \nmeasurement is a C -A-B-C loop, indicating the existence of antiferromagnetic interfacial \ncoupling . However, if there is no interfac ial exchange coupling, the loop will be like E -A-D-E, \nbecause it only depends on its coercivity of TmIG , that is, the forward and reverse minor loops \nshould basically coincide (just like individual TmIG Hysteresis loop). We can also get the \n-30 -20 -10 0 10 20 30-3.5-2.5-1.5-0.50.51.52.53.5\n Hysteresis Loop TmIG (350nm)/CoFeB (50nm)\n + Minor LoopM (memue)\nField (Oe) - Minor Loop\nIn-planeABC\nDE\nHex=3.94OeΔMs=1.975memue\n-30 -20 -10 0 10 20 30-3.5-2.5-1.5-0.50.51.52.53.5\n Hysteresis Loop TmIG (350nm)/CoFeB (50nm)\n + Minor LoopM (memue)\nField (Oe) - Minor Loop\nIn-plane(2)(1)\n(3)M (memu ) M (memu )\n(a) \n(b) \n20 \n similar results from the + minor loop with the similar method. \n21 \n \n \nSupplementary Figure 4 | Schematic of the magnetic insulator heterostructure with \nthickness of 𝒅𝟏 and 𝒅𝟐 , respectively. a, For the convenience of calculation, we set the \ninterface of bilayer as the interface of z=0. The direction of the static external magnetic field \n(𝐻𝑒𝑥𝑡) is set as the x -axis, and the dynamic magnetic field (ℎ𝑟𝑓) generated by CPW is \nperpendicular to the static magnetic field on the y -axis, where 𝐻𝑒𝑥𝑡≫ℎ𝑟𝑓 . b, Theoretical \ncalculation framework from MATLAB. We traverse the value of 𝐻𝑒𝑥𝑡 through the loop cycle \nand solve Eqs. (S17, S28) under each external magnetic field value to obtain the 𝑘1and 𝑘2, \nthen substitute them into the dispersion relationship to obtain the eigen frequency. \nStartStart Start Loop though StartSolve \nStartCalculate equation \nS16 to get Trune parameter?\nrangeStartStartYES\nNOTune parameter?\nHext range\n(a) \n(b) \n22 \n \nSupplementary Figure 5 | Numerical solution for the eigenfrequency when 𝑱=𝟎 𝒎𝑱/𝒎𝟐. \na, Frequencies of FMR and PSSW modes of TmIG (200 nm) and FMR mode of YIG (200 nm) \nas a function of H when 𝐽=0 𝑚𝐽/𝑚2. b-c, Numerical solutions 𝑘1,𝑘2 obtained by solving \ndispersion condition and boundary condition without interfacial exchange energy. When 𝐽=\n0 𝑚𝐽/𝑚2, we could obviously see that the crossing points are degenerate solution in a from \nthe shaded region, which indicates that 𝑘1,𝑘2 is qu antized solution with 𝑛𝜋/𝑑1 and 𝑛𝜋/𝑑2 \nrespectively . \n0 100 200 300 4000.00.51.01.52.02.53.0\n0.0 0.5 1.0 1.50123\n0 1 2 30.51.01.52.02.5Frequency (GHz)\nH (Oe) TmIG kittel \n YIG kittel \n TmIG n=1 \n TmIG n=2k2 (units of p/d2)\nk1 (units of i p/d1) Eq S28\n Eq S17\n·\n★TmIG n=0YIG n=0J=0 mJ/m2\nk2 (units of p/d2)\nk1 (units of i p/d1) Eq S28\n Eq S17\nTmIG n=1TmIG n=2J=0 mJ/m2\n◆▲\n(a) \n(b) \n (c) \n23 \n \nSupplementary Figure 6 | Numerical solution for the eigenfrequency when 𝑱=\n−𝟎.𝟎𝟓𝟕𝟑𝒎𝑱/𝒎𝟐 . a, Anti-crossing coupling f requencies of hybrid modes of TmIG (200 \nnm)/ YIG(200 nm) as a function of H when 𝐽=−0.0573 𝑚𝐽/𝑚2. b-c, Numerical solutions \n𝑘1,𝑘2 obtained by solving dispersion condition and boundary condition without interfacial \nexchange energy. With the interfacial exchange energy, we could see that the spin -wave vector \nsolution is no longer the integer of 𝑛𝜋/𝑑. It is noted that the evanescent wave emerges in the \nYIG layer when the frequency of the PSSW in TmIG layer is lower than the uniform mode in \nYIG layer ( 𝑘1 is pure imaginary part). \n \n \n ( ) \n b \n b \n \n ( ) \n \n \n ( ) \n \n \n \n(a) \n(b) \n (c) \n24 \n \n \nSupplementary Figure 7 | Full micromagnetic simulation based on frequency domain. a, \nColor -coded spin -wave spectra with frequency and magnetic field with the TmIG (200 nm) / \nYIG(200 nm) heterostructure. we can clearly see that TmIG(n=1,2) is coupled with YIG (n=0) \nto form anti -cross ing at 105 Oe and at 470 Oe. Then, w e use the value of change in 𝛿𝑚𝑧 as \nthe peak response. b, The spin spectrum curve at the minimum resonance separation of the first \nanti-crossing (H=105 Oe) . \n \n1234\n200 400 600Frequency (GHz)\nMagnetic Field (Oe)0.0 0.80\n0.011T 1.32GHz\n0.011T 1.56GHz\n1 2 3 40.00.30.71.0\n1234\n200 400 600Frequency (GHz)\nMagnetic Field (Oe)0.0 0.80\n0.011T 1.32GHz\n0.011T 1.56GHz\nAmplitude (Norm.)\nFrequency (GHz)Hext=105 Oe\nGraph19\n0.011T 1.56GHz2g=0.22GHz\n(a) \n(b) \n25 \n \nSupplementary Figure 8 | The two hybrid eigenmodes at 105 Oe . The 𝛿𝑚𝑧 distribution of \nthe normalized intensity in the z direction for the two hybrid modes at 1.33 GHz and 1.55 GHz \nrespectively. \n \n0 100 200200 300 400\n200 300 400 0100 200\ndmz (norm.)\nz (nm)\ndmz (norm.)A: 1.33GHz B: 1.55GHz \n26 \n \n \nSupplementary Figure 9 | The spin -wave absorption spectra for TmIG(100 nm)/YIG(100 \nnm) in comparison with YIG(100 nm)/GGG (grey dashed line) . Lower resonant magnetic \nfield 𝐻𝑟𝑒𝑠, or higher resonant frequency 𝜔𝑟𝑒𝑠, is observed for YIG( 100 nm)/GGG both before \nand after the avoided crossing. This shows that for the TmIG/YIG sample, the interfacial \nexchange field applied on YIG is opposite to the external magnetic field, which supports the \nantiferromagnetic coupling nature concluded in the main text. \n0 400 800 12002.03.04.05.0\n0 200 400 6001.02.03.04.0\n0 200 400 6001.02.03.04.00.080.120.16\n0.00 0.02 0.04 0.0624\n04008001200\n2.0 3.0 4.0 5.0Hext (Oe)\nw/2p (GHz)-2.8E-5 2.8E-5\n(a) (b) (c)\n(e) (f) (g)0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-1.1E-4 6.7E-5\n0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-2.8E-4 2.2E-4w/2p (GHz)\nHext (Oe)YIG(100nm)/TmIG(100nm) YIG(140nm)/TmIG(140nm) YIG(200nm)/TmIG(200nm)V (f,H) V (f,H) V (f,H)\n原始文件在 ‘画图全’中的folder1\nw/2p (GHz)\nHext (Oe)dw/2p (GHz)\nHext (Oe)\ng (GHz) Experiment\n Calculation0.08 0.10 0.12tYIG\n100 140 200 F1F1\n YIG single \n27 \n \nSupplementary Figure 10 | Magnetic insulator heterostructure thickness dependence of \nthe strong magnon -magnon coupling . a-c, Experimentally color -coded spin -wave absorption \nspectra with the YIG (100nm) /TmIG (100nm) (a), YIG (140nm) /TmIG (140nm) (b), YIG \n(200nm) /TmIG (200nm) (c). Inset magnification was performed at each anti -crossing region . \ne-g, Resonant absorption peaks of the two hybrid modes as a function of external magnetic \nfield with YIG (100nm) /TmIG (100nm) (e), YIG (140nm) /TmIG (140nm) (f), YIG \n(200nm) /TmIG (200nm) (g) bilayer . Solid curves show the numerical theory method fitting as \nhybri d modes using Supplementary note2 . Data points are extracted from experimental data by \nfitting the line shapes to two independent derivative Lorentzian functions from (a-c). The \nincrease of effective magnetization in YIG and TmIG and the stiffening of YIG and TmIG \nresonance frequency is due to the increase of different magnetic insulator thickness. \n0 400 800 12002.03.04.05.0\n0 200 400 6001.02.03.04.0\n0 200 400 6001.02.03.04.00.080.120.16\n0 400 800 12002.03.04.05.0\nHext (Oe)w/2p (GHz)-2.8E-5 2.8E-5\n(a) (b) (c)\n(e) (f) (g)0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-1.1E-4 6.7E-5\n0 200 400 6001.02.03.04.0\nHext (Oe)w/2p (GHz)-2.8E-4 2.2E-4w/2p (GHz)\nHext (Oe)YIG(100nm)/TmIG(100nm) YIG(140nm)/TmIG(140nm) YIG(200nm)/TmIG(200nm)V (f,H) V (f,H) V (f,H)\n原始文件在 ‘画图全’中的folder1\nw/2p (GHz)\nHext (Oe)dw/2p (GHz)\nHext (Oe)2g=\n0.31GHz2g=\n0.17GHz2g=\n0.06GHz\n2g=\n0.21GHz\ng (GHz) Experiment\n Calculation\n0.080.100.12\ntYIG100 140 200 \n28 \n \nSupplementary Figure 11 | Magnetic insulator heterostructure thickness dependence of \nthe coupling strength . a-c, Experimentally spin-wave spectra at the minimum resonance \nseparation with frequency with the TmIG (100 nm)/ YIG(100 nm) (a), TmIG (140 nm)/ YIG(140 \nnm) (b), TmIG (200 nm)/ YIG(200 nm) (c). At the same time, the red line is fitted by the Lorentz \npeak function to coupling strength . \n3.0 3.5 4.0-10-50\n2.5 3.0 3.5 4.0-505\n1.0 1.2 1.4 1.6-10-505Amptitude (V)\nFrequency (GHz) Experiment\n Lorentz Fitsignal ´106\n2g\nHext:\n500.3 Oe140nm/140nm YIG/TmIG\n Experiment\n Lorentz Fit\n1.56-1.40=2g\nk1=0.032/2\nk2=0.04821 /2\nC=16.59Amptitude (V)\nFrequency (GHz)signal ´106\n2g\nHext:\n574.5 Oe100nm/100nm YIG/TmIG\n Experiment\n Lorentz Fit\n3.49-3.18=2g\nk1=0.1888/2\nk2=0.0473/2\nC=10.763.51-3.30=2g\nk1=0.0673/2\nk2=0.0521/2\nC=12.577\nAmptitude (V)\nFrequency (GHz)2gHext:\n97.7 Oe200nm/200nm YIG/TmIG signal ´105\n(a) \n (b) \n (c) \n29 \n \nSupplementary Figure 12 | Thickness dependence of the Gilbert damping . a-c, \nExperimentally line-width with frequency with the TmIG (100 nm)/ YIG(100 nm) (a), \nTmIG (140 nm)/ YIG(140 nm) (b), TmIG (200 nm)/ YIG(200 nm) (c). Circle points with the \nerror bars represent linewidth extracted from experimental data by fitting the line shapes to \nthree independent derivative Lorentzian functions from (a-c). Dashed lines are linear fits away \nfrom the strong coupling region through equation 5 . The fitting effective damping constant s \nare summarized in the Supplementary Table S2. \n1.0 1.5 2.0 2.5 3.0 3.5 4.001020\n2.0 2.5 3.0 3.5 4.0 4.50204060\n1.0 1.5 2.0 2.5 3.0 3.5 4.005101520\n Hybrid 1\n Hybrid 2H (Oe)\nFrequency(GHz)YIG FMR\nTmIG FMR\n F\n Linear Fit of Sheet1 D\n Linear Fit of Sheet1 B\na\nYIG： 7.87e-4+-1.66e-4\nTmIG： 1.699e-3+-1.45e-4\nslope tmig FMR: 9.79046+- 0.83577\nslope YIG FMR: 3.53367+-0.74594140nm/140nm YIG/TmIG\n Fit\n FitStrong \nCoupling \nEquation y = a + b*x\nPlot B\nWeight Instrumental (=1/ei^2)\nIntercept 11.78418 ± 0.14368\nSlope 1.558 ± --\nResidual Sum of Squares 12.79439\nPearson's r 0.91465\nR-Square (COD) 0.65215\nAdj. R-Square 0.65215H (Oe)\nFrequency(GHz) Fit\n Fit\na\nYIG： 2.98e-3+-1.35e-4\nTmIG： 4.4e-3+-2.59e-4\nslope tmig FMR: 25.4005+- 1.4966\nslope YIG FMR: 13.82892+-0.60861100nm/100nm YIG/TmIG\nStrong \nCoupling Equation y = a + b*x\nPlot B\nWeight Instrumental (=1/ei^2)\nIntercept 6.75777 ± 0.03877\nSlope 2.2009 ± --\nResidual Sum of Squares 0.21698\nPearson's r 0.99552\nR-Square (COD) 0.97839\nAdj. R-Square 0.97839\nHybrid 1\nHybrid 2\nH (Oe)\nFrequency(GHz)YIG FMR\nTmIG FMR\n Hybrid 4\n Hybrid 3\n Linear Fit of Sheet1 D\n Linear Fit of Sheet1 B\na\nYIG： 6.66e-4+-0.42e-4\nTmIG： 1.07e-3+-1.16e-4\nslope tmig FMR: 6.17513+- 0.6707\nslope Hybrid 1: 2.98248+-0.19075200nm/200nm YIG/TmIG\n Fit\n FitStrong \nCoupling 2\ncell://[Book28]FitLinear6!No cell://[Book28]FitLinear\nPlot [Book28]FitLinear6!Par\n[Book28]FitLinear6!Notes. [Book28]FitLinear6!Not\nIntercept [Book28]FitLinear6!Par\nSlope [Book28]FitLinear6!Par\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n[Book28]FitLinear6!RegStat [Book28]FitLinear6!Reg\n原始文件在‘耦合系数'中的folder4Hybrid 1\nHybrid 2\n(a) \n (b) \n (c) \n30 \n \nSupplementary Figure 13 | Thickness dependence of the dissipation rate . a-c, Frequency \nsweep curve s away from the coupling region . Dissipation rate is obtained by Lorentzian peak \nfunction f itting. Red solid line presents YIG FMR, blue solid line presents TmIG FMR. \n2.0 2.5 3.0 3.5024681012 Amplitude (V)\nFrequency (GHz) 0.03068\n Lorentz Fit of Sheet1 AP\"0.03068\"\n 0.03216\n Lorentz Fit of Sheet1 AR\"0.03216\"\n 0.0344\n Lorentz Fit of Sheet1 AU\"0.0344\"\n 0.03588\n Lorentz Fit of Sheet1 AW\"0.03588\"\n 0.04927\n Lorentz Fit of Sheet1 BO\"0.04927\"\n 0.05001\n Lorentz Fit of Sheet1 BP\"0.05001\"\n 0.0515\n Lorentz Fit of Sheet1 BR\"0.0515\"\n 0.05076\n Lorentz Fit of Sheet1 BQ\"0.05076\"TmIG FMR peaks\nYIG FMR peaks30.0 mT32.1 mT34.4 mT35.8 mT49。2 mT50.0 mT50.7 mT51.0 mTTmIG（100nm）/YIG(100 nm)Signal´105\n1.0 1.5 2.0 2.5-40481216202428 Amplitude (V)\nFrequency (GHz)10.1 mT12.1 mT14.2 mT16.3 mT18.3 mT20.0 mTTmIG FMR \npeaksYIG FMR \npeaksYIG(140 nm)/TmIG(140 nm)Signal´105\n1.5 2.0 2.5 3.0-8-4048121620 Amplitude (V)\nFrequency (GHz)221Oe242Oe262Oe280Oe300Oe321Oe338Oe\nTmIG FMR \npeaks\nYIG FMR \npeaksSignal´105TmIG(200 nm)/YIG(200 nm)\n(a) \n (b) \n (c) \n31 \n \n \nSupplementary Figure 14 | Magnetic field dependency of resonant peaks for \nCoFeB 50nm/TmIG 350nm heterostructure. a, Experimentally color -coded spin -wave absorption \nspectra of the Co FeB50nm/TmIG 350nm for the first seven resonance modes of TmIG (n=0 -6) and \nthe uniform mode of CoFeB (n=0). b, Fitting of antiferromagnetic interfacial coupling energy \nby theoretical inversion. \n0 50 100 150 200 2501234frequency (Hz)\n-3E-4-2E-4-1E-4-7E-54E-68E-52E-42E-43E-4\nField (Oe)\n0 50 100 150 200 2501234frequency (Hz)\n-3E-4-2E-4-1E-4-7E-54E-68E-52E-42E-43E-4\nField (Oe) Theory fitting\n(a) \n(b) \n32 \n 1. Magnetic heterostructures: advances and perspectives in spinstructures and spintransport. (Springer \nVerlag, 2008). doi:10.1088/0031 -9112/23/4/020. \n2. Livesey, K. L., Crew, D. C. & Stamps, R. L. Spin wave valve in an exchange spring bilayer. Phys. Rev. \nB 73, 184432 (2006). \n3. Klingler, S. et al. Spin-Torque Excitation of Perpendicular Standing Spin Waves in Coupled YIG / Co \nHeterostructures. Phys. Rev. Lett. 120, 127201 (2018). \n4. Li, Y . et al. Coherent Spin Pumping in a Strongly Coupled Magnon -Magnon Hybrid System. Phys. Rev. \nLett. 124, 117202 (2020). \n5. Zhang, Z., Yang, H., Wang, Z., Cao , Y . & Yan, P. Strong coupling of quantized spin waves in \nferromagnetic bilayers. Phys. Rev. B 103, 104420 (2021). \n6. Hoffmann, F., Stankoff, A. & Pascard, H. Evidence for an Exchange Coupling at the Interface between \nTwo Ferromagnetic Films. Journal of Appl ied Physics 41, 1022 –1023 (1970). \n7. Hoffmann, F. Dynamic Pinning Induced by Nickel Layers on Permalloy Films. phys. stat. sol. (b) 41, \n807–813 (1970) \n8. Cochran, J. F. & Heinrich, B. Boundary conditions for exchange -coupled magnetic slabs. Phys. Rev. B \n45, 13096 –13099 (1992). \n9. Zhang, J., Yu, W., Chen, X. & Xiao, J. A frequency -domain micromagnetic simulation module based \non COMSOL Multiphysics. AIP Advance s 13, 055108 (2023). \n10. Zhang, X., Zou, C. -L., Jiang, L. & Tang, H. X. Strongly Coupled Magnons and Cavity Microwave \nPhotons. Phys. Rev. Lett. 113, 156401 (2014). \n11. Chen, J. et al. Strong Interlayer Magnon -Magnon Coupling in Magnetic Metal -Insulator Hybrid \nNanostructures. Phys. Rev. Lett. 120, 217202 (2018). \n12. Chen, J. et al. Excitation of unidirectional exchange spin waves by a nanoscale magnetic grating. Phys. \nRev. B 100, 104427 (2019). \n13. Khodadadi, B. et al. Conductivitylike Gilbert Damping due to Intraband Scattering in Epitaxial Iron. \nPhys. Rev. Lett. 124, 157201 (2020). \n14. Li, Y . et al. Giant Anisotropy of Gilbert Damping in Epitaxial CoFe Films. Phys. Rev. Lett. 122, 117203 \n(2019). \n15. Xiong, Y . et al. Probi ng magnon –magnon coupling in exchange coupled Y 3Fe5O12/Permalloy bilayers \nwith magneto -optical effects. Sci Rep 10, 12548 (2020). \n " }, { "title": "1901.09986v1.Spin_Hall_magnetoresistance_in_heterostructures_consisting_of_noncrystalline_paramagnetic_YIG_and_Pt.pdf", "content": "Spin Hall magnetoresistance in heterostructures consisting of noncrystalline\nparamagnetic YIG and Pt\nMichaela Lammel,1, 2,a)Richard Schlitz,3Kevin Geishendorf,1, 2Denys Makarov,4Tobias Kosub,4Savio Fabretti,3\nHelena Reichlova,3Rene Huebner,4Kornelius Nielsch,1, 2, 5Andy Thomas,1and Sebastian T.B. Goennenwein3,b)\n1)Institute for Metallic Materials, Leibnitz Institute of Solid State and Materials Science, 01069 Dresden,\nGermany\n2)Technische Universit at Dresden, Institute of Applied Physics, 01062 Dresden,\nGermany\n3)Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden, 01062 Dresden,\nGermany\n4)Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research,\n01328 Dresden, Germany\n5)Technische Universit at Dresden, Institute of Materials Science, 01062 Dresden,\nGermany\n(Dated: 30 January 2019)\nThe spin Hall magnetoresistance (SMR) e\u000bect arises from spin-transfer processes across the interface be-\ntween a spin Hall active metal and an insulating magnet. While the SMR response of ferrimagnetic and\nantiferromagnetic insulators has been studied extensively, the SMR of a paramagnetic spin ensemble is not\nwell established. Thus, we investigate herein the magnetoresistive response of as-deposited yttrium iron gar-\nnet/platinum thin \flm bilayers as a function of the orientation and the amplitude of an externally applied\nmagnetic \feld. Structural and magnetic characterization show no evidence for crystalline order or sponta-\nneous magnetization in the yttrium iron garnet layer. Nevertheless, we observe a clear magnetoresistance\nresponse with a dependence on the magnetic \feld orientation characteristic for the SMR. We propose two\nmodels for the origin of the SMR response in paramagnetic insulator/Pt heterostructures. The \frst model de-\nscribes the SMR of an ensemble of non-interacting paramagnetic moments, while the second model describes\nthe magnetoresistance arising by considering the total net moment. Interestingly, our experimental data are\nconsistently described by the net moment picture, in contrast to the situation in compensated ferrimagnets\nor antiferromagnets.\nSpin Hall magnetoresistance (SMR)1{3is commonly\nobserved in ferrimagnetic insulator (FMI)/normal metal\n(NM) heterostructures when the metal exhibits a large\nspin-orbit coupling. The SMR arises due to the interplay\nof the spin-transfer torque, the spin Hall e\u000bect (SHE) and\nthe inverse spin Hall e\u000bect at the FMI/NM interface.4{6\nWhile the SMR e\u000bect is usually discussed in terms of\nthe total (net) magnetization,1recent experimental work\nshowed that the SMR does not only probe the net magne-\ntization of FMIs, but is also sensitive to the contributions\nof the di\u000berent magnetic sublattices.7,8This observation\nis key to understand the SMR response of more com-\nplex magnetic systems, such as canted ferrimagnets7{9,\nantiferromagnets10{15, spin spirals16or helical phases.17\nTo date, SMR measurements have been performed ex-\ntensively in samples with di\u000berent long-range (sponta-\nneous) magnetic ordering.2,7,10,16{18In contrast, param-\nagnetic materials have not been in the focus of prior work\ndone for SMR measurements. However, the magnetore-\nsistive response of paramagnetic materials is an interest-\ning topic. For example, magnetoresistance measurements\nwere recently performed in a gated paramagnetic ionic\nliquid.19The presence of SMR has been reported by two\na)Electronic mail: m.lammel@ifw-dresden.de\nb)Electronic mail: sebastian.goennenwein@tu-dresden.degroups in di\u000berent magnetically ordered materials, in the\nparamagnetic phase above the ordering temperature.16,18\nSince the SMR is primarily studied in the magnetically\nordered phase in those works, the authors do not provide\na microscopic picture for the SMR in a randomly ordered\nspin ensemble. Therefore, in this work, we systematically\nstudy the SMR in a paramagnetic insulator (PMI)/spin\nHall metal bilayer and critically compare the experimen-\ntal results to the SMR expected from two di\u000berent micro-\nscopic models: one model assumes an ensemble of nonin-\nteracting moments, while the other model considers the\n(induced) net magnetization. More speci\fcally, we in-\nvestigate bilayers fabricated by sputtering of Y 3Fe5O12\n(YIG) and Pt at room temperature. These heterostruc-\ntures do not show a crystalline order of the YIG layer or\nspontaneous magnetization, such that we take the YIG\nlayer to be paramagnetic, but they nevertheless exhibit\na clear SMR-like magnetoresistive response.\nThe YIG/Pt bilayers were fabricated via sputtering\nat room temperature from 2 inch YIG and Pt tar-\ngets on commercially available (111)-oriented single-\ncrystalline yttrium aluminum garnet (Y 3Al5O12, YAG)\nsubstrates.2,7,9To rule out crystallization of the de-\nposited YIG layer on the YAG substrate due to the\nlow lattice mismatch, reference samples were fabricated\nin the same manner on (100)-oriented Si wafers ter-\nminated by a thermal oxide layer of 1 µm. The sub-\nstrates were immersed in isopropanol and ethanol andarXiv:1901.09986v1 [cond-mat.mes-hall] 28 Jan 20192\na) b)\nt jnc) d)YIGPt\nsub.\nFIG. 1. a) Schematic of the YIG/Pt bilayer sample. b) X-\nray di\u000braction \u0012-2\u0012scan of a typical noncrystalline YIG/Pt\nbilayer. The colored vertical lines give the expected positions\nfor di\u000berent crystalline di\u000braction peaks. c) Normalized mag-\nnetization of a noncrystalline YIG/Pt bilayer and two YAG\nsubstrates as a function of temperature. The light blue shad-\ning around the data indicates the scatter of two subsequent\nmeasurement runs. The observed moment is negative due to\nthe diamagnetic substrate. d) Electrical contacting scheme of\nthe patterned sample.\ncleaned in an ultrasonic bath prior to the deposition. The\nYIG layer was deposited via RF sputtering at 80 W for\n6000 s. Subsequently Pt was deposited using DC sput-\ntering for 73 s at 30 W without breaking the vacuum.\nA schematic of a typical stack is given in Fig.1a. The\nabove-mentioned sputtering parameters resulted in layer\nthicknesses of d YIG= (30\u00061) nm for the YIG layer and\ndPt= (2:5\u00060:5) nm for the Pt layer, as con\frmed by\nX-ray re\rectometry.\nX-ray di\u000braction measurements were performed using\na Bruker D8 Advanced di\u000bractometer equipped with a\ncobalt anode. As shown in Fig.1b, we do not observe\ndi\u000braction peaks that could be linked to YIG. We take\nthis as evidence that the YIG does not grow as small crys-\ntallites, but rather as an unordered \\amorphous\" layer.\nTherefore such YIG layers will be referred to as \\non-\ncrystalline\" in the following. In contrast, the Pt layer is\ntextured inh111i-direction which has been reported to\nbe the preferred orientation direction of Pt deposited at\nroom temperature.20The sharp step at 2 \u0012= 60 deg re-\nsults from the iron \flter that is used to suppress the Co K\f\nradiation. TEM studies on the YIG/Pt bilayers addition-\nally con\frm the noncrystallinity of the YIG layer while\nenergy-dispersive X-ray spectroscopy analyses show the\nYIG layer to be stoichometrically identical to the YAG\nsubstrate. For the magnetic characterization, a Quan-\ntum Design MPMS-XL7 SQUID magnetometer with re-\nciprocating sample option was used. Figure 1c shows themagnetization as a function of temperature measured at\n500 mT after cooling the sample in zero magnetic \feld.\nAs a reference, two substrates were measured by the iden-\ntical procedure. To ensure comparability and to account\nfor di\u000berences in sample size, the data were normalized to\nthe magnetization at 300 K. Comparing the normalized\nM(T) data from the YIG/Pt bilayer to the bare YAG\nsubstrates, we conclude that within the measurement\nerror, no (spontaneous) magnetization of the \flm can\nbe detected in our samples. Moreover, the samples ex-\nhibit only the negative magnetization expected for a dia-\nmagnetic substrate. The low temperature paramagnetic-\nlike behavior is likely caused by paramagnetic dopants\nthat were consistently observed in the commercial YAG\nsubstrates.21\nFor magnetotransport measurements, Hall bars with a\ncontact separation of l= 400 µm along the direction of\ncurrent \row and a width of w= 50 µm were de\fned by\nusing optical lithography and consecutive Ar ion etching.\nSubsequently, the samples were mounted into a chip car-\nrier and contacted by aluminum wire bonding. The elec-\ntric contacting scheme as well as the used coordinate sys-\ntem is given in Fig.1d. A current I= 90 µA was applied\nalong the Hall bar (along jdirection) utilizing a Keith-\nley 2450 sourcemeter. To decrease the noise level and to\nenhance the sensitivity, a current reversal technique was\nused.22The longitudinal voltage V, i.e. the voltage drop\nalong the direction of current \row, was measured by a\nKeithley 2182 nanovoltmeter.\nField orientation dependent magnetoresistance mea-\nsurements at di\u000berent temperatures and in three orthog-\nonal rotation planes were performed in a 3D vector mag-\nnetic \feld cryostat. The in-plane rotation of a constant\nexternal magnetic \feld Haround the surface normal nis\nherein denoted as ip (angle \u000b), the out-of-plane rotation\naround the current direction jas oopj (angle \f) and the\nout-of-plane rotation around the tdirection as oopt (an-\ngle\r), as is shown above Fig.2a, b and c, respectively.\nIn a model FMI/NM system with one single magnetic\nsublattice pointing along the magnetization unit vector\nm=M\nM, the SMR can be described by:1\n\u001a=\u001a0+ \u0001\u001a(1\u0000mt2)\n=\u001a0+ \u0001\u001a[1\u0000sin2(\u000b;\f)](1)\nwhere \u0001\u001a >0 gives the change of resistance as a func-\ntion of the projection of the magnetization unit vector\nmon the tdirection m t, as de\fned above. Thus, follow-\ning Eq.1, the resistance in a typical SMR measurement\nis minimal for mjjtand maximal for m?t. Figure 2\nshows the dependence of the magnetoresistance on the\nangles\u000b,\fand\r, at 200 K for di\u000berent amplitudes of\nthe external magnetic \feld. For each \feld amplitude, the\nvoltageVis recorded for clockwise and anticlockwise ro-\ntation of the magnetic \feld direction, and the data are\naveraged before normalization to account for slow tem-\nperature drifts. For the ip and the oopj rotation, the\ndata (open symbols) can be well described by a sin2(\u000b;\f)3\nt jn\nt jn\nt jn\na) b) c)ip oopj oopt\nFIG. 2. Magnetoresistance measurements as a function of\nmagnetic \feld orientation, recorded at 200 K in three or-\nthogonal rotation planes. The rotation geometries are dis-\nplayed above the corresponding panel. Measurements were\nperformed at di\u000berent, \fxed magnetic \feld strengths \u00160H=\n0:5 T, 1 T, 1:5 T and 2 T, which are given by the open blue\nrhombi, yellow triangles, red squares and gray circles, respec-\ntively. A sin2(\u000b;\f;\r ) \ft to the data is given by the solid line\nin the associated color (cf. Eq.1).\ndependence (solid line), whereas no modulation is vis-\nible in the oopt rotation. This angular dependence is\ncharacteristic for both the SMR1,2(cf. Eq.1) and the\nHanle magnetoresistance (HMR).23The HMR is micro-\nscopically ascribed to the dephasing of the spin accumu-\nlation at the Pt interface, also exists in pure Pt without\nthe adjacent magnetic layer, and scales with the exter-\nnal magnetic \feld H. In contrast, the SMR depends on\nthe magnetization orientation m, which is expected to be\n\feld dependent in a paramagnetic material. However, we\ndo not expect the HMR to contribute signi\fcantly to our\nresults, since the magnitude of the HMR for the external\nmagnetic \felds used in our measurements ( \u00160H\u00142 T)\nis reported to be \u0001 \u001aHMR=\u001a0\u00192:5\u000210\u00006and there-\nfore roughly one order of magnitude smaller than the\nMR ratio observed here23\u0000\u0001\u001a=\u001a0=\u00003\u000210\u00005. There-\nwith, \u0001\u001a=\u001a0of the noncrystalline YIG/Pt bilayers on\nYAG is roughly one order of magnitude smaller than in\ncomparable samples featuring a crystalline, ferrimagnetic\nYIG layer.2,3Similar measurements performed on refer-\nence non-crystalline YIG/Pt bilayers, in particular also\non the ones on Si/SiO 2substrates, show the same de-\npendencies, with a magnetoresistance in the same order\nof magnitude, further supporting the fact that it is in-\ndeed the non-crystalline YIG layer that is responsible\nfor the SMR. The angular dependence additionally dis-\nputes long-range antiferromagnetic ordering in our bi-\nlayers, since a shift of the extrema by 90 deg compared\nto the SMR introduced in Eq.1 would be expected for\nan AFM.10{12,18,24The existence of a magnetoresistance\nin PMI/Pt heterostructures has already been reported\nabove the Curie temperature for CoCr 2O4/Pt bilayers\na) b) c)FIG. 3. Field-dependent magnetoresistance measurements at\ndi\u000berent temperatures for Hjjj(panel a), Hjjt(panel b) and\nHjjn(panel c). Measurements were performed at di\u000berent\ntemperatures T = 10 K, 100 K, 200 K and 300 K, which are\ngiven by the open gray rhombi, blue triangles, violet squares\nand red circles, respectively.\nwith a MR ratio of \u0001 \u001a=\u001a0<2\u000210\u00006by Aqeel et al.16\nand above the Neel temperature for Cr 2O3/Pt bilayer\nstructures by Schlitz et al.18with \u0001\u001a=\u001a0>1\u000210\u00004.\nIt has also been reported that no MR was detectable in\nparamagnetic Gd 3Ga5O12(GGG)/Pt heterostructures at\nroom temperature. In contrast to the other systems, the\nmagnetic moment of GGG has its origin in the 4 felec-\ntrons (vs. 3 delectrons), which have been suggested not to\ncouple well to the spin accumulation of the Pt layer due\nto their strong localization.18Therewith, the reported\nvalues of the MR ratio in paramagnetic phases vary be-\ntween zero and nearly the MR of crystalline YIG/Pt\nbilayers.16,18Our data fall well within this range. We,\nhowever, address a real paramagnet and not the param-\nagnetic phase of a system that orders at lower tempera-\ntures.\nField-dependent measurements were performed on the\nsame sample and in the same experimental setup as de-\nscribed before. The resistivity ratio \u001a=\u001aH=0\u00001 forHjjj\n(Hjjt,Hjjn) is given in Fig.3a (b, c). Note that the re-\nsults are normalized with respect to the value at zero\nmagnetic \feld to ensure comparability of the data ac-\nquired for di\u000berent temperatures. An increase in the\nexternal magnetic \feld magnitude along the jandndi-\nrection leads to an increase in resistivity. For an ex-\nternal magnetic \feld applied along tdirection, how-\never, no substantial modulation of the resistivity is ob-\nserved. For these results a signi\fcant contribution of the\nHMR is again excluded since the reported resistivity ratio\n\u0001\u001a=\u001a\u00192:5\u000210\u00006at 2 T is one order of magnitude lower\nthan the values for the same \feld magnitude in Fig.3.23\nFor low temperatures, an increase in the external mag-\nnetic \feld does lead to a parabolic increase in the tand\nndirection that we ascribe to a Kohler's rule type ordi-\nnary magnetoresistance in the Pt layer.25Additionally,\nweak antilocalization has been reported to occur in Pt\nthin \flms on various substrates for temperatures below\n50 K which might give an additional contribution to the\n\feld dependency.23,26,27No saturation of the resistance\ncan be observed in our samples, even for the highest mag-\nnetic \felds that can be applied in our setup ( \u00062 T in the4\na) b) c)\nHsat -Hsat0d) e) f)\nHsat -Hsat0 Hsat -Hsat0jt\nn\nFIG. 4. Assuming that all magnetic moments in a param-\nagnet contribute individually to the SMR (i.e. hmt2i), one\nobtains the evolution of the resistivity sketched as a blue line\nforHjjj(a),Hjjt(b),Hjjn(c). Presuming that the SMR is\ndependent on the net magnetization (i.e. hmti2), the expected\nmagnetoresistive response is shown as a red line in panel d,\ne and f, for Hjjj,HjjtandHjjn, respectively. The external\nmagnetic \feld applied in each direction is indicated by the\narrows below the panels. The alignment of the magnetic mo-\nments for di\u000berent external \feld amplitudes is given by the\narrows on above and below the resistance curves.\njandtdirection and\u00066 T in the ndirection) and the\nlowest temperatures accessible (10 K).\nWe now compare the SMR expected from a model\nthat considers an ensemble of non-interacting moments\n(Fig.4a-c) with the SMR arising in a model that addresses\nthe total net moment (Fig.4d-f). To that end, we con-\nsider external magnetic \felds applied along j,tandndi-\nrection as schematically shown above the panels in Fig.4.\nIn a paramagnet, the moments at zero magnetic \feld are\nunordered and point in random directions, as schemati-\ncally sketched in Fig.4. For a su\u000eciently large magnetic\n\feld H!\u0006 Hsat, though, the majority of the magnetic\nmoments are aligned collinear to the external \feld. Since\nthe SMR is sensitive to m t2, one would expect the small-\nest resistivity (i.e. \u001a0) when all moments are parallel to\nthetdirection and the largest resistivity (i.e. \u001a0+ \u0001\u001a)\nwhen all moments are perpendicular to the tdirection\n(c.f. Eq.1). Thus, \u001a0is the value expected for open\nboundary conditions (i.e. no magnetic layer), while the\nintroduction of a magnetic layer to the system can only\nlead to an increase in the resistivity depending on its\nmagnetization orientation.1\nPresuming non-interacting moments in the PMI, the\ncontribution of each moment to the SMR is considered\nseparately. Thus, m t2in Eq.1 has to be understood as\nhmt2i. Applying su\u000eciently large magnetic \felds leads\nto a saturation of the resistivity at a minimum value \u001a0\nforHjjt(Fig.4b) and maximum value \u001a0+\u0001\u001aforHjjj;n\n(Fig.4a, c). In turn, this implies that for zero magnetic\n\feld an intermediate resistance \u001a(H = 0) is expected with\u001a0<\u001a(H = 0)<\u001a0+\u0001\u001a, since some but not all magnetic\nmoments are collinear to the tdirection (panel a, b and\nc of Fig.4 at H = 0).\nAssuming that the SMR depends on the total net mag-\nnetization in the paramagnetic layer instead, m t2in Eq.1\nhas to be understood as hmti2. Hence, no magnetic mo-\nment is expected for zero external magnetic \feld, which\nleads to a vanishing SMR and therefore to the minimum\nresistivity value \u001a0(Fig.4d-f). Applying su\u000eciently large\nexternal magnetic \felds along jandnleads to an increase\nin resistivity up to the saturation value of \u001a0+ \u0001\u001aas is\nshown in Fig.4d and e. In contrast, no change in the\nresistivity is expected for \felds applied along tdirection\n(Fig.4e).\nComparing the expected behavior from Fig.4 with the\n\feld-dependent measurements in Fig.3, shows that the\nmeasurements do not agree with the SMR stemming from\nan ensemble of non-interacting moments (see Fig.4a-c).\nInstead, the measurements corroborate that the SMR\nin the paramagnetic YIG/Pt bilayers is determined by\nthe total net magnetization of the system (cf. Fig.4d-e).\nThis result contradicts previous \fndings in compensated\ngarnets and antiferromagnets where the results were ex-\nplained by taking into consideration the magnetizations\nof the di\u000berent sublattices separately.7,8,10,11To con\frm\nthe saturation of the e\u000bect and to corroborate our ob-\nservations, further experimental work on di\u000berent PMI\nmaterials, as well as at higher external magnetic \felds\nand/or lower temperatures, is necessary.\nIn summary, we have studied the magnetoresistive\nresponse in noncrystalline, paramagnetic YIG/Pt het-\nerostructures. Upon rotating the external magnetic \feld\nat a \fxed magnitude in di\u000berent planes, we observe a\nmagnetoresistance with the characteristics of the spin\nHall magnetoresistance with a magnitude of j\u0001\u001a=\u001aj=\n3\u000210\u00005at\u00160H = 2 T and 200 K. Field-dependent mea-\nsurements show an increase of resistivity for an increasing\nmagnetic \feld along jandndirection, whereas no change\nof the resistance is observed for \felds applied along tdi-\nrection. No saturation is detected for the maximum mag-\nnetic \felds accessible in our setup. Furthermore, we pro-\npose two possible models for the origin of the SMR in a\nsimple paramagnetic insulator/Pt heterostructure taking\ninto consideration either an ensemble of non-interacting\nmoments (i.e.hmt2i), or the total net magnetization (i.e.\nhmti2). Comparing the experimentally observed signa-\nture with those models, we \fnd that the data are better\ndescribed in terms of the total moment picture. Thus,\nwe conclude that in a paramagnetic insulator, the mo-\nments do not contribute individually, but in a collective\nnet fashion to the SMR.\nWe thank K. Nenkov and B. Weise for technical\nsupport. We acknowledge \fnancial support by the\nDeutsche Forschungsgemeinschaft via SPP 1538 (project\nGO 944/4).\n1Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B.\nGoennenwein, E. Saitoh, and G. E. W. Bauer, Physical Review\nB87, 144411 (2013).5\n2M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Alt-\nmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross,\nD. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss,\nL. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and\nS. T. B. Goennenwein, Physical Review B 87, 224401 (2013).\n3H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kaji-\nwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi,\nR. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh,\nPhysical Review Letters 110, 206601 (2013).\n4D. Ralph and M. Stiles, Journal of Magnetism and Magnetic\nMaterials 320, 1190 (2008).\n5M. Dyakonov and V. Perel, Physics Letters A 35, 459 (1971).\n6J. E. 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Hou, Z. Qiu, J. Barker, K. Sato, K. Yamamoto, S. V\u0013 elez,\nJ. M. Gomez-Perez, L. E. Hueso, F. Casanova, and E. Saitoh,\nPhysical Review Letters 118, 147202 (2017).\n14J. H. Han, C. Song, F. Li, Y. Y. Wang, G. Y. Wang, Q. H. Yang,\nand F. Pan, Physical Review B 90, 144431 (2014).\n15Y. Ji, J. Miao, K. K. Meng, Z. Y. Ren, B. W. Dong, X. G.Xu, Y. Wu, and Y. Jiang, Applied Physics Letters 110, 262401\n(2017).\n16A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda,\nB. J. van Wees, and T. T. M. Palstra, Physical Review B 92,\n224410 (2015).\n17A. Aqeel, M. Mostovoy, B. J. van Wees, and T. T. M. Palstra,\nJournal of Physics D: Applied Physics 50, 174006 (2017).\n18R. Schlitz, T. Kosub, A. Thomas, S. Fabretti, K. Nielsch,\nD. Makarov, and S. T. B. Goennenwein, Applied Physics Letters\n112, 132401 (2018).\n19L. Liang, J. Shan, Q. H. Chen, J. M. Lu, G. R. Blake, T. T. M.\nPalstra, G. E. W. Bauer, B. J. van Wees, and J. T. Ye, Physical\nReview B 98, 134402 (2018).\n20J. Narayan, P. Tiwari, K. Jagannadham, and O. W. Holland,\nApplied Physics Letters 64, 2093 (1994).\n21The measured moment of the substrates corresponds to a con-\ncentration of paramagnetic dopants of \u001910\u000017cm\u00001. Besides,\none would expect a magnetic moment of \u001910\u00008Am2at 10 K as-\nsuming that all Fe moments of YIG are independently contribut-\ning whereas a magnetic moment of \u001910\u000014Am2is expected\nfor the YAG substrate with assuming a susceptibilty of YAG of\n\u001fYAG = 10\u00005.\n22S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn,\nM. Althammer, R. Gross, and H. Huebl, Applied Physics Letters\n107, 172405 (2015).\n23S. V\u0013 elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta,\nM. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and\nF. Casanova, Physical Review Letters 116, 016603 (2016).\n24L. Baldrati, A. Ross, T. Niizeki, R. Ramos, J. Cramer,\nO. Gomonay, E. Saitoh, J. Sinova, and M. Kl aui,\narXiv:1709.00910 (2017).\n25M. Kohler, Annalen der Physik 424, 211 (1938).\n26H. Ho\u000bmann, F. Hofmann, and W. Schoepe, Physical Review B\n25, 5563 (1982).\n27Y. Niimi, D. Wei, H. Idzuchi, T. Wakamura, T. Kato, and\nY. Otani, Phys. Rev. Lett. 110, 016805 (2013)." }, { "title": "1701.05320v1.Separation_of_inverse_spin_Hall_effect_and_anomalous_Nernst_effect_in_ferromagnetic_metals.pdf", "content": "arXiv:1701.05320v1 [cond-mat.mes-hall] 19 Jan 2017Separation ofinversespinHalleffect and anomalousNernst effect inferromagnetic metals\nHao Wu, Xiao Wang, Li Huang, Jianying Qin, Chi Fang, Xuan Zhan g, Caihua Wan, and Xiufeng Han∗\nBeijing National Laboratory for Condensed Matter Physics, Institute of Physics,\nUniversity of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China\nInverse spinHalleffect (ISHE)inferromagnetic metals(FM ) canalsobe used todetect the spin current gen-\neratedbylongitudinal spinSeebeck effect ina ferromagnet ic insulator YIG. However, anomalous Nernst effect\n(ANE)in FMitself always mixes inthe thermal voltage. Inthi s work, the exchange bias structure (NiFe/IrMn)\nisemployed toseparate these twoeffects. The exchange bias structure provides a shift fieldtoNiFe,which can\nseparate the magnetization of NiFe from that of YIG in M-Hloops. As a result, the ISHE related to magneti-\nzationof YIG and the ANE relatedtothe magnetization of NiFe can be separated as well. Bycomparison with\nPt, a relative spin Hall angle of NiFe (0.87) is obtained, whi ch results from the partially filled 3 dorbits and\nthe ferromagnetic order. This work puts forward a practical method to use the ISHE in ferromagnetic metals\ntowards future spintronic applications.\nSpincaloritronicsfocusesoncouplingheat,spinandcharg e\nin magnetic materials. [1] Spin Seebeck effect (SSE) ori-\ngins from the excitation of spin wave in magnetic materials\nby a temperature gradient, which can pump a spin current\ninto a contact metal. [2–5] In the past several years, SSE\nhas been achieved in magnetic metals[2], semiconductors[6 ]\nand insulators. [3, 4] Especially SSE in magnetic insulator s\ndraws many attentions since a pure spin current without any\nchargeflowisoneofthemostdesirablepropertiesfordevice s\nwith dramatically reduced power consumption. Transverse\nand longitudinal spin Seebeck effect are divided by differe nt\nexperimental configurations, while the detected spin curre nt\ncan be perpendicular or parallel to the temperature gradien t.\nEspecially longitudinal spin Seebeck effect (LSSE) in ferr o-\nmagneticinsulatorsiswidelyusedtopumpaspincurrentint o\ntheneighboringmaterials.\nInversespinHalleffect(ISHE)canconvertthespincurrent\nJsintothechargecurrent Je,whichcanbedetectedbyavolt-\nage signal: EISHE= (θSHρ)Js×σ, whereEISHEis the ISHE\nelectric field, θSHis the spin Hall angle, ρis the resistivity,\nandσis the unit vector of the spin. [7–9] It is generally be-\nlieved that the magnitude of spin Hall angle θSHdepends on\nthe strength of spin orbit coupling(SOC), and the strength o f\nSOC is proportionalto Z4, whileZis the atomic number, so\nheavymetals(HM)withlarge ZhavearelativelargespinHall\nangle. [10]\nSimilar to spin Hall effect (SHE) in non-magnetic metals,\n[11–13]anomalousHalleffect(AHE)inferromagneticmetal s\n(FM) comes fromthe spin dependentscattering of the charge\ncurrent. [14]Duetothespinpolarizationofthechargecurr ent\nin FM, the spin accumulationaccompaniedwith a charge ac-\ncumulationcanbegeneratedinthetransversedirection. Wh en\na pure spin current is injected to FM, as the inverse effect of\nAHE, ISHE in FM provides another potential application in\ndetectingthespincurrentbychargesignals.\nRecent works draw attention on using the ISHE in FM to\ndetect the spin currentgeneratedby LSSE in a ferromagnetic\ninsulatorY 3Fe5O12(YIG).[15–18] However,thetemperature\ngradient will also introduce additional anomalous Nernst e f-\n∗Email: xfhan@iphy.ac.cnfect (ANE)in FM: EANE∝ ∇Tz×M, whereEANEisthe ANE\nelectricfield, ∇Tzisthetemperaturegradientalongthethick-\nnessdirection,and MisthemagneticmomentofFM.[19,20]\nTherefore,the separationof ANE and ISHE in FM is in great\ndemand. Several works have used two magnetic materials\nwith different coercivity to separate the ISHE (related to t he\nmagnetization of YIG) and ANE (related to the magnetiza-\ntion of the ferromagnetic detector).[16, 17] However, ISHE\nandANEarestillmixedwitheachother,whichpreventsusto\ndirectlydetectthespin currentbyFM.\nIn this work, we designed the exchange bias structure\n(NiFe/IrMn) to detect the spin current generated by LSSE in\nYIG. A Cu layer with a negligible spin Hall angle is inserted\nbetween NiFe and YIG to reduce the magnetic coupling be-\ntweenYIGandNiFe,andthespincurrentcanalsopasswith-\nout too much loss at the same time. The exchangebias struc-\nture provides a bias field for NiFe, which can separate the\nmagnetizationswitching processof NiFe fromthat of YIG in\nM-Hloops. [21, 22] As a result, the ISHE related to magne-\ntization of YIG and the ANE related to the magnetization of\nNiFecanbeseparatedaswell. Moreimportantly,wecaneven\nobserve the only ISHE contribution in a field range which is\nsmaller than the exchange bias field that only the magneti-\nzation of YIG switches, while the magnetization of NiFe is\nfixed.\nTheexchangebiasstructureCu(5)/NiFe(5)/IrMn(12)/Ta(5 )\n(thickness in nanometers) was fabricated on polished 3.5 µm\nYIG films on GGG substrates by a magnetron sputtering\nsystem. In order to introduce the exchange bias effect in\nFM/AFM, an in-plane magnetic field was applied during the\ndepositionprocess. ThespinSeebeckvoltagewasmeasureby\nananovoltmeter(Keithley2182A)inaSeebeckmeasurement\nsystem with a Helmholtz coil, and the longitudinal temper-\nature difference along the thickness direction was measure d\nbetween the bottom of the GGG substrate and the top of the\nfilm. All datawasperformedatroomtemperature.\nFig. 1(a)showstheschematicdiagramofthemeasurement\nmethod and the physical process. In longitudinal spin See-\nbeck measurement, the temperature gradient ( ∇T) is applied\nalong the out-of-plane zdirection, and the magnetic field is\nscanned along xdirection (also the direction of the exchange\nbias field). According to EISHE= (θSHρ)Js×σ, the thermal\nvoltageshouldbe measuredalong ydirection. Firstly, aspre-2\nFIG. 1. Under the longitudinal temperature gradient, only t he inverse spin Hall effect (ISHE) exists in YIG/Pt sample, w hile both ISHE and\nthe anomalous Nernst effect (ANE)exist inYIG/Cu/NiFe/IrM n/Tasample. Once an insulating layer MgO is insertedbetwee n NiFe and YIG,\nthe spincurrent willbe blocked, soISHEvanishes whileANE s tillexists.\nFIG.2. Highresolutiontransmissionelectronmicroscopy( HRTEM)resultsoftheYIG/Cu/NiFe/IrMn/Tasample(a)andse lectedareaelectron\ndiffraction (SAED)patterns of the YIG region (b). M-Hloops measured inSi-SiO 2/Cu/NiFe/IrMn/Ta (c) and YIG/Cu/NiFe/IrMn/Ta(d), and\nthe magnetic fieldis appliedalong the directionof exchange bias field.\nviousworks,we use a Pt layer whichhas a relative largespin\nHallangleabout0.1[23–25]tomeasurethepurespinSeebeck\ninduced ISHE signal. Then, we changed the Pt with the ex-\nchangebiasstructureCu/NiFe/IrMn/Ta. ApartfromtheISHE\nsignal, ANE from NiFe itself will also contribute to the ther -\nmal voltage. Once we inserted an insulating layer MgO to\nblock the spin current injected from YIG to NiFe, ISHE sig-\nnalshouldvanishwhereonlyANEfromNiFe exists.\nFig. 2(a) shows the cross-sectional transmission electron\nmicroscopy (HRTEM) results of the YIG/Cu/NiFe/IrMn/Ta\nmultilayers,andtheinterfacebetweenCuandYIGisveryflat\nand clear. The selected area electron diffraction(SAED) pa t-\nternisshowninFig. 2(b),demonstratingthattheepitaxial di-\nrectionofYIGfilmcrystalisalongthe(111)directionandth elattice parameter is 12.4 ˚A. A Si-SiO 2/Cu/NiFe/IrMn/Ta ref-\nerencesampleisusedtochecktheexchangebiaseffect,wher e\na 220 Oe exchange bias field is obtained from the M-Hloop\n(alongxdirection) [Fig. 2(c)]. Then we measured the mag-\nnetic propertiesof YIG/Cu/NiFe/IrMn/Ta sample [Fig. 2(d) ],\nandthesaturationmagnetization MsofYIGis120emu/ccand\nthesaturationfieldofYIGislessthan10Oe. Fromthezoom-\ninfigurein Fig. 2(d),we cansee the magnetizationswitching\nrangeofNiFe/IrMnexchangebiasstructureisfrom150Oeto\n250Oe,whichisfarfromthe rangeofYIG.\nThen we measured the magnetic field dependence of the\nthermal voltage, and the longitudinal temperature differe nce\n∆Tkeeps 13 K during the measurement in Fig. 3(a)-(c).\nFirstly, as conventional LSSE measurement, a heavy metal3\nFIG. 3. (a)-(c) shows the spin dependent thermal volt-\nage measurement of YIG/Pt, YIG/Cu/NiFe/IrMn/Ta, and\nYIG/MgO/NiFe/IrMn/Ta samples, where the magnetic field is\napplied along xdirection (also the direction of the exchange bias\nfield).\nPt is used to measure the spin current generated by SSE in\nYIG, anda 0.4 µV ISHE voltagewhichis relatedto the mag-\nnetization of YIG is observed, as shown in Fig. 3(a). While\nin YIG/Cu/NiFe/IrMn/Ta structure, due to the exchange bias\neffect, ISHE (related to the magnetization of YIG) and ANE\n(related to the magnetization of NiFe) exist in different fie ld\nranges. The ISHE signal is around 0 Oe and the ANE signal\nis around 150 Oe, and the exchange bias field here is a little\nsmaller than that from the M-Hloop because the exchange\nbias decreaseswith the increasedtemperature,as seen in Fi g.\n3(b). It is worth noting that the polarityof ISHE and ANE in\nNiFe is the same, which is contrary to the result for CoFeB.\n[16] In order to prove that the ISHE voltage related to the\nmagnetizationofYIG indeedcomesfromthe spin currentin-\njection from YIG to NiFe, we insert an insulating layer MgO\ntoblockthisspincurrent. Asexpected,theISHEsignaldisa p-\npearswhiletheANEstill exists,asseeninFig. 3(c). ForSSE\ninheavymetal/ferromagnetstructures,themixingofmagne tic\nproximity effect [26, 27] (MPE) and ANE has a debate for a\nlong time, which means that magnetized Pt shows some fer-\nromagneticpropertiesintransportmeasurementsuchasANE\nandAHE.Inourwork,wedirectlyuseaFMtodetectthespin\ncurrent generatedby SSE, and our results show that although\nboth ISHE and ANE take place in the thermal voltage, how-\never,byusingtheexchangebiaseffect,ISHEandANEcanbe\nseparated in different field ranges. These results demonstr ate\nthat SSE and ANE share different physical origins, and ANE\nisnottheessential conditionofSSE.\nThen, we changed the temperature differences ∆Tfrom\n2.5 K to 13K, and the field dependent ISHE voltage for dif-\nFIG.4. (a)and(b)showthespindependentthermalvoltageme asure-\nment in YIG/Pt, YIG/Cu/NiFe/IrMn/Ta samples respectively under\nthe varied longitudinal temperature difference from 2.5 K t o 13 K,\nandtheoffsetvoltagehasbeenremovedtoobtainthefielddep endent\nISHE contribution. The magnetic field is applied along xdirection\n(also the direction of the exchange bias field), and the field r ange is\nsmallerthan the exchange bias field.\nferent∆Tis shown in Fig. 4(a) (YIG/Pt) and Fig. 4(b)\n(YIG/Cu/NiFe/IrMn/Ta). The ISHE voltages gradually in-\ncrease with increasing the temperature gradient in both sam -\nples, which are in accordance with the spin Seebeck mecha-\nnism. Under a ±80 Oe field range which is smaller than the\nexchangebiasfield, onlythe pureISHE signalswithout ANE\nshows that the comparableutility of FM (NiFe) with conven-\ntionalheavymetals(Pt)indetectingthespincurrent. Beca use\ninthiscaseonlythemagnetizationofYIGreverses,whileth e\nmagnetization of NiFe keeps fixed. And the optimization of\nFMwithlargespinHallanglewillbeanessentialsteptoward s\nfutureapplications.\nIn order to compare the spin Hall angle in Pt and NiFe,\nthe ISHE voltage is normalized by the resistance of the de-\ntecting electrode R, and the VISHE/R-∆Tcurve is fitted by the\nlinear shape, as seen in Fig. 5. VISHE/R=βθSH∆T, where\nβrepresents the efficiency from thermal current to the de-\ntected spin current. If we assume the same βin YIG/Pt and\nYIG/Cu/NiFe/IrMn/Ta samples, we can calculate the relativ e\nspin Hall angle of NiFe: θSH(NiFe)/θSH(Pt)≈0.87, which is\nclosetoourpreviousresult(0.98)bytransverseSSEmeasur e-\nment. [28]\nInconventionalunderstandingofSOC,thestrengthofSOC\nfollows a Z4dependence. While NiFe is composed of light\natoms, so SOC in NiFe should be small in this mechanism.\nHowever,previousworkshave also shownthat SOC not only\ndependson the atomic number Zbut also dependson the fill-\ning ofd-orbit, both Ni and Fe have partially filled 3 dorbits,4\nFIG. 5. The VISHE/R−∆Tcurves and the linear fitting curves mea-\nsured in YIG/Pt and YIG/Cu/NiFe/IrMn/Ta samples, and the IS HE\nvoltage has been normalized by the resistance of the detecti ng elec-\ntrode.\nsoSOCfromthe d-orbitfillingcouldtakeanimportantrolein\nNiFe. [29, 30] Moreover,ferromagneticorderinducedintri n-\nsic spin dependent scattering which is solely determined by\nthe electronic band structure can also contribute to the ISH E\nin FM, because the ISHE in FM is independentof its magne-tization. [31]\nIn conclusion, we have designed the exchange bias struc-\nture (NiFe/IrMn) to separate the ISHE and ANE in FM. As\nexpected, the ISHE related to magnetization of YIG and the\nANE related to the magnetization of NiFe can be separated\nin different ranges of magnetic field. By linear fitting the\nVISHE/R-∆Tcurves of NiFe and Pt, we calculated the relative\nspin Hall angle θSH(NiFe)/θSH(Pt)=0.87, and the partial filling\nof 3dorbits and the ferromagneticorder play important roles\nin thislargespin Hall angle ofNiFe. Thisworkdemonstrates\nthat ferromagnetic metals can also be used to detect the spin\ncurrentinspintronicsdevices.\nACKNOWLEDGMENTS\nThis work was supported by the 863 Plan Project of\nMinistry of Science and Technology (MOST) [Grant No.\n2014AA032904], the National Key Research and Develop-\nment Program of China [Grant No. 2016YFA0300802], the\nNationalNaturalScienceFoundationofChina(NSFC)[Grant\nNos. 11434014, 11404382], and the Strategic Priority Re-\nsearch Program (B) of the Chinese Academy of Sciences\n(CAS)[GrantNo. XDB07030200].\n[1] G. E. W. Bauer, E. Saitoh, and B. 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B 94, 020403 (2016)." }, { "title": "1612.08142v1.Concomitant_enhancement_of_longitudinal_spin_Seebeck_effect_with_thermal_conductivity.pdf", "content": "Concomitant enhancement of longitudinal spin Seebeck e\u000bect\nwith thermal conductivity\nRyo Iguchi,1,\u0003Ken-ichi Uchida,1, 2, 3, 4Shunsuke Daimon,1, 5and Eiji Saitoh1, 4, 5, 6\n1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n2National Institute for Materials Science, Tsukuba 305-0047, Japan\n3PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan\n4Center for Spintronics Research Network,\nTohoku University, Sendai 980-8577, Japan\n5WPI Advanced Institute for Materials Research,\nTohoku University, Sendai 980-8577, Japan\n6Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\nAbstract\nWe report a simultaneous measurement of a longitudinal spin Seebeck e\u000bect (LSSE) and thermal\nconductivity in a Pt/Y 3Fe5O12(YIG)/Pt system in a temperature range from 10 to 300 K. By\ndirectly monitoring the temperature di\u000berence in the system, we excluded thermal artifacts in\nthe LSSE measurements. It is found that both the LSSE signal and the thermal conductivity\nof YIG exhibit sharp peaks at the same temperature, di\u000berently from previous reports. The\nmaximum LSSE coe\u000ecient is found to be SLSSE>10\u0016V=K, one-order-of magnitude greater\nthan the previously reported values. The concomitant enhancement of the LSSE and thermal\nconductivity of YIG suggests the strong correlation between magnon and phonon transport in the\nLSSE.\n\u0003iguchi@imr.tohoku.ac.jp\n1arXiv:1612.08142v1 [cond-mat.mtrl-sci] 24 Dec 2016A spin counterpart of the Seebeck e\u000bect, the spin Seebeck e\u000bect (SSE) [1], has attracted\nmuch attention from the viewpoints of fundamental spintronic physics [2{4] and future\nthermoelectric applications [5{7]. The SSE converts temperature di\u000berence into a spin\ncurrent in a magnetic material, which can generate electrical power by attaching a conductor\nwith spin{orbit interaction [1, 5]. The SSE originates from thermally-excited magnons, and it\nappears even in magnetic insulators. In fact, after the pioneer work by Xiao et al. [8], the SSE\nhas been discussed in terms of the thermal non-equilibrium between magnons in a magnetic\nmaterial and electrons in an attached conductor. Recent experimental and theoretical works\nhave been focused on the transport and excitation of magnons contributing to the SSE\nin the magnetic material [9{17], whose importance can be recognized in the temperature\ndependence, magnetic-\feld-induced suppression, and thickness dependence of SSEs [18{\n23]. Most of the SSE experiments have been performed by using a junction comprising a\nferrimagnetic insulator Y 3Fe5O12(YIG) and a paramagnetic metal Pt since YIG/Pt enables\npure driving and e\u000ecient electric detection of spin-current e\u000bects; a YIG/Pt junction is now\nrecognized as a model system for the SSE studies.\nFigure 1(a) shows a schematic illustration of the SSE in an YIG/Pt-based system in\na longitudinal con\fguration, which is a typical con\fguration used for measuring the SSE.\nIn the longitudinal SSE (LSSE) con\fguration, when a temperature gradient rTis applied\nalong thezdirection, it generates a spin current across the YIG/Pt interface [5, 8, 24, 25].\nThis thermally-induced spin current is converted into an electric \feld ( EISHE) by the inverse\nspin Hall e\u000bect (ISHE) in Pt according to the relation [26, 27]\nEISHE/Js\u0002\u001b; (1)\nwhere Jsis the spatial direction of the spin current and \u001bis the spin-polarization vector of\nJs, which is parallel to the magnetization Mof YIG [see Fig. 1(a)]. When Mis along the\nxdirection, the LSSE is detected as a voltage, VLSSE =R\nEISHEdy, between the ends of the\nPt layer along the ydirection.\nIn the LSSE research, temperature dependence of the voltage generation has been essen-\ntial for investigating its mechanisms, such as spectral non-uniformity of magnon contribu-\ntions [21{23] and phonon-mediated e\u000bects [18, 19, 28]. The recent studies demonstrated that\nthe LSSE voltage in a single-crystalline YIG slab exhibits a peak at a low temperature, and\nthe peak temperature is di\u000berent from that of the thermal conductivity of YIG [20{22, 29].\n2MYIG slab\nPt film (bottom)Pt film (top)\n x zy\n∇T\nTT\nJsAu electrode\nEISHE (a)\n(b) (c)\nthermal grease (Apiezon N)Au electrode\nHsapphire plate\nheat bathheat bath\nPt/YIG/Pt sample\nsapphire plateIn \n∆T\nT 5 K (fixed)TH\nTLTave∇TTHset\nTLset\nH\nAu wire\nzFIG. 1. (a) A schematic illustration of the Pt/YIG/Pt sample. rT,H,M,EISHE, and Js\ndenote the temperature gradient, magnetic \feld (with the magnitude H), magnetization vector,\nelectric \feld induced by the ISHE, and spatial direction of the thermally generated spin current,\nrespectively. The electric voltage VH(VL) and resistance RH(RL) between the ends of top (bottom)\nPt layers were measured using a multimeter. (b) Experimental con\fguration for applying rT. The\nthickness and width of the sapphire plates are 0.33 and 2.0 mm, respectively. (c) A schematic plot\nof temperature pro\fle along the zdirection.\nThe di\u000berence in the peak temperatures was the basis of the recently-proposed scenario that\nthe LSSE is due purely to the thermal magnon excitation, rather than the phonon-mediated\nmagnon excitation [9, 12, 15].\nIn this paper, we report temperature dependence of LSSE in an YIG/Pt-based system\nfree from thermal artifacts and its strong correlation with the thermal conductivity of YIG.\nThe LSSE signal and thermal conductivity were simultaneously measured without changing\nthe experimental con\fguration. The intrinsic temperature dependence of the LSSE shows\nsigni\fcant di\u000berence from the prior results, while the thermal conductivity shows good\nagreement with the previous studies; the peak temperatures are found to be exceedingly\nclose to each other. These data will be useful for comparing experiments and theories\nquantitatively and for developing comprehensive theoretical models for the SSE.\nThe quantitative measurements of the thermoelectric properties are realized by directly\nmonitoring the temperature di\u000berence between the top and bottom surfaces of a single-\n3crystalline YIG slab. To do this, we extended a method proposed in Ref. 23, which is based\non the resistance measurements of Pt layers covering the top and bottom surfaces of the slab\n[Fig. 1(a)]. The lengths of the YIG slab along the x,y, andzdirections ( w,l, andt) are 1.9\nmm, 6.0 mm, and 1.0 mm, respectively. After polishing the x-ysurfaces [(111) plane] of the\nYIG slab, the 10-nm-thick Pt \flms were sputtered on the whole of the surfaces. The Pt \flms\nare electrically insulated from each other [30]. Au electrodes were formed on the edges of the\nPt layers, of which the gap length l0is 5.0 mm. The thermoelectric voltage and resistance\ninside the gap were measured by a multimeter. The sample is put between heat baths with\ntwo sapphire plates with a length of l0for electrical insulation. For thermal connection\nbetween them, thermal grease is used [Fig. 1(b)]. During the LSSE measurements, we set\nTset\nH=Tset\nL+5 K with Tset\nH(L)being the temperature of the top (bottom) heat bath (hereafter,\nwe use the subscripts H and L to represent the corresponding quantities of the top and\nbottom Pt \flms, respectively). In this condition, we monitored the temperature di\u000berence\nbetween the top and bottom of the sample \u0001 T=TH\u0000TLby using the Pt \flms not only as\nspin-current detectors but also as temperature sensors [Fig. 1(c)] [23][31] . A magnetic \feld\nwith the magnitude His applied in the xdirection.\nImportantly, the above method allows us to estimate the thermal conductivity \u0014YIGof the\nYIG slab at the same time as the LSSE measurements. This is realized simply by recording\nthe heater power PHeater in addition to \u0001 T;\u0014YIGcan be calculated as\n\u0014YIG=t\nwlPHeater\nf(l0) \u0001T: (2)\nin a similar manner to the steady heat-\row method, where f(l0) = 0:88 denotes a form factor\nfor adjusting the measured \u0001 Tto the temperature di\u000berence averaged over the sample length\n[32]. Note that the in\ruence of the thermal resistances of the Pt layers [33] and YIG/Pt\ninterfaces [34] are negligibly small for the \u0014YIGestimation. The simultaneous measurements\nenable quantitative comparison of the LSSE and \u0014YIG.\nThe inset to Fig. 2(a) shows the estimated values of \u0001 Tbased on the resistance measure-\nments. We found that the \u0001 Tvalue is smaller than the temperature di\u000berence applied to\nthe heat baths ( Tset\nH\u0000Tset\nL= 5 K in this study) at each temperature and strongly decreases\nwith decreasing the temperature. This behavior can be explained by dominant consumption\nof the applied temperature di\u000berence by the thermal grease layers; typical thermal resis-\ntance of 10- \u0016m-thick thermal grease is comparable to that of the YIG slab at 300 K and\n4(a)\n(b)THset-TLset\nTH-TL\n0.1110 ∆T (K) \n300 200 1000\nT (K)\n2VLSSE\n-5 05VH (µV) \n-0.5 0.0 0.5\nµ0H (T)THset=305K100\n50 \n0κYIG (Wm -1 K-1 )\n300 250 200 150 100 50 0\nT (K)50 \n25 \n0VLSSE /∆T (µV/K) \n VH/∆ T\n VL/∆ TVH/( THset-THset)FIG. 2. Temperature ( T) dependence of the thermal conductivity \u0014YIGestimated from \u0001 Tand the\nheater power PHeater (a) , andVLSSE=\u0001T(b). The inset to (a) shows \u0001 Testimated from RHand\nRLunder the temperature gradient. The inset to (b) shows Hdependence of VHatTset\nH(L)= 305\n(300) K. The peak temperatures are determined by parabolic \ftting of \fve points around the\nmaximums.\nmuch greater than that at low temperatures as \u0014YIGincreases at low temperatures [29][35].\nConsequently, the actual temperature di\u000berence (\u0001 T), and resultant VLSSE measured with\n\fxingTset\nH\u0000Tset\nL, strongly decrease. This result indicates that the conventional method,\nwhich monitors only Tset\nH\u0000Tset\nL, cannot reach the intrinsic temperature dependence of the\nLSSE.\nFigure 2(a) shows \u0014YIGas a function of Tave= (TH+TL)=2, estimated from Eq. (2).\nThe temperature dependence and magnitude of \u0014YIGare well consistent with the previous\nstudies [29], supporting the validity of our estimation. The \u0014YIGvalue exhibits a peak at\naround 27 K and reaches 1 :3\u0002102Wm\u00001K\u00001at the peak temperature; this temperature\ndependence can be related to phonon transport, i.e., the competition between the increase\nof the phonon life time due to the suppression of Umklapp scattering and the decrease of\nthe phonon number with decreasing the temperature [29].\nThe inset to Fig. 2(b) shows the Hdependence of VHin the Pt/YIG/Pt sample at\n5Tset\nH(L)= 305 K (300 K). The clear LSSE voltage was observed; the voltage shows a sign\nreversal in response to the reversal of the magnetization direction of the YIG slab [5, 24].\nWe extracted the LSSE voltage VLSSE from the averaged values of VL(VH) in the region of\n0:2 TΓ\nand−i(γ0+Γ)±i√\nΓ2−∆2(broken anti-PT) for |∆|<Γ. The\nbehavior of the real and imaginary parts of these eigenvalue s\nis provided in figure 2 (a), (b). As long as the stability cri-\nterion is fulfilled, the responses in (4) are inversely relat ed to\n(ωd−λ+)(ωd−λ−)=−γ0(2Γ+γ0)−∆2. The broken anti-PT\nphase brings in real singularities at ωd=1\n2(ωa+ωb) in the\nlimitγ0→0, which is evidenced by the resonant inhibition\nin the imaginary part of λ+, as marked by the point X in fig-\nure 2 (b). The extreme condition γ0=0 holds when none of\nthe modes suffers spontaneous losses to its independent sur-\nrounding, all the while interacting with the mediating rese r-\nvoir. Therefore, by harnessing dissipative coupling betwe en\ntwo modes and optimizing the e ffect of VIC, we observe a pre-\ncipitous divergence in the linear response under steady-st ate\nconditions.\nThe PT-symmetric configuration for coherently coupled\nsystems is conformable with the parameter structure ∆=γ0=\nΓ= 0. The constraintγb=−γaimplies that a loss in mode\namust be offset by a commensurate gain in mode b. With\nthe corresponding eigenvalues given as ∆0±i/radicalBig\nγ2\nab−g2for\n|γab|>gand as∆0±/radicalBig\ng2−γ2\nabfor|γab|12γ2. However, throughout this manuscript, we operate\nat adequately low drive powers to ward o ffbistable signature.\nNow, in the limitγ0→0 andδ→0,βbecomes vanishingly\nsmall, and the first two terms in Eq. (6) recede in importance,\nfor a given Rabi frequency Ω. Consequently, in the neighbor-\nhood ofδ=0, the response becomes highly sensitive to vari-\nations in U. To be more precise, for su fficiently low values of\nthe detuning, the response mimics the functional dependenc e\nx≈(I/4U2)1/3. A tenfold decrease in U, therefore, scales\nup the peak intensity of bby a factor of 4.64. In this con-\ntext, it is useful to strike a correspondence with the sensit ivity\nin eigenmode splitting around an EP which is typically em-\nployed in PT symmetric sensing protocols [15, 16, 19]. For\ntwo mode systems, where the EP is characterized by a square\nroot singularity, this splitting δωscales as the square root of\nthe perturbation parameter ǫimplying a sensitivity that goes\nas/vextendsingle/vextendsingle/vextendsingleδω\nδǫ/vextendsingle/vextendsingle/vextendsingle∝|ǫ|−1/2. However, in our setup, the sensitivity to Uin\nthe response is encoded as/vextendsingle/vextendsingle/vextendsingleδx\nδU/vextendsingle/vextendsingle/vextendsingle∝|U|−5/3.\nThe importance of the above result in the context of sens-\ning is hereby legitimized for dissipatively coupled system s.\nGuided by the recent experiments on dissipatively coupled h y-\nbrid magnon-photon systems [45–50], we apply these ideas\nto the specific example of Kerr nonlinearity in a YIG sam-\nple [57]. However, the bulk of these works have restricted\ntheir investigations to the linear domain. Here, we transce nd\nthis restriction and study the nonlinear response to an exte r-\nnal drive. We consider an integrated apparatus comprising\nan optical cavity and a YIG sphere, both interfacing with a\none-dimensional waveguide. The direct coupling between th e\ncavity and the magnon modes can be neglected. However,\nthe interaction with the waveguide would engender an indi-\nrect coupling between them. In order to excite the weak Kerr\nnonlinearity of the YIG sphere, a microwave laser is used to\ndrive the spatially uniform Kittel mode. The full Hamiltoni an\nin presence of the external drive can be cast exactly in the\nform of Eq. (1), with bsuperseded by the magnonic operator\nm[58],\nHeff//planckover2pi1=ωaa†a+/bracketleftBig\nωmm†m+U(m†2m2)/bracketrightBig\n+\niΩ/parenleftBig\nm†e−iωdt−meiωdt/parenrightBig\n.(7)\nAs discussed earlier, the mediating e ffect of the waveguide\nis reflected as a dissipative coupling between the two modes,\nwhich instills VIC into the system. With the anti-PT sym-\nmetric choices∆a=−∆m=δ/2,γa=γb=γ0, and the\nredefinitionγ0+Γ =γ, we recover Eq. (6) in the steady\nstate, with the obvious substitution b→mandx=|b|2de-\nnoting the spin current response. We now expound the utility\nof engineering a lossless system in sensing weak Kerr non-\nlinearity. To that end, we zero in on the parameter subspace\nΓ=γ=2π×10 MHz. Sinceβ=δ2/4, the contributions\nfrom the first two terms in Eq. (6) taper o ffas resonance is\napproached. As outlined earlier, we find that for all practi-\ncal purposes, the nonlinear response can be approximated as4\nFIG. 3: a) The spin current plotted against δat two dif-\nferent nonlinearities; b) spin currents away from the VIC\ncondition, compared against the lossless scenario, at di ffer-\nent drive powers and at U/2π=7.8 nHz - for ease of\ncomparison, the blue and maroon curves have been scaled\nup by 10; c) contrasting responses observed at a drive\npower of 1 mW for two different strengths of nonlinearity.\nx≈(I/4U2)1/3in the regionδ/2π< 1 MHz, which demon-\nstrates its stark sensitivity to U. A lower nonlinearity begets a\nhigher response, as manifested in figure 3 (a), where plots of x\nagainstδare studied at differing strengths of the nonlinearity.\nEven at Dp=1µW, we observe a significant enhancement in\nthe induced spin current of the YIG around δ=0. The result\nis a natural upshot of the VIC-induced divergent response in\nan anti-PT symmetric system in the linear regime. Quite con-\nveniently, the inclusion of nonlinearity dispels the seemi ngly\nabsurd problem of a real singularity noticed in the linear ca se.\nIfΓ<γ, a strong quenching in the response is observed, as\ndepicted in figure 3 (b). The sensitivity to variations in Ualso\nincurs deleterious consequences. Nevertheless, we can cou n-\nteract this decline by boosting the drive power. A drive powe r\nclose to 1 mWcan bring back the augmented response and the\npronounced sensitivity to U(figure 3 (c)). This mechanism\ncan serve as an efficient tool to sense small anharmonicities\npresent in a system. The fact that even a minute pump power\nin the range of 1µWto 1mW generates substantial enhance-\nment in the spin current makes it all the more robust.\nAs evident from the preceding discussion, it is imperative\nthat the waveguide-mediated coupling overshadows the ef-\nfect of spontaneous emissions. Moreover, the protocol hing es\non the anti-PT symmetric character and eigenmodes of H,\nwhich largely control the dynamics at low drive powers. At\nlarger drive powers ( ∼0.1W), the nonlinear correction in (3)\nbecomes important and activates new coherences. A theoreti -\ncal explanation of this phenomenon can be spelled out by lin-\nearizing the dynamics of the mode mabout its steady-state\nFIG. 4: Nonlinearity induced coherences and higher dimen-\nsionality: (a) Imaginary parts of the eigenmodes of an ef-\nfectively four-dimensional system at a drive power of 0 .1\nW- aroundδ/γ=−3.2, an extreme linewidth narrow-\ning is observed; b) the spin current response, with a peak\nnear toδ/γ=−2.9, highly skewed due to a stronger drive.\nvalue ensuing from (6). This linearization yields a higher-\ndimensional eigensystem, portrayed in figure 4 (a). The new\ncoherences are closely correlated with the extreme linewid th\nnarrowing manifested in the higher-dimensional model. Fig -\nure 4 (b) exemplifies new VIC-induced peaks that emerge\neven when spontaneous decays become comparable to the dis-\nsipative coupling. Details on this calculation are provide d in\nthe supplementary material [58].\nIn summary, we have proposed an optical test bed that\nshows enhanced sensitivity to Kerr nonlinearity in the mode\nresponses to an external field, hence qualifying it as a proto -\ntypical agency to gauge the strength of anharmonic perturba -\ntions under optimal conditions. The physical origin of this\npeculiar behavior lies in an e ffective coupling induced be-\ntween the cavity and the magnon modes in the presence of\na shared ancillary reservoir. Such a coupling is purely an ar -\ntifact of third-party mediation, and is dissipative in natu re as\nthe two modes synergistically drive up the energy being chan -\nneled into the interposing reservoir. Optimal results vis- ` a-vis\nthe estimation of nonlinearity are obtained when VIC strong ly\ndominates, i.e when spontaneous emissions from the modes\nto the surrounding environments become negligible in com-\nparison to the waveguide-mediated coupling. Since dissipa -\ntively coupled systems do not require the synthetic introdu c-\ntion of gain for efficient sensing, our setup o ffers a clear edge\nover PT-symmetric systems which rely on a balanced trade-\noffbetween gain and loss. At higher drive powers, we observe\nskewed VIC peaks as a testimony to strongly anharmonic re-\nsponses, even when decays into the environment become sig-\nnificant. These nonlinearity-induced VICs could bear rele-\nvance in other contexts and merits further investigation. 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B 94, 224410 (2016).\n[58] Refer to supplementary material." }, { "title": "1309.4841v2.Tuning_Magnetotransport_in_PdPt_Y3Fe5O12__Effects_of_magnetic_proximity_and_spin_orbital_coupling.pdf", "content": "arXiv:1309.4841v2 [cond-mat.mtrl-sci] 10 Oct 2013Tuning magnetotransport in PdPt/Y 3Fe5O12: Effects of\nmagnetic proximity and spin orbital coupling\nX. Zhou, L. Ma, Z. Shi, and S. M. Zhou∗\nShanghai Key Laboratory of Special Artificial Microstructu re and Pohl Institute\nof Solid State Physics and School of Physics Science and Engi neering,\nTongji University, Shanghai 200092, China\n(Dated: June 17, 2018)\nAbstract\nAnisotropic magnetoresistance (AMR) ratio and anomalous H all conductivity (AHC)\nin PdPt/Y 3Fe5O12(YIG) system are tuned significantly by spin orbital couplin g strength\nξthrough varying the Pt concentration. For both Pt/YIG and Pd /YIG, the maximal\nAMR ratio is located at temperatures for the maximal suscept ibility of paramagnetic Pt\nand Pd metals. The AHC and ordinary Hall effect both change the s ign with temperature\nfor Pt-rich system and vice versa for Pd-rich system. The pre sent results ambiguously\nevidence the spin polarization of Pt and Pd atoms in contact w ith YIG layers. The global\ncurvature near the Fermi surface is suggested to change with the Pt concentration and\ntemperature.\nPACS numbers: 72.25.Mk, 72.25.Ba, 75.47.-m, 75.70.-i\n∗Correspondence author. Electronic mail: shiming@tongji.edu.cn\n1Generation, manipulation, and detection of pure spin current are p opular topic\nin the community of spintronics because of its prominent advantage of negligible\nJoule heat in spintronic devices1–5. Pure spin current can be generated by spin Hall\neffect, spin Seebeck effect (SSE), and etc. By spin Hall effect, the pure spin current\ncan be achieved in semiconductors due to strong spin orbital couplin g (SOC). In\nthe SSE approach, the spin current is produced in ferromagnetic m aterials with\na temperature gradient and injected into another nonmagnetic lay er through the\ninterface. In general, the pure spin current cannot be probed by conventional\nelectric approach. Instead, it is detected by inverse spin Hall effec t6,7.\nWith strong SOC in Pt layers and long spin diffusion length in Y 3Fe5O12(YIG)\ninsulator layers, the Pt/YIG systems are particularly suitable for d esign and\nfabrication of spintronic devices8–17. In studies of the SSE phenomena of Pt/YIG\nsystem, the SEE and the anomalous Nernst effect were argued to b e entangled9,\nwhere the latter comes from the spin polarization due to the magnet ic proximity\neffect (MPE) of the nearly ferromagnetic Pt layers. Many attempt s have been made\nto study the MPE in Pt/YIG system. Since the atomic magnetic momen t of Pt is\ntoo small to be measured by magnetometry, anisotropic magnetor esistance (AMR)\nand anomalous Hall effect (AHE) have been investigated intensively a s a function\nof the Pt layer thickness and sampling temperature ( T)10–16. Up to date, however,\nmagnetotransport results are controversial. The AMR ratio of Pt /YIG system\nexhibits an angular dependence different from the conventional AM R in magnetic\nfilms, and it changes nonmonotonically with the Pt layer thickness. Alt hough these\nphenomena were attributed to spin Hall magnetoresistance (SMR)14,15instead of\nthe conventional AMR, the nonmonotonic variation of the AMR with Tcannot be\nunderstood in the SMR model13. Moreover, the ferromagnetic ordering in Pt layers\nwas proved by the anomalous Hall effect (AHE) in Pt/YIG system9. In particular,\nthe mechanism of the observed sign change of the AHE with Tis still unclear. Very\nrecently, the x-ray magnetic circular dichroism measurements hav e been performed\nby different groups and experimental results are still controvers ial possibly due\nto either small atomic magnetic moments of Pt or antiparallel alignmen t of spins\nbetween neighboring Pt atoms11,13,18. Therefore, an alternative ideal experimental\napproach must be taken to reveal the MPE in Pt/YIG system.\n2In this work, we will study the SOC effect on the AMR and AHE by using\nPd1−xPtx(PdPt)/YIG systems. Here, Pd and Pt atoms are isoelectric elemen ts\nwith different atomic order numbers such that the effective SOC str ength can be\nsignificantly adjusted by modifying x19. It is surprising that the AHE and AMR can\nbe tuned significantly for xfrom 0 to 1.0. Meanwhile, the nonmonotonic variation\nof the AMR with Tis revealed to be caused by the unique Tdependence of the\nspin polarization of Pt and Pd metals. The sign change of AHE with Tis found\nin Pt-rich samples and vice versa in Pd-rich systems. The intriguing ph enomena\nprovide strong evidence for the MPE. Meanwhile, the Ttuning effects on the\ncurvature near Fermi surface of polarized Pt and Pd layers are illus trated.\nA series of PdPt/YIG bilayers were fabricated by pulse laser deposit ion and\nsubsequent magnetron sputtering in ultrahigh vacuum on (111)-o riented, single\ncrystalline Gd 3Ga5O12(GGG) substrates. The 70 nm thick YIG thin films were\nepitaxially grown via pulsed laser deposition from a stoichiometric polyc rystalline\ntarget using a KrF excimer laser. Secondly, PdPt layers were depos ited on YIG\nthin films by magnetron sputtering. The thickness of the YIG and Pd Pt layers was\ndetermined by the X-ray reflection (XRR) as shown in Fig. 1(a). Figu re 1(b) shows\nthat the x-ray diffraction (XRD) peaks at 2 θ= 51 degrees for (444) orientations\nin GGG substrate and YIG films overlap each other. The epitaxial gro wth of the\nYIG films was confirmed by Φ and Ψ scan with fixed 2 θfor the (008) reflection\nof GGG substrates and YIG films, as shown in Fig. 1(c). In-plane mag netization\nhysteresis loops of the YIG films were measured at room temperatu re by vibrating\nsample magnetometer in Fig. 1(d). The measured magnetization of 1 34 emu/cm3\nis almost equal to the theoretical value, and the coercivity is as sma ll as 6.0 Oe. In\nexperiments, half width at half height of the ferromagnetic resona nce absorption\npeak is about 3 Oe17. Therefore, high quality epitaxial YIG films are achieved in\nthe present work.\nBefore measurements, the films were patterned into normal Hall b ar, and then\nAMR and AHE were measured from 10 to 300 K. Figure 2(a) shows the longitudinal\nresistivity ρxxversus the external magnetic field Hat room temperature. At the\nsaturation state, at the angle between the magnetization and the sensing current\nφH= 0 theρxxis larger than that of ρxxatφH= 90 degrees, similar to the conven-\n3tional AMR in thick magnetic metallic films such as permalloy13. Figure 2(b) shows\nthe in-plane angular dependence of the AMR at room temperature c an be fitted by\na linear function of cos2φH, exhibiting a similar attribute in permalloy films. The\nAMR ratio depends on both the sampling Tandx, as shown in Fig. 2(c). For both\nPt/YIG13and Pd/YIG systems, the ∆ ρxx/ρxxshows nonmonotonic variations with\nT; whereas for most ferromagnetic materials it changes monotonica lly. The max-\nimal value is located near 120 K and 60 K for Pt/YIG and Pd/YIG, resp ectively.\nRemarkably, the susceptibility of paramagnetic Pt and Pd was early o bserved to\nhave broad peaks almost at the same temperatures20,21. For nonmagnetic transition\nmetals, the enhanced susceptibility χ=χ0/(1−IN(EF)), where χ0,I, andN(EF)\nrefer to the susceptibility without the presence of Coulomb interac tion, the Stoner\nparameter, and the density of states (DOS) near Fermi level, res pectively. Both\nthe Stoner parameter and the MPE induced magnetic moment are als o expected\nto change nonmonotonically22,23. Therefore, the nonmonotonic dependence of the\nAMR in Pt/YIG and Pd/YIG systems should stem from their unique Tdependence\nof the induced magnetic moments of both Pt and Pd layers. The mono tonic change\nof the AMR ratio for intermediate xmay be due to the contribution of the impurity\nscattering, as proved below by the large ρxxat intermediate x. It is noted that the\npresent AMR ratio in Pd/YIG is much larger than the values reported by Linet\nal24, possibly due to the weak spin polarization of Pd atoms.\nIn experiments, the Hall resistivity ρxywas measured as a function of Hin\nthe out-of-plane geometry. The anomalous Hall resistivity ρAHwas extrapolated\nfrom the linear dependence of ρxyat large H. Figure 3 shows the Hall loops\nfor Pt/YIG and Pd/YIG at 10 K and 300 K. For Pt/YIG, both ρAHand the\nordinary Hall coefficient R0are negative at room temperature but positive at\n10 K. In contrast, they are negative at both 10 K and 300 K for Pd/ YIG\nsamples24. Figure 4(a) shows the σAHas a function of Tfor all samples, where\nσAH=ρAH/(ρ2\nxx+ρAH∗ρxx)≃ρAH/ρ2\nxxsinceρAH≪ρxx. TheσAHchanges\nfrom the positive to negative for Pt-rich samples10; whereas it is always negative\nin the measured Tregion for small x. Intriguingly, Figure 4(b) shows that the\nR0also changes the sign for large xwhereas no sign change occurs for small x.\nApparently, the Tdependence of σAHis correlated with that of R0as a function of\n4x. Figure 4(c) shows at high T,ρxxof all samples increases approximately linearly\nwithTand deviates from the linear dependence at low T. The residual resistivity\nchanges nonmonotonically as a function xwith a maximum near x= 0.6, verifying\nalmost random location of Pt and Pd atoms25.\nFor spherical Fermi surface, the R0sign is directly determined by the numbers\nof electrons and holes. For nonspherical Fermi surface, howeve r, it is also strongly\nrelated to the curvature near the Fermi surface. As the integra tion of the Berry\ncurvature over the Brillouin zone, the intrinsic AHC of magnetic tran sition metals\nis naturally determined by the curvature near the Fermi surface. For paramagnetic\nPt, the DOS near the Fermi surface changes sharply with the ener gy23and theR0\nchanges the sign nearthe Fermi level26. Due to the exchange splitting and SOC in\npolarized Pt, not only the numbers of electrons and holes but also th e curvature\nnear Fermi surface are significantly different from those of param agnetic ones27.\nTherefore, the R0(at lowT) is positive for polarized Pt (in Fig. 4(b)), opposite to\nthat of paramagnetic one26,28,29. With weak exchange splitting and SOC at high\nT, theR0in polarized Pt is negative, like paramagnetic Pt, as shown in Fig. 4(b).\nWith the prominent Teffect on the Berry curvature near the Fermi surface, the\nintrinsic contribution to the σAHin polarized Pt is expected to change the sign\nwithT. As well known, the σAHconsists of the skew scattering, side-jump, and\nintrinsic terms30. Since the magnitude of the skew scattering term (proportional t o\nσxx) changes slightly with T, theσAHfor Pt-rich systems is also expected to change\nthe sign, as shown in Fig. 4(a). The co-occurrent sign changes of both R0andσAH\nstrongly verify the globally varying curvature near the Fermi surf ace. In contrast,\nfor Fe and Mn 5Ge3films, the R0changes from the negative to positive near 80 K\nwhereas the σAHis always positive below room temperature, and the sign change\nis attributed to the change of the conductivity ratio of dandsbands instead of\nthe global curvature change near the Fermi surface31–33. For NiPt thin films, the\nσAHrather than R0changes the sign with T, which is attributed to other reasons\nrather than the global change of the curvature near the Fermi s urface34. Due to\nweak SOC in polarized Pd, the global curvature near the Fermi surf ace changes\nless prominently, compared with that of paramagnetic one and ther eforeR0in the\nmeasuring Tregion is always negative, like the paramagnetic one. Meanwhile,\n5neitherR0norσAHchanges the sign with Tas shown in Figs. 4(a)& 4(b). Similarly,\nfor pure Ni films neither R0norσAHchanges the sign below room temperature35.\nThe present correlation between the σAHandR0with the Pt concentration verifies\nthat they are largely determined by the curvature near the Fermi surface. The T\ntuning effect on the electronic band structure is also demonstrate d in Pt/YIG.\nSignificant SOC effect on the magnetotransport properties in PdPt /YIG is\nillustrated. Figures 5(a)& 5(b) show the ∆ ρxx/ρxxandσAHat 10 K as a function of\nx, respectively. The AMR ratio is 8 ×10−4and 1×10−4for Pt/YIG and Pd/YIG\nbilayers, respectively. It is enhanced in magnitude by a factor of ab out one order\nfromx= 0 to 1.0. In principle, the AMR in ferromagnetic materials arises from\nthes-dscattering, and it is theoretically predicted to be proportional to t he square\nof the SOC strength ξ2if the resistivity ratio of spin-up and spin-down channels is\nfixed according to the perturbation theory36. Since the ξof Pt is about 3 times that\nof Pd19, it is experimentally proved that the ratio of the AMR between Pt/YI G and\nPd/YIG is close to that of ξ2. For intermediate x, the AMR ratio deviates from\nthe quadratic dependence due to large contribution of the impurity scattering as\nshown in Fig. 5(c). With increasing x, theσAHat 10 K changes from the negative\nto positive. For Pt/YIG and Pd/YIG, it is about 3.0 and -1.0 (S/cm), r espectively,\nand the magnitude ratio is close to that of ξbetween two elements19,33. Therefore,\nthe sign and magnitude of σAHin PdPt/YIG system are tuned by changing ξwith\nvariousx.\nIn summary, the AMR ratio in PdPt/YIG system can be enhanced by a factor\nof about one order from x= 0 tox= 1. It changes nonmonotonically with T\ndue to similar Tdependence of the atomic magnetic moment. At 10 K, the AHC\nmagnitude of x= 1 is about 3 times that of x= 0. For Pt-rich samples both\ntheR0and AHC change their signs with Tand vice versa for Pd-rich system,\ndue to the global change of the curvature near the Fermi surfac e withx. The\nSOC tuning effects on the magnetotransport properties can be un derstood based\non the perturbation theory. All present phenomena directly evide nce the MPE\nin the PdPt/YIG. The Ttuning effect on the electronic band structure is also\ndemonstrated in polarized PdPt layers. The present work will also be helpful for\noptimizing the spintronics devices.\n6Acknowledgments This work was supported by the National Science Founda-\ntion of China Grant Nos.11374227, 51331004, 51171129, and 5120 1114, the State\nKey Project of Fundamental Research Grant No.2009CB929201, and Shanghai\nNanotechnology Program Center (No. 0252nm004).\n71M. I. D’Yakonov and V. I. Perel, Phys. Lett. 35A, 459(1971)\n2J. E. Hirsch, Phys. Rev. Lett. 83, 1834(1999)\n3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. 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Coleman, Phys. Rev. B 10, 2915(1974)\n32M. G. Cottam and R. B. Stikchombe, J. Phys. C: Solid State Phys .1, 1052(1968)\n33C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Phys. Rev. Lett. 96, 037204(2006)\n34T. Golod, A. Rydh, P. Svedlindh, and V. M. Krasnov, Phys. Rev. B87, 104407(2013)\n35S. P. McAlister and C. M. Hurd, J. Appl. Phys. 50, 7526(1979)\n36A. P. Malozemoff, Phys. Rev. B 34, 1853(1986)\n9FIGURE CAPTIONS\nFigure 1 (color online): For typical Pt/YIG films, x-ray refle ctivity at small angles (a)\nand XRD diffraction at large angles (b), Φ and Ψ scan with fixed 2 θfor the (008)\nreflection of GGG substrate and YIG film (c). In (d) is shown the room temperature\nin-plane magnetization hysteresis loop of the YIG layer. In (a) black and red lines\ncorrespond to YIG and Pt layers, respectively.\nFigure 2 (color online): For Pt (1 nm)/YIG films, AMR curves at φH= 0 and 90 degrees\n(a) and angular dependent AMR at H= 10 kOe (b). For PdPt (1 nm)/YIG films, the\nAMR ratio versus Tfor various x(c).\nFigure 3 (color online): For Pt (1 nm)/YIG (a, b) and Pd (1 nm)/ YIG (c, d) films, ρxy\nversusHat 10 K (a, c) and 300 K(b, d).\nFigure 4 (color online): For PdPt (1 nm)/YIG films, σAH(a),R0(b), and ρxx(c) versus\nTfor various x.\nFigure 5 (color online): For PdPt (1 nm)/YIG films, AMR (a), σAH(b), and ρxx(c) at\n10 K versus x. Solid lines serve a guide to the eye.\n101 2 3 4 5\n40455055600\n30\n60\n90\n120\n150180210240270300330\n40\n45\n50\n55\n6049 50 51 52 53\n(d)(c)(b)Intensity (a.u)\n2θ (deg)(a)\n75-2\nYIG (444)\n2θ (deg)\nGGG (444)\n-5\n-50 0 50-101M (102 emu/cm3)\nH (Oe)\nFIG. 1:\n11-90 0 90 180 270389.8390.0390.2\n0 100 200 300110-600 -400 -200 0 200 400 600389.9390.0390.1390.2\n(b)\nφH (deg) Rxx (Ω)\n(c)\n x=1\n 0.8\n 0.7\n 0.5\n 0.25\n 0.13\n 0104 Δρ/ρ\nT (K)Rxx(Ω)\nH (Oe)(a)\nFIG. 2:\n12-202\n-0.50.00.5\n-20 0 20-1.0-0.50.00.51.0\n-20 -10 0 10 20-0.50.00.5102 ρxy (µΩ⋅cm)(a)\n(d)(c)\n(b)102 ρxy (µΩ⋅cm)\nH (kOe) H (kOe)\nFIG. 3:\n13-50510-2024\n0 100 200 300253035106 R0 (µΩ cm/Oe)x=1.0\n 0.8\n 0.7\n 0.5\n 0.25\n 0.13\n 0 \n \n \n \n \n \n \n(b)(a)σAH (S/cm)\nT (K)ρxx (µΩ cm)(c)\nFIG. 4:\n14-10123\n0.0 0.2 0.4 0.6 0.8 1.02530048\n σAH(S/cm)(b)\nx (c)\n ρxx (µΩ cm)10 K(a)\n 104 Δρ/ρ\nFIG. 5:\n15" }, { "title": "2311.01711v1.Cryogenic_spin_Peltier_effect_detected_by_a_RuO__2__AlO__x__on_chip_microthermometer.pdf", "content": "arXiv:2311.01711v1 [cond-mat.mes-hall] 3 Nov 2023Cryogenic spin Peltier effect detected by a RuO 2-AlOxon-chip microthermometer\nTakashi Kikkawa,1,∗Haruka Kiguchi,2,1Alexey A. Kaverzin,1,3Ryo Takahashi,2and Eiji Saitoh1,3,4\n1Department of Applied Physics, The University of Tokyo, Tok yo 113-8656, Japan\n2Department of Physics, Ochanomizu University, Tokyo 112-8 610, Japan\n3Institute for AI and Beyond, The University of Tokyo, Tokyo 1 13-8656, Japan\n4WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan\n(Dated: November 6, 2023)\nWe report electric detection of the spin Peltier effect (SPE) in a bilayer consisting of a Pt film\nand a Y 3Fe5O12(YIG) single crystal at the cryogenic temperature Tas low as 2 K based on\na RuO 2−AlOxon-chip thermometer film. By means of a reactive co-sputteri ng technique, we\nsuccessfully fabricated RuO 2−AlOxfilms having a large temperature coefficient of resistance (TC R)\nof∼100% K−1at around 2 K. By using the RuO 2−AlOxfilm as an on-chip temperature sensor\nfor a Pt/YIG device, we observe a SPE-induced temperature ch ange on the order of sub- µK, the\nsign of which is reversed with respect to the external magnet ic fieldBdirection. We found that\nthe SPE signal gradually decreases and converges to zero by i ncreasing Bup to 10 T. The result is\nattributed to the suppression of magnon excitations due to t he Zeeman-gap opening in the magnon\ndispersion of YIG, whose energy much exceeds the thermal ene rgy at 2 K.\nI. INTRODUCTION\nOne of the important features in spintronics is that\nvarious phenomena have been found at room tempera-\nture in simple stacked structures, leading to their prac-\ntical device applications1–4. Meanwhile, exploring the\nspintronic phenomena at low temperatures often resulted\nin a discovery of new functional properties with both\nfundamental and practical prospects5–11. A typical ex-\nample is the spin Seebeck effect (SSE), which refers to\nthe generation of a spin current as a result of a tem-\nperature gradient in a magnetic material, and has been\nobserved at room temperature in a variety of magnetic\nmaterials, including garnet- and spinel-ferrites with high\nmagnetic ordering temperatures12–14. When SSEs are\nmeasured at low temperatures in certain systems how-\never, intriguing physics comes to the surface. Majorfind-\nings include the signal anomalies induced by hybridized\nmagnon-phonon excitations14–19, unconventional sign re-\nversal due to competing magnon modes having oppo-\nsite spin polarizations20, observationofa spin-superfluid-\nmediated nonlocal SSE signal21, and SSEs driven by\nparamagnetic spins22,23and exotic elementary excita-\ntions in quantum spin systems24–26. Furthermore, re-\ncently, a nuclear SSE has been observed in an antiferro-\nmagnet having strong hyperfine coupling14,27. The sig-\nnal increases down to ultralow temperatures on the order\nof 100 mK, which is distinct from conventional thermo-\nelectric effects in electronic (spin) systems14,27, and may\noffer an opportunity for exploring thermoelectric science\nand technologies at ultralow temperatures, an important\nenvironment in quantum information science.\nIn contrast to the intense research on SSEs, the spin\nPeltier effect28–39, the reciprocal of the SSE, remains to\nbe explored at low temperatures below 100 K because\nof its experimental difficulty. The SPE modulates the\ntemperature of a junction consisting of a metallic film\nand a magnet in response to a spin current29, and has\nbeen detected usually by means of lock-in thermogra-phy (LIT)29,30,32,36and thermocouples28,31,34. The LIT\nmeasures the infrared intensity emitted from the sample\nsurface based on a combination of the lock-in with tem-\nperature imaging technique, whose intensity is in pro-\nportion to the fourth power of the absolute tempera-\ntureT(the Stefan–Boltzmann law32,38). This results in\na typical resolution of 0.1 mK at room temperature38,\nwhich is sufficient to measure a SPE in a prototypical\nPt/Y3Fe5O12(YIG) system at higher temperatures ( ∼\nroom temperature and above)29,30. However, the LIT\nmay not be applicable for detecting the low-temperature\nSPE, because its sensitivity is dramatically reduced with\ndecreasing temperature32,38. Furthermore, a thermocou-\nple micro-sensor with a high resolution of ∼5µK was\nused to measure a SPE down to 100 K in Ref. 34. How-\never, it was found to be difficult to conduct the measure-\nments below 100K as the sensitivity of the thermocouple\ndecreases with decreasing T. It is therefore important to\nestablish an alternative experimental method for detect-\ning cryogenic SPEs4. An ultimate goal in this direction\nwould be to find cryogenic SPEs driven by nuclear and\nquantum spins that can be activated even at ultralow\ntemperatures, toward future possible cooling- and heat-\npump technologies in such an environment.\nIn this study, we have explored the SPE at a cryo-\ngenic temperature below the liquid-4He temperature in\na prototypical Pt/YIG system. There are three cru-\ncial requirements for practical realization of such mea-\nsurement that are (1) the high temperature-resolution\nof∼sub-µK-order or better at low temperatures, (2)\nability to detect a temperature change of a metallic\n(Pt) thin film (which implies for contact-mode mea-\nsurements sufficient thermal coupling and low heat ca-\npacity), and (3) reliability under a high magnetic-field\nenvironment. To realize the thermometry that meets\nthese requirements, we adopted a RuO 2-based micro-\nthermometer40–49(RuO2−AlOxcomposite film in our\ncase). In general, RuO 2-based resistors show a high\ntemperature-sensitivity due to their large negative tem-2\nperature coefficient of resistance. Besides, they show\nreasonably small magnetoresistance and can be made in\na thin-film form. Owing to these advantages, in fact,\nRuO2-based chip resistors have widely been used as tem-\nperaturesensorsatcryogenictemperatures40,41. We have\nfabricatedRuO 2−AlOxfilms bymeans ofa co-sputtering\ntechnique and found the optimal fabricationcondition by\ncharacterizing their electric transport properties. By us-\ning a RuO 2−AlOxfilm as an on-chip temperature sensor\nfor a Pt-film/YIG-slab system, we successfully measured\na SPE-induced temperature change on the order of sub-\nµK atT= 2 K. Our results provide an important step\ntoward a complete physical picture of the SPE and es-\ntablishment of cryogenic spin(calori)tronics37.\nII. EXPERIMENTAL PROCEDURE\nA. Fabrication of RuO 2−AlOxfilms\nWe have fabricated RuO 2−AlOxcomposite films as\na micro-thermometer by means of d.c. co-sputtering\ntechnique from RuO 2(99.9%, 2-inch diameter) and Al\n(99.999%, 2-inch diameter) targets under Ar and O 2at-\nmosphere. To obtain the most suitable thermometer film\nfor the SPE at low temperatures, a series of co-sputtered\nRuO2−AlOxfilms on thermally-oxidized Si substrates\nwas first prepared at several d.c. power values for the\nRuO2target (PRuO2= 25, 26, 27, 28, and 30 W) and the\nfixedd.c. powerfortheAltarget( PAlOx= 25W)undera\nsputtering gasofAr + 7.83vol.%O 2at a pressureof 0.13\nPa at room temperature. Here, the values of Ar −O2gas\namount and the d.c. power of PAlOx= 25 W were cho-\nsen such that highly-insulating AlO xfilms are obtained\nwith a reasonable deposition rate ( ∼1 nm/min) when\nAlOxis sputtered solely from the Al target. We note\nthat, if the O 2gas amount exceeds an onset value, the\nd.c. sputtering rate suddenly decreases due to the sur-\nface oxidization of the Al target50,51, whereas if the O 2\ngas amount is insufficient, the resultant AlO xfilm may\nshow finite electrical conduction. We found that the in-\ntroduction ofO 2by itselfdoes not playan important role\nin the temperature variation of the resistance for pure\nRuO2films (for details, see APPENDIX A). To keep the\nsputtering conditions and resultant films’ quality as con-\nsistent as possible through repeated deposition cycles, we\nintroduced common pre-sputtering processes just before\nactual depositions. To remove a possible oxidized top\nlayer of the Al target, it was pre-sputtered at a relatively\nhigh power of PAlOx= 30 W for 600 s without introduc-\ning O2gas, and then the RuO 2and Al targets were pre-\nsputtered for 60 s under the actual deposition conditions\n(i.e., Ar + 7.83 vol.% O 2)52. For electric transport mea-\nsurements of the RuO 2−AlOxfilms, they were patterned\nintoaHall-barshapehavingthelength, width, andthick-\nness of 1 .0 mm, 0.5 mm, and ∼100 nm, respectively, by\nco-sputtering RuO 2−AlOxthrough a metal mask. The\nRuO2content in the RuO 2−AlOxfilms under the differ-ent RuO 2sputtering power PRuO2wasevaluated through\nscanning electron microscopy with energy dispersive X-\nrayanalysis(SEM-EDX)andthesurfaceroughnessofthe\nfilms was characterized through atomic force microscopy\n(AFM).\nB. Fabrication of SPE device\nTo investigate the SPE below the liquid-4He tem-\nperature, we have prepared devices consisting of\na Pt-film/YIG-slab bilayer, where a 100-nm-thick\nRuO2−AlOxfilm with Au/Ti electrodes is attached on\nthe top surface of the Pt film to detect its SPE-induced\ntemperature change ∆ T[see the schematic illustrations\nand the optical microscope image of a typical SPE device\nshowninFigs. 1(a)–1(c)]. Threephotolithographysteps\nwere employed to make the SPE devices, where all the\nfilm depositions were performed at room temperature.\nFirst, a 5-nm-thick Pt wire with the width of 200 µm was\nformed on the (111) surface of a single-crystalline YIG\nslabwiththesizeof5 ×5×1mm3byd.c. magnetronsput-\ntering in a 0 .1 Pa Ar atmosphere under the d.c. power\nof 20 W. In the next photolithography step, a 70-nm-\nthick insulating AlO xlayer was formed at the area of\n230×350µm2[300×500µm2forthe deviceshownin Fig.\n1(c)] on top of the Pt/YIG layer to electrically isolate\nthe RuO 2−AlOxfilm from the Pt layer. Here, the AlO x\ndeposition was done by r.f. magnetron sputtering from\nan Al2O3target (99.99%, 2-inch diameter) under the r.f.\npower of 150 W and a sputtering gas of Ar + 1.0 vol.%\nO253at a pressure of 0.6 Pa. We later confirmed that\nthe AlO xfilm shows a high electric resistance on the or-\nder of 1−10GΩ along the out-of-plane direction at room\ntemperature. Subsequently, a100-nm-thickRuO 2−AlOx\nthermometer film was deposited on top of the AlO xlayer\nat the area of 230 ×350µm2[300×500µm2for the de-\nvice shown in Fig. 1(c)] through the co-sputtering un-\nder the d.c. sputtering power of PRuO2= 28 W and\nPAlOx= 25 W. Here, the dimensions and sputtering\npower for the RuO 2−AlOxfilm were chosen such that\nthe resistance Rof the resulting film is several tens of kΩ\nat2Kanditssensitivitymonotonicallyincreaseswithde-\ncreasing Tdown to 2 K54[as shown in Figs. 3and4(d)\nand discussed in Sec. IIIA]. We then proceeded with the\nfinal photolithography step for Au(150 nm)/Ti(20 nm)\nelectrodes, where the numbers in parentheses represent\nthe thicknesses of the deposited films. Each Au/Ti elec-\ntrode wire on the RuO 2−AlOxfilm has the 30- µm width\nand is placed at 50- µm intervals. To reduce the contact\nresistance between the RuO 2−AlOxand Ti films, Ar-\nion milling was performed directly before depositing the\nAu/Ti film. Both the Ti and Au layers were formed by\nr.f. magnetron sputtering in succession without breaking\nvacuum. The first lithography process for the Pt layer\nwas done using a single-layer photoresist (AZ5214E) fol-\nlowed by a lift-off process, whereas the second and third\nprocesses for the AlO x/RuO2−AlOxand Au/Ti layers3\nB(d) Input:\n(e) Output:RTM Rhigh\nRlowRoffsetJc/g39Jc\n/g16/g39 Jc0\nTime tTime t\ndata acquisition \n/g87delay0Resistance change /g39RTM \n/g39RTM \nRhigh\nRlowRoffset\n0/g39RTM /g39TRTM \nTemperature T (K) (f) /g39RTM /g3/g111/g3 /g39T conversion \n via RTM -T curve YIGPt \nxzy~105 times repeated\n for each B valueI+\nI-V+\nV-Ti/Au RuO 2-AlOx\n thermometer \ninsulating\nAlOx layer (a)\nRuO 2-AlOx (100 nm)\nAlOx (70 nm)\nPt (5 nm)\nYIG slab (1 mm)Ti(20 nm)/Au(150 nm) (b)\nxzy\n(c)\nTi/Au\nPt \nYIG300 μmRuO 2-AlOx on \ninsulating AlO xSquare-wave \ncharge current Jc\nJc\nFIG. 1: (a) A schematic illustration of the SPE device con-\nsisting of a Pt-film/YIG-slab bilayer, on top of which a\nRuO2−AlOxthermometer (TM) film is attached for the de-\ntection of the SPE-induced temperature change ∆ Tin the Pt\nfilm. Besides, in this device, Au/Ti electrodes are formed on\nthe RuO 2−AlOxfilm for the 4 terminal resistance measure-\nments and an AlO xfilm is inserted between the RuO 2−AlOx\nand Pt films for the electrical insulation between them. (b) A\nschematic side-view image of the SPE device, where the num-\nbers in parentheses represent the thickness. (c) An optical\nmicroscope image of a typical SPE device. (d) Input signal:\nA square-wave charge current Jcwith amplitude ∆ Jcapplied\nto the Pt film. (e) Output signal: A resistance RTMin the\nRuO2−AlOxfilm that responds to the change in the Jcpolar-\nity, ∆RTM(≡Rhigh−Rlow) originating from the SPE-induced\n∆Tof the Pt film ( ∝∆Jc)31,34. Here, the Joule-heating-\ninduced temperature change ( ∝∆J2\nc) is constant in time,\nand does not overlap with ∆ RTM. (f) A schematic illustra-\ntion of the temperature Tdependence of RTM, from which\nthe ∆RTMvalue can be converted to the temperature change\n∆T.\nwere done using a double-layered photoresist (LOR-3A\nand AZ5214E) to provide an undercut structure for a\nbetter success rate of the lift off process.\nC. SPE and SSE measurements\nFigure1(a) shows a schematic illustration of the SPE\ndevice and the experimental setup in the present study.\nThe SPE appears as a result of the interfacial spin and\nenergy transfer between magnons in YIG and electron\nspins in Pt28–31. Suppose that the magnetization Mofthe YIG layer is oriented along the + ˆ zdirection by the\nexternal magnetic field B||+ˆ z, as shown in Fig. 1(a).\nWith the application of a charge current Jc=Jcˆ yto the\nPtfilm, thespinHalleffect(SHE)55,56inducesanonequi-\nlibrium spin, or magnetic moment, accumulation at the\nPt/YIG interface28–31,34. ForJc||+ˆ y(Jc|| −ˆ y), the\naccumulated magnetic moment δmsat the interfacial Pt\norientsalongthe −ˆ z(+ˆ z)direction30,57, whichisantipar-\nallel (parallel) to the Mdirection in Fig. 1(a). Through\nthe interfacial spin-flip scattering, δmscreates or annihi-\nlates a magnon in YIG; the number of magnons in YIG\nincreases (decreases) when δms|| −M(δms||M)4,37.\nBecause of energy conservation, this process is accompa-\nnied by a heat flow Jqbetween the electron in Pt and the\nmagnon in YIG4,37. The temperature of Pt (YIG) thus\ndecreases(increases) when δms||−MunderJc||+ˆ yand\nB||+ˆ z[Fig.1(a)], whereas the temperature of Pt (YIG)\nincreases (decreases) when δms||Mby reversing either\nJcorBin Fig.1(a)29–31,34. The SPE-induced tempera-\nture change ∆ Tsatisfies the following relationship29–31\n∆T∝δms·M∝(Jc×M)·ˆ x. (1)\nFor the electric SPE detection based on the on-chip\nthermometer (TM), we utilized the highly-accurate re-\nsistance measurement scheme called the Delta mode, a\ncombination of low-noise current source and nanovolt-\nmeter(KeithleyModel6221and2182A31,34). Weapplied\na square-wave charge current Jcwith amplitude ∆ Jcto\nthe Pt film [Figs. 1(a) and1(d)] and measured the 4 ter-\nminal RuO 2−AlOxresistance RTMthat responds to the\nchange in the Jcpolarity, ∆ RTM≡Rhigh−Rlow, where\nRhigh(Rlow) represents the RTMvalue for Jc= +∆Jc\n(−∆Jc) and was measured under the sensing current of\n100 nA applied to the RuO 2−AlOxfilm [see Figs. 1(a)\nand1(e)]31. Here, the ∆ RTMvalue isfree fromthe Joule-\nheating-induced resistance change ( ∝∆J2\nc) that is inde-\npendent of time, which thereby only contributes to the\noffset resistance Roffsetof the RuO 2−AlOxfilm shown in\nFig.1(e)58. During the SPE measurement, the magnetic\nfieldB(with magnitude B) was applied in the film plane\nand perpendicular to the Pt wire, i.e., B||ˆ zin Fig.1(a),\nexcept for the control experiment shown in Fig. 4(b),\nwhereB||ˆ x. The resistance Rhigh,lowwas recorded af-\nter the time delay τdelayof 50 ms [except for the τdelay\ndependence shown in Fig. 4(e)] during the data acqui-\nsition time τsensof 20 ms59, and then was accumulated\nby repeating the process of the Jc-polarity change ∼105\ntimes for each Bpoint [see Fig. 1(d)] to improve the\nsignal-to-noise ratio. ∆ RTMcan be converted into the\ncorresponding temperature change ∆ T(= ∆RTM/S) by\nusing the sensitivity S≡ |dRTM/dT|of the RuO 2−AlOx\nfilm [see Figs. 1(f) and4(d)].\nTo compare the Bdependence of the SPE signal with\nthat of the SSE, we also measured the SSE at T= 2 K\nusing the same device, for which all SPE results pre-\nsented in this paper were obtained, but in a different\nexperimental run from the SPE measurement. Here,4\nthe SSE measurement was done by means of a lock-\nin detection technique19,22,27and the RuO 2−AlOxlayer\nwas used as a resistive heater; an a.c. charge current\nIc=√\n2Irmssin(ωt) with the amplitude of Irms= 5.48µA\nand the frequency of ω/2π= 13.423 Hz was applied to\nthe RuO 2-AlOxfilm, and the second harmonic voltage\nin the Pt layer induced by a spin current (driven by a\nheat current due to the Joule heating of the RuO 2-AlOx\nfilmPheater=RTMI2\nrms) was detected. During the SSE\nmeasurement, the externalfield Bwasapplied in the film\nplane and perpendicular to the Pt wire, i.e., B||ˆ zin Fig.\n1(a).\nIII. RESULTS AND DISCUSSION\nA. Electrical conduction in RuO 2−AlOxfilms\nWe first characterize the electrical conduction of the\nRuO2−AlOxfilms on thermally-oxidized Si substrates.\nFigure2(a) shows the Tdependence of the resistivity ρ\nfor the films grown under the several (fixed) sputtering\npower values for the RuO 2(Al) target PRuO2(PAlOx).\nFor all the films, ρincreases with decreasing Tin the\nentire temperature range, showing a negative tempera-\nture coefficient of resistance (TCR). Both ρand its slope\n|dρ/dT|increase significantly at low temperatures and\nmonotonically by decreasing the RuO 2sputtering power\nPRuO2. The overall ρ–Tcurve shifts toward the upper\nright by decreasing PRuO2. The result shows that the\nρversusTcharacteristics of the RuO 2−AlOxfilms can\nbe controlled simply by changing the sputtering power\nPRuO2. SEM-EDX analysis reveals that the RuO 2/AlOx\nratio decreases by decreasing PRuO2[Fig.2(c)], which\nleads to the ρincrease in the electric transport. We\nalso characterized the RuO 2−AlOxfilms by means of\nAFM andfoundthat atypicalroot-mean-squaredsurface\nroughnessis Rrms∼1nm, muchsmallerthan their thick-\nness∼100 nm [see the AFM image of the RuO 2−AlOx\nfilms grown under PRuO2= 28 W and PAlOx= 25 W\n(the RuO 2content of 41%) shown in Fig. 2(d)].\nThe electrical conduction at sufficiently low tem-\nperatures for RuO 2-based thermometers has often\nbeen analyzed by the variable-range hopping (VRH)\nmodel for three dimensional (3D) systems proposed by\nMott41–46,60,61,\nρ=ρ0exp/parenleftbiggT0\nT/parenrightbigg1/4\n, (2)\nwhereρ0is the resistivity coefficient and T0is the char-\nacteristic temperature related to the electron localization\nlengtha. To discuss our result in light of the VRH, we\nplot lnρversusT−1/4for the RuO 2−AlOxfilms in Fig.\n2(b). We found that ln ρscales linearly with T−1/4at\nlow-Tranges, and the ln ρ–T−1/4data is well fitted by\nEq. (2) [see the black solid lines in Fig. 2(b)], sug-\ngesting that the low- Telectrical conduction is indeed\n(a) (b)\n0.6\n0.4\n0.2\n0 20 40 60 80 100\nT (K)ρ (Ωcm) 300 50 25 10 5 2T (K) \nT-1/4 (K-1/4 )ln[ ρ (Ωcm)] \n0.2 0.4 0.6 0.802\n-4-2\n-6PRuO 2 = 25 W\n= 26 W\n= 27 W\n= 28 W\n= 30 WPAlO x = 25 W (fixed) RuO 2 37%\nRuO 2 38%\nRuO 2 39%\nRuO 2 41%\nRuO 2 43%3D VRH fittingRuO 2 content (%) \nRuO2 sputtering power PRuO 2 (W) (c) \n25 26 27 28 30 PAlO x = 25 W (fixed) \n36 \n34 \n32 42 \n40 \n38 44 (d)\n Surface image of RuO 2-AlO x \n500 nm 7.8\n0\n(nm) \nFIG. 2: (a) Tdependence of the resistivity ρfor the\nRuO2−AlOxfilms fabricated on thermally-oxidized Si sub-\nstrates under the several d.c. sputtering power for the RuO 2\ntarget (PRuO2) and the fixed d.c. power for the Al target\n(PAlOx). (b) ln ρversusT−1/4for the RuO 2−AlOxfilms.\nThe black solid lines are obtained by fitting Eq. ( 2) (the\n3D Mott VRH model) to the experimental data. (c) Rela-\ntionship between the RuO 2content in the RuO 2−AlOxfilms\nand the RuO 2sputtering power PRuO2determined by SEM-\nEDX. Using this correspondence, the figure legends in (b)\nand also Fig. 3are described in terms of the RuO 2content.\n(d) A typical AFM image of the RuO 2−AlOxfilm grown un-\nderPRuO2= 28 W and PAlOx= 25 W (the RuO 2content\nof 41%), where the root-mean-squared surface roughness is\nRrms= 1.2 nm. The white scale bar represents 500 nm.\ngoverned by the VRH. From the fitting, the T0val-\nues are obtained as 2 .58×105, 1.41×105, 8.95×104,\n5.73×103, and 2.60×102K for the RuO 2−AlOxfilms\ngrown under PRuO2= 25, 26, 27, 28, and 30 W, respec-\ntively. We note that, at all the Tranges adopted for\nthe VRH fitting, the average hopping distance ( Rhop) is\nlargerthan the electron localizationlength ( a) that is the\nrequirement for the VRH model to be valid: Rhop/a=\n(3/8)(T0/T)1/4>162–66. Besides, the Mott hopping en-\nergyEhop= (1/4)kBT(T0/T)1/4(kB: the Boltzmann\nconstant) obtained for the present films is larger than (or\ncomparable to) the thermal energy kBT, allowing for the\nelectron hopping62–66. The above argument further con-\nfirms the validity of the 3D Mott VRH model to describe\nthe conduction mechanism in the RuO 2−AlOxfilms.\nWe here discuss the T-dependent thermometer char-\nacteristics of the RuO 2−AlOxfilms. Figure 3shows the\nTdependence of (a) the resistance R, (b) the sensitivity\nS≡ |dR/dT|,(c)thetemperaturecoefficientofresistance\n(TCR)ST≡ |(1/R)dR/dT|, and (d) the dimensionless\nsensitivity SD≡ |(T/R)dR/dT|=|d(lnR)/d(lnT)|for5\n1 10 100 \nT (K)\n1 10 100\nT (K)1 10 100 \nT (K)1 10 100 \nT (K)\n10 -310 -210 -110 010 310 410 510 6\n10 310 410 510 6\n10 010 110 2\n10 -110 1\n10 0R (Ω) \n|dR / dT | (Ω K-1) |(1/ R) dR / dT | (K -1)\n |( T/R) dR / dT | T-coefficient of resistance (TCR) ST Dimensionless sensitivity SDSignal sensitivity S Resistance R (a) (b)\n(c) (d)RuO 2 37%\nRuO 2 38%\nRuO 2 39%\nRuO 2 41%\nRuO 2 43%RuO 2 37%\nRuO 2 38%\nRuO 2 39%\nRuO 2 41%\nRuO 2 43%\nRuO 2 37%\nRuO 2 38%\nRuO 2 39%\nRuO 2 41%\nRuO 2 43%\nRuO 2 41% \n(RuO 2-AlO x film \n on AlO x/Pt/YIG) RuO 2 41% \n(RuO 2-AlO x film \n on AlO x/Pt/YIG) RuO 2 37%\nRuO 2 38%\nRuO 2 39%\nRuO 2 41%\nRuO 2 43%\nFIG. 3: Tdependence of (a) the resistance R, (b) the sen-\nsitivityS≡ |dR/dT|, (c) the temperature coefficient of re-\nsistance (TCR) ST≡ |(1/R)dR/dT|, and (d) the dimension-\nless sensitivity SD≡ |(T/R)dR/dT|=|d(lnR)/d(lnT)|for\nthe RuO 2−AlOxfilms with different RuO 2content fabricated\non thermally-oxidized Si substrates. The films are patterne d\ninto a Hall-bar shape having the length, width, and thick-\nness of 1 .0 mm, 0 .5 mm, and ∼100 nm, respectively, by\nco-sputtering RuO 2−AlOxthrough a metal mask. In (c) and\n(d), the ST(T) andSD(T) results for the RuO 2−AlOxther-\nmometer film on the Pt/YIG sample are coplotted (red star\nmarks).\nthe RuO 2−AlOxfilms. Here, the sensitivity Sis an es-\nsential quantity when the thermometer is used as an ac-\ntual temperature-sensor device in its original form. The\nTCRSTis the normalized sensitivity Sby the measured\nresistance R, given that Sis geometry dependent (i.e.,\ndR/dTscales with R)40. The dimensionless sensitivity\nSDis a measure often used to compare the performance\nof the thermometers made of different materials, regard-\nless of their size40,67–69. For the present RuO 2−AlOx\nfilms with low RuO 2content ( <40%), the sensitivity S\ntakes a high value on the order of 104−106Ω/K below\n∼10 K. For such a low- Trange, however, their resis-\ntanceRvalues are highly enhanced, and exceed 1 MΩ at\n2 K, which is too high to use such films as thermometers\nin their originaldimensions below the liquid-4He temper-\nature. Besides, their TCR values start to show a satu-\nration behavior by decreasing Tin such a low- Tenvi-\nronment. By contrast, the RuO 2−AlOxfilm with the\nRuO2content of 41% (fabricated under PRuO2= 28 W\nandPAlOx= 25 W) shows a moderate R(S) value of\n104−105Ω (104−105Ω/K) and the best TCR char-\nacteristic of ∼100% K−1around 2 K. We therefore\nadopt its growth condition for our SPE device. Overall,\ntheS, TCR, and SDvalues of the present RuO 2−AlOx\nfilms are comparable to those of commercially avail-able Cernox ™zirconium oxy-nitride sensors40,69, carbon\ncomposites41,70,71, and AuGe films68commonly used at\na similar Trange.\nB. Observation of SPE based on RuO 2−AlOx\non-chip thermometer\nWe are now in a position to demonstrate a cryogenic\nSPEinthePt/YIGsamplebasedontheRuO 2−AlOxon-\nchip thermometer. Figure 4(a) shows the Bdependence\nof the RuO 2−AlOxresistance change ∆ RTMmeasured\natT= 2 K and a low- Brange of |B| ≤0.2 T. With the\napplication of the charge current ∆ Jc(= 0.15 mA) to the\nPt film, a clear ∆ RTMsignal appears with a magnitude\nsaturated at ∼30 mΩ72and its sign changes depending\non theB(||±ˆ z) direction. The signal disappears either\nwhen ∆Jcis essentially zero [gray diamonds in Fig. 4(a)]\nor when Bis applied perpendicular to the Pt/YIG in-\nterface (B|| ±ˆ x) [Fig.4(b)]. We also confirmed that\ntheBdependence of ∆ RTMis consistent with that of\nthe SSE in the identical Pt/YIG device [see Fig. 4(c)].\nThese are the representative features of the SPE29–38.\nFurthermore, the sign of ∆ RTMagrees with the SPE-\ninduced temperature change29,30. As shown in Fig. 4(a),\nthe measured ∆ RTMvalue is positive for B >0, mean-\ning that the resistance RTMincreases (decreases) when\nJc||+ˆ y(Jc||−ˆ y), for which the orientation of the SHE-\ninduced magnetic moment at the interfacial Pt layer is\nδms|| −ˆ z(δms||+ˆ z) in Fig. 1(a). According to the\nnegative TCR of the RuO 2−AlOxfilm, this implies that\nthe temperatureofthe Ptfilm decreases(increases)when\nδms|| −ˆ z(δms||+ˆ z) underM||B||+ˆ z. This cor-\nrespondence between the sign of the temperature change\n∆Tand the relative orientation of δmswith respect to\nMis consistent with the scenarioof the SPE described in\nSec.IIC. We thus conclude that we succeeded in mea-\nsuring a cryogenic SPE using the RuO 2−AlOxon-chip\nthermometer film.\nTo convertthe ∆ RTMvalue to the temperature change\n∆T, we measured the RTM–Tcurve for the RuO 2−AlOx\nfilm. As shown in Fig. 4(d), similar to the results de-\nscribed in Sec. IIIA, its resistance RTMincreases dra-\nmatically with decreasing Tat low temperatures, and\nthe sensitivity S=|dRTM/dT|is as large as 55 .3 kΩ/K\nat 2 K [The TCR STand dimensionless sensitivity SD\nfor the film are plotted in Figs. 3(c) and3(d), respec-\ntively, and ρand|dρ/dT|are plotted in Figs. 7(a) and\n7(b) in APPENDIX B, respectively, together with the\nresults for the RuO 2−AlOxfilms grown on thermally-\noxidized Si substrates]. In Fig. 4(c), we replot the Bde-\npendence of the SPE in units of the temperature change\n∆T(= ∆RTM/S) using the above Svalue. We evaluate\nthe magnitude of the SPE-induced temperature change\nto be ∆TSPE= 482±39 nK, by averaging the ∆ Tval-\nues for 0 .08 T≤B≤0.2 T, at which the magnetization\nMof the YIG slab fully orients along the Bdirection74\n[see the dashed line in Fig. 4(c)]. The standard devia-6\n100 0 200 300 \nT (K)40 \n30 \n20 RTM (kΩ) (d)\n10 2 3 4 5\nT (K)Sensitivity |dR TM / dT | (kΩ K-1)\n60 \n40 \n20 \n0-0.2 0 0.2\nB (T) 40 \n20 /g39RTM (mΩ) \n-40 -20 T = 2 K, B x<(b)\n/g39Jc = 150 μA \n0\n-0.2 -0.1 0 0.1 0.2\nB (T) T = 2 K, B z< 1.0\n0.5\n0/g39T (μK) \n-1.0-0.530 ms\n50 ms\n80 ms/g87delay :(e)-0.2 -0.1 0 0.1 0.2\nB (T) 40 \n20 \n0/g39RTM (mΩ) \n-40 -20 T = 2 K, B z\n0.15 mA \n 0 mA <(a)\n/g39Jc :\n1.0\n0.5\n0/g39T (μK) \n-1.0-0.5\n-0.2 -0.1 0 0.1 0.2\nB (T) T = 2 K, B z<\nSPE\nSSE(c) \n10 \n0\n-10 VSSE /Pheater (mV W -1)/g39TSPE\n1.0\n0.5\n0/g39TSPE (μK) (f)\n50 100 \n/g87delay (ms) \nFIG. 4: (a) Bdependence of the SPE-induced ∆ RTMat\nT= 2 K and B≤0.2 T (B||ˆ z) under ∆ Jc= 0.15 and\n0.00 mA and τdelay= 50 ms. The dashed lines connect adja-\ncent plots. (b) Bdependence of ∆ RTMforB||ˆ x(B≤0.3 T)\nunder ∆Jc= 0.15 mA. Note that the applied Bis larger than\nthe out-of-plane ( B||ˆ x) saturation field for bulk YIG, which\nis∼0.2 T73. (c) Comparison between the Bdependence of\nthe SPE-induced temperature change ∆ T(blue filled circles)\nand the SSE-induced voltage normalized by heating power\nVSSE/Pheater(orange solid curve) at T= 2 K and B||ˆ z. The\nSPE data shown here is the same as that plotted in (a), but\nthe left vertical axis is converted from ∆ RTMto ∆Tvia the\nRTM–Tcalibration curve plotted in (d). For details of the\nSSE measurement, see Sec. IIC. (d)Tdependence of RTM\n(main) and |dRTM/dT|(inset) for the RuO 2−AlOxfilm on\nthe Pt/YIG sample. (e) Bdependence of the SPE-induced\n∆TatT= 2 K under ∆ Jc= 0.15 mA and several τdelay\nvalues. (f) τdelaydependence of the magnitude of the SPE-\ninduced temperature change ∆ TSPE, where ∆ TSPEis evalu-\nated by averaging the ∆ Tvalues for 0 .08 T≤B≤0.2 T [see\nalso (c)]. The dashed line represents the averaged value. Al l\nthe ∆RTMand ∆Tdata were anti-symmetrized with respect\nto the magnetic field B.\ntion of 39 nK shows that our measurement scheme based\non the RuO 2−AlOxon-chip thermometer can resolve an\nextremely small ∆ Ton the order of several tens of nK\n(which is a value achieved by repeating the process of\ntheJc-polarity change of 7 ×104times at each B). The\n∆Tresolution is much higher than that reported in the\nprevious SPE measurements based on lock-in thermog-\nraphy, lock-in thermoreflectance, and thermocouples, forwhich the typical resolution is 100, 10 −100, and 5 µK,\nrespectively34,38,39. We found that the magnitude of\n∆TSPEnormalized by the charge-currentdensity ∆ jcap-\nplied to the Pt wire is ∆ TSPE/∆jc= 3.2×10−15Km2/A,\nwhich is two orders of magnitude smaller than the cor-\nresponding value for Pt/YIG systems measured at room\ntemperature30,31. The low- Tsignal reduction of the SPE\nis consistent with that found in the SSE17,75–77, and is\nattributed mainly to the reduction of the thermally acti-\nvated magnons contributing to these phenomena at cryo-\ngenic temperatures. Besides, there can be a finite tem-\nperature gradient across the insulating AlO xfilm, be-\ntweenthe Pt andRuO 2−AlOxlayers,resultingin further\ndecreaseofthedetected∆ Tsignal. Wealsomeasuredthe\ndelay time τdelaydependence of the SPE and found that\nthe ∆TSPEtakes almost the same value in the present\nτdelayrange (30 ms ≤τdelay≤80 ms) [see Figs. 4(e) and\n4(f)], showing that all the data were obtained under the\nsteady-state condition31,35.\nWe also explored the high magnetic field response of\nthe SPE signal. Figure 5(a) displays the ∆ TversusB\ndata measured at T= 2 K and B≤10 T (B||ˆ z). We\nfound that ∆ Texhibits a maximum at a low B(/lessorsimilar0.2 T)\nand, by increasing B, gradually decreases and is eventu-\nally suppressed. The Bdependence of the SPE agrees\nwell with that of the SSE measured with the identical\ndevice [see Fig. 5(a)]. We note that the magnetoresis-\ntance (MR) ratio of the RuO 2−AlOxfilm is as small as\n∼3.7% forB≤10 T at T= 2 K, so that the device\ncan be used reliably under the high- Brange. The ob-\nserved ∆ T(B) feature is explained in terms of the sup-\npression of magnon excitations by the Zeeman effect, as\nestablished in the previous SSE research17,27,75–77[see\nFig.5(b)]. By increasing B, the magnon dispersion\nshifts toward high frequencies due to the Zeeman inter-\naction (∝γB). AtB= 10 T, the Zeeman energy /planckover2pi1γB\nis∼13.5 K in units of temperature, which is greater\nthan the thermal energy kBTat 2 K [see Fig. 5(b)], re-\nsulting in an insignificant value of the Boltzmann factor:\nexp(−/planckover2pi1γB/kBT)∼10−3≪1, where γand/planckover2pi1represent\nthe gyromagnetic ratio and Dirac constant, respectively.\nTherefore, the thermal magnons that can contribute to\nthe SPE at a low Bare gradually suppressed with the\nincrease of Band, atB∼10 T, are hardly excited by\nthe strong Zeeman gap in the magnon spectrum [Fig.\n5(b)], which leads to the suppression of the SPE in the\nlow-Tand high- Benvironment. We also compared the\nexperimental result with a calculation for the interfacial\nheat current induced by the SPE JSPE\nqand spin current\ninduced by the SSE JSSE\ns, which are expressed as\nJSPE\nq∝/integraldisplayd3k\n(2π)3ω2∂nBE\n∂ω,\nJSSE\ns∝ −/integraldisplayd3k\n(2π)3ωT∂nBE\n∂T,(3)\nrespectively75,78–81. Here, ω=Dexk2+γBis the\nparabolic magnon dispersion for YIG with the stiff-7\n10 \n0\n-10 VSSE /Pheater (mV W -1)\n-10 -5 0 5 10 \nB (T)0.2\n0/g39T (μK) \n-0.2(a)\nT = 2 K, B z<\nSPE\nSSE\n0.2\n0ω/2 /g83 (THz) (b) Magnon dispersion \n0.3\n0.1\nk (10-8 m -1)1 2kBT at 2 KZeeman gap ( /g118/g3 γB )\nhγB /kB ~ 13.5 K \n at 10 T B = 10 T\n= 0 T\nmagnon \nfreeze-out \nmagnon Experiment\nCalculation\n(arb. units)\nFIG. 5: (a) Comparison between the high magnetic field B\nresponse of the SPE-induced temperature change ∆ T(blue\nfilled circles) and the SSE-induced voltage normalized by\nheating power VSSE/Pheater(blue solid curve) at T= 2 K\nandB≤10 T (B||ˆ z). The SPE data was obtained un-\nder ∆Jc= 0.15 mA and τdelay= 50 ms. For details of the\nSSE measurement, see Sec. IIC. The orange dashed curve\nshows the numerically calculated result based on Eq. ( 3) for\nT= 2 K. (b) Magnon dispersion relations for YIG15atB= 0\nand 10 T, at which the magnon-excitation gap values are ∼0\nand 13.5 K in units of temperature, respectively, where krep-\nresents the wavenumber. The thermal energy ( kBT) level of\n2 K is also plotted with a green dashed line, above which\nthermal excitation is exponentially suppressed.\nness constant of Dex= 7.7×10−6m2/s15andnBE=\n[exp(/planckover2pi1ω/kBT)−1]−1is the Bose −Einstein distribu-\ntion function. Note that the relation ω∂nBE/∂ω=\n−T∂nBE/∂Tensures the Onsager reciprocity between\nthe SSE and SPE81, which makes the above expressions\nto be of the same form in terms of the Bdependence.\nAs shown by the orange dashed curve in Fig. 5(a), the\ncalculated result based on Eq. ( 3) well reproduces the\nexperiment. This result further supports the origin of\nthe measured ∆ Tsignal and provides additional clues\nfor further understanding of the physics of the SPE.\nIV. CONCLUSIONS\nIn this study, we have fabricated RuO 2−AlOxfilms\nby means of a d.c. co-sputtering technique and charac-\nterized their electrical conduction and sensitivity at low\ntemperatures. Thesensitivitywasfoundtobetuned sim-\nply bythe relativesputtering powerapplied forthe RuO 2\nand Al targets, and the TCR value reaches ∼100% K−1\nfor the RuO 2−AlOxfilms with the moderate RuO 2con-\ntent (/greaterorsimilar41%). By using the RuO 2−AlOxfilm as an on-\nchip micro-thermometer, we successfully measured the\nSPE-induced temperature change ∆ Tin a Pt-film/YIG-\nslab system at the low temperature of 2 K based on the\nso-called Delta method, which can resolve an extremely\nsmall ∆Tvalue of several tens of nK. We also measured\nthe high Bresponse of the SPE at T= 2 K up to\nB= 10 T, and found that, by increasing B, the SPE\nsignal gradually decreases and is eventually suppressed.100 200 3000.51.01.5\n0Ar only \nAr + 33.3 vol.% O 2\nT (K) R / R (300 K) R-T characteristics of RuO 2 film\nFIG. 6: Tdependenceof R/R(T= 300 K)for thepure RuO 2\nfilms grown under only Ar gas flow and also under a large\namount of O 2gas flow (Ar + 33.3 vol.% O 2), for which the\nresistivity ρvalues at T= 300 K are evaluated as 3 .21×10−4\nand 3.96×10−4Ωcm, respectively.\nTheBdependence can be interpreted in terms of the\nfield-induced freeze-out of magnons due to the Zeeman-\ngap opening in the magnon spectrum of YIG. We an-\nticipate that our experimental methods based on an on-\nchip thin-film thermometer will be useful for exploring\nlow-Tthermoelectric heating/cooling effects in various\ntypes of micro devices, including a system based on two-\ndimensional van der Waals materials82–84. Besides, with\nan appropriate optimization of the resistance and sensi-\ntivity of the RuO 2−AlOxfilms by controllingthe content\nof RuO 2, our results can be extended toward even lower\ntemperature ranges below 1 K, where they can be used\nto detect unexplored cryogenic spin caloritronic effects\ndriven by nuclear and quantum spins.\nACKNOWLEDGMENTS\nWe thank S. Daimon, R. Yahiro, J. Numata, K. K.\nMeng, H. Arisawa, T. Makiuchi, and T. Hioki for valu-\nable discussions. This work was supported by JST-\nCREST (JPMJCR20C1 and JPMJCR20T2), Grant-in-\nAid for Scientific Research (JP19H05600, JP20H02599,\nand JP22K18686) and Grant-in-Aid for Transformative\nResearch Areas (JP22H05114) from JSPS KAKENHI,\nMEXT Initiative to Establish Next-generation Novel In-\ntegratedCircuits Centers (X-NICS) (JPJ011438),Japan,\nMurata Science Foundation, Daikin Industries, Ltd, and\nInstitute for AI and Beyond of the University of Tokyo.\nAPPENDIX A: ELECTRICAL CONDUCTION IN\nPURE RUO 2FILMS\nTo check the effect of O 2gas introduction during sput-\nteringontheRuO 2film, wealsofabricatedpristine(poly-\ncrystalline) RuO 2films under only Ar gas flow and also\nunder a large amount of O 2gas flow (Ar + 33.3 vol.%8\n1 10 100\nT (K)1 10 100\nT (K)(on AlOx/Pt/YIG) ρ (Ωcm) \n10 -310 -210 -110 010 1Resistivity slope Resistivity ρ (a) (b)\n10 -310 -210 -110 0|dρ / dT | (Ωcm K-1)\n10 -510 -4\nPRuO 2 = 28 WPRuO 2 = 25 W\n= 26 W\n= 27 W\n= 28 W\n= 30 W\n(on AlO x/Pt/YIG) PRuO 2 = 28 W(on SiO 2/Si) PRuO 2 = 25 W\n= 26 W\n= 27 W\n= 28 W\n= 30 W\n(on SiO 2/Si) \nFIG. 7: Tdependence of (a) the resistivity ρand (b) its\nslope|dρ/dT|for theRuO 2−AlOxfilms onthermally-oxidized\nSi substrates (filled circles) and on the Pt/YIG sample (red\nstar marks) grown under the several d.c. sputtering power\nfor the RuO 2target (PRuO2) and the fixed d.c. power for the\nAl target ( PAlOx= 25 W). Note that ST≡ |(1/R)dR/dT|=\n|(1/ρ)dρ/dT|andSD≡ |(T/R)dR/dT|=|(T/ρ)dρ/dT|, the\nTdependences of which are shown in Figs. 3(c) and 3(d),\nrespectively.\nO2, which is ∼5 times greater than that used for the\nRuO2−AlOxdeposition)andmeasuredtheir R–Tcurves.\nHere, the RuO 2films were patterned into a Hall-bar\nshape having the length, width, and thickness of 2 .0 mm,\n0.3 mm, and ∼10 nm, respectively, by sputtering RuO 2\nthrough a metal mask. Figure 6shows the Tdependence\nofRnormalized by the value at 300 K for each RuO 2film. For both the films, the R–Tcurve shows almost\nthe same characteristics; Rgradually increases with de-\ncreasing Tand the R(T)/R(300 K) value at T= 2 K\n(the temperature of interest) deviates only ∼2% with\neach other. This result shows that the effect of oxygen\non the RuO 2deposition does not play an essential role\nin theRversusTcharacteristics of RuO 2.\nAPPENDIX B: COMPARISON OF ρ−TCURVES\nBETWEEN RUO 2−ALOxFILMS ON SIO 2/SI\nSUBSTRATES AND ON PT/YIG DEVICE\nFigures7(a) and7(b) show the double logarithmicplot\nof (a) the resistivity ρand (b) its slope |dρ/dT|versus\ntemperature Tfor the RuO 2−AlOxfilms on thermally-\noxidized Si substrates (filled circles) and on the Pt/YIG\nsample (red star marks) grown under the several d.c.\nsputtering power for the RuO 2target (PRuO2) and the\nfixed d.c. power for the Al target ( PAlOx= 25 W). 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J.\nTelfordet al., The Magnetic Genome of Two-Dimensional\nvan der Waals Materials, ACS Nano 16, 6960 (2022)." }, { "title": "1408.2972v2.Quantitative_Temperature_Dependence_of_Longitudinal_Spin_Seebeck_Effect_at_High_Temperatures.pdf", "content": "arXiv:1408.2972v2 [cond-mat.mtrl-sci] 30 Sep 2014Quantitative Temperature Dependence of Longitudinal Spin Seebeck Effect\nat High Temperatures\nKen-ichi Uchida,1,2,∗Takashi Kikkawa,1Asuka Miura,3Junichiro Shiomi,2,3and Eiji Saitoh1,4,5,6\n1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n2PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan\n3Department of Mechanical Engineering, The University of To kyo, Tokyo 113-8656, Japan\n4WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan\n5CREST, Japan Science and Technology Agency, Tokyo 102-0076 , Japan\n6Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan\n(Dated: August 16, 2018)\nThis article reports temperature-dependent measurements of longitudinal spin Seebeck effects\n(LSSEs) in Pt/Y 3Fe5O12(YIG)/Pt systems in a high temperature range from room tempe rature\nto above the Curie temperature of YIG. The experimental resu lts show that the magnitude of\nthe LSSE voltage in the Pt/YIG/Pt systems rapidly decreases with increasing the temperature\nand disappears above the Curie temperature. The critical ex ponent of the LSSE voltage in the\nPt/YIG/Pt systems at the Curie temperature was estimated to be 3, which is much greater than\nthat for the magnetization curve of YIG. This difference high lights the fact that the mechanism of\nthe LSSE cannot be explained in terms of simple static magnet ic properties in YIG.\nPACS numbers: 85.75.-d, 72.25.-b, 72.15.Jf, 73.50.Lw\nI. INTRODUCTION\nThe Seebeck effect converts a temperature difference\ninto electric voltage in conductors [1]. Since the discov-\nery of the Seebeck effect nearly 200 years ago, it has been\nstudiedintensivelytorealizesimpleandenvironmentally-\nfriendly energy-conversion technologies [2]. The See-\nbeck effect has been measured using various materials\nin a wide temperature range to investigate thermoelec-\ntricconversionperformanceandthermoelectrictransport\nproperties. Temperature-dependent measurements in a\nhigh temperature range are especially important in the\ninvestigation of the Seebeck effect, since thermoelectric\ndevices are often used above room temperature [3, 4].\nIn the field of spintronics [5–7], a spin counterpart of\nthe Seebeck effect —the spin Seebeck effect (SSE)— was\nrecently discovered. The SSE converts a temperature\ndifference into spin voltage in ferromagnetic or ferrimag-\nnetic materials. When a conductor is attached to a mag-\nnet under a temperature gradient, the thermally gener-\nated spin voltage in the magnet injects a spin current\n[8–10] into the conductor. Since the SSE occurs not only\nin metals [11–14] and semiconductors [15, 16] but also\nin insulators [13, 17–32], it enables the construction of\n“insulator-based thermoelectric generators” [21] in com-\nbination with the inverse spin Hall effect (ISHE) [33–37],\nwhich was impossible if only the conventional Seebeck\neffect was used.\nThe observation of the SSE in insulators has been re-\nported mainly in a longitudinal configuration [13, 18–\n32]. The sample system for measuring the longi-\ntudinal SSE (LSSE) is a simple paramagnetic metal\n(PM)/ferrimagnetic insulator (FI) junction system. In\nmany cases, Pt and Y 3Fe5O12(YIG) are used as PMand FI, respectively. When a temperature gradient ∇T\nis applied to the PM/FI system perpendicular to the in-\nterface, the spin voltage is thermally generated and in-\njects a spin current into the PM along the ∇Tdirection\nowing to thermal spin-pumping mechanism [38–46]. This\nthermally induced spin current is converted into an elec-\ntric field EISHEby the ISHE in the PM according to the\nrelation\nEISHE∝Js×σ, (1)\nwhereJsis the spatial direction of the thermally induced\nspin current and σis the spin-polarization vector of elec-\ntrons in the PM, which is parallel to the magnetization\nMof FI [see Fig. 1(a)]. By measuring EISHEin the PM,\none can detect the LSSE electrically.\nIn the experimental research on the SSE, temperature-\ndependent measurements also have been used for inves-\ntigating various thermo-spin transport properties, such\nas phonon-mediated effects [15, 16, 39, 47], correlation\nbetween the SSE and magnon excitation [29, 48], and ef-\nfects of metal-insulator phase transition [13]. However,\nall the experiments on the SSE to date have been per-\nformedaroundandbelowroomtemperature. Inthisarti-\ncle, we report quantitative temperature-dependent mea-\nsurements of the LSSE in Pt/YIG systems in the high\ntemperature range from room temperature to above the\nCurie temperature of YIG.\nII. EXPERIMENTAL PROCEDURE\nThe sample system used in this study consists of a\nsingle-crystalline YIG slab covered with Pt films. One\ndifference from conventional samples is that the Pt films2\nVL\nVH\nYIG slab Pt film (Pt_H) \nPt film (Pt_L) (a) (b)\n x z\ny\n(c) (d)\nset temperature of heat baths P1 \nP2 \nP3 \nestimate THPt and TLPt under ∇T\nby combining P1 and P2 data n + 1 → n\nn <_ Nyes no measure RH and RL of Pt films measure RH and RL of Pt films \nset THset = THn, TLset = TLn ( THn > TLn)set THset = TLset = TLn\nmeasure LSSE ( H dep. of VH and VL)\nn = 1 n = 2 n = 3TL1 \ntime THset\nTLsetTL2 \nTH1 TL3 \nTH2 TL4 \nTH3 P1 P2 P3 HEISHE \nJs\n∇TJs\n(step) Msapphire plate\nheat bath ( TLset)heat bath ( THset)\nPt/YIG/Pt sampleVH\nVLthermal grease Au/Ti contacts∇T H Pt wiresmetal paste\nFIG. 1: (a) A schematic illustration of the Pt/YIG/Pt sample .∇T,H,M,EISHE, andJsdenote the temperature gradient,\nmagnetic field (with the magnitude H), magnetization vector, electric field induced by the ISHE, and spatial direction of\nthe thermally generated spin current, respectively. The el ectric voltage VH(VL) and resistance RH(RL) between the ends of\nPtH (PtL) were measured using a nanovolt/micro-ohm meter (Agilent 34420A) in the ‘DC-voltage’ and ‘2-wire-resistance’\nmodes, respectively. The DC-voltage (2-wire-resistance) mode corresponds to the switch-off (switch-on) state in this schematic\nillustration. (b) Experimental configuration for measurin g the LSSE used in the present study. (c) A flow chart of the\nmeasurement processes. (d) A schematic graph of the set temp eratures, THnandTLn, of the heat baths as a function of time\nor the measurement step number n. The process P1 was performed under the isothermal conditio n, while the processes P2 and\nP3 were under the temperature gradient. Before starting the processes P1 and P2, we waited for ∼30 minutes at each nto\nstabilize the Tset\nHandTset\nLvalues.\nare put on both the top and bottom surfaces of the\nYIG slab [Fig. 1(a)], while only the top surface of\nYIG is covered with a Pt film in conventional samples\n[18, 23, 27, 32]. The lengths of the YIG slab along the\nx,y, andzdirections are 3 mm, 7 mm, and 1 mm, re-\nspectively. 10-nm-thick Pt films were sputtered on the\nwhole of the 3 ×7-mm2(111) surfaces of the YIG. The\ntop and bottom Pt films are electrically insulated from\neach other because YIG is a very good insulator. Since\nYIG has the largechargegap of2.7eV [49], thermal exci-\ntation of charge carriers in YIG is vanishingly small even\nat the high temperatures.\nTo attach electrodes to both the Pt films symmetri-\ncally and to generate a uniform temperature gradient,\nwe made the configuration shown in Fig. 1(b). Here,\nthe Pt/YIG/Pt sample was sandwiched between two 0.5-mm-thick sapphire plates of which the surface is covered\nwith two separated Au/Ti contacts. The distance be-\ntween the two Au/Ti contacts is ∼6 mm. To extract\nvoltage signals in the Pt films, both the ends of the Pt\nfilms are connected to the Au/Ti contacts via sintering\nmetal paste, which can be used up to 900◦C, and thin Pt\nwires with the diameter of 0.1 mm were attached to the\nend of the contacts [see Fig. 1(b)]. The sapphire plates\nare thermally connected to heat baths of which the tem-\nperatures are controlled with the accuracy of <0.6 K by\nusingPID(proportional-integral-derivative)temperature\ncontrollers. We attached thermal paste to both the sur-\nfaces of the sapphire plates, except for the regions of the\nAu/Ti contacts, to improve the thermal contact [see Fig.\n1(b)]. During the measurements of the LSSE, the tem-\nperature of the upper heat bath, Tset\nH, is set to be higher3\nthan that of the lower one, Tset\nL. According to the direc-\ntion of the temperature gradient, hereafter, the top and\nbottom Pt films of the Pt/YIG/Pt sample are referred to\nas ‘PtH’ and ‘Pt L’, respectively. In this setup, we can\nmeasure the electric voltage VH(VL) and resistance RH\n(RL) between the ends of Pt H (PtL) without changing\nelectrodes, wiring, and measurement equipment [see also\nthe caption of Fig. 1]. An external magnetic field H\n(with the magnitude H) was applied along the xdirec-\ntion.\nTo investigate the temperature dependence of the\nLSSE quantitatively, it is important to estimate the tem-\nperature difference between the top and bottom of the\nYIG sample. However, in the conventional experiments,\nthe temperatures of the heat baths, not the sample it-\nself, were usually monitored [20]. Therefore, the mea-\nsured temperature difference includes the contributions\nfrom the interfacial thermal resistance between the sam-\nple and heat baths and from small temperature gradients\nin the sample holders. To avoid this problem, in this\nstudy, we used the Pt films not only as spin-current de-\ntectors but also as temperature sensors [28, 31]; we can\nknow the temperatures of the Pt films from the temper-\nature dependence of the resistance of the films, enabling\nthe estimation of the temperature difference between the\ntop and bottom of the sample during the LSSE measure-\nments.\nFigure 1(c) shows the flow chart of the measure-\nment processes. The measurement comprises the fol-\nlowing three processes P1-P3, and the processes are re-\npeatedNtimes. Hereafter, THnandTLndenote the\nset temperatures for each measurement step number\nn(= 1,2,..., N) (see Table I, where the values of THn\nandTLnfor each nare shown). The process P1 is the\nmeasurement of the resistance of Pt H and Pt L,RH\nTABLE I: Set temperatures of the heat baths for each mea-\nsurementstep number n(≤N). Here, the THnandTLnvalues\nare increased by 10 K with every nincrease, while THn−TLn\nis fixed at 8 K.\nStep number n T Ln(K)THn(K)\n1 290 298\n2 300 308\n3 310 318\n4 320 328\n5 330 338\n· · ·\n· · ·\n· · ·\n31 590 598\n32 600 608\n33 610 618\n34 (=N) 620 628andRL, in the isothermal condition, where the tempera-\ntures of the heat baths are set to the same temperature:\nTset\nH=Tset\nL=TLn. In this isothermalcondition, the tem-\nperature of the Pt/YIG/Pt sample is uniform and very\nclose to the set temperature of the heat baths irrespec-\ntive of the presence of the interfacial thermal resistance.\nNext, weapplyatemperaturegradienttothePt/YIG/Pt\nsample by increasing the temperature of the upper heat\nbath, where Tset\nH=THnandTset\nL=TLnwithTHn> TLn\n(see Table I). After waiting until the temperatures are\nstabilized, we measure the resistance of the Pt films un-\nderthe temperature gradient: this is the processP2. The\nprocesses P1 and P2 are performed without applying H.\nImmediately after the process P2, we proceed to the pro-\ncess P3; the LSSE, i.e. the Hdependence of VHandVL,\nis measured with keeping the magnitude of Tset\nH−Tset\nL\nconstant. After finishing the LSSE measurements, we go\non to the next step and increase nby 1, where the values\nofTHnandTLnare increased as shown in Table I. These\nmeasurement processes are summarized in Fig. 1(d).\nThe calibration method of the sample temperatures is\nas follows. From the process P1, we obtain the tem-\nperature ( TLn) dependence of RHandRLunder the\nisothermal condition. By comparing the resistance un-\nder the temperature gradient, obtained from the process\nP2, with the isothermal RH,L-TLncurves, we can cali-\nbrate the temperature of the Pt films, TPt\nHandTPt\nL, un-\nder the temperature gradient, allowingus to estimate the\naveragetemperature Tav(= (TPt\nH+TPt\nL)/2) and the tem-\nperature difference ∆ T(=TPt\nH−TPt\nL) in the YIG slab\nduring the LSSE measurements. The Tavand ∆Tvalues\nare free from the contributions from the interfacial ther-\nmal resistance between the sample and heat baths and\nfrom the temperature gradients in the sapphire plates,\nenabling the quantitative evaluation of the LSSE at var-\nious temperatures. Although the experimental protocol\nproposed here cannot estimate the contributions of the\ninterfacial thermal resistance at the Pt/YIG interfaces\nand temperature gradients in the Pt films, they are neg-\nligible compared with the temperature difference applied\nto the YIG slab [26, 32].\nIII. RESULTS AND DISCUSSION\nFigures 2(a) and 2(b) respectively show the Hdepen-\ndence of VHandVLin the Pt/YIG/Pt sample for each\nstepnumber n, measuredwhen THn−TLn= 8K. Around\nroom temperature, we observed clear voltage signals in\nboth the Pt films; the signs of VHandVLare reversed\nin response to the reversal of the magnetization direction\nof the YIG slab. Since the contribution of anomalous\nNernst effects induced by proximity ferromagnetism in\nPt [50] is negligibly small in Pt/YIG systems, the volt-\nage signals observed here are due purely to the LSSE\n[23, 27, 32]. The sign of the LSSE voltage in Pt H was4\n020406080100\n510 15\n300400500600300 400 500 600\nTLn (K) \nTav (K) ∆T (K) \n10 20 30 0\nnRH,L ( Ω)\n-1 0 1\nH (kOe) Pt_H Pt_L \nn = 1\nn = 29 n = 25 n = 21 n = 17 n = 13 n = 9n = 5\nn = 33 Step \n-1 0 1\nH (kOe) Pt_H Pt_L \n10.0 10.0 (c)\n(d)VH ( µV) (a)\n10 20 30 \nnRH.L ( Ω)\n020 60 \n40 80 \nPt_H Pt_L VLSSE VLSSE VL ( µV) (b)\nTHset = TLset = TLn\nTHset = THn\nTLset = TLnn = 1\nn = 29 n = 25 n = 21 n = 17 n = 13 n = 9n = 5\nn = 33 Step \n200 400 VBG ( µV)\n10 20 30 0\nn(e)\nBackground\nPt_H Pt_L \nFIG. 2: (a) Hdependence of VHfor PtH in the Pt/YIG/Pt sample for various values of n. (b)Hdependence of VLfor\nPtL. The LSSE voltage VLSSEfor PtH (PtL) is defined as VH(VL) atH= 1 kOe. (c) TLndependence of RH(RL) for Pt H\n(PtL) in the isothermal condition: Tset\nH=Tset\nL=TLn. (d) The average temperature Tavand temperature difference ∆ Tof the\nPt/YIG/Pt sample during the LSSE measurements as a function ofn. The inset to (d) shows RH(RL) for Pt H (PtL) as a\nfunction of nunder the temperature gradient: Tset\nH=THnandTset\nL=TLnwithTHn> TLn. (e) The background voltage VBG\nfor PtH and Pt L as a function of nunder the temperature gradient.\nobserved to be the same as that in Pt L, a situation con-\nsistent with the scenario of the SSE [38, 40, 44, 51]. Here\nwe note that, although the large backgroundvoltage VBG\nappears due to unavoidable thermopower differences in\nthe wires with increasing n[Fig. 2(e)], the noise level\nin theVHandVLsignals does not increase and the drift\nofVBGis small [Figs. 2(a) and 2(b)]. Therefore, we\ncan extract the LSSE voltage simply by measuring the\nHdependence of VHandVL. We also found that the\nmagnitude of the LSSE voltage in the Pt/YIG/Pt sam-\nple monotonically decreases with increasing the temper-\nature.\nTo quantitatively discuss the temperature dependence\nof the LSSE voltage in the Pt/YIG/Pt sample, we esti-\nmateTavand ∆Tat each step number nby using the\nmethod explained in Sec. II. Figure 2(c) shows the TLn\ndependence of RHandRLfor the Pt films measured un-\nder the isothermal condition. By combining the isother-\nmalRH,L-TLncurves with the RHandRLdata under the\ntemperature gradient [the inset to Fig. 2(d)], we obtain\ntheTavand∆Tvaluesateach n[Fig. 2(d)]. Importantly,\nthe calibrated values of ∆ Tare dependent on nand al-\nways smaller than the temperature difference applied to\nthe heat baths due to the interfacial thermal resistance\nand temperature gradients in the sapphire plates (notethatTHn−TLn= 8 K for all the measurements as shown\nin Table I).\nFigure 3(c) shows the LSSE voltage normalized by\nthe calibrated temperature difference applied to the\nPt/YIG/Pt sample, VLSSE/∆T, as a function of Tav.\nHere,VLSSEdenotesVH(VL) for Pt H (PtL) atH=\n1 kOe. We confirm again that the magnitude of\nVLSSE/∆Tmonotonically decreases with increasing the\ntemperatureanddisappearsabovetheCurietemperature\nTcof YIG, where Tcof our YIG slab was experimentally\nestimated to be 553 K (see Appendix A). This behav-\nior was observed not only in one sample but also in our\ndifferent samples as exemplified in Fig. 3(c). Interest-\ningly, the temperature dependence of VLSSE/∆Tis sig-\nnificantly different from the magnetization (4 πM) curve\nof the YIG slab [compare Figs. 3(a) and 3(c)]; the mag-\nnitude of VLSSE/∆Trapidly decreases with a concave-\nup shape, while the magnetization curve of YIG exhibits\na standard concave-down shape. We also checked that\nthe strong temperature dependence of the LSSE voltage\nin the Pt/YIG/Pt sample cannot be explained by the\nweak temperature dependence of the thermal conductiv-\nity of YIG (see Appendix B). Similar difference in the\ntemperature-dependent data between the ISHE voltage\nand magnetization wasobservedalso in Pt/GaMnAs sys-5\nTc of YIG 4πMs4πM at H = 2 kOe \nFitting with Eq. (2) \n300 400 500 600\nT (K)\n300 400 500 600\nTav (K)Tc of YIG Sample A \nSample B Pt_H \nPt_L \nPt_H \nPt_L \nFitting with Eq. (3) 12\n04πM (kG) 3(a)\n0123VLSSE /∆T ( µV/K) (c)10 110 210 3\n10 110 210 3\nTc − Tav (K)10 -1 10 04πMs (kG) 10 1 (b)\n10 -2 10 -1 10 010 1VLSSE /∆T ( µV/K) (d)Tc − T (K)\n∝ ( Tc − T)3∝ ( Tc − T)0.5\nFIG. 3: (a) Tdependence of the bulk magnetization 4 πMof the YIG slab. The green curve shows the 4 πMdata atH= 2 kOe,\nmeasured with a vibrating sample magnetometer (VSM). The gr ay circles show the saturation magnetization 4 πMsof the YIG\nslab. The gray curve was obtained by fitting the 4 πMsdata with Eq. (2). The values of 4 πMsand Curie temperature\nTc(= 553 K) of the YIG slab were estimated by using the Arrott plo t method (see Appendix A). (b) Double logarithmic plot\nof theTc−Tdependence of 4 πMsof the YIG slab. (c) Tavdependence of VLSSE/∆Tfor two different Pt/YIG/Pt samples\nA (circles) and B (triangles). The experimental results sho wn in Fig. 2 were measured using the Pt/YIG/Pt sample A. The\norange curve was obtained by fitting the VLSSE/∆Tdata with Eq. (3). (d) Double logarithmic plot of the Tc−Tavdependence\nofVLSSE/∆Tfor the Pt/YIG/Pt samples A and B.\ntems in the measurement of the transverse SSEs [15, 16].\nThe behavior of physical quantities near continuous\nphase transitions can be described by critical exponents\nin general. Here, we compare the critical exponents for\nthe observed temperature dependences of the LSSE volt-\nage in the Pt/YIG/Pt sample and the magnetization of\nYIG. First, we checked that the magnetization curve of\nYIG is well reproduced by a standard mean-field model\n[52]:\n4πMs=A(Tc−T)0.5, (2)\nwhere the critical exponent is fixed at 0.5 and Ais an\nadjustable parameter [see Figs. 3(a) and 3(b)]. The crit-\nical exponent γfor the LSSE was estimated by fitting\nthe experimental data in Fig. 3(c) with the following\nequation:\nVISHE\n∆T=S(Tc−T)γ, (3)\nwhere both Sandγare adjustable parameters and Tav\nis regarded as Tfor the LSSE data. We found that the\nobserved temperature dependence of VLSSE/∆Tfor the\nPt/YIG/Pt sample is well fitted by Eq. (3) with γ= 3,which is much greater than the critical exponent for the\nmagnetization curve [see also the double logarithmic plot\nin Fig. 3(d)]. This big difference in the critical exponents\nbetweentheLSSEandmagnetizationemphasizesthefact\nthat the LSSE is not attributed solely to static magnetic\nproperties in YIG.\nHere, we qualitatively discuss the origin of the temper-\nature dependence of the LSSE voltage in the Pt/YIG/Pt\nsample. According to the thermal spin-pumping mech-\nanism [38, 44] and phenomenological calculation of the\nISHE combined with short-circuit effects [53], the mag-\nnitude of the LSSE voltage is determined mainly by the\nfollowing factors: the spin-mixing conductance [54–57]\nat the Pt/YIG interfaces, spin-diffusion length and spin-\nHall angle of Pt, and difference between an effective\nmagnon temperature in YIG and an effective electron\ntemperature in Pt. Since the LSSE voltage is propor-\ntional to the spin-mixing conductance [38], it can con-\ntribute directly to the observed temperature dependence\nof the LSSE voltage. Recently, Ohnuma et al.formu-\nlated the relation between the spin-mixing conductance\nand interface s-dinteraction at paramagnet/ferromagnet\ninterfaces, and predicted that the spin-mixing conduc-\ntance is proportional to (4 πMs)2of the ferromagnet [58].6\nBy combining this prediction with Eq. (2), the spin-\nmixing conductance is proportional to Tc−T, of which\nthe critical exponent (= 1) is greater than that for the\nmagnetization curve. Although the temperature depen-\ndence of the spin-mixing conductance can explain the\nfactsthat the LSSEvoltagemonotonicallydecreaseswith\nincreasingthe temperatureanddisappearsat Tc, itisstill\nmuch weaker than the observed ( Tc−T)3dependence\nof the LSSE voltage. Furthermore, if the spin-diffusion\nlength of Pt decreases with increasing the temperature\n[59], it can also contribute to reducing the LSSE volt-\nage at high temperatures, while the spin-Hall angle of Pt\nwas shown to exhibit weak temperature dependence [60]\n(note that the magnitude of the ISHE voltage monoton-\nically decreases with decreasing the spin-diffusion length\nwhen the spin-Hall angle is constant [53]). The effec-\ntive magnon-electron temperature difference could also\nbe an important factor, but there is no clear framework\nto determine its temperature dependence at the present\nstage. Therefore, more elaborate investigations are nec-\nessaryforthe completeunderstandingofthetemperature\ndependence of the LSSE voltage.\nIV. CONCLUSION\nIn this study, we reported the longitudinal spin See-\nbeck effects (LSSEs) in Y 3Fe5O12(YIG) slabs sand-\nwiched by two Pt films in the high temperature range\nfrom room temperature to above the Curie temperature\nTcof YIG. To investigate the temperature dependence of\nthe LSSE quantitatively, we used the Pt films not only\nasspin-currentdetectorsbut alsoastemperaturesensors.\nThe measurement processes used here enabled accurate\nestimation of the average temperature and temperature\ndifference of the sample, being free from thermal arti-\nfacts. We found that the magnitude of the LSSE in the\nPt/YIG/Pt sample rapidly decreases with increasing the\ntemperature and disappears above Tcof YIG; the ob-\nserved LSSE voltage exhibits the ( Tc−T)3dependence\nof which the critical exponent (= 3) is much greater than\nthatofthemagnetizationofYIG(= 0 .5). Althoughmore\ndetailed experimental and theoretical investigations are\nrequired to clarify the microscopic origin of this discrep-\nancy, we anticipate that the quantitative temperature-\ndependent LSSE data at high temperatures will be help-\nful for obtaining full understanding of the mechanism of\nthe LSSE.\nACKNOWLEDGMENTS\nThe authors thank S. Maekawa, H. Adachi, Y.\nOhnuma, N. Yokoi, K. Sato, and J. Ohe for valuable dis-\ncussionsandY.Zhangforhisassistanceinmagnetometry\nmeasurements. This work was supported by PRESTO-JST “Phase Interfaces for Highly Efficient Energy Uti-\nlization”, CREST-JST “Creation of Nanosystems with\nNovel Functions through Process Integration”, Grant-\nin-Aid for Young Scientists (A) (25707029), Grant-in-\nAid for Challenging Exploratory Research (26600067),\nGrant-in-Aid for Scientific Research(A) (24244051)from\nMEXT, Japan, LC-IMR of Tohoku University, the Sum-\nitomo Foundation, the Tanikawa Fund Promotion of\nThermal Technology, the Casio Science Promotion Foun-\ndation, and the Iwatani Naoji Foundation.\nAPPENDIX A: ESTIMATION OF CURIE\nTEMPERATURE OF YIG\nThe Curie temperature Tcof the YIG slab used in the\npresent study was estimated from vibrating sample mag-\nnetometry and Arrott-plot analysis [52, 61]. The inset\nto Fig. 4(a) shows the Hdependence of the magnetiza-\ntion 4πMof the YIG slab for various values of T. From\nthis result, we obtained the Arrott plots, i.e. H/4πM\ndependence of (4 πM)2, of the YIG slab [see Fig. 4(a)].\nThe saturation magnetization 4 πMsof the YIG at each\ntemperature was extracted by extrapolating the (4 πM)2\ndatain the high-magnetic-fieldrangeto zerofield [see red\ndotted lines in Fig. 4(a)]. As shown in Fig. 4(b), the T\ndependence of (4 πMs)2of the YIG slab is well fitted by\na linear function; the horizontal intercept of the linear fit\nline corresponds to Tc. The fitting result shows that the\nCurie temperature of our YIG slab is Tc= 553 K, which\nis consistent with literature values [62, 63].\nAPPENDIX B: TEMPERATURE DEPENDENCE\nOF THERMAL CONDUCTIVITY OF YIG\nFigure 5(a) shows the thermal conductivity κof the\nYIG slab used in the present study as a function of T.\nTheκvalues were obtained by the combination of ther-\nmaldiffusivitymeasuredbyalaser-flashmethodandspe-\ncific heat Cmeasured by a differential scanning calorime-\ntry. Here, we measured the thermal diffusivity along the\n[111] direction of the single-crystallineYIG slab, which is\nparalleltothe ∇Tdirectionin theLSSEsetup. Asshown\nin Fig. 5(b), the measured Cvalues are consistent with\ntheDulong-Petit(DP) law[1]; thedifferenceofthe Cval-\nuesfrom the DPspecific heatofYIG, CDP= 0.676J/gK,\nis less than 10 % of CDPforT >350 K. The observed\nTdependence of κis well fitted by κ∝T−1, indicating\nthat the thermal conductivity of the YIG is dominated\nby phonons in this temperature range [see also the inset\nto Fig. 5(a)]. The κvalue at 300 K is consistent with\nliterature values [64–66]. We also confirmed that the T\ndependence of κis much weaker than that of the LSSE\nvoltage in the Pt/YIG/Pt sample [compare Figs. 3(c)\nand 5(a)] and shows no anomaly around Tc.7\nTc = 553 K 0 2 4 6 823\n300 400 500 600\nT (K)(4 πMs)2 (kG 2)\n013\n2H/ 4πM (Oe/G) (4 πM)2 (kG 2)\n1T = 295.3 K \n359.6 K \n400.4 K \n450.5 K \n501.7 K \n550.7 K (a)\n-1 0 1-2-10124πM (kG) \nH (kOe)601.5 K 295.3 K\n550.7 K\n(b)\nHYIG slab \nxz\ny\nFIG. 4: (a) Arrott plots of the YIG slab for various values of\nT. The inset to (a) shows the Hdependence of 4 πMof the\nYIG slab for various values of T, measured with VSM. The\nlengths along the x,y, andzdirections of the YIG slab used\nfor the magnetometry measurements are 3 mm, 7 mm, and 1\nmm, respectively. Hwas applied along the xdirection. (b)\nTdependence of (4 πMs)2of the YIG slab.\n∗Electronic address: kuchida@imr.tohoku.ac.jp\n[1] N. W. Ashcroft and N. D. Mermin, Solid State Physics\n(Saunders College, Philadelphia, 1976).\n[2] S. B. Riffat and X. 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Hofmeister, Thermal Diffusivity of Garnets at High\nTemperature , Phys. Chem. Minerals 33, 45 (2006)." }, { "title": "1906.12288v1.Spin_transport_in_an_insulating_ferrimagnetic_antiferromagnetic_ferrimagnetic_trilayer_as_a_function_of_temperature.pdf", "content": "Spin transport in an insulating\nferrimagnetic-antiferromagnetic-ferrimagnetic trilayer as a\nfunction of temperature\nYizhang Chen,1Egecan Cogulu,1Debangsu Roy,1Jinjun Ding,2Jamileh Beik\nMohammadi,1Paul G. Kotula,3Nancy A. Missert,3Mingzhong Wu,2and Andrew D.\nKent1,a)\n1)Center for Quantum Phenomena, Department of Physics, New York University,\nNew York, New York 10003, USA\n2)Department of Physics, Colorado State University, Fort Collins, Colorado 80523,\nUSA\n3)Sandia National Laboratories, Albuquerque, New Mexico 87185,\nUSA\n(Dated: 1 July 2019)\nWe present a study of the transport properties of thermally generated spin currents\nin an insulating ferrimagnetic-antiferromagnetic-ferrimagnetic trilayer over a wide\nrange of temperature. Spin currents generated by the spin Seebeck e\u000bect (SSE) in a\nyttrium iron garnet (YIG) YIG/NiO/YIG trilayer on a gadolinium gallium garnet\n(GGG) substrate were detected using the inverse spin Hall e\u000bect in Pt. By studying\nsamples with di\u000berent NiO thicknesses, the NiO spin di\u000busion length was deter-\nmined to be 4.2 nm at room temperature. Interestingly, below 30 K, the inverse\nspin Hall signals are associated with the GGG substrate. The \feld dependence of\nthe signal follows a Brillouin function for a S=7/2 spin (Gd3+) at low temperature.\nSharp changes in the SSE signal at low \felds are due to switching of the YIG mag-\nnetization. A broad peak in the SSE response was observed around 100 K, which\nwe associate with an increase in the spin-di\u000busion length in YIG. These observa-\ntions are important in understanding the generation and transport properties of\nspin currents through magnetic insulators and the role of a paramagnetic substrate\nin spin current generation.\nKeywords: spin current, spin transport, spin Seebeck e\u000bect, spin valve\nI. INTRODUCTION\nA spin current, or a \row of spin angular momentum, can be carried by conduction\nelectrons1,2or spin waves3,4. In a material with large spin-orbit coupling, like Pt, a spin\ncurrent can be converted into a measurable voltage by the inverse spin Hall e\u000bect (ISHE)5.\nSpin currents can be generated by the spin Hall e\u000bect (SHE)6{8, spin pumping9, or the spin\nSeebeck e\u000bect10{13. The spin Seebeck e\u000bect refers to the generation of spin currents when\na temperature gradient is applied to a magnetic material and has potential applications in\nconverting waste heat into electricity.\nA conventional spin valve consists of two ferromagnetic metals separated by a non-\nmagnetic metal14,15. Recently, a new spin valve structure based on an antiferromagnetic\ninsulator (AFI) sandwiched between two ferromagnetic insulators (FI) was proposed16. An\nAFI can conduct both up and down spins due to the degeneracy of its magnon spectrum at\nzero \feld. The predicted valve e\u000bect associated with thermally induced spin currents has\nbeen observed by controlling the relative orientations of Y 3Fe5O12(YIG) magnetization in a\nYIG/NiO/YIG structure17. YIG is a ferrimagnetic insulator with low magnetic dissipation,\nhighly e\u000ecient spin current generation18, and long-distance magnon transport19. Nickel\na)Electronic mail: andy.kent@nyu.eduarXiv:1906.12288v1 [cond-mat.mes-hall] 28 Jun 20192\nFIG. 1. A schematic of the sample and cross-sectional characterization of the sample by scanning\ntransmission electron microscopy energy-dispersive X-ray spectroscopy (STEM-EDS). (a) Sample\ngeometry showing the layers, the electrical contacts and the applied magnetic \feld. ~jcis the density\nof the charge current applied in the x-direction. Vxyis the voltage measured in the transverse\ndirection, and 'is the angle between the applied magnetic \feld and the current. GGG, YIG, NiO,\nand Pt are represented as purple, yellow, green, and grey, respectively. (b) Sample cross section\ncharacterized by STEM-EDS. GGG, YIG, NiO, and Pt layers are colored in blue, red, green, and\ndark gray, respectively.\nFIG. 2. Angular dependence and NiO thickness dependence of V2!\nxymeasured with an in-plane\nmagnetic \feld of 0 :4 T at room temperature. (a) Angular dependence of V2!\nxywith an AC density\nofjAC= 1:5\u00021010A=m2. \u0001V2!\nxyis extracted by \ftting the curve with a cosine function. (b) \u0001 V2!\nxy\nas a function of the NiO thickness. The curve is \ftted with V=V0e\u0000t=\u0015NiO, whereV0= 182 \u000644 nV\nand the spin di\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm.\nOxide (NiO) is an antiferromagnetic insulator used to decouple the two ferrimagnetic layers\nwhile conducting thermally generated spin currents.\nTo understand the generation, transmission, and detection of spin currents through a mul-\ntilayer consisting of di\u000berent magnetic insulators, transport measurements were performed\nin samples consisting of GGG(500 \u0016m)/YIG(20 nm)/NiO(t nm)/YIG(15 nm)/Pt(5 nm).\nHere GGG (Gd 3Ga5O12) is the standard substrate used to grow epitaxial YIG. Above the\nspin-glass transition temperature ( \u0018\u00000:18 K), GGG is paramagnetic with no long-range\nmagnetic order20,21. In addition, GGG has been shown to have a SSE, with a magnitude\ncomparable to the SSE that of YIG at low temperatures22.\nIn this article, room-temperature measurements were \frst performed to characterize the\nspin di\u000busion length of NiO. Then experiments were conducted over a broad range of tem-\nperature from 5 to 300 K. These revealed a strong enhancement of the SSE below 30 K that\noriginates from the GGG substrate. Further, \feld-dependent experiments show behavior3\nFIG. 3. Second harmonic response V2!\nxymeasured at several temperatures with an applied magnetic\n\feld\u00160H= 1:0 T for NiO thicknesses of 2.5, 5, and 10 nm. (a) Angular dependence of V2!\nxy\nmeasured from 5 to 300 K. Note that the angle 'is de\fned in Fig. 1(a). (b) \u0001 V2!\nxymeasured as a\nfunction of the temperature. Inset: a broad peak is observed around 100 K for the sample with a\n5 nm NiO thickness.\nassociated with switching of YIG magnetization and paramagnetism of GGG. Furthermore,\na broad peak in the SSE response around 100 K was observed, which may originate from\nthe temperature dependence of the spin di\u000busion length in YIG.\nII. SAMPLE FABRICATION AND MEASUREMENT TECHNIQUES\nThe sample was fabricated in the following way. First, a 20 nm YIG layer was grown\nepitaxially on a (111)-oriented GGG substrate (500 \u0016m) at room temperature and annealed\nin O 2at high-temperature23. An Ar plasma was used to clean the surface of the samples\nbefore depositing NiO via radio frequency (RF) sputtering in another chamber. Afterward,\na 15 nm YIG layer was grown on top with the same growth conditions of the \frst layer. Then\nthe sample was capped with a 5 nm Pt layer. For transport and SSE measurements, the\nPt was patterned into Hall bar structures using electron beam lithography and Ar plasma\netching. The Hall bar has a width of 4 \u0016m and the length between the two longitudinal\ncontacts is 130 \u0016m. An alternating current (AC) with a frequency of 953 Hz was used. As\nthe temperature gradient induced by the AC oscillates at twice the frequency, the second\nharmonic Hall voltage V2!\nxymeasured by a lock-in ampli\fer is proportional to the amplitude\nof the SSE-produced spin current24,25. Room-temperature measurements were performed\nwith a 0.4 T magnetic \feld applied in-plane. Temperature-dependent measurements are\ncarried out in the Quantum Design PPMS system also with an in-plane applied magnetic\n\feld.\nIII. EXPERIMENTAL RESULTS\nFigure 1(a) is a schematic of the GGG/YIG/NiO/YIG/Pt sample. A magnetic \feld is\napplied in-plane at an angle 'with respect to the current. The cross section of the sample\nis characterized by scanning transmission electron microscopy with energy-dispersive X-ray\nspectroscopy, shown in Fig. 1(b). Both the top and bottom YIG layers are crystalline, with\nthickness of 15 nm and 20 nm. NiO is polycrystalline, with a thickness of 5 nm for this\nsample (see Fig. S1 in the supplemental materials).\nFirst,V2!\nxywas measured as a function of 'for samples with NiO thicknesses of 2.5, 5,\n7.5, and 10 nm. V2!\nxyreaches a maximum at '= 180oand minimum at '= 0o. This is\nconsistent with the ISHE symmetry of the Hall voltage VISHE/~js\u0002^\u001b/rT\u0002^m/cos('),4\nwhere~jsis the spin current, ^ \u001bis the spin polarization direction, rTis the temperature\ngradient, and ^ mis a unit vector in the direction of magnetization. The angular dependence\nof theV2!\nxywas \ftted with a cosine function and the amplitude \u0001 V2!\nxyis plotted as a function\nof the NiO thickness (Fig. 2). \u0001 V2!\nxydecays rapidly as the NiO thickness increases and\nis \ftted to an exponentially decaying function V=V0e\u0000t=\u0015NiO. The characteristic spin\ndi\u000busion length of NiO is \u0015NiO\u00194:2\u00061:1 nm, close to what has been found in previous\nwork on YIG/NiO/Pt structures26.\nTo further understand the generation and transport of thermally generated spin currents\nthrough the heterostructure, the angular dependence of V2!\nxywas measured from 5 to 300 K\nwith an applied magnetic \feld of 1.0 T (Fig. 3(a)). The amplitude \u0001 V2!\nxyis extracted by the\nsame method discussed above and is plotted as a function of the temperature (Fig. 3(b)).\nFor the 5 nm thick NiO sample, as temperature decreases from 300 to 100 K, \u0001 V2!\nxyincreases\nsteadily from 150 to 271 nV. From 100 to 50 K, \u0001 V2!\nxyslightly decreases to 257 nV, forming\na broad peak around 100 K, shown in the inset of Fig. 3(b). However, as temperature\ndecreases below 30 K, \u0001 V2!\nxyincreases dramatically from 297 to 994 nV. The enhancement\nbelow 30 K was observed for all samples.\nAs has been previously noted, the SSE depends on the magnon population, the spin\ndi\u000busion length, and the interfacial spin-mixing conductance in the heterostructure. In\norder to understand the correlation between the SSE signal and the magnetization of the\nsamples, \feld-dependent measurements of V2!\nxywere performed in the sample with 2.5 nm\nthick NiO. Fig. 4(a) shows V2!\nxyas a function of the applied magnetic \feld between -5.0\nand 5.0 T with temperature ranging from 5 to 50 K. At 50 K, as the applied \feld goes\nfrom -5.0 to -1.0 T, V2!\nxy\u0019530 nV, almost independent of the applied \feld. As the applied\n\feld increases from -1.0 T to 20 mT, V2!\nxydecreases slowly to 108 nV. Then V2!\nxydrops\nsharply to -156 nV as the applied \feld increases from 20 to 100 mT. As the applied \feld\nincreases to 1.0 T, V2!\nxydecreases slowly to -540 nV and again is nearly constant thereafter.\nThe sharp switching steps observed in the V2!