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Nonlinear erasing of propagating spin-wave pulses in thin-film Ga:YIG D. Breitbach,1,a)M. Bechberger,1,a)B. Heinz,1A. Hamadeh,1J. Maskill,1K. O. Levchenko,2B. Lägel,1C. Dubs,3Q. Wang,4R. Verba,5and P. Pirro1 1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau, D-67663 Kaiserslautern, Germany 2)Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria 3)INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany 4)School of Physics, Huazhong University of Science and Technology, 430074 Wuhan, China 5)Institute of Magnetism, Kyiv 03142, Ukraine (*Electronic mail: dbreitba@rptu.de) (Dated: 2 February 2024) Nonlinear phenomena are key for magnon-based information processing, but the nonlinear interaction between two spin-wave signals requires their spatio-temporal overlap which can be challenging for directional processing devices. Our study focuses on a gallium-substituted yttrium iron garnet film, which exhibits an exchange-dominated dispersion relation and thus provides a particularly broad range of group velocities compared to pure YIG. Using time- and space- resolved Brillouin light scattering spectroscopy, we demonstrate the excitation of time-separated spin-wave pulses at different frequencies from the same source, where the delayed pulse catches up with the previously excited pulse and outruns it due to its higher group velocity. By varying the excitation power of the faster pulse, the outcome can be finely tuned from a linear superposition to a nonlinear interaction of both pulses, resulting in a full attenuation of the slower pulse. Therefore, our findings demonstrate the all-magnonic erasing process of a propagating magnonic signal, which enables the realization of complex temporal logic operations with potential application, e.g., in inhibitory neuromorphic functionalities. The last decade has seen an increasing interest in alterna- tive computing schemes such as neuromorphic computing1,2, particularly in the context of the growing computational de- mands and the physical limitations of current semiconductor technologies3. Contemporary research has focused consider- able attention on wave-based logic concepts, which offer addi- tional degrees of freedom, a non-Boolean data carrier, and ef- fects such as interference4,5. In this context, spin-wave based information processing stands out6–8, not least because spin waves exhibit efficient intrinsic nonlinear effects, fundamen- tal to any logic operation. This feature in conjunction with their low power footprint makes spin waves an efficient data carrier for future information processing schemes such as neu- romorphic computing9,10. To take advantage of nonlinear effects in the spin-wave do- main requires the spatio-temporal overlap, i.e., the superposi- tion, of spin-wave signals, which can be challenging for direc- tional processing with propagating spin-waves. Existing spin- wave logic typically makes use of structural design to achieve superposition, e.g., by using waveguides11–14or delay struc- tures such as ring resonators15to recouple previous signals. However, spin waves typically show a pronounced dispersion and therefore offer intrinsic means to achieve signal superpo- sition in a more elegant way. Here, we study the superpo- sition and interaction of separate spin-wave signals emitted from the same source with a time delay, by exploiting their difference in group velocity, i.e. their dispersive properties. For this purpose, the ultralow spin-wave damping material yt- trium iron garnet16–18(YIG) is well suited since it provides sufficiently long propagation distances as well as functional a)D. Breitbach and M. Bechberger contributed equally to this work.nonlinear properties. Moreover, it features a positive nonlin- ear frequency shift when magnetized out-of-plane, a property that facilitates powerful logic functionalities19. However, a drawback of this system is the limited range of group veloc- ities in the wavevector regime accessible via standard means of RF excitation. Furthermore, operation in the out-of-plane magnetization configuration poses a challenge for the optical detection of the spin-wave signals. Here we circumvent these limitations by employing a Gallium-substituted YIG20–25film instead. This material exhibits a nearly compensated magnetic configuration, re- sulting in a low saturation magnetization of MS=20.2mT, which leads to an almost vanishing dipolar contribution to the spin-wave dispersion. More importantly, Ga:YIG shows a large perpendicular magnetic anisotropy (PMA) and an ex- change length which is several times larger compared to pure YIG25,26. Therefore, the spin-wave dispersion in this mate- rial is exchange-dominated even at low wavevectors and of almost isotropic, quadratic shape, which is analogous to the dispersion of a non-relativistic particle in quantum mechanics. Thus, the group velocity has a linear wavevector dependence and covers a broad range with fine tunability compared to pure YIG. This allows for the excitation of spin-wave signals with a particularly broad difference in group velocities. Further- more, due to the PMA, the material shows a positive nonlin- ear frequency shift coefficient27when magnetized in the in- plane configuration, easily accessible by optical means such as Brillouin light scattering spectroscopy (BLS). In this work, we utilize these unique features to demonstrate the superpo- sition of propagating spin-wave signals which are excited by the same source but with a time delay. Furthermore, we study their nonlinear interaction with regard to the functionality of spin-wave erasing - a process where one temporal signal an- nihilates another.arXiv:2311.17821v2 [cond-mat.mtrl-sci] 1 Feb 20242 25 µm 267 nm485 nmCPW Focal spot Detector LaserTandem Fabry-Pérot interferometer λ = 457 nmBLS time- traces 0 50 100 150 200BLS intensity (a. u.) Time t (ns) H H FIG. 1. Colorized SEM micrograph of the CPW antenna under investigation, placed on top of the Ga:YIG film and a schematic of the applied time-resolved BLS microscope. In the experiment, an in-plane bias magnetic field of µ0Happ≈86mT is applied either par- allel or perpendicular to the scan direction to magnetize the Ga:YIG in-plane. Timetraces of an exemplary BLS measurement of a propa- gating spin-wave pulse excited at f=1.4GHz are shown to illustrate the measurement process. Figure 1 shows the investigated structure and the experi- mental setup. The magnonic material under study is an 59 nm thin Ga:YIG film, grown by liquid phase epitaxy (LPE)16,17 adjusted for the deposition of sub-100-nm, single crystalline Y3Fe5 –xGaxO12films. Its material properties have been thor- oughly studied in a previous work25. It is characterized by a strong PMA of µ0Hu= (94.1±0.5)mT, an exchange length of λex=91.9nm, which is substantially larger com- pared to pure YIG and the low Gilbert-damping parameter of α= (6.1±0.6)×10−4. Unless stated otherwise, for all pre- sented results an external in-plane magnetic field of µ0H⊥= (86±2)mT is applied to ensure that the magnetization of the film is aligned in the film plane (see supplementary material, Sec. I). On top of the film, a coplanar waveguide (CPW) antenna is structured for spin-wave excitation, composed of Ti/Au (10 nm/70 nm). The antenna is fed by a microwave setup con- sisting of two microwave sources connected to two individual microwave switches which are separately triggered by a pulse generator. The two circuits are then merged by a Wilkinson type power combiner such that the backward isolation is en- sured. Using this setup, short microwave pulses of approxi- mately τrf≈15ns are generated to excite a propagating spin- wave packet with a certain carrier frequency. The accessi- ble frequencies are defined by the excitation efficiency of the CPW antenna, which is a function of the wavevector kof the Wavevector k (rad/µm)Group velocity v (µm/ns)G Frequency f (GHz) 0 5 10 15 201.01.52.02.53.0 DE 88.54 mT BV 85.76 mT Measurement DE Measurement BV 0.000.250.500.751.001.25 Wavevector k (rad/µm) -1 Wavevector k (rad µm)Wavevector k (rad/µm) Wavevector k (rad/µm)-1 Group velocity v (µm ns) G -1 Group velocity v (µm ns ) G-1 Group velocity v (µm ns ) G -1 Group velocity v (µm ns) GFrequency f (GHz) 0 00 05 55 510 1010 1015 1515 1520 2020 201.01.52.02.53.0 Experiment ExperimentCalculation 0.000.000.00 0.000.25 0.250.25 0.250.50 0.500.50 0.500.750.750.75 0.751.00 1.001.00 1.001.25 1.251.25 1.25H H H ExperimentH H H Frequency f (GHz)Frequency f (GHz) Frequency f (GHz)(a)(a) (a) (b)(b) (b) 1.5 1.01.5 2.02.0 2.52.5 3.03.0 1.52.02.53.0 CalculationH ExperimentH H CalculationCalculationH H H FIG. 2. (a)Group velocity as a function of the spin-wave wavevec- tork, theoretical curve (black) and experimental data (points). Each datapoint is extracted from a 1D spatial BLS scan, respectively. The pulses were excited using an excitation pulse length of τRF≈10ns and varying microwave powers to account for the excitation effi- ciency of the CPW (see supplementary material, Sec. III). The datapoints were converted from excitation frequency to the corre- sponding wavevector using the dispersion relation shown in panel (b), calculated according to Ref25. The external field values are µ0H⊥=88.54mT and µ0H∥=85.76mT. excited spin waves (see supplementary material, Sec. II). The experimental investigation of the spin-wave propagation is studied by applying time- and space-resolved BLS28, focusing a laser beam of λ=457nm with a laser power of P=3.0mW on the sample. After the spin-wave pulse is excited, it propa- gates away from the CPW antenna. As schematically depicted in Fig. 1, time-resolved BLS measurements are performed at different distances to the antenna, allowing to track the spin- wave pulse in time and space and to extract its group velocity. The resulting group velocities vgare shown in Fig. 2 (a) as a function of the excited spin-wave wavevector k. The data was collected under two field configurations, k⊥M andk∥M. In addition, the dispersion relation of both cases is depicted in Fig. 2 (b). It can be seen from these two configurations at approximately the same field values Happ that the spin-wave properties are highly isotropic, confirming the exchange-dominated character of this system. Figure 2 also shows the quadratic shape of the dispersion relation ω(k). This results in the group velocity vg=∂ω ∂kbeing almost linearly dependent on the spin-wave wavevector k, which is in good agreement with the obtained data. Due to this property, the system exhibits a particularly large range of group velocities in the low-wavevector regime compared to pure YIG, ranging from approximately vg=0 atk=0 to above vg=1µmns−1atk=15radµm−1. This property allows for an interesting experiment: spin-wave pulses with strongly different group velocities excited from the same source could still come into superposition even when excited at different3 f = 2200 MHz B050100 0.040.060.080.10.20.40.60.81BLS intensity (arb. u.) 050100 Time t (ns) f+fA B 0 5 10 15 20 25050100 Position x (µm) f = 1110 MHz A 050100 f A f BTime t (ns) Time t (ns) Time t (ns)(a) (b) (c) (d)510152025Position x (µm) 510152025Position x (µm) 510152025Position x (µm)0.040.060.080.10.20.40.60.81BLS intensity (arb. u.) f = 2.20 GHz B f+fA Bf = 1.1 1 GHz A 25 f+fA B(a) (b) (c) (d) 0510152025 0 25 50 75 100 125Position x (µm) Time t (ns)v= 0.30 µm/ns g v= 0.96 µm/ns g fBLSI BLSfBLSI BLSfBLSI BLSfBLSI BLS FIG. 3. Time-resolved BLS measurements of two short pulses Aand Bexcited at different frequencies fA=1.11GHz and fB=2.2GHz and with a time delay of ∆t0≈32ns. (a)Only excitation of pulse A.(b)Only excitation of pulse B.(c)Combined excitation of pulses AandBwith same parameters as in previous panels. All color plots have the same color scale and normalization. (d)Same as (c)but only BLS frequencies around fAare shown (marked in red in the inset). Parameters: PA=0dBm and PB=15dBm. times. Due to the high isotropy of the system, the following investigations are carried out in k⊥Mconfiguration. Figure 3 (a)shows the time- and space-resolved measure- ment of a spin-wave pulse excited close to the band bot- tom at fA=1.11GHz, which corresponds to a low wavevec- tor of kA≈(5±1)radµm−1. On the other hand, Fig. 3 (b) depicts the measurement of a spin-wave pulse that was ex- cited with a frequency of fB=2.2GHz and a wavevector of kB≈(15.0±0.3)radµm−1. It is noted that while the fre- quency fAis required to be close to the band bottom, the ex- act choice of fBis not crucial to the effect under study, but rather an experimental trade-off between a high group veloc- ity and a feasible excitation efficiency of the CPW antenna (see supplementary material, Sec. IV). Close to the CPW antenna, the pulses have a duration (FWHM) of τA=17ns andτB=14ns. The pulse timings and their respective width (FWHM) are marked on the color plot as extracted by fitting remaining intensity Pulse A 15 Off 2 dBm 8 dBmPower P -5 0 5 100.00.20.40.60.81.0 Power P of pulse B (dBm)B Level without fast pulse Level without any pulse 0 50 100 1500.00.51.0 BLS intensity (arb. u.) Time t (ns) BLS intensity of pulse A (arb. u.)B(a) (b) fBLSI BLSFIG. 4. (a)BLS intensity of slow pulse A, extracted for BLS fre- quencies around fA, as a function of the applied microwave power PBof fast pulse B(see also supplementary material, Sec. IV). (b) Timetraces of pulse Aextracted for BLS frequencies around fAfor different microwave powers PB. Parameters: PA=0dBm, position: x=20.5µm. the time-dependent data with Gaussian functions. In the fol- lowing, these two pulses will be referred to as pulse AandB, respectively. It is evident from this measurement that pulse AandBsig- nificantly differ in their group velocity. While pulse Awith vg=0.30µmns−1is much slower, pulse Bpropagates with vg=0.96µmns−1, which is also the main reason for their dif- ferent decay behavior (see supplementary material, Sec. V). In the following, both pulses are combined in a single ex- periment while all other parameters are kept constant. The re- sult is shown in Fig. 3 (c)and demonstrates that, even with a time delay of ∆t0=32ns in their excitation, pulse Bcatches up with pulse Aapproximately 14 ns after its excitation, leading to a crossing point. However, pulse Avanishes when being passed by the faster pulse B, indicating a direct interaction be- tween both pulses. This becomes evident in Fig. 3 (d), which shows the combined measurement of both pulses ( fA+fB), but extracted only for BLS frequencies around fA. The fast pulse Bprovides a moving spatial barrier for spin waves in the low frequency range and erases the slow pulse Aincluding its pulse tail, which would otherwise reach the furthest point of the measurement (see also supplementary material, Sec. VI). This means that spin-wave signals generated at different times but from the same source can still be brought to superposition and even nonlinearly interact. To study the nonlinear interaction between the pulses, the remaining intensity of pulse Aat position x=20.5µm is considered as a function of the applied microwave power PB of the fast pulse B, see Fig. 4 (a)and(b). As PBincreases, the intensity of the slow pulse decreases from being almost unaffected to almost full attenuation. While pulse Aretains around 90% of its original intensity for PB=−5dBm, it is reduced to only 10% at PB=13dBm. Varying the excitation power of the pulse Btherefore enables either an almost linear superposition of both pulses or a nearly complete erasure of the slower pulse A. The results of Figs. 3 and 4 are confirmed by micromagnetic simulations (see supplementary material, Sec. VII).4 0.51.01.52.02.53.0 Time t (ns) BLS frequency f (GHz)BLS 0.000.050.100.150.30 BLS intensity (arb. u.) 0 25 50 75(c) >0.35(b) Power P (dBm) cw−20 −15 −10 −5 0 50.81.01.21.41.61.8 Minimum spin-wave frequencyBLS frequency f (GHz)BLS f = 1.1 1 GHz A c = 0B c = 0.1 1 = c B crit c = 0.15 > cB crit f = 1.1 1 GHz A(a) 0 2 4 6 8 100.81.01.21.41.61.8Frequency f (GHz) Wavevector k (rad/µm)xBLS x(cw)fB fA P = 15.0 dBmB 0.25 0.20 FIG. 5. Effect of the nonlinear frequency cross-shift of the mode kB=15radµm−1(not shown) on the linear spin-wave dispersion. (a) Calculated spin-wave dispersion relation for the linear case (red) and cases with increased amplitude cBat critical (blue) and overcritical level (green). Calculated for µ0Heff=88.5mT which includes effects of crystal anisotropy to match the experiment. (b)Measured minimal frequency of the thermal spin-wave signal (BLS) as a function of the applied, continuous microwave power PcwatfB=2.2GHz. Measured at position x(cw)=4.0µm. (c)Time-resolved BLS measurement of pulses AandBbefore the pulse crossing at the power levels PA=0dBm and PB=15dBm, measured at position x=4.0µm. To investigate the physical process behind this mechanism, the effect of the nonlinear frequency shift on the linear dis- persion is considered. The four-magnon interaction between two waves with wavevectors kandk′results, in particular, in a total shift of the spin-wave frequency of mode k29, ˜ωk=ωk+Tk|ck|2+2Tkk′|ck′|2, (1) where ωk=2πfkis the linear spin-wave frequency, ckand ck′are the spin-wave amplitudes of the two modes, Tkis the self nonlinear frequency shift coefficient and Tkk′is the cross nonlinear frequency shift. Calculations with the Hamiltonian formalism30result in a positive cross-shift of TkAkB/(2π)≈ 3.6GHz, as expected for an in-plane magnetized film with high PMA. This contribution is crucial for the comprehension of the experimental results – pulse Bshifts the entire disper- sion to higher frequencies, in particular, the bottom of the dis- persion, see Fig. 5 (a). The dispersion relations shown are calculated according to Eq. (1) using different spin wave am- plitudes cB:fk=f(k,ck→0,cB). The dispersion in the ab- sence of pulse B, i.e. the linear dispersion, is shown in red. In the presence of pulse B, it is increasingly shifted to higher frequencies for an increasing amplitude cB. When a critical value of cB= 0.11 is reached, corresponding to about 9 de- gree magnetization precession angle, the minimal frequency of the dispersion relation is shifted up to fA=1.1 GHz. For higher amplitudes cB, this low-frequency mode no longer ex- ists and pulse Abecomes nonresonant. Hence, as long as the frequency is conserved, this prevents further propagation of the slow pulse A. This mechanism occurs when the fast pulse passes the slower pulse with a spin-wave amplitude cB>ccrit, which results in the cut-off of pulse A. Note that according to Eq. (1) this mechanism does not depend on the phase of pulse B, but only on its intensity |cB|2. Due to the inaccessibility of absolute spin-wave amplitudes ck by means of BLS spectroscopy, only qualitative comparisons can be made between the experiment and this calculation. This can be achieved using a continuous microwave excita-tion at the frequency fBwith increasing excitation amplitude. This allows to investigate the effect of the nonlinear shift on the thermal spin-wave population, i.e. incoherent, thermally excited spin waves that can be detected by BLS at room tem- perature. Fig. 5 (b)shows the measured minimal frequency of the thermal spin-wave spectrum as a function of the con- tinuous excitation power. The minimum frequency increases tofAat a power of Pcw≈0dBm and increases even further for higher amplitudes. This experiment demonstrates that us- ing experimentally feasible spin-wave amplitudes, the linear spin-wave frequency can be shifted by several 100MHz. Since the previous experiments were done using pulsed RF excitation, the power PBcannot be directly compared to the continuous power levels in Fig. 5 (b). Fig. 5 (c)shows a frequency- and time-resolved BLS spectrum of the pulses A andBfor an excitation power of PB= 15 dBm before the pulse crossing. At this power level, pulse Bcan almost completely erase pulse A, as was shown in Fig. 4. It is visible that the thermal spin-wave spectrum shifts to frequencies higher than fAin the presence of pulse B, which prevents further propa- gation of the slow pulse. This interpretation is in agreement with the power-dependent attenuation in Fig. 4, as the nonlin- ear frequency shift is proportional to the spin-wave intensity, which increases with the applied microwave power (see Fig. 5(b)). This mechanism occurs when the fast pulse passes the slower one, which results in the cut-off of pulse A. Effectively, the faster pulse is an eraser for a spin-wave signal that was sent earlier. It remains an open question, however, how the energy of pulse Ais dissipated or redistributed during the erasing pro- cess. Potential channels for this could include reflection or even scattering perpendicular to the propagation direction, or scattering to the whole spin-wave spectrum as the mode be- comes nonresonant. In our current study, no clear signature of such energy transfer could be found. The difficulty to unravel the precise mechanisms lies not only in the two-dimensional extension of the investigated film, but also the wavevector de- pendent and -limited BLS sensitivity, which can effectively5 obscure energy scattering to different frequency channels. It is noted that in Fig. 5 (c), spin-waves at intermediate frequen- cies occur during the presence of pulse B. However, these are caused by the intense pulse Band are not related to an energy transfer of pulse A, as they also occur in absence of pulse A. The clarification of this question requires further experi- ments and extensive simulations as well as studying a confined waveguide. In conclusion, we demonstrated that temporally separated spin-wave signals excited by the same source can be brought into direct superposition by exploiting a systems intrinsic dis- persive properties. We achieved this by employing Ga:YIG, a low-damping material with an exchange-dominated disper- sion relation and a high range of excitable group velocities with fine tunability. Utilizing a considerable difference in group velocity, two temporally separated spin-wave pulses can be excited such that the second pulse catches up with the first, bringing them into direct superposition. As a proof-of-principle for the interaction between the pulses, we have demonstrated the tunable nonlinear erasing of the slow spin-wave pulse by the faster pulse. This is mediated by a positive nonlinear frequency shift induced by the fast pulse, which locally shifts the spin-wave dispersion to higher frequencies. This renders the slow pulse nonresonant, hence hindering its further propagation. Our work demonstrates the direct nonlinear interaction of signals from separated temporal inputs which is key to the realization of complex temporal logic operations and is essential for recurrent neuromorphic computing functionalities such as the fading memory. Furthermore, the ability to erase previous informa- tion is potentially applicable as an inhibitory component in neuromorphic computing. See the supplementary material for a additional data on the in-plane magnetization of the Ga:YIG film, on the exci- tation efficiency of the CPW antenna and for details on the selected power levels for the group velocity measurements. Furthermore, the supplementary material contains details on the choice of excitation frequencies, analysis of the spin-wave decay and additional time-domain data on the erasing process. Lastly, comparative micromagnetic simulations are shown. ACKNOWLEDGMENTS This research was funded by the European Research Coun- cil within the Starting Grant No. 101042439 "CoSpiN", by the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) within the Transregional Collaborative Research Center—TRR 173–268565370 “Spin + X” (project B01) and the project 271741898. The authors acknowledge support by the Max Planck Graduate Center with the Johannes Gutenberg-Universität Mainz (MPGC). R.V . acknowledges support by MES of Ukraine (project 0124U000270). K. L. ac- knowledges the Austrian Science Fund FWF for the support through Grant ESP 526-N "TopMag". Q. W. acknowledges support from the National Key Research and Development Program of China (Grant Nos. 2023YFA1406600) and Na- tional Natural Science Foundation of China, the startup grantof Huazhong University of Science and Technology (Grants No. 3034012104). AUTHOR DECLARATIONS Conflict of Interest The authors declare no competing interests. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1D. Markovi ´c, A. Mizrahi, D. Querlioz, and J. Grollier, Nat. Rev. Phys. 2, 499 (2020). 2J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V . Cros, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier, Nature 547, 428 (2017). 3M. M. Waldrop, Nature 530, 144 (2016). 4T. W. Hughes, I. A. D. Williamson, M. Minkov, and S. Fan, Science Ad- vances 5(2019), 10.1126/sciadv.aay6946. 5B. J. Shastri, A. N. Tait, T. Ferreira de Lima, W. H. P. Pernice, H. Bhaskaran, C. D. Wright, and P. R. Prucnal, Nat. Photonics 15, 102 (2021). 6A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V . Chumak, S. Hamdioui, C. Adelmann, and S. Cotofana, Journal of Applied Physics 128, 161101 (2020). 7A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Gräfe, C. Adelmann, S. Cotofana, A. Naeemi, V . I. Vasyuchka, B. Hillebrands, S. A. Nikitov, H. Yu, D. Grundler, A. V . Sadovnikov, A. A. Grachev, S. E. Sheshukova, J.-Y . Duquesne, M. Marangolo, G. Csaba, W. Porod, V . E. Demidov, S. Urazhdin, S. O. Demokritov, E. Albisetti, D. Petti, R. Bertacco, H. Schultheiss, V . V . Kruglyak, V . D. Poimanov, S. Sahoo, J. Sinha, H. Yang, M. Münzenberg, T. Moriyama, S. Mizukami, P. Lan- deros, R. A. Gallardo, G. Carlotti, J.-V . Kim, R. L. Stamps, R. E. Camley, B. Rana, Y . Otani, W. Yu, T. Yu, G. E. W. Bauer, C. Back, G. S. Uhrig, O. V . Dobrovolskiy, B. Budinska, H. Qin, S. van Dijken, A. V . Chumak, A. Khitun, D. E. Nikonov, I. A. Young, B. W. Zingsem, and M. Winkl- hofer, Journal of Physics: Condensed Matter 33, 413001 (2021). 8G. Csaba, Ádám Papp, and W. Porod, Physics Letters A 381, 1471 (2017). 9A. V . Chumak, P. Kabos, M. Wu, C. Abert, C. Adelmann, A. O. Adey- eye, J. Åkerman, F. G. Aliev, A. Anane, A. Awad, C. H. Back, A. Bar- man, G. E. W. Bauer, M. Becherer, E. N. Beginin, V . A. S. V . Bitten- court, Y . M. Blanter, P. Bortolotti, I. Boventer, D. A. Bozhko, S. A. Bun- yaev, J. J. Carmiggelt, R. R. Cheenikundil, F. Ciubotaru, S. Cotofana, G. Csaba, O. V . Dobrovolskiy, C. Dubs, M. Elyasi, K. G. Fripp, H. Fu- lara, I. A. Golovchanskiy, C. Gonzalez-Ballestero, P. Graczyk, D. Grundler, P. Gruszecki, G. Gubbiotti, K. Guslienko, A. Haldar, S. Hamdioui, R. Her- tel, B. Hillebrands, T. Hioki, A. Houshang, C.-M. Hu, H. Huebl, M. Huth, E. Iacocca, M. B. Jungfleisch, G. N. Kakazei, A. Khitun, R. Khymyn, T. Kikkawa, M. Kläui, O. Klein, J. W. Kłos, S. Knauer, S. Koraltan, M. Kostylev, M. Krawczyk, I. N. Krivorotov, V . V . Kruglyak, D. Lachance- Quirion, S. Ladak, R. Lebrun, Y . Li, M. Lindner, R. Macêdo, S. Mayr, G. A. Melkov, S. Mieszczak, Y . Nakamura, H. T. Nembach, A. A. Nikitin, S. A. Nikitov, V . Novosad, J. A. Otálora, Y . Otani, A. Papp, B. Pigeau, P. Pirro, W. Porod, F. Porrati, H. Qin, B. Rana, T. Reimann, F. Riente, O. Romero-Isart, A. Ross, A. V . Sadovnikov, A. R. Safin, E. Saitoh, G. Schmidt, H. Schultheiss, K. Schultheiss, A. A. Serga, S. Sharma, J. M. Shaw, D. Suess, O. Surzhenko, K. Szulc, T. Taniguchi, M. Urbánek, K. Us- ami, A. B. Ustinov, T. van der Sar, S. van Dijken, V . I. Vasyuchka, R. Verba, S. V . Kusminskiy, Q. Wang, M. Weides, M. Weiler, S. Wintz, S. P. Wolski, and X. Zhang, IEEE Transactions on Magnetics 58, 1 (2022).6 10A. N. Mahmoud, F. Vanderveken, C. Adelmann, F. Ciubotaru, S. Hamdioui, and S. Cotofana, IEEE Transactions on Magnetics 57, 1 (2021). 11P. Pirro, V . I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Reviews Materials 6, 1114 (2021). 12T. Fischer, M. Kewenig, D. A. Bozhko, A. A. Serga, I. I. Syvorotka, F. Ciub- otaru, C. Adelmann, B. Hillebrands, and A. V . Chumak, Applied Physics Letters 110, 152401 (2017). 13A. Khitun, M. Bao, and K. L. Wang, Journal of Physics D: Applied Physics 43, 264005 (2010). 14Q. Wang, P. Pirro, R. Verba, A. Slavin, B. Hillebrands, and A. V . Chumak, Science Advances 4, e1701517 (2018). 15Q. Wang, A. Hamadeh, R. Verba, V . Lomakin, M. Mohseni, B. Hillebrands, A. V . Chumak, and P. Pirro, npj Computational Materials 6, 192 (2020). 16C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. Hübner, and E. Wendler, Phys. Rev. Mater. 4, 024416 (2020). 17C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Brückner, and J. Del- lith, Journal of Physics D: Applied Physics 50, 204005 (2017). 18A. Serga, A. Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010). 19Q. Wang, R. Verba, B. Heinz, M. Schneider, O. Wojewoda, K. Davídková, K. Levchenko, C. Dubs, N. J. Mauser, M. Urbánek, P. Pirro, and A. V .Chumak, Science Advances 9, eadg4609 (2023). 20Boyle, J. W., Booth, J. G., Boardman, A. D., Zavislyak, I., Bobkov, V ., and Romanyuk, V ., J. Phys. IV France 07, C1 (1997). 21J. Guigay, J. Baruchel, D. Challeton, J. Daval, and F. Mezei, Journal of Magnetism and Magnetic Materials 51, 342 (1985). 22P. Hansen, P. Röschmann, and W. Tolksdorf, Journal of Applied Physics 45, 2728 (1974). 23P. Görnert and C. G. D’ambly, physica status solidi (a) 29, 95 (1975). 24P. Roschmann, IEEE Transactions on Magnetics 17, 2973 (1981). 25T. Böttcher, M. Ruhwedel, K. O. Levchenko, Q. Wang, H. L. Chumak, M. A. Popov, I. V . Zavislyak, C. Dubs, O. Surzhenko, B. Hillebrands, A. V . Chumak, and P. Pirro, Applied Physics Letters 120, 102401 (2022). 26S. Klingler, A. V . Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, Journal of Physics D: Applied Physics 48, 015001 (2014). 27J. J. Carmiggelt, O. C. Dreijer, C. Dubs, O. Surzhenko, and T. van der Sar, Applied Physics Letters 119, 202403 (2021). 28T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, Frontiers in Physics 3, 35 (2015). 29V . S. L’vov, Wave Turbulence under Parametric Excitation (Springer- Verlag, New York, 1994). 30P. Krivosik and C. E. Patton, Phys. Rev. B 82, 184428 (2010).

2023-11-29

Nonlinear phenomena are key for magnon-based information processing, but the nonlinear interaction between two spin-wave signals requires their spatio-temporal overlap which can be challenging for directional processing devices. Our study focuses on a gallium-substituted yttrium iron garnet film, which exhibits an exchange-dominated dispersion relation and thus provides a particularly broad range of group velocities compared to pure YIG. Using time- and space-resolved Brillouin light scattering spectroscopy, we demonstrate the excitation of time-separated spin-wave pulses at different frequencies from the same source, where the delayed pulse catches up with the previously excited pulse and outruns it due to its higher group velocity. By varying the excitation power of the faster pulse, the outcome can be finely tuned from a linear superposition to a nonlinear interaction of both pulses, resulting in a full attenuation of the slower pulse. Therefore, our findings demonstrate the all-magnonic erasing process of a propagating magnonic signal, which enables the realization of complex temporal logic operations with potential application, e.g., in inhibitory neuromorphic functionalities.

Nonlinear erasing of propagating spin-wave pulses in thin-film Ga:YIG

2311.17821v2

arXiv:2112.07264v1 [cond-mat.mtrl-sci] 14 Dec 2021Robust perpendicular magnetic anisotropy in Ce substitute d yttrium iron garnet epitaxial thin films Manik Kuila,1Archna Sagdeo,2,3Lanuakum A Longchar,4R.J.Choudhary,1S.Srinath,4and V.Raghavendra Reddy1,a) 1)UGC-DAE Consortium for Scientific Research, University Cam pus, Khandwa Road, Indore 452001, India. 2)Synchrotrons Utilization Section, RRCAT, Indore India. 3)Homi Bhabha National Institute, Training School Complex, A nushakti Nagar, Mumbai 400094, India. 4)School of Physics, University of Hyderabad, Hyderabad-500 046, India Cerium substituted yttrium iron garnet (Ce:YIG) epitaxial thin films a re prepared on gadolinium gallium garnet (GGG) substrate with pulsed laser deposition (PLD). It is ob servedthat the films grown on GGG(111) substrate exhibit perpendicular magnetic anisotropy (PMA) as com pared to films grown on GGG(100) sub- strate. The developed PMA is confirmed from magneto-optical Ker r effect, bulk magnetization and ferromag- netic resonance measurements. Further, the magnetic bubble do mains are observed in the films exhibiting PMA. The observations are explained in terms of the growth directio n of Ce:YIG films and the interplay of various magnetic anisotropy terms. The observed PMA is found to b e tunable with thickness of the film and a remarkable temperature stability of the PMA is observed in all the s tudied films of Ce:YIG deposited on GGG(111) substrate. Keywords: Garnet thin films, epitaxial thin films, perpendicular magn etic anisotropy, MOKE. I. INTRODUCTION Cerium substitute yttrium iron garnet (Ce:YIG) is a widelyknownmaterialbecauseofitsimportantmagneto- optical (MO) properties over a wide range of wavelength spectrum (infrared to near UV-visible)1–4. The strong MO activity and high transparency usually promotes Ce:YIG as a promising candidate for non-reciprocal de- vice applications2,5–7. Therefore, most of the studies were being focused for improving MO properties, such as Faraday and Kerr rotation in polycrystalline and/or epitaxial Ce:YIG films8,9. However, another interesting aspect in these kind of films is the development of per- pendicular magnetic anisotropy (PMA). The fact that the garnet films are insulating in nature and exhibit low magnetic losses, the development of garnet films with PMA has opened immense applications in the fields of spintronics, magneonic, spin caloritronics etc10–14. For example, in pure and substituted YIG pure spin currents are observeddue to the inverse spin Hall effect in a heavy metal over layer15, an electrical control of magnetization has also been realized in Tm 3Fe5O12(TmIG) with ad- jacent heavy metal layer due to spin-orbit-torque (SOT) induced effect16etc. Recently we have studied epitaxial Ce:YIG films in re- flection mode with longitudinal magneto-optical Kerr ef- fect (MOKE)andshownthatonecantune theMOactiv- ity by preparing the films in different O2partial pressure (OPP) during deposition and also by varying the thick- ness (deposition time) at a given OPP17. However, we could not observe PMA in these films. In fact, there a)Electronic mail: varimalla@yahoo.com; vrreddy@csr.res. inseems to be only one report dealing with the PMA in Ce:YIG films in literature, wherein the PMA is reported to be stable only at temperatures below 150 K18. In view of this, in the present work we have undertaken the developmentofPMAin Ce:YIG films byfurther optimiz- ingthe depositionconditionswith pulsed laserdeposition (PLD), especially the substrate temperature during de- position and the substrate orientation. Moreover, it is observed that the PMA is stable over a wide temper- ature range ( ≤400 K) enabling the possibility of room temperature based applications of these materials. II. EXPERIMENTAL DETAILS Pulsed laser deposition (PLD) technique is used to ab- late stoichiometric Ce 1Y2Fe5O12(Ce:YIG) target, the sametargetthathasbeenusedinourearlierworkdealing with the Ce:YIG films prepared at different OPP17. The films are grownin optimized O 2partial pressure of about 110mTorrandfixedsubstratetemperatureof780oC.The other parameters i.e., laser influence, repetition rate and target to substrate distance are 2.5 J/cm2, 6 Hz, 5 cm, respectively. All these conditions are kept same for ev- ery deposition except deposition time to vary the thick- ness. Three films of about 10, 30 and 65 nm thickness are prepared on Gd 3Ga5O12(GGG) substrates. For ev- ery deposition, both the (100) and (111) oriented GGG substrates are mounted side by side, so that the films are depositedinsameconditions. Thecrystallinestructureof thefilmsareanalyzedusingx-raydiffraction(XRD) mea- surements carried out at angle dispersive X-ray diffrac- tion (ADXRD) beamline (BL-12) of Indus-2, RRCAT In- dia. The x-ray reflectivity (XRR) and reciprocal space mapping (RSM) are carried out using Bruker D8 diffrac-2 TABLE I. Notations of the prepared films and the obtained structural parameters. For example, Ce:YIG10-111 stands f or thefilm ofabout 10nmthickness anddeposited on GGG(111) substrate. All the films are prepared at the same conditions except the deposition time i.e., film thickness. ’t’ is the th ick- ness (±0.1 nm), ρis the roughness ( ±0.1 nm). aoutis the out-of-plane lattice parameter ( ±0.0001 nm) obtained from Fig. 4. Notation Substrate t ρ a out (nm) (nm) (nm) Ce:YIG10-111 GGG(111) 10.0 0.5 1.2383 Ce:YIG30-111 GGG(111) 30.0 0.9 1.2364 Ce:YIG65-111 GGG(111) 65.7 1.0 1.2350 Ce:YIG10-100 GGG(100) 9.2 0.5 - Ce:YIG30-100 GGG(100) 29.9 0.9 - Ce:YIG65-100 GGG(100) 64.0 0.2 - tometer equipped with LynxEye detecotr, Goebel mirror andEuleriancradle. Furthersurfaceroughnessaredeter- mined byatomicforcemicroscopy(AFM) measurements. Domain structure and magnetic hysteresis loops of the films are simultaneously captured with Kerr microscopy system of M/s Evico magnetics, Germany equipped with white light LED source. Room temperature ferromag- netic resonance (FMR) measurements are carried out us- ingJEOL-FA200ElectronSpin ResonanceSpectrometer. FMR spectra have been recorded as a function of polar angle (θH)19. /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48 /s48/s46/s50/s53 /s48/s46/s51/s48/s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s49/s49/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s46/s41 /s113 /s122/s45/s49 /s32/s40/s65/s45/s49 /s41/s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49 /s40/s98/s41/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s48/s48 /s40/s97/s41 /s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s48/s48 /s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s48/s48/s82/s101/s102/s108/s101/s99/s116/s105/s118/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s46/s41 /s113 /s122/s45/s49 /s32/s40/s65/s45/s49 /s41 FIG. 1. X-ray reflectivity (XRR) data of the indicated films. Symbols represent the experimental data points and the soli d line is thebest fittothedata. Obtainedparameters are shown in Table-1. III. RESULTS Fig. 1 show XRR patterns of Ce:YIG films deposited on GGG(111) and GGG(100) substrates. Fitting to the experimental data using Parratt formalism gives the pa- rameter such as thickness, roughness and the obtained parameters are shown in table-I. Further, the surface /s40/s99/s41/s40/s97/s41 /s40/s98/s41 FIG. 2. Atomic force microscopy (AFM) images of (a) Ce:YIG10-111 (b) Ce:YIG66-111 films. morphology of the films is studied with atomic force mi- croscopy and Fig. 2 shows the AFM images 10 and 66 nm thick films deposited on GGG(111) substrate. The data indicate a smooth surface with rmsroughness less than 1 nm, corroborating the XRR data. Since the lattice parameter of the Ce:YIG films de- posited at higher OPP and substrate temperature is ex- pected to be very close to that of GGG substrate17, the film peaks are merged with that of substrate in the 2 θ- ωscans (not shown) of all the films measured using lab based x-ray diffractometer. However no other reflections are observed in the 2 θ-ωscans indicating that there are no secondary phases present for all the samples. This is consistent with our earlier work in which the film Bragg peaks are observed to shift close to GGG substrate peak at 110 mTorr OPP17. Further, reciprocal space map- ping (RSM) data across symmetric and asymmetric re- flections are carried out and the representative data of Ce:YIG66-111 film across (444) and (642) reflections is shown in Fig. 3. Since the present RSM measurements are not done in high-resolution mode with lab based x- ray diffractometer (as a result one can see the substrate α2reflection also) the film and substrate reciprocal lat- tice points are overlapping with each other, even though the data confirms that the films are epitaxial in nature. However, for samples deposited on GGG(111) substrates (which exhibit PMA as discussed below) the 2 θ-ωscans are carried out using synchrotron radiation and the data is shown in Fig. 4. /s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53/s53/s46/s53/s54/s53/s46/s53/s56/s53/s46/s54/s48/s53/s46/s54/s50/s53/s46/s54/s52/s53/s46/s54/s54/s40/s52/s52/s52/s41/s113 /s122/s32/s47/s47/s91/s49/s49 /s49 /s93/s32/s40/s110/s109/s45/s49 /s41 /s113 /s120 /s32/s47/s47/s32/s91/s49/s48 /s49 /s93/s32/s40/s110/s109/s45/s49 /s41/s48/s46/s49/s48/s48/s48/s46/s51/s48/s50/s48/s46/s57/s49/s49/s50/s46/s55/s53/s56/s46/s51/s48/s50/s53/s46/s49/s55/s53/s46/s54/s50/s50/s56/s54/s56/s57/s50/s46/s48/s56/s69/s43/s48/s51/s54/s46/s50/s56/s69/s43/s48/s51/s49/s46/s57/s48/s69/s43/s48/s52/s53/s46/s55/s50/s69/s43/s48/s52 /s50/s46/s50/s52 /s50/s46/s50/s56 /s50/s46/s51/s50 /s50/s46/s51/s54/s53/s46/s53/s54/s53/s46/s53/s56/s53/s46/s54/s48/s53/s46/s54/s50/s53/s46/s54/s52/s53/s46/s54/s54/s40/s54/s52/s50/s41/s113 /s122/s32/s47/s47/s32/s91/s49/s49/s49/s93/s32/s40/s110/s109/s45/s49 /s41 /s113 /s120 /s32/s47/s47/s32/s91/s49/s48 /s49 /s93/s32/s40/s110/s109/s45/s49 /s41/s48/s46/s49/s48/s48/s48/s48/s46/s50/s56/s53/s49/s48/s46/s56/s49/s50/s54/s50/s46/s51/s49/s55/s54/s46/s54/s48/s52/s49/s56/s46/s56/s51/s53/s51/s46/s54/s55/s49/s53/s51/s46/s48/s52/s51/s54/s46/s49/s49/s50/s52/s51/s51/s53/s52/s52/s49/s46/s48/s49/s48/s69/s43/s48/s52/s50/s46/s56/s56/s48/s69/s43/s48/s52 FIG. 3. Reciprocal space mapping (RSM) data of Ce:YIG66- 111measuredacrosstheindicatedreflections. Thetwointen se peaks (red color) are substrate α1andα2reflections.3 /s51/s46/s52 /s51/s46/s54 /s53/s46/s54 /s53/s46/s56 /s54/s46/s56 /s55/s46/s48 /s55/s46/s50/s40/s56/s56/s48/s41/s40/s52/s52/s52/s41 /s40/s56/s56/s56/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s113/s32/s40/s65/s45/s49 /s41/s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49 /s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49 /s32/s67/s101/s58/s89/s73/s71/s49/s48/s45/s49/s49/s49 FIG. 4. 2 θ-ωscans of the Ce:YIG films deposited on GGG(111) substrate mea sured with synchrotron radiation. The data is measured across symmetric (444), (888) reflections & asymme tric (880) reflection. The small vertical lines are to indica te the film peak. Fig. 4 shows the XRD measured around symmetric (444), (888) & asymmetric (880) reflections of all the films deposited on GGG(111) substrate. All the films exhibit pronounced Laue oscillations which clearly indi- cate that the films are highly smooth, uniform and single crystalline in nature. It may be noted that the thickness of the films calculated from Laue oscillations match ex- cellently with the XRR data. The out-of-plane (OOP) lattice parameter ( aout) is estimated from the (888) re- flectiondataandistabulatedintable-1. Usuallyonemay expect that the film peak shifts to higher (lower) angles with respect to the substrate, below a critical thickness of the film, due to the expected in-plane tensile (com- pressive) epitaxial strains in the case of lattice mismatch between substrate and film20. However, it is observed that for 10 nm thick sample, the film peak is overlapping with that of substrate, which could be due to the ex- cellent lattice matching between the film and substrate. With further increase in thickness, the OOP lattice pa- rameter is observed to decrease indicating the OOP lat- tice compression, unlike most of the cases in which films are expected to relax as thickness increases. The plau- sible reason for this could be either due to changes in the relative concentration of Ce3+, Ce4+and associated issues related to charge neutralization in the system with thickness17,21. Fig. 5 show the MOKE data of all the films along with the bulk magnetization data measured using SQUID- VSM for few selected samples. MOKE measurements are carried out in two geometries viz., longitudinal (L- MOKE) in which the applied magnetic field is in the film plane and polar (P-MOKE) in which the applied field is out-of-plane of the sample. P-MOKE measurements are sensitive to out-of-plane component of the magneti-zation and the L-MOKEis expected to be sensitive to in- planecomponentofmagnetization22. However,itmaybe noted that L-MOKE geometry can also have a sensitiv- ity of out-of-plane magnetization component in addition to the in-plane component due to the oblique incidence of light22. Perusal of Fig. 5, for all the films deposited on GGG(100) substrate the observed L-MOKE data is typical of films exhibiting in-plane magnetization. However, for the films deposited on GGG(111) sub- strates, an anomalous kind of L-MOKE loops are ob- servedas shown in Fig. 5. The data indicates two switch- ing fields (Fig. 5(d),(e),(f)). As one can see that at mid- dle portion of the L-MOKE loops, a clear switching of magnetization with finite H Cis observed in addition to the saturation at sufficiently high magnetic fields as in- dicated by vertical dashed line in Fig. 5(d),(e),(f), the magnetization saturates along the field direction. The central switching part could be due to the presence of considerable OOP magnetization component or domains with non-zero OOP field components during application of in-plane external magnetic field. In view of this, P- MOKE measurements are also carried out on films de- posited on GGG(111) substrate. Fig. 5 shows the room temperature P-MOKE hysteresis loops. It is very inter- esting to note that the observed squareness (ratio of re- manence/saturation, M R/MS) is close to unity for all the threefilmsintheP-MOKEdata. Inordertoconfirmthis, bulk magnetization measurementswith SQUID-VSM are alsocarriedout on these three films with the applied field along the out-of-plane direction. The SQUID-VSM data is observed to match excellently with P-MOKE data ex- cept small variation in the coercivity (H C). Therefore, both P-MOKE and SQUID-VSM data clearly suggest that at room temperature the easy magnetization axis4 /s40/s103/s41/s75/s101/s114/s114/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41 /s75/s101/s114/s114/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41/s67/s101/s58/s89/s73/s71/s47/s71/s71/s71/s40/s49/s49/s49/s41 /s54/s53/s32/s110/s109 /s51/s48/s32/s110/s109 /s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s49/s48/s32/s110/s109/s40/s105/s41 /s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48 /s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48 /s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48 /s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s101/s109/s117/s47/s99/s109/s51 /s41/s40/s97/s41/s75/s101/s114/s114/s32/s114/s111/s116/s97/s116/s105/s111/s110/s32/s40/s65/s114/s98/s46/s32/s85/s110/s105/s116/s115/s41/s54/s53/s32/s110/s109 /s40/s104/s41/s40/s98/s41 /s70/s105/s101/s108/s100/s32/s40/s109/s84/s41/s51/s48/s32/s110/s109/s67/s101/s58/s89/s73/s71/s47/s71/s71/s71/s40/s49/s48/s48/s41 /s40/s99/s41 /s49/s48/s32/s110/s109 FIG. 5. (a)-(c) L-MOKE loops of Ce:YIG films deposited on GGG(100) substrate. (d)-(f) L-MOKE loops of Ce:YIG films deposited on GGG(111) substrate. The vertical dotted lines of L-MOKE loops is to indicate the in-plane saturation field (H Sat). (g)-(i) P-MOKE (black color) and SQUID-VSM (blue color) loops of Ce:YIG films deposited on GGG(111) substrate. Thickness of the films is indicated in the respect ive frame. is found to be along out-of-plane direction for the films deposited on GGG(111) substrate. Development of PMA is further substantiated with magnetic microstructure of the films by capturing Kerr images. It may be noted that when the preferred (easy) direction of magnetization is perpendicular to the plane of the film, usually magnetic bubble domains are ex- pected to form23. Bubble domains are cylindrical mag- netic domains that may occur in a thin plate of a mag- netic material exhibiting PMA. Fig. 6 shows room tem- perature Kerr images of Ce:YIG10-111 and Ce:YIG66- 111 films captured in P-MOKE geometry. The samples are initially saturated by applying a sufficient negative field and the images are captured at the indicated pos- itive field, which is close to H Cand one can clearly see the bubble domain formation confirming the presence of PMA in the Ce:YIG films deposited on GGG(111) sub- strate. In order to further confirm the PMA, the FMR measurements are carried out on two films of each that are deposited on GGG(100) and GGG(111) substrates and the results are discussed as following. Fig. 7 shows the FMR derivative spectra of the indi- cated films measured asa function ofpolar angle, θH, be- tween the normal of the film plane and the applied field. FMR signal for 10 nm thick films was weak and hence is not shown. The resonance field values ( Hr) extracted from the FMR data as a function of θHis also shown FIG. 6. Room temperature Kerr micrograph of the indicated films captured in P-MOKE geometry. Scale bar is same for both the images./circledottext and/circlemultiplytext denote out-of-plane (OOP) and in to the plane respectively. /s32/s100/s97/s116/s97 /s32/s70/s105/s116 /s68 /s72 /s80/s80/s61/s49/s49/s46/s48/s32/s109/s84 FIG. 7. (a),(b) Ferromagnetic resonance (FMR) data of the indicated films. Inset of (b) shows the fitting of one represen - tative FMR data to extract resonance field (H r) and typical line width (∆ Hpp). IP and OOP denote in-plane and out-of- plane configuration respectively. (c),(d) show the variati on of Hrwith OOP angle ( θH), the solid line is guide to eye only. in Fig. 7. One can clearly see from the data that for thefilmsdepositedonGGG(111)substratetheresonance field (H r) values are the smallest in OOP configuration (θH=0◦) and the largest in IP configuration ( θH=90◦) whereas for films deposited on GGG(100) substrate, the trend is opposite. This indicates that easy axis magneti- zation is out of plane (i.e., PMA) for the films deposited onGGG(111)substrateandit isin-planeforthe filmsde- posited on GGG(100) substrate. The observed peak to peak lines width (∆ Hpp) forstudied samples arefound to be larger as compared to the pure YIG films24–26. This5 indicate in the studied Ce:YIG samples have higher mag- netic damping as also expected due to lattice dilation and different environments of the Fe cations due to the Ce doping9,27. It is also to be mentioned that spin orbit couplingpresent in the rareearthiron garnetgivesrise to larger magnetic damping than pure YIG. However, the magnetic damping in our Ce:YIG samples is found to be comparable or lesser than TmIG and other rare earth garnet those are showing PMA28,29. The above observations clearly indicate that the PMA is developed in films that are deposited on GGG(111) substrates, whereas the films deposited on GGG(100) substrate exhibit only in-plane magnetiza- tion. One can understand this by considering various magnetic anisotropy terms and their origin such as in- trinsic magneto-crystalline anisotropy (K MC), magneto- elastic (K ME), shape anisotropy (K Shape) and induced anisotropy as discussed in the following. In general in YIG and most of the other RIG systems, because of negative cubic anisotropy constant (K 1), easy axis of magnetization is always along /angbracketleft111/angbracketrightdirection30, which is an inherent magneto-crystalline anisotropy fac- tor (K MC). In addition to this one can also under- stand the easy axis of magnetization as /angbracketleft111/angbracketrightin terms of magneto-striction coefficient ( λ) i.e., magneto-elastic anisotropy (K ME). Depending upon λ, either compres- siveortensilestraindue tolattice mismatchbetween film and substrate can produce large uniaxial magneto-elastic anisotropy to overcome shape anisotropy for inducing PMA20,31,32. For example, considering YIG, the λvalues are given by λ111=-2.73×10−6andλ100=-1.25×10−6, which are usually small and negative30. However, larger in magnitude of λ111thanλ100make/angbracketleft111/angbracketrightdirection as a favorableeasy axis than /angbracketleft100/angbracketright, when a compressivestress is applied along the respective direction. Along these lines, it may be noted that from our XRD data, the pres- ence of in-plane tensile strain is argued for higher thick- ness films which require a negative λvalue to generate PMA in the present samples. However, it is reported in literature that the Ce substitution in YIG makes λfrom negativeto positivewith increasingCe concentration33,34 indicating that in addition to this magneto-elasticcontri- bution there could be other contribution for the observed TABLE II. M sis the saturation magnetization obtained from SQUID measurements. H Kand K Udenote out-of-plane (OOP) effective anisotropy field and anisotropy energy den- sity, respectively. K MEis the magneto elastic anisotropy density. H C(PMA)is the coercivity from the P-MOKE data (Fig. 5) , Sample 4 πMSHK KUKMEHC(PMA) (Gauss) (Oe) ( kJ/m3) (kJ/m3) (mT) Ce:YIG10-111 200 ±50 4200±200 3.4 - 0.8 Ce:YIG30-111 500 ±25 2500 ±50 5.0 0.50 3.6 Ce:YIG65-111 630 ±12 3130 ±40 7.8 0.82 14.5PMA in the present work. For this we estimated quan- titatively magneto-elastic anisotropy (K ME) as discussed in the following. The effective PMA field (H K) values can be calcu- lated using equation H Sat= HK-4πMS, where H Satis IP saturation field which is obtained from L-MOKE loops and M Sis the saturation magnetization deduced from SQUID-VSM M-H data (Fig. 5). The anisotropy K Uis related through the equation K U= (HKMS)/224. The observed value of K Uof the three studied PMA films are tabulated in the table-II. The K MEis calculated for the Ce:YIG30-111 and Ce:YIG65-111 films using equation K ME=(9/4)λ111C44 (π/2-β), whereC44isthe shearstiffness constantand βis the shear distortion angle. The β35can be deduced from IP and OOP lattice parameter values as obtained from x-ray diffraction data (Fig. 4), where the IP lattice pa- rameter of the film is considered to be equivalent to that of substrate due to coherent growth. The observed βare found to be 90◦(i.e., no distortion), 90.06◦and 90.1◦ for Ce:YIG10-111,Ce:YIG30-111 and Ce:YIG65-111, re- spectively. Substitution of thus obtained β,λ111=-2.73 ×10−6andC44=76.6 GPa results in K MEas 0.50±0.02, and 0.82±0.02,kJ/m3for Ce:YIG30-111 and CeYIG65- 111 films, respectively. We considered λ111andC44val- ues of YIG24,36in the above calculation mainly due to the unavailability of these values for Ce:YIG samples, but this calculation is expected to give an upper limit for the values of values of K ME. /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s56 /s45/s52 /s48 /s52 /s56 /s45/s52 /s45/s50 /s48 /s50 /s52 /s45/s52 /s45/s50 /s48 /s50 /s52 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s49/s48 /s48 /s49/s48 /s45/s56 /s45/s52 /s48 /s52 /s56 /s45/s50 /s45/s49 /s48 /s49 /s50 /s45/s56/s48 /s45/s52/s48 /s48 /s52/s48 /s56/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48 /s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s51/s48 /s45/s49/s53 /s48 /s49/s53 /s51/s48/s50/s48/s48/s75/s50/s52/s48/s75 /s51/s52/s48/s75 /s52/s48/s48/s75/s75/s101/s114/s114/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41 /s70/s105/s101/s108/s100/s40/s109/s84/s41 /s70/s105/s101/s108/s100/s40/s109/s84/s41/s70/s105/s101/s108/s100/s40/s109/s84/s41 /s70/s105/s101/s108/s100/s40/s109/s84/s41 FIG. 8. Temperature dependent P-MOKE data of Ce:YIG films deposited on GGG(111) substrate (top) 10 nm thickness (middle) 30 nm thickness (bottom) 66 nm thickness at the indicated temperatures. Clearly, experimentally observed OPP anisotropy K U (ref table-II) is one order of magnitude large than what is obtained from respective K ME. Even the magneto-6 elastic field H ME, defined as H ME=2KME/MSis not even 50% of the demagnetizing fields (4 πMS) of the respec- tive films and therefore K MEcannot be solely respon- sible for the observed PMA. An extra anisotropy term is therefore required to be introduced, which is often called growth induced anisotropy term (K GROWTH ) as reported by Soumah et al., in Bi substituted YIG films10. In view of the small value of K MC, KUis attributed to be sum of two contributions i.e., K MEand K GROWTH for Ce:YIG30-111 and Ce:YIG30-111 films whereas in case of Ce:YIG10-111 film, due to absence of strain, only KGROWTH term is expected to be responsible for the ob- served PMA. However the exact microscopic origin of KGROWTH and its dependence on factors such as compo- sition of the film, thickness, orientation etc., is beyond the scope of the present work. /s32/s67/s101/s58/s89/s73/s71/s51/s48/s45/s49/s49/s49 /s32/s67/s101/s58/s89/s73/s71/s54/s53/s45/s49/s49/s49 FIG. 9. Temperature variation of coercivity (H C) obtained from Fig. 8 of all the Ce:YIG films deposited on GGG(111) substrate. FIG. 10. Kerr micrograph of Ce:YIG films at the indicated temperatures captured in P-MOKE geometry. Scale bar is same for all the images. The stability of the developed PMA in Ce:YIG films deposited on GGG(111) is further studied with temper-ature. Temperature dependent P-MOKE measurements are carried out over a temperature range of 80 to 460 K and the data for all the three films is shown in Fig. 8. It is observed that the easy axis of magnetization remain perpendicular to the film plane irrespective of film thick- nesses much above the room temperature as shown in Fig. 8. The apparent curved shape of the loops is due to the Faraday contribution of the objective lens of the microscope at higher fields. This is a remarkable stabil- ity of the PMA in these kind of films as very few reports could show this kind of robust stability of PMA in garnet epitaxial thin films. For example, reorientationfrom out- of-plane to in-plane easy axis at about 173 K is reported in similar Ce:YIG epitaxial films, prepared at different growth conditions18. Thecoercivity(H C)isestimatedfromthedataofFig.8 forallthe filmsandthe variationofH Cisshownin Fig.9. The increase of H Cwith decreasing temperature is ob- served for all the films as expected. The temperature dependent Kerr images are captured near H Cduring the high temperature P-MOKE measurements and the data is shown in Fig. 10. For Ce:YIG65-111 film, the bubble nucleationpointand the sizeofthe the bubble isfound to be almost same irrespective of temperature. However for lower thickness film (Ce:YIG10-111) the size of magnetic bubble is found to increase with temperature. IV. CONCLUSIONS In conclusion, the present work reports the stabiliza- tion of perpendicular magnetic anisotropy (PMA) in cerium substituted yttrium iron garnet (Ce:YIG) epitax- ial thin films on GGG substrate by suitably optimizing the deposition conditions with pulsed laser deposition. The developed PMA is confirmed from bulk and surface sensitive magnetic measurements, which is further cor- roborated by ferromagnetic resonance and magnetic mi- crostructure. The observed stability of PMA over a wide range of temperatures makes the studied Ce:YIG films suitable for many practical applications. The cerium substitution in YIG seems to be a better choice for the insulating garnet films as the possibility of enhancing magneto-optical activity17and stabilization of PMA are demonstrated with suitably optimized deposition condi- tions. V. ACKNOWLEDGMENTS Dr.R.Venkatesh and Mr. Mohan Gangrade are thanked for the AFM measurements. MK thank Dr.Zaineb Hussain for the discussions and Mr.Rakhul Raj for the help during MOKE measurements. VRR thank Prof. Ajay Gupta for the discussions and encour- agement.7 VI. REFERENCES 1M. C. Onbasli, L. Beran, M. Zahradn´ ık, M. Kuˇ cera, R. Antoˇ s , J. Mistr´ ık, G. F. Dionne, M. Veis, and C. A. Ross, “Optical and magneto-optical behavior of cerium yttrium iron garnet thin filmsat wavelengths of 200–1770 nm,” Scientific Reports 6,23640 (2016). 2L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimer- ling, and C. Ross, “On-chip optical isolation in monolithic ally integrated non-reciprocal optical resonators,” Nature Ph otonics 5, 758 (2011). 3S. Shahrokhvand, A. Rozatian, M. Mozaffari, S. Hamidi, and M. Tehranchi, “Preparation and investigation of Ce:YIG thi n films with a high magneto-optical figure of merit,” Journal of Physics D: Applied Physics 45, 235001 (2012). 4T. Goto, Y. Eto, K. Kobayashi, Y. Haga, M. Inoue, and C. Ross, “Vacuum annealed cerium-substituted yttrium iron garnet fi lms on non-garnet substrates for integrated optical circuits, ” Journal of Applied Physics 113, 17A939 (2013). 5Y. Shoji and T. Mizumoto, “Magneto-optical non-reciprocal de- vices in silicon photonics,” Science and technology of adva nced materials (2014). 6B. J. Stadler and T. Mizumoto, “Integrated magneto-optical ma- terials and isolators: a review,” IEEE Photonics Journal 6, 1–15 (2013). 7Y. Zhang, Q. Du, C. Wang, W. Yan, L. Deng, J. Hu, C. A. Ross, and L. Bi, “Dysprosium substituted ce: Yig thin films wi th perpendicular magnetic anisotropy for silicon integrated optical isolator applications,” APL Materials 7, 081119 (2019). 8H. Kim, A. Grishin, and K. Rao, “Giant faraday rotation of blu e light in epitaxial CexY3−xFe5O12films grown by pulsed laser deposition,” Journal of Applied Physics 89, 4380–4383 (2001). 9A. Kehlberger, K. Richter, M. C. Onbasli, G. Jakob, D. H. Kim, T. Goto, C. A. Ross, G. G¨ otz, G. Reiss, T. Kuschel, et al., “Enhanced magneto-optic kerr effect and magnetic propertie s of CeY2Fe5O12epitaxial thin films,” Physical Review Applied 4, 014008 (2015). 10L. Soumah, N. Beaulieu, L. Qassym, C. Carr´ et´ ero, E. Jacque t, R. Lebourgeois, J. B. Youssef, P. Bortolotti, V. Cros, and A. Anane, “Ultra-low damping insulating magnetic thin films get perpendicular,” Nature communications 9, 1–6 (2018). 11C. O. Avci, “Current-induced magnetization control in insu lating ferrimagnetic garnets,” Journal of the Physical Society of Japan 90, 081007 (2021). 12H. Wu, L. Huang, C. Fang, B. Yang, C. Wan, G. Yu, J. Feng, H. Wei, and X. Han, “Magnon valve effect between two magnetic insulators,” Physical review letters 120, 097205 (2018). 13A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands , “Magnon spintronics,” Nature Physics 11, 453–461 (2015). 14K.-i. Uchida, J. Xiao, H. Adachi, J.-i. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, et al., “Spin see- beck insulator,” Nature materials 9, 894–897 (2010). 15O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C. Der an- lot, P. Bortolotti, et al., “Inverse spin hall effect in nanometer- thick yttrium iron garnet/pt system,” Applied Physics Lett ers 103, 082408 (2013). 16C. O. Avci, A. Quindeau, C.-F. Pai, M. Mann, L. Caretta, A. S. Tang, M. C. Onbasli, C. A. Ross, and G. S. Beach, “Current- induced switching in a magnetic insulator,” Nature materia ls16, 309–314 (2017). 17M. Kuila, U. Deshpande, R. Choudhary, P. Rajput, D. Phase, and V. Raghavendra Reddy, “Study of magneto-optical activi ty in cerium substituted yttrium iron garnet (ce: Yig) epitaxi al thin films,” Journal of Applied Physics 129, 093903 (2021). 18E. Lage, L. Beran, A. U. Quindeau, L. Ohnoutek, M. Kucera, R. Antos, S. R. Sani, G. F. Dionne, M. Veis, and C. A. Ross, “Temperature-dependent faraday rotation and magnetizati on re-orientation in cerium-substituted yttrium iron garnet thi n films,” APL Materials 5, 036104 (2017). 19B. K. Hazra, S. Kaul, S. Srinath, and M. M. Raja, “Uniaxial anisotropy, intrinsicand extrinsicdampinginco2fesiheu sleralloy thin films,” Journal of Physics D: Applied Physics 52, 325002 (2019). 20E. R. Rosenberg, L. Beran, C. O. Avci, C. Zeledon, B. Song, C. Gonzalez-Fuentes, J. Mendil, P. Gambardella, M. Veis, C. Garcia, et al., “Magnetism and spin transport in rare-earth- rich epitaxial terbium and europium iron garnet films,” Phys ical Review Materials 2, 094405 (2018). 21R. Sifat, J. C. Beam, and A. P. Grosvenor, “Investigation of factors that affect the oxidation state of ce in the garnet-ty pe structure,” Inorganic chemistry 58, 2299–2306 (2019). 22W. Kuch, R. Sch¨ afer, P. Fischer, and F. U. Hillebrecht, Magnetic microscopy of layered structures (Springer, 2015). 23A. Hubert and R. Sch¨ afer, Magnetic domains: the analysis of magnetic microstructures (Springer Science & Business Media, 2008). 24J. Ding, C. Liu, Y. Zhang, U. Erugu, Z. Quan, R. Yu, E. McCol- lum, S. Mo, S. Yang, H. Ding, et al., “Nanometer-thick yttrium iron garnet films with perpendicular anisotropy and low damp - ing,” Physical Review Applied 14, 014017 (2020). 25P. C. Van, S. Surabhi, V. Dongquoc, R. Kuchi, S.-G. Yoon, and J.-R. Jeong, “Effect of annealing temperature on surface morphology and ultralow ferromagnetic resonance linewidt h of yttrium iron garnet thin film grown by rf sputtering,” Applie d Surface Science 435, 377–383 (2018). 26C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. H¨ ubner, and E. Wendler, “Low damp- ing and microstructural perfection of sub-40nm-thin yttri um iron garnetfilms grownby liquidphase epitaxy,” Physical Review Ma- terials4, 024416 (2020). 27H.B. Vasili, B. Casals, R. Cichelero, F. Maci` a, J. Geshev, P .Gar- giani, M. Valvidares, J. Herrero-Martin, E. Pellegrin, J. F ontcu- berta,et al., “Direct observation of multivalent states and 4f →3d charge transfer in Ce-doped yttrium iron garnet thin films,” Physical Review B 96, 014433 (2017). 28C. Tang, P. Sellappan, Y. Liu, Y. Xu, J. E. Garay, and J. Shi, “Anomalous hall hysteresis in t m 3 f e 5 o 12/pt with strain- induced perpendicular magnetic anisotropy,” Physical Rev iew B 94, 140403 (2016). 29V. H. Ortiz, B. Arkook, J. Li, M. Aldosary, M. Biggerstaff, W. Yuan, C. Warren, Y. Kodera, J. E. Garay, I. Barsukov, et al., “First and second order magnetic anisotropy and damp- ing of europium iron garnet under high strain,” arXiv prepri nt arXiv:2111.15142 (2021). 30P. Hansen, “Anisotropy and magnetostriction of gallium- substituted yttrium iron garnet,” Journal of Applied Physi cs45, 3638–3642 (1974). 31A. J. Lee, A. S. Ahmed, B. A. McCullian, S. Guo, M. Zhu, S. Yu, P. M. Woodward, J. Hwang, P. C. Hammel, and F. Yang, “Inter- facial rashba-effect-induced anisotropy in nonmagnetic-m aterial– ferrimagnetic-insulator bilayers,” Physical Review Lett ers124, 257202 (2020). 32H. Wang, C. Du, P. C. Hammel, and F. Yang, “Strain-tunable magnetocrystalline anisotropy in epitaxial y 3 fe 5 o 12 thin films,” Physical Review B 89, 134404 (2014). 33R. Comstock and J. Raymond, “Magnetostriction of ytterbium and cerium in yig,” Journal of Applied Physics 38, 3737–3739 (1967). 34K. Belov, N. Volkova, V. Rajtsis, and A. Y. Chervonenkis, “Th e magnetostriction of the ce-substituted yttrium ferrite-g arnets,” Fizika Tverdogo Tela 14, 1850–1852 (1972). 35N. M. Vu, P. B. Meisenheimer, and J. T. Heron, “Tunable mag- netoelastic anisotropy in epitaxial (111) tm3fe5o12 thin fi lms,” Journal of Applied Physics 127, 153905 (2020). 36Y. A. Burenkov and S. Nikanorov, “Temperature effect on the elastic properties of yttrium garnet ferrite y 3 fe 5 o 12,” Ph ysics of the Solid State 44, 318–323 (2002).