\nxy\u0000Hcurves around\u000650 mT occur at the\ncoercive \feld of the YIG, which is smaller than 50 mT at room temperature (see Fig. S2\nin the supplemental materials). Only one magnetization reversal can be identi\fed between\n-200 and 200 mT in the V2!\nxy\u0000Hcurves. As temperature decreases from 50 to 5 K, V2!\nxy\nincreases from 535 to 1970 nV, while the low-\feld step does not change signi\fcantly. A\nclear correlation between the V2!\nxyand the magnetization of a paramagnet can be seen by\ncomparing the V2!\nxy\u0000Hcurves with the Brillouin function of a S = +7/2 spin (Gd3+),\nshown in Fig. 4(b).\nIV. DISCUSSION\nThe SSE voltages decay rapidly as NiO thickness increases, as presented in Fig. 2. This\nindicates that spin currents were generated not only from the top YIG layer but also from\nthe bottom YIG or GGG layer. The NiO spin di\u000busion length is close to what has been\nfound before at room temperature in YIG/NiO/Pt structures26.\nA dramatic enhancement of the SSE voltages has been observed below 30 K, which\nis likely associated with GGG. The same enhancement has been observed in GGG(500\n\u0016m)/YIG(20 nm)/Pt(5 nm), shown in Fig. S3 in the supplemental materials. A previous\nstudy has shown that the spin current jsgenerated by paramagnetic SSE in GGG/Pt bilayer\nhas aT\u00001temperature dependence, associated with GGG susceptibility, which follows the\nCurie-Weiss law \u001f=C=(T\u0000\u0002CW), whereCis the Curie constant and \u0002 CWCurie-\nWeiss temperature22. At low temperatures, the GGG thermal conductivity kGGG has a\nT3temperature dependence. Therefore, the temperature gradient generated by a constant\npower isrT/1=kGGG/1=T\u00003. The resulting SSE voltage goes as VSSE/js\u0001rT/T\u00004.\nIn addition, a broad peak of the SSE signal observed around 100 K in the YIG/Pt structure\nsuggests that the spin di\u000busion length in YIG has a strong temperature dependence27. As\nspin currents generated and transmitted through YIG layers, the temperature-dependent5\nFIG. 4. (a) Field dependence of the V2!\nxymeasured between 5 and 50 K, with '= 0o. The sample\nis GGG (500 \u0016m)/YIG(40 nm)/NiO(2.5 nm)/YIG(20 nm)/Pt (5 nm). The magnetic \feld is swept\nbetween -5 and +5 T. The o\u000bset of V2!\nxyhas been removed. (b) Brillouin function of an S=+7/2\nspin.\nspin di\u000busion length in YIG would have a signi\fcant e\u000bect on the ISHE voltage generated\nin Pt. So the broad peak observed around 100 K in Fig. 3 may be associated with the\ntemperature dependence of spin di\u000busion length in YIG. However, further experiments\nare needed to understand how spin currents are transmitted through bulk GGG, NiO,\nGGG/YIG, YIG/NiO, and NiO/YIG interfaces at di\u000berent temperatures.\nComparing the \feld dependence of SSE voltages and the Brillouin function from 5 to 50\nK, it is clear that there is a contribution to SSE from GGG at low temperatures. At 5 K,\nthe SSE voltage follows the Brillouin function as the magnetic \feld swept from -5.0 to 5.0\nT. As temperature increases, the SSE voltages start deviating from the Brillouin function\n(Fig. 4 and Fig. S4). The underlying physics is not yet fully understood, since the role\nplayed by GGG, YIG, NiO and their corresponding interfaces vary with temperature.\nV. SUMMARY\nIn summary, the spin transport properties of an insulating trilayer based on two ferrimag-\nnetic insulators separated by a thin antiferromagnetic insulator were presented. The spin\ndi\u000busion length of NiO was found to be \u0015NiO'4:2 nm at room temperature. In addition,\na large increase of the SSE signal was observed below 30 K, revealing the dramatic e\u000bects\nof paramagnetic SSE from the GGG substrate. The \feld dependence of the SSE shows the\nswitching of YIG magnetization at low \feld as well as paramagnetic behavior associated\nwith GGG. Furthermore, the SSE voltages show a broad peak around 100 K, a feature\nthat may be related to the temperature dependence of spin di\u000busion length in YIG. This\nexperimental study provides information on how spins can be generated, transported and\ndetected in a heterostructure consisting of paramagnetic, ferrimagnetic and antiferromag-\nnetic insulators.\nSUPPLEMENTARY MATERIAL\nThe supplementary material provides the details of sample characterization by scanning\ntransmission electron microscopy (SEM), Vibrating Sample Magnetometer (VSM), trans-\nport measurements of a GGG/YIG/Pt sample, and \feld-dependent measurements of a\nGGG/YIG/NiO/YIG/Pt sample above 50 K.6\nACKNOWLEDGEMENTS\nThis work was supported partially by the MRSEC Program of the National Science\nFoundation under Award Number DMR-1420073. The instrumentation used in this research\nwas support in part by the Gordon and Betty Moore Foundations EPiQS Initiative through\nGrant GBMF4838 and in part by the National Science Foundation under award NSF-DMR-\n1531664. ADK received support from the National Science Foundation under Grant No.\nDMR-1610416. 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Hueso2,3, Jürgen Faßbender1, Fèlix \nCasanova2,3, Denys Makarov1 \n \n1 Helmholtz -Zentrum Dresden -Rossendorf e.V ., Institute of Ion Beam Physics and Materials Research , 01328 Dresden, \nGermany \n2 CIC nanoGUNE, 20018 Donostia -San Sebastian, Basque Country, Spain \n†Present address: Department of Materials, ETH Z ürich, 8093 Zürich, Switzerland \n3 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain \n \n \n \nAnomalous Hall -like signals in platinum in contact with magnetic insulators are \ncommon observations that could be explained by either proximity magnetization \nor spin Hall magnetoresistance. In this work, longitudinal and transverse \nmagnetoresistance s are measured in a pure g old thin film on the ferrimagnetic \ninsulator Y3Fe5O12 (Yttrium Iron Garnet, YIG) . We show that both the longitudinal \nand transverse magnetoresistance s have quantitatively consistent scaling in \nYIG/Au and in a YIG/Pt reference system when applying the Spin Hall \nmagnetoresistance framework . No contribution of an anomalous Hall effect due \nto the magnetic proximity effect is evident . \n \n \nThroughout the last few years, systems of magnetic insulators and nonmagnetic metals (MI/NM) have \nseen significant attention from the spintronics community [1–10]. Such systems general ly display Spin \nHall magnetoresistance (SMR) [9–19] enabled by the eponymous Spin H all effect [20,21] of the metal \nwhich also offers elegant charge -spin-interconversion . One of the most popular metals in such \nsystems is platinum (Pt) due to its strong spin orbit coupling, large spin Hall angle and its benign \nchemical properties. Still, a larger variety of well -studied metals is urgently needed in the field of \ninsulator spintronics for two reasons: Firstly, some metals – Pt being a prot otypical example – are \nclose to the Stoner criterion for ferromagnetism and can thus show a strong magnetic proximity \neffect [22]. When Pt becomes a magnetic conductor in this way, its strong SHE becomes an \nanomalous Hall effect (AHE) [20] that would mirror the magnetization of a nearby MI [23–26]. In \ncontrast, the same phenomenon could be attributed solely due to the transverse SMR [27–29] or the \nnonlocal AHE [30] creating an ambiguity about the physics of the anomalous Hall -like signal in MI/NM \nsystems . Secondly, studying more diverse systems can probe the applicability limits of the SMR \ntheory [31]. The longitudinal and transverse SMR magnitudes are typically explained in terms of the \nreal and imaginary components of the interfacial spin -mixing conductivity 𝐺↑↓=𝐺𝑟+𝑖𝐺𝑖 [32,33] , but \nhigher order effects are already known [34]. \n \nHere, we report longitudinal as well as transverse signatures of the spin Hall magnetoresistance for a \ngold (Au) layer on YIG and compare this to a YIG/Pt reference system . Au shares or exceeds the \nexcellent chemical and electrical properties of Pt. At the same time, static proximity magnetization is \nnot expected for Au [35] and we take care to avoid dynamic proximity magnetization due to thermal \nspin pumping [Suppl. Information , 1]. As a result, we exclude possible influences from proximity \nmagnetization and its associated anomalous Hall effect. Using empirical data, we quantitatively predict \nthe longitudinal and transverse spin Hall magnetoresistance in YIG/Au and confirm their respective \nmagnitudes by measurements. Hence, the key finding of this work is the experimental observation of \nthe transverse SMR in Au. \n \nWe prepar ed a YIG/Au(10nm) and a YIG/Pt(2nm) system using DC sputtering under identical \nconditions on top of 3.5 -µm-thick liquid phase epitaxy YIG /GGG (gallium gadolinium garnet) \nsubstrates from Innovent e.V., Jena, Germany . The larger thickness for the Au metal fi lm was chosen \nto guarantee the continuity of the film [Suppl. Information, 2] . Hall bars 100 µm wide and 800 µm long \nwere patterned by e -beam lithography and Ar ion milling for b oth systems . \n 2 \nFigure 1: Magnetoresistance measurement geometries: (a) Longitudinal resistivity measurement (𝜌𝑥𝑥) while rotating the \nmagnetic field along the angles 𝛼 (green), 𝛽 (blue) and 𝛾 (brown) and while current is flowing in 𝑥-direction. (b) Transverse \nresistivity m easurement (𝜌T) while sweeping the magnetic field along the 𝑧-direction and current alternatingly in the 𝑥- and 𝑦-\ndirections (Zero -Offset Hall measurement) . (c,e) Longitudinal ADMR for YIG/Pt and YIG/Au, respectively , at a magnetic field \nstrength of 1 T. (d,f) Zero-Offset Hall measurement after subtracting the normal Hall effect for YIG/Pt and YIG/Au, respectively. \nGrey lines indicate the saturation of the YIG and the red line is a data fit. The insets show the data before the subtraction of the \nnormal Hall signal. \n \nThe longitudinal resistance in 𝑥-direction 𝜌𝑥𝑥 and its magnetic field angle dependent \nmagnetoresistance (ADMR) were measured using a conventional Kelvin contact layout [ Figure 1(a)] in \na l-He cryostat with 9 T field and 360° sample rotation capabilities. The transvers e resistance and its \nmagnetoresistance were obtained by out -of-plane field Zero-Offset Hall measurements [ Figure 1(b)] \nusing an integrated measurement device from HZDR innovation s GmbH [more details in Suppl. \nInformation, 2] . \n \nIn the context of SMR , the resistivit y tensor of the NM layer in lab coordinates is given by [12]: \n \n𝜌𝑥𝑥=𝜌0−∆𝜌1 𝑚𝑦2 \n𝜌𝑦𝑦=𝜌0−∆𝜌1 𝑚𝑥2 \n𝜌𝑥𝑦=∆𝜌1𝑚𝑥 𝑚𝑦−∆𝜌2 𝑚𝑧 \n (1) \nwhere 𝐦(𝑚x,𝑚y,𝑚z)=𝐌/𝑀s are the normalized projections of the magnetization of the YIG film the \nthree main axes and 𝑀s is the saturation magnetization of the MI layer. 𝜌0 is the Drude resistivity and \n3 ∆𝜌1 and ∆𝜌2 are the characteristic amplitudes of the SMR. The first term in the expression for 𝜌𝑥𝑦 is a \nplanar Hall effect resulting from the anisotropy of the longitudinal resistivity ( 𝜌PHE =𝜌𝑥𝑥−𝜌𝑦𝑦). In the \nfollowing, w e do n either discuss nor measure planar Hall effects, which are universally rejected by the \nZero-Offset Hall measurement technique [36] due to their origin in the longitudinal tensor components \n[Suppl. Information, 3]. In our measurement, the transvers e signature of the SMR thus simplifies to \n \n𝜌T=−∆𝜌2 𝑚𝑧 (2) \n \nThe longitudinal and transvers e amplitudes of the SMR, ∆𝜌L≡∆𝜌1 and ∆𝜌T≡∆𝜌2, are related to the \nspintronic properties of the MI/NM bilayer via: \n \n∆𝜌L≡∆𝜌1=2 𝜌0 𝜃SH2 𝜆\n𝑡 𝑅𝑒[𝜆 𝐺↑↓ tanh2 (𝑡\n2𝜆)\n1\n𝜌0+2 𝜆 𝐺↑↓ coth (𝑡\n𝜆)] (3) \n∆𝜌T≡∆𝜌2=−2 𝜌0 𝜃SH2 𝜆\n𝑡 𝐼𝑚[𝜆 𝐺↑↓ tanh2 (𝑡\n2𝜆)\n1\n𝜌0+2 𝜆 𝐺↑↓ coth (𝑡\n𝜆)] (4) \n \n \nwhere 𝜆 and 𝜃 are the spin diffusion length and the spin Hall angle of the NM layer , and 𝐺↑↓ is the spi n \nmixing conductiv ity of the MI/NM interface. Taking into account that 𝐺r𝐺i⁄ ≫1 [11,33] , we can \ncombine Eqs. (3) and (4) to obtain \n \n𝐺r\n𝐺i≈∆𝜌L\n∆𝜌T 1\n1+2 𝜌0 𝜆 𝐺rcoth (𝑡\n𝜆) (5) \n \nThe SMR amplitude ∆𝜌L was evaluated in YIG/Pt and YIG/Au as shown in Figure 1(a,c,e ) from \nlongitudinal ADMR measurements at a field of 1 T to keep YIG saturated . ∆𝜌T was evaluated in the \nsame samples from out -of-plane field Zero -Offset Hall measurements by driving the YIG film into \nmagnetic saturation along the 𝑧-direction as shown in Figure 1(b,d,f ). \n \nFor YIG/Pt , the longitudinal ADMR reveals the behavior expected from the SMR [Eq. (1)] , namely a \nresistivity change of ∆𝜌L in the 𝛼 and 𝛽 angles and no modulation in the 𝛾 angle [ Figure 1(c)]. The \nslight effect seen in 𝛾 is consistent with a sample misalignment of less than 0.1°. From the measured \nresistivity 𝜌0=516 nΩm in our Pt thin film, the values 𝜆Pt=(1.2±0.2) nm and 𝜃Pt=0.09±0.01 can \nbe inferred from the empirical relationships fo und for Pt thin films [37]. Together with the measured \n∆𝜌L=(0.351 ±0.004 ) nΩm this leads to (𝐺r)YIG /Pt=(3.8±1.0)∙1014 Ω−1 m−2 via Eq. ( 3). This value \nis in good agreement with previous reports in the same system [9,11,13,15,18,19] . When applying a \nmagnetic field along the 𝑧-direction, the YIG film is gradually brought into saturation, while a \nproportional transverse magnetoresistance develops [ Figure 1(d)]. The value ∆𝜌T corresponds to the \nsaturation value of the resistivity change. Eq. ( 5) then yields (𝐺r𝐺i⁄)YIG /Pt=22±3 for the YIG/Pt \nreference sample [ Table 1]. This ratio is in good agreement with the theoretical calculations of 𝐺r𝐺i⁄ ≈\n20 [33], as well as experimental values of 𝐺r𝐺i⁄ =16±4 [18] and 𝐺r𝐺i⁄ =33 [11]. \n \nThe experimental values of ∆𝜌L,Au=(1.05±0.12) pΩm and ∆𝜌T,Au=(11.2±1.7) fΩm of YIG/Au are \nobtained in the same manner as for the YIG/Pt reference sample [see Figure 1(e,f)]. To actually \nmeasure a transverse magnetoresistance at this level of approximately 1 μΩ, special care must be \ntaken to isolate the anomalous Hall signal from the much larger background contributions [ inset in \nFigure 1(f)], which we accomplish by eliminating the longitudinal resistance by Zero -Offset Hall [36] \nand by accounting for the nonlinearity of the normal Hall effect itself [Suppl. Information , 4]. \n 4 \nFigure 2: Reported spin diffusion length s 𝜆 (a) and spin Hall angle s 𝜃 (b) for Au thin films as a function of the film resistivities 𝜌0, \nand fits in context of the Elliott -Yafet relation (a) and resistivity dependence on the intrinsic spin Hall effect (b) shown as red \nlines with 1𝜎 confidence bands. The data points are empirical data taken f rom: Kimura [38] (solid diamond ), Brangham [39] \n(hollow squares ), Vlaminck & Obstbaum [40,41] (hollow cir cle), Isasa [42] (hollow triangle), Niimi [43] (solid triangle), Mosendz & \nObstbaum [44,45] (solid square) , Laczkowski [46] (hollow diamond) and Qu [47] (solid circle) . 𝜃 values from Refs. [42,43] have \nbeen multiplied by 2 for a proper comparison. The solid blue line denotes the resistivity of the Au film in the present YIG/Au \nsystem; dashed blue lines show the obtained spintronic quantities 𝜆 and 𝜃 for our Au film . \n \nIn the following, we will provide an independent calculation of the SMR magnitud es for YIG/Au. First, \nwe have to estimate the spintronic quantities 𝜆Au and 𝜃Au of our Au film. Concerning 𝜆, it is well \nestablished that the spin relaxation in metals is dominated by the Elliott -Yafet mechanism ( 𝜆∝\n𝜌−1) [37,48 –53]. Figure 2(a) illustrates how we obtain 𝜆Au=(25−8+12) nm based on the measured \nresistivity of the Au layer and empirical data [38–44,46,47] . This corresponds to a 𝜌- 𝜆-product of \n(2.1−0.7+1.0) 10−15 Ωm2. Regarding 𝜃, different mechanism have been suggested to contribute in \nAu [39,42,43] , but the origin of its SHE is not well established, yet. In order to estimate a reasonable \nvalue, we consider for simplicity that the intrinsic scattering contribution dominates the spin Hall angle . \nIn this case, 𝜃=𝜎SHint×𝜌 holds , where the intrin sic spin Hall conductivity 𝜎SHint depends on the band \nstructure and is thus constant for a given metal [21]. By taking reported data on Au [39–43,45 –47], we \nestimate 𝜎SHint=(2.0−1.0+2.0) [ℏ\n2𝑒] 105 Ω−1m−1, touching the upper end of theoretical predictions ranging \nfrom (0.22⋯0.9) [ℏ\n2𝑒] 105 Ω−1m−1 [47,54,55] . The fitted 𝜎SHint value leads to 𝜃Au=0.017−0.008+0.016 for our \nAu film as shown in Figure 2(b). \n \nIn addition, we assume identical interface spin mixing conductivities in our reference Pt system and \nthe Au system 𝐺YIG /Pt≡𝐺YIG /Au, owing to the identical fabrication conditions, similar chemical qualities \nof the metals and similar Fermi energies and Sha rvin conductivities of the metals [12]. Given these \nvalues, we can estimate the SMR magnitudes ∆𝜌L,Au=(0.7−0.4+2.0) pΩm and ∆𝜌T,Au=(7−5+25) fΩm for our \nYIG/Au sample . The calculated values are quantitatively consistent with the measured values. The \nuncertainty of the calculation will decrease in the future when the Elliott -Yafet scaling constant and \nintrinsic spin Hall conductivity are better understood for Au, as well as for other metals. \n \nTable 1: Overview of the obtained quantities for YIG/ Pt and YIG/Au studied here: Resistivity 𝜌0, the re lative longitudinal and \ntransverse amplitudes of the spin Hall magnetoresistance ∆𝜌L and ∆𝜌T, spin diffusion length 𝜆, spin Hall angle 𝜃 and real and \nimaginary spin mixing conductivities 𝐺r, 𝐺i. * 𝜆 and 𝜃 are derived from empirically observed scaling. * * Spin mixing conductivit ies \nare calculated for YIG/Pt and assumed to be identical for YIG/Au. \n \n Pt(2nm) Au(10nm) \n𝜌0 (nΩm) 516 84.4 \n∆𝜌L/𝜌0 6.8×10−4 1.3×10−5 \n∆𝜌T/𝜌0 2.2×10−5 1.3×10−7 \n𝜆 (nm) 1.2±0.2 * 25−8+12 * \n𝜃 0.09±0.01 * 0.017−0.008+0.016 * \n𝐺r (1014 Ω−1m−2) 3.8±1.0 3.8 ** \n𝐺r𝐺i⁄ 22±3 22 ** \n \n5 We conclude that the observed magnitudes of the longitudinal and transvers e magnetoresistance in \nour two systems, YIG/Au and YIG/Pt, are consistent with the same physical picture. Namely, the \ntransvers e magnetoresistance in these M I/NM systems can be fully understood as emergent from the \ntransvers e part of the spin Hall magnetoresistance due to th e imaginary component of the spin -mixing \nconductivity. No evidence points to a contribut ion of proximity magnetization via the anomalous Hall \neffect. Au is prototypical for materials with a low resistivity and intermediate spin Hall angle, which \nindicates that the conventional SMR theory applies in this regime. In addition, the possibility of \nmeasuring both the longitudinal and the transverse spin Hall magnetoresistance amplitudes for a wide \nrange of MI/NM interfaces provide s an elegant way to study the spin -mixing conduct ivity and its \nfundamental dependencies . \n \n \nSupplementary Material \n \nSupplementary material contains further details on the transport measurements including the \napproach to compensate for the normal Hall effect a nd remark on the absence of the planar Hall effect \nin Zero -Offset Hall measurements. Electron microscopy imaging of the samples in cross section is \nreported as well. Additional measurements are reported to assess the contribution of the anomalous \nHall effe ct due to the thermal spin pumping. The data includes hysteresis loops measured at different \ncurrent densities as well as transport data taken at higher harmonics. \n \n \nAcknowledgements \n \nSupport by the Structural Characterization Facilities Rossendorf at the Ion Beam Center (IBC) at the \nHZDR is greatly appreciated. The work was financially supported in part via the German Research \nFoundation (DFG) Grant MA 5144/9 -1, the BMBF project GUC -LSE (federal research fun ding of \nGermany FKZ: 01DK17007), the BMWi project WiTenso (03THW12G01), by the Spanish MINECO \nunder the Maria de Maeztu Units of Excellence Programme (MDM -2016 -0618) and under Project No. \nMAT2015 -65159 -R and by the Regional Council o f Gipuzkoa (Project No. 100/16). J.M.G. -P. thanks \nthe Spanish MINECO for a Ph.D. fellowship (Grant No. BES -2016 -077301). \n \n \nReferences \n \n[1] T. Kosub, M. Kopte, R. Hühne, P. Appel, B. Shields, P. Maletinsky, R. Hübner, M. O. Liedke, J. \nFassbender, O. G. Schmidt, and D. 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Inoue, Intrinsic \nspin Hall effect and orbital Hall effect in 4d and 5d transition metals , Phys. Rev. B 77 165117 \n(2008). \n \n 8 \nSupplementary Information \n \n \n1. Thermal spin pumping induced anomalous Hall effect \n \nIt has been reported that thermal gradients can lead to thermal spin pumping, which in turn can lead to \nproximity magnetization of Au thin films on YIG depending on the strength of the thermal gradient [1]. \nSuch proximity magnetization will also lead to an anomalous Hall effect but would be not due to the \nspin Hall magnetoresistance. We argue that the contribution of thermal spin pumping induced \nproximity magnetization is negligible in our present study. \n \nSeveral references and statements within D. Hou et al. [1] confirm tha t Au thin films are not expected \nto reveal any static magnetic proximity effect. This is also exemplified by the experiments in that work, \nwhich were performed without a thermal gradient resulting in no detected AHE signal. The authors \nargue that thermal s pin pumping leads to magnetization in Au. As our Au film is Joule heated due to \ntransport experiments, we expect a thermal gradient from Au towards YIG also in our samples. \n \nD. Hou et al. used an inverted notation of Hall voltage which is evident from their positive slope of the \nnormal Hall effect in their Figure 2(b). Therefore, the positive slope anomalous Hall effect (AHE) \ncurves in that work are identically signed as our negative slope AHE curves. Taking into account our \ntwice lower Au thickness, we woul d be able to explain our AHE signal magnitude with a purported \nthermal gradient of about 3.5 K/mm. If this thermal gradient is indeed present and is the origin of our \nHall signal, the expectation from the cited work would be an increase in the magnitude of AHE curves \nwhen larger currents are used to probe the Au film. \n \n \nFigure S1: Variation of the transverse magnetoresistance upon employing different probe current amplitudes 𝐼amp. \n \nWe did measure our sample at various levels of current. Due to our narrow st ripe pattern these \ncurrents lead to varying extents of Joule heating and thus thermal gradients orthogonal to the film \nplan. We did not observe an increasing trend of the AHE magnitude with increasing probe current. In \nfact, despite a more than 7 -fold vari ation of the Joule heating dissipation, we did not observe any \nsignificant change in the magnitude of the AHE effect [Figure S1]. The slight (but non -significant) \nvariation of the AHE magnitude is due to the processing of the nonlinear background, which is slightly \ndifferent scan to scan due to long term nature of the scans. Even if the variation of the AHE magnitude \nwith the current amplitude is taken to be real, it is opposite to that expected from thermal spin pumping \ndescribed by D. Hou et al. Therefore , we conclude that the thermal spin pumping induced AHE is \nnegligible for the total magnitude of our observed transversal magnetoresistance. \n \nFurthermore, it is also possible to separate the contributions from the transverse spin Hall \nmagnetoresistance and from thermal spin pumping AHE to the total transverse magnetoresistance \nthrough harmonic measurements. The signal in Figure S1 follows from the transverse voltage at the \nexcitation current frequency (first harmonic) and can contain co ntributions from both effects. In \ncontrast, the voltage at the third harmonic frequency should contain only contributions from thermal \n9 spin pumping: When we supply a low frequency ( 𝑓) AC current to probe our Au films, we expect to \nheat the Au film not onl y to an equilibrium value – which we discussed above – but the Au film would \nalso exhibit temperature variations with a 2𝑓 periodicity. The temperature would peak every time the \nprobe current sine wave is at its tips and temperature would dip below the av erage value at the zero -\ncrossings of the sine wave. According to D. Hou et al. such a temperature behavior would result in \nmore transient magnetization being present in Au while the probe current sine wave is near its tips. \nTherefore, the resulting AHE vol tage would be larger than a pure sine wave near the tips of the sine \nwave probe current resulting in an additional third harmonic Hall signal. This third harmonic Hall signal \nshould exhibit the same behavior as the magnetization of YIG, i.e. an antisymmetr ical curve saturating \nat about 0.15 T . \n \n \nFigure S2: Third harmonic Hall signal using 𝐼amp =5.61 mA. \n \nWe indeed also probed this third harmonic signal and found no significant YIG -like variation of it as we \nswept the field [Figure S2]. Fitting a YIG like si gnal to the third harmonic Hall signal of Figure S2 leads \nto a thermal spin pumping AHE of about 2 fΩm. This is a) much lower than the first harmonic \ntransverse magnetoresistance magnitude we observe and b) is questionable to exist at all in light of \nthe d ata in Figure S2 which could be also just a linear trend due to the normal Hall effect and slightly \nnon-sinusoidal excitation. \n \nJudging from the signal levels reported by D. Hou et al., we should have definitely observed these \ntrends using our very sensitiv e methodology if the thermal gradient was sufficient and the effect as \nstrong as reported. We hypothesize that the absence of the thermal gradient AHE signals in our study \ncould be due to either a lower thermal gradient or the different preparation conditi ons. Regarding the \nlatter, a lthough D. Hou et al. did not provide structural information on their YIG/Au interface, we \nbelieve that it is of more coherent quality due to their etching and in -situ-prenannealing procedure prior \nto Au deposition. Instead, to observe the SMR of Au/YIG, no interface optimization was found \nnecessary. As a result, we deposited Au on untreated YIG substrates, yielding dense continuous films, \nbut no epitaxy probably because of amorphous surface termination of the YIG surface due to \nadsorbates. These adsorbates could also render the thermal spin pumping inefficient in our samples. \n \n \n2. Sample preparation and measurement details \n \nThe reason we chose 10 nm Au as our main sample is because we wanted to avoid discontinuous \nmetal films. Such discontinuities appear due to the bad wetting of the noble metals on the untreated \nYIG, especially for metals with low melting temperatures, such as Cu, Ag, Au [Figure S3]. \nDiscontinuous layers can give rise to spurious effects like wrong resistivity values when assuming \nnominal thicknesses which would be problematic for our determination of the spin Hall angle and the \nspin diffusion length. \n \n10 \n \nFigure S3: Cross -sectional STEM image of 10 nm film of Au on an untreated liquid phase epitaxy YIG -on-GGG sub strate. The \nAu film is fully continuous and has a homogenous thickness. \n \nOn the other hand, films thicker than 10 nm would produce so much shunting of the interfacial SMR \neffect (or magnetic proximity effect), that the detection of the transversal signal would be no longer \npossible. We want to stress that the presented measurements have been very challenging even in this \n10 nm Au sample. Even though w e used a precisely patterned Hall cross with only about 0.3 % \ncontact skew, the residual longitudinal resistance background amounted to roughly 11 mΩ, which is \nroughly 105 over the minimum necessary precision to detect the AHE curves of about 100 nΩ and \nplaces high demands for continuous dynamic measurement range. In addition, to achieve this \nprecision at a noise density of about 2 µΩ√Hz⁄ (at 2 mA current amplitude) required approximately \n400 seconds of integration for each of the 100 field bins, amounting to an integration time of \napproximately half a day. Therefore, although our purpose made measurement device unites great \ncontinuous dyn amic range and extremely low noise, we had to employ sophisticated drift \ncompensation methods to achieve a low enough corner frequency to actually resolve the AHE effect in \nour samples. \n \n \n3. Planar Hall effect \n \nWhen measuring the transverse voltage drop develop ed by a slab of anisotropically conducting \nmaterial, a planar Hall effect is generally observed. In context of polycrystalline metal films used in \nspintronics, such anisotropy commonly appears as a result of e.g. the anisotropic \nmagnetoresistance [2,3] or spin Hall magnetoresistance [4]. When a voltage is applied to this \nanisotropically conducting thin film, the current will in general not flow collinearly w ith the voltage \ngradient, but experience a transverse deflection towards the high conductivity axis, which results in an \nequilibrium transverse voltage drop. In a stationary measurement layout, this planar Hall effect only \nvanishes for a) perfectly isotrop ic conduction or b) conduction exactly along the high or low \nconductivity axes. \n \nHowever, despite its name, the planar Hall effect is fundamentally different from other Hall effects such \nas the normal and an omalous Hall effects (real Hall effects) . When averaging all in -plane current \ndirections, the planar Hall effect assumes both positive and negative values as current experience \nalternatingly left -handed and right -handed deflection. In contrast , the real Hall effects persist as they \nexperience the same handedness of deflection for any in -plane current direction . Therefore, planar \nHall effects disappear in Zero -Offset Hall measurements [5], while the real Hall effects are maximally \npreserved. \n \n \n4. Normal Hall effect \n \nIn order to study tiny anomalous Hall-like signals, it is mandatory to dynamically reject the influence of \nthe longitudinal resistance, which – even for carefully patterned – Hall cross structures can be much \nlarger than the Hall signal of interest [5]. While this is elegantly achieved using the Zero -Offset Hall \nmeasuremen t scheme, the normal Hall effect cannot be rejected using this approach. D. Hou et al. \nreported an approach to reject the normal Hall effect via a lock -in method [1], but this is possible only \nwhen the origin of the anomal ous Hall effect can be switched on and off, which is not the case in our \nsample. \n \n11 We remove the normal Hall effect after the measurement by subtracting it from the measured \ntransverse resistance. The observed normal Hall effect is not completely linear in the magnetic field for \ntwo reasons: a) our magnetic field reading is not fully linear over the actual magnetic field mainly due \nto nonlinearities of the Si hall probe. b) the normal H all effect of the thin metal ( Pt or Au) films can be \nslightly non -linear itself. These influences cause a total nonlinearity of the normal Hall effect in the \nrange of 10−3 to 10−2, which is non -negligible in our study. The function we use to model the norm al \nHall effect is composed of three contributions: \n \n𝜌NHE =𝐴1𝐻𝑧+𝐴2(1−sech (𝐻𝑧\n𝐻NL))+𝐴3tanh (𝐻𝑧\n𝐻NL) (1) \n \nwhere 𝐻𝑧 is the magnetic field applied out -of-plane, 𝐻NL is a characteristic field determining the shape \nof the nonlinearity and 𝐴1,2,3 are scaling constants. The first contribution 𝐴1𝐻𝑧 is the largest by far, \nwhile the sech and tanh function s capture smaller even and odd nonlinearities, respectively . One \nmotivation for this model ch oice is that it cannot accidently introduce YIG -like signals, when 𝐻NL is \nsufficiently large, namely 𝐻NL≳2𝐻sat,YIG with 𝜇0𝐻sat,YIG≈0.16 T. When this is fulfilled, the tanh \ncontribution is essentially linear in the relevant field range. For the present ed measurements, we used \n𝜇0𝐻NL≈1.1 T to model and subtract the normal Hall contribution to 𝜌T [Figure 1(d,f) of the main text]. \n \n \nReferences \n \n[1] D. Hou, Z. Qiu, R. Iguchi, K. Sato, E. K. Vehstedt, K. Uchida, G. E. W. Bauer, and E. Saitoh, \nObservation of temperature -gradient -induced magnetization , Nat. Commun. 7 12265 EP (2016). \n[2] K. M. Seemann, F. Freimuth, H. Zhang, S. Blügel, Y. Mokrousov, D. E . Bürgler, and C. M. \nSchneider, Origin of the Planar Hall Effect in Nanocrystalline Co 60Fe20B20., Phys. Rev. Lett. 107 \n086603 (2011). \n[3] P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R. P. Campion, V. Novák, K. Olejník, F. \nMaccherozzi, S. S. Dh esi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. \nKuneš, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and \nT. Jungwirth, Electrical switching of an antiferromagnet , Science 351 587 (2016). \n[4] H. Nakayama, M. Althammer, Y. -T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. \nGeprägs, M. Opel, S. Takahashi, and others, Spin Hall Magnetoresistance Induced by a \nNonequilibrium Proximity Effect , Phys. Rev. Lett. 110 206601 (2013). \n[5] T. Kosub, M. Kopte, F. Radu, O. G. Schmidt, and D. Makarov, All-electric access to the magnetic -\nfield-invariant magnetization of antiferromagnets , Phys. Rev. Lett. 115 097201 (2015). \n \n " }, { "title": "2209.04291v1.A_spinwave_Ising_machine.pdf", "content": "A spinwave Ising machine\nArtem Litvinenko1\u0003, Roman Khymyn1, Victor H. Gonz ´alez1, Ahmad A. Awad1,\nVasyl Tyberkevych2, Andrei Slavin2and Johan ˚Akerman1\u0003\n1Department of Physics, University of Gothenburg, Fysikgr ¨and 3, 412 96 Gothenburg, Sweden\n2Department of Physics, Oakland University, 48309, Rochester, Michigan, USA,\n\u0003To whom correspondence should be addressed; E-mails:\nartem.litvinenko@physics.gu.se, johan.akerman@physics.gu.se\nWe demonstrate a spin-wave-based time-multiplexed Ising Machine (SWIM),\nimplemented using a 5 \u0016m thick Yttrium Iron Garnet (YIG) film and off-the-\nshelf microwave components. The artificial Ising spins consist of 34–68 ns\nlong 3.125 GHz spinwave RF pulses with their phase binarized using a phase-\nsensitive microwave amplifier. Thanks to the very low spinwave group veloc-\nity, the 7 mm long YIG waveguide can host an 8-spin MAX-CUT problem and\nsolve it in less than 4 \u0016s while consuming only 7 \u0016J. Using a real-time oscillo-\nscope, we follow the temporal evolution of each spin as the SWIM minimizes\nits energy and find both uniform and domain-propagation-like switching of\nthe spin state. The SWIM has the potential for substantial further miniatur-\nization, scalability, speed, and reduced power consumption, and may become a\nversatile platform for commercially feasible optimization problem solvers with\nhigh performance.\n1arXiv:2209.04291v1 [cond-mat.mes-hall] 9 Sep 2022Introduction\nAs Moore’s law comes to an end due to physical limitations, while the amount processing data\nis constantly growing, novel analog and digital computing paradigm are being investigated.\nLarge part of data processing tasks consist of combinatorial optimization problems and require\nspecial purpose accelerators for power efficient and fast computing. Combinatorial optimiza-\ntion is essential to select the optimal path from an enormous range of choices and appears\nin various social and industrial fields such as optimizing financial transactions ( 1), logistics, in-\ncluding travel ( 2) and channel assignment for wireless communications ( 3), genetic engineering\nand molecular design for drug discovery ( 4). Conventional computers based on V on Neumann\narchitectures are inefficient in solving hard combinatorial optimization problems due to the fac-\ntorial growth of all possible combinations to be evaluated using brute force. Fortunately, Ising\nmachines have emerged as a promising non-von-Neumann computing scheme that can acceler-\nate computation of NP-hard optimization problems.\nAn Ising machine maps an NP-problem onto the Ising Hamiltonian of an array of Nbina-\nrized and interacting physical entities, commonly referred to as spins si=\u00061.\nH(s1;:::;s N) =\u0000X\ni=< b†\n±Q>=/radicalbig\nN0/2, (4)corresponding to a condensate wave-function Ψ Q(y)∝\ncos(Qy). More generally Ψ Q(y) is multiplied by a phase\nfactoreiφ(r,t)//planckover2pi1, which is crucial to the BEC dynamics,\nbut not necessary for a stability analysis of the BEC.\nWe make a Holstein-Primakoff magnon expansion [9]\nup to 4th-order in magnon operators, including contribu-\ntions from both (2 in - 2 out) and (3 in - 1 out) magnon\nscattering processes [10] (we ignore multiple-scattering\ncontributions here, which are ∼O(Ud/Jo)∼10−3rela-\ntive to the leading terms). Then, taking quasi-averages,\nwe can write the Hamiltonian in the form\nH=/planckover2pi1/summationdisplay\nqωq(b†\nqbq+1\n2)+/hatwideVp\nint+/hatwideV−p\nint (5)\nwhere the interaction term /hatwideVp\ninttakes the form\n/hatwideVp\nint=n0(Γ0+ΓS\n4)/summationdisplay\np300 ns in Fig. 2 a, 2b) corresponding to a magnon system in equilibrium. Both, \nthe measured BLS frequency of the fundamental mode 𝑓𝑓FM and the chemical potential obtained \nexperimental ly with help of Eq. 11 are shown in Fig. 2b. In addition, the black line in Fig. 2 b \npresents the time evolution of the BLS intensity integrated over the frequency range at the bottom \nof the magnon spectrum . \nAs can be seen from Fig. 2 b, a good agreement with the simplified theoretical model in the \nmiddle panel of Fig. 1 a is visible. The BLS intensity at the bottom of the magnon spectrum \nincreases after the pulse. In addition, the accompanying increase of the chemical potential is verified experimentally. At its maximum, the chemical potential coincides within the error bars with the lowest magnon frequency. (Please note that lowest magnon frequency slightly shifts down during the current pulse due to the heating.) Hence, the criterion for the BEC – the equality of the \nchemical potential and the minimum energy of the system – is fulfilled. \nPlease note that an increase in the chemical potential is also observed during the pulse. \nThis increase can be attribut ed to the s pin Seebeck effect\n17,24,25. As it is reported by I. Barsukov 12 \n and I. N. Krivorotov at a variety of international conferences, the spin Seebeck effect serv s as the \nmain mechanism of magnon BEC in their experiments17. However, in our experiments, t he \nthermally induced spin injection from the Pt layer is not sufficient to reach the BEC condition. \nEven though the experiment was performed at various voltages and pulse durations, we observed \nthe BEC always after the pulse, i.e., at times at which the gr adient has already disappeared, and \nnever during the pulse. The same experiment in the continuous regime also did not show any \nmagnon generation in the lowest or any other observable magnon state. This can be related \nTherefore, in our experiment, the s pin Seebeck effect can be seen just as a supportive mechanism \nfor the BEC while the key mechanism is rapid cooling. 13 \n References : \n1. Anderson, M. H. , Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. \nObservation of Bose –Einstein condensation in a dilute atomic vapor. Science 269, 198–\n201 (1995). \n2. Santra, B. et al. Measuring finite -range phase coherence in an optical lattice using Talbot \ninterferometry. Nat. 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Parallel pumping for magnon spintronics: \nAmplification and manipulation of magnon spin currents on the micron- scale. \nPhys. Rep. 699, 1–34 (2017). \n17. Safranski C. et al. Spin caloritronic nano- oscillator. Nat. Commun. 8, 117 (2017). \n18. Gurevich, A. G. & Melkov, G. A. Magnetization oscillations and waves (CRC, 1996). \n19. Hüser, J. Kinetic theory of ma gnon Bose -Einstein condensation, PhD thesis, Westfälische \nWilhelms -Universität Münster, Münster, Germany (2016). \n20. Dubs, C. et al. Sub- micrometer yttrium iron garnet LPE films with low ferromagnetic \nresonance losses . J. Phys. D: Appl. Phys. 50, 204005 (2017). 14 \n 21. Sebastian T., Schultheiss, K., Obry, B., Hillebrands, B. & Schultheiss, H. Micro -focused \nBrillouin light scattering: ima ging spin waves at the nanoscale. Front. Phys. 3, 35 (2015). \n22. Cherepanov, V., Kolokolov, I. & L'vov, V. The saga of YIG. Phys. Rep. 229, 81–144 \n(1993). \n23. Olsson, K. S. et al. 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Nature 432, 691–694 (2004). 15 \n Acknowledgements: \nThis research has been supported by ERC Starting Grant 678309 MagnonCircuits, \nERC Advanced Grant 694709 Super -Magnonics, by the DFG in the framework of the Research \nUnit TRR 173 “Spin+X” (Projects B01 and B07) and Project DU 1427/2 -1, by the grants Nos. \nEFMA -1641989 and ECCS -1708982 from the National Science Foundation of the USA, and by \nthe DARPA M3IC grant under the contract W911- 17-C-0031. \n \nAuthor contributions: \nM.S. and D.B. performed the measurements and analyzed the experimental results. T.B. \nand A.V.C. supervised the measurements. T.B., P.P., A.A.S., A.V.C. planned the experiment. \nM.S., Bj.H., T.M. developed the experimental set -up. V.L. performed FMR characterizations and \npreliminary experiments. C.D. has grown the LPE YIG film. S.K. and E.Th.P. deposited the Pt \noverlayer. M.S., Bj.H., B.L., T.L., D.B. fabricated the structures under investigation. V.S.T. \ndeveloped the quasi -analyti cal model of the magnon spectral redistribution . D.A.B., H.Yu.M.- S., \nV.S.T., A.N.S. performed the theoretical calculations. M.S. and F.H. performed the COMSOL \nsimulations. Q.W. and P.P. performed the MuMax3 simulations. B.H. and A.V.C. led the project. \nAll authors discussed the results and wrote the manuscript. \n \nCompeting interests: None declared. \n \nAffiliations: \n1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität \nKaiserslautern, D -67663 Kaiserslautern, Germany. \n2Department of P hysics and Energy Science, University of Colorado at Colorado Springs, \nColorado Springs, CO 80918, USA \n3Graduate School Materials Science in Mainz, Staudingerweg 9, D -55128 Mainz, Germany. \n4THATec Innovation GmbH, Bautzner Landstraße 400, D -01328 Dresden, Germany. \n5Nano Structuring Center, Technische Universität Kaiserslautern, D -67663 Kaiserslautern, \nGermany. \n6INNOVENT e.V. Technologieentwicklung, Prüssingstraße 27B, D -07745 Jena, Germany. \n7Department of Physics, Oakland University, Rochester, MI 48309, USA. \n8Faculty of Physics, University of Vienna, Boltzmanngasse 5, AT -1090 Wien, Austria. \n*Corresponding author. Email: chumak@physik.uni -kl.de \n 16 \n Supplementary Information \n \n \nExtended Data Fig. 1 | Detailed results of the numerical modelling of the BEC by rapid \ncooling. a, Time evolution of the temperature of the phonon system (top panel), the magnon \nchemical potential (middle panel) and the magnon population at the lowest energy state (bottom panel). The moment in time of the instant cooling is marked by t\nOff and the vertical dashed line. \nb, Frequency distributions of the quasi -particles density for different times. The red lines in panels \n1–4 show the snapshots of the magnon densities (for the times marked by the pink dots in panel a \ncalculated using the dynamics equations. Dashed blue lines show the steady -state room -\ntemperature distribution calculated using Eq. (1) in the manuscript. The green dashed lines denote \nthe distribution when the chemical potential µ is equal to the minimal magnon energy hf min. Panel \n1: Stationary heated case before the instant cooling. Panel 2: Distribution just after the instant \ncooling. Panel 3: Snapshot of magnon density when the chemical potential µ reaches the minimal energy hf\nmin, resulting in BEC. Panel 4: Final room temp erature equilibrium state. c, Time \nevolution of the chemical potential and the magnon population at the lowest energy state calculated for a heating temperature of 510 K , which exceeds the threshold value of 440 K. The long time \npinning of the chemical pot ential µ at f\nmin is clear ly visible for the case of the stronger heating. \n17 \n \nParameter Value \nWidth of the strip 500 nm 1000 nm \nCurrent pulse duration τ p 120 ns 300 ns \nApplied voltage U 0.9 V 1.05 V \nResistance R 600 Ohm 403 Ohm \nRoom temperature 0T 288 K 288 K \nElevated temperature 1T 440 K 490 K \nGilbert damping constant Gα 1.5×10-3 1.5×10-3 \nMaximal frequency f max 600 GHz 600 GHz \nFrequency step f∆ 250 MHz 250 MHz \nFour -magnon scattering \nefficiency C 0.7 0.7 \nExtended Data Table 1 | Parameters used for the calculations of the magnon density . The \ntable shows the parameters according to the developed quasi -analytical theoretical model for two \ndifferent experimentally investigated strips. \n \n Spin- wave dispersion in the YIG nano -structures . The dispersion relations were obtained by \nmeans of micromagnetic simulations, which were performed using MuMax3\n1 – see Extended Data \nFig. 2 . The following parameters were used for the micromagnetic simulations. The size of the \nwaveguide is 20 µm × 500 nm × 70 nm (length × width × thickness). The mesh was set to \n10 nm × 10 nm × 70 nm. The following parameters for YIG were used: Saturation magnetization \nMs = 123 kA/m, exchange constant A = 8.5 pJ/m, and Gilbert damping α = 2 × 10-4. In the \nsimulations, the damping at the ends of the waveguides was set to increase exponentially to 0.5 in \norder to eliminate spin -wave reflections. The external field is 188 mT for bot h, the Backward \nVolume and the Damon -Eshbach geometries18. In order to excite spin- wave dynamics in the \nwaveguide, a sinc -function- shaped field pulse was applied to a 50 nm wide area in the center of \nthe waveguide. To gain access to both, odd and even spin- wave- width modes, this area was slightly \nshifted away from the center along the short axis of the waveguide. The sinc field is given by bz = \nb0 sinc (2πfct), with an oscillation field b 0 = 1 mT and a cutoff frequency f c = 20 GHz. The out -of-\nplane component Mz (x, y, t ) of each cell was collected over a period of t = 50 ns and stored in t s = \n12.5 ps intervals, which results in a frequency resolution of Δ f = 1/t = 0.02 GHz, whereas the highest \nresolvable frequency was f max = 1/(2ts) = 40 GHz. The fluctuations in m z (x, y, t ) were calculated for \nall cells via m z (x, y, t) = Mz (x, y, t) − Mz (x, y, t = 0), where Mz (x, y, t = 0) corresponds to the ground \n \n1 Vansteenkiste, A., et al. The design and verification of MuMax3, AIP Adv. 4, 107133 (2014). 18 \n state. To obtain the spin- wave dispersion curves, a two- dimensional fast Fourier transformation in \nspace and time has been performed2. \nTo visualize the dispersion curve, Extended Data Fig. 2 shows a 3D grey-scale map of the \nspin- wave intensity, which is proportional to mz2 (kx, f ), in logarithmic scale as a function of f and \nkx. In both configurations, several width modes are visible, of which the first three a re marked in \nthe graphs in Extended Data Fig. 2. Since the first three width modes are at similar frequencies, \nthey cannot be distinguished in the experiment. The energy minimum, where the generation of the Bose Einstein condensate is expected, is in both cases within the wave vector range accessible \nwith Brillouin -Light Scattering (BLS) spectroscopy ( k\nx = 13 rad/µm for B ||y and kx = 0 rad/µm for \nB||x). The frequency of the minimum of the magnon spectrum coincides with the ferromagnetic \nresonance frequency ( k = 0) in the case of the Damon -Eshbach B||y geometry and differs by a value \nof 380 MHz in the case of the Backward Volume geometry B ||x. This difference is on the ord er of \nmagnitude of the frequency resolution of the BLS spectroscopy. In general, the calculated frequency at the bottom of the magnon spectrum agrees well with the frequency of the BEC peak observed in the experiment taking into account the decrease of the saturation magnetization due to \nthe increase in temperature within the YIG nano -structure. \n \n \n2 Venkat G. , et al. Proposal of a standard micromagnetic problem: Spin wave dispersion in a \nmagnonic waveguide, IEEE Trans. Magn. 49, 524 – 529 (2013). \nExtended Data Fig. 2 | Simulated magnon dispersions. a, Simulated spin -wave intensity \nshown in a logarithmic black -to-white scale as a function of the frequency f and the \nwavenumber kx along the waveguide for B ||x. b, Simulated spin -wave intensity as a function \nof the frequency f and the wavenumber k x along th e waveguide for B ||y. \n \n \n 19 \n Simulation of time evolution of YIG/Pt structure by COMSOL Multiphysics . The \nnumerically calculated time evolution of the temperature of the YIG/Pt strips was determined \nby solving a 3D heat -transfer model of the experimental set -up with the COMSOL Multiphysics \nSoftware using the heat transfer module and the electric currents m odule. \nHereby, the conventional heat conduction differential equation and the differential \nequations for current conservation are solved taking into consideration the boundary conditions \napplied to the model as well as the material parameters of the mater ials used. The model \ncomprises a 9 µm × 4.5µm × 6.5 µm large volume which includes half of the strip, exploiting \nthe symmetry of the system with respect to its long axis. The simulated geometry includes the \nYIG/Pt strip, a part of the Au leads, which have a width of 1000 nm where they meet the strip \nand a width of 8 µm at a distance of 1 µm to the strip. In addition, the removal of the material \nby focused Ion -Beam with a width of 4 µm at the sides of the strip with a depth of 300 nm is \ntaken into account. T he used material parameters can be found in Table Extended Data Table 2. \nThe electrical conductivity and the corresponding temperature coefficient of Pt were measured \nexperimentally, whereas the other parameters are taken from the referenced literature or from \nthe COMSOL library. For the heat -transfer model, the boundary on the edges were set to the \nsymmetry boundary condition for the according edges and to open boundary for the others. For \nthe electric- currents model they were set to electric insulation, e xcept for the two faces of the \nleads where the electric potential is applied. The Joule heating is modelled for all metallic layers \nin the system. The simulation differs from the experiment by the fact that the convective cooling \ndue to the surrounding air as well as the laser heating are not implemented. \nExtended Data Fig. 3 shows the obtained temperature profile in the center of the Pt \noverlayer. The profile was obtained for a 300 ns long pulse with an amplitude of U Sim = 1.872 V \n(corresponding to a set v oltage of U = 1.05 V at the pulse generator for a 50 Ohms load resistance \nin the experiment) and transition times of 1 ns at the edges. It can be seen that the temperature \nincreases rapidly just after the current pulse is applied to Pt but tends to saturat e. After a time of \napproximately 300 ns the temperature still grows but the heating has already strongly decelerated \nallowing for the formation of the quasi -equilibrium between the magnon and phonon \ntemperatures3 considered in the theoretical model. The maximal temperature reached at the end \nof the applied current pulse in this particular case is 491 K. \n \n \n \n \n3 Agrawal, M. , et al. Direct measurement of magnon temperature: New insight into magnon -\nphonon coupling in magnetic insulators. Phys. Rev. Lett. 111, 107204 (2013). 20 \n Parameter Material Value/ Source \nDensity YIG 5170 kg m-3 \nClark, A. E. & Strakna , R. E. Elastic constants of single -\ncrystal YIG, J. Appl. Phys. 32, 1172 (1961) \nHeat conductivity YIG 6 W m-1K-1 \nHofmeister, A. M. Thermal diffusivity of garnets at high \ntemperature, Phys. Chem. Minerals 33, 45 -62 (2006) \nHeat capacity YIG 570 J kg-1 K-1 \nGuillot, M., Tchéou, F., Marchand, A., Feldmann, P. & \nLagnier, R., Specific heat in Erbium and Yttrium Iron \ngarnet crystals, Z. Phys. B – Condensed Matter 44, 53- 57 \n(1981) \nDensity GGG 7080 kg m-3 \nHofmeister A. M. Thermal diffusivity of garnets at high \ntemperature, Phys. Chem. Minerals 33, 45 -62 (2006) \nHeat conductivity GGG 7080 kg m-3 \nHofmeister A. M. Thermal diffusivity of garnets at high \ntemperature, Phys. Chem. Minerals 33, 45 -62 (2006) \nHeat capacity GGG 7080 kg m-3 \nHofmeister A. M. Thermal diffusivity of garnets at high temperature, Phys. Chem. Minerals 33, 45 -62 (2006)\n \nDensity Pt 21450 kg m-3 \nLide, D. CRC Handbook of Chemistry and Physics, 89th ed. \n(Taylor & Francis, London, 2008) \nElectrical conductivity Pt 1.41 × 106 S m-1 (Extended Data Fig. 3), 1.90 × 106 S m-1 \n(Extended Data Fig. 4), 1.79 × 106 S m-1 (Extended Data \nFig. 5a) \nMeasured \nResistance- Temperature \nCoefficient Pt 7.135 × 10-4 K-1 \nMeasured \nThermal Conductivity Pt 22 W m-1 K-1 \nYoneoka, S., et al. Electrical and thermal conduction in \natomic layer deposition nanobridges down to 7 nm \nthickness, Nano Lett. 12, 683- 686 (2012) \nHeat capacity Pt 130 J kg-1 K-1 \nFurukawa, G. T., Reilly, M. L., Gallagher, J. S. Critical \nAnalysis of Heat Capacity data and evaluation of \nthermodynamic properties of Ruthenium, Rhodium, \nPalladium, Iridium, and Platinum from 0 to 300K. A survey \nof the literature data on Osmium, J . Phys. Chem. Ref. Data \n3, 163 (1974) \nDensity Au 1900 kg m-3 \nCOMSOL Library 21 \n Electrical conductivity Au 4.1 × 107 S m-1 \nCOMSOL Library \nThermal Conductivity Au 190 W m-1 K-1 \nLanger, G., Hartmann, J. & Reichling M. Thermal \nconductivity of thin metallic films mea sured by \nphotothermal profile analysis , Rev. Sci. Instrum. 68, 3 \n(1997) \nHeat capacity Au 130 J kg-1 K-1 \nGeballe, T. H., & Giauque, W. F. The heat capacity and entropy of Gold from 15 to 300°K, J. Am. Chem. Soc. 1952, \n74)\n \nDensity Ti 4500 kg m-3 \nCOMSOL Library \nElectrical conductivity Ti 2 × 106 S m-1 \nCOMSOL Library \nThermal Conductivity Ti 22 W m-1 K-1 \nHo, C. Y., Powell, R. W. & Liley, P. E. Thermal conductivity of the elements, J. Phys. Chem. Ref. Data 1, 2 (1972)\n \nHeat capacity Ti 521.4 J kg-1 K-1 \nKothen, C. W. & Johnston, H. L. Low Temperature Heat \nCapacities of Inorganic Solids. XVII. Heat Capacity of \nTitanium from 15 to 305°K. J. Am. Chem. Soc. 75, 3101 \n(1953) \nHeat capacity Air 1047.6366 - 0.3726 T + 9.4530 × 10-4 T2 - 6.0241 × 10-7 \nT3+1.2859 × 10-10 T4 J kg-1 K-1 \nCOMSOL Library \nThermal Conductivity Air -0.0023 + 1.1548 × 10-4 T -7.9025 × 10-8 T2 + 4.1170 × 10-11 \nT3-7.4386 × 10-15 T4 W m-1 K-1 \nCOMSOL Library \nDensity Air 8.53 T-1 kg m-3 \nCOMSOL Library \n Extended Data Table 2 | Parameters used for the C OMSOL Simulations. \n \n \n \n \n 22 \n As it is seen in Extended Data Fig. 3a , switching off the current pulse results in a fast \ndecrease in temperature of the YIG/Pt nano -structure due to the thermal diffusion in the quasi -\nbulk surroundings. The cooling rate i s not constant and decreases with time. Moreover, our \nsimulations further revealed (not shown) that the cooling rate depends on the duration of the \napplied current pulse which is associated with the increase in temperature of the surroundings \nof the nano -structure for long current pulses. \nTo prove the crucial role of the fast cooling for the BEC, additional measurements were \nperformed where the current pulses were switched off with fall times of 50 ns and 100 ns – see \nFig. 2 in the main text of the manuscr ipt. It is evident that the condensate disappears for long fall \ntimes, i.e. slow cooling rates. For reference, Extended Data Fig. 3 shows the corresponding \nsimulated temperature evolutions in the phonon system. The maximal cooling rates indicated in \nExtend ed Data Fig. 3b clearly show that a cooling rate of 2 K/ns is already too slow to trigger the \nmagnon BEC, while a rate of 20.5 K/ns is still fast enough. \n \nExtended Data Fig. 3 | Results of the COMSOL simulations for different fall times. \na, Simulated temperature as a function of time in the middle of the Pt overlayer for a 300 ns long \npulse with a voltage of 1.05 V applied to a 1 µm wide waveguide. The fall time of the pulse was \n1 ns. The dashed blue lines indicate the time when the pulse is present. The room temperature is \nmarked with the dashed pink line. b, Simulated temperature as a function of time for different \nfall times τ Fall = 1 ns (green line), τ Fall = 50 ns (blue line), τ Fall = 100 ns (red line). The maximal \ntime derivatives are ma rked at their respective time of occurrence. \n \nTo support the COMSOL simulations an additional experiment is performed, measuring \nthe temperature of the Platinum layer time -resolved by means of electrical measurements. First, \nτP = 300 ns long DC -pulses with an amplitude of U = 0.85 V are applied to a 1 µm wide YIG/Pt \nwaveguide. The corresponding BLS -spectrum is shown in Extended Data Fig. 4b. The frequency \nshift during the pulse indicates the heating of the magnetic system, whereas the magnon ΒEC is \nobserved after the pulse is switched off. To get access to the time -resolved resistivity (which depends \non the temperature) during the applied pulse instead of the DC -pulse, a RF -pulse with the same \nduration and a frequency of 12.4 GHz is applied to heat up the sample. The RF -power was adjusted \nin such a way, that Joule heating results in the same steady state temperature as for a continuous \napplied voltage of U = 0.85 V. Thus, the resistance of the Pt layer was found to increase from \n319.5 Ohm to 372.1 Ohm when either a continuous DC -voltage of 0.85V or a continuous RF signal \nwith a power of 18.84 dBm and a frequency of 12.4 GHz is applied. Hence, the RF -power of \n18.84 dBm results in the same Joule heating of the Platinum layer as a DC -voltage of 0.85 V. Then, \n23 \n a continuous DC -voltage of 30 mV, which is used to measure the resistance change and a 300 ns \nlong RF -pulse are applied in parallel using a DC -block and a RF -circulator. From the changed \nvoltage drop on the sample, which is measured with an oscilloscope, the temperature is calculated. \nThe resulting temperature profile is shown in Extended Data Fig. 4a (black curve). In addition, we \nhave performed the same COMSOL simulation as for all other shown simulated temperature \nprofiles. Only the cor responding conductivity of the Pt -layer was matched to the measured resistivity \nof the specific structure. The results are as well depicted in Extended Data Fig. 4a (blue curve). The perfect agreement with the measured temperature of the Platinum layer shows that the used \nCOMSOL model can precisely describe the temperature dynamics of the investigated structures. \nFurther, from the BLS measurement the temperature is derived by considering the \nfrequency shift determined by BLS and the Kittel equation. The re sulting temperature profile is \nshown in Extended Data Fig. 4 a (red points) and deviates from the simulated and electrically \nmeasured temperature during the pulse and directly after the pulse. At times the BEC already disappeared the temperature profiles co incide. This difference can be since the BLS -frequency \nduring the pulse is not only given by the number of magnons, but also by the flattening of the \ndispersion, due to a temperature induced change of the magnetic parameters. Further, it might be influence d by a strong thermal gradient, which potentially changes the mode profiles and \nfrequencies. At the time the BEC occurs, a too high temperature is measured by BLS, which \nindicates ones more, that magnons condense to the lowest energy state. \n \nExtended Data Fig. 4 | Temperature profiles derived by electrical measurements, \nCOMSOL and BLS | a, Temperature as a function of time measured electrically (black curve) \nby means of resistivity change for an applied RF -pulse with a frequency of f = 12.4 GHz, a power \nof PRF = 18.84 dBm and a duration of τ P = 300 ns, obtained by COMSOL simulations (blue curve) \nfor an applied voltage of U = 0.85 V and a pulse duration of τ P = 300 ns, derived from the shift \nof the BLS -frequency (red data points) b, Time resol ved BLS spectrum for the temperature \nprofiles shown in a, for the case when a pulse of an amplitude of U = 0.85 V and a duration of \nτP = 300 ns is applied. \n \n \nThermal dynamics and the spin Seebeck effect. Another possible contribution to the observed \nBEC c an be the spin- orbit -mediated spin Seebeck effect as it was shown in Ref. 17. To clarify the \nrole of the SSE, we extracted the temperature profile across the YIG/Pt structure and simulated the \ntemporal evolution of the temperature and temperature gradient in the YIG film – see Extended \n24 \n Data Fig. 5. The simulation was performed for t he experiment shown in Fig. 2b in the main \nmanuscript. The current pulse duration was set to τP = 120 ns. The pulse voltage was of U Sim = \n1.662 V, which corresponded to a set voltage of U = 0.9 V at the pulse generator for a 50 Ohms \nload resistance in the experiment. A maximum temperature of T = 443 K was reached at the end \nof the pulse. One can see that the gradient can reach values up to 3.5 × 108 K/m. However, the \ntemporal evolution of the gradient shows that it is formed within few nanoseconds after the current \npulse is applied, stays practically constant during the pulse and, finally, disappears within few \nnanoseconds after the pulse is switched off. Thus, at the moment of BEC in our experiment, there \nis no thermal gradient present as well as a possible contribution from the SSE. At the same time, \nwe can assume that the SSE contributes to the increase in the magnon chemical potential during \nthe current pulse is applied17. However, in course of time, such a contribution should decrease with \nincrease in the YIG temperature due to a strong rapid temperature -dependent reduction in the SSE \nmagnitude (see Fig. 3c in Ref. 27). \n \n \nExtended Data Fig. 5 | Temperature profile and time evolution of temperature gradient. \nSimulated temperature (red curve, left axis) and temperature gradient (black curve, right axis) \nas a function of time. The profiles were simulated with COMSOL for the parameters \ncorrespondi ng to the experiment shown in Fig. 2b in the main manuscript. \n \n \n \nDependence of the BEC on the applied magnetic field . The described magnon BEC is driven by \nthe change of the temperature of the phonon system. In order to prove this, a set of measurements \nusing different structure sizes of width of 500 nm and 1000 nm, of different scanning laser powers \nin the range from 0.75 mW to 3.3 mW, and different scanning points on the structure were performed. \nThe experimental finding of the BEC, which manifests itself as a strong magnon peak at the bottom \nof the spectrum, could always be confirmed. In particular, the threshold for BEC is found to be \nindependent of the applied external field. Measurements of the threshold for a pulse duration of \nτP = 18 ns are performed in the range from 110 mT to 270 mT. The maximal integrated BLS \nintensities of the fun damental mode can be seen in Extended Data Fig. 6. \n25 \n \n \n In addition, the inset exemplary shows time traces for different field values and an applied voltage of U = 1.5 V. From the threshold curves, it can be seen that the threshold of the BEC does not \ndepend on the external field, whereas the inset shows that t he magnon lifetime is decreased for \nhigher fields, as expected. The measured lifetime is 21 ns and is small compared to the lifetime of \nmagnons in YIG films of µm thicknesses, used in previous experiments\n12,14,15 ,36,37. It is known, \nthat for YIG films with thicknesses in the nm -range, the spin -wave damping is larger, and \nconsequently, the lifetime is smaller. Additionally, due to the spin pumping effect to the Pt layer \nand due to distortions induced in YIG sample by the structuring process the effective mag netic \ndamping is enhanced. \nFurthermore, the fact that there is no dependence of the phenomenon on the direction of the \nexternal field was confirmed by changing the direction of the applied field. Extended Data Fig. 7 \nshows the corresponding time resolved BLS spectra in the case that the external field is applied perpendicular ( Extended Data Fig. 7a, B||y) or parallel ( Extended Data Fig. 7c , B||x) to the long axis \nof the strip for similar pulse durations of τ\nP = 120 ns and τ P = 150 ns. Further, the integrated intensity \nof the fundamental mode is shown for both cases in Extended Data Fig. 7b (B||y) and 7d (B||x). \n \nExtended Data Fig. 6 | Dependency of the \nthreshold on the external field. Threshold \ncurves for different external fields in the range from 110 mT to 270 mT. The field is applied in-plane along the short axis of a 500 nm wide \nstrip. The inset shows exemplary time traces for a supercritical voltage of U = 1.5V and for \nthree different fields. The threshold is found to \nbe approximately constant over the measured \nfield range, whereas the magnon lifetime \ndecreases for higher values of the applied \nmagnetic field . 26 \n \nThe frequencies of the fundamental modes are slightly la rger if the magnetic field is applied \nparallel to the long axis of the strip. This is associated with the demagnetization fields that result \nin a smaller internal field when the strip is magnetized along its short axis. In addition, the \ninhomogeneity of the demagnetization fields leads to the formation of an edge mode with a frequency below the fundamental mode of the strip. The edge mode is visible in the experiment \n(see. Fig. Extended Data Fig. 7a ) as well as in the simulations (see Extended Data Fig. 2b). The \nmagnon BEC is observed in both cases at the bottom of the band of the waveguide modes, i.e., the \nlowest frequency of the fundamental mode. Even though the edge mode is the total energy \nminimum at room temperature equilibrium, it cannot be separated in the experiment from the \nfundamental mode during the BEC since, in contrast to the other modes, its frequency is not \nsignificantly decreased during the pulse. The reason for this is that the frequency of the edge mode \nstrongly depends on the strength of t he demagnetization fields. These fields are reduced as well \nwhen the saturation magnetization is decreased due to Joule heating. This behaviour was \nconfirmed by additional MuMax3 simulations (not shown). The BEC for both field directions \nexcludes a contrib ution from the Spin Hall Effect to the observed condensation, since it would \nonly lead to an efficient injection of spin current when the magnetization is perpendicular to the direction of the electric current\n26,28,29. Moreover, since the phenomenon does neither depend on \nExtended Data Fig. 7 | BEC by rapid cooling for different geometries of the external field. \na, BLS spectrum as a function of time. The BLS signal (color -coded, log scale) is proportional \nto the density of magnons. FM indicates the fundamental mode, EM – the edge mode, and 1st M \n– the first thickness mode. The vertical dashed lines indicate the star t and the end of the pulse \n(τP = 150 ns , U = 0.9 V). The external field was parallel to the short axis of the strip ( B||y). \nb, BLS spectrum as a function of time for the case when the external magnetic field is parallel \nto short axis of the strip ( B||x), (τP = 120 ns , U = 0.9 V). c, d, Normalized magnon intensity \nintegrated from 4.95 GHz to 8.1 GHz as a function of time for the cases in a and b. The insets \nshow the sample and measurement geometry. \n27 \n the direction nor on the value of the applied magnetic field (and thus also not on the frequency of \nthe BEC), we can conclude that the Oersted fields generated by the eclectic current in the Pt \noverla yer do not play any sizable role in the experiments. \n \nDependence of the BEC threshold on the current pulse duration. The threshold voltages are \ndetermined experimentally for different pulse durations 18 ns ≤ τP ≤ 175 ns. For each pulse duration, \nBLS measurements were performed for different voltages. The maximum of the BLS intensity \nintegrated over the frequency range of the fundamental mode was extracted and plotted as a function the applied voltage for each pulse dur ation. This yields graphs similar to the depiction in \nFig. 3 b in the main manuscript. The corresponding threshold voltage is determined by fitting the \nslope linearly and calculating the intercept with the averaged subcritical (thermal) BLS intensity. The extracted threshold voltages are shown in Extended Data Fig. 8. \nAs can be seen, the threshold voltage increases for the shortest pulse durations, whereas it \nsaturates for the longest ones. This is due to the finite timescale of the Joule heating, which is in agreement with our model. The threshold temperature (i.e., the critical magnon density) should be independent of the applied voltage. As expected, the COMSOL simulations yielded similar values for the threshold temperature for all τ\nP > 70ns (blue points in Extended Data Fig. 8). \n Extended Data Fig. 8 | Dependency of the \nthreshold on the pulse duration. Threshold \nvoltage as a function of the pulse duration \n(black squares) and corresponding simulated \ntemperatures at the end of the pulses (blue \nsquares). \n \nThe slight increase in the threshold temperature for the shorter pulses is likely related to \nincomplete thermalization of low energy magnons for such short times. Note, that in our \nexperiment the spin -lattice relaxation time for the low energy magnons is about 21 ± 4 ns. The \nsolid line in Extended Data Fig. 9 shows the calculated time evolution of the chemical potential for the current pulse of 20 ns, which heats the sample to the maximal temperature of 4 40 K. In \nsuch a case, the chemical potential induced by the rapid cooling might not reach the bottom of the \nmagnon spectrum hf\nmin and no BEC occurs. \nNevertheless, as it is supported by the experimental findings shown in Extended Data \nFig. 8, the incomplete equilibration of magnon and phonon systems before t he beginning of the \nfast cooling process does not impede the BEC phenomenon. The critical magnon density can still be achieved at a higher phonon temperature by application of a stronger heating current. The dashed line in Extended Data Fig. 9 show s that the BEC takes place for the 20 ns long pulse if the \nmaximal sample temperature is increased to 490 K. \nIt is worth mentioning, that the “rapid heating” of the magnons' environment just after the \napplication of the current pulse should act as an inverse of the rapid cooling process and, thus, lead to the decrease in the chemical potential on a time interval comparable with the magnon \n28 \n thermalization time. This effect is clearly visible in the Extended Data Fig. 9 but is not observe d \nin our experiments. The reason could be the following. In general, the spin Seebeck effect leads to \nan increase of magnon chemical potential and the magnon dynamics during the current pulse is \ndetermined by the interplay of two counter -acting processes: “rapid heating” and SSE. Proper \ntheoretical description of the magnon dynamics in this interval would require development of a \ncompletely new theory that would treat these effects on equal footing and is outside the scope of \nthe present work. However, simple estimation of the SSE -induced effects just after the current \npulse is switched on shows, that temperature gradient might lead to magnon chemical potential \nµ ~ 5.25 GHz and should rapidly decrease afterwards due to the increase in the YIG temperature27. \nThus, it looks reasonable that the “rapid heating” decrease in the chemical potential is compensated or even overcompensated by the pulsed -like contribution of the spin Seebeck effect. \n \nExtended Data Fig. 9 | Time evolution of \nthe chemical potential for a short heating \npulse. The pulse duration τ\nP = 20 ns. The \ninitial decrease in the chemical potential is caused by the rapid heating of the sample. The incomplete thermalization of the magnon sub-system (µ < 0) at the end of the pulse \nincreases the threshold temperature of the BEC formation from 440\n K to 490 K. \n \n \nDependence of the BEC threshold on the fall time of the current pulses. The influence of the cooling rate on the magnon BEC is investigated by applying τ\nP = 50 ns long current pulses at a \n1 µm wide YIG/Pt structure. The fall times of the applied pulses varied in the range of \n1 ns ≤ τFall ≤ 100 ns. Extended Data Figure 10 shows the integrated BLS intensity as a function of \nthe fall time. The strong increase for shorter fall times is clearly visible (above -threshold regime), \nwhereas for fall times larger than the spin wave lifetime the accumulation at the bottom of the \nspectrum vanishes indicating that the BEC threshold is not reached. This supports the conclusion \nfrom the main manuscript, that a rapid cooling of the phonon system is crucial t o achieve the \nmagnon BEC and the required cooling rate is mainly determined by the lifetime of the low energy \nmagnons. \n29 \n Extended Data Fig. 10 | Dependence of the \nmagnon BEC intensity on the fall time of the \napplied pulses. Integrated BLS intensity of the \nfundamental mode as a function of the fall time of the applied current pulses. Only for fall times shorter than the lifetime of magnons t\nm = 21 ± \n4 ns a clear accumulation at the bottom of the spectrum is observed. \n \nInfluence of thermally induced spin currents on the magnon BEC by rapid cooling. An \nadditional experiment using an Al (7 nm)/Au (5 nm) heating layer on top of a 2 µm wide and 34 nm \nthick YIG waveguide was performed to show that the main mechanism underlaying the BEC is the Rapid Cooling. Owing a small spin- orbit interaction in Al , a negligible spin current can be \nthermally injected into YIG from the heating layer by the spin Seebeck effect. The length of the heating area of 4 µm was the same as in the main experim ent. The magnetic field of 188 mT was \napplied in- plane perpendicular to the waveguide long axis. Extended Data Figure 11 shows the \naccumulation of magnons after the 50 ns long current pulse applied. A voltage of 3.1 V is applied during this time, whereas the resistance of the structure under investigation is around 2 kOhm. The clear peak in the integrated BLS intensity (see panel b) is observed at the bottom frequency of the fundamental mode after the current pulse is switched off. This experiment directly confirms that \nthe rapid cooling rather than spin- orbit mediated Spin Seebeck effect\n17 is responsible for the \nreported BEC of magnons. \n Extended Data Fig. 11| Rapid \ncooling BEC generated in an Au/Al/YIG structure. a, Time \nresolved BLS spectrum for the case when a 50 ns long pulse is applied b, \nIntegrated BLS intensity over the \nfrequency range shown in a.\n \n \n \n \n" }, { "title": "1509.04336v1.Spin_Transport_in_Antiferromagnetic_Insulators_Mediated_by_Magnetic_Correlations.pdf", "content": "1 \n Spin Transport in Antiferromagnetic Insulators Mediated by Magnetic \nCorrelations \nHailong Wang†, Chunhui Du†, P. Chris Hammel* and Fengyuan Yang* \nDepartment of Physics, The Ohio State University, Columbus, OH, 43210, USA \n†These authors made equal contributions to this work \n*Emails: hammel@physics.osu.edu; fyyang@physics.osu.edu \n \nWe report a systematic study of spin transport in antiferromagnetic (AF) insulators having a wide \nrange of ordering temperatures . Spin current is dynamically injected from Y3Fe5O12 (YIG) into \nvarious AF insulators in Pt/insulator/ YIG trilayers. Robust , long -distance spin transport in the \nAF insulators is observed , which shows strong correlation with the AF ordering temperatures . \nWe find a striking linear relationship between the spin decay length in the AFs and the damping \nenhancement in YIG, suggesting the critical role of magnetic correlations in the AF insulators as \nwell as at the AF/YIG interface s for spin transport in magnetic insul ators . \n \nPACS: 75.50.Ee , 75.70.Cn , 76.50.+g, 81.15.Cd \n 2 \n Spin current s carried by mobile charges in metallic and semiconducting ferromagnetic (FM) and \nnonmagnetic (NM) materials ha ve been the central focus of spintronic s for the past two decades \n[1]. However, s pin transport in AF insulators has been essentially unexplored due to the \ndifficulty in generating magnetic excitations in these insulators . Ferromagnetic resonance \n(FMR) and thermally driven spin pumping [2-15] have attracted intense interest in magnon -\nmediated spin current s, which can propagate in both conducting and insulating FM s and AF s. \nWe recently reported observation of high ly efficien t spin transport in AF insulator NiO with long \nspin decay length [16]. In this letter, we probe the mechanism s responsible for spin transport in \nAF insulators by investigati ng several series of Pt/insulator/YIG trilayers ; this study is enabled \nby the large inverse spin Hall effect (ISHE) signals in our YIG -based structures [9-15]. \nEpitaxial YIG (epi-YIG) films are grown on (111) -oriented Gd 3Ga5O12 (GGG) substrates \nby sputtering [9-17]. X-ray diffraction and a tomic force microscopy measurements reveal high \ncrystalline quality and smooth surfaces of the YIG films [18 ]. Figure 1(a) shows a n in-plane \nmagnetic hysteresis loop for a 20 -nm YIG film which exhibits a small coercivity ( Hc) of 0.40 Oe \nand sharp magnetic reversal, indicating high magnetic uniformity. Figure 1 (b) presents a FMR \nderivative absorption spectrum for a 20 -nm YIG film taken in a cavity at radio -frequency (rf) f = \n9.65 GHz and microwave power Prf = 0.2 mW with an in -plane magnetic field, which gives a \nnarrow linewidth ( H) of 7.7 Oe. All of these measurements are carried out at room \ntemperature. \nIn order to probe spin transport in insulators of various magnetic structures , we select six \nmaterials, including: 1) amorphous SrTiO 3, a diamagnet , 2) epitaxial Gd 3Ga5O12, a paramagnet \nwith a large magnetic susceptibility ( ), and four antiferromagnets , 3) Cr2O3 [19], 4) amorphous \nYIG ( a-YIG) [20], 5) amorphous NiFe 2O4 (a-NFO) [21], and 6) NiO [19]. All insulator layers 3 \n are deposited by off -axis sputtering. Lattice matched, s train-free Gd3Ga5O12 films are epitaxially \ngrown on YIG at high temperature; the remaining five insulators are grown at room temperature \nto avoid strain ing the epi -YIG films which can significantly alter the magnetic resonance in YIG . \nElectrical transport measurements confirm the highly insulating nature of all these films. Figur es \n1(a) and 1(b) indicate that the a-YIG film has negligible magnetization and FMR absorption (a-\nNFO films exhibit similar behavior ). Thus, the six insulators include a diamagnet , a paramagnet , \nand four AFs with a wide range of ordering temperatures , allowing us to probe magnetic \nexcitation s and spin propagation in insulators both above and below the AF ordering \ntemperatures , hence illuminating the roles of both static and dynamic magnetic correlations . \nBulk Cr 2O3 and NiO have Néel temperature s TN = 318 and 525 K, respectively [19]. \nBoth YIG and NiFe 2O4 are ferrimagnets when in crystalline form; however, amorphous YIG and \nNiFe 2O4 become AFs due to the lack of crystalline ordering required for ferrimagnetism [20, 21]. \nThe temperature ( T) dependence of exchange bias in FM/AF bilayers allows direct measure ment \nof the blocking temperature, Tb, of the AFs. Our YIG(20 nm)/NiO(20 nm) bilayer exhibits a \nclear exchange bias field, HE = 13.5 Oe and a n enhanced coercivity Hc = 19.2 Oe ( Hc = 0.40 O e \nfor a single YIG film) , demonstrat ing exchange coupling between YIG and NiO [18]. However, \nthe very large paramagnetic background of GGG substrates prohibit s the measurement of \nexchange bias at low temperatures needed for Cr 2O3, a-YIG, and a-NFO . To determine Tb for \neach AF stud ied here , we use Ni81Fe19 (Py) as the FM and measure exchange bias in Py(5 \nnm)/AF(20 nm) bilayers grown on Si . Figure 2 (a) shows the hysteresis loops of four Py/AF \nbilayers at T = 5 K after field cooling from above Tb. All four samples exhibit substantial \nexchange bias: HE = 646, 1403, 568, and 97 Oe for Py/NiO, Py/ a-NFO, Py/ a-YIG, and Py/Cr 2O3, \nrespectively. Figures 2 (b) and 2 (c) show the temperature dependenc ies of HE for the four 4 \n bilayers, from which we determine Tb = 20, 45, 70, and 330 K for Cr 2O3, a-YIG, a-NFO, and \nNiO, respectively. \nSpin currents in insulators propagat e via precessional spin wave modes , e.g., magnons in \nordered FMs and AFs . However, it is challenging to excite AF magnons which, for example, \nrequires THz frequency in NiO [22]. Furthermore, the AF ordering temperatures in thin films \ndecrease at lower thicknesses and conventional magnons cannot be sustained above the ordering \ntemperatures . Here, we leverage the established technique of FMR spi n pumping in YIG -based \nstructures to excite the AF insulators via exchange coupling to the precessing YIG magnetization \nand to probe spin transfer in these insulators . For each of the six insulators, we grow a series of \nPt(5 nm) /insulator( t)/epi-YIG(20 nm) trilayers with various insulator thickness es t on YIG films \ncut from the same YIG/GGG wafer to ensure consistency of the YIG quality. Since Pt is the \nonly conduct or in the trilayers , the voltage signals detected are exclusively from the ISHE \n(VISHE), which proportionally reflects the spin current s pumped into Pt across the insulat ors. \nRoom -temperature s pin pumping measurements [18] are conducted on all trilayers (~1 \nmm wide and ~5 mm long) in an FMR cavity at f = 9.65 GHz and Prf = 200 mW in an in-plane \nDC field ( H), as illustrated in Fig. 3(a). The mV-level ISHE voltage s provide a dynamic range \nof more than three orders of magnitude for detecting the decay of spin current across the \ninsulat ors. The rates at which VISHE decays with increasing insulator thickness (t) differ \ndramatically among the six spacers. A 0.5-nm SrTiO 3 [18] already suppresses VISHE by a factor \nof 17 from the corresponding Pt/YIG bilayer [10]. As we change the insulator from SrTiO 3 \nGd3Ga5O12 Cr2O3 a-YIG a-NFO NiO, the spin current s exhibit substantially \nincreasing propagat ion lengths . Figure 3(b) summarizes the t-dependencies of the normalized \npeak VISHE at YIG resonance , Hres, for all six series . From the linear relationship in the semi -log 5 \n plots, we extract the spin decay length s 𝜆 in the insulat ors by fitting to 𝑉ISHE(𝑡)/𝑉ISHE(0)=\n𝑒−𝑡/𝜆, which gives 𝜆 = 0.18 , 0.69 , 1.6, 3.9, 6. 3, and 9.8 nm for SrTiO 3, Gd 3Ga5O12, Cr 2O3, a-\nYIG, a-NFO, and NiO, respectively (Table I) . More surprisingly, Fig. 3(c) shows that VISHE \ninitially increases by a factor of 2.1 and 1.6 when a 1 - or 2-nm NiO and a-NFO, respectively, is \ninserted between YIG and Pt (the point for t = 0 is excluded from the exponential fit for NiO and \nNFO) . This dramatic variation in the spin current propagation length -scale most likely arise s \nfrom different magnetic characteristics of the six insulators. \nFor dynamically generated spin current to transmit across insulating spacers beyond the \ntunneling range (~1 nm), magn etic excit ations in the insulat ors are expected to play a major role. \nExcept SrTiO 3, all other five insulators have strong magnetic character , including paramagnetic \nGd3Ga5O12 and four AFs with various ordering temperatures . For the same AF material, Tb can \nvary significantly depending on the film thickness [ 23]. Among the four AFs, NiO is the most \nrobust AF with Tb = 330 K for our 20 -nm NiO film, while for very thin NiO layers (<5 nm) , Tb is \nexpected to be well below 300 K [ 24]. For a-NFO , a-YIG and Cr2O3, the AF ordering \ntemperatures are well below room temperature (Table I) . It is interesting to note that Gd3Ga5O12 \nalso exhibits magnetic order at very low temperatures [ 25]. Thus, magnetic correlation s amongst \nthermally fluctuating AF moment s are critically important for the observed spin transport in \ninsulators . \nThese results suggest that , at resonance, the precessing YIG magnetization generates \nmagnetic excitations in the adjacent insulat or (either with static AF ordering or fluctuati ng \ncorrelated moments ) via interfacial exchange coupling , which in turn e nhances magnetic \ndamping of the YIG. We measure the Gilbert damping constant [26] from the frequency \ndependencies of FMR linewidth s H for six insulator(20 nm)/YIG(20 nm) bilayers and a single 6 \n epi-YIG film using a microstrip transmission line . Figure 4(a) show s the linear frequency \ndependence of Δ𝐻 given by Δ𝐻=Δ𝐻0+4𝜋𝛼𝑓\n√3𝛾 for the seven samples , where Δ𝐻0 is the y-\nintercept and 𝛾 is the gyromagnetic ratio . From t he slope s of least -squares fits , we obtain = 8.1 \n 10-4, 8.6 10-4, 11 10-4, 12 10-4, 14 10-4, 17 10-4, 26 10-4, and 3 6 10-4 for the bare \nYIG, SrTiO 3/YIG, Gd3Ga5O12/YIG, Cr2O3/YIG, a-YIG/YIG, a-NFO/YIG, NiO/ YIG, and \nPt/YIG, respectively (Table I) . The diamagnetic SrTiO 3 does not enhance the damping of YIG \nwithin experimental uncertainty while its spin current decay s over an atomic length scale (𝜆 = \n0.18 nm) due to quantum tunneling [10]. The large paramagnetic moment s in Gd3Ga5O12 can \nabsorb angular momentum via exchange coupling to YIG and conduct spin current , resulting in a \nlonger 𝜆 = 0.69 nm. The four AFs show much longer spin d ecay lengths together with enhanced \ndamping of YIG due to strong magnetic correlations [23]. NiO more than triples the damping of \nYIG and its spin d ecay length is almost 10 nm , while clear spin current is detected over a NiO \nthickness of 100 nm . \nThe AF resonance frequency of NiO is about 1 THz [22] which is much higher than the \n9.65 GHz used in our FMR excitation of YIG. Despite the difference in the dispersion relation s \nof YIG and the AFs, our result clearly demonstrate s highly efficient spin transport across the \nAFs. Considering that strong AF spin correlation s have been o bserved well above TN for NiO \n[27], we believe the excitation s responsible for spin transport in AFs must be m agnon s in ordered \nAFs and AF fluctuations in insulators with low blocking temperatures. In either case, the \nstrongly correlated AF spins are excited via exchange coupling to the precessing YIG \nmagnetization (either the net or staggered ferrimagnetic moments) at the AF/YIG interface and \ntransfer the spin current across the insulator to the interface with Pt, where it is converted to a 7 \n spin-polarized electr on current in Pt. This is analogous to the predicted magnon current s in FM \ninsulators [28]. \nThe independent ly measure d spin d ecay length 𝜆 and damping enhancement ∆𝛼 both \nincrease monotonically following SrTiO 3 Gd3Ga5O12 Cr2O3 a-YIG a-NFO NiO \n(Table I) . Figure 4(b) further show s that 𝜆 and ∆𝛼 exhibit a nearly perfect linear relationship for \nall insulators excluding SrTiO 3. The excellent linear relationship between 𝜆 and ∆𝛼 of five \nsignificantly different insulat ors indicates that spin transfer across the YIG/AF interfaces \n(measured by ∆𝛼) and spin propagation inside the AF insulators (characterized by 𝜆) are tightly \nrelated. T he exchange coupling between YIG magnetization and AF spins at the interfaces and \nthe exchange interaction between adjacent AF spins within the AFs play a dominant role in spin \ntransport in insulators. \nLastly , the strength of magnetic correlation s depend s on the AF ordering temperature s. \nOur results shown in Figs. 2 to 4 indicate that the correlation strength increases following the \norder SrTiO 3 (diamagnet) Gd3Ga5O12 (paramagnet ) Cr2O3 (AF, Tb = 20 K) a-YIG (AF, \nTb = 45 K) a-NFO (AF, Tb = 70 K) NiO (AF, Tb = 330 K). As magnetic correlation \nincreases, exchange interaction becomes stro nger, which, 1) facilitates the propagation of spin \ncurrent s carried by magnetic excitation s in the insulators, and 2) enhances the magnetic damping \nof the underlying YIG films . The surprising enhancement of ISHE signals for the trilayers with \n1- or 2-nm NiO and a-NFO [Fig. 3(c)] indicates that the Pt/NiO/YIG and Pt/a-NFO/YIG trilayer \nstructures are highly efficient in spin transfer , while the underlying mechanism remains to be \nunderstood . \nIn summary, we observ e clear spin current s in AF insulat ors mediated by AF magnetic \ncorrelation s be they static or fluctuating . This result brings a large family of insulators, in 8 \n particular, AF insulators, into the exploration of spintronic applications utilizing pure spin \ncurrents. \nThis work was primarily supported by the U.S. Department of Energy (DOE), Office of \nScience, Basic Energy Sciences, under Grants No. DEFG02 -03ER46054 (FMR and spin \npumping characterization) and No. DE -SC0001304 (sample synthesis and magnetic \ncharacterization). This work was supported in part by the Center for Emergent Materials, an \nNSF-funded MRSEC, under Grant No. DMR -1420451 (structural characterization). Partial \nsupport was provided by Lake Shore Cryogenics, Inc., and the NanoSystems Labor atory at the \nOhio State University . 9 \n Reference s: \n1. I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). \n2. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B. I. Halperin, Rev. Mod. Phys. 77, 1375 \n(2005). \n3. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe , K. Uchida, M. Mizuguchi, H. Umezawa, H. \nKawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). \n4. E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). \n5. O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader, and A. Hoffmann, Appl. Phys. Lett. 96, \n022502 (2010). \n6. A. Hoffmann , IEEE Trans. Magn. 49, 5172 (2013). \n7. B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y. -Y. Song, Y. Y. Sun, and M. \nZ. Wu, Phys. Rev. Lett . 107, 066604 (2011) . \n8. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. 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Phys. 99, 093909 (2006). \n27. T. Chatterji, G. J. McIntyre, and P. A. Lindgard , Phys. Rev. B 79, 172403 (2009). \n28. S.-L. Zhang and S. F. Zhang , Phys. Rev. B 86, 214424 (2012) . \n \n 11 \n Table I . Type of magnetism , blocking temperatures (for AFs only), and spin d ecay length (𝜆) \nfor the s ix insulator s as well as the Gilbert damping constant ( ) for the six insulator s (20-nm) on \nYIG. The parameters for a single epitaxial YIG film are also included for comparison . \n \n \n Layer Magnetism Tb (K) 𝜆 (nm) \nepi-YIG ferrimagnet (8.1 ± 0.6) 10-4 \nSrTiO 3 diamagnet 0.18 ± 0.01 (8.6 ± 1.0) 10-4 \nGd 3Ga5O12 paramagnet 0.69 ± 0.02 (11 ± 1) 10-4 \nCr2O3 antiferromagnet 20 1.6 ± 0.1 (12 ± 1) 10-4 \na-YIG antiferromagnet 45 3.9 ± 0.2 (14 ± 1) 10-4 \na-NFO antiferromagnet 70 6.3 ± 0.3 (17 ± 2) 10-4 \nNiO antiferromagnet 330 9.8 ± 0.8 (26 ± 3) 10-4 12 \n Figure captions: \nFigure 1. (a) Room temperature in -plane magnetic hysteresis loop s of a 20-nm epitaxial YIG \nfilm (blue) and a 20-nm amorphous YIG film (red) grown on GGG , where the paramagnetic \nbackground is from the GGG substrate. Inset: low -field hysteresis loop of the epitaxial YIG film \nshowing a coercivity of 0. 40 Oe. (b) FMR derivative absorption spectra of an epitaxial (blue) \nand an amorphous (red) 20-nm YIG film on GGG . \nFigure 2. (a) Magnetic hysteresis loops of four Py( 5 nm)/AF(20 nm) bilayers at 5 K after field \ncooling, all demonstrating clear exchange bias. Temperature dependence of HE for the four Py(5 \nnm)/AF(20 nm) bilayers in (b) linear and (c) log y-scale give the AF blocking temperature Tb = \n20, 45, 70, and 330 K for Cr 2O3, a-YIG, a-NFO, and NiO, respectively. \nFigure 3. (a) Schematic of the ISHE measurement on various Pt/Insulator/YIG structures. (b) \nSemi -log plots of VISHE as a function of the insulator thickness for the six series normalized to \nthe values for the corresponding Pt/YIG bilayers , where the straight lines are exponential fits to \neach series, from which the spin d ecay length s are determined . (c) Details of behavior shown \nin (b) for insulators below 10 nm. \nFigure 4. (a) Frequency dependencies of FMR linewidth s of a bare epitaxial YIG film, \nSrTiO 3(20 nm)/YIG, Gd3Ga5O12(20 nm) /YIG, Cr2O3(20 nm) /YIG, a-YIG/(20 nm) /YIG, a-\nNFO(20 nm)/YIG, and NiO(20 nm) /YIG bilayers. (b) Excellent linear correlation between spin \ndecay length 𝜆 and Gilbert damping enhancement ∆𝛼=𝛼Insulator /YIG−𝛼YIG. The line is a \nleast-squares linear fit to all data points excluding SrTiO 3. \n 13 \n \nFigure 1 \n \nroughness: 0.105 nm-1-0.500.51\n-40 -20 0 20 40 M (memu)\nH (Oe)(a)epi-YIG/GGG\na-YIG/GGG\n-4-2024\n2600 2700dIFMR / dH (a.u.)\n H (Oe)epi-YIG/GGG\na-YIG/GGG (b)-0.200.2\n-1 01(b)14 \n \n \nFigure 2 \n \n05001000 HE (Oe)Py/a-NFO\nPy/Cr2O3Py/a-YIGPy/NiO (b)-101\n-3 -2 -1 0 1 2M/Ms\nH (kOe)(a)\nT = 5 K\nPy/a-NFO\nPy/Cr2O3Py/a-YIGPy/NiO\n10-1100101102103\n0 100 200 300\nT (K)Py/a-NFO\nPy/Cr2O3 Py/a-YIGPy/NiO (c)15 \n \nFigure 3 \n \n10-1100\n0 5 10\nInsulator thickness t (nm)SrTiO3\nGGGCr2O3a-YIGa-NFONiOVISHE(t) / VISHE(0) (c)\n10-310-210-1100\n0 10 20 30 40 50VISHE(t) / VISHE(0)\nInsulator thickness t (nm)(b)\nSrTiO3: = 0.18 nmGd3Ga5O12: = 0.69 nm Cr2O3: = 1.6 nma-YIG: = 3.9 nma-NFO: = 6.3 nmNiO: = 9.8 nm\n(a)16 \n \nFigure 4 \n \n0102030\n0 5 10 15 20 H (Oe)\nf (GHz)NiO/YIG\na-NFO/YIG\na-YIG/YIG\nCr2O3/YIG\nGd3Ga5O12/YIG\nsingle YIG filmSrTiO3/YIG(a)\n0510\n0 2 4 6 8 10\n (nm) (10-4 )\nGd3Ga5O12\nCr2O3\na-YIG\na-NFO\nNiO\nSrTiO3(b)" }, { "title": "1508.06130v1.Non_local_magnetoresistance_in_YIG_Pt_nanostructures.pdf", "content": "Non-local magnetoresistance in YIG/Pt nanostructures\nSebastian T. B. Goennenwein,1, 2, 3, \u0003Richard Schlitz,1, 3Matthias\nPernpeintner,1, 2, 3Matthias Althammer,1Rudolf Gross,1, 2, 3and Hans Huebl1, 2, 3\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany\n2Nanosystems Initiative Munich (NIM), Schellingstra\u0019e 4, M unchen, Germany\n3Physik-Department, Technische Universit at M unchen, Garching, Germany\n(Dated: September 3, 2018)\nWe study the local and non-local magnetoresistance of thin Pt strips deposited onto yttrium iron\ngarnet. The local magnetoresistive response, inferred from the voltage drop measured along one\ngiven Pt strip upon current-biasing it, shows the characteristic magnetization orientation dependence\nof the spin Hall magnetoresistance. We simultaneously also record the non-local voltage appearing\nalong a second, electrically isolated, Pt strip, separated from the current carrying one by a gap of a\nfew 100 nm. The corresponding non-local magnetoresistance exhibits the symmetry expected for a\nmagnon spin accumulation-driven process, con\frming the results recently put forward by Cornelissen\net al.[1]. Our magnetotransport data, taken at a series of di\u000berent temperatures as a function of\nmagnetic \feld orientation, rotating the externally applied \feld in three mutually orthogonal planes,\nshow that the mechanisms behind the spin Hall and the non-local magnetoresistance are qualitatively\ndi\u000berent. In particular, the non-local magnetoresistance vanishes at liquid Helium temperatures,\nwhile the spin Hall magnetoresistance prevails.\nMagneto-resistive phenomena are powerful probes for\nthe magnetic properties. The anisotropic magnetoresis-\ntance in ferromagnetic metals [2], or the giant magne-\ntoresistance [3] and the tunneling magnetoresistance [4]\nobserved in thin \flm heterostructures based on mag-\nnetic metals are widely used in sensing and data stor-\nage applications [5]. Heterostructures consisting of an\ninsulating magnetic layer (such as yttrium iron garnet\nY3Fe5O12(YIG)) and a heavy metal (such as platinum\n(Pt)) also exhibit a magnetoresistance [6{9]. This so-\ncalled spin Hall magnetoresistance (SMR) is due to spin\ntorque transfer across the magnetic insulator/metal in-\nterface [10]. Owing to the spin Hall e\u000bect [11, 12], a spin\naccumulation \u001barises in the metal, at the interface to the\nmagnet (cf. Fig. 1). Given that \u001bis not collinear with the\nmagnetization Min the magnet, the spin accumulation\ncan exert a torque proportional to M\u0002(M\u0002\u001b) onM. In\nother words, a \fnite spin current \row across the interface\nis possible if Mand\u001benclose a \fnite angle. Since the\nspin current \row across the interface represents a dissipa-\ntion channel for the charge transport in the metal layer,\nits resistance therefore will change with the magnetiza-\ntion orientation as \u0001 R/M\u0002(M\u0002\u001b) [6{9]. In order\nto experimentally resolve this SMR \fngerprint, magneto-\nresistance measurements as a function of magnetization\norientation in at least three di\u000berent planes are manda-\ntory [7].\nRecently, Cornelissen et al. [1] discovered a non-local\nmagneto-resistance e\u000bect in YIG/Pt heterostructures\nand attributed it to magnon accumulation and transport.\nWe will refer to this e\u000bect as magnon-mediated magneto-\nresistance (MMR) in the following. The MMR is ob-\nserved in two parallel Pt strips separated by a distance\nddeposited onto YIG, as sketched in Fig. 1. Driving a\ncharge current through the left Pt strip will generate a\nYIG\n(magnetic insulator)Pt(metal)JCJS JCJSσd w wFIG. 1. Sketch of the magnon-mediated magnetoresistance\n(MMR) following Cornelissen et al. [1]. Driving a charge cur-\nrentJcthrough the left strip of platinum (Pt) results in an or-\nthogonal spin current with propagation direction Jsand spin\npolarization \u001b. The spin accumulation in the metal induces\na magnon accumulation (wiggly red arrows) in the adjacent\nmagnetic insulator yttrium iron garnet (YIG). This magnon\naccumulation decays with increasing distance to the current-\ncarrying Pt injector strip (shaded region). If a second Pt\nstrip, electrically isolated from the \frst, is within the range of\nmagnon accumulation, a spin current will \row back from the\nmagnetic insulator into the second Pt strip and give rise to\nan inverse spin Hall charge current there. The two Pt strips\nof widthware separated by an edge-to-edge distance d.\nspin accumulation \u001bin Pt at the interfaces. As shown in\nFig. 1,\u001bis perpendicular to the direction of charge cur-\nrent \row Jcand orthogonal to the spin current \row Js\nacross the interface. This spin accumulation in particular\nalso induces a magnon (spin) accumulation in YIG [1] {\nan e\u000bect which usually is assumed small and ignored in\nthe treatment of SMR [10]. The non-equilibrium magnon\naccumulation di\u000buses out into the magnetic insulator, as\nindicated by the shading in Fig. 1. Given that the sec-\nond Pt electrode is close enough such that the di\u000busing\nmagnons can reach it before decaying, the magnon accu-arXiv:1508.06130v1 [cond-mat.mtrl-sci] 25 Aug 20152\nγH\njtn\nHjtn\nα\nβH\njtn(b)\nip(a)\n(c)\noopj\n(d)ooptI+\n-+\n-Vloc+\n-Vnl\n10µm\nFIG. 2. (a) Optical micrograph of a typical YIG/Pt nanos-\ntructure. The two bright thin vertical lines in the center of the\n\fgure are the two Pt strips under investigation, the YIG \flm\nbeneath appears black. A current source attached to the left\nPt strip supplies a constant current I. The voltage drop Vloc\nalong the same Pt strip, as well as the non-local voltage drop\nVnlalong the second Pt strip, are simultaneously recorded.\nPanels (b), (c), and (d) show the di\u000berent magnetic \feld ro-\ntation planes and the corresponding magnetic \feld orientation\nangles\u000b,\fand\r, respectively.\nmulation will drive a spin current back into the second\nPt electrode. In turn, this spin current then generates\nan inverse spin Hall charge current in the second Pt elec-\ntrode. A large non-local charge current is expected for\nMjj\u001b, since the magnons (the spin angular momenta) be-\nneath the \frst Pt strip then can di\u000buse across the gap\nto the second Pt strip. For M?\u001b, in contrast, the non-\nlocal signal should be signi\fcantly reduced, since now\nspin torque transfer suppresses the magnon accumula-\ntion and/or the magnon propagation. This picture of a\nnon-local magnon-based magnetoresistance, put forward\nby Cornelissen et al. in Ref. [1], to date only has been\ntested against non-local inverse spin Hall voltage data\ntaken as a function of magnetic \feld orientation for the\nmagnetic \feld in the plane of the YIG \flm. Neither a di-\nrect comparison of the non-local magnetoresistance with\nthe SMR, nor a study of the evolution of the non-local\nvoltage as a function of out-of-plane magnetization orien-\ntation, have been put forward. Note also that the MMR\nis di\u000berent from the non-local electrical detection of spin\npumping [13, 14] or magnetoresistance experiments in\nmetallic spin valves [15, 16], since the YIG (the mag-\nnetically ordered material) only passively acts as a 'spin\ntransport' medium, contacted with conventional metallic\nnano-electrodes.\nIn this letter, we systematically compare the\nmagnetization-orientation dependent evolution of the\n(non-local) MMR and the (local) SMR in YIG/Pt nanos-tructures. We have simultaneously measured the MMR\nand SMR as sketched in Fig. 2, rotating the externally\napplied magnetic \feld of \fxed magnitude in three mu-\ntually orthogonal planes. Our data taken close to room\ntemperature corroborate the picture that the MMR is\nmediated by magnon di\u000busion, and reveal a qualitatively\ndi\u000berent evolution of SMR and MMR as a function of\ntemperature.\nThe YIG/Pt bilayers investigated were obtained start-\ning from a commercially available, 3 \u0016m thick YIG \flm\ngrown onto GGG via liquid phase epitaxy. The as-\npurchased YIG \flms were cleaned in a so-called Piranha\netch solution (3 volumes H 2SO4mixed with 1 volume\nH2O2) for 120 seconds and annealed in 50 \u0016bar oxygen\nfor 40 minutes at 500\u000eC. Without breaking the vacuum,\nthe samples were then transferred to an electron beam\nevaporation chamber, where we deposited a 9 :6 nm thick\nPt \flm. After removing the sample from the vacuum\nchamber, the Pt strips were de\fned using a combina-\ntion of electron beam lithography and Argon ion beam\nmilling. The Pt strips studied here are 100 \u0016m long and\nhave a lateral width of w= 1\u0016m. We focus on a de-\nvice with a strip separation (edge to edge, see Fig. 2) of\nd= 200 nm in the following, but have also studied devices\nwithd= 500 nm and d= 1000 nm. For the magneto-\ntransport experiments, the YIG/Pt nanostructures were\nwire-bonded to a chip carrier and inserted into the vari-\nable temperature insert of a superconducting 3D vector\nmagnet cryostat, allowing to rotate the externally ap-\nplied magnetic \feld \u00160H\u00142 T in any desired plane with\nrespect to the sample. The magnetotransport data were\ntaken by current-biasing one Pt strip with I= 100\u0016A\nusing a Keithley 2400 sourcemeter, while simultaneously\nrecording the local voltage drop Vloc(along the strip car-\nrying the current) as well as the non-local voltage Vnl\nappearing along the second, nearby Pt strip using Keith-\nley 2182 nanovoltmeters as sketched in Fig. 2(a). To\nenhance sensitivity, we use the current reversal (delta\nmode) method [17]. We here discuss transport data taken\nas a function of magnetic \feld orientation, for \fxed \feld\nmagnitude H. To ensure full saturation of the YIG mag-\nnetization along the externally applied magnetic \feld, we\ntook all data using the maximum available magnetic \feld\nj\u00160Hj= 2 T. We rotated the \feld in three mutually or-\nthogonal planes, as sketched in Fig. 2(b),(c),(d). The\nrotation of Haround the direction nnormal to the \flm\nplane, such that the magnetic \feld always resides within\nthe \flm plane, is referred to as ip (Fig. 2(b)). In the oopj\nrotation depicted in Fig. 2(c), His rotated around the\ndirection j(along which the charge current \rows), while\nin the oopt rotation depicted in Fig. 2(d), His rotated\naround the direction t, which is orthogonal to jandn.\nFigure 3 exemplarily shows magneto-transport data\ntaken on the sample with d= 200 nm, with the vari-\nable temperature insert thermalized to T= 300 K.\nSince all data discussed in the following were taken with3\n370.7370.8\n-90 0 90 180 270-200-1000Vloc,0\n(b)(a) oopt\noopjVloc(mV)\nip∆VSMR\n300K,2T\n∆VnlVnl(nV)\nα,β,γ(deg)oopt\noopjip\nFIG. 3. The local voltage Vloc(panel (a)) and the non-local\nvoltageVnl(panel (b)), recorded in two Pt strips separated by\nd= 200 nm as a function of the orientation \u000b(ip),\f(oopj),\r\n(oopt) of the externally applied magnetic \feld H(see Fig. 2).\nThe data were taken at T= 300 K and \u00160H= 2 T.Vlocis\nessentially constant in the oopt rotation plane (blue triangles),\nand varies in a sin2-type fashion in the ip (black rectangles)\nand oopj (red circles) rotation planes, respectively. The sin2\nmodulation with amplitude \u0001 VSMR<0 (gray vertical arrow)\nis superimposed on a constant voltage of magnitude Vloc;0\n(horizontal dashed arrow). Vnlis qualitatively similar to Vloc.\nHowever,Vnlalways is negative, and does not show a constant\no\u000bset voltage but only a sin2-type modulation with amplitude\n\u0001Vnl.\nj\u00160Hj= 2 T which exceeds the anisotropy \felds in YIG\nby at least one order of magnitude, we assume MjjH\nand use the magnetic \feld orientations \u000b,\fand\r(see\nFig. 2(b),(c),(d)) synonymously for MandH. The lo-\ncal voltage Vlocdepicted in Fig. 3(a) exhibits the depen-\ndence on magnetization orientation characteristic of the\nSMR. Upon rotating the external magnetic \feld in the\nplane of the YIG \flm (ip, see Fig. 2(b)), or in the plane\nperpendicular to j(oopj, Fig. 2(c)), a sin2-like modu-\nlation ofVlocwith amplitude \u0001 VSMR<0 on top of a\nconstant level Vloc;0is observed. Vloc;0hereby is the\nvoltage level observed when the YIG magnetization is\nalong the tdirection (e.g., \u000b= 90\u000ein ip or\f= 90\u000e\nin oopj). The magnitude of the SMR j\u0001VSMRj=Vloc;0\u0019\n4:5\u000210\u00004agrees reasonably well with the SMR ampli-\ntude\u00196\u000210\u00004observed in YIG/Pt heterostructures in\nwhich the\u001910 nm thick Pt \flms were deposited in-situ,\n0501001502002503000-50-100-150-200-2500-50-100-150∆VSMR(µV)\n160200240280320360400(a)\n∆VSMR\nVloc,0(mV)\nVloc,0\n(b)∆Vnl(nV)\nTemperature(K)FIG. 4. Evolution of (a) the o\u000bset voltage Vloc;0(open\nsquares) and the SMR modulation voltage \u0001 VSMR (full\nsquares) recorded in the local geometry, and of (b) the non-\nlocal voltage change \u0001 Vnl(full circles) as a function of tem-\nperature for the d= 200 nm sample.\ndirectly after the YIG growth process [7]. The evolu-\ntion of \u0001VSMR andVloc;0with temperature is shown in\nFig. 4(a).j\u0001VSMR(T)jmonotonically decreases by about\na factor of 3 from T= 300 K to T= 5 K, very simi-\nlarly to the behaviour observed in other YIG/Pt samples\nfabricated at the Walther-Meissner-Institut [18]. Vloc;0\nonly decreases by about 15% in the same temperature\ninterval, showing that defect or surface scattering is very\nstrong in the thin Pt \flm.\nThe non-local voltage Vnlrecorded in the same experi-\nment is shown in Fig. 3(b). Vnlis qualitatively very simi-\nlar toVloc, showing a sin2-like modulation with amplitude\n\u0001Vnl. We would like to stress, however, that on the one\nhand, there is no \fnite constant o\u000bset in Vnl, such that\nVnl= 0 (to within the experimental noise) for the oopt\nrotation. On the other hand, Vnlinvariably assumes neg-\native values (Vnl\u00140). According to our wiring scheme\n(Fig. 2(a)), a negative non-local voltage implies that the\nnon-local inverse spin Hall charge current arising in the\nsecond Pt strip (due to a di\u000busion of the magnon accu-\nmulation generated beneath the \frst Pt strip) must \row\nin the same direction as the charge current in the \frst\nPt strip. Since we detect Vnlusing open circuit bound-\nary conditions, this non-local ISHE charge current is ex-\nactly balanced by an electric potential of opposite (that\nis negative) sign. The negative sign of \u0001 Vnlin our ex-\nperiments thus is consistent with the positive non-local\n\u0001R > 0 reported by Cornelissen et al. [1], since these4\nauthors use an inverted sign convention for the non-local\nvoltage signal. The data shown in Fig. 3 furthermore are\nconsistent with the notion that magnon accumulation is\nat the origin of Vnl, since one would expect maximum\nmagnon di\u000busion signal (maximum Vnl<0 in our exper-\niment) for Mjj\u001bjjtand minimal magnon di\u000busion signal\n(Vnl= 0) for M?\u001b, which translates to Vnl= 0 for Mjjj\nandMjjn. The non-local voltage observed in our exper-\niment in all three rotation planes indeed con\frms this\nexpectation. Note also that magneto-thermal (spin See-\nbeck) voltages cannot account for Vnl, since these have a\nqualitatively di\u000berent dependence on magnetization ori-\nentation [1, 19].\nThe magnitudej\u0001Vnlj\u0019250 nV of the magnetization-\norientation dependent modulation in the non-local volt-\nage atT= 300 K is about 1000 times smaller than\nthe localj\u0001VSMRj \u0019 150\u0016V. Fitting the \u0001 Vnlob-\nserved for pairs of strips with separation d= 200 nm,\nd= 500 nm and d= 1\u0016m, respectively, using \u0001 Vnl=\n(C=\u0015) exp(d=\u0015)=(1\u0000exp(2d=\u0015)) derived as Eq. (7) in\nRef. [1] for 1D spin di\u000busion, we obtain \u0015\u0019700 nm. This\nvalue of\u0015is about one order of magnitude smaller than\nthe value reported by Cornelissen et al. [1] for their sam-\nples. The discrepancy might be evidence for enhanced\nmagnon scattering owing to YIG surface damage caused\nby our fabrication process. In addition, \u0015\u0019700 nm is\nsmaller than the YIG \flm thickness of 3 \u0016m in our case,\nsuggesting that di\u000busion in more than one dimension\ncould be important. To conclusively resolve this point,\nmultiple samples with di\u000berent YIG \flm thicknesses and\na series of di\u000berent Pt strip separations dmust be sys-\ntematically compared, which is beyond the scope of this\nwork.\nInterestingly, the temperature dependencies of \u0001 Vnl\nand \u0001VSMRare very di\u000berent. As evident from Fig. 4(b),\nthe magnitude of \u0001 VSMR at lowTis only about a factor\nof 3 smaller than at room temperature, while \u0001 Vnl= 0\nforT\u001410 K. The strong decrease in \u0001 Vnl(T) can be\nrationalized considering an increase of the magnon prop-\nagation length \u0015with decreasing T. In a simple picture,\nthe non-equilibrium magnons generated at the YIG/Pt\ninterface spread across a volume Vmag/\u00153, such that the\nmagnon accumulation (viz. the non-equilibrium magnon\ndensity) scaling with 1 =Vmagdecreases with T. In the\nlimit of in\fnite \u0015, the magnon accumulation and thus\nalso \u0001Vnlvanishes. More sophisticated theoretical anal-\nyses corroborate this intuitive picture [20, 21]. Note also\nthat the \fnite SMR signal at low Tis direct evidence\nthat the spin Hall e\u000bect is only weakly temperature de-\npendent [18], such that the decrease of the MMR (viz. of\n\u0001Vnl) withTcannot be simply attributed to spin Hall\nphysics.\nIn conclusion, we have simultaneously measured the\nlocal and the non-local magnetoresistive response of two\nparallel Pt strips separated by a gap of a few 100 nm,\ndeposited onto yttrium iron garnet. The local mag-netoresistance (current-biasing one Pt strip and mea-\nsuring the magnetization-orientation dependent voltage\ndrop along this same Pt strip) shows the characteristic\n\fngerprint of spin Hall magnetoresistance, as expected\nfor a YIG/Pt heterostructure. We furthermore observe\na non-local voltage Vnlalong the second, electrically iso-\nlated Pt strip upon current biasing the \frst one. Our\ndata taken at room temperature con\frm the results put\nforward by Cornelissen et al. [1]. In addition, we have\nmeasuredVnlas a function of magnetization orientation\nin three mutually orthogonal rotation planes, and studied\nthe evolution of both the local and the non-local magne-\ntoresistance from room temperature down to 5 K. All our\nexperimental data can be consistently understood assum-\ning that the non-local magnetoresistance is mediated via\nmagnon accumulation.\nWe gratefully acknowledge discussions with\nG. E. W. Bauer and S. 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Filip, and B. van Wees, Nature 410, 345\n(2001).\n[16] H. X. Tang, F. G. Monzon, F. J. Jedema, A. T. Filip,\nB. J. van Wees, and M. L. Roukes, in Semiconduc-\ntor Spintronics and Quantum Computation , NanoScience\nand Technology, edited by D. D. Awschalom, D. Loss,\nand N. Samarth (Springer, Berlin, 2002) Chap. 2, pp.\n31{92.\n[17] D. R u\u000ber, Non-local Phenomena in Metallic Nanostruc-\ntures , Diploma thesis, Walther-Mei\u0019ner-Institut, Tech-nische Universit at M unchen (2009).\n[18] S. Meyer, M. Althammer, S. Gepr ags, M. Opel, R. Gross,\nand S. T. B. Goennenwein, Appl. Phys. Lett. 104, 242411\n(2014).\n[19] K. Uchida, H. Adachi, T. Ota, H. Nakayama,\nS. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505\n(2010).\n[20] J. Xiao and G. E. W. Bauer, arXiv:1508.02486 (2015).\n[21] S. A. Bender and Y. Tserkovnyak, Phys. Rev. B 91,\n140402 (2015)." }, { "title": "1412.1521v1.Proximity_induced_ferromagnetism_in_graphene_revealed_by_anomalous_Hall_effect.pdf", "content": "1 \n Proximity -induced f erromagnetism in grap hene revealed by anomalous Hall effect \nZhiyong Wang, Chi Tang, Ray mond Sachs, Yafis Barlas , and Jing Shi \nDepartment of Physics and Astronomy, University of California, Riverside, CA 92521 \n \n \nWe demonstrate the anomalous Hall effect (AHE) in single -layer graphene \nexchange -coupled to an atomically flat yttrium iron garnet (YIG) ferromagnetic thin film . \nThe anomalous Hall conductance has magnitude of ~0.09(2e2/h) at low temperatures and is \nmeasurable up to ~ 300 K. Our observations indicate not only proximity -induced \nferromagnetism in graphene /YIG with large exchange interaction , but also enhanced \nspin-orbit coupling which is believed to be inheren tly weak in ideal graphene. T he \nproximity -induced ferromagnetic order in graphe ne can lead to novel transport phenomena \nsuch as the quantized AHE which are potentially useful for spintronics. \n 2 \n Although pristine graphene sheets only exhibit Laudau orbital diamagnetism, l ocal \nmagnetic moments can be introduced in a variety of forms, e.g. along the edges of \nnanoribbons [1] around vacancies [2] and adatoms [3]. However, a long-range ferromagnetic \norder in graphene does not occur without exchange coupling between the local moments. In \ngeneral, i ntroducing local moments and the exchange interaction in bulk material s can be \nsimultaneously accomplished by doping atoms with unfilled d- or f-shells [4]. For graphene , \nscattering caused by random impurities could be detrimental to its high carrier mobility , a \nunique electronic property that should be preserved . By coupling the single atomi c sheet of \ncarbons with a magnetic insulator film, e.g. YIG, we may introduce ferromagnetism in \ngraphene without sacrificing its excellent transport properties. T he hybridization between the \n-orbitals in graphene and the nearby spin-polarized d-orbitals in magnetic insulators gives \nrise to the exchange interaction required for long -range ferromagnetic ordering . On the other \nhand, such proximity coupling does not bring unnecessary disorder to graphene. In addition, \nunlike ferromagnetic metals that could in principle mediate proximity exchang e coupling , the \ninsulating material does not shunt current away from graphene. In this work, we demonstrate \nferromagneti c graphene via the proximity effect and directly probe the ferromagnetism by \nmeasuring the anomalous Hall effect (AHE) . \nTo bring graphene in contact with YIG substrates , we apply a previously developed \ntransfer technique (see SM) that is capable of transferring pre -fabricated functional graphene \ndevices to any target substrates [5]. We first fabricate exfoliated single -layer graphene \ndevices on 290 nm -thick SiO 2 atop highly doped Si substrates using standard electron -beam \nlithography and Au electron -beam evaporation. Both longitudinal and Hall resistivities are 3 \n measured at room temperature to characterize the state of the pre -transferred devices. To \ntransfer selec ted devices, we spin -coat the chip with poly-methyl methacrylate (PMMA) \nfollowed by a hard bake at 170 °C for 10 minutes. The entire chip is then s oaked in 1 M \nNaOH solution for two days to etch away SiO 2 so that the device /PMMA layer is released \nfrom the substrate. The PMMA layer attached with the fully nano -fabricated graphene \ndevices is then placed on the target substrate. Finally, the PMMA is dissolved with acetone \nfollowed by careful rinsing and dryi ng, and the device is ready for electrical transport and/or \nRaman measurements. This technique was previously applied to fabricate graphene devices \non SrTiO 3, a high nominal dielectric constant pervoskite material [5,6]. The transfer steps are \nschematically shown in Fig . S-1 in SM. \nFor this study, ~ 20 nm thick atomically flat YIG films are grown epitaxially on 0.5 \nnm-thick gadolinium gallium garnet (GGG) substrates by pulsed lase r deposition as described \nelsewhere [7], which are then subsequently annealed in an oxygen -flow furnace at 85 0 °C for \n6 hours to minimize oxygen deficiency . Magnetic hysteresis loop measurements and atomic \nforce microscop y (AFM) are performed to characterize the magnetic properties and the \nmorphology of YIG films , respectively . The hysteresis loop s of a representative YIG/GGG \nsample are displayed in Fig. 1(a). The YIG film clearly shows in-plane magnetic anisotropy . \nThe in-plane coercive field and saturation field are both small (~ a few G and < 20 G, \nrespectively) , and t he out-of-plane loop indicate s a typical hard-axis behavior with a \nsaturation field ~2000 G , which can vary from 1500 to 2500 G in different YIG samples . Fig. \n1(a) inset shows the AFM topographic image of a typical YIG film. The nearly parallel lines \nare terraces separated by steps with the atomic height and the roughness on the terrace is ~ 4 \n 0.06 nm . The smooth ness of the YIG surface is not only critical to a strong induced proximity \neffect in graphene , but also favorable for maintaining high carrier mobility [8]. \nIn order t o effectively tune the carrier density in graphene/YIG, we fabricate a thin methyl \nmethacrylate ( MMA ) or PMMA top gate. Fig. 1(b) shows a false -colored optical image of a \ngraphene device on YIG/GGG before the top gate is fabricated . Room -temperature Raman \nspectroscopy i s performed at different stages of the device fabrication. Representative spectra \nare shown in F ig. 1(c) for the same graphene device on SiO 2 (before transfer ) and YIG (after \ntransfer), and for YIG/GGG only . Graphene/YIG show s both the characteristic E2g (~1580 \ncm-1) and 2D peaks (~2700 cm-1) of single -layer graphene as well as YIG’ s own peaks, \nsuggesting success ful transfer. We also note that the transfer process does not produce any \nmeasurable D peak ( ~1350 cm-1) associated with defects [9]. Fig. 1(d) is a schematic drawing \nof a top-gated transferred device on YIG/GGG . \nLow-temperature transport measurements are performed in Quantum Design’s Physical \nProperty Measurement System . Fig. 2(a) is a plot of the gate voltage dependence of the \nfour-terminal electrical conductivity scaled by the effective capacitance per unit area, Cs. \nSince different gate dielectric s are used in the back - and top-gated graphene devices, Cs is \ncalculated based on the quantum Hall data which agrees with the calculated value usi ng the \nnominal dielectric constant and the measured dielectric film thickness . Before transfer, the \nDirac point is at ~ -9 V and the field-effect mobility is ~ 6000 cm2/V∙s. After transfer, the \nDirac point is shifted to ~ -18 V . The slope of the σxx/Cs vs. Vg curve increases somewhat , \nindicating slight ly higher mobility , which suggest s that the transfer process, the YIG substrate, \nand the top -gate dielectric do not cause any adverse effect on graphene mobility . At 2 K, the 5 \n mobility improves further , exceeding 10000 cm2/V∙s on the electron side. Well-defined \nlongitudinal resistance peaks and quantum Hall plateau s are both present at 8 T as shown in \nFig. 2(b), another indication of uncompromised device quality after transfer . In approximately \n8 devices studied, we find that the mobility of graphene/YIG is either compar able with or \nbetter than that of graphene/SiO 2. \nTo s tudy the proximity -induced magnetism in graphene, we perform the Hall effect \nmeasurements in the field range where the magnetization of YIG rotates out of plane over a \nwide range of temperatures . Nearly all graphene/YIG devices exhibit similar nonline ar \nbehavior at low temperature s as shown in Fig. 2(c). Fig. 2(d) only shows the Hall data after \nthe linear ordinary Hall background (the straight line in Fig. 2(c)) is subtracted . In \nferromagnets, the Hall resistivity generally consists of two parts [10]: from the ordinary Hall \neffect and the anomalous Hall effect (AHE) , i.e. 𝑅𝑥𝑦=𝑅𝐻(𝐵)+𝑅𝐴𝐻𝐸(𝑀)=𝛼𝐵+𝛽𝑀, \nhere B being the external magnetic field, M being the magnetization component in the \nperpendicular direction , and and are two B- and M-independent parameters respectively . \nThe B-linear term results from the Lorentz force on one type of carriers. Higher order terms \ncan appear if there are two or more types of carriers present. The M-linear term is due to the \nspin-orbit coupling in ferromagnets [10]. The observed non-linearity in Rxy suggests the \nfollowing three possible scenarios: the ordinary Hall effect arising from more than one type \nof carriers in response to the external magnetic field, the same Lorentz force related ordinary \nHall effect but due to the stray magnetic field from the underlying YIG film, and AHE from \nspin-polarized carriers . The nonlinear Hall curves saturate at Bs ~ 230 0 G, which is \napproximately correlated with the saturation of the YIG magnetization in Fig . 1(a). This 6 \n behavior is characteristic of AHE , i.e. RAHE ∝ MG, where MG is the induced magnetization of \ngraphene . Since MG result s from the proximity coupl ing with the magnetization of YIG , MYIG, \nboth MG and MYIG should saturate when the external field exceeds some value . The saturation \nfield of YIG is primarily determined by its shape anisotropy , i.e. 4πM YIG, which should not \nchange significantly far below the Curie temperature (550 K) of YIG . On the other hand, if it \nis caused by the Lorentz force on two type s of carriers , the nonlinear feature would not have \nany correlation with MYIG. These experimental facts do not support the first scenario . To \nfurther exclude the ordinary Hall effect due to the Lorentz force from stray fields from YIG, \nwe fabricate graphene devices on Al 2O3/YIG, in which the 5 nm thick continuous Al2O3 layer \nshould have little effect on the strength of the stray field but effectively cut off the proximity \ncoupling. We do not observe any measurable nonlinear Hall signal similar to those in \ncompanion graphene/YIG devices (Figs. S -6 and S-7 in SM). It excludes the effect of the \nstray field. Therefore, we attribute the non linear Hall signal in graphene/YIG to AHE which \nis due to spin -polarized carriers in ferromagnetic graphene. Further evidence will be \npresent ed when the gate voltage dependence is discussed below . \nFig. 3(a) shows the AHE resistance , RAHE, vs. the positive out-of-plane magnetic field \ntaken from 5 to 250 K . All linear background has been removed. Fig. 3 (b) is the extracted \ntemperature dependence of the saturated AHE resistance . The AHE signal decre ases as the \ntemperature is increase d, but it stays finite up to nearly 300 K. We note that t he AHE \nmagnitude changes sharply in the temperature ran ge of 2 – 80 K , and then stays relatively \nconstant above 80 K before it approaches ~ 300 K, which defines the Curie temperature of \nMG. In conducting ferromagnets , the AHE resistance , RAHE, scales with the longitudinal 7 \n resistance , Rxx, in the power -law fashion [10], i.e. 𝑅𝐴𝐻𝐸∝𝑀𝐺𝑅𝑥𝑥𝑛. Thus t he temperature \ndependence of RAHE could originate from MG and/or Rxx. Here MG should be a slow -varying \nfunction of the temperature below 80 K; however, the temperature dependence of Rxx in 1T \nfield (inset of Fig . 3(b)) cannot account for the steep temperature dependence of RAHE either. \nTherefore, we attribute the discrepancy to possible physical distance change between the \ngraphene sheet and YIG either due to an increase in the vibration al amplitude or different \nthermal expansion coefficients between the top -gate dielectric and YIG/GGG . We have \nobserve d variations in both the Curie temperature Tc for MG and the maximum RAHE (see Fig . \nS-2 and S -3). Among all 8 devices studied, the highest Tc is ~ 300 K and the largest RAHE at 2 \nK is ~ 200 Ω. \nWith a top gate, we can control the position of the Fermi level in graphene at a fixed \ntemperature , not possible in ferromagnetic metals. By sweeping the top -gate voltage, Vtg, we \nsystematically vary both RAHE and Rxx and keep the induced magnetization and exchange \ncoupling strength unchanged . More importantly , by changing the carrier type, a sign reversal \noccurs in the ordinary Hall , i.e. the slope of the linear background signal . We remove this \ncarrier density dependent linear background for each gate voltage and obtain the AHE signal. \nFig. 4(a) is the AHE resistivity of a device measured at 20 K for several Vtg’s: 60 V (red \nsquares), 0 V (green circle s), and -20 V (blue triangle s), respectively. The inset shows the \nVtg-dependence of the resistivity . The Dirac point is at ~35 V; therefore, carriers are \npredominately electrons at 60 V with a density ~ 2.5x1011 cm-2, but predominately holes at \nboth 0 and -20 V . We deliberately avoid the region close to the Dirac point where both \nelectrons and holes coexist and the ordinary Hall signal acquires high -order terms in B. In the 8 \n gate dependence data, i t is important to note that the AHE sign remains unchanged regardless \nof the carrier type. This is strong evidence that the observ ed non linear Hall signal is not due \nto the ordinary Hall effect from two types of carriers , either from the external or stray field, \nbut due to the AHE contribution from spin -polarized carriers in ferromagnetic sampl e. In \naddition, t he resistance at 60 V is the highest among the three, followed by that at 0 V, and \nthen -20 V , and t he corresponding RAHE magnitude follows the same order. \nTo further reveal the physical origin of AHE, we now focus on the relationship between \nRAHE and Rxx as Vtg is tuned. Fig. 4(b) shows more gate -tuned AHE data in another top -gated \ndevice measured at 2 K . We also exclude the data close to the Dirac point (-14 V for this \ndevice) for the reason mentioned above . Starting from -10 V , RAHE is the largest . As Vtg is \nincreased , the electron density increases, and Rxx decreases accordingly, which is \naccompanied by a steady decrease in RAHE. Due to the negatively biased Dirac point, we \ncannot reach the completely hole-dominated region within the safe Vtg range (gate leakage \ncurrent < 10 nA) . On the hole side where the background is still influenced by the two -band \ntransport, we do not observe any evidence of a sign change in RAHE. In the inset we plot RAHE \nvs. Rxx as Vtg is varied . From the slope of the straight line in the log-log plot, we obtain the \nexponent of the power -law: n =1.9 ± 0.2 . The same exponent is also obtained in a different \ngate-tuned device (see Fig . S-4 and S -5). As in many ferromagnetic conductors, t he quadratic \nrelationship indicates a scattering -independent AHE mechanism, which is different from the \nskew scattering induced AHE[10] . \nIt is understood that a necessary ingredient for AHE is the presence of SOC along with \nbroken time reversal symmetry [10]. AHE can result from either intrinsic (band structure 9 \n effect) or extrinsic (impurity scattering) mechanisms . Haldane showed that for a honeycomb \nlattice (graphene) the presence of intrinsic SOC (which breaks time reversal symmetry) can \nlead to quantized AHE ( QAHE ) for spin -less electrons [11]. Since intrinsic SOC in graphen e \nis very weak (~10 μeV) [12], this effect has not been observed experimentally. \nHowever, an enhanced Rashba SOC is possible when graphene is placed on substrates \n[13,14] or subjected to hydrogenation [15] due to broken inversion symmetry. Recently, Qiao \net al. predicted that ferromagnetic graphene with Rashba SOC should exhibit QAHE [16,17]. \nIn this case , the Dirac spectrum opens up a topological gap with magnitude smaller than \ntwice the minimum of exchange and SOC energy scale (see SM). As the Fermi level is turned \ninto the gap, a decrease in the four-terminal resistance is expected along with a simultaneous \nquantization of the AHE conductivity approaching 2e2/h. In devices exhibiting AHE, the \nlargest AHE at 2 K is ~ 200 Ω. Using the corresponding Rxx of 5230 Ω , we calculate the AH E \ncontribution and obtain σAHE ≈ 7 μS ≈ 0.09(2e2/h), nearly one order of magnitude smaller than \nthe predicted QAHE conductivity 2e2/h. Clearly we have not reached the QAHE regime due \nto the intrinsic band structure effect , indicating that the Rashba SOC strength λR is smaller \nthan the disorder energy scale . From the minimum conductivity plateau , we estimate the \nenergy scale associated with the disorder Δdis = ħ/τ ≈ 12 meV , assuming long-ranged \nCoulomb scattering [18]. Therefore our experimental results suggest that λR < 12 meV . To \nobserve QAHE , it is important to further improve the quality of the devices or to strengthen \nthe Rashba SOC to fulfill λR > Δdis, both of which are highly possible. \nIn order to understand the physical origin of the observed unquantized AHE in our devices, \nwe calculate the intrinsic AHE (see SM ) at the relevant densities for λR < 12 meV . Our results 10 \n show that the intrinsic AHE conductivity at these densities is an order of magnitude smaller \nthan the observed value , which argues against the intrinsic mechanism . Since charged \nimpurity screening in graphene becomes extremely weak as the Dirac point is approached , it \nis likely that the ex trinsic mechanisms play a more important role here. We would like to \npoint out that gate tunability in ferromagnetic graphene allow s for the observation of Fermi \nenergy dependen ce of the AHE conductivity , which cannot be achieved in ordinary \nferromagnet metals . If the carrier density can be modulated by gating, b esides the exponent, \nthe Fermi energy dependence of the AHE conductivity can be experimentally determined \nover a broad range of energy [19]. This additional information can help further pinpoint the \nphysical origin of AHE in 2D Dirac fermion systems . \nWe thank Z.S. Lin, T. Lin, B. Barrios, Q. Niu, and W. Beyerman for their help and useful \ndiscussions. ZYW and JS were supported by the DOE BES award #DE-FG02 -07ER46351 , \nCT was supported by NSF/ECCS, and RS was supported by NSF/NEB. \n \n 11 \n FIG. 1. (a) Magnetic hysteresis loop s in perpendicular and in -plane magnetic field s. Inset is \nthe AFM topographic i mage of YIG thin film surface. (b) Optical image (without top gate) \nand (d) schematic drawing (with top gate) of the devices after transferred to YIG/GGG \nsubstrate (false color). (c) Room temperature Raman spectra of graphene/YIG (purple), \ngraphene/SiO 2 (red), and YIG/GGG substrate only (blue). \n \nFIG. 2. (a) The gate voltage dependence of the device conductivity scaled by the c apacitance \nper unit area for the pre -transfer (293 K, black) and transferred devices (300 K, red; 2 K, \ngreen) with the same graphene sheet . (b) Quantum Hall effect of transferred graphene/YIG \ndevice in an 8 T perpendicular magnetic field at 2 K. (c) The measured total Hall resistivity \ndata at 2 K with a straight line indicating the ordina ry Hall background. (d) The non linear \nHall resistivity after the linear background is removed from the data in (c). \n \nFIG. 3. (a) AHE resistance at different temperatures. (b) The temperature dependence of AHE \nresistance. Inset i s the longitudinal resistance at the Dirac point with no magnetic field (black) \nand a 1 T perpendicular magnetic field (red). \n \nFIG. 4. (a) AHE resistance with different carrier types and concentrations at 20 K. Inset, gate \nvoltage dependence at 20 K. Red squares, green circles, and blue triangles represent 60 V , 0 V , \n-20 V top gate voltages, respectively. The sharp noise -like field -dependent features are \nreproducible. (b) Top gate voltage dependence of the AHE resistance at 2 K. Inset is the \nlog-log plot of RAHE vs. Rxx. Red curve is a linear fit with a slope of 1.9 ± 0.2. 12 \n FIG. 1 \n \n \n \n13 \n FIG. 2 \n \n \n \n \n \n \n14 \n FIG. 3 \n \n \n \n \n \n15 \n FIG. 4 \n \n \n \n \n \n16 \n References \n[1] Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature 444, 347 (2006). \n[2] J.-H. Chen, L. Li, W. G. Cullen, E. D. Williams, and M. S. Fuhrer, Nat . Phys . 7, 535 (2011). \n[3] B. Uchoa, V. N. Kotov, N. M. R. Peres, and A. H. Castro Neto, Phys . Rev. 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Lett. 96, 086805 (2006). \n \n " }, { "title": "2005.14133v2.A_First_Principle_Study_on_Magneto_Optical_Effects_and_Magnetism_in_Ferromagnetic_Semiconductors_Y__3_Fe__5_O___12___and_Bi__3_Fe__5_O___12__.pdf", "content": "arXiv:2005.14133v2 [cond-mat.mtrl-sci] 2 Aug 2020A First Principle Study on Magneto-Optical Effects in Ferrom agnetic Semiconductors\nY3Fe5O12and Bi 3Fe5O12\nWei-Kuo Li1and Guang-Yu Guo1,2,∗\n1Department of Physics and Center for Theoretical Physics,\nNational Taiwan University, Taipei 10617, Taiwan\n2Physics Division, National Center for Theoretical Science s, Hsinchu 30013, Taiwan\n(Dated: August 4, 2020)\nThe magneto-optical (MO) effects not only are a powerful prob e of magnetism and electronic\nstructure of magnetic solids but also have valuable applica tions in high-density data-storage tech-\nnology. Yttrium iron garnet (Y 3Fe5O12) (YIG) and bismuth iron garnet (Bi 3Fe5O12) (BIG) are two\nwidely used magnetic semiconductors with significant magne to-optical effects. In particular, YIG\nhas been routinely used as a spin current injector. In this pa per, we present a thorough theoretical\ninvestigation on magnetism, electronic, optical and MO pro perties of YIG and BIG, based on the\ndensity functional theory with the generalized gradient ap proximation plus onsite Coulomb repul-\nsion. We find that YIG exhibits significant MO Kerr and Faraday effects in UV frequency range\nthat are comparable to ferromagnetic iron. Strikingly, BIG shows gigantic MO effects in visible\nfrequency region that are several times larger than YIG. We fi nd that these distinctly different MO\nproperties of YIG and BIG result from the fact that the magnit ude of the calculated MO conduc-\ntivity (σxy) of BIG is one order of magnitude larger than that of YIG. Inte restingly, the calculated\nband structures reveal that both valence and conduction ban ds across the semiconducting band gap\nin BIG are purely spin-down states, i.e., BIG is a single spin semiconductor. They also show that in\nYIG, Ysdorbitals mix mainly with the high lying conduction bands, le aving Fe dorbital dominated\nlower conduction bands almost unaffected by the SOC on the Y at om. In contrast, Bi porbitals\nin BIG hybridize significantly with Fe dorbitals in the lower conduction bands, leading to large\nSOC-induced band splitting in the bands. Consequently, the MO transitions between the upper\nvalence bands and lower conduction bands are greatly enhanc ed when Y is replaced by heavier Bi.\nThis finding suggests a guideline in search for materials wit h desired MO effects. Our calculated\nKerr and Faraday rotation angles of YIG agree well with the av ailable experimental values. Our\ncalculated Faraday rotation angles for BIG are in nearly per fect agreement with the measured ones.\nThus, we hope that our predicted giant MO Kerr effect in BIG wil l stimulate further MOKE ex-\nperiments on high quality BIG crystals. Our interesting find ings show that the iron garnets not\nonly offer an useful platform for exploring the interplay of m icrowave, spin current, magnetism, and\noptics degrees of freedom, but also have promising applicat ions in high density MO data-storage\nand low-power consumption spintronic nanodevices.\nI. INTRODUCTION\nYttrium iron garnet (Y 3Fe5O12, YIG) is a ferrimag-\nnetic semiconductor with excellent magnetic properties\nsuch as high curie temperature Tc[1], low Gilbert damp-\ningα∼6.7×10−5[2–4] and long spin wave propragating\nlength [5]. Various applications such as spin pumping re-\nquire a non-metallic magnet. YIG is thus routinely used\nfor spin pumping purposes [4]. It is also widely used as a\nmagnetic insulating substrate for purposes such as intro-\nducingmagneticproximityeffect whileavoidingelectrical\nshort-cut. [6] YIG has high Curie temperature, which is\ngood for applications across a wide temperature range.\nThe low Gilbert damping of YIG also makes it a good\nmicrowave material. YIG thus becomes a famous mate-\nrial in the field of spintronics, where coupling between\nmagnetism, microwave and spin current becomes possi-\nble.\nMagneto-optical (MO) effects are important examples\nof light-matter interactions in magnetic phases. [7, 8]\n∗gyguo@phys.ntu.edu.twWhen a linearly polarized light beam is shined onto a\nmagnetic material, the reflected and transmitted light\nbecomes elliptically polarized. The principal axis is ro-\ntated with respect to the polarization direction of inci-\ndentlight beam. The formerandlattereffects aretermed\nMO Kerr (MOKE) and MO Faraday (MOFE) effects,\nrespectively. MOKE allowes us to detect the magnetiza-\ntion locally with a high spatial and temporal resolution\nin a non-invasive fashion. Furthermore, magnetic ma-\nterials with large MOKE would find valuable MO stor-\nage and sensor applications [9, 10]. Thus it has been\nwidely used to probe the electronic and magnetic proper-\nties of solids, surface, thin films and 2D magnets [8]. On\nthe other hand, MOFE can be used as a time-reversal\nsymmetry-breaking element in optics [11], and its ap-\nplications such as optical isolators are consequenses of\ntime-reversal symmetry-breaking [12]. Magnetic materi-\nals with large Kerr or Faraday rotation angles have tech-\nnological applications.\nYIG is also known to be MO active [13]. Various ex-\nperiments have been carried out to study the MOKE and\nMOFE of iron garnets in the visible and near-UV regime\n[14, 15]. Substituting yttrium with bismuth results in2\nbismuth iron garnet (Bi 3Fe5O12) (BIG). BIG has ap-\nproximately 7 times larger Faraday rotation angles than\nthat of YIG. The effect of doping bismuth into YIG on\nthe MOFE spectrum was studied [16, 17]. The large ra-\ndius of bismuth atoms seems to make bulk BIG unstable.\nThus high quality BIG film is difficult to synthesize [18].\nThough numerous experimental studies have been done\non these systems, first-principle calculations are scarce.\nThis is probably due to the complexity of the structures\nofBIG and YIG. As shown in Fig. 1(a), they havea total\nof 80 atoms in the primitive cell. Although the electronic\nstructures of YIG and BIG have been theoretically stud-\nied [19, 20], no first principle calculation on the MOKE\nor MOFE spectra of YIG and BIG have been reported.\nTherefore, here we carry out a systematic first-principle\ndensity functional study on the optical and MO proper-\nties of YIG and BIG. The rest of this paper is organized\nas follows. A brief description of the crystal structures of\nYIG and BIG as well as the theoretical methods used is\ngiven in Sec. II. In Sec. III, the calculated magnetic mo-\nments, electronic structure, optical conductivities, MO\nKerr and Faraday effects are presented. Finally, the con-\nclusions drawn from this work are given in section IV.\nII. CRYSTAL STRUCTURE AND\nCOMPUTATIONAL METHODS\nYIG and BIG crystalize in the cubic structure with\nspace group Ia3d[21, 22], as illustrated in Fig. 1(a). In\neach unit cell, there are 48 oxygen atoms at the Wyck-\noff 96h positions, 8 octahedrally coordinated iron atoms\n(FeO) at the 16a positions, and 12 tetrahedrally coor-\ndinated iron atoms (FeT) at the 24d positions in the\nprimitive cell. In other words, there are two FeOions\nand three FeTions per formula unit (f.u.). The ex-\nperimental lattice constant a= 12.376˚A, and the ex-\nperimental Wyckoff parameters for oxygen atoms are\n(x,y,z) = (0.9726,0.0572,0.1492). [21] The experimen-\ntal lattice constant for BIG a= 12.6469˚A. [22] Accurate\noxygen position measurement for BIG is still on demand\nand under debate [18]. Therefore we use the experimen-\ntal lattice constant for BIG with the atomic positions\ndetermined theoretically (see Table I), as described next.\nWe use the experimental lattice constant and atomic po-\nsitions for all YIG calculations,\nOur first principle calculations are based on the den-\nsity functional theory with the generalized gradient ap-\nproximation (GGA) of the Perdew-Burke-Ernzerhof for-\nmula [23] to the electron exchange-correlation potential.\nFurthermore, we use the GGA + Umethod to have a\nbetter description for on-site interaction for Fe delec-\ntrons. [24] Here we set U= 4.0 eV, which was found to\nbe rather appropriate for iron oxides [25]. Indeed, as we\nwill show below, the optical and MO spectra calculated\nusing this Uvalue agree rather well with the available\nexperimental spectra. All the calculations are carried\nout by using the accurate projector-augmented wave [26]TABLE I. Structuralparameters of Y 3Fe5O12and Bi 3Fe5O12.\nFor YIG, experimental lattice constant a= 12.376˚A and\noxygen positions [21] are used. For BIG, experimental latti ce\nconstant a= 12.6469˚A [22] is used while the oxygen positions\nare determined theoretically.\nY3Fe5O12Wyckoff position x y z\nFeO16a 0.0000 0.0000 0.0000\nFeT24d 0.3750 0.0000 0.2500\nY 24c 0.1250 0.0000 0.2500\nO 96h 0.9726a0.0572a0.1492a\nBi3Fe5O12Wyckoff position x y z\nFeO16a 0.0000 0.0000 0.0000\nFeT24d 0.3750 0.0000 0.2500\nBi 24c 0.1250 0.0000 0.2500\nO 96h 0.0540 0.0300 0.1485\naRef. 21.\nmethod, as implemented in Vienna ab initio Simulation\nPackage (VASP). [27, 28] A large energy cutoff of 450 eV\nfor the plane-wavebasis is used. A 6 ×6×6k-point mesh\nis used for both systems in the self-consistent chargeden-\nsitycalculations. The densityofstates(DOS) calculation\nis performed with a denser k-point mesh of 10 ×10×10.\nWefirstcalculatethe opticalconductivitytensorwhich\ndetermine the MOKE and MOFE. We let the magnetiza-\ntion of our systems be along (001) ( z) direction. In this\ncase, our systems have the four-fold rotational symmetry\nalong the zaxis and thus the optical conductivity tensor\ncan be written in the following form [29]:\nσ=\nσxxσxy0\n−σxyσxx0\n0 0 σzz\n. (1)\nThe optical conductivity tensor can be formulated within\nthe linear response theory. Here the real part of the di-\nagonal elements and imaginary part of the off-diagonal\nelements are given by [29–31]:\nσ1\naa(ω) =πe2\n/planckover2pi1ωm2/summationdisplay\ni,j/integraldisplay\nBZdk\n(2π)3|pa\nij|2δ(ǫkj−ǫki−/planckover2pi1ω),\n(2)\nσ2\nxy(ω) =πe2\n/planckover2pi1ωm2/summationdisplay\ni,j/integraldisplay\nBZdk\n(2π)3Im[px\nijpy\nji]δ(ǫkj−ǫki−/planckover2pi1ω),\n(3)\nwhere/planckover2pi1ωis the photon energy, and ǫki(j)are the en-\nergy eigenvalues of occupied (unoccupied) states. The\ntransition matrix elements pa\nij=/angbracketleftkj|ˆpa|ki/angbracketrightwhere|ki(j)/angbracketright\nare thei(j)th occupied(unoccupied) states at k-pointk,\nand ˆpais the Cartesian component aof the momentum\noperator. The imaginary part of the diagonal elements\nand the real part of the off-diagonal elements are then\nobtained from σ1\naa(ω) andσ2\nxy(ω), respectively, via the\nKramers-Kronig transformations as follows:\nσ2\naa(ω) =−2ω\nπP/integraldisplay∞\n0σ1\naa(ω′)\nω′2−ω2dω′,(4)3\nFIG. 1. (a) 1/8 of BIG conventional unit cell. Oxygen atoms\nare shown as red balls; bismuth atoms are shown as purple\nballs; FeTatoms are shown as yellow balls; FeOatoms are\nshown as blue balls. (b) Brillouin zone of both YIG and\nBIG. The red lines denote the high symmetry lines where the\ncalculated energy bands will be plotted.\nσ1\nxy(ω) =2\nπP/integraldisplay∞\n0ω′σ2\nxy(ω′)\nω′2−ω2dω′, (5)\nwherePdenotes the principal value of the integration.\nWe can see that Eq. (2) and Eq. (3) neglect transitions\nacrossdifferent k-points since the momentum of the opti-\ncal photon is negligibly small compared with the electron\ncrystal momentum and thus only the direct interband\ntransitions need to be considered. In our calculations\npa\nijare obtained in the PAW formalism [32]. We use a\n10×10×10k-point mesh and the Brillouin zone inte-\ngration is carried out with the linear tetrahedron method\n(see [33] and references therein), which leads to well con-\nverged results. To ensure that the σ2\naa(ω) andσ1\nxy(ω) in\nthe optical frequency range (e.g., /planckover2pi1ω <8 eV) obtained\nvia Eqs. (4) and (5) are converged, we include the unoc-\ncupied states at least 21 eV above the Fermi energy, i.e.,\na total of 1200 (1300) bands are used in the YIG (BIG)\ncalculations.\nFor a bulk magnetic material, the complex polar Kerr\nrotation angle is given by [34, 35],\nθK+iǫK=−σxy\nσxx/radicalbig\n1+i(4π/ω)σxx. (6)Similarly, the complex Faraday rotation angle for a thin\nfilm can be written as [39]\nθF+iǫF=ωd\n2c(n+−n−), (7)\nwheren+andn−represent the refractive indices for left-\nand right-handed polarized lights, respectively, and are\nrelated to the corresponding dielectric function (or opti-\ncal conductivity via expressions n2\n±=ε±= 1+4πi\nωσ±=\n1 +4πi\nω(σxx±iσxy). Here the real parts of the optical\nconductivity σ±can be written as\nσ1\n±(ω) =πe2\n/planckover2pi1ωm2/summationdisplay\ni,j/integraldisplay\nBZdk\n(2π)3|Π±\nij|2δ(ǫkj−ǫki−/planckover2pi1ω),\n(8)\nwhereΠ±\nij=/angbracketleftkj|1√\n2(ˆpx±iˆpy)|ki/angbracketright. Clearly, σxy=1\n2i(σ+−\nσ−), and this shows that σxywould be nonzeroonly if σ+\nandσ−are different. In other words, magnetic circular\ndichroism is the fundamental cause of the nonzero σxy\nand hence the MO effects.\nIII. RESULTS AND DISCUSSION\nA. Magnetic moments\nHere we first present calculated total and atom-\ndecomposed magnetic moments in Table I. As expected,\nY3Fe5O12is a ferrimagnet in which Fe ions of the same\ntype couple ferromagnetically while Fe ions of different\ntypes couple antiferromagnetically. Since there are two\nFeOions and three FeTions in a unit cell, Y 3Fe5O12is\nferrimagnetic with a total magnetic moment per f.u. be-\ning∼5.0µB(see Table I). The calculated spin magnetic\nmoments of Fe ions of both types are ∼4.0µB, being\nconsistent with the high spin state of Fe+2(d5↑t1↓\n2g) ions\nin either octahedral or tetrahedral crystal field. We note\nthat the orbital magnetic moments of Fe are parallel to\ntheir spin magnetic moments. Nonetheless, the calcu-\nlated orbital magnetic moments of Fe are small, because\nof strong crystal field quenching. Interestingly, there is a\nsignificantspin magnetic moment on eachO ion, and this\ntogether with the spin magnetic moment of one net Fe\nion per f. u. leads to the total spin magnetic moment per\nf.u. of∼5.0µB. The calculatedFe magneticmomentsfor\nboth symmetry sites agree rather well with the measured\nones of∼4.0µB. [36] The calculated total magnetization\nof∼5.0µB/f.u. is also in excellent agreement with the\nexperiment. [36]\nBi3Fe5O12is also predicted to be ferrimagnetic, al-\nthough the calculated magnetic moments of both FeO\nand FeTions are slightly smaller than the corresponding\nones in Y 3Fe5O12(see Table I). The total magnetization\nand local magnetic moments of the other ions in BIG are\nalmost identical to that in YIG. However, the experimen-\ntalmtotfor BIG is only 4 .4µB, [37] being significantly\nsmaller than the calculated value. As mentioned before,4\nTABLE II. Total spin magnetic moment ( mt\ns), atomic spin magnetic moments ( mFe\ns,mO\ns,mY(Bi)\ns), atomic Fe orbital magnetic\nmoments ( mFe\no) and band gap ( Eg) of ferrimagnetic Y 3Fe5O12and Bi 3Fe5O12from the full-relativistic electronic structure\ncalculations. For comparison, the available measured opti calEgand total magnetization mt\nexpare also listed.\nstructure mt(mt\nexp)mFe(16a)\ns(mFe(16a)\no)mFe(24d)\ns(mFe(24d)\no) mO\ns mY(Bi)\ns Eg(Eexp\ng)\n(µB/f.u.) ( µB/atom) ( µB/atom) ( µB/atom) ( µB/atom) (eV)\nY3Fe5O124.999 (5.0a) -4.177 (-0.016) 4.075 (0.018) 0.067 0.005 1.81 (2.4b)\nBi3Fe5O124.996 (4.4c) -4.161 (-0.018) 4.068 (0.019) 0.066 0.005 1.82 (2.1d)\naRef. 36.bRef. 14.cRef. 37.dRef. 17.\nstablehigh qualityBIGcrystalsarehardto grow. Conse-\nquently, this notable discrepancy in total magnetization\nbetween the calculation and the previous experiment [37]\ncould be due to the poor quality of the samples used in\nthe experiment.\nB. Electronic structure\nHere we present the calculated scalar-relativistic band\nstructures of YIG and BIG in Fig. 2(a) and Fig. 3(a),\nrespectively. The calculated band structures show that\nYIG and BIG are both direct band-gap semiconductors,\nwhere the conduction band minimum (CBM) and va-\nlence band maximum (VBM) are both located at the Γ\npoint. For BIG, both CBM and VBM are purely spin-up\nbands. This means that BIG is a single-spin semiconduc-\ntor, which may find applications for spintronic and spin\nphotovoltaic devices. The origin of the MO effects is the\nmagnetic circular dichorism [see Eq. (8)], as mentioned\nabove, which cannot occur without the presence of the\nspin-orbit coupling (SOC). Therefore, it is useful to ex-\namine how the SOC influence the band structures. The\nfully relativistic band structures for YIG and BIG are\npresented in Fig. 2(b) and Fig. 3(b), respectively. First,\nwe notice that with the inclusion of the SOC, YIG and\nBIG are still direct band-gap semiconductors, where the\nCBM and VBM are both located at the Γ point. Second,\nFig. 3(b) indicates that when the SOC is considered, the\nBIG band structure changes significantly, while the YIG\nband structure hardly changes [see Fig. 2(b)]. For exam-\nple, the band gap for BIG decreases from 2.0 to 1.8 eV\nafter the SOC is included. Also, the gap, which was at\n3.4 to 3.7 eV above the Fermi energy [see Fig. 3(a)], now\nbecomes from 3.9 to 4.5 eV above the Fermi energy [see\nFig. 3(b)]. Interestingly, the substitution of yittrium by\nbismuth not only enhances the SOC but also changes the\nelectronic band structure significantly, as can be seen by\ncomparing Figs. 2 and 3.\nWe also calculate total as well as site-, orbital-, and\nspin-projected densities of states (DOS) for YIG and\nBIG, as displayed in Fig. (4) and (5), respectively. First,\nFigs. (4) and (5) show that in both YIG and BIG,\nthe upper valence bands ranging from -4.0 to 0.0 eV,\nare dominated by O p-orbitals with minor contributions\nfrom Fed-orbitals as well as Y d-orbitals in YIG and Bi\nFIG. 2. (a) Scalar-relativistic spin-polarized band struc ture\nand (b) fully relativistic band structure of Y 3Fe5O12.5\nFIG. 3. (a) Scalar-relativistic spin-polarized band struc ture\nand (b) fully relativistic band structure of Bi 3Fe5O12.\nsp-orbitals in BIG. Second, the lower conduction band\nmanifold, ranging from 1.8 to ∼3.9 eV in YIG (Fig. 4)\nand from 2.0 to 3.4 eV in BIG (Fig. 5), stems predom-\ninately from Fe d-orbitals with small contributions from\nOp-orbitals. Therefore, the semiconducting band gaps\nin YIG and BIG are mainly of the charge transfer type.\nFurthermore, on the FeTsites, the d-DOS in this conduc-\ntion band is almost fully spin-down [see Figs. 4(e) and\n5(e)]. On the FeOsites, on the other hand, the d-DOS in\nthis conduction band is almost purely spin-up [see Figs.\n4(d) and 5(d)]. Here, the DOS peak marked amostly\nconsists of t2gorbital while that marked babove peak a,\nis made up of mainly egorbital. The gap between peaks\nFIG. 4. Spin-polarized density of states (DOS) of Y 3Fe5O12\nfrom the scalar-relativistic calculation.\nFIG. 5. Spin-polarized density of states (DOS) of Bi 3Fe5O12\nfrom the scalar-relativistic calculation.6\nFIG. 6. Calculated optical conductivity of Y 3Fe5O12. (a)\nReal part and (b) imaginary part of the diagonal element;\n(c) imaginary part and (d) real part of the off-diagonal ele-\nment. All the spectra have been convoluted with a Lorentzian\nof 0.3 eV to simulate the finite quasiparticle lifetime effect s.\nRed lines are the optical conductivity derived from the expe r-\nimental dielectric constant. [14]\naandbis thus caused by the crystal field splitting.\nFigure 4 indicates that in YIG, the upper conduction\nbands from 4.4 to 6.0 eV are mainly of Y dorbital char-\nacter with some contribution from O porbitals. In BIG,\non the other hand, the upper conduction bands from 3.6\nto 6.0 eV are mainly the Bi and O porbital hybridized\nbands (seeFig. 5). Notably, there is sizableBi spDOS in\nthe lower conduction band region from 2.0 to 3.4 eV (see\nFig. 5(c)], indicating that the lower conduction bands in\nBIG are significantly mixed with Bi sporbitals, as no-\nticed already by Oikawa et al.[20], Since the SOC of the\nBiporbitals are very strong, this explains why the band\nwidth of the lower conduction bands in BIG increases\nfrom∼1.4 to 2.1 eV when the SOC is included (see Fig.\n3). In contrast, the band width of the lower conduction\nbands in YIG remains unaffected by the SOC (see Fig.\n2). This also explains why the MO effects in BIG are\nmuch stronger than in YIG, as reported in Sec. III.D.\nbelow.\nC. Optical Conductivity\nHere we present the optical and magneto-optical con-\nductivities for YIG and BIG which are ingredients for\ncalculating the Kerrand Faradayrotation angles [see Eq.\n(6) and Eq. (7)]. In particular, the MO conductivity\nFIG. 7. Calculated optical conductivity of Bi 3Fe5O12. (a)\nReal part and (b) imaginary part of the diagonal element;\n(c) imaginary part and (d) real part of the off-diagonal ele-\nment. All the spectra have been convoluted with a Lorentzian\nof 0.3 eV to simulate the finite quasiparticle lifetime effect s.\nRed lines are the optical conductivity derived from the expe r-\nimental dielectric constant. [17]\n(i.e., the off-diagonal element of the conductivity tensor\nσxy) is crucial, as shown by Eq. (8). Calculated optical\nconductivity spectra of YIG and BIG are plotted as a\nfunction of photon energy in Fig. 6 and Fig. 7, respec-\ntively. For YIG, the real part of the diagonal element of\nthe conductivity tensor ( σ1\nxx) starts to increase rapidly\nfrom the absorption edge ( ∼2.3 eV) to ∼4.0 eV, and\nthen further increases with a smaller slope up to ∼5.6\neV [see Fig. 6(a)]. It then decreases slightly until 6.6\neV and finally increases again with a much steeper slope\nup to∼8.0 eV. Similarly, in BIG, σ1\nxxincreases steeply\nfrom the absorption edge ( ∼2.0 eV) to ∼4.0 eV, and\nthen further increases with a smaller slope up to ∼6.0\neV [see Fig. 7(a)]. It then decrease steadily from ∼6.0\neV to∼8.0 eV. The behaviors of the imaginary part of\nthe diagonal element ( σ2\nxx) of YIG and BIG are rather\nsimilar in the energy range up to 5.0 eV [see Figs. 6(b)\nand 7(b)]. The σ2\nxxspectrum has a broad valley at ∼3.5\neV (∼3.0 eV) in the case of YIG (BIG). However, the\nσ2\nxxspectra of YIG and BIG differ from each other for\nenergy>5.0 eV. There is a sign change in σ2\nxxoccuring\nat∼5.8 eV for BIG, while there is no such a sign change\ninσ2\nxxof YIG up to 8.0 eV.\nThe striking difference in the off-diagonal element of\ntheconductivity( σxy)(i.e., magneto-opticalconductivity\nor magnetic circular dichroism) between YIG and BIG is\nthatσxyof BIG is almost ten times larger than that of7\nYIG (see Figs. 6 and 7). Nonetheless, the line shapes\nof the off-diagonal element of YIG and BIG are rather\nsimilar except that their signs seem to be opposite and\ntheir peaks appear at quite different energy positions. In\nparticular,inthelowenergyrangeupto ∼4.4eV,theline\nshape of the imaginary part of the off-diagonal element\n(σ2\nxy) of BIG looks like a ”W” [see Fig. 7(c)], while that\nof YIG in the energy region up to ∼7.0 eV seems to\nhave the inverted ”W” shape [see Fig. 6(c)], The main\ndifference is that the σ2\nxyof BIG decreases oscillatorily\nfrom 4.4 to 8.0 eV. On the other hand, the line shape\nof the real part of the off-diagonal element ( σ1\nxy) of BIG\nlooks like a ”sine wave” between 2.0 and 4.7 eV [see Fig.\n7(d)], while that of YIG appears to be an inverted ”sine\nwave”between 2.6and6.4eV[seeFig. 6(d)]. The largest\nmagnitude of σ2\nxyof YIG is ∼1.6×1013s−1at∼4.3 eV,\nwhile that of BIG is ∼1.9×1014s−1at∼3.1 eV. The\nlargest magnitude of σ1\nxyof YIG is ∼1.2×1013s−1at∼\n4.8 eV, while that of BIG is ∼1.9×1014s−1at∼2.6 eV.\nIn order to compare with the available experimental\ndata, we also plot the experimental optical conductivity\nspectra [14, 17] in Figs. 6 and 7. The theoretical spectra\nof the diagonalelement of the optical conductivity tensor\nfor both YIG and BIG match well with that of the exper-\nimental onesin the measuredenergyrange[see Figs. 6(a)\nand 6(b) as well as Figs. 7(a) and 7(b)]. Interestingly, we\nnote that the relativistic GGA+U calculations give rise\nto the band gaps of YIG and BIG that are smaller than\nthe experimental ones (see Table II), and yet the cal-\nculated and measured optical spectra agree rather well\nwith each other. This apparently contradiction can be\nresolved as follows. In YIG, for example, the lowest con-\nduction bands at E= 1.8∼2.4 eV above the VBM are\nhighly dispersive (see Fig. 2) and thus have very low\nDOS (see Fig. 4). This results in very low optical tran-\nsition. Therefore, the main absoption edge that appears\nin the optical spectrum ( σ1\nxx) is∼2.2 eV, which is close\nto the experimental absorption edge of 2.5 eV, instead\nof 1.8 eV as determined by the calculated band struc-\nture (see Table II). In contrast, no such highly dispersive\nbands appear at the CBM in BIG, Thus the calculated\nband gap agrees better with the measured band gap [17]\n(Table II).\nFigures 7(c) and 7(d) showthat the calculated σ1\nxyand\nσ2\nxyof BIG agree almost perfectly with the experimental\ndata [17]. The peak positions, peak heights and overall\ntrend of the theoretical spectra are nearly identical to\nthat of the experimental ones [17]. On the other hand,\nthe calculated σ1\nxyandσ2\nxyfor YIG do not agree so well\nwith the experimental data [14] [Figs. 6(c) and 6(d]. For\nexample, there is a sharp peak at ∼4.8 eV in the ex-\nperimental σ1\nxyspectrum, which seems to be shifted to a\nhigher energy at 5.6 eV with much reduced magnitude in\nthe theoretical σ1\nxyspectrum [seeFig. 6(c)]. Also, for σ2\nxy\nspectrum, there is a sharp peak at ∼4.5 eV in the exper-\nimentalσ2\nxyspectrum, which appears at ∼4.8 eV with\nconsiderably reduced height [see Fig. 6(d)]. Nonetheless,\nthe overall trend of the theoretical σxyspectra of YIGis in rather good agreement with that of the measured\nones [14].\nEquations(2), (3), and (8) indicate that the absorptive\nparts of the optical conductivity elements ( σ1\nxx,σ1\nzz,σ2\nxy\nandσ1\n±) are directly related to the dipole allowed in-\nterband transitions. Thus, we analyze the origin of the\nmain features in the magneto-optical conductivity ( σ2\nxy)\nspectrum by determining the symmetries of the involved\nband states and the dipole selection rules (see the Ap-\npendix for details). The absorptive optical spectra are\nusually dominated by the interband transitions at the\nhigh symmetry points where the energy bands are gener-\nally flat (see, e.g., Figs. 2 and 3), thus resulting in large\njoint density of states. As an example, here we consider\nthe interband optical transitions at the Γ point where\nthe band extrema often occur. Based on the determined\nband state symmetries and dipole selection rules (see Ta-\nble III in the Appendix) as well as calculated transition\nmatrix elements [Im( px\nijpy\nji)], we assign the main features\ninσ2\nxy[labelled in Figs. 6(c) and 7(c)] to the main inter-\nband transitionsat the Γ point asshown in Figs. 8 and 9.\nThe details of these assignments, related interband tran-\nsitions and transition matrix elements for YIG and BIG\nare presented in Tables IV and V in the Appendix, re-\nspectively. Since there are too many possible transitions\nto list, we present only those transitions whose transition\nmatrix elements |Im(px\nijpy\nji)|>0.010 a.u. in YIG (Table\nIV) and |Im(px\nijpy\nji)|>0.012 a.u. in BIG (Table V).\nFigure 8 shows that nearly all the main optical tran-\nsitions in YIG are from the upper valence bands to the\nupper conduction bands, and only one main transition\n(P3) to the lower conduction bands. Consequently, these\ntransitions contribute to the main features in σ2\nxyat pho-\nton energy >4.0 eV [see Fig. 6(c)]. In contrast, in BIG,\na large number of the main transitions (e.g., P1-5, P7,\nN1-4, N5-8) are from the upper valence bands to lower\nconduction bands (see Fig. 9). This gives rise to the\nmain features in σ2\nxyfor photon energy <4.0 eV [see\nFig. 7(c)], whose magnitudes are generally one order\nof magnitude larger than that of σ2\nxyin YIG, as men-\ntioned above. The largely enhanced MO activity in BIG\nstems from the significant hybridization of Bi p-orbitals\nwith Fed-orbitalsin the lowerconductionbands, asmen-\ntioned above. Since heavy Bi has a strongspin-orbit cou-\npling, this hybridization greatly increases the dichroic in-\nterband transitions from the upper valence bands to the\nlower conduction bands in BIG. As mentioned above, Y\nsdorbitals contribute significantly only to the upper con-\nduction bands in YIG, and this results in the pronounced\nmagneto-optical transitions only from the upper valence\nbands to the upper conduction bands (Fig. 8). Further-\nmore, Y is lighter than Bi and thus has a weaker SOC\nthan Bi.\nThe discussion in the proceeding paragraph clearly\nindicates that the significant hybridization of heavy Bi\nporbitals with Fe dorbitals in the lower conduction\nbands just above the band gap is the main reason for\nthe large MO effect in BIG. The magnetism in BIG is8\nFIG. 8. Relativistic band structures of Y 3Fe5O12. Horizontal\ndashed lines denote the top of valance band. The principal\ninterband transitions at the Γ point and the corresponding\npeaks in the σxyin Fig. 6 (c) are indicated by red and blue\narrows.\nmainly caused by the iron dorbitals which have a rather\nweak SOC. However, through the hybridization between\nBiporbitals and Fe dorbitals, the strong SOC effect\nis also transfered to the lower conduction bands. Large\nexchange splitting and strong spin-orbit coupling in the\nvalence and conduction bands below and above the band\ngaparecrucialforstrongmagneticcirculardichroismand\nhence large MO effects. Therefore, in search of materials\nwith strongMO effects, one shouldlook formagneticsys-\ntems that contain heavy elements such as Bi and Pt [38].\nD. Magneto-optical Kerr and Faraday effect\nFinally, let us study the polar Kerrand Faradayeffects\ninYIG andBIG.ThecomplexKerrandFaradayrotation\nangles for YIG and BIG are plotted as a function of pho-\nton energy in Figs. 8 and 9, respectively. First of all, we\nnotice that the Kerr rotation angles of BIG [Fig. 10(c)]\nare many times larger than that of YIG [Fig. 10(a)]. For\nexample, the positiveKerrrotationmaximumof0.10◦in\nYIG occurs at ∼3.6 eV, while that (0.80◦) for BIG ap-\npears at ∼3.5 eV. The negative Kerr rotation maximum\n(-0.12◦) of YIG occurs at ∼4.8 eV, while that (-1.21\n◦) for BIG appears at ∼2.4 eV. This may be expected\nbecause Kerr rotation angle is proportional to the MO\nconductivity ( σ1\nxy) [Eq. (6)], which in BIG is nearly ten\ntimes larger than in YIG, as mentioned in the proceed-\ning subsection. Similarly, the Kerr ellipticity maximum\n(0.16◦) of YIG occurs at ∼4.1 eV [Fig. 10(b)], whereas\nFIG. 9. Relativistic band structures of Bi 3Fe5O12. Horizontal\ndashed lines denote the top of valance band. The principal\ninterband transitions at the Γ point and the corresponding\npeaks in the σxyin Fig. 7 (c) are indicated by red and blue\narrows.\nthat (0.54◦) of BIG [Fig. 10(d)] appears at ∼1.9 eV.\nThe negative Kerr ellipticity maximum (-0.07◦) of YIG\noccurs at ∼5.7 eV while that (-1.16◦) of BIG is located\nat∼2.9 eV.\nLet us now compare our calculated Kerr rotation an-\ngles with some known MO materials such as 3 dtran-\nsition metal alloys and compound semiconductors. [8]\nFor magnetic metals, ferromagnetic 3 dtransition metals\nand their alloys are an important family. Among them,\nmanganese-basedpnictides areknownto havestrongMO\neffects. In particular, MnBi thin films were reported to\nhave a large Kerr rotation angle of 2.3◦. [39, 40] Plat-\ninum alloys such as FePt, Co 2Pt [38] and PtMnSb [41]\nalso possess large Kerr rotation angles. It was shown\nthat the strong SOC on heavy Pt in these systems is the\nmain cause of the strong MOKE. [38] Among semicon-\nductor MO materials, diluted magnetic semiconductors\nGa1−xMnxAs were reported to show Kerr rotations an-\ngle as large as 0.4◦at 1.80 eV. [42] Therefore, the strong\nMOKE effect in YIG and BIG could have promising ap-\nplicationsinhighdensityMOdata-storagedevicesorMO\nnanosensors with high spatial resolution.\nFigure 9 shows that as for the Kerr rotation angles,\nthe Faraday rotation angles of BIG are generally up to\nten times larger than that of YIG. The Faraday rotation\nmaximum (7.2◦/µm) of YIG occurs at ∼3.9 eV, while\nthat (51.2◦/µm) of BIG is located at ∼3.7 eV. The Fara-\nday ellipticity maximum (7.9◦/µm) for YIG appears at\n∼4.4 eV, whereas that (54.1◦/µm) of BIG occurs at\n∼2.3 eV. On the other hand, the negative Faraday rota-9\nFIG. 10. Calculated complex Kerr rotation angles (blue\ncurves). (a) Kerr rotation ( θK) and (b) Kerr ellipticity ( εK)\nspectra of Y 3Fe5O12; (c) Kerr rotation ( θK) and (d) Kerr el-\nlipticity ( εK) spectra of Bi 3Fe5O12. Red circles in (a) and (b)\ndenote the experimental values from Ref. [15].\ntion maximum (-5.7◦/µm) occurs at ∼5.4 eV, while that\n(-74.6◦/µm) for BIG appears at ∼2.7 eV. The negative\nFaraday ellipticity maximum (-3.6◦/µm) of YIG occurs\nat∼6.6 eV, while that (-70.2◦/µm) for BIG is located\nat∼3.2 eV. For comparision, we notice that MnBi films\nare known to possess large Faraday rotation angles of\n∼80◦/µm at 1.8 eV. [39, 40]\nFinally, we compare our predicted MOKE and MOFE\nspectrawiththeavailableexperimentsinFigs. 10and11.