2021-12-14

Cerium substituted yttrium iron garnet (Ce:YIG) epitaxial thin films are prepared on gadolinium gallium garnet (GGG) substrate with pulsed laser deposition (PLD). It is observed that the films grown on GGG(111) substrate exhibit perpendicular magnetic anisotropy (PMA) as compared to films grown on GGG(100) substrate. The developed PMA is confirmed from magneto-optical Kerr effect, bulk magnetization and ferromagnetic resonance measurements. Further, the magnetic bubble domains are observed in the films exhibiting PMA. The observations are explained in terms of the growth direction of Ce:YIG films and the interplay of various magnetic anisotropy terms. The observed PMA is found to be tunable with thickness of the film and a remarkable temperature stability of the PMA is observed in all the studied films of Ce:YIG deposited on GGG(111) substrate.

Robust perpendicular magnetic anisotropy in Ce substituted yttrium iron garnet epitaxial thin films

2112.07264v1

Spin current transport in hybrid Pt / multifunctional magnetoelectric Ga 0.6Fe1.4O3 bilayers Suvidyakumar Homkar,† Elodie Martin,‡ Benjamin Meunier,† Alberto Anadon -Barcelona ,‡ Corinne Bouillet,† Jon Gorchon ,‡ Karine Dumesnil ,‡ Christophe Lefèvre,† François Roulland ,† Olivier Copie ,‡ Daniele Preziosi ,† Sébastien Petit -Watelot ,‡ Juan -Carlos Rojas -Sánchez,*,‡ and Nathalie Viart *,† †Université de Strasbourg, CNRS, IPCMS, UMR 7504, F -67000 S trasbourg, France ‡Université de Lorraine, CNRS, IJL, F -54000 Nancy, France ABSTRACT The low power manipulation of magnetization is currently a highly sought -after objective in spintronics. Non ferromagnetic large spin -orbit coupling heavy metal (NM) / ferromagnet (FM) hetero structures offer interesting elements of response to this issue , by granting the manipulation of the FM magnetization by the NM spin Hall effect (SHE) generated spin current . Additional functionalities , such as the electric field control of the spin current generation , can be offered using multifunctional ferromagnets. We have studied the spin current transfer processes between Pt and the multifunctional magnetoelectric Ga 0.6Fe1.4O3 (GFO). In particular, via angular dependent magnetotransport measurements , we were able to differentiate between magnetic proximity effect (MPE) -induced anisotropic magnetoresistance (AMR) and spin Hall magnetoresistance (SMR). Our analysis shows that SMR is the dominant pheno menon at all temperatures and is the only one to be considered near room temperature, wi th a magnitude comparable to those observed in Pd/YIG or Pt/YIG heterostructures. These results indicate that magnetoelectric GFO thin films show promises for achieving an electric -field control of the spin current generation in NM/FM oxide -based heter ostructures . ABSTRACT GRAPHIC KEYWORDS: magnetic oxides thin films, gallium ferrite, platinum, spintronics, spin Hall magnetoresistance, magnetic proximity effects 1. INTRODUCTION Non ferromagnetic heavy metal (NM) / ferromagnet (FM) hetero structures are currently largely investigated to study viable routes to exploit the fascinating interplay between heat, charge and spin transport .1–3 In particular , recent developments point at a suitable low-power manipulation of magnetization in spintronic devices through spin-orbit torques (SOT) effects , which ground on the intrinsic spin-orbit coupling. Indeed , it has been shown th at the spin current generated upon the application of a charge current in the heavy metal nonmagnetic material via the spin Hall effect (SHE) can manipulate the magnetic moments in the adjacent ferromagnetic (FM) layer through SOT .4–10 This has triggered the development of a new Magnetic Random Access Memory (MRAM) -based technology, the spin -orbit torque MRAM (SOT -MRAM)8, and the possibility to integrate SHE -based spin -valves in standard CMOS -like networks has already been demonstrated via complementary spintronic logic (CSL) design methodology .11 The interest for NM/FM heterostructures has recently moved from all -metal systems to systems in which the FM is an insulating oxide (FMI), and display the interesting phenomenon of spin Hall magnetoresistance (SMR) .12–14 The most emblematic NM/FMI system is based on garnet ferrite, with Pt/YIG (Y 3Fe5O12).3,15 –22 However, spinel fe rrites have also been considered so far , with systems such as Pt/NiFe 2O4,3,18 Pt/Fe3O418 and Pt/CoFe 2O4.23–25 Recently, a study on Pt/Bi0.9La0.1FeO 3 bilayers26 made a step towards the exciting challenge of the electric field control of the spin Hall effect . This, by granting the control of the generation of spin current in engineered NM/FM -based devices, adds extra functionalities to the realization of future spintronics devices . Here we propose the use of another oxide, the multifunctional magnetoelectric gallium ferrite Ga2-xFexO3 (GFO x, 0.8 ≤x≤1.4),27,28,29 with a view to use its magnetoelectric character in the future for an electric field control of the spin current generation from NM/FM heterostructures. Ga2-xFexO3 crystallizes in the polar orthorhombic Pna2 1 (equivalently Pc2 1n) space group (S.G.#33) ,30,31 different from the usual perovskite structure adopted by most of the other magnetoelectric compounds, with a = 0.5086(2) nm, b = 0.8765(2) nm, and c = 0.9422(2) nm for x = 1.4 .32 The material is polar with a polarization of ca. 25 µC/cm2,33,34 and ferrimagnetic with a Curie temperature increasing with x and reaching values above room temperature for x=1.3 .28,35,36 Taking into consi deration the lattice matching possibilities, t hin films of GFO x (00l) can be epitaxially deposited on various substrates such as yttrium stabilized zirconia (001) (YSZ) ,37,38 YSZ (111),38 Pt (111) buffered YSZ (111)37 or strontium titanate SrTiO 3 (111) (STO ).38,39 For symmetry reasons, while the (100) YSZ substrates allow the growth of six in-plane variants, the YSZ (111), Pt (111) buffered YSZ (111) and STO (111) ones will only allow three , the lowest number observed until now. GFO1.4 films were shown to be ferroelectric at room temperature ,39 to have a magnetic Curie temperature of ca. 370 K , and a saturation magnetization of about 100 emu/cm3 at room temperature .37,40 2. MATERIAL S ELABORATION AND CHARACTERIZATION - EXPERIMENTAL DETAILS Here we show results obtained from Pt/GFO1.4 heterostructures which, for convenience, will be simply indicated as Pt/GFO. These heterostructures were deposited onto SrTiO 3 (111) (STO) substrates (Furuuchi Chemical Corporation, Japan, with rms -roughness lower than 0.15 nm) , by pulsed laser deposition using a KrF excimer laser (= 248 nm ) with a fluence of 4 J/cm² . The choice of STO as the substrate was dictated by the will to have the lowest numbe r of in -plane variants , and we have recently demonstrated the possibility to have a layer -by-layer growth of highly crystalline GFO on this substrate.40 The growth of such atomically flat GFO films is a prerequisite to high quality Pt/GFO interfaces. The GFO layer (ca. 30 nm thick) was deposited first, at 900°C, by ablati ng a sintered stoichiometric Ga 0.6Fe1.4O3 ceramic target at a repetition rate of 2 Hz of in a 0.1 mbar O 2 pressure, using an already optimized procedure desc ribed elsewhere .40 The overall composition of the film has been assessed by energy dispersive X-ray spectroscopy (EDX) coupled to a scanning electron microscopy technique (JEOL 6700 F). The analysis was performed at 5 keV, ensuring a large surface sensitivity of the EDX sign al. The Pt deposition wa s performed at a repetition rate of 10 Hz, under the system base pressure of 2 .10-8 mbar, and at room temperature to avoid any interdiffusion between the metal and oxide layers . The Pt thickness chosen here for generating spin curr ents was of 5 nm . This thickness has been evidenced by other works as optimum f or SHE -induced spin currents from its associated spin diffusion length (λ sd), spin memory loss (SML) at the interface41 and spin Hall torque efficiency per applied electric field unit .42 The surface of the STO//GFO/Pt heterostructure, observed by atomic force microscopy (AFM) with a Bruker ICON microscope operated in tapping mode, has a low rms roughness of 0.3 nm (Figure 1a). The roughness of a GFO layer of similar thickness (32 nm) deposited alone on a STO (111) substrate , without any Pt layer on top, had already been previously characterized, and is also of about 0.3 nm.40 Observations of the heterostructure cross section by transmission elect ron microscope (TEM) (JEOL 2100 F) allowed confirming the low roughness of the Pt/GFO interface . A composition profile was measured across the interface by energy dispersive X -ray spectroscopy (EDS) and showed that interdiffusion is limited in this system and restricted to a less than 2 nm wide zone (Figure 1b). X-ray diffraction was performed on the heterostructure with a Rigaku Smart Lab diffractometer equipped with a rotating anode (9 kW) and a monochromated copper radiation (1.54056 Å). Both GFO oxide a nd Pt meta l layers of the heterostructure are well crystallised (Figure 1c). GFO is oriented along its [001] direction and Pt, along its [ 111] direction, on the STO (111) . A layer -by-layer growth together with a smooth Pt/GFO interface are demonstrated by clear Laue oscillations for both GFO and Pt around their 004 and 111 reflections, respectively. Reflectivity measurements allowed us to determine th at th e precise thicknesses of the GFO and Pt layers are 36 and 5 nm, respe ctively. The magnetic properties of the heterostructure were studied with a superconducting quantum interference device vibrating sample magnetometer (SQUID VSM MPMS 3, Quantum Design). Temperature dependent measurements of the magnetization performed in f ield cooled and zero field cooled modes (not shown here) indicate a Curie temperature of 364 K. The sample indeed still shows a ferromagnetic behaviour at room temperature with a saturation magnetization of 100 emu/cm3, as expected for this Fe/Ga ratio.37 A highly anisotropic behaviour is evidenced from the in -plane and out -of-plane magnetization loops measurements (Figure 1d). The magnetization lies preferentially in -plane, in perfect agreement wit h the fact that the [100] and [001] crystallographic axes are, respectively, the easy and hard magnetic directions for GFO,28 and are observed from X -ray characterizations to lie, respectively, in - and out -of-plane. Figure 1. (a) TEM cross view of the STO//GFO(3 6 nm)/Pt(5 nm) heterostructure with (insert) the AFM image of its top surface indicating a rms roughness of 0.3 nm, (b) EDX mapping at the Pt/GFO interface, (c) X -ray diffractogram of the heterostructure in the θ -2θ mode, with a focus on the GFO 004 peak sh owing the Laue oscillations observed for both the GFO 004 and Pt layers 111 reflections, (d) Magnetization hysteresis loops of the heterostructure measured in both parallel and perpendicular modes at 300 K and (insert) 10 K. 3. MAGNETO TRANSPORT STUDY – RESULTS AND DISCUSSION The Pt layer was patterned, using standard optical lith ography, into double Hall bars to minimize the electrical resistance associated to the metallic contacts, with a longitudinal length L=38 µm, a width w=10 µm, and a thickness t=5 nm (Figure 2). The current is injected in the Pt bar along the STO[0 -11] () direction, that is , perpendicularly to the [100] easy magnetization direction of GFO. The plane of the films will be referred to as xy, and the normal to the film will be named z. Hz (Hx, Hy) refer s to the magnetic field applied along the z (x, y) direction. The electric current density Jc is applied along x, and t he longitudinal (transversal) resistivity ρxx (ρxy) is calculated from the measured voltage Vxx (Vxy). Figure 2. Schematics of the lithographed double Hall bars on STO//GFO(3 6 nm)/Pt(5 nm) heterostructures with indications of length (L), width (w) and thickness (t) . Magnetotransport in heavy metal ( NM) / insulating ferromagnet (FM I) heterostructures may involve contributions from two main origins: (1) the magnetic proximity effect (MPE) , and (2) the spin Hall related effects with the spin Hall magnetoresistance (SMR) . Figure 3 offers a schematic representation of the microscopic mec hanisms behind each contribution and for two different geometries of measurement, i.e. longitudinal and transverse. Going to details, MPE implies the existence of some induced interfacial magnetism in the non - magnetic Pt layer . As a result, anisotropic magnetoresistance (AMR) and anomalous Hall effects43 (AHE), ch aracteristic of metallic FM, can play a role. Instead, SMR contribution does not necessitate the existence of any MPE as already put forward by Nakayama et al.14 SMR effects arise as a combination of the spin Hall effect (SHE) and the inverse spin Hall effect (ISHE). When a charge current flows longitudinally in the nonmagnetic NM, a spin current is produced along the film normal direction by SHE. The spin polarization 𝜎⃗ of this spi n current is perpendicular to both charge and spin current densities, 𝐽𝑒⃗⃗⃗⃗ and 𝐽𝑠⃗⃗⃗, respectively, in agreement with 𝐽𝑠⃗⃗⃗=𝜃𝑆𝐻(𝜎⃗ × 𝐽𝑒⃗⃗⃗⃗ ), where 𝜃𝑆𝐻 is the spin Hall angle .44,45 The spin current can either be reflected or absorbed by the adjacent FM layer depending on whether 𝜎⃗ is parallel or perpendicular to the magnetization direction of the FM layer, respectively .18 The reflected spin current will produce an additional charge current through the inverse spin Hall effect (ISHE) which will lead to a decrease of the longitudinal resistivity. The resistivity of the NM layer will therefore strongly depend upon the orienta tion of the FM magnetization. If one neglects contributions from other phenomena such as the topological Hall effect46 (THE) or unidirectional magnetoresistance47,48 (UMR), very unlikely in this collinear -spins s ystem , the longitudinal ( 𝜌𝑥𝑥) and transvers e (𝜌𝑥𝑦) resist ivities of the studied heterostructure as a function of an external magnetic field can be described by the following equations :13,18 𝜌𝑥𝑥=𝜌0+ ∆𝜌𝑀𝑃𝐸 𝐴𝑀𝑅 ⋅ 𝑚𝑥2 + Δ𝜌||𝑆𝑀𝑅 ⋅ 𝑚𝑦2 (1) 𝜌𝑥𝑦=(∆𝜌𝑀𝑃𝐸 𝑃𝐻𝐸−Δ𝜌||𝑆𝑀𝑅)⋅𝑚𝑥⋅𝑚𝑦+ (Δ𝜌𝑀𝑃𝐸 𝐴𝐻𝐸 +Δ𝜌⊥ 𝑆𝑀𝑅) ⋅ 𝑚𝑧 (2) where 𝑚𝑥, 𝑚𝑦, and 𝑚𝑧 are the magnetization unit vector components in the x, y, and z directions, respectively , 𝜌0 is the resistivity of platinum in the absence of a magnetic field, ∆𝜌𝑀𝑃𝐸 𝐴𝑀𝑅 is the anisotropic magneto -resistivity due to the MPE induced anisotropic magnetoresistance (MPE AMR), and Δ𝜌𝑀𝑃𝐸 𝑃𝐻𝐸 (𝐴𝐻𝐸 ) is the planar ( anomalous ) Hall resistivity due to the MPE induced planar ( anomalous ) Hall effect in the NM. The AMR and AHE contributions, which depend on the orientation of the MPE -induced magnetic Pt , arise due to extrinsic spin -flip scattering mechanism and /or intrinsic mechanism depending on the band structure of Pt. In addition, one has to consider the SMR related phenome na. Δ𝜌||𝑆𝑀𝑅 represents the SMR effect and Δ𝜌⊥ 𝑆𝑀𝑅 is a Hall -effect -type resistivity that can be relevant in systems where the imaginary part of the spin mixing conductance is important, but it is usually smaller than Δ𝜌||𝑆𝑀𝑅.18 Figure 3. Schematics of the various physical phenomena which have to be considered for magnetotransport in a Pt/GFO hete rostructure , for both longitudinal , (a) and (b), and transverse , (c) and (d), modes involving the magnetic proximity effect s (MPE) with anisotropic magnetoresistance (AMR), anomalous Hall effect (AHE) and planar Hall effect (PHE), and the s pin Hall induced effects (SHE). Longitudinal magnetoresistance The longitudinal magnetoresistance (MR) is defined as 𝜌𝑥𝑥 (𝐻) − 𝜌𝑥𝑥 (𝐻=7𝑇) 𝜌𝑥𝑥 (𝐻=7𝑇) = ∆𝜌𝑥𝑥 𝜌𝑥𝑥(𝐻). The field dependence of the MR ( for a field applied along the z direction) is plotted in Figure 4a for various temperatures between 20 and 300 K . These curves are a clear evidence of the presence of non-zero longitudinal magnetoresistance (MR) . The MR decreases with increasing temperature, goes to zero at approximately 120 K, after what it keeps a negligible value (Figure 4b). This behavior of MR cannot be attributed to weak localization (WL) and weak anti - localization (WAL) mechanisms. If one cannot completely exclude the existence of a 2DEG at the STO/GFO interface, it will however not be possible to observe any of its transport properties in the Pt layer, where t he electric al contacts are made, because of the insulating character of the GFO layer between the STO and Pt layers. Moreover, the temperatures at which WL and WAL are at play are usually much lower than 120 K, since such quantum effects require that the coherence of the wave functions is kept. Possible origins behind the temperature evolution of the MR will be discussed later . The MR curves presented in Figure 4a do not saturate even at magnetic fields beyond the magnetic saturation of the heterostructure observed by SQUID (Figure 1d). This is probably caused by some independent moments at the interface.49 It can be related to the highly anisotropic nature of magnetism in the GFO films, which induces high anisotropy for the induced magnetic Pt as well, through a magnetic proximity effect , and will inhibit a saturation point even at high temperatures. The measured Pt resistivity value of 23 µ .cm at room temperature is in good agreement with values reported for Pt thin films deposited either by sputtering or molecular beam epitaxy.50,51 Its linear decrease with temperature is also in agreement with previous studies.52 Figure 4. Longitudinal magnetoresistance (as defined in the main text) . (a) Measurements at various temperatures , and (b) Temperature dependence of the longitudinal magnetoresistance estimated for H z= 0 T. Transverse magnetoresistance The magnetic field dependence of the transverse Hall resistivity 𝜌𝑥𝑦 measured at various temperatures between 20 and 300 K (the field is applied along the z direction) , is presented in Figure 5a , after correction from both the ordinary magneto -resistance (OMR) contribution due to the variation of the temperature and the ordinary Hall resistance (OHR) due t o the Lorentz force applied onto the carriers . For all temperatures, this corrected transverse resistivity 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 behaves as an odd function (opposed signs for opposed Hz fields). In the light of Equation (2), this means that it results from the c ontribution of the second term, (Δ𝜌𝑀𝑃𝐸 𝐴𝐻𝐸 +Δ𝜌⊥ 𝑆𝑀𝑅). The first one, associated to in -plane projections of magnetization, is not expected to reverse with reversing Hz fields. One can observe a sign reversal of the 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 signal with temperature on Figure 5a. A more precise estimation of the temperature at which this sign reversal operates can be done by plotting the temperature dependence of the values measured for 𝜌𝑥𝑦−𝑐𝑜𝑟𝑟 at 7 T (Figure 5b). The inversion tempera ture is of about 120 K, which is the same as the one at which the longitudinal resistivity 𝜌𝑥𝑥 goes to zero (Figure 4b). A sign inversion of the transverse resistivity has already been observed for Pt/YIG systems and assigned to AHE-induced by MPE in th e Pt.15,19,53 Similar temperature variations of the AHE were also observed in the absence of any FM layer, in ion -gated platinum thin films and analogously attributed to some induced ferromagnetic ordering on the Pt surface.16,54 The exp lanation given by Zhou et al.53 for such a sign inversion is that for paramagnetic Pt, both the density of states (DOS) and curvatur e near the Fermi surface importantly change with temperature. They indeed observed, for Pt/YIG heterostructures, a sign inversion of the ordinary Hall coefficient R0 of Pt with temperature, indicating a change of the carriers type. The raw resistivity curv es we measured , before OMR and OHR corrections, are shown in SI, Figure S1. The ordinary Hall coefficient of Pt, R 0, determined as the slope of the linear part of the measurements at high magnetic fields , is always negative and does not vary significantly with temperature. It allows determining a free electron density in Pt of 48 .1028 /m3 , in good agreement with values expected for a metal55 and already reported for other Pt thin films.15,16 The absence of any sign inversion of R0 in our case could originate from the fact that we have used a thicker Pt layer (5 nm) than the one used by Zhou et al. (1 nm).53 Shimizu et al. ,16 who have used 3.5 nm thick Pt layers , also observe a constant negative R 0. The modification of the DOS and curvature near the Fermi surface, induced by a MPE, is indeed expected to happen only at the interface between the NM and FM materials, and could be masked in our case, for which the bulk signature predominates . We cannot therefore be fully conclusive on the incidence of MPE -induced AHE on the temperature behaviour of transverse resistivity. In order to go further in the understanding of our system, we have sought to disentangle the MPE -based and SMR contributions in the longitudinal measurements . Figure 5. Transverse Hall resistivity measurements with (a) Transverse resistivity corrected from both the ordinary magnetoresistance and the linear contribution of the ordinary Hall effect (for a selection of temperatures) , (b) Temperature dependence of the transverse resistivity at Hz = 7 T . Angular dependence of the longitudinal resistivity One possible way to distinguish between the MPE -based AMR and SMR contributions is through the insertion of a nonmagnetic metal such as Cu or of an antiferromagnetic oxide such as NiO, which might eliminate the possibility of MPE effects , leaving only SMR effects to play a role . However, this introd uces additional interfacial issues and important modifications of the SMR , depending upon the interlayer thickness, have been reported in both Pt/Cu/Co/Pt56 and Pt/NiO/YIG57 heterostructures . An a lternative way to separate AMR and SMR contributions is by perfor ming angle -depend ent longitudinal measurements with th e magnetic field in either the xz or the yz planes, while the current density Jc and the measured resistivity ρxx are in the x direction49 as schematized in Figure 6 a. This can be understood from Equation 1: a rotation of the magnetic field within the yz (xz) plane will only have an effect on the SMR (AMR) and no effect on the AMR (SMR ) which only depends on Mx (My). As depicted in Figure 3, w hile the AMR results from the fact that M is parallel to the current direction, the SMR originates from the fact that M is parallel to the spins of the electrons of the SHE-generated spin current, which prevents the spins to be absorbed by the FM , and causes them to be reflected into Pt .14 We measured ρxx at H = 7 T while rotating the sample in the yz (xz) plane to study the () angle dependence of ρxx, at various temperatures. The orientation of the magnetic field is describe d through the α (within the xz plane) and (within the yz plane) angles. For the experiments, the α and angles were limited to a 90° rotation, and conventionally, the z direction was chosen