\nAllthepredictedMOKEandMOFEspectraareinrather\ngoodagreementwiththeexperimentalonesintheexperi-\nmental photon energyrange[14, 15, 17, 43]. Nonetheless,\nour theoretical predictions would have a better agree-\nment with the experiments if all the calculated spectra\nare blue-shifted slightly by ∼0.3 eV, thus suggesting that\nthe theoretical band gaps are slightly too small.\nIV. CONCLUSION\nTosummarize,wehavesystematicallystudiedthe elec-\ntronic structure, magnetic, optical and MO properties of\ncubic iron garnets YIG and BIG by performing GGA+U\ncalculations. We find that YIG exhibits significant MO\nKerr and Faraday effects in UV frequency range that are\ncomparable to cubic ferromagnetic iron. Strikingly, we\nfind that BIG shows gigantic MO effects in the visible\nfrequency region that are several times larger than YIG.\nIn particular, the Kerr rotation angle of BIG becomes\nFIG. 11. Calculated complex Faraday rotation angles (blue\ncurves). (a) Faraday rotation ( θF) and (b) Faraday ellipticity\n(εF) spectra of Y 3Fe5O12; (c) Kerr rotation ( θF) and (d) Kerr\nellipticity ( εF) spectra of Bi 3Fe5O12. Red dashed line in (a)\ndenotes the measured values from Ref. [15]. Black circles in\n(a) and (b) are the experimental values from Ref. [14]. Red\n(green) circles in (c) and (d) are the experimental values fr om\nRef. [43] ([17])\nas large as -1.2◦at photon energy 2.4 eV, and the Fara-\nday rotation angle for the BIG film reaches -75◦/µm\nat 2.7 eV. Calculated MO conductivity ( σ2\nxy) spectra re-\nveal that these distinctly different MO properties of YIG\nand BIG result from the fact that the magnitude of σ2\nxy\nof BIG is nearly ten times larger than that of YIG. Our\ncalculatedKerrandFaradayrotationanglesofYIG agree\nwell with the available experimental values. Our calcu-\nlatedFaradayrotationanglesofBIG areinnearlyperfect\nagreement with the measured ones. Thus, we hope that\nour predicted giant MO Kerr effect in BIG will stimulate\nfurther MOKE experiments on high quality BIG crys-\ntals.‘\nPrincipal features in the optical and MO spectra are\nanalyzed in terms of the calculated band structures espe-\ncially the symmetry of the band states and optical tran-\nsition matrix elements at the Γ point of the BZ. We find\nthat in YIG, Y sdorbitals mix mainly with the upper\nconduction bands that are ∼4.5 eV abovethe VBM, and\nthus leave the Fe dorbital dominated lower conduction\nbands from 1.8 to 3.8 eV above the VBM almost unaf-\nfected by the SOC on the Y atom. In contrast, Bi por-\nbitals in BIG hybridize significantly with Fe dorbitals in\nthe lower conduction bands and this leads to large SOC-\ninduced band splitting andmuch increasedband width of\nthe lowerconduction bands. Consequently, the MO tran-10\nsitions between the upper valence bands and lower con-\nduction bands are greatly enhanced when Y is replaced\nby heavier Bi. This finding thus provides a guideline in\nsearch for materials with desired MO effects, i.e., one\nshould look for magnetic materials with heavy elements\nsuch as Bi whose orbitals hybridize significantly with the\nMO active conduction or valence bands.\nFinally, our findings of strong MO effects in these iron\ngarnetsand alsosingle-spinsemiconductivityin BIGsug-\ngest that cubic iron garnets are an useful playground of\nexploring the interplay of microwave, spin current, mag-\nnetism, and optics degrees of freedom, and also have\npromising applications in high density semiconductor\nMO data-storage and low-power consumption spintronic\nnanodevices.\nACKNOWLEDGMENTS\nThe authors thank Ming-Chun Jiang for many valu-\nable discussions throughout this work. The authors ac-\nknowledge the support from the Ministry of Science and\nTechnology and the National Center for Theoretical Sci-\nences (NCTS) of The R.O.C. The authors are also grate-\nful to the National Center for High-performance Com-\nputing (NCHC) for the computing time. G.-Y. Guo also\nthanks the support from the Far Eastern Y. Z. Hsu Sci-ence and Technology Memorial Foundation in Taiwan.\nAPPENDIX: DIPOLE SELECTION RULES AND\nSYMMETRIES OF BAND STATES AT Γ\nIn this Appendix, to help identify the origins of the\nmainfeaturesinthe magneto-opticalconductivity σxy(ω)\nspectra of YIG and BIG, we provide the dipole selection\nrules and the symmetries of the band states at the Γ as\nwell as the main optical transitions between them.\nBothYIG andBIGhavethe Ia¯3dspacegroupandthus\nthey have the C4h(4/mm′m′) point group at the Γ point\nin the Brillouin zone. Based on the character table of the\nC4hpoint group [44], we determine the dipole selection\nrules for the optical transitions between the band states\nat the Γ point, as listed in Table III. We calculate the\neigenvalues for all symmetry elements of each eigenstate\nof the Γ point using the Irvspprogram [45] and then\ndetermine the irreducible representation and hence the\nsymmetry of the state. Based on the obtained symme-\ntriesof the band states and alsocalculatedoptical matrix\nelements [Im( px\nijpy\nji)] [see Eq. (3)], we assign the peaks\nin theσxy(ω) spectra of YIG [see Fig. 6(c)] and BIG [see\nFig. 7(c)] to the main optical transitions at the Γ point\n(see Fig. 8 and 9, respectively), as listed in Tables IV\nand V, respectively.\n[1] V. Cherepanov, I. Kolokolov, and V. Lvov, The saga\nof YIG: spectra, thermodynamics, interaction and relax-\nation of magnons in a complex magnet, Phys. Rep. 229,\n81 (1993).\n[2] S. Mizukami, Y. Ando, and T. Miyazaki, Effect of spin\ndiffusion on Gilbert damping for a very thin permal-\nloy layer in Cu/permalloy/Cu/Pt films, Phys. Rev. B 66,\n104413 (2002).\n[3] S. Chikazumi, Physics of Ferromagnetism , 2nd ed. (Ox-\nford University Press, Oxford, 1997)\n[4] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Transmission of\nelectrical signals by spin-wave interconversion in a mag-\nnetic insulator, Nature (London) 464, 262 (2010).\n[5] T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R.\nL. Stamps, and M. P. Kostylev, Realization of spin-wave\nlogic gates, Appl. Phys. Lett. 92(2), 022505 (2008).\n[6] Y. Sun, H. Chang, M. Kabatek, Y.-Y. Song, Z. Wang, M.\nJantz, W. Schneider, M. Wu, E. Montoya, B. Kardasz,\nB. Heinrich, S. G. E. te Velthuis, H. Schultheiss, and A.\nHoffmann, Damping in Yttrium Iron Garnet Nanoscale\nFilms CappedbyPlatinum, Phys.Rev.Lett. 111, 106601\n(2013).\n[7] P. M. Oppeneer, Chapter 1 Magneto-optical Kerr Spec-\ntra, pp. 229-422, in Handbook of Magnetic Materials ,\nedited by K. H. J. Buschow. Elsevier, Amsterdam,\n(2001).\n[8] V. Antonov, B. Harmon, and A. Yaresko. Elec-\ntronic structure and magneto-optical properties of solids .TABLE III. Dipole selection rules for the C4hpoint group at\nthe Γ point in the Brillouin zone of YIG and BIG.\npolarization Γ+\n6Γ+\n5Γ+\n8Γ+\n7Γ−\n6Γ−\n5Γ−\n8Γ−\n7\nz Γ−\n6Γ−\n5Γ−\n8Γ−\n7Γ+\n6Γ+\n5Γ+\n8Γ+\n7\nx+iy Γ−\n5Γ−\n8Γ−\n7Γ−\n6Γ+\n5Γ+\n8Γ+\n7Γ+\n6\nx−iy Γ−\n7Γ−\n6Γ−\n5Γ−\n8Γ+\n7Γ+\n6Γ+\n5Γ+\n8\nSpringer Science & Business Media, (2004).\n[9] J. P. Castera, in Magneto-optical Devices , Vol.9ofEncy-\nclopedia of Applied Physics , edited byG. L. Trigg (Wiley-\nVCH, New York, 1996), p. 133.\n[10] M. Mansuripur, The Principles of Magneto-Optical\nRecording (Cambridge Univ. Press, Cambridge, 1995).\n[11] F. D. M. Haldane and S. Raghu, Possible Realization\nof Directional Optical Waveguides in Photonic Crystals\nwith Broken Time-Reversal Symmetry, Phys. Rev. Lett.\n100, 013904 (2008).\n[12] L. J. Aplet and J. W. Carson, A Faraday effect optical\nisolator, Appl. Opt. 3, 544 (1964).\n[13] J. F. Dillon, Optical properties of several ferrimagne tic\ngarnets, J. Appl. Phys. 29, 539 (1958).\n[14] S. Wittekoek, T. J. A. Popma, J. M. Robertson, P. F.\nBongers, Magneto-optic spectra and the dielectric ten-\nsor elements of bismuth-substituted iron garnets at pho-\nton energies between 2.2-5.2 eV, Phys. Rev. B 12, 2777\n(1975).11\nTABLE IV. Main optical transitions between the states at\nthe Γ point of the Brillouin zone of YIG. Symbols in the\nfirst column denote the assigned peaks in the magneto-optica l\nconductivity ( σ2\nxy) spectrum (Figs. 6 and 8). iandjdenote\nthe initial and final states, respectively. Im( px\nijpy\nji) denote the\ncalculated transition matrixelement (inatomic units)[se e Eq.\n(3)].EiandEjrepresent the initial and final state energies\n(in eV), respectively. ∆ Eij=Ej−Eiis the transition energy.\nPeak state istatejIm(px\nijpy\nji) ∆EijEjEi\nP8 545 (Γ+\n6) 802 (Γ−\n5) 0.0103 6.928 4.358 -2.569\nP3 556 (Γ−\n5) 761 (Γ+\n8) 0.0115 5.438 3.010 -2.428\nP6 609 (Γ−\n7) 803 (Γ+\n6) 0.0102 6.207 4.442 -1.765\nN4 632 (Γ−\n5) 803 (Γ+\n6) -0.0273 5.825 4.442 -1.384\nP5 635 (Γ−\n7) 803 (Γ+\n6) 0.0251 5.818 4.442 -1.376\nN3 649 (Γ−\n5) 803 (Γ+\n6) -0.0146 5.567 4.442 -1.125\nP4 651 (Γ−\n7) 803 (Γ+\n6) 0.0195 5.564 4.442 -1.123\nP7 668 (Γ+\n6) 844 (Γ−\n5) 0.0136 6.744 5.994 -0.750\nN5 670 (Γ+\n8) 844 (Γ−\n5) -0.0119 6.739 5.994 -0.745\nP2 690 (Γ−\n6) 801 (Γ+\n5) 0.0116 4.537 4.280 -0.257\nN2 692 (Γ−\n8) 801 (Γ+\n5) -0.0107 4.535 4.280 -0.254\nP1 698 (Γ−\n6) 801 (Γ+\n5) 0.0431 4.292 4.280 -0.012\nN1 700 (Γ−\n8) 801 (Γ+\n5) -0.0452 4.280 4.280 0.000\nTABLE V. Main optical transitions between the states at the\nΓ point of the Brillouin zone of BIG. Symbols in the first col-\numn denote the assigned peaks in the magneto-optical con-\nductivity ( σ2\nxy) spectrum (Figs. 7 and 9). iandjdenote the\ninitial and final states, respectively. Im( px\nijpy\nji) denote the cal-\nculated transition matrix element (in atomic units) [see Eq .\n(3)].EiandEjrepresent the initial and final state energies\n(in eV), respectively. ∆ Eij=Ej−Eiis the transition energy.\nPeak state istatejIm(px\nijpy\nji) ∆EijEjEi\nN8 578 (Γ+\n8) 783 (Γ−\n5) -0.0130 5.234 2.304 -2.930\nP7 578 (Γ+\n8) 789 (Γ−\n7) 0.0123 5.331 2.401 -2.930\nN7 579 (Γ+\n6) 782 (Γ−\n7) -0.0139 5.228 2.298 -2.930\nN9 661 (Γ−\n6) 856 (Γ+\n7) -0.0136 5.333 3.452 -1.882\nP4 684 (Γ−\n7) 846 (Γ+\n6) 0.0127 4.685 3.242 -1.443\nP5 709 (Γ+\n8) 869 (Γ−\n7) 0.0139 4.839 3.825 -1.014\nP9 710 (Γ+\n6) 873 (Γ−\n5) 0.0149 5.453 4.467 -0.986\nN5 712 (Γ+\n8) 842 (Γ−\n5) -0.0121 4.153 3.175 -0.979\nP3 715 (Γ+\n7) 854 (Γ−\n6) 0.0135 4.271 3.336 -0.935\nP8 723 (Γ+\n7) 876 (Γ−\n6) 0.0142 5.374 4.571 -0.803\nN6 727 (Γ+\n8) 873 (Γ−\n5) -0.0145 5.206 4.467 -0.738\nP2 741 (Γ−\n6) 872 (Γ+\n5) 0.0122 4.254 3.917 -0.337\nP6 743 (Γ−\n5) 878 (Γ+\n8) 0.0127 5.305 4.991 -0.314\nN3 743 (Γ−\n5) 749 (Γ+\n6) -0.0184 2.113 1.799 -0.314\nN2 744 (Γ−\n8) 755 (Γ+\n5) -0.0160 2.037 1.973 -0.063\nN1 745 (Γ−\n7) 753 (Γ+\n8) -0.0142 1.980 1.917 -0.062\nN4 747 (Γ−\n7) 871 (Γ+\n8) -0.0138 3.891 3.891 0.000\nP1 715 (Γ+\n7) 760 (Γ−\n6) 0.0125 3.032 2.097 -0.935\n[15] F. 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Goennenwein3, 2\n1Department of Materials, ETH Z urich, 8093 Z urich, Switzerland\n2Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden\nand W urzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany\n3Department of Physics, University of Konstanz, 78457 Konstanz, Germany\n4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n5State Key Laboratory of Surface Physics and Institute for Nanoelectronic\nDevices and Quantum Computing, Fudan University, Shanghai 200433, China\n6Zhangjiang Fudan International Innovation Center, Fudan University, Shanghai 201210, China\n7AIMR and CSRN, Tohoku University, Sendai 980-8577, Japan\n8Zernike Institute for Advanced Materials, Groningen University, Groningen, The Netherlands\n9Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n10Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n11Munich Center for Quantum Science and Technology (MCQST), 80799 M unchen, Germany\n(Dated: February 8, 2022)\n\\Pumping\" of phonons by a dynamic magnetization promises to extend the range and functionality\nof magnonic devices. We explore the impact of phonon pumping on room-temperature ferromagnetic\nresonance (FMR) spectra of bilayers of thin yttrium iron garnet (YIG) \flms on thick gadolinium\ngallium garnet substrates over a wide frequency range. At low frequencies the Kittel mode hybridizes\ncoherently with standing ultrasound waves of a bulk acoustic resonator to form magnon polarons that\ninduce rapid oscillations of the magnetic susceptibility, as reported before. At higher frequencies,\nthe phonon resonances overlap, merging into a conventional FMR line, but with an increased line\nwidth. The frequency dependence of the increased line broadening follows the predictions from\nphonon pumping theory in the thick substrate limit. In addition, we \fnd substantial magnon-phonon\ncoupling of a perpendicular standing spin wave (PSSW) mode. This evidences the importance of the\nmode overlap between the acoustic and magnetic modes, and provides a route towards engineering\nthe magnetoelastic mode coupling.\nMagnons and phonons are, respectively, the elemen-\ntary excitations of the magnetic and atomic order in\ncondensed matter. They are coupled by weak magne-\ntoelastic and magnetorotational interactions, which can\noften simply be disregarded. However, recent experimen-\ntal and theoretical research reveals that the magnon-\nphonon interaction may cause spectacular e\u000bects in (i)\nferromagnets close to a structural phases transition such\nas Galfenol [1, 2] or (ii) magnets with exceptionally high\nmagnetic and acoustic quality such as yttrium iron gar-\nnet [3{9].\nMagnons are promising carriers for future low-power\ninformation and communication technologies [10, 11].\nThe magnon-phonon interaction can bene\ft the function-\nality of magnonic devices by helping to control and en-\nhance magnon propagation when coherently coupled into\nmagnon polarons [4, 9]. On the other hand, magnon non-\nconserving magnon-phonon scattering is the main source\nof magnon dissipation at room temperature [12, 13].\nThe study of magnon-phonon interactions in high-\nquality magnets has a long history [14{21]. The arrival of\ncrystal growth techniques, strongly improved microwave\ntechnology, and discovery of new phenomena such as the\nspin Seebeck e\u000bect, led to a revival of the subject in the\npast few years, with emphasis on ultrathin \flms and het-\nerostructures [4, 6, 22{30].High-quality yttrium iron garnet (YIG) is an excellent\nmaterial to study magnons and phonons. Thin \flms grow\nbest on single-crystal substrates of gadolinium gallium\ngarnet (GGG), a paramagnetic insulator that is mag-\nnetically inert at elevated temperatures. However, the\nacoustic parameters of GGG are almost identical to YIG\nsuch that phonons are not localized to the magnet and\nthus the substrate cannot be simply disregarded. Streib\net al. [3] pointed out that magnetic energy can leak into\nthe substrate by magnon-phonon coupling by a process\ncalled \\phonon pumping\" and predicted that it should\ncause an increased magnetization damping with a char-\nacteristic non-monotonous dependence on frequency.\nPhonon pumping has been experimentally observed in\nthe ferromagnetic resonance of YIG \flms on GGG sub-\nstrates [4, 29, 30]. These experiments revealed coher-\nent hybridization of the (uniform) Kittel magnon with\nstanding sound waves extended over the whole sample.\nIn YIG/GGG/YIG trilayers phonon exchange couples\nmagnons dynamically over mm distances [4, 9]. However,\nthe predicted increased damping due to phonon pump-\ning and the coupling of other than the macro-spin Kittel\nmagnon remained elusive. The direct detection of the\nincreased damping is challenging due to the presence of\ninhomogeneous FMR line broadening and the resulting\nchanges of the resonance line shape, in particular in thearXiv:2202.03331v1 [cond-mat.mes-hall] 7 Feb 20222\nlow frequency regime for thin \flms [31] or due to the pres-\nence of several modes in the resonance for thicker YIG\n\flms [32{35].\nIn this Letter, we report FMR spectra of YIG/GGG\nbilayers over a large frequency range, demonstrating the\ncoupling of magnons and phonons from the high coop-\nerativity to the incoherent regime. We reproduce the\nmagnon polaron \fne structure at low frequencies [4, 29],\nand evidence the presence of the acoustic spin pumping\ne\u000bect on the magnetic dissipation predicted in Ref. [3]at higher frequencies. The excellent agreement with an\nanalytical model allows us to extract the parameters for\nthe phonon pumping by the (even) Kittel mode in the\nstrong coupling regime, and provides insights into the\nstrong-weak coupling regime at higher frequencies. In\naddition, we observe that the magnon-phonon coupling\nstrength also is characteristically modulated for an (odd)\nperpendicular standing spin wave mode. This shows that\nthe overlap integral between magnon and phonon modes\ngoverns the coupling strength, thus opening a pathway\nfor controlling it.\n(a)\nhrf\nCPWVNA\nP1 P2\nz || H0x\ny\nlow high\n(g)3(e)\n(f)GGG\nGGG\nFIG. 1. (a) A thin YIG \flm on a thick paramagnetic GGG substrate (gray square) is placed face down on a coplanar\nwaveguide. The latter is connected to a vector network analyzer to obtain the transmission parameter S21as a function of\nfrequency!. The external static magnetic \feld H0is applied normal to the surface. (b-d) High resolution maps of jS21jfor\ndi\u000berent!andH0. Shear waves with sound velocity ctand wavelength \u0015pform standing waves across the full layer stack. The\nthree panels correspond to tYIG\u0018\u0015p=2;\u0015pand 3\u0015p=2, respectively. The fundamental (Kittel) mode and the \frst perpendicular\nstanding spin wave (PSSW) are marked with orange and blue dashed lines and arrows, respectively. (e,f) The thickness pro\fle\nof the magnetic excitation ml(z) is shown for the Kittel mode (e) and the \frst PSSW (f) in the YIG \flm for p= 0:5;together\nwith the eigenmodes of the acoustic strain @\u0018(z)=@zcorresponding to panels (b-d). The positive and negative contributions\nto the magnetoelastic exchange integral are shaded in red and gray, respectively. The magnetic excitation and thus the mode\noverlap vanishes in the GGG layer, whereas the phonons extend across full YIG/GGG sample stack. (g) The magnetoelastic\nmode coupling gmeis proportional to the overlap of the phonon and magnon modes and shows characteristic oscillations.\nOur sample consists of a 630 nm Y 3Fe5O12\flm on a\n560µm thick Gd 3Ga5O12substrate glued onto a copla-\nnar waveguide (CPW) with a center conductor width\nw= 110 µm. It is inserted into the air gap of an electro-\nmagnet with surface normal parallel to the magnetic \feld\n[cf. Fig. 1(a)]. We improve the magnetic \feld resolution\nto the 1 µT range by an additional Helmholtz coil pair\nin the pole gap of the electromagnet that is biased with\na separate power supply. We measure the complex mi-\ncrowave transmission spectra S21(!) by a vector network\nanalyzer for a series of \fxed magnetic \feld strengths over\na large frequency interval at room temperature.\nWe \frst address S21(!) in the strong-couplingregime [4] in the form of jS21jas a function of magnetic\n\feld and frequency, see Fig. 1(b). The FMR reduces\nthe transmission, emphasized by blue color and centered\nat the dashed orange line. Periodic perturbations in\nthe FMR at \fxed frequencies with period of \u00183:2 MHz\n(dashed white lines) correspond to the acoustic free spec-\ntral range of the sample\n\u0001!p\n2\u0019\u0019ct\n2tGGG\u00183:2 MHz; (1)\nwherect= 3570 m s\u00001is the transverse sound veloc-\nity of GGG [36]. These are the anticrossings of the\nFMR dispersion with \feld-independent standing acous-3\ntic shear wave modes across the full YIG/GGG layer\nstack [4, 29, 30, 37, 38]. In Fig. 1(b) we addition-\nally observe a resonance corresponding to the PSSW\n(dashed blue line) shifted to a lower magnetic \felds by\nexchange splitting \u00160\u0001H=D\u00192=t2\nYIG\u00181:4 mT, where\nD= 5\u000210\u000017Tm2is the exchange sti\u000bness of YIG [32{\n35], but without visible coupling to the phonons.\nAt 6:15 GHz [cf. Fig. 1(c)] the periodic anticrossings\nvanish for the Kittel mode resonance, which implies a\nstrongly suppressed magnon-phonon coupling. In con-\ntrast, the PSSW now exhibits clear anticrossings similar\nto that of the Kittel mode in panel (b). Increasing the\nfrequency further [cf. Fig. 1(d)] to around 9 :63 GHz, the\nperiodic oscillations in the PSSW vanish again, but the\nanticrossings of the Kittel mode do not recover.\nWe interpret the suppression of the magnon polaron\nsignal at higher frequencies around 9 GHz in terms of a\ntransition from the (underdamped) high cooperativity [4]\nto the (overdamped) weak coupling regime. In the latter,\nthe di\u000berent phonon modes overlap, leading to a constant\ncontribution of the phonon pumping to the magnon line\nwidth. As a consequence, the periodic magnon polaron\nsignatures vanish in favor of a slowly varying additional\nbroadening of the FMR line that was predicted theoret-\nically in the limit of thick GGG substrates [2, 3].\nThe coupling between the elastic and the magnetic sub-\nsystems in a con\fned magnet scales with the overlap in-\ntegral of the phonon and magnon modes [3, 29]. The\npro\fle of a PSSW with index lcan be modelled by\nml(z) =psin ([l+ 1]\u0019[z+tYIG]=tYIG) +\n(1\u0000p) cos (l\u0019[z+tYIG]=tYIG); (2)\nwherez2[\u0000tYIG;0] and 0\u0014p\u00141 interpolates be-\ntween free ( p= 0) and pinned ( p= 1) surface dynamics.\nAssuming free elastic boundary conditions, a shear wave\nacross the full layer stack with amplitude \u0018and frequency\n!creates a strain pro\fle (disregarding the standing wave\nformation and thus the \fnite free spectral range) in the\nYIG \flm that is given by\n@\u0018(z)\n@z=!\nectsin!(tYIG+z)\nect; (3)\nwhere@\u0018(z) is the local displacement and ect= 3843 m s\u00001\nis the transverse sound velocity of YIG. Note that the\nladder of modes is disregarded here for simplicity. The\noverlap integral of the fundamental (Kittel) mode with\nl= 0 (Fig. 1(e)) and the \frst PSSW with l= 1 (Fig. 1(f))\nenters the interaction magnetoelastic coupling gmeas [29]\ngme;l=s\n2b2\r\n!\u001aM stGGGtYIG\f\f\f\fZ0\n\u0000tYIGml(z)@\u0018(z)\n@zdz\f\f\f\f;(4)\nwhere the parameters for YIG at room temperature are\nthe magnetoelastic coupling constant b= 7\u0002105J=m3,the mass density \u001a= 5:1 g=cm3, the gyromagnetic ratio\n\r=2\u0019= 28:5 GHz T\u00001and the saturation magnetization\nMs= 143 kA m\u00001[4].gme;0andgme;1in Fig. 1(g) for\np= 0:5 (yellow and blue lines, respectively) reveal dif-\nferences in the magnetoelastic coupling of the di\u000berent\nmagnetic modes. In both cases the coupling oscillates as\na function of frequency, but the maxima for l= 0 and\nl= 1 are shifted by l\u0001ect=2tYIG\u00193 GHz. Note that this\nis true also for the higher standing spin wave modes, so\nthat even at high frequencies, strong magnon-phonon in-\nteractions can be realized. In other materials the results\nmay depend on the details of the interface and surface\nboundary conditions [23].\n(a)\nphoton phonon magnon\nFIG. 2. (a) The phonons and magnons in YIG/GGG bilay-\ners form a two-partite system that can be modeled as coupled\nharmonic oscillators that are driven by microwave photons [4].\nThe parameters are the resonance frequencies !m;!p, damp-\ning constants \rm;\rp;and coupling strength gme. The coplanar\nwaveguide with transmission amplitude a, phase\u000band elec-\ntric length\u001c, interacts with the magnon system parametrized\nby the coupling strength \u0011. (b)jS21jas a function of the\nfrequency for \u00160H\u0019259 mT. A Gaussian \ft (red line) deter-\nmines the FMR frequency !m(right dashed line). (c) Zoom-in\non the phonon line with !m\u0000!p= 22 MHz (marked by the\nleft dashed line in panel a). We obtain the line width \rpand\nthe amplitude hpof the phonon resonance by a Lorentzian\n\ft.\nA phonon and a magnon mode with discrete frequen-\ncies!pand!m[=\r\u00160(H\u0000Me\u000b) for the Kittel mode]\nand amplitudes ApandAm;respectively, behave as two\nharmonic oscillators coupled by the interaction gme[4]:\n\u0000Am(\rm+i(!m\u0000!))\u0000iApgme=2 +\u0011= 0 (5)\n\u0000Ap(\rp+i(!p\u0000!))\u0000iAmgme=2 = 0;(6)\nwhere\rmand\rpare the decay rates (in angular fre-\nquency units). \u0011parametrizes the coupling of the mag-\nnetic order to the external microwaves at frequency !,\nsee Fig. 2(a). The solution for the magnetic amplitude is\nAm=\u0011\u0014\u0010gme\n2\u00112 1\n\rp+i(!p\u0000!)+ (\rm+i(!m\u0000!))\u0015\u00001\n:\n(7)4\nThis resonator couples to a CPW according to [39]\nS21(!) =aexp(i\u000b) exp(\u0000i\u001c!) [1\u0000Am]: (8)\nin which the \frst part in the square brackets rep-\nresents the external circuit with frequency-dependent\namplitude and phase shift aand\u000b;respectively, and\n\u001cis an electronic delay time. We can \ft the un-\nknown parameters \u0011andgmeto the observed spec-\ntra in Fig. 2(b,c)]. \u0011can be extracted from the\namplitude of the ferromagnetic resonance by solving\nhm=jS21(!=!m)\u0011=0j\u0000jS21(!=!m)gme=0j \u0011f(\u0011).\nSimilarly, the data for a phonon resonance hp=\njS21(!=!p)gme=0j\u0000jS21(!=!p)j \u0011g(gme) can be\nsolved forgme.hpandhmcan be extracted from \fts\nto the experimental data.\nWe \ft the Kittel mode lines in jS21(!)jat di\u000berent\n\fxed magnetic \felds by a Gaussian to distill the reso-\nnance frequency !m, the amplitude hmand width \rm\n(cf. Fig. 2b). A good \ft by a Gaussian line shape in-\ndicates inhomogeneous broadening of the FMR, see be-\nlow. Next, we select an acoustic resonance at a frequency\n!p;0with (!m\u0000!p;0)=2\u0019 >2\rm=2\u0019\u001922 MHz, which is\nonly weakly perturbed by the magnon-phonon coupling,\nbut still has a signi\fcant oscillator strength. For a bet-\nter statistics, we independently \ft a total of six phonon\nresonances with frequencies below !p;0by Lorentzians\n[cf. Fig. 2(c)] to extract their average height hpand\nbroadening \rp.\n1012m/2 (MHz)\nG=1.7×104\nm,0/2=9.3 MHz\n(a)\n0.250.500.75p/2 (MHz)\n/2=5.2×106 GHz1\np,0/2=144 kHz\n(b)\n0 2 4 6 8 10 12\n/2 (GHz)\n02gme/2 (MHz)\n(c)\nFIG. 3. (a) Half width at half maximum (HWHM) obtained\nfrom Gaussian \fts to the FMR lines. (b) HWHM from the\nLorentzian \fts to the acoustic resonances. (c) Magnetoelastic\nmode coupling obtained from the harmonic oscillator model\nusing the parameters from the \fts shown in (a), (b) and in\nthe SM [40]. The maximum coupling strength is \u00182:2 MHz.\nThe resulting \ft parameters are summarized in Fig. 3\nand in the supporting material [40]. The line width of theFMR\rm=\rm;0+\u000bG!shown in panel (a) is dominated\nby inhomogeneous broadening \rm;0=2\u0019= 9:3 MHz, which\nwe associate to variations of the local (e\u000bective) magne-\ntization over the 6 mm long sample and across the thick-\nness pro\fle. The low Gilbert damping \u000bG\u00181:7\u000210\u00004\nis evidence for an intrinsically high quality of the YIG\n\flm. We associate the parabolic increase of the sound\nattenuation \rp=\u0010!2+\rp;0[cf. Fig. 3(b)] to thermal\nphonon scattering in GGG [41{43]. The inhomogeneous\nphonon line width \rp;0=2\u0019= 144 kHz may be caused by\na small angle\u00181°between the bottom and top surfaces\nof our sample [44], where the estimate is based on the\nphonon mean-free-path \u000e\u0018ct=\rp\u00194 mm [4]. We do not\nobserve a larger scale disorder in the substrate thickness\nthat would contribute a term \u0018!to the attenuation [42].\nIn the lower frequency regime !=2\u0019.10 GHz the\nphonon mean-free path \u000e > 1 mm is larger than twice\nthe thickness of the bilayer. At frequencies above 10 GHz,\nhowever, we enter the cross-over regime between high co-\noperativity and weak coupling in which the phononic free\nspectral range approaches its attenuation (\u0001 !p\u00182\rp).\nThe \ftting with individual phonon lines becomes increas-\ningly inaccurate, as the baseline of the FMR signal with-\nout contributions due to phonons cannot be established\nfrom the data. In this regime, the strongly overlap-\nping phonon lines thus lead to an average increase of\nthe FMR line width in addition to the rapidly oscillat-\ning contributions. If the frequency is increased further,\nthe mode overlap further increases and the thickness of\nthe stack becomes irrelevant so that we can take it to be\nin\fnite. In this limit, and for \fnite magnetoelastic cou-\npling, phonons just give rise to an average broadening\nof the FMR line. While indirect, our observations thus\ncon\frm the predicted damping enhancement by phonon\npumping in the incoherent limit [2, 3].\nThe oscillations observed in magnetoelastic mode cou-\nplinggmein panel 3(c) agree well with the model Eq. (4)\n(red line) for a YIG \flm with a thickness of tYIG =\n630 nm and a pinning parameter p= 0:5 (from Fig. 1(g))\nfor!=2\u0019.7 GHz. An alternative assessment based on\na full \ft of the experiments by the coupled equations\nfor the complex scattering parameter leads to a similar\ngme=2\u0019= 1:6 MHz at!=2\u0019\u00192:2 GHz (see SM [40]). The\nmodel likely overestimates the coupling strength, since\nthe inhomogeneous contributions to the line broadening\nare not considered independently here.\nIn summary, our high resolution FMR data taken over\na broad frequency range con\frm that magnon-phonon\ncoupling in con\fned systems depends not only on the\nmaterial parameters, but also qualitatively changes with\nthe mode overlap. This provides the option of tun-\ning the magnon-phonon coupling strength by the fre-\nquency, magnetic \feld variations and sample geometry.\nWe analyzed the magnon-phonon mode coupling over\na broad frequency range by a simple harmonic oscilla-\ntor model, revealing the oscillating nature of the acous-5\ntic spin pumping e\u000eciency as predicted theoretically [3].\nBroadband phonon pumping experiments in heterostruc-\ntures as presented here can thus be used as experimental\nplatform to study the in\ruence of the magnetic phase\ndiagram on the acoustic properties also in an adjacent\nmagnetic substrate, e.g. in the frustrated magnetic phase\nat very low temperatures in GGG [45].\nWe would like to thank O. Klein and A. Kamra for\nfruitful discussions. We acknowledge \fnancial support\nby the Deutsche Forschungsgemeinschaft via SFB 1432\n(project no. B06), SFB 1143 (project no. 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By means of the lock-in thermography techn ique, we measured the spatial distribution of\nthe SPE-induced temperature modulation in the Pt/YIG syste m with the tYIGgradation, allowing us to obtain\nthe accurate tYIGdependence of SPE with high tYIGresolution. Based on the tYIGdependence of SPE, we\nverified the applicability of several phenomenological mod els to estimate the magnon diffusion length in\nYIG.\n1Interconversion between spin and heat currents has been ext ensively studied in the field of spin\ncaloritronics [1, 2]. One of the spin-caloritronic phenome na is the spin Seebeck effect (SSE) [3],\nwhich generates a spin current as a result of a heat current in metal/magnetic-insulator junction\nsystems. The Onsager reciprocal of SSE is the spin Peltier eff ect (SPE) [4–11]. A typical system\nused for studying SPE and SSE is paramagnetic metal Pt/ferri magnetic insulator yttrium iron garnet\n(YIG) junction systems [3–5, 7, 12–15], where the spin and he at currents are carried by electron\nspins in Pt and magnons in YIG [16–19].\nSPE and SSE are characterized by their length scale includin g the magnon-spin diffusion length\nlmand the magnon-phonon thermalization length lmpin YIG [18, 20, 21]. These length parameters\nhave been investigated by measuring discrete YIG-thicknes stYIGdependence of SPE and SSE in\nseveral Pt/YIG junction systems with different tYIG[7, 13, 14, 21, 22]. However, different junctions\nmay have different magnetic properties, surface roughness, crystallinity, and interface condition.\nThese variations make it hard to analyze the fine tYIGdependence and obtain correct values of the\nlength parameters.\nIn this letter, we measured the tYIGdependence of SPE by using a single Pt/YIG system with\natYIGgradient ( ∇tYIG). Since SPE induces temperature change reflecting the local tYIGvalue, we\ncan extract the tYIGdependence of SPE from a temperature distribution. The spat ial distribution of\nthe SPE-induced temperature change was visualized by means of the lock-in thermography (LIT)\n[5]. The LIT method allows us to obtain the accurate tYIGdependence of SPE with high tYIG\nresolution. By means of the thermoelectric imaging techniq ue based on laser heating [23], we also\nmeasured the tYIGdependence of SSE in a single Pt/YIG system, which shows the s ame behavior\nas that of SPE. By analyzing the measured tYIGdependence of SPE and using phenomenological\nmodels, we estimated lmand determined the upper limit of lmpfor YIG.\nThe sample used for measuring SPE consists of a Pt film and an YI G film with ∇tYIG[Fig.\n1(a),(d)]. ∇tYIGwas introduced by obliquely polishing a single-crystallin e YIG (111) film grown\nby a liquid phase epitaxy method on a single-crystalline Gd 3Ga5O12(GGG) (111) substrate. The\nobtained ∇tYIGis almost uniform in the measurement range, which was observ ed to be 9.2µm\nper a 1 mm lateral length by a cross-sectional scanning elect ron microscopy [Fig. 1(f)]. After the\npolishing, a U-shaped Pt film with a thickness of 5 nm and width of 0.5 mm was sputtered on the\nsurface of the YIG film. The longer lines of the U-shaped Pt film were along the ∇tYIGdirection.\nIn the microscope image of the sample in Fig. 1(d), the yellow (gray) area above (below) the white\ndotted line corresponds to the YIG film with ∇tYIG(GGG substrate).\n2FIG. 1: (a) Schematic of the SPE measurement using a Pt/YIG/G GG sample by means of the lock-in\nthermography method. A charge current Jcis applied to the U-shaped Pt film fabricated on the YIG\nfilm with a thickness gradient ∇tYIG. (b),(c) Time tprofile of an input a.c. charge current Jcand output\ntemperature change ∆Tfor the (b) SPE and (c) Joule heating configurations. (d) An op tical microscope\nimage of the sample. The yellow (gray) area above (below) the white dotted line corresponds to the YIG\nfilm with ∇tYIG(GGG substrate). His an applied magnetic field. (e) An infrared image of the samp le. (f)\nAtYIGprofile and cross-sectional image of the sample obtained wit h a scanning electron microscope.\nSPE induces temperature modulation in the Pt/YIG/GGG sampl e in response to a charge current\nin the Pt film. When we apply a charge current Jcto the Pt film as shown in Fig. 1(a), a spin\ncurrent is generated by the spin Hall effect in Pt [24, 25]. The spin current induces a heat current\nacross the Pt/YIG interface via SPE. The heat current result s in a temperature change ∆Twhich\nsatisfies the following relation [4, 5]: ∆T∝(Jc×M)·n, where Mandnare the magnetization\nvector of YIG and the normal vector of the Pt/YIG interface pl ane, respectively. Significantly, the\nSPE-induced temperature change reflects local tYIGinformation because the temperature change\ninduced by SPE is localized owing to the formation of dipolar heat sources [5, 7]. Based on the\n∇tYIGvalue and the spatial resolution of our LIT system, we obtain ed the high tYIGresolution of\n92 nm.\nThe procedure of the LIT-based SPE measurements are as follo ws [5, 26–31]. To excite SPE,\nwe applied a rectangular a.c. charge current Jcwith the amplitude J0, frequency f, and zero offset\nto the Pt film [Fig. 1(b)]. By extracting the first harmonic res ponse of a temperature change ∆T1f\n3FIG. 2: (a) Lock-in image of an infrared light emission ∆I1finduced by SPE. (b) Lock-in amplitude of the\ninfrared emission Ainduced by the Joule heating. (c),(d) YIG-thickness tYIGdependence of (c) ∆I1fand\nthe emissivityǫand (d) the temperature change ∆T1finduced by SPE.\nin this condition (SPE configuration), we can detect the pure SPE signal free from a Joule heating\ncontribution [5, 7]. Here, ∆T1fis defined as the temperature change oscillating in the same p hase\nasJcbecause∆Tgenerated by SPE follows the Jcoscillation [Fig. 1(b)] and the out-of-phase\nsignal is negligibly small [5, 7, 32, 33]. In the LIT measurem ent, we detect the first harmonic\ncomponent of the infrared light emission ∆I1fcaused by∆T1f, where∆I1f=Acosφwith Aand\nφrespectively being the lock-in amplitude and phase. We conv erted∆I1finto∆T1fby considering\nspatial distribution of an infrared emissivity ǫof the sample [5, 7]. All measurements of SPE were\nperformed at room temperature and atmospheric pressure und er a magnetic field with a magnitude\nof 20 mT, where Maligns along the field direction.\nFigure 2(a) shows the ∆I1fimage from the Pt/YIG/GGG sample in the SPE configuration wit h\nJ0=8 mA and f=5 Hz. Clear signals appear only on the Pt film with finite tYIGbut disappears\non the Pt/GGG interface with tYIG=0 [compare Figs. 1(d) and 2(a)]. The sign of ∆I1fis reversed\nwhen Jcis reversed, which confirms that the observed infrared signa l comes from SPE [4, 5].\nThe spatial profile of ∆I1falong the ydirection is plotted in Fig. 2(c), where the ∆I1fvalues are\naveraged along the xdirection in the area surrounded by the dotted line in Fig. 2( a).∆I1fgradually\nincreases with small oscillation with increasing tYIG. The oscillation originates from the oscillation\n4ofǫof the sample due to multiple reflection and interference of t he infrared light in the YIG film\n[see the infrared light image of the sample in Fig. 1(e)] [7]. To calibrate theǫoscillation in the tYIG\ndependence of SPE, the infrared emission induced by the Joul e heating was measured [Fig. 2(b)],\nwhere Jcwith the amplitude ∆Jc=0.5 mA, frequency f=5 Hz, and d.c. offset Jdc=8.0 mA was\nused as an input for the LIT measurement [Fig. 1(c)] [5]. Sinc e the temperature change induced\nby the Joule heating is uniform on the Pt film, the lock-in ampl itude of the infrared emission A\non the Pt film depends only on the ǫdistribution. We found that the obtained tYIGdependence\nofǫ(∝Adue to the Joule heating) and the ∆I1fsignal due to SPE exhibit the similar oscillating\nbehavior [Fig. 2(c)]. By calibrating ∆I1fbyǫ, we obtained the tYIGdependence of∆T1finduced\nby SPE [Fig. 2(d)]. ∆T1fmonotonically increases with increasing tYIG. The saturated∆T1fvalue\nfortYIG>6µm was determined by using the same Pt/YIG/GGG sample coated w ith a black-ink\ninfrared emission layer with high emissivity larger than 0. 95.\nThe accurate tYIGdependence of SPE with high tYIGresolution allows us to verify the appli-\ncability of several phenomenological models used for discu ssing the behaviors of SPE. First of\nall, we found the obtained tYIGdependence of the SPE-induced temperature modulation cann ot be\nexplained by a simple exponential approximation [7, 13, 14, 22]. Based on the assumption that\nthe magnon diffuses in YIG with the magnon diffusion length lm, the simple exponential decay has\nbeen used for the analysis of the tYIGdependence:\n∆T∝1−exp(−tYIG/lm). (1)\nHowever, in general, this expression cannot be used for the s mall thickness region since the\nexponential function should be modulated by the boundary co nditions for the spin and heat currents.\nIn fact, the fitting result using Eq. (1) significantly deviat es from the experimental data when\ntYIG<4µm (see the green curve in Fig. 3). The observed continuous tYIGdependence of SPE\nthus requires advanced understanding of the spin-heat conv ersion phenomena.\nNext we focus on two phenomenological models proposed in Ref s. 19 and 34. The model in\nRef. 19 is based on the linear Boltzmann’s theory for magnons and the tYIGdependence of SPE is\ndescribed as\n∆T∝1\nScoth(tYIG/lm)+1, (2)\nwhere Sis atYIG-independent constant used as a fitting parameter in our anal ysis. The red curve in\nFig. 3 shows the fitting result based on Eq. (2). We found that E q. (2) shows the best agreement\nwith the experimental result and lmis estimated to be 3 .9µm. We also analyzed the experimental\n5FIG. 3: Experimental results of the tYIGdependence of∆T1finduced by SPE for the Pt/YIG/GGG sample\nand fitting curves using Eqs. (1)-(3).\nresults by the model based on the non-equilibrium thermodyn amics [34]:\n∆T∝cosh(tYIG/lm)−1\nsinh(tYIG/lm)+rcosh(tYIG/lm), (3)\nwhere ris atYIG-independent constant used as a fitting parameter in our anal ysis. In contrast to\nEq. (2), Eq. (3) is less consistent with the experimental res ults in the whole thickness range (see\nthe yellow curve in Fig. 3) and gives a shorter magnon diffusio n length of 0.6µm. From the fitting\nresults using Eqs. (1-3), we conclude that Eq. (2) is the best phenomenological model to explain\nthe experimental results on SPE.\nTo check the reciprocity between SPE and SSE [35], we also mea sured the tYIGdependence of\nSSE by using a single Pt/YIG/GGG sample with ∇tYIG. The YIG film used in the SSE measurement\nwas obtained from the same YIG/GGG substrate and the SSE samp le was prepared by the same\nmethod as that for the SPE sample. The ∇tYIGvalue of the SSE sample was determined to be\n12.2µm per a 1 mm lateral length. To obtain the tYIGdependence of SSE, the SSE signal was\nmeasured in the Pt/YIG/GGG sample by means of the thermoelec tric imaging technique based on\nlaser heating [23, 36–38]. As shown in Fig. 4(a), the sample s urface was irradiated by laser light\nwith the wavelength of 1 .3µm and the laser spot diameter of 5 .2µm to generate a temperature\ngradient across the Pt/YIG interface. The local temperatur e gradient induces a spin current across\nthe interface due to SSE. The spin current is then converted i nto a charge current via the inverse\nspin Hall effect in Pt [25]. To avoid the reduction of the spati al resolution of the SSE signal due\nto the thermal diffusion, we adopted a lock-in technique in th e laser SSE measurement, where\n6FIG. 4: (a) Schematic of the SSE measurement using a laser hea ting method. JSSE\ncis a charge current due to\nthe inverse spin Hall effect induced by SSE. The thickness of t he Pt film is 50 nm, which is thick enough to\nprevent light transmission; the Pt layer is heated by the las er light. (b) Infrared light image of the sample. (c)\nThe SSE signal SSSEimage induced by the laser heating. (d) tYIGdependence of SSSEand comparison with\nthat of the temperature change induced by SPE. Here, we used a n arbitrary unit for the SSE signal because\nthe temperature gradient induced by the laser heating canno t be estimated experimentally. Nevertheless, the\nrelative change of SSSEis reliable because the heating of the Pt layer is uniform ove r the sample.\nthe laser intensity was modulated in a periodic square wavef orm with the frequency f=5 kHz\nand the thermopower signal S1foscillating with the same frequency as that of the input lase r was\nmeasured. The measurements at the high lock-in frequency re alized high spatial resolution for\nthe SSE signal because temperature broadening due to the hea t diffusion is suppressed. Here,\nwe defined the SSE signal SSSEas/bracketleftbig\nS1f(+50 mT)−S1f(−50 mT)/bracketrightbig\n/2 to remove magnetic-field-\nindependent background. By scanning the position of the las er spot on the sample, we visualized\nthe spatial distribution of the SSE signal with high tYIGresolution of 64 nm.\nFigure 4(c) shows the spatial distribution of SSSEfor the Pt/YIG/GGG sample. In response to\nthe laser heating, the clear signal was observed to appear in the Pt film. The tYIGdependence of\nthe SSE signal is plotted in Fig. 4(d), where the SSSEvalues were averaged along the xdirection\n7in the area surrounded by the dotted line in Fig. 4(c). The SSE signal monotonically increases\nwith increasing tYIG. Significantly, the tYIGdependence of SSSEshows the same behavior as that of\nthe SPE-induced temperature modulation [Fig. 4(d)]. This r esult supports the reciprocity between\nSPE and SSE and strengthens our conclusion in the SPE measure ment.\nIn the recent study on SSE in Ref. 21, Parakash et al. reported non-monotonical increase of the\nSSE signal with tYIG. Since the SSE signal takes a local maximum at tYIG∼lmp, they estimated\nlmpas 250 nm from the maximum point. However, in our Pt/YIG sampl es, the SSE and SPE\nsignals monotonically increase with increasing tYIG. These results suggest that lmpis shorter than\nthetYIGresolution, 64 nm, in our experiments. The conclusion is con sistent with the theoretical\nexpectation of lmp∼1 nm [18].\nIn conclusion, we have discussed the length scale of the spin and heat transport by magnons\nin YIG by measuring the tYIGdependence of SPE in the Pt/YIG sample. This measurement was\nrealized by using the YIG film with the tYIGgradient and the LIT method, which allow us to\nobtain the continuous tYIGdependence of SPE in the single Pt/YIG sample. The experimen tal\nresult is well reproduced by the phenomenological model bas ed on the linear Boltzmann’s theory\nfor magnons referenced in Ref. 19 and lmis estimated to be 3 .9µm for our YIG sample. We\nalso measured the tYIGdependence of SSE. The SPE and SSE signals show the same behav ior\nin the tYIGdependence and monotonically increase as tYIGincreases. The monotonic increase\nimplies that lmpis shorter than 64 nm for our YIG sample. These results give cr ucial information\nto understand the microscopic origin of the spin-heat conve rsion phenomena.\nThe authors thank R. Iguchi, T. Kikkawa, M. Matsuo, Y. Ohnuma , and G. E. W. Bauer for\nvaluable discussions. This work was supported by CREST “Cre ation of Innovative Core Tech-\nnologies for Nano-enabled Thermal Management” (JPMJCR17I 1), PRESTO “Phase Interfaces for\nHighly Efficient Energy Utilization” (JPMJPR12C1), and ERAT O “Spin Quantum Rectification\nProject” (JPMJER1402) from JST, Japan, Grant-in-Aid for Sc ientific Research (A) (JP15H02012)\nand Grant-in-Aid for Scientific Research on Innovative Area “Nano Spin Conversion Science”\n(JP26103005) from JSPS KAKENHI, Japan, the Inter-Universi ty Cooperative Research Program\nof the Institute for Materials Research (17K0005), Tohoku U niversity, and NEC Corporation.\n∗Electronic address: daimon@ap.t.u-tokyo.ac.jp\n†Electronic address: UCHIDA.Kenichi@nims.go.jp\n8[1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012).\n[2] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Enviro n. Sci. 7, 885 (2014).\n[3] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. 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Appl. 7, 044004 (2017).\n10" }, { "title": "2101.09931v1.Nonreciprocal_Transmission_and_Entanglement_in_a_cavity_magnomechanical_system.pdf", "content": "arXiv:2101.09931v1 [quant-ph] 25 Jan 2021Nonreciprocal Transmission and Entanglement in a cavity-m agnomechanical system\nZhi-Bo Yang1, Jin-Song Liu1, Ai-Dong Zhu1, Hong-Yu Liu1,∗and Rong-Can Yang2,3†\n1Department of Physics, College of Science, Yanbian Univers ity, Yanji, Jilin 133002, China\n2Fujian Provincial Key Laboratory of Quantum Manipulation a nd New Energy Materials,\nand College of Physics and Energy, Fujian Normal University , Fuzhou 350117, China and\n3Fujian Provincial Collaborative Innovation Center for Opt oelectronic\nSemiconductors and Efficient Devices, Xiamen 361005, China\n(Dated: January 26, 2021)\nQuantum entanglement, a key element for quanum information is generated with a cavity mag-\nnomechanical system. It comprises of two microwave cavitie s, a magnon mode and a vibrational\nmode, and the last two elements come from a YIG sphere trapped in the second cavity. The\ntwo microwave cavities are connected by a superconducting t ransmission line, resulting in a linear\ncoupling between them. The magnon mode is driven by a strong m icrowave field and coupled to\ncavity photons via magnetic dipole interaction, and at the s ame time interacts with phonons via\nmagnetostrictive interaction. By breaking symmetry of the configuration, we realize nonreciprocal\nphoton transmission and one-way bipartite quantum entangl ement. By using current experimental\nparameters for numerical simulation, it is hoped that our re sults may reveal a new strategy to built\nquantum resources for the realization of noise-tolerant qu antum processors, chiral networks, and so\non.\nI. INTRODUCTION\nAlthough reciprocity is ubiquitous in nature, nonre-\nciprocity promptes diverse applications such as chiral\nengineering, invisible sensing, and backaction-immune\ninformation processing [ 1–3]. So far, electromagnetic\nnonreciprocal transmission [ 4,5] has been demonstrated\nwith various systems ranging from microwave [ 6–9], ter-\nahertz [10], optical [ 11,12] photons to x-rays [13]. As\na special type of nonreciprocity which only allows one-\nway transmission, unidirectional transmission of light is\nalso revealed with some hybrid systems, such as atomic\ngases [14], nonlinear devices [ 11,15–17], moving me-\ndia [18] and synthetic materials [ 19]. While in quantum\nregime, a recent scheme has been proposed, where nonre-\nciprocal entangled states is prepared by using an optical\ndiode with a high isolation rate [ 20], making it possi-\nble to swift a single device between classical isolator and\nquantum diode, or to protect quantum resources from\nbackscattering losses.\nCavity spintronics [ 21–35] is an emerging and\nrapidlydevelopinginterdisciplinarythatstudiesmagnons\nstrongly couple with microwave photons via magnetic\ndipole interaction. Such a hybrid system shows great ap-\nplication prospect in the field of quantum information\nprocessing, especially for quantum transducer [ 36–42]\nand quantum memory [ 43]. The reason is that magnetic\nmaterials have several distinguishing advantages such as\nlong lifetime, high spin density and easy tunability. Fur-\nthermore, a collective excitation of spins in these mate-\nrials(called magnon mode or Kittel mode )in magnetic\nmaterialscan easilybe coupled to a varietyofothertypes\nof systems. Thus, cavity spintronics seems to be a po-\n∗liuhongyu@ybu.edu.cn\n†rcyang@fjnu.edu.cntentialcandidatetostudymultiple quantumcorrelations,\netc. By using cavity spintronics, in this manuscript, we\npropose a scheme to realize an optical diode with both\nquantum and classicalcharacteristics, i.e., the implemen-\ntation of both nonreciprocal microwavetransmission and\nnonreciprocal bipartite quantum entanglement.\nIn this manuscript, we present a proposal to carry out\nnonreciprocal microwave transmission and one-way bi-\npartite entanglement by using an additional microwave\ncavity coupled to a cavity-magnon system to break sym-\nmetryofspatialinversion[ 9]. Thetwomicrowavecavities\nare linked by superconducting transmission lines [ 44] and\none of them interacts with magnon mode of a ferrimag-\nnetic yttrium iron garnet (YIG)sphere [21,24,45–49].\nSimultaneously, the magnon mode is coupled to phonons\ndue to vibration of YIG sphere induced by the mag-\nnetostrictive force [ 26,50,51]. In addition, magnons\nare driven by a strong microwave field at the first red\nsideband with respect to phonons because entanglement\nmainly survives with small thermal phonon occupancy.\nThe(anti-Stokes )process not only realizes phonon cool-\ning, a prerequisite for observing quantum effects in the\nsystem [52], but also also enhances the effective magnon-\nphonon coupling in orderfor the generationof magnome-\nchanical entanglement. If both microwave cavities are\nresonant with the red sideband, then entanglement be-\ntween the other two subsystems will be generatedthough\nsome entanglement is very small. For the same driving\npower, different driving direction induces different effec-\ntive magnetostrictive interaction, leading the generated\nsubsystem entanglement to exhibit rare nonreciprocity\nand unidirectional invisibility. Furthermore, most of bi-\npartite entanglement is robust against ambient tempera-\nture.\nThe manuscript is organized as follows. At first, we\npresenta generalmodel ofthe scheme, and then solvethe\nsystem dynamics by means of the standard Langevin for-2\nmalism with linearization treatment. Next, we illustrate\nnonreciprocity of microwave transmission and bipartite\nentanglement in the stationary state. Finally, we show\nhow to measure generated entanglement and analyze the\nvalidity of our model.\nII. MODEL AND EQUATION OF MOTION\nAs schematically shown in Fig. 1(a), we study a cavity\nmagnomechanical system which consists of two coupled\nmicrowave cavities and one of them coupled to a YIG\nsphere. The sphere is uniformly magnetized to satura-\ntion by a bias magnetic field with B0=B0ez, whereB0\nandezrepresent magnetic amplitude and the unit vector\n\ta\nKittel mode \n#\u0011side view top view #\u0011Ea Ec\nleft cavity right cavity Emsuperconducting \ntransmission \nlinea cout out\nYIG sphere\nvibrations mode\nωa ωd ωm ωc\nωd + ωb ωd - ωb-ΔaΔm-Δc\nκcκaκm\nω\tb\nz\nxy\nFIG. 1. (a) A schematic diagram of two-cavity magnome-\nchanic system, consisting of two microwave cavities and a\nYIG sphere, which is placed in the right cavity. The YIG\nsphere, which is magnetized to saturation by a bias magnetic\nfieldB0aligned along the z-direction, is mounted near the\nright cavity wall, where the magnetic field of the cavity mode\nis the strongest and polarized along x-direction to excite the\nmagnon mode in YIG. The magnon mode is driven by a mi-\ncrowave field along the y-direction. Three magnetic fields are\nmutuallyperpendicular ar thesite ofthe YIGsphere. In addi -\ntion, thesystemwill exhibitvaryingdegrees ofmagnomecha n-\nical interaction when the left or right cavity is driven alon e.\n(b) Mode frequencies and linewidths. The magnon mode (two\nMW cavity modes )with frequency ωm(ωaandωc)is driven\nby a strong MW field at frequency ωd, and the mechanical\nmotion of frequency ωbscatters the driving photons onto two\nsidebands at ωd±ωb. If the magnon mode is resonant with\nthe blue (anti-Stokes )sideband, and two cavity modes are\nresonant with the red (Stokes)sideband, all subsystems are\nprepared in entangled state. See text for more details.inzdirection, respectively [ 23]. At the same time, the\nYIG sphere is also directly driven by a microwave field\nwith driving strength Emand frequency ωd. In addition,\neither the first cavity or the second one is driven by a mi-\ncrowave light beam with driving strength EaorEcwith\nthe same frequency ωd. The two cavities are linked by a\nsuperconducting transmission line [ 44] and the magnon\nmodecouplestothesecondcavitymodeandavibrational\nmode [21] via the collective magnetic-dipole interaction\nand magnetostrictive interaction, respectively. The mag-\nnetostatic mode with finite wave number has a distinct\nfrequency different from the Kittel mode so that the se-\nlective excitation may be implemented through the driv-\ning field wavelength and cavity mode selection. When all\nof the driving fields are included, the total Hamiltonian\nof the system can be written as follows\nH//planckover2pi1=ωaˆa†ˆa+ωcˆc†ˆc+ωmˆm†ˆm+ωbˆb†ˆb+gac(ˆa†ˆc+ˆc†ˆa)\n+gcm(ˆc†ˆm+ ˆm†ˆc)+gmbˆm†ˆm(ˆb†+ˆb)\n+iEa(ˆa†e−iωdt−ˆaeiωdt)+iEc(ˆc†e−iωdt−ˆceiωdt)\n+iEm(ˆm†e−iωdt−ˆmeiωdt). (1)\nHere, ˆa, ˆc, ˆmandˆb(ˆa†, ˆc†, ˆm†andˆb†)are individu-\nally the annihilation (creation)operators of correspond-\ning cavities, magnons and phonons with resonance fre-\nquencyωa,ωc,ωmandωb. The magnon frequency is\ndetermined by the external bias magnetic field B0, i.e.,\nωm=γB0withγ/2π= 2.8 MHz/Oe being the gyromag-\nnetic ratio. gacis the photon coupling rate [ 53] andgcm\nrepresents the linear photon-magnon coupling strength\nwhich can be estimated by measuring the reflection spec-\ntrum of YIG sphere in right cavity and can be adjusted\nby varying the direction of the bias field or the position\nof the YIG sphere inside the right cavity [ 50]. The single-\nmagnon magnomechanical coupling rate gmbis typically\nsmall, but the magnomechanical interaction can be en-\nhanced if magnons are strongly driven [ 21,22,54,55].\nThe amplitude for each driving fields is Ei=√κiεiwith\ntheeffectivestrength εi=/radicalbig\nPi//planckover2pi1ωdwiththecorrespond-\ning driving power and driving frequency being Piand\nωd, respectively [ 25].κa/candκm/bseparately repre-\nsent the total loss rate of the first/second cavity and the\nmagnon/phonon mode.\nIn the frame rotating at the driving frequency ωd, the\nquantum Langevin equations (QLEs)describing the sys-\ntem are given by\n˙ˆa=−(i∆a+κa)ˆa−igacˆc+Ea+√2κaˆain,\n˙ˆc=−(i∆c+κc)ˆc−igacˆa−igcmˆm+Ec+√2κcˆcin,\n˙ˆm=−(i∆m+κm)ˆm−igcmˆc−igmbˆm(ˆb†+ˆb)+Em\n+√2κmˆmin,\n˙ˆb=−(iωb+κb)ˆb−igmbˆm†ˆm+√\n2κbˆbin, (2)\nwhere ∆ j=ωj−ωdand ˆain, ˆcin, ˆmin,ˆbinare input\nnoise operators with zero mean value acting on the cavi-\nties, magnon and mechanical modes, respectively, which\nare characterized by the following correlation functions:3\n/angbracketleftˆain(t)ˆain†(t′)/angbracketright=/angbracketleftˆcin(t)ˆcin†(t′)/angbracketright=/angbracketleftˆmin(t)ˆmin†(t′)/angbracketright=\nδ(t−t′),/angbracketleftˆbin(t)ˆbin†(t′)/angbracketright= (nb+ 1)δ(t−t′), and\n/angbracketleftˆbin†(t)ˆbin(t′)/angbracketright=nbδ(t−t′) where the equilibrium mean\nthermal phonon numbers nb= [exp(/planckover2pi1ωb\nkBT)−1]−1withkB\nthe Boltzmann constant and Tthe ambient temperature.\nBecause the magnon mode and microwavephotons are\nstrongly driven, they all have large amplitude, i.e., |/angbracketlefta/angbracketright|,\n|/angbracketleftc/angbracketright|,|/angbracketleftm/angbracketright|≫1, whichallowsustolinearizethedynamics\nof the system around the steady-state values by writing\nany a operator as ˆ o=/angbracketlefto/angbracketright+δo(o=a,c,m,b)and ne-\nglecting second order fluctuation terms. Then, we obtain\na set of differential equations for the mean values:\n˙/angbracketlefta/angbracketright=−(i∆a+κa)/angbracketlefta/angbracketright−igac/angbracketleftc/angbracketright+Ea,\n˙/angbracketleftc/angbracketright=−(i∆c+κc)/angbracketleftc/angbracketright−igac/angbracketlefta/angbracketright−igcm/angbracketleftm/angbracketright+Ec,\n˙/angbracketleftm/angbracketright=−(i˜∆m+κm)/angbracketleftm/angbracketright−igcm/angbracketleftc/angbracketright+Em,\n˙/angbracketleftb/angbracketright=−(iωb+κb)/angbracketleftb/angbracketright−igmb|/angbracketleftm/angbracketright|2, (3)\nand the linearized QLEs for the quantum fluctuations:\nδ˙a=−(i∆a+κa)δa−igacδc+√2κaˆain,\nδ˙c=−(i∆c+κc)δc−igacδa−igcmδm+√2κcˆcin,\nδ˙m=−(i˜∆m+κm)δm−igcmδc−1\n2Gmb(δb†+δb)\n+√2κmˆmin,\nδ˙b=−(iωb+κb)δb−1\n2Gmb(δm†−δm)+√2κbˆbin,\n(4)\nwhere˜∆m= ∆m+gmb(/angbracketleftb/angbracketright+/angbracketleftb/angbracketright∗)istheeffectivedetuning\nof the magnon mode including the frequency shift caused\nby the magnetostrictive interaction. Gmb=i2gmb/angbracketleftm/angbracketright\nis the effective magnomechanical coupling rate. If we\nconsider the time to be t→ ∞and the detunings to\nsatisfy|∆a|,|∆c|,|˜∆m| ≫κa,κc,κm, then/angbracketlefta/angbracketright,/angbracketleftc/angbracketright,/angbracketleftm/angbracketright\nand/angbracketleftb/angbracketrightcan be given by\n/angbracketlefta/angbracketright ≃iEmgacgcm−Eag2\ncm−Ecgac˜∆m+Ea∆c˜∆m\ng2cm∆a+g2ac˜∆m−∆a∆c˜∆m,\n/angbracketleftc/angbracketright ≃ −iEmgcm∆a+Eagac˜∆m−Ec∆a˜∆m\ng2cm∆a+g2ac˜∆m−∆a∆c˜∆m,\n/angbracketleftm/angbracketright ≃ −iEmg2\nac−Eagacgcm+Ecgcm∆a−Em∆a∆c\ng2cm∆a+g2ac˜∆m−∆a∆c˜∆m,\n/angbracketleftb/angbracketright=−igmb|/angbracketleftm/angbracketright|2/(iωb+κb).\n(5)\nInwhatfollows,wefirstshowthattheeffectoftherelated\nparameters on the nonreciprocal transmission of a mi-\ncrowave field with a forward and backward driving-field\ninput by solving the equations numerically, and then we\nshow that the classical nonreciprocity can also be used\nto prepare nonreciprocal subsystem entangled states.III. NONRECIPROCAL TRANSMISSION\nIn order to study the nonreciprocity of cavity output\nfield, we let the first cavity mode driven (Ea/negationslash= 0 &\nEc= 0)or the second cavity mode driven (Ea= 0\n&Ec/negationslash= 0)by a classical field, and measure the output\nfield of the other cavity [ 25]. For the sake of concision,\nwe denote the first (second)case as the driving-field in-\nput from the forward (backward )direction in the follow-\ning passages. Thus, the transmission coefficients for the\ntwo cases are separately defined as T12=|cout/εa|=\n|√κc/angbracketleftc/angbracketright/εa|andT21=|aout/εc|=|√κa/angbracketlefta/angbracketright/εc|with\nthe average number of the first (second)cavity|/angbracketlefta(c)/angbracketright|\ncalculated with Eq. ( 5). In addition, in order to de-\npict the nonreciprocality, we define the isolation Tiso=\n20×log10|T12/T21|which is given in decibels (dB)[9,25].\nFor the sake of simplicity, we set the power of the mi-\ncrowavesource focused on the first or second cavity to be\nthe same, i.e. Pa=Pc=P, and the two cavities to be\nidentical, i.e. κa=κc=κ. The transmission coefficient\nT12/21and the isolation Tisoas a function of the driving\npowerPareplotted in Fig. 2(a)and (c) with gac/ωb= 1,\nand (b) and (d) with gac/ωb= 0.32. The other param-\neters are chosen as ωa/2π= 10 GHz, ωb/2π= 10 MHz,\n˜∆m= 0.9ωb, ∆c= ∆a=−ωb,κ/2π=κm/2π= 1 MHz,\nandgcm/2π= 3.2 MHz [23,54].\nFrom Eq. ( 5), we find that when gac=ωbthe same\ntransmissioncoefficientisobservedforthe differentdirec-\ntions(forward or backward injection )of the driving-field\ninput. We regard gac=ωbasaimpedance-matchingcon-\ndition [53] for the same microwave transmission. From\nFig.2(a)withgac/ωb= 1, wecanclearlyseethatthetwo\ntransmission coefficients (T12andT21)are always equal.\nHowever, if we decrease the coupling rate gacbetween\nthe two cavities, then the two coefficients (T12andT21)\nwill be distinctly different when Pis small. For example,\nT12≃0.08,0 andT21≃0,0.02 when P≃10,110 mW\n(see Fig.2(b)). While the increaseof Pwill lead T12and\nT21to be closer and then to keep some difference. There-\nfore, if the impedance-matching condition is broken, the\ntransmission for both two directions will be distinct.\nThis is a typical nonreciprocal phenomenon, which cor-\nresponds to the classical nonreciprocal microwave trans-\nmission [ 9]. In addition, the degree of the nonreciprocal\ntransmission of a microwavefield can be described by the\nisolation as follows: Tiso= 20×log10|T12/T21|given in\ndecibels(dB). The calculation results for Tiso, plotted as\na function of driving power, is shown in Fig. 2(c) and\n(d). Fig. 2(c) shows that when the impedance-matching\ncondition is met, the microwave transmission coefficients\nin the two directions are almost the same, i.e., Tiso= 0.\nHowever, Fig. 2(d) gives us a clearer perspective for the\ndifferences between the transmission coefficients T12and\nT21. At this time, the impedance-matching condition is\nbroken, and the microwavetransmission in the two direc-\ntions has a larger transmission isolation ratio Tiso>70\ndB. It is worthnoting that when P= 0,Tiso/negationslash= 0 in Fig. 2\n(d), thisisbecausethemagnonsmodeisalwaysdrivenby4\nFIG. 2. (a) (b) Transmission coefficients T12andT21and\n(c) (d) transmission isolation ratio Tisoversus driving power\nPfor (a) (c) gac/ωb= 1 and (b) (d) gac/ωb= 0.32, where\nPm= 94.5 mW. Other parameters are ωa/2π= 10 GHz,\nωb/2π= 10 MHz, ˜∆m= 0.9ωb, ∆c= ∆a=−ωb,κ/2π=\nκm/2π= 1 MHz, and gcm/2π= 3.2 MHz which are mostly\nbased on the latest experimental parameters [ 23,54].\na microwave source with driving power P= 94.5 mW to\nmaintain a certain magnitude of magnomechanical cou-\npling.\nIV. DEFINITION OF ENTANGLEMENT\nWe thereby obtain the linearized QLEs for\nthe quadratures δXo,δYodefined as δXo=\n(δo+δo†)/√\n2,δYo= (δo−δo†)/i√\n2, can be\nwritten as ˙ σ(t) =Aσ(t) +n(t) with σ(t) =\n[δXa(t),δYa(t),δXc(t),δYc(t),δXm(t),δYm(t),δXb(t),\nδYb(t)]Tandn(t) = [√2κaδXin\na(t),√2κaδYin\na(t),√2κc\nδXin\nc(t),√2κcδYin\nc(t),√2κmδXin\nm(t),√2κmδYin\nm(t),√2κbδXin\nb(t),√2κbδYin\nb(t)]Tbeing the vectors for quan-\ntum fluctuations and noise, respectively. In addition,\nthe drift matrix Areads\nA=\n−κa∆a0gac0 0 0 0\n−∆a−κa−gac0 0 0 0 0\n0gac−κc∆c0gcm0 0\n−gac0−∆c−κc−gcm0 0 0\n0 0 0 gcm−κm˜∆m−Gmb0\n0 0 −gcm0−˜∆m−κm0 0\n0 0 0 0 0 0 −κbωb\n0 0 0 0 0 Gmbωb−κb\n.\n(6)\nIn this case, when the forwardinput direction ofthe driv-\ning field is taken into account (i.e.,Ea/negationslash= 0 and Ec= 0),magnomechanical coupling can be written as follows\nGmb,12=2gmb(Emg2\nac−Eagacgcm−Em∆a∆c)\ng2cm∆a+g2ac˜∆m−∆a∆c˜∆m.(7)\nHowever, when the backward input direction of the driv-\ning field is taken into account (i.e.,Ea= 0 and Ec/negationslash= 0),\nmagnomechanical coupling changes to\nGmb,21=2gmb(Emg2\nac+Ecgcm∆a−Em∆a∆c)\ng2cm∆a+g2ac˜∆m−∆a∆c˜∆m.(8)\nSince we are using linearized QLEs, the Gaussian nature\nof the input states will be preserved during the time evo-\nlution for the system. So, the quantum fluctuation is in a\ncontinuous four-mode Gaussian state which can be com-\npletelycharacterizedbya8 ×8covariancematrix (CM)V\nin the phase space 2 Vij(t,t′) =/angbracketleftσi(t)σj(t′)+σj(t′)σi(t)/angbracketright,\n(i,j= 1,···,8). Then the vector can be obtained\nstraightforwardly by solving the Lyapunov equation\nAV+VAT=−D (9)\nwithD= diag [κa,κa,κc,κc,κm,κm,(2nb+1)κb,(2nb+\n1)κb] defind through 2 Dijδ(t−t′) =/angbracketleftni(t)nj(t′) +\nnj(t′)ni(t)/angbracketright. In this manuscript, we use logarithmic neg-\nativity [58,59] to quantify the degree of the quantum\nentanglement for the four-mode Gaussian state, which is\ndefined as EN≡max[0,−2lnν−], where ν−= min eig\n| ⊕2\nj=1(−σy)PVP|withσyandP=σz⊕1 are respec-\ntivelyy-Pauli matrix and the matrix that realizes partial\ntransposition at the CM level [ 58,60].\nFIG. 3. (a) Eac, (b)Ecm, (c)Emband (d)Eabversus detun-\nings ∆ aand ∆ c. We take Gmb/2π= 2.5 MHz,κb/2π= 100\nHz,T= 20 mK and the other parameters are same as Fig. 2.5\nV. NONRECIPROCAL ENTANGLEMENT\nThe foremost taskof studying entanglementproperties\nbetweenanytwosubsystemsinsuchahybridsystemis to\nfind the optimal effective interaction among modes, i.e.,\ntofindoptimalfrequencydetuningthatcangeneratesub-\nsystem entanglement [ 21]. In Fig. 3, we show four types\nof subsystem entanglement (Eac,Ecm,EmbandEab)\nas the function of cavity detunings ∆ aand ∆ c, where\nEac,Ecm,EmbandEabdenote the cavity-cavity entan-\nglement, cavity-magnon entanglement, magnon-phonon\nentanglement, and cavity-phonon distant entanglement,\nrespectively. In addition, we choose Gmb/2π= 2.5 MHz,\nκa=κc=κm,κb/2π= 100 Hz, T= 20 mK and\nthe other parameters are chosen as the same as that in\nFig.2. Furthermore, we also set ˜∆m≃ωbwhich im-\nply that magnon mode is in the blue sideband with re-\nspect to the first cavity mode, which corresponds to the\nanti-Stokes process, i.e., significantly cooling the phonon\nmode. Thus, the elimination of the main obstacle for ob-\nserving entanglement is obtained [ 21]. It is noted that\nall results are satisfying with the condition of the steady\nstate guaranteed by the negative eigenvalues (real parts )\nof the drift matrix A. From Fig. 3, it is shown that a\nparameter regime exists, i.e., ∆ a= ∆c=−ωb, where\nthe entanglement within any two subsystems occurs (see\nFig.1(b)). This is similar to the realization of entangle-\nment between magnon modes in a magnomechanic sys-\ntem with two YIG spheres proposed in Ref. [ 61]. In or-\nder to obtain the entanglement between any two subsys-\ntems and keep the system stable at the same time, the\nthree coupling rates gac,gcmandGmbshould be on the\nsame order of magnitude and chosen as a separate and\nmoderate value. Based on it, we choose gacgcmGmb≪\n|∆a∆c˜∆m| ≃ω3\nbandGmb= 2.5 MHz which corresponds\ntoEm≃Gmbω3\nb/2gmb(ω2\nb−g2\nac)≃4.2×1013Hz and\nPm≃1.85 mW, when Ea=Ec= 0 in Fig. 3. All entan-\nglement in the system comes from the magnetostrictive\ninteraction between the magnon and the phonon [ 21].\nNext, we will show the realization of nonreciprocal sub-\nsystem entanglement by controlling the classical nonre-\nciprocal transmission, i.e., controlling the effective mag-\nnomechanical interaction.\nFor the sake of brevity, we mainly focus on the en-\ntanglement EabandEmbdue to they being larger than\nthe other types of entanglement. In Fig. 4(a) (b), we\nshowEmbandEabas the function of driving power Pon\nthe cavity aorc, whereEij,12(Eij,21)represents the en-\ntanglement between mode iand mode jwhen the direc-\ntion of driving microwave source is forward (backward ).\nAdditionally, gmb/2π= 0.3 Hz,κa=κc=κmand\nT= 20 mK are set and the other parameters are cho-\nsen as the same as that in Fig. 2. Due to the existence\nof magnon-phonon nonlinear coupling (magnetostrictive\nforce), the entanglement of the magnomechanical sub-\nsystem is generated. And because there is a direct or\nindirect linear state-swap interaction between the two\nmicrowave cavities and the magnon mode, the resultingmagnomechanical entanglement is distributed to other\nsubsystems. Therefore, controlling the optimal effective\nmagnomechanical interaction will be an effective opera-\ntion to control whether entanglement exists in each sub-\nsystem. From Fig. 4(a) we can see that the magnome-\nchanical subsystem entanglement in the two directions\nshows very different trends with the increase of the driv-\ning power. Therefore, entanglement of other subsystems\nwill be affected accordingly (see Fig.4(d)). The results\nin Fig.4(a) (b) demonstrate that the entanglement of\nmultiple subsystems in a hybrid system can be prepared\nin a highly asymmetric way. This is in good agreement\nwith our previous expectations. This is a clear signature\nof quantum nonreciprocity, which is fundamentally dif-\nferent from that in classical devices [ 62,63] showing only\nnonreciprocal transmission rates.\nNext, the difference between subsystem entanglement\nwith forward driving direction and backward driving\ndirection is extracted in the decibel scale (defined as\n20×log10|Eij,12/Eij,21|, similar to Tiso[3]), and we take\nits value as the entanglement isolation ratio Eij,iso(units\nof dB). For the case of reciprocal subsystem entangle-\nment, we have Eij,12/Eij,21= 1 and Eij,iso= 0. A\nnonzeroEij,isopresents nonreciprocal entanglement and\nthe greater the value of Eij,iso, the higher is the de-\ngree of the nonreciprocal entanglement. The calculation\nresults for Eij,iso, plotted as a function of the driving\npowerP, are shown in Fig. 4(c) (d), which gives us a\nclearer perspective for the differences between the entan-\nglements Eij,12andEij,21. Notice the cutoffin Fig. 4(d).\nThe position of the cutoff corresponds to the position of\nEab,21= 0 in Fig. 4(b), which shows the unidirectional\ninvisibility of subsystem entanglement. In the multilayer\nmicrowave integrated quantum circuit, we can further\nFIG. 4. (a) Emb, (b)Eab, (c)Emb,iso, and (d) Eab,isoversus\ndriving power P. We take gmb/2π= 0.3 Hz,κb/2π= 100 Hz,\nT= 20 mK and the other parameters are same as Fig. 2.6\ndesign superconducting transmission lines and intercon-\nnects based on these existing technologies to provide the\nlarge range of necessary couplings and to minimize any\nparasitic losses [ 44]. In fact, we can find from Eqs. ( 7–8)\nthat when gac=ωb, the system reaches the impedance\nmatching condition [ 25,53], i.e.,Gmb,12=Gmb,21. This\nrealizes the switch from nonreciprocal to reciprocal of\nsubsystem entangled state and shows the potential ad-\nvantage of our solution as a tunable quantum diode.\nFig.5,EmbandEabas a function of ambient tem-\nperature Tfor driving power P= 1 W, shows that the\ngenerated subsystem entanglements is robust to ambient\ntemperature and survives up to ∼100 mK, below which\nthe average phonon number is always smaller than 1,\nshowing that mechanical cooling is, thus, a precondition\nfor observing quantum entanglement in the system [ 21].\nCompared with the scheme that a strong squeezed vac-\nuum field proposed in Ref. [ 64] is used to generate en-\ntanglement between magnon modes, in our scheme due\nto the inherent low frequency of phonon modes, this ro-\nbustness is generally weak.\nVI. DISCUSSION AND CONCLUSION\nLastly, we discuss how to detect the entanglement and\nverify the effectiveness of two-cavitymagnomechanicsys-\ntem. The generated subsystem entanglements can be\ndetected by measuring the CM of two cavity output\nfields [21]. Such measurement in the microwave domain\nhas been realized in the experiments [ 65,66]. In addi-\ntion, for a 0.5-mm-diameter YIG sphere, the number of\nspinsN≃2.8×1017, andPm= 189 mW corresponds\ntoEm≃3×1014Hz, and |/angbracketleftm/angbracketright| ≃5.9×106, leading to\nFIG. 5. (a) Emband (b)Eabversus ambient temperature T.\nWe take P= 0.5 W,gmb/2π= 0.3 Hz,κb/2π= 100 Hz and\nthe other parameters are same as Fig. 2./angbracketleftm†m/angbracketright ≃3.5×1013≪5N= 1.4×1018which is well\nfulfilled. It is worth noting that two cavity modes is res-\nonant with the red-sideband (∆a= ∆c=−ωb)results in\na higher magnon excitation number in the stable state in\nthe presence of EaorEc, where the magnon number has\na simpler form /angbracketleftm†m/angbracketright ≃[Em(ω2\nb−g2\nac)+Eagacgcm]2/ω6\nb\nor/angbracketleftm†m/angbracketright ≃[Em(ω2\nb−g2\nac) +Ecωbgcm]2/ω6\nb. Therefore,\nunder the premise that the magnon mode is continu-\nously driven by the classical microwave field with driv-\ning power Pm= 94.5 mW,P= 1 W corresponds to\n/angbracketleftm†m/angbracketright12≃1.4×1015≪5N(Ea/negationslash= 0,Ec= 0)and\n/angbracketleftm†m/angbracketright21≃3.47×1015≪5N(Ec/negationslash= 0,Ea= 0), re-\nspectively. In order to keep the Kerr effect negligible,\nK|/angbracketleftm/angbracketright|3≪/summationtext\ni=a,c,mEimust hold [ 21]. Kerr coefficient\nKis inversely proportional to the volume of the sphere.\nIn this manuscript, we use a 5-mm-diameter YIG sphere,\nK/2π≃8×10−10Hz which corresponds to K|/angbracketleftm/angbracketright|3≃\n4.2×1013Hz≪Em+Ea≃1.3×1015(Pm= 189 mW,\nP= 2 W, and Ec= 0)andK|/angbracketleftm/angbracketright|3≃1.64×1014Hz\n≪Em+Ec≃1.3×1015(Pm= 189 mW, P= 2 W, and\nEa= 0). This implying that the nonlinear effects are\nnegligible and the linearization treatment of the model is\na good approximation.\nIn summary, we show how to use an asymmetric cav-\nity magnomechanic system to produce classical nonre-\nciprocal transmission, and extend this nonreciprocity to\nquantum states, thereby generating nonreciprocal sub-\nsystem entangled states, rather than by means of nonre-\nciprocal devices [ 62]. The introduction of an additional\ncavity mode successfully breaks the symmetry of spa-\ntial inversion. Our work opens up a range of exciting\nopportunities for quantum information processing, net-\nworking and metrology by exploiting the power of quan-\ntum nonreciprocity. The ability to manipulate quan-\ntum states or nonclassical correlations in a nonrecipro-\ncal way sheds new lights on chiral quantum engineering\nand can stimulate more works on achieving and operat-\ning quantum nonreciprocal devices, such as directional\nquantum squeezing [ 67,68], backaction-immune quan-\ntum sensing [ 69,70], and quantum chiral coupling of cav-\nity optomechanics devices to superconducting qubits or\natomic spins [ 21,71,72].\nVII. ACKNOWLEDGMENTS\nThis work is supported by the Science and Technol-\nogy project of Jilin Provincial Education Department of\nChina during the 13th Five-Year Plan Period (Grant No.\nJJKH20200510KJ )and the Fujian NaturalScienceFoun-\ndation(Grant No. 2018J01661 and No. 2019J01431 ).7\n[1] Y. Shoji and T. Mizumoto, Sci. Technol. Adv. Mater. 15,\n014602 (2014).\n[2] A. Li and W. Bogaerts, OPTICA 7, 7 (2020).\n[3] Z. B. Yang, J. S. Liu, A. D. Zhu, H. Y. Liu, and R. C.\nYang, Ann. Phys. (Berlin) 532, 2000196 (2020).\n[4] C. Caloz, A. 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Lett. 120, 093601 (2018)." }, { "title": "1710.04179v1.Approaching_quantum_anomalous_Hall_effect_in_proximity_coupled_YIG_graphene_h_BN_sandwich_structure.pdf", "content": "1 \n Approaching quantum anomalous Hall effect in proximity -coupled \nYIG/ graphene/h -BN sandwich structure \n \nChi Tang1, Bin Cheng1, Mohammed Aldosary1, Zhiyong Wang1, Zilong Jiang1, K. Watanabe3, T. \nTaniguchi3, Marc Bockrath1, 2, and Jing Shi1, a \n1Department of Physics and Astronomy, University of California, Riverside, C A 92521, USA \n2Department of Physics, The Ohio State University, Columbus, O H 43210, USA \n3Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-\n0044, Japan \n \n \nQuantum anomalous Hall state is expected to emerge in Dirac electron systems such as graphene \nunder both sufficiently strong exchange and spin -orbit interactions. In pristine graphene, neither \ninteraction exists; however, both interactions can be acquired by coupling graphene to a \nmagnetic insulator (MI) as revealed by the anomalous Hall effect. Here, we show enhanced \nmagnetic proximity coupling by sandwiching graphene between a ferrimagnetic insulator yttrium \niron garnet (YIG) and hexagonal -boron nitride (h-BN) which also serves as a top gate dielectric. \nBy sweeping the top -gate voltage, we observe Fermi level- dependent anomalous Hall \nconductance . As the Dirac p oint is approached f rom both electron and hole sides, the anomalous \nHall conductance reaches ¼ of the quantum anomalous Hall conductance 2e2/h. The exchange \ncoupling strength is determined to be as high as 27 meV from the transition temperature of the \ninduced magnetic phase. YIG/graphene/h -BN is an excellent heterostructure for demonstrating \nproximity -induced interactions in two -dimensional electron systems . \n 2 \n Long-range fer romagnetic order in two -dimensional electron systems (2DES) has long \nbeen sought to involve the spin degree of freedom in spatially confined quantum systems1-5; \nhowever, the transition metal doping approach failed to deliver high-temperature \nferromagnetism . Recently it became possible w ith the advent of 2D layered materials such as \ngraphene6,7 and other van der Waals (vdW) materials8-11. In the former, ferromagnetism is \nintroduced by proximity coupling in a heterostructure comprising of graphene and magnetic \ninsulator; while in the later, the material itself is a spontaneously ferromagnetically order ed 2D \nvdW crystal, either a 2D H eisenberg or an Ising ferromagnet. Such 2D ferromagnetic systems \nprovide unprecedented opportunities to create novel heter ostructures for exploiting spin-\ndependent phenomena in spatial ly confined electron systems . \nA particularly interesting quantum phenomenon in graphene was theoretically proposed \nby Qiao et al.12,13, namely the quantum anomalous Hall effect (QAHE) , which emerges in the \npresence of finite exchange interaction and spin- orbit coupling (SOC) . Under these two \ninteractions, a topological band gap or an exchange gap is opened up at the Dirac point which \nconsequently gives rise to a quantized anomalous Hall conductance of ± 2e2/h. However, neither \ninteraction is present in pristine graphene. F erromagnetic order has recently been demonstrated \nvia proximity coupling in both yttrium iron garnet(YIG)/graphene6 and EuS/graphene7, where \nYIG or EuS breaks the time reversal and inversion symmetries and therefore serves as the source \nof the se interaction s. To ultimately realize the QAHE in graphene, besides sufficiently strong \nexchange and SOC, graphene needs to have relatively weak disorder so that the energy scale \nassociated with the disorder is smaller than the exchange gap . Therefore, a main challenge is to \nmaximize the exchange gap and simultaneously minimize the disorder energy scale. Here, we \nshow an enhanced proximity exchange effect in a YIG/graphene/h -BN sandwich structure \nrevealed by the anomalous Hall effect (AHE) . In this study, t he h-BN layer replaces the poly -\nmethyl methacrylate (PMMA) layer used in previous devices6, which not only makes an efficient \ntop gate but also protects graphene underneath to preserve its high mobility14. As a result, the \nAHE conductance shows a clear gate voltage dependence as the Fermi level is tuned towards the \nDirac point, which is expected from a gapped Dirac spectrum. Moreover, the AHE conduc tance \nreaches ¼ of the QAHE conductance 2e2/h. The induced magnetic phase transition temperature \nin the heterostructure is as high as room temperature, with an exchange coupling strength of ~27 \nmeV. 3 \n We first epitaxially gro w ~ 20 nm thick YIG films using pulsed laser deposition on \n(111) -oriented gadolinium gallium garnet substrate15. The surface topography of the YIG films is \ncharacteriz ed by atomic force microscope (AFM) with a root -mean -square (rms) roughness of ~1 \n- 2 Ǻ over a 2 × 2 μm2 scan area, as sh own in Fig. 1(b). Due to the interfacial nature of the \nproximity coupling, t he atomically flat surface of YIG films is critical to create a strong \nexchange interaction . Standard Hall bar patterned graphene devices are fabricated on 290 nm -\nthick Si/SiO 2 substrates. Single -layer graphene flakes are loc ated under an optical microscope \nand confirmed by Raman spectroscopy . We use a 20 – 30 nm thick h- BN layer via a \nmicro mechanical transfer process16 to cover the entire graphene region, which simultaneou sly \nprotects graphene from chemical solvent or resist contaminations and serves as an effective top \ngate dielectric la yer. After top gate electrodes are fabricated and tested, we transfer the entire \nfunctional graphene/h- BN devices from Si/SiO 2 substrates to YIG utilizing a previously \ndeveloped transfer technique6,17. The optical images of graphene/h -BN devi ces before and after \ntransfer are illustrated in Fig. 1(c), which show a successful transfer. Transport measureme nts are \nperformed on the same graphene/h -BN devices before and after transfer to track the magneto -\ntransport response s on different substrates . \nThe soft magnetic hysteresis of a bare YIG film with the magnetic field applied along the \nout-of-plane direction shown in Fig. 2 (a) confirms the in -plane magnetic anisotropy arising \nprimarily from the shape anisotropy15. Additional growth- or interface strain -induced in-plane \nmagnetic anisotropy contributes to a higher saturation magnetic field (usually between 2000 and \n3000 Oe ) than the demagnetizing field 4πMs18-20. When graphene is placed on SiO 2, the Hall \nresistance is linear in external magnetic field , which originates from the ordinary Hall effect \n(OHE). A fter the linear background is subtracted, no net signal is left as shown in Fig. 2 (a) . In \nthe same device transferred to YIG, a clear nonlinear Hall curve stands out after subtraction of a \nlinear OHE background also shown in Fig. 2 (a) . Below the Hall curves, the magnetic hysteresis \nof the YIG film measured with the out-of-plane magnetic fields is displayed. It is clear that the \nnonlinear Hall curve resembles the magnetization of the underlying YIG layer. The observed \nnonlinear Hall behavior can arise from the following possible mechanisms : (i) the AHE response \nof graphene in which the carriers are spin polarized due to the interfacial exchange coupling with \nYIG; (ii) the Lorentz force induced nonlinear Hall effect either from the coexistence of two types \nof carriers in graphene21 or the stray magnetic field produced by YIG . As will be discussed, t he 4 \n gate dependence measurements exclude the latter possibility, which favors the first scenario, i.e. \ninduc ed ferromagn etism in graphene due to the exchange coupling with the YIG layer. Here we \nascribe the nonlinear Hall response to the AHE. \nTo quantitatively characterize the exchange coupling strength in YIG/graphene/h -BN, the \nAHE response is studied over a wide range of temperatures from 13 K to 300 K . Fig. 2(b) shows \nthe data taken at several selected temperatures with a top gate voltage of 0.9 V. The AHE \nresistance progressively decreases as the temperature increases , but persists up to 300 K. Note \nthat the AHE resista nce is proportional to the spontaneous magnetization and also has power -law \ndependence on the longitudinal resistance22. Since t he latter is a smooth function of temperature, \nnear the ferromagnetic transition temperature T c, the AHE resistance can be expressed as RAH ~ \n(T – Tc) β with the critical component β = 0.5. By fitting the temperature dependence data, we \nextract the transition temperature T c ~ 308 K, or kBTc ~ 27 meV in exchange energy, which is \nhigher than what was previously reported6. \nAs mentioned earlier, a trivial cause of the nonlinear Hall signal in YIG/graphene/h -BN \nis the Lorentz force, either from the coexistence of two types of carriers or from the stray \nmagnetic field pro duced by the YIG substrate . Such a nonlinear Hall response should reverse its \nsign when the carrier s change from electron - to hole -type or vice versa, whereas the AHE sign \ndoes not have to change as the carrier type switche s. Thus, we measure the nonlinear Hall \nresponse at each fixed gate voltage and repeat for a range of top gate voltages which set different \ncarrier densities on both hole and electron sides . Fig. 3(a) shows the sheet resistance of \nYIG/graphene/ h-BN as a function of the top gate voltage Vtg measured at 13 K. The Dirac point \nVDP of the graphene device is at 0.5 V, very close to zero. T he Hall mobility is as high as 18,400 \ncm2/Vs, about the same or der of magnitude as in SiO 2/graphene /h-BN, indicating no negative \neffect on carrier mobi lity due to the transfer process. The mobility is at least 3 times as high as \nthat in graphene on SiO 2 without h- BN. Due to the thinner h- BN sheet, only much smaller \napplied gate voltage s (decreased by a factor of 15) are needed to tune the carrier density over a \nwide range. \nTo avoid the region where electron s and hole s coexist in the vicinity of the Dirac point \nwhich can also produce a strong and complex nonlinear OHE , we measure the Hall voltages in \nYIG/graphene/h -BN only with 𝑉𝑡𝑔≤0 𝑉 and 𝑉𝑡𝑔≥0.9 𝑉, as indicated in Fig. 3(b). T he sign of \nthe anomalous Hall resistance is independent of the carrier type. If the Hall response were 5 \n generated by the Lorentz force, the sign of nonlinear Hall resistance would switch once the \ncarrier type changes . The observed same nonlinear Hall sign clearly excludes the Lorentz force \nrelated mechanism , and therefore unambiguously demonstrate s the anomalous Hall origin due to \nthe SOC in the proximity -induced ferromagnetic phase of graphe ne. As expected for QAHE \ninsulator s12,13,22,23, the intrinsic AHE from the Berry curvature indeed has the same sign on both \nelectron and hole sides in unquantized regions where the Fermi level is outside the exchange gap . \nWe calculate the anomalous Hal conductance 𝜎𝐴𝐻 as a function of the top gate voltage \nand plot it in Fig. 4. At large gate voltages on both sides, 𝜎𝐴𝐻 is relatively small. As the gate \nvoltage decreases to approach the Dirac point from both sides, 𝜎𝐴𝐻 gradually increases and \nfollows the trend predicted by Qiao et al .12,13 Here w e intentionally stay away from the two -\ncarrier dominated region near the Dirac point. Over the measured gate voltage range, 𝜎𝐴𝐻 is \nclearly not quantized. The maximum 𝜎𝐴𝐻 in our best device reaches ¼ of the quantum anomalous \nHall co nductance , 2e2/h. Both the magnitude and the clear gate voltage dependence of 𝜎𝐴𝐻 \nindicate much improved proximity -induced exchange and SOC strengths compared to the \nprevious YIG/graphene/PMMA devices, thanks to the h- BN that preserves the high quality of \ngraphene sheet and reduces the disorder strength (~ 13 meV) . However, the absence of the \nQAHE plateau suggests a small exchange gap which is smeared out by thermal fluctuation s and \ndisorder . The exchange gap is determined by the small interaction of the exchange and SOC. \nSince we have achieved relatively large exchange interaction, the small exchange gap is mainly \ndue to relatively weak SOC. To ultimately demonstrate the QAHE in graphene, greatly enhanced \nSOC is clearly required. Recent experiments indic ate that it is possible to drastically enhance the \nRashba SOC via proximity coupling with transition metal dichalcogenide materials such as \nWSe 224,25. \nIn summary, the observed AHE dem onstrates the existence of long -range ferromagnetic \norder in graphene proximity coupled with a magnetic insulator. The maximu m anomalous Hall \nconductance reaches ¼ of the QAHE conductance 2e2/h. The exchang e coupling strength is \ndetermined to be as high as 27 meV , indicating an effective role the h- BN played in promoting \nboth exchange and SOC and reducing the disorder strength. Further exploration of incorporating \na transition metal dichalcogenide layer into the sandwich structure to enlarge SOC in graphene \nand utilizing thulium iron garnet with robust perpendicular magnetic anisotropy26 is promising to \nrealize the QAHE at high temperatures and zero magnetic field . 6 \n YIG film growth and ch aracterizations, YIG/graphene heterostructur e device fabrication, \ntransport measurements, and data analysis were supported in part by DOE BES Award No. DE -\nFG02 -07ER46351. Construction of the transfer microscope and device characterizations were \nsupported b y NSF -ECCS under Awards No. 1202559 and NSF -ECCS and No. 1610447. \nExfoliation and transfer of h- BN were supported by DOE ER 46940- DE-SC0010597. \n 7 \n \n \n \n \n \n \nFig. 1. (a) Schematic view of YIG/graphene/ h-BN device; (b) AFM image of a typical YIG film \ngrown by pulsed laser deposition with roughness ~ 0.15 nm across a 2 × 2 μm2 scan area; (c) \nTop: exfoliated graphene on SiO 2 covered with h- BN. Bottom: transferred graphene/ h-BN \ndevices on YIG substrate. \n \n \n8 \n \n \n \nFig. 2. (a) Top: A nomalous Hall resistance of graphene/h- BN on YIG (orange) and SiO 2 (black) \nat 300 K after the OHE background is subtracted . No nonlinear response is left in graphene on \nSiO 2 whereas there is a clear anomalous Hall signal in graphene transferr ed onto YIG. Bottom: \nMagnetic hysteresis b ehavior of YIG film (green) when an external magnetic field is applied \nperpendicular to graphene . The nonlinear Hall resistance in YIG/graphene/h -BN follows the \nmagnetization of the YIG film. (b) A nomalous Hall resistance curves of YIG/graphene/h- BN at \nthe top gate voltage of 0.9 V measured from 13 to 300 K; (c) T emperature dependence of \nanomalous Hall resistance of YIG/graphene/h -BN. The magnetic phase transition temperature T c \nof the induced ferromagne tism in graphene is extracted to be 308 K by using R AH ~ (T – Tc) β \nwith the critical component β = 0.5. \n \n9 \n \n \n \n \nFig. 3. (a) T op gate voltage dependence of the sheet resistance of graphene sandwiched between \nYIG and h -BN measured at 13 K with a mobility of 18,400 cm2/Vs and the Dirac point near 0.5 \nV. Sever al top gate voltages are selected to show the anomalous Hall resistance at differ ent \ncarrier densities on both electron and hole dominated regions. (b) A nomalous Hall resistance in \nYIG/graphene/h -BN at different top gate voltages . The anomalous Hall resistance sign remains \nthe same for both electron and hole carrier types . \n \n \n \n \n \n \n10 \n \n \n \nFig. 4. T op gate voltage and carrier density dependence of anomalous H all conductance \nmeasured at 13 K. R ed squares are experimental data and dashed green curve is drawn for th e \npurpose of eye guidance. The green area marks the electron or hole -dominated region where \nclear AHE is observed. The largest anomalous Hall conductance in YIG/graphene/h -BN reaches \n¼ of the quantum anomalous Hall conductance. The white region is t he e- h coexisting region \nwhere the F ermi level is too close to the Dirac point and additional oscillatory features are \nobserved due to multi- carriers in this region. \n \n \n \n \n \n \n \n11 \n Reference: \n1 Kikkawa, J. M., Baumberg, J. J., Awschalom, D. D., Leonard, D. & Petroff, P. M. \nOptical Studies of Locally Implanted Magnetic Ions in Gaas. Phys. Rev. B 50, 2003- 2006 \n(1994). \n2 Smorchkova, I. P., Samarth, N., Kikkawa, J. M. & Awschalom, D. D. Spin transpor t and \nlocalization in a magnetic two -dimensional electron gas. Phys. Rev. Lett. 78, 3571- 3574 \n(1997). \n3 Haury, A. et al. Observation of a ferromagnetic transition induced by two- dimensional \nhole gas in modulation- doped CdMnTe quantum wells. Phys. Rev. Lett . 79, 511- 514 \n(1997). \n4 Kikkawa, J. M., Smorchkova, I. P., Samarth, N. & Awschalom, D. D. Room -temperature \nspin memory in two -dimensional electron gases. Science 277, 1284 -1287 (1997). \n5 Smorchkova, I. P., Samarth, N., Kikkawa, J. M. & Awschalom, D. D. Gia nt \nmagnetoresistance and quantum phase transitions in strongly localized magnetic two -\ndimensional electron gases. Phys. Rev. B 58, R4238- R4241 (1998). \n6 Wang, Z. Y., Tang, C., Sachs, R., Barlas, Y. & Shi, J. Proximity -Induced \nFerromagnetism in Graphene Rev ealed by the Anomalous Hall Effect. Phys. Rev. Lett. \n114, 016603 (2015). \n7 Wei, P. et al. Strong interfacial exchange field in the graphene/EuS heterostructure. Nat. \nMater. 15, 711 (2016). \n8 Huang, B. et al. Layer -dependent ferromagnetism in a van der Waal s crystal down to the \nmonolayer limit. Nature 546, 270- 273 (2017). \n9 Gong, C. et al. Discovery of intrinsic ferromagnetism in two -dimensional van der Waals \ncrystals. Nature 546, 265 (2017). \n10 Tian, Y., Gray, M. J., Ji, H. W., Cava, R. J. & Burch, K. S. Ma gneto -elastic coupling in a \npotential ferromagnetic 2D atomic crystal. 2d Mater 3, 025035 (2016). \n11 Xing, W. Y. et al. Electric field effect in multilayer Cr2Ge2Te6: a ferromagnetic 2D \nmaterial. 2d Mater 4, 024009 (2017). \n12 Qiao, Z. H. et al. Quantum Anomalous Hall Effect in Graphene Proximity Coupled to an \nAntiferromagnetic Insulator. Phys. Rev. Lett. 112, 116404 (2014). \n13 Qiao, Z. H. et al. Quantum anomalous Hall effect in graphene from Rashba and exchange \neffects. Phys. Rev. B 82, 161414 ( 2010). \n14 Dean, C. R. et al. Boron nitride substrates for high- quality graphene electronics. Nat. \nNanotechnol. 5, 722- 726 (2010). \n15 Tang, C. et al. Exquisite growth control and magnetic properties of yttrium iron garnet \nthin films. Appl. Phys. Lett. 108, 102403 (2016). \n16 Wang, L. et al. One-Dimensional Electrical Contact to a Two -Dimensional Material. \nScience 342, 614 -617 (2013). \n17 Sachs, R., Lin, Z. S., Odenthal, P., Kawakami, R. & Shi, J. Direct comparison of \ngraphene devices before and after transfer to different substrates. Appl. Phys. Lett. 104, \n033103 (2014). \n18 Wang, H. L., Du, C. H., Hammel, P. C. & Yang, F. Y. Strain- tunable magnetocrystalline \nanisotropy in epitaxial Y3Fe5O12 thin films. Phys. Rev. B 89, 134404 (2014). \n19 Krockenberger, Y. et al. Layer -by-layer growth and magnetic properties of Y3Fe5O12 \nthin films on Gd3Ga5O12. J. Appl. Phys. 106, 123911 (2009). 12 \n 20 Sellappan, P., Tang, C., Shi, J. & Garay, J. E. An integrated approach to doped thin films \nwith strain -tunable magnetic anisotropy: po wder synthesis, target preparation and pulsed \nlaser deposition of Bi:YIG. Materials Research Letters 5, 41- 47 (2017). \n21 Bansal, N., Kim, Y. S., Brahlek, M., Edrey, E. & Oh, S. Thickness -Independent \nTransport Channels in Topological Insulator Bi 2Se3 Thin F ilms. Phys. Rev. Lett. 109, \n116804 (2012). \n22 Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall \neffect. Rev. Mod. Phys. 82 , 1539- 1592 (2010). \n23 Chang, C. Z. et al. Experimental Observation of the Quantum Anomalous Hall Effect in a \nMagnetic Topological Insulator. Science 340, 167 -170 (2013). \n24 Wang, Z. et al. Strong interface -induced spin- orbit interaction in graphene on WS 2. Nat. \nCommun. 6, 8339 (2015). \n25 Yang, B. W. et al. Tunable spin- orbit coupling and symmetry -protected edge states in \ngraphene/WS 2. 2d Mater 3, 031012 (2016). \n26 Tang, C. et al. Above 400- K robust perpendicular ferromagnetic phase in a topological \ninsulator. Sci Adv 3, e1700307 (2017). \n " }, { "title": "2307.06047v1.Quantum_information_diode_based_on_a_magnonic_crystal.pdf", "content": "Quantum information diode based on a magnonic crystal\nRohit K. Shukla\nDepartment of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi -\n221005, India\nLevan Chotorlishvili\nE-mail: levan.chotorlishvili@gmail.com\nDepartment of Physics and Medical Engineering, Rzeszow University of Technology, 35-959\nRzeszow Poland\nVipin Vijayan\nDepartment of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi -\n221005, India\nHarshit Verma\nCentre for Engineered Quantum Systems (EQUS), School of Mathematics and Physics, The\nUniversity of Queensland, St Lucia, QLD 4072, Australia\nArthur Ernst\nMax Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany\nInstitute of Theoretical Physics, Johannes Kepler University Alterger Strasse 69, 4040 Linz,\nAustria\nE-mail: Arthur.Ernst@jku.at\nStuart S. P. Parkin\nMax Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany\nSunil K. Mishra\nE-mail: sunilkm.app@iitbhu.ac.in\nDepartment of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi -\n221005, India\nAbstract. Exploiting the effect of nonreciprocal magnons in a system with no inversion\nsymmetry, we propose a concept of a quantum information diode, i.e., a device rectifying\nthe amount of quantum information transmitted in the opposite directions. We control the\nasymmetric left and right quantum information currents through an applied external electric\nfield and quantify it through the left and right out-of-time-ordered correlation (OTOC). To\nenhance the efficiency of the quantum information diode, we utilize a magnonic crystal.\nWe excite magnons of different frequencies and let them propagate in opposite directions.arXiv:2307.06047v1 [quant-ph] 12 Jul 2023Quantum information diode based on a magnonic crystal 2\nNonreciprocal magnons propagating in opposite directions have different dispersion relations.\nMagnons propagating in one direction match resonant conditions and scatter on gate magnons.\nTherefore, magnon flux in one direction is damped in the magnonic crystal leading to an\nasymmetric transport of quantum information in the quantum information diode. A quantum\ninformation diode can be fabricated from an yttrium iron garnet (YIG) film. This is an\nexperimentally feasible concept and implies certain conditions: low temperature and small\ndeviation from the equilibrium to exclude effects of phonons and magnon interactions. We\nshow that rectification of the flaw of quantum information can be controlled efficiently by an\nexternal electric field and magnetoelectric effects.Quantum information diode based on a magnonic crystal 3\n1. Introduction\nA diode is a device designated to support asymmetric transport. Nowadays, household electric\nappliances or advanced experimental scientific equipment are all inconceivable without\nextensive use of diodes. Diodes with a perfect rectification effect permit electrical current to\nflow in one direction only. The progress in nanotechnology and material science passes new\ndemands to a new generation of diodes; futuristic nano-devices that can rectify either acoustic\n(sound waves), thermal phononic, or magnonic spin current transport. Nevertheless, we note\nthat at the nano-scale, the rectification effect is never perfect, i.e., backflow is permitted, but\namplitudes of the front and backflows are different [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].\nIn the present work, we propose an entirely new type of diode designed to rectify the quantum\ninformation current. We do believe that in the foreseeable future the quantum information\ndiode (QID) has a perspective to become a benchmark of quantum information technologies.\nThe functionality of a QID relies on the use of magnonic crystals, i. e. , artificial\nmedia with a characteristic periodic lateral variation of magnetic properties. Similar to\nphotonic crystals, magnonic crystals possess a band gap in the magnonic excitation spectrum.\nTherefore, spin waves with frequencies matching the band gap are not allowed to propagate\nthrough the magnonic crystals [15, 16, 17, 18, 19, 20, 21]. This effect has been utilized earlier\nto demonstrate a magnonic transistor in a YIG strip [15, 16]. The essence of a magnonic\ntransistor is a YIG strip with a periodic modulation of its thickness (magnonic crystal). The\ntransistor is complemented by a source, a drain, and gate antennas. A gate antenna injects\nmagnonic crystal magnons with a frequency ωGmatching the magnonic crystal band gap. In\nthe process, the gate magnons cannot leave the crystal and may reach a high density. Magnons\nemitted from a source with a wave vector ksflowing towards the drain run into the magnonic\ncrystal. The interaction between the source magnons and the magnonic crystal magnons is\na four-magnon scattering process. The magnonic current emitted from the source attenuates\nin the magnonic crystal, and the weak signal reaches the drain due to the scattering. The\nrelaxation process is swift if the following condition holds [16, 22]\nks=m0π\na0, (1)\nwhere m0is the integer, and a0is the crystal lattice constant. The magnons with wave vectors\nsatisfying the Bragg conditions Eq. (1) will be resonantly scattered back, resulting in the\ngeneration of rejection bands in a spin-wave spectrum over which magnon propagation is\nentirely prohibited. Experimental verification of this effect is given in Ref. [16].\n2. Results\n2.1. Proposed set-up for QID\nA pictorial representation of a QID is shown in Fig. 1. A magnonic crystal can be fabricated\nfrom a YIG film. Grooves can be deposited using a lithography procedure in a few nanometer\nsteps, and, for our purpose, we consider parallel lines in width of 1 µm spaced with 10 µm\nfrom each other. Therefore, the lattice constant, approximately a0=11µm,i. e., is muchQuantum information diode based on a magnonic crystal 4\nFigure 1. Illustration of a quantum information diode: A plane of a YIG film with grooves\northogonal to the direction of the propagation of quantum information. In the middle of the\nQID, we pump extra magnons to excite the system. A quantum excitation propagates toward\nthe left, and the right ends asymmetrically. To describe the propagation process of quantum\ninformation, we introduce the left and right OTOC CL(t)andCR(t). Because the left-right\ninversion is equivalent to D→− D, meaning Ey→− Ey, we can invert the left and right OTOCs\nby switching the applied external electric field.\nlarger than the unit cell size a=10nm used in our coarse-graining approach. Due to the\ncapacity of our analytical calculations, we consider quantum spin chains of length about\nN=1000 spins and the maximal distance between the spins ri j=d(in the units of a),\nd=i−j=40. In what follows, we take k(ω)a≪1. The mechanism of the QID is\nbased on the effect of direction dependence of nonreciprocal magnons [23, 24, 25]. In the\nchiral spin systems, the absence of inversion symmetry causes a difference in dispersion\nrelations of the left and right propagating magnons, i. e., ωs,L(k)̸=ωs,R(−k). Due to the\nDzyaloshinskii–Moriya interaction (DMI), magnons of the same frequency ωspropagating\nin opposite directions have different wave vectors [26]: a(k+\ns−k−\ns) =D/J, where Jis the\nexchange constant, and Dis the DMI constant. Therefore, if the condition Eq.(1) holds for\nthe left propagating magnons, it is violated for the right propagating magnons and vice versa.\nThese magnons propagating in different directions decay differently in the magnonic crystal.\nWithout loss of generality, we assume that the right propagating magnons with k+\nssatisfy\nthe condition Eq.(1), and the current attenuates due to the scattering of source magnons by\nthe gate magnons. The left propagating magnons k−\nsviolate the condition Eq.(1), and the\ncurrent flows without scattering. Thus, reversing the source and drain antennas’ positions\nrectifies the current. Following ref. [16], we introduce a suppression rate of the source to\ndrain the magnonic current ξ(D) =1−n+\nD/n−\nD, where n+\nD0, spin\ncurrent is opposite to that under FMR [27], leading to an\nopposite sign in the ISHE voltage compared to the FMR.\nUnder open circuit conditions the inverse spin Hall e\u000bect\n[Eq. (2)] leads to a charge separation and an electrostatic\n\feld\nEEEs=\u0000e\u001a\nA\u0012SH[JJJs;^mmm\u0002^mmm]; (10)\nwhereAis the area of the ferromagnet jmetal interface\nand\u001ais the resistivity of the metal layer. This corre-\nsponds to an electromotive force E=\u0000EEEs\u0001lll, wherelll\nis the length vector from the Loto the Hicontact in\nFig. 1(c) and (d).\nThe sign of an applied magnetic \feld is related to the\ncurrent direction according to Ampere's right hand law\nas depicted in Fig. 1(b). In practice, it is convenient to\nuse a compass needle for comparison with the Earth's\nmagnetic \feld. Fig. 1(b) de\fnes the positive \feld di-\nrection from the antarctic to the arctic, i.e. along the\ngeomagnetic \feld. Typical experimental setups for spin\npumping [(c)] and spin Seebeck experiments [(d)] on yt-\ntrium iron garnet jplatinum thin \flm (YIG jPt) bilayers\nare also sketched in Fig. 1. In the former, a ferromagnet\n(F)jnormal metal (N) stack is exposed to microwave ra-\ndiation with frequency f(typically in the GHz regime),\nwhile in spin Seebeck experiments the bilayer is exposed\nto a thermal gradient. Sample parameters used by the\ndi\u000berent groups are listed in the third column of Fig. 2.\nFor details on the sample fabrication we refer to Refs. 33\nand 34 (WMI), Ref. 18 (RUG), Refs. 35 and 36 (UniKL)\nand Ref. 37 (IMR).\nFig. 2 summarizes the results of the participating\ngroups. Note that in each group both the spin pumping\nand spin Seebeck experiments were performed on the\nsame sample, without changing the setup.\nAt the WMI, FMR experiments (\frst row) were car-\nried out in a microwave cavity with \fxed frequency\nfres= 9:82 GHz as a function of an applied magnetic\n\feldHextleading to resonance at \u00160Hext\u0018=270 mT. A\nsource meter was used to drive a large ( Ih=\u000620 mA) dccharge current through the platinum \flm ( RPt= 197 \n)\nin order to generate a temperature gradient (hot Pt, cold\nYIG) [23]. By summing voltages recorded for opposite\nIhdirections, the magnetoresistive contributions cancel\nout, such that only the spin Seebeck signal remains. The\nISHE voltages for both FMR and the spin Seebeck exper-\niments were measured by the same, identically connected\nnanovoltmeter with microwave and heating current sep-\narately turned on.\nResults from the UniKL are shown in the second row\nin Fig. 2. A microwave with fres= 7 GHz fed into a\nCu stripline on top of the Pt \flm excites the FMR at\nan external magnetic \feld of \u00160Hext\u0018=175 mT. The mi-\ncrowave current amplitude was modulated at a frequency\noffmod= 500 Hz to allow for lock-in detection of the\ninduced voltages [36] that are measured by a nanovolt-\nmeter. The spin pumping data show a small o\u000bset be-\ntween positive and negative magnetic \felds, which stems\nfrom Joule heating in the Pt layer by the microwaves.\nPeltier elements on the top and bottom (separated by\nan AlN layer) generated a thermal gradient for the spin\nSeebeck experiments that were reversed for cross checks,\nas shown in the right graph.\nThe third row in Fig. 2 shows the results obtained\nat the IMR. Here, the sample is placed on a coplanar\nwaveguide such that at fres= 3:8 GHz FMR condition\nis ful\flled for \u00160Hext\u0018=70 mT. The thermal gradient\nfor the spin Seebeck measurements was generated by an\nelectrically isolated separate heater on top of the Pt.\nThe fourth row in Fig. 2 shows the RUG results. A\ncoplanar waveguide on top of the YIG was used to excite\nthe FMR at a magnetic \feld of \u00160Hext\u0018=6 mT. The\nspin Seebeck e\u000bect was detected using an ac-variant of\nthe current heating scheme. The spin Seebeck voltage\ncan thereby be detected as described above in the second\nharmonic of the ac voltage signal.\nIn spite of di\u000berences in samples and measurement\ntechniques, all experiments agree on the sign for spin\npumping and spin Seebeck e\u000bect. We all measure neg-\native spin pumping and positive spin Seebeck voltages\nfor positive applied magnetic \felds that all change sign\nwhen the magnetic \feld is reversed, consistent with the\ntheoretical expectations [1, 2, 26, 27].\nWe can now address the absolute sign of the spin\nHall angle. The results in Fig. 2 were obtained with\nmeasurement con\fgurations equivalent to the one\ndepicted in Figs. 1(c) and (d). With external magnetic\n\feldHextpointing in the ^ zdirection ^ m=^ zand^JJJs=^yyy\nfor FMR spin pumping. According to Eq. (10), when\n\u0012SH>0^EEEs=\u0000^ x, which leads to a negative (positive)\ncharge accumulation at the \u0000x(+x) edge of the Pt\n\flm and a negative spin pumping voltage is expected\nas well as observed. In the spin Seebeck experiments\nwith Pt hotter than YIG, the spin current \rows in\nthe opposite direction ( ^JJJs=\u0000^ y), and the voltage is\ninverted. Therefore, the spin Hall angle of Pt is positive\nif de\fned as above. The nature of the spin Hall e\u000bect\nin Pt is likely to be governed by its electronic band4\n-0.20 .0+ 0.2-3-2-10+ 1+ 2+ 3-\n10 + 1-10+ 1V\n/VsatS\nSEΔ\nT > 0l\nength: 4 mmw\nidth: 3 mmlength: 3 mmw\nidth: 1 mmG\nGG (500 µm)Y\nIG (4.1 µm)P\nt (10 nm) G\narchingspin pumpings pin Seebecks ample parametersG\nGG (500 µm)Y\nIG (160 nm)P\nt (7 nm)V/VmaxS\nP fres = 9.82 GHzµ\n0Hres = 268 mT -\n10 + 1-10+ 1 \n \n fres = 1 GHzµ\n0Hres = 5.5 mT fres = 7 GHzµ\n0Hres = 175 mT \n V/VmaxS\nP-\n0.30 .0+ 0.3-3-2-10+ 1+ 2+ 3 \nΔT < 0 \nV/VsatS\nSEK\naiserslauternΔ\nT > 0-\n10 + 1-10+ 1H\next/Hres \n V/VmaxS\nP-\n10 + 1-3-2-10+ 1+ 2+ 3 \nHext/Hreslength: 600 µmw\nidth: 30 µmGGG (500 µm)Y\nIG (200 nm)P\nt (6 nm)ΔT > 0G\nroningen V/VsatS\nSE -10 + 1-10+ 1 fres = 3.8 GHzµ\n0Hres = 70 mT \n V/VmaxS\nP-\n0.50 .0+ 0.5-3-2-10+ 1+ 2+ 3Δ\nT > 0 \nV/VsatS\nSE G\nGG (500 µm)Y\nIG (4 µm)P\nt (10 nm)length: 6 mmw\nidth: 2 mmS\nendai\nFigure 2. Measured voltage signals for the FMR spin pumping (left column) and spin Seebeck (middle column) experiments\nobtained by the contributing groups. The voltage signals have been normalized to a maximum modulus of unity while the applied\nmagnetic \felds are in units of the FMR resonance \feld Hresgiven in the insets. The temperature di\u000berence \u0001 T=TPt\u0000TYIG\nis positive. The third column lists sample layer thicknesses and dimensions. The sign of the observed voltages is consistent\nbetween the individual groups.\nstructure [38], but it should be a helpful to know that\nthe sign is identical to that caused by negatively charged\nimpurities.\nIn summary, we present spin pumping and spin See-\nbeck experiments for various samples and experimental\nconditions leading to gratifying agreement of the results\nobtained by di\u000berent groups. By carefully accounting for\nthe signs of all experimental parameters and de\fnitions\nwe were able to determine both the relative and the\nabsolute signs of both e\u000bects, linking the positive spin\nHall angle of Pt to a simple physical model of negative\nscattering centers. The relative signs of spin pumping\nand spin Seebeck e\u000bect are consistent with theoretical\npredictions [14, 27{29]. The techniques and samplesused in this letter are representative for a large number\nof spin pumping and spin Seebeck experiments and\nshould serve as a reference for other materials or sample\ngeometries.\nWe thank M. Wagner, M. Althammer and M. 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Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Appl.\nPhys. Lett. 103, 092404 (2013).\n[38] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa,\nPhys. Rev. Lett. 100, 096401 (2008)." }, { "title": "2103.10595v2.Remote_magnon_entanglement_between_two_massive_ferrimagnetic_spheres_via_cavity_optomagnonics.pdf", "content": "Remote magnon entanglement between two massive ferrimagnetic spheres\nvia cavity optomagnonics\nWei-Jiang Wu,1Yi-Pu Wang,1Jin-Ze Wu,1Jie Li,1,\u0003and J. Q. You1\n1Interdisciplinary Center of Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device,\nand State Key Laboratory of Modern Optical Instrumentation,\nDepartment of Physics, Zhejiang University, Hangzhou 310027, China\n(Dated: August 18, 2021)\nRecent studies show that hybrid quantum systems based on magnonics provide a new and promising platform\nfor generating macroscopic quantum states involving a large number of spins. Here we show how to entan-\ngle two magnon modes in two massive yttrium-iron-garnet (YIG) spheres using cavity optomagnonics, where\nmagnons couple to high-quality optical whispering gallery modes supported by the YIG sphere. The spheres\ncan be as large as 1 mm in diameter and each sphere contains more than 1018spins. The proposal is based\non the asymmetry of the Stokes and anti-Stokes sidebands generated by the magnon-induced Brillouin light\nscattering in cavity optomagnonics. This allows one to utilize the Stokes and anti-Stokes scattering process, re-\nspectively, for generating and verifying the entanglement. Our work indicates that cavity optomagnonics could\nbe a promising system for preparing macroscopic quantum states.\nI. INTRODUCTION\nDuring the past decade, cavity magnonics [1–6] has been\nemerged and developed as a new and active platform for the\nstudy of strong interactions between light and matter [7]. It\nconsists of microwave photons which reside in a resonant cav-\nity and interact with magnons (i.e., collective spin excitations)\nin a ferrimagnetic material, e.g., yttrium iron garnet (YIG).\nThe system exhibits its unique features and advantages, which\nlie in the large frequency tunability and low damping rate of\nthe magnon mode, as well as its excellent ability to coher-\nently interact with other systems, including microwave [1–\n6] or optical photons [8–12], phonons [13–15], and super-\nconducting qubits [16–18]. These hybrid cavity magnonic\nsystems promise potential applications in quantum informa-\ntion processing and quantum sensing [19]. A variety of in-\nteresting phenomena have been explored in cavity magnon-\nics, including magnon gradient memory [20], exceptional\npoints [21, 22], manipulation of distant spin currents [23],\nbi-[24] and multi-stability [25], level attraction [26–29], non-\nreciprocity [30], anti-PT symmetry [31, 32], among others.\nIn addition, it has been suggested that cavity-magnon polari-\ntons could be used as an ultra-sensitive magnetometer [33–\n35], and for searching dark matter axions [36] and detecting\nhigh-frequency gravitational waves using the gravitomagnetic\ne\u000bect [37].\nIn this article, we study an emerging field of cavity opto-\nmagnonics [8–10, 12], where a YIG sphere simultaneously\nsupports optical whispering gallery modes (WGMs) and a\nmagnetostatic mode of magnons. The WGM photons are scat-\ntered by the GHz magnons in the form of generating sideband\nphotons with the frequency shifted by the magnon frequency,\nand the high-quality WGM cavity drastically enhances this\nmagnon-induced Brillouin light scattering (BLS). The nature\nof the spin-orbit coupling of the WGM photons, combined\n\u0003jieli6677@hotmail.comwith the geometrical birefringence of the WGM resonator,\nleads to a pronounced nonreciprocity and asymmetry in the\nStokes and anti-Stokes sidebands generated by the magnon-\ninduced BLS [8–10, 12]. This is the result of the selection\nrule [38–41] imposed by the angular momentum conservation.\nBecause of this, this kind of BLS requires a change in optical\npolarization, distinctly di \u000berent from the light scattering in\ncavity optomechanics [42]. The asymmetry nature of the BLS\nallows us to select, on demand, the Stokes or anti-Stokes scat-\ntering event to occur, corresponding to the process of creat-\ning or annihilating magnons. Such a mechanism can be used\nfor the manipulation of magnons, and has been adopted for\npreparing nonclassical states of magnons. This includes the\nproposals for cooling the magnons [43], preparing magnon\nFock states [44], a magnon laser [45], an optomagnonic Bell\ntest [46], and an opto-microwave entanglement mediated by\nmagnons [47], etc.\nBased on this novel system and its unique properties, we\nprovide a scheme to entangle two magnon modes in two mas-\nsive YIG spheres, which can be separated remotely, mani-\nfesting the nonlocal nature of the macroscopic entanglement.\nSpecifically, a weak laser pulse with a certain polarization is\nsent into an optical interferometer formed by two 50 /50 beam\nsplitters (BSs). Each arm of the interferometer contains one\ncavity optomagnonic device, i.e., a YIG sphere supporting op-\ntical WGMs and a magnon mode [8–10, 12], and the magnon\nmode is cooled to its ground state using a dilution refrigerator.\nThe pulse is detuned to be resonant with the WGM cavities in\nthe two arms and to activate the Stokes scattering event, which\nyields a single magnon residing in one of the YIG spheres\nand a lower-frequency photon with a changed polarization in\nthe same arm. Since the BSs are 50 /50 and the two devices\nare assumed identical, the probabilities of the Stokes scatter-\ning event in each arm are thus equal. A single-photon detec-\ntion in the output of the interferometer then projects the two\nmagnon modes onto a path-entangled state, in which the two\nYIG spheres share a single magnon excitation.\nThe scheme can be regarded as the DLCZ protocol [48] ap-\nplied to the system of cavity optomagnonics. The DLCZ pro-arXiv:2103.10595v2 [quant-ph] 17 Aug 20212\ntocol for a cavity optomechanical system has been realized\nusing GHz mechanical resonators [49]. We would like to note\nthat the entanglement between two magnon modes of massive\nferrimagnets has been extensively studied in cavity magnon-\nics [51–57]. Such entangled states involving a large number\nof spins are genuinely macroscopic quantum states, and are\nthus useful for the study of the quantum-to-classical transi-\ntion and the test of unconventional decoherence theories [50].\nIn most studies, the magnon entanglement essentially orig-\ninates from the nonlinearity of the system, which can be\nachieved from, e.g., the magnetostrictive interaction [51], the\nmagnon Kerr e \u000bect [52], or the coupling to a superconduct-\ning qubit [57]. Alternatively, entanglement can be obtained\nby feeding a squeezed vacuum microwave field into the cav-\nity [55, 56]. However, by now proposals for entangling two\nmagnon modes by means of cavity optomagnonics are still\nmissing. The magnon entanglement achieved in the present\nwork utilizes the nonlinearity of the magnon-induced BLS,\nwhich is a unique property of the cavity optomagnonic sys-\ntem.\nII. BASIC INTERACTIONS IN OPTOMAGNONICS\nWe start with the description of two basic interactions in\ncavity optomagnonics, which are key elements for realizing\nour protocol. They are the optomagnonic two-mode squeezing\nand beamsplitter interactions, which are used, respectively,\nto prepare and verify the magnon entanglement of two YIG\nspheres.\nThe magnon-induced BLS in a cavity optomagnonic sys-\ntem [8–10, 12] is intrinsically a three-wave process, which\ncan be described by the Hamiltonian\nH=H0+Hint; (1)\nwhere H0is the free Hamiltonian of two WGMs and a magnon\nmode\nH0=~=!1ay\n1a1+!2ay\n2a2+!mmym; (2)\nwith ajandm(ay\njandmy,j=1;2) being the annihilation\n(creation) operators of the WGMs and magnon mode, respec-\ntively, and!i(i=1;2;m) being their resonance frequencies,\nwhich satisfy the relation !m\u001c!jandj!1\u0000!2j=!m,\nimposed by the conservation of energy in the BLS. The inter-\naction Hamiltonian Hintof the three modes is given by\nHint=~=G0\u0000ay\n1a2my+a1ay\n2m\u0001; (3)\nwhere G0is the single-photon coupling rate. This coupling\nis weak owing to the large frequency di \u000berence between the\nWGM and the magnon mode, but it can be significantly en-\nhanced by intensely driving one of the WGMs. To maximize\nthe BLS scattering probability, we resonantly pump the WGM\na1(a2) to activate the anti-Stokes (Stokes) scattering, which\nis responsible for the optomagnonic state-swap (two-mode\nsqueezing) interaction. Note that the selection rule [38–41]\ncauses di \u000berent polarizations of the two WGMs. Without lossof generality, we assume a2(a1) mode to be the transverse-\nmagnetic (TM) (transverse-electric (TE)) mode of a certain\nWGM orbit, and !2(TM)>! 1(TE) due to the geometrical bire-\nfringence of the WGM resonator [8].\nWe now consider that the WGM a2is resonantly pumped by\na strong optical field. In this case, the strongly driven mode a2\ncan be treated classically as a number \u000b2\u0011ha2i=pN2(\u000b2be-\ning real for a resonant drive), with N2the intra-cavity photon\nnumber, which is determined by the pump power and the de-\ncay rate of the WGM. The linearized interaction Hamiltonian\ncan then be obtained\nHSt:\nint=~G2\u0000ay\n1my+a1m\u0001; (4)\nwhere G2=G0\u000b2is the e \u000bective coupling rate. This Hamil-\ntonian is responsible for the two-mode squeezing interaction\nbetween the WGM a1and magnon mode m, and can be used\nto prepare optomagnonic entangled states. This corresponds\nto the Stokes scattering process, where a TM polarized photon\nconverts into a lower-frequency TE polarized photon by creat-\ning a magnon excitation. Under this Hamiltonian, the WGM\na1and magnon mode mare prepared in a two-mode squeezed\nstate (unnormalized)\nj ioptomag =j00ia1;m+p\nPj11ia1;m+Pj22ia1;m+O(P3=2);(5)\nwhere Pis the probability for a single Stokes scattering event\nto occur, andO(P3=2) denotes the terms with more excitations\nwhose probabilities are equal to or smaller than P3. The scat-\ntering probability Pincreases with the strength of the driv-\ning field. For a su \u000eciently weak driving field, P\u001c1 can\nbe achieved, e.g., in analogous cavity optomechanical experi-\nments, P'0:7% [49] and P'3% [58] were achieved using\nvery weak laser pulses. In this case, the probability of creating\ntwo-magnon /photon statej2im=aand higher excitation states\nis negligibly small. Such a low Stokes scattering probability\nof generating an entangled pair of single excitations, accom-\npanied with very weak laser pulses, is vital for realizing the\nDLCZ(-like) protocols [48, 49]. This is exactly what we shall\nutilize and apply to optomagnonics, in Sec. III, to generate an\nentangled pair of a single magnon and a TE polarized photon.\nSimilarly, when mode a1is resonantly pumped by a strong\nfield, one obtains the following linearized interaction Hamil-\ntonian\nHA:\u0000St:\nint =~G1\u0000a2my+ay\n2m\u0001; (6)\nwhere G1=G0\u000b1and\u000b1=pN1, with N1the intra-cavity pho-\nton number of WGM a1. This Hamiltonian leads to the state-\nswap interaction between the WGM a2and magnon mode m,\nand can be used to read out the magnon state by measuring the\ncreated anti-Stokes field a2. This anti-Stokes scattering corre-\nsponds to the process where a TE polarized photon converts\ninto a higher-frequency TM polarized photon by annihilating\na magnon. As will be shown in Sec. IV, we shall use this\nanti-Stokes process to verify the magnon entanglement.3\nFIG. 1: (a) Sketch of the system used for generating the magnon\nentanglement between two YIG spheres. It consists of an interfer-\nometer formed by two 50 /50 BSs and in each arm of the interfer-\nometer there is a cavity optomagnonic device, in which the magnon-\ninduced Brillouin light scattering occurs. A weak laser pulse with a\ncertain polarization is sent into the interferometer and a subsequent\nsingle-photon detection in the output of the interferometer projects\nthe two magnon modes onto an entangled state. (b) Mode frequen-\ncies of the Stokes light scattering by magnons. A \u001b+-polarized pho-\nton of frequency !\u001b+is converted into a \u0019-polarized Stokes photon\nof frequency !\u0019by creating a magnon of frequency !m. The in-\nput pulse couples to the \u001b+-polarized WGM and the generated \u0019-\npolarized photon goes into the detector, and meanwhile, the magnon\nmode gains a single excitation.\nIII. THE PROTOCOL\nWe now proceed to describe our protocol. The schematic\ndiagram of the protocol is depicted in Fig. 1. Two cavity op-\ntomagnonic devices are placed in two arms of an optical in-\nterferometer formed by two 50 /50 BSs. In each device, a YIG\nsphere supports a magnon mode and high- Qoptical WGMs.\nIn Refs. [8, 10, 12], a roughly 1-mm-diameter sphere was\nused. The YIG sphere is placed in a bias magnetic field along\nthezdirection, while the WGMs propagate along the perime-\nter of the sphere in the x-yplane [8–10, 12]. The frequency of\nthe magnon mode can be adjusted by varying the strength of\nthe bias magnetic field. The two devices and two optical paths\nare assumed identical to erase the ‘which-device /path’ infor-\nmation of the scattered photons at the output of the interferom-\neter (though some mismatches can be compensated via optical\noperations [49]), which is required by the DLCZ protocol. As\na particular advantage of the system, the two magnon-mode\nfrequencies can be tuned to be equal by altering the external\nbias fields. This is technically more di \u000ecult for optomechan-\nical systems as one has to fabricate a large number of samples\nand find a pair of nearly identical mechanical resonators [49].\nAt the end of the interferometer, two single-photon detectors\nare placed in the outputs of the 50 /50 BS, and in each output a\npolarizer is placed in front of the detector, to select the photon\nwith a certain polarization to be detected.We now describe the scheme first by neglecting any optical\nlosses, including the propagation loss and the detection loss\ndue to the nonunity detection e \u000eciency, as well as the loss as-\nsociated with the magnon modes. Also, we further assume\nthat the two magnon modes are prepared in their quantum\nground state. We then analyse the e \u000bects of various experi-\nmental imperfections, and finally provide a strategy for veri-\nfying the entanglement.\nA laser pulse is sent into one input port of the first BS of\nthe interferometer. The energy of the pulse is so weak that the\nmean photon number is much smaller than 1 [48, 49]. It is thus\nin a weak coherent state j\u000bi(j\u000bj\u001c1), with the probability of\nall the n-photon components jni(n>1) being negligible, i.e.,\nj\u000bi'j0i+ppj1i, where p=j\u000bj2\u001c1 is the probability of the\npulse being in the single-photon state. Since the BS is 50 /50,\nthe single photon goes, with equal probability, into one of the\ninterferometer arms (the two arms are termed as path A and\nB, see Fig. 1(a)), thus having the state after the BS\nj\u001eiopt'j00iAB+pp\np\n2\u0000j01iAB+j10iAB\u0001: (7)\nThe pulse is sent through a fiber polarization controller and\ncoupled to the WGM resonator via a tapered silica optical\nnanofiber [8], or a prism coupler [10, 12]. The polarization\nand the frequency of the pulse is tuned to couple to a certain\nTM WGM, e.g., the \u001b+-polarized TM mode. Therefore, the\nstate of the system, soon after the TM mode is excited, is\nj\u001e0iopt'j00i\u001b+\nA\u001b+\nB+pp\np\n2\u0000j01i\u001b+\nA\u001b+\nB+j10i\u001b+\nA\u001b+\nB\u0001; (8)\nwhere the subscript \u001b+\nj(j=A, B) denotes the \u001b+-polarized\nTM mode of the WGM resonator in path j. The\u001b+-polarized\nphoton is then scattered by creating a magnon, and generates a\n\u0019-polarized Stokes photon at frequency !\u0019=!\u001b+\u0000!minto the\nTE WGM, see Fig. 1(b). This is just what we introduced, in\nSec. II, the low Stokes scattering probability of generating an\nentangled pair of a single magnon and a TE polarized photon.\nThis magnon-induced Stokes BLS leads to the following state\nj\u001eioptomag'j0000imAmB\u0019A\u0019B\n+pp\np\n2\u0000j0101imAmB\u0019A\u0019B+j1010imAmB\u0019A\u0019B\u0001;(9)\nwherej0101imAmB\u0019A\u0019Bdenotes the coexistence of a magnon re-\nsiding in the YIG sphere and a \u0019-polarized Stokes photon in\npath B, and similarly for j1010imAmB\u0019A\u0019B. Note that this BLS\noccurs only between the TM and TE modes with the same\nWGM index, owing to the angular momentum conservation\nof photons. Because of the geometrical birefringence, which\nimposes a restriction on the frequencies of the TM and TE\nmodes (of the same mode index), i.e., !TM>! TE, the Stokes\nscattering is preferred in the BLS, while the anti-Stokes scat-\ntering is prohibited, in which a \u001b\u0000-polarized photon is con-\nverted into an anti-Stokes photon at frequency !\u0019=!\u001b\u0000+!m\nby annihilating a magnon [8]. As discussed later, in the part\nof entanglement verification, we shall send a pulse that is cou-\npled to a TE WGM to activate the anti-Stokes scattering. Such4\nan asymmetry nature of the BLS by the magnons is the cor-\nnerstone of realizing our scheme, which o \u000bers the possibility\nof separately implementing the entangling operation and the\nreadout operation of the scheme.\nThe generated \u0019-polarized Stokes photon, with equal prob-\nability in path A or B, then couples to the nanofiber (or the\nprism coupler) and enters the second 50 /50 BS. The polariz-\ners in the outputs of the BS select the TE Stokes photon over\nthe TM photons that failed to e \u000bectively activate the Stokes\nscattering (in practice, the experiment will be repeated many\ntimes due to the low scattering probability in the current opto-\nmagnonic weak coupling regime [8–10, 12]). A single-photon\ndetection in the output of the BS, which realizes the mea-\nsurement M\u0006=\u0000j01i\u0019A\u0019B\u0006j10i\u0019A\u0019B\u0001y, then projects the two\nmagnon modes onto the state\nj\u001eimag=1p\n2\u0000j01imAmB\u0006j10imAmB\u0001: (10)\nThis is a path-entangled state of the two magnon modes in\npaths A and B, and ‘ \u0006’ correspond to the detection of the TE\nphoton in di \u000berent outputs of the BS.\nIn deriving the entangled state (10), we have neglected the\noptical losses (e.g., the propagation loss and the detection\nloss), and the magnon loss. Also, we have assumed that the\nmagnon modes are initialized to their quantum ground state\nj00imAmBby eliminating the residual thermal excitations. We\nnow analyse these e \u000bects one after another.\nFor both the optical propagation loss and the detection loss\ndue to the nonunity detection e \u000eciency, their e \u000bect only re-\nduces the probability of obtaining the desired state (10) but\ndoes not destroy the state [59], which implies longer mea-\nsurement time. This is only true for a su \u000eciently weak pulse\n(being in a weak coherent state j\u000biwithj\u000bj\u001c1) that produces\nat most a single Stokes photon in the output of the interferom-\neter. Optical losses thus disable the single-photon detection.\nThis experiment will be disregarded such that there is no ac-\ntual impact. However, the two-photon component in j\u000bican\nindeed result in unwanted additional states, such as j02imAmB,\nj20imAmB,j11imAmB, in the final magnon state (10). This is also\ntrue for the case without su \u000bering any optical loss as the de-\ntectors are assumed to be photon-number non-resolving in our\nscheme. Nevertheless, the probability of the two-photon state\nis much smaller than that of the single-photon state for a weak\ncoherent state with j\u000bj\u001c1. As long as this is satisfied, those\nadditional states are negligible.\nFor the dissipation of the magnon modes, since the\ntimescale at which our scheme is realized (laser pulses with\nduration of tens of nanoseconds were used in Refs. [49, 58])\nis much shorter than the magnon lifetime (typically of a mi-\ncrosecond [8–10, 12]), during a complete run of the exper-\niment the magnon modes can be assumed to have negligi-\nble dissipation. However, the magnon modes cannot be per-\nfectly initialized to their ground state j00imAmBat typical cryo-\ngenic temperatures. We now study the impact of the residual\nmagnon thermal excitations. Since the frequencies of the two\nmagnon modes are tuned to be equal, the two magnon modesare in the same thermal state under the same temperature\n\u001ath\nmag=(1\u0000S)1X\nn=0Snjnihnj; (11)\nwhere S=¯n=(¯n+1), with ¯ n=\u0002exp(~!m=kBT)\u00001\u0003\u00001being\nthe equilibrium mean thermal magnon number at the temper-\nature T. In general, quantum states of macroscopic objects re-\nquire very low environmental temperatures. For the magnon\nmode with frequency of about 7 GHz [8–10, 12], the ther-\nmal occupation ¯ n'0:036 at T=100 mK. For ¯ n=0:036,\nS'0:035 and S2'0:001, therefore, high-excitation terms\njniwith n>1 can be safely neglected, and we can then ap-\nproximate it as \u001ath\nmag'(1\u0000S)\u0000j0ih0j+Sj1ih1j\u0001. The two\nmagnon modes are thus in a mixed state of a probabilistic mix-\nture of four pure states ji jimAmB(i(j)=0;1). The ratio of the\nprobabilities is 1 : S:S:S2for the two magnon modes being\ninitially inj00imAmB,j01imAmB,j10imAmB, andj11imAmB, respec-\ntively. The ground state j00imAmBis what we have assumed for\nobtaining the desired state j\u001eimagin (10). Combining the other\ninitial states, we obtain the final magnon state (unnormalized),\nconditioned on the single-photon detection, which is\n\u001afinal\nmag'j\u001e00ih\u001e00j+S\u0000j\u001e01ih\u001e01j+j\u001e10ih\u001e10j\u0001+S2j\u001e11ih\u001e11j;\n(12)\nwherej\u001e00i\u0011j\u001eimag, and\nj\u001e01i=1p\n2\u0000j02imAmB\u0006j11imAmB\u0001;\nj\u001e10i=1p\n2\u0000j11imAmB\u0006j20imAmB\u0001;\nj\u001e11i=1p\n2\u0000j12imAmB\u0006j21imAmB\u0001;(13)\ncorresponding to the magnon modes being initially in\nj01imAmB,j10imAmB, andj11imAmB, respectively. The states in\n(13) reduce the fidelity of the desired state (10), as F=\nh\u001e00j\u001afinal\nmagj\u001e00i ' 1=(1+2S+S2). Nevertheless, these addi-\ntional states can be well suppressed if S\u001c1. This is well\nfulfilled at the temperature of 100 mK (50 mK), since S'\n0:035 (0:001)\u001c1. Under this temperature, the fidelity of the\nstate (10) isF ' 0:93 (0.998). Therefore, the impact of the\nresidual thermal excitations under a temperature below 100\nmK will be negligibly small.\nIV . VERIFICATION OF THE ENTANGLEMENT\nLastly, we show how to verify the generated magnon en-\ntanglement. The magnon state can be read out by using the\nanti-Stokes process of the BLS. Specifically, as depicted in\nFig. 2, a weak read pulse is sent into the interferometer with\nits polarization and frequency tuned to couple to a TE WGM\nto activate the anti-Stokes scattering, where a \u0019-polarized\nphoton is converted into a \u001b+-polarized anti-Stokes photon\nby annihilating a magnon, satisfying the frequency relation\n!\u001b+=!\u0019+!m[8]. This can be regarded as the inverse pro-\ncess of the Stokes scattering used for preparing the entangle-\nment. The generated \u001b+-polarized anti-Stokes photon then5\nFIG. 2: (a) Sketch of the system for verifying the magnon entangle-\nment. A weak read pulse is sent into the interferometer soon after the\nentangling pulse and couples to a \u0019-polarized WGM to activate the\nanti-Stokes scattering. The created \u001b+-polarized anti-Stokes photon,\ncontaining the magnon state information, enters one of the detec-\ntors after passing through a polarization rotator (PR) and a polarizer\n(P). An electro-optic modulator (EOM) is added in one arm to re-\nalize the phase o \u000bset\u0001\u001e. (b) Mode frequencies of the anti-Stokes\nlight scattering by magnons. A \u0019-polarized photon is converted into\na\u001b+-polarized anti-Stokes photon by annihilating a magnon that was\nproduced in the preceding entangling stage.\ncouples to the nanofiber /prism coupler and enters the second\nBS. In this circumstance, the polarizers in the outputs of the\nBS select the TM anti-Stokes photon over the TE photons that\ndisable the anti-Stokes scattering. In practice, the read pulse is\nsent after the entangling pulse with a time delay much shorter\nthan the magnon lifetime. Within the delay, the polarizers\nmust be quickly switched in order to select the TM photon for\nverifying the entanglement. Alternatively, one could keep the\npolarizer fixed and put a polarization rotator before the po-\nlarizer, which quickly rotates the polarization of the photon\n(TM$TE). This can be realized via a high-speed waveguide\nelectro-optic polarization modulator [60].\nIn view of the similarity of optomagnonics and optome-\nchanics [61], we adopt the following witness for the magnon\nentanglement [49, 64]\nRm(\u0001\u001e;j)=4g(2)\nA1;Sj(\u0001\u001e)+g(2)\nA2;Sj(\u0001\u001e)\u00001\n\u0010\ng(2)\nA1;Sj(\u0001\u001e)\u0000g(2)\nA2;Sj(\u0001\u001e)\u00112; (14)where g(2)\nAi;Sj=hAy\niSy\njAiSji=\u0000hAy\niAiihSy\njSji\u0001is the second-\norder coherence between the TE Stokes photons (with Sjand\nSy\njthe annihilation and creation operators for the Stokes pho-\ntons going to detector j,j=1;2) and the TM anti-Stokes pho-\ntons (with AiandAy\nithe annihilation and creation operators\nfor the anti-Stokes photons going to detector i,i=1;2), and\n\u0001\u001eis the phase o \u000bset added to the read pulse in one of the in-\nterferometer arms via an electro-optic modulator (EOM), see\nFig. 2. The witness gives Rm(\u0001\u001e;j)\u00151 for all separable states\nof the two magnon modes for any \u0001\u001eandj. Therefore, if there\nis any \u0001\u001eandjwith which Rm(\u0001\u001e;j)<1, the two magnon\nmodes are then entangled.\nV . CONCLUSION\nWe present a scheme for entangling two magnon modes of\ntwo massive ferrimagnetic spheres in an optical interferom-\neter configuration, containing two optomagnonic devices, by\nusing short optical pulses in a cryogenic environment. The\nscheme is based on the asymmetry of the Stokes and anti-\nStokes sidebands in the magnon-induced Brillouin light scat-\ntering. The entanglement is generated on the condition of the\ndetection of single photons with a certain polarization. We\nanalyse the e \u000bects of various experimental imperfections and\nprovide a strategy for verifying the entanglement based on the\nsecond-order coherence between the Stokes and anti-Stokes\nphotons with di \u000berent polarizations. 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Lett.\n107, 123601 (2011)." }, { "title": "2006.10313v1.Current_induced_in_plane_magnetization_switching_in_biaxial_ferrimagnetic_insulator.pdf", "content": "1 \n Current -induced in -plane magnetization switching in biaxial \nferrimagnetic insulator \nYongjian Zhou1,4, Chenyang Guo2,4, Caihua Wan2, Xianzhe Chen1, Xiaofeng Zhou1, \nRuiqi Zhang1, Youdi Gu1, Ruyi Chen1, Huaqiang Wu3, Xiufeng Han2, Feng Pan1 and \nCheng Song1* \n1Key Laboratory of Advanced Materials (MOE), School of Materials Science and \nEngineering, Tsinghua University, Beijing 100084, China \n2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, \nUniversity of Chinese Academy of Sciences , Chinese Academy of Sciences, Beijing \n100190, China \n3Institute of Microelectronics, Tsinghua U niversity, Beijing 100084, China\n \n4These authors contributed equally: Y. Zhou, C. Guo. \n*songcheng@ mail. tsinghua.edu.cn 2 \n Ferrimagnetic insulators (FiMI) have been intensively used in microwave and \nmagneto -optical devices as well as spin caloritronics, where their magnetization \ndirection plays a fundamental role on the device perform ance. The \nmagnetization is generally switched by applying external magnetic fields. Here \nwe investigate current -induced spin -orbit torque (SOT ) switch ing of the \nmagnetization in Y 3Fe5O12 (YIG)/Pt bilayers with in -plane magnetic anisotropy, \nwhere the switch ing is detected by spin Hall magnetoresistance. Reversible \nswitching is found at room temperature for a threshold current density of 107 A \ncm–2. The YIG sublattices with antiparallel and unequal magnetic moments are \naligned parallel/antiparallel to the direction of current pulses, which is consistent \nto the N éel order switching in antiferromagnetic system. It is proposed t hat such \na switching behavior may be triggered by the antidamping -torque acting on the \ntwo antiparallel sublattices of FiMI . Our finding not only broadens the \nmagnetizatio n switching by electrical means and promote s the understanding of \nmagnetization switching, but also paves the way for all -electrically modulated \nmicrowave devices/ spin caloritronics with low power consumption. \nI. INTRODUCTION \nYttrium iron garnet (Y 3Fe5O12, YIG), a ferrimagnetic insulator (FiMI) with \nultralow Gilbert damping and high permeability , has long term been applied in the \nmicrowave and magneto -optical devices. It has drawn increasing interest in \nspintronics field, such as the investigations of the spin Seebe ck effect [1,2], spin Hall \nmagnetoresistance (SMR) [3–5], spin pumping [4,6], non-local magnon transport [7], \ncavity magnon polariton [8] and magnon valves [9]. In particular, YIG is an ideal \nmagnetic mat erial for pure spin current transport because charge current and \ncorresponding Seebeck effect and Nernst effect can be completely eliminated [2], 3 \n providing a promising candidate for electronics with low energy dissipation. Note that \nthe magnitude or even o n/off state of signals and resultant device performance are \nstrongly modulated by the magnetization direction in YIG-based applications , such as \nin spin Seebeck effect [2] and magnon valve [9]. From the application of spintronics, \nYIG is always treated as a simple “ferromagnet” with a single net moment, though it \nis a typical FiMI with two sublattices, especially a magnetic primitive cell of YIG \ncontains 20 Fe moments and a complicated spin structure [10]. In such case, the \nexternal magnetic fields are generally used to switch the magnetization direction of \nYIG. A remarkable miniaturization trend on electronics calls for the switching of \nFiMI with a more convenient and efficient method. \nThe spin -orbit torque (SOT) in ferromagnet/heavy metal bilayers , where the \nangular momentum of spin current induced by charge current with spin Hall effect [11] \nis transferred into magnetic layer in the form of magnetic torque , provides an effective \nelectrical means for manipulating magnetic dynamics and switching the uniform \nmagnetization [12]. Previous studies concentrated on the SOT in metallic systems, \nincluding out -of-plane [13–16] and in -plane [17–19] magnetization switching, \nmagnetic oscillations [20,21 ], domain wall motion [22,23 ] and skyrmions [24,25 ]. \nThese concepts are transferred to the FiMI system, and there are several recent works \nreporting on the SOT switching in FiMI/heavy metal bilayers with perpendicular \nmagnetic anisotropy [26–29], because of the lower energy dissipation and easy \nreadout of th e switching signal, such as by anomalous Hall effect. However, almost all \nof the microwave and spintronics applications of YIG, as mentioned above, are based \non YIG with in -plane magnetization, therefore the electrical switching of in -plane \nmagnetized YIG is strongly pursued (TABLE I ). The experiments described here \ninvestigate the SOT switching of in -plane magnetized YIG (001) in YIG/Pt bilayer s, 4 \n where the two anti -parallel magnetic moments are set parallel/antiparallel to the \ndirection of writing current. \nTABLE I . SOT switching in ferromagnet (FM), ferrimagnetic metal (FiM), \nferrimagnetic insulator (FiMI), antiferromagnetic alloy (AFM) and \nantiferromagnetic insulator (AFMI) with out -of-plane and in -plane magnetic \nanisotropy. Out-of-plane and in -plane SOT s witching was extensively studied in FM \nand FiM. Out -of-plane switching of FiMI and in -plane switching of AFMI were \nrealized recently. This work reports on the in -plane SOT switching of FiMI YIG. \nMagnetic anisotropy FM FiM FiMI AFM AFMI \nOut-of-plane Refs. \n[13,14 ] Refs. \n[15,16 ] Refs. \n[26–29] – – \nIn-plane Refs. \n[17,18 ] Ref. [19] This work Refs. \n[30–34] Refs. \n[35–37] \nII. MRTHODS \n The YIG films with in -plane magnetic anisotropy were deposited on GGG(001) \nsubstrates using a sputtering system with a base vacuum of 1 × 10–6 Pa. After the \ndeposition, high temperature annealing with oxygen atmosphere was carried out to \nfurther improve the crystalline quality and epitaxial relation between YIG film and \nGGG substrate [9]. The YIG thickness was determined using a pre -calibrated growth \nrate. The crystal structure was measured by x -ray diffraction (XRD). The in -plane \nmagnetic anisotropy was recorded by vibrating sample magnetometer (VSM). The \nannealed YIG films were then tra nsferred into another high -vacuum magnetron \nsputtering chamber to ex-situ deposit 5 nm Pt top layer at room temperature. \nThe YIG/Pt bilayers were patterned into eight -terminal devices with channel \nwidth of 5 μm through standard photolithography and Ar ion etching. The current 5 \n induced magnetization switching measurements were carried out at room temperature \nby applying current pulses of 1.4 × 107 A cm−2 with the width of 1 ms, then the \ntransverse Hall resistance was recorded with a reading current of 1.2 × 1 06 A cm−2. \nAnd the spin Hall magnetoresistance experiments with different current densities and \nmagnetic fields were conducted with physical properties measurement system \n(PPMS ). \nIII. RESULTS AND DISSCUSION \nA series of YIG films ( t = 15, 20, 30, 60 nm) were grown on Gd 3Ga5O12 (GGG) \n(001) substrates by magnetron sputtering. In the following we focus primarily on data \nobtained from 20 -nm-thick YIG films at room temperature. X -ray diffraction spectra \nin Fig. 1(a) shows that an addition al peak from YIG (008) emerges in YIG/GGG \nsample, besides the diffraction peak from the GGG substrate, indicating that the YIG \nexhibit (001) -orientation, which serves as the basis of magnetization easy -plane (001). \nFigure 1 (b) presents hysteresis loops of YIG with the magnetic field ( H) applied along \nfour in -plane directions of [100], [010], [110], and [ 1̅10], as well as out -of-plane \ndirection of [001]. A comparison of the squared in -plane loops and slanted \nout-of-plane loop shows that [001] is a hard -axis. The saturation field is ~15 Oe when \nH is applied along [110] and [ 1̅10], in contrast to ~75 Oe along [10 0] and [010]. This \nobservation reflects that the YIG films possess fourfold in -plane magnetic anisotropy \nwith easy -axes along [110] and [ 1̅10] and hard -axes along [100] and [010], which \nresults from the cubic anisotropy of bulk YIG [38]. The saturation magn etization ( MS) \nis around 115 emu cm–3, which is lower than the bulk value (140 emu cm–3) [39]. \nTo perform current -induced in -plane magnetization switching measurements, the \nYIG were covered by 5 -nm-thick P t and then fabricated into eight -terminal devices \nwith the channel width of 5 μm, where the writing current pulse channels are along 6 \n easy-axes of [ 1̅10] and [110] [Fig. 1(c)] . The in -plane switching measurements were \ncarried out in the following way: five successive pulses (current density J = 1.4 × 107 \nA cm–2, 1-ms-width) were applied along [ 1̅10] (write 1), and then along its orthogonal \ndirection [110] (write 2) at zero external magnetic field. After each writing current \npulse, a small reading current ( J = 1.2 × 106 A cm–2) was applied and the transvers e \nHall resistance variation (Δ Rxy) was recorded. Δ Rxy is intrinsically the spin Hall \nmagnetoresistance of YIG/Pt system, where the spin polarization and relevant \nresistance in Pt can reflect the alignment of YIG moments [3]. Concomitant Δ Rxy for \nthe magnet ization switching is displayed in Fig. 1(e), where the red and blue circles \ncorrespond to the red (write 1) and blue (write 2) arrows, respectively (consistent \ncorrespondence for the following results). The current pulse of 1.4 × 107 A cm–2 and \n1-ms-width is the threshold current density for the switching [40]. The most eminent \nresult is the two writing current pulses along [ 1̅10] and [110] lead to the variation of \nΔRxy between low and high resistance states. Further inspection shows that Δ Rxy \nshows a sudden decrease (increase) and is almost saturated whe n the first write 1 \n(write 2) current pulse is applied , and the following four pulses cause a negligible \nvariation in ΔRxy. This observation indicates step -like switching of in -plane \nmagnetized YIG, which is most likely due to the small in -plane magnetic anisotropy \nof the present YIG films. Similar switching features were observed in \nantiferromagnetic α-Fe2O3 [42] and Mn 2Au for the switching from hard - to easy -axis \n[34], but different from the multidomain switching in NiO/Pt [35]. The situation \ndiffers dramatically when the current pulses are applied along in -plane hard -axes \n([100] and [010]) [Fig. 1(d)] . The Δ Rxy remains constant when the current pulses of \n1.4 × 107 A cm–2 are alternatively applied along [100] and [010] [Fig. 1(f)] , indicating \nthe negligible in -plane switching between the hard -axes. A similar behavior also 7 \n occurs in YIG (111) films with in -plane isotropy [40]. It is then concluded that the \nmagnetic anisotropy of YIG plays a fundamental role during in -plane switching \nprocess and the magnetic moments of in -plane biaxial YIG can be switched between \nthe two easy -axes by a current pulse. \n \nFIG. 1. Crystal structure, magnetism and current -induced magnetization switching \nof 20 -nm-thick YIG. (a) X-ray diffraction spectra of GGG substrate and YIG/GGG \nsample. The peak from YIG (008) is marked. (b) Hysteresis loops at room \ntemperature with magnetic field (H) applied along out -of-plane direction [001] and \nfour in -plane directions, where [100] and [010] are in -plane hard -axes, while [110] \nand [ 1̅10] are in -plane easy -axes. The axe s are marked with the same colo r as their \ncorresponding hysteresis loops. (c),(d) Measurement configurations of \ncurrent -induced magnetization switching with writing current applied along in -plane \neasy- (c) and hard -axes (d), respectively. (e),(f) ΔRxy as a function of the number of \nwriting current pulses applied as depicted in (c) and (d), respectively. \n \nThe external magnetic field would provide an additional tool to modulate the \n8 \n current -induced switching of YIG. Figure 2 (a) shows Δ Rxy as a function of current \npulses with different additional fields ( H = 0, 5, 10, 50, and 75 Oe ). The measurement \nconfiguration is identical to the one used in Fig. 1(c), except the H is applied along \nhard-axis [100]. When H is fixed at a quite low value of 5 and 10 Oe, which is just \nbelow and above the coercivity of YIG [Fig. 2(b)] , there is no obv ious difference as \ncompared with the data at H = 0. The scenario turns out to be different when H \nincreases to 50 Oe. The amplitude of Δ Rxy variation is reduced but still evident as H is \nup to 50 Oe, which is close to the saturation field as shown in Fig. 2(b), suggesting \nthat the current -induced in -plane magnetization switching is partially suppressed by H. \nOnce H increases to the saturation field of 75 Oe, there is negligible change of Δ Rxy \nwhen the current pulses alter their directions between write 1 an d write 2, indicating \nthat the current -induced magnetization switching is completely suppressed. This \nmagnetic field modulated switching signals support that the present Δ Rxy variation is \nindeed ascribed to the SOT -induced in -plane switching of YIG ferrimagnetic \nsublattice magnetization , which is similar to the true Né el order switching in α-Fe2O3 \nwith magnetic field [42]. It indicates that thermal artifacts [42–44], which are at least \nnot sensitive to a low magnetic field , have a negligible effect in our experiments . \nMoreover, both the unambiguous current -induced switching signals and artifacts are \nfound in YIG/Cu/Pt trilayer [40 ]. 9 \n \nFIG. 2. SOT-induced switching in YIG/Pt and Co/Pt. (a) Summary of SOT -induced \nΔRxy as a function of writing current pulses with different additional fields H applied \nalong [100] in YIG/Pt bilayers . Results under different H are separat ed by regions of \ndifferent colo rs. (b) Hysteresis loop with H applied along [100]. The typical H \nemployed i n SOT switching measur ements are denoted by dashed arrows. (c) ΔRxy as \na function of the number of writing current pulses in Co/Pt bilayers with H = 0. The \nmeasurement configuration is identical to that of YIG/Pt sample. The polarity of Δ Rxy \nvariation of Co/Pt is opposite to th at of YIG/Pt. (d) Schematic of current -induced \nmagnetization switching in YIG/Pt, where M1 and M2 denote magnetic moments in \ntwo sublattices. M1 and M2 are switched toward the writing current direction. \n \nWe then turn toward the current -induced switching mechanism of YIG/Pt. Note \nthat only one ferrimagnetic insulator layer YIG is used, hence SOT rather than \nspin-transfer torque exists in our case. Remarkably, the switching polarity of Δ Rxy in \nFig. 1(e), negative for w rite 1 and positive for write 2, reveals that the sublattice \nmagnetizations in YIG are aligned parallel/antiparallel to the direction of writing \n10 \n current according to SMR theory [41 ]. This feature is also supported by the \nlongitudinal resistance variation [40]. These experimental observations unravel that \nthe current -induced in-plane sublattice magnetizations switching of bi-axial \nferrimagnet YIG/Pt is in analogy to the Né el order switching toward the writing \ncurrent direction in antiferromagnet/Pt systems [35,37 ], where the antidamping -torque \ndominates. It is also reasonable to propose that the antidamping -torque may induce \nthe ferrimagnetic moments switching of bi-axial YIG, where two sublattices with \nantiparallel magnetic moments is strongly antiferromag netic coupled , though \nuncompensated. Note that the polarity of current -induced switching in ferrimagnetic \namorphous metal CoGd [19], which possesses negligible magnetic anisotropy, is \nopposite to our results in bi -axial YIG. This indicates that the magneti c anisotropy in \nferrimagnet has a vital effect on current -induced in -plane switching, which is \nsupported by the absence of SOT -induced switching in our YIG(111) samples [40]. \nTo further study the SOT switching in biaxial ferrimagnetic YIG where the \nmagnetization is aligned parallel/antiparallel to the current axis, we performed contro l \nexperiment with ferromagnet and compare d the current -induced switching polarity of \nΔRxy using Co (2 nm)/Pt (5 nm) bilayers where the magnetization shoul d be aligned \nperpendicular to the current axis [17,18,45 ]. Although Co is a conductor, which is not \nperfect as a comparison of YIG, Co/Pt bilayer possesses considerable SMR signal \n[46], from which the direction of magnetization can be readout easily. Therefore, \nCo/Pt bilayer us ed as a control sample is reasonable . The measurement configuration \nis identical to that of YIG/Pt [Fig. 1(c)] . The variation of Δ Rxy in Fig. 2(c) is opposite \nto the YIG/Pt case. Once an external field of 10 kOe is applied on the devices, the \nswitching signals vanishes [40], indicating the variation of Δ Rxy is indeed due to the \nswitching of Co moments. In the case of current -induced in -plane switching of 11 \n ferromagnet, the magnetic moment should be switched to the direction of spin \npolarization [17,18 ,45], which is perpendicular to the writing current direction. This \nprocess can be understood by the transfer of spin angular moment from the \nspin-polarized current to the magnetization, similar to the spin -transfer -torque \nscenario. This control experiment further confirms that the sublattice magnetizations \nof YIG are aligned parallel/antiparallel to the writing current direction, rather than \nbeing aligned along spin polarization direction. This may indicate the importance of \nantiferromagnetic coupled sublattices during current -induced in -plane switching in \nbi-axial ferrimagnet. The charge current in the Pt layer produces spin current and the \nresultant spin accumulation by spin Hall effect [11], exerting a t orque on the two \nanti-parallel magnetic sublattices ( M1 and M2) and then resulting in their switching \nparallel or antiparallel to the writing current direction [Fig. 2(d)] . The intriguing \nin-plane switching feature in our case discloses that as a typical FiMI, the \nanti-paralleled magnetic sublattice in YIG may play an important role , which is \nsimilar to the N éel order switching in antiferromagnets to some extent at least from \nthe current -induced in -plane magnetization switching viewpoint. More experiments or \nsimulation s in different FiMI are needed for further understanding of current -induced \nin-plane switching in FiMI system s. Based on those results and analyse s, the magnetic \nfield manipulated true current -induced switching and co -existence of artifacts [40] \nmake in -plane bi -axial ferrimagnet YIG a model for investigating the current -induced \nswitching, which could promote the application of ferrimangetic spintronics. \nIn addition to the SOT -induced magnetization switching, we have explored \nSOT-induced magnetic moments tilting toward the current direction by SMR \nmeasurements with different J and H. For these experiments , Hall resistance ( Rxy) was \nrecorded when the current was applied along one of the easy -axes [110] and the 12 \n magnetic field was in -plane rotated from the current ( I) direction (angle αH = 0 for H \n// I), as depicted in Fig. 3(a). Figure 3 (b) shows the angle αH dependent Rxy with \ndifferent current densities ( J = 1.2, 2, 6, 8, and 10 × 106 A cm–2) and H = 50 Oe. \nRemarkably, as the current density increases gradually from 1.2 to 10 × 106 A cm–2, \nwithin 0 –90° scale the high Hall resistance state (peak) shifts to a high rotation angle. \nOn the basis of the SOT -switching results described above, magnetization is tilted \ntoward the current direction, therefore M1 (αM) and M2 deviate from the field direction \nand toward the current direction ( αM = 0) [Fig. 3(c)] . Thus it is necessary to use a \nmagnetic field with a larger rotation angle αH to compensate the tilting tendency \ninduced by SOT, which results in the shift of SMR curves. Also visible is that in the \nrange of 90–180° the low Hall resistance state (valley) shifts to a low rotation angle, \nbecause the SOT induces magnetization switching toward the current direction ( αM = \n180° ) and then a magnetic field with a smaller rotation angle αH is employed. The \nvariation tendencies in 180 –270° (peak) and 270–360° (valle y) scales are similar to \nthese two scenarios, respectively. In general, the SMR curve exhibits a high and low \nHall resistance states for αH = 45º /225º and 135º /315º , respectively [41]. Note that the \nslight deviation of the observed peak and v alley with low current density [Fig. 3(b)] \nfrom these theoretical angles is due to a low field of 50 Oe used, which is below the \nsaturation field of the YIG film. Such a deviation vanishes when a high magnetic field \nis used, such as H = 5000 Oe [40]. \nThe a ngle shift of the SMR curves as a function of J with different external fields \n(H = 50, 100, 1000, and 5000 Oe) is summarized in Fig. 3(d), where Δ αH is the angle \ndifference between the valley and peak (in 0 –180° ) of Rxy. There are two striking \nfeatures in the figure: (i) The angle difference Δ αH is reduced (angle shift is enhanced) \nwith increasing J; (ii) The modulation of Δ αH is suppressed with increasing magnetic 13 \n field, which almost vanishes with a high H, e.g., 5000 Oe [40]. It is easy to understand \nthat a larger current induces stronger magnetization titling toward the current direction, \nresulting in a larger “compensated angle” for the field, which reduces ΔαH. With \nincreasing magnetic field, the SOT -induced magnetization tilting is negligible because \nthe magnetization is always retained along the field. The magnetization switching \nphenomena deduced from these SMR results with different J and H are consistent \nwith switching responses in Fig. 2. Thereby the current -dependent SMR \nmeasurements support SOT -induced in -plane sublattice magnetization s switching \ntoward parallel/antiparallel to the writing current direction. In addition, we quantify \nthe SOT -field equivalence through the summary of Δ αH resulting from SOT induced \nSMR tilting. An extra angle αSOT,J produced by SOT with current density J is \nintroduced in transverse SMR equation: \n 𝑅xy=∆𝑅sin2(𝛼H−𝛼SOT ,J), 0°<𝛼<90° (1) \n𝑅xy=∆𝑅sin2(𝛼H+𝛼SOT ,J), 90° <𝛼<180° (2) \nwhere Rxy and Δ R are transverse resistance and amplitude of transverse SMR, \nrespectively. When αH = 45º +αSOT,J (135º –αSOT,J), Rxy is high (low) resistance state, \nnamely peak (valley) of SMR in 0 –180°. Therefore, Δ αH in Fig. 3(d) equals 90º –\n2αSOT,J. The variation of Δ αH with different J under H = 50 Oe are fitted by linear \nfunctions [dashed line in Fig. 3(d)], and the slope k is arou nd –4.1 deg/(106 A cm–2). \nBased on the equation of slope k: \n 𝑘=(90°−2𝛼SOT ,J1)−(90°−2𝛼SOT ,J2)\n𝐽1−𝐽2=−2𝛼SOT ,J1+2𝛼SOT ,J2\n𝐽1−𝐽2=−2∆𝛼SOT ,J\n∆𝐽 (3) \nwhere αSOT,J1(J2) , ΔαSOT,J and ΔJ are angles induced by SOT with current density J1(J2), \nthe variation of αSOT,J and the change of current density J, respectively, the value of \n∆𝛼SOT ,J\n∆𝐽 is calculated to be around 2.1 deg/(106 A cm–2). Based on the above analyses , 14 \n the sublattice magnetization is switched toward the direction of current, hence the \nSOT induced equivalent field HSOT is set along current axis here [40]. On the basis of \nmagnetic field vector addition, the SOT -field equival ence in YIG is determined to be \n2.6 Oe/(106 A cm–2) [40], which is comparable with or slightly larger than the value \nobtained in typical ferrimagneti c insulators, such as TmIG [0.6 Oe/(106 A cm–2)] [26]. \n \nFIG. 3. Current -induced magnetization tilting during SMR measurements . (a) \nMeasurement configurations of SMR. The current I is applied along [110]. The \nmagnetic field H rotates in the plane of device . (b) SMR curves with different current \ndensities (marked in the inset) and H = 50 Oe. The dashed arrows are a guide to \nreflect the shift of peak /valley positions. (c) Schematic of the magnetization tilting of \ntwo sublattices of YIG induced by applied current. αM (αH) denotes the angle between \nI and M1 (H). The red ( M1) and blue ( M2) arrows denote the magnetization of two \nmagnetic sublattices, and the thin purple arrow represents the tilting direction . (d) \nSummary of the angle difference Δ αH between the valley and peak (in 0 –180° ) of \nSMR with different J and H. The typical H used are marked. The error bars are \nestimated from standard deviation of three SMR measurements. The dashed line is \nlinear fitting of Δ αH with different current densities under H = 50 Oe. \n \n15 \n We then focus on the YIG -thickness (t) dependent transport measurements. Figure \n4(a) presents the SOT -induced Δ Rxy variation for 15, 20, 30, and 60 nm-thick YIG. All \nof the YIG/Pt samples exhibit reversible Δ Rxy variation. A comparison of the Δ Rxy \nshows that the magnitude of the Δ Rxy is greatl y enhanced with increasing t to 30 nm, \nand then Δ Rxy is saturated and keeps almost unchanged even up to 60 nm. On the \nother side, the angle αH dependent SMR curves measured with H = 5000 Oe for \ndifferent t is shown in Fig. 4(b). Remarkably, the magnitude of SMR signals are \nenhanced with increasing t and saturated at t = 30 nm, which coincides with the \nthickness -dependence of SOT -induced Δ Rxy variation. Since SOT -induced switching \nresults in the present case are dependent on two factors, the current -based writing \nefficiency and the SMR -based readout capability, it is significant to exclude the \ninfluence of thickness -dependent SMR when exploring the switching efficiency. In \nthis scenario, the ratio of Δ Rxy/ΔSMR , where ΔSMR is the Rxy difference between the \npeak and valley of SMR curves in Fig. 4(b), is introduced to reflect the switching \nefficiency in our case. The ratio of Δ Rxy/ΔSMR as a function of t is displayed in Fig. \n4(c), which shows a gradual enhancement with increasing t and is almost saturated at t \n= 30 nm. Meanwhile, the saturation magnetization ( MS) [40] of the YIG films is also \npresented in Fig. 4(c) for a comparison. It is found that both of them show a similar \nthickness -dependence, suggesting that the switching efficiency is relevant to MS. \nBecause of the enhancement of MS and corresponding interfacial exchange interaction, \nmore spin current can flow into the YIG [28], which enhances switching efficiency of \nYIG and result ant Δ Rxy/ΔSMR. 16 \n \nFIG. 4. SOT-induced switching and SMR measurements in YIG/Pt with different \nYIG thicknesses ( t). (a) ΔRxy as a function of pulse numbers in YIG/Pt bilayers with t \n= 15, 20, 30, and 60 nm. Results of different t are separate d by regions of different \ncolor s. (b) SMR results with different t under H = 5000 Oe. The Rxy differences \nbetween peak and valley ( ΔSMR ) are denoted by (grey) arrows in the inset, and the \nvalues are 3.0, 3.2, 3.8, and 3.8 m Ω for t = 15, 20, 30, and 60 nm, respectively. (c) \nΔRxy/ΔSMR and MS versus t. Both Δ Rxy/ΔSMR and MS show a similar \nthickness -dependence and saturate at t = 30 nm. The error bars of Δ Rxy/ΔSMR and MS \nare estimated from standard deviation of reversible switching and three magnetization \nmeasurements, respec tively. \n \nIV. CONCLUSION \nIn summary, we have demonstrated the reversible in -plane magnetization \nswitching of YIG in YIG/Pt bilayers by SOT. The switching signal is readout by spin \nHall magnetoresistance. The sublattice magnetizations of YIG are found to be aligned \nparallel/antiparallel to the direction of writing current, and may be ascribed to the \nantidamping -torque for the two strongly antiferromagnetic coupled sublattices, which \nis similar to the N éel order switching in antiferroma gnetic system to some extent . This \n17 \n phenomenon indicates that anti-paralleled sublattices in ferrimagnetic insulator may \nplay an important role during the current -induced in -plane magnetization switching \nprocess, and more studies with different materials are needed to further explore \nswitching in ferrimagnetic insulator s in the future. Our finding not only promotes the \nunderstanding of current -induced switching , but also accelerate s combination of \ncurrent -induced magnetization switching and previous microwave devices/spin \ncaloritronics to realize high energy efficient spintronic applications based on magnetic \ninsulators modulated by all -electrical means. \n \nACKNOWLEDGMENTS \nWe thank D r. D. Z. Hou for helpful discussions. C.S. acknowledges the support of \nthe Beijing Innovation Center for Future Chips, Tsinghua University and the Young \nChang Jiang Scholars Programme. 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Phys. 11, 570 (2015). \n " }, { "title": "2312.10660v2.Cryogenic_hybrid_magnonic_circuits_based_on_spalled_YIG_thin_films.pdf", "content": "Cryogenic hybrid magnonic circuits based on spalled YIG thin films\nJing Xu∗,1, 2Connor Horn∗,1Yu Jiang,3Xinhao Li,2Daniel Rosenmann,2\nXu Han,2Miguel Levy,4Supratik Guha†,1and Xufeng Zhang‡3\n1Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA\n2Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL 60439, USA\n3Department of Electrical and Computer Engineering,\nNortheastern University, Boston, MA 02115, USA\n4Department of Physics, Michigan Technological University, Houghton, MI 49931, USA\n(Dated: December 20, 2023)\nYttrium iron garnet (YIG) magnonics has sparked extensive research interests toward harnessing\nmagnons (quasiparticles of collective spin excitation) for signal processing. In particular, YIG\nmagnonics-based hybrid systems exhibit great potentials for quantum information science because\nof their wide frequency tunability and excellent compatibility with other platforms. However, the\nbroad application and scalability of thin-film YIG devices in the quantum regime has been severely\nlimited due to the substantial microwave loss in the host substrate for YIG, gadolinium gallium\ngarnet (GGG), at cryogenic temperatures. In this study, we demonstrate that substrate-free YIG\nthin films can be obtained by introducing the controlled spalling and layer transfer technology\nto YIG/GGG samples. Our approach is validated by measuring a hybrid device consisting of a\nsuperconducting resonator and a spalled YIG film, which gives a strong coupling feature indicating\nthe good coherence of our system. This advancement paves the way for enhanced on-chip integration\nand the scalability of YIG-based quantum devices.\nThe field of YIG magnonics [1] is a rapidly evolving re-\nsearch area dedicated to studying the collective spin exci-\ntations (magnons) in YIG (yttrium iron garnet) crystals.\nIn recent years, it has shown extensive potential in hybrid\ninformation systems [2–5]. Thanks to its low magnetic\ndamping, high spin density, and excellent compatibility\nwith various physical platforms, YIG has been consid-\nered as an ideal magnonic platform for hybrid quantum\ninformation processing. Researchers are actively explor-\ning various YIG-based hybrid systems such as electro-\nmagnonics [6–13], optomagnonics [14–16], and magnome-\nchanics [17–20] for different applications. As demand\ngrows for scalable quantum systems, thin-film YIG de-\nvices are highly desired for on-chip integration over bulk\nYIG spheres used in earlier research.\nHowever, the development of thin film YIG devices at\ncryogenic temperature regimes has been severely limited.\nOne major obstacle is the undesirable properties of the\nsubstrate used for the growth of YIG thin film. The\nbest growth method for high quality single crystalline\nYIG films is epitaxial growth on gadolinium gallium gar-\nnet (GGG) substrates which has a matched lattice con-\nstant with YIG, yielding a room-temperature magnon\nlinewidth close to that of single-crystal YIG spheres.\nHowever, at cryogenic temperatures such YIG films ex-\nhibit very high microwave losses because the host sub-\nstrate GGG undergoes a phase transition into a geomet-\nrically frustrating spin-liquid state below 5 Kelvin [21].\nIn this state, the short-range ordered spins in the GGG\nsubstrate shows strong absorption to external energy, a\nproperty that has found application in commercial adia-\nbatic demagnetization cooling [22, 23]. The presence of\nthis spin-liquid state in GGG degrades the lifetime of spin\nexcitations in the YIG layer, as indicated by the larger\nFMR linewidth [24–26] compared with spheres made ofpure YIG, posing a significant impediment to its integra-\ntion in cryogenic quantum systems. For instance, electro-\nmagnonic systems involving strongly coupled microwave\nphotons and magnons have been extensively studied in\nthe past few years [6–13, 27–29], which have enabled ad-\nvanced functionalities such as entanglement with super-\nconducting qubits [30–32], but most of previous demon-\nstrations are based on YIG spheres while YIG thin films\nhave been rarely used.\nOne promising solution to this grand challenge is us-\ning YIG thin films without the GGG substrate, which in-\ncludes two technical approaches. The first approach is to\ngrow YIG thin films on substrates other than GGG such\nas silicon [25, 33–35]. This approach is straightforward\nand more favorable from the aspect of device integration;\nhowever, the quality of YIG thin films are usually low be-\ncause of the lattice mismatch between YIG and the new\nsubstrate. The second approach involves growth of YIG\non GGG substrates with post-processing to detach YIG\nfrom the GGG substrate [36–39]. This method produces\nhigh quality YIG but the separation processes for YIG\nare usually challenging due to the similar physical and\nchemical properties of YIG and GGG, hindering wide\napplication of such technologies. In this work, we show\nour investigation on a new method that can simplify the\nYIG detaching process, providing a new direction for the\ndevelopment of YIG thin film devices.\nThe approach we used for detaching YIG from GGG\nis based on controlled mechanical spalling [40], as shown\nby the procedures in 1(b). The substrate we used is a\ncommercially available single-crystal YIG film [200 nm\nthick, (111)-oriented] grown on a 500 µm-thick GGG\nsubstrate by liquid phase epitaxy (LPE). After cleaning\nthe sample, a layer of 10-nm-thick chromium followed\nby 70-nm-thick gold are deposited on the YIG surfacearXiv:2312.10660v2 [cond-mat.mes-hall] 19 Dec 20232\nFIG. 1. (a) A schematic of an integrated hybrid quantum device using the substrate-free YIG thin film. (b) Schematics of\nthe YIG film spalling process: (I) (Optional) helium ion implantation process with implantation depth around 7 µm in the\nYIG/GGG substrate. (II) A stack of films comprising 10 nm of chromium, 70 nm of gold, and 7 µm of nickel deposited on a 200\nnm YIG/500 µm GGG substrate. (III) The separation point occurs at approximately 2 µm beneath the YIG/GGG substrate’s\ntop surface, where stress accumulates, achieved by using a thermal release tape. (IV) Materials remaining on the tape, from\nbottom to top: 7 µm of nickel, 70 nm of gold, 10 nm of chromium, 200 nm of YIG, and 2 µm of GGG. (V) Subsequent removal\nof metal layers is accomplished using nickel, gold, and chromium etchants in sequence. (c) Optical image of the spalled sample\nafter step II. Right side: substrate-free film adhered to the thermal release tape. Left side: the remaining GGG substrate after\nthe spalling process. (d) Confocal microscope measurement image of the surface of the remaining GGG substrate. The outline\nof the spalled area is marked by the yellow dashed line.\nthrough magnetron sputtering. Using the gold layer as a\nseed layer, a thick layer of Ni is electroplated with a fi-\nnal thickness of 7 µm using the electroplating conditions\nin reference [41], which yields an intrinsic tensile stress\nof 700 MPa. The tensile stress and thickness of the Ni\ndefines an equilibrium depth within the YIG/GGG sub-\nstrate at which steady state crack propagation can take\nplace [42]. Using a thermal release tape, the top layer\nstack (Ni/Au/Cr/YIG/GGG) is carefully spalled, result-\ning in a continuous substrate-free film, as shown in Figure\n1(c). Considering that this depth is larger than the thick-\nness of the YIG layer, the spalled YIG is still attached to\na thin layer of GGG. However, by choosing YIG wafers\nwith larger YIG thickness (e.g., 10 µ) and optimizing the\nthickness and tensile stress of the YIG layers, it is pos-\nsible to obtain a spalled layer of pure YIG without any\nresidual GGG. The final substrate-free device is obtained\nby removing the nickel, gold, and chromium layers with\nappropriate wet chemical etchants.\nWe noticed that with our current conditions, the\nspalling process is also largely affected by ion implan-\ntation in the substrate. When the substrate is intrinsic\nYIG/GGG sample, the resulting spalling depth is typi-\ncally 5-10 um, whereas if the YIG/GGG sample is treated\nwith helium ion implantation (following the conditions\nfrom Refs. [36, 43]), thinner spalling depths are obtainedwith smoother surfaces, as shown by height scan of the\nremaining GGG substrate using a 3D laser scanning con-\nfocal microscope [1(d)] which reveals a spalling depth of\naround 2-3 µm. This may be attributed to the fact that\nthe GGG layer above the ion implantation depth (7 µm)\nis damaged by the high-energy helium ions during the im-\nplanting process and becomes easier to break under the\nelastic stress from the Ni layer, resulting in a shallower\nspalling depth.\nTo characterize the microwave performance of the\nspalled YIG thin film at room temperature, it is flip-\nbonded to a rectangle split ring resonator (RSRR) made\nof copper [13] to test the ferromagnetic resonance (FMR)\nresponse, as schematically illustrated by Fig. 2(a). The\nreflection spectrum of the RSRR is obtained using a vec-\ntor network analyzer. When an out-of-plane bias mag-\nnetic field is tuned to sweep the FMR frequency, it is\nexpected that the magnon mode becomes visible in the\nspectrum when it is tuned very close to the microwave\nresonator frequency. However, when a 500 ×300×2µm3\nflake from the spalled YIG film (which has been treated\nwith helium ion implantation) is bonded on the RSRR,\nno magnon modes can be observed from the RSRR reflec-\ntion spectra. This is speculated as the result of the large\nmagnon linewidth in the YIG flake and the small volume\nof the YIG flake (accordingly small coupling with the3\nFIG. 2. (a) Schematics of a substrate-free YIG thin film\nbonded onto an RSRR chip. (b) Optical image of a piece\nof spalled YIG (approximately 500 µm×300µmaffixed to a\nKapton tape. (c) Measured reflection spectra for the RSRR\ndevice with the YIG/GGG chip attached (prior to annealing).\n(d) Measured reflection spectra for the RSRR device with the\nspalled YIG (after annealing).\nRSRR resonator). To verify this speculation, we tested\na larger piece (roughly 5mm lateral size) of unspalled\nYIG that was treated through the same ion implantation\nprocess, using the same RSRR reflection measurement.\nWith the increased YIG volume, the magnon mode is\nsuccessfully observed, but the measured data [Fig. 2(c)]\nshows a notably high dissipation rate κm/2π= 39.1 MHz\nat 11.6 GHz, which is one order of magnitude higher\nthan previously reported values on single-crystal YIG\nthin films [13]. Such elevated disspation rates can be\nattributed to two possible sources of dissipation: (1) The\ndamage to the crystalline structure caused by the high-\nenergy ions penetrating the YIG layer, and (2) The ac-\ncumulated helium ions in the YIG layer. Both effects\ncan be mitigated using an annealing process, which will\nrestore the damaged lattice structure and repel the ac-\ncumulated helium ions from YIG. We carried out an an-\nnealing process for multiple flakes of the substrate-free\nYIG film in ambient air, reaching a maximum temper-\nature of 850◦C. The samples is gradually heated from\nroom temperature to 850◦C over a period of 6 hours,\nfollowed by a 3-hour hold at the maximum temperature,\nand then a slow cooling process over a span of 14 hours\nto room temperature, providing ample time for the re-\npair of the YIG crystalline lattice. After the annealing\nprocess, the magnon mode shows up on a device with a\nsmall YIG flake (around 500 ×300×2µm3), as shown in\nFig. 2(d). Numerical fitting shows a dissipation rage of\nκm/2π= 2.08 MHz at 10.5 GHz, which is comparable\nwith those of high-quality LPE YIG thin films reported\nin previous articles [13, 24, 44].\nTo further demonstrate its potential for cryogenic\nFIG. 3. (a) Layout depicting the superconducting resonator\ncoupled to a bus transmission line. (b) Zoomed-in view of the\nlumped element resonator design, showing the interdigital ca-\npacitors on the sides and the inductor line in the middle. (c)\nOptical image displaying a superdoncuting resonator loaded\nwith the spalled YIG thin film (measuring approximately 300\nµm×200µm). The irregular shaded area is the GE varnish\nfor chip bonding. (d) The measured spectrum of the chip\nbonded with the spalled YIG at 200 mK, revealing two mi-\ncrowave resonances at 5.05 GHz and 9.34 GHz, respectively.\n(e) The measured transmission spectrum of another super-\nconducting resonator device at 200 mK with (red curve) and\nwithout (black curve) the unspalled YIG/GGG substrate.\nquantum operations, the spalled YIG thin films are fur-\nther characterized at millikelvin (mk) temperature. A\nsuperconducting resonator is used to couple with the\nYIG flake, which is fabricated using 100-nm-thik nio-\nbium through photolithography and dry etching. To en-\nhance the magnon-photon coupling, a lumped-element\nresonator with interdigital capacitors is used, which is\ninductively couples to a bus transmission line, as shown\nby the layout plot in Figs. 3(a) and (b). The YIG flake\nis flip-bonded using GE varnish to cover the center in-\nductor line where the microwave magnetic field is the\nstrongest, which further enhances the coupling strength\nbetween the magnon and photon modes. An optical im-\nage of the assembled device is shown in Fig. 3(c)], where\nthe position of the spalled YIG flake is outlined by the\nred dashed curve.\nFigure 3(d) depicts the measured microwave transmis-\nsion of our device at a temperature of 200 millikelvin\nperformed within an adiabatic demagnetization refriger-\nator (ADR). Two resonance modes can be observed as\nthe two sharp dips in the transmission spectrum, simi-\nlar to what is observed in Ref.[28]. The inset highlights\nthe fundamental mode at 5.05 GHz which has a lower\ndamping rate and smaller extinction ratio. The higher-4\nFIG. 4. (a) A heatmap of the measured transmission spec-\ntrum for the superconducting resonator device shown in Fig.\n3(c), showing the avoid-crossing feature. The overlaid red\ncircles represent the calculated frequencies of magnon-photon\nmodes. (b) A heatmap of the calculated transmission spec-\ntrum using Eq (1) with the fitted magnon linewidth. (c)\nTransmission data at 1300 G where the magnon mode is far\ndetuned. The photon linewidth is extracted as κc/2π= 3.9\nMHz. (d) Transmission data at 1000 G where the magnon\nand the photon modes are on resonance and fully hybridized.\nThe hybrid linewidth is fitted to be κm/2π= 15 MHz.\norder mode at 9.34 GHz has a larger extinction ratio and\nlarger dissipation rate but it couples weakly with the YIG\nmagnon mode and thus will not be discussed further. Im-\nportantly, the clear observation of two high-quality res-\nonances on the YIG-loaded superconducting resonator\nindicates that the effect of the GGG substrate has been\nsignificantly suppressed. As a comparison, we measured\nanother superconducting resonator device at the same\ntemperature (200 mK), which is loaded with a regular\nYIG thin film with the 500 −µm-thick GGG substrate\nstill attached. From the measurement results in Fig. 3(e)\n(red curve), no clear microwave resonances can be ob-\nserved, indicating the significantly increased loss on the\nsuperconducting resonator, while prior to the YIG/GGG\nbonding two resonances (at 7.4 GHz and 8.5 GHz) are\nclearly visible [black curve in Fig. 3(e)].\nTo further investigate the performance of the device\nshown in Fig. 3(c), a series of transmission spectra are col-\nlected while a magnetic field is swept to tune the magnon\nfrequency. The magnetic field is applied parallel to the\nsurface of the superconducting resonator chip, aligning\nwith the direction of the inductor wire. A clear avoid-\ncrossing feature is observed in the measured spectra, as\nshown in Fig. 4(a), indicating that our device has entered\nthe strong coupling regime, where the magnon-photon\ncoupling strength exceeds the dissipation rate of each in-\ndividual mode.\nUsing numerical fitting based on rotating-wave ap-\nproximation (RWA) [45], the magnon-photon couplingstrength gis extracted as 62 MHz, with detailed proce-\ndures described in Section II of the supplemental mate-\nrial [46]. The two branches of the calculated magnon-\nphoton mode frequencies, represented by the red dots in\nFig. 4(a), match well with the measured spectrum. At\na magnetic field of 1300 G where the magnon mode is\nlarge detuned from the microwave resonance, the dissipa-\ntion rate of the microwave modeis extracted as κc= 1.95\nMHz, which leads to a quality factor Q= 2πfc/κc=\n5.05GHz/ 1.95MHz ≈2600. Considering that the YIG\nflake is positions sufficiently far from the bus transmis-\nsion line, its direct coupling with the bus waveguide is\nnegligible. Therefore, the magnon mode is observed only\nwhen its frequency is close to or on-resonance with the\nmicrowave resonance. We can estimate the magnon dis-\nsipation rate using the relation κh= (κm+κc)/2 when\nthe two modes are maximally hybridized (when magnon\nand photon modes are on resonance). Here κhis the\non-resonance linewidth of the hybrid mode, which is fit-\nted to be 15 MHz as presented in Fig. 4(d). Accordingly,\nthe dissipation rate of the magnon mode is determinate\nto be κm/2π= 13 MHz. Compared with previous de-\nvices which have YIG on the 500- µm-thick GGG, the\nlinewidth of the magnon has been significantly reduced.\nAlthough it is still higher than the linewidths measured\non bulk YIG samples, it may be attributed to residual\nGGG layer attached to the YIG thin film which can be\nremoved through futher optimization. Such calculation\nresult is consistent with our fitting results based on the\ninput-output theory [47–49], as described by equation\nS21=ˆaout\nˆain=κexi∆c+g2/(i∆m−κm)\ni∆c−κc+g2/(i∆m−κm),(1)\nwhere grepresent the magnon-photon coupling strength,\nκexrepresents the external coupling rate between the mi-\ncrowave resonator and the bus feeding line, κc(κm) repre-\nsents the total dissipation rate of the resonator (magnon)\nmode, ∆ c=ωd−ωcand ∆ m=ωd−ωmrepresent the\ndetuning of the driving frequency from the microwave\nresonance and magnon frequency, respectively. The cal-\nculated spectra using Eq. 1 are plotted in Fig. 4(b), which\nmatch well with the measured result in Fig. 4(a). For a\ncomprehensive exploration of the fitting procedures and\ndetailed results, we direct readers to the Supplemental\nMaterial [46].\nIn conclusion, we have demonstrated a new approach\nfor developing substrate-free YIG thin films, and vali-\ndated the our method by measuring the strong magnon-\nphoton coupling on a hybrid device. The direct compar-\nison with conventional YIG/GGG devices confirmed the\neffectiveness of our approach. Our discovery represents\nthe first application of the spalling technology on mag-\nnetic garnets, which have been known as very strong and\nhard to process. Compared with other approaches, our\nspalling-based method offers distinct advantages includ-\ning reduced material contamination and flexible thickness\ncontrol. Although the linewidth of our measured device\nat cryogenic temperatures is still higher compared with5\nbulk YIG, it can be further improved by completely re-\nmoving the GGG substrate. Upon further optimization,\nincluding the thickness of the original YIG film as well\nas the stress and thickness of the Ni layer, our approach\nis promising for wafer-scale production of magnonic de-\nvices for quantum applications and beyond. In particu-\nlar, when combined with recent experimental techniques\n[13, 16, 50] for YIG magnonic devices, our substrate-free\nYIG thin films may offer unique properties for magnons\nto couple with a broad range of degrees of freedom, in-\ncluding optics, mechanics, and magnetics.\nACKNOWLEDGMENTS\nThe authors thank R. Divan, L. Stan, C. Miller, and D.\nCzaplewski for support in the device fabrication. X.Z. ac-knowledges support from ONR YIP (N00014-23-1-2144).\nContributions by C.H. and S.G. were supported by the\nVannevar Bush Fellowship received by S.G. under the\nprogram sponsored by the Office of the Undersecretary of\nDefense for Research and Engineering and in part by the\nOffice of Naval Research as the Executive Manager for\nthe grant. Work performed at the Center for Nanoscale\nMaterials, a U.S. Department of Energy Office of Science\nUser Facility, was supported by the U.S. DOE Office of\nBasic Energy Sciences, under Contract No. DE-AC02-\n06CH11357.\n†guha@uchicago.edu\n‡xu.zhang@northeastern.edu\n∗J. Xu and C. Horn contributed equally to this work.\n[1] A. A. Serga, A. V. Chumak, and B. Hillebrands, YIG\nmagnonics, J. Phys. D: Appl. Phys. 43, 264002 (2010).\n[2] D. D. Awschalom, C. R. Du, R. He, F. J. Heremans,\nA. Hoffmann, J. Hou, H. Kurebayashi, Y. Li, L. Liu,\nV. 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Chumak, Prop-\nagation of Spin-Wave Packets in Individual Nanosized,\nNano Lett. 20, 4220 (2020).Cryogenic hybrid magnonic circuits based on spalled YIG thin films\nJing Xu∗,1, 2Connor Horn∗,1Yu Jiang,3Xinhao Li,2Daniel Rosenmann,2\nXu Han,2Miguel Levy,4Supratik Guha†,1and Xufeng Zhang‡3\n1Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA\n2Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL 60439, USA\n3Department of Electrical and Computer Engineering,\nNortheastern University, Boston, MA 02115, USA\n4Department of Physics, Michigan Technological University, Houghton, MI 49931, USA\n(Dated: December 20, 2023)\nI. RESONATOR DESIGN\nThe superconducting resonator incorporates a hanger structure to enable inductive coupling with the bus transmis-\nsion line, featuring two series of interdigital capacitor fingers flanking a central inductive thin wire, as illustrated in\nFig. S1. Simulated transmission power (S21) results, focusing on the first two modes of the designed superconducting\nresonator, reveal that the low frequency mode at 4.85 GHz exhibits a lower extinction ratio and higher quality factor,\nindicative of well-confined RF field within the resonator. The magnetic field distribution is depicted in Fig. S1(c) using\na heat map, with the red color on the central inductance wire highlighting the highly confined magnetic fields. In\ncontrast, the high frequency mode at 8.26 GHz displays a higher extinction ratio and broader linewidth, as indicated\nby the field distribution in Fig. S1(d), which shows less confinement of the microwave field at that frequency.\nFIG. S1. (a) Experimental and (b) simulated reflection spectrum of the superconducting chip at 200 mK, respectively. (c) and\n(d) Distribution of the magnetic component of the microwave field for the low and high resonator photon mode, respectively.arXiv:2312.10660v2 [cond-mat.mes-hall] 19 Dec 20232\nBased on these simulation results, precise placement of the YIG thin film on the inductance wire is imperative to\nensure maximal overlap between the magnon and photon modes. This arrangement will cause the magnon mode to\nhave stronger coupling with the low frequency resonator mode and weaker coupling with the high frequency mode.\nExperimental measurement, as shown in Fig. S1(a), confirms the presence of two microwave modes at 5.05 GHz\nand 9.34 GHz, aligning closely with the simulated values. The slight blueshift observed in the experimental data\ncompared to the simulated values may originate from minor deviations in the designed geometrical parameters during\nthe fabrication process.\nII. COUPLING STRENGTH FITTING\nThe system Hamiltonian governing the coupled magnon and photon can be described using the rotating-wave\napproximation (RWA) [1]:\nˆH=ℏωcˆc†ˆc+ℏωmˆm†ˆm+ℏg(ˆc†ˆm+ ˆm†ˆc), (1)\nwhere candmare associated with resonator photons and magnons, respectively. When expressed in matrix form\nand taken dissipations into account, the equation becomes:\nˆH=/bracketleftbigg\nℏωc0\n0ℏωm/bracketrightbigg\n+/bracketleftbigg\n−iκcg\ng��iκm/bracketrightbigg\n(2)\nIn this equation, ℏis the reduced Planck’s constant, while ωc(ωm) and κc(κm) are the frequencies and dissipation\nrates of the photon (magnon) modes, respectively.\nThe eigenfrequencies of the system Hamiltonian are given by:\nℏω±=(ωc+ωm)−i(κc+κm)±/radicalbig\n[(ωc−ωm)−i(κc−κm)]2+ 4g2\n2(3)\nThis equation can be utilized to fit the coupling strength of the coupled magnon-photon system. We extract the\nfrequencies of the measurement dips from Fig. 3(e) in the main text at each scanning magnetic field point. To perform\nthe fitting, we use QKIT [2], an open-source Python package developed by the Karlsruhe Institute of Technology.\nThe fitting procedures and results are presented in Fig. S2. In Fig. S2(a), the measurement dips are extracted by\nidentifying the minimum transmission power at specific magnetic fields, which are marked in red (for the upper\nbranch) and blue (for the lower branch) colors in Fig. S2(b), respectively. The curves in Fig. S2(c) are calculated using\nthe fitted coupling strength (62 MHz), which closely matches the extracted transmission dips from the experiment.\nFIG. S2. (a) Heat map of the transmission spectrum as a function of the magnetic field strengths. (b) Extracted frequencies\nof transmission dips at different magnetic fields overlaid with the measured spectra, with upper and lower branches denoted in\nred and blue, respectively. (c) Comparison between the fitted (curves) and extracted (dots) mode frequencies.3\nIII. MAGNON DISSIPATION RATE FITTING\nEquqation (1) in the main text, derived from the input-output theory [3], reveals that the transmission spectrum\ndepends on the coupling strength gbetween the magnon and photon modes, as well as the dissipation rates κcandκm.\nIn the preceding sections, we successfully fitted the microwave photon dissipation rate ( κc= 1.95 MHz) and coupling\nstrength ( g= 62 MHz). Consequently, we can extract the magnon dissipation rate κmthrough single-parameter fitting\nbased on the transmission spectrum. The simplest approach to calculate κmis to use the on-resonance relation:\nκh=κm+κc\n2(4)\nHere, κhrepresents the dissipation rate of the hybridized mode, which is the average of the magnon and photon\ndissipation rates. In Fig. S3(b), the line cut at the on-resonance position, with a bias magnetic field strength of 1000\nGauss, reveals a fitted dissipation rate of approximately 15 MHz, indicating κmis around 13 MHz. Given the low\nsignal-to-noise ratio of the transmission spectrum in Fig. S3(b), we performed additional calculations to validate the\nfitted magnon dissipation rate. In Fig. S3(c), we present a color plot of the simulated transmission spectra at various\nmagnetic fields using Eq. (1) from the main text, with the magnon dissipation rate κmset to 28 MHz. This calculated\ndata match well with the experimental measurements shown in Fig. S3(a).\nFIG. S3. (a) Heat map of the transmission spectrum as a function of magnetic field strength. (b) Transmission line cut at\nthe coupling center where the magnon mode resonates with the photon mode, indicated by the red dashed line in (a). The\nlinewidth of the on-resonance dip is determined to be 30 MHz. (c) Heat map displaying the calculated transmission spectrum\nusing Eq. (1) from the main text, with the magnon dissipation rate κmset to 28 MHz.\nIV. SIMULATION ON TRANSMISSION SPECTRUM\nTo assess the sensitivity of Eq. (1) in the main text with respect to the fitting parameter κm, we present a color\nmap of the calculated spectrum with varying κmvalues in Fig. S4. As κmincreases from 1 MHz in Fig. S4(a) to 50\nMHz in Fig. S4(f), the frequency regimes in which the hybridized dip is observable gradually diminish. It is evident\nthat Fig. S4(d), where κmis set to 13 MHz, closely resembles our experimental results.\n[1] R. W. Boyd, Nonlinear Optics (Elsevier, Academic Press, 2008).\n[2] qkitgroup, qkit (2023), https://github.com/qkitgroup/qkit.\n[3] D. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, Germany, 2008).4\nFIG. S4. Color plot displaying the calculated transmission spectrum using Eq.(2) in the main text. The magnon dissipation\nrateκmin Figures (a)-(e) are set to be 1 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, and 50 MHz, respectively." }, { "title": "2202.12696v1.Direct_probing_of_strong_magnon_photon_coupling_in_a_planar_geometry.pdf", "content": "Direct probing of strong magnon-photon coupling in a planar geometry\nMojtaba Taghipour Ka\u000bash,1Dinesh Wagle,1Anish Rai,1\nThomas Meyer,2John Q. Xiao,1and M. Benjamin Jung\reisch1,\u0003\n1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States\n2THATec Innovation GmbH, D-67059 Ludwigshafen, Germany\n(Dated: February 28, 2022)\nWe demonstrate direct probing of strong magnon-photon coupling using Brillouin light scattering\nspectroscopy in a planar geometry. The magnonic hybrid system comprises a split-ring resonator\nloaded with epitaxial yttrium iron garnet thin \flms of 200 nm and 2.46 \u0016m thickness. The Brillouin\nlight scattering measurements are combined with microwave spectroscopy measurements where both\nbiasing magnetic \feld and microwave excitation frequency are varied. The cooperativity for the 200\nnm-thick YIG \flms is 4.5, and larger cooperativity of 137.4 is found for the 2.46 \u0016m-thick YIG\n\flm. We show that Brillouin light scattering is advantageous for probing the magnonic character of\nmagnon-photon polaritons, while microwave absorption is more sensitive to the photonic character of\nthe hybrid excitation. A miniaturized, planar device design is imperative for the potential integration\nof magnonic hybrid systems in future coherent information technologies, and our results are a \frst\nstepping stone in this regard. Furthermore, successfully detecting the magnonic hybrid excitation\nby Brillouin light scattering is an essential step for the up-conversion of quantum signals from the\noptical to the microwave regime in hybrid quantum systems.\nThe emergent properties of hybrid systems are promis-\ning for a wide range of quantum information applica-\ntions. In particular, light-matter interaction has been at\nthe forefront of contemporary studies on hybrid quantum\nsystems. To this end, hybrid magnonic systems based on\nthe coupling of magnons, the elementary excitations of\nmagnetic media, and photons have gained increased at-\ntention [1{4]. Magnons display a highly tunable disper-\nsion, while they can be used for coherent up- and down\nconversion between microwave and optical photons [5{\n9]. In addition, magnons can serve in quantum memory\napplications owing to their collective behavior and ro-\nbustness [10].\nA critical requirement for coherent information trans-\nfer based on magnons is a high cooperativity, which\nmeans that the coupling between the two disparate types\nof excitations, i.e., the photonic and the magnonic sub-\nsystems, exceeds the loss rates of either subsystem. This\nis known as the strong coupling regime in the language\nof quantum information. In this strong coupling regime,\ninformation can be e\u000eciently exchanged, potentially en-\nabling e\u000ecient transduction applications. Another pre-\nrequisite for large scale quantum information processing\nand transfer applications is the conversion between opti-\ncal and microwave frequencies. Previous microwave-to-\noptical transduction studies based on ferromagnets ei-\nther employed Brillouin scattering of optical whispering\ngallery modes by magnetostatic modes [6{8] or coupling\nof the microwave \feld through a cavity mode concomi-\ntant with the coupling of the optical \feld through the\nKittel mode via Faraday and inverse Faraday e\u000bects [5].\nMost of these prior works relied on macroscopic samples\nmade of bulk yttrium iron garnet (YIG) crystals. Uti-\nlizing YIG is advantageous as it has a large spin density\n\u0003mbj@udel.eduand narrow linewidth [11{14]. However, scalable on-chip\nsolutions require device miniaturization. Therefore, pla-\nnar microwave resonators are advantageous for building\nhybrid magnonic networks and circuits [15]. They of-\nfer great \rexibility in terms of circuit design; they are\ncompatible with lithographic fabrication processes and\nthe prevalent complementary metal-oxide-semiconductor\n(CMOS) platform [16]. Furthermore, planar microwave\nresonators typically have a smaller e\u000bective volume than\ntheir three-dimensional counterparts and can provide an\nenhanced coupling with magnetic dipoles [17, 18]. In ad-\ndition, they potentially simplify the integration of optical\ncomponents [9] enabling simpli\fed optomagnonic device\nconcepts.\nHere, we demonstrate coherent microwave-to-optical\nup-conversion using strong magnon-photon coupling in\na split-ring resonator/YIG thin \flm hybrid circuit. We\ndirectly probe the coupling in YIG \flms of 200 nm and\n2.46\u0016m thickness by conventional and microfocused Bril-\nlouin light scattering (BLS) spectroscopy and compare\nthese optical results to microwave absorption measure-\nments. Clear avoided level crossings are observed evi-\ndencing the hybridization of the magnon and microwave\nphoton modes in the strong coupling regime. In addition,\nwe identify contributions of higher order magneto-static\nsurface spin waves. The cooperativity for the 200 nm-\nthick YIG \flms is 4.5 and 137.4 for the 2.46 \u0016m-thick\nYIG \flm. On the one hand, we \fnd that BLS is advan-\ntageous for probing the magnonic character of magnon-\nphoton polaritons, while microwave absorption is found\nto be more sensitive to the photonic character. On the\nother hand, detecting the magnonic hybrid excitation by\nBrillouin light scattering demonstrates a coherent con-\nversion of microwave to optical photons.\nThe coherent microwave-to-optical up-conversion pro-\ncess based on the strong magnon-photon coupling is il-\nlustrated in Fig. 1(a). The magnonic hybrid system com-arXiv:2202.12696v1 [cond-mat.mtrl-sci] 25 Feb 20222\nacBLS Lightbd\nFIG. 1. (a) Schematic illustration of the coupling process be-\ntween the microwave photon (MW) mode of the split-ring res-\nonator (SRR) with the magnon mode of the YIG \flm, where\n\u0014pand\u0014mare the dissipation rates of microwave photon\nand magnon, respectively, and ge\u000bis their mutual coupling\nstrength. Microwave-to-optical up-conversion is achieved by\ncoupling the incident microwave photons via the the SRR\nto the magnon mode that interacts with the BLS laser pho-\ntons. (b) A typical BLS spectrum with the Rayleigh peak\nat 0 GHz and the Stokes signal at around -5 GHz. The ver-\ntical red dashed lines show the region of interest (ROI). (c)\nExperimental setup: The resonator consists of a square SRR\npatterned next to the microwave feed line. The YIG \flm is\nplaced on the top of the SRR. An external biasing magnetic\n\feld (iny-direction) magnetizes the sample during the BLS\nand MW measurements. The probing BLS beam is focused\nonto the surface of the YIG \flm. (d) Top view of the SRR\nwith the dimensions as de\fned in the text.\nprises a split-ring resonator (SRR) loaded with epitaxial\nYIG thin \flms. The microwave photons interact with\nthe SRR mode that exhibits a dissipation rate of \u0014pat\nits resonance frequency. The SRR mode couples with the\nmagnon mode of the YIG sample with a coupling con-\nstant ofge\u000b, while the YIG sample dissipates its energy\nat the rate\u0014m. Finally, the excited magnons interact and\ncouple with the incident BLS probe beam.\nThe up-conversion process is realized by two sepa-\nrate sets of measurements: in-plane magnetic \feld de-\npendent microwave (MW) absorption measurements and\nBLS (both microfocused and conventional) of RF driven\nmagnetization dynamics. A typical BLS spectrum is\nshown in Fig. 1(b), where the region of interest (ROI)\nis limited to the frequency range of hybrid excitation\n(here: Stokes peak, I S). The elastically scattered light\nis centered at 0 GHz. The probing BLS laser beam\nis focused on the sample surface; therefore, we detect\nmagnons modes only in the top layer [19], while both\nthe top and bottom layers contribute in the MW absorp-\ntion measurements. However, since each sample is grown\nunder the same fabrication procedure, similar properties\nare expected from each YIG-\flm layer in each sample.\nFigure 1(c) depicts the experimental con\fguration con-\nsisting of the square SRR in the vicinity of an MW feed\na\nbdc\nSRRFMR\nminℎ!\"max\n𝐴/𝑚\nSRRFMRFIG. 2. (a) SRR resonance obtained by HFSS simulations\n(Qsim= 83:1) and corresponding experimentally realized res-\nonance (Qexp= 94:0). Data shown in blue, corresponding \fts\nare shown by in red dashed lines. (b) RFmagnetic \feld ( hrf)\ndistribution obtained by HFSS simulations. (c) MW absorp-\ntion measurements of the magnon-photon hybridization (here,\nYIG \flm thickness: 2 :46\u0016m), where the false color represents\nthe S 12transmission parameter. (d) S 12transmission param-\neter versus frequency fat selected biasing magnetic \felds as\nshown by white dashed lines in (c).\nline loaded with a YIG sample placed on the top and in\nthe presence of a biasing in-plane magnetic \feld applied\nalong they-axis. For the MW absorption measurement,\na vector network analyzer (VNA) is used to record the\n\feld-dependent transmission parameter S 12with an out-\nput power of +13 dBm connected to P1 and P2. We\nuse a continuous single-mode 532-nm wavelength laser\nfor the BLS measurements that is focused on the YIG\n\flm's surface [see Fig. 1(c)]. A MW generator provides\nthe a RF signal to the feed line (P1) with output pow-\ners of +20 dBm for microfocused BLS and +27 dBm for\nconventional BLS measurements. The BLS process can\nbe described by the inelastic scattering of laser photons\nwith magnons [20]. Since this process is energy and mo-\nmentum conserving, inelastically scattered photons carry\ninformation about the probed magnons [21], which we\nanalyze using a high-contrast tandem Fabry-P\u0013 erot inter-\nferometer. Two di\u000berent objective lenses are used for\nthe BLS measurements: for the microfocused measure-\nments, a high-numerical-aperture (NA = 0.75) objective\nlens with a working distance of 4 mm is used, while a\nlens with a focal lens of 40 mm and a diameter of 1 inch\nis used for the conventional measurement setup.\nWe designed and optimized the SRR via ANSYS HFSS\nto exhibit a resonance ( f0) at 5.1 GHz, which agrees with\nthe experimentally observed result (4.9 GHz) as shown in\nFig. 2(a). Figure 1(d) illustrates the top view of the SRR\nwith the following dimensions: the SRR's outer and inner\nwidths ofa= 4:5 mm andb= 1:5 mm, the gap between3\nabc\nFIG. 3. False color-coded spectra of the magnon-photon hybridization of the 2.46 \u0016m-thick YIG \flm. Results obtained by\n(a) microwave absorption measurements, where S12is plotted versus fand\u00160H, (b) microfocused BLS spectroscopy, and (c)\nconventional BLS spectroscopy. In the BLS measurements, the Stokes peak [compare to Fig. 1(b)] is plotted in logarithmic\nscale versus fand\u00160H. The black dashed lines are the \fts to Eqs. (1) and (2).\nthe SRR and the feed line g= 0:2 mm, and the feed line's\nwidth ofw= 0:4 mm. The SRR is fabricated by etching\none side of Rogers RO3010 laminate with a dielectric con-\nstant of 10.20 \u00060.30 and copper thickness of 35 \u0016m that is\ncoated on both sides of the substrate. By \ftting the res-\nonance data to a Lorentzian function with full-width at\nhalf maximum (FWHM), we determine the quality factor\n(Q=f0=\u0001fFWHM ) of the resonator to be Qsim=83.1 for\nthe simulation and Qexp=94.0 for the experiment [shown\nwith the red dashed lines in Fig. 2(a)]. The 2D pro\fle of\nthe modeled RF-magnetic \feld hrfon resonance is shown\nin Fig. 2(b). hrfis the most intense and uniform at the\ncenter of the SRR. The SRR is loaded with low-loss YIG\n\flms placed on the top of the center of the SRR [for\ndetails on broadband ferromagnetic resonance measure-\nments we refer to the supplemental material (SM)]. We\ncompare the results of two YIG-\flm thicknesses: the lat-\neral dimensions of the 2.46 \u0016m thick square-shaped sam-\nple is 5.3 mm \u00025.3 mm, while the 200 nm thick-sample is\nparallelogram-shaped with a base and height of 10 mm\nand 7.5 mm, respectively. Both samples are grown on\n500\u0016m-thick gadolinium gallium garnet substrates by\nliquid phase epitaxy on both sides of the substrates.\nFig. 2(c) shows a typical false color-coded microwave\nabsorption spectrum of the magnon-photon hybridiza-\ntion (here, YIG \flm thickness: 2 :46\u0016m), where the\ncolor represents the transmission parameter. In the\n\feld/frequency region where the uncoupled photon and\nthe magnon modes would cross, we observe the behavior\nof an e\u000bective two-level system, where the two disparate\nsubsystems couple electromagnetically with the coupling\nstrengthge\u000b. The coupling is quanti\fed by the cooper-\nativityC=g2\ne\u000b=\u0014m\u0014p. The mode coupling lies in the\nstrong regime if ge\u000bis larger than the loss rate of YIG,\n\u0014m, and the SRR, \u0014p, respectively [22]; thus, C > 1.This\nis shown more in detail in Fig. 2(d), where S 12is plotted\nversusffor di\u000berent \felds from 86 to 102 mT close tothe avoided crossing as indicated by white dashed lines\nin Fig. 2(c). At high \felds (e.g., at 102 mT), the higher\nfrequency mode (FMR mode) has a lower intensity than\nthe lower frequency mode (SRR mode). By sweeping\nthe \feld from higher to lower values, the FMR mode\napproaches the SRR mode. In this transition regime,\nthe modes switch the magnitude of their intensities: at\n94 mT, both modes have the same intensity, and the fre-\nquency gap between them is almost minimum. Further\ndecreasing the \feld magnitude to 86 mT results in the\nmodes switching their intensities and moving apart. This\nbehavior describes an avoided level crossing indicative of\nthe formation of magnon-photon polaritons [23, 24].\nWe model the photon-magnon hybridization using a\ncoupled two harmonic oscillator model with f\u0006repre-\nsenting the hybridized mode frequencies:\nf\u0006=fSRR+fFMR\n2\u0006s\u0012fSRR\u0000fFMR\n2\u00132\n+\u0010ge\u000b\n2\u00112\n;\n(1)\nwherege\u000bis the coupling strength, fSRRis the uncou-\npled SRR resonance, fFMR is the ferromagnetic resonance\nof YIG that increases as the \feld is increased and is given\nby the Kittel formula\nfFMR =\r\n2\u0019\u00160p\nH(H+Me\u000b); (2)\nwhere\ris the gyromagnetic ratio and Me\u000bis the e\u000bec-\ntive magnetization (see SM). The systematic deviations\nof the BLS data from the FMR \ftting are discussed in\nthe SM.\nThe microwave absorption measurement result of the\n2:46\u0016m-thick YIG \flm shows a clear avoided crossing\nwhich centers at 96 mT, Fig. 3(a). As is visible from\nthe \fgure, the signal is particularly strong before and\nafter the avoided crossing ( <82 mT and >110 mT).\nThis \feld-independent signal is the SRR resonance mode.4\nFIG. 4. Conventional BLS spectra of the 2.46 \u0016m-thick YIG\n\flm. Dashed lines represent \fts to Eq. (3). The black bold\ndashed line represents the k= 0 (n= 0) mode, which is the\nFMR mode, while the higher-lying dashed lines are MSSW\nmodes (n= 1,..., 12) with k=n\u0019=2l.\nHowever, a pronounced avoided crossing is observed\nwhen the \feld-dependent FMR mode of YIG approaches\nthe SRR resonance at 96 mT leading to the formation of a\nhybridization. The upper and lower frequency modes and\nthe uncoupled FMR mode are \ftted to the experimental\nresults according to Eq. (1) and Eq. (2), respectively.\nUsing a similar \feld/frequency sweep range as in\nthe microwave absorption measurements, we probe the\nmagnon-photon hybridized state by microfocused BLS\n[Fig. 3(b)]. Here, the Stokes BLS intensity in logarithmic\nscale is plotted. The \feld is swept from 110 to 82 mT in\n0.3 mT \feld steps after saturating at 200 mT. The MW\nfrequency excites the sample from 4.25 to 5.10 GHz in\n12 MHz steps. The two hybridized modes are detectable,\nsimilar to the MW absorption measurements. Note that,\nin addition to the coupled resonances, we detect a con-\ntribution of modes directly excited by the feed line [12]\nin the BLS experiments. We will discuss these modes\nbelow.\nBLS's successful detection of the strongly cou-\npled magnon-photon state demonstrates a coherent\nmicrowave-to-optical up-conversion based on the scheme\nshown in Fig. 1(a). Interestingly, the intensity distri-\nbution detected in BLS is reverse to the MW absorption\ntechnique: BLS is more sensitive in probing the magnonic\ncharacter of the magnon-photon polariton compared to\nMW absorption measurements shown in Fig. 3(a), which\nis more sensitive to the photonic character of the hybrid\nexcitation con\frming previous reports [9].\nBy \ftting the experimental data to Eq. (1), we ex-\ntract the magnon-photon coupling strength ge\u000b. Ferro-\nmagnetic resonance measurements (SM) yield the fol-\nlowing parameters: \u00160Me\u000b= 183:5 mT and \r=2\u0019=28:2 GHzT\u00001, which we use to \ft the microfocused BLS\nresults [Figs. 3(b,c)] to Eq. (1), we obtain ge\u000b=2\u0019=\n114:6 MHz. By calculating the dissipation rates of the\nmicrowave photon ( \u0014p=2\u0019= 25:8 MHz) and the magnon\n(\u0014m=2\u0019= 3:9 MHz), we can obtain a cooperativity of\nC= 137:4, which ful\fls the conditions C > 1 and\nge\u000b>\u0014p;\u0014m.\nWe compare the microfocused BLS experiments to con-\nventional BLS measurements as is shown in Fig. 3(c). We\nuse a lens with a smaller numerical aperture than the ob-\njective lens used in the microfocused setup. However, the\nlaser beam spot size is signi\fcantly larger, and hence, it\ncovers a larger area of the YIG \flm leading to a stronger\nsignal intensity. Due to the stronger signal strength, we\nare able to detect modes inaccessible by the microfocused\nsystem as further evidenced in Fig. 4. The additional\n\fne features revealed by the conventional BLS measure-\nments lie in the anticrossing region parallel to the Kittel\nmode. These modes are due to the excitations of higher-\norder wavenumber spin-wave modes directly excited by\nthe MW feed line. These modes occur at frequencies\nhigher than the Kittel mode for a given magnetic \feld\nand are identi\fed as magnetostatic surface spin waves\n(MSSWs) that propagate in the \flm plane in a direction\nperpendicular to the applied \feld [12, 25{27]. We model\nthem by:\nfMSSW =\r\n2\u0019\u00160q\nH(H+Me\u000b) +M2\ne\u000b(1\u0000e\u00002kd)=4;\n(3)\nwheredis the thickness of the sample, k=n\u0019=2lis the\nspin-wave wavevector, lis the length of the square-shaped\nsample and nis the mode number with n= 0 being the\nuniform Kittel mode. \r=2\u0019= 28:2 GHzT\u00001and\u00160Me\u000b\n= 183.5 mT both of which are obtained from the \ftting\nof the lowest lying mode, Eq. (2). The dashed lines above\nthe main modes in Fig. 4 shows \fts of the experimental\ndata to Eq. (3) for n= 0;1;:::;12. Here, the bold dashed\nline represents the k= 0 (n= 0) mode, which is the FMR\nmode, while the other dashed lines are the higher-order\nMSSW modes ( n= 1,..., 12). These higher-order MSSW\nmodes have wavevectors of k= 3:5\u000210\u00003rad/\u0016m for\nn= 12, which is within the detectable wavevector range\nof our conventional system ( kmax= 6:9 rad/\u0016m).\nWhile most recent works on strong-magnon photon\ncoupling utilized micrometer-thick-YIG or YIG spheres\n[28{31], sample miniaturization is imperative for a scal-\nable on-chip solutions. In the following, we demonstrate\nmagnon-photon coupling in a miniaturized 200 nm-thick\nYIG \flm. Using the identical 2D planar resonator as\nused for studying the 2.46 \u0016m \flm, we observe mode\nanti-crossing by the microwave absorption technique as\nshown in Fig. 5(a). As is shown Fig. 5(b), we are unable\nto detect a su\u000eciently strong signal of the hybridized\nexcitation in the microfocused measurements. Surpris-\ningly, the Kittel mode directly excited by the feed line is\nsigni\fcantly stronger than the magnon-photon coupled\nmodes. However, as we switch from the microfocused to\nthe conventional BLS setup, not only the Kittel mode5\nabc\nFIG. 5. A typical false color-coded spectrum of the magnon-photon hybridization for the 200 nm-thick YIG \flm using the\n(a) microwave absorption technique, where S12is plotted versus fand\u00160H, (b) microfocused BLS technique, and (c) BLS\ntechnique with a conventional objective lens. The dashed lines are the \ftted plots to Eqs. (1) and (2).\nbecomes more intense, but also the two hybridized mode\ncan be detected in the spectra [Fig. 5(c)]. Fits to the\nexperimental data agree reasonably well as shown by the\nblack dashed lines. From the combined optical and mi-\ncrowave experiments, we extract the following parame-\nters:\u00160Me\u000b= 187:3 mT,\r=2\u0019= 28:2 GHzT\u00001, and\nge\u000b=2\u0019= 37:3 MHz. The photon and magnon dissi-\npation rates are found to be \u0014p=2\u0019= 25:8 MHz and\n\u0014m=2\u0019= 12:1 MHz, respectively. Therefore, the coop-\nerativityC= 4:5, ful\flling both conditions C > 1 and\nge\u000b> \u0014 p;\u0014mand, hence, the mode hybridization is in\nthe strong coupling regime. Higher modes similar to the\nones observed in the 2.46 \u0016m are absent in the spectra\nof the 200 nm \flm since the \feld/frequency separation of\nthe higher-order modes decreases as the YIG thickness\ndecreases [32].\nIn summary, we showed direct probing of strong\nmagnon-photon coupling using Brillouin light scatter-\ning spectroscopy in a planar geometry. The optical\nmeasurements are combined with microwave spec-\ntroscopy experiments where both biasing magnetic\n\feld and microwave excitation frequency are varied.\nThe miniaturized YIG sample of 200 nm thickness\nexhibits a cooperativity of 4.5, while 2.46 \u0016m-thick \flm\nshowed a larger cooperativity of 137.4. We \fnd that\nBrillouin light scattering is advantageous for probing themagnonic character of magnon-photon polaritons, while\nmicrowave absorption is more sensitive to the photonic\ncharacter of the hybrid excitation. In addition, modes\ndirectly excited by the feed line signi\fcantly contribute\nto the optical measurements: they are detected in the\ngaped region between the two coupled magnon-photon\nmodes. The detection of the magnonic hybrid excitation\nby Brillouin light scattering can be understood as an\nup-conversion mechanism of signals from the optical to\nthe microwave regime in the magnonic hybrid systems.\nThe planar structure presented here enables spatially-\nresolved imaging of magnon-photon polaritons that can\nserve as a platform for studying magnonics strongly\ncoupled to microwave photons.\nACKNOWLEDGMENT\nWe thank Prof. Matthew Doty, University of\nDelaware, for valuable discussions. 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Azevedo2 \n1Departamento de Física, Universidade Federal de Viçosa, 36570 -900 Viçosa, MG, Brazil \n2Departamento de Física, Universidade Federal de Pernambuco, 50670 -901 Recife, PE, Brazil \n3Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078 -900 Natal, RN, Brazil \n4Centro Interdisciplinar de Ciências da Natureza, Universidade Federal da Integração Latino -\nAmericana, 85867 -970 Foz do Iguaçu, PR, Brazil \n \n \nWe report an investigation of the spin - to charge -current conversion in sputter -deposited films of \ntopological insulator Bi2Se3 onto single crystalline layers of YIG ( Y3Fe5O12) and polycrystalline \nfilms of Permalloy ( Py = Ni81Fe19). Pure spin current was injected into the Bi 2Se3 layer by means \nof the spin pumping process in which the spin precession is obtained by exciting the ferromagnetic \nresonance of the ferromagnetic film. Th e spin -current to charge -current conversion, occurring at \nthe Bi 2Se3/ferromagnet interface, was attribute to the inverse Rashba -Edelstein effect (IREE ). By \nanalyzing the data as a function of the Bi 2Se3 thickness we calculated the IREE length used to \ncharacterize the efficiency of the conversion process and found that 1.2 pm ≤|𝜆𝐼𝑅𝐸𝐸|≤ 2.2 pm . \nThese results support the fact that the surface states of Bi 2Se3 have a dominant role in the spin -\ncharge conversion process, and the mechanism based on the spin diffusion process plays a \nsecondary role. We also d iscovered that the spin - to charge -current mechanism in Bi 2Se3 has the \nsame polarity as the one in Ta, which is the opposite to the one in Pt. The combination of the \nmagnetic properties of YIG and Py , with strong spin -orbit coupling and dissipationless surface \nstates topologically protected of Bi 2Se3 might lead to spintronic devices with fast and efficient \nspin-charge conversion . \n \n \n*Corresponding author: Joaquim B. S. Mendes, joaquim.mendes@ufv.br \n \n \n \n \n \n \n The investigation of new materials with strong spin -orbit coupli ng (SOC) has improve d \nthe means for the generation and detection of spin currents in nonmagnetic materials. This study \ngave birth to the emergent subfield of spintronics , named spin orbitronics [1-3]. Despite being a \nsubject of interest for many years to the investigation of magnet ocrystalline anisotropy , the SO C \nhas been pivotal to the revolution that spintronics has undergone in the last decade . In particular, \nheavy metals, such as Pt and Pd, have been used as efficient materials for mutual conversion \nbetween spin and charge currents via direct and inverse spin Hall effects (SHE and ISHE , \nrespectively ) [4-7]. In the l ast decade, there has been significant progress towards developing \nmaterials with strong SOC , which can produce current -driven torques strong enough to switch the \nmagnetization of a ferromagnetic (FM) layer in a spin-valve structure. Such improvement in th e \nSHE has been observed in a wide variety of systems that include enhancement of the SOC driven \nby surface roughness and volume impurities [8-11], at 2D materials [12] and interfacial effects \n[13-15]. \nIndeed, many spintronics -phenomena driven by interface -induced spin-orbit interaction \nhave been extensively investigated over the last few years . For instance, the inverse Rashba -\nEdelstein effect [16,17] (IREE) was considered for converting spin into charge current [13 ] in \nmany interface systems [18-25]. Moreover , other materials with outstanding spintronics \nproperties, the topological insulators (TIs) , stand out for the mutual conversion between charge \nand spin due to the large SOC in surface states that locks spin to momentum [ 26-29]. TIs are a \nnew class of quantum materials that presen t insulating bulk, but metallic dissipationless surface \nstates topologically protected by time reversal symmetry, opening several possibilities for \npractical applications in many scientific arenas including spintronics, quantum co mputation, \nmagnetic monopoles, highly correlated electron systems , and more recently in optical tweezers \nexperiments [30, 31 –34]. It is known that in TIs the effects of SOC are maximized because the \nelectron ’s spin orientation is fix ed relative to its direction of propagation. Among the 3D TIs, \nBi2Se3 is a unique material with large bandgap of 0.35 eV and its surface spectrum consists of \nsingle Dirac cone roughly centered within the gap [ 30]. In spite of the fact that the first \ninvestigations of spintronic s properties of TIs were performed i n samples grown by the Molecular \nBeam Epitaxy (MBE) [29,35], the sputtering deposition technique has been successfully used to \ngrow high quality Bi 2Se3 [23, 28, 36]. \nThe spin Hall angle ( 𝜃𝑆𝐻), used to quantify the mutual conversion between spin and \ncharge current, has limited use in systems in which the cross -section of the charge -current -\ncarrying layer is reduced. Owing to the transverse nature of the spin transport phenomena, SH E \nis a bulk effect occurring within a volume limited by t he spin -diffusion length (𝜆𝑠𝑑) [15]. For \ninstance, when a 3D spin current density 𝐽𝑆 [𝐴𝑚2⁄] is injected through an interface with high \nSOC, it generates a 2D charge current density 𝐽𝐶 [𝐴𝑚⁄] by means of the IREE. In this case , the ratio 𝐽𝐶𝐽𝑆⁄=(2𝑒ℏ⁄)𝜆𝐼𝑅𝐸𝐸 defines a length (𝜆𝐼𝑅𝐸𝐸) that is used as a parameter to measure the \nefficiency of conversion between spin- to charge current [2,13]. Not only the absolute value of \n𝜆𝐼𝑅𝐸𝐸, but also its polarity must be of interest to understand the physics behind the interplay \nbetween spin and charge currents. \nHere we report an investigation of the spin - to charge current conversion in bilayers of \nBi2Se3(t)/YIG (6 m), (YIG = Y 3Fe5O12, Yttrium Iron G arnet ) by means of the ferromagnetic \nresonance driven spin pumping (FMR -SP) technique . While the Bi 2Se3 films were grown by DC \nsputtering, the single -crystal YIG films were grown by Liquid Phase Epitaxy (LPE) onto (111) \nGGG ( =Gd 3Ga5O12) substrates . The pure spin-current density ( 𝐽𝑆), which flows across the \nBi2Se3(t)/YIG interface due to the YIG magnetization precession , is converted in to a transversal \ncharge current density (𝐽𝐶) that is detected by measuring a DC voltage between two edge contacts. \nThe Bi 2Se3 samples were deposited on top of small pieces of YIG /GGG(111) cut from the same \nwafer , with thickness 6 µm, width of 1.5 mm and length of 3.0 mm. The YIG films have in -plane \nmagnetization and thus the magnetic proximity effect is expected to shift the Dirac cone sideways \nalong the momentum direction and does not open an exchange gap (i.e. in our heterostructures, \nthe Dirac cone of the TI film will be preserved). The Bi 2Se3/YIG interface has the advantage over \nthe Bi 2Se3/ferro magnetic -metal because it ensures cleaner interface and avoids current shunting \nas well as spurious spin rectification effects. Previously reported spin -to-charge current \nconversion experiments with sputtered Bi 2Se3/YIG were carried out in YIG grown by sputtering \nor MBE and , to the authors knowledge, there is no investigation about the polarity of 𝜆𝐼𝑅𝐸𝐸 [23, \n28]. \nX-ray diffraction (XRD) analysis was carried out by mean s of out -of-plane scan as well \nas grazing incidence X -ray diffraction (GIXRD), which is more valuable for assessing ultra -thin \nfilm structures. Fig. 1(a) shows the out of plane XRD θ -2θ scan pattern of the Bi2Se3(6 \nnm)/ YIG(6µm)/GGG sample over a 2θ range between 20° and 70°. The pattern shown in Fig. \n1(a) displays reflections associated with the (222) and (444) crystal planes of YIG, proving that \nthe present YIG film is epitaxially grown on the GGG substrate. In the inset, we can se e the XRD \nspectrum at high resolution detailing the double peak corresponding to the (444) Bragg reflections \nof the GGG substrate and the epitaxial YIG in the (444) plane. In order to optimize the scattering \ncontribution from the Bi2Se3 films, we used graz ing incidence X -ray diffraction (GIXRD) for \ninvestigating the Bi2Se3/YIG(6µm) /GGG samples. As shown in Fig. 1(b), the GIXRD data \nevidenced the diffraction peaks characteristic of the Bi2Se3 6 nm thick film, meaning that the film \nis polycrystalline and has a preferential texture oriented in the planes: (0 0 9), (0 0 15), (0 0 18), \n(0 0 21), which is in agreement with the literature [37, 38]. Figure 1(c) shows the X -ray reflectivity \n(XRR) data for Bi 2Se3(16.0 nm)/Si. The w ell-defined and the good periodicity of the Kiessig \nfringes allow an accurate determination of the thickness of Bi 2Se3 films. Figure 1(d ) shows an atomic force microscopy (AFM) image of the YIG film surface and confirms the uni formity of \nthe YIG film surface with very small roughness (~0.2 nm). On the other hand, Fig. 1( e) shows \nthe AFM image of the sputtered granular bismuth selenide thin film ( t = 4 nm) grown onto \nYIG/GGG substrate. The image shows that Bi 2Se3 film grown onto YIG favors the formation of \na granular film, with grain sizes up to ~ 0.3 μm, and has a root -mean -square (RMS) surface \nroughness of about 1.0 nm. The typical energy -dispersive x -ray (EDX) spectrum of Bi2Se3 on the \nYIG film can be seen in Fig. 1(f). The EDX spectrum taken from an arbitrary region of the sample \nshows the presence only of yttrium (Y), iron (Fe), oxygen (O) of the YIG film; bismuth (Bi) and \nselenium (Se) of Bi 2Se3. The additional peak of the carbon (C) in the EDX spectrum is due to the \npresence of carbon tape used as support on which the samples are prepared for analysis. In the \nfigure there are also the EDX -maps showing that the Bi and Se are evenly distributed over the \nentire surface of the film. Different regions of the sam ples were analyzed, in order to confirm the \nresults of the EDX measurements. \nFig. 1 (color online). (a) Out-of-plane XRD patterns (θ -2θ scans) of Bi2Se3 film grown on YIG/GGG \nsubstrate . The XRD spectrum at high resolution detailing the position s of the peaks of the YI G film and the \nGGG substrate is shown in the inset. (b) The GIXRD pattern of the Bi 2Se3(6 nm)/YIG/GGG sample. (c) \nXRR spectra of the Bi 2Se3 thin film (t ≈ 16 nm). The red solid line across the XRR data indicates the best \nfitting obtained for the thickness calibration. (d) AFM image of the YIG film surface. (e) AFM image of \nthe surface of the Bi2Se3(4nm)/YIG(6µm)/GGG sample . (f) EDX spectrum (top) from an arbitrary region \nin the sample of Bi2Se3(6nm)/YIG and EDX -maps (bottom) showing that the Bi and Se are evenly \ndistributed over the entire surface of the film. \nFigure 2 (a) illustrates the performed experiments of FMR -SP in which the sample w ith \nelectrodes at the edges is mounted on the tip of a polyvinyl chloride (PVC) rod and is inserted, \nvia a hole drilled at the bottom wall of a shorted X -band waveguide, in a position of maximum rf \nmagnetic field and zero electric field. The loaded wavegui de is placed between the poles of an \nelectromagnet that applies a DC magnetic field 𝐻⃗⃗ 0 perpendicular to the in -plane RF magnetic \nfield, ℎ⃗ 𝑟𝑓. Electric contacts of silver were sputtered at the edges perpendicular to the larger sample \nsize, so that the spin pumping voltage ( 𝑉𝑆𝑃) can be directly measured by means a nanovoltmeter. \nAs the DC and RF magnetic field s are perpendicular to each other, the sample, attached to a \ngoniometer, can be rotated so that we can investigate de angular dependence of both the \nferromagnetic resonance (FMR) as well as 𝑉𝑆𝑃 . Field scan spectra of the derivative 𝑑𝑃𝑑𝐻⁄ , at a \nfixed frequency of 9.5 GHz, are obtained by modulating the field 𝐻⃗⃗ 0 with a small sinusoidal field \nat 1.2 kHz and using lock -in amplifier detectio n. Figure 2 (b) shows the FMR spectrum of a bare \nYIG sample (3.0 mm x 1.5 mm x 6.0 µm) obtained with the in -plane field applied normal to the \nlarger length with an incident power of 54 mW . The strongest line corresponds to the uniform \nFMR mode ( 𝑘0≅0) in whic h the frequency is given by the Kittel’s equation 𝜔0=\n𝛾√(𝐻0+𝐻𝐴)(𝐻0+𝐻𝐴+4𝜋𝑀𝑒𝑓𝑓), where 𝛾=2𝜋×2.8 𝐺𝐻𝑧𝑘𝑂𝑒⁄ and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀+\n𝐻𝑠≅1760 𝐺 for YIG. While the lines to the left of the uniform mode correspond to hybridized \nstanding spin -wave su rface modes, the lines to the right correspond to the backward volume \nmagnetostatic modes with quantized wave number 𝑘, subjected to the appropriated boundary \nconditions. All modes have similar half -width-half-maximum linewidth (HWHM ) of ∆𝐻𝑌𝐼𝐺=\n1.4 Oe. As shown in Fig. 2(c), the deposition of a 4.0 nm thick film of Bi 2Se3 on the YIG layer \nincreases the FMR linewidth to ∆���𝐵𝑖2𝑆𝑖3𝑌𝐼𝐺⁄=1.7 Oe. This lin ewidth increas e is mostly due to \nthe spin pumping process that transports spin angular moment out of the YIG layer [39,40]. As \nthe YIG magnetization vector precesses, it injects a pure spin current density 𝐽 𝑆, that flows \nperpendicularly to the YIG/Bi 2Se3 interface with transverse spin polarization 𝜎̂, which is given \nby \n𝐽 𝑆=(ℏ𝑔𝑒𝑓𝑓↑↓4𝜋𝑀𝑠2⁄)(𝑀⃗⃗ (𝑡)×𝜕𝑀⃗⃗ (𝑡)𝜕𝑡⁄), (1) \nwhere 𝑀𝑠 and 𝑀(𝑡) are the saturation and time dependent magnetization, respectively, and 𝑔𝑒𝑓𝑓↑↓ \nis the real part of the spin interface mixing conductance, that takes into account the forward and \nbackward flows of the spin current [39]. It is important to mention that 𝐽𝑆 in Eq. (1) has units of \n(angular moment)/(time.area). As previously mentioned, 𝐽 𝑆 results in an increased magnetization \ndamping due to the outflow of the spin an gular moment, and due to the IRE E it generates a \ntransverse charge current in the Bi 2Se3 film. From the additional linewidth broadening, we can \nestimate the value of the spin mixing conductance 𝑔𝑒𝑓𝑓↑↓ of the Bi 2Se3/YIG interface. As 𝑔𝑒𝑓𝑓↑↓ is \nproportional to the additional linewidth broadening, i .e., 𝑔𝑒𝑓𝑓↑↓=(4𝜋𝑀𝑡𝐹𝑀ℏ𝜔⁄)(∆𝐻𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄−\n∆𝐻𝑌𝐼𝐺), where 𝜔=2𝜋𝑓 and 𝑡𝐹𝑀 is the ferromagnetic (FM) layer thickness for thin FM films (or \nthe coherence length for films such as the used here), and considering that for the Pt/YIG bilayer obtain ed with the same YIG, ∆𝐻𝑃𝑡𝑌𝐼𝐺⁄−∆𝐻𝑌𝐼𝐺=0.55 Oe and 𝑔𝑒𝑓𝑓↑↓(𝑃𝑡𝑌𝐼𝐺⁄)=1014cm−2, we \nobtain 𝑔𝑒𝑓𝑓↑↓(𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄)≈5.4×1013cm−2. \n Figure 2 (color online) . (a) Schematic s of the FMR -SP technique in which we highlights the spin -current \nto charge -current conversion process at the interface . Field scan FMR absorption derivative , for a bare YIG \nfilm with thickness of 6 m (b) and (c) the bilayer of Bi 2Se3 (4 nm) / YIG(6 m). (d) F ield scan of the spin \npumping voltage measured for the bilayer Bi2Se3 (4 nm)/ YIG(6 m) at three different in -plane angles as \nillustrated in the inset, with an incident microwave power of 157 mW. \n \nThe measurement of 𝑉𝑆𝑃 is carried out by sweeping the DC field with no AC field \nmodulation and directly measuring 𝑉𝑆𝑃 that is generated between the two electrodes due to the \nspin-to-charge current conversion. Fig. 2(d) shows the spin pumping voltage, measured directly \nby a n anovoltmeter , in a bilayer of Bi 2Se3(4 nm)/YIG as function of the applied field for three in -\nplane directions given by 𝜙=0°,90°and 180° , as illustrated in the inset. As expected from the \nequation 𝐽 𝐶=𝜃𝑆𝐻(2𝑒ℏ⁄)(𝐽 𝑆×𝜎̂), where 𝜎̂∥𝐻⃗⃗ , the charge current flows in -plane so that the \nvalue of 𝑉𝑆𝑃 is maximum for 𝜙=0° and 𝜙=180° for blue and red curves, respectively. While \nit is null for 𝜙=90°, as shown by the black curve. The asymmetry between the positive and \nnegative peaks is similar to that observed in other bilayer systems and can be attributed to a \nthermoelectric effect [41]. \nWhile Fig. 3(a) shows the field scans of 𝑉𝑆𝑃 for 39 𝑚𝑊 ≤𝑃𝑟𝑓≤157 𝑚𝑊, Fig. 3(b) \nshows the RF -power dependence of the peak voltage measured at 𝜙=180° . The linear \ndependence of the 𝑉𝑆𝑃 as a function of 𝑃𝑟𝑓 confirms that we are exciting the FMR in the linear \nregime. On the other hand, the dependence of the peak voltage as a function of the Bi 2Se3 layer \nthickness ( 𝑡𝐵𝑖2𝑆𝑒3) exhibits a more challe nging behavior. It decreases as 𝑡𝐵𝑖2𝑆𝑒3 increases in a \nclear opposition with results shown by materials in which the spin - to charge current conversion \noccurs in the bulk, as in Pt, for example. This decrease in the peak voltage was also observed in \ncrystalline Bi 2Se3 grown by MBE [35]. We could try to explain the origin of the voltage in \nBi2Se3/YIG as due to the spin pumping ISHE mechanism, by means spin diffusion model where \nthe spin pumping voltage is given by [42 -44], \n𝑉𝑆𝑃(𝐻)=𝑅𝑁𝑒𝜃𝑆𝐻𝜆𝑁𝑤𝑝𝑥𝑧𝜔𝑔𝑒𝑓𝑓↑↓\n8𝜋tanh(𝑡𝑁\n2𝜆𝑁)(ℎ𝑟𝑓\n∆𝐻)2\n𝐿(𝐻−𝐻𝑅)cos𝜙. (2) \nHere, 𝑅𝑁, 𝑡𝑁, 𝜆𝑁 and w, are respectively the resistance, thickness, spin diffusion length and width \nof the Bi 2Se3 layer, considering the microwave frequency = 2f, and 𝑝𝑥𝑧 is a factor that \nexpresses the ellipticity and the spatial variation of the rf magnetization of the FMR mode. Also, \nℎ𝑟𝑓 and ∆𝐻 are the applied microwave field and FMR linewidth, and 𝐿(𝐻−𝐻𝑅) represents the \nLorentzian function. By assuming that 2𝜆𝑁≫𝑡𝑁, thus tanh(𝑡𝑁2𝜆𝑁��)≈𝑡𝑁2𝜆𝑁⁄ . Therefore, Eq. \n(2) can be written as 𝑉𝑆𝑃=(𝑅𝑁𝑓𝑒𝜃𝑆𝐻𝑤𝑝𝑥𝑧𝑔𝑒𝑓𝑓↑↓𝑡𝑁8⁄)(ℎ𝑟𝑓∆𝐻⁄)2. This expression does not \ndepend on 𝜆𝑁, as expected for TIs, so that 𝑡𝑁 can be interpreted as an effective thickness attributed \nto the Bi 2Se3. From the measured quantities for the bilayer 𝐵𝑖2𝑆𝑒3(4 𝑛𝑚)𝑌𝐼𝐺⁄ , 𝑅𝑁=173 𝑘Ω, \n𝑔𝑒𝑓𝑓↑↓≈5.4×1013𝑐𝑚−2, ℎ𝑟𝑓=0.055 𝑂𝑒, ∆𝐻=1.7 𝑂𝑒, 𝑉𝑆𝑃=44.7 𝜇𝑉 and 𝜃𝑆𝐻≅0.11 [as \nreported in Ref. [27] for average value of 𝜃𝑆𝐻], the effective thickness of the Bi 2Se3 layer is 𝑡𝑁=\n0.46 Å. This small value is certainly unphysical for an effective layer that converts a 3D spin \ncurrent density in a 3D charge current, as happens in the SHE effect. However, it provides an \nevidence that the spin-to-charge current conversion is dominated by surface states of the sputtered \nBi2Se3 layer. \nTo further verify that the spin - to charge -current conversion in 𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄ is dominated \nby the surface states, we can calculate the effective length 𝜆𝐼𝑅𝐸𝐸=(ℏ2𝑒⁄)𝐽𝐶𝐽𝑆⁄, where 𝑉𝐼𝑅𝐸𝐸=\n𝑅𝐵𝑖2𝑆𝑒3𝑤𝐽𝐶 and 𝐽𝑆=(𝑒𝜔𝑝𝑥𝑧𝑔𝑒𝑓𝑓↑↓16𝜋⁄)(ℎ𝑟𝑓∆𝐻⁄)2𝐿(𝐻−𝐻𝑅), with 𝑝11=0.31, see Ref . [45]. \nTherefore, 𝜆𝐼𝑅𝐸𝐸 is given by, \n𝜆𝐼𝑅𝐸𝐸=4𝑉𝐼𝑅𝐸𝐸\n𝑅𝐵𝑖2𝑆𝑒3𝑒𝑤𝑓𝑔𝑒𝑓𝑓↑↓𝑝𝑥𝑧(ℎ𝑟𝑓∆𝐻⁄)2. (3) Figure 3 (color online). (a) Field scans of 𝑉𝑆𝑃 for several values of the incident microwave power. (b) Peak \nvoltage value as a function of the incident microwave p ower measured for the bilayer of Bi 2Se3(4 nm)/YIG . \n(c) Peak voltage value measured as a function of the Bi 2Se3 thickness for an incident power of 157 mW. \nThe inset shows the dependence of the spin pumping current ( 𝐼𝑆𝑃=𝑉𝑆𝑃𝑝𝑒𝑎𝑘𝑅⁄). (d) Field scans of VSP for \nthe bilayer of Ta(2nm) /YIG obtained at same experimental configuration used to measure VSP in \nBi2Se3/YIG . By comparing Fig. 3(d) with Fig. 2(d) we concluded that the VSP polarization of Bi 2Se3 is the \nsame as in Ta. \n \nUsing the physical quantities for the bilayer 𝐵𝑖2𝑆𝑒3(4 nm)𝑌𝐼𝐺⁄ , given above, we obtained \n|𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=4 nm)=(2.2±0.4)×10−12 m. For the other two bilayers we obtained, \n|𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=6 nm)=(2.0±0.5)×10−12 m, and |𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=8 nm)=(1.2±\n0.1)×10−12 m. Where o nly th ree parameters varied from sample to sample, which are: \nresistance (R), average voltage <𝑉𝑆𝑃>, and the FMR linewidth ∆𝐻𝐵𝑖2𝑆𝑒3/YIG. The error bars \nwere incorporated in 𝜆𝐼𝑅𝐸𝐸 by taking into account the variation of 𝑉𝑆𝑃 measured at 𝜙=0° and \n180°. Therefore, we found values that varies in the range of 0.012 𝑛𝑚≤|𝜆𝐼𝑅𝐸𝐸|≤0.022 𝑛𝑚, \nand in the literature there are values reported in the range of 0.01 𝑛𝑚<𝜆𝐼𝑅𝐸𝐸≤0.11 𝑛𝑚 \n[23,35]. Although we cannot rule out the spin diffusion mechanism, the values of 𝜆𝐼𝑅𝐸𝐸 strongly \nsupport the role played by the surface states in the spin - to charge -current conversion process \noccurring in sputtered Bi 2Se3 layers. Indeed, granular Bi 2Se3 films grown by sputtering keep the \ntopological insulator properties even in the nanometer size regime. The basic mechanisms \nexplaining the existence of topological surface states in granular films of Bi 2Se3 is based on the \nelectron tunneling between grain surfaces. Also, the electron quantum confinement in nanometer \nsized grains, has been considered as the reason of the high charge -to-spin conversion effect in \ngranular TIs [36]. \n \n \nFigure 4 (color online). (a) Sketch of the bilayer sample Py/Bi2Se3. In order to minimize shunting effect s, \nthe Py film partially covers the Bi 2Se3 film surface. (b) Derivative FMR field scan for the bilayer of Py (12 \nnm)/ Bi2Se3 (4 nm) obtained by inserting the sample in a microwave rectangular cavity operating at 9.4 \nGHz. The inset show s the derivative FMR absorption field scan for the bare Py (12 nm) film. The increase \nof the linewidth (HMHM ) for the bilayer Py (12 nm)/ Bi2Se3 (4 nm) is mostly due to the spin pumping \nprocess. (c) Field scan of VSP for two in -plane angles, measured at the same experimental configuration. \nThe result confirms that the sign of the VSP in the bilayer Py (12 nm)/ Bi2Se3 (4 nm), is the same as the one \nmeasured in Bi 2Se3/YIG. (d) Decomposition of the symmetric and antisymme tric components of VSP \nobtained by fitting the data of (c). The inset shows the dependence of the peak value of the symmetric \ncomponent as a function of the microwave power, measured at 𝜙=180° . \n \nTo further study the polarization of the spin -to-charge current conversion process in \nBi2Se3, we investigated the spin -pumping voltage in the bilayer of Bi 2Se3(4 nm)/Py(12 nm), where \nPy is Permalloy (Ni 81Fe19). The investigated sample is illustrated in Fig. 4(a), where the layer of \nPy partially covers the Bi 2Se3 surface, so that the electrodes are attached out of the Py layer. The \nsample was sputter grown onto SiO 2(300nm)/Si(001) where a passivation layer of MgO(2nm) \nwas grown underneath the Bi 2Se3 layer. Fig. 4(b) shows the FMR spectrum of the bilayer Py(12 \nnm)/ Bi2Se3(4 nm) in which the sample is placed in a microwave cavity resonating at 9.4 GHz, \nwith 𝑄≈2000 and an incident microwave power of 17 mW. Due to the spin pumping effect, the \nFMR linewidth (HWH M) increased to 32 Oe in comparison with the linewidth of 27 Oe of a bare \nPy(12 nm) layer, shown in the inset of Fig. 4(b). Figure 4(c) shows the spin pumping voltage \nmeasured between the electrodes for 𝜙=0° (blue curve) and 𝜙=180° (red curve), with an \nincident power of 170 mW. The 𝑉𝑆𝑃 lineshape is descri bed by the sum of symmetric and \nantisymmetric components, 𝑉𝑆𝑃(𝐻)=𝑉𝑠(𝐻−𝐻𝑅)+𝑉𝐴𝑆(𝐻−𝐻𝑅), where 𝑉𝑠(𝐻−𝐻𝑅) is the \n(symmetric) Lorentzian function and 𝑉𝐴𝑆(𝐻−𝐻𝑅) is the (antisymmetric) Lorentzian derivative \ncentered at the FMR resonance field ( 𝐻𝑅). Figure 4(d) s hows the corresponding symmetric (red) \nand antisymmetric (green) components of the 𝑉𝑆𝑃 line shape for 𝜙=180° , obtained by fitting \nthe data (black symbols) with a sum of a Lorentzian function and Lorentzian derivative (given by \nthe cyan curve). The inset of Fig. 4(d) shows the linear dependence of the peak value of the \nsymmetric component as a function of the i ncident power. The symmetric component of 𝑉𝑆𝑃 in \nPy/Bi2Se3, whic h is attributed to the spin -to-charge current conversi on process, has the same \npolarity as the one observed for 𝑉𝑆𝑃 measured in Bi 2Se3/YIG and Ta/YIG bilayers. \nIn conclusion, we report an investigation of the spin - to charge -current conversion process \nin bilayers of YIG/Bi 2Se3 and Py/Bi 2Se3, where the Bi 2Se3 layer was grown by sputtering. The \nresults obtained by means of the ferromagnetic resonance driven spin pumping technique has shed \nlight in some aspects not investigated by previous papers. We discovered that the spin - to charge -\ncurrent mechanism in topological insulator Bi 2Se3 has the same polarity as the one of Ta, and \nopposite to the one in Pt. By interpreting the spin pumping voltage as due to the inverse Rashba -\nEdelstein effect , we calculated the value of 𝜆𝐼𝑅𝐸𝐸 as a function of B i2Se3 thickn ess and the values \nfound demonstr ate that the surface states have a dominant role in the spin -charge conversion \nprocess . Thus, the spin -charge conversion mechanism based on the spin diffusion process plays \na secondary role. We expect that our results will be useful for app lications in spintronic devices \nand understanding the spin - to charge -current mechanism in sputter -deposited films of topological \ninsulator Bi 2Se3. \nThis research was supported by Conselho Nacional de Desenvolvimento Cient ífico e Tecnol ógico \n(CNPq), Coordena ção de Aperfei çoamento de Pessoal de N ível Superior (CAPES), Financiadora \nde Estudos e Projetos (FINEP), Fundaç ão de Amparo à Ciência e Tecnologia do Estado de \nPernambuco (FACEPE), Fundaç ão Arthur Bernardes (Funarbe), Fundação de Amparo à Pesquisa \ndo Estado de Minas Gerais (FAPEMIG) - Rede de Pesquisa em Materiais 2D and Rede de \nNanomagnetismo. \n References : \n \n1. A. Manchon, A. Belabbes, Solid State Physics, Volume 68, Chap. 1, (2017). \n2. A. Soumyanarayanan, N. Reyren, A. Fert, C. Panagopoulos, Nature. 539 (7630) 509 –\n1517 (2016) . \n3. A. Hoffmann and S. D. Bader, Phys. Rev. Appl., 4, 047001 (2015). \n4. M. I. Dyakonov, and V. I. Perel, Phys. 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B 98, 144431 (2018). \n " }, { "title": "1604.07025v1.Who_pumps_spin_current_into_nonmagnetic_metal__NM__layer_in_YIG_NM_multilayers_at_ferromagnetic_resonance_.pdf", "content": "1Whopumpsspincurrentintononmagnetic ‐metal(NM)layerin\nYIG/NMmultilayers atferromagnetic resonance? \n\nYun Kang1, Hai Zhong1, Runrun Hao1, Shujun Hu1, Shishou Kang1★, Guolei Liu1, Y. Zhang2, \nX. R. Wang2★, Shishen Yan1, Yong Wu3, Shuyun Yu1, Guangbing Han1, Yong Jiang3 and \nLiangmo Mei1 \n \n1School of physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan, \nShandong, 250100, China. \n2Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, \nKowloon, Hong Kong, China. \n3State Key Laboratory for Advanced Metals and Materials, School of Materials Science and \nEngineering, University of Science and Technology Beijing, Beijing 100083, China. \n★email: skang@sdu.edu.cn; phxwan@ust.hk 2Spin pumping in Yttrium-iron-garnet (YIG)/nonmagnetic-metal (NM) layer \nsystems under ferromagnetic resonance (FMR ) conditions is a popular method of \ngenerating spin current in the NM l ayer. A good understanding of the spin \ncurrent source is essential in extracti ng spin Hall angle of the NM and in \npotential spintronics applications. It is widely believed that spin current is \npumped from precessing YIG magnetization into NM layer. Here, by combining microwave absorption and DC-voltage measurements on YIG/Pt and YIG/NM1/NM2 (NM1=Cu or Al, NM2=Pt or Ta), we unambiguously showed that \nspin current in NM came from the magnet ized NM surface (in contact with YIG) \ndue to the magnetic proximity effect (MPE), rather than the precessing YIG \nmagnetization. This conclusion is reached through our unique detecting method where the FMR microwave absorpti on of the magnetized NM surface, \nhardly observed in the conventional FMR experiments, was greatly amplified \nwhen the electrical detection circuit was switched on. \nSpin current generation, de tection, and manipulation i nvolve fundamental science \nas well as the key-technologies\n1,2 in spintronics. Spin pumping from precessing \nmagnetization under ferromagnetic resonanc e (FMR) conditions is an attractive \nmethod for generating cohere nt pure spin current3-8. Pure spin current is the key \nresource in spintronics as well as a base for studying the inverse spin Hall effect \n(ISHE) characterized by the spin Hall angle αSH. Ferromagnetic-insulator \n(FI)/nonmagnetic-metal (NM) bilayers are believed to be the ideal settings for \nmeasuring αSH of the non-magnetic metals. One of the widely studied such systems is \nYttrium-iron-garnet (YIG)/Pt bilayer9-16 because YIG [Y3Fe2(FeO4)3] is a 3well-known insulating magnetic material and Pt is a well-studied heavy metal with a \nstrong spin-orbit interacti on. The conventional understanding of the system under \nFMR conditions is that magnetization of YI G precesses coherently, and the precessing \nmagnetization pumps pure spin current into Pt layer9-12 across the YIG/Pt interface. \nThe spin current in the Pt layer is then c onverted into a transver se (normal to spin \nflow direction and spin polarization) char ge current that can be detected as an \nelectrical voltage signal. YIG/ Pt bilayer is believed to be a clean system for studying \nspin pumping and ISHE since YIG cannot conduct the electric current so that \nwhatever the DC-voltage measured in Pt must be from the ISHE. The signature of \nspin pumping is the broadening of the FMR peak width of YIG as well as a detected \nDC-voltage in the Pt layer. Controversially, the extracted values of αSH from different \ngroups differ by two orders of magnitude , inconsistent with each other10,16,17 in this \n“clean” system. It arises the question of who is responsible for the spin pumping, YIG \nor other magnetic sources? The answer to the question requires a good understanding \nof the interfacial phenomena between the YIG and a NM. \nHere, we use YIG(16nm)/Pt(10nm) bila yer and YIG(16nm)/Cu(5nm)/Pt(10nm), \nYIG(16nm)/Cu(5nm)/Ta(10nm), YIG(16 nm)/Al( 5nm)/Pt(10nm) trilayer systems in a \nrectangular cavity to investigate these issues by a combined measurements of \nmicrowave absorption and DC-voltage. Cont rary to the popular belief, the spin \ncurrent in Pt or Ta was not pumped fro m the precessing YIG magnetization, but from \nthe magnetized NM surface (in contact with YIG) originated from the magnetic proximity effect (MPE)\n18-24. Interestingly and surprisingl y , the FMR signal from the \nmagnetized NM surface was greatly amplif ied when the electrical measurement 4circuit was connected (otherwise, the signa l could hardly be observed). When the \nMPE was absent, such as in YIG(16nm)/A l(5nm)/Pt(10nm) samples, no DC-voltage \nsignal was observed in Pt. Our experiments showed unambiguously that spin pumping from the insulating YIG layer into the metallic Pt or Ta layer was not efficient and \neffective, in comparison with that from a magnetized metallic surface into Pt or Ta. \nResults \nYIG(16nm)/Pt(10nm) systems. The black circles in Fi g. 1a are the derivative \nmicrowave absorption spectrum ( dI/dH ) of a pure YIG sample as a function of \nexternal magnetic field H. The lineshape of the FMR derivative absorption spectrum \nof the pure YIG sample follows a standard di fferential Lorentzian line with FMR peak \nat H=2.497 kOe, and peak width of Γ=10 Oe which shows high quality of our YIG \nsamples16. \nThe green/blue circles in Fig. 1a are th e FMR derivative absorption spectra of a \nYIG(16nm)/Pt(10nm) bilayer strip when the electrical detection circuit is switched \non/off (see the left inset of Fig. 1a and the methods below). In contrast, the FMR \nderivative microwave absorption spectrum of YIG/Pt bilayer sample appears to shift \nto a lower field with a seem ingly broaden peak width that was observed in previous \nstudies5,16,25 and was used as an essential evidence of spin pumping from YIG. More \nstrikingly, the microwave absorption signals are substantially different when the \nelectrical detection circuit was switched on (green circles) and off (blue circles). \nObviously, the absorption signal was greatly amplified when the electrical detection circuit was switched on. A more careful examination showed that the absorption \ncurves of switch-off circuit are better described by two independent FMR signals. 5One of them with a relative amplitude of A1=60.5% was from the free YIG because it \nhas the same peak position and peak width of H1=2.497 kOe and Γ1=10 Oe as the free \nYIG. The other with peak position H2=2.488 kOe , peak width Γ2=12 Oe, and relative \namplitude A2=39.5% was naturally attr ibuted to the YIG covered by Pt. Because Pt \nmodifies magnetic pr operties of YIG25, the peak position and peak width of the \nsecond FMR signal differ slightly from those of the free YIG . \nIt is worthy to note that the FMR absorpti on curves of switch-on circuit were best \nfitted by three independent FMR signals as shown in Fig. 1b. Among the three signals in Fig. 1b, two signals (with A\n1=54.5% and A2=36.4% ) are exactly those of free YIG \nand Pt-covered YIG, and the third signal of H3=2.477 kOe, Γ3=14 Oe, and A 3=9.1% \ncame from the amplification of a very weak signal (hidden in the blue circles) \noriginated from the MPE-induced magnetized Pt surface that was in contact with YIG. \nFurthermore, the corresponding DC-voltage detected in Pt was from the magnetized \nPt surface since their peak positions and peak widths match exactly with each other as \nshown in Fig 1b. \nTo substantiate this interp retation, we fabricated also YIG/Pt bilayer samples in \nwhich YIG was fully covered by Pt layer. As shown in Fig. 1c, the blue and green \ncircles are the FMR absorption signals when the electrical meas urement circuit was \nswitched on (green) and off (blue). As e xpected, the signal from the free YIG was \nabsent. In Fig. 1d, the FMR absorption curves of switch-on circuit now consist of two \nsignals respectively from the Pt-covered YI G and the magnetized Pt surface. Again, \nthe DC-voltage relates to the signal of the magnetized Pt surface since their peak \npositions and widths match exactly with each other, and cannot be from the spin 6pumping of YIG. \nThe above conclusion could also be reach ed from the change of the shape of \nvoltage-H curves as angle θ between ac-magnetic field and sample long edge varies. \nThe DC-voltage originated from the spin pumping of YIG should follow a Lorentzian \nlineshape since the FMR absorption is described by the Lorentzian function26. Thus \nDC-voltage lineshape would be symmetric about its peak for any angle θ if the spin \npumping was from YIG. However, as plotted in Fig. 2a, it is clear that most spectra \nconsist of a superposition of a Lorentz- and a dispersive-type resonance lineshape26,27 \nFor a given θ, the DC-voltage curve was fitted to Eq. (1) so that both symmetric and \nasymmetric components of the DC-voltages Usym and Uasy were obtained. Their \nangle-dependences were plotted in Fig. 2b that fit well with theoretical prediction of \nEq. (2) (see the Methods below). V oltages s\nSRU (for symmetric component) and a\nSRU \n(for asymmetric component) due to the spin rectification are 0.065 V=s\nSRU μand \n0.568 V=a\nSRU μ. V oltage U sp from the ISHE due to spin pumping is 1.02 V=spU μ. \nThus it shows that substantial amount of the DC-voltage came from the AMR and \nAHE of a ferromagnetic metal that resulted in an asymmetric lineshape, and the only \npossibility is that the Pt surface in contact with YIG wa s magnetized and generated a \nspin rectification voltage28. Obviously, the spin pumping effect generated a larger \nDC-voltage than the spin rectification. Th ese results further confirmed the MPE and \nspin precession of magnetized Pt surface in YIG/Pt system. \nFigure 3 is the H -dependence of the derivative microwave absorption with the \nswitch-on circuit and DC voltage of a t ypical YIG/Pt strip sample for various \nfrequencies at θ=0o. The DC voltages (Fig. 3a) have a symmetric Lorentzian shape 7while the microwave absorption has multi- peaks (Fig. 3b) due to different FMR \nsources. This implies that only one FMR source pumped spin current into Pt and \ngenerated the DC-voltage. Clearly, the p eak position and width of the DC-voltage \nmatch well with those of the tiny FMR signa l. Furthermore, Fig. 3c shows that the \npeak field of the DC-voltage increased with the frequency that fits well with the Kittel \nformula29,30, f=(γ/2π)(H res*(H res+4πMs))1/2, with the gyromagnetic ratio γ= \ngμB/h=1.738×1011 T−1 s−1 and the saturation magnetization Ms =0.248 T . I t g a v e a \nLander factor g=2.08 for magnetized Pt surface. The fitted Ms of magnetized Pt is \nvery closed to the observed value in Ni/Pt system by XMCD measurement19. Thus, \nthis result further supports our assertion that the precessing magnetization of YIG did \nnot pump spin current to Pt laye r, contrary to the popular belief5,14-16,31. \nYIG(16nm)/Cu(5nm)/Pt(10nm) and YIG(16nm)/Cu(5nm)/Ta(10nm) systems. \nTo further substantiate our claim that the DC -voltage in YIG/Pt bilayer is due to the \nspin pumping from the MPE-induced magnetized Pt layer, a 5nm thick Cu was inserted between YIG and Pt (Ta) so that Pt (Ta) surfaces were not in contact with \nYIG and no MPE is possible for Pt (Ta). The upper panel of Fig. 4 is the typical FMR \nderivative microwave absorption spectrum of one of our YIG/Cu/Pt samples. The blue \ncircles are the results when the electrical detection circuit was switched off while the \ngreen circles are those when the circ uit was switched on. The signal from the \nmagnetized Pt surface was obviously absent , and was replaced by a new FMR signal \nat an even lower field of H=2.46 kOe, far below YIG resonance peak, and with a peak \nwidth of \nΓ=12 Oe. Although this new signal was seen in a 50 times enlarged figure as \nshown by the black circles in the top panel, it was extremely weak when the electrical 8detection circuit was switched off. Similar to YIG/Pt bilayer samples, an extra signal \ncan be clearly observed when the electric al detection circuit was switched on. As \nexpected this time, the DC voltage of YI G/Cu/Pt and/or YIG/Cu/Ta (middle and lower \npanels of Fig. 4) was observed at H=2.46 kOe, exactly corresponding to the new \nFMR signal. The peak widths of the FMR a nd DC-voltage signals matched again with \neach other as shown in the middle (for YIG/ Cu/Pt) and lower (for YIG/Cu/Ta) panels. \nIn contrast, there were no DC-voltage signa ls at the YIG resonant fields, confirming \nthat the DC-voltage was not due to the spin pumping of YI G, but due to that of the \nmagnetized Cu surface (in contact with YIG) 32,33,34,35. The signs of the DC-voltages \nof YIG/Cu/Pt and YIG/Cu/Ta samples are opposite due to the opposite sign of spin \nHall angle for Pt (positive) and Ta (negative)16. \nAgain, above experiments do not support th e general belief that precessing YIG \nmagnetization at FMR pumps spin current into Pt or Ta in YIG/Pt, YIG/Cu/Pt, and \nYIG/Cu/Ta systems. Our results are consiste nt with the assertion that it was the \nmagnetized NM surface pumping spin current into the Pt or Ta layer. The clear \nevidences include 1) DC-voltage peaks were far from the FMR peaks of the YIG, but \nwere exactly overlapped with the FMR peak of MPE-induced magnetized NM surface; \n2) the angle-dependence of DC-voltage lin eshape that shows big contribution from \nAMR and AHE of a magnetized metal. Furthe rmore, the MPE of both Pt and Cu were \nconfirmed by our first-principl e calculations (see the Me thods). It was found that a \nfew Cu atomic layers adjacent to Ni wa s magnetized with average moment about \n-0.02 μB/atom, which was only about 1/5 of the average moment of Pt (0.11 μB/atom) \nin Ni(111)/Pt system19. If one assumes that Cu/Ni and Cu/YIG (Pt/N i and Pt/YIG) 9have the similar MPE, then the precession of this small moment can pump spin \ncurrent into Pt or Ta layer. We can naturally interpret our observed DC-voltage as the ISHE. This smaller magnetized Cu moment explains the much smaller voltage than \nthat in YIG/Pt system as shown in Fi gs. 1 and 2. The small negative Cu magnetic \nmoment is also consistent with lower reso nance field for the magnetized Cu due to the \nnegative exchange field at FMR\n29. \nYIG(16nm)/Al(5nm)/Pt(10nm) systems. To further verify the assertion that spin \ncurrent in NM layer(s) was not from YI G in YIG/NM1 bilayer or YIG/NM1/NM2 \nmultilayer samples, but from magnetized NM 1 surface (in contact with YIG), we did \na controlled experiment with YIG(16nm)/Al( 5nm)/Pt(10nm) samples. Al has no MPE, \nin consistent with our first-principle calcu lations on Ni/Al system. As shown in Fig. 5, \nthe derivative microwave absorption spect rum of a typical YIG/Al/Pt samples does \nnot change whether the electri cal detection circuit was switch ed on or off, in contrast \nto that for YIG/Pt or YIG/Cu/Pt(Ta) system s. Consistent with our assertion, there was \nno spin current in Pt layer since YIG could not pump detectable spin current and no \nDC-voltage was observed as shown in the bottom panel of Fig. 5. \nDiscussion \nA magnetic metallic film at its FMR can gene rate not only a DC signal but also a \nradio-frequency ac field28,36 by the AHE and the AMR. As illustrated in Fig. 6 (see the \nMethods) when the sample is connected to Cu connection-pads through two Al wires, \nthe whole structure becomes a patch ante nna and a high fre quency pass filter. \nAccording to the patch antenna theory37,38, the ac signals from the AMR and AHE of \nthe magnetic metallic layer and from ISHE of nonmagnetic metallic layer can be 10radiated through fringing fiel ds at the radiating edges. This will result in the \namplification of the FMR signal of the metallic film. Here we termed this \namplification as “antenna effect” when the Al wires were connected to Cu-Pads (see \nexperimental setup, switch-on ci rcuit). Fig. 6d show s clearly this antenna effect since \nthe microwave absorption of a 3nm-thick Pe rmalloy (Py) film sample was greatly \nenhanced when Al wires were connected to Cu-Pads. \nOne possible reason, that the precessing YIG could not inject a detectable spin \ncurrent into Pt layer, might be due to the mismatch in the electronic structures of YIG and Pt, resulting in an inefficient angular momentum transfer. The FMR linewidth \nbroadening and additional damping mechan ism observed previously may be due to \nthe overlap of resonant peak s of both YIG and magnetized Pt\n12,14. Thus one should \nextract the spin Hall angle αSH by taking into the account of the new findings reported \nhere. \nConclusion \nIn summary, our experiments on YIG/Pt bilayer and YIG/Cu/Pt (YIG/Cu/Ta) \ntrilayer samples showed that the FMR microwave absorption was mainly from three sources: free YIG, YIG covered by a NM, and the magnetized NM surface arising from the MPE. Interestingly, the FMR microwave absorption signal from the magnetized NM layer was pronounced only when the electrical detection circuit was \nswitched on. The electrical detection circuit acted as an antenna for the FMR signal of the magnetized NM surface. Surprisingly, the DC-voltages were from the spin rectification effects and spin pumping of the magnetized NM layers, instead of spin \npumping of YIG alone. Thus, contrary to th e popular belief, our studies suggest that 11precessing magnetization of YIG does not pump detectable spin current into the NM \nlayer. Our findings are very important for properly extracting the spin Hall angle and for a better understanding of the c oncept of interface mixing conductance\n9-11,16. \nMethods \nSample preparation and experimental procedure. YIG [Y3Fe2(FeO 4)3] films (16 nm) \nwere fabricated on Gd 3Ga5O12 (GGG) wafers by pulsed laser deposition (PLD). X-ray \ndiffraction (XRD) and atomic force microscopy (AFM) showed that our YIG are high \nquality (See supplementary Figure 1). Py, Pt , Cu, Ta or Al with high purity (4N) was \nthen deposited on YIG by magnetron sputtering to create a NM strip with a mask of \n0.2 mm × 2.3 mm. All samples were cut into 1 mm × 3 mm for DC-voltage and \nmicrowave absorption measurements in a homemade X-band microwave absorption \nspectrometer. \nThe experimental setup is shown in Fig. 6. FMR microwave absorption and \nDC-voltage were measured at frequency f=9.7 GHz of TE10 mode in the X-band \ncavity. The sample with size of 1 mm x 3 mm was mounted in th e middle of a shorted \ncopper plate at one end of the cavity that can rotate in the XY-plane as illustrated in \nFig. 6. The angle between the Y-axis and the long edge of the sample is denoted as θ. \nTwo thin rectangular copper sheets of 1.5 mm × 8 mm were symmetrically placed on \nthe both sides ( 1 mm away from sample) of sample in the cavity as illustrated in Fig. \n6. These two small copper sheets were isol ated from the shorted copper plate and \nacted as electrical connecti on pads that connected to a SR-530 lock-in amplifier of \nStanford Research Systems or Keithley- 2182 nanovoltmeter. It should be pointed out \nthat these two thin copper sheets did not affect the X-band microwave distribution 12from the angular dependence measurements of FMR for Py (see Supplementary Fig.2). \nTwo Al wires of diameter about 30 micromet ers were attached to two long-edge ends \nof the sample. As illustrated in Fig. 6, the electrical measurement circuit was switched on when the other ends of Al wires were c onnected to the two Cu-pads. The in-plane \nexternal field and ac microwave field were always orthogonal with each other in order \nto have a maximal precessing magnetization. \nAngular dependence: The DC-voltage from ISHE, AMR and/or AHE of a \nferromagnetic metal near the FMR has a symmetric Lorentz-lineshape and an \nasymmetric dispersive-lineshape \n26,27,28\n00 (H, H , ) U (H, H , )sym asy UUL D=Γ + Γ \n2\n0 22\n0(H, H , )(H H )LΓΓ=−+Γ (1) \n0\n0 22\n0(H H )D(H, H , )(H H )Γ−Γ=−+Γ \nHere, H 0 and Γ are respectively the resonance field and resonant peak width. Usym \nand Uasy are the voltages of the symmetry a nd asymmetry components of DC-voltage \nthat depend on angle θ as39 \nsin(2 )sin( ) U cos( )s\nsym SR SPUU θθθ =+ (2) \nsin(2 )sin( )a\nasy SRUU θθ = \nHere,s\nSRU, a\nSRUare the voltages due to th e spin rectification. U sp is the voltage from \nthe ISHE due to spin pumping. \nFirst-Principle calculations: Because our computation resources do not allow us to \nperform reliable calculations on YIG/Pt a nd/or YIG/Cu systems due to the huge unit \ncells, we performed the calculations on Ni(111)/Cu systems instead by using the same 13method in Reference 19 for calculating MPE of Pt in Ni(111)/Pt systems. \nReferences \n1. Wolf, S. A. et al. Spintronics: A spin -based electronics vi sion for the future. Science 294, 1488 \n(2001). \n2. Źutić, I., Fabian, J. & Sarma, S. D. Spintronics: Fundamentals and applications. Rev. Mod. Phys. \n76, 323 (2004). \n3. Urban, R., Woltersdorf, G. & Heinrich, B. Gilbert damping in single and multilayer ultrathin \nfilms: Role of interfaces in nonlocal spin dynamics. Phys. Rev. Lett. 87, 217204 (2001). \n4. Mizukami, S., Ando,Y . & Miyazaki, T. Effect of spin diffusion on Gilbert damping for a very \nthin permalloy layer in Cu/permalloy/Cu/Pt films. Phys. Rev. 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Observation of spin rectification in Pt/yttrium iron garnet bilayer. J. Appl. Phys. \n117, 17C725 (2015). \n32. Carbone, C. et al. Exchange split quantum well states of a noble metal film on a magnetic \nsubstrate. Phys. Rev. Lett. 71, 2805 (1993). \n33. Garrison, K., Chang, Y . & Johnson, P. D. Spin polarization of quantum well states in copper \nthin films deposited on a Co(001) substrate. Phys. Rev. Lett. 71, 2801 (1993). \n34. Pizzini, S., Fontaine, A., Giorgetti, C. & Dartyge, E. Evidence for the spin polarization of \ncopper in Co/Cu and Fe/Cu multilayers. Phys. Rev. Lett. 74, 1470 (1995). \n35. Hirai, K. X-ray magnetic circular dichroism at the K-edge and proximity effects in Fe/Cu \nmultilyers. Physica B 345, 209 (2004). \n36. Jiao. H. & Bauer, G. E. W. Spin backflow and ac voltage generation by spin pumping and the \ninverse spin Hall effect. Phys. Rev. Lett. 110, 217602 (2013). \n37. Carver, K. R. & Mink, J. W. Microstrip antenna technology. IEEE Trans. Antennas Propag. 39, \n2-24 (1981). \n38. Stutzman,W. L. & Thiele, G. A. Antenna Theory and Design (Wiley, 1981). \n39. Azevedo, A. et al. Spin pumping and anisotropic magnetoresistance voltages in magnetic \nbilayers: Theory and experiment. Phys. Rev. B 83, 144402 (2011). \n \nAcknowledgements \nThis work was supported by the National Basic Research Program of China under \nGrant No. 2015CB921502 and 2013CB922303, the National Natural Science \nFoundation of China under Grant Nos 11474184, 11174183 and 11504203, and the 111 project under Grant No. B13029. YZ a nd XRW were supported by the Hong \nKong RGC Grants No. 163011151 and No. 605413. YW and YJ were supported by \nthe National Natural Science Foundation of China under Grant No. 51501007. \nAuthor contributions \nX.R.W. and S.K. contributed to project de sign and manuscript writing; Y .K., H.Z. and \nR.H. carried out the experiments; S.H. perf ormed the first principle calculations; Y .W. 15and Y .J. fabricated YIG. All author s participated in data analysis. \n 16 \nFigure legends \nFig. 1 (a) The FMR derivative absorption spectra of free YIG and YIG/Pt bilayer strip with θ=0o(θ \nis the angle between microwave field and the sample long edges). The lower panel is the \ncorresponding DC-voltage (bottom) spectra of YIG/Pt bilayer schematically illustrated on the right. The left inset is the schematic diagram of the electrical. The right inset shows the sample structure YIG/Pt stripe. (b) The fit of FMR spectrum of YIG/ Pt strip obtained with antenna effect by three \nFMR signals respectively for free YIG, YIG covere d with Pt, and magnetized Pt surface in contact \nwith YIG. The DC signal agrees with the a ssertion of spin pumping from the magnetized Pt \nsurface. (c) The FMR derivative microwave absorpti on (upper) and DC-voltage (lower) spectra of \nfully covered YIG/Pt bilayer (illustrated in the lower left) with \nθ=170o. The inset shows the YIG \nsample fully covered by Pt. (d) The FMR spectr um of fully covered YIG/Pt bilayer with the \nantenna effect is best fitted by two FMR sign als from YIG covered with Pt and magnetized Pt \nsurface. The corresponding DC-voltage si gnal was from the magnetized Pt surface. \n \nFig. 2 (a) The DC-voltage spectra of a YIG(16nm)/Pt (10nm) strip at various angle θ. The symmetrical \n(b) and asymmetrical (c) components of DC-voltages defined in Eq. (1) were extracted. The solid lines \nare the best fits of Eq. (2) with 0.065 V,=s\nSRU μ 0.568 V=a\nSRU μand 1.02 V=spU μ. The \ninset illustrates experimental setup and angle θ. \n \nFig. 3 The H-dependence of DC-voltage (a) and FMR spectra (b) of YIG/Pt stripe line at θ=0o and \nfor various frequencies with antenna effect. (c) The frequency dependence of peak position of \nDC-voltages. The solid line is the fit to the Kittel formula. Fig. 4 The FMR spectra of a YIG/Cu bilayer sample (upper panel) and DC-voltage spectra of \nYIG/Cu/Pt (middle panel) and YI G/Cu/Ta (lower panel) with \nθ=0o. In the upper panel, the green \n(blue) circles are the FMR derivative microwave ab sorption spectra when the electrical detection \ncircuit is switched on (off). A weak signal, as shown by the black circles that is the zoom-in (50 \ntimes enlarged) of the blue circles inside the red rectangle, was amplified when the electrical circuit was switched on. The DC-voltage in Pt (middle panel) and Ta (lower panel) can be fitted well by the Lorentzian function (solid lines).\n \n Fig. 5 Top: The FMR derivative microwave absorption spectra of a YIG/Al(5nm)/Pt(10nm) sample when the electrical detection circuit was switched on (red) and off (blue). No MPE-induced magnetized Al was observed. Bottom: No DC-voltage signal in Pt was observed.\n \n Fig. 6 (a) The experimental setup for the microwave absorption measurement and electrical detection of FMR in which DC-voltage along the long edge of the sample was measured. (b) The zoom-in of sample and copper-pads. The total th ickness of the sample including GGG substrate is \nabout 1 mm. (c) Th e in-plane rotation geometry of sample. The microwave of frequency \nf=9.7 GHz propagated along the Z-axis, and the external field H was along the X-axis. e and h are \nthe electric and magnetic components of the microwave, respectively along the X-axis and the \nY-axis. The angle between the Y-axis and the long edge of sample was denoted as \nθ that varies as \nthe shorted copper plate rotates in the XY-plane. (d) FMR signals of 3 nm Py thin film strip when \nthe Al wires were connected/disconnected to the Cu-pads (with/without antenna effect). The inset is the equivalent circuit with antenna effect. 17\n \n \nFig. 1 Kang et al . \n 18\n \n \n \nFIG. 2 Kang et al . \n 19\n \n \n \nFIG. 3 Kang et al . \n \n \n \n \n 20\n \nFIG. 4 Kang et al . \n \n 21 \n \n \n \nFIG. 5 Kang et al . \n \n \n 22 \n \n \n \n \nFIG 6 Kang et al . " }, { "title": "1311.1262v1.Investigation_of_Magnetic_Proximity_Effect_inTa_YIG_Bilayer_Hall_Bar_Structure.pdf", "content": "Investigation of Magnetic Proximity Effect inTa/YIG Bilayer Hall \nBar Structure \n \n \nYumeng Yang1,2, Baolei Wu1, Kui Yao2,Santiranjan Shannigrahi2, Baoyu Zong3, \nYihong Wu1, a) \n \n \n1Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, \nSingapore 117583 \n \n2Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 \nResearch Link, Singapore 117602 \n \n3Temasek Laboratories, National University of Singapore, T -Lab B uilding, 5A, Engineering Drive 1, #09-02, \nSingapore 117411 \n \n \nIn this work, the investigation of magnetic proximity effect was extended to Ta which has been reported \nto have a negative spin Hall angle. Magnetoresistance and Hall measurements for in- plane a nd out -of-\nplane applied magnetic field sweeps were carried out at room temperature. The size of the MR ratio \nobserved (~10\n-5) and its magnetization direction dependence are similar to that reported in Pt/YIG, both \nof which can be explained by the spin Hall magnetoresistance theory . Additionally, a flip of \nmagnetoresistance polarity is observed at 4 K in the temperature dependent measurements, which can be \nexplained by the magnetic proximity effect induced anisotropic magnetoresistance at low temperature. \nOur findings suggest that both magnetic proximity effect and spin Hall magnetoresistance have \ncontribution to the recently observed unconventional magnetoresistance effect. \n \n \n \n \n \n \na) Author to whom correspondence should be addressed: elewuyh@nus.edu.sg \n1 \n I. INTRODUCTIO N \n \nPlatinum (Pt) has been investigated as a promising spin current detector in recent reports on spin \npumping effect1-5 and spin Seebeck effect6-8 involving ferromagnet (FM)/non- ferromagnet (NM) \nstructures. In these investigations, the generated spin current is converted into an electromotive force via \ninverse spin Hall effect (ISHE), which can then be detected electrically. On the other hand, recently \nanisotropic -magnetoresistance (AMR) -like effect in Pt on the insulating ferromagnet yttrium iron garnet \n(YIG) was reporte d by Weiler et al .9 and Huang et al .10, which is attributed to the induced magnetic \ndipole moment in Pt due to magnetic proximity effect (MPE). This “pseudo- magnetization” in Pt was \nlater supported by X- ray magnetic circular dichroism (XMCD) measurement11. Further investigation by \nNakayama et al.12 found that the magnetization direction dependence of this resistance change is distinct \nfrom any of the known MR effect. They proposed a spin Hall magnetoresistance (SMR) theory13 to \nexplain the experimental observations by taking into account the influence of spin current to the charge \nresistance through spin -orbit coupling (SOC). Some of the follow- up works14-19 are in supportive of this \nmodel, in the way that the governing parameters such as spin mixing conductance of the interface, spin \ndiffusion length and spin Hall angle of Pt could be extracted and appeared to be comparable to those of \nprevious studies. \nAlthough the presence of unconventional MR effect is confirmed, its underly ing mechanism \nremains to be unveiled. Contradictory XMCD results showing the presence of only negligible Pt magnetic polarization in Pt/YIG structure\n20,21 cast some doubts o n the strength of MPE. On the other \nhand, the incompleteness of SMR theory was also indicated by the absence of similar effect in Au/YIG22 \nand its preservation in Pt/surface modified YIG23 structures. Therefore, in order to reveal the true \nmechanism of spintronic effects at FM/NM interf aces, it is necessary to extend the study to different \nFM/NM material systems. So far, with most investigations focusing on Pt/YIG, only a few have extended the bilayer structure to Au/YIG\n22, Ta/YIG17, Pt/Py24, Py/YIG23, Fe 3O4/Pt, NiFe 2O4/Pt14, and CoFe 2O4/Pt19. \nIn this work, we focus on the Ta/YIG interface and discuss for the first time the Hall measurement \nresults. The MR effect observed at room temperature is similar to that observed in Pt/YIG. The out -of-\nplane Hall resistance was found to be influenced by YIG substrate’s magnetization at magnetic field \nbelow its saturation value of around 2000 Oe. Both observations could be explained by the SMR theory \n2 \n However, we observed an additional dip centered at zero- field between around - 200 Oe and 200 Oe, \nwhich could not be attributed to YIG’s in -plane magnetization rotation15. The shape of the in- plane Hall \nresistance is different from that predicted by the SMR theory nor the MPE -induced planar Hall effect \n(PHE), but instead similar to that of the MR curve. The additional dip and shape difference might be due \nto the possible influence from MR. In the MR measurement with out -of-plane magnetic field, a flip of the \nMR polarity was observed at 4 K, implying the enhancement of MPE at low temperature. The results of \nthe temperature dependent Hall measurements were similar to those of the room temperature measurements. Our findings suggest that both the SMR and the MPE have a contribution in the recently \nobserved MR effect. \nII. SAMPLE FABRICATION AND CHARACTERIZATION \n \nSamples were fabricated on (111) oriented YIG (8 μm)/ GGG (500 μm) single crystal wafer with a \nsize of 5mm × 5mm. Fig. 1(a) shows the X- ray diffraction (XRD) pattern of an unpatterned reference Ta \nthin film (10 nm) deposited on YIG/GGG substrate. Comparison with the XRD pattern of bare substrate \n(inset of Fig. 1(a)) confirms that the Ta layer is in β -phase. To f abricate the Hall bars, the substrates were \nproperly cleaned with acetone and isopropyl alcohol before coated with a Microposit S1805/PMGI SF6 \nbilayer resist. The resist -coated substrate was subsequently exposed using a Microtech laser writer to form \na Hal l bar pattern. The size of the central area of the Hall bar is 0.2 mm × 2.3 mm, and that of the \ntransverse electrodes is 0.1 mm × 1 mm. A Ta (5 nm, 7 nm, 10 nm) layer was subsequently deposited \nusing a sputter with a base pressure below 5× 10\n-5 Pa, and in an Ar working pressure of 5× 10-1 Pa, \nfollowed by liftoff to form the Hall bar. Before electrical measurements, the as -fabricated samples were \nbonded to chip carriers using a wire bonder. The resistivity of the samples (5 nm, 7 nm, 10 nm) falls into \nthe range of 6.6× 102 to 1.1× 103 µΩ cm, which is in good agreement with the reported values for β -phase \nTa17. \nFigs. 2(a) -(b) and Figs. 3(a)- (b) are the schematic configurations for MR and Hall measurements. \nFor room temperature electrical measurement s, the samples were placed in the ambient environment with \nin-plane ( x, y directions) and out -of-plane ( z direction) magnetic field applied, respectively. Low \ntemperature measurements from 4 K to 250 K were performed in a variable temperature cryostat. A standard lock- in technique with 100 µA, 163 Hz AC current was used for the MR measurement, while the \n3 \n Hall measurement was performed using a DC technique with a current of 100 µ A. The hysteresis loops \nfor YIG/GGG were measured using vibrating sample magnetom eter (VSM) as a reference to assist the \ninterpretation of electrical transport data. \n \nIII. RESULTS AND DISCUSSION \n \nA. Magnetic properties of YIG \nFig. 1(b) shows the in- plane and out -of-plane M -H loops for YIG/GGG substrates. The observation \nof an in- plane saturati on field below 100 Oe and out -of- plane saturation field of around 2000 Oe suggests \nan in -plane anisotropy for (111) -oriented YIG/GGG. The inset shows a superimposition of the in- plane \nM-H loop and the M -H loop obtained with a 90° in -plane rotation of the sample with respect to its original \nposition. The close overlap of two loops suggests no preferred easy axis in the (111) plane. Notably, for \nboth cases, the coercivity (H c) for YIG is below 10 Oe, which is reasonable for a ferrimagnetic material. \nB. Room te mperature MR and Hall measurements \nFigs. 2(c), (e) and (g) are the MR (hereafter referred to as the MR observed in the measurements of \nthis work regardless of its origin) ratio of Ta (5 nm)/YIG for three applied magnetic field directions (H x, \nHy, H z), respectively. Conventionally, for NM, its resistance can increase slightly as the applied field \nincreases due to the Lorentz force felt by the conduction electrons25. This effect is named as ordinary \nmagnetoresistance (OMR) with a positive polarity. In the present case, since the field applied is small \n(maximum 500 Oe), OMR is negligible in the field sweeps. However, a dip or peak is observed during all \nsweeps in the field region between -50 Oe and +50 Oe, indicating the presence of interfacial coupling in \nTa/YIG. The MR ratio of 2 ×10-5 is comparable to that reported in Pt/YIG10. If we only focus on the MR in \nx and y directions, it seems that it can be explained by the proximity effect, i.e., the anisotropic \nmagnetoresistance (AMR) arises from the magnetized Ta layer (hereafter named as MPE -induced AMR). \nHowever, if this is the case, the MR in z direction should be negative as well instead of being positive as \nobserved experimentally. Instead of the MPE -induced AMR, the experimental data can be understood as \nfollows using the SMR theory. According to this th eory, the MR is determined by the angle between spin \npolarization of electrons in the Ta and YIG’s magnetization direction. In the present case, as current is applied in x direction, the spin polarization induced by SOC should be dominantly in y direction. In this \nsense, the resistance response to applied field in x and z direction should be similar except that the MR \n4 \n curve in z direction is broader than that in x direction due to shape anisotropy of the YIG layer. \nFigs. 2(d), (f) are the Hall resistance of Ta (5 nm)/YIG for in -plane applied magnetic field \ndirections (H x, H y), respectively. Conventionally, for in -plane field, the Hall effect in NM should vanish \nwith only th e presence of a background offset resistance coming from electrodes asymmetry. Similar to \nthe longitudinal resistance, we observed a dip or peak in the in- plane Hall resistance located at around -50 \nOe and +50. The magnitude of around 10- 20 mΩ is on the sa me order of those reported by N. Vlietstra et \nal.15. However, its shape is different from that predicted by SMR or MPE -induced PHE, which should be \nan odd functi on of the applied magnetic field. One possible reason is that due to the relatively large size of \nthe pattern and current distribution in the electrodes, the contribution from the longitudinal MR to the Hall \nresistance cannot be excluded. As the longitudinal MR is one order of magnitude larger than the Hall \nresistance, it is possible that the MR signal covers the true Hall signal. This can also be inferred from the \nsimilarity between the shape of the MR and Hall curves. \n The out -of-plane Hall resi stance of Ta (5 nm)/YIG , as shown in Fig. 2(h), is different from the \nlinear ordinary Hall effect for NM. It has a nonlinear region between -2000 Oe and 2000 Oe, which \ncoincides with YIG’s saturation field. This is presumably caused by the fact that the total field \nexperienced by electrons inside Ta is the sum of externally applied field and the stray field from the YIG. \nThe latter is large when the in -plane magnetization of YIG is oriented into the vertical direction by an \nexternal field, while it is relat ively small when it is saturated in -plane. The additional stray field dominates \nin the field range below saturation, causing the nonlinear region. While, when the field is above the \nsaturation, the contribution from the applied field dominates, resulting i n the linear region. Noticeably, an \nadditional dip is observed in the center between - 200 Oe and 200 Oe. N. Vlietstra et al .15 related it to the \nin-plane magnetization rotation at H c, which is unlikely the origin of the dip observed here as the H c of \nYIG i s below 10 Oe, as shown in the M -H loop. Considering its similar shape and field range as that of \nthe out -of-plane MR curve in Fig. 2(g), we believe that it is also related to the influence of MR signal as \ndiscussed above. \nC. Temperature -dependent out -of-plane MR and Hall measurements \nSo far, it seems that most of our results at RT can be explained by the SMR theory. However, we \nstill cannot completely exclude the MPE contribution in Ta/YIG bilayers, as the proximity effect may be \n5 \n enhanced at low temperature. To this end, temperature -dependent MR and Hall measurements were \nperformed in temperatures from 4 K to 300 K. As discussed in Part B in the room temperature \nmeasurements, the polarity of MPE -induced AMR and SMR differs from each other only when the \nmagneti c field is applied in z direction. This means that if the magnetic field is applied in x or y direction, \neven if both types are present at low temperature, it may still be difficult to separate them because of the \nsame polarity. Therefore, we chose to apply the field in the out -of-plane direction ( z direction) in the \ntemperature dependent measurem ents. As our measurement system does not allow in -situ rotation of \nsample direction, for the temperature -dependent measurements in z -direction, we have changed the 5- nm-\nthick sample to samples with a thickness of 7 nm and 10 nm, respectively. As all the sa mples show similar \nMR and Hall behavior at room temperature, the change of samples will not compromise the consistency \nof discussion. \nFig. 3(c) is the out -of-plane MR of Ta (10 nm)/YIG at temperatures from 4 K to 300 K (similar \nresults are obtained for Ta (7 nm)/YIG which are not shown here). The curves are clearly a superposition \nof two types of effects. For the curves at 10 - 300 K, a small positive MR is observed at large field region. \nThis background MR comes from OMR as discussed in Part B, which i s a result of the Lorentz force felt \nby the conduction electrons. Additionally, sharp central dips are observed in the range between - 2000 Oe \nand 2000 Oe, which are consistent with the polarity and field range predicted by the SMR theory. This \nlarge positi ve MR is the result of the SMR caused by the change in YIG’s magnetization direction below \nits saturation field. Therefore, the MR curves obtained at 10 - 300 K is dominantly a superposition of the \nOMR and SMR effect. For the curve at 4 K, the central dip polarity remains the same, indicating again the \nSMR contribution. However, the background MR polarity is flipped to be negative. This negative MR \nfollows the polarity predicted by MPE, i.e., the AMR arises from the magnetized Ta layer, suggesting the \nlarge enhancement of MPE at low temperature. In this sense, the MR curve at 4 K is dominantly a \nsuperposition of the MPE -induced AMR and SMR effect. A schematic illustration of the above scenario is \nshown in Fig. 3(e). Same MR measurement was performed on Pt (5 nm) /Ta (5 nm) multilayer structure on \nSiO 2/Si to exclude the contribution from magnetic impurities in the as -sputtered Ta. The polarity remains \nthe same for this sample at 4 K, indicating the flip in Ta/YIG is due to the presence of the ferrimagnetic \nYIG and MPE from YIG. Based on these different origins, the background MR is subtracted, and \n6 \n temperature -dependent SMR ratio is shown in Fig. 3(f) for both the 7 nm and 10 nm Ta samples. The size \nof the SMR ratio (~10-5), is comparable to room temperature value for the 5 nm Ta sample. Despite some \nfluctuations, the overall trend for the SMR ratio is that it increases when the temperature and Ta thickness \ndecrease. The former might be due to the enhancement of spin diffusion length at low temperature, while \nthe latter confirms the nature of SMR as an interface effect. \nFig. 3(d) is the out -of-plane Hall resistance of Ta (10 nm)/YIG at temperatures from 4 K to 300 K \n(similar results are obtained for Ta (7 nm)/YIG which are not shown here). The shapes of the curve s are \nsimilar to that of the 5 nm Ta sample at room temperature. After subtracting the linear ordinary Hall effect \ncontribution, the Hall resistance ratio is shown in Fig. 3(g). The size (~10-6) is one order of magnitude \nsmaller than the SMR ratio, and has a similar temperature and thickness dependence as that of the SMR \nratio, indicating the same origin for both MR and Hall resistance. \nIV. SUMMARY \nIn conclusion, unconventional MR effect was observed in Ta/YIG bilayer with a comparable size \nto Pt/YIG. Our MR and Hall results show that the electron transport property in the Ta overlayer is greatly \ninfluenced by YIG’s magnetization, which could be explained by the SMR theory. Additionally, the \nobservation of a flip of polarity in the temperature dependent MR measurements suggests that MPE -\ninduced AMR is enhanced at low temperature. Our findings suggest that both MPE and SMR may \ncontribute to the unconventional MR effect, in particular at low temperatures. A theory combining the \nfeatures of MPE and SMR may be h elpful in fully understanding the experimental results observed in \nNM/YIG bilayers. \nACKNOWLEDGM ENTS \nThis work was supported by the National Research Foundation of Singapore (Grants No. NRF -G-CRP \n2007 -05 and R -263-000-501-281). The authors wish to acknowledge Wei Zhang from National University \nof Singapore for her assistance during sample preparation, and Wei Ji, Meysam Sharifzadeh Mirshekarloo \nfrom Institute of Materials Research and Engineering (IMRE) for their assistances during structural \ncharacterization. K. Y. and S. S. wish to acknowledge the support from IMRE under project number \nIMRE/10 -1C0107. \n \n7 \n \n \n \nREFER ENCES \n \n1 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). \n2 K. Uchida, T. An, Y. Kajiw ara, M. Toda, and E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011). \n3 M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. \n108, 176601 (2012). \n4 Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizu guchi, H. Umezawa, H. Kawai, K. \nAndo, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). \n5 K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. 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Potter, IEEE Trans. on Mag. 11 , 1048 (1975). \n \n8 \n \n \n \n \n \nFIGURE CAPTIONS: \nFIG. 1. (a) Grazing incidence angle XRD pattern for the unpatterned reference Ta (10 nm)/YIG/GGG sample; (b) in -\nplane and out -of-plane M -H loop for YIG/GGG substrate. Inset in (a): XRD pattern for YIG/GGG wafer with (111) \norientation; inset in (b): Superimposition of in-plane M -H loop and M -H loop obtained with a 90o in-plane rotation of \nthe sample with respect to its original position. \n \nFIG. 2. (a) - (b) Schematic configuration for MR and Hall measurements; MR and Hall resistance measured with field \napplied in different directions for Ta (5 nm)/YIG: MR in x-direction (c), Hall effect in x -direction (d), MR in y -\ndirection (e), Hall effect in y -directi on (f), MR in z -direction (g), Hall effect in z -direction (h). An offset resistance of \n3-4 mΩ is subtracted in Hall resistance. All measurements were performed at room temperature. \n FIG. 3. (a)- (b) Schematic configuration for temperature -dependent MR an d Hall measurements; (c) Temperature-\ndependence of MR with field applied in z direction for Ta (10 nm)/YIG; (d) Temperature -dependence of Hall \nresistance with field applied in z direction for Ta (10 nm)/YIG; (e) Schematic illustration for the change of MR origin \nat 4 K; (f) Temperature -dependence of SMR ratio for Ta (7 nm)/YIG and Ta (10 nm)/YIG; (g) Temperature -\ndependence of Hall resistance ratio for Ta (7 nm)/YIG and Ta (10 nm)/YIG. An offset resistance of 3 -4 m Ω i s \nsubtracted in Hall resistance and the MR and Hall curves are vertically shifted for clarity. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n9 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIG. 1 \n Journal of Applied Physics \n Yumeng Yang \nTa \nYIG \nGGG \nYIG \nGGG \n-3.0 -1.5 0.0 1.5 3.0-1.0-0.50.00.51.0\nHy\nHx\nHz\nHxM/Ms\nApplied Field (kOe)(b)\n-0.2 0.0 0.2-101 \n \n 35 40 45 50 550.000.030.060.09\nYIG (444)\nGGG (444)Ta (312)Intensity (a. u.)\n2θ (degree )Ta (510)(a)\n50.5 51.0 51.5101102103104105 \n \n \n10 \n \n \n \n \nFIG. 2 \nJournal of Applied Physics \nYumeng Yang \n \n \n \n (a) (b)\n(h) (g)(f) (e)(d) (c)I\nVHz\nHxHy\n-300 -150 0150 300-12-8-404\n0102030Hx\nApplied Field (Oe)Ta(5nm)/YIG\nTa(5nm)/YIG\nHyRxy (mΩ)\n-200 -100 0100 200-3-2-101\n-10123Hx\nApplied Field (Oe)Ta(5nm)/YIGTa(5nm)/YIG\nHy∆Rxx/R (×10−5)\n-500 -250 0250 500-3-2-101\nTa(5nm)/YIG\n Applied Field (Oe)∆Rxx/R (×10−5)HzHz\nI\nVHxHy\n-4 -2 0 2 4-120-60060120\n Applied Field (kOe)Rxy (mΩ)HzTa(5nm)/YIG\n11 \n \n \n \n \n \n \n \n \n \n \n \nFIG. 3 \nJournal of Applied Physics \nYumeng Yang \n (g)(f)(b)\nI\nVHz(e)\n(c)(a)\n0 100 200 300234567\nTa (10nm)/YIGTa (7nm)/YIGSMR Ratio (×10-5)\nTemperature (K)\n-10 -5 0 5102548254925502551\n150 K120 K100 K50 K\n80 K\n200 K10 K\n300 K20 KRxx (Ω)\nApplied Field (kOe)4 KHz\nI\nV\n-10 -5 0 510-1000-800-600-400-2000 20 K10 K\n150 K\n200 K\n300 K100 K\n120 K50 K4 KRxy (mΩ)\nApplied Field (kOe)80 K\n(d)\n0 100 200 30012345\nTa (10nm)/YIGTa (7nm)/YIGRxy Ratio (×10-6)\nTemperature (K)Polarized Ta\nYIGTaOMR +SMR AMR +SMR\nTa\nT: 300K 4 K\nYIG\n12 \n " }, { "title": "1609.01613v2.Chiral_charge_pumping_in_graphene_deposited_on_a_magnetic_insulator.pdf", "content": "Chiral charge pumping in graphene deposited on a magnetic insulator\nMichael Evelt,1Hector Ochoa,2Oleksandr Dzyapko,1Vladislav E. Demidov,1Avgust Yurgens,3Jie Sun,3\nYaroslav Tserkovnyak,2Vladimir Bessonov,4Anatoliy B. Rinkevich,4and Sergej O. Demokritov1;4\n1Institute for Applied Physics and Center for nanotechnology,\nUniversity of Muenster, 48149 Muenster, Germany\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n3Department of Microtechnology and Nanoscience-MC2,\nChalmers University of Technology, SE-41296, Gothenburg, Sweden\n4Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russia\nWe demonstrate that a sizable chiral charge pumping can be achieved at room temperature in\ngraphene/Yttrium Iron Garnet (YIG) bilayer systems. The e\u000bect, which cannot be attributed to\nthe ordinary spin pumping, reveals itself in the creation of a dc electric \feld/voltage in graphene as a\nresponse to the dynamic magnetic excitations (spin waves) in an adjacent out-of-plane magnetized\nYIG \flm. We show that the induced voltage changes its sign when the orientation of the static\nmagnetization is reversed, clearly indicating the broken spatial inversion symmetry in the studied\nsystem. The strength of e\u000bect shows a non-monotonous dependence on the spin-wave frequency, in\nagreement with the proposed theoretical model.\nI. INTRODUCTION\nThe generation of spin-polarized currents is of techno-\nlogical interest in order to transmit information encoded\nin the spin degrees of freedom. A spin current can be in-\njected into a nonmagnetic conductor by spin pumping.1,2\nIn this process, the reservoir of the angular momentum is\na ferromagnet where a ferromagnetic resonance (FMR)\nis excited by a microwave \feld. The spin pumping with-\nout net charge transport stems from the exchange of\nangular momentum between the itinerant spins in the\nmetal and the collective magnetization dynamics in the\nadjacent ferromagnet. The dynamical generation of spin\ncurrents is of special relevance since alternative meth-\nods based on driving an electrical current through the\ninterface are limited by the conductance mismatch.3The\nspin pumping-induced currents have been detected as the\nlong-ranged dynamic interaction between two ferromag-\nnets separated by a normal metal4,5or as a dc-voltage\nsignal in bilayer systems.6In the case of nonmagnetic\nmetals with a strong spin-orbit coupling, the spin cur-\nrent through the interface may engender a charge current\nalong the metallic layer as result of the inverse spin-Hall\ne\u000bect.7,8As the thickness of the metallic layer is reduced,\nadditional interfacial e\u000bects must be considered. The dif-\nferent contributions may be identi\fed by analyzing the\nsymmetries of the voltage signal with respect to di\u000ber-\nent adjustable parameters, such as the orientation of the\nsaturated magnetization controlled by the applied static\n\feld.9Additionally, we can use propagating spin waves\ninstead of FMR and vary the direction of propagation.\nAmong other materials, the unique candidate for inves-\ntigations of interface-induced phenomena is single layer\ngraphene (SLG). Due to its unique mechanical, optical,\nand electronic properties, graphene has attracted enor-\nmous attention since its discovery in 2004.10,11Nowa-\ndays, one can produce large-area high-quality SLG\nby using, e.g., chemical vapor deposition on metalcatalysts.12{14For the observation of spin pumping ef-\nfects, SLG should be brought in contact with a mag-\nnetic material. Yttrium Iron Garnet (YIG) holds a spe-\ncial place among all magnetic materials. Speci\fcally, it\nshows an unprecedentedly small magnetic damping re-\nsulting in the narrowest known line of the FMR and en-\nabling propagation of spin waves over long distances.15,16\nDue to these unique characteristics, YIG \flms have been\nrecently considered as a promising material for spintronic\nand magnonic applications.17,18By combining YIG and\nSLG in a bilayer, one obtains a unique model system for\ninvestigations of the spin-wave induced interfacial e\u000bects.\nHere, we report the experimental observation of a chi-\nral pumping e\u000bect in out-of-plane magnetized YIG/SLG\nbilayer and show that spin waves propagating in the YIG\n\flm induce a sizable electric voltage in the adjacent SLG.\nWe study the symmetry of the observed phenomenon by\nreversing the static magnetic \feld and the direction of\nspin-wave propagation and \fnd that the corresponding\nchanges in the sign of the electric voltage clearly indicate\na broken re\rection symmetry about the plane normal to\nthe interface. We associate this symmetry breaking with\nthe presence of screw dislocations in the crystalline lat-\ntice of YIG, which are expected to have a strong e\u000bect\non the two-dimensional electron gas in SLG. We present\na theoretical model, which shows that such crystalline\ndefects can result in a chiral pumping e\u000bect with the\nobserved symmetry. It is worth mentioning that the\nconventional spin pumping-induced voltages obtained for\nin-plane magnetized \flms19{21are rooted in the re\rec-\ntion symmetry breaking about the plane of the interface.\nThey present di\u000berent symmetries and are therefore unre-\nlated to the signal reported here. Moreover, in agreement\nwith the experiment, our model predicts a non-monotonic\ndependence of the induced voltage on the spin-wave fre-\nquency.\nThe structure of the manuscript is the following: We\npresent the microwave and voltage measurements in\nSec. II. Special attention is paid to the inversion sym-arXiv:1609.01613v2 [cond-mat.mes-hall] 7 Sep 20162\nmetry breaking, con\frmed by further FMR experiments\nin YIG, revealing the chiral nature of the e\u000bect. The the-\noretical model is discussed in Sec. III. Technical details of\nthe model are set-aside in the Appendix. We summarize\nour \fndings in Sec. IV.\nII. FERROMAGNETIC RESONANCE\nEXPERIMENTS\nA. Sample preparation and characterization\nThe experimental layout is shown in Fig. 1(a).\nGraphene was placed on top of a monocrystalline YIG\n\flm of 5.1\u0016m thickness grown by means of liquid phase\nepitaxy on 0.5 mm thick gallium gadolinium garnet\n(GGG) substrate. The saturation magnetization of the\nYIG \flm de\fning the saturation \feld in the out-of-plane\ngeometry was 4 \u0019MS= 1:75 kG. The SLG sample was\ngrown on a 50 \u0016m thick 99.99% pure copper foils in a\ncold-wall low-pressure CVD reactor (Black Magic, AIX-\nTRON). After the Cu foil had been cleaned in acetone\nand then shortly etched in acetic acid to remove surface\noxide, it was placed on a graphitic heater inside the reac-\ntor and annealed at 1000\u000eC during 5 min in a \row of 20\nsccm H 2and 1000 sccm Ar. Then, a \row of pre-diluted\nCH4(5% in Ar, 30 sccm) was introduced during 5 min to\ninitiate the growth of graphene. After the growth, the gas\n\row was shut down, the system evacuated to <0:1 mbar\nand cooled down to room temperature.22SLG was trans-\nferred to YIG/GGG by using the bubbling delamina-\ntion technique23,24and lithographically patterned into a\nmulti-terminal Hall-bar structure with Au(100 nm)/Cr(5\nnm) pads at the edges. The lateral dimensions of the\nsample were 1.5 by 25 mm. The Hall-e\u000bect mobility of\nthe resulting devices was found to be in the range from\n600 to 1300 cm2(V\u0001s)\u00001at T=10 K. The local resistance\nmaximum at zero magnetic \feld H0\u00190, usually ascribed\nto the weak localization e\u000bect, allowed for an estimation\nof the phase coherence length L\u001e\u0018100 nm at the same\ntemperature.\nB. Microwave and voltage measurements\nThe YIG/SLG sample was built onto a massive metal\nholder whose temperature was controlled and kept con-\nstant at 295\u00060:02 K. The holder with the sample was\nplaced between the poles of an electromagnet, which cre-\nates a static magnetic \feld H0with the inhomogeneity\nbelow 2\u000210\u00005over 1 mm3. In all described experi-\nments we kept H0>2 kOe ensuring the collinear ori-\nentation of H0andMS. To excite spin waves in the\nYIG layer, broadband microstrip lines ending with 50\n\u0016m wide antennas were used, which were attached to the\nsample. Microwave (MW) \feld with a \fxed frequency in\nthe rangef= 3\u00008 GHz was applied to the antennas.\nA directional coupler was used to monitor the re\rected\nb)\nc)a)FIG. 1: Chiral charge pumping in YIG/SLG. a) Schematic of\nthe experiment. Single layer graphene (SLG) is transferred\nto yttrium-iron-garnet (YIG) \flm grown on gallium gadolin-\nium garnet (GGG) substrate. A, B, C, D are gold contact\npatches for electric measurements. The system is magnet-\nically saturated normally to the sample plane by means of\nan applied magnetic \feld H0. Spin waves are excited by mi-\ncrowave (MW) current \rowing in one of the two narrow an-\ntennae, producing a local MW \feld. b) Field dependence of\nthe MW absorption for the up(H+) and down (H\u0000) orien-\ntation of the static \feld. Di\u000berent peaks on the dependences\nindicate excitation of di\u000berent non-uniform spin-wave modes.\nNote the similarity of the two curves. c) Field dependence of\nthe dc-voltage detected between the pads A and B for the up\n(H+) and down (H\u0000) orientation of the applied \feld. Note\ndi\u000berent signs of the voltages for the two orientation of the\n\feld. The red curves in (b) and (c) are shifted vertically for\nthe sake of clarity.\nMW power, allowing for the determination of the reso-\nnance absorption in the sample due to the excitation of\nthe spin waves. By keeping fconstant and varying H0,\nwe recorded Pabs(H0) { curves characterizing the \feld-\ndependent excitation of spin waves in YIG, see Fig. 1 b).\nThe curves exhibit several maxima corresponding to the\nFMR, higher-order standing, as well as propagating spin3\nFIG. 2: Symmetry properties of the observed e\u000bect. Both the induced electric \feld Eand the wavevector of the spin wave\nksware in the plane of the sample, while the applied \feld H0and the YIG-magnetization MSare oriented normally to the\nsample plane. The + and \u0000signs indicate the polarity of the measured dc-voltage. Rotation of the system by 180\u000earound\nthez-axis reverses the directions of both Eandkswin agreement with the experiment. Mirroring with respect to the vertical\nplane (y!\u0000y) does not change Eandksw, whileH0andMSare reversed, in contrast to the experimental \fndings.\nwaves in YIG.\nSimultaneously with the microwave measurements, we\nrecorded the dc-voltage between the electrodes. A lock-\nin technique was applied in order to detect the dc-voltage\ncaused by the spin waves propagating in the YIG \flm.\nMicrowaves were modulated by a square wave with the\nrepetition frequency of 10 kHz and the voltage di\u000ber-\nence between di\u000berent leads was measured by the lock-\nin ampli\fer. During the measurements, the static mag-\nnetic \feld was swept and the dc-voltage along with the\nre\rected MW power signal as a function of H0were\nrecorded. Figure 1(c) shows a typical dependence of\nVAB(H0) { the voltage between the electrodes A and\nB. Comparing Figs. 1(b) and 1(c), one sees that the\ndc-voltage in SLG is induced at the same values of H0,\nwhere the e\u000ecient excitation of spin waves takes place.\nThis clearly indicates its direct connection to the high-\nfrequency magnetization precession in YIG.\nC. Symmetry breaking\nThe most striking feature of the observed phenomenon\nis its unusual symmetry with respect to the inversion of\nH0. As seen from Fig. 1(c), the detected voltage changes\nits sign if the orientation of H0(andMS) is reversed.\nWe emphasize that this inversion is not accompanied by\nvisible changes in the microwave absorption curve (Fig.1(b)). Further measurements show that the induced volt-\nage is the same for both sides of the sample, VAB=VCD.\nIt is proportional to the intensity of the spin wave (the\nabsorbed microwave power) and changes sign if the di-\nrection of the spin-wave propagation is reversed.\nTo obtain better insight into the underlying physics,\nwe consider the symmetry of the phenomenon in more\ndetail. As shown in Fig. 2, the induced electric \feld E\nis oriented in the sample plane parallel to the direction\nof the spin-wave propagation indicated in Fig. 2 by a\nvector ksw, whereas the magnetization/magnetic \feld is\nperpendicular to the sample plane. The + and \u0000signs in-\ndicate the polarity of the measured dc-voltage at a given\norientation of the magnetization. As seen in Fig. 2, rota-\ntion by 180\u000earound the normal to the \flm plane reverses\nboth the direction of the spin-wave propagation and that\nof the electric \feld, whereas the direction of the magneti-\nzation stays unchanged, all in agreement with the exper-\nimental \fndings. However, mirroring of the system with\nrespect to the vertical plane parallel to the direction of\nthe spin-wave propagation ( y!\u0000y) does not change nei-\nther the direction of the spin-wave propagation nor that\nof the electric \feld, since they are real vectors, whereas\nthe magnetization, which is an axial vector, is reversed.\nWe emphasize that this contradicts to the experiment\n(see Fig. 1 c)). In other words, the phenomenon respon-\nsible for the appearance of the voltage in the YIG/SLG\nbilayer requires that the inversion symmetry is broken.4\nMore speci\fcally, the observed e\u000bect would be forbidden\nif the clockwise and counter-clockwise directions in the\nsample plane were equivalent, which reveals its chiral na-\nture.\nTo check whether the observed non-equivalence is a\ncharacteristic feature of the YIG \flm itself, we performed\nhigh-precision FMR measurements on YIG \flms of small\nlateral dimensions (5.1 \u0016m thick circle with the 0.5 mm\nradius and 6 \u0016m thick 0:5\u00020:5 mm2square) without\nSLG. In these experiments, the FMR in the YIG sample\nwas excited by a quasi-uniform dynamic magnetic \feld.\nThe entire test device was mounted on a non-magnetic ro-\ntatable sample holder, which was placed in a static mag-\nnetic \feld with a homogeneity better than 0.1 Oe, with\nthex-axis being the rotation axis of the holder. Note here\nthat the actual rotation axis (the x-axis) in the described\nexperiments di\u000bers from the rotation axis of the thought\nexperiment (the z-axis) shown in Fig. 2. The sample\nholder was designed to keep the position of the sample\nconstant with an accuracy of 0.2 mm over the entire ro-\ntation. By sweeping the microwave frequency at a \fxed\nmagnetic \feld, the transmission coe\u000ecient T=Pout=Pin\nwas measured as a function of the frequency. Then, the\nholder with the sample was rotated by 180\u000e, and the\nmeasurements were repeated. The results of the mea-\nsurements for the two orientations of the sample were\naveraged over several measurement cycles. The obtained\nFMR curves for the directions of the static magnetic \feld\nparallel (H+) and antiparallel ( H\u0000) to the normal to the\n\flm surface demonstrating a clear frequency di\u000berence of\n\u0001f= 5 MHz are shown in Fig. 3 (a). The value of \u0001 f\nis signi\fcantly larger compared to the estimated uncer-\ntainty of 0.5 MHz originating from the inhomogeneity of\nthe static \feld and the stray \felds of the sample holder.\nThe measurements were repeated for di\u000berent values of\nthe applied magnetic \feld resulting in the \feld depen-\ndence of \u0001fshown in Fig. 3 (b). One clearly sees that \u0001 f\nsystematically increases as H0approaches 4 \u0019MS= 1:75\nkG. Qualitatively similar results were obtained for di\u000ber-\nent samples except that the maximum values \u0001 fwere\nfound to vary from 6 to 13 MHz.\nThe results presented in Fig. 3 show that the frequency\nof the FMR in YIG \flms depends on whether the static\nmagnetic \feld is parallel or antiparallel to the normal to\nthe \flm surface ^z, i.e. it depends on the direction (clock-\nwise or counter-clockwise with respect to ^z) of the mag-\nnetization precession. This indicates that the inversion\nsymmetry is broken, since the clockwise and the counter-\nclockwise directions in the plane of the \flm are not equiv-\nalent. Although the microscopic origin of this breaking\nis not yet fully clear, it should be connected with the de-\nfects in the crystallographic structure of YIG, since the\nideal high-symmetry cubic structure of YIG is de\fnitely\nincompatible with this symmetry breaking. It is known\nthat the dominating defects in high-quality epitaxial YIG\n\flms are growth dislocations. Their typical lateral den-\nsity of about 108cm\u00002is connected with the typical mis-\n\ft between YIG and GGG lattices of 10\u00003(Refs. 15,25).\nFIG. 3: Ferromagnetic resonance (FMR) in a YIG \flm. a)\nFMR curves measured at a constant H0for the up(H+) and\ndown (H\u0000) orientation of the \feld. Note a di\u000berence in the\nFMR frequencies \u0001 f. b) Field dependence of \u0001 f. The dash\nline is a guide for the eye.\nIt is also known that such dislocations strongly in\ru-\nence the magnetic dynamics in YIG.26Possible mech-\nanisms of the defect-mediated symmetry breaking can\ninclude a growth-induced misbalance between clockwise\nand counter-clockwise screw dislocations and e\u000bects sim-\nilar to the antisymmetric surface Dzyaloshinskii-Moriya-\nlike interactions27in combination with a dislocation.\nWe emphasize that, although the symmetry breaking\nis present in YIG \flms without SLG on top, its in\ruence\non the magnetization dynamics is very small and can\nonly be detected in high-precision measurements. This\nis not surprising, since the symmetry breaking appears\nto be a surface phenomenon, which has vanishing in\ru-\nence on the magnetization in the bulk of the \flm. On the\ncontrary, this symmetry breaking is expected to have sig-\nni\fcant in\ruence on conduction electrons in SLG placed\non the surface of the YIG \flm.\nIII. PHENOMENOLOGICAL MODEL\nNext, we provide a phenomenological model for the ob-\nserved e\u000bect based on the existence of a \fnite density of\nscrew dislocations in YIG. The voltage signal is related\nwith the electromotive forces induced by the magnetiza-\ntion dynamics. A screw dislocation creates a distortion\nof the graphene lattice as shown in Fig. 4 (a), which cou-\nples to the electron spin through the spin-orbit interac-\ntion. As a result, the exchange \feld seen by the itinerant\nelectrons is tilted with respect to the magnetization in\nYIG, Fig. 4 (b), modifying the longitudinal response in\nthe non-adiabatic regime. We emphasize that this mech-\nanism must be taken as a suggestive explanation for the\nobserved phenomenon since there is no direct observation\nof the proposed mechanical deformations.\nA. Hamiltonian\nWe assume that the exchange interaction couples the\nspins of itinerant electrons of SLG to the localized mag-5\n)b )a\nFIG. 4: Distorted SLG near a dislocation in a YIG-\flm. a)\nTop view of the distortion induced by a screw dislocation\n(the dislocation line is represented by the black dot). The\nred arrows illustrated the in-plane component of the e\u000bective\nexchange \feld. b) Isometric illustration of the tilt of the ef-\nfective exchange \feld (red arrows) along a path encircling the\ndislocation line (black arrow). The original \feld points along\nthe normal to the sample plane, and the tilting is generated\nby the spin-lattice coupling.\nnetic moments in YIG. The Hamiltonian reads as\nH=~vF\u0006\u0001p+ \u0001 exm(t;r)\u0001s+Hs-l: (1)\nThe \frst term is the Dirac Hamiltonian describing elec-\ntronic states with momentum paround the two inequiv-\nalent valleys K\u0006in graphene, where \u0006= (\u0006\u001bx;\u001by) is a\nvector of Pauli matrices associated to the sub-lattice de-\ngree of freedom of the wave function and vFis the Fermi\nvelocity. The second term corresponds to the exchange\ncoupling, \u0001 ex, where s= (sx;sy;sz) are the itinerant\nspin operators and m(t;r) is a unit vector along the lo-\ncal magnetization in YIG. The last term is a spin-lattice\ninteraction that couples the mechanical degrees of free-\ndom with the spins of electrons.\nAs it is well known,11graphene can support large me-\nchanical distortions that change dramatically the dynam-\nics of Dirac electrons. When an SLG is placed on top of\ndisturbed YIG, the carbon atoms near a screw disloca-\ntion can be expected to follow its distorted pro\fle, which\nintroduces a torsion in the honeycomb lattice, as repre-\nsented in Fig. 4 (a). This deformation couples to the\nelectron spin as\nHs-l= \u0001 so(@xuy\u0000@yux)(\u001bxsx+\u001bysy); (2)\nwhere@xuy\u0000@yuxis the anti-symmetrized strain tensor\n{u= (ux;uy) represents the displacements of the carbon\natoms { that parameterizes the torsion of the graphene\nlattice and can on average be related to the density of\nscrew dislocations. The form of this coupling is dictated\nby the C 6vsymmetry of the honeycomb lattice, and its\nmicroscopic origin resides in the hybridization of \u0019and\u001b\norbitals of carbon atoms, which enhances the spin-orbit\ne\u000bects within the low-energy bands.28Note here that the\nproposed deformation just models the observed symme-\ntry breaking to provide a proof of principle for the chiral\npumping e\u000bect. We expect our model to be qualitatively\ncorrect regardless of the actual deformations of the lattice\nclose to dislocations.The e\u000bective exchange \feld seen by itinerant electrons,\nn(t;r), is tilted away from the local magnetization fol-\nlowing the torsion of the lattice, as illustrated in Fig.\n4b. This is implemented by the spin-lattice coupling in\nEq. (2), which can be understood on geometry grounds\nas the non-abelian connection that transports the spin\nquantization axis along the distorted graphene lattice.\nThe details are provided in the Appendix. To the lowest\norder in \u0001 so, we obtain\nn(t;r)\u0019m(t;r)\u00002a\u0001so\n~vFb(r)\u0002m(t;r);(3)\nwhere b(r) is de\fned as\nb(r) =a\u00001Zr\nr0dr0(@xuy\u0000@yux): (4)\nHere r0is the origin of the dislocation and we have in-\ntroduced a microscopic length scale ain order to make\nb(r) dimensionless.\nB. Electromotive forces\nThe magnetization dynamics induces electromotive\nforces in SLG along the direction of propagation of spin\nwaves. At low frequencies, majority electrons along the\nquantization axis de\fned by n(t;r) experience an e\u000bec-\ntive electric \feld of the form29{31\nEi=~\n2e_n\u0001(n\u0002@in+\f@in): (5)\nThe \frst term is purely geometrical, strictly valid in the\nlimit \u0001 ex\u001d~j_nj,~vFj@inj, when the electron spins fol-\nlow adiabatically the direction of e\u000bective exchange \feld.\nThe\fcorrection is related to a slight misalignment due\nto spin relaxation caused by the spin-orbit interaction.\nThe electromotive force in the adiabatic limit, which\nis rooted in the associated geometrical Berry phase, re-\nspects the structural symmetries of the device. Indeed,\nthe correction due to the magnetization tilting expressed\nin Eq. (3) averages to 0 when integrated upon the period\nof the spin wave excitation. The introduction of screw\ndislocations in YIG breaks the structural symmetries of\nthe transport signals in SLG owing only to non-adiabatic\ncorrections to the electromotive force. These appear as\nhigher order expansions in~\n\u0001ex_nand spatial gradients\n~vF\n\u0001ex@inthat capture spin-orbital e\u000bects. The new terms\ncompatible with C 6vsymmetry read\nEi=~3vF\n2e\u00012ex(\f1@in+\f2n\u0002@in)\u0001^z\u0002r(_n)2;(6)\nwhere\f1;2are dimensionless phenomenological parame-\nters. The tilting of the exchange \feld translates the e\u000bect\nof the symmetry breaking in YIG into the electronic re-\nsponse of graphene, driving a voltage along the direction6\n02 4 6 8 10-0.20.00.20.40.60.81.06.2GHz5\n.8GHz5\n.0GHz8\n.0GHzV(µV)P\nabs (mW)a)4\n.0GHz4\n5 6 7 8 0.00.10.20.3V / Pabs ( µV / mW )F\nrequency (GHz)b)\nFIG. 5: Frequency dependence of the e\u000bect. a) The detected\ndc-voltage as a function of the absorbed MW-power for dif-\nferent frequencies as indicated. The dash lines are linear \fts\nof the experimental data used for calculation of the e\u000eciency\ncoe\u000ecientV=P abs. b) The e\u000eciency coe\u000ecient V=P absas a\nfunction of the spin-wave frequency. Note that a sizable ef-\nfect is observed between 5 and 7 GHz only. The dash line is\na guide for the eye.\nof propagation of the spin waves of the form32\nV=\f1~2\u0001so\n\u00012exZ\ndx(@xuy\u0000@yux)^z\u0001m@x(_m)2:(7)\nAs seen from this equation, the proposed theoretical\nmodel reproduces the experimentally observed inversion\nof the sign of the induced voltage accompanying the\nreversal of the magnetization ( ^z\u0001m! \u0000 ^z\u0001mwhen\nH0!\u0000H0) or the direction of the spin-wave propaga-\ntion (@x(_m)2!\u0000@x(_m)2whenksw!\u0000ksw).\nThe contribution in Eq. (7) should saturate when the\ntheory crosses over to the strongly non-adiabatic regime,\nat frequencies of the order of \u0001 ex=~. We expect for the\nelectromotive force to be suppressed at larger frequencies\ndue to a dynamic averaging e\u000bect, similarly to the mo-\ntional narrowing in the spin di\u000busion problem: the mag-\nnetization precession is so fast that its time-dependent\ncomponent e\u000bectively averages out from the view of elec-\ntron spins.\nC. Frequency dependence\nThe model predicts that the e\u000bect should exist only\nwithin a certain range of spin-wave frequencies around\n\u0001ex=~. Although \u0001 excannot be directly measured in\nthe experiment, the theoretical prediction can be veri\fed\nby analyzing the frequency dependence of the observed\ne\u000bect. Figure 5(a) shows the dependences of the voltage\nonPabsproportional to the intensity of the excited spin\nwaves, recorded for di\u000berent spin-wave frequencies. As\nseen from Fig. 5(a), the voltage is linearly proportionalto the spin-wave intensity and the proportionality coef-\n\fcientV=Pabsis indeed strongly frequency dependent.\nMoreover, in agreement with the theoretical results, the\nfrequency dependence of V=Pabs(Fig. 5(b)) clearly ex-\nhibits a resonance-like behavior with the maximum at\nabout 6 GHz. The relatively small exchange constant\n\u0001ex=~\u001930 eV obtained from the observed resonant fre-\nquency nicely agrees with the recently determined ex-\nchange \feld of 0.2 T.33\nIV. CONCLUSIONS\nIn conclusion, we experimentally observe a chiral\ncharge pumping in YIG/SLG bilayers caused by the sym-\nmetry breaking at the YIG surface. The e\u000bect provides a\nnovel, e\u000ecient mechanism for detection of magnetization\ndynamics for spintronic and magnonic applications. The\ndeveloped theoretical model, taking into account the ex-\nchange interaction between localized magnetic moments\nin YIG and itinerant spins in graphene, predicts that the\nstrength of the e\u000bect can be increased by making use\nof two-dimensional conductors with a strong spin-orbital\ncoupling, such as MoS 2.34Additionally, the found sym-\nmetry breaking can result in other, not yet observed,\ne\u000bects. In particular, it can enable the reciprocal ef-\nfect consisting in the excitation of spin-waves of unusual\nsymmetry via an electric current in graphene. These ef-\nfects provide essentially new opportunities for the direct\nelectrical detection and manipulation of magnetization in\ninsulating magnetic materials and open new horizons for\nthe emerging technologies.\nAcknowledgements\nThis work was supported in part by the Deutsche\nForschungsgemeinschaft, the Swedish Research Coun-\ncil, the Swedish Foundation for Strategic Research, the\nChalmers Area of Advance Nano, the Knut and Alice\nWallenberg Foundation, the U.S. Department of Energy,\nO\u000ece of Basic Energy Sciences under Award No. DE-\nSC0012190 (H.O. and Y.T.), and the program Megagrant\nNo. 14.Z50.31.0025 of the Russian Ministry of Education\nand Science. J. S. thanks the support of STINT, SOEB,\nand CTS. S.O.D is thankful to L.A. Melnikovsky for fruit-\nful discussions and to R. Sch afer for domain mapping in\nYIG.\nAppendix: Tilting of the exchange \feld\nThe spin rotational symmetry of graphene Hamilto-\nnian is expressly broken by the spin-lattice coupling in\nEq. (2), but it can be approximately restored in certain\nlimit by means of a local unitary rotation of the spinor7\nwave function. Such a gauge transformation reads as\nU= Exp [\u0000i\u0010b(r)\u0001s]\u0019I\u0000i\u0010b(r)\u0001s; (A.1)\nwhere b(r) is an axial vector, de\fned in Eq. (4), and \u0010is\na small, dimensionless parameter that we identify with\n\u0010=a\u0001so\n~vF\u001c1: (A.2)\nNotice that there is a gauge ambiguity in the de\fnition of\nU. In Eq. (4), the gauge is \fxed by choosing the origin of\nthe dislocation as the lower limit of integration, but this\nis arbitrary. As a result, physical e\u000bects induced by the\ntilting of the e\u000bective exchange \feld appear as derivatives\nofb(r), making this approach fully consistent.\nNext, we see that the unitary transformation in\nEq. (A.1) gauges away the last term in the Hamiltonian\nof Eq. (1), generating new terms that are second order\nin\u0010and we can safely neglect. First, we have that\nUyHs-lU\u0019H s-l\u0000i\u0010[Hs-l;b\u0001s] +O\u0000\n\u00102\u0001\n; (A.3)\nbut note that the strength of Hs\u0000lis already \frst order\nin\u0010and therefore the second term in this last equation\nis actually a second order term; then, we can write\nUyHs-lU\u0019H s-l+O\u0000\n\u00102\u0001\n: (A.4)\nOn the other hand, if we apply the unitary transforma-\ntion to the kinetic term in Eq. (1) we obtain\n\u0000i~vFUy\u0006\u0001@U\u0019\u0000~vF\na\u0010(@xuy\u0000@yux) (\u0006xsx+ \u0006ysy)\n+O\u0000\n\u00102\u0001\n=\u0000Hs-l+O\u0000\n\u00102\u0001\n: (A.5)\nThen, the subsequent gauge \feld cancels out the spin-\nlattice coupling. To the leading in \u0010\u001c1 (therefore, inthe strength of the interaction), the spin-lattice coupling\nin Eq. 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B 77,\n134407 (2008).\n32By plugging Eq. (3) into Eq. (6), the \f1term generates the\nlongitudinal voltage in Eq. (7), whereas the \f2term leads\nto a transverse voltage. The latter is even in the applied\nmagnetic \feld, in agreement with symmetry arguments.\nThis prediction is in fact supported by preliminary exper-\niments. A careful study of the induced transverse voltage\nwill be a topic our further investigations.\n33C. Leutenantsmeyer, A. A. Kaverzin, M. Wojtaszek, and\nB. J. van Wees, arXiv:1601.00995 [cond-mat.mes-hall].\n34K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz,\nPhys. Rev. Lett. 105, 36805 (2010).\n35The action of the SO(3) generators over a vector vsatisfy\nthe following property: ( u\u0001`)v= 2iu\u0002v." }, { "title": "1512.07773v5.Ultra_High_Cooperativity_Interactions_between_Magnons_and_Resonant_Photons_in_a_YIG_sphere.pdf", "content": "Ultra-High Cooperativity Interactions between Magnons and Resonant Photons in a\nYIG sphere\nJ. Bourhill,1,\u0003N. Kostylev,1M. Goryachev,1D. L. Creedon,1and M.E. Tobar1\n1ARC Centre of Excellence for Engineered Quantum Systems,\nUniversity of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia\n(Dated: September 6, 2018)\nResonant photon modes of a 5mm diameter YIG sphere loaded in a cylindrical cavity in the 10-\n30GHz frequency range are characterised as a function of applied DC magnetic \feld at millikelvin\ntemperatures. The photon modes are con\fned mainly to the sphere, and exhibited large mode \flling\nfactors in comparison to previous experiments, allowing ultrastrong coupling with the magnon spin\nwave resonances. The largest observed coupling between photons and magnons is 2g=2\u0019= 7:11GHz\nfor a 15.5 GHz mode, corresponding to a cooperativity of C= 1:51\u00060:47\u0002107. Complex modi\fcations\nbeyond a simple multi-oscillator model, of the photon mode frequencies were observed between 0 and\n0.1 Tesla. Between 0.4 to 1 Tesla, degenerate resonant photon modes were observed to interact with\nmagnon spin wave resonances with di\u000berent couplings strengths, indicating time reversal symmetry\nbreaking due to the gyrotropic permeability of YIG. Bare dielectric resonator mode frequencies were\ndetermined by detuning magnon modes to signi\fcantly higher frequencies with strong magnetic \felds.\nBy comparing measured mode frequencies at 7 Tesla with Finite Element modelling, a bare dielectric\npermittivity of 15:96\u00060:02of the YIG crystal has been determined at about 20mK.\nINTRODUCTION\nHybrid photon-magnon systems in ferromagnetic\nspheres have recently emerged as a promising approach\ntowards coherent information processing [1{8]. Due to\nthe large exchange interaction between spins in ferromag-\nnets, they will lock together to form a macrospin that can\nbe utilised for coherent information processing protocols\n[8, 9]. The quantised excitation of the collective spin is\nreferred to as a magnon. Yttrium Iron Garnet (YIG)\nbased magnon systems are attractive due to very high\nspin density, resulting in signi\fcant cooperativity as well\nas relatively narrow linewidths [4, 5, 10, 11]. Further-\nmore, due to the possibility of coupling magnon modes to\nphotons at optical frequencies [10{13], magnon systems\nmay be considered as a candidate for coherent conver-\nsion of microwave and optical photons [10, 11]. In addi-\ntion, magnons interact with elastic waves [14, 15] open-\ning a window for combining mechanical and magnetic\nsystems. These systems therefore possesses great poten-\ntial as an information transducer that mediates inter-\nconversion between information carriers of di\u000berent phys-\nical nature thus establishing a novel approach to hybrid\nquantum systems [9, 16{18].\nAmong all magnon systems the central role is de-\nvoted to YIG, a material that possesses exceptional mag-\nnetic and microwave properties and has been used in\nmicrowave systems such as tuneable oscillators and \fl-\nters for many decades [19, 20]. Although, only recently\nSoykal and Flatt\u0013 e proposed and modelled the photon-\nmagnon interaction based on YIG nano-spheres with ap-\nplication to quantum systems [21, 22]. As predicted by\nthe authors, extremely large coupling rates, g, could be\nachieved in YIG spheres, which is favourable for coher-\nent information exchange and has been demonstrated ex-perimentally later [4, 5, 23]. For these experiments, the\ninteraction is observed between photon and magnon reso-\nnances created correspondingly by photon cavity bound-\nary conditions and spin precession under external DC\nmagnetic \feld. A commonly used method is to place a\nrelatively small YIG sphere in a local maxima of the mag-\nnetic \feld inside a much larger microwave cavity. This is\ndone to achieve quasi uniform distribution of the cavity\n\feld over the sphere volume to avoid spurious magnon\nmodes. Cavities can take oval [3, 4] or spherical shapes\n[21, 22, 24], and even re-entrant cavities with multiple\nposts have been used in an attempt to focus the mi-\ncrowave energy over the sphere [5, 23]. In this work\nwe investigate a completely di\u000berent regime in which the\nmagnon and photon wavelengths are comparable, lead-\ning to considerably larger coupling strengths, but addi-\ntional couplings to higher order modes. In general, for\nthis case the strength of the photon-magnon interaction\nwill be determined by an overlap integral of the two re-\nspective mode shape functions. Given the magnon mode\nshape is limited to the sphere's volume, this integral will\nbe maximised when the photon mode is con\fned to the\nsame volume. To achieve the latter, we utilise an ex-\nceptionally large YIG sphere with diameter d= 5 mm,\nmatching magnon and microwave photon mode volumes,\nunlike previous microwave cavity experiments.\nIn order to investigate this regime we use common mi-\ncrowave spectroscopy techniques [5, 23, 25{28] to directly\nobserve the mode splitting caused by the magnon inter-\naction to determine the coupling values. Similar systems\nhave been extensively utilised not only in the \feld of spin-\ntronics to investigate the interaction between microwave\nphotons and paramagnetic spin ensembles [25{29], but\nalso to realise optical comb generation [30], ultra-low\nthreshold lasing [31], cavity-assisted cooling, control andarXiv:1512.07773v5 [quant-ph] 26 Apr 20162\n573061.38YIG sphereSapphiresubstrateMicrowavecoaxial cableSMA connectorMagnetic loopprobeCopper cavityzx\nApplied Magnetic Field\nFIG. 1: (Color online) Cross section of the copper cavity\nthat houses the 5 mm YIG sphere. The sphere sits on a\nsapphire disk, and microwaves are coupled in and out of it\nvia loop probes which produce and detect an H\u001ecomponent.\nA variable DC magnetic \feld is applied along the zaxis.\nmeasurement of optomechanical systems [32, 33], and ex-\ntremely stable cryogenic sapphire oscillator clock technol-\nogy [34, 35]. To date, exciting internal, highly-con\fned\nphotonic modes in a YIG sphere has only recently been\ndemonstrated in the optical regime using Whispering\nGallery Modes (WGM) [10, 11] but has never before been\nachieved in the microwave domain. This is due to the\ntypical sub-millimetre diameter of the spheres. As such,\ninteractions with magnons must be observed via Bril-\nlouin scattering [36], which has yielded high quality fac-\ntors and also demonstrated a pronounced nonreciprocity\nand asymmetry in the sideband signals generated by the\nmagnon-induced scattering.\nExtremely large mode splittings ( g=! > 0:1) cause si-\nmultaneous coupling to a higher density of modes, with\nan overlap of avoided level crossings. Therefore, the\nmodel proposed by Soykal and Flatt\u0013 e [21] becomes no\nlonger applicable, as it assumes the interaction occurs\nbetween a single photonic and magnon mode. More re-\ncently, a paper by Rameshti et al. [24] simulated a sim-\nilar scenario of the presented experiment, in which the\nferromagnetic sphere is itself the microwave cavity. Our\nobserved results may appear to be in good agreement\nwith this work's predictions, however, what is apparent is\nthat in this specialised case, one must consider more than\njust the magnetostatic, uniform Kittel magnon mode, a\nlimitation of [24]. Indeed, due to the nonuniformity of\nboth the microwave mode magnetic \feld energy density\nacross the sphere, which is unique to this experiment, and\nthe nonuniformity of the sphere parameters arising due\nto cryogenic cooling, the assumption that only the uni-\nform Kittel magnon resonance participates is no longer\nvalid. Despite this, in this paper we use a two mode\nmodel to obtain estimations of coupling strengths, and\ndemonstrate how this results in inconsistent susceptibil-\nity values.\nFrequency (GHz)\nMagnetic Field (T)00.20.40.60.811.21015202530\n101520\n5Normalised S21 (dB)\n02530FIG. 2: (Color online) Transmission data as magnetic \feld\nis swept.\nPHYSICAL REALIZATION\nThed= 5 mm YIG sphere was manufactured by Ferri-\nsphere, Inc. with a quoted room temperature saturation\nmagnetisation of \u00160M= 0:178 T. It is placed on a small\nsapphire disk, with a concavity etched out using a dia-\nmond tipped ball grinder, to keep the sphere from rolling\nout of position, and reduce dielectric losses that would\narise if the YIG were in direct contact with the conduc-\ntive copper housing. Sapphire was chosen over te\ron as\nan intermediary between the YIG and copper to improve\nthe thermal conductivity to the sphere.\nTogether, the sapphire and YIG are housed inside a\ncopper cavity with dimensions speci\fed in \fgure 1. A\nloop probe constructed from \rexible subminiature ver-\nsion A cable launchers is used to input microwaves and\na second is used to make measurements, allowing the de-\ntermination of Sparameters. The entire cavity is cooled\nto about 20 mK by means of a dilution refrigerator (DR)\nwith a cooling power of about 500 \u0016W at 100 mK. The\ncavity is attached to a copper rod bolted to the mixing\nchamber stage of the DR that places it at the \feld center\nof a 7 T superconducting magnet, whose applied \feld is\noriented in the zdirection of the cavity. The magnet is\nattached to the 4 K stage of the DR, with the copper cav-\nity mounted within a radiation shield of approximately\n100 mK that sits within the bore of the magnet.\nEXPERIMENTAL OBSERVATIONS\nThe transmission spectrum of the YIG was recorded\nfor DC magnetic \felds swept from 0 { 7 T using a vec-\ntor network analyser (VNA), with partial results shown\nin \fgure 2. A host of magnon resonances/higher or-3\nder magnon-polaritons can be observed originating from\n(0 T, 0 GHz) with an approximate gradient of 28\nGHz/T. The more-or-less horizontal lines approaching\nthe magnon resonances from either side correspond to\nresonant photon modes of the sphere. Importantly, we\ncan observe that in the dispersive regime, far removed\nfrom any microwave resonant mode, there still exist mul-\ntiple magnon modes. We observe that the anticrossing\ngaps are populated by unperturbed modes, which are\nremnant \\tails\" of both \\higher\" and \\lower\" mode in-\nteractions, as predicted by Rameshti et al. in the ultra-\nstrong coupling regime [24].\nFor the remainder of this paper, we will focus on the six\nlowest frequency photon modes, whose resonant frequen-\ncies may only be accurately determined at large magnetic\n\felds, when the entire spin ensemble has been detuned,\nas shown by \fgure 3.\nThe modes have been categorised into three distinct\nclasses: mode \\ x\" is the lowest frequency and lowest Q\nfactor mode, the two highest Qfactor modes; \\ i\" and\n\\ii\", and the three remaining highest frequency modes;\n\\1\", \\2\" and \\3\". Their asymptotic frequencies as B!7\nT are summarised in table I.