as the 90° angle. Both 𝜌(𝛽)−𝜌𝑧 𝜌𝑧, and 𝜌(𝛼)−𝜌𝑧 𝜌𝑧 calculated quantities are shown in Figure 6b for various temperatures. The -dependent measurements show that the resistivity value decreases when going from Hz to Hy at all temperatures, with a 180° periodic oscillation. On the other side, t he α measurem ents also show that the resistivity value decreases when going from Hz to Hx but with a smaller change in resistivity, and no unambiguous periodic oscillation. By fitting the measurements with cos2𝛽 (SI, Figure S 2), the 𝜌𝑦 −𝜌𝑧 𝜌𝑧 SMR values can be extracted. The measurements could not unambiguously be fitted with cos2𝛼, and the 𝜌𝑥 −𝜌𝑧 𝜌𝑧 MPE AMR values are extracted from the 𝜌(𝛼)−𝜌𝑧 𝜌𝑧 measurements at = 0°. This procedure is comforted by the fact that the values of 𝜌𝑦 −𝜌𝑧 𝜌𝑧 (SMR ), extracted from the measurements , have the same values as 𝜌(𝛽)−𝜌𝑧 𝜌𝑧 for = 0°. Hence, assuming that the measurements are periodic as well , we extracted the 𝜌𝑥 −𝜌𝑧 𝜌𝑧 (AMR) values from 𝜌(𝛼)−𝜌𝑧 𝜌𝑧 measurements at = 0°. The SMR and AMR values obtained as explained are plotted in Figure 6c. Both contributions globally decrease with increasing temperatures. SMR shows a minimum at about 120 K, which is also the temperature at which AMR goes to zero. For all temperatures, the SMR contribution dominates over the AMR one, and it is the only one present above 120 K. The predominance of the SMR mechanism over the entire temperature range has also been observed for Pt/YIG and Pd/YIG samples. The SMR value measured for the Pt/GFO heterostru ctures is about 2 .10-4 at 300 K and 4.5 .10-4 at 20 K. These values are similar to what is observed in Pd/YIG heterostructures, and only slightly smaller than the ones observed in Pt/YIG (4 .10-4 at 300 K and 6 .10-4 at 20 K).49 If the AMR contribution we observe is very similar to the one reported in other works19,49,58,59: same positive sign, similar amplitude of ca. 10-4, and a decrease with increasing temperature wh ich leaves it practica lly insignificant after ab out 1 00 K, we however highlight some differences stemming from the temperature dependence of the SMR contribution. Studies performed under a high magnetic field, such as the present one of Pt/GFO performed at 7 T or the study of Pt/YIG under 100 kOe,49 show V -shaped SMR curves, with first a decrease and then an increase with the increasing temperature, the position of the minimum varying between 20 and 120 K from one system to another. The measurements performed in lower magnetic fields (10 kOe)53,59 globally show an inversed tendency, with first an increase and then a decrease, with a maximum at about 100 K. Figure 6. Identification of the S MR and AMR contributions to the longitudinal effects, (a) Geometries of the measurements, (b) Angular measurements at various temperatures (selection of temperatures) in both geometries, (c) Temperature dependence of SMR an d AMR as deduced from the ρ(β)−𝜌𝑧 𝜌𝑧 and ρ(α)−𝜌𝑧 𝜌𝑧 values measured at zero degree angle for β and α, respectively. The magnetic state of the magnetic material is probably to be considered to explain such differences. Recently, a theoretical study has shown that the orbital hybridization of the magnetic material plays a role in the magnetoresistance, most probably in relation with spin - orbit coupling.60 The minimum in SMR we observe here could thus be put in perspective with the modifi cation of the spin -orbit coupling observed in bare GFO thin films near 120 K via our XMCD study .61 In fact, since both SMR and MPE -induced m agnetoresistive contributions stem from the interaction between the charge current flowing in the Pt layer and the magnetic properties of the FM, the modification of the spin-orbit coupling at 120 K for GFO could contribute to the observed temperature vari ation s of both longitudinal and transverse magnetoresistance measurements in the 90-140 K temperature range (Figures 4b and 5b ). 4. CONCLUSION The multifunctional magnetoelectric GFO oxide has successfully been introduced in FM I/NM (with Pt as the NM) heterostructures of high crystalline quality. The Pt/ GFO interface is sharp and t his makes the heterostructure suitable for spin currents transparency. The interactions between the spin Hall current from Pt and the GFO magnetic orientation have been eviden ced by magnetotransport measurements. SMR has been shown to be the dominant phenomenon at all temperatures , and it is the only one to be considered near room temperature, wi th a magnitude comparable to those observed in the classically studied Pd/YIG or Pt/YIG heterostructures. This study therefore validates the use of GFO as a multifunctional magnetoelectric material in NM/ FMI heterostructures with a view to control their spin current generation by an electric field. SUPPORTING INFORMATION Transverse Hall resistivity measurements, Fitting of the longitudinal angular dependent measurements AUTHOR INFORMATION Corresponding Authors *E-mail: juan -carlos.rojas -sanchez@univ -lorraine.fr (J.-C.R.-S.) *E-mail: viart@unistra.fr (N.V.) ACKNOWLEDGEMENTS This work was funded by the French National Research Agency (ANR) through the ANR -18- CECE24 -0008 -01 ‘ANR MISSION’ and, within the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, it was supported by IdEx Unistra (ANR 10 IDEX 0002), and by SFRI STRAT’US project (ANR 20 SFRI 0012) and ANR -11-LABX -0058_NIE and ANR -17-EURE -0024 under the framework of the French Investments for the Future Program. The authors wish to thank D. Troadec (IEMN, Lille, France) and A. -M. Blanchenet (UMET, Lille, France) for the preparation of the TEM FIB lamellae, as well as the XRD , MEB -CRO, and TEM platforms of the IPCMS. We acknowledge partial support from the French PIA project “Lorraine Université d’Excellence ”, reference ANR - 15IDEX -04-LUE. Devices in the present study were patterned at MiNaLor clean -room platform which is parti ally supported by FEDER and Grand Est Region through the RaNGE project. REFERENCES (1) Slachter, A.; Bakker, F. L.; Adam, J. -P.; van Wees, B. J. Thermally Driven Spin Injection from a Ferromagnet into a Non -Magnetic Metal. Nat. Phys. 2010 , 6 (11), 879 –882. https://doi.org/10.1038/nphys1767. (2) Ando, K.; Takahashi, S.; Ieda, J.; Kajiwara, Y .; Nakayama, H.; Yoshino, T.; Harii, K.; Fujikawa, Y.; Matsuo, M.; Maekawa, S.; Saitoh, E. Inverse Spin -Hall Effect Induced by Spin Pumping in Metallic System. J. Appl. Phys. 2011 , 109 (10), 103913. https://doi.org/10.1063/1.3587173. (3) Meier, D.; Reinha rdt, D.; van Straaten, M.; Klewe, C.; Althammer, M.; Schreier, M.; Goennenwein, S. T. B.; Gupta, A.; Schmid, M.; Back, C. H.; Schmalhorst, J. -M.; Kuschel, T.; Reiss, G. Longitudinal Spin Seebeck Effect Contribution in Transverse Spin Seebeck Effect Experim ents in Pt/YIG and Pt/NFO. Nat. Commun. 2015 , 6 (1), 8211. https://doi.org/10.1038/ncomms9211. (4) Kajiwara, Y.; Harii, K.; Takahashi, S.; Ohe, J.; Uchida, K.; Mizuguchi, M.; Umezawa, H.; Kawai, H.; Ando, K.; Takanashi, K.; Maekawa, S.; Saitoh, E. Transmi ssion of Electrical Signals by Spin - Wave Interconversion in a Magnetic Insulator. Nature 2010 , 464 (7286), 262 -U141. https://doi.org/10.1038/nature08876. (5) Liu, L.; Lee, O. J.; Gudmundsen, T. J.; Ralph, D. C.; Buhrman, R. A. Current -Induced Switching of Perpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect. Phys. Rev. Lett. 2012 , 109 (9), 096602 –096602. (6) Liu, L.; Pai, C. -F.; Li, Y.; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin -Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 2012 , 336, 555. (7) Manchon, A.; Železný, J.; Miron, I. M.; Jungwirth, T.; Sinova, J.; Thiaville, A.; Garello, K.; Gambardella , P. Current -Induced Spin -Orbit Torques in Ferromagnetic and Antiferromagnetic Systems. Rev. Mod. Phys. 2019 , 91 (3), 035004. https://doi.org/10.1103/RevModPhys.91.035004. (8) Mihai Miron, I.; Garello, K.; Gaudin, G.; Zermatten, P. -J.; Costache, M. V.; Au ffret, S.; Bandiera, S.; Rodmacq, B.; Schuhl, A.; Gambardella, P. Perpendicular Switching of a Single Ferromagnetic Layer Induced by In -Plane Current Injection. Nature 2011 , 476 (7359), 189 -U88. https://doi.org/10.1038/nature10309. (9) Rojas -Sánchez, J. -C.; Laczkowski, P.; Sampaio, J.; Collin, S.; Bouzehouane, K.; Reyren, N.; Jaffrès, H.; Mougin, A.; George, J. -M. Perpendicular Magne tization Reversal in Pt/[Co/Ni] 3/Al Multilayers via the Spin Hall Effect of Pt. Appl. Phys. Lett. 2016 , 108 (8), 082406. (10) Pham, T. H.; Je, S. -G.; Vallobra, P.; Fache, T.; Lacour, D.; Malinowski, G.; Cyrille, M. C.; Gaudin, G.; Boulle, O.; Hehn, M.; Rojas -Sánchez, J. -C.; Mangin, S. Thermal Contribution to the Spin - Orbit Torque in Metallic -Ferrimagnetic Systems. Phys. Rev. A ppl. 2018 , 9 (6), 064032. https://doi.org/10.1103/PhysRevApplied.9.064032. (11) Kang, W.; Wang, Z.; Zhang, Y.; Klein, J. -O.; Lv, W.; Zhao, W. Spintronic Logic Design Methodology Based on Spin Hall Effect -Driven Magnetic Tunnel Junctions. J. Phys. -Appl. P hys. 2016 , 49 (6), 65008 –65008. (12) Chen, Y. -T.; Takahashi, S.; Nakayama, H.; Althammer, M.; Goennenwein, S. T. B.; Saitoh, E.; Bauer, G. E. W. Theory of Spin Hall Magnetoresistance. Phys. Rev. B 2013 , 87 (14). https://doi.org/10.1103/PhysRevB.87.144411. (13) Chen, Y. -T.; Takahashi, S.; Nakayama, H.; Althammer, M.; Goennenwein, S. T. B.; Saitoh, E.; Bauer, G. E. W. Theory of Spin Hall Magnetoresistance (SMR) and Related Phenomena. J. Phys. Condens. Matter 2016 , 28 (10), 103004. https://doi.org/10.1088/09 53-8984/28/10/103004. (14) Nakayama, H.; Althammer, M.; Chen, Y. T.; Uchida, K.; Kajiwara, Y.; Kikuchi, D.; Ohtani, T.; Gepraegs, S.; Opel, M.; Takahashi, S.; Gross, R.; Bauer, G. E. W.; Goennenwein, S. T. B.; Saitoh, E. Spin Hall Magnetoresistance Induce d by a Nonequilibrium Proximity Effect. Phys. Rev. Lett. 2013 , 110 (20), 206601. https://doi.org/10.1103/PhysRevLett.110.206601. (15) Huang, S. Y.; Fan, X.; Qu, D.; Chen, Y. P.; Wang, W. G.; Wu, J.; Chen, T. Y.; Xiao, J. Q.; Chien, C. L. Transport Magneti c Proximity Effects in Platinum. Phys. Rev. Lett. 2012 , 109 (10), 107204. https://doi.org/10.1103/PhysRevLett.109.107204. (16) Shimizu, S.; Takahashi, K. S.; Hatano, T.; Kawasaki, M.; Tokura, Y.; Iwasa, Y. Electrically Tunable Anomalous Hall Effect in Pt Thin Films. Phys. Rev. Lett. 2013 , 111 (21), 216803. https://doi.org/10.1103/PhysRevLett.111.216803. (17) Vlietstra, N.; Shan, J.; Castel, V.; van Wees, B. J.; Ben Youssef, J. Spin -Hall Magnetoresistance in Platinum on Yttrium Iron Garnet: Dependence on P latinum Thickness and in -Plane/out -of- Plane Magnetization. Phys. Rev. B 2013 , 87 (18), 184421. https://doi.org/10.1103/PhysRevB.87.184421. (18) Althammer, M.; Meyer, S.; Nakayama, H.; Schreier, M.; Altmannshofer, S.; Weiler, M.; Huebl, H.; Geprägs, S.; Op el, M.; Gross, R.; Meier, D.; Klewe, C.; Kuschel, T.; Schmalhorst, J. -M.; Reiss, G.; Shen, L.; Gupta, A.; Chen, Y. -T.; Bauer, G. E. W.; Saitoh, E.; Goennenwein, S. T. B. Quantitative Study of the Spin Hall Magnetoresistance in Ferromagnetic Insulator/Norma l Metal Hybrids. Phys. Rev. B 2013 , 87 (22), 224401. https://doi.org/10.1103/PhysRevB.87.224401. (19) Zhou, X.; Ma, L.; Shi, Z.; Fan, W. J.; Zheng, J. -G.; Evans, R. F. L.; Zhou, S. M. Magnetotransport in Metal/Insulating -Ferromagnet Heterostructures: Spin Hall Magnetoresistance or Magnetic Proximity Effect. Phys. Rev. B 2015 , 92 (6), 060402. https://doi.org/10.1103/PhysRevB.92.060402. (20) Evelt, M.; Safranski, C.; Aldosary, M.; Demidov, V. E.; Barsukov, I.; Nosov, A. P.; Rinkevich, A. B.; Sobotkiewich, K.; Li, X.; Shi, J.; Krivorotov, I. N.; Demokritov, S. O. Spin Hall -Induced Auto - Oscillations in Ultrathin YIG Grown on Pt. Sci. Rep. 2018 , 8 (1), 1269. https://doi.org/10.1038/s41598 -018-19606 -5. (21) Miao, B. F.; Huang, S. Y.; Qu, D.; Chien, C. L. Physical Origins of the New Magnetoresistance in Pt/YIG. Phys. Rev. Lett. 2014 , 112 (23), 236601. https://doi.org/10.1103/PhysRevLett.112.236601. (22) Vélez, S.; Golovach, V. N.; Bedoya -Pinto, A.; Isasa, M.; Sagasta, E.; Abadia, M.; Rogero, C.; Hueso, L. E.; Bergeret, F. S.; Casanova, F. Hanle Magnetoresistance in Thin Metal Films with Strong Spin -Orbit Coupling. Phys. Rev. Lett. 2016 , 116 (1), 01 6603. https://doi.org/10.1103/PhysRevLett.116.016603. (23) Amamou, W.; Pinchuk, I. V.; Trout, A. H.; Williams, R. E. A.; Antolin, N.; Goad, A.; O’Hara, D. J.; Ahmed, A. S.; Windl, W.; McComb, D. W.; Kawakami, R. K. Magnetic Proximity Effect in Pt/CoFe 2O4 Bilayers. Phys. Rev. Mater. 2018 , 2 (1), 011401. https://doi.org/10.1103/PhysRevMaterials.2.011401. (24) Isasa, M.; Bedoya -Pinto, A.; Velez, S.; Golmar, F.; Sanchez, F.; Hueso, L. E.; Fontcuberta, J.; Casanova, F. Spin Hall Magnetoresistance at Pt/CoFe 2O4 Interfaces and Texture Effects. Appl. Phys. Lett. 2014 , 105 (14), 142402. https://doi.org/10.1063/1.4897544. (25) Vasili, H. B.; Gamino, M.; Gàzquez, J.; Sánchez, F.; Valvidares, M.; Gargiani, P.; Pellegrin, E.; Fontcuberta, J. Magnetoresistance in Hybrid Pt/CoFe 2O4 Bilayers Controlled by Competing Spin Accumulation and Interfacial Chemical Reconstruction. ACS Appl. M ater. Interfaces 2018 , 10 (14), 12031 –12041. https://doi.org/10.1021/acsami.8b00384. (26) Gao, R.; Fu, C.; Cai, W.; Chen, G.; Deng, X.; Zhang, H.; Sun, J.; Shen, B. Electric Control of the Hall Effect in Pt/Bi 0.9La 0.1FeO 3 Bilayers. Sci. Rep. 2016 , 6 (1), 20330. https://doi.org/10.1038/srep20330. (27) Roy, A.; Mukherjee, S.; Gupta, R.; Prasad, R.; Garg, A. Structure and Properties of Magnetoelectric Gallium Ferrite: A Brief Review. Ferroelectrics 2014 , 473 (1), 154 –170. https://doi.org/10.1080/0015019 3.2014.923704. (28) Arima, T.; Higashiyama, D.; Kaneko, Y.; He, J. P.; Goto, T.; Miyasaka, S.; Kimura, T.; Oikawa, K.; Kamiyama, T.; Kumai, R.; Tokura, Y. Structural and Magnetoelectric Properties of Ga 2-xFexO3 Single Crystals Grown by a Floating -Zone Met hod. Phys. Rev. B 2004 , 70 (6), 064426. (29) Rado, G. T. Mechanism of Magnetoelectric Effect in an Antiferromagnet. Phys. Rev. Lett. 1961 , 6 (11), 609 -. (30) Abrahams, S. C.; Reddy, J. M.; Bernstein, J. L. Crystal Structure of Piezoelectric Ferromagnetic Gallium Iron Oxide. J. Chem. Phys. 1965 , 42 (11), 3957 –3968. (31) Bertaut, E. F.; Bassi, G.; Buisson, G.; Chappert, J.; Delapalme, A.; Pauthenet, R.; Rebouillet, H. P.; Aleonard, R. Etude Par Effet Mössbauer, Rayons X, Diffraction Neutronique et Mesures Magnétiques de Fe 1.15Ga 0.85O3. J. Phys. 1966 , 27, 433. (32) Lefevre, C.; Roulland, F.; Thomasson, A.; Meny, C.; Porcher, F.; Andre, G.; Viart, N. Magnetic and Polar Properties’ Optimization in the Magnetoelectric Ga 2-xFexO3 Compounds. J. Phys. Chem. C 2013 , 117 (28), 14832 –14839. https://doi.org/10.1021/jp403733b. (33) Stoeffler, D. First Principles Study of the Electric Polarization and of Its Switching in the Multiferroic GaFeO 3 System. J. Phys. -Condens. Matter 2012 , 24 (18), 185502. (34) Kundys, B. ; Roulland, F.; Lefevre, C.; Meny, C.; Thomasson, A.; Viart, N. Room Temperature Polarization in the Ferrimagnetic Ga 2-xFexO3 Ceramics. J. Eur. Ceram. Soc. 2015 , 35 (8), 2277 – 2281. (35) Remeika, J. P. GaFeO 3: A Ferromagnetic -Piezoelectric Compound. J. Ap pl. Phys. 1960 , 31 (5), 263S. (36) Roulland, F.; Lefevre, C.; Thomasson, A.; Viart, N. Study of Ga(2 -x)FexO3 Solid Solution: Optimisation of the Ceramic Processing. J. Eur. Ceram. Soc. 2013 , 33 (5), 1029 –1035. https://doi.org/10.1016/j.jeurceramsoc.2012.1 1.014. (37) Trassin, M.; Viart, N.; Versini, G.; Barre, S.; Pourroy, G.; Lee, J.; Jo, W.; Dumesnil, K.; Dufour, C.; Robert, S. Room Temperature Ferrimagnetic Thin Films of the Magnetoelectric Ga 2-xFexO3. J. Mater. Chem. 2009 , 19 (46), 8876 –8880. https://d oi.org/10.1039/b913359c. (38) Song, S.; Jang, H. M.; Lee, N. -S.; Son, J. Y.; Gupta, R.; Garg, A.; Ratanapreechachai, J.; Scott, J. F. Ferroelectric Polarization Switching with a Remarkably High Activation Energy in Orthorhombic GaFeO 3 Thin Films. NPG Asia Mater 2016 , 8, e242. https://doi.org/10.1038/am.2016.3. (39) Thomasson, A.; Cherifi, S.; Lefevre, C.; Roulland, F.; Gautier, B.; Albertini, D.; Meny, C.; Viart, N. Room Temperature Multiferroicity in Ga 0.6Fe1.4O3:Mg Thin Films. J. Appl. Phys. 2013 , 113 (21), 214101. https://doi.org/10.1063/1.4808349. (40) Homkar, S.; Preziosi, D.; Devaux, X.; Bouillet, C.; Nordlander, J.; Trassin, M.; Roulland, F.; Lefèvre, C.; Versini, G.; Barre, S.; Leuvrey, C.; Lenertz, M.; Fiebig, M.; Pourroy, G.; Viart, N. Ultrathin Regime Growth of Atomically Flat Multiferroic Gallium Ferrite Films with Perpendicular Magnetic Anisotropy. Phys. Rev. Mater. 2019 , 3 (12), 124416. https://doi.org/10.1103/PhysRevMaterials.3.124416. (41) Rojas -Sánchez, J. -C.; Reyren, N.; Laczkowski, P.; Savero, W.; Attané, J. -P.; Deranlot, C.; Jamet, M.; George, J. -M.; Vila, L.; Jaffrès, H. Spin Pumping and Inverse Spin Hall Effect in Platinum: The Essential Role of Spin -Memory Loss at Metallic Interfaces. Phys. Rev. Lett. 2014 , 112 (10), 106602. (42) Nguyen, M. -H.; Ralph, D. C.; Buhrman, R. A. Spin Torque Study of the Spin Hall Conductivity and Spin Diffusion Length in Platinum Thin Films with Varying Resistivity. Phys. Rev. Lett. 2016 , 116 (12), 126601. https://doi.org/10.1103/PhysRevLett.116.126601. (43) Nagaosa, N.; Sinova, J.; Onoda, S.; MacDonald, A. H.; Ong, N. P. Anomalous Hall Effect. Rev. Mod. Phys. 2010 , 82 (2), 1539 –1592. https://doi.org/10.1103/RevModPhys.82.1539. (44) Hoffmann, A. Spin Hall Effects in Metals. IEEE Trans. Magn. 2013 , 49 (10), 5172 –5193. https://doi.org/10.1109/TMAG.2013.2262947. (45) Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C. H.; Jungwirth, T. Spin Hall Effects. Rev. Mod. Phys. 2015 , 87 (4), 1213 –1259. https://doi.org/10.1103/RevModPhys.87.1213. (46) Zeiss ler, K.; Finizio, S.; Shahbazi, K.; Massey, J.; Ma’Mari, F. A.; Bracher, D. M.; Kleibert, A.; Rosamond, M. C.; Linfield, E. H.; Moore, T. A.; Raabe, J.; Burnell, G.; Marrows, C. H. Discrete Hall Resistivity Contribution from Néel Skyrmions in Multilayer Na nodiscs. Nat. Nanotechnol. 2018 , 13 (12), 1161 –1166. https://doi.org/10.1038/s41565 -018-0268 -y. (47) Avci, C. O.; Garello, K.; Ghosh, A.; Gabureac, M.; Alvarado, S. F.; Gambardella, P. Unidirectional Spin Hall Magnetoresistance in Ferromagnet/Normal Metal Bilayers. Nat. Phys. 2015 , 11 (7), 570–575. https://doi.org/10.1038/nphys3356. (48) Guillet, T.; Zucchetti, C.; Barbedienne, Q.; Marty, A.; Isella, G.; Cagnon, L.; Vergnaud, C.; Jaffrès, H.; Reyren, N.; George, J. -M.; Fert, A.; Jamet, M. Observation of L arge Unidirectional Rashba Magnetoresistance in Ge(111). Phys. Rev. Lett. 2020 , 124 (2), 027201. https://doi.org/10.1103/PhysRevLett.124.027201. (49) Lin, T. Magnetic Insulator Thin Films and Induced Magneto -Transport Effect at Normal Metal / Magnetic Ins ulato r Interface, UC Riverside, 2013. https://escholarship.org/uc/item/8r22h4rt . (50) Guillemard, C.; Petit -Watelot, S.; Andrieu, S.; Rojas -Sánchez, J. -C. Charge -Spin Current Conversion in High Quality Epitaxial Fe/Pt Systems: Isotropic Spin Hall Angle a long Different in - Plane Crystalline Directions. Appl. Phys. Lett. 2018 , 113 (26), 262404. https://doi.org/10.1063/1.5079236. (51) Sagasta, E.; Omori, Y.; Isasa, M.; Gradhand, M.; Hueso, L. E.; Niimi, Y.; Otani, Y.; Casanova, F. Tuning the Spin Hall Effect of Pt from the Moderately Dirty to the Superclean Regime. Phys. Rev. B 2016 , 94 (6), 060412. https://doi.org/10.1103/PhysRevB.94.060412. (52) Vila, L.; Kimura, T.; Otani, Y. Evolution of the Spin Hall Effect in Pt Nanowires: Size and Temperature Effects. Phys. Rev. Lett. 2007 , 99 (22), 226604. https://doi.org/10.1103/PhysRevLett.99.226604. (53) Zhou, X.; Ma, L.; Shi, Z.; Guo, G. Y.; Hu, J.; Wu, R. Q.; Zhou, S. M. Tuning Magnetotransport in PdPt/Y3Fe5O12: Effects of Magnetic Proximity and Spin -Orbit Coupling. Appl. Phys. Lett. 2014 , 105 (1), 012408. https://doi.org/10.1063/1.4890239. (54) Min, K. -J.; Chae, D. -H.; Kim, D.; Jeon, J.; Kim, T.; Joo, S. Electrically Controlled Magnetoresistance in Ion -Gated Platinum Thin Films. J. Korean Phys. Soc. 2019 , 75 (5), 398 – 403. https://doi.org/10.3938/jkps.75.398. (55) Ashcroft, N.; Mermin, N. Solid State Ph ysics , New édition.; Brooks/Cole: New York, 1976. (56) Anadón, A.; Guerrero, R.; Jover -Galtier, J. A.; Gudín, A.; Díez Toledano, J. M.; Olleros - Rodríguez, P.; Miranda, R.; Camarero, J.; Perna, P. Spin -Orbit Torque from the Introduction of Cu Interlayers i n Pt/Cu/Co/Pt Nanolayered Structures for Spintronic Devices. ACS Appl. Nano Mater. 2021 , 4 (1), 487 –492. https://doi.org/10.1021/acsanm.0c02808. (57) Luan, Z. Z.; Chang, F. F.; Wang, P.; Zhou, L. F.; Cooper, J. F. K.; Kinane, C. J.; Langridge, S.; Cai, J. W.; Du, J.; Zhu, T.; Wu, D. Interfacial Coupling and Negative Spin Hall Magnetoresistance in Pt/NiO/YIG. Appl. Phys. Lett. 2018 , 113 (7), 072406. https://doi.org/10.1063/1.5041865. (58) Lin, T.; Tang, C.; Shi, J. Induced Magneto -Transport Properties at P alladium/Yttrium Iron Garnet Interface. Appl. Phys. Lett. 2013 , 103 (13), 132407. https://doi.org/10.1063/1.4822267. (59) Lin, T.; Tang, C.; Alyahayaei, H. M.; Shi, J. Experimental Investigation of the Nature of the Magnetoresistance Effects in Pd -YIG Hyb rid Structures. Phys. Rev. Lett. 2014 , 113 (3), 037203. https://doi.org/10.1103/PhysRevLett.113.037203. (60) Ko, H. -W.; Park, H. -J.; Go, G.; Oh, J. H.; Kim, K. -W.; Lee, K. -J. Role of Orbital Hybridization in Anisotropic Magnetoresistance. Phys. Rev. B 2020 , 101 (18), 184413. https://doi.org/10.1103/PhysRevB.101.184413. (61) Meunier, B.; Homkar, S.; Lefèvre, C.; Roulland, F.; Pourroy, G.; Salluzzo, M.; Preziosi, D.; Viart, Nathalie. Spin Reorientation Transition in Multiferroic Gallium Thin Films Eviden ced by Temperature Dependent X -Ray Magnetic Dichroism Experiments. Manuscript under preparation .

2021-12-20

The low power manipulation of magnetization is currently a highly sought-after objective in spintronics. Non ferromagnetic large spin-orbit coupling heavy metal (NM) / ferromagnet (FM) heterostructures offer interesting elements of response to this issue, by granting the manipulation of the FM magnetization by the NM spin Hall effect (SHE) generated spin current. Additional functionalities, such as the electric field control of the spin current generation, can be offered using multifunctional ferromagnets. We have studied the spin current transfer processes between Pt and the multifunctional magnetoelectric Ga0.6Fe1.4O3 (GFO). In particular, via angular dependent magnetotransport measurements, we were able to differentiate between magnetic proximity effect (MPE)-induced anisotropic magnetoresistance (AMR) and spin Hall magnetoresistance (SMR). Our analysis shows that SMR is the dominant phenomenon at all temperatures and is the only one to be considered near room temperature, with a magnitude comparable to those observed in Pd/YIG or Pt/YIG heterostructures. These results indicate that magnetoelectric GFO thin films show promises for achieving an electric-field control of the spin current generation in NM/FM oxide-based heterostructures.