\nThe behaviour of these modes as the magnon reso-\nnances are tuned via the applied magnetic \feld is shown\nin \fgure 4. It has been shown previously [5] that a\nstandard model of two interacting harmonic oscillators\ncan accurately determine the coupling values from such\navoided crossings. However, we observe strong distortion\naround 0 T, and also an asymmetry of the mode split-\ntings about the central magnon resonances due to the\nultrastrong coupling of the photon modes to the magnon\nmodes, as was observed previously in Ruby [29]. There-\nFrequency (GHz)1013161514121112Magnetic Field (T)34567\nxi123ii\n1618Normalised S21 (dB)2021412108640Normalised S21 (dB)\nB = 7 Txi1ii23(a)(b)\nFIG. 3: (Color online) (a) Asymptotic frequency values of\nthe six lowest order photon modes as B!7 T. (b)\nTransmission spectra at B= 7 T, from which mode\nlinewidths may be measured.Mode!jjB!7 T=2\u0019\u0000j=\u0019gj=\u0019Cjgj=!j\n(GHz) (MHz) (GHz) (\u0002105) (%)\nx 12.779 11.84 4.79 5.97\u00061.85 18.7\ni 15.506 1.029 7.11 151\u000647.0 22.9\nii 15.563 1.197 4.19 45.2\u000614.0 13.5\n1 15.732 5.355 6.15 21.8\u00066.76 19.5\n2 15.893 2.965 3.04 9.60\u00062.98 9.56\n3 15.950 2.965 0.78 0.632\u00060.196 2.45\nTABLE I: Measured and calculated results for each\nmode showing couplings gjand cooperativities, Cj.\nFrequency (GHz)Magnetic Field (T)00.10.20.30.40.50.60.70.80.91020\n131619181715141211xi1ii23i’1’2’3’ii’x’\nFIG. 4: (Color online) Two harmonic oscillator model\n\ftting to modes i,ii, 1, 2 and 3. From the curved\nlineshapes, one can determine the coupling value g.\nfore we \ft only the curves to the right of the magnon res-\nonance. These \fts are shown as the dashed lines in \fgure\n4. From these \fts we can approximate the values of gfor\neach mode, as summarised in table I. The linewidths, \u0000 j\nand frequencies, !j=2\u0019of the photon modes are deter-\nmined from the transmission spectra taken at high \feld\nvalues (\fgure 3 (b)), whilst the magnon linewidth, \u0000 mag\ncan be determined by analysing the transmission spec-\ntra in the dispersive regime. We take a frequency sweep\natB= 0:2475 T from 5.75{9 GHz in order to view the\nmagnon resonance peaks far away from any interaction\nwith the dielectric microwave modes, as shown in \fg-\nure 5. There is a level of variation amongst the magnon\nlinewidths as calculated by \ftting the peaks with Fano\nresonance \fts, as shown in \fgure 6. This variation and\nthe presence of multiple peaks demonstrates the presence\nof higher order magnon modes. Taking the average and\nstandard deviation of these linewidths gives a \fnal es-\ntimate of magnon linewidth as \u0000 mag=\u0019= 3:247\u00060:493\nMHz. Cooperativity is calculated as Cj=g2\nj=\u0000mag\u0000j.\nThe cooperativity values in table I demonstrate that\nall modes are strongly coupled to the magnons, and all\nwith the exception of modes 2 and 3 are in the ultrastrong\ncoupling regime (i.e. gj=!j\u00150:1 [24]). The largest co-4\nFIG. 5: S21 transmission spectra showing a host of\nmagnon resonance peaks at B= 0:2475 T.\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=2.95 MHz6.1006.1026.1046.1066.1086.11025303540\nFrequency(GHz)Normalised S21(dB)\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=4.12 MHz6.3606.3656.3706.3756.3806.385152025303540\nFrequency(GHz)Normalised S21(dB)\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=2.81 MHz6.5206.5256.5306.5356.54051015202530\nFrequency(GHz)Normalised S21(dB)\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●\n●●●●\n●●\n●●●●●\n●●\n●●●●●●●●●●\n●●●●●●2Γmag/2π=2.89 MHz6.6406.6456.6506.6556.6606.6656.670-5051015\nFrequency(GHz)Normalised S21(dB)\n●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=3.46 MHz6.836.846.856.866.87-10-50510\nFrequency(GHz)Normalised S21(dB)\nFIG. 6: Fitting the magnon resonances with Fano \fts.\noperativity value obtained is that of mode i, which is, to\nthe authors' knowledge, the largest value ever reported\nto date in any previously studied spin system.\nA transmission spectrum taken at B= 0:6425 T is\nshown in \fgure 7, demonstrating the mode splitting of\nmode 1, symmetric about the magnon resonance. Over-\nlaid in red is the bare photon resonance at 7 T, i.e.\nthe microwave mode unperturbed by the magnon modes.\nFrom this red curve, 2\u0000 1=2\u0019is determined to be 5.355\nMHz, as shown in table I. When one takes the average\nof 2\u0000 mag=2\u0019= 3:247 MHz and 2\u0000 1=2\u0019, one obtains the\nline width of the resulting hybrid state when the magnon\nresonance is tuned coincident in frequency with the pho-\nton mode, as depicted by the dashed blue curve in \fgure\n7, i.e.,\u00184.4 MHz. This excellent agreement indicates\nthat at this particular B\feld, mode 1 exists as a hybrid\nmagnon-polariton.\nAroundB= 0 T, we observe a severe distortion of the\ncavity mode frequency dependence on magnetic \feld as\ndemonstrated by Fig. 8. Around 16 GHz, we see there\nexist \fve modes, corresponding to modes i,iiand 1{\n3, on the \\left\" side of the magnon resonances. These\nmodes have been given a primed nomenclature to indi-\ng/!≈ 6.2 GHz\nTransmission spectrum for H = 0.6425 TBare dielectricresonance for H = 7 T (overlay)Magnon-polariton fit●●●●●●●●●●●●●●●●●●●●●●●●●12.70512.71512.725354045505560●●●●●●●●●●●●●●●●●18.8418.8518.863638404244464850\n15.7415.72\n2Γ=2Γmag+2ΓcavB = 0.6425 T4.4 MHz\n2Γ=2Γmag+2Γcav2Γ=2Γmag+2Γcav4.4 MHz2Γcav5.4 MHz22Γ=2Γmag+2ΓcavFIG. 7: (Color online) Transmission spectrum at\nB= 0:6425 T. At this applied magnetic \feld the magnon\nresonance is tuned coincident in frequency with mode 1, and\nthe strong coupling between the two results in a mode\nsplitting of 6.2 GHz. The high density of resonant peaks in\nthe centre of the \fgure suggest a large number of higher\norder magnon modes are present in this system.\ncate their existence at B\felds lower than that required to\ntune the magnons to their frequencies. This phenomenon\nhas been previously observed in single crystal YAG [37]\nhighly doped with rare-earth Erbium ions, and is ex-\nplained by the in\ruence on the ferromagnetic phase of\nthe impurity ions on degenerate modes. The e\u000bect can\nbe explained by the in\ruence of the ensemble of strongly\ncoupled spins on the centre-propagating waves of the near\ndegenerate mode doublet. For large spin-photon inter-\nactions, tails of Avoided Level Crossings (ALCs) from\nthe positive half plane ( B > 0) should still exist on the\nMagnetic Field (T)00.10.20.150.05-0.1-0.2-0.15-0.05Frequency (GHz)2021\n1619181715142223\n5Normalised S21 (dB)\n01015202530\n-5i’1’2’3’ii’\nFIG. 8: (Color online) Behaviour of the photon modes\naroundB= 0 T, a result of the internal magnetisation of\nYIG.5\nnegative half plane ( B < 0) and vice a versa. Although,\ninstead of gradual change of direction, the system demon-\nstrates an abrupt transition to a \\no coupling\" state. It is\nworth mentioning that such e\u000bect has not been observed\nin photonic systems interacting with paramagnetic spin\nensembles [25, 26, 38]. In the present case, the e\u000bect\nis much more pronounced, with fractional frequency de-\nviations and the magnetic \feld range of the e\u000bect both\norders of magnitude larger than observed previously [37],\na result of the magnetic spin density.\nDISCUSSION\nCOMSOL 3.5's electromagnetic package was used to\nmodel the system. A 3D model was used so as to analyse\nthe degeneracies in the \u001eaxis of the dielectric modes.\nThe internal copper wall of the cavity is modelled as a\nperfect electrical conductor, which for the purposes of the\ndesired eigenfrequency study, is an appropriate simpli\f-\ncation.\nThe results of the FEM using a value of \u000fYIG=\u000f0=\n15:965 andr0= 3:71 mm, where r0is the radius of curva-\nture of the sapphire support's concavity, are summarised\nin \fgure 10 and in table II. The measured frequency of\nthe doublet modes has been taken as the average of the\ntwo constituent's frequencies at B= 7 T.\nFrom the FEM and the analytical mode shapes of\nspherical dielectric resonances described by [39], we can\nidentify mode xas ann= 0 mode with no degener-\nacy. Therefore it is present as a singular resonance. The\nother \fve modes appear as n= 1 modes. There should\nonly exist a 2 n+ 1 fold degeneracy for resonant spheri-\ncal photon modes, which can be broken by internal im-\npurities or by asymmetric boundary conditions set by\na cylindrical enclosure, microwave loop probes, and the\nsapphire substrate, collectively termed \\backscatterers\".\nThis degeneracy arises from a Legendre polynomial in\nthe mode's HandE\feld analytical expressions of the\nformPm\nn(cos\u0012)n\ncos(m\u001e)\nsin(m\u001e)o\n, wherem= 0;:::;n . The in-\ntegersmandnrepresent the number of maxima of the\nmode's energy density in the \u001edirection over 180\u000e, and\nthe number in the \u0012direction over 180\u000e, respectively.\nThis would imply that for n= 1 we should observe three\ndistinct modes corresponding to a single ( n;m) = (1;0)\nand two (1;1) modes, rather than \fve modes. However,\nModefmeas (GHz)fsim(GHz) (n;m)\nx 12.779 12.785 (0,0)\ni&ii 15.534 15.286 (1,1)\n1 15.732 15.736 (1,0)\n2 & 3 15.922 15.921 (1,1)\nTABLE II: Comparison of FEM and measured\nfrequencies and hence mode identi\fcation.the FEM demonstrates that the use of the sapphire sup-\nport base introduces a further degeneracy to the (1 ;1)\nmodes depending on the amount of \feld that permeates\nthe sapphire. The modelling predicts four (1 ;1) modes,\nexisting as two sets of two, which are separated by ap-\nproximately 500 MHz. This is in fair agreement with the\nseparation of modes i,iiwith modes 1{3. Therefore it\nis apparent that modes iandiiare a doublet pair with\n(n;m) = (1;1).\nGiven that modes 2 and 3 approach relatively similar\nfrequencies at high magnetic \felds, it is reasonable to as-\nsume that these modes correspond to the second (1 ;1)\ndoublet pair, which FEM predicts will have a larger pro-\nportion of microwave \feld inside the sapphire support.\nThis means that mode 1 must be the (1 ;0) single mode.\nFrom \fgure 4, we can see that both the doublet pairs\ndemonstrate a gyrotropic response when interacting with\nthe magnon resonances, i.e., one mode interacts more\nthan its doublet pair. This is a common occurrence in\nspin ensemble systems and has been observed in para-\nmagnetic systems such as Fe3+in sapphire [26, 27, 40].\nThis asymmetric interaction strength for doublet pairs\nhas also been observed in ferromagnets by Krupka et al.\n[41, 42] and predicted by Rameshti et al. [24], with the\nlatter stating that gn;m=n> gn;m=\u0000n, where a di\u000ber-\nent notation to that used here is employed, in which\nm=\u0000n;:::; 0;:::;n . The notations are equivalent as\nanm=\u0006ndoublet in [24] corresponds to an\ncos(m\u001e)\nsin(m\u001e)o\ndoublet pair here.\nThe gyrotropic response is a result of the anisotropy of\na ferromagnet's permeability tensor; the same reason why\nthese materials are used in circulators. The permeability\ntensor, containing o\u000b diagonal terms appears as\n~ \u0016=\u001600\n@1 +\u001f\u0000i\u00140\ni\u0014 1 +\u001f0\n0 0 11\nA; (1)\nwhere\u00160is the permeability of free space, and \u001fis\nthe magnetic susceptibility of the ferromagnet, which\nis related to the magnetic permeability tensor by ~ \u0016=\n\u00160\u0010\n~1 +~ \u001f\u0011\n.\nWhen any resonant photonic mode exists as a doublet,\nit is because then\ncos(m\u001e)\nsin(m\u001e)o\ndegeneracy has been broken\nby some backscatterer, and the two resulting modes ex-\nist as counter propagating travelling waves [26, 40]. The\noverall e\u000bect is that one travelling wave will see an e\u000bec-\ntive permeability of \u0016+=\u00160(1 +\u001f+\u0014), whilst the other\nwill see\u0016\u0000=\u00160(1 +\u001f\u0000\u0014), which can be rewritten as\n\u0016\u0006=\u00160(1+\u001f\u0006), and we can state that ( \u001f++\u001f\u0000)=2 =\u001f,\nwhere\u001fis the \\unperturbed\" magnetic susceptibility\nthat a standing wave would observe.\nThe e\u000bective susceptibility that a mode experiences\nwill determine the interaction strength of that mode with6\na magnon resonance according to [5]:\ng2\ni=\u001fe\u000b!2\u0018; (2)\nwhere\u0018is the total magnetic \flling factor of the mode;\ni.e., the proportion of magnetic \feld within the ferro-\nmagnetic material compared to the entire system. This\nparameter is used in an attempt to quantify the overlap\nof the magnon and photon modes and is calculated as\n\u0018=RRR\nVYIG\u00160~H\u0003~HdV YIGRRR\nV\u00160~H\u0003~HdV: (3)\nIt should be noted that typically it is only the magnetic\n\feld energy density perpendicular to the external mag-\nnetic \feld that is considered to interact with the spin\nsystem [5, 21]. However, the interaction of mode 1 is far\nlarger than its perpendicular \flling factor of 0.075 would\nsuggest. So, in an attempt to account for the interac-\ntion with nonuniform magnon modes, the total magnetic\n\flling factor has been used. These have been calculated\nfrom the FEM and the resulting values of \u001fe\u000bare dis-\nplayed in table III.\nGiven our assumption that mode 1 represents the (1,0)\ndielectric mode, which will exist as a standing wave given\nno possible degeneracy, the calculated \u001fe\u000bvalue for this\nmode should represent the unperturbed magnetic suscep-\ntibility of the YIG. Taking the average of the \u001fe\u000bvalues\nfor the doublet modes i(\u001f+) andii(\u001f\u0000) yields a value\nof\u001f= 0:0595; in reasonable agreement with the value\nobtained from mode 1.\nThe FEM predicts that modes x, 2 and 3 will each\ncontain a signi\fcant proportion of magnetic \feld energy\nwithin the sapphire support, so one would expect these\nmodes to observe a lower e\u000bective magnetic susceptibil-\nity, which would appear true for the latter two modes\n(their average susceptibility yields an unperturbed sus-\nceptibility of\u00180:01). However, mode xdemonstrates a\nmuch larger coupling strength than what should be af-\nforded a mode with its \flling factor, hence a \u001fe\u000bvalue\napproximately three times larger than the unperturbed\nvalue obtained from modes i,iiand 1. This suggests that\nour approximation of using the total magnetic \flling fac-\ntor to quantify the overlap of the magnon and photon\nMode!jjB!7 T=2\u0019gj=\u0019\u0018j\u001fe\u000b\n(GHz) (GHz)\nx 12.779 4.79 0.221 0.159\ni 15.506 7.11 0.594 0.0885\nii 15.563 4.19 0.594 0.0305\n1 15.732 6.15 0.728 0.0525\n2 15.893 3.04 0.493 0.0185\n3 15.950 0.78 0.493 0.00121\nTABLE III: Calculated magnetic \flling factors, \u0018jand\ne\u000bective magnetic susceptibilities, \u001fe\u000bfor each of the\nphoton modes.\nδf1\nδf 2&δf 3\nδf xϵ/ϵ0=15.965\n15.4 15.6 15.8 16.0 16.2 16.4-0.2-0.10.00.10.20.3\nϵ/ϵ0δf=f sim-f meas (GHz)FIG. 9: Frequency di\u000berence between simulated and\nmeasured results as the relative permittivity of YIG is\nvaried in the FEM software. The radius of curvature of\nthe sapphire support used here was r0= 3:7 mm, which\nfor mode 1 is largely irrelevant, but for modes x, 2 and\n3 gives good agreement.\nmodes is not entirely accurate. To accurately explain\nthe origins of the di\u000bering interaction strengths of each\nmode, knowledge of higher order, nonuniform magnon\nmode shapes are required, in order to replace the \fll-\ning factor approximation with an overlap value. Un-\nlike Zhang et al. 's [4] ultrastrong coupling results with\nad= 2:5 mm YIG sphere, in which higher order magnon\nmodes mostly couple weakly with the microwave cavity,\nhere we excite internal, nonuniform electromagnetic res-\nonances, so it is more likely than not that these modes\nwill couple more strongly to nonuniform magnon modes\nif their mode shapes match up well spatially. The de-\nrived values of susceptibility in table III agree within an\norder of magnitude to previously measured results [41]\nbut have been underestimated due to the use of \flling\nfactor as opposed to a mode overlap integral.\nFinally, we can use the predicted mode frequencies\nof the FEM to determine the permittivity of the YIG\nsample, by varying \u000fYIG=\u000f0until the frequencies match\nthe asymptotic values measured at high magnetic \felds.\nAt these magnetic \feld values, the matrix in equation\n(1) becomes the identity matrix [41]. By measuring the\ndepth of the sapphire concavity and its width at the\nsurface, the radius of curvature was determined to be\nr0= 3:71\u00060:2 mm. With this information, an iterative\nsimulation was conducted mapping mode frequencies\nversus relative permittivity of YIG. It was found that\nmode 1 is relatively insensitive to the radius of curvature\nof the sapphire support. This is due to the absence of\nelectric \feld density outside the YIG for this particular\nmode. Given that r0contains a signi\fcant amount\nof uncertainty, this mode is used to match fsimwith\nfmeas. A plot of \u000ef=fsim\u0000fmeas versus permittivity\nis shown in \fgure 9. From this result, we can state\nthat\u000fYIG=\u000f0= 15:96\u00060:02. This value agrees well7\nwith previous measurements taken using the so-called\n\\Courtney\" technique with YIG samples [42].\nIn conclusion, we observe ultrastrong coupling between\ninternal dielectric microwave resonances and magnons in-\nside ad= 5 mm YIG sphere. The large diameter of the\nsphere results in not only an increased number of spins,\nbut also the accessibility of the internal electromagnetic\nresonances due to their existence below K-band frequen-\ncies. The use of internal microwave modes instead of an\nexternal cavity resonance results in far larger magnetic\n\flling factors than ever before achieved in such an exper-\niment, hence the coupling values and cooperitivity values\nobserved are, to the authour's knowledge, the largest ever\nreported, with a maximum g=\u0019= 7:11 GHz, or\u00187000\nmode linewidths, and C= 1:5\u0002107. This implies an ex-\ntremely high level of coherence in this system. Most im-\nportantly however, the numerous resonant magnon peaks\nin the dispersive regime and the discrepancies in calcu-\nlated susceptibilities suggest that higher order magnon\nmodes participate in this system. This implies that the\npreviously theoretically analysed models of such systems\nare incomplete.\n\u0003jeremy.bourhill@uwa.edu.au\n[1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. 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Tabuchi, R. Ya-\nmazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura,\nand Y. Nakamura, \\Cavity optomagnonics with spin-\norbit coupled photons,\" arXiv:1510.03545 (2015).\n[12] Y. R. Shen and N. Bloembergen, \\Interaction between\nlight waves and spin waves,\" Phys. Rev. 143, 372{384\n(1966).\n[13] S.O. Demokritov, B. Hillebrands, and A.N. Slavin, \\Bril-\nlouin light scattering studies of con\fned spin waves: lin-\near and nonlinear con\fnement,\" Physics Reports 348,\n441 { 489 (2001).\n[14] C. Kittel, \\Excitation of spin waves in a ferromagnet by\na uniform rf \feld,\" Phys. Rev. 110, 1295{1297 (1958).\n[15] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, \\Cavity\nmagnomechanics,\" arXiv:1511.02680v2 (2015).\n[16] L. Tian, P. Rabl, R. Blatt, and P. Zoller, \\Interfacing\nquantum-optical and solid-state qubits,\" Phys. Rev. Lett.\n92, 247902 (2004).\n[17] J. Verd\u0013 u, H. Zoubi, Ch. Koller, J. Majer, H. Ritsch, and\nJ. 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Goryachev, and M. E. Tobar, \\Super-\nstrong coupling of a microwave cavity to yig magnons,\"\narXiv:1508.04967 (2015).\n[24] Babak Zare Rameshti, Yunshan Cao, and Gerrit E. W.\nBauer, \\Magnetic spheres in microwave cavities,\" Phys.\nRev. B 91, 214430 (2015).\n[25] Warrick G. Farr, Daniel L. Creedon, Maxim Goryachev,\nKarim Benmessai, and Michael E. Tobar, \\Ultrasensi-\ntive microwave spectroscopy of paramagnetic impurities\nin sapphire crystals at millikelvin temperatures,\" Phys.\nRev. B 88, 224426 (2013).\n[26] Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon,\nand Michael E. Tobar, \\Spin-photon interaction in a cav-\nity with time-reversal symmetry breaking,\" Phys. Rev. B\n89, 224407 (2014).\n[27] J. Bourhill, K. Benmessai, M. Goryachev, D. L. Cree-8\nFIG. 10: Spherical coordinate \feld components of the six lowest dielectric modes in the YIG/sapphire/air system.\nEach mode is viewed parallel to the z-axis (top row) and in the x;yplane (bottom row), except where no \feld is\npresent. We can readily identify modes xand 1 as (0,0) and (1,0) spherical dielectric modes, respectively. These two\nmodes appear as \\pure\" dielectric modes containing only three \feld components. The two doublet modes ( i,ii, 2\nand 3) appear to contain energy density in all six \feld components, and are greatly a\u000bected by the sapphire support,\nwhich is what appears to lead to the additional degeneracy; splitting two modes in to four modes. From the radial\ncomponents we can identify these modes as being modi\fed (1,1) spherical dielectric modes.9\ndon, W. Farr, and M. E. Tobar, \\Spin bath maser in\na cryogenically cooled sapphire whispering gallery mode\nresonator,\" Phys. Rev. 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Tobar, \\Precision\nmeasurement of a low-loss cylindrical dumbbell-shaped\nsapphire mechanical oscillator using radiation pressure,\"\nPhys. Rev. A 92, 023817 (2015).\n[34] Eugene N Ivanov and Michael E Tobar, \\Microwave\nphase detection at the level of 10(-11) rad.\" Rev Sci In-\nstrum 80, 044701 (2009).\n[35] E.N. Ivanov and M.E. Tobar, \\Low phase-noise sapphire\ncrystal microwave oscillators: current status,\" Ultrason-\nics, Ferroelectrics, and Frequency Control, IEEE Trans-\nactions on 56, 263{269 (2009).\n[36] A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B.\nJung\reisch, B. Hillebrands, A. Kreisel, P. Kopietz, and\nM. P. Kostylev, \\Brillouin light scattering spectroscopyof parametrically excited dipole-exchange magnons,\"\nPhys. Rev. B 86, 134403 (2012).\n[37] Warrick G. Farr, Maxim Goryachev, Jean-Michel\nle Floch, Pavel Bushev, and Michael E. Tobar, \\Evi-\ndence of dilute ferromagnetism in rare-earth doped yt-\ntrium aluminium garnet,\" Applied Physics Letters 107,\n122401 (2015), http://dx.doi.org/10.1063/1.4931432.\n[38] Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon,\nand Michael E. Tobar, \\Controlling a whispering-gallery-\ndoublet-mode avoided frequency crossing: Strong cou-\npling between photon bosonic and spin degrees of free-\ndom,\" Phys. Rev. A 89, 013810 (2014).\n[39] Jean-Michel le Floch, James David Anstie, Michael Ed-\nmund Tobar, John Gideon Hartnett, Pierre-Yves Bour-\ngeois, and Dominique Cros, \\Whispering modes in\nanisotropic and isotropic dielectric spherical resonators,\"\nPhysics Letters A 359, 1 { 7 (2006).\n[40] Karim Benmessai, Michael Edmund Tobar, Nicholas\nBazin, Pierre-Yves Bourgeois, Yann Kersal\u0013 e, and Vin-\ncent Giordano, \\Creating traveling waves from standing\nwaves from the gyrotropic paramagnetic properties of\nfe3+ions in a high- qwhispering gallery mode sapphire\nresonator,\" Phys. Rev. B 79, 174432 (2009).\n[41] J. Krupka, \\Measurements of all complex permeability\ntensor components and the e\u000bective line widths of mi-\ncrowave ferrites using dielectric ring resonators,\" Mi-\ncrowave Theory and Techniques, IEEE Transactions on\n39, 1148{1157 (1991).\n[42] Jerzy Krupka, Stephen A Gabelich, Krzysztof Derza-\nkowski, and Brian M Pierce, \\Comparison of split post\ndielectric resonator and ferrite disc resonator techniques\nfor microwave permittivity measurements of polycrys-\ntalline yttrium iron garnet,\" Measurement Science and\nTechnology 10, 1004 (1999)." }, { "title": "2203.04018v1.Interplay_between_nonlinear_spectral_shift_and_nonlinear_damping_of_spin_waves_in_ultrathin_YIG_waveguides.pdf", "content": "1 \n Interplay between nonlinear spectral shift and nonlinear damping of spin \nwaves in ultrathin YIG waveguides \nS. R. Lake1 , B. Divinskiy2*, G. Schmidt1,3, S. O. Demokritov2, and V. E. Demidov2 \n1Institut für Physik, Martin -Luther -Universität Halle -Wittenberg, 06120 Halle, Germany \n2Institute of Applied Physics, University of Muenster, 48149 Muenster, Germany \n3Interdisziplinäres Zentrum für Material wissenschaften , Martin -Luther -Universität Halle -\nWittenberg, 06120 Halle, Germany \n \nWe use the phase -resolved imaging to directly study the nonlinear modification of the \nwavelength of spin waves propagating in 100-nm thick , in-plane magnetized YIG waveguides. \nWe show that, by using moderate microwave powers, one can realize spin waves with large \namplitudes corresponding to precession angles in excess of 10 degrees and nonlinear \nwavelength variation of up to 18% in this system . We also find that , at large precession angles , \nthe propagation of spin waves is strongly affected by the onset of nonlinear damping, which \nresults in a strong spati al dependence of the wavelength. This effect leads to a spatially -\ndependent controllability of the wavelength by the microwave power . Furthermore, it leads to \nthe saturation of nonlinear spectral shift’ s effects several micrometers away from the excitation \npoint. These findings are important for the development of nonlinear , integrated spin-wave \nsignal -processing devices and can be used to optimize their characteristics. \n \n \n*Corresponding author, e -mail: b_divi01@uni -muenster.de 2 \n I. INTRODUCTION \nNonlinear phenomena accompanying propagation of spin waves in magnetic films have \nbeen known for many decades to enable implementing a large variety of advanced signa l-\nprocessing devices [ 1,2]. One fundamental nonlinear phenomen on that is frequently applied is \nthe nonlinear transformation of the spin wave’s dispersion spectrum that occurs with increasing \nspin wave intensity . This phenomenon can be regarded as a frequency shift of the dispersion \nspectrum of the spin waves . This shift results from the decrease in the magnitude of the static \nmagnetization caused by the increase in the magnetization precession angle [1,2]. Because of \nthis frequency shift, the wavelen gth of a spin wave at a given frequency becomes dependen t \non its amplitude, which allows one to implement , for example, nonlinear magnonic phase \nshifters [ 3,4], interferometers [ 5], couplers [ 6,7], and switches [ 8]. Additionally, these effects \nhold promise for implementation of spin -wave logic devices [ 9-11] and neuromorphic \ncomputing with spin waves [12,13 ]. \nThe vast majority of previous work on nonlinear spin waves utilized millimeter -scale \nmagnetic structures fabricated from low -loss magnetic insulator – yttrium iron garnet (YIG) \n[14]. In recent years, the miniaturization of spin -wave devices down to micrometer and sub -\nmicrometer dimensions has become the main trend and challenge in the field of magnonics. \nSuch m iniatur ization has been greatly advanced by the advent of high -quality ultrathin YIG \nfilms [ 15-17] that can be structured on the nanometer scale [18-20]. This breakthrough not only \nprovide s the possibility to implement low -loss propagation of spin waves in the linear regime , \nbut also brings novel opportunities to investigate dynamic nonlinear phenomena in ultrathin \nfilms and their utilization in signal -processing applications . In particular, the strong \nquantization of the spectrum of magnetic excitations in microscopic YIG structures can \nsubstantially suppress the spin-wave scattering effects and help achieve very large amplitudes \nof magnetic dynamics [21] that have not been observed on the macroscopic scale. 3 \n Although nonlinear spin -wave dynamics in microscopic YIG structures remain largely \nunexplored, it has already been demonstrated experimentally that the nonlinear shift of the \ndispersion spectrum of propagating spin waves can be used, for example, to control switching \nin coupled spin -wave waveguides via intensity [7] and to indirectly excite propagati ng spin \nwaves [22]. It has also been theorized to be the underlying physical phenomenon for the \nrealization of nonlinear nano -ring resonators [ 23] and nanoscale neural networks [ 13]. \nHowever, until now it has remained unclear how large nonlinear shift is practically achievable \nand which factors can limit this effect in real devices . \nIn this work, we study the propagation of intense spin waves in microscopic ultrathin -\nYIG waveguide s using a wide range of power for the excitation signal . By using high -\nresolution , phase -sensitive , magneto -optical detection, we directly measure the spin wave’s \nspatial dependencies of intensity and wavelength. Our experimental findings indicate that the \nnonlinear spectral shif t is strongly affected by the onset of the nonlinear magnetic damping: at \nlarge amplitudes, spin waves start to exhibit strongly enhanced spatial attenuation leading to a \nspatially -dependent magnitude of the nonlinear spectral shift. As a result, the wavel ength \n(wavevector) of intense spin waves varies strongly along the propagation path . While the \nwavelength exhibits good controllability by the applied microwave power at the beginning of \nthe propagation path , this controllability is almost completely lost several micrometers away \nfrom the excitation point. O ur findings provide insight into nonlinear propagation of spin waves \nin microscopic waveguides and are of decisive importan ce for the development of efficient \nmagnonic devices utilizing the effect of the nonlinear spectral shift. \nII. EXPERIMENT \nFigure 1(a) shows the schematics of our experiment. We study propagation of spin \nwaves in a 2-m wide spin-wave waveguide patterned from a 100 -nm thick YIG film. The YIG 4 \n film is characterized by a saturation magne tization of 4π MS = 1.75 kG and a Gilbert damping \nconstant α = 4×10-4, as determined from ferrom agnetic resonance measurements. The spin \nwaves are excited by using a 500 -nm wide and 150 -nm thick inductive Au antenna carrying \nmicrowave current with a frequen cy f and a power P. The YIG waveguide is magnetized to \nsaturation by a static magnetic field, H0=1000 Oe , which is applied in-plane along the Au \nantenna. \nFor patterning , a double -layer PMMA resist is deposited on a <111> oriented GGG \nsubstrate. The resist is exposed by e -beam lithography using a RAITH Pioneer system and \ndeveloped in isopropanol. Subsequently 110 nm of YIG are deposited at room temperature by \npulsed laser deposition using a recipe by Hauser et al. [17]. After lift -off in acetone , the sample \nis annealed in a pure oxygen atmosphere. Wet etching in Phosphoric acid is used to remove 10 \nnm of YIG to smoothen the edges of the remaining structures. The process is completed by \npatterning a microstrip antenna (10 nm Ti, 150 nm Au) on top of the structures using electron \nbeam lithography, e -beam evaporation , and lift -off. \nWe study the propagation of spin waves in the YIG waveguide with spatial and phase \nresolution by using micro -focus Brillouin light scattering (BLS) spectroscopy [ 24]. We focus \nthe probing laser light with a wavelength of 473 nm and a power of 0.25 mW into a diffraction -\nlimited spot on the sample surface [Fig. 1(a)] and analyze the modulation of the probing light \ndue to its interaction with the magnetization dynamics . The intensity of this modulation , or \nBLS intensity, is proportional to the intensity of spin waves at t he position of the probing spot. \nThis allows us to record two -dimensional maps of the spin -wave intensity by rastering the spot \nover the sample surface . Additionally, we use the interference of the modulated light with the \nreference light to measure the spatial maps of cos( ), where is the phase diffe rence between \nthe spin wave and the microwave signal applied to the antenna. Analysis of the phase maps 5 \n provides information about the wavelength of spin waves at a given frequency f and allows us \nto directly address the effect of the nonlinear spectral sh ift. \nFirst, we characterize the expected effect by using micromagnetic simulations (Figs. \n1(b) and 1 (c)). We calculate amplitude -dependent dispersion curves using the simulation \nsoftware MuMax3 [ 25] and the approach developed in Ref. [26]. We consider a 2-m wide \nand 100 -nm thick waveguide with the length L=20 m discretized into 10 nm 10 nm 10 \nnm cells with periodic boundary conditions at the ends. The standard for YIG exchange \nconstant of 3.66×10-7 erg/cm is used. The Gilbert damping parameter is se t to an artificially \nsmall value 10-12 to fix the amplitude of the magnetization dynamics at the chosen level. We \nexcite magnetization dynamics by initially deflecting magnetic moments from their \nequilibrium orientation by an angle . The deflection is spa tially periodic with the period L/n \n(where n is an integer) , which defines the wavelengt h of the excited wave. By analyz ing the \nfree dynamics of magnetization, we determine the frequency corresponding to the given spatial \nperiod and obtain the frequency vs wavenumber relations (solid curves in Fig. 1( b)) for a given \nspin-wave amplitude characterized by the precession angle . \nAs seen from the data of Fig. 1(b), the increase in the angle from 0.1 to 15° leads to \na noticeable shift of the dispersion cu rve down toward smaller frequencies and a slight overall \ndecrease in its slope. Both effects are consistent with the changes expected fo r a decrease in \nthe static component of magnetization , MST, with the increase of the amplitude of the \nmagnetization precession: 𝑀ST=√(𝑀S2−𝑚2)≈𝑀S−1\n2𝑀Ssin2(𝜃), where MS is the \nsaturation magnetization, m is the amplitude of the dynamic magnetization, and is the mean \nprecession angle. Because MST enters the expression for the frequency of ferromagnetic \nresonance: 𝑓FMR=𝛾√𝐻0(𝐻0+4𝜋𝑀ST), its decrease causes the overall decrease in the \nfrequency of spin waves. Additionally, the decrease of MST is known to lead to a decrease in \nthe group velocity of spin waves [2], which is consistent with the overall decrease of the slope 6 \n of the dispersion curve in Fig. 1(b). The nonlinea r transformation of the spectrum can be \nquantitatively characterized by the variation of the wavenumber d k due to the increase in \nfrom 0.1 to 15° (dashed curve in Fig. 1(b)). These data indicate that the dispersion’s shift \nbecomes significantly stronge r as wavelength decreases . As can be seen from the data of Fig. \n1(c), dk varies approximately linearly with sin2(𝜃), which in turn is proportional to the spin-\nwave intensity. This linear dependence makes it straight forward to calibrate the precession \nangle achieved in the experiment. We also emphasize that the dispersion curve calculated for \n = 0.1° coincides well with that measured by BLS at low excitation power P = 0.1 mW. This \nprovides clear proof of the validity of the results obtained from the micromagnetic simulations. \nIII. RESULTS AND DISCUSSION \nTo experimentally address the effect of the nonlinear spectral shift, we perform phase -\nresolved BLS measurements at a fixed frequency of the excitation signal f and analyze the \nvariation of the wavele ngth of spin waves with the increase in the power P. Figure 2(a) shows \nrepresentative spin-wave phase maps recorded at f = 4.8 GHz and P = 0.1 and 3 mW , whereas \nFig. 2(b) shows the spatial dependence of the wavelength of spin waves , obtained from the \nFourier analysis of these maps. As seen from these data, at P = 0.1 mW , the spin waves exhibit \na well -defined constant wavelength = 1.38 m, which is in good agreement with the \ndispersion curve calculated for the small precession angle = 0.1° (Fig. 1(b)). At P = 3 mW, \nthe situation changes drastically. First, the wavelength is reduced , explained by the effect of \nthe spectral shift. Second, the wavelength strongly changes in space exhibiting the maximum \nreduction of about 18% close to the antenn a, which becomes as small as 3% at the propagation \ndistance of 18 m. This spatial variation could be associated with the decrease of the intensity \nof the spin wave along the propagation path due to the damping. However, taking into account \nthe linear dependence in Fig. 1(c), this would require that the intensity decrease s by more than 7 \n a factor of five over the propagation interval x = 2 – 18 m, which is inconsistent with the small \ndamping in the YIG film. \nWe quantify t he spatial dec ay of spin waves in the waveguide by analyzing the spin -\nwave intensity maps (Fig. 3(a)). The direct comparison of the maps recorded at P = 0.1 mW \nand 3 mW clearly shows that the spatial decay becomes significantly stronger, as the power is \nincreased. In Fi g. 3(b) , we show in the log -linear coordinates the spatial dependence of the BLS \nintensity integrated over the waveguide width. At small power, P = 0.1 mW, this dependence \nis exponential and is characterized by the decay length = 29 m, where is the distance, at \nwhich the amplitude decreases by a factor of e . This value agrees reasonably well with = 34 \nm obtained from micromagnetic simulations using experimentally determined α = 4��10-4. In \ncontrast to these simple behaviors, a t large powers, the s patial decay is not described by a single \nexponential and strongly exceeds that observed in the linear propagation regime. This fact \nexplains the strong spatial variation of the wavelength in Fig. 2(b). \nTo characterize the modification of the decay with the increase of P in detail, we plot \nin Fig. 4(a) the power dependence of the BLS intensity recorded at x = 0 and 20 m. We note \nthat, in the linear propagation regime, the se dependence s are expected to grow linearly with \npower . As seen from the data of Fig. 4(a), at x = 0, the BLS intensity remains proportional to \nP at P < 1 mW and then starts to saturate. At approximately the same power, the intensity \nrecorded at x = 20 m exhibits a drop reflecting an increase in the spatial decay. This indicates \nan onset at the threshold power P = 1 mW of the so -called nonlinear damping [ 26-32], which \nis associated with the energy transfer from the coherent propagating spin wave into other \nincoherent sh ort-wavelength spin -wave modes . These sh ort-wavelength spin -wave modes do \nnot contribute to the BLS intensity and cannot be directly accessed in the experiment. \nThe contribution of this phenomenon relative to the normal linear damping can be \nestimated based on the data presented in Fig. 4(b) . The figure shows the decrease of the spin 8 \n wave intensity I over a propagation length of 1 µm expressed by an attenuation factor I(x)/I(x+1 \nµm). This factor is plotted over the used range of microwave power for positions x = 0 and 20 \num. At large distance s from the antenna ( x = 20 m), the attenuation factor is about 1.08 and \nremains approximately constant within the entire studied interval of P (point -down triangles in \nFig. 4(b)) . This value is consistent with that expected for the effects of the linear da mping for \nα = 4×10-4. We note that it does not increase at large P, because even for P = 3 mW, the strong \nnonlinear decay at the initial propagation stage has already severely diminished the intensity \nof spin waves at x = 20 m. This initial strong decay is well characterized by the attenuation \nfactor measured at x = 0 (point -up triangles in Fig. 4(b)) . In the linear propagation regime ( P < \n1 mW), the attenuation factor remains constant and coincides with the observed value at x= 20 \nm. However, at P > 1 mW, it increases significantly and reaches the value of 1.93 at P = 3 \nmW. Comparing this value with 1.08, one can conclude that the nonlinear energy transfer into \nshort -wavelength spin -wave modes causes the increase of the effective damping of the coherent \nwave by nearly a factor of two. \nThe results presented above suggest that, due to the effects of the nonlinear damping, \nthe efficient controllability of the wavelength (wavenumber) by the microwave power can only \nbe achieved near the excitation point, while at large propagation distances this controllability \nbecomes relatively poor . This is evidenced by the data in Fig. 5, which shows how the change \nin wavenumber, d k, depends on the applied power for various distances from the antenna. Clos e \nto the antenna, d k changes li nearly with the microwave power reaching the value of 0. 9 m-1 \nat P = 3 mW (point -up triangles in Fig. 5) . In contrast, a t a distance x = 8 m (circles in Fig. \n5), dk saturates to the value of about 0. 4 m-1 and remains nearly constant for P > 1.5 mW . \nFinally, at x = 18 m (point -down triangles) , the maximum achieved shift never exceeds 0.1 \nm-1 within the entire range of P. 9 \n It is instructive to discuss approaches to suppress the detrimental effects of the nonlinear \ndamping. The dominating mechanism of the nonlinear damping is the energy transfer from a \ncoherent spin wave into incoherent modes possessing the same frequency. This process can be \ncontrolled by varying the geometrical parameters of the waveguide , which allows one to \nmodify the spin -wave dispersion spectrum using the effects of spin -wave quantization and \navoid the detrimental spectral degeneracy (see, e.g., Refs. 32, 33). In particular, this can be \nachieved by reducing the width of the waveguide b elow a certain critical value. For the 100 -\nnm thick YIG film used in our study, we estimate the critical width of about 200 nm. Reduction \nof the width below this value can allow a significant suppression of the nonlinear damping \nwithin the addressed range of the wavelength of spin waves. However, this reduction is also \nexpected to result in an increase of the linear damping due to the increasing contribution of the \nspin-wave scatting caused by the roughness of the waveguide edges. We note that, due to the \ncompeting effects of the dipolar and the exchange interaction, the critical width weakly \ndepends on the thickness of the YIG film. It remains nearly unchanged within the practically \nimportant range of thicknesses 50 -200 nm. Additionally, since the nonlinear damping is caused \nby the parametric interactions of spin -wave modes, its threshold can be increased by increasing \nthe linear damping. However, this approach is impractical, since it leads to a faster decay of \nspin waves. Finally , the threshold power of the nonlinear damping can be increased by reducing \nthe ellipticity of the magnetization precession, which can be achieved by using magnetic films \nwith perpendicular magnetic anisotropy [26,34]. \nWe now turn to the estimation of characteristic precession angl es in our experiment s. \nWe base our analysis on the comparison of the experimental dependence d k(P) measured close \nto the antenna (point -up triangles in Fig. 5) with the dependence d k(sin2) obtained from \nmicromagnetic simulations ( red curve in Fig. 1(c)). Because both of these relationships are \nlinear with their respective variable , one can conclude that sin2 is proportional to the excitation 10 \n power P. Additionally, the intensity of the spin wave excited by the antenna is also proportio nal \nto P at P < 1 mW, where the nonlinear damping is not active (point -up triangles in Fig. 4(a)). \nThese facts allow us to estimate the critical precession angle corresponding to the onset of the \nnonlinear damping at the threshold power P=1 mW: 9°. At this angle, the nonlinear shift of \nthe wavevector d k = 0.27 m-1, which corresponds to the modification of the wavelength by \nabout 6%. \nAt powers P > 1 mW , the estimation of precession angles is less straightforward. Due \nto the influence of the nonlinear d amping, the energy becomes transferred from the directly \nexcited coherent spin wave into incoherent short -wavelength spin -wave modes each of them \nreducing the saturation magnetization accordingly . Under these conditions, the reduction of the \nstatic magneti zation causing the spectral shift is determined not only by the intensity of the \ncoherent wave but also by those of the incoherent modes. Therefore, we introduce an effective \ntotal precession angle , tot, which is generally larger than the coherent spin-wave precession \nangle, SW. The former can be directly found from the analysis of the shift of the dispersion \nspectrum (point -up triangles in Fig. 5) : at the maximum power P=3 mW, the effective total \nangle tot17°. The angle SW can be estimated based on the analysis of the power dependence \nof the intensity of the coherent spin wave (point -up triangles in Fig. 4(a)). W e extrapolate the \nlinear fit of the experimental data (line in Fig. 4(a)) to P=3 mW and find the ratio of this value \nto the experimentally observed intensity at this power , 1.7. This ratio is approximately equal \nto sin2(tot)/sin2(SW), where sin2(SW) is proportional to the measured BLS intensity while \nsin2(tot) is proportional to the microwave power. This allows us to estimate SW13°. As seen \nfrom the data of Figs. 4(a) and Fig. 5, further increase in the excitation power above 3 mW is \nexpected to result in the further increase of tot, while SW shows a clear tendency to saturation . \nWe emphasize that, although the spectral shift is det ermined by tot, the linear increase of its 11 \n value with the increase in P is only observed near the antenna, while at distance s of several \nmicrometers , this angle also exhibits saturation ( circles and point -down triangles in Fig. 5). \nWe finally discuss possible effects of the heating of the sample by the intense \nmicrowave radiation. According to the theory of spin -wave interactions (Ref. 2), the increase \nof the temperature is not expected to noticeably affect the nonlinear damping. Thi s was also \nshown experimentally in, e.g., Ref. 35. However, the heating can potentially contribute to the \nobserved shift of the spin -wave spectrum. Since the heating results in an increase of the \nintensities of incoherent spin -wave modes, it contributes to the reduction of the saturation \nmagnetization and causes an additional frequency shift. Thermally induced shift can be \ndistinguished based on its slow temporal dynamics. Therefore, we performed additional time -\nresolved measurements, where the excitation w as applied in the form of pulses with the \nduration of 5 -50 s and the spectral shift was analyzed in the time domain with the resolution \ndown to 2 ns. The measurements at the largest microwave power of 3 mW did not reveal any \nslowly varying contribution su ggesting that the heating effects are negligible in the studied \nsystem. \nIV. CONCLUSIONS \nIn conclusion, we have shown that the effect of the nonlinear spectral shift enabling the \ncontrollability of the wavelength of spin waves by their intensity is a compl ex phenomenon, \nwhich is strongly affected by the nonlinear spin -wave damping. This effect becomes \npronounced at precession angles exceeding 9° and results in a spatially -dependent \ncontrollability of the wavelength. T he efficient controllability can only be achieved at small \ndistances from the excitation point, wh ereas the controllability becomes relatively poor several \nmicrometers away . Additionally, the fast spatial decay caused by the nonlinear damping results \nin the strong spatial variation of the wavele ngth near the excitation point. 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(b) Solid curves – calculated dispersion curves of \nspin waves in the YIG waveguide corresponding to different angles of magnetization \nprecession, as labeled. Dashed curve – variation of the wavenumber caused by the increas e in \nthe precession angle from 0.1 to 15°. Symbols – dispersion curve measured by BLS at low \nexcitation power P = 0.1 mW. (c) Dependence of the nonlinear variation of the wavenumber \non the precession angle calculated at different frequencies, as label ed. T he data are obtained \nat H0=1000 Oe. \n18 \n \n \n \n \n \n \nFIG. 2. (a) Representative spin -wave phase maps recorded by BLS at f = 4.8 GHz and P = 0.1 \nand 3 mW , as label ed. (b) Spatial dependence of the wavelength of spin waves obtained from \nthe Fourier analysis of the pha se maps. Symbols – experimental data. Curves – guide for the \neye. The data are obtained at H0=1000 Oe. \n \n19 \n \n \n \n \n \nFIG. 3. (a) Spin -wave intensity maps recorded by BLS at f = 4.8 GHz and P = 0.1 and 3 mW , \nas label ed. (b) Normalized s patial dependence of the BLS intensity integrated over the \nwaveguide width. Note the logarithmic scale of the vertical axis. Symbols – experimental \ndata. Curves – exponential fit of the data obtained at P = 0.1 mW and double -exponential fit \nof the data obt ained at P = 3 mW. The data are obtained at H0=1000 Oe. \n \n \n20 \n \n \n \n \n \nFIG. 4. (a) Power dependence of the BLS intensity recorded at x = 0 and 20 m, as labe led. \nLine – linear fit of the experimental data at P < 1 mW . (b) Power dependence of the factor, \nby which the spin -wave intensity is attenuated over a propagation distance of 1 m, at x = 0 \nand 20 m, as label ed. Vertical dashed line in (a) and (b) marks the threshold power, at which \nthe nonlinear damping emerges . The data are obtained at H0=1000 Oe. \n \n \n21 \n \n \n \n \n \n \n \n \nFIG. 5. Power dependences of the nonlinear variation of the wavenumber of spin waves at \ndifferent dista nces from the antenna, as label ed. Symbols – experimental data. Solid line – \nlinear fit of the data ob tained at x = 2 m. Horizontal dashed lines mark the saturation values \nat x = 8 and 18 m. The data are obtained at H0=1000 Oe. \n" }, { "title": "2206.14588v2.Mechanical_Bistability_in_Kerr_modified_Cavity_Magnomechanics.pdf", "content": "Mechanical Bistability in Kerr-modified Cavity Magnomechanics\nRui-Chang Shen,1Jie Li,1,\u0003Zhi-Yuan Fan,1Yi-Pu Wang,1,yand J. Q. You1,z\n1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation,\nand Zhejiang Province Key Laboratory of Quantum Technology and Device,\nSchool of Physics, Zhejiang University, Hangzhou 310027, China\nBistable mechanical vibration is observed in a cavity magnomechanical system, which consists of a mi-\ncrowave cavity mode, a magnon mode, and a mechanical vibration mode of a ferrimagnetic yttrium-iron-garnet\n(YIG) sphere. The bistability manifests itself in both the mechanical frequency and linewidth under a strong\nmicrowave drive field, which simultaneously activates three di \u000berent kinds of nonlinearities, namely, magne-\ntostriction, magnon self-Kerr, and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr\nnonlinearity is first predicted and measured in magnomechanics. The system enters a regime where Kerr-\ntype nonlinearities strongly modify the conventional cavity magnomechanics that possesses only a radiation-\npressure-like magnomechanical coupling. Three di \u000berent kinds of nonlinearities are identified and distinguished\nin the experiment. Our work demonstrates a new mechanism for achieving mechanical bistability by combining\nmagnetostriction and Kerr-type nonlinearities, and indicates that such Kerr-modified cavity magnomechanics\nprovides a unique platform for studying many distinct nonlinearities in a single experiment.\nIntroduction.— Bistability, or multistability, discontinuous\njumps, and hysteresis are characteristic features of nonlin-\near systems. Bistability is a widespread phenomenon that\nexists in a variety of physical systems, e.g., optics [1–3],\nelectronic tunneling structures [4], magnetic nanorings [5],\nthermal radiation [6], a driven-dissipative superfluid [7], and\ncavity magnonics [8]. Its presence requires nonlinearity in\nthe system. To date, bistability has been studied in vari-\nous mechanical systems, including nano- or micromechani-\ncal resonators [9–11], piezoelectric beams [12], mechanical\nmorphing structures [13], and levitated nanoparticles [14].\nBistable mechanical motion finds many important applica-\ntions: It is the basis for mechanical switches [15, 16], mem-\nory elements [17, 18], logic gates [19], vibration energy har-\nvesters [12, 20], and signal amplifiers [11, 14], etc.\nDi\u000berent mechanisms can bring about nonlinearity in the\nsystem leading to bistable mechanical motion. Most com-\nmonly, a strong drive can induce bistability of a mechanical\noscillator, of which the dynamics is described by the Du \u000b-\ning equation [11, 21–23]. Mechanical bistability can also be\ncaused by the Casimir force [9], nanomechanical e \u000bects on\nCoulomb blockade [10], magnetic repulsion [24], and intrin-\nsic nonlinearity in the optomechanical coupling [25], etc.\nHere we introduce a mechanism to induce mechanical\nbistability, distinguished from all the above mechanisms, by\nexploiting rich nonlinearities in the ferrimagnetic yttrium-\niron-garnet (YIG) in cavity magnomechanics (CMM). In\nthe CMM [26–29], magnons are the quanta of collective\nspin excitations in magnetically ordered materials, such as\nYIG. They can strongly couple to microwave cavity pho-\ntons by the magnetic-dipole interaction, leading to cavity po-\nlaritons [30–35]. They can also couple to deformation vi-\nbration phonons of the ferrimagnet via the magnetostrictive\nforce [26, 28, 36]. Such a radiation-pressure-like magnome-\nchanical coupling provides necessary nonlinearity, enabling a\nnumber of theoretical proposals, including the preparation of\nentangled states [37–41], squeezed states [42–44], mechanical\nquantum-ground states [45–47], slow light [48, 49], thermom-etry [50], quantum memory [51, 52], exceptional points [53],\nand parity-time-related phenomena [54–57], etc. In contrast,\nthe experimental studies on this system are, by now, very lim-\nited: Magnomechanically induced transparency and absorp-\ntion [26] and mechanical cooling and lasing [28] have been\ndemonstrated.\nIn this Letter, we report an experimental observation of\nbistable mechanical vibration of a YIG sphere in the CMM.\nWe show that both the frequency and linewidth of the me-\nchanical mode exhibit a bistable feature as a result of the\ncombined e \u000bects of the radiation-pressure-like magnetostric-\ntive interaction [26, 28], the magnon self-Kerr [58, 59], and\nthe magnon-phonon cross-Kerr nonlinearities. Three di \u000berent\nkinds of nonlinearities are simultaneously activated by apply-\ning a strong drive field on the YIG sphere. Their respective\ncontributions to the mechanical frequency and linewidth are\ndiscussed.\nKerr-modified CMM.— The CMM system consists of a mi-\ncrowave cavity mode, a magnon mode, and a mechanical vi-\nbration mode, see Fig. 1. In the experiment, we use the\noxygen-free copper cavity with dimensions of 42 \u000222\u00028 mm3.\nThe cavity TE 101mode has a frequency !a=2\u0019=7:653 GHz,\nand a total decay rate \u0014a=2\u0019=2:78 MHz. The cavity decay\nrates associated with the two ports are \u00141;2=2\u0019=0:22 MHz\nand 1:05 MHz, respectively. The magnon and mechanical\nmodes are supported by a 0.28 mm-diameter YIG sphere.\nThe frequency of the magnon mode can be tuned by ad-\njusting the bias magnetic field B0via!m=\rB0, with\n\rbeing the gyromagnetic ratio. The magnon dissipation\nrate is\u0014m=2\u0019=2:2 MHz. The magnon mode couples\nto the cavity magnetic field by the magnetic-dipole inter-\naction with the coupling strength gma=2\u0019=7:37 MHz,\nand to a vibration mode by the magnetostrictive (radiation-\npressure-like) interaction with the bare magnomechanical\ncoupling strength gmb=2\u0019=1:22 mHz. Here we consider the\nlower-frequency mechanical mode (with a natural frequency\n!b=2\u0019=11:0308 MHz and linewidth \u0014b=2\u0019=550 Hz) in\nour observed two adjacent mechanical modes, which has aarXiv:2206.14588v2 [quant-ph] 3 Sep 20222\nx\nyzPort 1\ngmbgma\nB0MW\nPort 2\nVNA\nx\ny\nminmaxB0\nFIG. 1. Device schematic. Left panel: Schematic of the CMM system. A\n0.28 mm-diameter YIG sphere is placed (free to move) in a horizontal 0.9\nmm-inner-diameter glass capillary and at the antinode of the magnetic field\nof the cavity mode TE 101. The cavity has two ports: Port 1 is connected to\na microwave source (MW) to load the drive field, and Port 2 is connected to\na vector network analyzer (VNA) to measure the reflection of the probe field\nwith the power of \u00005 dBm. We set the direction of the bias magnetic field B0\nas the zdirection, and the vertical direction as the xdirection. Right panel:\nSchematic of the coupled three modes.\nstronger coupling gmb. The magnomechanical coupling can\nbe significantly enhanced by applying a pump field on the\nmagnon mode [37]. In our experiment, this is realized by\nstrongly driving the cavity, which linearly couples to the\nmagnon mode. In Ref. [60], we provide a list of parameters\nand the details of how they are extracted by fitting the experi-\nmental data.\nUnder a strong pump, the Hamiltonian of the CMM system\nis given by [60]\nH=~=!aaya+!mmym+!bbyb+gma(aym+amy)\n+gmbmym\u0010\nb+by\u0011\n+HKerr=~+p\u00141\"d\u0010\naye\u0000i!dt+H:c:\u0011\n;\n(1)\nwhere a,m, and b(ay,my, and by) are the annihilation (cre-\nation) operators of the cavity mode, the magnon mode, and the\nmechanical mode, respectively. The last term is the driving\nHamiltonian, where \u00141is the cavity decay rate associated with\nthe driving port (Port 1), and \"d=pPd=(~!d), with Pd(!d)\nbeing the power (frequency) of the microwave drive field. The\nnovel part of the Hamiltonian, with respect to the conventional\nCMM, is the Kerr nonlinear term HKerractivated by the strong\npump field [60]\nHKerr=~=Kmmymmym+Kcrossmymbyb; (2)\nwhere Kmis the magnon self-Kerr coe \u000ecient, and Kcrossis the\nmagnon-phonon cross-Kerr coe \u000ecient. The magnon self-Kerr\ne\u000bect is caused by the magnetocrystalline anisotropy [58, 59],\nand the cross-Kerr nonlinearity originates from the magne-\ntoelastic coupling by including the second-order terms in the\nstrain tensor [61, 62],\n\u000fi j=1\n20BBBBB@@ui\n@lj+@uj\n@li+X\nk@uk\n@li@uk\n@lj1CCCCCA; (3)where uiare the components of the displacement vector, and\nli=i(i=x;y;z). The first-order terms lead to the conven-\ntional radiation-pressure-like interaction Hamiltonian [63],\n~gmbmym\u0010\nb+by\u0011\n. Under a moderate drive field, the second-\norder terms are negligible [26, 28], but can no longer be ne-\nglected when the drive becomes su \u000eciently strong, as in our\nexperiment, yielding an appreciable magnon-phonon cross-\nKerr nonlinearity. As will be seen later, the cross-Kerr nonlin-\nearity is indispensable in the model for fitting the mechanical\nfrequency shift.\nBoth the magnon self-Kerr and magnon-phonon cross-\nKerr terms, as well as the radiation-pressure-like term, cause\na magnon frequency shift \u000e!m=2KmjMj2+KcrossjBj2+\n2gmbRe[B], where O=hoi(o=m;b;a) denote the average of\nthe modes. In our experiment, the dominant contribution is\nfrom the self-Kerr nonlinearity [60], which gives a bistable\nmagnon frequency shift [8]. Also, the cross-Kerr nonlinearity\ncauses a mechanical frequency shift \u000e!b=KcrossjMj2. Using\nthe Heisenberg-Langevin approach, we obtain the equation for\nthe steady-state average M[60]\n\u0011a\u00141g2\nma\"2\nd=jMj2\u0002\n\"\u0010\n\u0001m\u0000\u0011ag2\nma\u0001a+2KmjMj2\u00112+\u0012\u0014m\n2+\u0011ag2\nma\u0014a\n2\u00132#\n;(4)\nwhere \u0001a (m)=!a (m)\u0000!d,\u0011a=1\n\u00012a+(\u0014a=2)2. In deriving Eq. (4),\nwe neglect contributions from the mechanical mode to the\nmagnon frequency shift, because of a much smaller mechan-\nical excitation number compared to the magnon excitation\nnumber for the drive powers used in this work. It is a cubic\nequation of the magnon excitation number jMj2. In a suitable\nrange of the drive power, there are two stable solutions, lead-\ning to the bistable magnon and phonon frequency shifts by\nvarying the drive power.\nThe radiation-pressure-like coupling gives rise to an e \u000bec-\ntive susceptibility of the mechanical mode [60]\n\u001fb;e\u000b(!)=\u0010\n\u001f\u00001\nb(!)\u00002ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001\u0011\u00001;(5)\nwhere\u001fb(!) is the natural susceptibility of the mechanical\nmode, but depends on the modified mechanical frequency\n˜!b=!b+KcrossjMj2, which includes the cross-Kerr induced\nfrequency shift. The e \u000bective coupling Gmb=gmbM, and\n\u001fma(!)=h\n\u001f\u00001\nm(!)+g2\nma\u001fa(!)i\u00001, where\u001fm(!) and\u001fa(!) are\nthe natural susceptibilities of the magnon and cavity modes,\nwith the magnon detuning in \u001fm(!) modified as ˜\u0001m= \u0001 m+\n2KmjMj2, which includes the dominant magnon self-Kerr in-\nduced frequency shift. See [60] for the explicit expressions of\nthe susceptibilities.\nThe e \u000bective mechanical susceptibility yields a frequency\nshift of the phonon mode (the so-called “magnonic spring”\ne\u000bect [28], in analogy to the “optical spring” in optomechan-\nics [69])\n\u000e!b=\u0000Reh\n2ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001i\n+KcrossjMj2;(6)\nwhere we write together the frequency shift induced by the\ncross-Kerr nonlinearity. Moreover, it leads to a mechanical3\n-6-4-20\n7.64 7.66\nFrequency (GHz)-6-4-20\n-11.04 -11.03 -11.02\npd (MHz)-0.2-0.10-6.2-6-5.8-4-20\n-0.24-0.23\n7.64 7.66\nFrequency (GHz)-8-6-4-20 Reflection (dB)\n11.02 11.03 11.04-8-6-4\npd (MHz)(a)\n(b)\n Reflection (dB)4.7 dBm\n29.7 dBm\nωm\nωa\nωaωbωb\nZoom-inZoom-in\n4.7 dBm\n23.7 dBmωmωb\nωb\nωbωb\nωb\nωb\nFIG. 2. (a) Left panel: Measured reflection spectra under a red-detuned\ndrive. The frequency of the drive field !d=2\u0019=7:645 GHz (black ar-\nrow). By increasing the drive power, the magnon frequency shift is neg-\native:!L\nm=2\u0019=7:658 GHz (light green dashed line) at a lower power\nPd=4:7 dBm (upper panel) and !H\nm=2\u0019=7:640 GHz (dark green dashed\nline) at Pd=29:7 dBm (lower panel). The green arrow indicates the direc-\ntion in which the magnon frequency shifts by increasing the power. Right\npanel: Zoom-in on the red shaded areas in the left panel shows detailed spec-\ntra of the magnomechanically induced resonances, where \u0001pd=!p\u0000!d. The\nblack lines are the fitting curves. (b) Left panel: Measured reflection spec-\ntra under a blue-detuned drive. The drive frequency !d=2\u0019=7:660 GHz.\nBy adjusting the bias magnetic field, the magnon frequency is tuned close\nto the drive frequency !L\nm=2\u0019'!dat the power Pd=4:7 dBm. Increas-\ning the power to 23 :7 dBm, the magnon frequency !H\nm=2\u0019=7:645 GHz.\nRight panel: Zoom-in on the blue shaded areas in the left panel shows de-\ntailed spectra of the magnomechanically induced resonances. We observe\ntwo adjacent mechanical modes with the frequencies !b=2\u0019=11:0308 MHz\nand!0\nb=2\u0019=11:0377 MHz. Due to their similar behaviors, we focus on the\nlower-frequency mode in the text.\nlinewidth change\n\u000e\u0000b=Imh\n2ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001i\n: (7)\nClearly, this linewidth change is only caused by the radiation-\npressure-like coupling, distinguished from the frequency shift\ncaused by the self-Kerr or cross-Kerr nonlinearity. By apply-\ning a red- or blue-detuned drive field, we can choose to operate\nthe system in two di \u000berent regimes, where either the mag-\nnomechanical anti-Stokes or Stokes scattering is dominant.\nThis yields an increased ( \u000e\u0000b>0) or a reduced ( \u000e\u0000b<0)\nmechanical linewidth, corresponding to the cooling or ampli-\nfication of the mechanical motion [28, 69].\nIn our system, due to the strong coupling gma> \u0014 m;\u0014a,\nthe magnon and cavity modes form two cavity polariton (hy-\nbridized) modes (Fig. 2, left panels) [30–35]. Here, a red\n(blue)-detuned drive means that the drive frequency is lower\n(higher) than the frequency of the cavity-like polariton mode,\ni.e., the “deeper” polariton in the spectra close to the cavity\nresonance. For the red (blue)-detuned drive, we show the anti-Stokes (Stokes) sidebands associated with two mechanical\nmodes for two drive powers in the zoom-in plots of Fig. 2(a)\n(Fig. 2(b)). When the detuning between the drive field and the\ndeeper polariton matches the mechanical frequencies, the anti-\nStokes (Stokes) sidebands are manifested as the magnome-\nchanically induced transparency (absorption) [26].\nRed-detuned drive.— To implement a red-detuned drive,\nwe drive the cavity with a microwave field at frequency\n!d=2\u0019=7:645 GHz. By adjusting the bias magnetic field,\nwe tune the magnon frequency to be !L\nm=2\u0019=7:658 GHz\nat the drive power Pd=4:7 dBm (see Fig. 2(a)). We have\nthe [110] axis of the YIG sphere aligned parallel to the static\nmagnetic field, which yields a negative self-Kerr coe \u000ecient\nKm=2\u0019=\u00006:5 nHz. An increase in power thus results in a\nnegative magnon frequency shift \u000e!m=2KmjMj2, and the\nmagnon frequency reduces to !H\nm=2\u0019=7:640 GHz when the\npower increases to Pd=29:7 dBm (Fig. 2(a)), which yields\nan e\u000bective coupling Gmb=2\u0019=45:8 kHz. Under these con-\nditions, the magnon excitation number jMj2shows a bistable\nbehavior through variation of the power.\nEquation (6) indicates that the radiation-pressure-like cou-\npling results in a mechanical frequency shift, and so does\nthe cross-Kerr nonlinearity. This is confirmed by the exper-\nimental data in Fig. 3(a). It shows that the cross-Kerr plays\na dominant role because of a large magnon excitation num-\nber, and both the frequency shifts caused, respectively, by\nthe cross-Kerr and the radiation-pressure-like coupling show\na bistable feature with the forward and backward sweeps of\n0 5 10 15 20\nDrive power (dBm)-6-4-20b (kHz)\n0 5 10 15 20\nDrive power (dBm)04080b (Hz)Scan forward\nScan backward\nRadiation-pressure-like \nCross-Kerr\nSum\nScan forward\nScan backward\nTheory(a)\n(b)\nFIG. 3. Bistable mechanical frequency and linewidth under a red-detuned\ndrive. (a) The mechanical frequency shift versus the drive power. The red\n(green) curve is the fitting of the frequency shift induced by the radiation-\npressure-like coupling (cross-Kerr e \u000bect) using Eq. (6), and the black curve\nis the sum of the two contributions. (b) The mechanical linewidth variation\nversus the drive power. The black curve is the fitting of the linewidth change\nusing Eq. (7). In both figures, the blue (orange) triangles are the experimental\ndata obtained via forward (backward) sweep of the drive power.4\nthe drive power. This is because both of them originate from\nthe bistable magnon excitation number jMj2, c.f. Eq. (6). Note\nthat for the spring e \u000bect, the bistability of jMj2is mapped to\nthe magnon frequency shift ˜\u0001m, then to the polariton suscep-\ntibility\u001fma(!), and finally to the mechanical frequency.\nBy increasing the drive power from 4 :7 dBm to 19 :7 dBm,\nthe cross-Kerr causes a maximum frequency shift of \u00004:6 kHz\n(Fig. 3(a), the green line). The fitting cross-Kerr coe \u000e-\ncient is Kcross=2\u0019=\u00005:4 pHz. Under the red-detuned drive,\nthe magnonic spring e \u000bect yields a negative frequency shift\n\u000e!b=Reh\n\u001f\u00001\nb;e\u000b(!)\u0000\u001f\u00001\nb(!)i\n<0 (Fig. 3(a), the red line), and\nthe maximum frequency shift is \u0000200 Hz. Adding up these\ntwo frequency shifts gives the total mechanical frequency shift\n(Fig. 3(a), the black line), which fits well with the experimen-\ntal data (Fig. 3(a), triangles) when the power is not too strong.\nAnother interesting finding is the bistable feature of the me-\nchanical linewidth (Fig. 3(b)). The magnomechanical backac-\ntion leads to the variation of the mechanical linewidth \u000e\u0000b=\n\u0000Imh\n\u001f\u00001\nb;e\u000b(!)\u0000\u001f\u00001\nb(!)i\n. For a red-detuned drive, the anti-\nStokes process is dominant, resulting in an increased mechan-\nical linewidth \u000e\u0000b>0 and the cooling of the motion. The\nbistable mechanical linewidth is also induced by the bistable\njMj2(see Eq. (7)), similar to the mechanical frequency. The\ntheory fits well with the experimental results, and the discrep-\nancy appears only in the high-power regime. This is because\nthe considerable heating e \u000bect at strong pump powers can\nbroaden the mechanical linewidth [70, 71], which is not in-\ncluded in our model.\nBlue-detuned drive.— When a blue-detuned drive is applied,\nthe system enters a regime where the Stokes scattering is dom-\ninant. The magnomechanical parametric down-conversion\namplifies the mechanical motion with the characteristic of a\nreduced linewidth. Furthermore, the mechanical frequency\nshift induced by the spring e \u000bect will move in the opposite\ndirection compared with the red-detuned drive.\nTo implement a blue-detuned drive, we drive the cavity\nwith a microwave field at frequency !d=2\u0019=7:66 GHz, and\ntune the magnon frequency close to the drive frequency (see\nFig. 2(b)). We attempted to make the Stokes sideband of the\ndrive field resonate with the “deeper” polariton at a high pump\npower , such that the Stokes scattering rate is maximized and\nthe magnomechanical coupling strength Gmbbecomes strong.\nHowever, to meet the drive conditions for a bistable magnon\nexcitation number jMj2, the drive frequency is restricted to\na certain range, which hinders us to make the Stokes side-\nband and the “deeper” polariton resonate. Therefore, we only\nachieve this at lower drive powers, giving a faint magnome-\nchanically induced absorption (Fig. 2(b), upper panels).\nFrom the red to the blue detuning, we only adjust the\nmagnon and the drive frequencies. Because the direction of\nthe crystal axis is unchanged, the magnon self-Kerr coe \u000e-\ncient Kmis still negative, so again a negative frequency shift\nby increasing the power (green arrow in Fig. 2(b)). For the\npower up to 23.7 dBm, which yields Gmb=2\u0019=42:7 kHz,\nthe frequency of the cavity-like polariton is always lower than\nthe drive frequency, so the system is operated under a blue-\n0 5 10 15 20 25\nDrive power (dBm)-8-6-4-20b (kHz) Scan forward\nScan backward\nRadiation-pressure-like \nCross-Kerr\nSum\n0 5 10 15 20 25\nDrive power (dBm)-120-80-400b (Hz)\nScan forward\nScan backward\nTheory(a)\n(b)FIG. 4. Bistable mechanical frequency and linewidth under a blue-detuned\ndrive. (a) The mechanical frequency shift and (b) the mechanical linewidth\nchange versus the drive power. The curves and the triangles are shown in the\nsame manner as in Fig. 3.\ndetuned drive.\nFigure 4 displays the bistable mechanical frequency shift\n\u000e!band linewidth change \u000e\u0000b. For the frequency shift,\nboth the contributions from the cross-Kerr and the radiation-\npressure-like coupling should be considered, but the former\nplays a dominant role (Fig. 4(a), the green line), as in the case\nof the red-detuned drive, yielding a frequency shift of \u00006:5\nkHz at the power of 23.7 dBm. Di \u000berently, the spring ef-\nfect induced frequency shift (370 Hz at 23 :7 dBm) is positive\n(Fig. 4(a), the red line). The opposite frequency shifts by the\nspring e \u000bect in the blue and red-detuned drives agree with the\nfinding of Ref. [28], but no bistability was observed in their\nwork.\nThe reduced mechanical linewidth \u000e\u0000b<0 under a blue-\ndetuned drive is confirmed by Fig. 4(b). However, unlike the\nbistable curve in the red-detuned drive case (Fig. 3(b)), \u000e\u0000b\nmanifests the bistability in an alpha -shaped curve by sweep-\ning the drive power. This is the result of the trade-o \u000bbe-\ntween the growing coupling strength Gmb(which enhances\nthe Stokes scattering rate, yielding an increasing j\u000e\u0000bj) and the\nlarger detuning between the Stokes sideband and the “deeper”\npolariton (Fig. 2(b)) (which reduces the Stokes scattering rate,\nresulting in a decreasing j\u000e\u0000bj) by raising the drive power.\nThese two e \u000bects are balanced when the power is in the range\nof 12 dBm to 15 dBm.\nConclusions.— We have observed bistable mechanical fre-\nquency and linewidth and the magnon-phonon cross-Kerr\nnonlinearity in the CMM system. The mechanical bistability\nresults from the magnomechanical backaction on the mechan-\nical mode and the strong modifications on the backaction due\nto the magnon self-Kerr and magnon-phonon cross-Kerr non-5\nlinearities. The e \u000bects of the magnon self-Kerr, the magnon-\nphonon cross-Kerr, and the radiation-pressure-like interac-\ntions can be identified by measuring primarily the magnon\nfrequency shift, the mechanical frequency shift, and the me-\nchanical linewidth, respectively. The new mechanism for\nachieving bistable mechanical motion revealed by this work\npromises a wide range of applications, such as in mechanical\nswitches, memories, logic gates, and signal amplifiers.\nAcknowledgments. This work was supported by the\nNational Natural Science Foundation of China (Grants\nNos. 11934010, U1801661, 12174329, 11874249), Zhejiang\nProvince Program for Science and Technology (Grant No.\n2020C01019), and the Fundamental Research Funds for the\nCentral Universities (No. 2021FZZX001-02).\n\u0003jieli007@zju.edu.cn\nyyipuwang@zju.edu.cn\nzjqyou@zju.edu.cn\n[1] P. D. Drummond, and D. F. Walls, Quantum theory of optical\nbistability. I. Nonlinear polarisability model. J. Phys. 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SYSTEM PARAMETERS\nQuantity Symbol Value\nGyromagnetic ratio \r 2\u0019\u00022.8 MHz /Oe\nFrequency of the cavity TE 101mode !a 2\u0019\u00027.653 GHz\nFrequency of the magnon mode !m \rB0\nFrequency of the lower-frequency phonon mode !b 2\u0019\u000211.0308 MHz\nFrequency of the higher-frequency phonon mode !0\nb2\u0019\u000211.0377 MHz\nTotal cavity decay rate \u0014a 2\u0019\u00022.78 MHz\nCavity decay rate via Port 1 \u00141 2\u0019\u00020.22 MHz\nCavity decay rate via Port 2 \u00142 2\u0019\u00021.05 MHz\nLinewidth of the magnon mode \u0014m 2\u0019\u00022.2 MHz\nLinewidth of the lower-frequency phonon mode \u0014b 2\u0019\u0002550 Hz\nLinewidth of the higher-frequency phonon mode \u00140\nb2\u0019\u0002180 Hz\nCavity-magnon coupling strength gma 2\u0019\u00027.37 MHz\nBare magnomechanical coupling strength (for the lower-frequency mode) gmb 2\u0019\u00021.22 mHz\nBare magnomechanical coupling strength (for the higher-frequency mode) g0\nmb2\u0019\u00020.62 mHz\nMagnon self-Kerr coe \u000ecient Km -2\u0019\u00026.5 nHz\nMagnon-phonon cross-Kerr coe \u000ecient Kcross -2\u0019\u00025.4 pHz\nTABLE S1. List of system parameters.\nTable S1 provides a list of the system parameters. In the experiment, the frequencies and the dissipation rates (linewidths)\nof the cavity and magnon modes and their coupling strength gmaare extracted by fitting the cavity-magnon polariton in the\nreflection spectra using Eq. (S30). The mechanical frequency and linewidth and the magnomechanical coupling strength gmb\nare obtained by fitting the spectra of the magnomechanically induced resonances using also Eq. (S30). The cavity decay rates \u00141\nand\u00142associated with the two ports need to be fitted by measuring the reflection spectrum through the two ports, respectively.\nThe magnon self-Kerr coe \u000ecient Kmof the crystal axis [110] is determined by measuring the magnon frequency shift, which\nis consistent with the value calculated from Eq. (S4). The magnon-phonon cross-Kerr coe \u000ecient Kcrossis obtained by fitting\nthe mechanical frequency shift \u000e!busing Eq. (S26). Specifically, the first term of Eq. (S26) is the frequency shift caused by\nthe magnon-phonon radiation-pressure-like coupling, which can be calculated (e.g., -200 Hz under a red-detuned drive with the\npower 19.7 dBm). The magnon excitation number jMj2can also be calculated by using Eq. (S20). With all these at hand, the\ncross-Kerr coe \u000ecient Kcrosscan then be determined by fitting the mechanical frequency shift.\nII. DERIV ATION OF THE HAMILTONIAN FOR KERR NONLINEARITIES\nIn this section, we provide a detailed derivation of the Hamiltonian HKerrassociated with the two Kerr nonlinear terms, namely,\nthe magnon self-Kerr nonlinearity and the magnon-phonon cross-Kerr nonlinearity. These two terms become appreciable in the\nsystem when the pump field is su \u000eciently strong, such that the system enters a regime where the Kerr-type nonlinearities strongly\nmodify the conventional cavity magnomechanics (CMM), which we term as the Kerr-modified CMM. Here the conventional\nCMM means that there is only a radiation-pressure-like coupling between magnons and vibration phonons [26, 28].\nA. Magnon self-Kerr nonlinearity\nThe magnon self-Kerr nonlinearity originates from the anisotropic field. When the bias magnetic field is aligned along the\n[110] axis of the YIG sphere, the anisotropic field is given by [64]\nHan=3KanMx\n\u00160M2ex+9KanMy\n4\u00160M2ey+KanMz\n\u00160M2ez; (S1)\nwhere Kanis the first-order magnetocrystalline anisotropy constant, and for the YIG at room temperature Kan=\u0000610 J=m3.\nM=(Mx;My;Mz) denotes the magnetization of the YIG sphere, Mis the saturation magnetization, and \u00160is the permeability8\nof vacuum. The anisotropy Hamiltonian reads\nHan=\u0000\u00160\n2Z\nVmM\u0001Hand\u001c;\n=\u0000KanVm\n8M2\u0010\n12M2\nx+9M2\ny+4M2\nz\u0011\n;(S2)\nwhere Vmis the volume of the YIG sphere. By using the relation S=MVm=~\r\u0011(Sx;Sy;Sz) [65], with Sbeing the macrospin\noperator and \rbeing the gyromagnetic ratio, the anisotropy Hamiltonian Hancan be written as\nHan=~=\u00003~Kan\r2\n2M2VmS2\nx\u00009~Kan\r2\n8M2VmS2\ny\u0000~Kan\r2\n2M2VmS2\nz: (S3)\nUsing the Holstein-Primako \u000btransformation [66]: S+=p\n2S\u0000mymm,S\u0000=myp\n2S\u0000mym, and Sz=S\u0000mym, where Sis\nthe total spin number of the macrospin and my(m) is the creation (annihilation) operator of the magnon mode, we obtain\nHan=~\u0019\u000013~S K an\r2\n8M2Vmmym+13~Kan\r2\n16M2Vmmymmym: (S4)\nThe first term of the anisotropy Hamiltonian will modify the magnon frequency !0\nm=!m\u000013~S K an\r2\n8M2Vm, and the second term\naccounts for the magnon self-Kerr nonlinearity, which can be written in the form of Hself–Kerr=~=Kmmymmym, with the self-\nKerr coe \u000ecient Km=13~Kan\r2\n16M2Vm. Note that in deriving the Hamiltonian (S4), we have omitted the constant term and the higher\norder terms.\nB. Magnon-phonon cross-Kerr nonlinearity\nThe magnetoelastic coupling describes the interaction between the magnetization and the elastic strain of the magnetic ma-\nterial. Depending on the distance between magnetic atoms (or ions), there are di \u000berent kinds of interactions: the spin-orbital\ninteraction, the exchange interaction between magnetic atoms (or ions), and the magnetic dipole-dipole interaction [64]. In a\ncubic crystal, the magnetoelastic energy density is given by [67]\nfme=b1\nM2\u0010\nM2\nx\u000fxx+M2\ny\u000fyy+M2\nz\u000fzz\u0011\n+2b2\nM2\u0010\nMxMy\u000fxy+MxMz\u000fxz+MyMz\u000fyz\u0011\n; (S5)\nwhere b1andb2are the magnetoelastic coupling constants, and the strain tensor \u000fi jis given in the nonlinear Euler-Bernoulli\ntheory by [61, 62]\n\u000fi j=1\n20BBBBB@@ui\n@lj+@uj\n@li+X\nk@uk\n@li@uk\n@lj1CCCCCA; (S6)\nwith uibeing the components of the displacement vector. The two first-order terms in \u000fi jlead to the magnon-phonon radiation-\npressure-like coupling (see [63] for a strict derivation). In typical CMM experiments using a moderate drive field [26, 28], the\nsecond-order terms are neglected. However, for an intense drive field as used in our experiment, those terms will produce a\nnoticeable e \u000bect. As we will derive below, those second-order terms in the strain tensor are responsible for the magnon-phonon\ncross-Kerr nonlinearity.\nIn the magnetoelastic energy density (S5), the second term accounts for the parametric magnon generation when the phonon\nfrequency is twice the magnon frequency, or the linear magnon-phonon coupling when they are nearly resonant [26, 63]. Thus,\nthis term is negligible for our system with a low mechanical frequency !b\u001c!m, and we can only consider the first term in (S5).\nIntegrating over the whole volume of the YIG sphere, the interaction Hamiltonian can be written as\nH1=b1\nM2Z\ndl3\u0010\nM2\nx\u000fxx+M2\ny\u000fyy+M2\nz\u000fzz\u0011\n;\n'b1\nM~\r\nVmmymZ\ndl3\u0010\n\u000fxx+\u000fyy\u00002\u000fzz\u0011\n:(S7)\nTo quantize the above Hamiltonian, we express the displacement vector uias a superposition\n~u=X\nn;m;ld(n;m;l)~\u001f(n;m;l)(x;y;z); (S8)9\nwhere~\u001f(n;m;l)(x;y;z)is the displacement eigenmode with the corresponding amplitude d(n;m;l), and the superscript n;m;ldenote\nthe mode indices. The mechanical displacement can be quantized as d(n;m;l)=d(n;m;l)\nzpm (bn;m;l+by\nn;m;l), where d(n;m;l)\nzpm is the amplitude\nof the zero-point motion, and bn;m;l(by\nn;m;l) is the bosonic annihilation (creation) operator of the mechanical mode. Substituting\n(S8) into the Hamiltonian (S7), we obtain\nH1'X\nn;m;l~g(n;m;l)\nmbmym\u0010\nbn;m;l+by\nn;m;l\u0011\n+X\nn;m;l~K(n;m;l)\ncross mymby\nn;m;lbn;m;l;(S9)\nwhere we use the rotating-wave approximation by neglecting the fast-oscillating terms in deriving the second term. This is valid\nwhen!b\u001dKcrossjMj2, which is well satisfied in our experiment. The first term describes the magnon-phonon radiation-pressure-\nlike interaction, and the second term accounts for the cross-Kerr interaction between the magnon and mechanical modes. The\nradiation-pressure-like coupling strength g(n;m;l)\nmband the cross-Kerr coe \u000ecient K(n;m;l)\ncross are given by\ng(n;m;l)\nmb=b1\nM\r\nVmZ\ndl3d(n;m;l)\nzpm0BBBBB@@\u001f(n;m;l)\nx\n@x+@\u001f(n;m;l)\ny\n@y\u00002@\u001f(n;m;l)\nz\n@z1CCCCCA;\nK(n;m;l)\ncross =b1\nM\r\nVmZ\ndl3d(n;m;l)2\nzpmX\nk2666666640BBBBB@@\u001f(n;m;l)\nk\n@x1CCCCCA2\n+0BBBBB@@\u001f(n;m;l)\nk\n@y1CCCCCA2\n\u000020BBBBB@@\u001f(n;m;l)\nk\n@z1CCCCCA2377777775:(S10)\nWhen considering a specific mechanical mode as in our experiment, the interaction Hamiltonian takes a simple form of\nH1=~=gmbmym\u0010\nb+by\u0011\n+Kcrossmymbyb: (S11)\nIII. HAMILTONIAN OF THE KERR-MODIFIED CA VITY MAGNOMECHANICAL SYSTEM\nThe CMM system under study consists of a microwave cavity mode, a magnon (Kittel) mode, and a mechanical vibration\nmode. The magnon mode couples to the cavity mode by the magnetic-dipole interaction, and to the mechanical mode by the\nmagnetostrictive interaction. There is no direct coupling between the cavity and the mechanics. The drive field is applied to the\ncavity mode via the Port 1 of the cavity, and the probe field is sent via the Port 2 of the cavity. Under a strong pump, the magnon\nself-Kerr and magnon-phonon cross-Kerr nonlinearities are activated in the system. The total Hamiltonian of the CMM system\nis given by\nH=~=!aaya+!mmym+!bbyb+gma(aym+amy)+gmbmym\u0010\nb+by\u0011\n+Kmmymmym\n+Kcrossmymbyb+p\u00141\"d\u0010\naei!dt+aye\u0000i!dt\u0011\n+p\u00142\"p\u0010\naei!pt+aye\u0000i!pt\u0011\n;(S12)\nwhere\u00141(\u00142) is the cavity decay rate associated with the driving (probe) port, \"d=q\nPd\n~!dand\"p=q\nPp\n~!p, with Pd(Pp) and!d\n(!p) being the power and frequency of the drive (probe) microwave field. Since in the experiment the probe field is of a much\nsmaller power than that of the drive field and thus can be treated as a perturbation to the system, we therefore omit the probe\nterm in the Hamiltonian, giving rise to the Hamiltonian (1) that is provided in the main text\nH=~=!aaya+!mmym+!bbyb+gma(aym+amy)+gmbmym\u0010\nb+by\u0011\n+Kmmymmym+Kcrossmymbyb+p\u00141\"d\u0010\naei!dt+aye\u0000i!dt\u0011\n:(S13)\nIV . DETERMINATION OF THE MAGNON EXCITATION NUMBER\nFrom the Hamiltonian (S13), we can obtain the Heisenberg-Langevin equations by including the dissipation and input noise\nof each mode. In the frame rotating at the drive frequency, they are given by\nda\ndt=\u0000\u0012\ni\u0001a+\u0014a\n2\u0013\na\u0000igmam\u0000ip\u00141\"d+p\u0014aain;\ndm\ndt=\u0000\u0014\ni\u0010\n\u0001m+2Kmmym+Km+Kcrossbyb\u0011\n+igmb(b+by)+\u0014m\n2\u0015\nm\u0000igmaa+p\u0014mmin;\ndb\ndt=\u0000\u0014\ni\u0010\n!b+Kcrossmym\u0011\n+\u0014b\n2\u0015\nb\u0000igmbmym+p\u0014bbin;(S14)10\nwhere \u0001a=!a\u0000!d, and \u0001m=!m\u0000!d, while\u0014a,\u0014mand\u0014b(ain,min, and bin) are the dissipation rates (input noises) of the three\nmodes. Since the cavity mode is strongly driven, this leads to a large amplitude jhaij\u001d 1 in the steady state, and further due\nto the cavity-magnon coupling, the magnon mode also has a large amplitude jhmij\u001d 1. This allows us to linearize the system\ndynamics around the classical average values by writing the mode operators as a\u0011A+\u000ea,m\u0011M+\u000em, and b\u0011B+\u000eb, and\nneglecting small second-order fluctuation terms [68]. Substituting these mode operators into Eq. (S14), the equations are then\nseparated into two sets of equations, respectively, for classical averages ( A,M,B) and for quantum fluctuations ( \u000ea,\u000em,\u000eb).\nThe equations for the classical averages in the steady state are as follows:\n\u0012\n\u0001a\u0000i\u0014a\n2\u0013\nA+gmaM+p\u00141\"d=0;\n\u0014\n\u0001m+2KmjMj2+Km+KcrossjBj2+gmb(B+B\u0003)\u0000i\u0014m\n2\u0015\nM+gmaA=0;\n\u0012\n!b+KcrossjMj2\u0000i\u0014b\n2\u0013\nB+gmbjMj2=0:(S15)\nFrom the first equation of Eq. (S15), we get\nA=\u0000\u0011agma\u0012\n\u0001a+i\u0014a\n2\u0013\nM\u0000\u0011ap\u00141\u0012\n\u0001a+i\u0014a\n2\u0013\n\"d; (S16)\nwhere\u0011a=1\n\u00012a+(\u0014a\n2)2. Substituting Ainto the second equation of Eq. (S15), we obtain\n\u0014\n\u0001m\u0000\u0011ag2\nma\u0001a+2KmjMj2+Km+KcrossjBj2+2gmbRe[B]\u0000i\u0012\u0014m\n2+\u0011ag2\nma\u0014a\n2\u0013\u0015\nM+\u0011ap\u00141gma(\u0001a+i\u0014a\n2)\"d=0: (S17)\nMultiplying Eq. (S17) with its complex conjugate, we obtain the equation for the magnon excitation number\n\"\u0010\n\u0001m\u0000\u0011ag2\nma\u0001a+2KmjMj2+Km+KcrossjBj2+2gmbRe[B]\u00112+\u0012\u0014m\n2+\u0011ag2\nma\u0014a\n2\u00132#\njMj2=\u0011a\u00141g2\nma\"2\nd: (S18)\nSimilarly, we get the equation for the phonon excitation number\n\"\u0010\n!b+KcrossjMj2\u00112+\u0012\u0014b\n2\u00132#\njBj2=g2\nmbjMj4: (S19)\nIn our experiment, the drive power is scanned from 4.7 dBm to 23.7 dBm, which gives the magnon excitation number jMj22\n[1012;1015], and the phonon excitation number jBj22[106;1010]. Thus, the phonon excitation number is much smaller than\nthat of the magnon. Using our parameters gmb=2\u0019=1:22 mHz, Km=2\u0019=\u00006:5 nHz, and Kcross=2\u0019=\u00005:4 pHz, the magnon\nfrequency shift \u000e!m=2KmjMj2+Km+KcrossjBj2+2gmbRe[B]\u00192KmjMj2. Therefore, Eq. (S18) is reduced to\n\"\u0010\n\u0001m\u0000\u0011ag2\nma\u0001a+2KmjMj2\u00112+\u0012\u0014m\n2+\u0011ag2\nma\u0014a\n2\u00132#\njMj2=\u0011a\u00141g2\nma\"2\nd: (S20)\nThis is a cubic equation of the magnon excitation number jMj2, and given as Eq. (4) in the main text. Under certain conditions,\nall the three solutions of jMj2are real, among which there are two stable solutions. The stable solutions can be measured in the\nexperiment, and it shows a hysteresis loop by varying the drive power.\nV . EFFECTIVE SUSCEPTIBILITY OF THE MECHANICAL MODE\nThe magnon-phonon radiation-pressure-like coupling gives rise to the magnomechanical backaction on the mechanical mode,\nwhich is manifested as the mechanical frequency shift (i.e., the magnonic spring e \u000bect) and the increased (reduced) linewidth\nassociated with the cooling (amplification) of the mechanical mode. The frequency shift and the linewidth variation can be eval-\nuated from the e \u000bective susceptibility of the mechanical mode. In what follows, we show in detail how the e \u000bective mechanical\nsusceptibility is derived.\nThe linearization of the Langevin equations (S14) yields a set of linearized quantum Langevin equations for the quantum\nfluctuations ( \u000em;\u000ea;\u000ex;\u000ep), where\u000ex=(\u000eb+\u000eby)=p\n2 and\u000ep=i(\u000eby\u0000\u000eb)=p\n2 denote the fluctuations of two mechanical11\nquadratures (position and momentum). By taking the Fourier transform, we obtain the following equations in the frequency\ndomain:\n\u0000i!\u000em=\u0000\u0012\ni˜\u0001m+\u0014m\n2\u0013\n\u000em\u0000igma\u000ea\u0000ip\n2Gmb\u000ex+p\u0014mmin;\n\u0000i!\u000emy=\u0000\u0012\n\u0000i˜\u0001m+\u0014m\n2\u0013\n\u000emy+igma\u000eay+ip\n2G\u0003\nmb\u000ex+p\u0014mmy\nin;\n\u0000i!\u000ea=\u0000\u0012\ni\u0001a+\u0014a\n2\u0013\n\u000ea\u0000igma\u000em+p\u0014aain;\n\u0000i!\u000eay=\u0000\u0012\n\u0000i\u0001a+\u0014a\n2\u0013\n\u000eay+igma\u000emy+p\u0014aay\nin;\n\u0000i!\u000ex=˜!b\u000ep;\n\u0000i!\u000ep=\u0000˜!b\u000ex\u0000\u0014b\u000ep\u0000p\n2\u0010\nG\u0003\nmb\u000em+Gmb\u000emy\u0011\n+\u0018;(S21)\nwhere ˜\u0001m'\u0001m+2KmjMj2includes the magnon frequency shift dominantly caused by the magnon self-Kerr e \u000bect, and ˜!b=\n!b+KcrossjMj2includes the mechanical frequency shift due to the magnon-phonon cross-Kerr e \u000bect.Gmb=gmbMis the e \u000bective\nradiation-pressure-like coupling strength. Note that we adopt an equivalent model of dealing with the mechanical damping\nand input noise, where the damping rate \u0014band the Hermitian Brownian noise operator \u0018are added only in the momentum\nequation [68]. Solving separately the two equations for each mode, we obtain the following equations:\n\u000em=\u001fm(!)\u0010\n\u0000igma\u000ea\u0000ip\n2Gmb\u000ex+p\u0014mmin\u0011\n;\n\u000emy=\u001f\u0003\nm(\u0000!)\u0010\nigma\u000eay+ip\n2G\u0003\nmb\u000ex+p\u0014mmy\nin\u0011\n;\n\u000ea=\u001fa(!)\u0010\n\u0000igma\u000em+p\u0014aain\u0011\n;\n\u000eay=\u001f\u0003\na(\u0000!)\u0010\nigma\u000emy+p\u0014aay\nin\u0011\n;\n\u000ex=\u001fb(!)\u0010\n\u0000p\n2G\u0003\nmb\u000em\u0000p\n2Gmb\u000emy+\u0018\u0011\n;\n\u000ep=\u0000i!\n˜!b\u000ex;(S22)\nwhere we define the natural susceptibilities of the magnon, cavity, and mechanical modes as\n\u001fm(!)=1\ni\u0010˜\u0001m\u0000!\u0011\n+\u0014m\n2; \u001f\u0003\nm(\u0000!)=1\n\u0000i\u0010˜\u0001m+!\u0011\n+\u0014m\n2;\n\u001fa(!)=1\ni(\u0001a\u0000!)+\u0014a\n2; \u001f\u0003\na(\u0000!)=1\n\u0000i(\u0001a+!)+\u0014a\n2;\n\u001fb(!)=˜!b\n˜!2\nb\u0000!2\u0000i\u0014b!:(S23)\nSolving the first four equations in Eq. (S22) for \u000emand\u000emy, and inserting their solutions into the equation of \u000ex, we obtain\n\u000ex=\u001fb;e\u000b(!)0BBBBBB@\u0018+ip\n2G\u0003\nmb\u001fm(!)\u0010\nip\u0014mmin+p\u0014agma\u001fa(!)ain\u0011\n1+g2ma\u001fa(!)\u001fm(!)\u0000ip\n2Gmb\u001f\u0003\nm(\u0000!)\u0010\n\u0000ip\u0014mmy\nin+p\u0014agma\u001f\u0003\na(\u0000!)ay\nin\u0011\n1+g2ma\u001f\u0003a(\u0000!)\u001f\u0003m(\u0000!)1CCCCCCA; (S24)\nwhere\u001fb;e\u000bis the e \u000bective mechanical susceptibility, defined as\n\u001fb;e\u000b(!)=\u0010\n\u001f\u00001\nb(!)\u00002ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001\u0011\u00001; (S25)\nwith\u001fma(!)=1\n\u001f\u00001m(!)+g2ma\u001fa(!). The change of the mechanical frequency and linewidth can be extracted from the e \u000bective suscep-\ntibility: the real part of \u001f\u00001\nb;e\u000b(!)\u0000\u001f\u00001\nb(!) corresponds to the mechanical frequency shift\n\u000e!b=\u0000Reh\n2ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001i\n+KcrossjMj2; (S26)\nwhere we write together the frequency shift due to the cross-Kerr e \u000bect, and the imaginary part of \u001f\u00001\nb(!)\u0000\u001f\u00001\nb;e\u000b(!) yields the\nvariation of the mechanical linewidth\n\u000e\u0000b=Imh\n2ijGmbj2\u0000\u001fma(!)\u0000\u001f\u0003\nma(\u0000!)\u0001i\n: (S27)12\nVI. REFLECTION SPECTRUM OF THE PROBE FIELD\nHere we show how to derive the reflection spectrum of the probe field under the strong drive field. The Hamiltonian including\nthe probe field is given in Eq. (S12). The reflection spectrum can be conveniently solved by including the strong pump e \u000bects\ninto the linearized Langevin equations. Following the linearization approach used in Sec. IV and Sec. V , the Hamiltonian (S12)\nleads to the following Langevin equations for the classical averages in the frequency domain:\n\u0000i!M=\u0000\u0012\ni˜\u0001m+\u0014m\n2\u0013\nM\u0000igmaA\u0000ip\n2GmbX;\n\u0000i!A=\u0000\u0012\ni\u0001a+\u0014a\n2\u0013\nA\u0000igmaM\u0000ip\u00142\"p\u000e(!p\u0000!d\u0000!);\n\u0000i!X=˜!bP;\n\u0000i!P=\u0000˜!bX\u0000\u0014bP\u0000p\n2G\u0003\nmbM;(S28)\nwhere X=(B+B\u0003)=p\n2 and P=i(B\u0003\u0000B)=p\n2 denote the classical averages of the mechanical position and momentum. Note\nthat, same as Eqs. (S14) and (S21), the above equations are provided in the frame rotating at the drive frequency !d.\nSolving the above equations, we obtain\nA(!)=ip\u00142\u0010\n1\u00002ijGmbj2\u001fb(!)\u001fm(!)\u0011\n\u0000g2\nmb(!)\u001fm(!)\u0000\u001f\u00001a(!)\u00001\u00002ijGmbj2\u001fb(!)\u001fm(!)\u0001\"p: (S29)\nUsing the input-output theory, Aout=\"p+ip\u00142A, we therefore achieve the reflection spectrum of the probe field\nr(!)\u0011Aout\n\"p=1\u0000\u00142\u0010\n1\u00002ijGmbj2\u001fb(!)\u001fm(!)\u0011\n\u0000g2\nmb(!)\u001fm(!)\u0000\u001f\u00001a(!)\u00001\u00002ijGmbj2\u001fb(!)\u001fm(!)\u0001: (S30)" }, { "title": "1707.03754v1.Superconductivity_Induced_by_Interfacial_Coupling_to_Magnons.pdf", "content": "Superconductivity Induced by Interfacial Coupling to Magnons\nNiklas Rohling, Eirik L\u001chaugen Fj\u001arbu, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim\nWe consider a thin normal metal sandwiched between two ferromagnetic insulators. At the in-\nterfaces, the exchange coupling causes electrons within the metal to interact with magnons in the\ninsulators. This electron-magnon interaction induces electron-electron interactions, which, in turn,\ncan result in p-wave superconductivity. In the weak-coupling limit, we solve the gap equation nu-\nmerically and estimate the critical temperature. In YIG-Au-YIG trilayers, superconductivity sets\nin at temperatures somewhere in the interval between 1 and 10 K. EuO-Au-EuO trilayers require a\nlower temperature, in the range from 0.01 to 1 K.\nThe interactions between electrons in a conductor and\nordered spins across interfaces are of central importance\nin spintronics [1, 2]. Here, we focus on the case in which\nthe magnetically ordered system is a ferromagnetic insu-\nlator (FI). The interaction at an FI-normal metal (NM)\ninterface can be described in terms of an exchange cou-\npling [3{6]. In the static regime, this coupling induces\ne\u000bective Zeeman \felds near the boundary [7{10]. The\nmagnetization dynamics caused by the coupling can be\ndescribed in terms of the spin-mixing conductance [4{\n6]. Such dynamics can include spin pumping from the\nFI into the NM [11, 12] and its reciprocal e\u000bect, spin-\ntransfer torques [5, 13]. These spin-transfer torques en-\nable electrical control of the magnetization in FIs [14].\nOne important characteristic of FIs is that the Gilbert\ndamping is typically small. This leads to low-dissipation\nmagnetization dynamics [15], which, in turn, facilitates\ncoherent magnon dynamics and the long-range transport\nof spin signals [5, 13]. These phenomena should also en-\nable other uses of the quantum nature of the magnons.\nHere, we study a previously unexplored e\u000bect that is\nalso governed by the electron-magnon interactions at FI-\nNM interfaces but is qualitatively di\u000berent from spin\npumping and spin-transfer torques. We explore how\nthe magnons in FIs can mediate superconductivity in a\nmetal. The exchange coupling at the interfaces between\nthe FIs and the NM induces Cooper pairing. In this\nscenario, the electrons and the magnons mediating the\npairing reside in two di\u000berent materials. This opens up\na wide range of possibilities for tuning the superconduct-\ning properties of the system by combining layers with the\ndesired characteristics. The electron and magnon disper-\nsions within the layers as well as the electron-magnon\ncoupling between the layers in\ruence the pairing mecha-\nnism. Consequently, the superconducting gap can also be\ntuned by modifying the layer thickness, interface quality,\nand external \felds.\nSince the interactions occur at the interfaces, the con-\nsequences of the coupling are most profound when the\nNM layer is thin. We therefore consider atomically thin\nFI and NM layers. This also reduces the complexity of\nthe calculations. For thicker layers, multiple modes exist\nalong the direction transverse to the interface ( x), withdi\u000berent e\u000bective coupling strengths. We expect a qual-\nitatively similar, but somewhat weaker, e\u000bect for thicker\nlayers.\nParamagnonic [16] or magnonic [17] coupling may ex-\nplain experimental observations of superconductivity co-\nexisting with ferromagnetism in bulk materials [18{20].\nParamagnons [16, 21] and magnons [17, 22] are predicted\nto mediate triplet p-wave pairing with equal and antipar-\nallel spins, respectively.\nHigh-quality thin \flms o\u000ber new possibilities for super-\nconductivity [23]. Consequently, the emergence of super-\nconductivity at interfaces has recently received consid-\nerable attention [23{31]. Theoretical studies have been\nconducted on interface-induced superconductivity medi-\nated by phonons [27{29], excitons [32], and polarizable\nlocalized excitations [31, 33].\nA model of interface-induced magnon-mediated d-wave\npairing has been proposed to explain the observed super-\nconductivity in Bi/Ni bilayers [34]. A p-wave pairing\nof electrons with equal momentum|so-called Amperean\npairing|has been predicted to occur in a similar sys-\ntem [35]. Importantly, the electrons that form pairs in\nthese models reside in a spin-momentum-locked surface\nconduction band.\nBy contrast, we consider a spin-degenerate conduction\nband in an FI-NM-FI trilayer system. We \fnd interfa-\ncially mediated p-wave superconductivity with antipar-\nallel spins and momenta. These pairing symmetries are\ndistinct from those of the 2D systems mentioned above.\nWe assume that the equilibrium magnetization of the left\n(right) FI is along the ^ z(\u0000^z) direction; see Fig. 1. We\nconsider matching square lattices, with lattice constant\na, in all three monolayers. The interfacial plane com-\nprisesNsites with periodic boundary conditions. The\nHamiltonian is\nH=HA\nFI+HB\nFI+HNM+Hint; (1)\nwhere we use A(B) to denote the left (right) FI.\nThe Heisenberg Hamiltonian\nHA\nFI=\u0000J\n~2X\niX\nj2NN(i)SA\ni\u0001SA\nj (2)arXiv:1707.03754v1 [cond-mat.mes-hall] 12 Jul 20172\nFI\nNM\nFI\nFIG. 1. A trilayer formed of a normal metal between two fer-\nromagnetic insulators. The magnetizations are antiparallel.\nAt the interfaces, conduction electrons couple to magnons.\nThis results in e\u000bective electron-electron interactions in the\nmetal.\ndescribes the left FI. Here, iis an in-plane site, NN( i) is\nthe set of its nearest neighbors, Jis the exchange interac-\ntion, and SA\niis the localized spin at site i. The expression\nforHB\nFIis similar.\nFor the time being, we assume that the conduction\nelectron eigenstates in the NM are plane waves of the\nformcq;\u001b=P\njexp(irj\u0001q)cj\u001b=p\nN. Here,c(y)\nj\u001bannihi-\nlates (creates) a conduction electron with spin \u001bat site\njin the NM, and qis the wavevector. For now, the\nNM Hamiltonian is HNM=P\nqP\n\u001bEqcy\nq\u001bcq\u001b, and the\ndispersion is quadratic,\nEq=~2q2=(2m): (3)\nHere,mis the e\u000bective electron mass. Below, when esti-\nmating the coupling JIat YIG-Au interfaces, we consider\nanother Hamiltonian with di\u000berent eigenstates and a dif-\nferent dispersion.\nWe model the coupling between the conduction elec-\ntrons and the localized spins as an exchange interaction\nof strength JI:\nHint=\u00002JI\n~X\n\u001b\u001b0X\njX\nL=A;Bcy\nj\u001b\u001b\u001b\u001b0cj\u001b0\u0001SL\nj;(4)\nwhere \u001b= (\u001bx;\u001by;\u001bz) is a vector of Pauli matrices.\nAfter a Holstein-Primako\u000b transformation, we expand\nthe Heisenberg Hamiltonian given in Eq. (2) up to second\norder in the bosonic operators and diagonalize it. We rep-\nresent SA\njbySA\njx+iSA\njy=~p\n2saj,SA\njx\u0000iSA\njy=~p\n2say\nj,\nandSA\njz=~(s\u0000ay\njaj), wheresis the spin quantum num-\nber of the localized spins and a(y)\njis a bosonic annihila-\ntion (creation) operator at site j. The magnons in layer\nA, with the form ak=P\nj2Aexp(irj\u0001k)aj=p\nN, are the\neigenstates of the resulting Hamiltonian. Analogously,\nthe magnons in layer Bare denoted by bk. The magnon\ndispersion is\n\"k= 4sJ[2\u0000cos(kya)\u0000cos(kza)]: (5)\nWe disregard second-order terms in the bosonic operatorsfrom the interfacial coupling and obtain\nH=X\nk\"k(ay\nkak+by\nkbk) +X\nq\u001bEqcy\nq\u001bcq\u001b\n+X\nkqV(akcy\nq+k;#cq\"+bkcy\nq+k;\"cq#) + h.c.;(6)\nwhereV=\u00002JIps=p\n2Nis the coupling strength be-\ntween the electrons in the NM and the magnons in the\nFI layers.\nThere is no induced Zeeman \feld in the NM since the\nmagnetizations in the FIs are antiparallel. Analogously\nto phonon-mediated coupling in conventional supercon-\nductors, the magnons mediate e\u000bective interactions be-\ntween the electrons. For electron pairs with opposite mo-\nmenta, we obtain\nHpair=X\nkk0Vkk0cy\nk#cy\n\u0000k\"c\u0000k0\"ck0#; (7)\nwith the interaction strength\nVkk0= 2jVj2\"k+k0\n\"2\nk+k0\u0000(Ek\u0000Ek0)2: (8)\nWe de\fne the gap function in the usual way: \u0001 k=P\nk0Vkk0hc\u0000k0\"ck0#i. The gap equation becomes\n\u0001k=\u0000X\nk0Vkk0\u0001k0\n2~Ek0tanh ~Ek0\n2kBT!\n; (9)\nwhere ~Ek=p\n(Ek\u0000EF)2+j\u0001kj2andEFis the Fermi\nenergy.\nIn the continuum limit, we replace the discrete sum\nover momenta kwith integrals over E=Ekand the\nangle', where k=k[sin(');cos(')]. We assume that\nonly the conduction electrons close to the Fermi surface\nform pairs. The magnon energy that appears in Eq. (8)\nis then given by \"k+k0\u0019\"('0;'), where\n\"('0;') =4sJf2\u0000cos(kFa[sin'+ sin'0])\n\u0000cos(kFa[cos'+ cos'0])g:(10)\nHere,kF=p2mEF=~is the Fermi wavenumber. We\nassume that the NM is half \flled, kF=p\n2\u0019=a. We\nintroduce the energy scale E\u0003= 4sJk2\nFa2= 8\u0019sJ, which\nis associated with the FI exchange interaction. Then, we\nscale all other energies with respect to E\u0003:\u000e= \u0001=E\u0003,\n\u001c=kBT=E\u0003,x= (E\u0000EF)=E\u0003, ~x=~E=E\u0003, and\u000f=\n\"=E\u0003. In this way, the gap equation presented in Eq. (9)\nsimpli\fes to\n\u000e(x;') =\u0000p\n2\u000b\n\u0019xBZ\n\u0000xBdx02\u0019Z\n0d'0\u000f('0;')\u000e(x0;'0) tanhh\n~x0\n2\u001ci\n~x0[\u000f2('0;')\u0000(x\u0000x0)2];\n(11)\nwith the dimensionless coupling constant \u000b=\nJ2\nI=(16p\n2\u0019EFJ) =J2\nIma2=(16p\n2\u00192~2J). In Eq. (11),3\nwe have restricted the energy integral to the range\n[EF\u0000xBE\u0003;EF+xBE\u0003]. We choose xB|based on\nthe value of \u000b|in the following way. xBmust be suf-\n\fciently large that all contributions to the gap from re-\ngions outside this range are vanishingly small. In the\nweak-coupling limit ( \u000b\u001c1), the gap function has a nar-\nrow peak near x= 0, and therefore, xBcan be much\nsmaller than 1.\nTo gain a better understanding, we \frst assume a\nquadratic dispersion for the magnons, which matches\nthat of Eq. (5) in the long-wavelength limit. Conse-\nquently, the dimensionless magnon energy \u000f('0;') be-\ncomes\u000fq('0;') = 1 + cos( '0\u0000'). Below, we numerically\ncheck the correspondence between the solutions result-\ning from the full dispersion versus the solutions obtained\nwith the quadratic approximation assumed here. For the\nquadratic magnon dispersion, the gap equation has a so-\nlution with p-wave symmetry, \u000e(x;') =f(x) exp(\u0006i').\nApplying this ansatz to Eq. (11), we calculate the in-\ntegral over the angle '0in the weak-coupling limit [36].\nThe gap equation becomes\nf(x) =\u000bxBZ\n\u0000xBdx0V(x\u0000x0)f(x0) tanh\u0014p\nx02+f(x0)2\n2\u001c\u0015\np\nx02+f(x0)2;\n(12)\nwhereV(x\u0000x0)\u00191=p\njx\u0000x0j\u00002p\n2.\nUsing a Gaussian centered at x= 0 as an initial\nguess, we solve Eq. (12) numerically through iteration\n[37]. Fig. 2 shows the results. For a \fxed coupling \u000b,\nthe maximum value occurs when x= 0 and\u001c= 0. The\ndimensionless critical temperature \u001ccis the temperature\nat which the gap vanishes. As in the BCS theory, the gap\nequation can also be solved analytically by approximat-\ningV(x) as a constant with a cuto\u000b centered at x= 0. In\nthis constant-potential approximation, the ratio fmax=\u001cc\nis approximately 1 :76, which is slightly lower than what\nwe \fnd numerically; see Fig. 2 (c).\nLet us check that the numerical solutions to Eq. (12),\nfor the quadratic magnon energy, resemble the solutions\nto Eq. (11) for the full magnon energy of Eq. (10). To this\nend, we numerically iterate Eq. (11), starting from the\nsolution to Eq. (12) as the initial guess [38]. We consider\nthe case of zero temperature, \u001c= 0. The symmetries\n\u000e(x;') =\u000e(\u0000x;') =i\u000e(x;'+\u0019=2) =\u000e\u0003(x;\u0000'), where\n\u000e\u0003is the complex conjugate of \u000e, imply that we need to\nconsider only x > 0 and 0< ' < \u0019= 4. We show the\nresults of these iterative calculations in Fig. 3. The third\niteration of \u000eis shown in Fig. 3 (a,b). After only three\niterations, the di\u000berences between consecutive functions\nare already nearly imperceptible; see Fig. 3 (c,d). The\ngap as a function of energy still exhibits a peak at the\nFermi energy. Compared with the results obtained for\na quadratic magnon dispersion, this peak is of a similar\nshape but is slightly lower and narrower; see the inset of\nFig. 3 (c). There are also additional features of \u000e(x;')\n0510\nx×1030246f×104 (a)\n0123\nτ×1040246f(x=0)×104\n(b)\n-3-2-1\nlog10(α)1.922.1fmax/τc (c)\n-3-2-1\nlog10(α)-4-3-2-1log10(fmax)(d)FIG. 2. Numerical solutions to the gap equation (12) deter-\nmined through iteration. (a) Gaussian-shaped initial guess\n(dashed line) and the results of the \frst eight iterative cal-\nculations of the gap f(x) (from light blue to red) when the\ndimensionless temperature is \u001c= 0 and the coupling constant\nis\u000b= 0:005. Note that f(\u0000x) =f(x) and that the energy\ncuto\u000bxB\u00190:03 lies outside the range of the plot. (b) Gap\nfat energyx= 0 as a function of \u001cfor\u000b= 0:005. (c) Ratio\nbetween the maximum gap value, fmax, and the dimensionless\ncritical temperature \u001ccas a function of \u000b. (d)\u000bdependence of\nfmax. The gray line corresponds to a quadratic dependence,\nfmax\u0018\u000b2.\nat positions ( x;') = (\u000f('0;');') in the parameter space\nwhere the derivative of \u000f('0;') with respect to '0van-\nishes.\nNext, we estimate the critical temperatures Tcfor two\npossible experimental realizations, one in which the FI\nis yttrium-iron-garnet (YIG) and one in which the FI is\neuropium oxide (EuO). The NM layer is gold in both\ncases. We consider the YIG-Au-YIG trilayer \frst.\nFor the FIs, we assume|encouraged by the results pre-\nsented in Fig. 3|that the low-energy magnons dominate\nthe gap. The relevant magnons can therefore be well de-\nscribed by a quadratic dispersion. Our model assumes\nthat the FI and NM layers have the same lattice struc-\nture. However, in reality, the unit cell of YIG is much\nlarger than that of Au. To capture the properties of YIG\nin our model, we \ft the parameters such that the FIs\nhave the same exchange sti\u000bness ( D=kB= 71 K nm2[39])\nand saturation magnetization ( Ms= 1:6\u0001105A/m [39])\nas those of bulk YIG. We assume that each YIG layer\nhas a thickness equal to the bulk lattice constant of YIG\n(aYIG\u001912\u0017A [39]). We use the thickness, the saturation\nmagnetization and the electron gyromagnetic ratio \reto4\n0246\n00.10.20.3\nx0π\n8π\n4ϕ\n(a)|δ3|×104\n0π2π\n00.10.20.3\nx0π\n8π\n4 ϕ\n(b)arg/braceleftbig\nδ3/bracerightbig\n00.050.1\nx0246|δi(x,ϕ=0)|×104\n(c)\n0.002.0050246\n0π/8π/4\nϕ0246 |δi(x=0,ϕ)|×104\n(d)\n0π\n8π\n4\narg/braceleftbig\nδi(x=0,ϕ)/bracerightbig\nFIG. 3. Numerical iteration of the gap equation (11), starting\nfrom the solution to Eq. (12) as an initial guess, for \u001c= 0\nand\u000b= 0:005. (a,b) Absolute value and phase of \u000e3(x;'),\nwhere the index 3 indicates the number of iterations. (c)\nj\u000ei(x;' = 0)jfori= 0 (orange line), i= 2 (black dashed\nline), andi= 3 (gray circles). (d) j\u000ei(x= 0;')j(left axis)\nfori= 0;2;3, with the same colors as in (c), and the phase\nof\u000ei(x= 0;') (right axis) for i= 0 (purple), i= 2 (blue,\ndashed),i= 3 (cyan, wide). Note that the di\u000berence from\nthe second to the third iteration is nearly indiscernible.\nestimate the spin quantum number s=MsaYIGa2=(~\re).\nUsing the quadratic dispersion approximation, we deter-\nmine the exchange interaction to be J=D=(2a2s). The\nlattice spacing aremains undetermined.\nIn the bulk, gold has an fcc lattice and a half-\flled con-\nduction band. We use experimental values of the Fermi\nenergy (EB\nF= 5:5 eV [40]) and the Sharvin conductance\n(gSh= 12 nm\u00002[6]) to determine the e\u000bective mass,\nm= 2\u0019gSh~2=EB\nF. We assume that the monolayer is half\n\flled and has the same e\u000bective electron mass as that of\nbulk gold. We consider the case in which the monolayer\nlattice constant ais equal to the lattice constant atof\na simple cubic tight-binding model for gold. atis ap-\nproximately 20% smaller than the bulk nearest-neighbor\ndistance of actual gold.\nWe calculate the interfacial exchange coupling JIfor\na YIG-Au bilayer in terms of the spin-mixing conduc-\ntance, which has been experimentally measured. In\ndoing this calculation, we use the same model for the\nYIG as in the trilayer case; however, for the gold,\nwe employ a tight-binding model of the form Ht=\n\u0000ttP\n\u001bP\niP\nj2NN(i)cy\ni\u001bcj\u001b, with a simple cubic lat-\ntice. The Hamiltonian of the bilayer is HB=Ht+\nHA\nFI+Hint. We assume that JIs\u001ctt, which al-lows us to disregard the proximity-induced Zeeman \feld.\nThe energy eigenstates ct\nq\u001band the dispersion Et\nq=\n4tt(3\u0000cos(qxat)\u0000cos(qyat)\u0000cos(qzat)) ofHtare well\nknown. Under the assumption of half \flling, we \fnd that\ntt=EB\nF=12 andat=p\n0:63=gSh. We use the same ex-\nperimental values for EB\nFandgSh(from Ref. 6 and 40)\nas before.\nWe set the lattice constant of the trilayer, a, equal to\nthe lattice constant of the bilayer, at. This ensures that\nboth models have the same lattice structure at the in-\nterface and, consequently, that the interfacial exchange\ninteraction Hamiltonian Hinthas the same form in both\ncases. To \frst order in the bosonic operators, Hint=P\nkqVtakcty\nq+k;#ct\nq\". The coupling strength Vtis propor-\ntional to the amplitudes of the tight-binding-model eigen-\nstates at the interface: Vt= 2Vsin(qxat) sin([kx+qx]at).\nThe spin-mixing conductance can now be calculated for\nthe ferromagnetic resonance (FMR) mode, resulting [41]\ning\"#= 4a2\ntV0sN=(2\u0019)2, where\nV0=ZZ\njVj2sin(qxat)2sin(q0\nxat)2\u000e\u0000\nqy\u0000q0\ny\u0001\n\u000e(qz\u0000q0\nz)\u000e\u0000\nEt\nq\u0000EF\u0001\n\u000e\u0000\nEt\nq0\u0000EF\u0001\nd3qd3q0:(13)\nWe numerically evaluate V0and estimate the bilayer in-\nterfacial exchange coupling JI=p\n(2\u0019)2g\"#t2\nta2\nt=(9:16s2)\nusing measured values of the spin-mixing conductance\ng\"#. We assume that JIhas the same value in the tri-\nlayer case. Using E\u0003= 8\u0019sJ, we \fnd that E\u0003is approx-\nimately 1:5 eV. We \fnd the coupling constant \u000bfrom\nthe relation \u000b=J2\nIma2=(16p\n2\u00192~2J). The reported ex-\nperimental values for the spin-mixing conductance range\nfrom 1:2 nm\u00002to 6 nm\u00002[42{44]. In turn, this implies\nthat\u000blies in the range of [0 :0014{0:007]. The corre-\nsponding critical temperatures range from 0 :5 K to 10 K.\nNext, we consider a EuO-Au-EuO trilayer. Europium\noxide has an fcc lattice structure with a lattice constant of\n5:1\u0017A, a spin quantum number of s= 7=2 and a nearest-\nneighbor exchange coupling of J=kB= 0:6 K [45]. The\nnodes on a (100) surface of an fcc lattice form a square\nlattice in which the lattice constant is equal to the dis-\ntance between nearest neighbors in the bulk. We assume\nthat the monolayer has the same structure and therefore\nsetaequal to the distance between nearest neighbors in\nbulk EuO. We use the same e\u000bective mass as for the YIG-\nAu-YIG trilayer. Then, the Fermi energy is EF= 1:8 eV,\nand the energy scale E\u0003=kBis approximately 53 K. Val-\nues on the order of 10 meV have been reported for the\ninterfacial exchange coupling strengths JI[46] in EuO/Al\n[7], EuO/V [8], and EuS/Al [9, 10]. These estimates were\nbased on measurements of a proximity-induced e\u000bective\nZeeman \feld. Under the assumption that JIis in the\nrange of [5{15] meV, we \fnd a wide range of values of\n[0:004{0:03] for\u000b. We estimate the corresponding critical\ntemperatures numerically using the quadratic dispersion\napproximation. Finally, we \fnd a range of [0 :01{0:4] K\nas possible values for Tc.5\nIn conclusion, interfacial coupling to magnons induces\np-wave superconductivity in metals. The critical temper-\natures are experimentally accessible in the weak-coupling\nlimit. The gap size strongly depends on the magnitude of\nthe interfacial exchange coupling. 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Laboratory for Neutron Scattering and \nImaging, Paul Scherrer Institut, CH -5232 Villigen, Switzerland. 4 INNOVENT e.V., Technologieentwicklung , Pruessingstrasse. 27B , D-\n07745 Jena , GERMANY The Final Chapter I n The Saga O f YIG \n \nA. J. Princep1*, R. A. Ewings2, S. Ward3, S. T óth3, C. Dubs4, D. Prabhakaran1, A. T. Boothroyd1 \n \nThe magnetic insulator Y ttrium Iron Garnet can be grown with exceptional quality , has a ferrimagnetic transition \ntemperature of nearly 600 K, and is used in microwave and spintronic devices that can operate at room \ntemperature1. The most accurate prior measurements of the magnon spectrum date back nearly 40 years, but cover \nonly 3 of the lowest energy m odes out of 20 distinct magnon branches2. Here we have used time -of-flight inelastic \nneutron s cattering to measure the full magnon spectrum throughout the Brillouin zone . We find that the existing \nmodel of the excitation spectrum , well known from an earlier work titled “The Saga of YIG” 3, fails to describe the \noptical magnon modes . Using a very general spin Hamiltonian , we show that the magnetic interactions are both \nlonger -ranged and more complex than was previously understood . The result s provide the basis for accurate \nmicroscopic models of the finite temperature magnetic properties of Yttrium Iron Garnet, necessary for next -\ngeneration electronic devices. \n \nYttrium Ir on G arnet (YIG) is the ‘miracle material’ of microwave magnetics. Since its synthesi s by Geller and Gilleo in \n19574, it is widely acknowledged to have contributed more to the understanding of electronic spin -wave and magnon \ndynamics than any other substance3. YIG (chemical formula Y3Fe5O12, crystal structure depicted in Fig. 1 ) is a \nferrimagnetic insula ting oxide with the lowest magnon damping of any known material. Its exceptionally narrow \nmagnetic resonance linewidth — orders of magnitude lower than the best polycrystalline metals — allows magnon \npropagation to be obse rved over centimetre distances. This makes YIG both a superior model system for the \nexperimental study of fundamental aspects of microwave magnetic dynami cs5 (and indeed, general wave and quasi -\nparticle dynamics6,7), and an ideal platform for the development of microwave magnetic technologies , which have \nalready resulted in the creation of the magnon transistor and the first magnon logic gate5,8. \n \nThe unique p roperties of YIG have underpinned the recent emergence of new fields of research, including magnonics : \nthe study of magnon dynamics in magnetic thin-films and nanostructures5, and magnon spintronics , concerning \nstructures and devices that involve the interconversion between electronic spin currents and magnon currents1. Such \nsystems exploit the established toolbox of electron -based spintronics as well as the ability of magnons to be \ndecoupled from th eir environment and efficiently manipulated both magnetically and electrically5,9,10. Significant \ninterest has also developed in quantum aspects of magnon dynamics, using YIG as the basis for new solid -state \nquantum measurement and info rmation processing technologies including cavity-based QED, optomagnonics, and \noptomechanics11. It has also recently been realised that one can stimulate strong coupling between the magnon \nmodes of YIG and a superconducting qubit, potentially as a tool for q uantum informatio n technologies12. Spin \n \n Figure 1. Crystal structure and magnetic exchange paths in YIG. Left: First octant of the unit cell of YIG, indicating the two \ndifferent Fe3+ sites , with the tetrahedral sites in green and the octahedral sites in blue . Exchange pathways used in the \nHeisenberg effective Hamiltonian are labelled. Right: Unit cell of YIG, with the majority tetrahedral sites in green and the \nminority octahedral sites in blue . Black spheres are yttrium, red spheres are oxygen. \n \nFigure 2. Neutron scattering intensity maps of the magnetic excitation \nspectrum of YIG. a-c) Measured magnon spectrum along (H,H,4), (H,H,3), \nand (H,H,H) directions in reciprocal space , recorded in absolute units of \nmb sr-1 meV-1 f.u-1. d-f) resolution convoluted best fit to the model \npresented in the text , currently the basis for theoretical models of YIG. No \nscaling factors were used in the model. \ncaloritronics has also recently emerged as a potential application of YIG, utilising the spin Seebeck effect (SSE) and the \nspin Peltier effect (SPE) to interconvert between magnon and thermal currents, either for efficient large -scale energy \nharvesting, or the generation of spin cu rrents using thermal gradients13. \n \nIf the research into classical and quantum aspects of spin wave propagation in YIG is to achieve its potential, it is \nabsolutely clear that the community requires the deep understanding of its mode structure , which only neutron \nscattering measurements can offer. In many theories and experiments , YIG is treated as a ferromagnet with a single, \nparabolic spin wave mode14,15, simply because the influence of YIG ’s complex electronic and magnetic structure on \nspin transport is not known in sufficient detail . Such approaches must break down at high temperature when the \noptical modes are appreciably populated and a detailed knowl edge of the structure of the optic al modes is a \nnecessary first step in any realistic model of the magnetic properties of YIG in this operational regime. Despite this, \nsurprisingly little data exists relating to the detail of its magnon mode structure. The key previous work in this area is \ndue to Plant et. al. 2, and dates back to the 70 s. Using a triple -axis spectrometer , these early measurements were able \nto record 3 of the spin wave modes up to approximately 55 meV, but crucially there are 20 such modes and they are \npredicted to extend up to approximately 90meV (22 THz) 3. \n \nData were collected (see methods section) as a large, 4 -dimensional hypervolume in frequency and momentum space, \ncovering the complete magnon dispersion over a large number of Brillouin zones. Fig. 2 shows two -dimensional \nenergy -momentum slices from this hyper volume with the wave -vector along three high -symmetry directions , \nnormalised to a measurement on vanadium (see methods section) . A large number of modes can be seen up to an \nenergy of 80meV , whilst data in other slices show modes extending up to nearly 100 meV. The spectrum is dominated \nby a strongly dispersing and well-isolated acoust ic mode at low energies (the so -called ‘ferromagnetic ’ mode), and a \nstrongly dispersing optical mode separated from this by a gap of approximately 30meV at the zone centre. \nIntersecting this upper mode is a large number of more weakly dispersing optic al modes in the region of 30 -50meV . \n \nWe model the data using a Heisenberg \neffective spin Hamiltonian, appropriate to YIG \nas it is both a good insulator and the Fe3+ ions \n(S = 5/2, g = 2) possess a negligible magnetic \nanisotropy due to the quenched orbital \nmoment. \n𝐻=∑𝐒𝑖𝑇𝐽𝑖𝑗𝐒𝑗\n𝑖,𝑗+∑𝐒𝑖𝑇𝐴𝑖𝐒𝑖\n𝑖 \nWe nevertheless include a magnetic anisotropy \nAi in our analysis to take into account crystal \nfield effects, but find this term to be \nvanishingly small, consistent with previous \nresults. The exchange matrix Jij is a general 3x3 \nmatrix, whose elements are restricted by the \nsymmetry of the bond connecting the spins Si \nand Sj. Following common practice, Jij is \nrestricted to having only identical diagonal \ncomponents (i.e. isotropic exchange) since \nanisotropic and off -diagonal contributions are \nlikely to be small due to the lack of significant \norbital angular momentum . \n \nThe spin Hamiltonian was diagonalized using \nthe SpinW software package16 and th e \ncalculated magnon dispersion was fit ted by a \nconstrained nonlinear least squares method to \n1D cuts taken through the 2D intensity slices . \nWe do not include any scaling factors for the \nmagnon intensity, so the agreement between \nthe model and the data in terms of absolute \nunits is indicative of the quality of the model. \nOur final /best -fit model includes isotropic \nexchange interactions up to th e 6th nearest neighbour, labelled J 1-J6 in Fig 1. The exchanges in this work can be mapped to the exchanges commonly considered \nfor YIG as follows: J ad=J1, Jdd=J2, and J aa ={J 3a,J3b}, where the subscript refers to the majority tetrahedral (d) and minority \noctahedral (a) sites . Due to the extremely large number of magnetic atoms (20) within the primitive cell, and the \nconsideration of so many exchange pathways, this analysis would be impossible without the use of sophisticated \nsoftware such as SpinW as the construction of an analytic model would be prohibitively time consuming. During the \nfitting process, f eatures in the spectrum were weighted so that weak but meaningful features in the data were \nconsidered as significant as strong features. The Spin W model output is then convoluted with the calculated \nexperimental resolution of the MAPS spectrometer, including all features of the neutron flight path and associated \nfocussing/defocussing effects, as well as the detector coverage and effects from symme trisation (see Supplementary \nInformation for details). The final fitted values of the exchanges are listed in Table 1. \n \nAn important difference between our results and those of previous authors2,3 is that there are two symmetry -distinct \n3rd-nearest -neighb our bonds (the so -called J aa in the literature , which we label J 3a and J 3b) which have identical length \nbut differ in symmetry . The J 3b exchange lies precisely along the body -diagonal of the crystal , and thus has severely \nlimited symmetry -allowed components owing to the high symmetry of the bond (point group D3). The J 3a exchange \nconnects the same atoms with the same radial separation, but represents a different Fe -O-Fe exchange pathway as a \nresult of the different point group symmetry (point group C2), so it is distinguished from J 3b by the environment \naround the Fe atoms. Models including anisotropic exchange or Dzyaloshinskii -Moriya interactions on the 1st -4th \nneighbour bonds were tested, but such interactions were found to destabilise the magnetic structure for arbitrarily \nsmall perturbations . We also find that J2 (Jdd) is much smaller than previously supposed — the main effect of this \nexchange is to increase the bandwidth and split the optic modes clustered around 40 meV in a way contradicted by \nthe data. \n \nIt has been pointed out17 that the magnetic structure of YIG is \nincompatible with the cubic crystal symmetry, although to date no \nmeasurements have found any evidence for departures from the \nideal cubic structure. Nevertheless, it is necessary to refine the \nmagnetic structure in a trigonal space group ( a symmetry that is \nexperimentally observed in terbium rare earth garnets where the \nmagnetoelastic coupling is much stronger18), in order to obtain a \nsatisfactory goodness of fit, and a magnetic moment which agrees \nwith bulk magnetome try19. Treating the unit cell of YIG in this \nfashion for the purposes of the SpinW simulation would be feasible, \nbut would introduce a large number of free parameters which \n(given the very small size of the departure from cubic symmetry) \nwould nevertheless be expected to change very little from a cubic \nmodel. This expectation is borne out by the excellent agreement \nbetween the data and the cubic spinwave model (see Fig. 2 and the \nsupplementa ry materials for more details) . \n \nFeatures absent from the data which are relevant to technological applications (such as the conversion of microwave \nphotons into magnons) include any strong indications of magnon -phonon or magnon -magnon coupling. The data are \nwell described by a linear spinwave model, although the size o f the 5th-neighbour exchange is perhaps indicative of \nsome small deviations not easily captured without such couplings . A strong magnon -phonon coupling would be \nexpected to cause both broadening and anomalies in the dispersion of the magnon modes20. We do not observe any \nsuch effects, although our measurements would not be sensitive to any magnon -phonon coupling that shifts or \nbroadens th e spin wave signal by less than 3meV (the instrumental resolution) . \n \nOur results require a substantial revision of the impact of the optical modes on the room -temperature magnetic \nproperties . The differences compared with the existing model are illustrated in Fig. 3, in which we plot the \nantisymmetric combination of transverse scattering function s Sxy(Q,ω) − Syx(Q,ω), which is proportional to the sign \nand magnitude of the measured spin -Seebeck effect arising from the associated magnon mode21. Most strikingly the \nabsence of spectral weight in the flat mode at ~35meV, as well as a compression and shift of the ‘positively’ polarised \n(red) optical modes i.e. those modes which would precess counterclockwise with respect to an applied field . As has \nrecen tly been shown, the thermal population, broadening , and softening of these modes at elevated temperatures \nsubstantially modifies the magnitude of the measured spin-Seebeck effect, which places limitations on device \nperformance and determines the optimum operating temperature . Our results show that the distribution of optic al \nmodes is very different from what had previously been assumed , which has consequences for the temper ature -Table 1 . Fitted exchange parameters for YIG and \ntheir statistical uncertainties. The exchange \nconstants are defined in Fig. 1 \nExchange This work \n(meV) Ref. 3 (meV) \nJ1 6.8(2) 6.87 \nJ2 0.52(4) 2.3 \nJ3a 0.0(1) 0.65 \nJ3b 1.1(3) 0.65 \nJ4 -0.07(2) - \nJ5 0.47(8) - \nJ6 -0.09(5) - \n dependent broad ening21. Our new measurements can therefore be used as the basis for a precise microscopic model \nof the temperature dependent dynamical magnetic properties in YIG. \n \nWe have also estimated the parabolicity of \nthe lowest lying ‘ferromagnetic’ magnon \nmode i.e. the point where the error of a \nquadratic fit becomes greater than 5 %. We \nfind this region to extend 14.8% of the way \ntowards the Brillouin zone boundary along the \nH direction ( the (0,0,1) direction in t he \ncentred unit cell) , represe nting about 0.3% of \nthe entire B rillouin zone . The departure from \na purely parabolic acoustic magnon \ndispersion, as well as the population of optic \nmagnon modes directly generates the \ntemperature dependence of the spin Seebec k \neffect and our model can be used to fully \nunderstand such effects even at elevated \ntemperatures through extension to a multi -\nmagnon picture following the procedure in \nRefs. 21, and 22. \n \nWe have presented the most detailed and \ncomp lete measurement of the magnon \ndispersion of YIG in a pristine, high quality \ncrystal. Using linear spin -wave theory analysis \nwe are able to reproduce the entire magnon \nspectrum across a large number of brillouin \nzones including a reproduction of the absolute \nintensities of the mo des. We confirm the importance of the near est-neighbour exchange, but are forced to radically \nreinterpret the nature and hierarchy of longer -ranged interactions. Our work has uncovered substantial discrepancies \nbetween previous models and the measured disp ersion of the optical magnon modes in the 30 -50 meV region, as well \nas the total magnon bandwi dth, and the detailed nature of the magnetic exchanges . Through a detailed consideration \nof the symm etries of the exchange pathways and long -ranged interactions , we are able to fully reproduce the entire \nmeasured spectrum . Technological applications of YIG, particularly those utilising the spin Seebeck effect, are very \nsensitive to the optical magnons in the region of 30 -40meV. This work overturns 40 years of estab lished work on \nmagnons in YIG and will be an essential tool for accurate modelling of the optical magnon modes in the room \ntemperature regime. \n \nAcknowledgements \nThis work was supported by the Engineering and Physical Sciences Research Council of the Unite d Kingdom (grant nos. \nEP/J017124 /1 and EP/M020517/1 ). We wish to acknowledge useful discussions with A. Karenowska , B. Hillebrandts , \nand T Hesje dal. We are grateful to S. Capelli (ISIS Facility) for use of the SXD instrument to characterise the crystal s \nused in the experiments . Furthermore , C.D. would like to acknowledge the technical assistance of R. Meyer in the \ncrystal growth of YIG . Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beamtime \nallocation from the United Kingdom Science and Technology Facilities Council. \n \nAuthor Contributions \nDP produced the preliminary optical floating zone crystal, and CD grew the flux crystal used in the final experiment. \nAJP and RAE performed the neutron experiments, with AJP responsible for the data reduction and RAE performed the \ninstrumental resolution convolution . SW and AJP performed the data analysis and spinwave simulation, with advice \nand input from ST. AJP prepared the manuscript with input from all the co -authors. ATB supervised the project and \nassisted in the planning and editing of the manuscript. \n \nReferences\n1 A.V. Chumak, V. I. Vasyuchka, A. A. Serga & B. Hillebrands . \nMagnon Spintronics. Nature Physics 11, 453–461 ( 2015 ); \n[2] J.S. Plant., Spinwave dispersion curves for yttrium iron garnet . \nJ. Phys. C . 10, 4805 -4814 (1977) [3] V. Cherepanov , I. Kolokolov & V. L’Vov . The saga of YIG: \nSpectra, thermodynamics, interaction and relaxation of magnons \nin a complex magnet . Phys. Reps. 229, 81 - 144 (1993) \n[4] S. Geller & M. A. Gilleo . Structure and ferrimagnetism of \nyttrium and rare -earth -iron garnets . Acta Cryst. 10 (1957) 239 \nFigure 3: Simulations of the magnon dispersion in YIG. (Right) \nthe previous model3,21 and ( left) our new model of the magnon \ndispersion, w here the colour and intensity correspond to the sign \nand magnitude of the correlator Sxy(Q,ω) − Syx(Q,ω), which is \nresponsible for the spin Seebeck effect21. The horizontal line \nindicates k BT at room temperature. \n \n[5] A. A. Serga, A.V. Chumak, & B Hillebrands . YIG Magnonics. J. \nPhys. D: Appl. Phys . 43 264002 (2010) \n[6] A.V. Chumak , , A. A. Serga & B.Hillebrands . Magnon transistor \nfor all -magnon data processing . Nat. Commun . 5 4700 (2014) \n[7] A. A. Serga et. al. Bose –Einstein condensation in an ultra -hot \ngas of pumped magnons. Nat. Commun . 5 3452 (2014) \n[8] Y. Kajiwara et. al. Transmission of electrical signals by spin -\nwave interconversion in a magnetic insulator . Nature, 464 262-266 \n(2010) \n[9] M. B. Jungfleisch et. al . Temporal evolution of inverse spin Hall \neffect voltage in a magnetic insulator -nonmagnetic metal \nstructure . Appl. Phys. Lett ., 99 182512 (2011) \n[10] J. Flipse et. al. Observation of the Spin Peltier Effect for \nMagnetic Insulators . Phys. Rev. Lett. 113 027601 (2014) \n[11] D. Zhang et. al. Cavity quantum electrodynamics with \nferromagnetic magnons in a small yttrium -iron-garnet sphere . NPJ \nQuant. Inf . 1, 15014 (2015) \n[12] Y. Tabuchi, et. al. Coherent coupling between a ferromagnetic \nmagnon and a superconducting qubit . Science 349 405-408 (2015) \n[13] A. Hoffmann & S. D. Bader . Opportunities at the Frontiers of \nSpintronics . Phys. Rev, Appl . 5 047001 (2015) \n[14] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, & S. Maekawa , \nTheory of magnon -driven spin Seebeck effect . Phys. Rev. B 81, \n214418 (2010) \n[15] U. Ritzmann, D. Hinzke, & U. Nowak . Propagation of thermally \ninduced magnonic spin currents . Phys. Rev. B 89, 024409 (2014) \n[16] S. Toth & B. Lake . Linear spin wave theory for single -Q \nincommensurate magnetic structures . J. Phys.: Condens. Matt . 27, \n166002 (2015). \n[17] D. Rodic, M. Mitric, R. Tellgren, H. Rundlof, A. Kremenovic . \nTrue magnetic structure of the ferrimagnetic garnet Y 3Fe5O12 and \nmagnetic moments of iron ions . J. Magn. Magn. Mat. 191 137-145 \n(1999) \n[18] R. Hock, H. Fuess, T. Vogt, M. Bonnet. Crystallographic \ndistortion and magnetic structure of terbium iron garnet at low \ntemperatures . J. Solid State Chem. 84 39-51 (1990) . \n19 G. Winkler. Magnetic Garnets , 5th ed. (Vieweg, \nBraunschweig/ Wiesbaden, 1981). \n[20] P. Dai et al. Magnon damping by magnon -phonon coupling in \nmanganese perovskites . Phys Rev B 61 9553 (2000) \n[21] J. Barker & G. E. W. Bauer Thermal Spin Dynamics of Yttrium \nIron Garnet . Phys. Rev. Lett . 117 217201 (2016) \n[22] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, & J.P. Heremans . \nEffect of the magnon dispersion on the longitudinal spin Seebeck \neffect in yttrium iron garnets . Phys. Rev. B 92, 054436 (2015) \n \nMethods \nCrystal Growth. YIG crystal growth was carried out in \nhigh -temperature solutions a pplying the slow cooling \nmethod23. Starting compounds of yttrium oxide \n(99.999%) and iron oxide (99.8%) as solute and a \nboron oxide - lead oxide solvent were placed in a \nplatinum crucible and melted in a tubular furnace to \nobtain a high -temperature solution24. Using an \nappropriate temperature gradient only a few single \ncrystals nucleate spontaneously at the cooler crucible \nbottom and forced convect ion, obtained by \naccelerated crucible rotation technique (ACRT), allows \na stable growth which results in nearly defect -free \nlarge YIG crystals25. The YIG crystal used in this study \nexhibits a size of 25 mm x 20 mm x 11 mm and a \nweight of 12 g. It was confirmed by neutron and X -ray \ndiffraction that the YIG crystal was a single grain with a \ncrystalline mosaic of approximately 0.07 degree s \nFWHM. Preliminary measurements were made on a \ncrystal that was grown by the optical floating -zone \nmethod, starting fr om a pure powder of YIG. \n Neutron Scattering Data Collection and Reduction. \nData were collected on the MAPS time -of-flight \nneutron spectrometer at the ISIS spallation neutron \nsource at the STFC Rutherford Appleton Laboratory, \nUK. On direct geome try spectro meters such as MAPS , \nmonochromatic pulses of neutrons are selected using \na Fermi chopper with a suitably chosen phase. In our \nexperiment neutrons with an incident energy (E i) of \n120 meV were used with the chopper spun at 350 Hz, \ngiving energy resolution of 5.4 meV at the elastic line, \n3.8 meV at an energy transfer of 50 meV, and 3.1 meV \nat an energy transfer of 90 meV. The spectra were \nnormalized to the incoherent scattering from a \nstand ard vanadium sample measured with the same \nincident energy, enabling us to present the data in \nabsolute units of mb sr-1 meV-1 f.u. -1 (where f.u. refers \nto one formula unit of Y 3Fe5O12). Neutrons are \nscattered by the sample on to a large area detector on \nwhich their time of flight, and hence final energy , and \nposition are recorded. The two spherical polar angles \nof each detector element, time of flight, and sample \norientation allow the scattering function S( Q, ω) to be \nmapped in a four dimensional space (Q x,Qy,Qz,E). In \nour experimen t the sample was oriented with the \n(HHL) -plane horizontal, while the angle of the (00L) \ndirection with respect to the incident beam direction \nwas varied over a 120 degree range in 0.25 degree \nsteps . This resulted in coverage of a large number of \nBrillouin zones , which was essential in order to \ndisentangle the 20 different magnon modes. D ue to \nthe complex structure factor resulting from the \nnumber of Fe atoms in the unit cell, the mode \nintensity varies considerably throughout recip rocal \nspace . The large datasets recording the 4 D space of \nS(Q,ω), ~100 GB in this case, were redu ced using the \nMantid framework26, and both visualized and analyzed \nusing the Horace software package27. Taking \nadvantage of the cubic symmetry of YIG, t he data \nwere folded into a single octant of reciprocal space \n(H>0, K>0, L>0) , adding together data points that are \nequivalent in order to produce a better signal -to-\nnoise. 2D slices were taken from the 6 reciprocal space \ndirections depicted in Fi gure 2 and in Sup plementary \nInformation . \n23 P. Görnert & F. Voigt, in: Current Topics in Materials Science, \nVol. 11, Ed. E. Kaldis (North -Holland, Amsterdam, 1984) ch. 1 \n24] S. Bornmann, E. Glauche , P. Görnert , R. Hergt , & C. Becker , \nPreparation and Properties of YIG Single Crystals . Krist. Tech . 9, \n895 (1974) \n[25 C. Wende & P. Görnert . Study of ACRT Influence on Crystal \nGrowth in High -Temperature BY Solutions by the “High -Reso lution \nInduced Striation Method “. Phys. Stat. S ol. (a) 41, 263 (1977) \n[26] J. Taylor et al. Mantid, A high performance framework for \nreduction and analysis of neutron scattering data . Bulletin of the American Physical Society 57 (2012) \n[27] R. A. Ewings et. al. HORACE: software for the analysis of data \nfrom single crystal spectroscopy experiments at time -of-flight neutron instruments . Nucl. Inst rum. Meth . A 834 132 (2016) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary Information For: The Final Chapter In The Saga \nOf YIG \n \nA. J. Princep1*, R. A. Ewings2, S. Ward3, S. Tóth3, C. Dubs4, D. Prabhakaran1, A. T. Boothroyd1 \n \n1. Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom. \n2. ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot OX11 0QX, United Kingdom \n3. Laboratory for Neutron Scattering and Imaging, P aul Scherrer Institut, CH -5232 Villigen, Switzerland. \n4 INNOVENT e.V., Technologieentwicklung, Pruessingstrasse. 27B, D -07745 Jena, GERMANY \n \n \n \n \n \n \n \nContents: \n \n1) Additional information on the fitting procedure \n \n2) Symmetry of the 3rd neighbour Exchange \n \n3) Additional data slices simulated, and detailed comparison with previous model. \n \n4) Instrumental broadening and simulation \n \n5) Supplementary References \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n1. Additional information on the fitting procedure \n \nTo extract the exchange parameters fitting was performed by a bounded non -linear least squares fit to the following \ncuts of the experimental data set: Q-BASIS Q RANGE (R.L.U) ENERGY RANGE (MEV) \nH H 4 0.0 0.05 3.5 40 1.5 80 \n 2.5 0.05 5.0 08 1.5 90 \nH H H 0.0 0.05 3.5 20 1.5 60 \n 3.0 0.05 5.0 08 1.5 90 \nH H 3 0.0 0.05 5.0 09 1.5 90 \n2 2 L 0.0 0.05 1.0 40 1.5 50 \n 3.0 0.05 5.0 20 1.5 90 \n3 3 L 0.0 0.05 5.0 09 1.5 90 \n \nInitial fitting was performed on exchanges J 2, J3a, J3b, J4, J5 and J 6 where the results were subsequently used as starting \npoints for when the exchanges J 1, and the anisotropy parameter D were allowed to vary. Due to the exceptional \nquality of the data, the background signal was approximated as a constant value, which was un ique to each cut. As \nwell as this, a common intensity factor and convolution width was used for all cuts. The convolution width was not \nfitted in the initial procedure, rather it was fitted separately to a 1D cut from (H, H, 4) with integration 1.9-2.1 r.l .u and \nenergy binned between 9.0 and 90 meV in steps of 1.5 meV. The ferromagnetic features around (4, 4, 4) were found \nto be dominant over the intricate higher energy features in the basic fitting approach. Masking this feature led to \nunsatisfactory param eter convergence, so to overcome this a weighting factor was introduced, which allowed a \nparameter convergence describing both low and high energy features. \n \nDuring the fitting procedure parameters were allowed to vary between their boundary conditions: \n \nExchange Value Boundary condition \nJ1 6.7600 ( 0.2025) [3 10] \nJ2 0.5207 (0.0370) [0 1] \nJ3a 0.0001 (0.1362) [-2 2] \nJ3b 1.0539 ( 0.3216) [-2 2] \nJ4 -0.0686 (0.0152 ) [-1 1] \nJ5 0.4736 (0.0840) [0 1] \nJ6 -0.0930 (0.0499) [-1 1] \nD 0.01000 (0) [-1 1] \n \n \nStarting parameters were randomly selected within the limits and were free to vary within the boundary conditions. It \nwas found that only those around the presumed starting parameters gave a meaningful convergence. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n2. Symmetry of the 3rd neighbour Exchange \n \nThe difference between J 3a and J 3b can be understood easily from a projection of the crystal structure along the axis of \nthe bond, depicted in figure S1. \n \n \nFigure S1. The left image depicts the projection of the crystal stru cture along the J3a bond, from which the 2 -fold \nsymmetry can be seen. The right image depicts the projection of the crystal structure along the J3b bond, from which \nthe symmetry can be seen to obey the higher symmetry D3 point group. These images were gene rated using VESTAS1. \n \n3. Additional data slices simulated, and detailed comparison with previous model. \n \nFurther 2D slices to the data that were used for fitting are found below in figure S2. Figure S3 depicts a detailed \ncomparison of the optical magnon modes in the region 30 -50meV, contrasting the model presented in this work with \nthat of previous authors3. \n \n \n \nFigure S2 . Supplementary neutron scattering intensity maps along (H,H,4), (H,H,3), and (H,H,H) directions in recipr ocal \nspace. Left: Experimental data. Right: Model. \n \nFigure S3. Zoomed figures emphasising comparison between the model in this paper (middle column) and the model \nowing to previous work3 (right column). The left column s hows the corresponding data in the same region of \nreciprocal space. \n \n4. Instrumental broadening and simulation \n \nThe broadening of the signal due to instrumental resolution arises due to several effects. The main contributions are: \n Deviations in neutron arriva l time (at both the chopper and detectors) due to the finite depth of the source \nmoderator and the angle of the moderator face with respect to the beamline collimation, and hence \ndeviations in the departure time of the neutron burst compared to the notiona l zero time. \n Deviations in neutron arrival time due to the finite sweep time, length and slit width of the monochromating \nFermi chopper. \n The divergence of the scattered beam due to the finite size of the sample subtended at each detector \nelement. \n The divergence of the incident neutron beam due to the finite size of the beam and the geometry of the \nbeamline collimation \nThese effects are computed in the reference frame of the spectrometer and then converted into the reference frame \nof the sample and convoluted. The convolution is performed by Monte -Carlo sampling over the resolution width of \neach of the terms described above. In general each Q-energy bin in the images shown in Fig. 2 of the main article (as \nwell as supplementary figures S2 and S3) con tains data from many detector elements taken with many different \nsample orientations. The resolution for each element was accounted for in the corresponding simulations shown in \nthe same figures. \n \n4. Supplementary references \n[S1] K. Momma & F. Izumi. VESTA: a three -dimensional visualization system for electronic and structural analysis . J. \nAppl. Crystallogr. , 41, 653 -658 (2008) \n" }, { "title": "1705.02491v3.Scaling_of_the_spin_Seebeck_effect_in_bulk_and_thin_film.pdf", "content": " \n 1 Scaling of the spin Seebeck effect in bulk and thin film \n \nBy K. Morrison,1* A.J Caruana,1,2 C. Cox1 & T.A. Rose1 \n \n[*] Dr K. Morrison \n1Department of Physics, Loughborough University, \nLoughborough LE11 3TU (United Kingdom) \n2ISIS Neutron and Muon Source, Didcot, Oxfordshire, OX11 0QX \nE-mail: k.morrison@lboro.ac.uk \n \nKeywords: Magnetic materials, thermoelectrics, thin films \nPACS: 72.25.Mk, 72.25.-b, 75.76.+j \n \nWhilst there have been several reports of the spin Seebeck effect to date, comparison of the absolute \nvoltage(s) measured, in particular for thin films, is li mited. In this letter we de monstrate normalization of the \nspin Seebeck effect for Fe 3O4:Pt thin film and YIG:Pt bulk samp les with respect to the heat flux Jq, and \ntemperature difference ΔT. We demonstrate that the standard normalization procedures for these \nmeasurements do not account for an une xpected scaling of the measured volta ge with area that is observed in \nboth bulk and thin film. Finally, we present an alterna tive spin Seebeck coefficien t for substrate and sample \ngeometry independent characterization of the spin Seebeck effect. \n\nI. INTRODUCTION \n \nThe spin Seebeck effect (SSE) is defined as the \nproduction of a spin polarized current in a magnetic material subject to a thermal gradient. It was first \nhighlighted by Uchida et al. [1], and led to the \ndevelopment of a new field of magnetothermal effects (such as spin Peltier and spin Nernst) now grouped together under spin caloritronics [2]. Observation of the spin Seebeck effect is \ntypically achieved by placing a heavy metal such as Pt in \ncontact with the magnetic material, where the spin current generated in the magnetic layer is injected into the Pt layer and converted to a useable voltage by the inverse spin Hall effect [3,4]. The advantage of the SSE is that it can be used as a source of pure spin polarized \ncurrent for spintronics applications, paving the way for a \nhost of new devices (such as spin Seebeck based diodes) [5–8]. In addition to its potential applications in spintronics, it has been pr esented as an alternative to \nconventional thermoelectrics for harvesting of waste \nheat, due to decoupling of the electric and thermal \nconductivities that typically dominate the efficiency [9] [10] [11]. This is in part due to a completely different device architecture, such as that shown in Figure 1, where the magnetic layer can be \nchosen to have low thermal conductivity independent of \nthe second paramagnetic layer, which in turn can be selected to have low resistivity. In this work, we discuss the longitudinal SSE (LSSE) geometry (Figure 1(a)), since it lends itself readily to energy harvesting. An important factor to \nconsider in this geometry is the impact of other \nmagneto-thermal contributions to the measured voltage, \nV\nISHE, when assessing the LSSE. This includes the anomalous Nernst effect (ANE) and proximity induced \nANE (PANE) in the Pt detection layer. For semiconducting or metallic thin films, such as presented here, both the ANE and PANE should be considered. For insulating magnetic materials, such as YIG, the \nPANE is the only potential artefact of the measurement. \n Whilst there have been extensive studies of the spin Seebeck effect over the last 5-10 years, this has \npredominantly been in the YI G:Pt system, where for thin \nfilms, the substrate is often GGG [14] [15]. Reports of \nthe spin Seebeck coefficient fo r this material system can, \nhowever, vary significantly [9]. Whilst this is sometimes attributed to the quality of the interface [14] there are indications that it could also be affected by an artefact of the measurement of ΔT [16], [17], [18]. \n Despite indications of the importance of heat \nflux in measurements [16], [17] [19], typically, LSSE \nmeasurements are still characterized by measuring the voltage generated across two contacts (of a spin convertor layer), as the temperature difference across the substrate, magnetic layer, and Pt is monitored. This is \nproblematic, especially for thin films, where the \ntemperature difference across the active material is not directly measured. If such spin Seebeck devices are to be considered for power conversion there needs to be a shift \nin metrology so that meaningful comparison of the \nmaterial parameters can be made. This is of particular importance if we consider the impact that the quality of the magnet:Pt interface can have on the observed voltage (i.e. efficiency of spin injection) [14]. An intermediate step is to use a simple thermal model to estimate the \ntemperature difference across the active (magnetic) layer, \nhowever, this is limited by the unknown thermal conductivity of the thin films (likely to differ from the \n 2 bulk), and thermal interface resistances. By measuring \nthe voltage generated with respect to the heat flux, J Q, \nwe can obtain a robust measurement of the temperature \ngradient across the active material ( ܶ ൌ ܬ ொ/ߢ .)In this \ncase we are only limited by our knowledge of the \nthermal conductivity, κ. \n In this paper we present measurement of the spin \nSeebeck voltage with respect to the temperature \ndifference, ΔT, and heat flux, JQ. We demonstrate that \nthe JQ method is indeed more reliable than the ΔT \nmethod and can be used to compare thin films with varying substrate thicknesses and thermal conductivity. \nIn addition we find the surprising result that the \nmeasured spin Seebeck voltage, V\nISHE, is proportional \nnot only to the contact separation, Ly, and temperature \ngradient, ܶሺൌ ܬ ொ/ߢሻ, but also the sample area, AT. This \nmanifests as an increase in VISHE/Lyܶ in both thin film \nand bulk samples with area, suggesting that the spin Seebeck coefficient typically used to define the magnitude of the effect in a bilayer system needs to be redefined. We present examples of this scaling in both thin film and bulk samples and put forward a new \ncoefficient suitable for normalizing the measured \nvoltage for generic sample geometries. \nII. EXPERIMENTAL METHOD \nThe thin film samples tested here were part of a series of 80:5 nm thick Fe\n3O4:Pt bilayers deposited using pulsed \nlaser deposition (PLD) in ultra-high vacuum (base pressure 5 x 10\n-9 mbar) onto 0.15 mm, 0.3 mm and 0.5 \nmm thick borosilicate glass and 0.6 mm or 0.9 mm fused silica substrates. The laser fluence and substrate \ntemperature for Fe\n3O4 and Pt deposition were 1.9 ± 0.1 J \ncm-2 and 3.7 ± 0.2 J cm-2, and 400 °C and 25 °C, \nrespectively. Fe 3O4 was chosen due to its large spin \npolarization (~80% [20]), low thermal conductivity (at 300 K: κ\nthin film ~2-3.5 W m-1 K-1, [21] κbulk ~ 2-7 W m-1 K-\n1 [22]), and relative abundance of the constituent \nelements. Of particular note are the following: X-ray \nreflectivity data indicated a typical Pt thickness across \nthe series of 4.5 0.5 nm and a roughness σ = 1.5 ± 0.2 \nnm. Fe 3O4 thickness was varied from 20 to 320 nm. In \naddition, these films have been shown to demonstrate \nhighly textured, columnar growth, with grain sizes of the order of 100 μm. Further details are given in [19] and \nthe supplementary information. Bulk YIG was prepared by the solid state \nmethod [23]. Stoichiometric amounts of Y\n2O3 and \nFe2O3 starting powders (Sigma Aldrich 99.999% and \n99.995% trace metals basis, respectively) were ground and mixed together before calcining in air at 1050 ᵒC for \n24 hours. Approximately 0.5g of the calcined powder was then dry pressed into a 13 mm diameter, 1.8±0.2 \nmm thick cylindrical pellet. The pellet was then sintered \nat 1400 ᵒC for 12 hours, after which, it was checked by \nXRD prior to sputtering 5 nm of Pt onto the as prepared surface using a benchtop Quor um turbo-pumped sputter coater. Samples were cut to size thereafter, using an \nIsoMet low speed precision cutter. Average grain size for this polycrystalline sample (determined using \nscanning electron microscopy) was 14.5 μm, with an \nopen porosity of 30% measured by the Archimedes method. The thermal conductivity of this sample was measured using a Cryogenic Ltd Thermal Transport Option and found to be 3 W/K/m. Further details are given in the supplementary information. Magnetic \ncharacterization was obtained using a Quantum Design \nMagnetic Property Measurement System (SQuID) as a function of temperature and field. Magnetometry of the Fe\n3O4 films at room temperature typically exhibited a \ncoercive field Hc = 197±5 Oe, saturation magnetization \nMs = 90 emu g-1, and a remanent moment of Mr = 68 \nemu g-1. Magnetometry of the YIG pellet exhibited a \ncoercive field Hc = 7 Oe, saturation magnetization Ms = \n25 emu g-1, and a remanent moment of Mr = 3 emu g-1 \n Spin Seebeck measurements were obtained from \na set-up similar to that of Sola et al. [16] and optimized \nfor 12x12, and 40x40 mm samples. The thin film was sandwiched between 2 Peltier ce lls, where the top Peltier \ncell (1) acted as a heat source, and the bottom Peltier cell (2) monitored the heat Q, passing through the sample. \nAdditional measurements we re obtained in set-ups \noptimized for 10x10, 20x20 and 40x40 mm samples, \nwhere both the top and bottom Peltiers monitored the heat passing through the sample, and the heat source was a resistor mounted to the top side of the top Peltier. The Peltier cells were calibrated by monitoring the Peltier \nvoltage V\nP, generated as a current was passed through a \nresistor fixed to the top side of the Peltier cell (where the sensitivity S\np of the set-ups varied from 0.11 to 0.264 V \nW-1 at 300K). Two type E thermocouples mounted on \nthe surface of the Peltier cells (such that they are in contact with the sample during the measurement) \nmonitored the temperature difference across the sample, \nΔT; a schematic of this set-up is given in Figure 1(c). \nFurther information on calibration is given in the supplementary information. The voltage generated by the inverse spin Hall \neffect, V\nISHE, was determined by taking the saturation \nvalues at positive and negative fields, as shown in Figure 1(f) and demonstrated in detail in the supplementary information. This was repeated at several different heating powers, where V\nISHE is expected to increase \nlinearly with ΔT and Q. Finally, ΔT and Q were plotted \nas a function of one another (see the example in Figure \n1(e)), where non-linearity starts to appear if radiation losses become significant. The impact of the ANE and PANE on the measured voltage was also ta ken into consideration. For \nthe Fe\n3O4:Pt thin films we demonstrated that these \ncontributions were negligible in a previous work by measuring thin films with a Au spacer layer between the Fe\n3O4 and the Pt. In this case, the V ISHE was similar to \ncorresponding PM layer thicknesses without the spacer \n 3 layer[19]. In addition, a separate work by Ramos et al. , \nhas shown that the ANE contribution was ~3% of the total signal in their epitaxially grown Fe\n3O4 thin films \nand obtained an upper limit of contribution due to PANE \nof 7.5 nV/K.[12] This was an order of magnitude smaller \nthan their observed VISHE, which is comparable to the \nmeasurements shown here. For the bulk YIG measurements, the ANE is considered negligible due to the insulating nature of the YIG and only the PANE in \nthe Pt layer is a potential contribution to V\nISHE. In this \ncase, it has been shown by several authors that the PANE in a >3 nm Pt layer on YIG is negligible with respect to the LSSE, by, for example, measurement in \nalternate geometries to separate the two \ncontributions [13] or by introducing spacer layers \nbetween the YIG and the Pt. [24]\n \n \nIII. THEORY \n \nIt is useful at this point to note some key characteristics \nof the SSE measurement. First, the voltage has been shown (for a single device) to increase linearly with temperature difference ( ΔT) [10] or heat flux ( J\nQ) [16]. \nSecondly, it is known to increase linearly with contact separation ( L\ny) [10]. Lastly, for thin films, there have \nbeen indications of a length scale of the order of the magnon free path length above which the voltage \ngenerated saturates [25] [26]. In response to these observations, some of the \nfirst attempts to quantify the SSE were to normalize the \nvoltage measured, V\nISHE, to the temperature difference \nΔT, \n \n 1ISHEVST (1) \n \nwhere the units are ( μV K-1). More accurately, this could \nalso be normalized to contact separation Ly, \n \n2ISHE\nyVSLT (2) \n \nwith units of ( μV K-1 m-1). \n Given that the spin Seebeck effect is often defined in terms of the thermal gradient across the magnetic \nmaterial ܶ ,the spin Seebeck coefficient is more often \ndefined as, \n \nܵ\nଷൌିாೄಹಶ\n்ൌೄಹಶ\n∆் (3) \n \n \nFigure 1. (a) Longitudinal spin Seebeck measurement geometries. Thermally generated spin current Js, is produced in the \nmagnetic layer (dark grey) and converted to a measureable voltage ( EISHE) in the spin convertor layer (light grey) by the \ninverse spin Hall effect. (b) Typical longitudinal sp in Seebeck thin film device; the length scales Lx, Ly, and Lz denote device \ndimensions with respect to the thermal gradient (alo ng z axis). Individual thicknesses of each layer { d1, d2, d3} are also \nindicated. (c) Schematic of the experimental set-up used in this work. (d) Top view of a typical thin film device where active \nmaterial may not cover the entire substrate. In this case, the thermal contact area is quantified by lengthscales LT\nx and LT\ny, \nactive material width by Lx, and contact separation by Ly±σs, where σs denotes contact size. (e) Example of the linear \nrelationship between heat flux, JQ, and temperature difference, ΔT, in these measurements. (f) Example spin Seebeck data, \nVISHE (symbols) from 80 nm Fe 3O4:Pt thin film plotted alongside corre sponding SQuID magnetometry (line). \n \n 4 where the units are ( μV K-1), and the thermal gradient \nܶ can be described by th e temperature difference ΔT, \ndivided by the thickness of the sample Lz. \n It was shown recently by Sola et al. , that there is \nthe added complication of thermal resistance between \nthe sample and the hot and cold baths (i.e. the interface across which ΔT is measured) [17]. Here they showed in \nmeasurements of the same sample in an experimental set-up at both INRIM and Bielefield that the \nmeasurement as a function of ΔT was unreliable – \ndiffering by a factor of 4.6 (gave S\n3 = 0.231 µV K-1 and \n0.0496 µV K-1, respectively), whereas by normalizing to \nheat flux, both set-ups obtained the same value to within 4% (S\n3 = 0.685 and 0.662 µV K-1, respectively). \n In this work they measured the heat flux, JQ, \nusing a calibrated Peltier cell and initially determined \nthe normalized voltage generated per unit of heat flux, \n4ISHE\ny\nTVS\nQLA\n (4) \n \nwhere Q is the heat passing through a sample with cross-\nsectional area surface AT, and the units are ( μV m W-1). \nGiven that the thermal conductivity can be defined by, \n \n/\nTzQT\nA L \n (5) \n \nwhere JQ = Q/A T, they argued that S3 (equation 3) could \nbe estimated by multiplying S4 (equation 4) by κ. This \nwas based on using a simple linear model that assumed \nthat the thermal conductivity of the substrate and \nmagnetic film were well matched. For samples where this is not the case (the thermal conductivity is not well matched), we showed that the substrate can play a significant role in determining the value of ΔT across the active material – \nthe magnetic layer – in a thin film device [19]. This \nhighlights a significant disadvantage of using the spin Seebeck coefficients outline d in equations (1)-(3) for \nthin film devices, where the measurement of ΔT is a \npoor indicator of the temperature gradient across the \nactive material. To circumvent this problem, we argued \nthat in the equilibrium condition a simple thermal model can be used to determine the heat flux through the entire sample, \n3 12\n123TATQd dd\n (6) \n \nwhere { d1,d2,d3} and {1,2,3} are the thicknesses and \nthermal conductivities of the top layer (1), FM layer (2) and substrate (3), respectively; and ΔT is the temperature \ndifference across the entire device. It can be seen from this, that the temperature gradient is a function of the \nthermal conductivities of each layer and that the \ntemperature difference across the ‘active’ magnetic layer can be thermally shunted by substrates with relatively low thermal conductivities (for example, see the comparison made between SrTiO\n3 and glass Fe 3O4:Pt \nin [19]). Note that this treatment of the thermal profile \nis, an oversimplification as it does not take into account \nany temperature drops at interfaces. Given that J\nQ will be constant across the sample \nonce thermal equilibrium has been reached, and that it will be proportional to the temperature gradient across \nthe active magnetic layer, of thickness d\n2, we can write \nthe thermal gradient as, \n2\n22 2 2QQJJd TTdd \n (7) \n \nwhere for the magnetic layer Lz=d2 and we now have a \ndirect relationship between the thermal gradient, ܶ ,and \nthe heat flux JQ in the active material . \n Finally, whilst there has been limited discussion \nof the scaling of the spin Seeb eck effect with area, it was \nargued by Kirihara et al. , that the spin Seebeck effect \nmay also scale with area [27] based on the change in internal resistance of the paramagnetic layer, \n \nܴ\nൌ\tఘ\nೣ௧ು (8) \n \nwhere ρ is the resistivity, tPt is the thickness, and Lx is \nthe width of the Pt layer. This suggests that the maximum power extracted is, \nܲ\n௫∝మ\nோబ∝ܮ௫ܮ௬ (9) \n \nNote that this observation would only indicate an \nincrease in power output of the bilayer (not the voltage). \n \nIV. RESULTS AND DISCUSSION \n \nTo test normalization of the spin Seebeck measurements for both heat flux and sample area, we prepared several \nFe\n3O4:Pt samples on glass substrates of varying \nthickness, where the Pt thickness was kept constant at 5 \nnm (0.5 nm) in order to minimize any variation in the \nobserved voltage, VISHE, due to the inverse spin Hall \neffect [19]. VISHE was then measured for various thermal \ngradients as a function of applied magnetic field. We first took one of the Fe\n3O4:Pt samples from \nour study (22x22x0.5 mm glass substrate, 80 nm Fe 3O4, \n5 nm Pt) and measured it in various orientations, as \nsummarized in Figure 2(a)&(e). \n 5 To increase the area of the measured sample, a \nbuffer layer between the two Peltier cells was introduced \n(substrate of same thickness as the sample – on the assumption that control of J\nQ by the magnetic layer is \nlargely negligible). This effectively increased the \nthermal contact area AT, by increasing { LT\nx, LT\ny} whilst \n{Lx, Ly, Lz} were fixed (see Figure 1(d)). Secondly, the \n22x22 mm sample was cleaved in two and measured \nalong the long and short sides, thus changing AT, the \naspect ratio and Ly. Errors were determined from the \ncombination of uncertainty in contact separation Ly, thermal contact area AT, and the noise floor of the VISHE \nvoltage. \n Figure 2(a) shows the calculated spin Seebeck coefficients S\n2 (left) and S4 (right axis) as the area, AT, \nwas increased. Initially this data suggests that there appears to be an increase in the coefficient measured by \nthe heat flux method ( S\n4), but not by the temperature \ndifference method ( S2). In other words, for the same \nsample, the aspect ratio and total area seem to have an impact on the determination of S\n4. In addition, by \nplotting S4 against various combinations of LT\nx, LT\ny, and \nFigure 2 Summary of normalization measurements for thin film Fe 3O4:Pt and bulk polycrystalline YIG:Pt. (a)&(b) S2 (open \nsymbols) and S4 (closed symbols) as a function of AT (as Lz is fixed S2 and S3 are equivalent). The line s are guides for the eye. \n(c) S2 (open symbols) and S4 (closed symbols) as a function of A T0.5 for the YIG sample. (d) Simulated change in the \nmeasurement of S 2 as a function of A T0.5 when the interfacial thermal resistance is varied. (e) & (f) S4 as a function of AT0.5 for \nthe thin film and bulk sample, respectively. Inset sketches indicate aspect ratio for the individual data points. For the thin film \nmeasurements a buffer layer (same thickness glas s substrate) was inserted in order to increase AT; as indicated by the inset \nsketch with dotted outline. Where x-axis errors are not visible they are smaller than the symbol used. \n \n 6 AT (as shown in Appendix A, Figure A1), we found that \nthe most likely way to re duce the measurement to a \nconstant value was to divide through by AT0.5. This \nsuggests that the data requires the following \nnormalization relation in order to produce a geometry independent coefficient, S\n5, \n \nܵହൌೄಹಶ\n൬ೂ\nඥಲ൰ ( 1 0 ) \n \nThis is further demonstrated by the linear dependence \nobserved for the thin film data in Figure 2(e). Note that for this series of measurements, as A\nT exceeded 484 mm2 \nthe area of active material was no longer increasing (as \nAT was further increased by introducing a buffer layer \nwith matched thermal conductivity). So whilst the heat flux across the active material was constant (by definition of S\n4 in equation (4)) there was an apparent \nincrease in the voltage per unit heat flux. It could be \nargued that this is simply due to heat losses, however, as \nwill be shown later, this trend was still observed for \nsamples where AT was varied and measured in a setup \nwith matching Peltier surface area. Given that in the steady state, S\n4κ=S2 (equations \n2-7), the mismatch between the values of S2 and S4 \ndetermined for the thin film samples indicates that there is a measurement artefact that needs to be resolved. To \ntest for this, we also measured bulk YIG samples as a function of A\nT, where the temperature difference across \nthe active material would now be an order of magnitude \nlarger (than our thin films) and thus, less prone to errors such as interfacial thermal resi stance. In addition, due to \nthe insulating nature of YIG, any contribution to V\nISHE \ndue to the anomalous Nernst effect (ANE) will no longer be present. \n Figure 2(b), (c) & (f), shows the same \nnormalization measurements for the bulk YIG:Pt sample as a function of A\nT. Here, the sample was measured in \nboth the 12x12 mm and 40x40 mm sample holders, and cut from the original 13 mm pellet to various sizes. \nNotice that the data from the 12x12 mm and 40x40 mm \nmeasurement set-ups are the same within error, and that it still indicates possible scaling with A\nT0.5. For this \ndataset, plotting S4 as a function of LT\nx also indicated a \nlinear trend (as seen in Appendix A, Figure A2), but this is for the case where A\nT=LxLy (as LxT=Lx and LyT = Ly), i.e. \nit does not account for non-standard geometries, where \nthe contacts are not necessarily at the edges of the sample. In either case, th ere is still a pronounced \nincrease in the measured voltage as the sample size increases that cannot be resolved by normalizing simply \nby contact separation and/or resistance. \n For the bulk samples, however, there is now an indication of scaling of the temperature difference method ( S\n2) with AT. This can be seen in Figure 2(b) and \n(c) where S 2 is plotted alongside S4 as a function of AT \nand AT0.5. The difference between these measurements and that of the thin films is firstly that the measurement \nof ΔT across the (active) magnetic sample is now direct, \nand secondly, that the increased thickness of the sample \n(1.8 mm rather than 0.15 – 0.9 mm) limits the impact of \ntemperature drops at the Peltier:sample interface due to thermal resistance. We argue that these are the reasons why the increase of S\n2 with AT is now obvious for the \nbulk samples. In order to demonstrate this, in Figure 2(d), we \npresent a simple model of the expected trend for S\n2, as \nthe sample area is increased. We first rewrite equation (2) to include the systematic error in measurement of ΔT, \n \nܵ\nଶ′ൌೄಹಶ\nሺ∆்ೞା∆்ሻ ( 1 1 ) \n \nwhere ΔTs is the actual temperature drop across the \nsample and ΔTi is the temperature drop at the \nsample:thermocouple interface(s). (For the true measurement of S\n2, the temperature offset, ΔTi, should \nbe subtracted.) We then make the assumption that ΔTi \nwill be approximately constant (for the same Q). We \nargue that if the individual measurements are well \ncontrolled (i.e. similar sample mounting, use of thermal grease and comparable force when clamping the sample between the two Peltiers), then this is a reasonable assumption as it is likely driven by the thermal \nproperties of the interface to which the thermocouples \nare attached (i.e. an effective thermal conductance). If this were not the case, then there would be considerably more scatter in the data for measurement of S\n2, both here \nand in the literature. In this case, according to equation (5), we can \ndefine the heat, Q, passing through the sample and the \ninterface as follows, \nܳൌ\n∆்\nௗܣ ( 1 2 ) \nܳൌ∆்ೞೞ\nௗೞܣ௦ ( 1 3 ) \n \nwhere κi & κs are the thermal conductivities, Ai & As are \nthe cross sectional areas, and di & ds are the thicknesses \nof the interface and the sample, respectively. If we combine equations (12) & (13), we can write ΔT\ni in \nterms of ΔTs as, \n \n∆ܶൌ∆ ܶ௦ቀௗೞೞ\nௗೞቁ ( 1 4 ) \n \nand finally the measured temperature difference, ΔT, as, \n \n∆ܶ ൌ ∆ܶ ௦\t∆ܶൌ∆ ܶ௦ቀ1ௗೞೞ\nௗೞቁ (15) \n \nFrom equation (15) it should be clear that as the sample area is increased, the influence of the interfacial resistance (for a given measurement setup, where κ\niAi/di \nremains approximately constant) increases. Similarly, \n 7 for the thin film samples, as ds is of the order of 80 nm, \nthe impact of ΔTi will be more pronounced. This was \nshown previously, where ΔTs was found to be 0.01% of \nthe total measured ΔT in our thin film Fe 3O4. [19] \n Equation (15) was used to simulate S2’, i.e. how \nthe temperature dependent spin Seebeck coefficient ( S2) \nis modified as a result of ΔTi. In this case we used the \nmeasured values of S4, and the known thermal \nconductivity and thickness of the sample (3 W/K/m and 1.8 mm) to estimate S\n2 where ΔTi was negligible . We \nthen multiplied this by ΔTs/ (ΔTs + ΔTi) to obtain S2’ for \nvarious values of κeff (=κiAi/di). Note that as AT is \nincreased, S2’ appears to saturate, and that this occurs \nsooner for higher values of κeff (as seen in Figure 2(d)). \n To summarize, there is competition between an increase in S\n2 as AT increases (which is observed with \nthe heat flux method), and a decrease in S2 due to ΔTi. \nAs stated earlier, the advantage of defining a spin \nSeebeck coefficient dependent on JQ rather than ΔT is \nthat it will be independent of thermal contact resistance [16], and the substrate. Finally, to confirm that the observed trend is not a result of heat losses, when the Peltier area was not \nmatched to the sample area, we repeated the thin film \nmeasurement for 3 separate measurement set-ups, where the Peltier area was 10x10mm, 20x20mm, and 40x40mm and a single film was cleaved to match this. In this case, the temperature gradient was driven by a \nresistor (R) mounted onto the top surface of the top \nPeltier so that heat flow could be monitored either side of the sample. The sample used here was deposited onto a 50x50 mm substrate, where we might expect a thickness variation of the Pt layer of approximately 10% across the sample area. To quantify the impact of this \nthickness variation on the measurement, we took 3 \n10x10 mm pieces from corners of the sample, to measure the scatter in the spin Seebeck coefficient. This \ncan be seen in Figure 3, where for one of the 10x10 mm \nsamples S\n4 = 51.63 nV.m/K compared to 30.1 and 37.43 \nnV.m/K, and gives an upper limit of the expected variation in the film (see supplementary information for \nmore information). The result of these measurements is given alongside a tabulated example of the key figures \nfor a subset of the measurements (i.e. one heat flux for \neach area) in Figure 3\n. Notice here, that the difference in \nS4 is greater than a factor of 3 between the smallest and \nlargest sample(s). Even when scaling by the power supplied to the resistive heater (where we expect heat \nlosses), this increase in S\n4 is observed. \n In addition, we also performed measurements on a sample with fixed area, A\nT, but modifying Lx and Ly by \nscribing away areas of the sample (so that it can be considered as thermally connected but electrically isolated). In this case, V\nISHE always scaled with Ly, as \ntypically expected fo r a measurement where AT does not \nchange. \nTo summarize, we observed, for fixed thickness of \nthe magnetic and Pt layers: \n1. At constant A\nT and JQ: Linear scaling of VISHE with \nLy, which is independent of sample geometry (as \nexpected). \n2. At constant AT and Ly: Linear scaling of VISHE with \nΔT and JQ (as expected). \n3. At constant AT and JQ: No observable change, when \nareas of the sample had been rendered inactive by scribing a line between the magnetic layers (i.e. scribing away the electric contact and modifying the resistance R and charge current in that section, \nI\nc). \nFigure 3 - Spin Seebeck measurement(s) of Fe 3O4:Pt 50x50 mm thin film which was cleaved to match measurement areas \nof 10x10, 20x20, and 40x40mm Peltier cells. The line is a guide for the eye and x-error was smaller than the symbols \nused. The table shows a set of values used to obtain this plot where Sp is the Peltier calibration coefficient, Iresistor , the \ncurrent supplied to a resistor, R, mounted onto the top surf ace of the top Peltier cell. Qresistor is the resulting heat source \ndue to the resistor, Vp is the measured voltage for the top Peltier cell, and Qtop the measured heat from the top Peltier \n(=Vp/Sp). \n \n 8 4. At constant JQ: An increase in the spin Seebeck \ncoefficient, S4 (=VISHEAT/LyQ) when AT was \nincreased. \n5. At constant JQ and constant Ly: An increase in the \nspin Seebeck coefficient, S4 (=VISHEAT/LyQ) when \nAT was increased. \n The above observations indicate that there is an additional parameter that is being overlooked when \nevaluating the magnitude of voltage generation due to \nthe inverse spin Hall effect, or thermal pumping of spin current, I\ns, into the Pt layer. At this stage it is not clear \nwhat this mechanism is, but we speculate that it could be due to the assumption that the spin current density, J\nS, \ninjected in the Pt layer at the Ferromagnet:Paramagnet \ninterface is directly proportional to JQ. Finally, this \nsuggests that S5, as defined in Equation (10), is the most \nappropriate coefficient to use when comparing different bilayer systems, with arbitrary area, A\nT. \n Overall, for the thin film Fe 3O4 samples we can \ncompare our results to that of Ramos et al. ,[12]. In this \nwork they reported S2 = 150 μV/K/m for 50 nm Fe 3O4 \ndeposited epitaxially onto a 0.5 mm thick, 8x4 mm SrTiO\n3 substrate, with a 5 nm Pt layer. If we assume that \nthe temperature gradient is controlled by the thermal \nconductivity of the substrate so that S4 = S2κ/d \n(kSrTiO3 =11.9 W/m/K for SrTiO 3) this would give S4 = 6.3 \nnV.m/W. For the 0.5 mm thick SiO 2 substrate sample in \nFigure 2 of this work (80 nm Fe 3O4, 5 nm Pt, 10x10 \nmm) applying a similar approach ( kSiO2~1 W/m/K, d = \n0.5 mm) would give S4 ~ 12 nV.m/W, where we \nmeasure approximately 30 nV.m/W by the heat flux method. This data shows that for these Fe\n3O4:Pt bilayers \nS5 = 3±0.2 µV W-1. The difference between the values \nobtained by the heat flux and temperature difference methods are not unexpected as Sola et al .[17] showed \nthat the ΔT method can routinely underestimate the spin \nSeebeck coefficient. However, it does demonstrate the comparable quality of our thin films. \nFor bulk YIG measurements we obtain S\n5 = S4/AT0.5 = \n4.8±0.34 μV/W and S2/AT0.5 = 8±0.24 mV/K/m2 (using κ \n= 3 W/m/K and d = 1.8 mm). These values are \nreasonable given the porosity of this YIG pellet. For comparison, measurements of bulk polycrystalline YIG by Saiga et al. , [28] found S\n1 of up to 5 μV/K, when \nannealing to improve the interface equality. Given the dimensions L\nx = 5 mm, Ly = 2 mm, Lz = 1 mm, this \namounts to S2 = 1000 μV/K/m and S2/AT0.5 = 316 \nmV/K/m2, or S5 = 45 μV/W (assuming κ = 7 W/m/K). If \nwe compared to direct heat flux measurements by Sola et al. , with thin film YIG on GGG [17], they measured \nS\n4 = 0.11 μV.m/W for a 5x2 mm sample. Hence, S5 = 34 \nμV/W. Whilst these values are an order of magnitude \nlarger than our measurement, for these examples care was taken to maximize the in terface quality. In addition, \nthe Pt thickness was smaller ( t\nPt < 3 nm), which would \nbe expected to increase VISHE further. To further demonstrate the impact of thermal \nresistance on the measurement of S2, we show in Figure \n4, results of spin Seebeck measurements for our \nFe3O4:Pt bilayers as a function of substrate and Fe 3O4 \nthickness ( tsubstrate and tFe3O4 respectively). Note that \nthere was a scatter in our individual data points for S2 \nthat can be attributed to varying thermal resistance \nbetween repeated measurements for the same sample (see supplementary information for individual \nmeasurements); this led to larger error bars. In Figure \n4(a) we show a summary of all measurements as a function of substrate thickness. As can be seen, S\n2 \ndecreased linearly with increasing thickness, as a larger proportion of ΔT was shunted across the substrate. This \nis not unexpected. As soon as the data was normalized \naccording to Equation (10), as is also shown in Figure 4(a) we obtained a relatively constant value (within error, and due to variations that might be expected from changing interface quality [14] or small differences in t\nPt \nand the resistivity). \n With regards to the thickness dependence of the spin Seebeck effect (Figure 4(b)) we also see collapse of \n \nFigure 4 Summary of measurements as a function of \nsubstrate and Fe 3O4 thickness, tsubstrate and tFe3O4 , \nrespectively. (a) Average value for S2 (open symbols) and \nS5 (=VISHEAT0.5/LyQ) (closed symbols) as a function of the \nsubstrate thickness, tsubstrate . (d) S5 as a function of tFe3O4. \nDashed line shows a fit according to linear response \ntheory[25], with a magnon accumulation length of 17 nm. \n \n 9 the data onto the expected saturation at around 80 nm. \nAgain, this is not unusual: saturation of the signal would be expected as the thickness of the Fe\n3O4 layer increases, \ngiven that the volume magnetization also saturates at \naround 80 nm (see supplementary information, Fig. S3). The dashed line in this figure shows the corresponding \nfit to this data using the approach of Anadón et al. ,[25] \nwho found a magnon accumulation length of 17 (± 3) nm at 300K from this fit. The slight decrease in the spin \nSeebeck coefficient for the th ickest sample could be due \nto a change in the morphology of the thicker film and is the focus of current work. \n V. CONCLUSION \n \nTo summarize, we have demonstrated that the heat flux \nmethod is suitable for obtaining substrate independent measurements of the spin Seebeck effect, where the \nthermal conductivity of the thin film need not be known, \nand have proposed an alternative spin Seebeck \ncoefficient ( S\n5) for comparison of different material \nsystems. By measuring the spin Seebeck effect as the thickness of the substrate, Fe\n3O4 layer and sample \ndimensions { Lx, Ly, AT} were varied, we demonstrated \nthat for the same material system (assuming tPt is \nconstant at 5 nm) this method reliably returned a value \nof 3.0±0.2 µV W-1. Since this result holds for different \nvalues of tsubstrate , the heat flux normalization method \ncould, therefore, be used to compare similar samples where the substrate thickness or type were varied. This \nwould thus be a useful metric for comparison of \nprospective material systems (for application). Lastly, we demonstrated unexpected scaling of the spin Seebeck coefficient (as typically defined in the literature). These results indi cate that the voltage is \nproportional to the available energy per unit length ( Q), \nwhich we speculate could be a result of thermal spin \ninjection into the paramagne tic layer before detection. \nThe exact nature of this scaling will be the topic of future studies. \nACKNOWLEDGEMENTS \nThis work was supported by EPSRC First Grant (EP/L024918/1) and Fellowship (EP/P006221/1) and the Loughborough School of Science Strategic Major Operational Fund. KM would also like to thank M.D. Cropper for their help with use of the PLD system for \nsample fabrication and M Greenaway and J Betouras for \nuseful discussions. Supporting data will be made available via the Loughborough data repository under doi 10.17028/rd.lboro.5117578. \nAPPENDIX A: ADDITIONAL PLOTS OF S\n4 FOR \nTHE THIN FILM AND BULK SAMPLES Figures A1 and A2 show additional plots of the data \npresented in Figure 2 (e) & (f), as AT was varied. This \nincludes parameters such as the horizontal and vertical \nlengths, LxT, LyT, their ratio, and the area AT. The dashed \nlines are guides for the eye. \nAPPENDIX B: HEAT LOSS CONSIDERATIONS \nIn an ideal case, the sample area would be chosen such \nthat it matches the area of the hot and cold bath(s) in the \nmeasurement. Given that this paper studies the impact \nof sample size (where A\nT is varied), it could be argued \nthat for samples where AT is less than the area of the top \nand bottom Peltier cells, there are significant thermal losses (radiation and convection). This was monitored \nduring measurement by plotting the change in the \nmeasured J\nQ as a function of ΔT (see Figure 1(e)). It was \nalso confirmed by obtaining comparable measurements in air and under vacuum, as detailed in the supplementary information. \n \nWe consider here the potential impact of radiative losses, as defined by: Q = εσA(T\nh4- Tc4) ( 1 6 ) \n \nwhere ε is the emissivity of the radiating surface ( ε = 1 \nfor a perfect radiator), σ is the Stefan-Boltzman constant \n(σ = 5.67x10-8 W/m2/K4), A is the radiating surface area, \nand Th and Tc are the temperatures of the hot and cold \n(ambient) surfaces respectively. For the sample \nenvironment used in these measurements there will be 3 \nsources of radiative heat loss: 1) Top surface of the Peltier cell where the sample is not \nconnected (forming a conductive path). \n2) Bottom surface of the Peltier cell where sample is not \nconnected. \n3) The sides of the sample. \n Sources 1&2 can be considered simultaneously given that A will be the same, and it is reasonable to assume \nthat the majority of the heat lost from the top surface \nwill be measured by the bottom surface (i.e. overestimating J\nQ). Thus for 1&2: \n Q\nrad = εσ(AP-AT)(T h4-Tc4) (17) \n \nwhere Ap is the Peltier area. \n For the worst case scenar io for radiative losses \n1&2 the smallest samples measured in the 40x40mm2 \nPeltier setup had AT=100 mm2. In this case, AP-AT=1500 \nmm2. Inserting this into equation (17), the radiative heat \nloss would be Q = 0.045 W for a measured heat flow of \n2.5 W (1.8%). \n 10 The worst case scenario for radiative loss via the \nsides of the sample was for the thickest samples, with \nlargest area. The thickness of the measured thin films were 0.17-1.34 mm, with a maximum radiative area (from the sides) of 171.2 mm\n2. For a typical temperature \ndifference of 5 K this would amount to radiative loss of \napproximately 0.0045 W (0.2% of Q). For the YIG \nsamples, with thickness = 1.8 mm, and temperature differences of up to 25 K the radiative loss would be 0.012 W (0.5% of Q). In this context, for these \ntemperature differences, radiative loss is negligible with \nrespect to the changes seen as a function of A\nT0.5. \n As a general rule of thumb, radiative heat losses in this system can be quickly assessed by monitoring J\nQ \nvs. ΔT. Given equations (5) and (16), for the ideal case the heat flux through the sample should be linear with \nrespect to ΔT. As soon as the radiative heat losses \nbecome significant with respect to the heat flow through the sample, this linearity breaks down as a larger J\nQ will \nbe observed per unit ΔT. \n Conversely, for much thicker samples, where \nradiative loss from the sides of the sample starts to \nbecome significant, such as is the case with the bulk \nYIG samples, heat loss would result in a lower measured J\nQ per unit ΔT as well as non-linearity of \nJQ(ΔT). We only present data for values of ΔT where this \nlinearity was preserved. \n \n \n \nFigure A1 (a) – (f) Spin Seebeck measurement(s) for the Fe 3O4:Pt thin film as AT (=LT\nx.LT\ny), Lx and Ly were varied, plotted \nagainst LT\nx, LT\ny, LT\ny/LT\nx, AT/LT\ny, AT and AT0.5. Where x-axis errors are not visible th ey are smaller than the symbol used. \n \n 11 REFERENCES \n[1] K. Uchida, S. Takahashi, K. Harii, J. 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Cox1, T.A Rose1 \n1 Department of Physics, Loughborough Univ ersity, Loughborough, LE11 3TU, United Kingdom \n2 ISIS Neutron and Muon Source, Didcot, Oxfordshire, OX11 0QX \n \nKeywords Magnetic materials, thermoelectrics, thin films \n \n1 Introduction Extensive experimental details and additiona l characterization of the films presented in \n“Scaling of the spin Seebeck effect in bulk and thin film” is given in this supplementary information. \nAdditional X-ray diffraction (XRD), resistivity and X-ra y reflectivity (XRR) data is used to demonstrate \nthe quality of Fe 3O4 thin films – exhibiting the characteristic Verw ey transition for this phase[1], as well as \nthe expected magnetic characteristics as a function of te mperature. This is followed by additional XRD, and \nXRR data for the Fe 3O4:Pt bilayers to further demonstrate the magne tic and structural quality of the films, as \nwell as XRD and SEM data of the YIG pellet chosen for this study. Finally, further details of the spin \nSeebeck measurements are given, alongsid e the additional ‘scribing’ study data. \n2 Experimental Details \n2.1 Sample preparation \nThe series of films prepared for this study are listed in Table S1 . They were prepared in vacuum (base \npressure of 5x10-9 mbar) by pulsed laser deposition (PLD) using a frequency doubled Nd:YAG laser (Quanta \nRay GCR-5) with a wavelength of 5 32 nm and 10 Hz repetition rate. The films were deposited onto 10 x 10 \nmm and 22 x 22 mm glass slides (Agar Scientific bor osilicate coverglasses), or 24 x 32 and 50 x 50 mm \nfused silica, which were baked out at 400 °C prior to the growth of the Fe\n3O4 layer at the same \nsubstrate temperature. The samples were then left to cool in vacuum until they reached room temperature, \nat which point the Pt layer was deposited. The \nFe\n3O4 and Pt layers were deposited from \nstoichiometric Fe 2O3 (Pi-Kem purity 99.9%) and \nelemental Pt (Testbourne purity 99.99%) targets using ablation fluences of 1.9±0.1 and 3.7±0.2 J cm\n-\n2 respectively. The target to substrate distance was \n110 mm. The Pt layer thickness was kept constant at 5±0.5 nm for all films, whilst the substrate and Fe\n3O4 thicknesses were varied. \n Bulk YIG was prepared by the solid state \nmethod. Stoichiometric amounts of Y 2O3 and Fe 2O3 \nstarting powders (Sigma Aldrich 99.999% and \n99.995% trace metals basis, respectively) were ground and mixed together before calcining in air at 1050 ᵒC for 24 hours. Approximately 0.5g of the \ncalcined powder was then dry pressed into a 13 mm \ndiameter, 1.8±0.2 mm thick cylindrical pellet. The \npellet was then sintered at 1400 ᵒC for 12 hours, \nafter which, it was checked by XRD prior to sputtering 5 nm of Pt onto the as prepared surface using a benchtop Quorum turbo-pumped sputter \ncoater. Samples were cut to size thereafter, using an \nIsoMet low speed precision cutter. \n \n2.2 Structural characterization \nXRD was obtained using a Bruker D2 Phaser in the \nFIG. S1 Schematic of the spin Seebeck measurement \nsystem. Top and bottom Peltier cells act as heat source \nand heat flux monitor, respec tively. Copper plates on the \nsample facing side of the Peltier cells provide high thermal conductivity at th e interface, thus reducing \nlateral temperature gradients. GE varnish and mica are \nused to electrically isol ate the copper surface from \nvoltage contacts. Thermocouples monitor the \ntemperature (with respect to the cold sink) at the top and \nbottom of the sample. Finally, the voltage, V\nISHE is \nmonitored as Q and B are varied. Note that the thermal \ncontact area, A T, is defined as the area of sample \nconnecting the top and bottom Peltier cells. \n2 \n standard Bragg-Brentano geometry using Cu K radiation; a Ni filter was used to remove the Cu K line. The \nsamples were rotated at a constant 15 rpm throughout the scan to increase the number of crystallites being \nsampled. XRR measurements were taken using a modified Siemens D5000 diffractometer comprising a \ngraphite monochromator reflecting only Cu K into the detector. The incident beam was collimated with a \n0.05 mm divergence slit, while the reflected beam w as passed through a 0.2 mm anti-scatter slit followed by \na 0.1 mm resolution slit. XRR data was fitte d using the GenX software package [2]. \n \n2.3 Magnetic and electric characterization \nMagnetization as a function of applied magnetic fiel d was measured using a Quantum Design Magnetic \nProperties Measurement System (SQUID). Several bilayer samples were characterized as a function of field \nat room temperature, whilst the Fe 3O4 film (without Pt contact) was measu red at 5 K, 100-130 K, and 295 K \nin order to observe the characteristic magnetic Verwey transition at 115 K. \nRoom temperature sheet resistance was obtained using the Van der Pauw method with a Keithley \n6221/2182 nanovoltmeter and current source in conjunction with a Keithley 705 scanner. \n \n2.4 Thermoelectric characterization \nSpin Seebeck measurements were obtained from a set-up similar to that of Sola et al. [3], and outlined in \nFigure S1. There were 4 separate set-ups, optimi zed for 10x10, 12x12, 20x20 and 40x40 mm samples. The \nmajority of data presented in the main manuscr ipt were obtained using the 12x12 and 40x40mm setups \nTable S1 Summary of the thin film samples measured. The contact separation and corresponding thermal contact \narea, AT, are also indicated. There were 3 distinct series of sample: (a) where the Fe 3O4 and Pt thicknesses, tFe3O4 and tPt \nrespectively, were kept constant (at a nominal 80 nm:5 nm) and the substrate thickness tsubstrate , was varied, (b) where \nthe Fe 3O4 thickness was varied from 20 – 320 nm, whilst the Pt thickness was kept constant, and (c) a large area \nsample (50x50 mm) that was cleaved into sections for measurement in the 10x10, 20x20, and 40x40 mm setups. Film thickness(es) were determined from fitting XRR data, except where t\nFe3O4 exceeded 80 nm. \nSam ple A T (mm2) L y (mm) Substrate t substrate (mm) t Pt (nm) t Fe3O4 (nm) \nFP1 768 28.8 Glass 0.17 ~4.8 80 \n \nFP3 \n(G6) \n(FP3 frag) \n136.8 \n203 \n8.47 12.4 \nGlass \n0.3 \n4.9 \n80 \n \n \nFP5 (G34) 484, 684, \n880, 1530 \n214 270 \n 19.3 \n19.3 7.62 \n19.44 \nGlass \n0.5 \n5.5 \n80 \n \nFP6 772 28.56 Fused silica 0.67 \n \n4.8 \n \n80 \n 0.67+0.67 \n \nFP9 983 36.66 Fused silica 0.9 5.4 80 \n \n \nF320P \n(G27) 66.6 6.58 Glass 0.3 4.4 320 \n \nF40P \n(G30) 100 6.26 Glass 0.3 5.2 40 \n \nF160P \n(G31) 100 5.5 Glass 0.3 4.5 160 \n \nF20P \n(G32) \n 100 7.25 Glass 0.3 5 20 \nG48 \n 2500 \n1400 \n400 \n100 - \n37 20 \n8.5 \n \n \nGlass \n \n0.5 \n \n5 \n \n80 3 \n unless otherwise stated. The thin film was sandwiched between 2 Peltier cells, where the top Peltier cell (1) \nacted as a heat source, and the bottom Peltier cell (2 ) monitored the heat Q, through the sample. A copper \nsheet was affixed to the surface of the Peltie r cells to promote uniform heat transfer. \n Two type E thermocouples were mounted on the su rface of these copper sheets such that they were \nin contact with the sample during the measurement. These thermocouples were connected in differential \nmode to monitor the temperatur e difference across the sample, ΔT. The Peltier cells were calibrated by \ncomparing the voltage generation, V p, in two scenarios: \n \n(1) where the Peltiers were clamped together, with a resist or attached to the top of one, and the heat flow \npassing through both was assumed to be constant, such that: Q\ntop = Q bottom ( S . 1 ) \n \n(2) where the resistor was clamped between the two Pe ltier cells and the total power measured by the \nPeltier cells was assumed to be equal to the heat output of the resistor, such that:. \n \nQtop + Q bottom = P resistor ( S . 2 ) \n \nSolving these simultaneous equations produced the sensitivity, Sp, which was found to be 0.11 - 0.26 V/W at \n300K. The heat flux, JQ, is then determined by: \n \nJQ = V p/(SpAT) ( S . 3 ) \n where A\nT is the contact area between the top and bottom Pe ltier cells. This can be quickly related to the \nmeasured temperature difference ΔT, assuming that there are no major thermal losses or temperature drops at \nthe sample:Peltier interface(s): \n \nJQ = k effΔT/L z ( S . 4 ) \n where k\neff is the effective thermal conductivity of the sample. \n Given that this paper studies the impact of sample size (thus A T is varied), it could be argued that for \nsamples where A T is less than the area of the top and bottom Pe ltier cells, there are signi ficant thermal losses \n(radiation and convection). This is monitored by plotting the change in the measured J Q as a function of ΔT, \nas was seen in Figure 1. In addition, Figure 6 of the main manuscript (and Fig. S7 here) shows data where \nthe size of the sample was matched to the size of the Peltier cell for measurement of J Q. Finally, Figure S9 \nhere shows similar spin Seebeck measurements in ai r and vacuum, where both Peltier cells are used to \nmonitor heat flow. \n \n3 Results \n3.1 Characterization of the Fe 3O4 layer \nAdditional characterization data for the Fe 3O4 thin films (deposited under id entical conditions as the devices \nmeasured, but without the Pt top layer) is given in Fig. S2. \nThe Verwey transition is a characteristic feature of the Fe 3O4 phase of iron oxide and manifests as a sharp \nincrease in resistivity and hysteresis below the Verw ey transition temperature (typically between 115 and \n120 K)[1]. This feature is clearly seen in Fig. S2 (a), where magnetometry shows a sudden increase in hysteresis below 120 K. \nIn addition, XRD indicates a preference for <111> text ure that is moderately sensitive to the direction of \nthe plume during PLD (see repeated film depositions of Fig. S2 (c)). Due to the instability of the Fe\n3O4 phase, \nsome Fe is also present. Finally, an example of the X-ray Reflectivity (XRR) fits used to obtain film \nthickness is given in Fig. S2 (d). \n \n3.2 Characterization of the Fe 3O4:Pt bilayers \nAdditional characterization data for some of the Fe 3O4:Pt bilayers is given in Fig. S3. Previous work \nshowed that for t Pt>2 nm a thin continuous paramagnetic Pt layer deposited on top of the Fe 3O4 layer follows \nthe wavy surface of the Fe 3O4 layer. The Pt stacks on top of the Fe 3O4 (111) planes and the two grains have \nan orientation relationship of [011]Pt//[011]Fe 3O4, (111)Pt//(111)Fe 3O4, which minimizes the interface \nenergy due to minimal lattice mismatch of d(111) Pt (0.226 nm; JCPDS card 4-802) and d(222) Fe3O4 (0.242 nm; \nJCPDS card 19-629). \nXRD data for the sample series where the Fe 3O4 and substrate thickness was va ried is given in Fig. S3. \nFor the substrate study, XRR (S3b) indicated similar Pt thickness (low frequency ‘bumps’ in the data) as 4 \n well as some variation in the interface quality (whether high frequency fringes can be observed indicates the \nrelative roughness of the substrate, Fe 3O4 and Pt layers). Fig. S3 also shows the XRD and magnetization \nmeasurements of the sample series where the Fe 3O4 thickness was varied. Of particular note is the saturation \nof the magnetic moment of Fe 3O4 measured at 1 T for thicknesses > 80 nm. \n \n3.3 Characterization of YIG \nXRD of the YIG pellet presented here is given in Figure S4. No evidence of unreacted starter powders \n(Fe 2O3, Y 2O3) or potential secondary phase YIP (YFeO 3) was seen, as highlighted in Figure S4(b). Scanning \nelectron microscopy images show grain structure with a porosity expected from 30% estimated open porosity \nas measured by the Archimedes method. \n3.4 Additional spin Seebeck data \nFig. S5 shows some of the raw data for FP5 (analysed datapoints shown in Figure 2&3 of the main article). \nThe heat source for these measurements was the same Peltier, with a maximum power output of approximately 1W. Fig. S6 shows some of the raw data for the YIG pellet as it was cut to different sizes. Fig. \nS7 shows more detailed data for the large area thin film measurement given in Figure 6 of the main text. Fig. \nS8 shows the results of the scribing study, where A\nT was fixed, but L x, L y (not L xT, L yT) were varied by \nscribing away sections of the thin film, as illustrated in Fig. S8(a). In this case, the voltage always scaled \nwith the contact separation, L y, indicating that the behavior we see is not due to a change in the film \nproperties (such as resistance), but scaling of the h eat flux. Figure S9 shows additional spin Seebeck \nmeasurements where the rig was altered to monitor heat flux above and below the sample, with a resistor as \nthe heat source. In this example, the Peltier area was 20x20 mm and the resistor was 57 Ohms. \nMeasurements taken in air and under vacuum indicate a difference of approximately 3.9% due to convective \nlosses (from the sides of the sa mple). Differences between Q top and Q bottom indicate radiative losses of the \norder of 1.3%, which is comparable to the estimate given in the main text. Finally, Figure S10 shows \nexpanded data from Figure 6 of the main manuscript, where scatter in individual datapoints due to poor \nthermal contact, or variation in substrate thickness or area, which was not accounted for. \n \nReferences \n[1] F. Walz, J. Phys. Condens. Mat . 14, R285-R340 (2002). \n[2] M. Björck, and G. Andersson , J. Appl. Cryst. 40, 1174-1178 (2007). \n[3] A. Sola, M. Kuepferling, V. Basso, M. P asquale, T. Kikkawa, K. Uchida, and E. Saitoh, J. Appl. Phys. \n117, 17C510 (2015) \n 5 \n FIG. S2 Characterization of the Fe 3O4 film. a) SQUID magnetometry above and below the Verwey transition, TV. \nb) Resistivity as a function of temperature. c) XRD of a set of 4 separately prepared Fe 3O4 films. The inset shows \na close-up of the (311), (222) peaks. d) Example XRR data (symbols) and fit (solid line), indicating thickness = 79 \nnm, roughness = 1.5 nm. \n \nFIG. S3 Characterization of the Fe 3O4:Pt films, where Fe 3O4 and substrate thickness was varied. a) XRD for Fe 3O4 \nthickness study, b) XRR for substrate study, and c) SQUID magnetometry as a function of Fe 3O4 thickness at \n300K. d) Saturation magnetisation, M, as a function of Fe 3O4 thickness. \n6 \n \n \n \n \nFIG. S4 (a)&(b) X-ray diffraction data for the YIG pelle t. (a) Normalized data alongside reference for Y 3Fe5O12 (YIG). (b) \nZoomed in data selec tion plotted alongside possible impurity phases Fe 3O4 and Y 2O3 (starting powders) and YFeO 3 (YIP). (c) \nScanning Electron Microscope image of the pellet, wh ere grain size was found to be an average of 14.5 μm, with open porosity \nof 30%. \n7 \n \n \n \nFIG. S5 Raw voltage measurements for FP5 at 3 different areas, A T. The contact separation, area, temperature difference, heat \nflux, JQ, and resultant spin Seebeck coefficient, S 4, are shown for each sample. Data has been offset for clarity. \n8 \n \n \nFIG. S6 Raw voltage measurements for the bulk YIG pellet as it was cut down to different sizes, A T. The contact separation, \narea, temperature difference, heat flux, J Q, and resultant spin Seebeck coefficient, S 4, are shown for each sample. Inset shows \nthe linear relationship between heat flux, J Q, and temperature difference ΔT, which starts to break down for ΔT>10K due to \nradiation losses. \n9 \n \n \nFIG. S7 Summary of the voltage measured as a function of heat flux for the (a) – (c) 10x10, (d) 20x20, (e) 40x40 mm Peltier \ncell measuremen ts. (f) S4 plotted as a function of A T0.5 for these samples. Note the scatter for the 100mm2 samples is indicative \nof thickness variation over the 50x50 mm film, which was sample d by selecting pieces from 3 corners of the sample. Due to \nplume direction during the pulsed laser deposit ion process, we usually expe ct thickness variation of th e Pt layer to differ by 10-\n15% at one edge of the film and this is evident by the larger voltage seen in (b), where the Pt thickness was approximately 0.5 -1 \nnm thinner. To avoid this difference in standard measur ements we constrain the sample deposition area to 40x40mm. \n10 \n \n \n \nFIG. S8 Results of the scribing study. As long as A T was unchanged, as L x and L y were varied (by scribing away the sample \nsuch that it is still thermally connected, but electrically isolat ed), the voltage always scaled the same. Large noise is due t o \nreducing contact separation and additional electrical noise during some of the measuremen ts (due to a loose wire). \n \nFIG. S9 Additional spin Seebeck meas urements for the 20x20 mm Peltier arrangement. (a) & (b) Final calibration data for the \nPeltier cells in air and under vacuum (P~1x10-4 mBar), respectively. Q top is the heat measured by the top Peltier (W), and Q \nbottom is the heat measured by the bottom Peltier. This data indi cates a heat loss of less than 6% across the stack. (c) & (d) Raw \nvoltage measurements (data offset for clarity) at various heating powers (driven by a resistive heater) in air and vacuum, \nrespectively. (e) schematic of the altered measurement. (f) V ISHE as a function of heat flux (extracted from data in (c) and (d)) \nfor the air and vacuum measurements. The fit to this data indicates a deviation of approximately 2%. \n11 \n \n \n \nFIG. S10 Data from the substrate and Fe 3O4 thickness study plotted to show scatter in datapoints. (a) Impact of poor thermal \ncontact on S 2, (b), Impact on S 4 of the variation of aspect ratio for the substrate measurements. (c) S 2 measured for the Fe 3O4 \nthickness measurements. (d) S 4 (blue symbols) versus S 5 (black symbols) for the substrate measurements. \n" }, { "title": "2306.14029v2.Magnon_confinement_in_an_all_on_chip_YIG_cavity_resonator_using_hybrid_YIG_Py_magnon_barriers.pdf", "content": "Magnon confinement in an all-on-chip YIG\ncavity resonator using hybrid YIG/Py magnon\nbarriers†\nObed Alves Santos∗,‡,¶and Bart J. van Wees‡\n‡Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen,\nNijenborgh 4, Groningen, AG 9747, The Netherlands\n¶Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom\nE-mail: oa330@cam.ac.uk\nAbstract\nConfining magnons in cavities can introduce new functionalities to magnonic devices, en-\nabling future magnonic structures to emulate established photonic and electronic components.\nAs a proof-of-concept, we report magnon confinement in a lithographically defined all-on-chip\nYIG cavity created between two YIG/Permalloy bilayers. We take advantage of the modified\nmagnetic properties of covered/uncovered YIG film to define on-chip distinct regions with\nboundaries capable of confining magnons. We confirm this by measuring multiple spin pump-\ning voltage peaks in a 400 nm wide platinum strip placed along the center of the cavity. These\npeaks coincide with multiple spin-wave resonance modes calculated for a YIG slab with the\ncorresponding geometry. The fabrication of micrometer-sized YIG cavities following this tech-\nnique represents a new approach to control coherent magnons, while the spin pumping voltage\nin a nanometer-sized Pt strip demonstrates to be a non-invasive local detector of the magnon\nresonance intensity.\n†Keywords: Magnonics, Spin pumping, Spin waves, YIG resonators, On-chip cavity magnonics.\n1arXiv:2306.14029v2 [cond-mat.mes-hall] 6 Oct 2023Magnonics aims to transmit, store, and process information in micro- and nano-scale by means\nof magnons, the quanta of spin waves.1,2Typical studies in this field use the ferrimagnet Yttrium\nIron Garnet (Y 3Fe5O12- YIG) as a key material, due to its desirable properties, such as very\nlow magnetic losses,2large applicability in mainstream electronics,3long magnon spin relaxation\ntime,4and high magnon spin conductivity.5,6The magnetic proprieties of YIG can be modified by\nthe presence of strong exchange/dipolar coupling caused by the deposition of a ferromagnetic layer\nonto the YIG film.7,8These hybrid magnonic systems, such as YIG/Py,8–13YIG/CoFeB,14,15and\nYIG/Co,16,17draw attention from fundamental and applied physics towards the control of coherent\nmagnon excitations for information processing.18Alternatively, systems in which the magnetic\nmaterial strongly interacts with electrodynamic cavities also provide a good platform for next-\ngeneration quantum information technologies, using the dual magnon-photon nature to enable new\nquantum functionalities.19,20The field called cavity magnonics, among many applications, can\noffer good compatibility with CMOS, room temperature operation, and GHz-to-THz spin-wave\ntransducers.21–24\nThe possibility of confining magnons in cavities may enable future magnonic devices that\nemulate established electronic and photonic components, such as magnonic quantum point con-\ntacts, magnonic crystals, magnonic quantum bits, magnonic frequency combs, among others.25–33\nBuilding upon similar ideas, recent theoretical results suggest all-on-chip structures to produce\nmagnonic cavities by magneto-dipole interaction with a chiral magnonic element,34using an array\nof antiferromagnets on a ferromagnet,35or by proximity with superconductors.36\nOne approach to confining magnons involves the fabrication of rectangular or circular nano-\nand micro-structures using YIG.37–40Many of these structures are made by etching the YIG film\nor sputtering YIG from a target on defined micro-structures, adding an extra step in the fabrication\nprocess and limiting the options of available structures. Recently, Qin et al. , demonstrated the (par-\ntial) confinement of magnons in a region of the YIG film covered by a ferromagnetic metal.29This\nconfinement arises from the difference in magnetic properties between the covered and uncovered\nregions of the YIG film, resulting in magnon reflections at the boundaries. Such phenomena en-\n2abled the fabrication of an on-chip nanoscale magnonic Fabry-Pérot cavity,28,29observed by the\ntransmission of magnons through the YIG film.30\n(c)\n𝑽𝒓𝑺𝑷+\n−\n𝑽𝒄𝑺𝑷+\n−\n𝒉𝒓𝒇z\nxy(b) \nlw𝑽𝒑𝑺𝑷\n+\n−𝑩\n𝑩(a)\n10𝜇mw\nPt\nPyPtPy Py\nFigure 1: (a) Schematic illustration of the lateral view of the cavity formed by confining the YIG\nfilm between two YIG/Py bilayers. The Pt strip in the middle of the cavity can be used as a local\nmagnon detector. (b) Schematic illustration of the waveguide stripline and sample arrangement\nof the microwave excitation setup for the FMR absorption and spin pumping measurements. (c)\nOptical image of the fabricated devices, and the electrical connections used in the experiments.\nThe cavity is formed in the YIG region constrained between the Py squares, defined by w×l.\nMultiple peaks are observed with the platinum strip placed along the center of the cavity, VSP\nc, and\nabsent in both Pt strips placed outside of the cavity, VSP\nrandVSP\np.\nIn this letter, we complement these studies by achieving an order of magnitude higher magnon\nconfinement in a YIG film region which is not covered by the ferromagnetic metal, preserving\nthe optimal magnetic properties of YIG within the cavity. We report, as a proof-of-concept, the\nfabrication of an all-on-chip micrometer-sized YIG cavity, by partially covering 100 nm thick LPE-\ngrown YIG film with two square-shaped 30 nm thick permalloy (Py) layers. The exchange/dipolar\n3interactions in the YIG/Py bilayer define on-chip, magnetically distinct regions, effectively forming\nreflecting boundaries for magnons, resulting in a magnonic cavity. We confirm this by measuring\nmultiple spin pumping voltage (VSP)peaks with a 400 nm wide Pt strip placed along the center of\nthe cavity. This voltage is proportional to the intensity of the FMR-excited magnon resonances in\ntheuncovered YIG film, indicating the formation of standing-wave resonance modes. We assign\nthese peaks to multiple standing backward volume spin wave modes (BVSWs) and magnetostatic\nsurface spin wave modes (MSSWs), calculated from the spin-wave theory for a YIG slab with\nsimilar dimensions. Multiple spin pumping voltage peaks were not observed for Pt strips placed\noutside of the cavity. All the measurements were performed at room temperature.\nThe presence of BVSWs and MSSWs modes in micrometer YIG cavities has already been\nobserved.37,38,41–44However, the YIG cavities were produced by sputtering or wet-etching tech-\nnique, and the modes were measured by a local FMR antenna or time-resolved magneto-optical\nKerr effect. To the best of our knowledge, this work represents the first observation of cavity res-\nonant modes measured by spin pumping voltages using all-on-chip hybrid magnonic structures,\nillustrated in Figure 1 a). The device was fabricated on a high-quality 100 nm thick YIG film\ngrown by liquid phase epitaxy on a GGG substrate. Electron beam lithography was used to pattern\nthe device, consisting of multiple strips of Pt and two Py squares with edge-to-edge distance of\nw=2µm, Figure 1 b) and c). The square shape was chosen to avoid effects of shape anisotropy in\nthe Py film.45,46The sample was placed on top of a stripline waveguide and connected to a vector\nnetwork analyzer. The stripline waveguide was then placed between two poles of an electromagnet\nin such a way that the DC external magnetic field, B=µ0H, where µ0is the vacuum permeability,\nand the microwave field, hr f, were perpendicular to each other and both were in the plane of the\nYIG film in all the measurements, see Figure 1 a) and b). See supporting information section I for\nmore details on sample fabrication and experimental setup.47\nTheB-field scan of the microwave absorption, S21, which measures the overall magnetic re-\nsponse of 4 ×3 mm sized YIG film, is shown in the top panel of Figure 2 a), for 1 to 9 GHz. The\nFMR absorption peak of the 100 nm thick YIG film has a typical linewidth of ≈0.2 mT, demon-\n40.00.51.01.5\n 1 GHz 5 GHz 8 GHz 9 GHz\n 2 GHz 6 GHz\n 3 GHz 7 GHz\n 4 GHzSP voltage ( mV)\n0 50 100 150 200 25002468\nMagnetic field (mT) Kittel equation\n Remote strip Frequency (GHz)(a)\n(b)\n-0.2-0.10.0\n 1 GHz \n 2 GHz 5 GHz\n 3 GHz 6 GHz\n 4 GHz 7 GHz 8 GHz 9 GhzS21(a.u)\n0 50 100 150 200 25002468\nMagnetic field (mT) Kittel equation\n m0HFMR of bulk YIGFrequency (GHz)\n2 4 6 8 100.00.10.20.3Linewidth (mT)\nFrequency (GHz)Figure 2: Comparison between FMR absorption of the bulk YIG and the spin pumping voltage in\ntheremote strip. (a) Top panel shows B-field scan of the microwave transmission absorption, S21.\nThe bottom panel shows the field position of FMR peaks for different microwave frequencies. The\ninset shows the linewidth vs.resonance frequency. Note that we do not see the magnon spectra\nbecause the magnetic field is uniform on a relatively long scale. Therefore, only the uniform mode\nis excited and measured. (b) The top panel shows the B-field scans of the VSPof the remote Pt strip\nfor different microwave frequencies. The bottom panel shows the field position of the maximum\nVSP\nras a function of the microwave frequency.\n5strating the high-quality of the YIG film. The FMR spectra are fit using an asymmetric Lorentizan\nfunction, obtaining the B-field value of the FMR peak and the linewidth (FWHM), see details in\nSupporting Information section I. The bottom panel of Figure 2 a) shows the microwave frequency\nversus the B-field value of the FMR peak. The solid blue curve corresponds to the best fit to the\nKittel equation, f=γµ0/2π/radicalbig\nH(H+M),3where γis the gyromagnetic ratio. The best fit was\nobtained for γ/2π=27.2±0.1 GHz/T and M=142.4±0.8 kA/m. The inset in Figure 2 a) shows\nthe linewidth as a function of the microwave frequency. The solid blue curve corresponds to the\nbest linear fit, obtaining a Gilbert damping of α≈5.0×10−4and the inhomogeneous linewidth of\nµ0∆H0=0.06 mT. In this letter, we address the uniform FMR resonance of the YIG film as \"bulk\"\nYIG resonance, or µ0HFMR, to distinguish the FMR absorption measurement of the mm-range size\nYIG film from the local spin pumping voltage measurements for different platinum strips. The\nupper panel of Figure 2 b) shows the B-field scan of the spin-pumping voltage for the remote Pt\nstrip, VSP\nr, for different microwave frequencies. The bottom panel presents the magnetic field of\ntheVSP\nrpeak for different rffrequencies. Again, we fit the results using the Kittel equation with\nγ/2π=27.2±0.1 GHz/T and M=140.2±0.9 kA/m.\nIt is important to emphasize that albeit Figures 2 a) and b) look effectively identical, they cor-\nrespond to two completely different experiments. Both measure the intensity of the ferromagnetic\nresonance, but in one case we measure the FMR absorption of the bulk YIG film, and in the other\nwe measure the local spin pumping voltage, as a result of the injection of spin current by means\nof the spin pumping effect,48and the conversion of the spin current into charge current in the Pt\nstrip by the inverse spin Hall effect.49,50We did not observe a significant peak broadening caused\nby the spin absorption due to the presence of the Pt layer. This is because the width of the strip is\n400 nm, such that it covers only a fraction of the YIG film resulting in a spin pumping response\nproportional to the FMR absorption of the bulk YIG.51These results show that the platinum strip\nis a local and non-invasive intensity detector of the magnon excitation.\nFigure 3 compares the B-field scan of the spin pumping voltage measured on the remote strip,\nVSP\nr, solid black lines, and on the cavity strip, VSP\nc, solid blue lines, for different rffrequencies,\n640 45 50 55 60 653 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV\n100 105 110 115 120 125 1305 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV\n175 180 185 190 195 200 2057 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV\n215 220 225 230 235 2408 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV\n250 255 260 265 270 275 2809 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV\n15 20 25 30 352 GHzSpin Pumping voltage\nMagnetic field (mT) Cavity strip\n Remote strip\n BVSWs\n MSSWs1 mV(a) (b) (c)\n(d) (e) (f)Figure 3: The B-field scan of the spin pumping voltage measured with the remote strip VSP\nrand\ncavity strip VSP\ncfor different frequencies is shown from (a) to (f). The resonance modes of the\ncavity strip occur before the FMR of the bulk YIG resonance field for low frequency, 2 GHz, and\nare present after the FMR bulk resonance for 9 GHz. The resonance frequencies of the BVSWs\nand MSSWs modes obtained using equations 1 and 2 is shown as blue and pink star-symbol,\nrespectively, for each frequency. One can notice a secondary broad peak in the remote strip at 7\nGHz. This peak is less evident or absent in other remote strip. We discuss more on that in the\nsupporting information section III.47\n7from 2 GHz to 9 GHz. The remote strip shows a single resonance peak, while the cavity strip\nshows multiple resonances. The average linewidth of the spin pumping voltage peaks on the cavity\nstrip is slightly broader than the remote strip, suggesting an additional damping contribution. Note\nthat there is no pronounced peak on the cavity strip B-field scan corresponding to the bulk YIG\nresonance at 2 GHz, and the corresponding peak is small for 8 and 9 GHz, Figure 3 a), e) and\nf), respectively. This indicates that the spin pumping voltage on the cavity strip is dominated by\nmagnons excited inside the cavity itself, not by magnons generated outside of the cavity, corre-\nsponding to bulk FMR values, which could be transmitted into the cavity. As mentioned above,\nVSP\ncis proportional to the resonance intensity of the YIG film in the region between the Py squares.\nThis means that the series of resonance modes present underneath the Pt strip, indicate the exis-\ntence of a cavity supporting standing magnonic waves.\nWe can explain these modes by calculating the spin-wave dispersion relation for a YIG slab\nwith dimensions w×lwith magnetization in the plane of the film, given by52–54\nf=γµ0\n2π/parenleftig/parenleftbig\nH+Ha+λexk2\ntotM/parenrightbig/parenleftbig\nH+Ha+λexk2\ntotM+MF/parenrightbig/parenrightig1/2\n, (1)\nwhere λex=2A/µ0M2is the exchange constant with A=3.5 pJ/m, ktot2=kn2+km2is the total\nquantized wavenumber defined by kn=nπ/wandkm=mπ/l, where nandmare the mode numbers\nalong the width and length of the cavity, respectively. The function Fcan be written as\nF=P+/parenleftbigg\n1−P/parenleftbig\n1+cos2(φk−φM)/parenrightbig\n+MP(1−P)sin2(φk−φM)\nH+Mλexktot2/parenrightbigg\n, (2)\nwhere P=1−1−edktot\ndktot, for a YIG film with thickness d. In Eq. 2, φk=arctan (km/kn), and φM\nis the angle between the magnetization and the spin-wave propagation or direction of ⃗k. Two\nmagnetostatic modes can be accessed considering the symmetry of the device, the magnetostatic\nsurface spin wave modes (MSSWs), where ⃗k⊥⃗M,i.e.,φM=π/2 and the backward volume spin\nwaves modes (BVSWs) where ⃗k∥⃗M,i.e.,φM=0.\nThe best fit to Eq. 1 and 2 reproducing the majority of the spin pumping peaks for different\n8frequencies was obtained using M=130 kA/m, d=100 nm, w=2.5µm,l=30µm,µ0Ha=10\nmT, and γ/2π=26.5 GHz/T. The calculated values for (m=1)of the n= (1,2,3,...6)mode\nof the BVSWs and first and second mode of MSSWs are shown in Figure 3 a) to f), blue, and\npink star symbols, respectively. We obtained a good agreement between the peaks present in VSP\nc\nand the calculated modes. Since l≫w, modes with m>1 are hard to distinguish since they\nare superimposed on the m=1 mode due to their proximity in frequency. One important feature\nobtained from the fit is an anisotropy field of µ0Ha=10 mT. The anisotropy may originate from\nthe stray fields produced by the magnetization of Py, leading to a local increase of the effective\nDC-magnetic field applied on the cavity.55Py stray fields may also be responsible for inducing\neven modes in the cavity, observed in our results.55These even modes are not expected for a YIG\nslab when a homogeneous DC magnetic field and rffield are applied.37,38\nOne can see that the modes appear in Figure 3 at a lower field than the bulk YIG resonance\nfor 2 GHz and a higher field than the bulk resonance at 9 GHz. We can analyze the frequencies of\nthe cavity resonance modes by simultaneously plotting the SP-FMR resonance field of the remote\nstrip (black circles) and the resonance field of each magnon mode in the cavity strip (vertical blue\nstripes), Figure 4 a). Overall, the resonance field distribution of cavity modes evolves linearly in\nfrequency, with a slope of ≈28 GHz/T, solid red curve in Figure 4 a). The linear behavior was\nalso reported in previous YIG cavity results.38,43,56\nTo emphasize the localized magnon detection characteristic of the Pt strip, we normalize the\nspin pumping voltage of the cavity strip, VSP\nc, between 0 and 1, based on the lowest and highest\nspin pumping voltage value in each B-field scan. We centered the B-field scans with respect to\nthe resonance field of the bulk YIG obtained from the FMR measurements, µ0HFMR, for each\nfrequency. Figure 4 b) shows the dispersion of the resonant modes of the cavity strip compared\nwith FMR of the bulk YIG. The BVSWs modes, up to (n=6), and the first and second modes of\nthe MSSWs calculated from Eq. 1 and 2 are shown in Figure 4 b) as solid black and dashed pink\nlines, respectively. The model of Eq. 1 and 2 is very useful for confirming the mode position as a\nfunction of field and frequencies, and the peak spacing between each peak. Presenting the voltages\n9-20 -15 -10 -5 0 5 10 15 2023456789 BVSWs\n MSSWsFrequency (GHz)\nm0H - m0HFMR (mT)01\n0 50 100 150 200 2500123456789 Magnon modes in the cavity strip\n Remote strip\n Kittel equation\n 28 GHz/TFrequency (GHz)\nMagnetic field (mT)(a)\n(b)Figure 4: (a) Distribution of the identified magnon modes in the cavity as a function of the mag-\nnetic field for each frequency, plotted as vertical blue stripes. The solid red curve corresponds to\n28 GHz/T. The resonance of the remote strip and the best fit of the Kittel equation are addressed\nas a black symbol and a dashed black line, respectively. (b) Spin pumping intensity spectra of the\nPt strip within the cavity. The solid black and dashed pink lines correspond to the resonant modes\nfor a YIG slab with similar dimensions as the build cavity.\n10as intensity spectra demonstrates that the 400 nm wide Pt strip can be used as a localized FMR\ndetector in future magnonic cavity studies. The spectrum at 1 GHz is not shown in Fig. 4 b) due\nto the absence of a prominent spin-pumping voltage peak. Additional intensity spectra with other\ncavity widths are shown in the supporting information section II,47confirming the reproducibility\nof the fabrication technique.\nWe do not observe multiple voltage peaks with the proximity strip, VSP\np, placed close to the Py\nsquare but still outside of the cavity. This confirms that the Pt layer alone does not induce sufficient\nmagnetic changes in the YIG to create a cavity. Additionally, we do not observe multiple peaks in a\ndevice where the Py film was replaced with gold, ruling out microwave artifacts and confirming the\nrequirement of confinement by YIG/Py bilayers on both sides to create the cavity, see supporting\ninformation section III.47We hypothesize that the difference between the magnetization dynamics\nof covered and uncovered YIG regions by the Py film creates a magnon barrier, as ilustrated in\nFigure 1 a), and discussed in previous reports.28,29\nIn analogy to a (lossy) cavity resonator, we can estimate a finesse given by Φm=∆Bspacing /∆B,\nwhere ∆Bspacing is the B-field peak spacing and ∆Bis the linewidth of the resonance peak.57As a\nfigure of merit, the frequency dependence of the average ∆Bspacing and calculated finesse is shown\nin Figure 5 a). Although the peak spacing increases as a function of frequency, the finesse fluctuates\naround a value of Φm≈10, about one order of magnitude higher than the previous report.29This\nvalue of finesse corresponds to a reflectance of R≈0.73, approximately constant through the entire\nfrequency range. Figure 5 b) shows the average peak spacing and the finesse as a function of the\ncavity width, calculated from the B-field scans at 5 GHz, see supporting information section II.47\nAlthough only three sets of data are presented, a clear correlation between the average peak spacing\nand the finesse is observed. This correlation arises because the cavity is formed in a uncovered YIG\nregion, preserving the optimal magnetic properties of YIG. In fact, the peak linewidth decrease\nwith decreasing cavity width, w, indicating that the additional broadening may stem from the\nsuperposition of higher modes ( m>1) along the cavity length, see supporting information section\nII.47The frequency spacing between these modes increases as a function of the aspect ratio (l/w).\n112 3 4 5 6 7 8 91234\nFrequency (GHz)Avarage peak spacing (mT)\n0510152025\nFinesse, Fm(a)\n1.5 2.0 2.5 3.0 3.5 4.01234\nCavity width ( mm)Avarage peak spacing (mT)\n0510152025\nFinesse, Fm (b)\n𝑤=2𝜇m5GHzFigure 5: (a) Frequency dependence of the average ∆Bspacing and the finesse Φm. The error bar\nis calculated as the standard deviation of ∆Bspacing . (b) Average peak spacing and finesse as a\nfunction of cavity width calculated from the B-field scan at 5 GHz.\nUltimately, a maximum finesse of Φm≈21, or R≈0.86, is achieved for a cavity with w=1.6µm,\nwhich demonstrates a high potential for magnon confinement.\nWe consider Eq. 1 and 2 as an approximate model since it describes a YIG slab with di-\nmensions w×l, where the magnetization amplitude is minimum at the boundary, i.e., an infinite\npotential well. In our case, the cavity is a consequence of the exchange and dipolar interaction in\nthe YIG/Py bilayer.7,8,29,58This means that a finite height potential well would better describe the\nsystem. This discrepancy can be the origin of a minor deviation between the measured VSP\ncpeaks\nand the calculated cavity modes. Accurate modeling of the modes should be performed using\nmicromagnetic simulations, taking into account the exchange and dipolar interaction with Py in\nfurther investigation. The potential height of the cavity barriers should be dependent on the thick-\nness of the YIG film and the exchange/dipolar interaction with the top ferromagnetic layer. Further\ninvestigation using the present technique should be performed for different adjacent ferromagnetic\nlayers in which strong exchange interaction has already been reported, such as YIG/CoFeB15and\nYIG/Co.16YIG films thinner than 100 nm with the damping below than 5 .0×10−4are good can-\ndidates to produce magnonic cavities with higher reflectance factors keeping the magnetic losses\nclose to those reported in this letter.59\nIn summary, we take advantage of the difference in the magnetic dynamics between the YIG\n12film and the YIG/Py bilayers, to fabricate an all-on-chip magnonic cavity supporting standing\nmagnon modes in a uncovered YIG film between two YIG/Py bilayers. This approach enables\nthe confinement of magnons while preserving the optimal magnetic properties of the YIG cavity.\nThe spin pumping voltage of a 400 nm wide Pt strip proved to be a reliable technique to detect\nthe magnon resonance modes of the cavity. Following this idea, 1D and 2D magnonic crystals\ncould be obtained by having a regular array of magnetic strips onto YIG, with the possibility of\nmeasuring the magnon modes locally by means of spin pumping. Moreover, further investigations\nshould involve designing coupled cavities by placing two cavities side by side, where the coupling\nstrength could be controlled by the width of the central YIG/Py bilayer. This cavity fabrication\nprocess opens new possibilities for investigating and characterizing micron-sized YIG cavities\nwith a wide range of arbitrary shapes. It also allows for the implementation of on-chip magnonic\ncomputation structures, serving as a printed circuit board for magnons. These results demonstrate a\npromising combination of hybrid magnonics and cavity magnonics, which has the potential to drive\nthe integration of future all-on-chip magnonic devices into mainstream microwave electronics.\nAcknowledgement\nWe acknowledge the technical support from J. G. Holstein, T. J. Schouten and H. de Vries, F. A. van\nZwol, A. Joshua. We are grateful to A. Azevedo, C. Ciccarelli and C. M. Gilardoni for the valuable\ndiscussion. We acknowledge the financial support of the Zernike Institute for Advanced Materi-\nals and the Future and Emerging Technologies (FET) programme within the Seventh Framework\nProgramme for Research of the European Commission, under FET-Open Grant No. 618083(CN-\nTQC). This project is also financed by the NWO Spinoza prize awarded to Prof. B. J. van Wees by\nthe NWO, and ERC Advanced Grant 2DMAGSPIN (Grant agreement No. 101053054).\n13Supporting Information Available\nDevice fabrication and experimental setup details (section I). 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Ultra Thin Films of\nYttrium Iron Garnet with Very Low Damping: A Review. physica status solidi (b) 2020 ,257,\n1900644.\n20TOC Graphic\nw\n215 220 225 230 235 2400.00.20.40.6\n8 GHzSpin Pumping voltage ( mV)\nMagnetic field (mT) magnon cavity modes\n10𝜇m𝑉𝑆𝑃\n𝐵𝑒𝑥\n21SUPPORTING INFORMATION\nMagnon confinement in an all-on-chip YIG cavity resonator using hybrid\nYIG/Py magnon barriers\nObed Alves Santos∗\nPhysics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4,\nGroningen, AG 9747, The Netherlands and\nCavendish Laboratory, University of Cambridge,\nCambridge, CB3 0HE, United Kingdom\nBart J. van Wees\nPhysics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, Nijenborgh 4,\nGroningen, AG 9747, The Netherlands\n1arXiv:2306.14029v2 [cond-mat.mes-hall] 6 Oct 2023I. SAMPLE FABRICATION AND EXPERIMENTAL SETUP\nThe samples consists of a high-quality 100 nm thick YIG film (Y3Fe5O12)grown by liquid\nphase epitaxy on a GGG substrate, obtained commercially from Matesy GmbH, measuring\napproximately (4×3)mm. Electron beam lithography (EBL) was used to pattern the device,\nwhich consists of multiple strips of Pt with 35 µm length and 400 nm wide. The Permalloy (Py)\nsquares have dimensions (30×30µm2), the square shape was chosen to avoid shape anisotropies\nof the Py film.[1, 2] The Pt and Py layers were deposited by DC sputtering in an Ar+ plasma with\nthicknesses of 8 nm and 30 nm, respectively. The deposition of Ti(5 nm)/Au(75 nm) leads is\nmade by e-beam evaporation. The last step consists of mounting the sample onto a non-resonant\nstripline waveguide, which can be used from 0.1 to 9 GHz with a characteristic impedance of 50\nΩ, 0.030\" RO4350, GCPWG, with a signal line 45 mil (1.14 mm) wide. Figure S1 a) show an\noptical image of the final YIG film sample containing multiple devices, while b) shows a zoom-in\nimage of an individual device. The wire bonding is made using AlSi (Al 99%, Si 1%) wires on\nthe sample holder and connected to a lock-in amplifier.\n(a)\n(b)\n1 mm\nFIG. S1. a)Show the optical image of the final YIG sample with multiple devices. b)Show a zoom-in\noptical image of an individual device.\n∗Correspondence should be addressed:\noa330@cam.ac.uk, or obed.alves.santos@gmail.com\n2The microwave field excitation is sent connecting the stripline waveguide to a vector network\nanalyzer (VNA). The FMR absorption of the YIG film is obtained by scanning the absorption (S21)\nor reflection (S11) in the VNA for a fixed microwave frequency as a function of the magnetic field.\nTo obtain the µ0HFMR value and the FMR linewidth (∆H), each S21 B-field scan is fitted using\nan asymmetric Lorentzian function, L(H−HFMR) =S∆H2/[(H−HFMR) +∆H2] +A[∆H(H−\nHFMR]/[(H−HFMR) +∆H2]. Where SandAare the symmetric and antisymmetric amplitudes.\nThe spin pumping voltage is measured at the end of the Pt strips by a lock-in amplifier, triggered\nwith the VNA. The microwave is then switched from low power Plow\nr f=−25 dBm to high power\nPhigh\nr f=16 dBm, with a modulation frequency of 27 .71 Hz. The (waveguide stripline + sample) is\npositioned between two poles of an electromagnet such that the external magnetic field (H)and\nthe microwave field (hr f)are perpendicular to each other, and both are in applied in the plane of\nthe YIG film. All the measurements were performed at room temperature.\n5 6 7 8 9 10-0.04-0.03-0.02-0.010.000.01S21 (a.u)\nMagnetic field (mT) FMR at 1 GHz\n Asymmetric Lorentzian\n0 20 40 60 80 100012345\n Bulk YIG FMR\n Kittel equationFrequency (GHz)\nMagnetic field (mT)(a) (b)\n113 114 115 116 117 118-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.020.000.020.04S21 (a.u)\nMagnetic field (mT) FMR at 5 GHz\n Asymmetric Lorentzian (c)\nFIG. S2. a)Field resonances of the bulk YIG FMR for low frequency. b)andc)are S21 measurements of\nthe bulk YIG FMR at 1 GHz and 5 GHz, respectively. Our external magnetic field step corresponds to 0.1\nmT. The error bar corresponds to the standard deviation from the multiple VNA readings.\nIn spintronics, the FMR drives the spin pumping effect (SPE),[3] where a flow of spin current,\noccurs from the ferromagnetic/ferrimagnetic layer towards an adjacent layer at the peak of the\nmicrowave absorption during the FMR process.[4] Therefore, the YIG film injects a spin current\nby means of the SPE into the Pt strip.[3] That spin current can be expressed as\nJs=g↑↓\ne f f¯hω\n4π/parenleftbigghr f\n∆H/parenrightbigg2\nL(H−HFMR), (S1)\nwhere g↑↓\ne f fis the effective spin mixing conductance, ω=2πfis the r ffrequency, ¯his the reduced\nPlanck constant, and L(H−HFMR)is the FMR absorption, usually a Lorentizian-like line shape.\nThe spin current along ˆ zwith spin polarization ⃗σalong ˆ xis converted into charge current along ˆ y\n3direction by the inverse spin Hall effect (ISHE), following ⃗Jc∝θPt(⃗Js×⃗σ).[5] Where θPtis the\nspin Hall angle, which quantifies the conversion efficiency between spin and charge currents. The\ntotal voltage build-up at the edge of the Pt strip can be expressed by[5, 6]\nVSP=θPtRPtλPtwPt\ntPt/parenleftbigg2e\n¯h/parenrightbigg\ntanh/parenleftbiggtPt\n2λPt/parenrightbigg\nJs, (S2)\nwhere RPt,tPt,wPtandλPtare the resistance, thickness, width, and the spin diffusion length of the\nPt strip.\nThe spin pumping process typically leads to an increase in the FMR linewidth as it introduces\nan additional component to the magnetic losses of the ferromagnetic layer. However, in our\nmeasurements, we did not observe any significant broadening of the linewidth in the spin\npumping voltages measured on the Pt strip. This absence of pronounced linewidth broadening in\nVSP\nrcan be attributed to the fact that the Pt strip only covers a fraction of the YIG film. Cheng et\nal.demonstrated that for Pt strips with widths w<5µm, the spin pumping voltage is dominated\nby the spin current injected in the proximity of the Pt strip, within a YIG region unaffected by the\npresence of the Pt layer. In this region, the damping and linewidth correspond to the bulk values.\nHence, the spin pumping voltage measured with the Pt strip effectively acts as a localized FMR\nabsorption detector or an rf antenna.[7]\nOne can realize that an additional spin pumping voltage could be detected by the Pt strip within\nthe field range corresponding to the resonance of the Py layer, resulting from spin pumping from\nPy towards the YIG film. However, in our measurements, we observed that the voltage remained\nbelow the noise level throughout the resonance field of Py. As a result, our main focus in this study\nis on the resonance of the YIG film.\nII. ADDITIONAL CA VITIES MEASUREMENTS\nAdditional spin pumping measurements were conducted on several other cavities to\ndemonstrate the reproducibility of the method. Optical microscope images of the cavities with\ndifferent widths are shown in Figure S3 a) and b). The main results presented in the text are\nbased on the cavity denoted as \"Cavity A,\" which has a width of w=2.0µm. Subsequently, we\npresent results for two separate cavities: \"Cavity B\" and \"Cavity C,\" both with a width of w=1.6\nµm, as well as \"Cavity D\", which has a width of w=3.7µm. Figure S3 c) shows the average\npeak spacing, average peak linewidth, and finesse as a function of the inverse square of the cavity\n4105 110 115 120 125 130 1350.00.51.01.52.0SP voltage ( mV)\nMagnetic field (mT) Cavity strip\n105 110 115 120 125 130 1350.00.51.01.5SP voltage ( mV)\nMagnetic field (mT) Cavity strip\n105 110 115 120 125 130 1350.00.51.01.5SP voltage ( mV)\nMagnetic field (mT) Cavity strip𝑤=1.6𝜇m 𝑤=3.7𝜇m\n(a) (b)10 𝜇m 10 𝜇m\n(d) (e) (f)\n𝑤=2.0𝜇m𝑤=2.0𝜇m 𝑤=3.7𝜇m 𝑤=1.6𝜇m(c)\n0.0 0.1 0.2 0.3 0.41234Average peak spacing (mT)\n1/w2(mm-2)0.10.20.30.4\nAverage peak linewidth (mT)\n0510152025\nFinesse, Fm\n3.7 µm 2.0 µm 1.6 µmFIG. S3. a)andb)show optical images from different devices fabricated with distinct distance cavity\nwidths, indicated in the figure. c)Average peeks spacing in (mT), average peak linewidth, and calculated\nfinesse as a function of the inverse square of the cavity width (w).d),e)andf)shows B-field scan of the\nspin pumping voltage in the central platinum strip at 5 GHz, for a cavity width of w=1.6µm,w=2.0µm,\nandw=3.7µm, respectively.\nwidth. It can be observed that the peak spacing shows a linear dependence as function of 1 /w2\nwithin the error bar, following the expected behavior for a magnonic cavity.[8, 9] The linear\nbehavior is also evident in the average peak linewidth, suggesting that increasing the aspect ratio\nof the cavity ( l/w) leads to a more homogeneous distribution of cavity modes. However, it should\nbe noted that our results are limited to only three different cavity widths, making it challenging to\nmake a definitive statement about the observed behavior. Figure S3 d) to f) show the B-field scan\nof the spin pumping voltage for three cavities widths for 5 GHz. As the width decreases, a few\nobservations can be made. Firstly, the peak linewidth decreases, indicating a narrower spectral\ndistribution of the spin pumping voltage. Secondly, the peak spacing increases, suggesting a\nlarger separation in corresponding excitation frequencies. Finally, the spin pumping voltage\nheight becomes more equal for the cavity with w=1.6µm.\nFigure S4 a) to c), shows the B-field spin pumping voltage for the remote cavity and cavities\nB and C. All three measurements were obtained simultaneously, with each cavity strip connected\nto an independent lock-in amplifier. Both cavities, B and C, have been designed with the same\n555 60 65 70 75 80SP Voltage ( mV)\nMagnetic field (mT) Remote strip\n Cavity strip B\n Cavity strip C1 mV3.5 GHz\n140 145 150 155 160 165SP Voltage ( mV)\nMagnetic field (mT) Remote strip\n Cavity strip B\n Cavity strip C6.0 GHz\n1 mV\n180 185 190 195 200 205SP Voltage ( mV)\nMagnetic field (mT) Remote strip\n Cavity strip B\n Cavity strip C1 mV7.0 GHz(a) (b) (c)\n(d) (e)\nCavity strip B Cavity strip C\n-20 -15 -10 -5 0 5 10 15 2023456789Frequency (GHz)\nm0H - m0HFMR (mT)01\n-20 -15 -10 -5 0 5 10 15 2023456789Frequency (GHz)\nm0H - m0HFMR (mT)01FIG. S4. a)toc)shows the B-field scan of the spin pumping voltage obtained for the remote strip and\nthe Pt strips placed along the center of the cavity B and C, respectively. d)ande)shows the spin pumping\nintensity spectra for two different cavities with the same width distance of w=1.6µm.\nwidth of w=1.6µm. This explains why the field position of almost every peak aligns with each\nother. Once again, it is evident that the corresponding peak for the bulk YIG resonance is absent\nor reduced to the noise level at 6 GHz and 7 GHz. This observation highlights that the voltage\npeaks are primarily determined by the magnon modes confined within the cavity. One can also\nobserve a small alternation of the intensity between consecutive peaks in Figure S4 a) to c). A\nsimilar trend seams to be present in Figure 3 of the main text. This behavior might occur due to\nthe difference in microwave excitation corresponding to odd and even modes. However, we cannot\nexplain them in detail without more detailed micromagnetic simulations, taking into account the\nexchange and dipolar interactions in the YIG/Py bilayer, and then correctly addressing each peak\nto the corresponding magnon mode and intensity.\n6The spin pumping intensity spectra for cavities B and C are present in Figure S4 d) and e), with\na frequency spacing of 0.5 GHz. Although the peak between 5.5 GHz and 6.5 GHz in cavity C is\ndifficult to identify due to the noise, both cavities exhibit a similar peak dispersion, with almost\nevery peak matching at each frequency. The solid black lines in Figure S4 d) and e) are the modes\ncalculated by equation (1) and (2) from the main text, up to n=6 using the following parameters:\nw=1.6µm,l=30µm,M=130 kA/m, µ0H=14 mT, and γ/2π=26.5 GHz/T. We believe the\nsmall diminishment in the gyromagnetic ratio could be due to other factors and effects not included\nin the model, such as the boundary interface with the YIG/Py bilayer. These results confirm the\nreproducibility of the technique.\nIII. CONTROL SAMPLES\nIn Figure 3 d) on the main text, one can identify a secondary peak at 7 GHz. This secondary\n“peak” at the left side of the spin-pumping-FMR peak might be caused by the excitation of the\nfinite kbulk-magnons, corresponding to the spin wave modes in the whole YIG sample. The\ncorresponding peak is also not observed in the FMR absorption. This secondary structure is less\nevident or absent in another remote Pt strip using a different YIG sample, shown in Figure S5, as\nwell as in Figure S6. We believe this structure could be due to the magnon interferences in the\nbulk-YIG region. This does not affects the analysis of the resonance modes of the cavity.\n110 112 114 116 118 1200.00.30.60.91.2SP voltage ( mV)\nMagnetic field (mT) 5 GHz\n178 180 182 184 186 1880.00.30.60.91.2SP voltage ( mV)\nMagnetic field (mT) 7 GHz\n214 216 218 220 222 2240.00.30.60.91.2SP voltage ( mV)\nMagnetic field (mT) 8 GHz(a) (b) (c)\nFIG. S5. Less intense or not present \"secondary\" peak in a Remote Pt strip on a different 100 nm thick YIG\nsample at a)5 GHz, b)7 GHz, and a)8 GHz. Further experiments using rectangular microwave waveguides\nmay avoid surface mode excitations in the bulk YIG film.\nFigure S6 a) shows the absence of magnetostatic modes in a control device where an Au film\nreplaces the Py film. This result rules out the possibility of (multiple) peaks originating from\n70 50 100 150 200 2500246810\n Bulk FMR\n Spin pumping of remote strip\n Spin pumping of gold squares\n Kittel curveFrquency (GHz)\nMagnetic field (mT)\n140 160 180 200 220 2400.00.20.40.60.8Spin Pumping voltage ( mV)\nMagnetic field (mT) 6 GHz\n 7 GHz\n140 160 180 200 220 2400.00.20.40.60.8Spin Pumping voltage ( mV)\nMagnetic field (mT) 7 GHz\n 8 GHzSide Pt strip 𝑉𝑝𝑆𝑃Py replaced by Au(a) (b) (c)FIG. S6. a)Absence of the multiple peaks when the Py squares are replaced with gold. b)Comparison\nof Kittel fitting between FMR of YIG film, spin pumping voltage in remote strip, and when Py square is\nreplaced with Au. c)Spin pumping voltage with a proximity strip, no evidence of cavity resonant modes\nwas observed.\nmicrowave artifacts caused by the proximity of a metallic film to the Pt strip.. Figure S6 b) shows\nthe direct comparison of the Kittel equation for FMR of the YIG bulk, the spin pumping voltage of\ntheremote Pt strip, and the spin pumping voltage of the Pt strip when Py is replaced with gold. The\nadjusted Kittel curves show only a slight deviation. These results also suggest that the magnetic\nnature of the exchange/dipolar interaction between the YIG and Py films is crucial for constructing\nthe cavity through engineered-design lithography.\nThe spin pumping voltage was also measured in a Pt strip located near the left side of the Py\nsquare, VSP\npin Figure 1 b) of the main text. However, despite the higher noise level, the spin\npumping voltage as a function of the B-field did not exhibit any evidence of (multiple) peaks, as\nshown in Figure S6 c). It becomes evident that the presence of YIG|Py interfaces on both sides is\nnecessary to create the magnon resonant modes within the cavity. In Figure 1 c) of the main text,\none can identify a Pt strip placed underneath the middle of the Py square. The electrical resistance\nof that strip was in the M Ωrange, much higher than the usual 3 .5kΩfor 400 nm wide, 35 µm\nlong, and 8 nm thick Pt strip. This suggests that the Pt strip may have been damaged during the\nsputtering and lift-off process involved in the fabrication of the Py square. As a result, we do not\nhave reliable data for the electrical contact of that particular strip.\n[1] M. Mruczkiewicz, P. Graczyk, P. Lupo, A. Adeyeye, G. Gubbiotti, and M. Krawczyk, Spin-wave\nnonreciprocity and magnonic band structure in a thin permalloy film induced by dynamical coupling\n8with an array of Ni stripes, Physical Review B 96, 104411 (2017).\n[2] A. Talapatra and A. Adeyeye, Linear chains of nanomagnets: engineering the effective magnetic\nanisotropy, Nanoscale 12, 20933 (2020).\n[3] Y . Tserkovnyak, A. Brataas, and G. E. Bauer, Enhanced gilbert damping in thin ferromagnetic films,\nPhysical Review Letters 88, 117601 (2002).\n[4] S. Mizukami, Y . Ando, and T. Miyazaki, Ferromagnetic resonance linewidth for NM/80NiFe/NM films\n(NM= Cu, Ta, Pd and Pt), Journal of Magnetism and Magnetic Materials 226, 1640 (2001).\n[5] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Spin Hall effects, Reviews of\nModern Physics 87, 1213 (2015).\n[6] A. Azevedo, L. Vilela-Leão, R. Rodríguez-Suárez, A. L. Santos, and S. Rezende, Spin pumping and\nanisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment, Physical Review\nB83, 144402 (2011).\n[7] Y . Cheng, A. J. Lee, G. Wu, D. V . Pelekhov, P. C. Hammel, and F. Yang, Nonlocal uniform-mode\nferromagnetic resonance spin pumping, Nano Letters 20, 7257 (2020).\n[8] T. Yu and G. E. W. Bauer, Efficient gating of magnons by proximity superconductors, Physcal Review\nLetters 129, 117201 (2022).\n[9] Y . W. Xing, Z. R. Yan, and X. F. Han, Magnon valve effect and resonant transmission in a one-\ndimensional magnonic crystal, Physical Review B 103, 054425 (2021).\n9" }, { "title": "2302.01141v2.Leveraging_symmetry_for_an_accurate_spin_orbit_torques_characterization_in_ferrimagnetic_insulators.pdf", "content": "1\nLeveraging symmetry for an accurate spin-orbit torques \ncharacterization in ferrimagnetic insulators \n \nMartín Testa-Anta1,*, Charles- Henri Lambert2, Can Onur Avci1,* \n1Institut de Ciència de Materials de Barcel ona (ICMAB-CSIC), Campus de la UAB, \n08193 Bellaterra, Spain \n2Department of Materials, ETH Züri ch, Hönggerbergring 64, CH-8093 Zürich, \nSwitzerland \n \nAbstract \nSpin-orbit torques (SOTs) have emerged as an e fficient means to electrically control the \nmagnetization in ferromagnetic het erostructures. Lately, an in creasing attention has been \ndevoted to SOTs in heavy metal (HM)/magnetic insulator (MI) bilayers owing to their \ntunable magnetic properties and insulating natu re. Quantitative characterization of SOTs \nin HM/MI heterostructures are, thus, vita l for fundamental understandi ng of charge-spin \ninterrelations and designing novel devices. Howe ver, the accurate det ermination of SOTs \nin MIs have been limited so far due to sma ll electrical signal outputs and dominant \nspurious thermoelectric effects caused by Joule heating. Here, we report a simple \nmethodology based on harmonic Hall voltage det ection and macrospin simulations to \naccurately quantify the damping-like and field-lik e SOTs, and thermoelectric contributions \nseparately in MI-based systems. Experiment s on the archetypical Bi-doped YIG/Pt \nheterostructure using the developed method yi eld precise values for the field-like and \ndamping-like SOTs, reaching -0.14 and -0.15 mT per 1.7 ൈ1011 A/m2, respectively. We \nfurther reveal that current-induced Joule heat ing changes the spin transparency at the \ninterface, reducing the spin Hall m agnetoresistance and damping-like SOT, \nsimultaneously. These results and the devis ed method can be beneficial for fundamental \nunderstanding of SOTs in MI- based heterostructures and designing new devices where \naccurate knowledge of SOTs is necessary. \n \n \n \n 2\nI. INTRODUCTION \n \nSpin-orbit torques (SOTs) are the generic nam e given to the current -induced torques in \nferro-/ferrimagnetic heterostructures with la rge spin-orbit couplin g and broken inversion \nsymmetry mainly driven by, but not limited to, bulk spin Hall (SHE) and interfacial Rashba-\nEdelstein effects.1 During the past decade, SOTs have become the state-of-the-art \nmagnetic manipulation met hod by electrical currents and enabled magnetization \nswitching,2,3,4 domain wall and skyrmion motion,5,6,7 steady-state magnetic oscillations,8,9 \nand magnon generation/suppression10,11 in convenient device geometries. These \nexperimental achievements have given rise to a multitude of spintronics device concepts \nwith memory, logic, signal transmission and co mputing functionalities suitable for the \ncomplementary metal-oxide-se miconductor (CMOS) industry and the post-CMOS \ncomputing era.12,13,14,15 SOTs have been originally discove red and extensively studied in \nall-conducting heavy metal (HM)/ferromagnetic (F M) bilayers such as Pt/Co, Ta/CoFeB \nand W/CoFeB.16,17,18,19,20 However, the research has rapidly expanded into other \nmaterials such as antiferromagnets,21,22,23 ferrimagnets,24,25,26 topological insulators27,28,29 \nand magnetic insulators (MIs)4,30,31, among others. \nRecently, the interest in MIs in the SOTs context has rapidly grown.5,32,33,34 MIs offer many \nadvantages over their conducting counterparts thanks to thei r tunable magnetization and \nanisotropy, low Gilbert damping35, and long spin diffusion lengths36. These attributes, \ntogether with their insulating nature leading to reduced Ohmi c losses when integrated into \nmicroelectronic circuits, make MIs a highly a ttractive material pl atform for low power \nmagnonic and spintronic applications.37,38,39 Within the family of MIs, yttrium iron garnet \n(Y3Fe5O12, YIG) is particularly appealing due to its record low Gilbert damping (~10-4 - 10-\n5), making it ideal for nano-oscillators,40,41 and magnonic devices,42,43,44,45 with SOT-\nenabled operation. Despite the vi tal importance of SOTs in the integration of MIs in \npotential spintronic concepts, their accurate and straightforward characterization by \nsimple electrical methods is intrinsi cally difficult. Harmonic Hall voltage (HHV)4,16,17,46 and \nspin-torque ferromagnetic resonance47 measurements are the tw o established methods \nin this context but rely on the Hall resi stance and magnetoresistance output signals, \nrespectively, which are orders of magnitude sm aller in HM/MI systems with respect to all-\nmetallic systems. The few existing studies have considered only the damping-like (DL)-\nSOT characterization,4,46,48,49,50 but the separate quantification of the field-like (FL)-SOT, \nequally important for the SOT- driven magnetization dynamics,51,52,53,54,55,56 and a proper \naccount of current-induced Joule heating in t hese measurements and interfacial spin \ntransport properties remained elusive thus far. \nIn this article, we show a simple method fo r an accurate SOTs quanti fication in MIs relying \non the combination of HHV m easurements and macrospin simu lations. We develop, and \ntest with simulations, a HHV measurement scheme using a non-standard geometry, \nwhich allows us to exploit simple symmetry arguments to disentangle the effective fields \ndue to the damping-like ( 𝐵) and field-like ( 𝐵ி) SOTs, and thermoelectric contributions \nprecisely. The quantification of both SOTs components and ther moelectric effects yield 3\nan intrinsic theoretical error as low as ~0.2% on the simulated data. As a proof-of-\nconcept, we apply the proposed a pproach on a Bi-doped YIG/Pt bilayer with in-plane (IP) \nmagnetic anisotropy obtaining 𝐵 = -0.15 mT and 𝐵ி = -0.14 mT and a thermoelectric \nfield of 𝐸 = 0.27 V/m per 𝑗 = 1.7ൈ1011 A/m2 injected current, all values in the range \nexpected of such systems. Finally, current -dependent measurements show that Joule \nheating reduces the spin mixing conductance ( 𝐺↑↓) at the HM/MI and results in a \nsystematic reduction of the spin Hall magnetoresistance (SMR) and 𝐵 up to ~15%, \nsimultaneously. \n \nII. EXPERIMENTAL DETAILS \n \nSample preparation . An ~18-nm-thick Bi-doped YIG (Bi:YIG from hereon) layer was rf-\nsputtered from a stoichiometric target ont o a single-crystal Sc-substituted gadolinium \ngallium garnet substrate (Gd 3Sc2Ga3O12, GSGG) with a base pressure <5 ൈ10-8 Torr. The \ngrowth temperature and the Ar partial pressure during th e Bi:YIG deposition were 800 ℃ \nand 1.5 mTorr, respectively. After the deposit ion, the samples were annealed for 30 min \nat the deposition temperatur e and subsequently cooled down to room temperature in \nvacuum. In order to have a clean interface, a 4 nm-thick Pt was dc-sputtered in-situ at \nroom temperature without break ing the vacuum. The Ar pre ssure during the Pt deposition \nwas 3 mTorr. After the Pt deposition, the c ontinuous Bi:YIG/Pt layers were patterned into \nHall bar structures by means of standar d photolithography followed by top-down ion \nmilling. The lateral dimensi ons of the Hall bars were 10 μm (current line width) ൈ 30 μm \n(distance between two Hall crosses). \nHall effect measurements . HHV measurements were performed by applying an AC \nvoltage along the current line modulated at 𝜔/2𝜋 = 1092 Hz and simultaneously \nmeasuring the first and second harmonic Hall vo ltage responses with a lock-in amplifier. \nThe current amplitude through t he device under test was determined by connecting a 10 \nΩ resistor in series and reading the voltage drop across with a digita l multimeter before \nor during the measurements. The acquired HHVs were converted into Hall resistances \nusing the relation 𝑅\nఠு = 𝑉ఠு/𝐼 for comparability with the liter ature. For the field scans, an \nout-of-plane (OOP) or in-plane (IP) DC magnetic field was swept in the േ600 mT or േ200 \nmT range, respectively, while keeping the azimuthal angle ( 𝜑) fixed. For the polar ( 𝜃) \nangle scans, the sample was rotated in the pres ence of a constant DC field in the 0º-360º \nrange using a motorized rotation stage at an angular speed of 3 degrees/second. All \nmeasurements were performed at room temperature and a ll the data presented were \naveraged over five scans to impr ove the signal-to-noise ratio, unless specified otherwise. \n \n \n 4\nIII. RESULTS AND DISCUSSION \nA. Structural and ma gnetic characterization \nA structural characterization of the as-dep osited Bi:YIG/Pt heterostructure was carried \nout by means of high-resolution X-ray di ffraction (XRD). Figure 1a shows a symmetric 𝜃-\n2𝜃 scan around the (444) diffraction peak of the GSGG substrate, occurring at 50.44º. \nThe data do not show a clear emergence of t he Bragg reflection for the Bi:YIG phase, but \nonly a small shoulder at the right side of the substrate peak. Such observation may be \nattributed to a reduced crystallinit y of the magnetic film or, more likely in our case, to the \noverlap between the substrate and Bi:YIG (444) reflections. Indeed, in the absence of \ninterfacial strain, Bi-substitution leads to an increase in the cubic lattice parameter, \nresulting in a lower 2 𝜃 position compared to the bul k YIG (expected at 51.09º)57. \nAdditionally, the fact th at the Bi:YIG peak is not shifted to lower 2 𝜃 values with respect to \nthe substrate indicates that: i) its OOP lattice parameter is smaller than that of GSGG, \nand ii) the tetragonal distortion induced by the latti ce mismatch with th e substrate is not \nsignificant.58 Owing to the negative (111) magnetostricti on constant for this particular iron \ngarnet,59 this absence of moderate interfacial strain will favor the occurrence of IP \nmagnetic anisotropy. X-ray reflectivity (XRR) analysis was performed in order to check \nthe thickness of Bi:YIG confirming the antici pated value of 18 nm. The sample topography \nwas addressed through atomic force micr oscopy (AFM) before and after the Pt \ndeposition. Figure 1b is a repr esentative AFM image acquired from the Bi:YIG/Pt sample. \nThe sample surface is rather flat with an RMS roughness of 0.66 േ0.13 nm, similar to the \nroughness before Pt deposition (<1 nm, not s hown). This number was estimated by \naveraging measurements of five different regions on the sa mple, which highlights the \noverall good quality of the films. \nThe IP hysteresis loop recorded at 300 K us ing the superconducting quantum interference \ndevice (SQUID) magnetometry is displa yed in Figure 1c. The large remanence \naccompanied by a negligible coercivity (<1 mT) are characteristic of IP magnetized \ngarnets due to their small magnet ocrystalline anisotropy. Prefer ential IP magnetization of \nBi:YIG was also corroborated by magneto-optical Kerr effect (MOKE) measurements as \nshown in Figure 1d. Data recorded in polar (signal proportional to the OOP component of \nthe magnetization) and longit udinal (signal proportional to the IP component of the \nmagnetization) MOKE c onfigurations with corresponding fi eld sweeps clearly show that \nthe IP and OOP are the easy and hard axes, res pectively. Additional longitudinal MOKE \nmeasurements at different az imuthal angles showed a negligible difference (data not \nshown), hence, the sample is hereafter assu med to exhibit easy-plane anisotropy. \nNotably, the experimental value of the saturation magnetization ( 𝑀௦~85 emu/cm3) lies \nwell below that of the bulk YIG ( ~140 emu/cm3 at 300 K).60,61 Such reduction can be \nexplained considering the Bi3+ ↔ Y3+ substitution at the dodecahedral sites, as this will \nexpand the crystal lattice and weaken the superexchange interact ions between the Fe3+ \ncations located at the tetrahedral and octahedr al sites, primarily responsible for the \nferrimagnetic behavior in iron garnets.62 Another possibility re lates to an oxygen off-5\nstoichiometry, occurring due to the annealing at very low O 2 pressures. Under these \nconditions, the formation of oxygen va cancies has been reported to proceed via Fe3+ to \nFe2+ reduction, giving rise to a decrease in magnetization.63 Nevertheless, our results are \nconsistent with literature values for Bi :YIG thin films grown on GGG or GSGG \nsubstrates,62,64 and exploring the underlying cause of the reduced magnetization is \nbeyond the scope of the present study. \n \nFigure 1. Structural and magnetic characterization of the Bi:YIG/Pt sample: (a) 𝜃-2𝜃 scan \nof the Bi:YIG (18 nm)/Pt (4 nm) bilayer deposited onto a (111)-oriented GSGG substrate; \n(b) Representative AFM image of the prev ious film, revealing a rms roughness ( 𝑆) value \nof 0.66േ0.13 nm (averaged over different regi ons in the sample); (c) In-plane \nmagnetization as a function of an external magnetic field (a fter subtraction of the linear \nparamagnetic background), obtained by SQUID magnetometry; (d) MOKE response as a \nfunction of an in-plane (red, top axis) or out-of-plane (bl ue, bottom axis) magnetic field in \ntheir respective measurement ge ometries (i.e., longitudinal and polar), which confirms the \nin-plane anisotropy of the prepared sample. \n \n \n6\nB. First harmonic Hall characterization \nThe magnetotransport pr operties of Bi:YIG/Pt were in ferred through SM R measurements \nand reported in Figure 2. In Figure 2a, a curr ent injection through t he Pt layer along the \nx-axis produces a pure spin current due to the SHE and other po ssible charge-spin \nconversion mechanisms with polarization along the y-axis and flowing along the z-axis \nreaching Bi:YIG. The absorption and reflection of this spin current at the Bi:YIG/Pt \ninterface modulates the longitudinal and transverse resistance of Pt depending on the \nrelative orientation between the magnetization and spin accu mulation, which lies at the \norigin of SMR in HM/MI bilayers.65 The SMR manifests itself on the Hall voltage (which \nwe mainly use in this study) wit h signals proportional to both IP ( ∝𝑚௫𝑚௬) and OOP ( ∝\n𝑚௭) components of t he magnetization.65,66 Therefore, these two SM R contributions exhibit \nthe very same symmetry as the planar and anomalous Hall resistances in conducting \nmagnetic materials, respectively, but with ty pical values several orders of magnitude \nlower. \nAccording to the angle definitions given in Figure 2a, the Hall resistance at the first \nharmonic of the current frequency, 𝑅ఠு, can be expressed as follows: \n𝑅ఠுൌ𝑅ௌெோsinଶ𝜃ெsinሺ2𝜑ெሻ𝑅ௌெோିுா cos𝜃ெ𝑅ைுா𝐵௫௧cos𝜃 ሺ1ሻ \nwhere 𝑅ௌெோ, 𝑅ௌெோ ,ுா and 𝑅ைுா represent the resistance m odulation due to SMR, SMR-\ninduced anomalous Hall effect (SMR-AHE ) and the ordinary Ha ll effect (OHE), \nrespectively. 𝐵௫௧ and 𝜃 are the magnitude and polar angle of the external magnetic field \nwhereas 𝜃ெ and 𝜑ெ are the polar and azimuthal angles of the magnetizat ion vector. Note \nthat the OHE is t he property of the Pt itse lf and has, in principle, no relationship with the \nmagnetic layer underneath. \nFigure 2b shows the first harmo nic Hall resistance in a swept OOP field. Besides the \nlinear OHE contribution, the signal difference at positive and negative field at high \namplitudes reflect the magnetiz ation vector switching betw een the up and down states. \nBy symmetry operations with respect to 𝐵௭ = 0, we find that the magnetization saturates \nalong the z-axis at around 𝐵௭ = 45 mT, which we denote as 𝐵௦௧. Since the Bi:YIG layer \npossesses in-plane anisotropy, this value is equivalent to the su m of the demagnetizing \nfield and the effective perpendicular anisotropy field possibly due to the reminiscent OOP \nanisotropy driven by a small lattice mism atch undetected in the XRD measurements. \nFrom the experimental data we find 𝑅ௌெோିுா = -1.36 m Ω for an rms cu rrent density of \n1.7ൈ1011 A/m2, which yields 𝜌ௌெோିுா = -5.4ൈ10-12 Ωꞏm. A symmetric component is \nlikewise visible in the OOP fi eld-scan, responsible for t he inverse U-shape signal and \nspikes observed at low fields. This additional contri bution is ascribed to the reorientation \nof the magnetization vector within the film plane for 𝐵௭<𝐵௦௧ and possibly domain \nformation at very small fields le ading to extra contributions in 𝑅ఠு due to the non-zero \n𝑚௫𝑚௬ components. 7\nIP field sweep measurements at different azimuthal angles ( 𝜑) are depicted in Figure \n2c. The modulation of 𝑅ఠு due to SMR shows a maximum (minimum) at 𝜑 = 45º (135º), \nconsistent with the positive 𝑅ௌெோ coefficient in YIG/Pt.4 The 𝜑-dependence of the data \nis displayed in Figure 2d. The fitting following eq. 1 reveals a 𝑅ௌெோ value of 11.23 m Ω, \ncorresponding to 𝜌ௌெோ = 4.5ൈ10-11 Ωꞏm. Note that we assume 𝜑ൎ𝜑ெ due to easy-plane \nanisotropy, which is further corroborated by t he similar saturation fields observed for all \nIP field-scans. Our results show therefore that 𝑅ௌெோ is much larger than 𝑅ௌெோିுா (a factor \n8.25), as customary in MI/Pt bilayers. \n \n \nFigure 2. Harmonic Hall characterization of the Bi:YIG/Pt sample: (a) Schematic \nrepresentation of the device geometry and corresponding coordinate system; Transverse \nHall resistance measured upon sweeping an exte rnal (b) OOP or (c ) IP (at different 𝜑 \nangles) magnetic field; (d) 𝜑-dependence of the transverse resistance measurements \nsummarized in c, fitted to a sin2 𝜑 function in accordance with eq. 1. All measurements \nwere conducted at a current density (rms) of 𝑗 = 1.7ൈ1011 A/m2. Note that a constant \noffset has been subtracted fr om the raw data shown in b and c. \n8\nC. Description of the SOTs and macrospin model \nMacrospin simulations are a useful tool not onl y to verify that the experimental data can \nbe reproduced with existing theoretic al models, but also to a scertain the symmetry of the \ndifferent contributions to the current-induced fields. It is esta blished that SOTs consist of \ntwo distinct components acting on the magnetization such as a field-like torque with \nsymmetry 𝐓𝐅𝐋∝𝐦ൈ𝐲 , and a damping-like torque defined as 𝐓𝐃𝐋∝𝐦ൈሺ𝐲ൈ𝐦ሻ. Here, \n𝐦 stands for the unit m agnetization vector and 𝐲 the in-plane axis transverse to the \ncurrent flow (see Figure 2a). The effect of the 𝐓𝐅𝐋 is therefore equivalent to an in-plane \nfield 𝐁𝐅𝐋 acting along 𝐲, whereas that of 𝐓𝐃𝐋 is an effective field 𝐁𝐃𝐋 perpendicular to the \nmagnetization, with rotati onal symmetry within the 𝑧𝑥-plane. Upon applying an AC \ncurrent, these current-induced SOTs will induce periodic oscillations to the magnetization \nabout its equilibrium position at the same AC frequency, dict ated by the balance between \nthe demagnetizing ( 𝐁𝐝𝐞𝐦), anisotropy ( 𝐁𝐚𝐧𝐢) and external ( 𝐁𝐞𝐱𝐭) fields. In a macrospin \napproximation, we can then write the total torque ( 𝐓𝐭𝐨𝐭) as: \n𝐓𝐭𝐨𝐭ൌ𝐓𝐝𝐞𝐦𝐓𝐚𝐧𝐢𝐓𝐞𝐱𝐭𝐓𝐅𝐋𝐓𝐃𝐋\nൌ 𝑀௦ሺ𝐦ൈ𝐁 𝐝𝐞𝐦𝐦ൈ𝐁 𝐚𝐧𝐢𝐦ൈ𝐁 𝐞𝐱𝐭െ𝐵ி𝐦ൈ𝐲െ𝐵 𝐦ൈ𝐲ൈ𝐦 ሻ ሺ2ሻ \nIn the present work, we simulated the equiva lent first and second harmonic signals using \nthe Scilab open source software for numerical computations.67 More specifically, we \nperformed a polar angle scan of the external field ( 𝜃) in the 0º-360º range beyond the \nOOP saturation field of the magnetizat ion, applied at a fi xed azimuthal angle 𝜑. The \nequilibrium configuration ( 𝜃ெ, 𝜑ெ) is calculated for each set of ( 𝜃, 𝜑, 𝐁𝐞𝐱𝐭) by minimizing \neq. 2. Subsequently, the Hall resistance is com puted according to the following relations: \n𝑅ூାுൌ𝑅ௌெோsinଶ𝜃ெsin2𝜑ெ𝑅ௌெோିுா cos𝜃ெ𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ3ሻ \n𝑅ூିுൌെ 𝑅ௌெோsinଶ𝜃ெsin2𝜑ெെ𝑅ௌெோିுா cos𝜃ெ𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ4ሻ \nwhere 𝑅ூାு and 𝑅ூିு constitute the Hall resistances corresponding to a positive and \nnegative DC current, respectively. Here we note that reversing the current direction \nchanges the sign of the Hall effect coefficients but not the last term related to the current-\ninduced thermoelectric contribution, which re mains constant irrespective of the current \ndirection. The unavoidable occurrence of J oule heating primarily creates a vertical \ntemperature gradient, owing to the large t hermal conductivity differences between the \nsubstrate and the air surrounding the device (see Figure 3a, right panel). This will create \nspin Seebeck effect (SSE) in the MI68 and subsequently an inve rse SHE voltage in Pt, \nwhose contribution to the Hall resist ance is quantified through the parameter 𝑟ௌௌா and \nadded to eq. 3 and 4. We then find the equival ent first and second harmonic resistances \nby applying the followi ng operations to the a bove computed signals: \n𝑅ఠுൌ1\n2ሺ𝑅ூାுെ𝑅ூିுሻ ሺ5ሻ \n𝑅ଶఠுൌ1\n2ሺ𝑅ூାு𝑅ூିுሻ ሺ6ሻ 9\nD. Simulations of the second harmonic response \nBased on the model detailed above, we first discuss the harmonic signals occurring \nduring 𝜃-scans at 𝜑 = 90º, which is the standard geomet ry for quantifying SOTs in MI-\nbased heterostructures.4,17 The simulations were carried out using the experimental \nvalues of 𝐵௦௧, 𝑅ௌெோ and 𝑅ௌெோିுா (previously indicated in Section III-B), and, for \nsymmetry comparison, the individual DL -SOT, FL-SOT and SSE contributions are \nseparately depicted in Figures 3b- d (black traces). In this geometry, the SSE contribution \nvanishes since it is proportional to cos𝜑ெ, making thus 𝜑 = 90º an ideal geometry for the \nSOTs quantification. In this configuration 𝐵ி (𝐵) drives the OOP (IP) magnetization \noscillations, such that they scale with 𝑅ௌெோିுா (𝑅ௌெோ), respectively (see Figure 3a). This \njustifies the much smaller contribution of 𝐵ி to the second harmonic response, which is \nabout one order of magnitude lower than that of 𝐵 even though the same field \namplitudes are inputted. In terms of symmetry, they both display a similar 𝜃-dependence \nas explained below. \nIn the limit of small oscill ations of the magnetization and assuming a linear relationship \nbetween the field and the current, the sec ond harmonic resistance (for an angle-scan) \ncan be expressed as follows:17 \n𝑅ଶఠுൌሾ𝑅ௌெோିுா െ2𝑅ௌெோcos𝜃ெsinሺ2𝜑ெሻሿ𝑑cos𝜃ெ\n𝑑𝜃𝐵ூఏ\ncosሺ𝜃െ𝜃ெሻ𝐵\n𝑅ௌெோsinଶ𝜃ெ𝑑sinሺ2𝜑ெሻ\n𝑑𝜑𝐵ூఝ\nsin𝜃cosሺ𝜑െ𝜑ெሻ𝐵௫௧𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ7ሻ \nwhere 𝐵ூఏ (𝐵ூఝ) represents the component of the current-induced field ( 𝐵ூ) that induces a \nchange in 𝜃ெ (𝜑ெ). The first term in eq. 7 accounts for the OOP oscill ations and will be \ncounteracted by the demagnetiz ing field, hence the effect ive field is defined as 𝐵ൌ\n𝐵௫௧𝐵ௗ. Note that, due to the negligible ani sotropy field of the Bi:YIG sample, 𝐵ௗ \nis herein approximated as ൎ𝐵௦௧. \nDuring the simulation of the 𝜃-scans, we set 𝐵௫௧ൌ120 mT, which is beyond the OOP \nsaturation field of 45 mT. Thus, the magnetizat ion can be assumed to be saturated along \n𝐵௫௧ throughout the entire angle-scan ( 𝜃ெൎ𝜃). Recalling as well that the sample \ndisplays easy-plane anisotropy ( 𝜑ெൎ𝜑), in the vicinity of 𝜑 = 90º eq. 7 reads: \n𝑅ଶఠுൌ𝑅ௌெோିுா𝑑cos𝜃ெ\n𝑑𝜃ெ𝐵ூఏ\n𝐵௫௧𝐵ௗ𝑅ௌெோsin𝜃ெ𝑑sinሺ2𝜑ெሻ\n𝑑𝜑ெ𝐵ூఝ\n𝐵௫௧ ሺ8ሻ \nSince 𝐁𝐅𝐋 = 𝐵ி𝐲 and 𝐁𝐃𝐋 = 𝐵𝐲ൈ𝐦 , in the framework of a spherical coordinate system \n(defined by unit vectors 𝐞𝐫, 𝐞𝛉 and 𝐞𝛗) 𝐵ூఏ and 𝐵ூఝ are given by: \n𝐵ூఏൌcos𝜃ெsin𝜑ெ𝐵ிcos𝜑ெ𝐵 ሺ9ሻ \n𝐵ூఝൌcos𝜑ெ𝐵ிെcos𝜃ெsin𝜑ெ𝐵 ሺ10ሻ 10\nAgain, for an azimuthal angle 𝜑 = 90º one obtains 𝐵ூఏ = cos𝜃ெ𝐵ி and 𝐵ூఝ = െcos𝜃ெ𝐵. \nTaking these relations into account and computi ng the derivative terms in eq. 8, it follows \nthat: \n𝑅ଶఠுൌെ1\n2𝑅ௌெோିுா sinሺ2𝜃ெሻ𝐵ிାை\n𝐵௫௧𝐵ௗ𝑅ௌெோsinሺ2𝜃ெሻ𝐵\n𝐵௫௧ ሺ11ሻ \nEquation 11 shows that, for the standar d geometry discussed above, both the 𝐵ி \n(including the Oersted field) and 𝐵 components acquire a sinሺ2𝜃ெሻ dependence, which \nis well reproduced by the macrospin simulati ons. An accurate discrimination between the \nDL and FL-SOT signals at the 𝜑 = 90º geometry is thus impossible by symmetry \nconsiderations. We note that the small di fference in the field dependences are very \ndifficult to discriminate due to low signal output leve ls typical of this system. Hence, the \nreported approaches typi cally assume that 𝐵ிൎ𝐵.4,46 On that basis, and considering \nthat for most MIs 𝑅ௌெோିுா ≪𝑅ௌெோ, the second harmonic resist ance is approximated as \narising solely due to the DL-S OT, leaving FL-SOT unquantified. The estimation of the FL-\nSOT can be normally achieved by using the other standard geometry 𝜑 = 0º, where 𝐵ி \ndrives the IP oscillatio ns and its contribution to 𝑅ଶఠு is maximum. However, since the SSE \ncontribution is proportional to cos𝜑ெ, its contribution dominates the second harmonic \nsignal making the 𝐵ி quantification highly inaccurate, if not impossible. \nIn the above context, we suit ably find the Hall signals at 𝜑 = 85º and 95º as an alternative \nto circumvent the aforementione d issues. Indeed, deviations from 𝜑 = 90º induce a \nsignificant asymmetry in the 𝐵ி contribution, thereby provid ing a convenient tool for its \nquantification. We note that this will occur at the expense of nonzero thermoelectric \neffects, so that a reasonabl e compromise is found at 𝜑 = 85º and 95º fo r the parameter \nset considered in this study. 11\n \nFigure 3. Symmetry-based toolbox herein proposed to disentangle the interplay between \nthe DL, FL and SSE components. (a) Schemati c representation depicting the symmetry \nof the current-induced fields at 𝜑 = 90º geometry. Owing to t heir different symmetry with \nrespect to 𝜑 = 90º, the second harmonic oscillati ons originating from the previous \nindividual contributions were simulated as a function of the azimuthal angle at 𝜑 = 85º \nand 95º (top panel, b-d), and their average (m iddle panel, e-g) and difference (bottom \npanel, h-j) are herein proposed as reference si gnals for spin-orbit torque quantification. \n \n \n12\nE. Separation of the DL-SOT, FL-SOT and thermoelectric signals \nFigures 3b-d show simulated second harmonic 𝜃-scan curves at 𝜑 = 85º, 90º and 95º \n(red, black, and blue lines). We discussed the standard case of 𝜑 = 90º in the previous \nsection and the problems associ ated with its consideration fo r the SOT quantification. For \nthe 𝜑 = 85º and 95º case, the first important obse rvation is the nearly insensitivity of the \nDL-SOT component (Fig.3b) to small 𝜑 variations around 𝜑 = 90º. These changes \naccount for 0.6% and will fall well below the detection limit in any given harmonic \nmeasurement. The contribution from the FL-SOT is, however, significantly distorted with \nrespect to the 𝜑 = 90º data and is amplified by a factor of 2.7. Moreover , and importantly, \nit has an opposite sign considering the 𝜃 ~90º and 270º data points as reference. Finally, \nthe SSE contribution becomes now finite and has an opposite sign between 𝜑 = 85º and \n95º. \nIn an actual measurement, the signal will be a convolution of these three contributions. \nTo separate the individual components, we proceed with the addition/ subtraction of the \ntransverse signals at 𝜑 = 85º and 95º as the key step to disentangle the signals with \ndifferent origins. The average (difference) of the second ha rmonic response at these two \n𝜑 angles, hereafter denoted as 𝑅ଶఠ଼ହାଽହ (𝑅ଶఠ଼ହିଽହ) for simplicity, is depicted in Figures 3e-\ng (Figures 3h-j). As expected, 𝑅ଶఠ଼ହାଽହ exhibits the same sy mmetry and magnitude as in \nthe 𝜑 = 90º case for all three components, with a calculated variation as small as 0.6% \nand 0.3% for the DL and FL-SOT contributions. On the other hand, 𝑅ଶఠ଼ହିଽହ effectively \nsuppresses the contribution from the DL-SOT, resulting in a combined response of the \nFL-SOT and SSE effects. It can be observed that 𝑅ଶఠ଼ହିଽହ displays a maximum (minimum) \nat 𝜃 = 90º (270º) for the latter two components. Note that in the SSE case it is a direct \nconsequence of the sin𝜃ெ dependence as 𝜃ெൎ𝜃 holds throughout the entire scan. \nFurther separation between the FL-SOT and SSE can be achieved analyzing the 𝑅ଶఠ଼ହିଽହ \nvariation with the external field amplitude. For increasing 𝐵௫௧, the amplitude of the 𝐵ி-\ninduced oscillations are reduc ed, as the magnetization beco mes more strongly coupled \nto the field direction. The SSE constitutes instead a static effect, which depends on the \nmagnetization orient ation but not on t he field amplitude.17 The exact external field \ndependence of 𝑅ଶఠ଼ହିଽହ at 𝜃 = 90º can be ascertained from the second harmonic analysis. \nIn the vicinity of 𝜃 = 90º, eq. 7 reduces to: \n𝑅ଶఠுൌ𝑅ௌெோିுா𝑑cos𝜃ெ\n𝑑𝜃ெ𝐵ூఏ\n𝐵௫௧𝐵ௗ𝑅ௌெோ𝑑sinሺ2𝜑ெሻ\n𝑑𝜑ெ𝐵ூఝ\n𝐵௫௧𝑟ௌௌாcos𝜑ெ ሺ12ሻ \nAt this 𝜃 angle, the angular components of the curr ent-induced fields in eq. 12 are given \nby 𝐵ூఏ = cos𝜑ெ𝐵 and 𝐵ூఝ = cos𝜑ெ𝐵ி. Upon calculating the derivative terms, we obtain: \n𝑅ଶఠுൌെ 𝑅ௌெோିுா cos𝜑ெ𝐵\n𝐵௫௧𝐵ௗ2𝑅ௌெோcosሺ2𝜑ெሻcos𝜑ெ𝐵ிାை\n𝐵௫௧𝑟ௌௌாcos𝜑ெ\nൌ𝑅ଶఠ𝑅ଶఠி𝑅ଶఠௌௌா ሺ13ሻ 13\nNote that the contributions of the DL-S OT, FL-SOT and SSE to the transverse second \nharmonic resistance ( 𝑅ଶఠ, 𝑅ଶఠி and 𝑅ଶఠௌௌா, respectively) appear as separate terms. \nSummarizing these relations, and taking into account that cosሺ90°െ𝑥ሻ = െcosሺ90°𝑥ሻ, \nthe following expression is derived for the 𝑅ଶఠ଼ହିଽହ parameter: \n𝑅ଶఠ଼ହିଽହൌെ 𝑅ௌெோିுா𝐵\n𝐵௫௧𝐵ௗcos85°2𝑅ௌெோ𝐵ிାை\n𝐵௫௧ሺ2cosଷ85°െcos85°ሻ\n𝑟ௌௌாcos85° ሺ14ሻ \nThe discrimination between the DL and FL-SOT can be then accomp lished through their \ndifferent dependencies on 𝐵௫௧. In fact, since 𝑅ௌெோିுா ≪2𝑅ௌெோ and the effective field \nacting against 𝑅ଶఠ is larger than that of 𝑅ଶఠி because of 𝐵ௗ, the second term in eq. 14 \nsignificantly dominates over the first one. This leads to a linear relationship between \n𝑅ଶఠ଼ହିଽହ and 1/𝐵௫௧, according to which the slope is modulated by the product of 𝑅ௌெோ and \n𝐵ி, and 𝑟ௌௌா provides a constant offset independent of the external field. \nWe verified the above calculations by m eans of macrospin simulations. Figure 4a \nsummarizes the magnetic field dependence of 𝑅ଶఠ଼ହିଽହ, simulated with parameters of 𝐵 \n= 𝐵ி = 1 mT and 𝑟ௌௌா = 1 mΩ. For a better appreciation, a clos e-up view of the same plot \nabout 𝜃 = 90º is displayed in Figure 4b. T he amplitude modulation of the peak at 𝜃 = \n90º with 𝐵௫௧ is predominantly due to the FL-SOT. The SSE contribution to the data is \nconstant at 𝜃 = 90º or 270º and only the width of the peak changes due to stronger \ncoupling of the magnetizat ion vector with the exte rnal field at higher 𝐵௫௧ (see \nsupplementary Figure S1c). Exploiting this particular behavior, the peak amplitude is \nplotted as a function of the inverse of the external field in Figur e 4c. In agreement with \neq. 14 the data can be fitt ed to a straight line and 𝐵ி can be extracted by dividing the \nslope by 2𝑅ௌெோሺ2cosଷ85°െcos85°ሻ. The y-axis intercept of the linear fitting yields 𝑟ௌௌா \nupon normalizing over cos85° . The quantification of 𝐵 can be finally accomplished \nthrough the 𝑅ଶఠ଼ହାଽହ data or the second harmoni c resistance measured at 𝜑 = 90º (𝑅ଶఠଽ). \nIn both data the 𝑟ௌௌா contribution vanishes, such that 𝐵 can then be deduced by \nquantitatively comparing the macrospi n simulations (per formed with fixed 𝐵ி and \nvariable 𝐵 parameters) with the experimental data. The as-described methodology will \nhold for moderate 𝐵/𝐵ி ratios, as demonstrated by t he error estimation in these two \nparameters (included in Figur e S1e). Nevertheless, when 𝐵>>𝐵ி a strong non-linearity \nwill be observed in the 𝑅ଶ��଼ହିଽହ vs. 1/𝐵௫௧ data, leading to an increas ing systematic error in \nthe determination of 𝐵ி. For such scenario an iterative approach should be followed, so \nthat the calculated 𝐵 is recursively used as an input parameter for 𝐵ி quantification via \nfitting of the 𝑅ଶఠ଼ହିଽହ data according to eq. 14. This pr ocedure allows for an intrinsic error \nin 𝐵ி of only 0.6% even when consideri ng the unfavorable sc enario in which 𝐵/𝐵ி=10 \n(refer to Figure S1f for estima tion errors upon conducting the pr evious iterative analysis). 14\n \nFigure 4. (a) External field dependence of 𝑅ଶఠ଼ହିଽହ and (b) a close-up view of the same \nplot around 𝜃= 90°; (c) 𝑅ଶఠ଼ହିଽହ (measured at 𝜃= 90°) as a function of the inverse external \nfield. A linear fit to this data allows for the quantification of 𝐵ி and 𝑟ௌௌா by dividing the \nslope and intercept over 2𝑅ௌெோሺ2cosଷ85°െcos85°ሻ and cos85° , respectively. \n \nF. Experimental results and discussion \n \na. Second harmonic measurement s of SOTs in Bi:YIG/Pt \nAs a proof-of-concept, the described met hodology has been applied to the Bi:YIG/Pt \ndevice described in Section III-A. The seco nd harmonic Hall resistance was measured \nwhile sweeping the polar field angle 𝜃 for a fixed magnitude of 120 mT and a current \ndensity of 𝑗 = 1.7ൈ1011 A/m2. The measurements were performed at a 𝜑 angle of 85º or \n95º, as shown in Figure 5a (in red and blue, respectively). The lineshape of the as-\nmeasured signals greatly differs from that simulated in Fi gure 3. In fact, from the \nsimulations it can be seen that all current-i nduced effects are antisymmetric with respect \nto 𝜃 = 180º. This distinctive property allows us to identify and separate relevant SOT and \nSSE signals from other spurious signals based on symmetry operations. Figures 5b,c \nshow the antisymmetric and symmetric component s of the raw data where only the former \nis considered for the HHV analysis. The li neshape of the additional (spurious) symmetric \ncomponent resembles that of the firs t harmonic signal originating from 𝑅ௌெோିுா and \n𝑅ைுா. Therefore, we believe that the signal in the second ha rmonic is related to an in-\nplane temperature gradient in the device along the current injection line, producing the \nthermal counterparts of 𝑅ௌெோିுா and 𝑅ைுா. \nThe external field dependence of 𝑅ଶఠ଼ହିଽହ, determined from the ex perimental data in Figure \n5b, is displayed in Figure 5d. Notice that the sign of 𝑅ଶఠ଼ହିଽହ is opposite to the simulations \nand that the 𝜃 = 90º peak amplitude becomes less negative as increasing 𝐵௫௧. This is \nconsistent with negative 𝑟ௌௌா and positive 𝐵ிାை parameters. The dependence of this \npeak amplitude as a function of 1/ 𝐵௫௧ is plotted in Figure 5e, whose linear fitting reveals \n𝑟ௌௌா and 𝐵ிାை values of -0.39( േ0.01) mΩ and 0.29( േ0.04) mT respectively. Upon \n15\nsubtraction of the Oersted field, which is as sumed to be linearly pro portional to the current \n(𝐵ை = 0.43 mT), we obtain 𝐵ி = -0.14( േ0.04) mT. The quantification of 𝐵 was then \naddressed by measuring the se cond harmonic resistance at 𝜑 = 90º (i.e. 𝑅ଶఠଽ, see Figure \n5f). Rescaling the simulated 𝜃-scans for the DL and FL components (shown in Figures \n3b,c respectively) with the as-calculated 𝐵ி and variable 𝐵 parameters, a quantitative \ncomparison of the simulations wi th the experimental data reveals a 𝐵 value of -0.15 mT. \nThis result was also verified with the 𝑅ଶఠ଼ହାଽହ data (not shown). Ignoring 𝐵ி instead results \nin a non-negligible overestimation of 𝐵 of about 6.7%, proving the necessity and \nrelevance of the hereby described method. \n \nFigure 5. Quantification of the SO Ts via second harmonic Ha ll measurements: (a) Raw \nsecond harmonic transverse resistance as a function of the azimuthal angle (at 𝜑 = 85°, \n90° and 95°), applying an external field of 120 mT; (b) Antisymmetric and (c) symmetric \ncomponents of the data shown in a, being the former hereafter employed for further \nquantitative analysis; (d) Variation of the 𝑅ଶఠ଼ହିଽହ signal and (e) its amplitude (at 𝜃= 90°) \nwith the inverse of the exter nal field. (f) Fitting of the 𝑅ଶఠு response at 𝜑 = 90° taking into \naccount the damping-like and field-like cont ributions simulated in Figures 3b,c. All \nmeasurements were conducted at a current density (rms) of 𝑗 = 1.7ൈ1011 A/m2. \nThe values of 𝐵 and 𝐵ி reported in Section III-F,a are significantly lower than those \nreported in metallic films16,17,69 and in the specific case of 𝐵, it is somewhat lower than \nthose found in other MI/Pt systems4,46. For quantitative comparison, we convert 𝐵 into \nan effective spin Hall angle (SHA) assuming t he SHE as the sole spin current generating \nmechanism:70 \n16\n𝜃ௌுൌ2𝑒\nℏ𝑀௦𝑡:ூீ𝐵\n𝑗 ሺ15ሻ \nwhere 𝑒 stands for the electron charge, ℏ the reduced Planck constant, and 𝑀௦ and 𝑡:ூீ \nthe saturation magnetization and thickness of the Bi:YIG layer. We note that eq.15 does \nnot take into account the effect of spin diffusion length and assume full absorption of the \nspin current without cancelation effect due to the opposite spin accumulation at the \ncounter interface. We find a SHA value of ~0.4%, which is lower than the common values \nreported for YIG/Pt bilayers.35 Assuming that Pt deposited in our chamber has \ncomparable bulk properties to the systems r eported in literature, a relatively small \neffective SHA in our system can have multiple interfacial origins. One common difference \nis due to the inefficiency of the conversion of the spin current into damping-like spin-\ntorque. This can be due to a low spin mixi ng conductance or large spin memory loss71,72 \nat the Bi:YIG/Pt interface. Another potential orig in is related to the density of magnetic \nions at the interface. It has been specul ated that a reduced magnetization would also \nreduce the capability of the magnetic layer to absorb the spin current and convert it into \nspin-torque.48,73,74 Due to Bi3+ doping and consequently low 𝑀௦ value, it is plausible that \nwe obtain a lower SHA with respect to other garnet systems studied thus far. \nAnother remarkable observation is that the magnitude of 𝐵ி is comparable to that of 𝐵. \nTypically, the damping-like and field-like torques are associ ated with the real ( 𝐺↑↓) and \nimaginary ( 𝐺↑↓) parts of the spin mixing conductances, re spectively. In this picture, judging \nfrom the 𝑅ௌெோ (proportional to 𝐺↑↓) and 𝑅ௌெோିுா (proportional to 𝐺↑↓), 𝐵 should be \nabout a factor 8 larger than 𝐵ி. The large 𝐵ி in our system suggests that, either this \ncommonly accepted picture should be revised (if SHE is assumed to be the sole SOT \nsource) or some additional SOT-generating mec hanisms exist at the Bi:YIG/Pt interface. \nPlausible mechanisms include the occurrence of the Rashba-Edelst ein effect generating \nadditional SOTs with a stronger field-like contribution or magnet ic proximity effect in Pt \nacting as an additional source of spin scatte ring near the interface favoring field-like SOT \ngeneration.75 Nevertheless, investigating the SOT anomalies and deviations with respect \nto the literature are beyond t he scope of the present work. \n b. Current dependence of SOTs in Bi:YIG/Pt \nIn typical metallic SOTs systems (e.g., Pt/C o), current-induced Joule heating plays a \nminor role in the first and second harm onic responses, hence generally neglected in \nmeasurements with moderat e current densities ( 𝑗 < 2ൈ1011 A/m2).16 It is typically assumed \nthat the parameters giving rise to the firs t harmonic response are independent of the \ncurrent, meanwhile the SOTs and thermoelectric effects (expressed in resistance) \ndepends linearly in current. Howeve r, in MI/Pt, comparable cu rrent densities can create \na larger Joule heating due to lowe r thermal conductivity of the MI and the substrate, which \ncan cause a significant modulation to t he SMR parameters thr ough the temperature-17\ndependence of interfacial sp in mixing conductance,76 as well as a decrease of the \nmagnetization, magnetocr ystalline or magnetoelastic anisotropies.77 \nIn Figure 6a, we report the current-dependence of the 𝑅ௌெோ, 𝑅ௌெோିுா and 𝐵 in \nBi:YIG/Pt in the current range of 𝑗 = 0.5-2ൈ1011 A/m2. Assuming that the first data point \nat 𝑗 =0.5ൈ1011 A/m2 is representative of room tem perature values, we find that the \nmagnitude of 𝑅ௌெோ progressively decreases by about 16% when increasing the current \ndensity to 2 ൈ1011 A/m2, whereas 𝑅ௌெோିுா increases by 43% in the same current range. \nSurprisingly, 𝐵 departs from its expec ted linear trend at about 𝑗 = 1.25ൈ1011 A/m2 and \nfollows a decreasing tendency at elevated cu rrent densities. At first approximation, 𝑅ௌெோ \nand 𝐵 are both directly related to the real part of the spin-mixing conductance of the \nBi:YIG/Pt interface. In order to compare them on equal grounds, we subtract a linear \nbackground from the 𝐵 values using the data points corresponding to 𝑗 ≤ 1.0ൈ1011 A/m2 \nand then normalize both 𝐵 and 𝑅ௌெோ dataset by dividing them by the value obtained at \nthe lowest current density, i.e., 𝑗 = 0.5ൈ1011 A/m2. Figure 6c shows the comparison of the \nnormalized data displaying a strong correla tion between these two sets of data and \npinpointing their expec ted common origin. \nThese results collectively show that the sp in-dependent paramet ers of Bi:YIG/Pt critically \ndepend on the measurement te mperature even for modest current densities. This \nbehavior is tentatively attri buted to the strong sensitiv ity of magnetic and spin-dependent \ninterfacial properties of Bi:YIG to temper ature variations created by Joule heating. \nEspecially, in light of the literature,78,79 we believe that the decreasing spin-mixing \nconductance upon increasing temper ature is the likely cause of the reduced damping-like \nSOT efficiency and 𝑅ௌெோ collectively. \n \nFigure 6. (a) Modulation of the 𝑅ௌெோିுா and 𝑅ௌெோ parameters as a func tion of the current \ndensity (in rms) due to Joule heating and (b) as-derived current dependence of 𝐵. A \nsmall deviation from linearit y is observed at large 𝑗, even after considering the current-\ncorrected values of 𝑅ௌெோିுா and 𝑅ௌெோ, which highlights a non- negligible temperature-\ndependence of 𝐵. Note that the linear fitting in b has been performed considering only \nthe experimental data for 𝑗 ≤ 1.0ൈ1011 A/m2. (c) Correlation between the current \n18\nmodulation of the 𝑅ௌெோ coefficient and the linear deviation in 𝐵. For that, the 𝑅ௌெோ and \n𝐵 parameters were normalized over the experimental val ues obtained at 𝑗 = 0.5ൈ1011 \nA/m2 (𝑅ௌெோ , and 𝐵,, respectively), where no moderat e Joule heating is expected. \n \nIV. CONCLUSIONS \nIn summary, the study herein presented offe rs an alternative scheme for SOT vector \ncharacterization in MIs by shi fting to a non-standard geometry at 𝜑 = 85º and 95º. \nSupported by macrospin simulations and sy mmetry arguments, we have developed a \nconsistent methodology that allows to accu rately quantify the damping-like and field-like \nSOTs and thermoelectric contributions separat ely, with an inherent error on the simulated \ndata well below 1% for all three components. As a proof-of-concept, this strategy was \ntested on an IP-magnetized Bi:YIG/P t heterostructure, obtaining 𝐵 and 𝐵ி values of -\n0.15 and -0.14 mT per 𝑗 = 1.7ൈ1011 A/m2 injected current, respectively. Despite the \nappropriateness of the devised method for this sample, we further show how a recursive \nprocedure can be equally applied without compromi sing the accuracy of the resultant data \nto systems displaying higher 𝐵/𝐵ி ratios, where the 𝐵ி quantification becomes \nremarkedly challenging and inaccurate. This helps broaden its applicability to a wide range of materials and confi rms the robustness of the pr oposed methodology, also found \nto be stable against moderate azimuthal angle misalignments. Finally, we demonstrate \nthat Joule heating can modulate the spin transparency at th e interface, even for modest \ncurrent densities. This leads to a systematic reduction of the SMR and the DL-SOT, which \nis often neglected in most studies up to date. Ov erall, this study reinforces the need of an \nadequate SOT characterization, which is of paramount importance fo r the design of MI-\nbased spin Hall nano-osc illators and novel magnoni cs devices, among others. \n ACKNOWLEDGEMENTS \nThe authors acknowledge funding from the European Research Council (ERC) under the \nEuropean Union’s Horizon 2020 research and innovation programme (project \nMAGNEPIC, grant agreement No . 949052) and from the Spanish Ministry of Science and \nInnovation through grant reference No. PI D2021-125973OA-I00. M. T.-A. acknowledges \nfinancial support from the Spanish Ministry of Science and I nnovation under grant \nFJC2021-046680-I. \n \nCONFLICT OF INTEREST The authors declare no conflict of interest. \n 19\nREFERENCES \n(1) Manchon, A.; Železný, J. ; Miron, I. M.; Jungwirth, T. ; Sinova, J.; Thiaville, A.; \nGarello, K.; Gambardella, P. Curr ent-Induced Spin-Orbit Torques in \nFerromagnetic and Antiferromagnetic Systems. Rev. Mod. 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R.; Ali, M.; McLaren, M.; Williams, D. A. ; Hickey, B. J. Temperature \nDependence of Spin Hall Magnetoresis tance in Thin YIG/Pt Films. Phys. Rev. B \n2014 , 89 (22), 220404. \n \n \n\n\n26\nSupporting Information \n \nLeveraging symmetry for an accurate spin-orbit torques \ncharacterization in ferrimagnetic insulators \n \nMartín Testa-Anta1,*, Charles-Henri Lambert2, Can Onur Avci1,* \n1Institut de Ciència de Materials de Barcel ona (ICMAB-CSIC), Campus de la UAB, \n08193 Bellaterra, Spain \n2Department of Materials, ETH Züri ch, Hönggerbergring 64, CH-8093 Zürich, \nSwitzerland \n \n \n \nFigure S1 . External field dependence of 𝑅ଶఠ଼ହିଽହ considering the individual (a) DL, (b) FL \nand (c) SSE contributions; (d) Zoom- out of the same plot shown in a, displaying the \nresidual contribution to 𝑅ଶఠ଼ହିଽହ due to the OOP oscillations induced by 𝐵. This \ncontribution was neglected in the SOT analysis, as being ~20 and ~120 times smaller \nthan those of the FL-SOT and SSE, respectively. (e) Estimated errors in the field-like and \n27\ndamping-like fields for different 𝐵/𝐵ி ratios owing to t he contribution shown in d. \nConsidering a system where 𝐵 = 𝐵ி = 1 mT and 𝑟ௌௌா = 1 mΩ, the estimated error is \ncalculated to be 0.26, 3.54 and 0.13% for 𝐵, 𝐵ி and 𝑟ௌௌா respectively, which is expected \nto be hidden within the experimental noise. Note that the error in 𝐵 (𝐵ி) will slightly \ndecrease (significantly increase) as increasing the 𝐵/𝐵ி ratio, as depicted in e. For \nsuch scenario an iterative process is proposed, in which the calculated 𝐵 is used as an \ninput parameter for the estimation of 𝐵ி via fitting of the 𝑅ଶఠ଼ହିଽହ vs. 𝐵௫௧ experimental \ndata (according to eq. 14 in the manuscript) . The remarked reduction of the estimated \nerror in the 𝐵ி parameter after conducting this iterative process is shown in f. \n \n \n \n \nFigure S2. (a) 𝑅ଶఠ଼ହିଽହ, (b) 𝑅ଶఠ଼ହାଽହ and (c) 𝑅ଶఠଽ as a function of the azimuthal angle for \ndifferent 𝜑-angle off-centerings. A critical fact or when registering the second harmonic \nresponse is the Hall bar alignment with respect to the external field. Owing to the large \nSSE intrinsic to the Bi:YIG laye r, small off-centerings from 𝜑 = 90º (even <<1º) will lead \nto a significant SSE contribution that will mani fest through changes in the relative intensity \nof the peaks at 𝜃 ~ 28º and 208º, as shown in c. An identical parasitic sin𝜃ெ contribution \nis observed in the 𝑅ଶఠ଼ହାଽହ response for different off-centerings (see graph b), provided that \nthe measurements at the two 𝜑 angles span a constant 10º range. In any case, in light \nof the antisymmetric behavior of the SOTs with respect to 𝜃 = 180º, the experimental \noccurrence of a 𝜑 off-centering can be corrected in both 𝑅ଶఠଽ and 𝑅ଶఠ଼ହାଽହ by taking the \naverage of the peaks at ���� ~ 28º and 208º as the actual am plitude of the second harmonic \nresistance. This average is indica ted by horizontal dotted lines in b and c. Alternatively, \n𝑅ଶఠ଼ହିଽହ sees a negligible influence on a potential off-centering (see graph a), meaning that \nneither 𝐵ி nor 𝑟ௌௌா will be affected by the misalignment. \n \n \n28\n \n \nFigure S3. Current-dependence of the (a) firs t and (b) second harmonic resistance \nmeasured at 𝜑 = 90º (under a 120 mT exte rnal field). At this c onfiguration, the second \nharmonic resistance encompas ses the DL and FL-SOT contributions, without the \ninfluence of the thermoelectric SSE effect. A remarked deviation from linearity can be \nobserved at high 𝑗, which stems from the J oule heating modulation of 𝑅ௌெோ and, \nsubsequently, of the in-plane oscillations. \n \n \n \n \n \n \n \n \n \n29\nSciLab script for macrospin simulations \n \nclear; \nfunction T_tot=T_min(angle) \n theta_M=angle(1); phi_M=angle(2); \n zeta_M=angle(3); \n \n//---Current-induced fields scaling with the injected current--------------- \n \nH_FL=I_DC*HFL; H_DL=I_DC*HDL; \n \n//---Torques acting on the magnetization--------------- \n T_ext=[H*M_s*(sin(theta_M)*sin( phi_M)*cos(theta_H(ii))-cos(theta_M) *sin(phi_H)*sin(theta_H(ii))); \n H*M_s*(-sin(theta_M)*cos(phi_M)*co s(theta_H(ii))+cos(theta_M)*cos(phi_H)*sin(theta_H(ii))); \n H*M_s*(sin(theta_M)*cos(phi_M)*sin(phi_H)* sin(theta_H(ii))-sin(t heta_M)*sin(phi_M)*sin(theta_H(ii))*cos(phi_H))]; \n \nT_dem=[-0.5*H_dem*M_s*sin(2*theta_M)*sin(phi_M); 0.5* H_dem*M_s*sin(2*theta_M)*cos(phi_M); \n H_dem*M_s*0]; \n \nT_FL=[H_FL*M_s*(cos(theta_M)); \n H_FL*M_s*0; -H_FL*M_s*sin(theta_M)*cos(phi_M)] \n \nT_DL=[-H_DL*sin(theta_M)^2*sin(phi_M)*cos(phi_M); H_DL*(cos(t heta_M)^2+sin(theta_M)^2*cos(phi_M)^2); \n -H_DL*cos(theta_M)*sin(theta_M)*sin(phi_M)]; \n T_tot(1)=T_ext(1)+T _dem(1)+T_FL(1)+T_DL(1); \n T_tot(2)=T_ext(2)+T _dem(2)+T_FL(2)+T_DL(2); \n T_tot(3)=T_ext(3)+T _dem(3)+T_FL(3)+T_DL(3); \n \nendfunction \n \n//---External parameters--------------- \n \nI_app=1; //Scaling parameter related to the applied current \ntheta_Hdeg=[0:0.1:360]; //Out-of-plane field angle with respect to z-axis \nphi_H=95*%pi/180; //In-plane field angle with respect to current inje ction direction \nHext=[120]; //External field amplitude (in mT) \n \n//---Magnetic, electrical and thermal parameters--------------- \n H_dem=45; //Demagnetizing field (in mT) \nM_s=1; //Scaling parameter related to the saturation magnetization \nSMR=11.23; //Spin Hall magnetoresistance (in mOhm) \nSMR_AHE=-1.36; //Anomalous Hall-like spin Hall magnetoresistance (in mOhm) \nHFL=1; //Field-like SOT effective field (in mT) \nHDL=1; //Damping-like SOT effective field (in mT) \nr_SSE=1; //SSE coefficient (in mOhm) for a temperature gradient along z-axis \n 30\n//---Data initialization--------------- \n theta_H=[]; \n R_xy1=[]; R_xy2=[]; //Transverse resistance for I+ and I- \nRth_xy1=[]; Rth_xy2=[]; //Thermal contribution to transverse resistance for I+ and I- \n Ph1=[]; Ph2=[]; //Equilibrium phi_M \nTh1=[]; Th2=[]; //Equilibrium theta_M \n R_xy=[]; //First harmonic transverse resistance \nDelta_R_xy=[]; //Second harmonic transverse resistance \n \ninit1=theta_Hdeg(1)*%pi/180; //Initial value of theta_M \ninit2=phi_H; //initial value of phi_M \n //---Computation--------------- \n \nfor hh=1:length(Hext) //Loop for different external field amplitude values \nH=Hext(hh); \n for ii=1:length(theta_Hdeg); //Loop for different external field theta angle values \ntheta_H(ii)=theta_Hdeg (ii)*%pi/180; \n \n//--- Computation for positive current I+ --------------- \n \nI_DC=+I_app; \nfor jj=0:10; \n [t]=fsolve([init1;init2;0],T_min,1e-15); \n init1=t(1); init2=t(2); \nend \n \nPh1(ii)=2*t(2); \nTh1(ii)=t(1); \nRth_xy1(ii)= r_SSE*sin(t(1))*cos(t(2)); \n R_xy1(ii)=I_app*(SMR*(sin(t(1))^2)*sin(2*t(2)))+ SMR_AHE *cos(t(1))+Rth_xy1(ii); \n \n//--- Computation for negative current I- --------------- \n I_DC=-I_app; \n for kk=0:10; \n [t]=fsolve([init1;init2;0],T_min,1e-15); \n init1=t(1); init2=t(2); \nend \n \nPh2(ii)=2*t(2); Th2(ii)=t(1); \n \nRth_xy2(ii)=-(r_SSE*si n(t(1))*cos(t(2))); \n \nR_xy2(ii)=I_app*(SMR*(sin(t(1))^2)*sin(2*t(2)))+SMR_AHE*cos(t(1))+Rth_xy2(ii); 31\n//--- Computing the average and the difference between I+ and I- --------------- \n R_xy(ii)=(R_xy1(ii)+R_xy2(ii))/2; //Results for the first harmonic transverse resistance \nDelta_R_xy(ii)=(R_xy1(ii)-R_xy2(ii))/2; //Results for the second harmonic transverse resistance \n \nend \n \n//---Data saving - Output--------------- \n \nout=[theta_Hdeg' R_xy Delta_R_xy]; writefile='Data saving directory\\filename.dat' \nfprintfMat(writefile,out,'%.10g'); \n \nend \n \n" }, { "title": "1103.3764v2.Spin_transfer_torque_on_magnetic_insulators.pdf", "content": "arXiv:1103.3764v2 [cond-mat.mtrl-sci] 15 Sep 2011epl draft\nSpin transfer torque on magnetic insulators\nXingtao Jia1, Kai Liu1andKe Xia1 (a)and Gerrit E. W. Bauer2,3\n1Department of Physics, Beijing Normal University, Beijing 100875, China\n2Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n3Delft University of Technology, Kavli Institute of NanoSci ence, 2628 CJ Delft, The Netherlands\nPACS72.25.Mk – Spin transport through interfaces\nPACS72.10.-d – Theory of electronic transport; scattering mechanisms\nPACS85.75.-d – Magnetoelectronics; spintronics: devices exploiting sp in polarized transport or\nintegrated magnetic fields\nAbstract –Recent experimental and theoretical studies focus on spin -mediated heat currents\nat interfaces between normal metals and magnetic insulator s. We resolve conflicting estimates\nfor the order of magnitude of the spin transfer torque by first -principles calculations. The spin\nmixing conductance G↑↓of the interface between silver and the insulating ferrimag net Yttrium\nIron Garnet (YIG) is dominated by its real part and of the orde r of 1014Ω−1m−2,i.e.close to\nthe value for intermetallic interface, which can be explain ed by a local spin model.\nIntroduction. – It has recently been reported that\nthe magnetism of insulators can be actuated electrically\nand thermally by normal metal contacts [1,2]. The mate-\nrial of choice is the ferrimagnet Y 3Fe5O12(YIG), because\nof its extremely small magnetic damping [3–5]. The low-\nlying excitations of magnetic insulators are spin waves,\nwhich carry heat and angular momentum [6]. Existing\nexperiments use Pt contacts, which by means of the in-\nverse spin Hall effect are effective spin current detectors\n[7]. Slonczewski [8] reports that the thermal spin transfer\ntorquein magneticnanopillars[9,10] canbemuch moreef-\nficientthanthe electricallygeneratedspintorquein metal-\nlic structures.\nThe electrical and thermal injection of spin and heat\ncurrents into insulating magnets is governed by the spin\ntransfer torque at the metal |insulator interface [11,12],\nwhich is parameterized by the spin-mixing conductance\nG↑↓=e2Tr/parenleftBig\nI−r†\n↑r↓/parenrightBig\n/h,whereIandrσare the unit\nmatrix and the matrix of interface reflection coefficients\nfor spinσspanned by the scattering channels at the Fermi\nenergy of the metal [13]. Crude approximations such as\na Stoner model with spin-split conduction bands [11] and\nparameterized exchange between the itinerant metal elec-\ntrons and local moments of the ferromagnet [1,8,12] have\nbeen used to estimate G↑↓for YIG interfaces1. Experi-\n(a)E-mail: kexia@bnu.edu.cn\n1The spin mixing conductance is governed by the reflection coe ffi-\ncients only and remains finite when the transmission coefficie nts van-ments and initial theoretical estimates found very small\nspin torques that are at odds with Slonczewski’s predic-\ntions [8].\nHere we report calculations of the spin mixing conduc-\ntanceforthe Ag |YIGinterfacebasedonrealisticelectronic\nstructures. Silver is a promising material[14] fornon-local\nspin current detection [15], which should be more efficient\nthan the inverse spin Hall effect in nanostructures. We\ndemonstrate that the calculated G↑↓for the Ag |YIG inter-\nface is much larger than expected from the Stoner model\nand better described by local-moment exchange fields.\nFree-electron model. – We start with a reference\nstructureconsistingofanAg |FI|Ag(001)junction inwhich\nthe ferromagnetic insulator (FI) is modeled by a spin-\nsplit vacuum barrier, i.e.,the free-electron Stoner model.\nThe vacuum potential is chosen to be spin-split by 0 .3\nand 3.0 eV, whereas the barrier height is adjusted to 0.3,\n1.4, 2.6 and 2 .85 eV, respectively. The barrier thick-\nness (1.2 nm) is chosen here such that electron transmis-\nsion is negligible. Table 1 lists the corresponding G↑↓of\nAg|FI|Ag. Both Re G↑↓and ImG↑↓decrease with increas-\ning barrier height, as expected [11].\nBand structure. – We calculate the electronic struc-\nture of YIG using the tight-binding linear-muffin-tin-\nish. This is not a breach of the scattering theory of transpor t,since\nthe incoming and outgoing scattering states are well defined as prop-\nagating states in the metallic contacts.\np-1X. Jiaet al.\nTable 1: Spin-dependent and spin mixing conductances of a\nAg|FI|Ag(001) junction with different barrier heights and spin\nsplitting ∆ = 0 .3 and 3.0 eV. The mixing conductances for\nthe (111) orientation differs by less than 20%. The Sharvin\nconductance (GSh) of Ag(001) is 4 .5×1014Ω−1m−2.\nBarrier G↑/GshG↓/GshReG↑↓/GshImG↑↓/Gsh\n∆ = 0.3eV\n0.3 6.3E-5 5.1E-6 0.009 -1.1E-1\n1.4 3.3E-8 7.1E-9 0.003 -7.4E-2\n2.6 3.5E-9 1.3E-9 0.001 -4.0E-2\n2.85 0* 0 0.001 -5.1E-2\n∆ = 3.0 eV\n0.3 7.0E-6 0 0.15 -0.45\n1.4 7.4E-10 0 0.08 -0.35\n2.6 0 0 0.05 -0.28\n2.85 0 0 0.04 -0.27\n* 0 means a transmission probability of less than 10−10\norbital code in the augmented spherical wave approxi-\nmation as implemented in the Stuttgart code [16–18] us-\ning the generalized gradient correction (GGA) to the lo-\ncal density approximation (LDA). The cubic lattice con-\nstanta= 12.2˚A is chosen 1.6% smaller than the exper-\nimental one [19]. We use 136 additional empty spheres\n(ES) for better space filling and reduced overlap between\nneighboring atomic spheres. YIG is a ferrimagnetic insu-\nlator with band gap of 2 .85 eV [20,21]. Magnetism is\ncarried by majority and minority spin Fe atoms (tetrago-\nnal Fe(T) and octahedral Fe(O) sites in Fig.1(a), respec-\ntive.) with a net magnetic moment of 5 µBper formula\nunit [19,22–24]. The magnetic moments are 3.95 and\n−4.06µBfor majorityand minority spin Fe atoms, respec-\ntively. Both Y and O atoms show small positive magnetic\nmoments of 0.03 and 0.09 µB, respectively, while those on\nthe empty spheres do not exceed 0.007 µB. The common\nproblem of density-functional theory to predict the energy\ngap of insulators can be handled by an on-site Coulomb\ncorrection (LDA/GGA+U) [25,26] or a scissor operator\n(LDA/GGA+C) [27]. Figure 1(b) is a plot of the band\nstructureofGGA with a fundamental band gapof0 .33 eV\nbetween the valence band edge of the majority-spin chan-\nnel and conductance band edge of minority-spin channel.\nThe GGA+C method can be used to increase the band\ngap depending on the scissor parameters C. A GGA+C\nband structure with a band gap of ∽1.25 eV is shown in\nFig. 1(c). The GGA+U method applied to the YIG band\nstructure using the parameters from Ref. [25,26] leads to\nthe band structure plotted in 1(d) with the same energy\ngap∽1.25 eV.\nWhile a visual comparison of the band structures in\nFigs. 1(c) and(d) assurestheequivalenceofthe twometh-\nods, we can assess the differences quantitatively by com-\nparing the effective masses at the band edges as shown inTable 2: Band gap ( Eg) and effective masses (in unit of me) of\nthe band structure of YIG at the Γ point as calculated by the\nGGA, GGA+U, and GGA+C methods. CB and VB denote\nconductance and valence bands, respectively.\nEg(eV) Majority-spin Minority-spin\nVB CB VB CB\nGGA 0.33 0.10 0.52 0.40 0.17\nGGA+Ua1.25 0.13 0.60 0.37 0.19\nGGA+Cb1.25 0.17 1.00 0.31 0.27\nGGA+Cc1.4 0.17 1.00 0.28 0.31\nGGA+Cd1.4 0.18 1.46 0.25 0.25\naU= 3.5 eV,J= 0.8 eV\nbC(Fe,Y) = 6.1 eV,C(ES) = 3.05 eV\ncC(Fe,Y) = 7.2 eV\ndC(Fe,Y) = 7.5 eV,C(ES) = 3.75 eV\nTable 2. For band gaps of ∽1.25 eV the effective mass at\nthe conductance band edge of majority-spin as obtained\nby the GGA+U and GGA+C methods differ by up to\n67%. This seems significant, but the effects on the mixing\nconductance, which is the quantity of our main interest\nhere, is small, as discussed in the next section.\nAg|YIG interface. – We study the spin mixing con-\nductance in Ag |YIG|Ag with a 3 ×3 and 6×6 lateral su-\npercell of fcc Ag to match a cubic YIG unit cell along the\n(001) and (111) directions with lattice mismatch ∼1%.\nInterfacescanbeclassifiedaccordingtotheirmagneticsur-\nface properties into three types of terminations. For the\n(001) texture, one cut is terminated by Y as well as ma-\njority and minority spin Fe atoms with compensated mag-\nnetic moment (“YFe-termination”). Another cut yields\nonlymajorityFeatomsattheinterface(“Fe-termination”)\nwith total magnetic moment of 7 .90µBper lateral unit\ncell. The third interface covered by O atoms is obtained\nby removing Fe and Y atoms from the YFe-termination.\nThe oxygen layer is separated from adjacent Fe atoms by\nonly∼0.3˚A. Including the latter, the “O-termination”\nalso corresponds to a net interface magnetic moment of\n7.90µB. The interfaces for the (111) direction can be clas-\nsified analogously. The “YFe-termination” cut has now\na net interface magnetic moment of 23 .70µB. The Fe-\ntermination contains now minority-spin Fe atoms with net\nmagnetic moment of −16.24µB, while the O-terminated\nsurface has the same magnetic moment when including\nthe shallowly buried Fe layer.\nWe chose a YIG film of 4 unit cell layers, because its\nelectric conductance does not exceed 10−10e2/hper unit\ncell.G↑↓is therefore governedsolely by the single Ag |YIG\ninterface.\nFirst, we inspect G↑↓of Ag|YIG interfaces computed\nwith and without scissor corrections. We find that the\ndifference of Re G↑↓is as small as 21% when increasing\nthe band gap of YIG from its GGA value of 0 .33 eV to\np-2Spin transfer torque on magnetic insulators\nFig. 1: (a): 1/8 of the cubic YIG cell; the full structure can b e\nobtained by symmetry operations. Here, Fe(T) and Fe(O) are\nFeatoms at tetragonal and octahedronal sites, respectivel y. (b-\nd): Band structures of YIG with GGA, GGA+C, and GGA+U\nmethod with band gap of 0.33, 1.25, and 1 .25 eV, respectively.\na GGA+C band gap of 2 .1 eV as shown in Table 3. We\nconclude that the precise band gap is a parameter that\nhardly affects the G↑↓of the Ag |YIG interface.\nTable 3: Spin mixing conductance of Ag |YIG(001) with differ-\nent YIG band gaps modulated by the GGA+C methods. We\npin the Fermi level of Ag at mod-gap of YIG. 0 .33 eV is the\nband gap of GGA (without a scissor operator).\nEg(eV) 0.33 0.65 0.95 1.4 1.8 2.1\nReG↑↓\n(1014Ω−1m−2) 3.46 3.94 3.43 3.01 2.82 2.74\nBy scanning the Fermi energy of Ag (or the YIG work\nfunction), we can obtain information similar to that when\nvaryingthe band gap. Scanning the Ag Fermi energyfrom\nthe valence to conductance band edges for a band gap of\n1.4 eV, we obtain results equivalent to a mid-gap Fermi\nenergy and band gaps varying from zero to 2 .8 eV, but\nwithout changing the details of the band dispersion. In\nFigure 2 we plot the mixing conductance of Ag |YIG(001)\nwith YFe termination as a function of YIG’s work func-\ntion. Here we consider two kinds of band dispersions with\nthe same band gap of 1.4eV obtained by different scissor\noperator implementations as shown in Table.2. We find\nthat the mixing conductance does not depend sensitively\non (i) the YIG work function or interface potential bar-\nrier as well as (ii) the band dispersion when fixing the\nFermi energy of Ag in the middle of the band gap of YIG;\nthe difference in effective mass of 46% causes changes in\nFig. 2: Effect of band dispersion and band alignment on\nthe spin mixing conductance of Ag |YIG|Ag(001) with YFe-\ntermination. We use YIG with same band gap of 1.4eV with\ndifferent implementations of scissor operator (a) C(Fe,Y) =\n7.2 eV; (b) C(Fe,Y) = 7 .5 eV, and C(ES) = 3 .75 eV to see the\neffect of band dispersion. We fix the Fermi energy of Ag while\nscanning the YIG work function.\nReG↑↓of only 13%. These deviations are within the error\nbars due to other approximations (see below). We there-\nfore conclude that the transport properties in the present\nsystem are sufficiently well represented by the scissor op-\nerator or on-site Coulomb correction methods for the gap\nproblem.\nBesides the band alignment discussed in the previous\nparagraph, two more properties are difficult to compute\nself-consistently for large unit cells, viz. the atomic inter-\nface configuration and the ferromagnetic proximity effect:\n(i): We determine the distance between Ag |YIG by mini-\nmizing ASA overlap while keeping the space filled. We es-\ntimate that the differences in G↑↓for configurations with\nmaximum and minimum ASA overlap is less than 30%\n(ii): We assess the ferromagnetic proximity effect by us-\ning the self-consistent electronic structure of Ag atom at\nthe Ag|Fe interface. We find that the Ag atoms closest\nto Fe acquire a magnetic moment of 0.025 µBand the ef-\np-3X. Jiaet al.\nFig. 3: Spin mixing conductance of Ag |YIG(001) (left) and\nAg|YIG(111) (right) with a YIG band gap of 1 .4 eV. We fix\nthe Fermi energy of Ag while scanning the YIG work function.\nfect is observable up to the 4th Ag layer. The spin mixing\nconductance is found to be enhanced by about 10% in this\nsystem. In the following wedisregardsuch an effect. From\nvarious checks of these and other issues, the magnitude of\na possible systematic error in the mixing conductance is\nestimated to be <40%.\nResults. – Fig. 3 summarizes our results for G↑↓\nof Ag|YIG(111) and Ag |YIG(001) junction with different\nYIG interface-terminations. We find G↑↓≃1014Ω−1m−2\nfor both Ag |YIG(111) and Ag |YIG(001), with the real\npart dominating over the imaginary one, in stark con-\ntrast to the Stoner model (cf. Table 1). G↑↓depends\nonly weakly on exposing different YIG(111) surface cuts\ntoAg, whichweattributetothehomogeneousdistribution\nof magnetic atoms. The YFe termination of YIG(001) is a\nnearly compensated magnetic interface, but we still calcu-\nlate a large spin mixing conductance. Finally, our results\nare two orders of magnitude larger than the experimental\nvalue found for Pt |YIG(111) [1]! The difference between\nAg and Pt cannot account for this discrepancy: one would\nratherexpect a larger G↑↓for Pt because of its higher con-\nduction electron density.\nThe difference between the Stoner model and the\nfirst-principles calculations indicate that the spin-transfer\ntorquephysicsat normalmetal interfaceswith YIG is very\ndifferent from those with transition metals. Spin-transfer\nis equivalent to the absorption of a spin current at an in-\nterface that is polarized transversely to the magnetization\ndirection. Magnetism in insulators is usually described in\na local moment model. The physical picture of spin trans-\nferappropriateformetals, viz.thedestructiveinterference\nofprecessingspinsintheferromagnet,thenobviouslyfails.\nWhen the spin transfer acts locally on the magnetic ions,\nwe expect no difference for the spin absorbed by a fully\nordered interface with a large net magnetic moment or a\nFig. 4: Re G↑↓as a function of interface magnetic moment den-\nsity in Ag |YIG compared with that of Fe atoms at the Ag |Vac\ninterface. Insert: Crystal-planeresolvedspindensity /angbracketleftσy/angbracketrightinar-\nbitraryunitsfor Ag |YIG(001) withYFe-terminationwhenfully\npolarized electrons are injected from theAgside at mid-gap en-\nergy. The maximum magnetic moment density is 39 µB/nm2as\nestimated from a full monolayer of Fe at the interface, in whi ch\nthe Fe atoms adopt the Ag structure with magnetic moment\nof 2.81µB.\ncompensated one, in which the local moments point in op-\nposite directions, as is indeed born out of our calculations.\nIn orderto test the localmoment paradigm, we consider\nnon-conducting Ag |Vac(4L)|Ag(111) junctions. We now\nsprinkle one vacuum interface randomly with Fe atoms.\nAt low densities the Fe atoms are weakly coupled and\nform local moments. The electronic structure is gener-\nated using the Coherent Potential Approximation (CPA)\nfor interface disorder. A 10 ×10 lateral supercell with\n100 atoms in one principle layer is used to model a mag-\nnetic impurity range from 1% to 80%. The high den-\nsity limit is a monolayer of Fe atoms in the fcc struc-\nture: Ag|Fe(1L)|Vac(3L)|Ag(111)with totalmagneticmo-\nment of 2 .81µBper Fe atom. So, the maximum mag-\nnetic moment density here is 39 µB/nm2. The results for\nthe mixing conductances is summarized in Fig. 4. We\nfind that the ratio of G↑↓to the (Ag) Sharvin conduc-\ntances monotonically increases with the Fe density at the\nAg|Vac interface. The increase is linear at small densi-\nties and saturates around 30 µB/nm2due to interactions\nbetween neighboring moments. We find that Re G↑↓of\nAg|YIG and Ag |Fe|vacuum agrees well for corresponding\nFe densities at the interface, in strong support of the local\nmoment model.\nSince the mixing conductance is dominated by the local\nmoments at the interface, we understand that the results\nare relatively stable against the difficulties density func-\ntional theory has for insulators. The variation of the band\ngap of the insulator as well as the band alignment with\nrespect to normal metal changes the penetration of the\np-4Spin transfer torque on magnetic insulators\nspin accumulation, but since only the uppermost layers\ncontribute this is of little consequence.\nTable 4: G↑↓of a disordered Ag |YIG(001) interface with YFe-\ntermination. Directional disorder is introduced by flippin g\nthree majority Fe spins in the 2 ×2 super cell.\nReG↑↓(1014Ω−1m−2) ImG↑↓(1014Ω−1m−2)\nclean 3.010 0.302\ndisorder 3.145 0.382\nTable 4 shows the effect of directional disorder\nof magnetic moments on the mixing conductance for\nAg|YIG(001) with YFe-termination, for which the in-\ntegrated surface magnetic moment density is close to\nzero. Here, we use a 2 ×2 lateral YIG supercell in\nwhich three magnetic moments are flipped to a negative\nvalue, amounting to a total surface magnetic moment of\n−23.7µBper lateral unit cell. The directional disorder\nof magnetic moments at the interface slightly enhances\nReG↑↓(around 5%), as indeed expected from the local\nmoment picture.\nFig. 5: Spin mixing conductance of Ag |Fen|YIG|Fen|Ag(001)\nwith YFe termination for (a) ballistic and (b) diffusive tran s-\nport (i.e. in the presence of Schep correction), where nis the\nnumber for Fe monolayers inserted between Ag and YIG.\nInserting a thin ferromagnetic metallic layer betweenthe normalmetalandYIG shouldenhancethe spinmixing\nconductance. In Fig. 5 (a), we show that inserting Fe\natomic layers indeed increases Re G↑↓by 40-65% up to\nthe intermetallic Ag |Fe value, which is close to the Ag\nSharvin conductance.\nIn these calculation, the Ag reservoirshas been assumed\nto be ballistic. When the spin mixing conductanceis small\nrelative to the Sharvin conductance, this is a valid approx-\nimation, but otherwise the diffusive nature of transport\nmay not be be neglected. Since Re G↑↓turns out to be\nof the same order as GSh\nAgwe have to introduce the diffu-\nsive transport correction as introduced by Schep et al.as\n[28,29]\n1\n˜G↑↓=1\nG↑↓−1\n2GSh\nAg. (1)\nThe results are shown in Fig. 5(b). We observe that the\n”Schep” correction enhances the spin mixing conductance\nby 20% for for the and about 90% for 4 nonolayers Fe\ninsertions between Ag and YIG.\nThe spin transfer can be maximized by a high den-\nsity of magnetic ions at the interfaces. In YIG we could\nnot identify interface directions or cuts that are espe-\ncially promising, but this could be different for other\nmagnetic insulator, such as ferrites [8]. Slonczewski [8]\nuses a local moment model with a somewhat smaller ex-\nchange splitting (0.5eV) than found here; when defined\nas△/parenleftBig\n/vectorR/parenrightBig\n=/integraltext\nΩWS/parenleftBig\nV↓\nxc/parenleftBig\n/vectorR,/vector r/parenrightBig\n−V↑\nxc/parenleftBig\n/vectorR,/vector r/parenrightBig/parenrightBig\nρ/parenleftBig\n/vectorR,/vector r/parenrightBig\nd/vector r,\nwhereρ/parenleftBig\n/vectorR,/vector r/parenrightBig\nis the density of the evanescent wave func-\ntion in YIG at mid-gap energy disregarding its spin split-\nting [30], Ω WSthe Wigner-Seitz sphere at the lattice site\n/vectorR, andV↑(↓)\nxcdenotes the exchange-correlation potentials\nfor spin-up (down) electronsthe exchange splitting felt by\nthe Ag conduction electrons at the YIG interface is up to\n∼3.0 eV. Since Slonzcewski focusses on the magnetiza-\ntion dynamics of the magnetic insulator we cannot carry\nout a quantitative comparison with his model here.\nConclusion. – In conclusion, we computed the spin\nmixing conductance G↑↓of the interface between sil-\nver and the insulating ferrimagnet Yttrium Iron Garnet\n(YIG). Re G↑↓is found to be ofthe orderof1014Ω−1m−2,\nwhich is much larger than expected for a Stoner model,\nwhich indicates the importance of the local magnetic ex-\nchange field at the interface. On the other hand, G↑↓is\nnot very sensitive to crystal orientation and interface cut.\nReG↑↓can be enhanced to around 40-65% of the fully\nmetallic limit by inserting a monolayers of iron between\nAg and YIG. The discrepancy between the measured and\ncalculated mixing conductance might indicate previously\nunidentified interfacecontaminationsthat, when removed,\nwould greatly improve the usefulness of magnetic insula-\ntors in spintronics.\n∗∗∗\nWe would like to thank Burkard Hillebrands, Eiji\np-5X. Jiaet al.\nSaitoh, and Ken-ichi Uchida for stimulating discussions.\nThis work was supported by National Basic Research\nProgram of China (973 Program) under the grant No.\n2011CB921803 and NSF-China grant No. 60825404, the\nEC Contract ICT-257159 “MACALO” and the Dutch\nFOM foundation. This research was supported in\npart by the Project of Knowledge Innovation Program\n(PKIP) of Chinese Academy of Sciences, Grant No.\nKJCX2.YW.W10.\nAdditional remark: After first submission of our\nmanuscript arXiv:1103.3764, a manuscript was submitted\nand accepted by Physical Review Letters (Heinrich B. et\nal., Phys. Rev. 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In this article we extend the ex-\nisting theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R)\n(2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal\nspin conductance non-perturbatively (but in the low-frequency limit) and at \fnite frequency (but in\nthe limit of low interface transparency). For the paradigmatic spintronic material system YIG jPt,\nwe \fnd that non-perturbative e\u000bects lead to a longitudinal spin conductance that is ca. 40% smaller\nthan the perturbative limit, whereas \fnite-frequency corrections are relevant at low temperatures\n.100 K only, when only few magnon modes are thermally occupied.\nI. INTRODUCTION\nIn magnetic insulators, transport of angular momen-\ntum is possible via spin waves, collective wave-like\nexcursions of the magnetization from its equilibrium\ndirection.1{3A spin wave | or its quantized counter-\npart, a \\magnon\" | carries both an oscillating angu-\nlar momentum current with polarization perpendicular\n(transverse) to and a non-oscillating angular momentum\ncurrent with polarization parallel (longitudinal) to the\nmagnetization direction. The magnitude of the trans-\nverse spin current is proportional to the amplitude of the\nspin wave; the magnitude of the longitudinal spin current\nis quadratic in the spin wave amplitude, i.e., it scales pro-\nportional to the number of excited magnons.4{6\nBoth components of the spin current couple to con-\nduction electrons at the interface between a ferro-\n/ferrimagnetic insulator (F) and a normal metal (N).\nMicroscopically, the coupling of the transverse compo-\nnent can be understood in terms of the interfacial spin\ntorque and spin pumping,7{11which both give an an-\ngular momentum current perpendicular to the magneti-\nzation direction, see Fig. 1 (left). A longitudinal spin\ncurrent across the interface is obtained from the spin\ntorque acting on or spin pumped by the small thermally-\ninduced transverse magnetization component.12Alterna-\ntively and equivalently, a longitudinal interfacial spin cur-\nrent results from magnon-emitting or -absorbing scat-\ntering at the interface, as shown schematically in Fig.\n1 (right). The transverse component of the interfacial\nspin current is relevant for coherent e\u000bects, such as the\nspin-torque diode e\u000bect13,14or the spin-torque induced\nferromagnetic resonance.15{17The longitudinal compo-\nnent governs incoherent e\u000bects, such as the interfacial\ncontribution to the spin-Seebeck e\u000bect,18{21the spin-\nPeltier e\u000bect,22or non-local magnonic spin-transport\ne\u000bects.23{25The spin-Hall magnetoresistance26{31de-\npends on a competition between both components of the\nFIG. 1. Illustration of the microscopic mechanisms underly-\ning the transverse (left) and longitudinal (right) components\nof the spin current through the interface between a ferro-\n/ferrimagnetic insulator F and a normal metal N. The trans-\nverse spin current is mediated by the spin torque and spin\npumping involving electrons (red) with spins perpendicular\nto the magnetization and spin waves (blue) with frequency\n!equal to the frequency at which the spin accumulation \u0016s\nin N is driven. The longitudinal component arises from spin-\n\rip scattering of conduction electrons (red), combined with\nthe creation or absorption of thermal magnons of frequency\n\n (blue). (The thermal magnon frequency \n is not related\nto the driving frequency !.) Alternatively, the longitudinal\ncomponent can be seen as arising from the spin torque exerted\non/spin pumped by the transverse magnetization component\ninduced by thermal \ructuations in F (not shown schemati-\ncally).\nspin current.32,33\nIn the linear-response regime, the transverse spin cur-\nrent density jx\ns?through the FjN interface (directed from\nN to F) is proportional to the di\u000berence of the trans-\nverse spin accumulation \u0016s?in N and the time deriva-\ntive of the transverse magnetization amplitude m?at thearXiv:2208.01420v1 [cond-mat.mes-hall] 2 Aug 20222\ninterface,11\njx\ns?(!) =g\"#\n4\u0019[\u0016s?(!) +~!m?(!)]: (1)\nThe coe\u000ecient of proportionality g\"#is complex and\nknown as the \\spin-mixing conductance\" per unit area.34\nOmitting Seebeck-type contributions that depend on the\ntemperature di\u000berence across the F jN interface, the lon-\ngitudinal spin current density jx\nskis proportional to the\ndi\u000berence of the longitudinal spin accumulation \u0016skin N\nand the \\magnon chemical potential\" \u0016m,35\njx\nsk(!) =gsk\n4\u0019[\u0016sk(!)\u0000\u0016m(!)]: (2)\nIn the limit of weak coupling across the F jN interface the\nlongitudinal interfacial spin conductance is proportional\nto the real part of the spin-mixing conductance,12,35,36\ngsk=4 Reg\"#\nsZ\nd\n\u0017m(\n)\n\u0012\n\u0000dfT(\n)\nd\n\u0013\n: (3)\nHeresis the spin per volume in F, \u0017m(\n) the density\nof states (DOS) of magnon modes at frequency \n, and\nfT(\n) the Planck distribution at temperature Tof the\nmagnons.\nThe availability of high-quality THz sources, combined\nwith spin-orbit-mediated conversion of electric into mag-\nnetic driving, as well as of femtosecond laser pulses for\npump-probe spectroscopy has made it possible to exper-\nimentally access spin transport across F jN interfaces on\nultrafast time scales.37{43Whereas Eq. (1) is valid for fre-\nquencies small in comparison to the frequencies of acous-\ntic magnons at the zone boundary,10which reach well\ninto the THz regime, Eq. (3) requires driving frequencies\nmuch smaller than the frequencies of thermal magnons,\ni.e.,!=2\u0019.kBT=h\u00196:3 THz for 300 K.12At room\ntemperature, the two conditions roughly coincide for the\nmagnetic insulator YIG, which is the material of choice\nfor many experiments, or for ferrites, such as CoFe 2O4\nand NiFe 2O4, see Refs. 44 and 45. But at low temper-\natures, the condition for the applicability of Eq. (3) is\nstricter and may be violated for su\u000eciently fast driving\nfor these materials.46An example of a magnetic mate-\nrial for which the two conditions do not coincide already\natroom temperature is Fe 3O4(magnetite), for which the\nfrequency of acoustic magnons at the zone boundary is\nwell above the frequency of thermal magnons at room\ntemperature.47\nIn this article, we present two calculations of the lon-\ngitudinal interfacial spin conductance gsk(!) per area\nthat go beyond the low-frequency weak-coupling regime\nof validity of Eq. (3): (i) We calculate gskin the low-\nfrequency limit, but without the assumption of weak cou-\npling across the F jN interface, and (ii) we calculate the\n\fnite-frequency longitudinal spin conductance gsk(!) per\narea in the weak-coupling limit. Our \fnite-frequency re-\nsult is applicable in the same frequency range as Eq.\n(1), i.e., within the entire frequency range of acousticmagnons. Additionally, the temperature Tmust be low\nenough such that only acoustic magnons are thermally\nexcited. For YIG this condition amounts to the require-\nment thatT.300 K.48Comparing our non-perturbative\nlow-frequency calculation to the weak-coupling result\nin Eq. (3), we \fnd that the latter is a good order-\nof-magnitude estimate for most material combinations,\nwhereas quantitative deviations are possible.\nThis article is organized as follows: In Sec. II we report\nour non-perturbative calculation of the longitudinal spin\nconductance gskat zero frequency, using scattering the-\nory for the re\rection of spin waves from the F jN interface.\nIn Sec. III we present our perturbative calculation of the\n\fnite-frequency longitudinal spin conductance gsk(!), us-\ning the method of non-equilibrium Green functions. We\ngive numerical estimates for material combinations in-\nvolving the magnetic insulator YIG in Sec. IV and we\nconclude in Sec. V. Appendices A and B contain further\ndetails of the calculations.\nII. NON-PERTURBATIVE CALCULATION AT\nZERO FREQUENCY\nCentral to our non-perturbative calculation is the am-\nplitude\u001a(\n) that a magnon with frequency \n incident on\nthe FjN interface is re\rected back into F. The \\transmis-\nsion coe\u000ecient\"j\u001c(\n)j2= 1\u0000j\u001a(\n)j2is the probability\nthat the magnon is not re\rected and, instead, transfers\nits angular momentum ~to the conduction electrons in\nN. As we show below, knowledge of \u001a(\n) is su\u000ecient for\nthe calculation of the longitudinal interfacial spin con-\nductancegsk(!) per area in the low-frequency limit.\nMagnon re\rection amplitude \u001a.|To keep the notation\nsimple, we describe our calculation for a one-dimensional\ngeometry and switch to three dimensions in the presen-\ntation of the \fnal results. We consider an F jN interface\nwith coordinate xnormal to the interface and a magnetic\ninsulator F for x>0, see Fig. 1. Magnetization dynamics\nin F is described by the Landau-Lifshitz equation\n_m=!0ek\u0002m+1\n~s@\n@xjx\ns; (4)\nwhere mis a unit vector pointing along the direction of\nthe magnetization, !0is the ferromagnetic resonance fre-\nquency, ekthe equilibrium magnetization direction, and\njx\ns=\u0000~sDexm\u0002@m\n@x(5)\nthe spin current density, with Dexthe spin sti\u000bness of di-\nmension length2\u0001time\u00001. (We recall that the gyromag-\nnetic ratio is negative, so that the angular momentum\ndensity corresponding to the magnetization direction m\nis\u0000~sm.) The spin current density through the F jN3\ninterface is10,11,34,49,50\njx\ns=\u00001\n4\u0019(Reg\"#m\u0002+ Img\"#) [(m\u0002\u0016s) +~_m]\n+~r\nReg\"#\n2\u0019m\u0002h0; (6)\nwhereg\"#is the complex spin-mixing conductance51and\nh0is proportional to a stochastic magnetic \feld repre-\nsenting the spin torque due to current \ructuations in N.\nIf the normal metal is in equilibrium at temperature TN,\nthe correlation function of the stochastic term h0is given\nby the \ructuation-dissipation theorem,49\nhh0\n\u000b(\n0)\u0003h0\n\f(\n)i= \nfTN(\n)\u000e(\n\u0000\n0)\u000e\u000b\f; (7)\nwherefT(\n) = 1=(e~\n=kBT\u00001) is the Planck function\nand the Fourier transform is de\fned as\nh0(t) =1p\n2\u00191Z\n\u00001d\nh0(\n)e\u0000i\nt: (8)\nWe parameterize the magnetization direction mas\nm(x;t) =p\n1\u00002jm?(x;t)j2ek\n+m?(x;t)e?+m?(x;t)\u0003e\u0003\n?; (9)\nwhere the complex unit vectors e?and e\u0003\n?span the di-\nrections orthogonal to the equilibrium magnetization di-\nrection ekand satisfy the condition e?\u0002ek=ie?. The\nsolution of the Landau-Lifshitz equation (4), up to linear\norder in the magnetization amplitude m?, then reads\nm?(x;t) =1Z\n\u00001d\ne\u0000i\nt\np4\u0019sD exkx\n\u0002\u0002\nain(\n)e\u0000ikxx+aout(\n)eikxx\u0003\n;(10)\nwhere\nkx(\n) =r\n\n\u0000!0\nDex(11)\nandain(\n) andaout(\n) are \rux-normalized amplitudes\nfor spin waves moving towards the F jN interface at x= 0\nand away from it, respectively. (The amplitudes ain(\n)\nandaout(\n) may be interpreted as magnon annihilation\noperators in a quantized formulation.) The spin current\ndensity jx\nscan be decomposed into transverse and longi-\ntudinal contributions analogous to Eq. (9),\njx\ns(x;t) =jx\nsk(x;t)ek+jx\ns?(x;t)e?+jx\ns?(x;t)\u0003e\u0003\n?:(12)\nIn the same way, the spin accumulation \u0016sand the\nstochastic term h0can be decomposed into transverse\nand longitudinal contributions.\nWe \frst consider the transverse spin current density\njx\ns?to linear order in the magnetization amplitude m?.\nFrom Eqs. (5) and (10), one \fnds that the magnonictransverse spin current density jx\ns?(0;t) at the FjN inter-\nfacex= 0 is\njx\ns?(0;t) =i~sDex@m?(x;t)\n@x(13)\n=~\n4\u00191Z\n\u00001d\ne\u0000i\ntp\n4\u0019sD exkx(\n)\n\u0002[ain(\n)\u0000aout(\n)]:\nEquation (6) implies that the transverse spin current den-\nsity through the interface is given by\njx\ns?(0;t) =g\"#\n4\u0019[\u0016s?(t) +i~_m?(0;t)\u0000\u0016sk(t)m?(0;t)]\n\u0000i~r\nReg\"#\n2\u0019h0\n?(t): (14)\nImposing continuity of the transverse spin current at the\nFjN interface allows us to express the amplitude aoutof\nmagnons moving away from the interface in terms of the\namplitude ainof incident magnons and the stochastic\n\feldh0\n?. Inserting Eqs. (10) and (13) into the bound-\nary condition (14), we get\naout(\n) =\u001a(\n)ain(\n) +\u001a0(\n)h0\n?(\n); (15)\nwith\n\u001a(\n) =4\u0019sD exkx(\n)\u0000(\n\u0000\u0016sk=~)g\"#\n4\u0019sD exkx(\n) + (\n\u0000\u0016sk=~)g\"#;\n\u001a0(\n) =2p\n4\u0019sD exkx(\n) Reg\"#\n4\u0019sD exkx(\n) + (\n\u0000\u0016sk=~)g\"#: (16)\nThe coe\u000ecient \u001a(\n) is the amplitude that a magnon with\nfrequency \n incident on the F jN interface is re\rected.\nOne therefore may interpret\nj\u001c(\n)j2= 1\u0000j\u001a(\n)j2\n= (\n\u0000\u0016sk=~)j\u001a0(\n)j2(17)\nas the probability that a magnon is annihilated at the\nFjN interface while exciting a spinful excitation in N.\nLongitudinal interfacial spin conductance.| The lon-\ngitudinal spin current is quadratic in the magnetization\namplitude. From Eqs. (5) and (12) one \fnds\njx\nsk(0;t) =m?(0;t)\u0003jx\ns?(0;t) +jx\ns?(0;t)\u0003m?(0;t);(18)\nso that continuity of jx\ns?at the FjN interface to linear\norder inm?also ensures continuity of jx\nsk. In terms of\nthe magnon amplitudes, we \fnd from Eqs. (5) and (10)\nthat\njx\nsk(0;t) =~1Z\n\u00001d!\n2\u0019e\u0000i!t1Z\n\u00001d\n (19)\n\u0002[aout(\n\u0000)\u0003aout(\n+)\u0000ain(\n\u0000)\u0003ain(\n+)];4\nwhere we abbreviated \n \u0006= \n\u0006!=2 and omitted terms\nthat drop out in the limit !!0. The correlation func-\ntion of the magnon amplitudes is given by the (quantum-\nmechanical) \ructuation-dissipation theorem,52\nhain(\n\u0000)\u0003ain(\n+)i=fTF(\n\u0000\u0016m=~)\u000e(!): (20)\nHereTFis the (magnon) temperature of the magnetic\ninsulator and fTF(\n) = 1=(e~\n=kBTF\u00001)\u0002(\n\u0000!0) the\nPlanck function, with \u0002 the Heaviside step function. To\nobtain the correlation function of the stochastic \feld h0\nin the presence of a spin accumulation \u0016s=\u0016skek, we\nuse the equilibrium result in Eq. (7) and make use of\nthe fact that a spin accumulation \u0016scan be shifted away\nby transforming to a spin reference frame that rotates at\nangular frequency !=\u0016s=~, see App. A. Denoting the\nstochastic \feld in the rotating frame by ~h0, we then have\n~h0\n?(\n) =h0\n?