Spin current transport in hybrid Pt / multifunctional magnetoelectric Ga0.6Fe1.4O3 bilayers

2112.10406v1

arXiv:1604.03272v2 [cond-mat.mtrl-sci] 22 Apr 2016The effect of inserted NiO layer on spin-Hall magnetoresista nce in Pt/NiO/YIG heterostrucures T. Shang,1,∗H. L. Yang,1Q. F. Zhan,1,†Z. H. Zuo,1Y. L. Xie,1L. P. Liu,1S. L. Zhang,1Y. Zhang,1H. H. Li,1B. M. Wang,1Y. H. Wu,2S. Zhang,3,‡and Run-Wei Li1,§ 1Key Laboratory of Magnetic Materials and Devices &Zhejiang Province Key Laboratory of Magnetic Materials and Application Techn ology, Ningbo Institute of Material Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, China 2Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3 117 583, Singapore 3Department of Physics, University of Arizona, Tucson, Ariz ona 85721, USA (Dated: August 25, 2021) We investigate the spin-current transport through antifer romagnetic insulator (AFMI) by means of the spin-Hall magnetoressitance (SMR) over a wide temper ature range in Pt/NiO/Y 3Fe5O12 (Pt/NiO/YIG) heterostructures. By inserting the AFMI NiO l ayer, the SMR dramatically decreases bydecreasing thetemperature down totheantiferromagneti cally ordered state ofNiO, which implies that the AFM order prevents rather than promotes the spin-cu rrent transport. On the other hand, the magnetic proximity effect (MPE) on induced Pt moments by Y IG, which entangles with the spin-Hall effect (SHE) in Pt, can be efficiently screened, and p ure SMR can be derived by insertion of NiO. The dual roles of the NiO insertion including efficient ly blocking the MPE and transporting the spin current from Pt to YIG are outstanding compared with other antiferromagnetic (AFM) metal or nonmagnetic metal (NM). Spin current, the motion of spin angular moment, has attracted intense interest due to the prospects of low consumption spintronic devices1,2. Several experimen- tal techniques have been developed to generate and ma- nipulate spin current, e.g., spin pumping3–5, spin See- beck effect6–8, and also spin-Hall effect (SHE)9–11. Re- cently, the generation and propagation of spin current in antiferromagnets (AFMs) including insulators or met- als have been extensively investigated by various tech- niques12–22. Especially, the thermally injected or dy- namically pumped spin current from ferromagnetic(FM) YIG layer can flow into the NiO or CoO AFM insulator (AFMI) layer and reach the Pt or Ta nonmagnetic metal (NM) layer where it can be converted into charge current by means of inverse spin-Hall effect (ISHE)18–22. By in- serting thin AFMI layer, the ISHE voltage is largely en- hancedandexhibitsnonmonotonictemperatureorAFMI thickness dependence, which reaches a maximum near the N´ eel temperature of AFMI or with the AFMI thick- ness of∼1-2 nm, respectively18–22. Several theoretical models have been proposed for the propagation of injected spin current through AFMs in NM/AFMs/FM heterostructures23–26. These models de- scribe the spin-current transport and its enhancement by assuming that the AFMs are ordered at room tem- peratures, while the AFM ordering temperatures of thin AFMs are well below the room temperature23–26. How the spin current interacts with AFM order is still beyond understood. In most of these experimental or theoretical investigations, the spin currentcarried by spin waves, are producedin YIG layerand flowinto theAFMI layer18–26. While there is another type of spin current, which is car- ried by conduction electrons via SHE in strong spin-orbit coupling (SOC) NM9–11. However, the investigationsof spin-current transport from SOC NM into AFMI are rare, which can help us to further understand the mech- anism of spin-current propagation in AFMs. Herein, we investigated the spin-current transport in Pt/NiO/YIG heterostructures by injecting the spin current from top Pt layer. Our results show that when approaching the antiferromagnetically ordered state of NiO, the spin- Hall magnetoresistance (SMR) amplitude is largely sup- pressed, implying that the AFM order prevents rather than promotes the spin-current transport. The Pt/NiO/YIG heterostructures were prepared in a combined ultra-high vacuum (10−9Torr) pulse laser deposition (PLD) and sputter system. The high-quality YIG films were epitaxially deposited on (111)-orientated Gd3Ga5O12(GGG) substrates via PLD technique. The thin NiO films were deposited on YIG films after the growth of YIG film. The top Pt layers were sputtered in anin situprocess. In this study, the thicknesses of YIG and Pt films are fixed at 60 nm and 3 nm, respectively, while the NiO thickness ranges from 0 to 8 nm. Figure 1(a) plot representative room-temperature x-ray diffrac- tion (XRD) patterns for epitaxial YIG/GGG film near the (444) reflections. Clear Laue oscillations indicate the flatness and uniformity of the epitaxial films. As shown in the insets of Fig. 1(b), no indication of impurities or misorientation was detected in the range of 10 to 80 de- gree. The atomic force microscope surface topography of Pt/NiO(3)/YIG heterostructure over an area of 3 µm ×3µm in Fig. 1(c) reveals a root-mean-square surface roughness of 0.14 nm (The number in the brackets rep- resents the thickness of NiO layer in nm unit), indicating atomically flat of prepared films, with the other films showing similar surface topology. A representative ferro- magnetic resonance (FMR) derivative absorption spec-/s48/s46/s54/s110/s109 /s48/s46/s48/s110/s109 /s50/s46/s53 /s109 /s32 /s50/s46/s48 /s49/s46/s53 /s49/s46/s48 /s48/s46/s53 /s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53 /s109 /s32/s32/s80/s116/s47/s78/s105/s79/s47/s89/s73/s71/s40/s99/s41 /s50/s48/s54/s48 /s50/s48/s56/s48 /s50/s49/s48/s48 /s50/s49/s50/s48 /s50/s49/s52/s48 /s50/s49/s54/s48/s100/s73 /s70/s77/s82/s47/s100/s72/s32/s40/s97/s46/s117/s46/s41 /s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s72 /s32/s126/s32/s56/s79/s101/s40/s100/s41/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s40/s52/s52/s52/s41 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s40/s100/s101/s103/s114/s101/s101 /s41/s40/s50/s50/s50/s41/s40/s98/s41 /s52/s57/s46/s53 /s53/s48/s46/s48 /s53/s48/s46/s53 /s53/s49/s46/s48 /s53/s49/s46/s53 /s53/s50/s46/s48 /s53/s50/s46/s53/s40/s97/s41 /s71/s71/s71 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s40/s100/s101/s103/s114/s101/s101 /s41/s89 /s73/s71 FIG. 1. (Color online) (a) A representative 2 θ-ωXRD pat- terns for YIG/GGG film near the (444) reflections. (b) The XRD patterns from 20 to 80 degree. (c) Three dimensional plot of the atomic force microscope surface topography for Pt/NiO(3)/YIG heterostructure over an area of 3 µm×3 µm. (d) A representative FMR derivative absorption spec- trum of YIG film with. FIG. 2. Anisotropic magnetoresistance at various tempera- tureswiththemagnetic fieldvariedwithin xy(top-rowpanel), xz(middle-row panel), and yz(bottom-row panel) planes. The right panel shows the schematic plots of longitudinal an d transverse resistance measurements. The magnetic fields ar e applied with angles θxy,θxz, andθyzrelative to the y-,z-, andz-axes. The electric current is applied along the x-axis. The AMR for Pt/YIG are shown in (a)-(c), while the results of Pt/NiO(1)/YIG are shown in (d)-(f). trum of YIG film (60 nm) shown in Fig. 1(d) exhibits a line width ∆H of 8 Oe, which was measured at radio frequency 9.39 GHz and power 0.1 mW with an in-plane magnetic field at room temperature. All the above prop- erties indicate the excellent quality of prepared films. The recent proposed SMR built on the combined SHE and ISHE was applied to investigate the spin-current transport from SOC NM into AFMI in Pt/NiO/YIGheterostructures15,27–30. As shown in the right panel of Fig. 2, all the films were patterned into Hall-bar ge- ometry by using mask to measure the longitudinal and transverse Hall resistance. The anisotropic magnetore- sistance (AMR) were measured in a magnetic field of 20 kOe for both Pt/NiO/YIG and Pt/NiO/MgO het- erostructures. The absence of AMR in Pt/NiO/Mgo implies that the NiO moments are robust against such magnetic field. Figures 2(a)-(c) plots the AMR for Pt/YIG at various temperatures, while the results of Pt/NiO(1)/YIG are shown in Figs. 2(d)-(f). Both the Pt/YIG and Pt/NiO(1)/YIG heterostructures demon- strate clear SMR, with the amplitudes reaching 6.1 × 10−4and 4.5×10−4at room temperature, respectively [see Figs. 2(c) and (f)], implying that the spin current generated by SHE in Pt can transport through the NiO to interact with the YIG. Apparently, the spin current can pass through NiO from both sides, i.e., Pt →NiO→ YIG or YIG →NiO→Pt18,19,22. For Pt/YIG, as shown in Fig. 2(b), the magnetic proximity effect (MPE) in- duced conventional AMR (CAMR) always coexists with SMR and its maximum amplitude of 2.2 ×10−4is com- parable to the SMR. However, for Pt/NiO(1)/YIG, there is no clear CAMR even down to lowest temperature [see Fig. 2(e)], and the observed AMR are almost attributed to the SMR. As shown in Figs. 2(d)(f), the θxyandθyz scansexhibit similarbehaviors, with theiramplitudes be- ingalmostidenticaltoeachother. TheinsertedNiOlayer between Pt and YIG efficiently suppresses the MPE at the interface, which also can be revealed by anomalous- Hall resistance (AHR). For instance, the AHR at 70 kOe for Pt/NiO(1)/YIG is 4.6 mΩ at 5 K, which is 6 times smaller than the Pt/YIG (27.6 mΩ). As further increas- ing the NiO thickness, the AHR is negligible for tNiO≥ 3 nm. Compared to the Pt/IrMn/YIG, Pt/Cu/YIG, or Pt/Au/YIG heterostructures, only 26% of the SMR is lost by insertion of NiO, while over 80% of the SMR is suppressed by insertion of IrMn, Cu, or Au30–32. Thus, the dual roles of the inserted NiO are outstanding, which efficiently blocks the MPE and transports the spin cur- rent. All the AMR amplitudes are summarized in Fig. 3 as a function of temperature. For Pt/YIG, the SMR exhibits nonmonotonic temperature dependence and ac- quires its maximum value of 6.9 ×10−4around 150 K. While for θxyscan, where both CAMR and SMR con- tribute the total AMR, the amplitude is almost identi- cal to the difference between SMR and CAMR [see star symbols in Fig. 3(a)], i.e., |∆ρ|/ρ0(θxy) =|∆ρ|/ρ0(θyz) -|∆ρ|/ρ0(θxz), indicating that the MPE induced CAMR at Pt/YIG interface competes with the SMR. However, after inserting NiO, the SMR exhibits significantly dif- ferent behaviors. As shown in Fig. 3(b), the SMR captures the AMR for θxyscan due to the absence of MPE, whose amplitude is almost identical to the θyz scan, i.e., |∆ρ|/ρ0(θxy) =|∆ρ|/ρ0(θyz). Similar results were also reported previously in MPE-free Rh/YIG bi- layers33. Moreover, the SMR undergoes a sharp decrease 2/s48/s51/s48/s54/s48/s57/s48 /s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s48/s52/s48/s47 /s48/s40/s49/s48/s45/s53 /s41 /s120 /s121/s32 /s121/s122/s45/s32 /s120 /s122/s32 /s120 /s122/s32 /s121/s122/s40/s97/s41/s32/s80/s116/s47/s89/s73/s71 /s84/s32/s40/s75/s41/s40/s98/s41/s32/s80/s116/s47/s78/s105/s79/s47/s89/s73/s71/s47 /s48/s40/s49/s48/s45/s53 /s41 /s84 /s98 FIG. 3. Temperature dependence of the AMR amplitudes for (a) Pt/YIG and (b) Pt/NiO(1)/YIG heterostructures. The cubic, circle, and triangle symbols stand for θxy,θxz, andθyz scans, respectively. The star symbols represent the differe nce between SMR and CAMR ( θyz-θxz). The solid lines are guides to the eyes. as decreasing the temperature and changes its sign be- low 70 K [see Figs. 2(d)(f)], with its amplitude being temperature independent for T <70 K. According to magnetization results, the appearance of the exchange bias field around the blocking temperature Tb∼70 K in Pt/NiO(1)/YIG suggests that the moments in NiO layer become antiferromagnetically ordered when approaching this temperature(see detailsin Fig. 4). Since the signre- versal happens simultaneously around the blocking tem- perature, it is likely associated with the interactions be- tweenthe antiferromgneticallyalignedNiO moments and spin current. For Pt/NiO(1)/YIG, the SMR amplitude is almost 10 times smaller for T < T bthan the room- temperature value. Most of the spin current are blocked by NiO beforereachingthe NiO/YIG interfacewhen NiO moments are ordered. Since the AFM transition temperature for very thin AFMs is expected to be well below the room tempera- ture, we also measured the field dependence of magne- tization for Pt/NiO/YIG at various temperatures, from which the AFM transition temperature can be roughly trackedby blockingtemperature from exchangebias field HE. As shownin Fig. 4(a), thenormalizedmagnetichys- teresis loops M/Msfor Pt/NiO(1)/YIG at various tem- peratures are presented, with the other heterostructures showing similar behaviors. The derived HEas a function of temperature are summarized in Fig. 4(b), from which theTbare approximately determined to be around 70 K, 110 K, and 150 K for Pt/NiO(1)/YIG, Pt/NiO(3)/YIG, and Pt/NiO(5)/YIG, respectively. Similar blocking tem- peratures were previously reported in Co/NiO/(Co/Pt) heterostructures34. Moreover, as shown in the inset of/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s48/s48/s45/s50/s48/s45/s52/s48/s45/s54/s48/s45/s56/s48/s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s49/s110/s109/s78/s105/s79 /s51/s110/s109 /s53/s110/s109/s72 /s69/s32/s40/s79/s101/s41 /s84/s32/s40/s75/s41/s40/s98/s41 /s72 /s99/s32/s40/s79/s101/s41 /s84/s32/s40/s75/s41 /s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s77/s47/s77 /s83 /s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s50/s75 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s49/s48/s48 /s51/s48/s48/s80/s116/s47/s78/s105/s79/s40/s49/s110/s109/s41/s47/s89 /s73/s71/s40/s97/s41 FIG. 4. (a) Field dependence of normalized magnetization M/Msfor Pt/NiO(1)/YIG at various temperatures. (b) The in-plane exchange bias field HEfor Pt/NiO/YIG with differ- ent NiO thicknesses versus temperature. The arrows indicat e the AFM block temperatures Tb. The inset plots the coerciv- ity field HCfor Pt/NiO(1)/YIG. Fig. 4(b), the coercivity HCof Pt/Ni(1)/YIG also ex- hibitsastep-likeincreasenearthe Tbduetotheenhanced magnetic exchangecoupling between NiO and YIG layer. Different from the spin pumping or spin Seebeck tech- niques, where the spin current are produced at the AFM/FM interface by the precessing magnetization or thermal gradient in the YIG layer and flow through the AFMI layer18–22, the direction of spin-current transport is reversal in our experiments. According to the SMR results, the spin current generated in Pt via SHE can transport through the NiO to interact with the YIG. However, the SMR amplitude is strongly suppressed be- low the blocking temperature of NiO. Based on the SMR model, the reflected spin current interact with NiO mo- ments again before converting into charge current in Pt, which is more complicated than spin pumping or spin Seebeckprocess. Severaltheoreticalmodelsincludingthe coherent magnetization dynamics and incoherent ther- mal magnons have been proposed to explain the spin- current transport in AFMI and its enhancement by in- sertion of AFMI23–26. In the magnetic dynamics model, the processed magnetization in YIG layer exerts a torque on the AFM moments and promotes the magnetization dynamics propagation to the AFM/FM interface23–25. The other model describe the spin-current transport by considering the diffusion of incoherent thermal magnons which is caused by the accumulation of magnons at the AFM/FM interface26. According to these theoretical calculations, the coupling of the spin excitations at the NiO/YIG interface is much larger than at Pt/NiO or at Pt/YIG, and the spin mixing conductance of AFM/FM interface also depends on the AFMI layer thickness and linearlyscaleswiththeenhancementofspincurrent,both of which can explains the initial increase of spin current transport by increasing the AFMI thickness22–26. In our case, the spin current is generated via SHE in Pt layer and is independent of YIG/NiO interface. However, the 3enhanced coupling of spin excitation at the NiO/YIG in- terface can efficiently enhance the spin pumping or spin Seebeck signals23–26. So far, all these theoretical models assume the AFMI are ordered at room temperature23–26, where all the simulations have been done, but the order- ing temperatures of thin AFMs are well below the room temperature. Further theoretical models which include the temperature degree of freedom are highly desirable to understand the spin-current transport in AFMs. In summary, we investigated the spin-current trans- port properties by means of SMR in Pt/NiO/YIG het- erostructures over a wide temperature range. Different from Pt/YIG, the SMR in Pt/NiO/YIG is significantly suppressedwhen the temperatureapproachesthe antifer- romagnetically ordered state of NiO and becomes tem-perature independent below the blocking temperature. However, the dual roles of the NiO insertion includ- ing efficiently screening the MPE at Pt/YIG interface and transporting the spin current from Pt to YIG lay- ers are outstanding. These experimental results extend the knowledge for the further theoretical investigations on the spin-current propagation in AFMs. WethankthehighmagneticfieldlaboratoryofChinese Academy of Sciences for the FMR measurements. This work is financially supported by the National Natural Science foundation of China (Grants No. 11274321, No. 11404349, No. 51502314, No. 51522105, No. 11374312) and the Key Research Program of the Chinese Academy of Sciences (Grant No. KJZD-EW-M05). S. Zhang was partially supported by the U. S. National Science Foun- dation (Grant No. ECCS-1404542). ∗Present address: Swiss Light Source & Laboratory for Sci- entific Developments and Novel Materials, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland †zhanqf@nimte.ac.cn ‡zhangshu@email.arizona.edu §runweili@nimte.ac.cn 1S. Maekwa, Spin Current (Oxford University press, Ox- ford, 2012). 2M. Z. Wu and A. Hoffmann, Recent Advances in Magnetic Insulators - From Spintronics to Microwave Applications (Academic Press, San Diego, Vol 64, 2013). 3B.Heinrich, C.Burrowes, E.Montoya, B.Kardasz, E.Girt, Young-Yeal Song, Y. Sun, and M. Z. Wu, Phys. Rev. Lett. 107, 066604 (2011). 4S. M. Rezende, R. L. Rodr´ ıguez-Su´ arez, M. M. Soares, L. H. Vilela-Leao, D. Ley Dom´ ınguez, and A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013). 5Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 6K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 7K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010). 8G. E.W.Bauer, E. Saitoh, andB. J.vanWees, Nat.Mater. 11, 391 (2012). 9J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 10J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 11Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). 12J. B. S. Mendes, R. O. Cunha, O. Alves Santos, P. R. T. Ribeiro, F. L. A. Machado, R. L. Rodr´ ıguez-Su´ arez, A. Azevedo, and S. M. Rezende, Phys. Rev B 89, 140406(R) (2014). 13W. Zhang, M. B. Jungfleisch, W. Jiang, J. E. Pearson, A. Hoffmann, F. Freimuth, and Y. Mokrousov, Phys. Rev. Lett.113, 196602 (2014).14L. Frangou, S. Oyarz´ un, S. Auffret, L. Vila, S. Gambarelli, and V. Baltz, Phys. Rev. Lett. 116, 077203 (2016). 15J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015) and reference therein. 16X. Zhou, L. Ma, Z. Shi, W. J. Fan, J. G. Zheng, R. F. L. Evans, and S. M. Zhou, Phys. Rev. B 92, 060402(R) (2015). 17S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi, M. Nakamura, Y.Kaneko, M. Kawasaki, andY.Tokura, Phys. Rev. Lett. 115, 266601 (2015). 18C. Hahn, G. de. Loubens, O. Klein, M. Viret, V. V. Nale- tov, and J. B. Youssef, Europhys. Lett. 108, 57005 (2014). 19H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 113, 097202 (2014). 20H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 91, 220410(R) (2015). 21Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. NDiaye, A. Tan, K. Uchida, K. Sato, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, arXiv: 1505.03926. 22W. Lin, K. Chen, S. Zhang, and C. L. Chien, arXiv: 1603.00931. 23R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett.113, 057601 (2014). 24S.Takei, T. Moriyama, T. Ono, andY.Tserkovnyak, Phys. Rev. B92, 020409(R) (2015). 25R. Khymyn, I. Lisenkov, V. S. Tiberkevich, A. N. Slavin, B. A. Ivanov, arXiv: 1511.05785. 26S. M. Rezende, R. L. Rodr´ ıguez-Su´ arez, and A. Azevedo, Phys. Rev. B 93, 014425 (2016). 27H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen- wein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 28Y. T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). 29M. Isasa, A. B. Pinto, S. V´ elez, F. Golmar, F. S´ anchez, L. E. Hueso, J. Fontcuberta, and F. Casanova, Appl. Phys. Lett.105, 142402 (2014). 30Y. T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, J. 4Phys. Condens. Matter 28, 103004 (2016) and reference therein. 31T. Shang, H. L. Yang, Q. F. Zhan, Z. H. Zuo, Y. L. Xie, L. P. Liu, S. L. Zhang, Y. Zhang, H. H. Li, B. M. Wang, Y. H. Wu, S. Zhang, Run-Wei Li, arXiv: 1603.03578. 32M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. M. Schmal-horst, G. Reiss, L. M. Shen, A. Gupta, Y. T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B87, 224401 (2013). 33T. Shang, Q. F. Zhan, H. L. Yang, Z. H. Zuo, Y. L. Xie, H. H. Li, L. P. Liu, B. M. Wang, Y. H. Wu, S. Zhang, and Run-Wei Li, Sci. Rep. 5, 17734 (2015). 34A. Baruth and S. Adenwalla, Phys. Rev. B 78, 174407 (2008). 5

2016-04-12

We investigate the spin-current transport through antiferromagnetic insulator (AFMI) by means of the spin-Hall magnetoressitance (SMR) over a wide temperature range in Pt/NiO/Y$_3$Fe$_5$O$_{12}$ (Pt/NiO/YIG) heterostructures. By inserting the AFMI NiO layer, the SMR dramatically decreases by decreasing the temperature down to the antiferromagnetically ordered state of NiO, which implies that the AFM order prevents rather than promotes the spin-current transport. On the other hand, the magnetic proximity effect (MPE) on induced Pt moments by YIG, which entangles with the spin-Hall effect (SHE) in Pt, can be efficiently screened, and pure SMR can be derived by insertion of NiO. The dual roles of the NiO insertion including efficiently blocking the MPE and transporting the spin current from Pt to YIG are outstanding compared with other antiferromagnetic (AFM) metal or nonmagnetic metal (NM).

The effect of inserted NiO layer on spin-Hall magnetoresistance in Pt/NiO/YIG heterostructures