(\n +\u0016sk): (21)\nIn the rotating frame there is no spin accumulation in N,\nso that the correlation function of ~h0\n?is given by Eq. (7).\nIt follows that\nhh0\n?(\n\u0000)\u0003h0\n?(\n+)i= (\n\u0000\u0016sk=~)\n\u0002fTN(\n\u0000\u0016sk=~)\u000e(!):(22)\nInserting this result as well as Eqs. (15), (17), and (20)\ninto Eq. (19), we \fnd for the longitudinal spin current\njx\nsk=~\n2\u00191Z\n!0d\nj\u001c(\n)j2\n\u0002[fTN(\n\u0000\u0016sk=~)\u0000fTF(\n\u0000\u0016m=~)]: (23)\nEquation (23), together with Eq. (17) for j\u001c(\n)j2, illus-\ntrates the equivalence of the two pictures of longitudinal\nspin transport mentioned in the introduction: as aris-\ning from magnon-emitting/absorbing scattering at the\nFjN interface (see \frst line in Eq. (17)) as well as from\nstochastic spin torques due to thermal \ructuations (see\nsecond line in Eq. (17)).\nIn three dimensions the calculation of the longitudinal\nspin current density involves an integration over modes\nwith transverse wavenumbers ( ky;kz). For each trans-\nverse mode the previous calculation applies, but with\nkx(\n) replaced by\nkx(\n;k?) =r\n\n\u0000!0\nDex\u0000k2\n?; (24)\nwithk2\n?=k2\ny+k2\nz. In particular, the mode-dependent\nre\rection amplitude \u001a(\n;k?) and transmission coe\u000ecient\nj\u001c(\n;k?)j2are found by substituting kx(\n;k?) forkx(\n)\nin Eq. (16). For the steady-state longitudinal spin current\ndensity we then \fnd\njx\nsk=~\n2(2\u0019)21Z\n!0d\nkx(\n)2Tm(\n)\n\u0002[fTN(\n\u0000\u0016sk=~)\u0000fTF(\n\u0000\u0016m=~)]; (25)wherekx(\n) is given by Eq. (11) and Tm(\n) is the mode-\naveraged magnon transmission coe\u000ecient,\nTm(\n) =2\nkx(\n)2kx(\n)Z\n0dk?k?j\u001c(\n;k?)j2: (26)\nThe validity of Eqs. (23) and (25) is not restricted to\nlinear response or to weak coupling across the F jN in-\nterface. For comparison with the literature and with the\nperturbative calculation of the next section, it is never-\ntheless instructive to expand Eqs. (23) and (25) to linear\norder in the interfacial spin-mixing conductance, which\ngives\njx\nsk=1\n\u0019sReg\"#1Z\n!0d\n\u0017m(\n)(~\n\u0000\u0016sk)\n\u0002[fTN(\n\u0000\u0016sk=~)\u0000fTF(\n\u0000\u0016m=~)]; (27)\nwhere\u0017m(\n) is the magnon density of states, which\nequals\u00171D\nm(\n) = 1=2\u0019Dexkx(\n) in the one-dimensional\ncase and \u00173D\nm(\n) =kx(\n)=4\u00192Dexin the three-\ndimensional case. One veri\fes that this expression is\nconsistent with Eq. (3) to linear order in \u0016sk\u0000\u0016m.\nIII. PERTURBATIVE CALCULATION AT\nFINITE FREQUENCIES\nIn this section we again consider the longitudinal spin\ncurrent density jx\nskthrough the interface between a ferro-\n/ferrimagnetic insulator F and a normal metal N, but\nnow with a time-dependent spin accumulation \u0016sk(t) in\nN. We calculate jx\nskto leading order in the spin-mixing\nconductance per unit area, g\"#. To keep the notation\nsimple, we present the calculation for a one-dimensional\nFjN junction. To generalize to the three-dimensional case\nit is su\u000ecient to replace the magnon density of states\n\u0017m(\n) by\u00173D\nm(\n).\nStarting point of our calculation is the Hamiltonian\ncoupling conduction electrons in N and magnons in F,\n^H=J^ y\n\"^ #^a+J\u0003^ y\n#^ \"^ay: (28)\nHere ^ \u001bis the annihilation operator for a conduction\nelectron with spin \u001bat the FjN interface, Jis the (suit-\nably normalized) interfacial exchange (s-d) interaction\nstrength, and the raising and lowering operators ^ ayand\n^adescribe the transverse magnetization amplitude at the\nFjN interface at x= 0. (They are the Fourier transforms\nof the second-quantization counterparts of the amplitude\nain(\n) +aout(\n) of the previous section.) The spin cur-\nrent through the F jN interface is\n^jx\nsk=i[J^ y\n\"^ #^a\u0000J\u0003^ y\n#^ \"^ay]: (29)5\nCalculating the expectation value jx\nskto leading order\ninJusing Fermi's Golden rule, one \fnds\njx\nsk= 2\u0019jJj2\u001721Z\n\u00001d\"1Z\n!0d\n\u0017m(\n) (30)\n\u0002fn\"(\")[1\u0000n#(\"\u0000~\n)][1 +fTF(\n\u0000\u0016m=~)]\n\u0000[1\u0000n\"(\")]n#(\"\u0000~\n)fTF(\n\u0000\u0016m=~)g;\nwheren\u001bis the distribution function of electrons with\nspin\u001bin N,\u0017the electron density of states at the Fermi\nenergy, and \u0017m(\n) the magnon density of states at the in-\nterface. (We assume that the electronic density of states\nis constant within the energy window of interest.) Tak-\ning a Fermi-Dirac distribution with chemical potential \u0016\u001b\nand temperature TNfor the electron distribution function\nn\u001band performing the integration over the electron en-\nergy\", one obtains\njx\nsk= 2\u0019jJj2\u001721Z\n!0d\n\u0017m(\n)(~\n\u0000\u0016sk)\n\u0002\u0002\nfTN(\n\u0000\u0016sk=~)\u0000fTF(\n\u0000\u0016m=~)\u0003\n;(31)\nwhere\u0016sk=\u0016\"\u0000\u0016#andfTis the Planck distribution as\nbefore. This result is identical to Eq. (27) if we identify12\njJj2\u00172=Reg\"#\n2\u00192s: (32)\nTo obtain the spin current density for an oscillating\nspin accumulation, we set\n\u0016\u001b(t) = \u0016\u0016\u001b+1Z\n\u00001d!\u000e\u0016\u001b(!)e\u0000i!t; \u0016 m(t) = \u0016\u0016m;(33)\nwith\u000e\u0016\u001b(!) =\u000e\u0016\u001b(\u0000!)\u0003. Hence, we impose oscillat-\ning chemical potentials \u000e\u0016\u001bon top of a time-independent\nbackground \u0016 \u0016\u001bin N and a time-independent background\n\u0016\u0016min F. We use the method of non-equilibrium Green\nfunctions to calculate the expectation value jx\nskin the\npresence of the chemical potentials of Eq. (33). To linear\norder in\u000e\u0016sk(!) =\u000e\u0016\"(!)\u0000\u000e\u0016#(!), we \fnd (see App. B\nfor details)\njx\nsk(t) =\u0016jx\nsk+1Z\n\u00001d!\u000ejx\nsk(!)e\u0000i!t; (34)\nwith \u0016jx\nskequal to the steady-state spin current density of\nEq. (31) with \u0016m= \u0016\u0016m;\u0016sk= \u0016\u0016skand\n\u000ejx\nsk(!) =gsk(!)\n4\u0019\u000e\u0016sk(!): (35)Heregsk(!) is the \fnite-frequency longitudinal spin con-\nductance per unit area,\ngsk(!) =i2 Reg\"#\n\u0019s1Z\n\u00001d\n (36)\n\u0002fD(\n) [fTF(\n\u0000\u0016\u0016m=~)\u0000F N(\n;!)]\n\u0000D(\n)\u0003[fTF(\n\u0000\u0016\u0016m=~)\u0000F N(\n;\u0000!)]g;\nwhere we de\fned\nFN(\n;!) =1\n~!\u0002\n(~\n\u0000\u0016\u0016sk)fTN(\n\u0000\u0016\u0016sk=~) (37)\n\u0000(~\n\u0000~!\u0000\u0016\u0016sk)fTN(\n\u0000!\u0000\u0016\u0016sk=~)\u0003\n;\nand where\nD(\n) =1Z\n\u00001d\n0\u0017m(\n0)\n\n +i\u0011\u0000\n0(38)\nis the (retarded) magnon Green function, with \u0011a posi-\ntive in\fnitesimal. One veri\fes that Eq. (36) reproduces\nthe perturbative result in Eq. (27) for the limit !!0\nand that it satis\fes the Kramers-Kronig relation\ngsk(!) =1\ni\u0019Z\nd!0Regsk(!0)\n!0\u0000!\u0000i\u0011: (39)\nIV. DISCUSSION\nZero-frequency limit.| We evaluate the results of our\ncalculations in Secs. II and III for the paradigmatic\nspintronic material combination YIG jPt. Longitudinal\nspin transport through the F jN interface is expected to\nplay an important role for the ferrimagnetic insulator\nYIG, since at room temperature the longitudinal spin\nconductance gskis comparable to the (transverse) spin-\nmixing conductance g\"#for this material. (This leads,\ne.g., to a prediction of a remarkable frequency depen-\ndence of the spin-Hall magnetoresistance for this mate-\nrial combination.33) To facilitate a comparison with the\nliterature, we use the same material parameters as Cor-\nnelissen et al. in Ref. 35 (if applicable). We summarize\nthe material parameters in Tab. I.\nOur non-perturbative calculation of the longitudinal\nspin conductance uses the magnon dispersion of the\nLandau-Lifshitz equation (4). This is a good approxima-\ntion at long wavelengths, for which the magnon disper-\nsion is quadratic as in Eq. (11). The use of the quadratic\napproximation to the magnon dispersion is justi\fed if\nkBT\u001c~\nmax, where \n maxis the frequency of acoustic\nmagnons at the zone boundary,\n\nmax\u0019!0+12Dex\na2m; (40)\nwithamthe size the of the magnetic unit cell. For YIG,\none has \n max=2\u0019\u00191013Hz,48,55so that the condition6\nmaterial experimental parameters ref.\nYIG !0=2\u0019= 7:96\u0001109Hz29\nam= 1:2\u000110\u00009m35\nDex= 8:0\u000110\u00006m2s\u00001 35\ns= 5:3\u00011027m\u00003 35\nYIGjPt (e2=h)Reg\"#= 1:6\u00011014\n\u00001m\u00002 29,53\n(e2=h)Img\"#= 0:08\u00011014\n\u00001m\u00002\nTABLE I. Typical values for the relevant material parameters\nof YIG and YIGjPt interfaces considered in this article. The\nlast column states the references used for our estimates. The\nspin density s=S=a3\nm, whereS= 10 is the magnitude of\nthe spin in each magnetic unit cell with lattice constant am.\nThe frequency of acoustic magnons at the zone boundary is\n\nmax=2\u0019\u0019(12Dex=a2\nm)=2\u0019\u00191:0\u00021013Hz. The imaginary\npart of the spin-mixing conductance is found from the esti-\nmate Img\"#=Reg\"#\u00190:05, see Refs. 31 and 54.\nkBT\u001c~\nmaxis only weakly obeyed at room tempera-\nture.\nThe result (26) for the mode-averaged transmission co-\ne\u000ecientTm(\n), which is the probability that a magnon\nis annihilated at the F jN interface and excites a spin-\nful excitation of the conduction electrons in N, is shown\nin Fig. 2 for \u0016sk= 0. At the lowest magnon fre-\nquency!0, the magnon wave vector k= 0 ( i.e.,kx=\nk?= 0) and thus the re\rection coe\u000ecient \u001a(!0;k?) =\n\u00001, so that Tm(!0) = 0. However, upon increas-\ning \n above !0,j\u001a(\n;k?)j\frst very quickly drops to\napproximately 0 and then reaches a maximum; corre-\nspondingly, Tm(\n) \frst features a maximum and then\nreaches a minimum upon increasing the magnon fre-\nquency \n above !0. The maximum is at a frequency\n(\n\u0000!0)=!0\u0019!0jg\"#j2=(4\u0019s)2Dex\u001c1; the minimum\nis at \n\u00192!0. Upon further increasing the frequency,\nTm(\n) increases monotonously with \n. In this frequency\nrange, a good approximation for Tm(\n) is obtained by\nexpandingj\u001c(\n;k?)j2to \frst order in g\"#, which gives\nT(p)\nm(\n) =8\u0019Reg\"#\ns(\n\u0000\u0016sk=~)\u00173D\nm(\n)\nkx(\n)2(41)\nas shown by the blue dashed curve in Fig. 2. The\nperturbative approximation for Tm(\n) remains valid for\n\n\u001c(4\u0019s)2Dex=jg\"#j2, a condition that is obeyed as long\nas \n\u001c\nmax. (The condition \n \u001c(4\u0019s)2Dex=jg\"#j2\nbecomes equal to the condition \n \u001c\nmax if one\nuses the Sharvin approximation for the spin-mixing\nconductance34,56and takes the Fermi wavelength of elec-\ntrons in N to be of the same order of magnitude as the\nsizeamof the magnetic unit cell, so that jg\"#j\u00191=a2\nm.)\nNow we are ready to discuss the di\u000berential longitudi-\nnal spin conductance per unit area\ngsk= 4\u0019@jx\nsk\n@\u0016sk: (42)\nFrom Eq. (25) we \fnd for T=TN=TFand\u0016=\u0016sk=\n10101011101210130.00.20.40.60.81.0\n7.95 8.00 8.05\nΩ/2π (GHz)0.00.51.0\n0.0 0.2 0.4 0.6 0.8 1.0\nΩ/2π (Hz)0.00.20.40.60.81.0TmFIG. 2. Mode-averaged magnon transmission coe\u000ecient\nTm(\n) at a YIGjPt-interface. The solid red curve shows\nthe non-perturbative result of Eq. (26) and the blue dashed\ncurve the weak-coupling approximation T(p)\nm(\n) of Eq. (41).\nBoth curves are based on the quadratic approximation to the\nmagnon dispersion, which breaks down for magnon frequen-\ncies \n\u0019\nmax, which is the frequency of acoustic magnons at\nthe zone boundary. Parameter values are taken from Tab. I.\n\u0016m, that\ngsk=1\n2\u00191Z\n!0d\nkx(\n)2Tm(\n)\u0012\n\u0000@fT(\n\u0000\u0016=~)\n@\n\u0013\n:(43)\nIn the perturbative limit of small g\"#this result simpli\fes\nto\ng(p)\nsk=4Reg\"#\ns1Z\n!0d\n\u00173D\nm(\n)(\n\u0000\u0016=~)\n\u0002\u0012\n\u0000@fT(\n\u0000\u0016=~)\n@\n\u0013\n: (44)\nThe perturbative result for the ratio gsk=Reg\"#depends\non the magnetic properties of bulk YIG only and not on\nthe choice of the normal metal N or the transparency of\nthe interface, whereas the non-perturbative result shows\na (quantitative, but not qualitative) dependence on the\ninterface properties. The results of Eqs. (43) and (44)\nare shown in Fig. 3 as functions of temperature Tfor\nthe material parameters of a YIG jPt interface, see Tab.\nI. (We assume no temperature dependence of the spin\ndensitysand the spin sti\u000bness Dex.) The green dashed\nstraight line in Fig. 3 is the perturbative result with the\nadditional approximation ~!0\u001ckBT, which gives35\ng(p0)\nsk=cReg\"#\ns\"\u0012kBTF\n\u0019~Dex\u00133=2\n+1\n2\u0012kBTN\n\u0019~Dex\u00133=2#\n(45)\nwithc= (1=2)\u0010(3=2)\u00191:31. The di\u000berence between\nthe perturbative and non-perturbative results increases7\nwith temperature and reaches a factor \u00191:7 at room\ntemperature, whereby the non-perturbative result for gsk\nis always below the small- g\"#approximation, see Fig. 3\n(upper left inset).\nSince the perturbative \fnite-frequency expression for\nthe longitudinal spin conductance, discussed below, can\nnot be evaluated using a magnon density of states \u0017m(\n)\nof a continuum magnon model, we compare the zero-\nfrequency longitudinal spin conductance for a quadratic\nmagnon dispersion (as is used in the main panel of Fig.\n3) with that for a magnon dispersion of a Heisenberg\nmodel on a simple cubic lattice (see Eq. (46) below).\nThis comparison is shown in the lower right inset of Fig.\n3. Whereas the di\u000berence between the two cases is small\nfor low temperatures and near room temperature, the\nHeisenberg model leads to a longitudinal spin conduc-\ntance that is up to a factor \u00191:45 larger than that of the\nquadratic approximation at intermediate temperatures.\nThis is consistent with the absence of van Hove peaks in\nthe magnon density of states in the quadratic approxi-\nmation.\nIn principle, the di\u000berential longitudinal spin conduc-\ntance per unit area, gsk, also depends on the chemical\npotentials\u0016skand\u0016m. Such dependence governs the in-\nterfacial spin current beyond linear order in \u0016sk\u0000\u0016m.\nBecause the driving potentials \u0016skand\u0016mmust remain\nbelow ~!0| otherwise the magnon system is unstable\n|, the range of admissible values for \u0016skand\u0016mremains\nwell below kBTat most temperatures, so that apprecia-\nble nonlinear e\u000bects can be found only for extremely low\ntemperatures T.1 K. At those low temperatures ther-\nmal magnons are as good as absent, so that the longitudi-\nnal spin conductance is negligibly small in comparison to\nthe transverse spin conductance. For a further discussion\nwe refer to the discussion of nonlinear e\u000bects in the con-\ntext of the \fnite-frequency longitudinal spin conductance\nbelow.\nFinite-frequency longitudinal spin transport.| For a\ndiscussion of the \fnite-frequency longitudinal spin con-\nductance per unit area, gsk(!), the quadratic approxi-\nmation of the magnon dispersion is not su\u000ecient even\nat temperatures kBT\u001c~\nmax. The reason is that\nat \fnite frequencies, gsk(!) acquires a \fnite imaginary\npart, which depends on the full magnon spectrum. (The\nreal part of gsk(!), which describes the dissipative re-\nsponse, can still be calculated within the quadratic ap-\nproximation.) For temperatures of the order of room\ntemperature and below and for frequencies !.\nmax\nit is su\u000ecient to consider the lowest-lying magnon band\nand neglect higher magnon bands in YIG.48The lowest\nmagnon band can be described e\u000bectively by a Heisen-\nberg model of spins on a simple cubic lattice with nearest-\nneighbor interactions.55,57The resulting dispersion rela-\ntion is given by\n\n(k) =!0+2Dex\na2mX\n\u000b=x;y;z(1\u0000cos(k\u000bam)); (46)\nwith maximal magnon frequency \n max, given in Eq.\n10010110210-510-410-310-210-1100\n1001011020.51.0gs\ng(p0)\ns\n1001011021.001.25g(pH)\ns\ng(p)\ns\n0.0 0.2 0.4 0.6 0.8 1.0\nT (K)0.00.20.40.60.81.0gs/Reg\nFIG. 3. Zero-frequency longitudinal spin conductance per\nunit area,gsk, at a YIGjPt-interface as function of the temper-\natureT=TN=TFfor\u0016=\u0016sk=\u0016m= 0. The red solid curve\nshows the non-perturbative result gsk=Reg\"#of Eq. (43), the\nblue dashed curve the perturbative result g(p)\nsk=Reg\"#of Eq.\n(44), and the thin green dot-dashed curve the approximation\ng(p0)\nsk=Reg\"#of Eq. (45). The upper left inset shows the ratios\ngsk=g(p0)\nsk(red solid curve) and g(p)\nsk=g(p0)\nsk(blue dashed curve).\nThe lower right inset shows the ratio g(pH)\nsk=g(p)\nsk, whereg(pH)\nsk\nis the result of Eq. (44) for the magnon density of states ob-\ntained from a Heisenberg model, see Eq. (46), and g(p)\nskthat\nof Eq. (44) for the quadratic approximation of the magnon\ndispersion. Parameter values are taken from Tab. I.\n(40), and agrees with the quadratic approximation for\n\n\u001c\nmax. The \fnite magnon bandwidth regular-\nizes the integrations for the imaginary part of gsk(!).\nIn the numerical evaluations of the real and imaginary\nparts ofgsk(!) that are discussed below we therefore\nuse the magnon density of states corresponding to the\ndispersion in Eq. (46). We veri\fed that as long as !,\nkBT=~\u001c\nmaxthe results for real and imaginary parts\nofgsk(!) depend only weakly on the precise form of the\nmagnon density of states at frequencies !\u001dkBT=~.\nFigures 4 and 5 show the real and imaginary parts of\nthe \fnite-frequency spin conductance gskat an FjN in-\nterface with F=YIG as function of the driving frequency\n!and for di\u000berent temperatures T=TN=TFand\n\u0016\u0016sk= \u0016\u0016m= 0. In the perturbative regime, the ratio\ngsk=Reg\"#is independent of the choice of the normal\nmetal N or the quality of the F jN interface.\nFor driving frequencies !\u001ckBT=~, the real part\nRegskapproaches the zero-frequency limit discussed\nabove. (Note that there may be small deviations between\nthe zero-frequency limit obtained from the quadratic\napproximation of the magnon dispersion and from the\nmagnon dispersion of Eq. (46), see Fig. 3, lower right in-\nset.) ForT= 300 K, the real part Re gskdoes not show an\nappreciable frequency dependence. At this temperature,\nthe Planck distribution fTNmay be well approximated8\n10101011101210131014101510-610-510-410-310-210-1100\nT=0KT=3KT=10KT=30KT=100KT=300K\n101210140.00.20.40.60.81.0\n0.0 0.2 0.4 0.6 0.8 1.0\nω/2π (Hz)0.00.20.40.60.81.0Regs/Reg\nFIG. 4. Real part Re gskof the \fnite-frequency longitudi-\nnal spin conductance of an F jN interface in the perturbative\nregime of weak coupling as a function of the driving frequency\n!=2\u0019for various temperatures T=TN=TF(solid colored\nlines). Material parameters are taken for an F jN interface\nwith F=YIG and N an arbitrary normal metal, see Tab. I. The\nblack dashed curve shows the result of the Rayleigh-Jeans ap-\nproximation, see Eq. (47). The inset displays the same curves.\nThe time-independent background magnon chemical potential\nand spin accumulation have been set to zero, \u0016 \u0016m= \u0016\u0016sk= 0.\nby the Rayleigh-Jeans distribution\nfTN(\n) =kBTN\n~\n: (47)\nIn this limit one \fnds that FN(\n;!) = 0 in Eq. (36), so\nthatgsk(!) is independent of frequency !, temperature\nTN, and background spin accumulation \u0016 \u0016skin N. At lower\ntemperatures, Re gskshows an increase with frequency\nfor!&kBTN=~, followed by a saturation at !\u0019\nmax.\nOne may obtain an analytical expression for Re gsk(!) in\nthe limit!\u001dkBTN=~(setting \u0016\u0016sk= \u0016\u0016m= 0):\nRegsk(!)\u00192Reg\"#\ns2\n421Z\n!0d\n\u0017m(\n)fTF(\n\u0000\u0016\u0016m)\n+!+\u0016\u0016sk=~Z\n!0d\n\u0017m(\n)!\u0000\n\u0000\u0016\u0016sk\n!3\n75:(48)\n(To keep the notation simple, we drop the superscript\n\\(p)\" because all \fnite-frequency spin conductances are\nobtained in the perturbative limit of small g\"#.) The\n\frst line in Eq. (48) is a frequency-independent o\u000bset\nwhich depends on the temperature TFand magnon chem-\nical potential \u0016 \u0016mof the ferro-/ferrimagnetic insulator\nonly. Using the quadratic approximation for the magnon\ndispersion and assuming !0\u001ckBTF=~, this term is\nfound to be equal to the \frst term in Eq. (45). For !0,\nkBTN=~\u001c!\u001c\nmaxwe may also use the quadratic\napproximation for the magnon dispersion in the secondterm and \fnd\nRegsk(!)\u0019Reg\"#\ns\"\nc\u0012kBTF\n\u0019~Dex\u00133=2\n+8\n15\u0012!\nDex\u00133=2#\n;\n(49)\nwherec= (1=2)\u0010(3=2)\u00191:31 as below Eq. (45). In\nthe limit!\u001d\nmax(but stillkBT\u001c~\nmax) we \fnd\nsimilarly\nRegsk(!)\u0019Reg\"#\ns\"\nc\u0012kBTF\n\u0019~Dex\u00133=2\n+2\na3m#\n: (50)\nwhereamis the lattice constant of the magnetic unit cell.\n(Note that, up to a numerical factor of order unity in the\nsecond term, Eq. (50) is what one obtains when kBTN=~\nin Eq. (45) is replaced by \n max.)\nWith respect to the high-frequency limit !&\nmax\nand/or the high-temperature limit kBT&~\nmax, it\nshould be kept in mind that our calculation only consid-\ners the contribution from the lowest-lying magnon band.\nFor such high frequencies, other magnon bands are likely\nto contribute to gsk(!) as well and such a contribution\nis not included in our theory. Hence, Eq. (50) and anal-\nogously Eq. (53) for Im gsk(!) discussed below should\nbe interpreted as the contribution of the lowest-lying\nmagnon band to the longitudinal spin conductance only.\nThe imaginary part Im gsk(!) increases linearly\nwith!for small frequencies, reaches a maximum at\nmax(\n max;kBT=~), and decreases with !in the high-\nfrequency limit, see Fig. 5. The linear increase with !\nfor frequencies !.\nmaxis given by the expression\nImgsk(!)\u0019\u0000!2Reg\"#\n\u0019sZ\nd\n\u0017m(\n + \u0016\u0016sk=~)\n\nhTN(\n);\n(51)\nwith\nhT(\n) =1Z\n\u00001d\n0\n\n\u0000\n0@2\n@\n02[\n0fT(\n0)]: (52)\nThis function behaves as hT(\n)!1 for \n\u001dkBT=~\nandhT(\n)!0 for \n\u001ckBT=~. Hence, e\u000bectively\nonly frequencies \n &kBT=~contribute to the integra-\ntion in Eq. (51), which explains the decreasing slope\nof\u0000Imgsk(!) vs.!|i.e., the intercept with the ver-\ntical axis in Fig. 5 | with increasing temperature T.\nThe decay of Im gsk(!) in the limit of large frequencies\n!\u001d\nmaxis described by\nImgsk(!)\u0019\u00001\n!4Reg\"#\n\u0019sZ\nd\n\u0017m(\n + \u0016\u0016sk~)\n\u0002h\n1 + ln!\n\n+h0\nTN(\n)i\n; (53)\nwith\nh0\nT(\n) =1Z\n\u00001d\n0 \n0\n\n(\n0\u0000\n)[fT(\n0) + \u0002(\u0000\n0)];(54)9\n10101011101210131014101510-510-410-310-210-1\nT=0K, 3K, 10K, 30K\nT=100K\nT=300K\n0.0 0.2 0.4 0.6 0.8 1.0\nω/2π (Hz)0.00.20.40.60.81.0− Imgs/Reg\nFIG. 5. Same as Fig. 4, but for the imaginary part Im gskof\nthe \fnite-frequency spin conductance. The black dashed lines\nshow the limiting behavior for small and large !according to\nEqs. (51) and (53) for TN!0.\nwhere \u0002(\n0) is the Heaviside step function. The\ntemperature-dependent term proportional to h0\nTNis sub-\nleading for !\u001d\nmax, so that Im gsk(!) becomes e\u000bec-\ntively temperature-independent for su\u000eciently high fre-\nquency!, as seen in Fig. 5.\nThe role of the time-independent background magnon\nchemical potential \u0016 \u0016mand spin accumulation \u0016 \u0016skis ad-\ndressed in Fig. 6. The \fgure shows Re gsk(!) as function\nof!, as in Fig. 4, but for di\u000berent values of \u0016 \u0016mand\n\u0016\u0016sk, while satisfying the bound \u0016 \u0016m, \u0016\u0016sk<~!0. As the\nmagnon chemical potential and spin accumulation ap-\npear in Eq. (36) only in the combinations \u0016 \u0016m=kBTFand\n\u0016\u0016sk=kBTNand since ~!0is much smaller than kBTfor\nmost temperatures considered, we only show results for\nT=TN=TF= 0:03 K andT=TN=TF= 3 K. As\ncan be seen in Fig. 6, the dependence of Re gsk(!) on \u0016\u0016m\nand \u0016\u0016skdisappears, when ~!0\u001ckBT(as forT= 3K\nin Fig. 6) or when ~!becomes large in comparison to\n\u0016\u0016mand \u0016\u0016sk. The imaginary part of gsk(!) does not show\nany appreciable dependence on \u0016 \u0016skin the full parameter\nrange considered (not shown) and is independent of \u0016 \u0016m.\nSpin-Seebeck coe\u000ecient.| Our non-perturbative cal-\nculation of the longitudinal spin current through the F jN\ninterface also describes the longitudinal spin current in\nresponse to a temperature di\u000berence \u000eTacross the in-\nterface. We set TF=T+\u000eT,TN=T,\u0016m=\u0016sk, and\nexpandjx\nskin Eq. (25) to linear order in \u000eT, resulting in\njx\nsk=LSSE\nT\u000eT (55)\n10810101012101410-1410-1110-810-510-2\nT=3K\nT=0.03K\n0.0 0.2 0.4 0.6 0.8 1.0\nω/2π (Hz)0.00.20.40.60.81.0Regs/Reg\nFIG. 6. Real part Re gskof the \fnite-frequency spin con-\nductance of an FjN interface in the weak-coupling regime as\nfunction of the driving frequency for two values of the tem-\nperatureT=TN=TF. Material parameters are taken for\nan FjN interface with F=YIG and N is an arbitrary normal\nmetal, see Tab. I. Curves are shown for three combinations of\nthe time-independent background potentials \u0016 \u0016mand \u0016\u0016sk: The\nsolid colored curves correspond to \u0016 \u0016m= \u0016\u0016sk= 0; the dashed\ncurves correspond to \u0016 \u0016m= \u0016\u0016sk= 0:5~!0, and the dot-dashed\nones to \u0016\u0016m= \u0016\u0016sk=\u00000:5~!0.\nwith the spin-Seebeck coe\u000ecient LSSE58\nLSSE=~\n2(2\u0019)21Z\n!0d\nkx(\n)2Tm(\n)(\n\u0000\u0016sk=~)\n\u0002\u0012\n\u0000@fT(\n\u0000\u0016sk=~)\n@\n\u0013\n: (56)\nIn the weak-coupling limit of Eq. (27), one recovers the\nspin-Seebeck coe\u000ecient obtained by Cornelissen et al. ,35\nL(p)\nSSE=~Reg\"#\n\u0019sZ\nd\n\u00173D\nm(\n)(\n\u0000\u0016sk=~)2\n\u0002\u0012\n\u0000@fT(\n\u0000\u0016sk=~)\n@\n\u0013\n: (57)\nIn the limit !0\u001ckBT=~the frequency integration may\nbe performed and one \fnds35\nL(p0)\nSSE=c0Reg\"#kBT\n\u0019s\u0012kBT\n\u0019~Dex\u00133=2\n; (58)\nwithc0= 15\u0010(5=2)=32\u00190:63. All three expressions\nare evaluated in Fig. 7 as a function of Tfor mate-\nrial parameters of a YIG jPt interface. Like in the case\nof the longitudinal spin conductance, we observe that\nthere are small quantitative di\u000berences between the non-\nperturbative and perturbative results. These di\u000berences\nare small at low temperatures, but the perturbative\nweak-coupling result deviates from the non-perturbative\none at higher temperatures, the di\u000berence reaching a fac-\ntor\u00192:3 at room temperature, see the upper left inset\nof Fig. 7.10\n10010110210-2910-2710-2510-2310-21\n1001011020.51.0\nLSSE\nL(p0)\nSSE\n1001011020.51.01.5L(pH)\nSSE\nL(p)\nSSE\n0.0 0.2 0.4 0.6 0.8 1.0\nT (K)0.00.20.40.60.81.0LSSE/Reg (J)\nFIG. 7. Spin-Seebeck coe\u000ecient LSSEat a YIGjPt-interface\nin the linear-response regime as function of the temperature\nT=\u0016TN=\u0016TF. All three curves in the main panel are ob-\ntained using the parabolic approximation of the magnon dis-\npersion. The red solid curve shows LSSEaccording to the non-\nperturbative theory, see Eq. (56), the blue dashed curve the\nperturbative result L(p)\nSSEof Eq. (57), and the thin green dot-\ndashed curve includes the approximation L(p0)\nSSEof Eq. (58).\nThe upper left inset shows the ratios LSSE=L(p0)\nSSE(red solid\ncurve) and L(p)\nSSE=L(p0)\nSSE(blue dashed curve). The lower right\ninset shows the ratio L(pH)\nSSE=L(p)\nSSE, whereL(pH)\nSSE is the result\nof Eq. (57) for the magnon density of states obtained from\na Heisenberg model, see Eq. (46), and L(p)\nSSEthat of Eq. (57)\nfor the quadratic approximation of the magnon dispersion.\nParameter values are taken from Tab. I.\nV. CONCLUSIONS AND OUTLOOK\nThe spin angular momentum current from a normal\nmetal N into a ferro-/ferrimagnetic insulator F in general\nhas a component collinear with the magnetization, which\nis carried by thermal magnons in F. In this article, we pre-\nsented two calculations of the longitudinal interfacial spin\nconductance: At zero frequency, but for arbitrary trans-\nparency of the interface, and at \fnite frequencies, but to\nleading order in the interface transparency. In general,\none expects the longitudinal interfacial spin conductance\nto acquire a dependence on the driving frequency !, when\n!exceedskBT=~. In the case of typical parameters for\nthe material combination YIG jPt and at room tempera-\nture, we \fnd that the resulting frequency dependence of\nthe interfacial spin conductance is rather weak, not more\nthan a factor\u00191:1 between the low- and high-frequency\nlimits. Also, we \fnd that (at zero frequency) the di\u000ber-\nence between the spin conductance in a non-perturbative\ntreatment of the coupling across the F jN interface and the\nperturbative result to leading order in the spin-mixing\nconductance g\"#is not more than a factor \u00191:7 at room\ntemperature, despite the fact that g\"#of a good YIGjPt\ninterface (see Tab. I) is only slightly below the Sharvin\nlimit (e2=h)g\"#=\u0019e2=h\u00152\ne\u00196:8\u00011014\n\u00001m\u00002,56where\u0015eis the Fermi wavelength of Pt.59{61In that sense, for\nFjN interfaces involving the ferrimagnetic insulator YIG,\nour two calculations may seen as a con\frmation of the ex-\nisting low-frequency weak-coupling theory.12,23,35A sim-\nilar conclusion applies to the interfacial spin-Seebeck\ncoe\u000ecient, for which we compared the existing weak-\ncoupling zero-frequency theory35,36,62with a calculation\nnon-perturbative in the interface transparency.\nOf course, one may turn the question around and ask,\nunder which experimental conditions or for which mate-\nrial combinations a frequency dependence of the inter-\nfacial longitudinal spin conductance or a deviation from\nthe perturbative weak-coupling approximation will be-\ncome signi\fcant. To see an appreciable frequency de-\npendence of gsk, it is necessary that the temperature is\nsigni\fcantly below the maximum energy ~\nmaxof acous-\ntic magnons. For YIG, this means that the temperature\nmust be well below room temperature. Our numerical\nestimates based on material parameters for YIG indicate\nthatgsk(!) may increase by a factor &10 between low-\nand high-frequency regimes if T.30 K and that the\ne\u000bect can be larger at lower temperatures, whereas the\nfrequency dependence of gsk(!) is small for T&100 K.\nAn experimental technique to measure these e\u000bects is\nthe spin-Hall magnetoresistance, which depends on the\ncompetition of longitudinal and transversal spin trans-\nport across the F jN interface. Measurements of the\nspin-Hall magnetoresistance up to the lower GHz range63\nhave already been performed. Since the longitudinal and\ntransversal interfacial spin conductances are of compa-\nrable magnitude in the high-frequency limit, one may\nthus expect a visible frequency dependence of the spin-\nHall magnetoresistance e\u000bect for frequencies in the THz\nrange, if the temperature is low enough that not all\nmagnon modes are thermally excited. (This e\u000bect is ad-\nditional to a frequency dependence of the spin-Hall mag-\nnetoresistance in the GHz range predicted in Ref. 33.)\nHowever, since the spin-Hall magnetoresistance e\u000bect in-\nvolves the di\u000berence of two contributions of comparable\nmagnitude, a more precise material-speci\fc modeling is\nrequired to reach a \frm prediction.\nAnother experimental platform in which the lon-\ngitudinal interfacial spin conductance plays a role is\nthat of non-local magnonic spin transport.23{25In this\ncase, the interfacial spin conductance directly determines\nthe coupling between the magnon system in a ferro-\n/ferrimagnetic insulator and the electrical currents in ad-\njacent normal-metal contacts used to excite and detect\nthe magnon currents. Our predictions directly trans-\nlate to a frequency dependence of the electron-to-magnon\nand magnon-to-electron conversion in such experiments.\nFurthermore, the di\u000berence between the weak-coupling\nand strong-coupling predictions may quantitatively af-\nfect estimates of the spin-mixing conductance based on\na measurement of the longitudinal spin conductance or\nthe spin-Seebeck coe\u000ecient.28{31,53,64,65.\nWe predict that the longitudinal spin conductance\ndepends on the temperatures TFandTNof the ferro-11\n/ferrimagnetic insulator and the normal metal in di\u000ber-\nent ways, see, e.g., Eqs. (45) and (50). Whereas the lon-\ngitudinal spin current in F is carried by thermal magnons\nif F and N are close to equilibrium, the longitudinal spin\nconductance does not vanish if TF= 0, as long as TN\nis non-zero. In this case, the spin current is carried by\nmagnons in F excited by spin-\rip scattering of thermally\nexcited electrons at the F jN interface. Apart from the\ndi\u000eculty that a large temperature di\u000berence between F\nand N is di\u000ecult to realize experimentally, a large tem-\nperature di\u000berence across an F jN interface also leads to\na large steady-state spin current via the interfacial spin-\nSeebeck e\u000bect. However, this DC spin current can be\neasily distinguished experimentally from the AC signal,\nwhich is caused by time-dependent driving of the spin\naccumulation in N.\nAt the interface between a normal metal and a ferro-\n/ferrimagnetic metal, there are two contributions to the\nlongitudinal spin current: A contribution from conduc-\ntion electrons in the ferro-/ferrimagnetic metal and a\nmagnonic contribution. The results derived in this article\nalso apply to the magnonic contribution at such an inter-\nface. However, at metal-metal interfaces, the magnonic\ncontribution to the spin current is typically much smaller\nthan the electronic contribution so that the frequency\nand temperature dependence of the magnonic contribu-\ntion is a sub-leading e\u000bect at such interfaces.\nWe close with two remarks on possible further ex-\ntensions of our work. An important limitation of our\ntheory is the restriction to the lowest magnon band.\nOn the one hand, this limitation enters into our non-\nperturbative calculation for low frequencies, because the\ncalculation relies on the continuum limit of the Landau-\nLifshitz-Gilbert equation. On the other hand, this lim-\nitation enters both calculations, because the boundary\ncondition at the F jN interface implicitly assumes that\nthe coupling between electronic degrees of freedom in N\nand the magnonic degrees of freedom in F at the inter-\nface is local. For acoustic magnons at the zone boundary\nand for higher magnon bands, electrons in N re\recting\no\u000b the ferro-/ferrimagnetic insulator F penetrate F suf-\n\fciently deep such that they are in\ruenced by the non-\nuniformity of m, violating the assumption of a local cou-\npling between magnonic and electronic degrees of free-\ndom. The \frst problem can be partially addressed by\nreplacing the quadratic magnon dispersion by the dis-\npersion of a Heisenberg model on a simple cubic lattice,\nas we have done in Sec. IV, but this replacement does\nnot account for the non-uniformity of the magnetization\nnear the interface. A rough estimate for the frequency at\nwhich the non-uniformity becomes relevant is the max-\nimum frequency \n maxof acoustic magnons, where for\nYIG \n max=2\u0019\u00191013Hz. It is an open task for the\nfuture to extend our theory to appropriate couplings be-\ntween electron spins and short-wavelength magnons, op-\ntical magnons, and antiferromagnons in antiferromagnets\nand ferrimagnets.\nOur \fnite-frequency calculations assume that it is onlythe electronic distribution in the normal metal N that is\ndriven out of equilibrium. Experiments exciting directly\nthe phonons of an insulating magnet F such as YIG, e.g.,\nby an ultrashort THz laser pulse, might also create a\ntime-dependent magnon chemical potential in F on ul-\ntrafast time scales.35Time-dependent magnon chemical\npotentials may also appear in ultrafast versions of non-\nlocal magnon transport experiments or in the ultrafast\nspin-Hall magnetoresistance e\u000bect with an ultrathin mag-\nnetic insulator F.33Investigating the ultrafast response\nto a change of magnon chemical potential is another in-\nteresting avenue for future research.\nAcknowledgements.| We thank T. Kampfrath, T. S.\nSeifert, U. Atxitia, F. Jakobs, and S. M. Rouzegar for\nstimulating discussions. This work was funded by the\nGerman Research Foundation (DFG) via the collabora-\ntive research center SFB-TRR 227 \\Ultrafast Spin Dy-\nnamics\" (project B03).\nAppendix A: Transformation to a rotating frame\nHere we consider the transformation to a reference sys-\ntem for the spin degree of freedom that rotates with an-\ngular frequency \u0016!(t) = \u0016!(t)ek. We discuss how the lon-\ngitudinal spin accumulation \u0016skin N, the magnetization\namplitudem?, and the stochastic transverse spin current\njs?transform upon passing to the rotating frame. We re-\nstrict the discussion to linear response in \u0016skand\u0016!. We\nuse a prime to denote creation and annihilation operators\nand observables in the rotating reference system.\nWe \frst consider the transformation to a frame rotat-\ning at constant angular velocity \u0016!= \u0016!ek. The transfor-\nmation relation for the electron annihilation operators in\nN is\n^ 0(t) =ei\u0016!\u0001\u001bt=2^ (t); (A1)\nwhere (t) is a two-component column spinor for the\nwavefunction of the conduction electrons. Solving for\nthe annihilation operator c(\") in energy representation,\nwe have\n^ 0(\") =^ (\"+~\u0016!\u0001\u001b=2) (A2)\nand, similarly,\n^ 0(\")y=^ (\"+~\u0016!\u0001\u001b=2)y: (A3)\nIt follows that the distribution function f0(\") in the ro-\ntating frame is\nf0\nTN(\") =fTN(\"+~\u0016!\u0001\u001b=2); (A4)\nwherefTNis the distribution function in the original ref-\nerence frame. We thus conclude that, in linear response,\nupon transforming to a rotating frame the spin accumu-\nlation changes as\n\u00160\ns=\u0016s\u0000~\u0016!: (A5)12\nAppendix B: Weak-coupling spin current at \fnite\nfrequency\nWe \frst discuss the expression (29) for the longitudi-\nnal spin current through the F jN interface. From the\nHeisenberg equation of motion, the spin current into the\nmagnetic insulator is\n^jx\nsk(t) =\u0000i\n2[^H;^N\"\u0000^N#]; (B1)\nwhere ^His the Hamiltonian and ^N\u001bis the number of\nconduction electrons with spin \u001b,\u001b=\",#. The only\ncontribution to ^Hthat does not commute with ^N\u001bis the\nterm (28) describing the coupling via the F jN interface.\nInserting Eq (28) into the above expression gives Eq. (29)\nof the main text.\nWe next turn to the calculation of the expectation\nvaluejx\nsk(t) of the interfacial longitudinal spin current.\nCalculating jx\nsk(t) to leading order in perturbation the-ory inJgives\njx\nsk(t) =ijJj2\n~Z\ncdt0[G\"(t0;t)G#(t;t0)D(t;t0)\n\u0000G\"(t;t0)G#(t0;t)D(t0;t)]; (B2)\nwheret0is integrated along the Keldysh contour ( i.e.,\nforward and backward integrations along the real time\naxis),\nG\u001b(t0;t) =\u0000ihTc^ \u001b(t0)^ y\n\u001b(t)i (B3)\nis the contour-ordered Green function for the conduction\nelectrons, evaluated at the interface, and\nD(t0;t) =\u0000ihTc^a(t0)^ay(t)i (B4)\nis the contour-ordered magnon Green function, again\nevaluated at the interface. Equation (B2) may be written\nas\njx\nsk(t) =ijJj2\n~1Z\n\u00001dt0h\n(GR\n\"(t0;t) +G<\n\"(t0;t))(GR\n#(t;t0) +G<\n#(t;t0))(DR(t;t0) +D<(t;t0))\n\u0000(GR\n\"(t;t0) +G<\n\"(t;t0))(GR\n#(t0;t) +G<\n#(t0;t))(DR(t0;t) +D<(t0;t))\n\u0000G>\n\"(t0;t)G<\n#(t;t0)D<(t;t0) +G<\n\"(t;t0)G>\n#(t0;t)D>(t0;t)i\n: (B5)\nIn this expression, the integration variable t0is a time,\nnot a contour time.\nWe \frst evaluate Eq. (B5) for the case that the three\nsubsystems | conduction electrons with spin up, con-\nduction electrons with spin down, and magnons | are\nseparately in equilibrium at chemical potentials \u0016\u001band\n\u0016mand temperatures T\u001bandTm, respectively. In this\ncase, all Green functions depend on the time di\u000berence\nt\u0000t0only. Changing to the integration variable t0\u0000tfor\nthe \frst term and third term and t\u0000t0for the second\nand fourth term in Eq. (B5), one \fnds that the \frst and\nthird terms in Eq. (B5) cancel, whereas the second and\nfourth terms give, after Fourier transform,\njx\nsk=\u0000ijJj2\n~Zd\"\n2\u0019Zd\n2\u0019h\nG>\n\"(\")G<\n#(\"\u0000\n)D<(\n)\n\u0000G<\n\"(\")G>\n#(\"\u0000\n)D>(\n)i\n: (B6)\nAccording to the \ructuation-dissipation theorem, one has\nG>\n\u001b(\") =\u00002\u0019i~(1\u0000n\u001b(\"))\u0017\u001b;\nG<\n\u001b(\") = 2\u0019i~n\u001b(\")\u0017\u001b: (B7)Similarly, for the magnon Green function, one has\nD>(\n) =\u00002\u0019i(fTF(\n\u0000\u0016m=~) + 1)\u0017m(\n);\nD<(\n) =\u00002\u0019ifTF(\n\u0000\u0016m=~)\u0017m(\n): (B8)\nHence, we \fnd that the spin current is\njx\nsk= 2\u0019jJj2\u0017\"\u0017#Z\nd\"Z\nd\n\u0017m(\n) (B9)\n\u0002[n\"(\")(1\u0000n#(\"\u0000~\n))(1 +fTF(\n)\u0000\u0016m=~)\n\u0000(1\u0000n\"(\"))n#(\"\u0000~\n)fTF(\n\u0000\u0016m=~)]:\nSettingT\"=T#=TNand performing the integration\nover\", one reproduces Eqs. (29) and (30) of the main\ntext, which was derived from Fermi's Golden Rule.\nWe now consider an additional oscillating component\nof the chemical potential as in Eq. (33) of the main text.\nIn the presence of the oscillating chemical potential the\nelectron Green function G(t;t0) reads13\nG\u001b(t;t0) =G\u001b0(t;t0)e\u0000iRt\nt0d\u001c\u000e\u0016\u001b(\u001c0)=~\n=G\u001b0(t;t0)\u001a\n1 +Z\nd!\u000e\u0016\u001b(!)\n~!e\u0000i!t[1\u0000ei!(t\u0000t0)]\u001b\n+:::; (B10)\nfor the greater and lesser Green functions, where, in the second line, the subscript \\0\" indicates the equilibrium Green\nfunction and the dots indicate terms of higher order in \u000e\u0016\u001b(!). Similarly, one has\nG\u001b(t0;t) =G\u001b0(t0;t)\u001a\n1\u0000Z\nd!\u000e\u0016\u001b(!)\n~!e\u0000i!t[1\u0000ei!(t\u0000t0)]\u001b\n+:::: (B11)\nTo \fnd the spin current, we \fnd it advantageous to cast the \frst two terms of Eq. (B5) into a di\u000berent form, making\nrepeated use of the identities GR+G<=GA+G>andDR+D<=DA+D>,\n\u000ejx\nsk(t) =ijJj2\n~Z\ndt0h\n(GA\n\"(t0;t) +G>\n\"(t0;t))(GR\n#(t;t0) +G<\n#(t;t0))(DR(t;t0) +D<(t;t0))\n\u0000(GR\n\"(t;t0) +G<\n\"(t;t0))(GA\n#(t0;t) +G>\n#(t0;t))(DA(t0;t) +D>(t0;t))\n\u0000G>\n\"(t0;t)G<\n#(t;t0)D<(t;t0) +G<\n\"(t;t0)G>\n#(t0;t)D>(t0;t)i\n: (B12)\nFor the linear-response correction to the spin current, we then obtain\n\u000ejx\nsk(!) =ijJj2\n~2!Z\ndt0[ei!(t\u0000t0)\u00001]\n\u0002n\n[G<\n\"0(t0\u0000t)GR\n#(t\u0000t0)\u000e\u0016\"(!)\u0000GA\n\"(t0\u0000t)G<\n#0(t\u0000t0)\u000e\u0016#(!)][DR(t\u0000t0) +D<(t\u0000t0)]\n+ [GA\n#(t0\u0000t)G<\n\"0(t\u0000t0)\u000e\u0016\"(!)\u0000G<\n#0(t0\u0000t)GR\n\"(t\u0000t0)\u000e\u0016#(!)][DA(t0\u0000t) +D>(t0\u0000t)]\n+G>\n\"0(t0\u0000t)G<\n#0(t\u0000t0)(\u000e\u0016\"(!)\u0000\u000e\u0016#(!))DR(t\u0000t0)\n+G<\n\"0(t\u0000t0)G>\n#0(t0\u0000t)(\u000e\u0016\"(!)\u0000\u000e\u0016#(!))DA(t0\u0000t)o\n: (B13)\nWriting the Green functions in terms of their Fourier representations, we write this as\n\u000ejx\nsk(!) =ijJj2\n~!Zd\"\n2\u0019Zd\n2\u0019\n\u0002nh\n[G<\n\"0(\"\u0000!)\u0000G<\n\"0(\")]GR\n#(\"\u0000\n)\u000e\u0016\"(!)\u0000GA(\"+ \n)[G<\n#0(\"+!)\u0000G<\n#0(\")]\u000e\u0016#(!)i\n[DR(\n) +D<(\n)]\n+h\nG<\n\"0(\"+!)\u0000G<\n\"0(\")]GA\n#(\"\u0000\n)\u000e\u0016\"(!)\u0000GR(\"+ \n)[G<\n#0(\"\u0000!)\u0000G<\n#0(\")]\u000e\u0016#(!)i\n[DA(\n) +D>(\n)]\n+ [G>\n\"0(\"\u0000!)\u0000G>\n\"0(\")]G<\n#0(\"\u0000\n)(\u000e\u0016\"(!)\u0000\u000e\u0016#(!))DR(\n)\n+ [G<\n\"0(\"+!)\u0000G<\n\"0(\")]G>\n#0(\"\u0000\n)(\u000e\u0016\"(!)\u0000\u000e\u0016#(!))DA(\n)o\n: (B14)\nAgain we use the \ructuation-dissipation theorem, see Eqs. (B7) and (B8). For the electrons we assume that the\nspectral density is independent of energy and we set GR\n\u001b(\") =\u0000GA\n\u001b(\") =\u0000i\u0019~\u0017\u001b. For the magnons we use that\nDR(\n) +D<(\n) =DA(\n) +D>(\n)\n=DR(\n)(fTF(\n\u0000\u0016\u0016m=~) + 1)\u0000DA(\n)fTF(\n\u0000\u0016\u0016m=~): (B15)\nWe then \fnd\n\u000ejx\nsk(!) =ijJj2\n2~!Z\nd\"Z\nd\n\u0017\"\u0017#f[[n\"(\"\u0000~!)\u0000n\"(\"+~!)]\u000e\u0016\"(!)\u0000[n#(\"\u0000~!)\u0000n#(\"+~!)]\u000e\u0016#(!)]\n\u0002[DR(\n)(fTF(\n\u0000\u0016\u0016m=~) + 1)\u0000DA(\n)fTF(\n\u0000\u0016\u0016m=~)]\u00002(\u000e\u0016\"(!)\u0000\u000e\u0016#(!))\n\u0002\u0002\n[n\"(\"\u0000~!)\u0000n\"(\")]n#(\"\u0000~\n)DR(\n)\u0000[n\"(\"+~!)\u0000n\"(\")][1\u0000n#(\"\u0000~\n)]DA(\n)\u0003\t\n=ijJj2\n~!\u000e\u0016sk(!)Z\nd\n\u001a\nDR(\n)\u0014\n![fTF(\n\u0000\u0016\u0016m=~) + 1]\u0000Z\nd\"[n\"(\"\u0000~!)\u0000n\"(\")]n#(\"\u0000~\n)\u0015\n+DA(\n)\u0014\n(\u0000!)[fTF(\n\u0000\u0016\u0016m=~) + 1]\u0000Z\nd\"[n\"(\"+~!)\u0000n\"(\")]n#(\"\u0000~\n)\u0015\u001b\n: (B16)14\nIfT\"=T#=Tthis may be further simpli\fed as\n\u000ejx\nsk(!) =ijJj2\n~!\u0017\"\u0017#\u000e\u0016sk(!)Z\nd\n\u0002\b\nDR(\n)\u0002\n(~\n\u0000\u0016\u0016sk\u0000!)fTN(\n\u0000!\u0000\u0016\u0016sk=~)\u0000(~\n\u0000\u0016\u0016sk)fTN(\n\u0000\u0016\u0016sk=~) +~!fTF(\n\u0000\u0016\u0016m=~)\u0003\n+DA(\n)\u0002\n(~\n\u0000\u0016\u0016sk+~!)fTN(\n +!\u0000\u0016\u0016sk=~)\u0000(~\n\u0000\u0016\u0016sk)fTN(\n\u0000\u0016\u0016sk=~)\u0000~!fTF(\n\u0000\u0016\u0016m=~)\u0003\t\n:\n(B17)\nThe retarded and advanced magnon Green functions can be obtained from the Krppmers-Kronig relations,\nDR(\n) =DA(\n)\u0003\n=Z\nd\n0\u0017m(\n0)\n\n +i\u0011\u0000\n0; (B18)\nwhere\u0011is a positive in\fnitesimal. 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Hammel,\nand F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014)." }, { "title": "2308.15826v1.Chiral_cavity_magnonic_system_for_the_unidirectional_emission_of_a_tunable_squeezed_microwave_field.pdf", "content": "arXiv:2308.15826v1 [quant-ph] 30 Aug 2023Chiral cavity-magnonic system for the unidirectional emis sion of a tunable squeezed\nmicrowave field\nJi-kun Xie, Sheng-li Ma,∗Ya-long Ren, Shao-yan Gao, and Fu-li Li\nMOE Key Laboratory for Non-equilibrium Synthesis and Modul ation of Condensed Matter,\nShaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices,\nSchool of Physics, Xi’an Jiaotong University, Xi’an 710049 , China\nUnidirectional photon emission is crucial for constructin gquantumnetworks and realizing scalable\nquantum information processing. In the present work an effici ent scheme is developed for the\nunidirectional emission of a tunable squeezed microwave fie ld. Our scheme is based on a chiral\ncavity magnonic system, where a magnon mode in a single-crys talline yttrium iron garnet (YIG)\nsphereisselectively coupledtooneofthetwodegeneratero tatingmicrowave modesinatorus-shaped\ncavity with the same chirality. With the YIG sphere driven by a two-color Floquet field to induce\nsidebands in the magnon-photon coupling, we show that the un idirectional emission of a tunable\nsqueezed microwave field can be generated via the assistance of the dissipative magnon mode and a\nwaveguide. Moreover, the direction of the proposed one-way emitter can be controlled on demand by\nreversing the biased magnetic field. Our work opens up an aven ue to create and manipulate one-way\nnonclassical microwave radiation field and could find potent ial quantum technological applications.\nI. INTRODUCTION\nChiral quantum optics has become a burgeoning field\ndue to its potential applications in quantum informa-\ntion processing and quantum networks [ 1–5]. It seeks\nto exploit new approaches and systems exhibiting chi-\nral light-matter interactions [ 6–16]. On this subject,\nnanophotonic systems, such as nanoscale waveguides\nand whispering-gallery-mode(WGM) resonators[ 17–19],\nhave emerged as attractive candidates, where the light\nfield is strongly transversely confined in a subwavelength\nspaceandexhibitstheopticalspin-orbitcoupling[ 20,21].\nThe chiral interaction can be achieved by coupling the\nspin-momentum-locked light to quantum emitters with\npolarization-dependent dipole transitions [ 22–26]. And a\nlarge number of chiral devices have been proposed the-\noretically and demonstrated experimentally, like single-\nphoton diodes [ 27,28], single-photon routing [ 29–31], cir-\nculators [ 28,32], isolators [ 32–34] and etc. These ele-\nments are key components for building large-scale quan-\ntum networks.\nIn parallel, the chiral light-matter interactionscan also\nbe utilized to control the direction of photon emission\n[35–38]. By designing the suitable chiral coupling be-\ntween quantum emitters and evanescent fields, the de-\nterministic and the highly directional photon emission\ncan be achieved along only one of the selected directions\n[17,22,38]. Recently, thechiral-waveguide-basedandthe\nchiral-cavity-based systems have been explored to gener-\nate the unidirectional emission of single photons [ 39–43].\nIn addition, the unidirectional laser emission has also\nbeen demonstrated through the non-Hermitian scatter-\ning induced chiralityin the WGM resonator[ 44,45]. The\nunidirectional photon emission can facilitate the free-\nspacecoupling ofenergy, improvethe collection efficiency\n∗msl1987@xjtu.edu.cnof weak optical signals and on-demand distribute quan-\ntum information [ 46–50].\nIn recent years, the hybrid quantum system of yttrium\niron garnet (YIG) and superconducting circuit has been\nconsidered as a powerful platform for quantum informa-\ntion applications [ 51–54]. Due to the high spin density\nand lowdamping rate ofthe ferrimagneticinsulator YIG,\nthe strong and even ultrastrong coupling between a mag-\nnetostatic mode in a YIG sphere and a microwave cavity\nmode have been experimentally observed [ 55–60]. Many\nintriguing phenomena have been studied in this hybrid\nsystem, such as the generation of various quantum states\n[61–72], nonreciprocity [ 73–75], non-Hermitian physics\n[76–78], and Floquet engineering [ 79–82]. Remarkably,\nmagnonsalsocarryangularmomentumor“spin”. Forin-\nstance, the Kittel mode magnetization precessescounter-\nclockwise around the applied magnetic field [ 83]. Anal-\nogous with the chiral coupling of spin-polarized atoms\nor quantum dots with spin-momentum-locked light [ 17–\n19], the Kittel mode can only couple to photons with\nthe same polarization in ring-shaped cavity or waveguide\n[84–88]. Therefore, this hybrid magnonic system can act\nas a promising architecture for studying chiral quantum\noptics [89–93].\nIn this paper, we develop a method for the unidirec-\ntional emission of a tunable squeezed microwave field\nbasedonachiralcavitymagnonicsystem. In ourscheme,\na torus-shaped microwave cavity that supports a pair\nof degenerate counter-rotating microwave modes is ex-\nploited to chirally coupled to a YIG sphere. The chiral\ninteraction can be achieved by locating the YIG sphere\non a special chiral line in the cavity, where the circular\npolarization of the cavity mode is locked to its propa-\ngation direction [ 86–90]. As a result, the Kittel mode\nwill only couple to one of the microwave modes with\nthe same polarization. By applying a two-color Floquet\ndriving field to the magnetic sphere to induce sideband\ntransitions, we can engineer a squeezing-type interaction\nbetween the magnon mode and the selected microwave2\nYIG\nwaveguide \nFIG. 1. (Color online) Schematic diagram of the hybridquan-\ntum setup. A torus-shaped cavity supports both CCW ( a)\nand CW ( b) rotating microwave modes, which are evanes-\ncently coupled to a nearby microwave waveguide. A small fer-\nromagnetic YIG sphere is placed inside the cavity and biased\nperpendicularly by a static magnetic field B0to establish the\nmagnon-photon coupling. Additionally, the magnon mode is\ndriven by a two-color Floquet driving field (not shown) along\nthe bias field direction.\nmode, which can be steered into a squeezed state with\nthe aid of large decay rate of the magnon mode. By\nfurther considering the cavity evanescently coupled to a\nmicrowave waveguide, we can generate a unidirectionally\nsqueezed microwave source. Notably, the emission direc-\ntion can be conveniently adjusted by reversing the bias\nmagneticfield, and the amountofsqueezingis tunable by\nchanging the external parameters. Moreover, the numer-\nical simulations indicate that the unidirectional emission\nis robust against the imperfect chiral coupling. Our work\ncould find a wide range of practical applications in quan-\ntum networks and quantum sensing.\nII. MODEL\nAs schematically shown in Fig. 1, the hybrid cavity\nmagnonic system under consideration consists of a torus-\nshaped microwave cavity with a small highly polished\nYIG sphere placed inside [ 87,88]. Due to the geomet-\nrical rotational symmetry, the cavity supports a pair of\ndegeneratecounterclockwise(CCW) andclockwise(CW)\nmicrowave modes, which are evanescently coupled to a\nnearbymicrowavewaveguide. Underastaticout-of-plane\nbias magnetic field at a strength B0, many magnetostatic\nmodes will be excited in the YIG sphere [ 55–57]. In our\nscheme, we focus only on the Kittel mode, which has\nthe uniform spin precessions in the whole volume of the\nmagnetic sphere. The two microwave modes are cou-\npled to the Kittel mode through the collective magnetic-\ndipoleinteraction,respectively. Now,wecangivethefree\nHamiltonianoftheKittelmode andthe cavitymodes(we\nset/planckover2pi1= 1 hereafter)\nH0=ωmm†m+ω0/summationdisplay\nα=a,bα†α. (1)Here,m(m†) is the boson operator of the magnon mode\nwith the resonance frequency ωm=γB0, where γ=\n2π×28 GHz/T is the gyromagnetic ratio. a(a†) and\nb(b†) represent the annihilation (creation) operators for\nthe CCW and CW microwave modes with the resonance\nfrequency ω0.\nWe then consider the chiral photon-magnon interac-\ntion, which takes the form [ 86–90]\nHint=/summationdisplay\nα=a,bgα(α†+α)(m†+m) (2)\nwithgα(α=a,b) describing the coupling strength be-\ntweenthe cavitymode αand the magnonmode m. To be\nspecific, the Kittel mode magnetization precesses around\nthe effective magnetic field in an anticlockwise manner\nand couples preferentially to photons with the same po-\nlarization. Here, the magnetic field of each TE mode is\ntransversallyconfinedinthecavity,sothatithasastrong\nlongitudinal polarization component along the propaga-\ntion direction [ 1,29,94]. Moreover, the longitudinal and\ntransverse polarization components oscillate ±90◦out of\nphase with each other, where + ( −) sign depends on the\npropagationdirection of the cavity modes. At the special\nradial locations, the magnetic field can be circularly po-\nlarized when the longitudinal and transversepolarization\ncomponents have the same amplitudes. Also, since the\npolarization direction is locked to the sign of their lin-\near momentum, the counterpropagated CW and CCW\nmodes can thereby possess mutually orthogonal polar-\nizations and opposite chiralities [ 86]. As a result, the\nmagnon mode can only couple to one of them with the\nsame chirality, i.e., ga/ne}ationslash= 0 and gb= 0, or gb/ne}ationslash= 0 and\nga= 0.\nIn addition, the chiral coupling can be well controlled\nbyreversingthedirectionofthebiasedmagneticfield, be-\ncause it is accompanied with the reversion of the preces-\nsionoftheKittelmode. Alternatively,eachofthetwode-\ngenerate counter-propagatingmicrowavemodes can have\nopposite chirality at different radial locations [ 87,88], so\nthe chiral coupling can also be controlled by shifting the\nmagnet position.\nFurthermore, a two-color Floquet driving field is ap-\nplied to the YIG sphere along the bias magnetic field\ndirection, under which the oscillating frequency of the\nmagnon mode is modulated accordingly. So the coupling\nbetween the Floquet driving field and the Kittel mode is\ndescribed by [ 79–82]\nHd(t) = [Ω 1cos(ωd1t)+Ω2cos(ωd2t+θ)]m†m,(3)\nwhere Ω j(j= 1,2) denotes the driving amplitudes with\nthedrivingfrequencies ωdj,θistherelativephase. Exper-\nimentally,thismanipulationcanbeimplementedthrough\na small coil being looped tightly around the magnetic\nsphere to modulate the bias magnetic field [ 79]. Under\nthis kind of periodical modulation, the desired paramet-\nric magnon-photon interaction can be sculpted to realize\nthe one-way squeezed microwave source.3\nFIG. 2. (Color online) Schematic diagram of sideband tran-\nsitions of the coupled magnon-photon modes, where ωd1=\nωm−ω0andωd2=ωm+ω0correspond to the red and blue\nsidebands, respectively.\nIII. UNIDIRECTIONAL EMISSION OF A\nTUNABLE SQUEEZED MICROWAVE FIELD\nIn this section, we will detail the procedure for unidi-\nrectionally creating a tunable squeezed microwave field\nbased on above introduced chiral cavity magnonic sys-\ntem. The chirality stems from the selective coupling of\nthe magnon mode to one of the two degenerate rotating\nmicrowavemodeswiththesamepolarization. Combining\nwith the Floquet driving to engineer a desired squeezing-\ntype interaction, we can generate a chiral squeezed mi-\ncrowave radiation under the assistance of a dissipative\nmagnon mode and a waveguide.\nA. The effective Hamiltonian for the generation of\na chiral squeezed state\nWe start to derive the effective Hamiltonian to selec-\ntively generate a chiral squeezed state; that is, only one\nof the two cavity modes will be squeezed. To this end,\nthe Floquet driving to the magnonmode plays a keyrole.\nWe define the following rotating transformation:\nU(t) =Texp{−i/integraldisplayt\n0[H0+Hd(τ)]dτ}\n= exp{−i(ωmm†m+ω0/summationdisplay\nα=a,bα†α)t\n−i[ξ1sin(ωd1t)+ξ2sin(ωd2t+θ)]m†m},(4)\nwhereTis the time order operator and ξj= Ωj/ωdj(j=\n1,2). Inthe rotatingframewith respectto U(t), the total\nHamiltonian Ht=H0+Hint+Hd(t) of the system will\nbecome\nHI=U†(t)HtU(t)+idU†(t)\ndtU(t)\n=/summationdisplay\nα=a,bgαm†ei[ωmt+ξ1sin(ωd1t)+ξ2sin(ωd2t+θ)]\n×(αe−iω0t+α†eiω0t)+H.c. (5)By using the Jacobi-Anger identity\neixsinφ=n=∞/summationdisplay\nn=−∞Jn(x)einφ(6)\nwith the nth Bessel functions of the first kind Jn(x), the\nHamiltonian can be rewritten as\nHI=/summationdisplay\nα=a,bgαm†eiωmt∞/summationdisplay\nn=−∞Jn(ξ1)einωd1t∞/summationdisplay\nk=−∞Jk(ξ2)\n×eik(ωd2t+θ)(αe−iω0t+α†eiω0t)+H.c. (7)\nTo produce the desired magnon-photon coupling, as\ndisplayed in Fig. 2, we choose the Floquet driving fre-\nquencies satisfying ωd1=ωm−ω0andωd2=ωm+ω0,\nwhich correspond to the red and blue sideband transi-\ntions, respectively. Furthermore, under the condition\ngα≪ {ωdj,ωm,ω0}, we can only keep the resonant terms\nin Eq. (7), but safely discard those fast oscillating terms\nby means of the rotating-wave approximation. Then, we\ncan obtain the effective Hamiltonian of the system\nHeff=/summationdisplay\nα=a,bGαm†(α+εe−iθα†)+H.c., (8)\nwhere we have Gα=gαJ−1(ξ1)J0(ξ2), and\nεe−iθ=J0(ξ1)J−1(ξ2)e−iθ/J−1(ξ1)J0(ξ2) is the ra-\ntio of the parametric-amplifier interaction to the\nJaynes-Cummings-type coupling. In what follows, we\nexclusively consider the situation of |ε|<1, which\nensures that Eq. ( 8) is stable. It is now clear that Heff\ndescribes a squeezing-type interaction, and can be used\nto produce the squeezed state. However, due to the\nchiral coupling, i.e., Ga/ne}ationslash= 0 and Gb= 0, orGb/ne}ationslash= 0 and\nGa= 0, we can only generate a squeezed state in one of\nthe two cavity modes.\nBy introducing a hybridized mode A= (Gaa+\nGbb)//radicalbig\nG2a+G2\nb, the Hamiltonian in Eq. ( 8) can be\nrewritten as\nHeff=/radicalBig\nG2a+G2\nb[m†(A+εe−iθA†)+H.c.].(9)\nTo clearly reveal the mechanism for generating the chiral\nsqueezed state, we perform the unitary squeezing trans-\nformation HS=S†\nA(ζ)HeffSA(ζ) [95], where SA(ζ) =\nexp[(ζA2−ζ∗A†2)/2] represents the single-mode squeez-\ningoperatorwith ζ=reiθ. Hererissqueezingparameter\ndefined by tanh r=εandθdenotes the squeezing angle,\nboth of which can be controlled on-demand by adjust-\ning the two-color Floquet driving field. In the squeezed\nframe, the effectiveHamiltonian ( 9) is transformedto the\nform\nHS=/radicalBig\n(G2a+G2\nb)(1−ε2)(m†A+mA†).(10)\nObviously, HSdescribes a beam-splitter interaction. If\nwe setGa/ne}ationslash= 0 and Gb= 0 (Gb/ne}ationslash= 0 and Ga= 0) via the4\nselective coupling rule, the mode A=a(A=b) can be\ncooled down to the vacuum state in the squeezed picture\nvia the dissipation of the magnon mode. Reversing the\nsqueezing transformation, the cavity mode a(b) is actu-\nally steered into a single-mode squeezed state. In this\nway, a chiral squeezed state is created. By further con-\nsidering the microwave cavity coupled to a waveguide,\nthe squeezed output field can be emitted along the right\n(left) direction of the waveguide. This is the basic idea\nfor the unidirectional emission of a squeezed source.\nB. Unidirectional squeezing emission\nWe now study the unidirectional squeezing emission\nby considering the coupling of the microwave cavity cou-\npled to a transmission line that acts the role of a waveg-\nuide. According to the Hamiltonian ( 8) and the stan-\ndardinput–outputtheory, thegeneralquantumLangevin\nequations of the system is given by\n˙α=−κ\n2α−iGα(m+εe−iθm†)+√κexαin,ex\n+√κ0αin,0, (11a)\n˙m=−γm\n2m−i/summationdisplay\nα=a,bGα(α+εe−iθα†)+√γmmin.(11b)\nWithout loss of generality, the cavity modes aandbare\nassumed to have the same damping rate κ=κ0+κex,\nwhereκ0is the intrinsic decay rate, and κexdenotes\nthe external coupling between the cavity mode and the\nwaveguide. In addition, αin,0andαin,ex(α=a,b) repre-\nsent the noise operators. minis the noise operator of the\nmagnon mode, and γmis the associated damping rate.\nThese noise operators obey the following correlation re-\nlations\n/angbracketleftBig\nαin,ex(t)α†\nin,ex(t′)/angbracketrightBig\n=δ(t−t′),(12a)\n/angbracketleftBig\nαin,0(t)α†\nin,0(t′)/angbracketrightBig\n=δ(t−t′),(12b)/angbracketleftBig\nmin(t)m†\nin(t′)/angbracketrightBig\n=δ(t−t′), (12c)\nwhere we have neglected the thermal effects. Because,\nat the experimental working temperature T= 20 mK,\nthe average thermal excitation number of a boson mode\nwith resonance frequency 6 .5 GHz is about 10−7, which\nis thus negligible.\nBy introducing the Fourier transformation o(t) =/integraltext+∞\n−∞o(ω)e−iωtdω/2πfor an arbitrary operator o, we can\nrewritethe Eqs. ( 11a)and(11b) in thefrequency domain\nas\nα(ω) =1\nκ\n2−iω{−iGα[m(ω)+εe−iθm†(−ω)]\n+√κexαin,ex(ω)+√κ0αin,0(ω)},(13a)\nm(ω) =1\nγm\n2−iω{−i/summationdisplay\nα=a,bGα[α(ω)\n+εe−iθα†(−ω)]+√γmmin(ω)}.(13b)\nCorrespondingly, the correlation functions of Eqs. ( 12a)-\n(12c) in the frequency domain yield\n/angbracketleftBig\nαin,ex(ω)α†\nin,ex(−ω′)/angbracketrightBig\n= 2πδ(ω+ω′),(14a)\n/angbracketleftBig\nαin,0(ω)α†\nin,0(−ω′)/angbracketrightBig\n= 2πδ(ω+ω′),(14b)\n/angbracketleftBig\nmin(ω)m†\nin(−ω′)/angbracketrightBig\n= 2πδ(ω+ω′).(14c)\nOur interest is the output fields of the cavity. Hence, in\nterms of the standard input-output relation\nαout,ex=√κexα−αin,ex, (15)\nwecanderiveoutthefrequencycomponentsoftheoutput\nfield operators\naout,ex(ω) = [Nb(ω)κex\nD(ω)−1]ain,ex(ω)+1\nD(ω){Nb(ω)√κexκ0ain,0(ω)+8GaGb(1−ε2)\n×[κexbin,ex(ω)+√κexκ0bin,0(ω)]+4iGaκ−√κexγm[min(ω)+εe−iθm†\nin(−ω)]},(16a)\nbout,ex(ω) = [Na(ω)κex\nD(ω)−1]bin,ex(ω)+1\nD(ω){Na(ω)√κexκ0bin,0(ω)+8GaGb(1−ε2)\n×[κexain,ex(ω)+√κexκ0ain,0(ω)]+4iGbκ−√κexγm[min(ω)+εe−iθm†\nin(−ω)]}(16b)\nwithκ−=κ−2iω,γ−=γm−2iω,Nα(ω) = 8G2\nα(ε2−1)−2κ−γ−andD(ω) = 4κ−(ε2−1)(G2\na+G2\nb)−κ2\n−γ−.\nNow, the spectrum of the output microwave field for the αmode is given by\nSα,out(ω) =1\n4π/integraldisplay+∞\n−∞dω′e−i(ω+ω′)t[/angbracketleftbig\nXθ\nα,out(ω)Xθ\nα,out(ω′)/angbracketrightbig\n+/angbracketleftbig\nXθ\nα,out(ω′)Xθ\nα,out(ω)/angbracketrightbig\n], (17)5\nwhereXθ\nα,out(ω) =αout,ex(ω)eiθ\n2+α†\nout,ex(−ω)e−iθ\n2is the associated quadrature operator of the output fields. Sub-\nstituting Eqs. ( 16a) and (16b) into Eq. ( 17), we can obtain the squeezing spectra of the output fields\nSa,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleNb(ω)κex\nD(ω)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleNb(ω)√κexκ0\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)κex\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)√κexκ0\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gaκ−(1−ε)√κexγm\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (18a)\nSb,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleNa(ω)κex\nD(ω)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingleNa(ω)√κexκ0\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)κex\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)√κexκ0\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gbκ−(1−ε)√κexγm\nD(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n. (18b)\nTo quantify the amount of squeezing of the output\nfields, we now define the quantum noise reduction\nFα,out(ω) =−10log10[Sα,out(ω)/Svac\nα,out(ω)] in dB units\nwithSvac\nα,out(ω) = 1 being the squeezing spectrum of the\nvacuum state. Note that if the output field is squeezed,\nthe quantum noise reduction Fα,out(ω) is larger than\nzero. Clearly, to achieve a perfect one-way squeezing\nemitter, the condition Fa,out(ω)>0 andFb,out(ω) = 0\n(Fb,out(ω)>0 andFa,out(ω) = 0) should be satisfied,\nwhich indicates that the squeezed field is only emitted\nalong the right (left) direction of the waveguide.\nTo well grasp the main result of this work, we start\nour analysis with the ideal chiral coupling, i.e., ga/ne}ationslash= 0\nandgb= 0. Then, the output squeezing spectra in Eqs.\n(18a) and (18b) yield\nSa,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleκ−γ−(2κex−κ−)−4κ−(1−ε2)G2\na\nκ2\n−γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle2κ−γ−√κexκ0\nκ2\n−γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gaκ−(ε−1)√κexγm\nκ2\n−γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,(19a)\nSb,out(ω) = 1, (19b)\nIt is now clear that the spectra Sa,out(ω)<1 and\nSb,out(ω) = 1 can be satisfied under the situation of per-\nfect chirality, which is equivalent to Fa,out(ω)>0 and\nFb,out(ω) = 0. So, a desired one-way squeezed source\ncan be created; that is, the squeezed microwave field can\nonly be emitted in one direction. To exhibit the unidi-\nrectional squeezing emission, we choose ε= 0.95, and\nshowFa,out(ω) andFb,out(ω) versus the dimensionless\nfrequency ω/γmin Fig. 3. It is observed that Fa,out(ω)\nis larger than zero within a large frequency range, where\nthe largest value of Fa,out(ω) is about 13 .45 dB at the\ncentral resonance frequency. On the contrary, Fb,out(ω)\nis zero over the whole frequency range. This means that\nwe can obtain a squeezed output field in the right side\nof the waveguide, but there is no squeezing of the output(a) (b)\nFIG. 3. (Color online) The squeezing spectra Fa,out(ω) [(a)]\nandFb,out(ω) [(b)] versus the dimensionless frequency ω/γm\nunder the different coupling ratios gb/ga= 0, 0.05, 0.1. The\nrelated parameters are chosen as Ga/γm= 2.5,ε= 0.95,\nκ0/γm= 0.05,κex/γm= 2.5 andγm/2π= 2 MHz.\nfield in the left side of the waveguide. Thus, the unidi-\nrectional emission of a squeezed source can be achieved.\nNotably, we can readily switch the emission direction of\nthe squeezed field by reversing the direction of the exter-\nnal biased magnetic field.\nIn practice, our system will inevitably suffer from the\nnon-ideal chiral coupling, i.e., ga/ne}ationslash= 0 and gb/ne}ationslash= 0, such\nthat that the completely one-way squeezing emission will\nbe spoiled. In the presence of non-ideal chiral coupling,\nwe know from Eq. ( 10) that the hybridized mode Awill\nbe steered into a squeezed state via the dissipation of\nthe magnon mode m, i.e., both the modes aandbare\nthereby squeezed. As a result, the squeezed fields will be\nemitted in both directions of the waveguide. With the\nincrease of gb, the associated output squeezing Fa,out(ω)\nwill be reduced, while the output squeezing Fb,out(ω) is\ngradually increased. Nonetheless, for gb/ga= 0.1, the\nvalue of Fa,out(ω) only has a negligible reduction, and\nthe maximal value of Fb,out(ω) is about 0 .04 dB [See Fig.\n3]. So, our scheme is robust against the imperfect chiral\ncoupling, indicating that we can still implement a high-\nperformance unidirectional squeezing emitter.\nFinally, we emphasize here that the proposed one-way\nemitter with a tunable amount of squeezing can also be6\n(a) (b)\nFIG. 4. (Color online) The squeezing degree Fa,out(0) [(a)]\nandFb,out(0) [(b)] versus the parameter εunder the different\ncoupling ratios gb/ga= 0, 0.05, 0.1. The other parameters\nare the same as in Fig. 3.\ngenerated in our scheme. Fig. 4 displays the Fa,out(0)\nandFb,out(0) versus the parameter ε. We can see that\nFa,out(0)behavesasanonmonotonicfunctionof ε,imply-\ning that a tunable squeezed output field can be created\nby tuning the parameter ε. This can be implemented by\nadjusting the Floquet driving parameters Ω j(j= 1,2).\nAs presented in Fig. 4(a), there exists an optimum\nvalue of ε, where a maximal amount of output squeez-\ning is produced. To illustrate this point, we recall that\nthe intra-cavity squeezing depends on the parameter ε.\nMeanwhile, according to the standard input-output re-\nlation, the intra-cavity squeezing determines the output\none. So, the output squeezing is also determined by the\nparameter ε. Based on Eqs. ( 9) and (10), we know that\nthe dissipative magnon mode can be used to cool the\ncavity mode down to a squeezed state. So, it seems like\nthat the intra-cavity squeezing will be continuously in-\ncreased with the increase of ε. However, as the param-\neterεincreasing, the effective magnon-photon coupling,\nwhich plays the role of cooling, is gradually decreased in\nthe squeezed representation [See Eq. ( 10)]. Besides, thecavity vacuum noise is transformed to the effective ther-\nmal noise, which is also be amplified. As a result, the\ncavity can not approach a squeezed vacuum state, which\nis actually a thermal squeezed state. Therefore, the com-\npetition ofthese two processesleads to an optimum value\nofεthat can generate a peak of output squeezing.\nIV. CONCLUSION\nIn summary, we have presented an approach for the\nunidirectional emission of a tunable squeezed microwave\nfield in a chiral cavity magnonic system of a YIG sphere\nand a torus-shaped cavity. The chirality comes from\nthe selective coupling between the Kittel mode to one of\nthe two counter-propagating microwave modes with the\nsame polarization. Under a bichromatic Floquet driv-\ning field to induce sidebands, the unidirectional emission\nof a tunable squeezed microwave source can be gener-\nated with the help of a dissipative magnon mode and a\nwaveguide. 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Onbasli2, Caroline A. Ross2**, Burkard Hillebrands1**, and Andrii V. Chumak1*** 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA * Present address: Univ. Grenoble Alpes, CNRS, CEA, INAC-SPINTEC, 17, rue des Martyrs 38054, Grenoble, France ** Fellow, IEEE *** Senior Member, IEEE Abstract— The temporal evolution of pulsed Spin Hall Effect – Spin Transfer Torque (SHE-STT) driven auto-oscillations in a Yttrium Iron Garnet (YIG) | platinum (Pt) microdisc is studied experimentally using time-resolved Brillouin Light Scattering (BLS) spectroscopy. It is demonstrated that the frequency of the auto-oscillations is different in the center and at the edge of the investigated disc that is related to the simultaneous STT excitation of a bullet and a non-localized spin-wave mode. Furthermore, the magnetization precession intensity is found to saturate on a time scale of 20 ns or longer, depending on the current density. For this reason, our findings suggest that a proper ratio between the current and the pulse duration is of crucial importance for future STT-based devices. Index Terms— Spin Transfer Torque, Spin Hall Effect, Spin Electronics, Magnetodynamics, Nanomagnetics I. INTRODUCTION The Spin-Transfer Torque (STT) effect [Slonczewski 1996, Berger 1996], which is induced by the Spin Hall Effect (SHE) [Dyakonov 1971, Hirsch 1999] in a heavy metal layer adjacent to a magnetic layer, attracts attention since it can be used for the compensation of magnetization precession damping [Ando 2008, Demidov 2011, Hamadeh 2014, Lauer 2016A] as well as for the excitation of magnetization auto-oscillations [Kajiwara 2010, Demidov 2012, Collet 2016, Demidov 2016] driven by a direct current (see also the review by Chumak et al. [2015]). First successful experiments on the excitation of auto-oscillations were performed on patterned all-metallic bilayers of NiFe/Pt [Demidov 2012] and, later, also on bilayers of the ferrimagnetic insulator Yttrium Iron Garnet (YIG) and Pt [Collet 2016, Demidov 2016]. YIG is known for its very low Gilbert damping and, thus, is important for fundamental research in magnonics and for potential future applications [Serga 2010, Chumak 2015]. Up to now, experiments have been typically performed by applying continuous direct currents to the Pt layer. To the best of our knowledge, only in the experiments of Demidov et al. [2011] the temporal behavior of SHE-STT-driven spin dynamics have been investigated so far. However, the time resolution of 20 ns, which was achieved in these experiments, is larger than the magnon life-time in metallic structures. Thus, it could not provide insight into a very important Corresponding author: Andrii V. Chumak (chumak@physik.uni-kl.de). regime of STT-driven dynamics. In particular, the question how fast the dynamical equilibrium of the auto-oscillations is reached is still open. This is an important aspect since pulsed excitations of the magnetization precession using a pulse duration of a few nanoseconds or shorter are more realistic to the working regime of future spintronic devices. Therefore, time-resolved investigations of the onset of auto-oscillations with a time resolution on the ns-scale are demanded. Here, we address the experimental investigation of pulsed SHE-STT-driven auto-oscillations (see the schematic of the effects in Fig. 1(a)) in a YIG/Pt microdisc by using time-resolved Brillouin Light Scattering (BLS) spectroscopy [Sebastian 2015] measurements with a time resolution of down to 1 ns. The focus lies on the temporal evolution of the spin dynamics in the microstructure as soon as the anti-damping STT overcompensates the intrinsic Gilbert damping in the system. It is found that the magnetization precession amplitude saturates on a time scale of a few tens of ns depending on the particular current density. Furthermore, both, the maximum intensity and the saturation time, saturate with increasing driving current. Fig. 1. (a) Configuration of the biasing field, the applied charge and the SHE-STT-generated spin current densities (jc and js) for the exertion of anti-damping STT on the magnetization in the YIG layer. (b) Sketch of the investigated YIG/Pt disc with tapered Au leads on top. The charge current is applied perpendicular to the biasing field. (c) Current density in the Pt disc calculated using the COMSOL Multiphysics® software. Only the in-plane component of the current that is perpendicular to the biasing magnetic field is taken into account. Right panel shows the cross-section of the current density distribution along the dashed line shown on top of the color map. II. STRUCTURE UNDER INVESTIGATION AND METHODOLOGY The probed YIG/Pt microdisc has a diameter of 1 µm, and the layer thicknesses of YIG and Pt are dYIG = 20 nm and dPt = 7 nm, respectively – see Fig. 1(b). In order to fabricate the microstructures, a YIG film was grown by pulsed laser deposition on a gadolinium gallium garnet substrate [Onbasli 2014]. Subsequently, after a conventional cleaning in an ultrasonic bath and a treatment of the YIG surface by an oxygen plasma [Jungfleisch 2013], a Pt film was deposited on top by means of sputter deposition. Microwave-based ferromagnetic resonance measurements yielded a Gilbert damping parameter of αYIG = 1.2·10-3 for the YIG film before the Pt deposition, and an increased damping value of αYIG/Pt = 5.7·10-3 for the YIG/Pt bilayer. The subsequent structuring of the microdisc was achieved by electron-beam lithography and argon-ion milling [Pirro 2014]. Eventually, tapered Au leads on top of the microdisc with a spacing of 50 nm (see Fig. 1(b)) were patterned using electron-beam lithography and physical vapor deposition to allow for the application of high current densities in the center of the disc. The total resistance of the structure under investigation is around 15 Ohm. Since the current density in the Pt disk is not uniform, a numerical simulation was performed using the COMSOL Multiphysics® software. A Pt conductivity value of 2.5·106 S/m was used in these calculations. The so obtained current density distribution is shown in Fig. 1(c). The in-plane density component that is perpendicular to the biasing field and, thus, contributes to the STT, is plotted in the right panel of Fig. 1(c) as a function of the coordinate along the disc as indicated by the white dashed line on top of the color map. One can see that the density shows a pronounced maximum in the disc between the nano-contacts. Unless otherwise stated, the current density values jc shown below represent the calculated density maximum in the disc center by taking into account the particular applied voltage, the resistance of the structure, and the density distribution in Fig. 1(c). Time-resolved BLS measurements were performed by using a probing laser with a wavelength of 491 nm and a power of 2 mW, focused down to a laser-spot diameter of approximately 400 nm on the structure. In the experiment, 75 ns long dc pulses having 5 ns rise and fall times are applied to the Au leads with a repetition period of 500 ns that result in a corresponding charge current flowing in the Pt layer of the microdisc. An external biasing field of µ0Hext = 110 mT magnetizes the microdisc perpendicular to the current flow direction. An exemplary configuration of the biasing field Hext relative to the charge current density jc, and the SHE-STT-generated spin current density js are depicted in Fig. 1(a) for the case of a resulting anti-damping STT on the magnetization in the YIG layer [Schreier 2015]. It should be mentioned that, in the present experiment, a threshold-like onset of auto-oscillations for a given field polarity is observed above a critical current density of jc,crit = 1.09·1012 A·m-2 only for the current direction which is expected to generate an anti-damping STT according to the theory of the SHE. Such a behavior is consistent with other experimental findings as, e.g., shown by Demidov et al. [2012, 2016], Lauer et al. [2016A], and Collet et al. [2016]. Moreover, these observations prove that, unlike in [Safranski 2016, Lauer 2016B], the auto-oscillations cannot originate from the spin Seebeck effect (SSE) due to a thermal gradient, since the SSE is known to excite magnetization precession for both current orientations. In our case, the highly heat-conducting Au leads on top of the Pt layer act as heat sinks and prevent the formation of sufficiently large thermal gradients required for triggering of auto-oscillations due to the SSE. Moreover, the whole structure was covered by a 220 nm thick layer of SiO2 that acts as an additional heat sink and reduces the influence of the sample heating on the studied phenomena. III. EXPERIMENTAL RESULTS Figure 2(a) shows the temporal evolution of the frequency-integrated intensity of the excited magnetization precession represented by the BLS intensity detected in the center of the microdisc for different applied current densities above the critical density value. A dynamic state is apparently excited during the pulse duration of 75 ns illustrated by the shaded areas in the graphs. We find that for current densities higher than 1.58·1012 A·m-2, the BLS intensity saturates within the pulse duration. Nonlinear magnon scattering processes are assumed to limit a further increase (the interplay between different spin-wave modes excited by the SHE-STT will be reported elsewhere). It is noteworthy that the saturation level is also a function of the applied current density (see Fig. 2(b)). Furthermore, the saturation time (as indicated by the dashed lines in Fig. 2(a)) is plotted in Fig. 2(c). This saturation time drops with the applied current and saturates at a value of approximately 23 ns at high currents. \n \n Fig. 2. (a) Frequency-integrated BLS intensity detected in the center of the microdisc as a function of time for different applied current densities above the threshold. The pulse duration of 75ns is depicted by the shaded areas. (b) The maximum BLS intensity reached during the pulse as a function of the applied current density. (c) The saturation time of the BLS intensity (indicated by dashed lines in (a)) within the pulse duration before reaching saturation as a function of the applied current density. Our findings suggest that the signal output of pulsed auto-oscillations may be optimized with respect to energy consumption by choosing a proper ratio between operating current and pulse duration (compare Figs. 2(b) and (c)). In particular, the importance of the appropriate current value is emphasized by the temporal behavior observed at rather low and rather high currents. For applied voltages that correspond to the current densities below 1.6·1012 A·m-2, which are still above the threshold shown in the upper left graph of Fig. 2(a), the pulse duration is too short to reach saturation. On the other hand, for high voltages, e.g., that correspond to the current density 3.5·1012 A·m-2 the BLS signal slowly drops within the pulse duration after reaching its maximum after 23 ns, as shown in the lower right graph of Fig. 2(a). This intensity decrease over time within the pulse duration is assumed to result from a decrease in the spin-mixing conductance due to the increase in the total temperature in the microdisc [Uchida 2014], which consequently leads to a reduced injection of the SHE-generated spin current, and to a reduction of the anti-damping STT. Thus, in the integrated structure, the pulse duration should not fall below 23 ns and the optimal operating current density is about 2.1·1012 A·m-2 in the center of the disc. These crucial features need to be considered for potential spintronics applications based on pulsed SHE-STT-driven nano-oscillators. In order to investigate the spatial distribution of SHE-STT-driven spin dynamics, a linescan at an applied current density of 2.1·1012 A·m-2 is performed across the disc through the center in the direction perpendicular to the current flow. The BLS intensity integrated during the pulse over all investigated BLS frequencies is plotted in Fig. 3(a) as a function of the position along the disc. It shows a maximum in the disc center, and moderate intensities at the disc edges. Taking into account the current density distribution shown in Fig. 1(c) and the threshold current density of 1.09·1012 A·m-2, we conclude that the density of the current at the edges of the disc is not high enough to reach the threshold of auto-oscillations. Nevertheless, the magnetization precession is detected also at the edges of the disc suggesting that a non-localized spin-wave mode is excited in the whole disc by the high current densities in the center. Additionally, the finite size of the laser spot, which is equal to almost half of the disc diameter, should be taken into account and influences the data shown in Fig. 3(a). In order to better understand the magnetization dynamics in the disc, we have performed frequency-dependent measurements of the signal detected by BLS spectroscopy in the center of the disc and at its edge. Figure 3(b) shows the BLS intensity time-integrated over the 75 ns long pulse as a function of the BLS frequency at different positions on the disc. Please note that the spectra shown in Fig. 3(b) are the result of the subtraction of the spectra with and without applied dc pulses and, thus, all points having intensity larger than zero are associated with STT-based generation of magnetization precession. One can see, that the linewidths of the generated frequency peaks are much larger than the frequency resolution of our BLS setup which is around 50 MHz. Furthermore, the BLS frequency of maximum intensity is lower in the disc center than at the disc edge. We associate this with a simultaneous SHE-STT excitation of at least two modes in the disc that can be simultaneously maintained during the pulse. In the center, a so-called bullet mode is excited that has a solitonic nature and spreads over an area of a few tens of nanometers [Demidov 2012]. This mode is known to have frequencies smaller than the ferromagnetic resonance frequency and is strongly localized in the disc center. Simultaneously, a non-localized mode of higher frequency distributed over the whole disc is excited [Collet 2016, Jungfleisch 2016]. The intensity of this mode is comparable in the disc center and at the edge – see vertical scales in Fig. 3(b). However, the intensity of the bullet mode in the disc center is larger than the intensity of the non-localized mode. \n Fig. 3: (a) Linescan perpendicular to the current flow direction showing the time- and frequency-integrated BLS intensity. (b) BLS intensity integrated over the pulse duration as a function of the BLS frequency, normalized to the BLS intensity when the pulse is switched off. The solid lines are Lorentzian fits. \nThe fact that we see a multi-mode generation, while most of the previously reported studies demonstrate a single-mode generation, leads us to the conclusion that only the application of SHE-STT for a relatively short time period allows for the simultaneous excitation of multiple modes. An application of electric current to the Pt layer for a longer time will probably result in the appearance of nonlinear mode competition mechanisms and in the subsequent survival of only one mode. IV. CONCLUSION In summary, we performed time-resolved BLS measurements to investigate the onset of pulsed SHE-STT-driven magnetization auto-oscillations in a YIG/Pt microdisc. The BLS intensity is found to saturate on a time scale of 25 ns or longer, dependending on the particular current density which originates from the applied voltage. Furthermore, both the maximum intensity and the saturation time saturate with increasing operating voltage, which we associate with nonlinear magnon-magnon interaction in the system. For this reason, our findings suggest that a proper ratio between the voltage and the pulse duration is of crucial importance for the power consumption of potential devices based on pulsed auto-oscillations. It was demonstrated that the peak frequency of the SHE-STT-excited magnetization is different in the disc center and at its edges. This is associated with the simultaneous STT excitation of a bullet mode and a non-localized spin-wave mode in the system. 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" }, { "title": "2002.00003v1.Temperature_dependence_of_spin_pinning_and_spin_wave_dispersion_in_nanoscopic_ferromagnetic_waveguides.pdf", "content": "TEMPERATURE DEPENDENCE OF SPIN PINNING AND SPIN-WAVE DISPERSION IN NANOSCOPIC FERROMAGNETIC WAVEGUIDES\n1B. HEINZ*,1, 2Q. WANG*,1R. VERBA,3V. I. VASYUCHKA,1M. KEWENIG,1P.\nPIRRO,1M. SCHNEIDER,1T. MEYER,1, 4B. L¨AGEL,5C. DUBS,6T. BR¨ACHER,1\nO. V. DOBROVOLSKIY,7A. V. CHUMAK**1, 8\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at\nKaiserslautern\n(Erwin-Schr¨ odinger-Stra ße 56, 67663 Kaiserslautern, Germany)\n2Graduate School Materials Science in Mainz\n(Staudingerwerg 9, 55128 Mainz, Germany)\n3Institute of Magnetism\n(Vernadskogo blvd. 36-b, 03142 Kyiv, Ukraine)\n4THATec Innovation GmbH\n(Augustaanlage 23, 68165 Mannheim, Germany)\n5Nano Structuring Center, Technische Universit¨ at Kaiserslautern\n(Erwin-Schr¨ odinger-Stra ße 13, 67663 Kaiserslautern, Germany)\n6INNOVENT e.V., Technologieentwicklung\n(Pr¨ ussingstra ße 27B, 07745 Jena, Germany)\n7Physikalisches Institut, Goethe-Universit¨ at Frankfurt\n(Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany)\n8Nanomagnetism and Magnonics, Faculty of Physics, University of Vienna\n(Boltzmanngasse 5, A-1090 Wien, Austria)\nTemperature dependence of spin pinning and spin-wave\ndispersion in nanoscopic ferromagnetic waveguides\nThe field of magnonics attracts significant attention due to the possibility of utilizing informa-\ntion coded into the spin-wave phase or amplitude to perform computation operations on the\nnanoscale. Recently, spin waves were investigated in Yttrium Iron Garnet (YIG) waveguides\nwith widths ranging down to 50 nm and aspect ratios thickness over width approaching unity.\nA critical width was found, below which the exchange interaction suppresses the dipolar pin-\nning phenomenon and the system becomes unpinned. Here we continue these investigations\nand analyse the pinning phenomenon and spin-wave dispersions as a function of temperature,\nthickness and material of choice. Higher order modes, the influence of a finite wavevector\nalong the waveguide and the impact of the pinning phenomenon on the spin-wave lifetime\nare discussed as well as the influence of a trapezoidal cross section and edge roughness of\nthe waveguides. The presented results are of particular interest for potential applications in\nmagnonic devices and the incipient field of quantum magnonics at cryogenic temperatures.\nKeywords : spin waves, yttrium iron garnet, Brillouin light scattering spectroscopy, low\ntemperatures\nc○B. HEINZ*, Q. WANG*, R. VERBA,\nV. I. VASYUCHKA, M. KEWENIG, P. PIRRO, M.\nSCHNEIDER, T. MEYER, B. L ¨AGEL, C. DUBS, T.\nBR¨ACHER, O. V. DOBROVOLSKIY,\nA. V. CHUMAK**arXiv:2002.00003v1 [physics.app-ph] 31 Jan 2020B. Heinz et al.\n1. Introduction\nThefieldofmagnonicsproposesapromisingapproach\nfor a novel type of computing systems, in which\nmagnons, the quanta of spin waves, carry the infor-\nmation instead of electrons [1–12]. Since the phase\nof a spin wave provides an additional degree of free-\ndom efficient computing concepts can be used result-\ning in a valuable decrease in the footprint of logic\nunits. Moreover, the scalability of magnonic struc-\ntures down to the nanometer scale and the possibility\nto operate with spin waves of nanometer wavelength\nare additional advantages of the magnonics approach.\nThe further miniaturization will, consequently, result\nin an increase in the frequency of spin waves used in\nthe devices from the currently employed GHz range\nup to the THz range. In classical magnonics, spin-\nwave modes in thin films or rather planar waveguides\nwith thickness-to-width aspect ratios 𝑎r=ℎ/𝑤≪1\nhave been utilized. In the case of a waveguide, edge\nmagnetostatic charges arise, which can be accounted\nfor by the introduction of boundary conditions [13].\nTherefore thin waveguides demonstrate the effect of\n\"dipolar pinning\" at the lateral edges, and for its the-\noretical description the thin strip approximation was\ndeveloped, in which only pinning of the much-larger-\nin-amplitudedynamicin-planemagnetizationcompo-\nnent is taken into account [14–19].\nThe recent progress in fabrication technology leads\nto the development of nanoscopic magnetic devices in\nwhich the width 𝑤and the thickness ℎbecome com-\nparable [20–27]. The description of such waveguides\nis beyond the thin strip model of effective pinning,\nbecause the scale of nonuniformity of the dynamic\ndipolar fields, which is described as \"effective dipo-\nlar boundary conditions\", becomes comparable to the\nwaveguide width. Additionally, both, in-plane and\nout-of-plane dynamic magnetization components, be-\ncome involved in the effective dipolar pinning, as they\nbecome of comparable amplitude. Thus, a more gen-\neral model should be developed and verified experi-\nmentally. In addition, such nanoscopic feature sizes\nimply that the spin-wave modes bear a strong ex-\nchange character, since the widths of the structures\nare now comparable to the exchange length [28]. A\nproper description of the spin-wave eigenmodes in\nnanoscopic strips which considers the influence of the\nexchangeinteraction, aswellastheshapeofthestruc-ture, was recently performed in [29] and is fundamen-\ntal for the field of magnonics.\nVery recently, the fields of quantum magnonics\nand magnonics at cryogenic temperatures were es-\ntablished. Among the highlights, one could mention\nthe first realization of a coherent coupling between a\nferromagnetic magnon and a superconducting qubit\n[30], the first observation of the interaction between\nmagnons and Abrikosov fluxes in supercondutor-\nferromagnet hybrid structures [31] and the investi-\ngation of the interplay of magnetization dynamics\nwith a microwave waveguide at cryogenic tempera-\ntures [32]. Thus, the understanding of the influence\nof temperature on spin pinning conditions and on\nspin-wavedispersionsinnano-structuresisofhighde-\nmand.\nIn this Article, we continue the investigation car-\nriedoutinPhys. Rev. Lett. 122, 247202(2019). The\nevolution of the frequencies and profiles of the spin-\nwave modes in Yttrium Iron Garnet (YIG) waveg-\nuides with a thickness of 39nm and widths rang-\ning down to 50nm are discussed in detail. The\nphenomenon of unpinning and the underlying the-\nory as well as the experimental proof are outlined.\nA throughout discussion of the effective width and\nthe critical width, at which the system becomes un-\npinned, in dependency of the thickness and the ma-\nterial of choice is presented. Moreover, the temper-\nature dependency is analysed theoretically. Higher\norder modes up to 𝑛= 2, the influence of a finite\nwavevector along the waveguide and the impact of\nthe pinning phenomenon on the spin wave lifetime\nare discussed. To account for the imperfections of a\nreal system, the influence of a trapezoidal cross sec-\ntion and edge roughness on the effective width and\nthe critical width are investigated.\n2. Methodology\n2.1.Sample fabrication\nA39nm thick Yttrium Iron Garnet (YIG) film has\nbeen grown on a 1 inch (111) 500𝜇m thick Gadolin-\niumGalliumGarnet(GGG)substratebyliquidphase\nepitaxy from PbO-B 2O3based high-temperature so-\nlutionsat 860∘Cusingtheisothermaldippingmethod\n(see e.g. Ref. [33]). A pure Y 3Fe5O12film with\na smooth surface was obtained by rotating the sub-\nstrate horizontally with a rotation rate of 100rpm.\nThe saturation magnetization of the YIG film is\n2Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nFig. 1. (a) Schematically depicted main steps in the nanos-\ntructuring process. (b) Sketch of the sample and the exper-\nimental configuration: a set of YIG waveguides is placed on\na microstrip line to excite the quasi-FMR in the waveguides.\nBLS spectroscopy is used to measure the local spin-wave dy-\nnamics. (c) and (d) SEM micrograph of a 1𝜇m and a 50-nm\nwide YIG waveguide of 39-nm thickness. The color code shows\nthe simulated amplitudes of the fundamental mode at quasi-\nferromagnetic resonance, i.e., 𝑘𝑥= 0, in the waveguides. The\nmode in the 50𝑛m waveguide is almost uniform across the\nwidth of the waveguide evidencing the unpinning directly. [29]\n1.37×105A/m and its Gilbert damping is 𝛼=\n6.41×10−4, as it was extracted by ferromagnetic res-\nonance spectroscopy [34].\nThe nanostructures were fabricated by utilising a\nhard mask ion milling procedure. The key steps\nin the fabrication process are shown in Fig. 1(a).\nFirst a double layer of polymethyl methacrylate\n(PMMA) was spin coated on the YIG film and a\nChromium/Titanium hard mask was fabricated using\nelectron beam lithography and electron beam evapo-\nration. This hard mask acts as a protective layer in\na successive Ar+ion milling step. In a final step, any\nresidualChromiumisremovedusinganacidthatYIG\nis inert to.\n2.2.Microfocused Brillouin Light Scattering\n(BLS) spectroscopy measurements\nBLS spectroscopy is a unique technique for measur-\ning the spin-wave intensities in frequency, space, andtimedomains. Itisbasedontheinelasticscatteringof\nan incident laser beam from a magnetic material. In\nourmeasurements,alaserbeamof 457nmwavelength\nand a power of 1.8mW is focused through the trans-\nparent GGG substrate on the center of the respective\nindividual waveguide using a 100×microscope objec-\ntive with a large numerical aperture ( NA = 0 .85).\nThe effective spot-size is 350nm. The scattered light\nwas collected and guided into a six-pass Fabry-P´ erot\ninterferometer to analyse the frequency shift.\n2.3.Numerical simulations\nThemicromagneticsimulationsofthespace-andtime\ndependent magnetization dynamics were performed\nbytheGPU-acceleratedsimulationprogramMumax3\nusing a finite-difference discretisation [35]. The struc-\nture is schematically shown in Fig. 1(b). The follow-\ning material parameters were used in the simulations:\nthe saturation magnetization 𝑀s= 1.37×105A/m\nand the Gilbert damping 𝛼= 6.41×10−4were\nextracted by ferromagnetic resonance spectroscopy\nmeasurements of the plain film before patterning [36].\nA gyromagnetic ratio of 𝛾= 175 .86rad/(ns·T) and\nan exchange constant of 𝐴= 3.5pJ/m for a stan-\ndard YIG film were assumed. An external field\n𝐵= 108 .9mT is applied along the waveguide long\naxis. Three steps were performed to calculate the\nspin-wave dispersion curve: (i) The external field was\napplied along the waveguide, and the magnetization\nwas allowed to relax into a stationary state (ground\nstate). (ii) A sinc field pulse 𝑏𝑦=𝑏0sinc(2 𝜋𝑓𝑐𝑡),\nwith oscillation field 𝑏0= 1mT and cut-off frequency\n𝑓𝑐= 10GHz, was used to excite a wide range of\nspin waves. (iii) The spin-wave dispersion relations\nwere obtained by performing the two-dimensional\nFast Fourier Transformation of the time- and space-\ndependent data. Furthermore, the spin-wave width\nprofileswereextractedfromthe 𝑚𝑧componentacross\nthe width of the waveguides using a single frequency\nexcitation.\n2.4.Quasi-analytical theoretical model\nIn order to accurately describe the spin-wave char-\nacteristics in nanoscopic longitudinally magnetized\nwaveguides, a more general semi-analytical theory\nis provided which goes beyond the thin strip ap-\nproximation [29]. Here, we assume a uniform spin\nwave mode profile across the waveguide thickness (no\n3B. Heinz et al.\n𝑧-dependency), which is valid for the fundamental\nthickness mode in thin waveguides. In a general case\nthez-dependencyofanyhigherthicknessmodecanbe\nincluded in a similar way as shown here. Also, please\nnote that the theory is not applicable in transversely\nmagnetized waveguides due to their more involved\ninternal field landscape [20]. The lateral spin-wave\nmode profile m𝑘𝑥(𝑦)and frequency can be found as\nsolutions of the linearized Landau-Lifshitz equation\n[37,38]\n−𝑖𝜔𝑘𝑥m𝑘𝑥(𝑦) =\u0016×(︁\n^Ω𝑘𝑥·m𝑘𝑥(𝑦))︁\n(2.1)\nwith appropriate exchange boundary conditions,\nwhich take into account the surface anisotropy at the\nedges. Here, \u0016is the unit vector in the static mag-\nnetization direction and ^Ω𝑘𝑥is a tensorial Hamilton\noperator, which is given by\n^Ω𝑘𝑥·m𝑘𝑥(𝑦) =(︂\n𝜔H+𝜔M𝜆2(︂\n𝑘2\n𝑥−𝑑2\n𝑑𝑦2)︂)︂\nm𝑘𝑥(𝑦)\n+𝜔M∫︀^G𝑘𝑥(𝑦−𝑦′)·m𝑘𝑥(𝑦′)𝑑𝑦′.\n(2.2)\nHere, 𝜔H=𝛾𝐵,𝐵is the static internal magnetic\nfield that is considered to be equal to the external\nfield due to the negligible demagnetization along the\n𝑥-direction, 𝜔M=𝛾𝜇0𝑀s,𝛾is the gyromagnetic ra-\ntio and ^G𝑘𝑥is the Green’s function (see next sub-\nsection).\nA numerical solution of Eq. (2.1) gives both, the\nspin-wave profile m𝑘𝑥and frequency 𝜔𝑘𝑥. In the fol-\nlowing, we will regard the ouf-of-plane component\n𝑚𝑧(𝑦)to show the mode profiles representatively. In\nthe past, it was demonstrated that in microscopic\nwaveguides, the fundamental mode is well fitted by\nthe function 𝑚𝑧(𝑦) =𝐴0cos(𝜋𝑦/𝑤 eff)with the am-\nplitude 𝐴0and the effective width 𝑤eff[16,17]. This\nmode, as well as the higher modes, are referred to as\n“partially pinned”. Pinning hereby refers to the fact\nthat the amplitude of the mode at the edges of the\nwaveguideisreduced. Inthatcase, theeffectivewidth\n𝑤effdetermines where the amplitude of the modes\nwould vanish outside the waveguide [9,16,27]. With\nthis effective width, the spin-wave dispersion relation\ncan also be calculated by the analytical formula [9]:\n𝜔0(𝑘𝑥) =\n√︁\n(𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑦𝑦\n𝑘𝑥))(𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑧𝑧\n𝑘𝑥)),(2.3)\nwhere 𝐾=√︀\n𝑘2𝑥+𝜅2and𝜅=𝜋/𝑤 eff. The ten-\nsor^𝐹𝑘𝑥=1\n2𝜋∫︁∞\n−∞|𝜎𝑘|2\n˜𝑤^N𝑘𝑑𝑘𝑦accounts for the dy-\nnamic magnetization, 𝜎𝑘=∫︁𝑤/2\n−𝑤/2𝑚(𝑦)𝑒−𝑖𝑘𝑦𝑦𝑑𝑦is\nthe Fourier-transform of the spin-wave profile across\nthe width of the waveguide and ˜𝑤=∫︁𝑤/2\n−𝑤/2𝑚(𝑦)2𝑑𝑦\nis the normalization of the mode profile 𝑚(𝑦).\n2.5.Numerical solution of the eigenproblem\nIn this sub-section we discuss the details of the nu-\nmerical solution of the eigenproblem. The eigenprob-\nlem Eq. (2.1) should be solved with proper boundary\nconditions at the lateral edges of the waveguide. We\nuse a complete description of the dipolar interaction\nvia Green’s functions:\n^G𝑘𝑥(𝑦) =1\n2𝜋∫︁∞\n−∞^N𝑘𝑒𝑖𝑘𝑦𝑦𝑑𝑘𝑦. (2.4)\nHere,\n^N𝑘=⎛\n⎜⎝𝑘2\n𝑥\n𝑘2𝑓(𝑘ℎ)𝑘𝑥𝑘𝑦\n𝑘2𝑓(𝑘ℎ) 0\n𝑘𝑥𝑘𝑦\n𝑘2𝑓(𝑘ℎ)𝑘2\n𝑦\n𝑘2𝑓(𝑘ℎ) 0\n0 0 1 −𝑓(𝑘ℎ)⎞\n⎟⎠,(2.5)\nwhere 𝑓(𝑘ℎ) = 1−(1−exp(−𝑘ℎ))/𝑘ℎ,𝑘=√︁\n𝑘2𝑥+𝑘2𝑦\nand it is assumed that the waveguides are infinitely\nlong.\nThe boundary condition (2.6) only accounts for the\nexchange interaction and surface anisotropy (if any)\nand reads [39]\nm×(𝜇0𝑀s𝜆2𝜕m\n𝜕n−∇M𝐸a) = 0 , (2.6)\nwhere nis the unit vector defining the inward normal\ndirection to the waveguide edge, and 𝐸a(m)is the en-\nergy density of the surface anisotropy. In the studied\ncase of a waveguide magnetized along its long axis,\nthe conditions (2.6) for the dynamic magnetization\ncomponents can be simplified to\n±𝜕𝑚𝑦\n𝜕𝑦+𝑑𝑚𝑦|𝑦=±𝑤/2= 0,𝜕𝑚𝑧\n𝜕𝑦|𝑦=±𝑤/2= 0,\n(2.7)\n4Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nwhere 𝑑=−2𝐾s/(𝑚0𝑀2\ns𝜆2)isthepinningparameter\n[18] and 𝐾sis the surface anisotropy constant at the\nwaveguide lateral edges. More complex cases like,\ne.g., diffusive interfaces can be considered in the same\nmanner [40].\nFor the numerical solution of Eq. (2.1) it is con-\nvenient to use finite element methods and to dis-\ncretize the waveguide into 𝑛elements of the width\nΔ𝑤=𝑤/𝑛, where 𝑤is the width of the wave-\nguide. The discretization step should be at least sev-\neral times smaller than the waveguide thickness and\nthe spin-wave wavelength 2𝜋/𝑘𝑥for a proper descrip-\ntion of the magneto-dipolar fields. The discretization\ntransforms Eq. (2.1) into a system of linear equations\nfor magnetizations m𝑗,𝑗= 1,2,3,···𝑛\n𝑖\u0016×((𝜔M+𝜔M𝜆2𝑘2\n𝑥)m𝑗\n−𝜔M𝜆2m𝑗−1−2m𝑗+m𝑗+1\nΔ𝑤2\n+𝜔M∑︀𝑛\n𝑗′=1^G𝑗−𝑗′·m𝑗′) =𝜔m𝑗(2.8)\nwhere dipolar interaction between the discretized el-\nements is described by\n^G𝑘𝑥,𝑗(𝑦) =\n1\nΔ𝑤∫︀Δ𝑤/2\n−Δ𝑤/2𝑑𝑦∫︀Δ𝑤/2\n−Δ𝑤/2𝑑𝑦′^G𝑘𝑥(𝑦−𝑦′−𝑗Δ𝑤).(2.9)\nThe direct use of Eq. (2.9) is complicated since the\nGreen’s function ^G𝑘𝑥(𝑦)is an integral itself. Using\nthe Fourier transform it can be derived as\n^G𝑘𝑥,𝑗(𝑦) =Δ𝑤\n2𝜋∫︁\nsinc(𝑘𝑦Δ𝑤/2)^N𝑘𝑒𝑖𝑘𝑦𝑗Δ𝑤𝑑𝑘𝑦,\n(2.10)\nwith sinc(𝑥) =sin(𝑥)\n𝑥. This can be easily calcu-\nlated, especially using fast Fourier transform. Equa-\ntion (2.8) is, in fact, a 2𝑛-dimensional linear alge-\nbraic eigenproblem (since m𝑗is a 2-component vec-\ntor),whichissolvedbystandardmethods. Thevalues\nm0andm𝑛+1in Eq. (2.8) are determined from the\nboundary conditions (2.7). In particular, for negligi-\nble anisotropy at the waveguide edges one should set\nm0=m1andm𝑛+1=m𝑛.\n3. Results and discussions\n3.1.Original experimental findings\nIn these studies, we consider rectangular magnetic\nwaveguides as shown schematically in Fig. 1(b). Inthe experiment, a spin-wave mode is excited by a\nstripline that provides a homogeneous excitation field\nover the sample containing various waveguides etched\nfrom a ℎ= 39nm thick YIG film. The widths of the\nwaveguides range from 𝑤= 50nm to 𝑤= 1𝜇m and\nthe length is 60𝜇m. The waveguides are uniformly\nmagnetized along their long axis by an external field\n𝐵(𝑥-direction). Figure 1(c) and 1(d) show scanning\nelectronmicroscopy(SEM)micrographsofthelargest\nand the narrowest waveguide studied in the experi-\nment. The intensity of the magnetization precession\nis measured by microfocused BLS spectroscopy [41]\n(see Methodology section) as shown in Fig. 1(b).\nBlack and red lines in Fig. 3(a) show the frequency\nspectra for a 1𝜇m and a 50-nm wide waveguide, re-\nspectively. No standing modes across the thickness\nwere observed in our experiment, as their frequen-\ncies lie higher than 20GHz due to the small thick-\nness. The quasi-FMR frequency is 5.007GHz for the\n1𝜇m wide waveguide. This frequency is compara-\nble to 5.029GHz, the value predicted by the classi-\ncal theoretical model using the thin strip approxima-\ntion [16–18]. In contrast, the quasi-FMR frequency is\n5.35GHz for a 50nm wide waveguide which is much\nsmaller than the value of 7.687GHz predicted by the\nsame model. The reason is that the thin strip approx-\nimation overestimates the effect of dipolar pinning in\nwaveguides with aspect ratio either 𝑎r= 1or close to\none, forwhichthenonuniformityofthedynamicdipo-\nlar fields is not well-localized at the waveguide edges.\nAdditionally, in such nanoscale waveguides, the dy-\nnamicmagnetizationcomponentsbecomeofthesame\norder of magnitude and both affect the effective mode\npinning, in contrast to thin waveguides, in which the\nin-plane magnetization component is dominant.\n3.2.Spin pinning in nano-structures\nIn the following, the experiment is compared to the\ntheory and to micromagnetic simulations.\nThe bottom panels of Fig. 2(a) and 2(b) show the\nspin-wave mode profile of the fundamental mode for\n𝑘𝑥= 0, which corresponds to the quasi-FMR, in a\n1𝜇m (a) and 50nm (b) wide waveguide which have\nbeen obtained by micromagnetic simulations (blue\ndots) and by solving Eq. (2.1) numerically (black\nlines) (higher width modes are discussed in the next\nsections). The top panels illustrate the mode pro-\nfile and the local precession amplitude in the wave-\n5B. Heinz et al.\nFig. 2. Schematic of the precessing spins and simulated pre-\ncession trajectories (ellipses in the second panel) and spin-wave\nprofile 𝑚𝑧(𝑦)of the quasi-FMR. The profiles have been ob-\ntained by micromagnetic simulations (blue dots) and by the\nquasi-analytical approach (black lines) for a (a) 1𝜇m and a (b)\n50nm wide waveguide. (c), (d): Corresponding normalized\nsquare of the spin-wave eigenfrequency 𝜔′2/𝜔2\nMas a function\nof𝑤/𝑤 effand the relative dipolar and exchange contributions.\n[29]\nguide. As it can be seen, the two waveguides feature\nquite different profiles of their fundamental modes:\nin the 1𝜇m wide waveguide, the spins are partially\npinned and the amplitude at the edges of the wave-\nguide is reduced compared to the maximal value of\n𝑚𝑧= 1. This still resembles the cosine-like profile of\nthe lowest width mode 𝑛= 0that has been well es-\ntablished in investigations of spin-wave dynamics in\nwaveguides on the micron scale [23,27,42] and that\ncan be well-described by the simple introduction of a\nfinite effective width 𝑤eff> 𝑤(𝑤eff=𝑤for the case\nof full pinning). In contrast, the spins at the edges of\nthe narrow waveguide are completely unpinned and\nthe amplitude of the dynamic magnetization 𝑚𝑧of\nthe lowest mode 𝑛= 0is almost constant across the\nwidth of the waveguide, resulting in 𝑤eff→∞.\nTo understand the nature of this depinning, it is in-\nstructive to consider the spin-wave energy as a func-\ntion of the geometric width of the waveguide normal-\nized by the effective width 𝑤/𝑤 eff. This ratio cor-\nresponds to some kind of pinning parameter taking\nvalues in between 1 for the fully pinned case and 0for the fully unpinned case. According to the Ritz’s\nvariational principle, the profiles of the spin wave\nmodes correspond to the respective minima of the\nspin wave frequency (energy) functional. Since only\none minimization parameter – 𝑤eff– is used, the min-\nimization as a function of 𝑤/𝑤 effyields only the ap-\nproximate spin wave profiles. Nevertheless, this is\nsufficient for the qualitative understanding. To illus-\ntrate this, Figs. 2(c) and 2(d) show the normalized\nsquare of the spin-wave eigenfrequencies 𝜔′2/𝜔2\nMfor\nthetwodifferentwidthsasafunctionof 𝑤/𝑤 eff. Here,\n𝜔′2refers to a frequency square, not taking into ac-\ncount the Zeeman contribution (𝜔2\nH+𝜔H𝜔M), which\nonly leads to an offset in frequency. The minimum\nof𝜔′2is equivalent to the solution with the lowest\nenergy corresponding to the effective width 𝑤eff. In\naddition to the total 𝜔′2(black), also the individ-\nual contributions from the dipolar term (red) and the\nexchange term (blue) are shown, which can only be\nseparated conveniently from each other if the square\nof Eq. (2.3) is considered for 𝑘𝑥= 0. The dipolar\ncontribution is non-monotonous and features a mini-\nmum at a finite effective width 𝑤eff, which can clearly\nbe observed for 𝑤= 1𝜇m. The appearance of this\nminimum, which leads to the effect known as “effec-\ntive dipolar pinning” [17,18], is a result of the inter-\nplay of two tendencies: (i) an increase of the volume\ncontribution with increasing 𝑤/𝑤 eff, as for common\nDamon-Eshbach spin waves, and (ii) a decrease of\nthe edge contribution when the spin-wave amplitude\nat the edges vanishes ( 𝑤/𝑤 eff>1). This minimum\nis also present in the case of a 50nm wide waveguide\n(red line), even though this is hardly perceivable in\nFig. 2(d) due to the scale. In contrast, the exchange\nleads to a monotonous increase of the frequency as a\nfunction of 𝑤/𝑤 eff, which is minimal for the unpinned\ncase, i.e., 𝑤/𝑤 eff= 0implying 𝑤eff→∞, when all\nspins are parallel. In the case of the 50nm wave-\nguide, the smaller width and the corresponding much\nlarger quantized wavenumber in the case of pinned\nspins would lead to a much larger exchange contribu-\ntion than this is the case for the 1𝜇m wide waveguide\n(please note the vertical scales). Consequently, the\nsystem avoids pinning and the solution with lowest\nenergy is situated at 𝑤/𝑤 eff= 0. In contrast, in the\n1𝜇m wide waveguide, the interplay of dipolar and ex-\nchange energy implies that the energy is minimized\nat a finite 𝑤/𝑤 eff.\n6Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nFig. 3. (a) Frequency spectra for 1𝜇m and 50nm wide waveguides measured for a respective microwave power of 6dBm\nand 15dBm. (b) Experimentally determined resonance frequencies (black squares) together with theoretical predictions and\nmicromagnetic simulations. (c) Inverse effective width 𝑤/𝑤 effas a function of the waveguide width. (d) The critical width\n(𝑤crit) as a function of thickness ℎ. (e) Inverse effective width 𝑤/𝑤 effas a function of waveguide width for different materials at\nfixed thickness of 39 nm. (f) The critical width 𝑤critas a function of exchange length 𝜆for different thicknesses. (a) - (c) [29]\n3.3.The dependence of the spin-wave\nfrequency on the spin pinning and the critical\nwidth of the exchange unpinning\nAs it is evident from Figs. 2(c) and 2(d), the pin-\nning and the corresponding effective width have a\nlarge influence on the spin-wave frequency. This al-\nlows for an experimental verification of the presented\ntheory, since the frequency of partially pinned spin-\nwave modes would be significantly higher than in the\nunpinned case. Black squares in Fig. 3(b) summa-\nrize the dependence of the frequency of the quasi-\nFMR on the width of the YIG waveguide. The ma-\ngenta line shows the expected frequencies assuming\npinned spins, the blue (dashed) line gives the reso-\nnance frequencies extrapolating the formula conven-\ntionally used for micron-sized waveguides [43] to the\nnanoscopic scenario, and the red line gives the result\nof the theory presented here, together with simula-\ntion results (green dashed line). As it can be seen,\nthe experimentally observed frequencies can be well\nreproduced if the real pinning conditions are taken\ninto account.As has been discussed alongside with Fig. 2, the\nunpinning occurs when the exchange interaction con-\ntribution becomes so large that it compensates the\nminimum in the dipolar contribution of the spin-wave\nenergy. Since the energy contributions and the de-\nmagnetization tensor change with the thickness of\nthe investigated waveguide, the critical width below\nwhich the spins become unpinned is different for dif-\nferent waveguide thicknesses. This is shown in Fig.\n3(c),wheretheinverseeffectivewidth 𝑤/𝑤 effisshown\nfor different waveguide thicknesses. Symbols are the\nresults of micromagnetic simulations, lines are calcu-\nlated semi-analytically. As can be seen from the fig-\nure, the critical width linearly increases with increas-\ning thickness. This is summarized in Fig. 3(d), which\nshows the critical width (i.e. the maximum width for\nwhich 𝑤/𝑤 eff= 0) as a function of thickness.\nThe critical widths for YIG, Permalloy, CoFeB and\nthe Heusler compound Co 2Mn0.6Fe0.4Si with differ-\nent thicknesses are investigated. Figure 3(e) shows\nthe inverse effective width 𝑤/𝑤 effas a function of\nthe waveguide width for these materials which can\nbe considered as typical materials used in magnonics.\n7B. Heinz et al.\nWidth coordinate (nm) y Width coordinate (nm) y Width coordinate (nm) yNormalized mzNormalized mzNormalized mz\nFig. 4. The spin-wave profile representatively depicted using the 𝑚𝑧component of the dynamic magnetization for the three\nlowest width modes obtained by micromagnetic simulation (black solid lines) and numerical calculation (red dots) for (a) 5𝜇m,\n(b)1𝜇m and (c) 50nm wide waveguides.\nFigure 3(f) shows the critical width ( 𝑤crit) as a func-\ntion of the exchange length 𝜆for different thicknesses.\nA simple empirical linear formula is found by fitting\nthe critical widths for different materials in a wide\nrange of thicknesses to estimate the critical width:\n𝑤crit= 2.2ℎ+ 6.7𝜆 (3.1)\nwhere ℎis the thickness of the waveguide and 𝜆is\nthe exchange length given by 𝜆=√︀\n2𝐴/(𝜇0𝑀2s)with\nthe exchange constant 𝐴, the vacuum permeability\n𝜇0, and the saturation magnetization 𝑀s.\n3.4.Profiles of higher-order width modes\nIn [29], only the profile of the fundamental mode\n(𝑛= 0) has been discussed, therefore the mode pro-\nfiles of higher width modes are shown in Fig. 4 for the\nwidthsofthewaveguidesof 5𝜇mcorrespondingtothe\npractically fully pinned case (Fig. 4(a)), 50nm rep-\nresenting fully unpinned case (Fig. 4(c)), and 1𝜇m\nwhich can be considered as an intermediate case (Fig.\n4(b)). For a 5𝜇m wide waveguide all higher width\nmodes are clearly partially pinned due to the large\nwidthandaninsufficientcontributionoftheexchangeenergy. In contrast to this the higher modes of a 1𝜇m\nwidewaveguideareclearlyunpinnedformodes 𝑛 >2.\nSince the fundamental mode is already unpinned for\na50nm wide waveguide also all higher width modes\nare fully unpinned.\n3.5.Temperature dependence of spin pinning\nand frequencies of the spin-wave modes\nIn the following, the quasi-analytic theory is used to\nstudy the influence of the temperature on the dis-\ncussed phenomena. There are two main parameters\nthat introduce the temperature dependence of the\nspin-wave dispersion, the pinning condition and the\npinning parameter: the saturation magnetization 𝑀s\nand the exchange constant 𝐴. Furthermore, the tem-\nperature dependence of the surface anisotropy con-\nstant 𝐾sat the lateral edges of the waveguide, can\nlead to an additional temperature dependence of the\npinning parameter 𝑑(see Eq. (2.7)). However, typi-\ncally this dependency is rather weak and is therefore\nneglected in the following. The calculated saturation\nmagnetization 𝑀sfor YIG is shown in Fig. 5(a) as a\nfunction of the temperature and was obtained using\n8Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nFig. 5. Temperature dependence of the saturation magnetization (a) and exchange constant (b) of YIG. (c) The temperature\ndependencies of the frequencies of the first three modes for a YIG waveguide with ℎ= 20nm,𝑤= 200nm and 𝐵0= 108 .9mT.\n(d) The temperature dependence of the inverse effective width (left axis) and critical width of the exchange unpinning (right\naxis).\nthe theoretical model developed in [44]. The exper-\nimentally measured temperature dependence of the\nexchange constant 𝐴taken from [45] is shown in Fig.\n5(b).\nFigure 5(c) shows the resulting temperature de-\npendency of the frequencies of the first three modes\nfor a YIG waveguide of thickness ℎ= 20nm, width\n𝑤= 200nm, and for an external magnetic field\n𝐵0= 108 .9mT applied along the stripe. One can\nclearly see that the frequencies of all modes decrease\nwith the increase in temperature due to the decrease\nin the saturation magnetization. The critical width,\nat which the unpinning takes place, depends on both\nthe saturation magnetization and the exchange con-\nstant as it can be seen e.g. from the empirical Eq.\n(3.1). The interplay between the both dependen-\ncies results in the increase of the critical width with\nthe increase in temperature from the value of around\n140nm for zero temperature up to around 200nm for\n500K - see Fig. 5(d). At the same time, the spin\npinning, which is shown in the same figure in terms\nof inverse effective width of the waveguide 𝑤/𝑤 eff, de-\ncreases with the increase in temperature. This hap-\npens due to the dominant contribution of the tem-perature dependence of the saturation magnetization\nwhich, consequently, defines the strength of the dipo-\nlar pinning phenomenon. To conclude, if one con-\nducts low temperature experiments which rely or re-\nquire a fully unpinned state of the system a careful\ndesign of the structure dimensions is necessary.\n3.6.Spin-wave dispersion in nano-strucutres\nand the dependence of spin pinning of\nspin-wave wavenumber\nUp to now, the discussion was limited to the spe-\ncial case of 𝑘𝑥= 0. In the following, the influence\nof finite wave vector will be addressed. The spin-\nwave dispersion relation of the fundamental ( 𝑛=\n0) mode obtained from micromagnetic simulations\n(color-code) together with the semi-analytical solu-\ntion (white dashed line) are shown in Fig. 6(a) for\nthe YIG waveguide of 𝑤= 50nm width. The figure\nalso shows the low-wavenumber part of the disper-\nsion of the first width mode ( 𝑛= 1), which is pushed\nup significantly in frequency due to its large exchange\ncontribution. Bothmodesaredescribedaccuratelyby\nthe quasi-analytical theory. As it is described above,\nthe spins are fully unpinned in this particular case.\n9B. Heinz et al.\nn\nn\nFig. 6.(a) Spin-wave dispersion relation of the first two width\nmodesfrommicromagneticsimulations(color-code)andtheory\n(dashed lines). (b) Inverse effective width 𝑤/𝑤 effas a function\nof the spin-wave wavenumber 𝑘𝑥for different thicknesses and\nwaveguide widths, respectively. [29]\nIn order to demonstrate the influence of the pinning\nconditions on the spin-wave dispersion, a hypothetic\ndispersion relation for the case of partial pinning is\nshown in the figure with a dash-dotted white line (the\ncase of 𝑤/𝑤 eff= 0.63is considered that would result\nfrom the usage of the thin strip approximation [16]).\nOne can clearly see that the spin-wave frequencies in\nthis case are considerably higher. Figure 6(b) shows\nthe inverse effective width 𝑤/𝑤 effas a function of the\nwavenumber 𝑘𝑥forthreeexemplarywaveguidewidths\nof𝑤= 50nm,300nm and 500nm. As it can be seen,\ntheeffectivewidthand,consequently,theratio 𝑤/𝑤 eff\nshows only a weak nonmonotonic dependence on the\nspin-wave wavenumber in the propagation direction.\nThis dependence is a result of an increase of the in-\nhomogeneity of the dipolar fields near the edges forlarger 𝑘𝑥, which increases pinning [18], and of the\nsimultaneous decrease of the overall strength of dy-\nnamic dipolar fields for shorter spin waves. Please\nnote that the mode profiles are not only important\nfor the spin-wave dispersion. The unpinned mode\nprofiles also greatly improve the coupling efficiency\nbetween two adjacent waveguides [9,46–48].\n3.7.Spin-wave lifetime in magnetic\nnanostructures\nThe spin-wave lifetime depends on the ellipticity of\nthe magnetization precession, and, thus, on the spin\npinning conditions. The top panel of Fig. 2(b) shows\nan additional feature of the narrow waveguide: as the\naspect ratio of the waveguides approaches unity, the\nellipticity of the precession, a well-known feature of\nmicron-sized waveguides which still resemble a thin\nfilm [27, 39], vanishes and the precession becomes\nnearly circular. In addition, in nanoscale waveguides,\nthe ellipticity is constant across the width, while it\ncan be different at the waveguide center and near its\nedges for a 1𝜇m wide waveguide. In general, the def-\ninition of the ellipticity 𝜖of the precession is given by\nthe ratio of the precession components as follows:\n𝜖= 1−𝑚min\n𝑚max, (3.2)\nwhere 𝑚minand𝑚maxdenote the respective ampli-\ntudes of the smaller and larger component of the pre-\ncession. Calculating the average relation between the\nmagnetization components 𝑚𝑦and𝑚𝑧it follows\n⃒⃒⃒⃒𝑚𝑦\n𝑚𝑧⃒⃒⃒⃒=⎯⎸⎸⎷(︃\n(𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑧𝑧\n𝑘𝑥)\n(𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑦𝑦\n𝑘𝑥)))︃\n, (3.3)\nfrom which the ellipticity can be calculated for any\nwidthindependencyofthespin-wavewavenumber 𝑘𝑥\nas it is shown in Fig. 7(a).\nThe relaxation lifetime 𝜏of the uniform precession\nmode in an infinite medium (without inhomogeneous\nlinewidth Δ𝐵0) is simply defined as 𝜏= 1/(𝛼𝜔),\nwhere 𝜔is the angular frequency of the spin wave\nand𝛼is the damping. However, the dynamic demag-\nnetizing field has to be taken into account in finite\nspin-wave waveguide. The lifetime can be found by\nthe phenomenological model [49–51]\n𝜏=(︂\n𝛼𝜔𝜕𝜔\n𝜕𝜔H)︂−1\n. (3.4)\n10Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nThe dispersion relation has been shown in the\nmanuscript (Eq. (2.3)). The demagnetization ten-\nsors are independent of 𝜔H. Differentiating Eq. (2.3)\nyields the lifetime as\n𝜏=(︂1\n2𝛼(2𝜔H+ 2𝜔M𝜆2𝐾2+𝜔M(𝐹𝑧𝑧\n𝑘𝑥+𝐹𝑦𝑦\n𝑘𝑥)))︂−1\n.\n(3.5)\nThis formula clearly shows that the lifetime of the\nuniform precession ( 𝑘𝑥= 0) depends only on the sum\nof the dynamic 𝑦𝑦and𝑧𝑧components of demagneti-\nzation tensors.\nFigure 7(b) shows the cross-section, spin precession\ntrajectory (red line) and the dynamic components\nof the demagnetization tensors of different sample\ngeometries. The spin precession trajectory changes\nfrom elliptic for the thin film ( 𝑎r≪1) to circular for\nthe nanoscopic waveguide ( 𝑎r= 1). The spin preces-\nsion trajectory in the bulk material is also circular (in\nthe geometry when spin waves propagate parallel to\nthe static magnetic field, the same geometry as stud-\nied for nanoscale waveguides). The dependence of the\nlifetime on the wavenumber is shown in Fig. 7(c) for\nYIG with a damping constant 𝛼= 2×10−4. The\ninhomogeneous linewidth is not taken into account.\nThelifetimeoftheuniformprecession( 𝑘𝑥= 0)forthe\nbulk material is much large than that in the thin film\nand nanoscopic waveguide, another consequence of\nthe absence of dynamic demagnetization in the bulk\n(𝐹𝑧𝑧\n0=𝐹𝑦𝑦\n0). Moreover, the lifetimes of the uniform\nprecession ( 𝑘𝑥= 0) for a thin film (red line) and for\na nanoscopic waveguide (black line) have the same\nvalue, because the lifetime depends only on the sum\nof the two components, which is the same for both\ncases.\nMoreover, the 𝑦𝑦and𝑧𝑧components of the de-\nmagnetization tensor decrease with an increase of the\nspin-wave wavenumber (instead, the 𝑥𝑥component,\nwhich does not affect the spin wave dynamic in our\ngeometry, increases). Thelifetimeisinverselypropor-\ntional to the square of the wavenumber and the sum\nof the dynamic demagnetization components. In the\nexchange region, the lifetime is, thus, dominated by\nthe wavenumber. Therefore, the lifetimes for short-\nwave spin-waves are nearly the same for the three\ndifferent geometries.\nFig. 7. (a) Ellipticity as a function of the waveguide width\nfor different spin-wave wavenumbers 𝑘𝑥for a thickness of ℎ=\n39nmandanexternalmagneticfieldof 𝐵0= 108 .9mT(b)The\nspin precession trajectories (red lines) and the components of\nthe demagnetization tensor 𝐹𝑦𝑦\n0and𝐹𝑧𝑧\n0for different sample\ngeometries. (c) The spin-wave lifetime as a function of the\nspin-wave wavenumber. The lines and dots are obtained from\nEq. (3.5) and micromagnetic simulation, respectively.\n3.8.Dependence of the spin pinning on a\ntrapezoidal form of the waveguides\nA perfect rectangular form is not achievable in the\nexperiment due to the involved patterning technique.\nAs a result of the etching, the cross-section of the\nwaveguides is always slightly trapezoidal. In this sec-\ntion, the influence of such a trapezoidal form on the\nspin pinningconditions isstudied. Inour experiment,\n11B. Heinz et al.\nFig. 8. (a) Trapezoidal cross section of the simulated wave-\nguide with the normalized spin-wave profile for the different\nlayers. (b) The inverse effective width 𝑤/𝑤 effas a function\nof the width of the waveguide for trapezoidal and rectangular\nform.\nthe trapezoidal edges extent for approximately 20nm\non both sides for all the patterned waveguides as it\ncan be seen from Fig. 1(d). We performed addi-\ntionalsimulationonwaveguideswithsuchtrapezoidal\nedges. The simulated cross-section is shown in the\ntop of Fig. 8. The thickness of the waveguide is di-\nvided into 5 layers with different widths ranging from\n90nm to 50nm. The steps at the edges are hard to\nbe avoided due to the finite difference method used\nin MuMax3. The spin-wave profiles in the different\n𝑧-layers are shown at the bottom of Fig. 8(a).\nThe results clearly show that the spin-wave profiles\nare fully unpinned along the entire thickness. This\nis due to the fact that the largest width ( 90nm) isstill far below the critical width. Hence, the influence\nof the trapezoidal form of the waveguide on the spin\npinning condition is negligible for very narrow waveg-\nuides. For large waveguides, it also does not have a\nlarge impact as the ratio of the edge to the wave-\nguide area becomes close to zero. Quantitatively, the\nquasi-ferromagnetic resonance frequency in a 50nm\nwide waveguide decreases from 5.45GHz for the rect-\nangular shape to 5.38GHz for the trapezoidal form\ndue to the increase of the averaged width which, in\nfact, even closer to the experiment results ( 5.35GHz).\nThe inverse effective width 𝑤/𝑤 effas a function of\nthe width of the waveguides is simulated for a trape-\nzoidal and a rectangular form and the result is shown\nin Fig. 8(b). Here, the width is defined by the min-\nimal width for the trapezoidal form, i.e., the width\nof the top layer. In the case of trapezoidal form, the\ninverse effective width is averaged over all 5layers.\nThe critical width slightly decreases from 200nm for\nthe rectangular cross-section to 180nm for the trape-\nzoidal form due to the increase of the averaged width.\nThe difference between the inverse effective widths\ndecreases with increasing width of the waveguide and\nvanishes when the width is larger than 300nm.\nFurthermore, it should be noted that the results of\nthe multilayer simulations demonstrate that the as-\nsumption of a uniform dynamic magnetization distri-\nbution across the thickness that is used in our analyt-\nical theory and micromagnetic simulations featuring\nonly one cell in the z dimension is valid.\n3.9.Influence of edge roughness on the spin\npinning\nPerfectly smooth edges are also hard to obtain in the\nexperiment. Therefore, we have considered the in-\nfluence of edge roughness on the spin pinning. We\nperformed additional simulations on waveguides with\nrough boundaries for a fixed thickness of 39nm. 5nm\n(for50nm to 100nm wide waveguides) or 10nm (for\n100nm to 1000nm wide waveguides) wide rectangu-\nlar nonmagnetic regions with a random length are\nintroduced randomly on both sides of the waveguides\nto act as defects. The introduction of roughness re-\nsults in a slight increase of the critical width from\n200nm to 240nm, as is shown in Fig. 9(a). These re-\nsults demonstrate that edge roughness does not have\na large influence on spin pinning condition.\n12Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides\nFig. 9.(a) Top: Schematic of the rough waveguide and close-\nup image of the introduced edge roughness. A single random-\nized defect pattern is generated for each structure width. Bot-\ntom: Inverse effective width 𝑤/𝑤 effas a function of the wave-\nguide width for rough and smooth edges. (b) The normalized\nspin-wave intensity as a function of the propagation length for\n50nm wide waveguides with smooth and rough edges.\nAdditional simulations are performed to study\nthe influence of a rough edge on the propagation\nlength of spin waves with frequency 6.16GHz ( 𝑘𝑥=\n0.03rad/nm). Figure9(b)showsthenormalizedspin-\nwave intensity as a function of propagation length for\nsmooth and rough edged waveguide of 450nm width.\nThe decay length slightly decreases from 15.96𝜇m for\nsmooth edges to 15.76𝜇m for rough edges. Since the\nspins in nanoscopic waveguides are already unpinned,\nthe effect of such an edge roughness is not too impor-\ntant anymore and the propagation length is essen-\ntially unaffected.4. Conclusions\nTo conclude, an in-detail investigation of the pinning\nphenomenon based on the theoretical description of\n[29] is presented and the quasi-analytical model is\noutlined. The dependency of the effective width on\nthe thickness and the material of choice is analysed\nand a simple empirical formula is found to predict\nthe critical width for a given system. In addition to\n[29], higher order width modes up to 𝑛= 2are anal-\nysed. An investigation of the effective width for finite\nwavevectors along the waveguide yields only a weak\nnonmonotonic dependence. It is shown, that assum-\ning a more realistic trapezoidal cross section of the\nstructures rather than the ideal rectangular shape re-\nsults in a small decrease of the quasi-FMR frequency\nand a slight reduction of the critical width. More-\nover, the influence of edge roughness is studied which\nshows a small increase of the critical width compared\nto the case of smooth edges. Here, also the impact on\nthe decay length of propagating waves is investigated\nand only a small reduction is found. The tempera-\nture dependence of the pinning phenomenon shows\nthat the dependencies of the saturation magnetiza-\ntion and exchange constant of YIG result in the de-\ncrease of spin pinning with the increase in tempera-\nture and in the increase in the critical width of the\nexchange unpinning. This assumes that low temper-\natures are favourable for the dipolar pinning and the\nsizes of the structures have to be decreased further\nin order to operate with fully-unpinned uniform spin-\nwave modes.\nThe presented results provide valuable guidelines\nfor applications in nano-magnonics where spin waves\npropagate in nanoscopic waveguides with aspect ra-\ntios close to one and lateral sizes comparable to the\nsizes of modern CMOS technology.\nAcknowledgement. The authors thank Burkard\nHillebrands and Andrei Slavin for valuable discus-\nsions. This research has been supported by ERC\nStarting Grant 678309 MagnonCircuits and by the\nDFG through the Collaborative Research Center\nSFB/TRR-173“Spin+X” (ProjectsB01)andthrough\ntheProjectDU1427/2-1. B.H.acknowledgessupport\nby the Graduate School Material Science in Mainz\n(MAINZ). R. V. acknowledges support by the Na-\ntional Academy of Sciences of Ukraine Grant No. 23-\n04/01-2019.\n13B. Heinz et al.\n*These authors have contributed equally to this\nwork.\n**chumak@physik.uni-kl.de\n1. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Magnon spintronics, Nat. Phys. 11, 453 (2015),\nDOI:http://dx.doi.org/10.1038/nphys3347.\n2. V. V. Kruglyak, S. O. Demokritov, and D. Grundler,\nMagnonics, J. Phys. D 43, 264001 (2010), DOI:http://dx.\ndoi.org/10.1088/0022-3727/43/26/264001.\n3. C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Cher-\ntopalov, M. T. Bryan, S. V. Grishin, D. A. Allwood,\nY. P. Sharaevskii, S. A. Nikitov, and V. V. 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Slavin, Damping oflinear spin-wave modes in magnetic nanostructures: Lo-\ncal, nonlocal, and coordinate-dependent damping, Phys.\nRev.B 98, 104408(2018), DOI:http://dx.doi.org/10.1103/\nPhysRevB.98.104408.\n16" }, { "title": "1904.04167v1.Quantum_entanglement_between_two_magnon_modes_via_Kerr_nonlinearity.pdf", "content": "arXiv:1904.04167v1 [quant-ph] 8 Apr 2019Quantum entanglement between two magnon modes via Kerr nonl inearity\nZhedong Zhang,1,∗Marlan O. Scully,1, 2, 3and Girish S. Agarwal1, 4,†\n1Institute for Quantum Science and Engineering, Texas A &M University, College Station, TX 77843, USA\n2Quantum Optics Laboratory, Baylor Research and Innovation Collaborative, Waco, TX 76704, USA\n3Department of Mechanical and Aerospace Engineering, Princ eton University, Princeton, NJ 08544, USA\n4Department of Biological and Agricultural Engineering,\nDepartment of Physics and Astronomy, Texas A &M University, College Station, TX 77843, USA\n(Dated: April 9, 2019)\nWe propose a scheme to entangle two magnon modes via Kerr nonl inear effect when driving the systems\nfar-from-equilibrium. We consider two macroscopic yttriu m iron garnets (YIGs) interacting with a single-mode\nmicrocavity through the magnetic dipole coupling. The Kitt el mode describing the collective excitations of large\nnumber of spins are excited through driving cavity with a str ong microwave field. We demonstrate how the Kerr\nnonlineraity creates the entangled quantum states between the two macroscopic ferromagnetic samples, when\nthe microcavity is strongly driven by a blue-detuned microw ave field. Such quantum entanglement survives\nat the steady state. Our work o ffers new insights and guidance to designate the experiments f or observing the\nentanglement in massive ferromagnetic materials. It can al so find broad applications in macroscopic quantum\neffects and magnetic spintronics.\nIntroduction.– Recent advance in ferromagnetic materials\ndraw considerable attention in the studies of quantum natur e\nin magnetic systems, as the limitations of electrical circu itry\nare reached. Thanks to the low loss of the collective exci-\ntations of spins known as magnons in magnetic samples, the\nmagnons offer a new paradigm for developing future gener-\nation of spintronic devices and quantum engineering [1–6].\nThe yttrium iron garnet (YIG) with the size of ∼100µm\nas fabricated in recent experiments provides new insights f or\nstudying the macroscopic quantum e ffects, such as entangle-\nment and squeezing that have raised widespread interest in\ndifferent branches of physics during decade [7–12]. Quan-\ntum entanglement between massive mirror and optical cav-\nity photons has been explored, in both theoretical and ex-\nperimental aspects [13–18]. Several ideas follow-on sugge st\nthe extension of such entangled quantum state towards the\nmagnons in microwave regime, due to their great potential fo r\nmacroscopic spintronic devices. Much experimental e fforts\nhave been devoted to the quantum nature of magnon states,\nthrough hybridizing the spin waves with other degrees of fre e-\ndoms, e.g., superconducting qubits and phonon modes [19–\n22]. Compared to atoms and photonics, magnonics holds the\npotential for implementing quantum states in more massive\nobjects. This can be seen from the 320 µm-diam YIG spheres\nimplemented in recent experiments [23].\nAs a powerful platform for investigating the light-matter i n-\nteraction [23–30], ferromagnetic materials are taking the ad-\nvantage of reaching strong and ultrastrong coupling regime s,\nalong with the fact of their high spin density as well as low\ndissipation rate. The strong coupling results in the cavity\nmagnon-polariton, serving as a potential candidate for im-\nplementing quantum information transducers and memories\n[30, 31]. To achieve the quantum regime in magnon polari-\ntons, the macroscopic quantum e ffects are essentially wor-\nthy of being explored. The most recent work using driven-\ndissipation theory suggest the magnon-photon-phonon enta n-\nglement and also the squeezing of magnon modes in whichboth the entanglement and squeezing are essentially trans-\nferred into the mechanical mode [32–34]. From a theoreti-\ncal view-point, this macroscopic quantum nature of magnon\nmodes stems from the nonlinearity that can be enhanced by\ndriving the systems far-from-equilibrium. Two prominent\nschemes are responsible for introducing such nonlinearity : the\nmagnetostrictive interaction and the Kerr e ffect, where the\nlatter results from the magnetocrystalline anisotropy. Ap art\nfrom the magnon-phonon interaction, Kerr nonlinearity pla ys\na significant role in magnon spintronics [5]. Recent exper-\niments in YIG spheres demonstrated the multistability and\nphoton-mediated control of spin current, due to the Kerr ef-\nfect [35, 36].\nIn this Letter, we propose a scheme of entangling magnon\nmodes in two massive YIG spheres via the Kerr nonlinearity.\nThe two magnon modes interact with a microcavity through\nthe beam-splitter-like coupling, which cannot produce any en-\ntanglement. Nevertheless, activating the Kerr nonlineari ty via\nstrong driving results in squeezing-like coupling which ma y\nlet magnon get entangled with cavity photons. The subse-\nquent entanglement transfer between photons and the other\nmagnon mode will lead to the entanglement between magnon\nmodes. The condition for optimizing the magnon-magnon en-\ntanglement is found and is confirmed by our numerical cal-\nculations. By taking into account the experimentally feasi ble\nparameters, we show the considerable magnon-magnon entan-\nglement can be created. Such entanglement is also shown to\nbe robust against cavity leakage. Our work o ffers new insight\nand perspective for studying the quantum e ffects in complex\nmolecules. These have been manifested by the excited-state\ndynamics in dye molecules and even bacterias implying the\nentangled quantum states when interacting with microcavit ies\n[37–41].\nModel and equation of motion.– We consider a hybrid\nmagnon-cavity system consisting of two bulk ferromagnetic\nmaterials and one microwave cavity mode. The ferromagnetic\nsample contains dispersive spin waves, in which only the spa -2\nFIG. 1: Schematic of cavity magnons. Two YIG spheres are\ninteracting with the basic mode of microcavity in which the\nright mirror is made of high-reflection material so that pho-\ntons leak from the left side. The static magnetic field for pro -\nducing Kittel mode is along z-axis whereas the microwave\ndriving and magnetic field inside cavity are along x-axis.\ntially uniform mode (Kittel mode [42]) is assumed to strongl y\ninteract with cavity photons. The full Hamiltonian of this c av-\nity magnonics system reads [43]\nH=−/integraldisplay\nMzB0dr−µ0\n2/integraldisplay\nMzHandr\n+1\n2/integraldisplay/parenleftigg\nε0E2+B2\nµ0/parenrightigg\ndr−/integraldisplay\nM·Bdr(1)\nwhere B0=B0ezis the applied static magnetic field and\nM=γS/Vmwithγ=e/medenoting the gyromagnetic ra-\ntio.Sstands for the collective spin operator and Vmis volume\nof ferromagnetic material. Hanis the anisotropic field due to\nthe magnetocrystalline anisotropy and has zcomponent only\nowing to the crystallographic axis being aligned along the a p-\nplied static magnetic field. Thereby the anisotropic field is\ngiven by Han=−2KanMz/M2where KanandMdenote the\ndominant 1st anisotropy constant and the saturation magnet i-\nzation, respectively. One can recast the Hamiltonian in Eq. (1)\ninto\nH=−γ2/summationdisplay\nj=1Bj,0Sj,z+γ22/summationdisplay\nj=1µ0K(j)\nan\nM2\njVj,mS2\nj,z\n+/planckover2pi1ωca†a−γ2/summationdisplay\nj=1Sj,xBj,x(2)\nby assuming the magnetic field inside cavity is along x-\naxis. The Holstein-Primako fftransform yields to Si,z=Si−\nm†\nimi,Si,+=(2Si−m†\nimi)1/2mi,Si,−=m†\ni(2Si−m†\nimi)1/2\nwhere Si,±≡Si,x±iSi,yandmirepresents the bosonic anni-\nhilation operator [44]. For the yttrium iron garnets (YIGs)\nwith diameter d=40µm, the density of ferrum ion Fe3+\nisρ=4.22×1027m−3, which leads to the total spin S=\n5\n2ρVm=7.07×1014. This is often much larger than the num-\nber of magnons, so that we can safely approximate Sj,+≃/radicalbig2Sjmj,Sj,−≃/radicalbig2Sjm†\nj. In the presence of external mi-\ncrowave driving field, the e ffective Hamiltonian of hybrid\nmagnon-cavity system is of the form\nHeff=/planckover2pi1ωca†a+/planckover2pi12/summationdisplay\nj=1/bracketleftig\nωjm†\njmj+gj(m†\nja+mja†)\n+∆ jm†\njmjm†\njmj/bracketrightig\n+i/planckover2pi1Ω(a†e−iωdt−aeiωdt)(3)\nwhere the rotating-wave approximation was employed and\ncavity frequency is denoted by ωc. The frequency of Kit-\ntel mode isωj=γBj,0withγ/2π=28GHz/T.gjgives the\nmagnon-cavity coupling and ∆j=µ0K(j)\nanγ2/M2\njVj,mgives the\nKerr nonlinearity, resulting from the on-site magnon-magn on\nscattering. The Rabi frequency Ω=/radicalbig\n2Pdγc//planckover2pi1ωdin last term\nquantifies the strength of the field inside microcavity drive n\nby the microwave magnetic field, where Pdandωdrepresent\nthe power and frequency of the microwave field, respectively .\nγcis the cavity leaking rate. In the rotating frame of mi-\ncrowave field, the dynamics of hyrid cavity-magnon system\nis governed by the quantum Langevin equations (QLEs)\n˙ms=−(iδs+γs)ms−2i∆sm†\nsmsms−igsa+/radicalbig\n2γsmin\ns(t)\n˙a=−(iδc+γc)a−i2/summationdisplay\nj=1gjmj+Ω+/radicalbig\n2γcain(t)(4)\nwhereγsquantifies the magnon dissipation. δs=ωs−\nωd, δc=ωc−ωd.min\ns(t) and ain(t) are the input noise op-\nerators having zero mean and white noise: /angbracketleftmin,†\ns(t)min\ns(t′)/angbracketright=\n¯nsδ(t−t′),/angbracketleftmin\ns(t)min,†\ns(t′)/angbracketright=(¯ns+1)δ(t−t′);/angbracketleftain,†(t)ain(t′)/angbracketright=\n0,/angbracketleftain(t)ain,†(t′)/angbracketright=δ(t−t′) where ¯ ns=[exp(/planckover2pi1ωs/kBT)−1]−1\ndenotes the Planck factor of the s-th magnon mode.\nSince the microcavity is under strong driving by the\nmicrowave field, the beam-splitter-like coupling between\nmagnons and cavity leads to the large amplitudes of both\nmagnon and cavity modes, namely, |/angbracketleftms/angbracketright|,|/angbracketlefta/angbracketright| ≫ 1. In\nthis case, one can safely introduce the expansion ms=\n/angbracketleftms/angbracketright+δms,a=/angbracketlefta/angbracketright+δain the vicinity of steady state,\nby neglecting the higher-order fluctuations of the operator s.\nWe thereby obtain the linearized QLEs for the quadratures\nδXs,δYs,δX,δYdefined asδX1=(δm1+δm†\n1)/√\n2, δY1=\n(δm1−δm†\n1)/i√\n2, δX2=(δm2+δm†\n2)/√\n2, δY2=(δm2−\nδm†\n2)/i√\n2,δX=(δa+δa†)/√\n2,δY=(δa−δa†)/i√\n2\n˙σ(t)=Aσ(t)+f(t) (5)\nwhereσ(t)=[δX1(t),δY1(t),δX2(t),δY2(t),δX(t),δY(t)]Tand\nf(t)=[/radicalbig\n2γ1Xin\n1(t),/radicalbig\n2γ1Yin\n1(t),/radicalbig\n2γ2Xin\n2(t),/radicalbig\n2γ2Yin\n2(t),/radicalbig\n2γcXin(t),/radicalbig\n2γ1Yin(t)]Tare the vectors for quantum fluctuations and3\n(a)\n(b)(c)\n(d)(e)\n(f)\nFIG. 2: 2D plots for (top) magnon-magnon entanglement Em1m2and (bottom) magnon-cavity entanglement Em1awhen\nturning offthe coupling between cavity and the 2nd sphere ( g2=0). (a) g1,2/2π=41MHz,δc/2π=−0.03GHz;\n(b)g1/2π=41MHz, g2=0,δc/2π=−0.03GHz; (c) F1,2=−0.048GHz, g1,2/2π=41MHz; (d) F1,2=\n−0.048GHz, g1/2π=41MHz, g2=0 and (e,f) F1,2=−0.048GHz,δc/2π=−0.03GHz. Other param-\neters areω1,2/2π=10GHz,δ1,2/2π=−1MHz,γ1,2/2π=8.8MHz,γc/2π=1.9MHz and T=10mK.\nnoise, respectively. The drift matrix reads\nA=F1−γ1˜δ1−G1 0 0 0 g1\n−˜δ1−G1−F1−γ1 0 0−g10\n0 0 F2−γ2˜δ2−G20 g2\n0 0−˜δ2−G2−F2−γ2−g20\n0 g1 0 g2−γcδc\n−g1 0−g2 0−δc−γc\n(6)\nwith magnetocrystalline anisotropy quantified by Gs=\n2∆sRe/angbracketleftms/angbracketright2,Fs=2∆sIm/angbracketleftms/angbracketright2and the effective detuning of\nmagnons ˜δs=δs+2/radicalbig\nG2s+F2s=δs+4∆s|/angbracketleftms/angbracketright|2, which in-\ncludes the frequency shift caused by Kerr nonlinearity. The\nmean/angbracketleftm1,2/angbracketrightare given by\n/angbracketleftm1/angbracketright=ig1Ω\n(˜δ1−iγ1)(δc−iγc)−g2\n1−g2\n2(˜δ1−iγ1)\n˜δ2−iγ2,\nand (1↔2)(7)\nBefore the study of entanglement, it is essential to elucida te\nthe mechanism for optimizing the entanglement via Kerr non-\nlinearity. To this end, we proceed via the e ffective Hamilto-nian for quantum fluctuations\nHqf=/planckover2pi12/summationdisplay\ns=1/bracketleftig˜δsδm†\nsδms+˜∆sδm†\nsδm†\ns+˜∆∗\nsδmsδms\n+gs/parenleftig\nδm†\nsδa+δmsδa†/parenrightig/bracketrightig\n+/planckover2pi1δcδa†δa(8)\nwhere ˜∆s= (Gs+iFs)/2. The quadratic terms\nδm†\nsδm†\ns, δmsδmsimply the effective magnon-magnon inter-\naction induced by the magnetocrystalline anisotropy, whic h\nmay be significantly enhanced by strong driving. This, in\nfact, is responsible for the entanglement. To make it elabo-\nrate, let us introduce the Bogoliubov transformation [45, 4 6]\nδβs=usδms−v∗\nsδm†\ns, δβ†\ns=−vsδms+u∗\nsδm†\nswhere us=/radicalig\n1\n2/parenleftig˜δs\nεs+1/parenrightig\n,vseiα=−/radicalig\n1\n2/parenleftig˜δs\nεs−1/parenrightig\n,α=arctan( Fs/Gs) and\nεs=/parenleftig˜δ2\ns−4|˜∆s|2/parenrightig1/2. Inserting these into Eq.(8) we find\nHqf=/planckover2pi12/summationdisplay\ns=1/bracketleftig\nεsδβ†\nsδβs+gs/parenleftig\n(vsδβs+usδβ†\ns)δa\n+(u∗\nsδβs+v∗\nsδβ†\ns)δa†/parenrightig/bracketrightig\n+/planckover2pi1δcδa†δa(9)\nwhich showsεs≃−δcis optimal for the entanglement, due\nto the magnon-photon squeezing term gs(vsδβsδa+v∗\nsδβ†\nsδa†).4\nThis will be confirmed by the latter numerical results when\ntaking into account of experimental parameters.\nEntanglement between magnon modes.– Since we are us-\ning the linearized quantum Langevin equations, the Gaussia n\nnature of the input states will be preserved during the time\nevolution of systems. The quantum fluctuations are thus the\ncontinuous three-mode Gaussian state, which is completely\ncharacterized by an 6 ×6 covariance matrix (CM) defined\nasCi j(t,t′)=1\n2/angbracketleftσi(t)σj(t′)+σj(t′)σi(t)/angbracketright; (i,j=1,2,···,6)\nwhere the average is taken over the system and bath degrees\nof freedoms. Suppose the drift matrix Ais negatively defined,\nthe solution to Eq.(5) is σ(t)=M(t)σ(0)+/integraltextt\n0M(s)f(t−s)ds\nwhere M(t)=exp(At). This enables us to find the equation\nwhich CM obeys\n˙C(t+τ,t)=AC(t+τ,t)+C(t+τ,t)AT+eAτD (10)\nforτ≥0. Thus the stationary CM can be straightforwardly\nobtained by letting τ=0,t→∞ in Eq.(10) that yields to the\nLyapunov equation\nAC∞+C∞AT=−D (11)\nwhere the diffusion matrix is D=diag[γ1(2¯n1+1),γ1(2¯n1+\n1),γ2(2¯n2+1),γ2(2¯n2+1),γc,γc] defined through/angbracketleftfi(t)fj(t′)+\nfj(t′)fi(t)/angbracketright=2Di jδ(t−t′). We adopt the logarithmic negativity\nENto quantify the magnon-magnon and magnon-photon en-\ntanglements by comupting the 4 ×4 CM related to the two\nmagnon modes. This can be achieved by defining EN=\nmax[0,−ln2v−] where v−=min|eig⊕2\nj=1(−σy)P12C∞P12|and\nσyis the Pauli matrix [47, 48]. The matrix P12=σz⊕1\nrealizes the partial transposition at the level of CM. In wha t\nfollows, we will work in the monostable scheme of magnons.\nFurthermore, we will focus on the case of two identical\nmagnons having G1,2=G,F1,2=F,˜δ1,2=˜δ,∆1,2=\n∆,g1,2=g.\nFig.2 shows the magnon-magnon entanglement versus\nsome key parameters of the system. Here we have taken\ninto account the experimentally feasible parameters [35]:\nω1,2/2π=10GHz,δ1,2/2π=−1MHz,γ1,2/2π=8.8MHz\nandγc/2π=1.9MHz for the YIG bulk at low temperature\nT=10mK. First of all we observe from Fig.2(a,b) that\nthe Kerr nonlinearity is responsible for creating the stead y-\nstate entanglement between two magnon modes, evident by\nthe fact that the entanglement dies out when G=F=0.\nThis results from the dominated beam-splitter-interactio n be-\ntween magnon mode and cavity photons, once G=F=0.\nThereby no magnon-cavity entanglement can be created, as\nseen in Fig.2(b). We take the condition εs≃ −δcfor op-\ntimizing the magnon-photon entanglement, as illustrated i n\nFig.2(d) whereεs≃/radicalbig\n3(G2s+F2s). The two-mode squeez-\ning term gs(vsδβsδa+v∗\nsδβ†\nsδa†) squeezes the joint state be-\ntween one magnon mode and cavity photons, which results in\nthe partial entanglement in between. Because the same type\nof interaction occurs when coupling the other magnon mode\nwith cavity, the two distanced magnon modes are expected(a) (b)\nFIG. 3: Entanglement between two magnon modes varies\nwith (a) cavity detuning and (b) driving power. (a,b) Solid\nblue, dotdashed purple and dashing red lines are for the\ncavity leakageγc/2π=1.9MHz,20MHz and 70MHz, re-\nspectively; g1,2/2π=41MHz. (a) Solid blue, dotdashed\npurple and dashing red lines also correspond to driving\npower Pd=393mW,38mW and 11mW, respectively; (b)\nδc/2π=−30MHz. Other parameters are the same as Fig.2.\nto be entangled. This is confirmed in Fig.2(c) manifesting\nthe optimal magnon-magnon entanglement in the vicinity of\nεs≃ −δc. The elaborate transfer from magnon-photon en-\ntanglement to magnon-magnon entanglement is subsequently\nevident by Fig.1S in supplementary material (SM) that the\nconsiderable reduction of magnon-photon entanglement as\nthe coupling of cavity to another sphere is turned on. Since\nthe biparticle entanglement is originated from the Kerr non -\nlinearity quantified by G1,2and F1,2, there must be the in-\nterplay between the couplings Gs,Fsandgswhich is de-\npicted in Fig.2(e,f). In Fig.2(c) we take g1,2/2π=41MHz\nand it impliesδc/2π≃−0.03GHz for the optimal entangle-\nment Em1m2. We then adopt the magnitude of δcfor plot-\nting Fig.2(a,b). Using√\nG2+F2=2∆|/angbracketleftm/angbracketright|2and Eq.(7) for\nthe 40µm-diam YIG spheres, the optimal entanglement with\n|G|=0.038GHz,|F|=0.028GHz (see Fig.2(a)) yields to the\nRabi frequencyΩ= 1.06×1015Hz, corresponding to the drive\npower Pd=314mW. Indeed, the stronger nonlinearity will\ncreate more entanglement between the magnon modes. But\nwe have to ensure the negatively defined matrix Agiven in\nEq.(6). Also, the experimental feasibility of ultrastrong drive\nusing microwave field needs the consideration.\nSince we are working with the strong driving, it is worthy\nof checking the validity of the results obtained above. The\nmagnon description for magnetic materials is e ffective only\nwhen/angbracketleftm†\njmj/angbracketright ≪ 2Njs=5Njwhere Nj=ρjVj,mdenotes\nthe total number of spins in the bulk material. For the 40 µm-\ndiam YIG sphere, Nj≃1.41×1014and the drive power Pd=\n393mW results in|/angbracketleftmj/angbracketright|≃2.3×106, giving/angbracketleftm†\njmj/angbracketright≃5.28×\n1012≪5N=7.07×1014. Hence the condition /angbracketleftm†\njmj/angbracketright≪\n2Njsis fulfilled.\nFig.3 illustrates the entanglement between two magnon\nmodes versus some controllable parameters by considering\nthe 40µm-diam YIG-sphere experiment, where ω1,2/2π=\n10GHz,δ1,2/2π=−1MHz,∆1,2/2π=1µHz, g1,2/2π=\n41MHz,γ1,2/2π=8.8MHz andγc/2π=1.9MHz have been5\ntaken according to Ref.[35]. We observe in Fig.3(a) that\nfor fixed driving power, the magnon-magnon entanglement\nis quite sensitive to cavity detuning δc≡ωc−ωd, reaching\nits maximum atδc/2π≃−0.03GHz. This is consistent with\nthe conditionεj≃−δcas clarified for optimizing the entan-\nglement. Fig.3(b) shows the considerable entanglement whe n\nthe system is driven far-from-equilibrium. This is reasona ble\nbecause the strong external driving significantly enhances the\nKerr nonlinearity that is responsible for both magnon-cavi ty\nsqueezing and entanglement, as elucidated in Eq.(7) and (8) .\nFurthermore, Fig.3 shows that the weaker magnon-magnon\nentanglement is observed when increasing the cavity leakag e.\nBy noting the magnitude, we can still obtain some entangle-\nment, even with a low-quality cavity showing weak magnon-\ncavity coupling where γc=8γ1,2>g1,2denoted by red dashed\nlines. This regime is crucial for detecting the entanglemen t\nused in Refs[17, 18]. in which an additional cavity has a\nbeam-splitter-like interaction with the magnon mode for re ad-\ning out the magnon states associated with the CM. The trans-\nferred entanglement can then be measured through the homo-\ndyne detection by sending a weak microwave probe. This ap-\nproach requires much larger cavity leakage than the magnon\ndissipation, namely, γc≫γ1,2, so that the magnon states can\nremain almost unchanged when switching o ffthe laser driv-\ning.\nThe time-resolved detection of the photons emitting o ffthe\ncavity axis may offer an alternative scheme for entanglement\nmeasurement. The quadrature information of magnon modes\ncan be transferred to the time-gated emitted photons, which\ncan be homodynely detected by interfering with an extra mi-\ncrowave field. This quantum-light-probe scheme may take the\nadvantage of being noninvasive detection for the entanglem ent\nmeasurement.\nConclusion and remarks.– In conclusion, we have proposed\na protocol for entangling the magnon modes in two mas-\nsive YIG spheres, through the Kerr nonlinearity that origi-\nnates from the magnetocrystalline anisotropy. We shew that\nsuch nonlinearity has to be essentially included, for produ c-\ning the entanglement. Our work demonstrated the stationary\nentanglement between two macroscopic YIG spheres driven\nfar-from-equilibrium, within the experimentally feasibl e pa-\nrameter regime. The amount of entanglement is quantified\nby the logarithmic negativity and surprisingly robust agai nst\nthe cavity leakage: entangled quantum state may persist wit h\nlow-quality cavity giving weak magnon-cavity coupling. Th is\nmay be helpful to the experimental design for the entangle-\nment measurement.\nWe should note that our idea for entangling magnon modes\nmay be potentially extensive to other complex systems, such\nas molecular aggregates and clusters, along with the fact\nof similar forms of nonlinear couplings b†bqand∆b†bb†b.\nWith the scaled-up parameters, the long-range entanglemen t\nin molecular aggregates would be anticipated, in that the\nexciton-exciton interaction is of several orders of magnit ude\nhigher than the Kerr nonlinearity resulting from the magne-\ntocrystalline anisotropy. For instance, the two-exciton c ou-pling in J-aggregate and light-harvesting antenna take the\nvalue of∼50cm−1which is∼0.3% of the exciton frequency.\nThis is much stronger nonlinearity than that in YIGs with Ker r\ncoefficient K∼0.1nHz that is∼10−11of its Kittel frequency.\nRecent development in both ultrafast spectroscopy and syn-\nthesis have revealed the important role of quantum coher-\nence which may significantly modify the functions of complex\nmolecules and may help the design of polaritonic molecular\ndevices as well as polariton chemistry. Hence entangling th e\nmolecular aggregates may help the studies of quantum phe-\nnomena in complex molecules.\nWe gratefully acknowledge the support of AFOSR Award\nFA-9550-18-1-0141, ONR Award N00014-16-1-3054 and\nRobert A. Welch Foundation (Award A-1261 & A-1943-\n20180324). We also thank Jie Li and Tao Peng for the useful\ndiscussions.\n∗zhedong.zhang@tamu.edu\n†girish.agarwal@tamu.edu\n[1] Y . 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Pan,1, 2,\u0003Y. Yang,2Z.H. An,1, 3, 4,yand C.-M. Hu2,z\n1State Key Laboratory of Surface Physics, Department of Physics,\nFudan University, Shanghai 200433, People's Republic of China\n2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2\n3Institute of Nanoelectronic Devices and Quantum Computing,\nFudan University, Shanghai 200433, People's Republic of China\n4Shanghai Qi Zhi Institute, 41th Floor, AI Tower, No. 701 Yunjin Road,\nXuhui District, Shanghai, 200232, People's Republic of China\n(Dated: June 6, 2022)\nDissipative coupling of resonators arising from their cooperative dampings to a common reservoir\ninduces intriguingly new physics such as energy level attraction. In this study, we report the nonlin-\near properties in a dissipatively coupled cavity magnonic system. A magnetic material YIG (yttrium\niron garnet) is placed at the magnetic \feld node of a Fabry-Perot-like microwave cavity such that\nthe magnons and cavity photons are dissipatively coupled. Under high power excitation, a nonlinear\ne\u000bect is observed in the transmission spectra, showing bistable behaviors. The observed bistabili-\nties are manifested as clockwise, counterclockwise, and butter\ry-like hysteresis loops with di\u000berent\nfrequency detuning. The experimental results are well explained as a Du\u000eng oscillator dissipatively\ncoupled with a harmonic one and the required trigger condition for bistability could be determined\nquantitatively by the coupled oscillator model. Our results demonstrate that the magnon damping\nhas been suppressed by the dissipative interaction, which thereby reduces the threshold for conven-\ntional magnon Kerr bistability. This work sheds light upon potential applications in developing low\npower nonlinearity devices, enhanced anharmonicity sensors and for exploring the non-Hermitian\nphysics of cavity magnonics in the nonlinear regime.\nI. INTRODUCTION\nNonlinearities are ubiquitous phenomena in various\nphysics \felds. For instance, Kerr nonlinearity and res-\nonant two-level nonlinearity [1], leading to anharmonici-\nties, have been widely investigated in the context of op-\ntics [2]. One signature of nonlinear dynamics is bista-\nbility for a given input exceeding a threshold power and\nmanifests itself as a foldover e\u000bect [3{7]. The e\u000bects of\nnonlinearity have technological implications in sophisti-\ncated optical devices for controlling light with light [8],\nnovel magnetic data storage devices [9], switches with\nbistable metamaterial [10] as well as developed mechani-\ncal devices for emergent applications like energy harvest-\ning [11, 12].\nHybrid quantum systems, which have aroused tremen-\ndous interest for application in quantum information pro-\ncessing [13, 14], have the potential to push the develop-\nment of the realm of nonlinearity. Among various hybrid\nsubsystems, cavity magnonics appears to be an excep-\ntional candidate (see, e.g., [15{30]), which utilizes fer-\nrimagnetic materials like YIG (yttrium iron garnet) to\ncreate collective spin excitations. This cavity magnon-\nics system has resulted in a variety of semiclassical and\nquantum phenomena including, cavity-magnon polari-\ntons [31{33], magnon bistability [29, 34, 35], bidirectional\nmicrowave-optical conversion mediated by ferromagnetic\n\u0003hpan19@fudan.edu.cn\nyanzhenghua@fudan.edu.cn\nzhu@physics.umanitoba.camagnons [36], magnon dark mode [19], synchronization\nvia spin-photon coupling [37], and non-Hermitian exotic\nproperties [25, 38, 39]. Recent work has attempted to\nutilize the photon-magnon coupling freedom to tune the\nnonlinearity. The magnon-polariton bistability has been\nsuccessfully observed in a coherent hybrid system [29]\nand the bistable behavior could also be directly measured\nwith the coupled cavity being pumped [35]. However, in\nthese reported works the photon-magnon are typically co-\nherently coupled, and their coherent coupling enhances\nthe damping of magnon states [40]. Due to the cubic\ndependence of the threshold for nonlinearity on the ef-\nfective damping of magnon [35], the coherent coupling\nof cavity-magnon hybrid system therefore increases the\nnonlinearity threshold which may impede the device ap-\nplication of nonlinearity.\nCoherent coupling originates from the direct spatial\noverlap between photon and magnon modes and forms\nhybrid states with repulsed energy levels and attracted\ndamping rates. In contrast, a dissipative form of magnon-\nphoton interaction [41] requires no direct mode over-\nlap and has been demonstrated to cause level attraction\nand a damping repulsion e\u000bect[42{44]. This dissipative\ncoupling is indirect as it is mediated through a shared\nreservoir, resulting in an imaginary spin-photon coupling\nstrength. Dissipatively coupled systems have important\napplications such as nonreciprocal transport [45], en-\nhanced sensing [46], and non-Hermitian singularities [47,\n48]. So far, however, reported work with dissipative cou-\npling have been focused on the linear regime and the\nnonlinearity with dissipative coupling have not been ex-\nperimentally examined, except a theoretical prediction ofarXiv:2206.01231v1 [cond-mat.mes-hall] 2 Jun 20222\na lower threshold if the (imaginary) coupling strength is\nsu\u000eciently large in an anti-PT regime [49].\nInspired by this, we study a dissipatively coupled\ncavity-magnon system where a YIG sphere is imbed-\nded in a Fabry-Perot-like cavity. The coupled system\nis directly pumped through the cavity and the pumping\npower is su\u000eciently high to create the nonlinear e\u000bect.\nWe experimentally demonstrate that, in the dissipatively\ncoupled nonlinear system, the threshold power for bista-\nbility is lower than the corresponding bound in coherent\nscenario. This is due to the suppression of magnon damp-\ning on-resonance induced by dissipative coupling. In ad-\ndition, such a dissipatively coupled hybridized system\nresults in di\u000berent magnetic \feld and power dependent\nbistable behaviors as the magnon frequency is detuned\nfor on and o\u000b cavity resonant frequency. By introducing\nthe model based on a Du\u000eng oscillator dissipatively cou-\npled with a harmonic oscillator, the experimental results\ncan be well explained. This method allows us to de-\ntermine the required experimental condition for trigger-\ning bistable behavior in a dissipatively coupled system.\nOur work shows that dissipative coupling in a hybrid sys-\ntem with a nonlinear e\u000bect may be utilized to engineer\na lower power threshold for nonlinearity and enhanced\nanharmonicity sensors.\nII. THEORETICAL MODEL\nWe begin with a model considering the nonlinear Kerr\ne\u000bect of the magnon excitation in a YIG sphere which\ndissipatively interacts with the cavity photons. Here\nthe cavity is directly pumped. This dissipatively cou-\npled nonlinear system is characterized by a Hamiltonian\n(the details shown in Appendix A and with reduced units\n~\u00111):\nHtot=!ca+a+!mb+b+Kb+bb+b+i\u0000\u0000\na+b+ab+\u0001\n+ \nd\u0000\na+e\u0000i!t+aei!t\u0001\n; (1)\nwherea+(a) andb+(b) correspond to the creation (an-\nnihilation) operators of the cavity photons at frequency\n!cand of the Kittel-mode magnons at !m, respectively.\nHere, the Kerr e\u000bect of magnons term Kb+bb+borig-\ninates from magnetocrystalline anisotropy in the YIG\nmaterial [29], in which Kis the Kerr coe\u000ecient and is\npositive in our experiment speci\fcally (see the Appendix\nA).\u0000is the dissipative coupling strength between the\ncavity photon and the magnon via their common reser-\nvoir. The last term in the above equation describes that\nthe cavity is pumped by the oscillating microwave \feld,\nwith an amplitude \n d[50].\nBy adopting the Heisenberg-Langevin approach [51],\nthe dynamics of the coupled cavity photon-magnon hy-brid nonlinear system can be described by the equations:\nda\ndt=\u0000i(!c\u0000i\u0014)a+ \u0000b\u0000i\nd; (2)\ndb\ndt=\u0000i(!m\u0000i\r)b\u0000i(Kbb+b+Kb+bb) + \u0000a: (3)\nUnder the mean-\feld approximation [46], the higher-\norder expectations can be decoupled, so that the non-\nlinear termhb+bbiandhbb+bican be simpli\fed as jbj2b,\nthen the dynamics of the dissipatively coupled magnon-\nphoton system follows as,\nda\ndt=\u0000i(!c\u0000i\u0014)a+ \u0000b\u0000i\nd; (4)\ndb\ndt=\u0000i\u0000\n!m+ 2Kjbj2\u0000i\r\u0001\nb+ \u0000a; (5)\nwhere\u0014=\u0014i+\u0014eis the total damping of the cavity mode,\nwith\u0014i(\u0014e) being the intrinsic (external) damping of the\ncavity mode. \ris the damping rate of the Kittel-mode.\nSuppose the photon and magnon mode have time depen-\ndence ofe\u0000i!t, then Eqs. (4) and (5) can be simpli\fed as\n(!\u0000!c+i\u0014)a\u0000i\u0000b= \nd; (6)\u0010\n!\u0000!m\u00002Kjbj2+i\r\u0011\nb=i\u0000a: (7)\nEq. (6) can be expressed as a= (i\u0000b+\nd)=(!\u0000!c+i\u0014).\nBy substituting such an expression into Eq. (7), we get\njbj\u0002\n\rm0\u0000i\u0000\n\u000e!m0\u00002Kjbj2\u0001\u0003\n=\u0000\nd\n\u000e!c+i\u0014: (8)\nHere,\u000e!m0=\u000e!m+\u0011\u000e!cwith\u000e!m=!\u0000!m,\u000e!c=\n!\u0000!cdenotes e\u000bective frequency shift, and\n\rm0=\r\u0000\u0011\u0014; (9)\ndenotes the e\u000bective damping of the magnon for the dis-\nsipative coupling system, where the negative sign shows\nthe suppression e\u000bect of the magnon damping. We note\nthat for the coherent coupling, the sign is positive rep-\nresenting the enhancement e\u000bect of the magnon damp-\ning [29, 35]. The coe\u000ecient \u0011= \u00002=(\u000e!2\nc+\u00142) stands for\nthe transfer e\u000eciency of the excitation power Pfrom the\ninput port into the magnon system, and \u0011is dependent\non the dissipative coupling strength \u0000, the frequency de-\ntuning\u000e!c, and the total damping \u0014of the cavity.\nTaking the squared modulus of Eq. (8) and de\fning\n\u0001m\u00112Kjbj2, which is the shift of the magnon fre-\nquency [34], we have\n\u0001mh\n\r0\nm2+ (\u000e!0\nm\u0000\u0001m)2i\n= 2\u0011K\n2\nd: (10)\nEquation (10) has a similar form to the uncoupled Du\u000b-\ning oscillator [3], except that \r0\nmand\u000e!0\nmare the e\u000bective\ndamping and frequency shift of the magnon, respectively,\nwhich result from dissipative coupling with the cavity\nphoton. The right term of Eq. (10) denotes the e\u000bective\ndrive \feld of the magnon via dissipative interaction with\nthe cavity photon as the \feld directly pumps the cavity\nrather than the YIG sphere.3\nA. Bistability and the transition point\nEquation (10) describes an oscillator with cubic nonlin-\nearity, called the Du\u000eng equation. As for this equation,\nthere are three real roots for a \fnite range of frequen-\ncies when \n dexceeds a speci\fc value called the critical\n\feld \n t. Among the three roots, one is unstable and the\nother two are stable, called bistability, which is a signa-\nture of the anharmonic oscillator. We will focus on the\ntwo boundaries (hereafter termed as up border and down\nborder) inbetween which bistability occurs. As there are\nabrupt transitions between two stable states at up and\ndown borders, the transition points of bistability are de-\ntermined by the condition d\nd=d\u0001 m= 0, i.e.,\n3\u00012\nm\u00004\u000e!0\nm\u0001m+\u000e!0\nm2+\r0\nm2= 0: (11)\nAccording to the root discriminant of the quadratic equa-\ntion, when 4 \u000e!0\nm2\u000012\r0\nm2= 0 , i.e.,\n\u000e!0\nm=\u0000p\n3\r0\nm; (12)\nthere is only one root of Eq. (11), which corresponds to\nthe critical condition of bistable behavior.