1604.03272v2

arXiv:1911.03708v1 [cond-mat.soft] 9 Nov 2019The magnonic superfluid droplet at room temperature Yu. M. Bunkov(a),∗A.Farhutdinov(b), A. N. Kuzmichev(a), T. R. Safin(a,b), P. M. Vetoshko(a), V. I. Belotelov(a), and M. S. Tagirov(b,c) aRussian Quantum Center, Skolkovo, 143025 Moscow, Russia bKazan Federal University, 420008, Kazan, Russia cInstitute of Applied Research of Tatarstan Academy of Scien ces, 420111, Kazan, Russia (Dated: November 12, 2019) We declare the observation of spin superfluid state in Yttriu m Iron Garnet (YIG) at room temper- ature. It is similar to a Homogeneous Precessing State (HPD) , observed earlier in antiferromagnetic superfluid3He-B. The formation of this state explains bythe repulsive i nteraction between magnons, which is required as a prior condition for the spin superfluid ity. It establishes an energy gap, which stabilizes the long range superfluid transport of magnetiza tion and determines the Ginzburg-Landau coherence length. This discovery paves a way to many quantum applications of supermagnonics at room temperature, such as magnetic Josephson effect, long di stance spin transport, Q-bit, quantum logics, magnetic sensors and others. PACS numbers: 67.57.Fg, 05.30.Jp, 11.00.Lm Keywords: Supermagnonics, spin supercurrent, magnon BEC, YIG film The coherent quantum state of matter - the super- fluid state was discovered in 1938 by P. L. Kapitza in liquid4He at temperature of 2 K [1]. The first theoreti- cal explanation of this state was done by F. London. He suggested that the superfluidity could have some connec- tion with Bose-Einstein condensation (BEC) [2]. In BEC theory by Einstein the coherent state of noninteracting particles was considered [3]. Recently it was successfully applied for the weakly interacted deluted gas of atoms at extremely low temperatures [4, 5]. But it can not be di- rectly applied for the system with a strong inter-particle interaction, like4He liquid. In 1941 L. D. Landau sug- gested that superfluidity can be understood in terms of atomic states, modified by interaction. The phenomeno- logical theory by L. D. Landau successfully explained the superfluid properties [6]. The depletion of condensate in 4He is very strong: in the limit of zero temperature only about 10% of particles occupy the state with zero mo- mentum. Nevertheless, BEC still remains the key mech- anism for the phenomenon of superfluidity in liquid4He: due to BEC the whole liquid (100% of4He atoms) forms a coherent quantum state at T= 0 and participates in the non-dissipative superfluid flow. Nowadays the super- fluid state is well known as a quantum state, governed by a single wave function. Indeed, the superfluid state may exist without BEC, like in BerezinskiiKosterlitzThouless transition in 2D materials. And opposite, BEC may ex- ist, but does not lead to a superfluid transport. It takes place when there is no energy gap and kinetic energy de- stroys the coherent state. In this case the critical current is equal to zero. The magnetically ordered materials are described by the ground state and a gas of bosonic excitations which are represented by magnons. At a condition of ther- mal equilibrium the density of magnons is always be- low the critical density, required for the Bose condensa- tion. Indeed, the density of magnons can be increasedup to about Avagadro number by a magnetic resonance methods. Usually the spin-spin interaction time is much shorter than the spin-lattice relaxation time and magnon gas may exist at a quasi equilibrium state. The magnon gasisa veryinterestingobjecttostudy the Bose-Einstein condensation (BEC) and formation of a spin superfluid states. Its properties are strongy depends from the type of spin-orbit interactions, which result in attractive or repulsive interaction between magnons. At the first case the formation of magnon BEC should be unstable. In- deed, the recent experiments with magnon BEC forma- tion shows a very interesting new dynamic effects at this case. [7, 8]. Contrary, at the repulsive interac- tion, the magnon BEC should be stable [9]. Further- more, at higher density of magnons the collective spin modes emerge. It is described by a common wave func- tion and shows the properties comparable with the su- perfluid component of4He liquid. It exhibit the long dis- tance spin supercurrent transport which is characterized by a Ginzburg-Lanadau coherence length. The first magnonic superfluid state was discovered in 1984 by A. S. Borovik-Romanov and Yu. M. Bunkov ex- perimental group and described theoretically by I. Fomin as a new state of magnetically ordered matter [10, 11]. It was observed as the spontaneously self-organized phase- coherent precession of spins in an antiferromagnetic su- perfluid3He-B. This state fulfills all criteria of coher- ence, suggested later by Snoke [12]. It is radically differ- ent from the conventional ordered states in magnets. It emergesonthebackgroundoftheorderedmagneticstate, andcanbedescribedintermsofthecondensationofmag- netic excitations to a superfluid coherent quantum state [13–15]. The spin superfluid state exhibit many phe- nomena analogues to other superfluid states, including longdistancespin supercurrentin the channelandphase- slippageatacriticalspinsupercurrent[16–18], Josephson spin-current effect [19, 20] as well as spin-current vortices2 [21,22]. TheGoldstonecollectiveexcitationsofthis state (the analog of the second sound in4He), were also ob- served[23,24]. Thestate, whichisveryneartoamagnon BEC state, was observed in a conditions of spatial trap. In this case the BEC signal may ring for about tenth of minutes at a frequency of 1 MHz [25, 26]. This new state can be described in a terms of a self trapping model, suggested earlier for the formation of elementary parti- cles [27, 28]. The spin superfluid state was also observed in3He-A in the conditions, when spin-orbit interaction was changed from attractive to repulsive [29–31]. It was also resently observed in other superfluid state -3He-P [32]. The discovery of spin superfluidity in3He has been recognized in 2004 by the Low Temperature community by a prestigious F. London prize [33, 34]. It is very important to note, that there is no any principal difference between the magnon gas in super- fluid3He and in solid magnetically ordered materials. The magnetic ordering is a quantum phenomenon. The model of spatially fixed magnetic moments, usually ap- plied at some theoretical considerations, describes only limited number of observations. For example, it fails to reproduce the temperature dependens of magnetiza- tion. The correct consideration follows from Holstein- Primakoff transformation model [35] in which magnons - the quanta of magnetic existations are not localized and may flow as a magnon liquid. That is why the nature of superflow of magnetization in superfluid3He is similar to one for solid magnetic materials. The difference is only thevalueofGilbertdamping, whichinsuperfluid3Hecan be as small as 10−8while in solid magnetic materials it is about 10−5in the best caseof the magnetic dielectric fer- rimagnets represented by yttrium iron-garnets. Indeed, thepropertiesofmagnonflow, magnonBECandmagnon supercurrent in antiferromagnetic superfluid3He and in YIG film are very similar. The first experimental obser- vations of the spin superfluid phenomenon in a YIG film at room temperature are presented in this article. The superfluid state is characterized by the off- diagonal long-range order (ODLRO) [36]. In superfluid 4He and in the coherent atomic systems the operators of creation and annihilations of atoms with momentum p= 0 have the time-dependent vacuum expectation value. For the creation operator /angb∇acketleftˆa0/angb∇acket∇ight=N1/2 0eiµt+iα, (1) whereN0is the number of particles in the coherent state andµandαarechemical potentialand the phaseofwave function, respectively. The analogy between spin precession and ODLRO in superfluids is seen if one compares the operator of cre- ation of particle ˆ a+ 0with the operator ˆS+of creation of spin projecton on axis z, whose expectation value is: /angbracketleftBig ˆS+/angbracketrightBig =Sx+iSy=/radicalBig S −ˆSzeiωt+iα.(2)This analogy suggests that in the coherent spin pre- cession the role of particle number Nis played by the projection of total spin on the direction of the external magnetic field Sz[14, 37]. The role of chemical potential is taken by the precession frequency. It is important to note, that the Zeeman energy should be incorporated to the chemical potential, since it may vary in space in the case of magnetic field inhomogeneity. The hydrodynamic equations for the magnon super- fluid are the Hamilton equations for the canonically con- jugated variables Nandα: ˙α=δF δN,˙N=−δF δα, (3) whereFis the Ginzburg-Landau free energy functional of the system [38]. This functional in a frame, rotating at a frequency ωhas the conventional form F−µN=/integraldisplay d3r/braceleftBig|∇Ψ|2 2m+/bracketleftbig ωL(r)−ω/bracketrightbig |Ψ|2+Fso(|Ψ|2)/bracerightBig . (4) HereωL(r) =γH(r) is the localLarmorfrequency, which plays the role of external potential U(r) in atomic con- densates. The last term Fso(|Ψ|2) contains nonlinear- ity which comes from the spin-orbit interaction. It is analogous to the 4-th order term in the atomic BEC, which describes the interaction between the atoms. The spin-orbit interaction provides the effective interaction between magnons, which can be attractive or repulsive. The spin-orbitinteractioncontainsquadraticand quartic terms in |Ψ|. Fso(|Ψ|2) =a|Ψ|2+b|Ψ|4+.... (5) While the quadratic term modifies the potential Uin the Ginzburg-Landau free energy, the quartic term simulates the interaction between magnons. We are able to rewrite Eq. (4) in the next form [14]: F −µN=/integraldisplay d3r/braceleftBig|∇Ψ|2 2m+/bracketleftbig ω0(r)−ω/bracketrightbig |Ψ|2+b|Ψ|4)/bracerightBig , (6) whereω0(r) =ωL(r)+a(r) isthe frequencyofprecession ata limit ofsmall excitation. Hereweregropeterms with |Ψ|2and|Ψ|4. The magnon number density Nis related to the deflection angle βvia N=|Ψ|2=M(1−cosβ) (7) and the frequency of magnetization precession is ωS(r) =ω0(r)+2b(1−cos(β(r))), (8) whereMis the magnetization. The gradientsof αexcites the spin supercurrent, which transports the longitudinal magnetization [38]: J=N∇α , (9)3 This long distance spin supercurrent was measured di- rectly in the experiments, described in [16–18]. It was confirmed that the critical current corresponds to the critical phase gradient, which is the inverse value of the Ginzburg-Landau coherence length ξGL: ∇αc= 1/ξGL=/radicalbig ω0(ωS−ω0)/cSW,(10) wherecSWis a spin wave velocity. It is determined by the competition between the energy of repulsive interac- tion (third term in Eq. 6) and kinetic energyofflow(first term in Eq. 6). Please note, that at the attractive inter- action the coefficient bis negative and spin supercurrents unstable. The remarkable consequence of the spin superfluid- ity is the formation of the homogeneous precession do- main (HPD) even in a strongly inhomogeneous magnetic field. It is formed because the gradient of magnetic field leads to a gradient of phase of magnetization precession which excites the spin supercurrent. The latter trans- ports magnons to the direction of smaller field. The precession frequency increases untill the gradient of pre- cession vanishes. Finally the equilibrium state with the precession frequency ωS(r) =constis established. Any perturbation of this state excites the spin supercurrent which restore the coherent precession. It was shown that the HPDstate appearsspontaneouslyat somedelayafter the pulseofmagneticresonanceexcitation. This statera- diates an extremely long living induction signal [10, 11]. It can persist for several minutes due to the very slow re- laxation(evaporation)ofquasiparticles[28,39]. Recently this type of the signal was considered as a time-crystal [40]. It is important to note that the HPD state is the eigen state of the ensemble of excited magnons. At relax- ation the number of magnons decreases, the frequency of HPD decreases but the magnons remain in the coherent state. The other distinctive feature of the HPD state is its permanence. In the case of the atomic BEC the state disappears because of the atom evaporation. In the case of HPD it is possible to replenish the losses(evaporation) of quasiparticles by excitation of new quasiparticles. It was shown experimentally [29, 41, 42], that a weak RF pumping at a frequency ωSstabilizes chemical poten- tial of magnon gas and keeps its density constant. The corresponding phase difference between the magnetiza- tion precession and the RF field appears automatically to compensate the energy losses. In this article we describe the first observation of HPD state in an out-of-plane magnetized YIG film. The ex- periments were performed on a YIG films of 6 and 1 µm thicknessinashapeofadisks0.5and0.3mmindiameter (see Methods). The magnetization precession in the out-of-plane mag- netized YIG film has dynamic properties very similar to ones in a superfluid3He-B. The repulsive interaction be- tween magnons leads to the dynamical frequency shift ofthe precession at its deflection on angle β[43] described by equation: ω−ω0=γ4πMS(1−cosβ), (11) whereωis magnonfrequency of the homogeneouspreces- sion (k=0) at the deflection angle β,ω0is magnon fre- quency at the limit of small excitation, γis gyromagnetic ratio,and MSissaturationmagnetization. It meansthat the local frequency grows with increasing magnon den- sity. As a result the resonance field decreases if we excite the system at a constant frequency. We haveinvestigatedthe FMR adsorptionsignals from the YIG film at frequency 9.26 GHz and different RF power of excitation. The signals are shown in Fig. 1. At first sight it looks like a well known signals of the non- linear (bi-stable) resonance, first described by Anderson et al. [44]. However this theory does not correspond well to the experimental results at relatively high excitation [45]. The reason for this discrepancy is that the Ander- son’s theory doesn’t take into account the spin super- currents, which play a very important role at a high an- gles of the magnetization deflection, when the density of magnonssurpasethe conditionsofmagnonBose-Einstein condensation, which is about 2-3◦for YIG film [46]. At a relatively small excitation power of 0.05, 0.1 and 0.4 mW (see Fig. 1B) the amplitude of absorption sig- nalgrowsproportionallyto the RF field. This propertyis correspondto an excitationofthe magnongas. At higher excitation the “capture” of a signal takes place. The sig- nals start to follow the field change (see curve c in Fig. 1B). This non-linear behavior starts at an angle of de- flection about 2-3◦, when the field shift of the resonance became bigger then the broadening of the resonance line. This angle of deflection is corresponds to the theoreti- cal approximation for magnon BEC formation [46]. At higher excitation the signal follows the magnetic field up to its change by 80 Oe, as follows from the curve d at the Fig. 1A. This field shift corresponds to a frequency shift of about 320 MHz and the angle of magnetization deflection (see Eq. 7) by about 20◦! At some critical field change the signal disintegrate. This critical field shift (CFS) stronglydepends on the ex- citing power. If we sweep the field up the signal restore at a some field, which is quite near to the field of disinte- gration (see dotted line d in Fig. 1A). The observed be- havior of the signal amplitude shows that the signals are generatedbya statewith coherentprecessionofmagneti- zation, which appears due to a spatial magnons redistri- bution by the spin supercurrent. Its properties are wery similar to one in antiferromagnetic superfluid3He, and particularlytothe signalsin3He-Ain aerogel[29, 30, 47]. According to a theoretical considerationfor the out-of- plane magnetized yttrium iron garnet film the density of excited magnons should reach the conditions of BEC at theangleofmagnetizationdeflectionabout β= 2,5◦[46].4 5040 5060 5080 5100 5120 5140 51600.000.010.020.030.040.050.06 5140 5145 5150 51550.0000.0010.002d c b Amplitude, arb. u. 10.0 mW (a) 20.0 mW (b) 40.0 mW (c) 80.0 mW (d) H0, Oea(A) (B) de c b a 0.05 mW (a) 0.1 mW (b) 0.4 mW (c) 0.8 mW (d) 1.0 mW (e) Amplitude, arb.u. H0, Oe FIG. 1: Amplitudes of the absorption signals at a frequency of 9.26 GHz for different RF pumping powers at a magnetic field sweep down. The curves a, b and c in Fig. B corre- sponds to a linear magnetic resonance, when the amplitude of the signal is proportional to the RF field. The curve c cor- respond to conditions, when the magnon BEC should forms. Curves at higher excitation correspond to the conditions to a non-linear behavior of signals when the magnon superflow plays an important role. It redistribute the magnetization in the sample and provide the stability of coherent precession . Curve d in Fig A corresponds to ahighest energy of excitation at this experiment. The field shift of the signal about 80 Oe was achieved. It corresponds to an angle of precessing mag- netization deflection of about 20◦. The curve d, which shown by dotted line corresponds to a signal we have observed at a sweep field up. It shows that the signal of non-linear mag- netic resonance restores at a small hysteresis. (The power o f 100 mW in these experiments corresponds roughly to the RF magnetic field of 0.1 Oe.) This angle well corresponds to a begining of signals non- linearity (see Fig. 1). At an excitation power higher then 0.4 mW for the conditions of our experiments the density of magnons increasesand strong repulsive interaction be- tween magnons modifies the spectrum of magnons. The spin dynamics can not be described anymoreby the BEC theory of weekly interacted magnons. The formation of coherent macroscopic states and long distance spin su- percurrent should be considered. In Fig. 2 the profile of effective magnetic field in the 500µm in diameter disk-shaped YIG film is shown. It was calculated by OOMMF micromagnetic software [48]. The effective field has a minimum at the disk center due to the demagnetization field. One may suggest that at a field sweep down the resonance conditions appear first at a central part. Then the magnetization deflects and the frequencyfollowstothe resonanceconditionsatasmaller field untill the signal breaks down. But why the signal restore at a field sweep up with a very small hysteresis. Let us consider the case, when the resonanse conditions appears only near the edge of the sample. If we apply the RF field at the frequency corresponding to the ferro- magnetic resonancenear the edge (at R = 200 µm in Fig. 2) the spin waves will be excited at this region (see the right inset Fig. 2 red dot line). If the excitation power N FIG. 2: Variation of the local effective magnetic field from th e center of the sample (red line) and the frequency shift due to themagnonchemical potential (blueline), whichcompensat es the inhomogeneity of magnetic field. The inset to the right shows schematically the spatial distribution of magnetiza tion deflection (sin β) at small excitation (point red line), at a higher excitation, when magnons start to superflow to lower field parts of the disk (dashed blue line) and the HPD state (solid black line). The inset to the left side shows the HPD droplet of coherent magnons. will be increased the deflection angle also increases and may surpasses the critical BEC angle. The gradient of phase of precession and spin supercurrent will appears due to the gradient of effective magnetic field. It will transport magnons in the direction of smaller field, to the central part of the sample (Fig. 2 dashed blue line). Finally, the angle of magnetization deflection in the cen- tralpartincreasesuptoavalue, whichcorrespondstothe precession frequency equal to RF field (black line). The spatial distribution of dynamic frequency shift is shown in Fig. 2 by a solid blue line. This shift compensates the inhomogeneity of effective magnetic field and all the magnetizationprecessesatthefrequencyofRFfield. The HPD state is the eigenstate in inhomogeneous magnetic field for a given number of magnons. This scenario was carefully investigated in the experiments with superfluid 3He and MnCO 3. [41, 49]. TheHPDstatehasaformofdroplet, artisticallyshown in the left inset in Fig. 2. Deflected magnetization pre- cesses coherently and homogeneously at a frequency: ω=γH(r)+∆ω(r) =const, (12) whereH(r) is the effective magnetic field at a point r. Furthermore, the spin supercurrent compensates the in- homogeneity of magnetization relaxation by redistribu- tion of magnons density. This state exists permanently in the case of RF pumping at the frequency of the HPD state, which compensates the evaporation of magnons. The superfluid state is supported by a permanent pumping of the RF magnetic field, HRF, which is trans- verse to the applied constant external magnetic field5 H0. The RF field prescribes the precession frequency, ω=ωRF, and thus fixes the chemical potential µ=ω. In the precession frame, where both the RF field and the deflected magnetization Mare constant, the interaction energy term is FRF=−HRF·M=−HRFM⊥cos(α−αRF),(13) whereHRFandαRFare the amplitude and the phase of the RF field. This term softly breaks the U(1)-symmetry and serves as a source of the mass of Nambu-Goldstone mode [50]. The phase difference between the condensate and RF field, ( α−αRF), is determined by the energy losses due to magnetic relaxation, which is compensated by the pumping power of the RF field: W=ωMHRFsinβsin(α−αRF).(14) The phase difference is automatically adjusted to com- pensate the losses. If dissipation is small, the phase shift is small ( α−αRF≪1) and can be neglected. The ne- glected (α−αRF)2term leads to the nonzero mass of the Goldstone boson – quantum of the second sound waves (phonons) in the magnonic superfluid [50]. The signal breaks down at the moment, when the RF power is not enough to compensate the magnons dissipation. Since the pumping (14) is proportional to sin βsin(α−αRF), a critical tilting angle βc, at which the pumping cannot compensate the losses, increases with increasing HRF. The breaks down occurs when the phase shift ( α−αRF) reaches 90◦. At this moment the adsorption signal cor- responds to a transverse magnetization of the sample. In Fig. 3 it is shown by points. The theoretical curve, showninFig. 3bysolidline, iscalculatedfortheassump- tion that the HPD droplet fills up all the region, where magnetic field is less than ωRF/γ. The theoretical curve shows a good agreement with the experimental points. The similar results were obtained in other samples. Let us compare the experimental results with the the- ory of non-linear resonance [44]. This theoretical ap- proach is based on the properties of a single non-linear oscillator. It supposes that at increase of the excitation power, the angle of deflection increases which leads to a frequency shift. This theory describes well the non-linear resonance in ferromagnets at a relatively small angles of deflection but strongy disagrees with the experimental results at higher angles [45]. A more sophisticated the- ory of autoresonance was suggested in [51], where a singe non-linear oscillator is also considered. This model is not applicable to real magnetically ordered materials since the spin system consists of many oscillators with differ- ent ground frequency due to inhomogeneity of magnetic field. If we try to modify the theory for a number of the non-interacting oscillators with different ground frequen- cies we should conclude that the break down points are different for different oscillators. As a result the exper- imental curve of signal break down should be broaden/s53/s48/s52/s48 /s53/s48/s54/s48 /s53/s48/s56/s48 /s53/s49/s48/s48 /s53/s49/s50/s48 /s53/s49/s52/s48 /s53/s49/s54/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s48/s46/s48/s54 /s53/s48/s52/s48 /s53/s48/s56/s48 /s53/s49/s50/s48 /s53/s49/s54/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52 /s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s32/s115/s105/s110/s32/s115/s105/s110 /s32/s83/s83/s44/s32/s109/s109/s50 /s72 /s48/s44/s32/s79/s101 /s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s44/s32/s97/s114/s98/s46/s32/s117/s46 /s72 /s48/s44/s32/s79/s101 FIG. 3: The absorption signal amplitude at the moment of breaks down of the HPD state (points). The theoretical curve (red solid line) was calculated for the signal from the magnonic droplet at these conditions. The inset shows the magnetization deflection (sin β) in the center of the magnon droplet and the film surface area, occupied by the droplet (S) . on the magnetic field. Indeed, experimentally the break down appears very abruptly. The even more important argument against this theory is the recovery of signal with a very small hysteresis at a sweep magnetic field up. The non-liner oscillator can not be excited by out of resonance excitation. It is clearly shown in Fig. 3 of Ref. [51]. It may be excited at a sweep field up only near the resonance, at a field shift of a homogeneous broadening of the line, which is about 2 Oe. Experimentally the ex- citation takes place at uncomparable big shift of 80 Oe. This can be explained only by a resonance excition of magnons at the edge and its redistribution throw all the sample by spin supercurrent. The comparison between the model of non-linear oscillator and HPD have been investigated in Ref. [52]. It is very important to note that the spin supercur- rent exists only at the conditions of repulsive interaction between magnons. In the case of attractive interaction the superfluid critical velocity is equal to zero and the superfluid state is unstable [53]. Finally, we have shown the big importance of spin su- percurrenttransportto explain the propertiesofthe non- linear magnetic resonance. We have demonstrated the formation of a macroscopic region with the coherent pre- cession of magnetization in an inhomogeneous effective magnetic field in the out-of-plane magnetized YIG film. This state is the first permanent superfluid state of con- densed matter demonstrated at room temperature. It is an idealplatform for developmentofmicrowavemagnetic technologies, which have already resulted in the creation of the magnon transistor and the first magnon logic gate [54, 55]. The YIG films can be used as the basis for6 new solid-state quantum measurement and information processing technologies including cavity-based QED, op- tomagnonics, and optomechanics [56]. A chain of YIG samples with excited HPD droplets may be considered as a Q-bits, interacting through Josephson junctions for aquantumcomputer. TheformationofthemagnonBEC in YIG and observation of the spin supercurrent, like in 3He, should lead to the development of a new branch of modern magnetism - supermagnonics. Methods All samples were prepared from yttrium iron gar- net films grown by liquid phase epitaxy on 500 µm thick GGG substrates with the (111) crystallo- graphic orientation [57]. To reduce the effect of cu- bic magnetic anisotropy, we used scandium substituted Lu1.5Y1.5Fe4.4Sc0.6O12iron garnet films; the introduc- tion of lutetium ions was necessary to match the param- eters of the substrate and film crystal gratings. It is known that the introduction of scandium ions in such an amount reduces the field of cubic anisotropy by more than an order of magnitude [58]. In addition, the used lutetium and scandium ions practically do not contribute to additional relaxation in the YIG. The samples were prepared in the form of a disk with diameters of 500 and 300µm and a thickness of 6 µm. The disk was made by photolithography. To avoid magnetic pinning on the surface the sample was etched in a hot phosphoric acid [59]. As a result, the edges of the disk have a slope of 45 degrees and had a smooth surface. The CW FMR experiments were performed on Varian E-12X-bandEPRspectrometeratthe roomtemperature and the frequency 9.26 GHz. The RF field was oriented inplaneofthesamples. Theamplitudeandthefrequency of magnetic field modulation were 0.05 Oe and 100 kHz, respectively. This frequency is much lower than the es- timated frequency of the second sound of the magnon BEC (The Goldston mode). That is why we may con- sider these conditions as stationary. The absorption sig- nals, presented here, were obtained after the integration of the original signals. Acknowledgments The authors wish to thank G. E. Volovik, V. P. Mi- neev V. L’vov and A. Serga for helpful comments and O. Demokritov for stimulating discussions. Financial sup- portby the RussianScienceFoundation within the Grant 19-12-00397Spin Superfluids is gratefully acknowledged.∗Electronic address: y.bunkov@rqc.ru [1] P Kapitza, “Viscosity of liquidhelium below the λ-point” Nature3558, 74 (1938). [2] F. London, “Superfluids” ,IIJohn Wilet and Sons, Inc., New York, (1954). [3] A. Einstein, ”Quantentheorie des einatomigen idealen Gases. PartI”. Sber. Preuss. Akad. Wiss. ,22,261(1924); ”Quantentheoriedeseinatomigen idealen Gases. PartII”. Sber. Preuss. Akad. Wiss. 1,3, (1925). [4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman & E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor”, Science269,198 (1995). [5] K. B. Davis, M. O. Mewes, M. R. Andrewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “ Bose- Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. ,75, 3969 (1995). [6] L. D. Landau, “Theory of helium 2 superfluidity” J. Phys. USSR ,5, p.71 (1941). [7] D. A. Bozhko, A. A. Serga, A. Pomyalov, V. S. Lvov andB.Hillebrands, “Bogoliubov wavesanddistanttrans- port of magnon condensate at room temperature”. Na- ture Comm. 10, 2460 (2019). [8] O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk, V. E. Demi- dov, S. O. Demokritov, B. Koene, A. Kirilyuk, T. Rasing, V. Tiberkevich and A. Slavin, “Magnon-magnon inter- actions in a room-temperature magnonic Bose-Einstein condensate”. Phys. Rev. B 96, 064438 (2017). [9] I. S. Tupitsyn, P. C. E. Stamp, A. L. Burin, “Stability of Bose-Einstein condensates of hot magnons in YIG film”, Phys. Rev. Lett. 100, 257202 (2008). [10] A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmitriev and Yu. M. Mukharskii, “Long-lived induction signal in superfluid3He-B”.JETP Lett. 40,1033 (1984). [11] I. A. Fomin, “Long-lived induction signal and spatiall y nonuniform spin precession in3He-B”.JETP Lett. 40, 1037 (1984). [12] D. Snoke, ”Coherent questions”. Nature443,403 (2006). [13] Yu. M. Bunkov “Spin Supercurrent” J. Mag. Mag. Mat , 310, 1476 (2007). [14] Yu. M. Bunkov and G. E. Volovik, “Bose-Einstein Con- densation of Magnons in Superfluid3He”.J. of Low Temp. Phys. 150,135 (2008). [15] Yu. M. Bunkov, “Spin superfluidity and magnons BoseE- instein condensation” Physics Uspekhi, 53,848 (2010). [16] Yu. M. Bunkov, “Spin Supercurrent in 3He-B”, Japan J. Appl. Phys ,26, 1809 (1987). [17] A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmitriev and Yu. M. Mukharskiy, “Phase Slipage Observations of Spin Supercarrent in 3He-B”, JETPh Lett., 45, 124 (1987). [18] A. S. Borovik-Romanov, Yu.M. Bunkov, V. V. Dmitriev, Yu. M. Mukharskiy and D. A. Sergatskov, “Investigation of Spin Supercurrent in 3He-B”, Phys.Rev.Lett. 62, 1631 (1989). [19] A. S. Borovik-Romanov, Yu.M. Bunkov, V. V. Dmitriev, V. Makroczyova, Yu. M. Mukharskii, D. A. Sergatskov, A. de Waard, “ The analog of the Josephson Effect in the Spin Supercurrent” Journal de Physique 49 (C8) 2067 (1988). [20] A.S.Borovik-Romanov, Yu.M.Bunkov, A.deWaard, V.7 V. Dmitriev, V. Makrotsieva, Yu. M. Mukharskiy and D. A. Sergatskov, “Observation of a Spin Supercarrent Ana- log of the Josephson Effec”, JETP Lett., 47, 478 (1988). [21] A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmitriev, Yu. M. Mukharskiy and D. A. Sergatskov, “Observation of Vortex-like Spin Supercurrent in 3He-B”, Physica B, 165, 649 (1990). [22] Yu. M. Bunkov and G. E. Volovik “Spin Vortex in magnon BEC of Superfluid 3He-B”, Physica C, 468, 600 (2008). [23] Yu. M. Bunkov, V. V. Dmitriev, and Yu. M. Mukharskii, “Torsional vibrations of a domain with uniform magne- tization precession in3He-B”.JETP Lett. 43,131-134 (1986). [24] Yu. M. Bunkov, V. V. Dmitriev, and Yu. M. Mukharskii, “Low frequency oscillations of the homogeneously pre- cessing domain in3He-B”.Physica B 178,196 (1992). [25] Yu.M. Bunkov,S.N.Fisher, A.M.Guenault, G.R.Pick- ett, “Persistent Spin Precession in 3He-B in the Regime of Vanishing Quasiparticle Density”, Phys. Rev. Lett. , 69, 3092 (1992). [26] Yu. M. Bunkov, “Persistent Signal; Coherent NMR state Trapped by Orbital Texture” J. Low Temp. Phys ,138, 753 (2005). [27] Yu. M. Bunkov and G. E. Volovik, “Magnon Condensa- tion into a Q Ball in 3He-B” Phys. Rev. Lett. ,98, 265302 (2007). [28] S. Autti, Yu. M. Bunkov, V. B. Eltsov, et al. Self- trapping of magnon Bose-Einstein condensates in the ground and excited levels: from harmonic to a box con- finement Phys. Rev. Lett. ,108, 145303 (2012). [29] T. Sato, T. Kunimatsu, K. Izumina, A. Matsubara, M. Kubota, T. Mizusaki, andYu.M. Bunkov“CoherentPre- cession of Magnetization in the Superfluid 3He A-Phase” Phys. Rev. Lett. ,101, 055301 (2008). [30] P. Hunger, Y. M. Bunkov, E. Collin and H. Godfrin “Ev- idence for Magnon BEC in Superfluid 3He-A” J. Low Temp. Phys. 158, 129 (2010). [31] Yu. M. Bunkov, and G. E. Volovik, “Magnon BEC in superfluid3He-A”.JETP Lett. 89, 306-310 (2009) [32] S. Autti, V. V. Dmitriev, J. T .Makinen, T. Rysti, A. A. Soldatov, G. E. Volovik, A. N. Yudin, V. B. Eltsov, “Bose-Einstein Condensation of Magnons and Spin Su- perfluidity in the Polar Phase of3He”,Phys.Rev.Lett. 121, 025303 (2018). [33] “The 2008 Fritz London Prize” J. Low Temp. Phys. 152, 1 (2008). [34] Yu. M. Bunkov, “Spin Supercurrent and coherent spin precession” London prize lecture, J. Phys.:Cond. Mat. , 21, 164201 (2009). [35] T. Holstein and H. Primakoff, “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet” Phys. Rev.58,1098 (1940). [36] C.N.Yang, “Concept ofoff-diagonal long-range orderan d the quantum phases of liquid He and of superconduc- tors”,Rev. Mod. Phys. ,34, 694 (1962). [37] Yu. M. Bunkov and G. E. Volovik, “Magnon Bose- Einstein condensation and spin superfluidity”. J. Phys.: Cond. Mat. 22,164210 (2010). [38] Yu. M. Bunkov and G. E. Volovik, “Spin Superfluidity and Magnon BEC” inNovel Superfluids Ch.4, (eds. Ben- nemann, K. H. & Ketterson, J. B. Oxford Univ. Press, Oxford, (2013). [39] D. J. Cousins, S. N. Fisher, A. I. Gregory, G. R. Pick-ett and N. S. Shaw, “Persistent coherent spin precession in superfluid3He-B driven by off-resonant excitation”, Phys. Rev. Lett. ,82, 4484 (1999). [40] S. Autti, V. B. Eltsov, and G. E. Volovik, “Observation of a time quasicrystal and its transition to a super fluid time crystal”, Phys. Rev. Letts. 120, 215301 (2018). [41] A. S. Borovik-Romanov, Yu.M. Bunkov, V. V. Dmitriev, Yu. M. Mukharskii, E. V. Poddyakova and O. D. Timo- feevskaya, “Distinctive Features of a CW NMR in3He-B due to a Spin Supercurrent”. JETP,69,542 (1989). [42] Yu. M. Bunkov, “Magnonics and Supermagnonics” Spin, 9, 1940005 (2019). [43] Yu. V. Gulyaev, P. E. Zilberman, A. G. Temiryazev and M. P. Tikhomirova, “Principal Mode of the Nonlinear Spin-Wave Resonance in Perpendicular Magnetized Fer- rite Films”. Physics of the Solid State 42,1062 (2000). [44] P. W. Anderson and H. Suhl, “Instability in the motion of ferromagnets at high microwave power levels”, Phys. Rev.,100, 1788 (1955). [45] Yu. K. Fetisov, C. E. Patton and V. T. Synogach, “Non- linear Ferromagnetic Resonance and Foldover in Yt- trium Iron Garnet Thin FilmsInadequacy of the Clas- sical Model”, IEEE Transactions on magnetics ,35,4511 (1999). [46] Yu. M. Bunkov and V. L. Safonov, “Magnon condensa- tion and spin superfluidity”. J. Mag. and Mag. Mat. 452, 30 (2018). [47] Bunkov, Yu. M. and Volovik, G. E. ”On the possibility of the Homogeneously Precessing Domain in Bulk3He-A”, Europhys. Lett. ,21,837 (1993). [48] https://math.nist.gov/oommf/software.html [49] Y. M. Bunkov, A. V. Klochkov, T. R. Safin, K. R. Safi- ullin and M. S. Tagirov, “Nonresonance excitation of magnon Bose-Einstain condensation in MnCO 3”,JETP. Lett.,109, 43 (2019). [50] G.E. Volovik, “Phonons in magnon superfluid and sym- metry breaking field”, JETP Lett. ,87,639 (2008) [51] M. A. Shamsutdinov, L. A. Kalyakin and A. T. Kharisov, “Autoresonance in a Ferromagnetic Film”, Technical Physics,55, 860 (2010). [52] L. V. Abdurakhimov, M. A. Borich, Yu. M. Bunkov, R. R. Gazizulin, D. Konstantinov, M. I. Kurkin and A. P. Tankeyev, “Nonlinear NMR and magnon BEC in antifer- romagnetic materials with coupled electron and nuclear spin precession” Phys. Rev. B ,97, 024425 (2018) [53] A. S. Borovik-Romanov, Yu.M. Bunkov, V. V. Dmitriev, Yu. M. Mukharskiy “Instability of Homogeneous Spin Precession in Superfluid 3He-A”, JETP Lett. ,39, 469 (1984). [54] A. V. Chumak, A. A. Serga and B.Hillebrands, “Magnonic crystals for data processing”. J. Phys. D: Appl. Phys. ,50,244001 (2017). [55] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa and E. Saitoh “Transmission of electrical signals by spin-wave interconversion in a mag- netic insulator”, Nature,464,262-266 (2010). [56] Dengke Zhang, Xin-Ming Wang, Tie-Fu Li, Xiao-Qing Luo, Weidong Wu, Franco Nori and J. Q. You “Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere”. NPJ Quant. Inf. , 1,15014 (2015). [57] The samples were provided by the company “M-Granat” (http://m-granat.ru/).8 [58] A. R. Prokopov, P. M. Vetoshko, A. G. Shumilov, A. N. Shaposhnikov, A. N. Kuz’michev, N. N. Koshlyakova, V. N. Berzhansky, A. K. Zvezdin and V. I. Belotelov, Epitaxial BiGdSc iron-garnet films for magnetophotonic applications, J. Alloys and Compounds ,671, 403 (2016). [59] P. M. Vetoshko, A. K. Zvezdina, V. A. Skidanov, I. I.Syvorotka, I. M. Syvorotka and V. I. Belotelov, “The Ef- fect of the Disk Magnetic Element Profile on the Satura- tion Field and Noise of a Magneto-Modulation Magnetic Field Sensor”. Technical Phys. Lett. ,41,458 (2015).

2019-11-09

We declare the observation of spin superfluid state in Yttrium Iron Garnet (YIG) at room temperature. It is similar to a Homogeneous Precessing State (HPD), observed earlier in antiferromagnetic superfluid $^3$He-B. The formation of this state explains by the repulsive interaction between magnons, which is required as a prior condition for the spin superfluidity. It establishes an energy gap, which stabilizes the long range superfluid transport of magnetization and determines the Ginzburg-Landau coherence length. This discovery paves a way to many quantum applications of supermagnonics at room temperature, such as magnetic Josephson effect, long distance spin transport, Q-bit, quantum logics, magnetic sensors and others.

The magnonic superfluid droplet at room temperature

1911.03708v1

Phase-resolved electrical detection of coherently coupled magnonic devices Yi Li,1Chenbo Zhao,1Vivek P. Amin,2, 3Zhizhi Zhang,1Michael Vogel,1, 4Yuzan Xiong,5, 1Joseph Sklenar,6 Ralu Divan,7John Pearson,1Mark D. Stiles,3Wei Zhang,5, 1Axel Homann,8and Valentyn Novosad1, 1Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USAy 2Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA 3Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 4Institute of Physics and Center for Interdisciplinary Nanostructure Science and Technology (CINSaT), University of Kassel, Heinrich-Plett-Strasse 40, Kassel 34132, Germany 5Department of Physics, Oakland University, Rochester, MI 48309, USA 6Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202, USA 7Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439, USA 8Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign Urbana, IL 61801, USA (Dated: May 25, 2021) We demonstrate the electrical detection of magnon-magnon hybrid dynamics in yttrium iron garnet/permalloy (YIG/Py) thin lm bilayer devices. Direct microwave current injection through the conductive Py layer excites the hybrid dynamics consisting of the uniform mode of Py and the rst standing spin wave ( n= 1) mode of YIG, which are coupled via interfacial exchange. Both the two hybrid modes, with Py or YIG dominated excitations, can be detected via the spin rectication signals from the conductive Py layer, providing phase resolution of the coupled dynamics. The phase characterization is also applied to a nonlocally excited Py device, revealing the additional phase shift due to the perpendicular Oersted eld. Our results provide a device platform for exploring hybrid magnonic dynamics and probing their phases, which are crucial for implementing coherent information processing with magnon excitations. Hybrid magnonic systems have recently attracted wide attention due to their rich physics and application in coherent information processing [1{15]. The introduc- tion of magnons has greatly enhanced the tunability in hybrid dynamics, the capability of coupling to di- verse excitations for coherent transduction [16{20], as well as the potential for on-chip integration [12{14]. Re- cently, thin-lm-based magnon-magnon hybrid systems have provided a unique platform for implementing hy- brid magnonic systems [21{30]. Coupling between mate- rials in the hybrid structure can arise through the inter- facial exchange interaction. Because magnon excitations are conned within the magnetic media, it is convenient to build up more compact micron-scale hybrid platforms compared with millimeter-scale microwave circuits. Fur- thermore, abundant spintronic phenomena, such as spin- torque manipulation and spin pumping, can be used to control and engineer the hybrid dynamics especially for magnetic thin-lm devices. One important aspect of hybrid magnonic systems is controlling and engineering the phase relation be- tween dierent dynamic components, leading to phenom- ena such as exceptional points [31, 32], level attraction [33, 34] and nonreciprocity [35, 36] in cavity spintron- ics. Phase resolved detection of individual magnetiza- tion dynamics has been extensively explored electrically, optically, and with advanced light sources. In particu- lar, electrical measurements of magnetic thin-lm devices via spin rectication eects [37{42] can directly trans- form microwave magnetic excitations into sizable dc volt- ages. This technique has been used to sensitively mea-sure nanoscale magnetic devices and, more importantly, the phase of magnetization dynamics in order to quan- tify the spin torque generated from charge current ow [43{54]. In this work, we establish the usefulness of electrical ex- citation and detection for the study of coherently coupled magnon-magnon hybrid modes in Y 3Fe5O12/Ni80Fe20 (YIG/Py) thin-lm bilayer devices. This approach dif- fers from the previous work on inductive microwave mea- surements [21, 23, 24] in its applicability to nanoscale devices and its phase sensitivity. The coupled YIG/Py magnetization dynamics are excited by directly apply- ing microwave current through the conductive Py layer. Only the Py layer contributes to the spin rectication sig- nal because the YIG is insulating, enabling clear phase- resolved detection of the Py component of the YIG-Py hybrid modes. We measure a constant phase for the Py- dominated hybrid modes and a phase oset across the avoided crossing for the YIG-dominated modes. From the slope of the phase shift we can determine the inter- layer coupling strength, agreeing with the measurement from the avoided crossing. We have also characterized a nonlocally excited YIG/Py sample, in which a phase o- set compared with the single bilayer device suggests the existence of a large perpendicular Oersted eld driving the dynamics. Our results open an avenue of building up, reading out and designing circuits of on-chip magnonic hybrid devices for the application of coherent magnonic information processing. YIG thin lms (50 nm, 70 nm and 85 nm) were sput- tered on Gd 3Ga5O12(111) substrate with lithographi-arXiv:2105.11057v1 [cond-mat.mes-hall] 24 May 20212 (a) YIGPyAu electrode Au electrodeBias-T VIrf Vdc (b) Irf (d) (c)45 o HB 50 μV(f) (e) Py (n=0) YIG (n=0) YIG (n=1) 8 GHz 11 GHz YIG (n=1)Py (n=0) +anti- crossing FIG. 1. (a) Illustration of electrical excitation and detection of YIG/Py bilayer devices. The in-plane external biasing eld is kept as 45from theIrfdirection along the Py devices. (b-c) Optical microscope images of the device and Au coplanar waveguide antenna for (b) single devices and (c) nonlocally excited device. (d-f) Spin rectication signals for the YIG(70 nm)/Py(9 nm) single device, with the mode anti-crossing between YIG ( n= 1) and Py ( n= 0) modes marked by the dashed box. (e) Zoom-in lineshape of (d) at 11 GHz where the extrapolated YIG ( n= 1) and Py ( n= 0) peaks are well separated. (f) Lineshape at 8 GHz where the YIG ( n= 1) and Py ( n= 0) modes are degenerate in eld. The red and green dashed curves in (f) denote the t of the two YIG-Py hybrid modes. cally dened device patterns, followed by lifto and an- nealing in air at 850C for 3 h [24, 55]. Then a second- layer Py device (8 nm or 9 nm) was dened on the YIG device with lithography and sputtering, with 1 min ion milling of YIG surface in vacuum right before deposition. Lastly a 200 nm thick Au coplanar waveguide (CPW) was fabricated which was in contact with the Py device for electrical excitations and measurements. Fig. 1(a) shows the schematic of the spin rectication measure- ment. The top-view optical microscope images of the devices are shown in Fig. 1(b) for the single devices and (c) for the nonlocally excited devices. The dimensions of the Py devices are 10 m40m in Fig. 1(b) and 6m20m in Fig. 1(c). The two Py devices in Figs. 1(c) are separated by 2 m, one for applying nonlocal excitation signals and the other for the spin rectication measurements. Throughout the measurements, the ex- ternal biasing eld is applied in the sample plane and tilted 45away from the microwave current direction, which is the commonly used conguration in spin recti- cation measurements for maximizing the output signals [51]. Fig. 1(d) shows the eld-swept spin rectication sig- nals of the YIG(70 nm)/Py(9 nm) single device at dif- ferent frequencies. We observe both the nominal Py and YIG uniform mode resonances, as reported previously [56, 57], even though the signals come from only the Py layer. For the Py uniform mode, the excitation is mainly due to a nite Oersted eld projection to the dynamic mode, which has also been observed in a single CoFe layer in our prior work [58]. For the YIG excitation, the interfacial exchange coupling creates coupled modes with nite amplitude on the Py, leading to a modulation of the Py resistance even for mode that is nominally the YIG uniform mode. In addition, the YIG ( n= 1) PSSW YIG(a) (b) (c) (d) 85 nm70 nm50 nmFIG. 2. (a-c) Extracted resonance peak positions of (a) YIG(50 nm)/Py(8 nm), (b) YIG(70 nm)/Py(9 nm) and (c) YIG(85 nm)/Py(9 nm) single devices. The mode degeneracy between the Py uniform mode and YIG ( n= 1) mode happens at!c=2= 19:5 GHz for (a), 7.9 GHz for (b) and 4.7 GHz for (c), denoted by vertical dashed lines. (d) Exchange eld for dierent tYIG. mode is also excited when it intersects with the Py uni- form mode, forming an avoided crossing between the two modes at!c=2= 7:9 GHz (Fig. 1f). The separation of the two hybrid peaks is 8.5 mT, which is larger than the extrapolated individual linewidths of the Py ( n= 0) and YIG(n= 1) modes ( 0
HPy= 5:5 mT,0
HYIG= 3:0 mT). Far away from the avoided crossing, the excitations of the YIG ( n= 1) mode are almost unnoticeable (Fig. 1e), which is due to the weak coupling of the uniform3 Oersted eld to the odd PSSW modes. Thus the drive of the YIG (n= 1) mode is dominated by excitation of the admixture of the Py mode due to the interfacial exchange [21{24]. To analyze the spin rectication signals, the measured lineshape for each peak can be tted to the following function: Vdc= ImV0ei
H (HBHr)i
H (1) whereHBis the biasing eld, Hris the resonance eld as a function of frequency,
His the half-width-half- maximum linewidth, V0is the peak amplitude, and  represents the mixing of the symmetric and antisymmet- ric Lorentzian lineshapes and re ects the phase evolu- tion of the Py component in the YIG-Py hybrid dynam- ics. The operator Im[] takes the imaginary part. This technique has been used to probe the dampinglike and eldlike torque components [43{46, 48{53] as well as in recent optical rectication experiments [59{64]. More- over, the single source of spin rectication signal from Py allows convenient theoretical analysis for studying the phase evolution of the hybrid dynamics, as will be shown below. Figs. 2(a-c) show the extracted Hras a function of frequency!fortYIG= 50 nm, 70 nm and 85 nm, re- spectively. The two hybrid modes, marked as blue and green circles, are formed between Py uniform and YIG (n= 1) modes. With dierent tYIG, the mode intersec- tion happens at dierent frequencies due to the eective exchange eld 0Hex= (2Aex=Ms)k2withk==tYIG, which shifts the YIG ( n= 1) mode towards higher fre- quencies. Fig. 2(d) plots the extracted 0Hexas a func- tion of (=tYIG)2; good linear dependence conrms the role of the exchange eld. By tting the data to the Kit- tel equation plus the exchange eld, we obtain similar values of magnetization in all lms as 0MPy= 0:81 T, 0MYIG= 0:19 T. From the linear ts in Fig. 2(d) we obtainAex= 4:7 pJ/m for the YIG lm. The mode anticrossing behaviors in Figs. 2(a-c) can be tted to the equation developed by the two coupled magnon resonances [24]: 0H=0HYIG+HPy 2s 0HYIGHPy 22 +g2 H (2) where0HYIG=p 2 0M2 YIG=4 +!2= 20MYIG=2 + 0Hexand0HPy=q 2 0M2 Py=4 +!2= 20MPy=2 are the solutions of the Kittel equation for the YIG ( n= 1) and Py modes, =2= (geff=2)27:99 GHz/T and gHis the interfacial exchange coupling strength in the magnetic eld domain. In our previous work [24], we de- rived thatgH=f(!)p J=M PytPy
J=M YIGtYIGwhereJ is the interfacial exchange coupling strength. The fac- torf(!)0:9 accounts for the nonlinearity due to thedemagnetizing eld. Data ttings to Eq. (2) yield an averagedgH= 8:7 mT andJ= 0:066 mJ/m2; the lat- ter is consistent with the reported value of 0.06 mJ/m2 for continuous thin lms [24]. We also double-check the value ofJby measuring the inductive ferromagnetic res- onance of a 200 m40m YIG stripe from the same fabrication as the YIG(50 nm)/Py(8 nm) device, with the peak dispersion shown as stars in Fig. 2(a). From the Kittel tting, we obtain a constant resonance eld oset of0Hk= 13:8 mT between the YIG stripe and the YIG/Py device. From this static oset, we extract J=0HkMYIGtYIG= 0:110 mJ/m2[24], in good agree- ment with the value of 0.112 mJ/m2obtained above from the avoided crossing for tYIG= 50 nm. The YIG/Py de- vice shows a higher resonance eld than YIG, conrming the antiferromagnetic exchange coupling between YIG and Py. Next, we show the evolution of for the hybrid modes, which are the main results of this work. Fig. 3(a) shows the extracted phases for the three modes in the YIG(70 nm)/Py(9 nm) single device, with the color cor- responding to the resonance eld plot in Fig. 2(b). The two hybrid modes exhibit a clear phase crossing where their resonance elds intersect at !c=2= 7:9 GHz (ver- tical dashed line). For the Py-dominated hybrid modes which are represented by the blue circles lower than !c and the green circles higher than !c, the phase stays at a constant level ( Py=0:23). This is expected in spin rectication measurements, where a consistent phase re- lation between the Py dynamics and the microwave cur- rent is maintained in a broad frequency domain. For ideal eldlike excitations as illustrated in Fig. 3(c) in the single device, we expect Py==2. Experimentally, the deviation of Pymay be due to the self spin torque providing a nite dampinglike drive component [65]. Al- ternatively, the phase oset may be also a re ection of the inhomogeneous mode prole of Py in the presence of the YIG/Py interfacial exchange boundary as well as the nonuniform current distribution across the thickness of Py. The phase of the YIG-dominated hybrid modes, on the other hand, evolve from below Pyto abovePywith an increment of nearly . As a rough explanation, by pass- ing through the avoided crossing, the frequency of the YIG-dominated mode evolves from below the Py reso- nance frequency to above it. This leads to a phase shift of for the Py susceptibility. Because the YIG dynamics is driven by the interfacial exchange from the Py excitation, a phase shift of is also expected in the YIG-dominated mode. Furthermore, due to the strong magnon-magnon coupling, the phase shift does not take a sharp transi- tion at!c, but takes a gradual transition with the transi- tion bandwidth determined by the coupling strength gH. To quantitatively understand the phase evolution of the hybrid mode, we follow the susceptibility tensor which has been derived in our prior work see Eq. (S-4 hrf z Irf Py YIG Py YIG hrf Irf VVrad rad (a) (b) (e) gH=1.0 mT gH=4.0 mT gH=7.0 mT Theory (c) (d) rad FIG. 3. Phase evolution of the spin rectication signals for (a) YIG(70 nm)/Py(9 nm) single device and (b) YIG(85 nm)/Py(9 nm) nonlocally excited device, with their microwave current ow and eld distribution illustrated in (c) and (d), respectively. The blue and green curves show the theoretical prediction from Eq. (5) with (b) gH= 4:0 mT for (a) and 5.3 mT for (b). The error bars indicate single standard deviation uncertainties that arise primarily from the tting of the resonances. (e) Theoretical plots of phase evolution from Eq. (5) using the Hrin (a) andPy= 0 for dierent gH. 4) in the Supplemental Materials of Ref. [24]. In the limit of weak damping and ignoring the precession ellip- ticity, the dynamics of the Py uniform and YIG ( n= 1) modes can be expressed as: ~mPy=~hx Py HBHPyi
HPyg2 H HBHYIGi
HYIG(3a) ~mYIG=gH~mPy HBHYIGi
HYIG(3b) where ~mPyand ~mYIGdenote the unitless transverse com- ponents for Py and YIG,
HPyand
HYIGdenote their linewidths. For the Py layer, the eective eld ~hx Pyis ex- erted from the microwave current owing through. For the YIG layer, the eective eld gH~mPyis provided by the interfacial exchange when the Py magnetization pre- cesses. Note that because YIG is an insulator, the spin rectication signal is only contributed by ~ mPy, which sig- nicantly simplies the theoretical analysis. Eq. (3a) can be rewritten as: ~mPy=~hx Py(HBHYIGi
HYIG) (HBH+i
H+)(HBHi
H)(4) where the values of Hare dened in Eq. (2) and
H are the linewidths for the two hybrid modes. Compared with Eq. (1), the phase for the Hresonance can be nally expressed as: =Py+ tan1
HYIG HHYIG tan1
H HH (5) In Eq. (5) the rst term comes from a nite phase oset between ~hx Pyand the microwave current, the second termcomes from the numerator and provides the phase shift, and the last term is usually close to zero in the strong coupling regime as the linewidth is much smaller than the resonance detuning. The calculation results of Eq. (5) are plotted in Fig. 3(a), which nicely reproduce the experimental data and the positive increment of phase for the YIG-dominated hybrid mode. We also plot the calculated phase evolution for dierent values of gHin Fig. 3(e). For small gH, the YIG-dominated mode shows a rapid phase shift near the mode crossing frequency. AsgHincreases, the phase transition regime broadens becausegHdenes how quickly the hybrid mode evolves to uncoupled individual modes. The phase-resolved spin rectication measurement of the hybrid modes are also repeated on a nonlocally ex- cited device. With the excitation and detection schemat- ics shown in Fig. 3(d), the microwave current ows through a nonlocal Py electrode, which provides an Oer- sted eld that is perpendicular to the Py device be- ing measured. For the detection, due to the induc- tive coupling between the two adjacent Py devices, a nite microwave current ows through the second Py device which leads to a measurable spin rectication voltage when the Py magnetization dynamics is excited. Fig. 3(b) shows the measured for the three modes. Above!c=2= 4:7 GHz, the YIG-dominated mode ex- hibits a phase advance close to =2 compared with the Py-dominated mode, which agrees with the theoretical prediction. For the Py-dominated mode, the extracted value ofPy=0:99also agrees with theoretical pre- diction ofPy=due to the additional =2 phase oset from the perpendicular Oersted eld from the non- local antenna. Below 4.7 GHz, the anomalous phase drift5 is accompanied with the linewidth drift and is due to the weak signals. Thus we consider this low-frequency phase drift as to be an artifact due to weak signals rather than a signicant eect. Note that the nonlocal excita- tion schematic should eliminate the spurious phase oset due to the complex excitation prole, because the out- of-plane Oersted eld is rather uniform. The YIG uniform modes exhibit a consistent phase of Py==2 in both Figs. 3(a) and (b). Note that we still usePyto represent the phase because the spin rec- tication signals come from the motion of the Py layer induced by the resonance of YIG via the interfacial ex- change [51, 66]. The value of Pysuggests a dominat- ing in-plane Oersted eld on the YIG layer from the mi- crowave current owing through the adjacent Py layer. For the YIG(70 nm)/Py(9 nm) single device, the only Py layer acts as an antenna which is highly ecient in exciting the YIG uniform mode [Fig. 3(c)]. For the YIG(85 nm)/Py(9 nm) nonlocally excited device, the un- changedPy==2 shows that the perpendicular eld from the nonlocal Py antenna is still insignicant com- pared with the induced microwave current in the Py de- vice being electrically measured, with the latter much more ecient in producing an in-plane Oersted eld on the YIG layer underneath. Note that the sign change ofPyfrom the Py-dominated uniform mode is caused by the negative value of gHfrom antiferromagnetic cou- pling, adding an additional phase to the YIG uniform mode. A similar observation has also been reported in Ref. [57]. In conclusion, we have demonstrated phase-resolved electrical measurements of YIG/Py bilayer devices with strong magnon-magnon coupling. The micron-wide and nanometer-thick devices serve as an on-chip miniatur- ized two-cavity hybrid system, where the two microwave cavities are composed of two exchange-coupled thin lay- ers of magnon resonators. Furthermore, the unique cou- pling mechanism and the conned magnon resonance al- low versatile geometric conguration, such as the non- local device, as well as convenient electrical excitation and detection. In the recent rapid development of cav- ity spintronics and magnon hybrid systems [67{71], lots of emerging physics and device engineering including ex- ceptional points [31, 32], level attraction [11, 33, 34] and nonreciprocity [35, 36] have utilized coherent interaction of dierent microwave ingredients. Our results provide a platform for implementing and realizing these ndings in geometrically conned, thin-lm based dynamic systems and for studying the driving and coupling interactions, which are critical for applications in coherent information processing. Work at Argonne on sample preparation and char- acterization was supported by the U.S. Department of Energy, Oce of Science, Basic Energy Sciences, Mate- rials Sciences and Engineering Division, while work at Argonne and National Institute of Standards and Tech-nology (NIST) on data analysis and theoretical modeling was supported as part of Quantum Materials for Energy Ecient Neuromorphic Computing, an Energy Frontier Research Center funded by the U.S. DOE, Oce of Sci- ence, Basic Energy Sciences (BES) under Award #DE- SC0019273. Use of the Center for Nanoscale Materials, an Oce of Science user facility, was supported by the U.S. Department of Energy, Oce of Science, Oce of Basic Energy Sciences, under Contract No. DE-AC02- 06CH11357. W. Z. acknowledges support from AFOSR under grant no. FA9550-19-1-0254. DATA AVAILABILITY The data that support the ndings of this study are available from the corresponding author upon reasonable request. novosad@anl.gov yYi Li and Chenbo Zhao contributed equally to this paper [1] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 127003 (2013). [2] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [3] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [4] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, Phys. Rev. Applied 2, 054002 (2014). [5] B. Bhoi, T. Cli, I. S. Maksymov, M. Kostylev, R. Aiyar, N. Venkataramani, S. Prasad, and R. L. Stamps, J. Appl. Phys. 116, 243906 (2014). [6] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015). [7] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Nature Communi. 6, 8914 (2015). [8] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [9] D. Lachance-Quirion, Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, and Y. Nakamura, Science Advances 3(2017), 10.1126/sciadv.1603150. [10] L. Bai, M. Harder, P. Hyde, Z. Zhang, C.-M. Hu, Y. P. Chen, and J. Q. Xiao, Phys. Rev. Lett. 118, 217201 (2017). [11] M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, Phys. Rev. Lett. 121, 137203 (2018). [12] Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Ho- mann, and W. Zhang, Phys. Rev. Lett. 123, 107701 (2019). [13] J. T. Hou and L. Liu, Phys. Rev. Lett. 123, 107702 (2019).6 [14] L. McKenzie-Sell, J. Xie, C.-M. Lee, J. W. A. Robinson, C. Ciccarelli, and J. A. Haigh, Phys. Rev. B 99, 140414 (2019). [15] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura, Science 367, 425 (2020). [16] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya- mazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Phys. Rev. Lett. 116, 223601 (2016). [17] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016). [18] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv. 2, e1501286 (2016). [19] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurd- equint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Phys. Rev. B 101, 060407 (2020). [20] C. Zhao, Y. Li, Z. Zhang, M. Vogel, J. E. Pearson, J. Wang, W. Zhang, V. Novosad, Q. Liu, and A. Ho- mann, Phys. Rev. Applied 13, 054032 (2020). [21] S. Klingler, V. Amin, S. Gepr ags, K. Ganzhorn, H. Maier-Flaig, M. Althammer, H. Huebl, R. Gross, R. D. McMichael, M. D. Stiles, S. T. B. Goennenwein, and M. Weiler, Phys. Rev. Lett. 120, 127201 (2018). [22] J. Chen, C. Liu, T. Liu, Y. Xiao, K. Xia, G. E. W. Bauer, M. Wu, and H. Yu, Phys. Rev. Lett. 120, 217202 (2018). [23] H. Qin, S. J. H am al ainen, and S. van Dijken, Sci. Rep. 8, 5755 (2018). [24] Y. Li, W. Cao, V. P. Amin, Z. Zhang, J. Gibbons, J. Skle- nar, J. Pearson, P. M. Haney, M. D. Stiles, W. E. Bailey, V. Novosad, A. Homann, and W. Zhang, Phys. Rev. Lett.124, 117202 (2020). [25] Y. Fan, P. Quarterman, J. Finley, J. Han, P. Zhang, J. T. Hou, M. D. Stiles, A. J. Grutter, and L. Liu, Phys. Rev. Applied 13, 061002 (2020). [26] Y. Xiong, Y. Li, M. Hammami, R. Bidthanapally, J. Skle- nar, X. Zhang, H. Qu, G. Srinivasan, J. Pearson, A. Ho- mann, V. Novosad, and W. Zhang, Sci. Rep. 10, 12548 (2020). [27] Y. Xiong, Y. Li, R. Bidthanapally, J. Sklenar, M. Ham- mami, S. Hall, X. Zhang, P. Li, J. E. Pearson, T. Sebastian, G. Srinivasan, A. Homann, H. Qu, V. Novosad, and W. Zhang, IEEE Trans. Magn. (2020), 10.1109/TMAG.2020.3013063. [28] D. MacNeill, J. T. Hou, D. R. Klein, P. Zhang, P. Jarillo- Herrero, and L. Liu, Phys. Rev. Lett. 123, 047204 (2019). [29] L. Liensberger, A. Kamra, H. Maier-Flaig, S. Gepr ags, A. Erb, S. T. B. Goennenwein, R. Gross, W. Belzig, H. Huebl, and M. Weiler, Phys. Rev. Lett. 123, 117204 (2019). [30] C. Dai, K. Xie, Z. Pan, and F. Ma, J. Appl. Phys. 127, 203902 (2020). [31] D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. You, Nature Commun. 8, 1368 (2017). [32] X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Phys. Rev. Lett. 123, 237202 (2019). [33] B. Bhoi, B. Kim, S.-H. Jang, J. Kim, J. Yang, Y.-J. Cho, and S.-K. Kim, Phys. Rev. B 99, 134426 (2019). [34] I. Boventer, C. D or inger, T. Wolz, R. Mac^ edo, R. Le- brun, M. Kl aui, and M. Weides, Phys. Rev. Research 2, 013154 (2020). [35] Y.-P. Wang, J. W. Rao, Y. Yang, P.-C. Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C.-M. Hu, Phys. Rev. Lett.123, 127202 (2019). [36] X. Zhang, A. Galda, X. Han, D. Jin, and V. M. Vinokur, Phys. Rev. Applied 13, 044039 (2020). [37] H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960). [38] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub- ota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005). [39] G. D. Fuchs, J. C. Sankey, V. S. Pribiag, L. Qian, P. M. Braganca, A. G. F. Garcia, E. M. Ryan, Z.-P. Li, O. Ozatay, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 91, 062507 (2007). [40] Y. S. Gui, N. Mecking, X. Zhou, G. Williams, and C.-M. Hu, Phys. Rev. Lett. 98, 107602 (2007). [41] J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman, and D. C. Ralph, Nature Phys. 4, 67 (2008). [42] H. Kubota, A. Fukushima, K. Yakushiji, T. Naga- hama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and Y. Suzuki, Nature Phys. 4, 37 (2008). [43] W. Chen, J.-M. L. Beaujour, G. de Loubens, A. D. Kent, and J. Z. Sun, Appl. Phys. Lett. 92, 012507 (2008). [44] D. Fang, H. Kurebayashi, J. Wunderlich, K. V yborn y, L. P. Z^ arbo, R. P. Campion, A. Casiraghi, B. L. Gal- lagher, T. Jungwirth, and A. J. Ferguson, Nature Nano. 6, 413 (2011). [45] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [46] L. Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlaminck, J. E. Pearson, S. D. Bader, and A. Homann, Phys. Rev. Lett. 111, 217602 (2013). [47] Y. Gui, L. Bai, and C.-M. Hu, Sci. China Phys. Mech. Astron. 56, 124{141 (2013). [48] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, Nature 511, 449 (2014). [49] W. Zhang, M. B. Jung eisch, W. Jiang, J. E. Pearson, A. Homann, F. Freimuth, and Y. Mokrousov, Phys. Rev. Lett. 113, 196602 (2014). [50] J.-C. Rojas-S anchez, N. Reyren, P. Laczkowski, W. Savero, J.-P. Attan e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and H. Jar es, Phys. Rev. Lett. 112, 106602 (2014). [51] J. Sklenar, W. Zhang, M. B. Jung eisch, W. Jiang, H. Chang, J. E. Pearson, M. Wu, J. B. Ketterson, and A. Homann, Phys. Rev. B 92, 174406 (2015). [52] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. Brown, and N. X. Sun, Phys. Rev. B 91, 214416 (2015). [53] M. B. Jung eisch, W. Zhang, J. Sklenar, J. Ding, W. Jiang, H. Chang, F. Y. Fradin, J. E. Pearson, J. B. Ketterson, V. Novosad, M. Wu, and A. Homann, Phys. Rev. Lett. 116, 057601 (2016). [54] M. Harder, Y. Gui, and C.-M. Hu, Phys. Rep. 661, 1 (2016). [55] S. Li, W. Zhang, J. Ding, J. E. Pearson, V. Novosad, and A. Homann, Nanoscale 8, 388 (2016). [56] P. Hyde, L. Bai, D. M. J. Kumar, B. W. Southern, C.-M. Hu, S. Y. Huang, B. F. Miao, and C. L. Chien, Phys. Rev. B 89, 180404 (2014). [57] W. L. Yang, J. W. Wei, C. H. Wan, Y. W. Xing, Z. R. Yan, X. Wang, C. Fang, C. Y. Guo, G. Q. Yu, and X. F. Han, Phys. Rev. B 101, 064412 (2020).7 [58] Y. Li, F. Zeng, S. S.-L. Zhang, H. Shin, H. Saglam, V. Karakas, O. Ozatay, J. E. Pearson, O. G. Heinonen, Y. Wu, A. Homann, and W. Zhang, Phys. Rev. Lett. 122, 117203 (2019). [59] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. B 93, 174427 (2016). [60] S. Yoon, J. Liu, and R. D. McMichael, Phys. Rev. B 93, 144423 (2016). [61] A. Capua, T. Wang, S.-H. Yang, C. Rettner, T. Phung, and S. S. P. Parkin, Phys. Rev. B 95, 064401 (2017). [62] Y. Li, H. Saglam, Z. Zhang, R. Bidthanapally, Y. Xiong, J. E. Pearson, V. Novosad, H. Qu, G. Srinivasan, A. Ho- mann, and W. Zhang, Phys. Rev. Applied 11, 034047 (2019). [63] Y. Li, F. Zeng, H. Saglam, J. Sklenar, J. E. Pearson, T. Sebastian, Y. Wu, A. Homann, and W. Zhang, IEEE Trans. Magn. 55, 6100605 (2019).[64] J. Wei, C. He, X. Wang, H. Xu, Y. Liu, Y. Guang, C. Wan, J. Feng, G. Yu, and X. Han, Phys. Rev. Applied 13, 034041 (2020). [65] W. Wang, T. Wang, V. P. Amin, Y. Wang, A. Radhakr- ishnan, A. Davidson, S. R. Allen, T. J. Silva, H. Ohldag, D. Balzar, B. L. Zink, P. M. Haney, J. Q. Xiao, D. G. Cahill, V. O. Lorenz, and X. Fan, Nature Nano. 14, 819 (2019). [66] T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev. Applied 2, 034003 (2014). [67] B. Bhoi and S.-K. Kim, Solid State Phys. 70, 1 (2019). [68] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Us- ami, and Y. Nakamura, Appl. Phys. Express 12, 070101 (2019). [69] S. V. Kusminskiy, arXiv 1911.11104. [70] Y.-P. Wang and C.-M. Hu, J. Appl. Phys. 127, 130901 (2020). [71] Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Ho- mann, and V. Novosad, J. Appl. Phys. 128, 130902 (2020).