\nWhen 4\u000e!0\nm2\u000012\r0\nm2>0, there are two real roots\nwhich correspond to the up border and down border of\nthe bistability. By adopting the approximationp\n3\r0\nm\u001c\n\u000e!0\nmto solve Eq. (11), then the upper border satis\fes\n\u000e!m= 33r\n\u0011KS2P\n2\u0000\u0011\u000e!c;(up) (13)\nand the down border satis\fes\n\u000e!m=2\u0011KS2P\n(\r\u0000\u0011\u0014)2\u0000\u0011\u000e!c;(down) (14)\nwhere \n d=Sp\nP, withSdescribing the conversion ef-\n\fciency from input power Pto the \feld \n ddriving the\ncavity resonance mode. The magnitude of Sdepends\non the frequency, phenomenologically introduced exter-\nnal loss of the cavity \feld, and the loss in the cable which\nconnects the device. Based on Eqs. (13) and (14), we\ncan come to conclusions that both the upper border and\ndown border have input power dependence, with the up-\nper border satisfying \u000e!m/P1=3compared with down\nborder satisfying \u000e!m/P. These dependencies are\ndi\u000berent from those of coherently coupled anharmonic\noscillators [35], where the upper border has \u000e!m/P\ndependence and the down border \u000e!m/P1=3depen-\ndence. The di\u000berence arises from that the magnon shifts\nto higher frequencies with negative Kerr coe\u000ecient K\nwhile that of our work shifts to lower frequencies with\npositive Kerr coe\u000ecient. In fact, the two cases can be\nuni\fed where the border with a large frequency shift has\naPdependence and the border with a small frequency\nshift has aP1=3dependence.\nThe above discussion is based on tuning the magnetic\n\feld to obtain bistability, called \feld bistability, and itstransition point has power dependence. On the other\nhand, we can adjust input power to obtain bistability,\ncalled power bistability, and its transition point has mag-\nnetic \feld dependence. According to Eqs. (13) and (14),\nwe can get borders of power bistability by regarding mag-\nnetic \feldHas an independent variable\nP=2(\u000e!m+\u0011\u000e!c)3\n27\u0011KS2;(up) (15)\nP=(\u000e!m+\u0011\u000e!c)(\r\u0000\u0011\u00142)\n2\u0011KS2:(down) (16)\nAs the magnon resonance detuning \u000e!m=\r0\u000eHwith\nde\fnition\u000eH=H\u0000Hr(Hris the resonant magnetic\n\feld and\r0is gyromagnetic ratio), the up and down bor-\nder of power bistability have respective j\u000eHj3andj\u000eHj\ndependence when up-sweeping and down-sweeping power\nas shown in Eqs. (15) and (16), respectively.\nB. Critical condition of bistability\nWhen Eq. (11) has only one real solution, the bista-\nbility vanishes because the two transition points collapse\nto one point. The critical driving \feld of the cavity cor-\nresponds to the threshold beyond which the bistability\nappears. Thus, by substituting the critical condition\nof bistability shown in Eq. (12) and the only solution\n\u0001m= 2=3\u000e!minto Eq. (10), the critical \feld \n t(critical\npowerPt) of \feld bistability is obtained:\n\n2\nt=4p\n3\n9K\r03\nm\n\u0011; (17a)\nPt=4p\n3\n9KS2\r03\nm\n\u0011: (17b)\nThese equations imply that the threshold has cubic de-\npendence on the e\u000bective damping of magnon \r0\nm, which\nis similar to that of coherently coupled nonlinear sys-\ntem [35]. The threshold of nonlinearity for the hybrid sys-\ntem have similar dependence with that of uncoupled ferri-\nmagnetic resonance which has cubic dependence upon the\ndamping of magnon (see Table I). Next, we compare the\nthreshold value for the dissipatively coupled system with\nthat of the coherently coupled system. For this purpose,\nthe threshold for a generic two-mode system involving\nboth coherent and dissipative coupling is generated by\nsubstituting \r0\nm=\r+\u0011\u0014with\u0011= (g+i\u0000)2=(\u000e!2\nc+\u00142)\ninto Eqs. (17a) and (17b) where gis the coherent cou-\npling strength. Supposing scenarios \u0000 = 0 and g= 0, we\nwill get the threshold for coherently coupled and dissipa-\ntively coupled systems, respectively. By comparing the\nthresholds in relation to the nature of coupling (setting\n\u000e!c= 0), we have\n(\ntd)2\n(\ntc)2=Ptd\nPtc=g2\n\u00002\f\f\f\f1\u0000\u00002=\u0014\r\n1 +g2=\u0014\r\f\f\f\f3\n; (18)4\nwhere we have assumed that the two kinds of coupling\nsystems have the same damping coe\u000ecient and \n td(Pd\nt),\n\ntc(Pc\nt) denote the critical \feld (power) of bistability for\ndissipative and coherent coupling respectively.\nTABLE I. Thresholds of the ferrimagnetic materials in the\ncoupled and uncoupled scenarios.\nsample threshold e\u000bective dampinga\nYIG sphere [52] \r2\n0h2\nt=8p\n3\n9\u001f\r3/\r3b\r=\r0\u0001Hc\nPy \flm [6] \r2\n0h2\nt=16p\n3\n9\r0Ms\r3/\r3\r=\r0\u0001H\nYIG (Coh.)d[35] \n2\nt=4p\n3\n9K\u0011\r03\nm/\r03\nm\r0\nm=\r+\u0011\u0014\nYIG (Dis.)e\n2\nt=4p\n3\n9K\u0011\r03\nm/\r03\nm\r0\nm=\r\u0000\u0011\u0014\naE\u000bective damping of magnon of each system.\nbHere,htis the critical drive magnetic \feld, and \u001fis related to\ncrystalline anisotropy [52].\nc\u0001His the linewidth of the ferromagnetic resonance.\ndYIG sphere is placed in a microwave cavity, and they interact\ncoherently.\neYIG sphere is placed in a microwave cavity, and they interact\ndissipatively.\nNotably, the expression of Eq. (18) is always less than\n1 for nonzero \u0000 and g, revealing a consistently lower\nthreshold in the dissipatively coupled system than that\nin the coherently coupled system. In order to make a fair\ncomparison, we assume identical magnitudes of coupling\nstrength, i.e., \u0000 =2\u0019=g=2\u0019= 21 MHz. Then, by substi-\ntuting the value of \r=2\u0019= 5:1 MHz and \u0014=2\u0019= 126 MHz\ninto Eq. (18), the dissipative-coherent threshold ratio be-\ncomes a \fnite value of 0 :0064, implying that the dissipa-\ntive coupling may lead to very low power threshold of the\nbistability. Such a low threshold arises intrinsically from\nthe suppression of the magnon damping on-resonance [see\nEq. (9)] and the cubic dependence of the threshold upon\nthe e\u000bective magnon damping [indicated by Eqs. (17a)\nand (17b)].\nOn the other hand, we can obtain the requirement\nto observe the power bistability. The only one root of\nEq. (11) described by Eq. (12) corresponds to the criti-\ncal magnetic \feld to generate power bistability. Combin-\ning the relation \u000e!m=\r0\u000eHand Eq. (12), the critical\nmagnetic \feld is given by\n\u000eH=1\n\r0[\u0000\u0011\u000e!c\u0000p\n3(\r\u0000\u0011\u0014)]: (19)\nCompared with the critical magnetic \feld condition H\u0000\nHr=p\n3\r=\r 0of uncoupled magnetic systems [6, 53],\nthe extra term of the magnetic \feld shift \u0000p\n3\u0011\u0014=\r 0in\nEq. (19) results from the e\u000bective damping of the magnon\nresonance near !=!c[18]. The factor \u0000\u0011\u000e!c=\r0rep-\nresents the additional resonance shift of the magnon\nwhich arises from the interaction between magnon and\ncavity.C. Transmission spectra with bistability\nThe bistability can be detected experimentally via mi-\ncrowave transmission spectra of the cavity. In this sec-\ntion, we show the magnon frequency shift \u0001m(due to the\nKerr nonlinearity) is observed in the transmission spec-\ntra of the cavity. By considering that the cavity mode\ncouples with the input energy from the port, the dynamic\nequation Eq. (4) can be rewritten as\nda\ndt=\u0000i(!c\u0000i\u0014)a+ \u0000b+p\u0014ecin; (20)\nwherecinis the input \feld. From Eq. (7), the amplitude\nof the cavity \feld can be derived\nb=i\u0000a\n!\u0000!m\u0000\u0001m+i\r: (21)\nAccording to the input-output theory [51], the relation\nof input-output \feld can be described as\ncout\u0000cin=\u0000p\u0014ea; (22)\nwherecoutis the output \feld. Combining Eqs. (20), (21),\n(22), with the de\fnition of transmission coe\u000ecient S21=\ncout=cin, we can obtain the transmission coe\u000ecient\nS21= 1 +\u0014e\ni(!\u0000!c)\u0000\u0014\u0000\u00002=[i(!\u0000!m\u0000\u0001m)\u0000\r]:\n(23)\nThe transmission of nonlinear dissipatively coupled sys-\ntem in Eq. (23) can be reduced to that of two cou-\npled linear oscillators if we perform the transformation\n!m+\u0001m!!m. Here,\u0001mis the nonlinear magnon res-\nonance frequency shift which is the solution of the Du\u000b-\ning equation as shown in Eq. (10). The nonlinear e\u000bect\ncan be observed through cavity transmission due to the\ninteraction between the cavity photon and magnon.\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\nA. The hybridized cavity-magnon mode in the\nlinear range\nThe experimental setup is sketched in Fig. 1(a). A pol-\nished YIG sphere with 1 mm diameter is placed at the\nmagnetic \feld node of a Fabry-Perot-like cavity, which is\nan assembled apparatus with a circular waveguide con-\nnecting to coaxial-rectangular adapters [32, 35], such that\nthe excited magnon and cavity photon are dissipatively\ncoupled. The cavity is pumped by a microwave gener-\nator, and the transmission is detected by a signal ana-\nlyzer. The embedded YIG sphere is placed in a static\nmagnetic \feld Hproduced by tunable electromagnets at\nroom temperature, which is not depicted in Fig. 1(a).\nWe \frst conduct the experiment under linear condi-\ntions, using a vector network analyzer to measure the5\ntransmission of the cavity with input power below the\nthreshold to create the nonlinear e\u000bect, so that the Kerr\nnonlinear e\u000bect is negligible. Our cavity resonance is\nat!c=2\u0019= 12:8888 GHz. The resonance frequency of\nKittel-mode in our experiments follows the dispersion\n!m=\r0(Hr+HA), where\r0=2\u0019= 29:86\u00160GHz/T is the\ngyromagnetic ratio, \u00160HA=\u00006:1 mT is the anisotropy\n\feld, andHris the biased static magnetic \feld at reso-\nnance. When the frequency of the Kittel-mode is tuned in\nresonance with the cavity microwave photons, the stan-\ndard level attraction of the hybridized modes, which is\nthe signature of dissipative coupling, was measured and\nis shown in Fig. 1(b). On the left side of this level attrac-\ntion, an additional mode split caused by the high order\nspin wave is not of immediate interest for the discussion\nof dissipatively coupled nonlinear bistable e\u000bect. The dis-\npersion of the hybridized cavity mode and magnon mode\nis shown in Fig. 1(c), where points A and E correspond to\nfar o\u000b-resonance condition with j!\u0000!cj\u001d2\u0000, and point\nC indicates on-resonance condition with j!\u0000!cj= 0, and\npoints B and D indicate an intermediate frequency condi-\ntion withj!\u0000!cj\u00182\u0000. The dissipative coupling strength\ncan be determined by the separated gap at !m=!c\nin Fig. 1(d), i.e., \u0000 =2\u0019= 21 MHz. The intrinsic and\nextrinsic linewidth of cavity-mode \u0014i=2\u0019= 2:58 MHz,\n\u0014e=2\u0019= 126 MHz are obtained by \ftting the transmis-\nsion coe\u000ecient spectra when j!c\u0000!mj\u001d2\u0000 where the\ncoupling e\u000bect is negligible. As seen from Fig. 1(d), the\n\ftting agrees well with the experimental results. The\nintrinsic and extrinsic dampings of the magnon are 1 :6\nMHz and 3 :5 MHz respectively. Hence, the total damp-\ning of magnon is \r=2\u0019= 5:1 MHz.\nB. Field foldover hysteresis loop\nIn this section, nonlinear e\u000bects in our coupled cavity-\nmagnon system for on- and far o\u000b-resonance frequencies\nare measured by sweeping the magnetic \feld. Here high\nmicrowave powers provided by a microwave generator are\nused to drive the large angle precession of the magnon,\nwhile the transmission signal is measured by a signal an-\nalyzer, as depicted in Fig. 1(a).\nWe start our measurements in the linear range by\nsetting the output power of the microwave generator\nto be 0:1 mW. The transmission was measured at an\non-resonance frequency !=2\u0019= 12:8888 GHz indicated\nby point C in Fig. 2(c), and o\u000b-resonance frequen-\ncies!=2\u0019= 12:6000 GHz, 13 :5000 GHz indicated by\npoints A and E in Fig. 2(c), when we perform up-\nsweeping and down-sweeping magnetic \feld, as shown\nin Figs. 2(a), 2(c) and 2(e), respectively. At conditions\n!=2\u0019= 12:6000 GHz and !=2\u0019= 13:5000 GHz, where\nthe magnon mode is dominant, the spectra show a min-\nimum transmission at the resonance condition H=Hr\nbecause of strong absorption due to the magnon exci-\ntation as shown in Figs. 2(a) and 2(e). In contrast,\nthe spectrum shows a maximum transmission at condi-\nFIG. 1. (a) Illustration of the experimental setup, where a\nYIG sphere is imbedded in an assembly cavity. Here, the\nYIG is at the node of microwave magnetic \feld. The cavity\nis pumped by the microwave generator and the transmission\nis measured by signal analyzer. (b) Transmission coe\u000ecient\nmapping of the hybridized cavity-magnon system, with level\nattraction implying dissipative coupling. White dashed line\nindicates the cut at the coupling point with !m=!c. (c)\nDispersion relation of hybridized cavity and magnon mode,\nwhere points A and E indicate o\u000b-resonance with j!\u0000!cj\u001d\n2\u0000, and point C indicates on-resonance with j!\u0000!cj= 0, and\npoints B and D indicate intermediate frequencies with j!\u0000\n!cj\u00182\u0000. (d) The \fxed \feld cut of transmission coe\u000ecient\nmapping at the coupling condition !m=!cindicated by\nwhite dashed line in (b), with symbols for the experimental\ndata and solid line for calculation based on Eq. (23).\ntion!=!cas shown in Fig. 2(c). This peak origi-\nnates from two factors: (1) the cavity strongly absorbs\nmicrowaves at resonance near !=!cwith the small\ntransmission producing the \rat background; (2) when\nthe cavity mode dissipatively couples to a YIG magnon\nmode at!m=!c, the hybrid system will thus result in\na maximum transmission signal at H=Hr. The peak\nof transmission spectra indicates half-photon and half-\nmagnon mode.\nIn order to study the nonlinear e\u000bect, we increase\npower above the threshold. By up- and down-sweeping\nthe magnetic \feld, we observe that two abrupt jumps of\ntransmission spectra occur at di\u000berent static magnetic\n\feld biases H, corresponding to abrupt transitions be-\ntween these two stable states. In the range of the tran-\nsition, a hysteresis loop is clearly seen in the up- and\ndown-sweeping traces of transmission spectra shown in\nFigs. 2(b), 2(d) and 2(f). This behavior can be explained\nby Eq. (10), which predicts the behavior of bistability and\ntransitions between the two stable states when the power\nis above the threshold. The hysteresis loop becomes more\nevident with the increasing power, because the transition\npoints will depart from each other as microwave power6\nincreases, as shown in Fig. 4.\nThe bistability of o\u000b- and on-resonance have distinctly\ndi\u000berent behaviors in our magnon-cavity system. When\nthe system is o\u000b-resonance, i.e., j!\u0000!cj\u001d2\u0000, the \feld\nhysteresis loops (Figs. 2(b) and 2(f)) are clockwise when\nconsidering the up- and down-sweeping direction of the\nstatic magnetic \feld. In contrast, when the system is\non-resonance, i.e., at !=!c, the \feld hysteresis loop in\nFig. 2(d) is counterclockwise. This behavior is quite dif-\nferent from the bistability of coherently coupled magnon-\nphoton system as measured in Ref. [35]. The work in\nRef. [35] demonstrated when K < 0 there is clockwise\nhysteresis for on-resonance (with peak background) and\nanti-clockwise hysteresis for o\u000b-resonance (with dip back-\nground).\nThe direction of hysteresis depends on the sign of K\nand the background shape of the resonance. For any\nnonlinear mode with a Lorentzian dip or peak resonance\nthat is excited with high power, the trace of the trans-\nmission spectrum will jump at the last transition point\nFIG. 2.jS21j2versus H with (a) P= 0:1 mW and (b) P=\n200 mW at o\u000b-resonant frequency !=2\u0019= 12:6000 GHz.\njS21j2versusHwith (c)P= 0:1 mW and (d) P= 200 mW\nat on-resonant frequency !=2\u0019= 12:8888 GHz.jS21j2versus\nHwith (e)P= 0:1 mW and (f) P= 200 mW at o\u000b-resonant\nfrequency!=2\u0019= 13:5000 GHz. Blue (orange) circle symbols\nare experimental data by up-sweeping (down-sweeping) static\nmagnetic \feld H. The solid curves are calculated. Dashed\ngreen lines with arrows indicate the transition process of bista-\nbility and dashed green lines without arrows indicate the un-\nstable state.\nFIG. 3.jS21j2versus H with (a) P= 0:1 mW and (b) P=\n200 mW at intermediate frequency !=2\u0019= 12:7688 GHz.\njS21j2versusHwith (c)P= 0:1 mW and (d) P= 200 mW at\nintermediate frequency !=2\u0019= 12:9288 GHz. Blue (orange)\ncircle symbols are experimental data by up-sweeping (down-\nsweeping) the static magnetic \feld H. The solid curves are\ncalculated. Dashed green lines with arrow indicate the tran-\nsition process of bistability and dashed green lines without\narrow indicate the unstable state.\nalong the sweeping direction because of the hysteresis\nphenomena. For instance, suppose K > 0, the trace\nof the transmission spectrum with peak background will\njump at the right transition point from a low to high am-\nplitude by up sweeping the magnetic \feld, while it will\njump at the left transition point from a high to low am-\nplitude by down sweeping the magnetic \feld. Thus, the\nhysteresis for K > 0 and peak background lineshape is\ncounterclockwise. By the same approach, the direction of\nhysteresis can be summarized in Table II, through which\nthe di\u000berence in the direction of hysteresis among our\nmeasurement and the work of [29, 35] is well explained.\nTABLE II. Direction of hysteresis.\npeak dip\nK > 0 counterclockwise clockwise\nK < 0 clockwise counterclockwise\nWhenj!\u0000!cj\u00182\u0000 indicated by points B and D shown\nin Fig. 2(c), a di\u000berent foldover hysteresis loop is seen\nat intermediate frequencies above and below !c. Gener-\nally the lineshape of transmission spectrum is symmetric\nwhen the power is below the nonlinear threshold, for in-\nstance, a typical Lorentzian peak characteristic at !=!c\nand a Lorentzian dip j!\u0000!cj>>2\u0000. However, the line-\nshape of transmission spectrum is asymmetric when the\nfrequency is tuned to the region among j!\u0000!cj\u00182\u0000,\nshown in Figs. 3(a) and 3(c) with input power 0 :1 mW.\nThis asymmetric lineshape, similar to that in the coher-7\nent scenario [35], is due to Fano-like resonance [54]. How-\never, their polarities are opposite because of di\u000berent\ncoupling mechanism. As microwave power is increased\nto 200 mW, in contrast to the general hysteresis loop\non- and far o\u000b-resonance, a butter\ry-like hysteresis loop\nappears and the polarities of the butter\ry-like hysteresis\nloop are opposite when the microwave frequency is set at\nintermediate frequencies above and below !cas shown\nin Figs. 3(b) and 3(d). This di\u000berence of shape results\nfrom the transition direction of bistability. For on- and\no\u000b-resonance frequency, the direction of two transitions\nare opposite, where one is from the low to high transmis-\nsion, and another is from the high to low transmission.\nWhile, for the intermediate frequencies, the direction of\ntwo transitions are the identical, i.e., both from high to\nlow transmission or from low to high transmission.\nThe e\u000bective dampings of the magnon are \r0\nm=2\u0019=\n4:50;0:20;0:10;0:15;5:10 MHz by \ftting the\ntransmission spectra when P= 0:1 mW for di\u000ber-\nent frequencies A-E, respectively. The \ftted e\u000bec-\ntive damping of the magnon reveals that the damp-\ning of the magnon is suppressed on-resonance due to\nthe dissipative interaction between the cavity and the\nmagnon, which is consistent with the result of Ref. [40].\nThen, with the damping parameters of the cavity and\nmagnon extracted from the linear process and the so-\nlution of Eq. (10), we obtain the \ftted parameter\nKS2= 4:5\u000210\u00009;6:2\u000210\u00008;8:7\u000210\u00008;9:9\u0002\n10\u00008;7:0\u000210\u00009GHz3=mW at cavity frequency !=2\u0019=\n12:6000;12:7688;12:8888;12:9688;13:5000 GHz, re-\nspectively. (Here, KandScan not be determined in-\ndividually.) In addition, the \ftted transmission spec-\ntrum as a function of magnetic \feld can reproduce the\n\feld-sweeping bistability as shown with green line in\nFigs. 2 and 3. This agreement veri\fes the validity of our\ngeneralized model which describes dissipatively coupled\nDu\u000eng oscillator and linear oscillator in this quasi-one-\ndimensional cavity [32].\nC. Power dependence of transition and threshold\nfor \feld foldover hysteresis loop\nWe have observed that the resonance gradually shifts\ntoward lower Hwhen increasing the microwave power be-\ncause the Kerr coe\u000ecient Kis positive (see Appendix A).\nWhile this is di\u000berent from the negative Kerr coe\u000ecient\nsystem where the resonance gradually shifts toward high\nH[35]. When the power is above the threshold, the bista-\nbility appears, and its two transitions will shift depending\non the power. Figures. 4(a)-4(e) show the jump positions\nas a function of the microwave power at !=2\u0019= 12:6000,\n12:7688, 12:8888, 12:9288, and 13 :5000 GHz, respectively.\nAs predicted by Eqs. (13) and (14), the up-sweeping tran-\nsition (purple symbols) follows a P1=3dependence (solid\nline) and the down-sweeping transition (blue symbols)\nfollows a linear Pdependence (solid line) with \ftted pa-\nrametersKS2= 4:3\u000210\u00009, 5:8\u000210\u00008, 8:9\u000210\u00008, 8:7\u0002\nFIG. 4. The jump position of \feld foldover hysteresis versus\nP at cavity frequencies (a) !=2\u0019= 12:6000 GHz, (b) !=2\u0019=\n12:7688 GHz, (c) !=2\u0019= 12:8888 GHz, (d) !=2\u0019=\n12:9288 GHz, (e) !=2\u0019= 13:5 GHz. Purple (blue) circle sym-\nbols are experimental results of forward (backward) H \feld\nsweeping. The yellow and cyan solid curves are \ftted using\nEqs. (13) and (14) for forward and backward sweeping, respec-\ntively. (f) The threshold for coherent and dissipative coupling\nsystem when o\u000b- and on-resonance. The threshold for co-\nherent coupling is from Ref. [35]. For simplicity, we consider\neach threshold relative to that of o\u000b-resonance in each system.\nHere we have utilized the relation \n2\nt=\n2\nt(off)=Pt=Pt(off),\nwhere \nt(off)andPt(off)are the critical driving \feld and\npower o\u000b-resonance, respectively.\n10\u00008, 6:5\u000210\u00009GHz3=mW, respectively. This KS2\nmagnitudes determined by \ftting the transition points\nwith the power dependence in Eq. (14) is comparable\nto that \ftted by transmission spectrum with Eq. (23).\nThe agreement of KS2magnitudes \ftted by two di\u000ber-\nent methods is gratifying in view of the error.\nThe power dependencies are di\u000berent from those of a\ncoherently coupled nonlinear system, where the down-\nsweeping jump follows a P1=3dependence and the up-\nsweeping jump follows a linear Pdependence [35].\nBecause the positive Kerr coe\u000bcient corresponds with\n\u000e!m<0 and negative Kerr coe\u000bcient corresponds with\n\u000e!m>0, these opposing frequency shifts will lead to the\nreversed power dependence of two transition points.\nFigures 4(a)-4(e) record the up-sweeping and down-\nsweeping transition point when increasing power for on-\nand o\u000b-resonance frequencies A-E. We can extract the8\nFIG. 5. The power foldover hysteresis with \fxed static magnetic \feld at cavity frequencies (a) !=2\u0019= 12:6000 GHz, (b) !=2\u0019=\n12:7688 GHz, (c) !=2\u0019= 12:8888 GHz, (d) !=2\u0019= 12:9288 GHz, (e) !=2\u0019= 13:5000 GHz. The recorded power foldover\nhysteresis transition points versus magnetic \feld detuning H\u0000Hrat cavity frequencies (f) !=2\u0019= 12:6000 GHz, (g) !=2\u0019=\n12:7688 GHz, (h) !=2\u0019= 12:8888 GHz, (i) !=2\u0019= 12:9288 GHz, (j) !=2\u0019= 13:5 GHz.\nthreshold for each frequency and plot them in Fig. 4(f) for\nthe dissipatively [marked by orange arrows in Figs. 4(a)-\n4(e)] and coherently (data from Ref. [35]) coupled sys-\ntems with coupling strength \u0000 =2\u0019= 21 MHz, g=2\u0019=\n18 MHz, respectively. The results in Fig. 4(f) reveal\nthat the threshold of bistability on-resonance is about 2/3\nof those o\u000b-resonance in a dissipatively coupled system.\nThis bears stark disparities with a coherently coupled\nsystem, where the threshold of the cavity on-resonance\nis 2.5-fold of that of o\u000b-resonance in Ref. [35]. The\nresults imply that the dissipative coupling indeed re-\nduces the threshold despite the fact that our coupled\nsystem deviates from anti-PT conditions. In fact, the\nobserved improvement of lower threshold originates from\nthe threshold's cubic dependence on e\u000bective magnon\ndamping which remains valid in the dissipative coupling\ncase (see Table I) and the suppressed magnon damping\nin our dissipative coupling condition [(see Eq. (9)].\nD. Power foldover hysteresis loop and critical\nmagnetic \feld for power bistability\nThe hysteresis loops can also be observed by up-\nsweeping and down-sweeping power at \fxed !and\nHas shown in the upper panels (a)-(e) of Fig. 5,\nwhere we set the biasing magnetic \feld \u000eHto be\n\u00000:80,\u00001:04,\u00000:54,\u00000:64,\u00000:76 mT at cavity fre-\nquency!=2\u0019= 12:6000, 2:7688, 12:8888, 12:9288,\n13:5000 GHz, respectively. In contrast to the case of\nK < 0 with opposite direction for power and \feld hys-\nteresis [35], here for K > 0, they have identical direction\nat each frequency A-E as shown in Figs. 2(b), 2(d), 2(f),\n3(b), 3(d), and 5(a)-5(e).\nThe bottom panels (f)-(j) of Fig. 5 show that the tran-\nsition points of power bistability have j\u000eHj3andj\u000eHj\ndependence for up-sweeping and down-sweeping power\nrespectively, which agree with the theoretical predictionof Eqs. (15) and (16). As seen in Figs. 5(f)-5(j), the two\ntransition points depart from each other, and the area\nof power hysteresis loops becomes larger when the bias\nmagnetic \feld is tuned to be away from Hr, and vice\nversa. At a critical magnetic \feld Hc, the power hystere-\nsis loops disappear. This phenomenon can be explained\nby Eq. (19), which implies the requirement of produc-\ning power hysteresis loops is that the biasing magnetic\n\feld should be below the critical magnetic \feld at each\nfrequency A-E.\nIV. CONCLUSIONS\nTo summarize, we have observed both the \feld and\npower bistability of the dissipatively coupled cavity-\nmagnon system. A theoretical model is studied in which\na Du\u000eng and linear oscillator are dissipatively coupled\nto explain the bistable behaviors. Such a dissipatively\ncoupled hybridized system results in distinctly di\u000ber-\nent bistable behaviors, like butter\ry-like, clockwise, and\ncounterclockwise hysteresis which are visualized through\ntransmission spectra. For the \feld bistability, the tran-\nsition points show P1=3andPrespective dependence\nwhen up-sweeping and down-sweeping the magnetic \feld.\nCorrespondingly, for the power bistability, the transi-\ntion points show j\u000eHj3andj\u000eHjdependence when up-\nsweeping and down-sweeping power, respectively. Mean-\nwhile, the critical condition required for observing \feld\nand power bistability is obtained. With the suppressed\nmagnon damping and therefore lowered threshold for\nbistability, our system may lay the foundation for wide\napplications of very low power nonlinearity devices. Be-\nsides, bene\fting from \rexible tunability with, e.g., the\nmagnon frequency, the interaction strength between cav-\nity and magnon, the drive power, the bistability of the\ncavity magnonics system may be potentially applied in\nemergent applications like memories and switches.9\nACKNOWLEDGMENTS\nThis work has been funded by NSERC Discov-\nery Grants and NSERC Discovery Accelerator Supple-\nments (C.-M. H.). Z.H. A. acknowledges the \fnancial\nsupport from the National Natural Science Foundation\nof China under Grant Nos. 12027805/11991060, and\nthe Shanghai Science and Technology Committee un-\nder Grant Nos. 20JC1414700, 20DZ1100604. H. Pan\nwas supported in part by the China Scholarship Council\n(CSC). The authors thank Garrett Kozyniak and Bentley\nTurner for discussions and suggestions.\nAppendix A: Hamiltonian of the coupled hybrid\nsystem\nThe cavity-magnon hybrid system shown in Fig. 1(a)\nincludes a small YIG sphere with Kerr nonlinearity which\nis dissipatively coupled to a Fabry-Perot-like cavity, and\nthe cavity is driven by a microwave \feld. The Hamil-\ntonian of such a system consists of four parts (setting\n~\u00111):\nHtot=Hc+Hm+HI+Hd; (A1)\nhereHc=!c\u000b+\u000bcorresponds to the bare Hamiltonian\nof the cavity mode, with the creation (annihilation) op-\nerator\u000b+(\u000b) at frequency !c.\nIn our experiment, we apply a uniform static magnetic\n\feldH=Hezorientating along the z-axis, and the YIG\nsphere has volume Vm. The static magnetic \feld is used\nto align the magnetization and tune the frequency of the\nmagnon mode. When Zeeman energy and magnetocrys-\ntalline anisotropy energy are included, the Hamiltonian\nof the YIG sphere reads:\nHm=\u0000\u00160Z\nVmM\u0001Hd\u001c\u0000\u00160\n2Z\nVmM\u0001Hand\u001c; (A2)\nwhere\u00160is the vacuum magnetic permeability, M=\n(Mx;My;Mz) is the macrospin magnetization of the YIG\nsphere, and Hanis the magnetocrystalline anisotropy\n\feld in the YIG crystal. We have neglected the con-\ntribution of demagnetization energy of the YIG sphere\nin Eq. (A2) as it is a constant term [34, 55].\nFor a uniformly magnetized YIG sphere, which is mag-\nnetized along z-axis with its anisotropy \feld along z-axis\nin our experiment, the anisotropy \feld can be written as\nHan=mM zez, wheremis dependent upon the domi-\nnant \frst-order anisotropy constant and the saturation\nmagnetization [56]. Since HA<0 in our experiment,\nwe can obtain that m < 0. Thus, the Hamiltonian of\nEq. (A2) turns out to be\nHm=\u0000\u00160HM zVm+1\n2\u00160mM2\nzVm: (A3)\nSince the relation between macrospin magnetization Mand macrospin operator S[29, 34, 57] is\nM=\u0000\r0S\nVm=\u0000\r0\nVm(Sx;Sy;Sz); (A4)\nwhere\r0is the gyromagnetic ratio. By inserting such\nrelation indicated by Eq. (A4) into Eq. (A3), we obtain\nHm=\u0000\u00160\r0HSz\u0000\u00160m\r2\n0S2\nz\n2Vm: (A5)\nThe \frst term of the above equation corresponds to Zee-\nman energy. The macrospin operators and the magnon\noperators are related via the Holstein-Primako\u000b transfor-\nmation [58]\nS+= (p\n(2S\u0000b+b))b; (A6)\nS\u0000=b+(p\n(2S\u0000b+b)); (A7)\nSz=S\u0000b+b; (A8)\nwhereSis the total spin number of the YIG sphere, b+(b)\nthe creation (annihilation) operator of the magnon at\nfrequency!m, andS\u0006\u0011Sx\u0006iSyare the raising and\nlowering operators of the macrospin. Through inserting\nEq. (A8) into Eq. (A5), the Hamiltonian Hmcan be writ-\nten as\nHm=!mb+b+Kb+bb+b; (A9)\nhere!m=\u00160\r0H+\u00160\r2\n0mS=V mdenotes the frequency of\nthe magnon mode and K=\u0000\u00160\r2\n0m=(2Vm) the Kerr co-\ne\u000ecient. Since m< 0 for our experiment, the Kerr coef-\n\fcient K is positive. Notably, the Kerr e\u000bect of magnons\ntermKb+bb+barises from magnetocrystalline anisotropy.\nThe Hamiltonian representing the interaction between\nthe magnon and the cavity mode is\nHI=i\u0000(b++b)(a++a); (A10)\nwhere \u0000 denotes the dissipative coupling strength be-\ntween the magnon and the cavity mode. With the\nrotating-wave approximation, we can neglect the fast os-\ncillating terms [51], and the cavity-magnon interaction\nHamiltonian can be reduced as\nHI=i\u0000(b+a+ba+): (A11)\nThe interaction between the cavity photon and the drive\n\feld can be expressed as [50]\nHd= \nd(a+e\u0000i!t+aei!t); (A12)\nwhere \n dis the amplitude of the driving \feld. 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Primako\u000b, Field dependence of the\nintrinsic domain magnetization of a ferromagnet, Phys.\nRev.58, 1098 (1940)." }, { "title": "2303.11303v2.Secondary_Excitation_of_Spin_Waves__How_Electromagnetic_Cross_Talk_Impacts_on_Magnonic_Devices.pdf", "content": "1\nSecondary Excitation of Spin-Waves: How Electromagnetic\nCross-Talk Impacts on Magnonic Devices\nJohannes Greil1, Matthias Golibrzuch1, Martina Kiechle1,´Ad´am Papp2,\nValentin Ahrens1, Gy¨orgy Csaba2, and Markus Becherer1\n1School of Computation, Information and Technology, Technical University of Munich, Germany\n2Faculty of Information Technology and Bionics, P ´azm´any P ´eter Catholic University, Budapest, Hungary\nThis work examines the impact of electromagnetic cross-talk in magnonic devices when using inductive spin-wave (SW) transducers.\nWe present detailed electrical SW spectroscopy measurements showing the signal contributions to be considered in magnonic device\ndesign. We further provide a rule of thumb estimation for the cross-talk that is responsible for the secondary SW excitation at\nthe output transducer. Simulations and calibrated electrical characterizations underpin this method. Additionally, we visualize the\nsecondary SW excitation via time-resolved MOKE imaging in the forward-volume conguration in a 100 nm Yttrium-Iron-Garnet\n(YIG) system. Our work is a step towards fast yet robust joint electromagentic-micromagnetic magnonic device design.\nIndex Terms—Magnonic Devices, Electromagnetic Cross-Talk, Spin-Wave Spectroscopy, Secondary Spin-Wave Excitation\nI. I NTRODUCTION\nMAGNONICS has become aeld of science in which\nthe focus is no longer only on fundamental research but\nalso on devices and their commercial usability. The promising\nproperties of magnonic devices—tunability, ultra-low losses,\nGHz-regime applications—make them attractive for being\nminiaturized, integrated, and manufactured on a large scale.\nDespite the convenient inductive excitation and pick-up of\nspin-waves (SWs), the electromagnetic cross-talk between the\nSW transducers is a major hurdle when it comes to device\ndesign and benchmarking [1]. Additionally, the secondary\nexcitation of SWs at the output transducer due to inductive\ncoupling deteriorates the overall device performance.\nAn essential step between the functional design of a\nmagnonic device and the layout of its electrical input/output\n(I/O) and matching network is the examination of the elec-\ntromagnetic cross-talk that inevitably arises using inductive\nSW transducers. The parasitic cross-talk results in a signi-\ncant modication of the transmitted SW signals in electrical\nmeasurements, most prominent in AESWS. While in magnon-\ntransport evaluation the cross-talk is calibrated out via refer-\nence measurements at no or off-resonance DC magnetic bias\neld, the absolute magnitude of the direct coupling is crucially\nimportant for the practical use in real-time signal processing.\nThis work focuses on the origin of signal contributions\nof typical all-electrical SW spectroscopy (AESWS) [2]–[4]\nmeasurements in the forward-volume (FV) conguration [5].\nThese contributions can be explained by a straightforward\ncross-talk model based on the Hertzian dipole [6] for mi-\ncrostrip line (MSL) transducers and easy-to-access 2D sim-\nulations for coplanar waveguides (CPWs) using the mag-\nnetostatics solver FEMM [7]. Further, we provide time-\nresolved magneto-optical Kerr effect (trMOKE) images con-\nrming the secondary SW excitation resulting from the\nelectromagnetic cross-talk.\nCorresponding author: J.Greil (email: johannes.greil@tum.de).3.1003.1253.1503.1753.2003.2253.2503.2753.3003.3253.3503.3753.400−2−10\nm0m1\n∆fp\nFrequency in GHz\n∆S-Parameters in dB∆S11\n∆S22\n4·∆S 21\n4·∆S 12\nFig. 1.∆S-Parameter for a CPW transducer pair with60µmcenter-to-center\ndistance on a100 nmYIGlm in the forward-volume conguration (see top\nview sketch in the inset). The cross-talk of about30 dBis subtracted. Both\nreections show a dip at3.285 GHzresulting from the power absorption in\nthe YIGlm. In transmission, the main signal contribution is still the power\nabsorption indicated by the large dip in∆S 21and∆S 12. The ripple on\ntop of those signals is due to the varying spin-wave group velocity during a\nfrequency sweep. Based on the calibrated VNA measurements it is possible\nto assess that the propagating SW signal has an amplitude of about0.2 dB.\nII. E LECTRICAL SPIN-WAVESPECTROSCOPY SIGNAL\nCONTRIBUTIONS\nTypical AESWS measurements in the FV conguration\nshow three main signal contributions (in descending order of\ntheir amplitude): First, the electromagnetic cross-talk between\nthe SW transducers due to the inductive coupling. Second,\nthe absorption of RF power from the input transducer into\nthe magnetic layer underneath. This absorption comes from\nthe excitation of SWs that lowers the electromagnetic cross-\ntalk measured via the transmission signals in the form ofS 21\norS 12. Third, a signal contribution corresponding to SWs\npropagating from the input to the output transducer.\nIn Fig. 1 the latter two contributions are shown for\nthe characterization of a100 nmthin plain YIGlm.\nThe almost constant cross-talk of40 dBin the evalu-\nated frequency span is subtracted for better visualization\n(∆S xx=S xx,H ext̸=0−S xx,H ext=0). The inset shows the2\nmeasurement conguration using a pair of shorted CPWs\nwith equally broad signal (S) and ground (G) lines and gaps\nwS=w G=w ap= 4µm, and a center-to-center distance of\nD= 60µm. The transducers have a length of1 mmand\nare tapered towards contact pads for bonding them to a\nchip carrier. The RF excitation frequencyf RFis swept from\n31 GHzto34 GHzwith a step size of625 kHzfor an out-of-\nplane (OOP) DC magnetic biaseldµH xt= 260 mT. The\nreection signals∆S 11and∆S 22are used to determine the\nfrequency of the most efcient SW excitation which is indi-\ncated by the dip at3285 GHzin both signals. Their different\nheights are caused by slightly varying bond connections to the\ncarrier. A comparison to the transmission signal shows that\nthe reection signal peaks are located at about2 MHzhigher\nfrequencies. The peak shift arises because in reection mea-\nsurements mainly thelm characteristics directly underneath\nthe CPW are probed, while in transmission the spin precession\nis enforced by additional but weakereld components up to\nseveral micron distances [6], [8]. Moreover, therst higher-\norder excitation modem 1of the CPW transducer—therst\nspatial harmonic of the cross-sectionaleld prole of the CPW\ngeometry—generates another small dip in the reection signals\nat334 GHz[1], [8], [9].\nThe characterization of propagating SWs is obtained via\nthe transmission parameters∆S 21and∆S 12. Besides the\nsubtracted electromagnetic cross-talk, the power absorption\ninto the YIGlm for exciting SWs is the strongest contribution\nand shows up as the large dip in∆S 21and∆S 12in Fig. 1. This\nimplies that one can estimate the overall absorbed power along\nthe transducer via calibrated VNA measurements. Especially\nin the FV geometry not more than half of the accepted\npower for SW excitation is transported towards the output\ntransducer due to the isotropic wave propagation. For the\nexample, from Fig. 1 we can determine that about07 dB,\nrelative to the input powerP in=−10 dBm, are absorbed\ninto thelm. The maximum amplitude of the ripple on top\nof the large dip is in turn02 dB. This ripple in∆S 21and\n∆S12denes the third signal contribution which represents the\npropagating SW signal. The ripple stems from the changing\nSW wavelength while sweeping the excitation frequency for\naxedH xtand constant transducer distanceDsuch that\nmore SWst between the transducers. From the frequency\nspacingf pbetween two adjacent peaks we can determine the\ngroup velocity via [3], [5], [9]\nv=dωSW\ndk≈∆f pD(1)\ntov ≈660 ms. Therst higher-order excitation of propagat-\ning spin-wavesm 1is also clearly visible in the transmission\nsignal at3325 GHz.\nSumming up, calibrated AESWS measurements allow for a\nqualitative and quantitative distinction between the discussed\nsignal contributions. Those device-specic characteristics set\nthe basis for highly optimized magnonic devices, whereas the\ncross-talk reduction is the core challenge for a practically\nuseful on-chip post processing of SW signals.III. E LECTROMAGNETIC CROSS -TALK BETWEEN\nSPIN-WAVETRANSDUCERS\nThe electromagnetic cross-talk is a severe signal contribu-\ntion in the electrical evaluation of magnonic devices since\nit buries almost all SW-related signals. Thus, it is of great\npractical use to tackle the electromagnetic cross-talk in a\nstraightforward and easy-to-use rule of thumb manner pre-\nsented in the following.\nThe SW excitation using inductive transducers is described\nvia the in-plane RF magneticelds that force the spins to\nprecess around the effective magneticeldH [5]. This\nexcitation is only applicable in very close proximity to the\ntransducer lines because, as with conventional antennas, these\neld components correspond to the so-called reactive neareld\nand decay fast compared to the far-eld components [6]. For\nSW transducers with lengths in the range of several ten to hun-\ndred microns, the near-eld character is preserved also for the\nelectromagnetic cross-talk since for conventional, electrically\nsmall antennas the far-eld distanced is dened as [10]\nd≈2λ0(2)\nIn the neareld, reactive power oscillates between the antenna\nstructure and the surrounding space such that the Poynting\nvector is zero and no radiation takes place. In contrast, in the\nfareld, a locally plane wave is formed because electric and\nmagneticeld components are in phase [6]. From (2) we see,\nthatd is not dependent on the physical size of the transducer\nbecause it acts as a point-like source for the electromagnetic\nwaves with comparably long wavelengthsλ 0in free-space.\nFor applications up to20 GHzthe fareld is thus reached\nrst atd ≈30 mmwhich is much larger than a reasonable\nmagnonic device dimension. The radiation characteristics of\nelectrically small rod antennas can be modeled via the Hertzian\nDipole (HD) if their cross-section is small compared to the\ntransducer length [6]. The azimuth magneticeld component\nforϑ= 90◦, i.e. in thexy-plane, of a HD aligned to thez-axis\nis written in polar coordinates as [6]\nHφ= ikIl\n4πr\n1 +1\nikr\ne−ikr,(3)\nwhereIis the impressed current,lis the length of the\ndipole,ris the radial distance to the origin, andkis the\n(electromagnetic) wave number in azimuth direction. When\nipping the HD over to be aligned with they-axis, the azimuth\neld component describes the OOPeld of an MSL in the\nyx-plane such thatH φ=Hz. From (3) we see that for small\nvalues ofrtheeld amplitude decays in good approximation\nwith1r3. In the fareld, it decreases with1ras the second\nsummand in (3) vanishes. Simulations in FEMM underpin\nthat the HD is a good approximation for electrically short\nMSLs used as SW transducers. The widely used shorted\nCPW SW transducer [2], [8], [9], [11] can be constructed\nby a superposition of three sucheld components, writ-\nten asH z,CPW ≈ −1\n2Hφ(x+w) +H φ(x)−1\n2Hφ(x−w),\nwherewis the center distance between the signal line and\nthe ground lines. The factors−12stem from the fact that\nhalf the signal-line current returns in each of the ground lines.\nThus, from the superposition, the near-eld amplitude of a3\nFig. 2. a) Cross-sectional view of CPW transducers withw S=w gap=w G= 4µmand a center-to-center distance ofD= 60µmfabricated on100 nm\nthin YIG. The SW channel between the transducers is removed over a width ofd= 35µm. b) TrMOKE picture for a repeated scan of the same area, both on\nthe input and output side, for input powers from−21 dBmto15 dBmat the supplied CPW (left). For the primarily excited SWs on the left the non-linear\nprecession starts aroundP in=−15 dBm, while for the secondarily excited SWsP in≈5 dBmis required. The non-linear excitation is visible by the\npower-dependent increase of the SW wavelength. The minimum input power to observe SWs is−35 dBm(not shown) and−5 dBmfor the primary and\nsecondary excitation, respectively.\nCPW resembles a decay proportional to1r4. While this\nrough approximation does not perfectly go in line with the\nsimulatedeld proles due to the neglected lateral dimension\nof the CPW, it offers a straightforward order-of-magnitude\nestimation of those.\nAs a consequence, the inductive coupling entails not only\na strong cross-talk but also the secondary SW excitation at\nthe output transducer. The shorted CPW forms two loops that\npick up not merely the SW-induced signal but also the perme-\nating magnetic out-of-planeelds from the input transducer.\nThe resulting magneticux through the loop surface can be\nestimated via Faraday’s law of induction\nΦ=\nAlBz·dA l,(4)\nwhereA l= 2w lis the area of the loop formed by the\nshorted CPW with lengthland gapsw . The induced voltage\ncomputes toU in=−dΦdt. Further, the ohmic resistance\nof the electrically small CPW lines can be estimated via the\ngeneral DC resistance model\nR=ρl\nA,(5)\nwith the specic electrical resistivityρand the cross-sectional\narea of a single lineA . Hence, the induced current in\nthe signal line isI S,in=U inR, where by denition\nIG,in=IS,in2for a shorted CPW. Finally, approximating\neach of the CPW lines again as a thin conductor rod, the\nmagnitude of the induced secondary IP magneticeld inthe magneticlm underneath the output transducer can be\nestimated via Amp `ere’s law\n|Hs,IP|≈Iin\n2πr,(6)\nwhereris the radial distance to the surface of the trans-\nducer line.\nLet us discuss a specic example: for a pair of CPW\ntransducers as shown in Fig. 3 with lengthl= 1 mm,\nwS=w =w G= 4µm, a center-to-center distanceD=\n60µm, a metallization thickness of300 nm, and an input\npowerP in=−10 dBmthe following values can be estimated:\nFor a remaining OOP magneticeld of about6µT(FEMM\nsimulation) at the output transducer forming two loops with an\noverall area ofA l= 2w l= 0008 mm2, the induced voltage\nUin≈30 mV. From (5) the induced current in300 nmthick\naluminum transducers is then6µAwhich in turn induces a\nsecondary IP magneticeld ofµ 0Hs,IP ≈20µTat half the\nthickness of the100 nmYIGlm. This secondarily induced IP\neld is sufcient to linearly excite SWs in the same manner as\nthe supplied transducer does but at much lower power levels.\nIn a perfect50ΩRF system this corresponds to an input power\nlevel ofP in≈ −34 dBmwhich is close to the minimum\nrequired input power in our trMOKE setup to resolve SW\nsignals. In summary, the secondary excitation of SWs can\nbe estimated in a rule-of-thumb manner via (4)-(6) showing\nreasonable results, also validated by the measurements in the\nfollowing section. Thereby, the loop areaA lof the CPW has\na major inuence on the strength of the secondary excitation.4\nFig. 3. Top view sketches of the sample settings discussed in Sec. IV. The\ndistancesD= 60µmandd= 35µmand the transducer lengthl= 1 mmare\nthe same for both congurations. The trMOKE picture in Fig.2 corresponds\nto conguration a) for an excitation at the left transducer. The measurements\nshown in Fig. 4 and Fig. 5 correspond to conguration b) for an excitation\nat the right and left transducer, respectively.\nIV. S PATIAL VISUALIZATION OF SECONDARILY EXCITED\nSPIN-WAVES USING TIME-RESOLVED MOKE\nA. Both Transducers on YIG\nTo substantiate the presented method, we have consid-\nered sample settings that can be used to quantify the sec-\nondary SW excitation resulting from the electromagnetic\ncross-talk via trMOKE images. Arst test conguration\nconsists of two parallel shorted CPWs with lengthl=\n1 mm,w S=w ap=w G= 4µmand a center-to-center dis-\ntance of60µmfabricated on two areas of YIG separated by\na35µmwide gap as shown in Fig. 3 a. Forf RF= 205 GHz\nandµH xt≈214 mT, sweeping the input powerP infrom\n−27 dBmto15 dBmin steps of2 dBmresults in the trMOKE\nimage shown in Fig. 2. The left transducer is supplied by an\nRF source while the right one is not supplied and open at\nthe tapered end. To avoid possible deviations arising from\npossiblelm inhomogeneities the same sample area is re-\npeatedly scanned for the different input powers. Due to the\nlarge range of input powers, it is necessary to underscale the\nhigh-amplitude Kerr signals to resolve weaker signals. For the\nprimary excitation, the minimum input power generating a\nresolvable Kerr signal is−35 dBm, while non-linear excitation\nbegins at aroundP in=−15 dBm. The transition to non-\nlinear excitation is determined by the increase in wavelength\ncompared to the scan atP in=−13 dBm. Looking at the\nsecondarily excited SWs on the right, it is possible to detect\nwave fronts starting fromP in≈ −5 dBmat the supplied\ntransducer. The non-linear secondary excitation starts at about\nPin= 5 dBm. In this conguration, the difference between\nthe accepted power for primarily and secondarily excited SWs\nis about10 dBwhich corresponds to the overall cross-talk\nloss between the transducers. This loss contains the radiated\nbut uncoupledeld components of the supplied transducer\nthat do not permeate the unsupplied CPW and the—in this\nsense unused—power that is absorbed at the left CPW for the\nprimary excitation of SWs.\n0 20 40 60 80 100 120 140\nx-position in µm-27-25-23-21-19-17-15-13-11-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 Power in dBm\n-36-16-4 0 4 16 36 Kerr signal in a.u.100 nm YIG\n500m GGGPin̸= 0\nFig. 4. TrMOKE picture of primarily excited SWs for an input power\nsweep from−21 dBmto15 dBmin steps of2 dBmat the right transducer\nin Fig. 3b. The lowest possibleP inis−35 dBm(not shown), while the\nnon-linear excitation starts atP in≈ −19 dBmdenoted by the increase\nin SW wavelengthλ SW. The intermediate bend ofλ SWis attributed to a\nconnement effect between the sharp YIG edge and the CPW.\nB. One Transducer on GGG, One Transdcuer on YIG\nA second conguration is shown in Fig. 3 b, where the\nleft transducer is fabricated directly on the GGG substrate\nleaving only a100 nmYIG layer underneath the right trans-\nducer. The CPW dimensions are the same as for the previous\nmeasurement, the excitation frequency and externaleld are\nset tof RF= 205 GHzandµH xt≈223 mT, respectively.\nThe trMOKE picture of the primarily excited SWs is shown\nin Fig. 4 for input powers of−27 dBmto15 dBmin steps\nof2 dBm. Similar to the former conguration a minimum\ninput power of−35 dBmgenerates a resolvable Kerr sig-\nnal, while the non-linear excitation now starts at around\nPin=−19 dBm. The intermediate bend of wavelengths for\npower levels between−7 dBmand−1 dBmis attributed to\na connement effect that occurs from the cavity-like structure\nformed by the sharp YIG edge and the CPW but is not yet\ninvestigated in detail.\nThe trMOKE picture of the secondarily excited SWs is\nshown in Fig. 5 for the same input power levels but this\ntime applied to the transducer on the bare GGG substrate.\nAgain, the same area is repeatedly scanned for all power levels\nto avoid inuences oflm inhomogeneities. The minimum\nrequired input power to detect SW signals is−27 dBm, while\nthe non-linear excitation begins atP in≈1 dBm. In this\nway we see that non-linearity is reached at4 dBless input\npower compared to the conguration in Sec. IV-A where both\ntransducers are fabricated on YIG. This difference shows in\nturn how much power is absorbed into the YIGlm for the\nprimary SW excitation at the left transducer in Fig. 2. It reects\nthe fact that in the FV geometry a maximum of50 %of the\nabsorbed power can be transported by the spin waves and at\nleast50 %propagate in other directions. Thus, comparing the5\n0 20 40 60 80 100 120 140\nx-position in µm-27-25-23-21-19-17-15-13-11-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 Power in dBm\n-36-16-4 0 4 16 36 Kerr signal in a.u.100 nm YIG\n500m GGGPin= 0\nFig. 5. TrMOKE picturce of secondarily excited SWs at the right CPW in\nFig. 3b for an input power sweep at the left CPW from−27 dBmto15 dBm\nin steps of2 dBm. The minimum required input power to detect a SW signal\nis around−23 dBm, while the non-linear excitation starts atP in≈1 dBm.\ncongurations in Fig. 3 a & b, we can measure the fraction\nof power that is absorbed for primary SW excitation and\npropagates away from the (removed) SW channel between the\ntransducers.\nC. Distant Excitation of Spin-Waves\nIn a third conguration we omit the right transducer in\nFig.3b such that only the left transducer is fabricated on the\nGGG substrate with distances of20µmto60µmto the YIG. In\nthis experiment, it was not possible to excite SWs or near-FMR\nmodes. It shows that there is not a distant FMR-like excitation\n[8] but the non-driven output transducer is necessary for the\nsecondary SW excitation. At those relatively long distances\nbetween the transducer and the YIG layer the IPelds are\ntoo much conned underneath the supplied CPW to be strong\nenough to force the spins in the YIG to precession. Following\nthe discussion in [12] e.g. an MSL in close proximity to the\nYIG edge could be a feasible transducer geometry to excite a\nrelatively broad spectrum of SWs without the need to fabricate\nthe transducer directly on YIG. However, to our knowledge the\nexcitation efciency of this method was not yet investigated\nexperimentally. Even more, the fact that the secondary SWs\nare excited by the output transducer and not by a distanteld\ncomponent of the input CPW is indicated by the equally long\nSW wavelengths in the linear regime: From Fig. 2 wend\nthatλSW≈7µmfor a linear excitation on both transducer\nsides. And from the comparison of Fig. 4 and Fig. 5 wend\nλSW≈10µm. The comparison shows that the wavelength\nfor linear excitation depends on the cross-sectional geometry\nof the CPW, whereas the difference between the measured\nwavelengths in the two experiments results from the changed\nHxtat axedf RF[11].V. C ONCLUSION\nOur measurements show that using inductive SW transduc-\ners leads to two major challenges when it comes to high-\nspeed signal transmission requiring lowest distortions: First,\nthe electromagnetic cross-talk is the strongest contribution to\nthe transmission signal and often buries the SW-related signals\nalmost completely. The commonly used cross-talk subtraction\nin AESWS measurements is somewhat sufcient for investi-\ngating magnon-transport phenomena but is not applicable for\nreal-time measurements in an I/O circuitry. And second, the\nelectromagnetic cross-talk is such strong that we observe a\nsecondary excitation of SWs at the output transducer due to the\ninductive coupling. These secondarily excited SWs interfere\nwith the intentionally excited SWs and lower the magnonic\nsignal-to-noise ratio as well as the overall detection sensitivity\nof the output transducer due to its own reaction on varying\nfrequency or magneticeld. Thus, one of the main goals for\npaving the way for magnonic devices towards integration is\nto minimize the electromagnetic cross-talk between inductive\nSW transducers. This can be achieved either by shielding\nthem from each other while maintaining the SW excitation\nefciency as well as the electrical RF properties. Or by\nintroducing asymmetric transducer structures, e.g. with a CPW\nas input and several loop antennas as outputs which are\noptimized for the magnonic device characteristics. A fully\ncoupled electromagnetic-micromagentic design approach is\ntherefore essential for practically useful magnonic devices.\nACKNOWLEDGMENT\nThe authors acknowledge funding from the European Union within\nHORIZON-CL4-2021-DIGITAL-EMERGING-01 (No. 101070536,\nMandMEMS), German Research Foundation (DFG No. 429656450),\nand the German Academic Exchange Service (DAAD, No.\n57562081).\nREFERENCES\n[1] D. A. Connelly, G. Csabaet al., “Efcient electromagnetic transducers\nfor spin-wave devices,”Scientic Reports, vol. 11, no. 1, Sep. 2021.\n[2] V. Vlaminck and M. Bailleul, “Current-induced spin-wave doppler shift,”\nScience, vol. 322, no. 5900, pp. 410–413, Oct. 2008.\n[3] S. Neusser, G. Duerret al., “Anisotropic propagation and damping of\nspin waves in a nanopatterned antidot lattice,”Physical Review Letters,\nvol. 105, no. 6, Aug. 2010.\n[4] J. Chen, T. Yuet al., “Excitation of unidirectional exchange spin waves\nby a nanoscale magnetic grating,”Physical Review B, vol. 100, no. 10,\nSep. 2019.\n[5] A. Prabhakar and D. D. Stancil,Spin Waves: Theory and Applications.\nSpringer US, 2009.\n[6] D. M. Pozar,Microwave Engineering 4th Edition. John Wiley & Sons,\n2011.\n[7] D. Meeker, “Finite element method magnetics.” [Online]. Available:\nhttps://www.femm.info\n[8] M. Sushruth, M. Grassiet al., “Electrical spectroscopy of forward\nvolume spin waves in perpendicularly magnetized materials,”Physical\nReview Research, vol. 2, no. 4, Nov. 2020.\n[9] V. Vlaminck and M. Bailleul, “Spin-wave transduction at the submi-\ncrometer scale: Experiment and modeling,”Physical Review B, vol. 81,\nno. 1, Jan. 2010.\n[10] H. A. Wheeler, “The Radiansphere Around a Small Antenna,”Proceed-\nings of the IRE, vol. 47, Aug. 1959.\n[11] J. Lucassen, C. F. Schipperset al., “Optimizing propagating spin wave\nspectroscopy,”Applied Physics Letters, vol. 115, no. 1, p. 012403, Jul.\n2019.\n[12] ´A. Papp, W. Porodet al., “Nanoscale spectrum analyzer based on spin-\nwave interference,”Scientic Reports, vol. 7, no. 1, Aug. 2017." }, { "title": "0903.0373v1.Magnetooptical_control_of_light_collapse_in_bulk_Kerr_media.pdf", "content": "arXiv:0903.0373v1 [nlin.PS] 2 Mar 2009Magneto-optical control of light collapse in bulk Kerr medi a\nY. Linzon∗,1K. A. Rutkowska,1,2B. A. Malomed,3and R. Morandotti1\n1Universit´ e du Quebec, Institute National de la Recherche S cientifique, Varennes, Quebec J3X 1S2, Canada\n2Faculty of Physics, Warsaw University of Technology, Warsa w PL-00662, Poland\n3Department of Physical Electronics, School of Electrical E ngineering,\nFaculty of Engineering, Tel Aviv University, Tel Aviv 69978 , Israel\n(Dated: November 3, 2018)\nMagneto-optical crystals allow an efficient control of the bi refringence of light via the Cotton-\nMouton and Faraday effects. These effects enable a unique comb ination of adjustable linear and\ncircular birefringence, which, in turn, can affect the propa gation of light in nonlinear Kerr media.\nWe show numerically that the combined birefringences can ac celerate, delay, or arrest the nonlinear\ncollapse of (2+1)D beams, and report an experimental observ ation of the acceleration of the onset\nof collapse in a bulk Yttrium Iron Garnet (YIG) crystal in an e xternal magnetic field.\nPACS numbers: 42.65.Sf, 42.65.Jx, 42.81.Gs, 78.20.Ls\nWaves propagating in multidimensional self-focusing\nmedia are subject to instabilities that lead to the catas-\ntrophic collapse after a finite propagation distance, fol-\nlowed by beam filamentation or material damage [1, 2,\n3, 4, 5, 6, 7, 8, 9, 10, 11]. In the (2+1)D [(2+1)-\ndimensional)] setting, above a certain threshold value\nof the input power, the critical collapse driven by fo-\ncusing nonlinearities is a universal scenario, observed in\nhigh-power excitations of plasmas [1, 2], hydrodynami-\ncal systems [3], Bose-Einstein condensates (BECs) [4, 5],\nand optical media [2, 6, 7]. In particular, in optical\npulse propagation through amorphous media and crys-\ntals without special symmetries, the dominant nonlin-\nearity is the Kerr (cubic) effect, which is modeled by the\nnonlinear Schr¨ odinger equation (NLSE) [12]. Collapsing\nbeams in Kerr media were studied in detail, especially in\nthe course of the past decade [6, 7, 8, 9, 10, 11].\nThe challenge to controlthe wave collapse (and in par-\nticular to mitigate its detrimental effects) has recently\ndrawn much attention [5, 9, 10, 11]. As recently demon-\nstrated, the collapse distance of ultra-intense laser pulses\nin air can be controlled by passive optical elements [9],\nand in BECs the collapse time is strongly affected by the\nso-calledFeshbach-resonancetechnique [5]. In condensed\noptical media, where the collapse occurs with pulse en-\nergies far below the creation of plasma [6, 7], a recently\nproposed scheme for collapse management relies on the\nuse of a layered structure with the nonlinearity strength\nalternating in the longitudinal direction [10] (a similar\nmechanism was proposed for the stabilization of BECs\nin the 2D case [11]). However, this scheme is difficult to\nimplement in bulk media, and such structures may give\nrise to linear losses induced by the reflection of light from\ninterfaces between the alternating layers.\nAn alternative approach to control the transition to\nthe collapse may be provided by optical birefringence,\nwhich promotes energy and phase transfer between the\npolarization components of the beam. The interplay be-\ntween birefringence and nonlinearity is known to inducecoupling of the polarization rotation and a characteristic\ntemporal evolution of solitons in optical fibers [13]. In\nthis paper, we explore the birefringence as a tool for the\n“management” of the collapse of (2+1)D beams, which\nmay be relatively easily implemented in affordable ex-\nperimental conditions by using magneto-optical(MO) ef-\nfects [14, 15, 16, 17], while avoiding reflection losses. To\nthis end, we present numerical and experimental studies\nof the combined effects of linear and circular birefrin-\ngences on the dynamics of collapsing beams in a bulk\nself-focusing Kerr medium. We find that the onset of\ncollapse can be accelerated, delayed, or even suppressed,\nat certain values of the combined birefringence strengths.\nWe also show that the required linear and circular bire-\nfringence can be induced in a transparent MO Yttrium\nIron Garnet (YIG, Y 3Al5O12) crystal by the application\nofan external dc magnetic field, thus opening up the per-\nspective of using MO effects to generate and control var-\nious nonlinear phenomena in optics. In our experiments,\nanadjustablebalanceoflinearandcircularbirefringences\nwasrealizedviaacombination[14]oftheCotton-Mouton\n(orVoigt) [15] and Faraday[16] MO effects in abulk YIG\ncrystal. Following the propagation of femtosecond pulses\nin the crystal, we observed a controllable decrease of the\nthreshold input power necessary for the onset of collapse\nat the output facet as a function of the magnetic field.\nThis also constitutes a first experimental study in non-\nlinear optics where birefringence can be switched on and\nvaried continuously in an adjustable fashion.\nThe evolution of the complex electric-field amplitudes,\nurandul, representing the right- and left-circular polar-\nizations (RCP and LCP), in the presence of a Kerr non-\nlinearity and combined linear and circular birefringences,\nobeys the coupled NLSEs, in the scaled form [13]:\ni∂ur\n∂z+1\n2∇2\n⊥ur+bur+cul+(|ur|2+2|ul|2)ur= 0,\ni∂ul\n∂z+1\n2∇2\n⊥ul−bul+cur+(|ul|2+2|ur|2)ul= 0\n(1)\nwherezis the propagation axis, ∇2\n⊥is the transverse2\nLaplacian, while bandcare the strengths of the circular\nand linear birefringences, respectively. As evident from\nEqs. (1) the linear and circular birefringences account\nfor, respectively, the ratesof linearamplitude mixing and\nphase shift between the RCP and LCP fields. In terms\nof the RCP and LCP, the ratio between the cross- and\nself-phase modulation coefficients (XPM/SPM) is 2 for\ncubically-symmetric crystals, including YIG [12, 13].\nAssuming vorticity-freesolutions with circular symme-\ntry, (2+1)D Eqs. (1) reduce to a (1+1)D form, in terms\nofzand the radial coordinate R. We consider input\nGaussian beams, ur(R,z= 0) =Aexp/parenleftBig\n−R2\n2ρ2/parenrightBig\ncosθand\nul(R,z= 0) =Aexp/parenleftBig\n−R2\n2ρ2/parenrightBig\nsinθ, with normalized in-\nput width ρ=1, amplitude A, and symmetricpolarization\ncontentθ=π/4. This choice corresponds to an unchirped\nGaussian profile launched with an horizontal linear po-\nlarization. Since the input does not carry vorticity, the\ncircular-symmetric structure of the solutions is not sub-\nject to an azimuthal modulational instability [7].\nFigure 1 shows propagation maps obtained by direct\nsimulations of Eqs. (1). In the absence of birefringence\n[Figs. 1(a)-(d)], the RCP and LCP components are cou-\npled only via the XPM term, which becomes significant\nonly close to the collapse point. While a low input power\nexcitation [Figs. 1(a,b)] results in beam diffraction, the\ncollapse occurs after a finite propagation distance [for\ninstance, z= 7.3 in Fig. 1(c)] when the input power ex-\nceeds the critical level [8]. A substantial converging por-\ntion of the phase fronts emerges near the collapse point,\nsee Fig. 1(d). The collapse can be accelerated by the in-\n0 2 4 6 8-10-50510\n0246810\n0 2 4 6 8-10-50510\n0246810\n0 2 4 6 8-10-50510\n0246810\n0 2 4 6 8-10\n-5\n0\n5\n10\n -3-2-10123\n0 2 4 6 8-10\n-5\n0\n5\n10\n -3-2-10123\n0 2 4 6 8-10-50510\n00.020.040.06\n(c)(a)\nzR (b)\n(d)\n$=1.13,b=0,c=0.2\n(e)$=1.13,b=0.08,c=0.2\n(f)\u0015S\n\u0010\u0015S\n>10\n>10 >10$=0.1,b=0,c=0\n$=1.13,b=0,c=0R\nz\nzR R\nz\nFIG.1: (Color online)Propagation mapsof ur(R,z)obtained\nfrom direct solutions of Eq. (1). (a)-(d): Zero birefringen ces,\nwith (a),(b) low and (c),(d) high input powers. The pairs of\npanels (a),(c) and (b),(d) display the evolution of the inte n-\nsity,|ur(R,z)|2, and phase of ur(R,z), respectively. (e), (f):\nIntensity maps in the presence of the birefringences. Numer -\nical parameters are indicated above each respective panel.\n00.10.20.30.40\n0.1\n0.2\n0.3\n0.4\n0.5\n5101520\n00.10.20.30.40\n0.1\n0.2\n0.3\n0.4\n0.5\n 345678910(b) (a)\nYIG\ncrystal\nc\nbc>10\nCollapseacceleratedCollapsedel\nayed/supp\nressed\nb0.5 0.5\nFIG. 2: (Color online) Evolution maps in the parameter\nspace (b,c) (see the text), as obtained from direct simulations\nof Eqs. (1), for A=1.13 and 0 ≤z≤50, up to z= 50 or the\nvaluezcoll<50 in which the collapse occurs. The solutions\nthat do not collapse up to z= 50 are represented in yellow\nareas. The dashed (green) curves show the calculated depen-\ndence between bandccorresponding to crystalline YIG in an\nexternal magnetic field. (a) zcoll, and (b) beam width at zcoll.\ntroduction of linear birefringence, as shown in Fig. 1(e).\nCircular birefringence, if acting alone, does not affect the\ncollapse, since the respective terms in Eqs. (1) can be\neliminated by a straightforward transformation. When\nboth of the birefringences are present, the propagation\ndistance necessary for the onset of collapse can be ex-\ntended, i.e., the onset of the collapse is delayed [Fig. 1(f)]\ndue to the interplay between amplitude and phase mix-\ning. For low bandcvalues, such as those used in Fig.\n1(e,f) and typically achievable in experiment, the differ-\nences in the evolution of urandulare marginal, i.e.,\nthe RCP and LCP beam components feature the same\ncollapse dynamics. With larger birefringence parameters\nthe differences between the components become substan-\ntial; however, such large birefringence values were not\naccessible in the current experiment, see below.\nThe results of systematic simulations are summarized\nin Fig. 2 by means of maps in the plane of ( b,c). Panels\n(a) and (b) show, respectively, the values of zcollwhere\nthe beam collapses, if zcoll<50, and the final values of\nthe beam width, ρ(z= 50); in cases where the collapse\noccurs at zcoll<50, the final width is set as ρ= 0 [black\nareas in (b)]. While Figs. 1 and 2 display the results for\nthe RCP component, the LCP maps are similar in the\nentire range considered. As seen in Fig. 2, for a given\ninputpowerthedominationofamplitudemixingbetween\nRCP and LCP ( c > b) usually results in an acceleration\nof the collapse, while dominant phase mixing ( b > c) can\nlead to the delay or effective suppression of the collapse.\nIn the experiment, a bulk YIG single-crystal was\nplaced with the easy crystallographic axis [100] paral-\nlel toz, cf. Ref. [17]. These cubic dielectric crystals are\nhighly transparent for optical signals in the near-infrared\nand exhibit large MO transmission coefficients, owing to\ntheir ferrimagnetic phase [15, 16]. We chose to work at\nthe wavelength of 1 .2µm, which offers an optimal trade-\noff between the magnitudes of the MO coefficients and\nthe absorption losses [16]. The temporal dispersion in\nYIG is normal and weak in the near-infrared [18], and\nhence a high power beam is free from the development of3\n\u0010V \u000eVC\nPOPA 800C \nLaser\nO=1.2 Pm\nND\n(a)\nTL2\nQWP(b)Vidicon\nCamera\n10PmL1\nEnergy Meter xyInput\npulseOutput pulse\nFIG. 3: (Color online) Experimental setup schematics. OPA,\noptical parametric amplifier laser; T, cylindrical telesco pe;\nQWP, quarter wave plate; P, polarizer; ND, neutral density\nattenuators; L1,L1, aspheric coupling and imaging lenses; C,\ncrystal sample. Insets: (a) Photograph of the magnet’s inte -\nrior with lenses L1, L2 and crystal C; (b) The beam’s waist\nprofile at the input facet of the crystal.\ntemporal modulational instabilitiess [7].\nThe application of an external magnetic field Hper-\npendicular to the propagation axis renders the crystal\noptically uniaxial, with the optical axis parallel to H\nand the respective linear birefringence proportional to H\n[14, 15]. The largest phase retardation associated with\nthe linear birefringence, for components of the wavevec-\ntorkperpendicular to the magnetic field, is 0 .45µm/cm\nin YIG [15] at a saturation field of H= 800 G. This cor-\nresponds to a saturation value of c= 0.1 in terms of Eq.\n(1) for our crystal. For wavevector components parallel\nto the magnetic field, the Faraday effect induces different\nrefractiveindices for the RCP and LCP waves(i.e., circu-\nlarbirefringence),witharespectivepolarizationphasere-\ntardation of 2 µm/cm at the saturation level [16]. While\nthe Faraday effect is predominant in the k/ba∇dblHgeome-\ntry [17], it is weaker in the present setting, as only near\nthe collapse point the wavevectors feature significant lat-\neral components. An estimation corresponding to the\ncase shown in Figs. 1(c),(d) yields an effective saturation\nvalue of b= 0.05. Below saturation, the Faraday coeffi-\ncient is proportional to the magnetic field, b∼H, while\nthe Cotton-Mouton coefficient depends on Hquadrati-\ncally, i.e. c∼H2[14, 15, 16], implying a parabolic de-\npendence betweenthe birefringencecoefficientsinagiven\nexternal magnetic field, b∼c2. Calculated relations be-\ntweenbandcfor the YIG crystal at different magnetic\nfields are shown by the dashed curves in Fig. 2.\nThe experimental setup is sketched in Fig. 3. We used\na Spectra Physics model 800C Optical Parametric Am-\nplifier laser system, delivering pulses of 200 fs duration\nat a repetition rate of 1 KHz with peak powers ≤50\nMW. The input beam was shaped by means of a cylin-\ndrical telescope (T), followed by the combination of a\nquarter wave plate (QWP) and a polarizer (P), which\nwere used to fix the input linear polarization parallel to\nH(i.e. horizontal). Variable neutral density filters (ND)\nwere used to control the input power. The YIG sample\n2.1 MW\n(a)2.3 MW 2.9 MW\n 3.6 MW\n(c)\n(d)\n(e)10Pm\nPosition ( Pm)\nNormalized intensity (a.u.)\n01\nxy\nFIG. 4: (Color online) (a) Output beam profiles as a func-\ntion of the input peak power, in the absence of an external\nmagnetic field. (b) Line-outs along x(full circles) and y(hol-\nlow rectangles) around the central spot of the collapsed (2 .9\nMW) output beam. (c)-(e) Output beam images, as function\nof the input power (horizontal) and the external magnetic\nfield (vertical), for the fields: H= 200 G (c), 400 G (d), 800\nG (e).\nwas placed inside an electromagnet (GMW model 3470).\nThe application of a driving voltage between the poles\n(±V) induced a uniform out-of-plane dc magnetic field\nin the crystal. A digital Tesla-meter (Group3 DTM-133)\nwas used to calibrate the free-space magnetic field and\nverify its homogeneity in the sample. A combination of\naspheric lenses, L1 and L2, was used for coupling and\nimaging, respectively. Importantly, the lenses and crys-\ntal were mounted on nonmagnetic pyrex holders, see Fig.\n3(a). Standard metallic mounts, which usually hold mi-\ncroscopeobjectivelenses, werenotused, aswefoundthat\nthey strongly affect the uniformity of the magnetic field\nwithin the sample. The sample input facet was placed at\nthe focal plane of L1. The beam’s profile at this plane is\nshown in Fig. 3(b). The FWHMs of this elliptical input\nbeam were 30 and 15 µm along the major ( x) and minor\n(y) axes. The beam profile at the sample output facet\nwas imaged by L2 onto a Vidicon infrared camera.\nExperimental results are summarized in Fig. 4. Im-\nages of the output beam as a function of the input peak\npower, in the absence of a magnetic field, are shown in\nFig. 4(a). At the critical power of 2 .9 MW, a central\nhigh-power spot appears in the diffracting background.\nThis spot is circularly symmetric [see Fig. 4(b)], al-\nthough the input beam profile was elliptical, with the\nabove-mentioned major-to-minor axes ratio of 2 : 1. The\nspot fits a Townes profile [6], which indicates that it is\ngenerated by the onset of the collapse. The correspond-\ning critical power complies with the theoretical predic-\ntion for collapse [8], using λ0= 1.2µm, the linear re-\nfractive index of YIG n0= 1.83 and its Kerr coefficient\nn2= 7.2×10−16cm2/W [18]. The corresponding col-\nlapse distance prediction with A= 1.13,zcoll= 7.3 [Figs.\n1(c),(d)], when rescaled back into physical units using\nthe well-known transformations [6, 7, 8] and assuming\na circular input beam of 10 µm diameter, matches the4\n0 200 400 600 8002.32.52.72.9 (a)\nMagnetic field (G)(b)Critical input power (MW)b\n2\ncrA\n0.00 0.01 0.02 0.03 0.04 0.05-0.10-0.08-0.06-0.04-0.020.00\nc(c)\nField (G)Critical power (MW)800 G0 G\nb\nFIG. 5: (Color online) (a) Measured critical power required\nfor the onset of the collapse in the output facet, as a func-\ntion of the applied magnetic field. (b) Numerically calculat ed\nsquared critical amplitude of the input beam ( A2\ncr) necessary\nfor the onset of the collapse at z= 7.3, as a function of the\nbirefringences corresponding to crystalline YIG. (c) The c al-\nculated dependence between bandcin YIG, used in (b).\nsample length, 3 mm. At higher powers, filaments are\nobserved in the output facet images, indicating that the\ncollapse occurred earlier in the crystal. Following the ap-\nplication of a magnetic field, the onset of collapse at the\noutput facet is observed at lowerpowers [Figs. 4(c-e)],\nwith the largest collapse-acceleration effect recorded at\na magnetic field of H= 400 G [see Fig. 4(d)], corre-\nsponding to half the saturation field of YIG [17]. This\nobservation agrees with the fact that, when YIG is ex-\ncited by light traveling perpendicular to the magnetic\nfield, the linear birefringence is stronger than its circu-\nlar counterpart[14, 15]. A qualitative characterizationof\nthe output beam polarization state has also shown that\nwithout an external magnetic field the beam remained\nlinearly polarized, while with an application of the field\nthe beam became elliptically polarized, with the ellip-\nticity growing with the field. This indicates the certain\npresence of magnetically-induced polarization dynamics\nin the crystal.\nFigure 5(a) shows the measured critical power, corre-\nsponding to the data of Fig. 4, that was required for the\nonset of collapse at the output facet, as a function of the\nmagnetic field. In Fig. 5(b), this is compared to results\nobtained by the numerical solutions of Eqs. (1), where\nfor each set of birefringence values (corresponding to a\ngiven magnetic field) we find the critical amplitude Acr\nof the input beam for which the collapse occurs at the\npropagation distance corresponding to the output facet\n(z= 7.3). Figure5(c)againdisplaystherelationbetween\nthe normalizedbirefringence parametersofthe YIG crys-\ntal used, cf. the dashed curves in Fig. 2. The behavior\nof the collapse detuning is similar in Figs. 5(a) and 5(b),\neven though the theoretical model did not take into con-\nsideration magnetically-induced losses. Specifically, the\nCotton-Mouton effect is always accompanied by linear\nmagnetic dichroism [15], and the Faraday effect entails\ncircular magnetic dichroism [16, 17], both of which are\nweak but present at λ= 1.2µm.\nIn conclusion, we have investigated the combined ef-\nfects of circular and linear birefringences on the propa-gation of collapsing (2+1)D beams in self-focusing bulk\nKerr media, and have shown that the onset of collapse\ncan be accelerated, delayed, or suppressed, depending on\nthe relative birefringence strengths. Experimentally, we\nhave demonstrated a controlled acceleration of the col-\nlapse at the output facet of a ferrimagnetic YIG crys-\ntal, following the application of an external magnetic\nfield which induces the birefringences. The direct ob-\nservation of magnetization-induced effects in collapsing\nbeams provides a unique demonstration of an all-optical\nmagnetically-controlled lensing mechanism, pioneering\nthe use of MO crystals in nonlinear optics experiments.\nFinally, sinceEqs. (1)arealsoGross-Pitaevskiiequations\ndescribing a binary BEC with linear interconversion [19],\naccountedforbythecoefficient c, similarphenomenamay\nalso be observed in nonlinear matter-wave dynamics.\nThis research was supported by NSERC and TeraXion\n(Canada). YL and KR respectively acknowledgesupport\nfrom FQRNT-MELS and IOF Marie Curie fellowships.\n∗Corresponding author: yoli@emt.inrs.ca.\n[1] P. A. Robinson, Rev. Mod. Phys. 69, 507 (1997).\n[2] L. Berg´ e, Phys. Rep. 303, 260 (1998).\n[3] B. W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop,\nNature (London) 403, 401 (2000).\n[4] E. A. Donley et al., Nature 412, 295 (2001); M. Greiner,\nO. Mandel, T. W. Hansch, and I. Bloch, Nature 419, 51\n(2002); T. 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Rev.Lett.78, 1607 (1997)." }, { "title": "2204.05482v1.Spin_Peltier_effect_and_its_length_scale_in_Pt_YIG_system_at_high_temperatures.pdf", "content": "1 Spin Peltier effect and its length scale in Pt/YIG system at high temperatures \nAtsushi Takahagi1, Takamasa Hirai2, Ryo Iguchi2, Keita Nakagawara2,a), Hosei Nagano1, and \nKen-ichi Uchida2,3* \n1Department of Mechanical Systems Engineer ing, Nagoya University, Nagoya 464-8601, Japan \n2National Institute for Materials Science, Tsukuba 305-0047, Japan \n3Institute for Materials Research, To hoku University, Sendai 980-8577, Japan \na)Present address: Graduate School of Scienc e, Tohoku University, Sendai 980-8578, Japan \nE-mail: UCHIDA.Kenichi@nims.go.jp \n \nThe temperature and yttrium-iron-garnet (YIG) thickness dependences of the spin Peltier effect (SPE) have been investig ated using a Pt/YIG junction system at \ntemperatures ranging from r oom temperature to the Curie temperature of YIG by the \nlock-in thermography method. By analyzin g the YIG thickness dependence using an \nexponential decay model, the characteristic length of SPE in YIG is estimated to be \n0.9 m near room temperature and almost cons tant even near the Curie temperature. \nThe high-temperature behavior of SPE is clearly different from that of the spin \nSeebeck effect, providing a clue for micros copically understandi ng the reciprocal \nrelation between them. \n \n \nIn recent years, the field of spin caloritronics1-3) has been rapidly de veloping by involving \nthermal effects in spintronics. In this field, various magneto-thermoelectric and thermo-spin effects unique to magnetic and spintronic materi als have been discovered and investigated. \nRepresentative thermo-spin effects are the spin Seebeck effect (SSE)\n4,5) and its reciprocal called \nthe spin Peltier effect (SPE)6,7) appearing in metal/magnetic-i nsulator junction systems. SSE \n(SPE) is the generation of a spin (heat) current fr om a heat (spin) current, where the spin current \nis carried by magnons in th e magnetic insulator. In combination with the inverse (direct) spin Hall \neffect,8,9) SSE (SPE) enables transverse heat-to-charge (charge-to-heat) current conversion in \nsimple insulator-based systems. Because of their intriguing mechanism and technological \nadvantages, SSE and SPE have attracted attent ion as next-generation energy harvesting and \nthermal management principles for spintronic devices, respectively.10-12) \nIn order to clarify the ph ysical mechanism of SSE and SPE, their temperature T, magnetic field \nH, and thickness dependences have been investig ated systematically using the junction systems \ncomprising a paramagnetic metal Pt and fe rrimagnetic insulator yttrium iron garnet (Y 3Fe5O12: 2 YIG).13-20) The T dependence of SSE in Pt/YIG systems has been measured quantitatively in a \nwide temperature range from <10 K to above the Curie temperature of YIG21) (Tc=550-560 K). In \nthe high-temperature range, it has been reported th at the inverse spin Hall voltage induced by SSE \n(SSE voltage) monotonically decreases with increasing T and the T dependence follows (TcT) \nwith being the critical exponent.21-23) The values of obtained from various experiments were \n1.5-3.0; the T dependence of the SSE voltage for the Pt/YIG systems is convex downward (note \nthat =3.0 was obtained when SSE was measured with applying a steady and uniform temperature \ngradient to the sample21)). However, the origin of this dependence is yet to be clarified. On the \nother hand, SPE has been measured only belo w room temperature. Investigating the T dependence \nof SPE at high temperatures quantitatively would help to clarify the origin of the thermo-spin \nconversion. Importantly, in 2020, Daimon et al. proposed a method to syst ematically measure the \nYIG thickness tYIG dependence of SPE using a Pt/wedged-YIG system.24) By applying this method \nto the measurements in the high-temperature range, the T dependence of the ch aracteristic length \nof SPE can also be investigated. \nIn this paper, we report systematic investigations on the T and tYIG dependences of SPE in the \nPt/YIG junction system. We measur ed the spatial distribution of the temperature change induced \nby SPE in the Pt/wedged-YIG system by us ing the lock-in thermography (LIT) method7,17,19,24) at \nvarious temperatures ranging fro m room temperature to around Tc. We obtained the tYIG \ndependence of SPE with high tYIG resolution at each te mperature and estimated the characteristic \nlength of SPE lSPE in the high-temperatur e range by analyzing the tYIG dependence. The obtained \nsystematic dataset will help to elucidate the detailed mechanisms of SSE and SPE. \nFigure 1(a) shows the schematic illustration of the sample used in this paper. The sample \nconsists of a Pt strip with a th ickness of 5 nm and width of 0.4 mm sputtered on an YIG/gadolinium \ngallium garnet (Gd 3Ga5O12: GGG) substrate with th e YIG thickness gradient tYIG. The substrate \nwith tYIG was prepared by ob liquely polishing a 25- m-thick single-crystalline YIG (111) grown \non a 0.5-mm-thick single-crystal line GGG (111) substrate by a liquid phase epitaxy method. The \nobtained tYIG was confirmed to be 6.4 m/mm by a cross-sectional scanning electron microscopy \n[Fig. 1(c)]. In order to investigate the T dependence, the sample was fi xed on a stage with a heater \nand a temperature sensor by heat-resistant ad hesive (Aron Ceramic D, Toagosei Co., Ltd.). \nConducting wires were electrically connected to the ends of the Pt strip using heat-resistant \nconductive paste (MAX102, Nihon Handa Co., Ltd.). An insulating black ink was coated on the \nsample surface to increase the infrared emissivi ty and to make the emis sivity uniform. To degas \nthe sample and stage, they were heated up to 600 K for an hour in a high vacuum before the measurement. The electrical resistivity of the Pt strip was unchanged after heating and its T 3 dependence was similar to that obtained in a previous study.25) \nWe measured the SPE-induced temperature cha nge by using the LIT me thod [Fig. 1(d)]. A \nsquare-wave-modulated AC charge current with the amplitude of J0=5 mA, frequency of f=5 Hz, \nand zero offset was applied to the Pt strip. To a lign the magnetization of YIG, an in-plane magnetic \nfield H with the magnitude 0|H|=100 mT was applied along the x direction, where 0 is the \nvacuum permeability. When the charge current is a pplied to the Pt strip, a spin current is generated \nby the spin Hall effect.8,9) This spin current is injected into YIG at the Pt/YIG interface, causing \nthe temperature change induced by SPE TSPE when the charge current direction is perpendicular \nto the magnetization of YIG.6,7,17) As demonstrated in the previous studies, TSPE can be detected \nby the LIT method because TSPE changes its sign depending on the charge current \ndirection.7,17,19,24) The first-harmonic component of the te mperature oscillati on was extracted from \nthermal images taken with an infrared camera by Fourier analysis and converted into the lock-in \namplitude A and phase images. Since TSPE shows the H-odd dependence, we calculated \nAodd=|A(+H)ei(+H)A(H)ei(H)|/2 and odd=arg[A(+H)ei(+H)A(H)ei(H)] from the thermal \nimages measured with applying positive and negative magnetic fields. The SPE-induced \ntemperature change was obtained by TSPE=Aoddcosodd since the time delay due to thermal \ndiffusion is negligibly small in the LIT-based SPE measurements.7,17,19) Figure 1(b) shows a \nsteady-state infrared image of the black-ink-coa ted sample at 314 K, where the YIG film with \ntYIG exists below the white dotted line. In the region above the line, no SPE signal should be \ngenerated above room temperatur e because of the absence of YIG. The area surrounded by an \norange dotted rectangle shows the position and size of the Pt strip. Based on the tYIG value, the \ntYIG resolution in our setup was obtained to be 30 nm/pixel. During the LIT measurements, the \ntemperature of the sample surface was monitored w ith the infrared camera in a high vacuum of \n<410-4 Pa through a CaF 2 window with high infrared transpar ency. The small re duction of the \ninfrared emission intensity from the sample due to the CaF 2 window was corrected based on a \ncalibration curve, which was measured by using a resistance temperature sensor as a dummy \nsample and by comparing the sensor and thermal image values. \nFigures 2(a) and 2(b) show the Aodd and odd images for the Pt/wedged-YIG sample at T=314 \nK. The current-induced temperature change appears only in the region with the Pt strip on the YIG \nfilm. Since Aodd and odd are the H-odd components, the contributi on from the field-independent \nPeltier effect is eliminated. The magnitude ( Aodd) and sign ( odd) of the observed temperature \nmodulation is consistent with the SPE signal reported previ ously in the Pt/YIG systems.24) No \ntemperature modulation is genera ted in the region without YIG, confirming that the ordinary 4 Ettingshausen effect in Pt is negligibly small. Th ese features indicate that this temperature change \nis due to SPE. Figures 2(c) a nd 2(d) show the profile of the Aodd and odd signals in the y direction. \nIn the region with finite tYIG, the SPE signal monotonically increases with increasing tYIG and \nsaturates when tYIG>5 m. The spatial distribution of the SPE signal is also consistent with the \nprevious result.24) \nFigures 3(a) and 3(b) show the Aodd and odd images for various values of T in the high-\ntemperature range. The images show that the SPE signal disappears at 552 K, around Tc of YIG \n[note that the electrical resistivity of the Pt strip exhib its no anomaly around Tc, as shown in the \ninset to Fig. 3(c)]. The T dependence of TSPE at each tYIG was calculated by averaging the Aodd \nand odd data along the x direction over a length of 0.4 mm. At tYIG=5 m, where the SPE signal \nreaches the saturation value, the magnitude of th e SPE signal is almost constant up to 400 K but \nmonotonically decreases with increasing T for T>400 K [Fig. 3(c)]. The similar T dependence of \nTSPE was obtained also in the small tYIG region in which the SPE signal does not saturate. \nTo investigate the T dependence of the characteristic length of SPE lSPE in YIG, we analyzed \nthe tYIG dependence of the SPE signal at each temp erature. Here, we ad opt a phenomenological \nexponential decay model as a simplest analysis: \n∆𝑇ୗ∝1െexp൬െ𝑡ଢ଼୍ୋ\n𝑙ୗ൰. ሺ1ሻ \nFigure 4(a) shows the experimental and fitting results for various values of T. When the SPE \nsignal is sufficiently large, the experimental resu lts are well fitted by Eq. (1) in the whole thickness \nrange, where the coefficient of determination is R2>0.8 for T<520 K. However, for T>520 K, the \nfitting accuracy is poor ( R2<0.8) in the small tYIG region because of the small magnitude of the \nSPE signal. Figure 4(b) shows the T dependence of lSPE and R2. The lSPE value near room \ntemperature (314 K) is estimated to be 0.9 µm, which is comparable to the values obtained in the \nprevious studies on SPE.24,26,27) We found that lSPE remains almost constant as the temperature \nincreases. \nNext, we compare the T dependence of the SPE signal with that of the SSE voltage. If the \nreciprocal relation between SPE and SSE and T-independent lSPE are assumed, the SSE voltage \nnormalized by the applied temperature difference can be compared with the factor proportional to \n(/T)(TSPE/jc), where is the thermal conductivity of YIG.20) Figure 4(c) shows the \n(/T)(TSPE/jc) values as a function of T at tYIG=5 m, where the T dependence of is estimated \nfrom the data in Ref. 21. The magnitude of ( /T)(TSPE/jc) almost linearly decreases with \nincreasing T, which is different from the behavior of the SSE voltage.21) In order to quantify the \nT dependence of SPE, ( /T)(TSPE/jc) is fitted by 5 𝜅\n𝑇∆𝑇ୗ\n𝑗ୡ∝ሺ𝑇ୡെ𝑇ሻఉ. ሺ2ሻ \nThe (/T)(TSPE/jc) data is well fitted with the critical exponent of =1.1 [Fig. 4(c)]. We observed \nthe same T dependence of th e SPE signal with =1.1 also in a Pt-film/Y IG-slab junction system \nwith the single-crystalline YIG slab grown by a flux method. This result is clearly different from \n=1.5-3.0 for SSE in the high-temperature range. \nFinally, we discuss the possi ble reason of the different T dependence of SPE and SSE. We \nemphasize again that the same T dependence of the SPE signal wa s obtained not only in the YIG \nfilm grown by the liquid phase ep itaxy method but also the YI G slab grown by the flux method, \nindicating that the difference in the growth method of YIG is irrelevant to the different behaviors \nbetween SPE and SSE. Now recall the fact that the thermo-spin conversion by SPE and SSE has \nthe magnon frequency dependence.13) In the case of SSE, a temperature gradient applied to the \nPt/YIG system induces magnon spin currents an d low-frequency subthermal magnons dominantly \ncontribute to the SSE voltage.13,14,28) On the other hand, in the case of SPE, the spin Hall effect in \nPt induces magnon spin currents in YIG and the magnon frequenc y dominantly contributing to \nthe SPE-induced temperature cha nge is unclear. If the magnon fre quency excited by the spin Hall \neffect in SPE is different from that exc ited by the temperature gradient in SSE, the T dependence \nand characteristic length of these phenomena can be different from each other. In fact, the \ncharacteristic length for SPE in Fig. 4(b) is co mparable to that estimated in the previous SPE \nexperiments24,26,27) but smaller than that estimated in the previous SSE experiments.15,29-33) This \nsituation indicates that the Onsa ger reciprocal relation between SPE and SSE is not simple and its \nmagnon frequency dependence should be taken into account. To clarif y the spectral na ture of SPE, \nin addition to the T and tYIG dependences, the high-magnetic-fie ld response of SPE and SSE should \nbe compared systematically in a wide temperatur e range, which is one of the remaining tasks in \nthe study of these phenomena. \nIn summary, we investigated the T and tYIG dependences of SPE by the LIT method. The SPE \nsignal in the Pt/YIG system was found to monotonically decrease with increasing T when T>400 \nK and disappear around the Curie temperat ure of YIG. By fitting the observed tYIG dependence of \nthe SPE signal by the exponential de cay model, the characteristic le ngth of SPE was estimated to \nbe 0.9 m near room temperature (314K) and be almost constant as the te mperature increases. \nThe SPE-related factor that can be compared with the SSE voltage, ( /T)(TSPE/jc), shows the \n(TcT)1.1 dependence, which is signi ficantly different from the T dependence of the SSE voltage \nin the Pt/YIG system: ( TcT) with =1.5-3.0. The results reported here highlight the difference \nin the thermo-spin conversion between SPE and SSE and suggest the magnon-frequency-6 dependent nature in the reci procal relation between them. \n \nAcknowledgments \nThe authors thank M. Isomura for technical supports. This work was supported by CREST \n\"Creation of Innovative Core Technologies for Nano-enable d Thermal Management\" (No. \nJPMJCR17I1) from JST, Japan; Grant-in-Aid fo r Scientific Research (B) (No. 19H02585) and \nGrant-in-Aid for Scientific Research (S) (No. 18H05246) from JSPS KAKENHI, Japan; NEC \nCorporation; and NIMS Jo int Research Hub Program. \n \nReferences \n1) G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). \n2) S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). \n3) K. Uchida, Proc. Jpn. Acad., Ser. B 97, 69 (2021). \n4) K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y . Kajiwara, H. \nUmezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010). \n5) K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. \n97, 172505 (2010). \n6) J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, \nPhys. Rev. 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Ujiie, Y . Hattori, R. Tsuboi, and E. Saitoh, Appl. Phys. Express 13, \n103001 (2020). \n25) K. Uchida, Z. Qiu, T. Kikkawa, R. I guchi, and E. Saitoh, Appl. Phys. Lett. 106, 052405 \n(2015). \n26) T. Yamazaki, R. Iguchi, T. Ohkubo, H. Nagano, and K. Uchida, Phys. Rev. B 101, 020415(R) \n(2020). \n27) S. S. Costa and L. C. Sampaio, J. Phys. D: Appl. Phys. 53, 355001 (2020). \n28) I. Diniz and A. T. Costa, New J. Phys. 18, 052002 (2016). \n29) A. Kehlberger, U. Ritzmann, D. Hinzke, E. J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. \nH. Kim, C. A. Ross, M. B. Jungfleisch, B. Hillebrands, U. Nowak, and M. Kläui, Phys. Rev. Lett. 115, 096602 (2015). \n30) L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 \n(2015). \n31) B. L. Giles, Z. Yang, J. S. Jamison, and R. C. Myers, Phys. Rev. B 92, 224415 (2015). \n32) T. Hioki, R. Iguchi, Z. Qiu, D. Hou, K. Uc hida, and E. Saitoh, Appl. Phys. Express 10, \n073002 (2017). \n33) B. L. Giles, Z. Yang, J. S. Jamison, J. M. Gomez-Perez, S. Vélez, L. E. Hueso, F. Casanova, \nand R. C. Myers, Phys. Rev. B 96, 180412(R) (2017). \n 8 \nFig. 1. (a) Schematic of the Pt/wedged-YIG system used for the SPE measurement. The YIG film \nhas a thickness gradient tYIG along the y direction. (b) Steady-st ate infrared image of the \nPt/wedged-YIG system. The area below the white dotted line corresponds to the area with the \nfinite YIG thickness tYIG. The area surrounded by the orange dot ted rectangle corresponds to the \narea with the Pt film. (c) tYIG profile and cross-sectional imag e of the wedged-YIG/GGG substrate \nwithout the Pt layer, measured by scanning electron microscopy. The samples used for the SPE \nand scanning electron microscopy measurements were cut from the same wafer. (d) Schematic of \nthe LIT method for measuring SPE. The base temperature T of the sample was controlled by a \nheater attached to the stage and monitored with an infrared camera. \n \n9 \nFig. 2. (a),(b) Lock-in amplitude Aodd and phase odd images for the Pt/wedged-YIG system at \nT=314 K, magnetic field of 0|H|=100 mT, and square-wave ch arge current amplitude of J0=5 \nmA. 0 is the vacuum permeability. (c),(d) y-directional profile and corresponding tYIG \ndependence of the Aodd and odd signals for the Pt/w edged-YIG system at T=314 K. The Aodd and \nodd profiles are obtained by averaging the raw data in the area defined by the white dotted squares \nin (a) and (b), respectively. \n \n10 \nFig. 3. (a),(b) Aodd and odd images for the Pt/wedged-YIG system for various values of T at \n0|H|=100 mT and J0=5 mA. (c) T dependence of TSPE=Aoddcosodd for various values of tYIG. \nThe data points are obtained by av eraging the raw data along the x direction over 0.4 mm on the \npositions of the white dotted lines depicted in the rightmost Aodd image in (a). The error bars \nrepresent the standard de viation of the data. The inset to (c) shows the T dependence of the \nresistivity of the Pt film. The values were measured using a 5-nm-thick Pt film sputtered on a \nsingle-crystalline YIG without tYIG by the four-terminal method. \n \n11 \nFig. 4. (a) tYIG dependence of TSPE (blue) and fitting curves (o range) for various values of T at \n0|H|=100 mT and J0=5 mA. (b) T dependence of the char acteristic length of SPE lSPE (orange) \nand coefficient of determ ination of the fitting R2 (green), obtain ed by fitting the tYIG dependence \nof TSPE at each temperature with Eq. (1). (c) T dependence of ( /T)(TSPE/jc), where and jc are \nthe thermal conductivity of YIG and charge current de nsity, respectively. The T dependence of \nis estimated from the data in Ref. 21. The green line shows the fitting results using Eq. (2). \n" }, { "title": "1910.05371v3.Simulation_of_sympathetic_cooling_an_optically_levitated_magnetic_nanoparticle_via_coupling_to_a_cold_atomic_gas.pdf", "content": "Simulation of sympathetic cooling an optically levitated magnetic nanoparticle via\ncoupling to a cold atomic gas\nT. Seberson1, Peng Ju1, Jonghoon Ahn3, Jaehoon Bang3, Tongcang Li1;2;3;4, F. Robicheaux1;2,\n1Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA\n2Purdue Quantum Science and Engineering Institute,\nPurdue University, West Lafayette, Indiana 47907, USA\n3School of Electrical and Computer Engineering,\nPurdue University, West Lafayette, Indiana 47907, USA and\n4Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA\n(Dated: June 1, 2022)\nA proposal for cooling the translational motion of optically levitated magnetic nanoparticles is\npresented. The theoretical cooling scheme involves the sympathetic cooling of a ferromagnetic YIG\nnanosphere with a spin-polarized atomic gas. Particle-atom cloud coupling is mediated through the\nmagnetic dipole-dipole interaction. When the particle and atom oscillations are small compared to\ntheir separation, the interaction potential becomes dominantly linear which allows the particle to\nexchange energy with the Natoms. While the atoms are continuously Doppler cooled, energy is\nable to be removed from the nanoparticle's motion as it exchanges energy with the atoms. The rate\nat which energy is removed from the nanoparticle's motion was studied for three species of atoms\n(Dy, Cr, Rb) by simulating the full N+ 1 equations of motion and was found to depend on system\nparameters with scalings that are consistent with a simpli\fed model. The nanoparticle's damping\nrate due to sympathetic cooling is competitive with and has the potential to exceed commonly\nemployed cooling methods.\nI. INTRODUCTION\nCooling the motion of an optically levitated nanopar-\nticle to the motional ground state has proven to be\na formidable experimental challenge. Limiting the\nnanoparticle temperature is ine\u000ecient detection of scat-\ntered light, laser shot noise, and phase noise, among\nothers. These limitations seen in conventional tweezer\ntraps have sparked theorists and experimentalists alike\nto explore new and hybrid levitated systems that may\no\u000ber alternative routes to the quantum regime. Pas-\nsive/sympathetic cooling schemes involving coupling dif-\nferent degrees of freedom or nearby particles has been ex-\nplored [1{6]. Cavity cooling has had success [7{9] where\nstrong coupling rates have been achieved through coher-\nent scattering with the addition of a tweezer trap and al-\nlowed cooling to the lowest reported occupation number\nofn<1 [10, 11]. Even in its beginning stages, all electri-\ncal or electro-optical hybrid systems utilizing electronic\ncircuitry are able to reach mK temperatures [12{16] with\none particular experiment reaching n= 4 through cold\ndamping [17]. The \feld has also recently seen magnetic\nparticles and traps being investigated [18{21], such as\nstudying the dynamics of a ferromagnetic particle levi-\ntated above a superconductor [22{24].\nIn the spirit of promising new systems, this paper in-\nvestigates a possible method of cooling the translational\nmotion of an optically trapped ferromagnetic nanopar-\nticle by coupling to a spin-polarized cold atomic gas.\nThe coupling arises from the magnetic dipole-dipole in-\nteraction and allows signi\fcant energy exchange between\nthe two systems. While the atom cloud is continuously\nDoppler cooled, energy is extracted from the nanopar-\nticle through this energy exchange. The coupling of ananoparticle to an atom cloud has been proposed previ-\nously with the coupling mediated by scattered light into\na cavity [6]. Sympathetic cooling a particular vibrational\nmode of a membrane was successfully demonstrated using\na similar technique [25{28], but with \fnal temperatures\nwell above the ground state. The scheme proposed here\ndoes not require optical cavities and has the potential to\ncool to the quantum regime.\nSimulations show that an atom cloud containing 106\nor more atoms is su\u000ecient to achieve damping rates that\nare competitive with or exceeding that of cold damping\nor parametric feedback cooling. The theoretical cooling\nscheme proposed is best suited for, but not limited to,\nnanoparticle frequencies in the 100 kHz range or larger.\nThis article is organized as follows. In Sec. II the the-\noretical proposal to couple a spin-polarized atomic gas to\na ferromagnetic nanosphere through the magnetic dipole-\ndipole interaction is given. In Sec. III, simulation results\nof the particle-atom cloud system with continuous atom\nDoppler cooling are provided with a discussion of the\ncooling results and extension to three dimensions. Lastly,\na comparison with other cooling methods is discussed as\nwell as experimental considerations.\nII. MODEL OF THE SYSTEM\nThe proposed physical system includes a ferromag-\nnetic nanosphere of radius Rand massMpharmonically\ntrapped in the focus of a laser beam traveling in the\n~k=2\u0019\n\u0015^zdirection (see Fig. 1). A ferromagnetic sphere\nwith dipole moment ~ mproduces a magnetic \feld [29]\n~Bs(r) =\u0010\u00160\n4\u0019\u0011\u00143 (~ m\u0001^r) ^r\nr3\u0000~ m\nr3\u0015\n; (1)arXiv:1910.05371v3 [physics.atom-ph] 9 Oct 20202\nFIG. 1. Illustration of the proposed model. A ferromagnetic\nnanosphere is trapped at the focus of a Gaussian beam. The\noscillation frequency for the nanoparticle in the ydirection is\n!p. A cloud of Natoms a distance y0away are trapped in a\nseparate, far red-detuned dipole trap oscillating at frequency\n!ain theydirection. An external magnetic \feld ~Bextorients\nthe magnetic moments of the nanoparticle and atoms.\nwhere~ ris directed outwards from the center of the\nsphere. The sphere's moment will align along the y-\naxis if a constant, uniform magnetic \feld ~Bext=B0^y\nis present, and a \feld distribution described by Eq. (1)\nwill surround the particle.\nA distance y0above the focus of the nanoparticle\ntrap, a cloud of N, non-interacting, spin-polarized atoms\neach with dipole moment ~ \u0016a=\u0000\u0016a^yand massMaare\ntrapped in a far red-detuned dipole trap with oscilla-\ntion frequency !a. The total particle-atom cloud poten-\ntial energy including the repulsive interaction Uint;j=\n\u0000~ \u0016a;j\u0001~Bs(rj) for each atom jis\nU=1\n2Mp!2\npr2\np+NX\nj=1\u00121\n2Ma!2\nar2\na;j+Uint;j\u0013\n;(2)\nwherera;j(rp) is the atom (particle) position. If both\nthe atoms and the nanoparticle undergo small oscilla-\ntions compared to the distance separating them, y0\u001d\n(rp;ra;j), the interaction Uint;jis quasi-one-dimensional\nUint;j\u0019g=(ya;j+y0\u0000yp)3; (3)\nwhereg= 2\u0016aj~ mj\u00160=4\u0019de\fnes the interaction strength.\nTheN+ 1 equations of motion for the ydegrees of free-\ndom are\nyp=\u0000!2\npyp\u0000NX\nj=13g=Mp\n(ya;j+y0\u0000yp)4; (4)\nya;1=\u0000!2\naya;1+3g=Ma\n(ya;1+y0\u0000yp)4;\n:::\nya;N=\u0000!2\naya;N+3g=Ma\n(ya;N+y0\u0000yp)4:(5)The focus from here on will be the dynamics associated\nwith theydegree of freedom. The equations of motion for\nthexandzdegrees of freedom have minimal coupling and\nare therefore largely harmonic oscillators. Extension to\nthree dimensions is possible by placing the atom trap at\na distance~ r0=hx0;y0;z0iwhile preserving anti-parallel\natom and nanoparticle dipole orientations. For simplic-\nity, the analysis in this paper focuses on one dimension.\nFurther discussion of three dimensional cooling can be\nfound in Sec. III C.\nIn what follows, the aim will be to study the possibility\nof removing motional energy from the nanoparticle sym-\npathetically by continuously cooling each atom. Doppler\ncooling was the method of choice for cooling the atoms\nin this paper. Under Doppler cooling each atom experi-\nences a momentum kick \u0016 hkin the ^kdirection if a photon\nis absorbed followed by a kick of the same magnitude in a\nrandom direction after spontaneous emission. The prob-\nability for absorption in each time interval dt\u001c1=!ais\nP=Rdtwith absorption rate [30]\nR=\u0000\n2=4\n\u0010\n\u0001 +~ va;j\u0001~k\u00112\n+ \n2=2 + \u00002=4: (6)\nHere, \n = \u0000p\nr=2 is the Rabi frequency, \u0000 the decay\nrate,r=I=Isatthe saturation intensity ratio, and \u0001 the\nlaser detuning. The resulting atom velocity after this\ntime interval can be written as\n_ya;j(t+dt) = _ya;j(t) +\u0016hk\nMa(\nsgn(^k)\u00061;absorbed\n0; otherwise:\n(7)\nOther factors contributing to the dynamics of the\ntrapped nanoparticle are collisions with the surround-\ning gas molecules and laser shot noise heating. For\nthe vacuum chamber pressures used in atom trapping,\n\u001810\u00008\u000010\u00009Torr, the a\u000bects due to laser shot noise\ndominate that of the surrounding gas. After a time in-\ntervaldtthe nanoparticle velocity becomes\n_yp(t+dt) = _yp(t) +q\n2_ETdt=MpW(0;1); (8)\nwhere _ETis the translational shot noise heating rate\n[31] andW(0;1) is a random Gaussian number with zero\nmean and unit variance.\nSection III presents the results of simulating Eqs. (4)\nto (8) with Eqs. (6) and (7) modeled using a Monte Carlo\nmethod. Subsections II A and II B explore the dynamics\nof Eqs. (4) and (5) analytically under a linear coupling\napproximation both with and without atom damping.3\nA. Dynamics under an approximate linear coupling\nIn the regime y0\u001d(yp;ya;j), Eq. (3) may be expanded\nUint;j\u0019g\ny3\n0\u0014\n1 +3\ny0(yp\u0000ya;j) +6\ny2\n0(yp\u0000ya;j)2+:::\u0015\n:\n(9)\nKeeping only terms to second order in Eq. (9) and\nde\fning the center of mass of the atom cloud as Ya\u0011\n1\nNPN\nj=1ya;j, the equations of motion may be written as\nyp=\u0000Nap\u0000\u0000\n!2\np+N\n2\np\u0001\nyp+N\n2\npYa; (10)\nYa=aa\u0000\u0000\n!2\na+ \n2\na\u0001\nYa+ \n2\nayp; (11)\nwhereai=\u0000\n3g=Miy4\n0\u0001\n,i= (p;a), is a constant accelera-\ntion that shifts the equilibrium position of the oscillator\nand \n2\ni=\u0000\n12g=Miy5\n0\u0001\nis a coupling constant as well as\na frequency shift in the harmonic potential. The \n ihere\nare di\u000berent from the Rabi frequency de\fned in Eq. (6).\nProvided the condition y0\u001d(yp;ya) is maintained so\nthat higher order terms in Eq. (9) do not contribute, the\nnanoparticle will exchange energy with the atom cloud\ndue to the linear coupling in Eqs. (10) and (11). Retain-\ning predominantly the lower order terms is only possible\nfor nanoparticle temperatures much smaller than room\ntemperature as the atoms' positions will increase dras-\ntically as they exchange energy with the particle. The\ncondition may also be satis\fed if the motion of the atoms\nis continuously cooled via a cooling mechanism such as\nDoppler cooling, which will be discussed in the next sub-\nsection.\nDue to the frequency shifts, \n2\ni, the nanoparticle\nand/or atom cloud trap frequencies need to be tuned\nto resonance for coherent energy exchange !2\na+ \n2\na=\n!2\np+N\n2\np=!2\n0. While on resonance, the rate at which\nenergy is exchanged from the particle to the atoms is\nsolved for by \fnding the normal mode frequencies of Eqs.\n(10) and (11) and identifying the beat frequency. This\n\"exchange frequency\" is\nfexch=\np\nap\nN\n\u0019!0: (12)\nNote that while the overall force due to the magnetic\ndipole-dipole interaction was chosen to be repulsive, the\ndynamics are similar for an attractive interaction since\nthe energy exchange e\u000bect is independent of the sign\nof the linear coupling term. Thus, although the atom\ncloud may be uniformly spin-polarized through the ex-\nternal magnetic \feld and optical pumping [32{34], there\nis no loss of coupling if atoms undergo spin \rips on time\nscales greater than the oscillation period.B. Sympathetic cooling with linear coupling\nIf each atom in the cloud is continuously Doppler\ncooled, motional energy can be removed from the\nnanoparticle. For the calculations in this section,\nDoppler cooling is modeled using Langevin dynamics and\nis valid for the time scales of consideration here, \u0000 \u001d!0.\nDoppler cooled atoms experience an e\u000bective damping\nforceFD;j=\u0000\u000b_ya;jwith damping rate \u0000 a=\u000b=Ma=\n\u0016hk2I=(I0Ma) when tuned to reach the Doppler tempera-\ntureTmin= \u0016h\u0000=2kB[30, 34, 35]. Excluding the constant\naccelerations in Eqs. (10) and (11), the equations of mo-\ntion with Doppler cooling on the atoms as well as laser\nshot noise on the nanoparticle become\nyp=\u0000!2\n0yp+N\n2\npYa+\u0018SN(t); (13)\nYa=\u0000!2\n0Ya+ \n2\nayp\u0000\u0000a_Ya+\u0018DC(t)=p\nN; (14)\nwhere\u0018SN(t) accounts for \ructuations due to laser shot\nnoise and1\nNPN\nj=1Fa;j(t)=Ma!\u0018DC(t)=p\nNare \ructu-\nations due to spontaneous emission during Doppler cool-\ning.\nWith the atoms continuously Doppler cooled, the \f-\nnal temperature of the nanoparticle, Tp, under sympa-\nthetic cooling can be estimated. One method is through\nintegration of the power spectral density (PSD) since\nTp/R\nSyy(!)d!=hy2\npi. Fourier transforming Eqs. (13)\nand (14) and solving for the nanoparticle's displacement\nin the frequency domain, eyp=Ffypg, gives\neyp=\"p\nN\n2\npe\u0018DC(!)\n\u0001+e\u0018SN(!)#\u00141\n\u000e2(!)\u0000N\n2p\n2a=\u0001\u0015\n;\n(15)\nwhere \u0001 = \u000e2(!)+i\u0000a!and\u000e2(!) =!2\n0\u0000!2. Shot noise\nadds an overall constant to the noise \roor of the spectrum\nand near!\u0018!0the a\u000bects are negligible, e\u0018SN(!)\u001c\nN\n2\npe\u0018DC(!)=\u0001, and may therefore be omitted. Using the\nsingle sided noise spectrum je\u0018DC(!)j2!4\u0000akBTmin=Ma,\nthe PSD of the nanoparticle's displacement is\nSyy(!) =N\n4\np(4\u0000akBTmin=Ma)\n\u000e4(!)h\u0000\n\u000e2(!)\u0000N\n2p\n2a=\u000e2(!)\u00012+ (\u0000a!)2i:\n(16)\nEquation (16) is exact in the absence of laser shot noise\non the nanoparticle and has been con\frmed through\nsimulation of Eqs. (13) and (14) with and without\n\u0018SN(t). The PSD is well described by two peaks lo-\ncated at!2\n\u0006=!2\n0\u0006p\nN\np\nain the weak coupling\nregime!2\n0>p\nN\np\na. In the strong coupling regime,\n!2\n0p\nN\np\na. As the cal-\nculations were performed using the full N+ 1 equations\nof motion, simulation time was the only constraint from\nobserving the e\u000bects for larger atom numbers, N > 104.\nThe average \fnal temperatures reached for each atom\nspecies (Dy,Cr,Rb) was 794 \u0016K;406\u0016K;and 158 mK,respectively, for a simulation time of t= 100 ms at\nN= 104. Numerical integration of Eq. (16) for the three\natom species (Dy, Cr, Rb) at N= 104gives an approxi-\nmated equilibrium temperature of Tp\u0019Mp!2\n0hy2\npi=kB=\n(2:002;0:584;0:572) mK, respectively. Since the particle\nis not a simple harmonic oscillator the values are approx-\nimate, but may serve as an upper bound for the parti-\ncle temperature in experiments. The \fnal temperature\nmay additionally be estimated using the rate equation\nEq. (19),Tp= (784;256;756)\u0016K atN= 104for (Dy,\nCr, Rb), respectively, which shows better agreement with\nthe simulation results for Dy and Cr. A simulation time\nof 100 ms was not long enough to observe the equilibrium\ntemperature of the particle using Rb atoms. The expres-\nsions used to estimate the \fnal temperature assume equi-\nlibrium has been reached, hence the discrepancy between\nthe simulation temperature and the estimates.\nBesides the number of atoms in the trap, Eq. (18)\npredicts that the cooling rate \rcool/g2depends on\nthe square of the magnetic coupling strength g=\n2\u0016aj~ mj\u00160=4\u0019. Using the data in Fig. 2(a) and the values\nfor\u0016ain Table I, \r/\u00162\nais con\frmed with a coe\u000ecient\nof determination r2= 0:997. Together with the linear\ndependence of \ronN, this con\frms the ability to de-\nscribe this non-linear system in an approximated linear\ncoupling regime as was done in Sec. II B.\nNote that\rin Fig. 2 includes the cooling rate \rcoolas\nwell as the competing shot noise heating rate _ET. The\ntwo parameters that these two quantities share are the\ndensity, which is predominantly \fxed, and the size (ra-\ndiusR) of the nanoparticle. Approximating j~ mj/R3\nwe \fnd\rcool/R3while _ET/R3shares the same\nRdependence [31]. However, shot noise heating is lin-\near in time while the cooling is exponential, indicating\nthat larger particles may provide faster cooling, but will\nnot in\ruence the \fnal temperature of the nanoparticle.\nThe in\ruence of shot noise heating may be further re-\nduced by cooling the degree of freedom in the laser po-\nlarization direction, ^ x, since the least amount of shot\nnoise is delivered to that degree of freedom for a particle\nin the Rayleigh limit [31]. The major source of heat-\ning in conventional levitated systems are collisions with\nthe surrounding gas for pressures above 10\u00006Torr with\n\u0000gas=2\u0019\u001810 kHz. Cold atom experiments typically op-\nerate with 10\u00008\u000010\u00009Torr chamber pressures, making\nlaser shot noise _ET=\u0016h!0= 7:5 kHz the dominant source\nof heating for the YIG particle in this proposal.6\nC. Discussion\nThe sympathetic cooling scheme can be extended to\ngive three dimensional cooling by placing the center of\nthe atom cloud a distance ~ r0=hx0;y0;z0iaway from\nthe nanoparticle and directing the magnetic \feld in that\ndirection ^Bext= ^r0so that the dipole moments align. In\nthis case all of the translational degrees of freedom would\nbe coupled with one another, as well as with all of the\ntranslational degrees of freedom of the atoms. Despite\nthe various couplings, simulations of the equations of mo-\ntion up to the same order as in Sec. II show that this\ndoes not cause instability and allows cooling of each de-\ngree of freedom so long as di\u000berent degrees of freedom are\nnot resonant with one another, !i;a=!i;p,!i;a6=!j;a\nwhere (i;j) = (x;y;z ) andi6=j. This condition is found\nexperimentally since generally tweezer traps have di\u000ber-\nent beam waists in the xandydirections causing the\nfrequencies to be separated.\nFor three dimensional cooling, it is possible to match\nall three frequencies. The radial frequencies ( !x;!y) can\nbe set by tuning the respective beam waists of the beam\nwhich can be controlled by either tightening the focus or\ntuning the lens. The axial frequency ( !z) is dependent\non the wavelength and the x;ybeam waists non-linearly.\nThe frequency of the atoms and the particle share the\nsame dependence on these parameters. To match the\natoms to the particle, each frequency has the same de-\npendence on the trap intensity, scaling proportionally.\nTo date, cold damping and cavity cooling by coherent\nscattering have proven to be the most e\u000bective methods\nof cooling a levitated nanoparticle with reported occupa-\ntion numbers of n= 4 andn < 1, respectively [11, 17].\nSimilar to parametric feedback cooling, cold damping\nprovides damping rates in the \u00181 kHz range while co-\nherent scattering has yielded rates in the 10 kHz range\n[10, 11, 17, 41]. The results above indicate that atom\nnumbers of the order N\u0018105(106), which corresponds\nto\r >1 kHz, would be su\u000ecient for the nanoparticle to\nreach the atom's Doppler temperature for atom species\nwith magnetic moment \u0016a=\u0016B= 10(1). The number of\natoms that have been trapped experimentally is in the\nrange\u0018106\u0000108for chromium, rubidium, and others\n[6, 34, 42]. The ground state energy of the nanoparticle\nisT= \u0016h!0=kB= 5\u0016K. Many of the commonly trapped\natom species have unit magnetic moment, large N, and\nare able to reach the \u00181\u000010\u0016K regime [43, 44]. Com-\nparing with the energy removal rate found for Rb in Fig.\n2(a), these parameters are su\u000ecient for motional ground\nstate cooling of the nanoparticle.\nAdmittedly, performing an experiment with the ex-\nact numbers as outlined in this paper may be a chal-\nlenging task with currently technology. Optical dipole-\ntraps allow for 100 kHz range trapping frequencies, large\natom numbers, as well as low temperatures [27]. How-\never, obtaining high enough densities to support an atom-\nnanoparticle separation of <1\u0016m may be di\u000ecult on long\ntime scales. To the best of our knowledge, atom densitiesof 1012\u00001015cm\u00003are achievable to date [45{49].\nThe general idea of the proposed scheme is to sym-\npathetically cool a nanoparticle utilizing the linear cou-\npling found in the dipole-dipole interaction, but is not\nlimited to the methods chosen here and allows for adapt-\nability. For example, Doppler cooling as the atom cooling\nmethod was chosen for simulation simplicity while retain-\ning physicality. Other atom cooling methods o\u000ber lower\ntemperatures such as sideband cooling, Sisyphus cool-\ning, or using a spin-polarized Bose-Einstein condensate,\nwhich are able to reach nK temperatures [50]. A charged\nYIG particle could also be trapped in an ion trap instead\nof an optical trap.\nBose-Einstein condensates may have potential as sym-\npathetic cooling candidates owing to their large densities\n(1015cm\u00003) and low temperatures [33]. Bose-Einstein\ncondensates have been shown to reach submicron sepa-\nrations from a SiN surface [51, 52]. In the experiment of\nRef. [51], with !a= 10 kHz (5 kHz) and N= 103, the ra-\ndius of the Rb BEC was 290 nm (430 nm). The BEC was\nable to resonantly couple with a cantilever at separations\nof\u00181\u0016m with the cantilever at room temperature and\nno active cooling applied to the system. Further limiting\nthe separation to the cantilever was the attractive BEC-\nsurface interactive potential /1=r4which distorted the\ntrapping potential at small separations. The magnetic\ndipole-dipole interaction in this paper is /1=r3, has no\nconcern with surface interactions, and assumes a system\nstabilized by active cooling. These considerations imply\nshorter separations are attainable. As BEC's are quan-\ntum in nature, a separate investigation of this possibility\nwould be in order, though the methods would be similar\nto those outlined in this classical investigation.\nIV. CONCLUSION\nA theoretical proposal to sympathetically cool a levi-\ntated ferromagnetic nanoparticle via coupling to a spin-\npolarized atomic gas was studied. While oscillating in\ntheir respective traps, the particle and atom cloud sys-\ntems would be coupled through the non-linear magnetic\ndipole-dipole interaction. For su\u000eciently large separa-\ntion between the particle and the atom cloud relative\nto their displacements, the nanoparticle and atom cloud\nwould exchange energy with one another via the linear\ncoupling term that is dominant in the magnetic force ex-\npansion. If the atoms are continuously Doppler cooled,\nenergy would be able to be removed from the particle's\nmotion.\nSimulations of the particle-atom cloud system were\nperformed using the full, non-linear, magnetic dipole-\ndipole interaction for three species of atoms and varying\nnumbers of atoms in the trap. The nanoparticle cooling\nrate was shown to be proportional to the number of atoms\nin the trap as well as the square of the magnetic moment\nof the atom, validating that it is possible to describe the\ndynamics using a linear approximation to the magnetic7\nforce. The rate at which energy is removed from the par-\nticle motion is signi\fcant for 104atoms in the trap when\nthe atoms are continuously Doppler cooled. It is expected\nthat the particle would reach the atom Doppler temper-\nature as the number of atoms increases. This method of\nsympathetic cooling has potential to cool the nanopar-\nticle to its motional ground state for atom species with\nlower Doppler temperatures. 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Hossain1,5,†\n1Department of Physics, Indian Institute of Technology, Kan pur 208016, India.\n2Institute of Physics, Bhubaneswar 751005, India.\n3Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany.\n4Homi Bhabha National Institute, Anushakti Nagar, Mumbai 40 0085, India.\n5Institute of Low Temperature and Structure Research, 50-42 2 Wroclaw, Poland.\n(Dated: September 27, 2021)\nWe report strong damping enhancement in a 200 nm thick yttriu m iron garnet (YIG) film due\nto spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet\n(GdIG) layer antiferromagnetically (AFM) exchange couple s with the rest of the YIG layer. The\nout-of-plane angular variation of ferromagnetic resonanc e (FMR) linewidth ∆ Hreflects a large in-\nhomogeneous distribution of effective magnetization ∆4 πMeffdue to the presence of an exchange\nspringlike moments arrangement in YIG. We probe the spin inh omogeneity at the YIG-GdIG inter-\nface by performing an in-plane angular variation of resonan ce fieldHr, leading to a unidirectional\nfeature. The large extrinsic ∆4 πMeffcontribution, apart from the inherent intrinsic Gilbert co n-\ntribution, manifests enhanced precessional damping in YIG film.\nI. INTRODUCTION\nThe viability of spintronics demands novel magnetic\nmaterials and YIG is a potential candidate as it ex-\nhibits ultra-low precessional damping, α∼3×10−5[1].\nThe magnetic properties of YIG thin films epitaxially\ngrown on top of Gd 3Ga5O12(GGG) vary significantly\ndue to growth tuning[ 2,3], film thickness[ 4], heavy met-\nals substitution[ 5–7] and coupling with thin metallic\nlayers[8–10]. The growth processes may also induce the\nformation of a thin interfacial-GdIG layer at the YIG-\nGGG interface[ 11–13]. The YIG-GdIG heterostructure\nderived out of monolithic YIG film growth on GGG ex-\nhibits interestingphenomenasuchasall-insulatingequiv-\nalent of a synthetic antiferromagnet[ 12] and hysteresis\nloop inversion governed by positive exchange-bias [ 13].\nThe radio frequency magnetization dynamics on YIG-\nGdIG heterostructure still remains unexplored and need\na detailed FMR study.\nThe relaxation of magnetic excitation towards equi-\nlibrium is governed by intrinsic and extrinsic mecha-\nnisms, leading to a finite ∆ H[14,15]. The former mech-\nanism dictates Gilbert type relaxation, a consequence of\ndirect energy transfer to the lattice governed by both\nspin-orbit coupling and exchange interaction in all mag-\nnetic materials[ 14,15]. Whereas, the latter mechanism\nis a non-Gilbert-type relaxation, divided mainly into two\ncategories[ 14,15]- (i) the magnetic inhomogeneity in-\nduced broadening: inhomogeneity in the internal static\nmagnetic field, and the crystallographic axis orienta-\ntion; (ii) two-magnon scattering: the energy dissipates in\nthe spin subsystem by virtue of magnon scattering with\nnonzero wave vector, k∝negationslash= 0, where, the uniform reso-\nnance mode couples with the degeneratespin waves. The\n∗ravindk@iitk.ac.in\n†zakir@iitk.ac.inangular variation of Hrprovides information about the\npresence of different magnetic anisotropies[ 4,6]. Most\nattention has been paid towards the angular dependence\nofHr[4,6], whereas, the angular variation of the ∆ H\nis sparsely investigated. The studies involving angular\ndependence of ∆ Hmay help to probe different contribu-\ntions to the precessional damping.\nInthispaper, theeffectsofintrinsicandextrinsicrelax-\nation mechanisms on precessionaldamping of YIG film is\nstudied extensively using FMR technique. An enhanced\nvalue of α∼1.2×10−3is realized, which is almost two\norders of magnitude higher than what is usually seen in\nYIGthinfilms, ∼6×10−5[1,2]. Theout-of-planeangular\nvariation of ∆ Hshows an unusual behaviour where spin\ninhomogeneity at the interface plays significant role in\ndefining the ∆ Hbroadening and enhanced α. In-plane\nangular variation showing a unidirectional feature, de-\nmandstheincorporationofanexchangeanisotropytothe\nfree energy density, evidence of the presence of an AFM\nexchangecoupling at the YIG-GdIG interface. The AFM\nexchange coupling leads to a Bloch domain-wall-like spi-\nralmoments arrangementin YIG and givesrise to a large\n∆4πMeff. This extrinsic ∆4 πMeffcontribution due to\nspin inhomogeneity at the interface adds up to the inher-\nent Gilbert contribution, which may lead to a significant\nenhancement in precessional damping.\nII. SAMPLE AND MEASUREMENT SETUPS\nWe deposit a ∼200 nm thick epitaxial YIG film on\nGGG(111)-substrate by employing a KrF Excimer laser\n(Lambda Physik COMPex Pro, λ= 248 nm) of 20 ns\npulse width. A solid state synthesized Y3Fe5O12target\nis ablated using an areal energy of 2.12 J.cm−2with a\nrepetition frequency of 10 Hz. The GGG(111) substrate\nis placed 50 mm away from the target. The film is grown\nat 800oC temperature and in-situpost annealed at the\nsame temperature for 60 minutes in pure oxygen envi-ronment. The θ−2θX-ray diffraction pattern shows epi-\ntaxial growth with trails of Laue oscillations (Fig. 3(a)\nof ref[3]). FMR measurements are performed using a\nBruker EMX EPR spectrometer and a broadband copla-\nnar waveguide (CPW) setup. The former technique uses\na cavity mode frequency f≈9.60 GHz, and enables us\nto perform FMR spectra for various θHandφHangu-\nlar variations. The latter technique enables us to mea-\nsure frequency dependent FMR spectra. We define the\nconfigurations Hparallel ( θH= 90o) and perpendicular\n(θH= 0o) to the film plane for rf frequency and angu-\nlar dependent measurements. The resultant spectra are\nobtained as the derivative of microwave absorption w.r.t.\nthe applied field H.\nIII. RESULTS AND DISCUSSION\nA. Broadband FMR\nFig.1(a) shows typical broadband FMR spectra in\na frequency frange of 1.5 to 13 GHz for 200 nm thick\nYIG film at temperature T= 300 K and θH= 90o.\nThe mode appearing at a lower field value is the main\nmode, whereas the one at higher field value represents\nsurface mode. We discuss all these features in detail in\nthe succeeding subsection IIIB. We determine the res-\nonance field Hrand linewidth (peak-to-peak linewidth)\n∆Hfrom the first derivative of the absorption spectra.\nFig.1(b) shows the rf frequency dependence of Hrat\nθH= 90oand 0o. We use the Kittel equation for fitting\nthe frequency vs. Hrdata from the resonance condi-\ntion expressed as[ 10],f=γ[Hr(Hr+4πMeff)]1/2/(2π)\nforθH= 90oandf=γ(Hr−4πMeff)/(2π) for\nθH= 0o. Where, γ=gµB/ℏis the gyromagnetic ratio,\n4πMeff= 4πMS−Haniis the effective magnetization\nconsisting of 4 πMSsaturation magnetization (calculated\nusing M(H)) and Hanianisotropy field parametrizing cu-\nbic and out-of-plane uniaxial anisotropies. The fitting\ngives 4πMeff≈2000 Oe, which is used to calculate the\nHani≈ −370 Oe.\nFig.1(c) shows the frequency dependence of ∆ Hat\nθH= 90o. The intrinsic and extrinsic damping contri-\nbutions are responsible for a finite width of the FMR\nsignal. The intrinsic damping ∆ Hintarises due to the\nGilbert damping of the precessing moments. Whereas,\nthe extrinsic damping ∆ Hextexists due to different non-\nGilbert-type relaxations such as inhomogeneity due to\nthe distribution of magnetic anisotropy ∆ Hinhom, or\ntwo-magnon scattering (TMS) ∆ HTMS. The intrinsic\nGilbert damping coefficient ( α) can be determined using\nthe Landau-Liftshitz-Gilbert equation expressed as[ 10],\n∆H= ∆Hin+ ∆Hinhom= (4πα/√\n3γ)f+ ∆Hinhom.\nConsidering the above equation where ∆ Hobeys lin-\nearfdependence, the slope determines the value of α,\nand ∆Hinhomcorresponds to the intercept on the ver-\ntical axis. We observe a very weak non-linearity in the\nfdependence of ∆ H, which is believed to be due to thecontribution of TMS to the linewidth ∆ HTMS. The non-\nlinearfdependence of ∆ Hin Fig.1(c) can be described\nin terms of TMS, assuming ∆ H= ∆Hin+ ∆Hinhom+\n∆HTMS. We put a factor of 1 /√\n3 to ∆Hdue to the\npeak-to-peak linewidth value extraction[ 14]. The TMS\ninduces non-linear slope at low frequencies, whereas a\nsaturation is expected at high frequencies. TMS is in-\nduced by scattering centers and surface defects in the\nsample. The defects with size comparable to the wave-\nlength of spin waves are supposed to act as scattering\ncentres. The TMS term at θH= 90ocan be expressed\nas[16]-\n∆HTMS(ω) = Γsin−1/radicalBigg/radicalbig\nω2+(ω0/2)2−ω0/2/radicalbig\nω2+(ω0/2)2+ω0/2,(1)\nwithω= 2πfandω0=γ4πMeff. The prefactor Γ\ndefines the strength of TMS. The extracted values are\nas follows: α= 1.2×10−3, ∆H0= 13 Oe and Γ = 2 .5\nOe. The Gilbert damping for even very thin YIG film\nis extremely low, ∼6×10−5. Whereas, the value we\nachieved is higher than the reported in the literature for\nYIG thin films[ 2]. Also, the value of Γ is insignificant,\nimplying negligible contribution to the damping.\nB. Cavity FMR\nFig.2(a) shows typical T= 300 K cavity-FMR\n(f≈9.6 GHz) spectra for YIG film performed at dif-\nferentθH. The FMR spectra exhibit some universal\nfeatures: (i) Spin-Wave resonance (SWR) spectrum for\nθH= 0o; (ii) rotating the Haway from the θH= 0o,\nthe SWR modes successively start diminishing, and at\ncertain critical angle θc(falls in a range of 30 −35o;\nshaded region in Fig. 2(b)), all the modes vanish except\na single mode (uniform FMR mode). Further rotation\nofHforθH> θc, the SWR modes start re-emerging.\nWe observe that the SWR mode appearing at the higher\nfield side for θH> θc, represents an exchange-dominated\nnon-propagating surface mode[ 17–19]. The above dis-\ncussed complexity in HrvsθHbehaviour has already\nbeen realized in some material systems[ 19], including a\nµ-thick YIG film[ 18]. The localized mode or surface\nspin-wave mode appears for H∝bardblbut not⊥to the film-\nplane[17–19]. WeassigntheSWR modesforthesequence\nn= 1,2,3,...., as it provides the best correspondence\ntoHex∝n2, where, Hex=Hr(n)−Hr(0) defines ex-\nchange field[ 20]. The exchange stiffness can be obtained\nby considering the modified Schreiber and Frait classical\napproach using the mode number n2dependence of res-\nonance field (inset Fig. 3(c))[ 20]. For a fixed frequency,\nthe exchange field Hexof thickness modes is determined\nby subtracting the highest field resonance mode ( n= 1)\nfrom the higher modes ( n∝negationslash= 1). In modified Schreiber\nand Frait equation, the Hexshows direct dependency on\nthe exchange stiffness D:µ0Hex=Dπ2\nd2n2(wheredis\n2/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 1. Room temperature frequency dependent FMR measureme nts. (a) Representative FMR derivative spectra for differen t\nfrequencies at θH= 90o. (b) Resonance field vs. frequency data for θH= 90oandθH= 0oare represented using red and\nblue data points, respectively. The fitting to both the data a re shown using black lines. (c) Linewidth vs. frequency data at\nθH= 90o. The solid red circles represent experimental data, wherea s the solid black line represents ∆ Hfitting. Inhomogeneous\n(∆Hinhom), Gilbert (∆ Hα) and two-magnon scattering (∆ HTMS) contributions to ∆ Hare shown using dashed green, solid\nyellow and blue lines, respectively.\nthe film thickness). The linear fit of data shown in the\ninset of Fig. 2(b) gives D= 3.15×10−17T.m2. The ex-\nchange stiffness constant Acan be determined using the\nrelationA=D MS/2. The calculated value is A= 2.05\npJ.m−1, which is comparable to the value calculated for\nYIG,A= 3.7 pJ.m−1[20].\nYIG thin films with in-plane easy magnetization ex-\nhibit extrinsic uniaxial magnetic and intrinsic magne-\ntocrystallinecubic anisotropies[ 21]. The total free energy\ndensity for YIG(111) is given by[ 21,22]:\nF=−HMS/bracketleftbigg\nsinθHsinθMcos(φH−φM)\n+cosθHcosθM/bracketrightbigg\n+2πM2\nScos2θM−Kucos2θM\n+K1\n12/parenleftbigg7sin4θM−8sin2θM+4−\n4√\n2sin3θMcosθMcos3φM/parenrightbigg\n+K2\n108\n−24sin6θM+45sin4θM−24sin2θM+4\n−2√\n2sin3θMcosθM/parenleftbig\n5sin2θM−2/parenrightbig\ncos3φM\n+sin6θMcos6φM\n\n(2)\nThe Eq. 2consists of the following different energy\nterms; the first term is the Zeeman energy, the second\nterm is the demagnetization energy, the third term is\nthe out-of-plane uniaxial magnetocrystalline anisotropy\nenergyKu, and the last two terms are the first and sec-\nond order cubic magnetocrystalline anisotropy energies\n(K1andK2), respectively. The total free energy density\nequation is minimized by taking partial derivatives w.r.t.\ntoθMandφMtoobtaintheequilibriumorientationofthe\nmagnetization vector M(H), i.e.,∂F/∂θ M=∂F/∂φ M=\n0. Theresonancefrequencyofuniformprecessionatequi-\nlibrium condition is expressed as[ 21,23,24]:\nωres=γ\nMSsinθM/bracketleftBigg\n∂2F\n∂θ2\nM∂2F\n∂φ2\nM−/parenleftbigg∂2F\n∂θM∂φM/parenrightbigg2/bracketrightBigg1/2\n(3)\nMathematica is used to numerically solve the reso-nance condition described by Eq. 3for the energy den-\nsity given by Eq. 2. The solution for a fixed frequency\nis used to fit the angle dependent resonance data ( Hr\nvs.θH) shown in Fig. 2(b). The main mode data\nsimulation is shown using a black line. The parame-\nters obtained from the simulation are Ku=−1.45×104\nerg.cm−3,K1= 1.50×103erg.cm−3, andK2= 0.13×103\nerg.cm−3. The calculated uniaxial anisotropy field value\nisHu∼ −223 Oe.\nThe ∆Hmanifests the spin dynamics and related re-\nlaxation mechanisms in a magnetic system. The intrinsic\ncontribution to ∆ Harises due to Gilbert term ∆ Hint≈\n∆Hα, whereas, the extrinsic contribution ∆ Hextconsists\nof line broadening due to ∆ Hinhomand ∆HTMS. The\nterms representing the precessional damping due to in-\ntrinsic and extrinsic contributions can be expressed in\ndifferent phenomenologicalforms. Figure 2(c) shows∆ H\nas a function of θH. TheθHvariation of ∆ Hshows\ndistinct signatures due to different origins of magnetic\ndamping. We consider both ∆ Hintand ∆Hextmag-\nnetic damping contributions to the broadening of ∆ H,\n∆H= ∆Hα+∆Hinhom+∆HTMS. The first term can\nbe expressed as[ 14]-\n∆Hα=α\nMS/bracketleftbigg∂2F\n∂θ2\nM+1\nsin2θM∂2F\n∂φ2\nM/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(2πf\nγ)\n∂Hr/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n.(4)\nThe second term ∆ Hinhomhas a form[ 14]-\n∆Hinhom=/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\nd4πMeff/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆4πMeff+/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\ndθH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆θH.(5)\nWhere, the dispersion of magnitude and direction of\nthe 4πMeffare represented by ∆4 πMeffand ∆θH, re-\nspectively. The ∆ Hinhomcontribution arises due to a\nsmall spread of the sample parameters such as thickness,\ninternal fields, or orientation of crystallites within the\nthin film. The third term ∆ HTMScan be written as[ 25]-\n3/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s50/s51/s32/s72\n/s101/s120/s40/s107/s79/s101/s41\n/s110/s50/s67/s114/s105/s116/s105/s99/s97/s108/s32/s97/s110/s103/s108/s101/s54/s53/s52/s51/s72\n/s114/s32/s40/s32/s107/s79/s101/s32/s41/s50/s110/s32/s61/s32/s49\n/s72/s32/s40/s32/s68/s101/s103/s114/s101/s101/s32/s41/s50 /s51 /s52 /s53 /s54/s68/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72 /s32/s40/s107/s79/s101/s41/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s32\n/s32/s69/s120/s112/s116/s46/s32/s68/s97/s116/s97\n/s32 /s72\n/s32 /s72\n/s32 /s77\n/s101/s102/s102\n/s32\n/s32 /s72\n/s84/s77/s83/s72 /s32/s40/s79/s101/s41/s32\n/s40/s100/s101/s103/s114/s101/s101 /s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 2. Room temperature out-of-plane angular θHdependence of FMR. (a) Derivative FMR spectra shown for diffe rentθH\nperformed at ≈9.6 GHz. (b) θHvariation of uniform mode and SWR modes of resonance field Hr. Inset: Exchange field\n(Hex) vs mode number square ( n2). (c)θHvariation of the linewidth (∆ H), where, the experimental and simulated data are\nrepresented by solid yellow circles and black line, respect ively. The different contributions ∆ Hα, ∆4πMeff, ∆θHand ∆HTMS\nare represented by gray, purple, green and red lines, respec tively.\n∆HTMS=/summationtext\ni=1Γout\nifi(φH)\nµ0γΦsin−1/radicalbigg√\nω2+(ω0/2)2−ω0/2√\nω2+(ω0/2)2+ω0/2,\nΓout\ni= Γ0\niΦA(θ−π/4)dHr(θH)\ndω(θH)/slashbigg\ndHr(θH=0)\ndω(θH=0)\n(6)\nThe prefactor Γout\nidefines the TMS strength and has\naθHdependency in this case. The type and size of the\ndefects responsible for TMS is difficult to characterize\nwhich makes it non-trivial to express the exact form of\nΓout\ni. Although, it mayhaveasimplified expressiongiven\nin Eq.6, where, Γ0\niis a constant; A(θ−π/4), a step\nfunction which makes sure that the TMS is deactivated\nforθH< π/4; anddHr(θH)/dω(θH), a normalization\nfactor responsible for the θHdependence of the Γout\ni.\nIn fig.2(c)the solid dark yellow circles and black solid\nline represent the experimental and simulated ∆ HvsθH\ndata, respectively. We also plot contributions of different\nterms such as ∆ Hα(blue color line), ∆4 πMeff(purple\ncolorline), ∆ θH(greencolorline) and ∆ HTMS(red color\nline). The fitting provides following extracted parame-\nters,α= 1.3×10−3, ∆4πMeff= 58 Oe, ∆ θH= 0.29o\nand Γ0\ni= 1.3 Oe. The precessional damping calculated\nfromthe∆ Hvs.θHcorroboratewiththevalueextracted\nfrom the frequency dependence of ∆ Hdata (shown in\nFig.1(c));α= 1.2×10−3. The ∆ Hbroadening and\nthe overwhelmingly enhanced precessional damping are\nthedirectconsequenceofcontributionsfromintrinsicand\nextrinsic damping. Usually, the Gilbert term and the\ninhomogeneity due to sample quality contribute to the\nbroadening of ∆ Hand enhanced αin YIG thin films. If\nwe interpret the ∆ HvsθHdata, it is clear that damping\nenhancement in YIG is arising from the extrinsic mag-\nnetic inhomogeneity.The role of an interface in YIG coupled with metals\nor insulators leading to the increments in ∆ Handαhas\nbeen vastly explored. Wang et. al. [ 9] studied a variety\nof insulating spacers between YIG and Pt to probe the\neffect on spin pumping efficiency. Their results suggest\nthe generation of magnetic excitations in the adjacent\ninsulating layers due to the precessing magnetization in\nYIG at resonance. This happens either due to fluctu-\nating correlated moments or antiferromagnetic ordering,\nvia interfacial exchange coupling, leading to ∆ Hbroad-\nening and enhanced precessional damping of the YIG[ 9].\nTheimpurityrelaxationmechanismisalsoresponsiblefor\n∆HbroadeningandenhancedmagneticdampinginYIG,\nbut is prominent only at low temperatures[ 16]. Strong\nenhancement in magnetic damping of YIG capped with\nPt has been observed by Sun et. al. [ 8]. They suggest\nferromagneticorderingin an atomically thin Pt layerdue\nto proximity with YIG at the YIG-Pt interface, dynam-\nically exchange couples to the spins in YIG[ 8]. In recent\nyears, some research groups have reported the presence\nof a thin interfacial layer at the YIG-GGG interface[ 11–\n13]. The 200 nm film we used in this study is of high\nquality with a trails of sharp Laue oscillations [see Fig\n3(a) in ref.[ 3]]. Thus it is quite clear that the observed\n∆Hbroadening and enhanced αis not a consequence of\nsample inhomogeneity. The formation of an interfacial\nGdIG layer at the YIG-GGG interface, which exchange\ncouples with the YIG film may lead to ∆ Hbroadening\nand increased α. Considering the above experimental ev-\nidences leading to ∆ Hbroadening and enhanced Gilbert\ndampingdueto couplingwithmetals andinsulators[ 8,9],\nit is safe to assume that the interfacial GdIG layer at the\ninterface AFM exchange couples with the YIG[ 11–13],\nand responsible for enhanced ∆ Handα.\nFig.3shows in-plane φHangular variation of Hr. We\n4/s32/s68/s97/s116/s97\n/s32/s84/s111/s116/s97/s108\n/s32/s69/s120/s99/s104/s97/s110/s103/s101/s72\n/s114/s32/s40/s79/s101/s41/s50/s48/s48/s32/s110/s109/s40/s97/s41\n/s40/s98/s41\n/s49/s48/s48/s32/s110/s109\n/s72/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFIG. 3. (a) In-plane angular φHvariation of Hr. The exper-\nimental data are represented by solid grey circles. Whereas ,\nthe simulated data for total and exchange (unidirectional)\nanisotropy are represented by black and red solid lines, re-\nspectively. (a) 200 nm thick YIG sample. (b) 100 nm thick\nYIG sample.\nsimulate the in-plane HrvsφHangular variation using\nthe free energy densities provided in ref. [ 26] and an\nadditional term, −KEA.sinθM.cosφM, representing the\nexchange anisotropy ( KEA). Even though φHvaria-\ntion ofHrshown in Fig. 3(a) is not so appreciable\nas the film is 200 nm thick, a very weak unidirectional\nanisotropy trend is visible, suggesting an AFM exchange\ncoupling between the interface and YIG. It has been\nshown that the large inhomogeneous 4 πMeffis a direct\nconsequence of the AFM exchange coupling at the inter-\nface of LSMO and a growth induced interfacial layer[ 27].\nThe YIG thin film system due to the presence of a hard\nferrimagnetic GdIG interfacial layer possesses AFM ex-\nchange coupling[ 11–13]. A Bloch domain-wall-like spiral\nmoments arrangement takes place due to the AFM ex-\nchange coupling acrossthe interfacial GdIG and top bulk\nYIG layer[ 11–13]. An exchange springlike characteris-\ntic is found in YIG film due to the spiral arrangement\nof the magnetic moments [ 11–13]. The FMR measure-\nment and the extracted value of ∆4 πMeffreflect inho-\nmogeneous distribution of 4 πMeffin YIG-GdIG bilayer\nsystem. The argument of Bloch domain-wall-like spiral\narrangement of moments is conceivable, as this arrange-\nment between the adjacent layers lowers the exchange\ninteraction energy[ 27]. To further substantiate the pres-ence of an interfacial AFM exchange coupling leading\nto spin inhomogeneity at YIG-GdIG interface, we per-\nformed in-plane φHvariation of Hron a relatively thin\nYIG film ( ∼100 nm with growth conditions leading to\nthe formation of a GdIG interfacial layer[ 13]). Fig.3(b)\nshows prominent feature of unidirectional anisotropydue\nto AFM exchange coupling in 100 nm thick film. It is\nevident that the interfacial layer exchange couples with\nthe rest of the YIG film and leads to a unidirectional\nanisotropy. We observethatthe interfacialexchangecou-\npling may cause ∆ Hbroadening and enhanced αdue to\nspin inhomogeneity at the YIG-GdIG interface, even in\na 200 nm thick YIG film.\nIV. CONCLUSIONS\nThe effects of spin inhomogeneity at the YIG and\ngrowth-induced GdIG interface on the magnetization dy-\nnamics of a 200 nm thick YIG film is studied extensively\nusing ferromagnetic resonance technique. The Gilbert\ndamping is almost two orders of magnitude larger\n(∼1.2×10−3) than usually reported in YIG thin films.\nThe out-of-plane angular dependence of ∆ Hshows\nan unusual behaviour which can only be justified after\nconsidering extrinsic mechanism in combination with the\nGilbert contribution. The extracted parameters from\nthe ∆HvsθHsimulation are, (i) α= 1.3×10−3from\nGilbert term; (ii) ∆4 πMeff= 58 Oe and ∆ θH= 0.29o\nfrom the inhomogeneity in effective magnetization and\nanisotropy axes, respectively; (iii) Γ0\ni= 1.3 Oe from\nTMS. The TMS strength Γ is not so appreciable,\nindicating high quality thin film with insignificant defect\nsites. The AFM exchange coupling between YIG and\nthe interfacial GdIG layer causes exchange springlike\nbehaviour of the magnetic moments in YIG, leading to a\nlarge ∆4 πMeff. The presence of large ∆4 πMeffimpels\nthe quick dragging of the precessional motion towards\nequilibrium. A unidirectional behaviour is observed in\nthe in-plane angular variation of resonance field due to\nthe presence of an exchange anisotropy. This further\nreinforces the spin inhomogeneity at the YIG-GdIG\ninterface due to the AFM exchange coupling.\nACKNOWLEDGEMENTS\nWe gratefully acknowledge the research support from\nIIT Kanpur and SERB, Government of India (Grant\nNo. CRG/2018/000220). RK and DS acknowledge the\nfinancial support from Max-Planck partner group. ZH\nacknowledges financial support from Polish National\nAgency for Academic Exchange under Ulam Fellowship.\nThe authors thank Veena Singh for her help with the\nangular dependent FMR measurements.\n[1] M. Sparks, Ferromagnetic-relaxation theory. , advanced\nphysics monograph series ed. (McGraw-Hill, 1964).[2] C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt,\n5M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. G. Ebbing-\nhaus, and G. Schmidt, Sci. Rep. 6, 20827 (2016) .\n[3] R. Kumar, Z. Hossain, and R. C. Budhani,\nJ. Appl. Phys. 121, 113901 (2017) .\n[4] H. Wang, C. Du, P. C. Hammel, and F. Yang,\nPhys. Rev. B 89, 134404 (2014) .\n[5] L. E. Helseth, R. W. 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B 101, 024408 (2020) .\n6" }, { "title": "2303.00936v1.Unidirectional_Microwave_Transduction_with_Chirality_Selected_Short_Wavelength_Magnon_Excitations.pdf", "content": "Unidirectional Microwave Transduction with Chirality Selected Short-Wavelength\nMagnon Excitations\nYi Li,1,\u0003Tzu-Hsiang Lo,2Jinho Lim,2John E. Pearson,1Ralu Divan,3Wei Zhang,4\nUlrich Welp,1Wai-Kwong Kwok,1Axel Ho\u000bmann,2,yand Valentine Novosad1,z\n1Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n2Department of Materials Science and Engineering, UIUC, Urbana, IL 61820, USA\n3Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439, USA\n4Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, USA\n(Dated: March 3, 2023)\nNonreciprocal magnon propagation has recently become a highly potential approach of developing\nchip-embedded microwave isolators for advanced information processing. However, it is challenging\nto achieve large nonreciprocity in miniaturized magnetic thin-\flm devices because of the di\u000eculty of\ndistinguishing propagating surface spin waves along the opposite directions when the \flm thickness\nis small. In this work, we experimentally realize unidirectional microwave transduction with sub-\nmicron-wavelength propagating magnons in a yttrium iron garnet (YIG) thin \flm delay line. We\nachieve a non-decaying isolation of 30 dB with a broad \feld-tunable band-pass frequency range up\nto 14 GHz. The large isolation is due to the selection of chiral magnetostatic surface spin waves with\nthe Oersted \feld generated from the coplanar waveguide antenna. Increasing the geometry ratio\nbetween the antenna width and YIG thickness drastically reduces the nonreciprocity and introduces\nadditional magnon transmission bands. Our results pave the way for on-chip microwave isolation\nand tunable delay line with short-wavelength magnonic excitations.\nRecent advances in information technologies such as\nquantum information [1], microelectronics [2] and 5G/6G\nnetworks [3] call for disruptive innovations in microwave\nsignal processing. In particular, circuit-integrated mi-\ncrowave isolators are highly desirable for many applica-\ntions as they \flter unwanted microwave back\row. In\nquantum information, \fltering environmental noise from\nthe output ports is essential for protecting quantum\nstates and entanglement [4]. Being able to embed iso-\nlators on chip will signi\fcantly reduce the device volume\ncompared to currently used bulk ferrite-based isolators,\nand enable non-Hermitian engineering of dynamic sys-\ntems at a circuit level [5, 6].\nMagnonics o\u000bers opportunities for implementing uni-\ndirectional microwave transduction with miniaturized ge-\nometry [7{9]. Because of their special dispersion relations\n[10], magnons support short-wavelength excitations down\nto nanometer scale at microwave bandwidth [11{16] along\nwith superior frequency tunability with an external mag-\nnetic \feld. In addition, magnons exhibit nonreciproc-\nity based on many unique properties of magnetic excita-\ntions, including intrinsic chirality selection in propagat-\ning magnetostatic surface spin waves (MSSW) [17{24],\nwavevector-dependent spin wave dispersion shifting [25{\n35], and non-Hermitian circuit engineering [36, 37]. This\nallows for compact integration of miniaturized, broad-\nband and highly tunable isolators in microwave circuits.\nFurthermore, the realization of miniaturized magnonic\nisolator is also important for spin wave computing, where\nmagnons are used to carry, transport and process infor-\nmation [38, 39]. Unidirectional \row of magnon informa-\ntion enables the isolation of input and output and ensures\ndeterministic logic output [40, 41].\nFIG. 1. (a) Optical microscope image of the YIG delay line\nwithtYIG= 100 nm, w= 200 nm and d= 10\u0016m. (b)\nNonreciprocal microwave transmission spectra of the device\nshown in (a) at \u00160HB= 0:22 T, measured by a vector network\nanalyzer. (c-d) Repeated measurements of (b) at di\u000berent\nmagnetic \felds, showing a broad band nonreciprocity with a\nconstant isolation of 30 dB.\nOne major challenge of chip-integrated magnonic mi-\ncrowave isolators is the degradation of nonreciprocity\nin magnetic thin-\flm devices. As the \flm thickness is\ndecreased down to nanometer levels, the + kand\u0000k\nMSSWs modes, which are localized at the top and bot-\ntom surfaces of the \flm, will permeate to the entire thick-\nness and be both excited by antennas at the top sur-arXiv:2303.00936v1 [cond-mat.mes-hall] 2 Mar 20232\nface, leading to suppressed isolation that are typically less\nthan 10 dB. Other approaches of realizing large magnon\nnonreciprocity usually su\u000ber from the narrow bandwidth\n[36, 42, 43]. Alternatively, spin wave nonreciprocity can\nbe achieved by chirality selection from well-designed mi-\ncrowave antennas [22, 44, 45], where + kand\u0000kMSSWs\nexhibit clockwise and counter-clockwise mode pro\fles de-\npending on the orientation of the magnetization vector.\nThis technique is not restricted by the \flm thickness and\nthus can be applied for highly e\u000ecient spin wave isolation\nwith proper geometric design.\nIn this work, we obtain unidirectional magnon exci-\ntations with over 30 dB isolation in yttrium iron gar-\nnet (YIG) thin \flm delay lines with thickness down to\n100 nm. We \fnd the leading mechanism of nonreciproc-\nity to be the Oersted \feld chirality generated from the\ncoplanar waveguide (CPW) antenna, which shows a sinc-\nfunction-like spatial pro\fle and selectively couples to the\nMSSW mode propagating unidirectionally. The isolation\nquickly decreases from 35 dB to 10 dB as the ratio of the\nantenna width to the YIG \flm thickness increases from\n1 to 5, which is due to the development of sharp kinks\nfor the Oersted \feld at the electrode edges. In addition,\nwe also utilize the time-domain functionality of the mea-\nsurement system to obtain the delay time, group velocity\nand the nature of the additional microwave transduction\nbands as the antenna and \flm geometries deviate from\nthe optimal condition. The demonstration and physi-\ncal understanding of high-isolation, high-tunability and\nthin-\flm-based compact microwave isolator are critical\nfor extending microwave engineering with magnonic de-\nvices, and bring new potential for on-chip noise reduction\nin quantum information applications.\nThe structure of the delay line is shown in Figure 1(a).\nA pair of nanofabricated ground-signal-ground (GSG)\nCPW antennas are patterned on top of a YIG thin-\flm\nstripe with a thickness of tYIG= 100 or 200 nm. The\nthree electrodes of the GSG antennas and the gaps be-\ntween them have an equal width of w= 200 or 500 nm\nand a length of 30 \u0016m, leading to the excitation of spin\nwaves with a wavelength around \u0015= 4w[46]. The sepa-\nration of the two CPWs is d= 5, 10 or 20 \u0016m. The ex-\nternal magnetic \feld is applied parallel to the two CPWs\nin order to de\fne the MSSW propagation modes between\nthe two antennas.\nFigure 1(b) shows the microwave transmission spec-\ntra fortYIG= 100 nm, w= 200 nm and d= 10\u0016m\nat\u00160HB= 0:22 T, which are measured with a vector\nnetwork analyzer (VNA). For magnons that propagate\nfrom left to right ( S21) as shown in Fig. 1(a), a single\nmicrowave transmission band is observed around 9 GHz\nand stands well above the microwave background. By\nreversing the measurement direction from right to left\n(S12), the transmission band disappears. The S21band\ncorresponds to the MSSW modes with the wavevector\nkorthogonal to the magnetization M. By \ripping thebiasing \feld, the magnon transmission direction is also\nreversed [47]. With di\u000berent external biasing \felds, the\ntransmission band can be tuned continuously along the\nfrequency axis, as shown in Fig. 1(c). A consistent iso-\nlation of 30 dB and a 3-dB bandwidth of 0.15 GHz are\nmeasured with negligible change in a broad frequency\nband from 8 to 15 GHz.\nTo further explore the role of geometric structure in\nnonreciprocity, we compare the transmission spectra for a\nfew di\u000berent antenna geometries, with the results shown\nin Figs. 2(a-d). Experimentally we see that by increas-\ningwfrom 200 nm to 500 nm, there is a major increase\nof the isolated band ( S12) from below the noise back-\nground to being clearly visible, leading to a suppressed\nnonreciprocity of 15 dB in Fig. 2(b) and 10 dB in Fig.\n2(d). In addition, a new side band appears at a higher\nfrequency which is marked as mode 2. This corresponds\nto the second harmonic of the main MSSW mode.\nIn order to understand such a large magnon nonre-\nciprocity, we calculate the Oersted \feld distribution of\nthe GSG antenna as well as the + kand\u0000kMSSW\nmagnon pro\fles with the experimental geometry. As\nshown in Fig. 2(e), the Oersted \feld and the + k\nmode share the same counter-clockwise spatial chiral-\nity, whereas the \u0000kmode exhibit a clockwise chirality\n[28]. This means the GSG antenna selectively excites the\n+kmode and is decoupled from the \u0000kmode, leading\nto chirality induced nonreciprocal magnon excitations.\nFurthermore, by increasing the aspect ratio between the\nantenna width and YIG thickness, w=tYIG, from 0.5 to 5,\nthe Oersted \feld pro\fle changes from a sinc-like smooth\npro\fle to a pro\fle with multiple sharp kinks at the edges\nof the antenna electrodes [Fig. 2(f)], leading to a \fnite\noverlap with the \u0000kmode and a reduction of nonre-\nciprocity. This e\u000bect also increases the e\u000eciency of the\nsecond harmonic excitation; see the Supplemental Infor-\nmation for more details [47].\nWe also conduct numerical simulations of the magnon\nisolations using Mumax in order to compare with exper-\niments. Interestingly, we \fnd a large discrepancy be-\ntween experiments and simulations, as plotted in Fig.\n2(g). The simulations show a slow reduction in isolation\nfrom 24 dB to 18 dB as the aspect ratio w=tYIGincreases\nfrom 0.5 to 5. However, the experimental results show\na much faster reduction in isolation from 36 dB to 10\ndB forw=tYIGincreasing from 1 to 5. We attribute the\ndisagreement to the di\u000berent simulation conditions from\nthe experiments as we used a single pixel through the\nentire thickness to speed up the calculation. This may\nlead to omission of subtle magnetic pro\fle distributions\nwhen the CPW antenna width is comparable to the \flm\nthickness. In most previous studies on CPW geometry\n[22{24], the antenna widths are usually much larger than\nthe \flm thickness and the chirality selection of the Oer-\nsted \feld has been suppressed. We also note that the\nskin e\u000bect can be ruled out because we do not observe3\nTransmission \nTransmission \nTransmission Transmission mode 1mode 1 \nmode 2 mode 1 mode 3 \nmode 2 mode 1 mode 3 (a) (b) (c) (d) w=200 nm\ntYIG =100 nmw=500 nm\ntYIG =100 nmw=200 nm\ntYIG =200 nmw=500 nm\ntYIG =200 nm\n+k \n-k irf -i rf /2 -i rf /2 \nhrf \nm(+k)\nm(-k)=0.5\n=5 (e) (f) (g) \nFIG. 2. (a-d) Nonreciprocal microwave transmission spectra with di\u000berent geometries: (a) w= 200 nm, tYIG= 100 nm; (b)\nw= 500 nm, tYIG= 100 nm; (c) w= 200 nm, tYIG= 200 nm; (d) w= 500 nm, tYIG= 200 nm. The biasing \felds are all\n\u00160HB= 0:22 T for comparison. (e) OOMMF simulations of the top CPW Oersted \feld chirality and the \u0006kMSSW mode\nchirality. (f) Oersted \feld distribution for top:w=tYIG= 0:5 and bottom:w=tYIG= 5. (g) Summary of experimental magnon\nisolations as a function of aspect ratio w=tYIG, along with MuMax simulation results.\nstrong frequency dependence of the isolation. The skin\ndepth of gold at 10 GHz is about 785 nm [48], which is\nalso much larger than the antenna widths of this work\nso evenly distributed microwave current in the antenna\nshould be a valid assumption.\nTo examine the performance of the nonreciprocal\nmagnonic delay line in the time domain, we also perform\ntime-domain analysis using inverse Fast Fourier Transfor-\nmation (IFFT) function of the VNA. IFFT simulates the\ninput of a sinc function pulse holding an equal weight of\nall the frequency components set by the frequency win-\ndow. As a demonstration, we focus on the delay lines\nwithw= 200 nm and tYIG= 100 nm. Figs. 3(a) and (b)\nshow the frequency and time domain measurements of\nthe major transmission band ( S21) at di\u000berent antenna\nseparations ( d= 5, 10 and 20 \u0016m). The frequency win-\ndow for the IFFT analysis of Fig. 3(b) is set to 3 to\n15 GHz in order to capture the transmission of the en-\ntire frequency band. In the frequency domain, nearly\nidentical transmission bands are measured at di\u000berent\nd, with only a decreasing transmission amplitude due to\nthe magnon decay with additional propagating distance.\nIn the time domain, the magnon transmission peaks are\nfollowed by an initial microwave pulse at 20 ns which is\ndue to the direct microwave radiation between the two\nantennas serving as the microwave background in 3(a).The time delay of the magnon signal from the microwave\nradiation pulse scales linearly as a function of d, yielding\na time delay of 96 ns at d= 20\u0016m or a magnon group\nvelocity ofvg= 208 m/s at !peak=2\u0019= 10 GHz. The val-\nues of the group velocities are also con\frmed from the S21\nphase changing rate in the frequency domain, as marked\nby the stars in Fig. 3(e); see the Supplemental Materials\nfor details [47]. No magnon signals are measured for the\nopposite microwave transmission direction ( S12).\nThe \feld dependence of the extracted magnon trans-\nmission parameters are plotted as a function of peak\ntransmission frequency ( !peak) in Figs. 3(c-f). For\nthe frequency domain, all the three devices show\nnearly frequency-independent peak transmission ampli-\ntudes (S21;peak) and linewidth (\u0001 !3dB) from 8 to 14 GHz.\nThed-dependence of S21;peakyields a magnon decay rate\nof 0.5 dB/\u0016m or an exponential decay length of 8.7 \u0016m.\nThe value of \u0001 !3dB=2\u0019is around 0.12 GHz. Note that it\nis a function of the GSG antenna geometry and is inde-\npendent of frequency. For the time domain, the extracted\ngroup velocities from time delays are consistent for the\nthree devices and show a slow decrease with frequency,\nwhich is the characteristic of MSSW modes. Comparing\nwith analytical calculations, the group velocity values fall\nbetween the dipolar and exchange regimes [47], which\nis in accordance with the range of sub-micron magnon4\npeak (a) (b) \n(c)\n(d)(e) \n(f) \npeak peak peak \nFIG. 3. (a) Comparision of microwave transmission spectra\nwithw= 200 nm,tYIG= 100 nm, for three di\u000berent antenna\nseparations d. (b) IFFT of (a) with a transformation window\nfrom 3 to 15 GHz. The biasing \felds for (a) and (b) are\nset as\u00160HB= 0:27 T. (c) Maximal transmission amplitude\nS21;peak as a function of the peak position !peak. (d) 3 dB\nfrequency linewidth \u0001 !3dBas a function of !peak. (c) and\n(d) are extracted from the measurements of (a) at di\u000berent\n\felds. (e) Group velocity obtained from the magnon time\ndelay, and plotted as a function of !peak. The stars are the\ngroup velocity obtained from the phase changing rate in the\nfrequency spectra measured at \u00160HB= 0:22 T, same as in\nFig. 1(b). (f) 3 dB linewidth of the IFFT peak as a function\nof!peak. (e) and (f) are extracted from the measurements of\n(b) at di\u000berent \felds.\nwavelengths.\nIFFT analysis also allow us to access the magnon pulse\nwidth (\u0001t3dB), which are in the range of 4-6 ns for the\nthree devices as shown in Fig. 3(f). Because the simu-\nlated sinc pulse has a much smaller width ( <0:1 ns for a\nfrequency window between 3 and 15 GHz), the measured\nvalues of \u0001 t3dBpose a fundamental limit of the pulse\nwidth of the delay line structure. The main source of\nbroadening is the \fnite spatial width of the antenna for\nthe magnon pulse to pass by. If we take the width of the\nentire antenna as 4 w= 800 nm and the group velocity\nvgas 200 m/s, the total traveling time of the magnon\npulse across the antenna is 4 w=vg= 4 ns, which is close\nto the lower bound of \u0001 t3dBin Fig. 3(f). The increase\nof \u0001t3dBwithdis due to the distribution of vgwithin\nthe \fnite bandwidth, \u0001 !3dB, leading to the expansion of\nthe magnon wave package as it propagates. As an esti-\nmate, the additional magnon pulse broadening \u0001 tkcan\nbe expressed as \u0001 tk= (\u0001!3dB=!)t; see the Supplemen-\ntal Materials for detailed discussion [47]. From Fig. 3(d),\n\u0001!3dB=!\u00191%. This yields \u0001 tk\u00191 ns by changing dfrom 5 nm to 20 nm, which agrees with the change of\n\u0001t3dBin Fig. 3(f).\nFinally, we use IFFT analysis to identify the nature\nof additional bands for di\u000berent antenna geometries in\nFigs. 2(a-d), which are marked as modes 1, 2 and 3. Fig.\n4 shows the time-domain pro\fles of di\u000berent modes for\neach geometry. The frequency windows are limited to\nthe edges of each magnon band in order to \flter out the\ncontributions from other bands. Mode 1 is known to be\nthe main transmission band. For mode 2 which appears\nin Figs. 4(b) and (d), the peak position has a longer\ndelay compared with mode 1. This supports that mode\n2 is the second harmonic because for MSSW modes the\n!-kdispersion curve softens at higher kand the group\nvelocity becomes smaller, resulting in longer travel time\nof propagating magnons. Mode 3 which appears in Fig.\n4(c) and (d) exhibits a long time span of more than 50\nns. From the S21phase changing rate as discussed in the\nSupplemental Materials [47], we determine mode 3 to be\nthe obliquely launched backward-volume magnetostatic\nspin waves (BVMSWs) with canted wavevector due to\nspin wave di\u000braction [49, 50]. From the phase analysis\nfor Fig. 2(d), the high-frequency edge of mode 3 exhibits\na near-zero group velocity due to the \rat !-kdispersion\nat a certain canted angle, causing the magnon pulse to\ntake long time to be transmitted [51{53]. We conclude\nfrom the time domain analysis that purer magnon modes\narise from narrower GSG antenna widths, which is desir-\nable for minimizing loss and decoherence during the pulse\nmicrowave processing with the nonreciprocal magnon de-\nlay line.\nIn summary, we present a systematic study of YIG-\nthin-\flm magnon delay lines with broad-band isolation\nabove 30 dB. We identify the source of nonreciprocity\nas the selective coupling of the chiral Oersted \feld from\nthe GSG antenna to the propagating MSSW modes in\nonly one direction. Using time-domain IFFT analysis,\nwe determine important parameters of the delay line, in-\ncluding time delay, group velocity, bandwidth and time\ndomain broadening. We also identify the nature of the\nadditional magnon transmission bands as the second har-\nmonic and the canted BVMSWs band, which can be sup-\npressed by limiting the aspect ratio of the antenna and\nthe YIG thickness. Our results show a new promise of\nnonreciprocal magnonic delay lines for processing pulse\nmicrowave signals with excellent noise isolation, which\nmay be implemented as chip-embedded microwave isola-\ntors for spin wave computing and quantum information\nprocessing. We note that our demonstration of nonrecip-\nrocal spin wave propagation can be also combined with\nthe recent work of spin-torque spin wave ampli\fcation\n[54] for implementing unidirectional magnonic microwave\nnano-ampli\fers.\nMethods. The devices were fabricated using YIG thin\n\flms grown on Gd 3Ga5O12(GGG) substrate by liquid\nphase epitaxy (LPE), which are commercially available5\n(a)\n(b)(c)\n(d)\nFIG. 4. Decomposition of magnonic mode traces with di\u000berent IFFT frequency windows. (a-d) are converted from IFFT of the\nfrequency domain spectra of Fig. 2(a-d), respectively, with \u00160HB= 0:22 T. (a) Mode 1 is obtained with a frequency window of\n8.5-9.5 GHz. (b) Modes 1 and 2 are obtained from frequency ranges as 1: 8.45 to 8.85 GHz, and 2: 8.85 to 9.2 GHz. (c) Modes\n1 and 3 are obtained from frequency ranges as 1: 9.0 to 10 GHz, and 3: 8.5 to 9.0 GHz. (d) Modes 1, 2, 3 are obtained from\nfrequency ranges as 1: 8.88 to 9.1 GHz, 2: 9.1 to 9.6 GHz, and 3: 8.5 to 8.88 GHz. The amplitudes are measured in voltage\n(V21).\n[55]. In the \frst step, chromium hard masks were litho-\ngraphically patterned on YIG \flms and used for etching\nthe YIG delay lines. Ar+ion milling was used for ef-\n\fciently etching the YIG layer. The Cr masks can be\nremoved by Cr wet etch without damaging the YIG de-\nvices. The edges of the two ends are 45 degree away\nfrom the long sides in order to minimize magnon re\rec-\ntion. The width of the YIG stripe is 56 \u0016m and the\nlengths of the CPW antennas are 30 \u0016m. In the sec-\nond step, nano-CPWs were fabricated on top of the YIG\ndevices using e-beam lithography, with Ti(5 nm)/Au(50\nnm) electrodes grown by e-beam evaporation to minimize\nside walls. In the third step, the extended CPWs were\nfabricated using photolithography. The electrode thick-\nnesses are Ti(5 nm)/Au(150 nm) for tYIG= 100 nm and\nTi(5 nm)/Au(250 nm) for tYIG= 200 nm in order to\nminimize the impedance mismatch across the side walls\nof the YIG devices.\nAcknowledgement. The work was supported by the\nU.S. DOE, O\u000ece of Science, Basic Energy Sciences, Ma-\nterials Sciences and Engineering Division, with parts of\nthe manuscript preparation, \frst-principle calculations,\ndevice design, and sample fabrication and characteriza-\ntion supported under contract No. DE-SC0022060. U.\nW, W.-K. K. and V. N. acknowledge support by the U.S.\nDOE, O\u000ece of Science, Basic Energy Sciences, Materi-\nals Sciences and Engineering Division. 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