2021-05-24

We demonstrate the electrical detection of magnon-magnon hybrid dynamics in yttrium iron garnet/permalloy (YIG/Py) thin film bilayer devices. Direct microwave current injection through the conductive Py layer excites the hybrid dynamics consisting of the uniform mode of Py and the first standing spin wave ($n=1$) mode of YIG, which are coupled via interfacial exchange. Both the two hybrid modes, with Py or YIG dominated excitations, can be detected via the spin rectification signals from the conductive Py layer, providing phase resolution of the coupled dynamics. The phase characterization is also applied to a nonlocally excited Py device, revealing the additional phase shift due to the perpendicular Oersted field. Our results provide a device platform for exploring hybrid magnonic dynamics and probing their phases, which are crucial for implementing coherent information processing with magnon excitations

Phase-resolved electrical detection of coherently coupled magnonic devices

2105.11057v1

Magnon Hall effect and a nisotropic thermal transport in NiFe and YIG ferrom agnets B.Madon1, Do Ch. Pham1, D. Lacour2, A. Anane3, R. Bernard3, V. Cros3, M. Hehn2 and J.-E. Wegrowe*1 1Ecole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palaiseau , France . 2Institut Jean Lamour UMR 7198 CNRS, Université de Lorraine, Vandoeuvre les Nancy France . 3Unité Mixte de Physique CNRS/Thales and Universit é Paris Sud, Palaiseau , France . *Correspondence to :jean-eric.wegrowe@polytechnique. edu. Abstract: The Righi -Leduc effect refers to the thermal analogue of the Hall effect, for which the electric current is replaced by the heat current and the electric field by the temperature gradient. In both cases, the magnetic field generates a transverse force that deviates the carriers (electron, phonon, magnon) in the direction perpendicular to the current. In a ferromagnet, the magnetization plays the role of the magn etic field, and the corresponding effect is called anomalous Hall effect . Furthermore, a second transverse contribution due to the anisotropy, the planar Hall effect, is superimposed to the anomalous Hall effect. We report experimental evidence of the ther mal counterpa rt of the Hall effect s in ferromagnet s, namely the magnon Hall effect (or equivalently the anomalous Righi -Leduc effect) and the planar Righi -Leduc effect , measured on ferromagnets that are either electrical conductor (NiFe) or insulator (YIG) . The study shows the universal character of these new thermokinetic effects, related to the intrinsic chirality of the anisotropic ferromagnetic degrees of freedom. 2 Intro duction We report a new effect that could be added to the large family of the rmokinetic transport phenomena . It consists in the observation of both anomalous Righi -Leduc effect - or magnon Hall effect [1,2]- and planar Righi -Leduc effect , measured on YIG and NiFe ferromagnets . The conventional Righi -Leduc effect is the thermal counterpart of the well -known Hall effect , and it accounts for the temperature gradient developed transversally to a heat current under a magnetic field. The adjectives anomalous and planar – that characterize the effect reported here - refer to the action of the magnetization axial vector (instead of a magnetic field) and the corresponding vector potential . The application of a magnetic axial vector result s in the partial break ing of two different symmetries . These symmetries are , on the one hand , the invariance under time reversal of the dynamical equations at the microscopic scale [3], and on the other hand, the rotational invariance (for an initially isotropic system ). However, the symmetry breaking is partial . Indeed, in the first case, the time reversal invariance is recovered by the application of a rotation to the magnetization , and i n the second case, the symmetry breaking is partial because the system is still invariant under any rotation around the magnetization . The consequence of the se reduced symmetries is to impose a specific form to the heat transport coefficients [4] (see Supplementary), so that the temperature gradient becomes a very specific function of the magnetization states (as shown in Eq. (1) below) . Since t he addition of a thin electrode in thermal contact with both edges of the ferromagnetic layer (see the set -up of Fig1) plays the role of a Seebeck thermometer (or thermocouple), the temperature difference T is converted to a voltage difference V, allowing 3 the measurement of a magneto -voltaic signal. Such a device defines the principle of a magneto - thermal sensor. The studies of magneto -voltaic signals measured in response to thermal excitations on ferromagnetic layers has attracted considerable a ttention in the last years, with the observation of similar signals in conductor (NiFe), semiconductor (GaMnAs), and insulator (YIG) [5-13]. Fig.1: Schematic views of typical device s including a ferromagnet (either a conductor or an insulator ) and two transversal non-ferromagnet ic electrode s (noted heater and probe) . In our study, the electrodes have a width of 200µm and are spaced 5mm apart. The direction of the magnetization is defin ed by the angles θ and φ. The anomalous Righi -Leduc effect has been predicted in various magnetic systems [14- 17] and it has been measured recently in peculiar insulating ferromagnetic materials that possess a chiral crystalline structure [1,2]. The study of the anomalous Righi -Leduc effect in usual ferromagnetic layers ( e.g. NiFe and YIG) has however been overlooked. On the other hand, the planar Righi -Leduc effect refers to the contribution of the anisotropy in the thermal conductivity. This anisotropy originates from the difference r between the t hermal resistivity measured along 4 the magnetization axis and the thermal resistivity perpendicular to the magnetization axis [18]. The comparative study between anomalous and planar Righi -Leduc effects, in NiFe and YIG ferromagnets , allows us to make a call in favor of a unifying interpretation in terms of anisotropic thermal transport (AThT), and to point out the universality of the phenomenon. Experimental angular dependences The sample s contained electrodes that have been fabricated using the same se t of shadow mask s in a sputtering deposition system . They are fixed on top of two different magnetic material s. The first sample include s a 20nm thick Ni80Fe20 conductor stripe while the second sample contains a 20nm thick ferromagnetic YIG insulator [19]. Electrode s composed of Platinum (Pt) are deposited on top of each magnetic layer (see fig . 1). An ac electric current I(t) = I0 cos(t) is injected into the heater electrode (the power is of the order of a fraction of Watt and the frequency is a fraction of Hz ). It produces a heat current 2/)1)2 (cos( )(2 0 t cRItJQ having twice the frequency of the electric current (c is a constant th at takes into account the power dissipation (see Supplementary ). The voltaic response ΔV y to the thermal excitation is measured using a lock-in method via a probe electrode placed 5mm away from the heater electrode . All the measurements are done under a magnetic field H of 1 Tesla that serves to rotate the magnetization . The voltage ),(H H yV has been recorded for the two aforementioned devices either by varying the azimuthal angle φH while keeping the polar angle θH fixed to 90° or by varying the polar angle keeping φH fixed to 90° (see fig. 2) . First we observe that in both cases (conductor and insulator ferro magnetic material ), π- periodic signals are measured in the magnetization in-plane (IP) configuration , while 2π -periodic signals are observed in the out-of-plane (OOP) configuration . Second w e find that the angular 5 voltage variations display opposite phase on NiFe/Pt and on YIG/Pt . Finally , a triangular rather than a sinusoidal feature is observed in the NiFe sample for the measurements under an out -of- plane field . Fig.2 Transverse voltages vs. direction of the 1T magnetic field H. The results collected in two different measu rement geometries are presented: In-plane configuration for which the polar angle θH is fixed and equal to 90° , the azimuthal angle φH is varied ( ); Out-of- plane configuration for which the azimuthal angle φH is fixed to 90° and the polar angle is varied ( ). The data corresponding to the ferromagnetic conductor case (Ni 80Fe20) and the ferromagnetic insulator case (YIG) are represented in the top graphs (purple color) and in the bottom graphs ( red colors) respectively. Anisotropic Thermal Transport (AThT) According to the AThT phenomenology (see Supplementary ), a heat current QJ injected inside the ferromagnet generates a thermal gradient ),(T related to the orientation of the magnetization (θ,φ). Its description , based on t he anisotropic Fourier equation, is valid both for 6 the electric conductor and for the insulator . The probe electrode serves as a thermocouple that convert s a local transverse temperature difference into a voltage rJSQ x. . The parameter stands for the difference between the Seebeck coefficients of the materials that compose the device . Considering that the heat current is along the x direction, the transverse voltage yV is given by the expression [4]: cos )2sin().(sin2.2 RLQ x y rrJS V Eq.(1), where is the planar Righi -Leduc coefficient and is the an omalous Righi -Leduc coefficient. From Eq.(1), the period s observed in Fig.2 can be easily understood . The term allows to explain the π -periodic ity in the IP configuration while the 2π -periodic signals are linked the term that occur only in the OP configuration . Moreover we can also predict from Eq.( 1), that the magnitude of the oscillations are equal to rJSQ x. for the IP configurations and equal to ARLQ xrJS. for the OOP configuration . Using an independent measurement setup, we have determined S for the NiFe and YIG based devices to be respectively -16.2µV.K-1 and 0.69µV.K-1 (see Supplementary ). The opposite signs of S provide a straightforward explanation for the aforementioned "antiphase" feature observed in Fig. 2 comparing the results on NiFe/Pt and YIG/Pt . Finally taking into account the magnetic propert ies of the ferromagnetic layers (see Supplementary ), we were able to fit al l measurements the only free parameters were either rJSQ x. (in the IP configuration ) or ARLQ xrJS. (in the OP configuration ). It can be seen on Fig.2 (grey lines) that all the experimental results are in excellent agreements with our interpr etation based on anisotropic thermal transport in the ferromagnet . The triangular profile (rather than sinusoidal ) exhibited by the Ni80Fe20 device in the OP configuration is simply due to 7 the fact that a 1 Tesla magnetic field i s not large enough to fully saturate the magnetization perpendicular to the plane of the film (see Supplementary ). To test the robustness of the AThT explanation , we have first varied the thickness of the Pt probe from 5nm to 100nm. Defining the maximum amplitude of the magneto -voltaic signal 0 180y y y V V V . A decrease of dVy as a function of the probe thickness is observed ( Fig.3a). Such a decrease is often interpreted as the effect of spin injection at the interface [20-22]. Here we demonstrate that the thermal shunt effect suffice s to explain the data . Not all the injected heat current Q xJ* contributes to the AThT effect since a part of this current is also flowing into the neutral Pt electrode. In order to evaluate the active part of the heat current, we can rewrite : Q x NiFe Th NiFePt ThNiFePt Th Q x Jd ddJ* . .. Eq.(2), assuming a simple scheme of two thermal conductors in parallel , as for anisotropic Hall measurements [23] (see Supplementary ). The dependence of the signal on the thickness d of the Pt probe is calculate d using the tabulated value s Py Th/1 =72W.m-1.K-1 for Ni 80Fe20 and Pt Th/1 = 46W.m-1.K-1 for Pt, and the value of ARLQ xrJS ..* presented in Fig. 2 (see Supplementary ). From the good agreement between the experimental curve and the prediction of Eq. (2) in Fig.3a) , we conclude that the sole thermal shunt effect suffices to reprodu ce the observed decrease (the electrical counter part of the shunt effect is also reproduced without adjustable parameter , as shown in of Supplementa ry Fig. S9. 8 Fig 3 . a) Voltages difference yV vs. thickness of the Pt electrode d (). The gray line presents the expected dependence taking into account only a thermal shunt effect. b) Transverse voltage yV vs. θH ( ) for a device composed of a C u electrode (No PE effect, no ISHE). Moreover, the AThT interpretation does not require to invoke the hypothesis on inverse spin Hall effect (ISHE) or of a proximity effect arising from induced magnetic moments [11, 13]. In order to verify this stat ement , we have replaced the Pt electrodes by ultra -pure copper ones (99.9999% purity target). Indeed, the use of Cu electrodes allows to test at the same time the ISHE and PE hypotheses since both effects are absent in Cu [24,25 ,11]. We observed the magneto -voltaic signal even wi th pure copper electrode (Fig .3 b), as expected for AthT . Discussion and Conclusion . We have observed the coexistence of both anomalous and planar Righi -Leduc contributions in NiFe and YIG, of comparable amplitude s (leading to a transverse temperature difference of the order of 10 mK ). 9 Although t he anomalous Righi -Leduc effect can simply be understood on the basis of the Onsager reciprocity relations [3] (Supplementary Equations S2), a microscopic description can also been performed with a dedicated vector potential – or the corresponding local gauge and Berry phase – associated to the ferromagnetic system under consideration [26-28,14 -17]. This problem generalizes sixty years of intensive theoretical development related to the anomalous Hall effect (starting with the work of Karplus and L uttinger in 1954 [29], and summarized e.g. in the review by Nagaosa et al [30]). Like the Lorentz force in the case of the conventional Hall effect, and like the spin -orbit scattering force in the case of anomalous Hall effect, the transversal force measured in this study can be derived from a vector potential . This force is thus neither conservative (it cannot be derived from a scalar potential) nor dissipative (no power can be extracted). A second transverse force is observed , which is generated by the anisotropy of the ferromagnetic excitations r ≠0. T he measurements show that the two forces are not independent: the anomalous Righi -Leduc coefficient is associated to the planar Righi -Leduc coefficient . The same ferromagnetic axial vector is indeed responsible for both the anisotropy of the heat resistance (r ≠ 0) and the breaking of the time invariance symmetry . In conclusion, our results show that the anomalous Righi -Leduc effect, which has already been observed in specific ferroma gnetic structures, is universal . This effect is observed in parallel to the planar Righi -Leduc effect. Both planar and anomalous Righi -Leduc effects should be present in any ferromagnetic materials in the same manner as anomalous and planar Hall effects can be expected a priori in any ferromagnetic conductors. 10 References and Notes: [1] Onose, Y. et al. Observa tion of the M agnon Hall effect. Science 329, 297-299 (2010). [2] Ideue , T. et al. Effect of lattice geometry on magnon Hall effect in ferromagnetic insulators. Phys. Rev. B 85, 134411 (2012). [3] Onsager , L. RECIPROCAL RELATION IN IRREVERSIBLE PROCESS II. Phys. Rev. 38, 2265 -2279 (1931). [4] Wegrowe, J.-E., Lacour, D., & Drouhin , H.-J. Anisotropic magnetothermal transport and spin Seebeck effect . Phys. Rev. B 89, 094409 (2014). [5] Uchida , K. et al. Observation of the spin Seebeck effect . Nature 455, 778-781 (2008). [6] Uchida , K. et al . Spin Seebeck insulator. Nature Mater. 9, 894 -897 (2010) . [7] Jaworsky , C. M. et al. Observation of the spin -Seebeck effect in a ferromagnetic semiconductor . Nature. Mater. 9, 898-903 (2010). [8] Huang, S. Y. , Wang, W. G. , Lee, S. F., Kwo, J. & Chien, C. L. Intrinsic Spin-Dependen t Thermal Transport. Phys. Rev. Lett. 107, 216604 (2011). [9] Avery, A. D. , Pufall, M. R. & Zink , B. L. Observation of the Planar Nernst effect in permalloy and Nickel thin films with in -plane thermal gradient. Phys. Rev. Lett. 109, 196602 (2012). [10] Schultheiss, H., Pearson , J. E., Bader, S. D. , & Hoffmann, A. Thermoelectric Detection of Spin Waves. Phys. Rev. Lett. 109, 237204 (2012). [11] Huang , S. Y. et al. Transport M agnetic Proximity Effects in P latinium. Phys. Rev. Lett. 109, 107204 (2012). [12] Schmid , M. et al. Transverse Spin Seebeck Effect versus Anomalous and Planar Nernst Effects in Permalloy Thin F ilms. Phys. Rev. Lett. 111, 187201 (2013). 11 [13] Kikkawa , T. et al. Longitudinal Spin Seebeck Effect Free from the Proximity Nernst Effect . Phys. Rev. Lett. 110, 067207 (2013). [14] Fujimoto , S. Hall Effect of Spin Waves in Frustrated M agnets. Phys. Rev. Lett. 103, 047203 (2009). [15] Katsura, H., Nagaosa, N., & Lee, P. A. Theory of the M agnon Hall E ffect in Quantum Magnets. Phys. Rev. Lett. 104, 066403 (2010). [16] Matsumoto, R. & Murakami , S. Theor etical Prediction of a Rotatin Magnon Wave Packet in F erromagn ets. Phys. Rev. Lett. 106, 197202 (2011). [17] Qin, T., Niu, Q. & Shi, J. Energy Magnetization and the Thermal Hall effect. Phys. Rev. Lett. 107, 236601 (2011). [18] Kimling, J., Gooth, J., & Nielsch, K. Anisotropic Magnetothermal Resistance in Ni Nanowires. Phys. Rev. B 87, 094409 (2013). [19] O. d’Allivy Kelly et al. Inverse spin Hall effect in nanometer -thick yttrium iron garnet/Pt system. Appl. Phys. Lett. 103, 082408 (2013). [20] Nakayama, H. et al. Geometry dependence on inverse spin Hall effect induced by spin pumping in Ni 81Fe 19/Pt films. Phys. Rev. B 85, 144408 (2012). [21] Althammer , M. et al. Quantitative study of the spin Hall magnetoresistance in ferromagnetic insulator/normal metal hybrids . Phys. Rev. B 87, 224401 (2013). [22] Castel, V., Vlietstra, N., Ben Youssef , J. & van Wees , B. J. Platinum thickness dependence of the inverse spin -Hall voltage from spin pumping in a hybrid yttrium iron garnet/platinum system. Appl. Phys. Lett. 101, 132 414 (2012). [23] Xu, W. J. et al. Scaling law of anomalous Hall effect in Fe/Cu bilayers. Eur. Jour. Phys. 12 B 65, 233-237 (2008). [24] Niimi, Y. et al. Extrinsic Spin Hall Effect Induced by Iridium Impurities in Copper. Phys. Rev. Lett. 106, 126601 (2011). [25] Harii , K., Ando , K., Inoue , H. Y., Sasage , K. & Saito h, E. Inverse spin -Hall effect and spin pumping in metallic films. J. App l. Phys. 103, 07F311 (2008). [26] Holstein , B. R. The adiabatic theorem and Berry’s phase . Am. J. Phys. 57, 1079 -1084 (1989 ). [27] Bruno , P. Nonquantized Dirac monopoles and strings in the Berry ph ase of anisotropic spin systems. Phys. Rev. Lett. 93, 247202 (2004 ). [28] Matsumoto, R., Shindou, R. & Murakami , S., Thermal Hall effect of magnon in magnets with dipolar interaction. Phys. Rev. B 89, 054420 (2014 ). [29] R. Karplus and J. M. Luttinger, Hall Effect in Ferromagnetics . Phys. Rev. 95, 1154 -1160 (1954 ). [30] Nagaosa, N., Sinova, J., S. Onoda, MacDonald, A. H. & Ong, N. P. Anomalous Hall Effect. Rev. Mod. Phys. 82, 1539 -1592 (2010). Acknowledgements The Authors acknowledge H. Molpeceres and A. Jacquet for their assistance and O. d’Allivy Kelly for fruitful discussion. Financial funding RTRA `Triangle de la physique’ Projects DEFIT n° 2009 -075T and DECELER 2011 -085T , the FEDER, France, La région Lorraine, Le grand Nancy, ICEEL and the ANR -12-ASTR -0023 Trinidad is greatly acknowleged . 13 Suppl ementary M aterials I Magnetic and electric charact erization of the 20 nm thick permalloy (Ni 80Fe20) samples . I -1. Ferrom agnetic quasi -static states. Due to the thin layer structure, the magnetization of t he Permalloy (Ni80Fe20 or Py) layer is single domain. As a consequence, the magnetization mM Ms is a vector of constant modulus Ms (magnetization at saturation) oriented along the unit vector m . The quasi -static magnetizati on states are given by the minimum of the ferromagnetic free energy. This energy depends on three parameters, namely the magnetization at saturation M s, the demagnetizing field H d, and the magnetocrystalline anisotropy field Han, confined in the plane of t he layer. The corresponding energy is the s um of the three terms: 2 2cos21sin21.s d a s an MH MH MH F Eq.(S1) where ),( mHan a is the angle between the magnetocrystalline anisotropy axis and the magnetization, and is the angle between t he vector n normal to the plane of the layer and the magnetization. The minimum of the energy F (Eq.(S1)) sets the position of the magnetization, i.e. the radial angle and the azimuthal angle as a f unction of the amplitude H and direction H andH of the applied field. The minimum is calculated through numerical methods (Mathematica® program). The magnetization states were characterized using anisotropic electric transport properties, with the use of three different experimental configurations , which correspond to anisot ropic magnetoresistance (AMR) (31), planar Hall effect (PHE), and anomalous Hall effect (AHE )(30). I-2. Electric properties The electric transport is described by the Ohm’s law that relates the electric field to the electric current eJ with the use of the conductivity tensor: eJ.ˆ (note that for convenience, the experiments are usual ly performed in a galvanostatic mode, i.e. with constant current distribution eJ). For a polycrystalline conducting ferromagnet, the conductivity tensor is defined by three parameters . If the reference frame is such that the unit vector is aligned along Oz, the parameters are the resistivity measured perpendicular to magnetization, the resistivity z measured parallel t o magnetization, and the Hall cross -coefficient H. According to Onsager reciprocity relation H xy = -yx and we have in the reference frame {x,y,z} : zHH 0 000 ˆ Accordingly, t he Ohm’s law can be expressed in an arbitrary reference frame, as (31): 14 e He zeJm mmJ J . . or, explicitly: e z ze y x H z ye x y H z xe z x H z ye y xe x z H y xe z y H z xe y z H y xe x x Jm Jm mm Jm mmJm mm Jm Jm mmJm mm Jm mm Jm 222 , Eq. (S2) where =z cos sinxm , sin sinym , coszm . The angle θ is the same radial angle as the one introdu ced in the magnetic free energy, is the azimuthal angle between the direction Ox and the projection of the magnetization in the film plane . After integration, Eq. ( S2) gives the magneto -voltaic signals that corresponds to the Anisotropic magnetoresistance (diagonal terms) , the anomalous magn etoresis tance (second term of the non -diagonal matrix elements) , and the planar magnetoresi stance (first term of the non -diagonal matrix elements) . The same line of reasoning is applied in section II-1 below for the transport of heat . I-2-1. Anisotropic magnetoresistance (AMR) For AMR measurements , the voltage is measured along the same axis as the current flow (see Fig.1). The voltage is given by the integration over x of the first line in Eq. (2) with 0e ye zJ J . Fig.S1: Resistance as a function of the amplitude of the external perpendicular field at = 0° for (a) =0 and (b) zoom for =5°, =23° and =50°. The points are the measured data and the line is the fit calculated from the minimization of the en ergy Eq.(S1) and Eq.(S2). Figure S1 shows the resistance as a function of the external perpendicu lar field at = 0. The fitted parameters are Hd = 1T and the AMR ratio is found to be R/R= 1.83% . Note that the saturation is not reached for H=1T. Consequently, the direction of the magnetization ( ,) does not exactly coincide with that of the external field (H,H): Indeed we have exploited this behavior in order to show that the magneto -voltaic signal is not a response to the external magnetic field (i.e. it is not the usual Nernst or Righi -Leduc effect) , but a response to the magnetization (i.e. it is either the anisotropic Nernst or the anisotropic Righi -Leduc effect) . On the other hand, t he in-plane magnetocrystalline anisotropy field Han is very weak , about 5.10- 4 T, but its effect is rather dramatic as shown in Fig. 2. In the vicinity of H = 0° (modulo 180°) , the magnetization suddenly switche s from its initial position imposed by the applied field from 15 H = 0° or = 90° to = 30° which is the direction of in plane anisotropy . This jump is well reproduced by the numerical simulation shown in Fig.S2. Fig.S2: Magnetoresistance ratio (AMR) as a function of the out -of-plane angle for an external field of H=0.2T at (black upper curve), and at (lower curve ). If is close to zero modulo the magnetization swi tches to the direction (which corresponds to the plane defined by the external field and the anisotropy field). I-2-2. Anomalous Hall effect (AHE) and Planar Hall effect (PHE) Fig.S3: Configuration for AHE and PHE measurements. For planar Hall e ffect (PHE) and anomalous Hall effect (AHE) , the electric current is injected along 0x axis, but the voltage is now measured on the transverse electrode , along 0y (see F ig.S3). The voltage is given by the integration along the electrode of the second line of equation (2) with 0e ye zJ J : cos 2sin sin2' 2 H x y BRRAI V Eq. S3 The first term is due to PHE while the second term is due to AHE. The coefficients A’ and B are fitting parameters of the order of L/A where L is the distance between the two co ntacts and A is 16 the section of the electrode (A’ and B also include the contact resistance , so that th ey differ slightly from one sample to the other ). The two contributions co -exist for an arbitrary direction of the magnetization , except if the configurat ions are fixed for the external magnetic field = 90° (in plane measurements as a function of for pure planar Hall effect) or at =0° or =90° (out-of plane measurements as a function of for pure anomalous Hall effect) . Figure S4 show s out-of-plane measurements (AHE) as a function of the angle , performed at (A) H=0.2T and H=1T. The calculated curve (continuous line s) follows closely the experimental data for H=0.2T. The jump of the magnetization for close to zero [ resp. 180°] is that described on the AMR measurements presented in Fig.S2. The deviation between calculation and experimental data in Fig. S4(B ) is explained by the metastable states due to the irreversible jump (the hysteresis loop is time dependent) , that are not taken into account in the calculation of the quasi -static states. Fig S4 : (A) Out-of-plane (AHE) voltage as a function the angle H for H=0.2T at H=0°. (B) Same c onfiguration for H =1T. The symbols are th e experimental data and the l ine is calculated based on Eq.(S3) and on minimization of Eq.(S1). Figure S5 shows the in plane measurement s with a saturation field of H=1T. The cur ve follows exactly the expected with a single adjustable parameter RH. Fig S5 : Planar Hall voltage as a function of the angle for an in-plane field (=H=90° of H=1T for the Cu and Pt electrodes. (a) Py(20nm)/Cu(5nm)/ Pt(10nm) and (b) Py(20nm) /Pt(10nm). The presence of Cu does not change the magnetization states. 17 Fig.S6: (a) Measurements of the Hall voltage as a function of the out -of-plane external magnetic field (=0) for different angle . (b) Calculation based on Eqn.S1 and Eqn.S3 Planar Hall effect dominates. Note the brutal reversal from H=180° to H=180.5° . It is the same as the one shown in Fig .S2 and Fig .S4. The m easurements presented in Fig .S6 show that the magnetization states are well characterized by the simulation based on Eqn.S1 and Eqn.S3, and using the parameters fitted as described previously (with in-plane and out -of-plane angular dependence ). I-2-3. AHE and PHE as a function of the thickness of the electrodes Figure S7 show s the dependence of both AHE (a) and PHE (b) as a function of electrodes thicknesses ranging from 5nm to 100nm under an applied field of H=1T. The profile of the curve is not changed by the variation of the thickness, which means that the magn etization states are not impacted by the electrode thickness. Fig .S7 shows that the a mplitude of the signal changes dramatically between 5 and 50 nm. Fig.S7: Measurement of the voltage for different thicknesses of the Pt electrode as a function of the a ngles at H =1T for (a) planar Hall effect ( H = ) and (b) anomalous Hall effect (H ≠ ). The signal V is defined as the voltage difference between the maxima and minima. In order to justify the thickness dependence of the AHE and PHE signals , we first take the assumption that the non -ferromagne tic electrode is passive . The effective current that flows 18 inside the ferromagnetic layer is not t he initial current but it is divided into t wo branches (Fig. S8). A first branch is defined by the resistance of the ferromagnetic layer (shunt effect )(23). Fig. S8: Illustration of the shunt effect that takes place at the level of the electrode. The effect is well described by a two resistor model R Pt and R Py. The thickness dependence is given by the coefficient such that I eff = I. We have: Pt Py Py PtPy Pt d dd Eq.S4 The Py thickness is dPy and that of the Pt electrode is dPt. The corre sponding resistivities are Py and Pt that have been determined by independent resistance measurements. Table.S1 : Parameters used for the calculation of Fig.S9 The typical profile s of the thickness dependence of both the AHE and ANE are presented in Fig.S9. The measured data follows perfectly the profile predicted taking into account the shunt effect. There is no adjustable parameter in the calculation . We took the mean value s of <R/R> and <RH> obtained by averag ing the parameters (Table.S1 ) over all samples. 19 Fig.S9: (A) Anomalous and (B) planar Hall signal s V as a function of the thickness d of the Py electrode. The points are the measured data and the line is the correction (coefficient ) due to the shunt ing effect Eq.(4 ). The excellent agreement between experiments and the predictions show n in Fig. S9 bring as a clear conclusion that the typical thickness dependence is only due to the shunt ing effect. II) Anisotropic Thermal Thransport (AThT) II-1 The anisotropic Fourier equation In order to describe the transport of heat in a ferromagnetic system, we follow an equivalent approach of the one used in section I -2. Indeed, b oth electric and t hermal transport phenomena obey the same symmetry properties, namely the rotational invariance of the system through any rotation around the magnetization axis and the time reversal invariance associated to the rotation . The Fourier law takes th us the same form as the Ohm’s law (the electric current is replaced by a heat current and the electric field by a gradient of temperature). Fourier law relates the gradient of the temperature T to the electric current QJrT ˆ . The conductivity tensor rˆ of a polycrystalline conducting ferromagnet (this is the case of the NiFe samples) is defined by three parameters . If th e reference frame is such that the unit vector m is along Oz, we define the thermal resistance r measured perpendicular to the magnetization, the thermal resistance measured parallel to the magnetization, and the Righi -Leduc cross - coefficient rARL. According to Onsager reciprocity relation, we have in the reference frame {x,y,z} : zARLARL rr rr r r 0 000 ˆ The Fourier ’s law can then be expressed in an arbitrary reference frame, as (4): Q ARLQ QJmrmmJrr JrT . .// where r = rz – r. Explicitly: 20 Q z zQ y x ARL zyQ x y ARL zxQ z x ARL zyQ y xQ x z ARL yxQ z y ARL zxQ y z ARL yxQ x x Jrm Jmr mrm Jmr mrmJmr mrm Jrm r Jmr mrmJmr mrm Jmr mrm Jrm r T 222 , Eq.(S5) where cos sinxm , sin sinym , coszm , θ is the same as the one introduced in the magnetic free energy and is the angle between the direction Ox and the projection of the magnetization in the plane of the sample. The temperature difference Ty can be measured between the two edg es of the ferromagnetic layer along 0y, thanks to the thermocouple effect. The voltage is given by the Seebeck coefficient S, such that Vy = S Ty (see II -3 below) . Since the heat current is mainly along 0x, we obtain the main equation used in this stud y: cos 2sin sin22 ARLQ x y rrSJ V Eq.(S6) The second term in the r ight hand side of Eq.( S6) – proportional to cos - defines the anomalous Righi-Leduc coefficient rARL , that can be measured directly with setting =0 or =90° (out-of- plane measurements) . On the other hand, the first term in the right hand side of Eqn.S6 – proportional to sin(2) (in-plane angle) – defines the planar Righi -Leduc coefficient r, that can be measured directly with se tting (in-plane measurements) . II-2 AThT on NiFe sample In complement to the measurements on NiFe ferromagnet presented in the main text, complementary results obtained with a Cu(5nm)/Pt(10nm) electrode are shown in Fig. S10 and Fig.S11. We observe that the results are identical to that corresponding to the Pt(5 0nm) presented in Fig.2 of the main text (after correction due to the shunting effect ). We can conclude that the Cu(5nm) electrode deposited between the ferromagnet and the Pt does not modify the signals significantly , in agreement with Eq.( S6). The angular dependence s (radial and a zimuthal) for H=1T are plotted in Fig.S10, with the numerical simulation , according to equation ( S6). Fig.S10 and Fig.S11 display supplementary measurements with Cu electrodes. 21 Fig.S10 :(A-C) Transverse voltages vs. dire ction of the 1T magnetic field . Cu(5nm)/Pt(10nm) electrode: (A) In-plane configuration at θH = 90°and (B) out-of-plane configuration for φH = 90°. (C) Cu(20nm) electro de: in -plane configuration (see Fig3B for the out -of-plane configuration). The line s correspond to the calculation of Eq.(6) with minimization of the energy Eq.(1). The Anosotropic Thermal Transport (AThT) signals have been measured a function of the magn etic field (see Fig.11(a) ) for three values of the direction of the applied field ( ). The numerical simulations (continuous lines) are in excellent agreement with the experimental results . The magnetization reversal at small field is shown in the inset . The out -of plane angular variation at H = 0° for a medium magnetic field (H = 0.18T) is plotted in Fig. S11(B) . The irreversible jump of the magnetization (presented in Fig. S2 and Fig. S4) is clearly observed , and descri bed by the numerical simulations . 22 Fig.S11. Transverse voltages on Py/Cu/Pt electrode as a function of (A) amplitude of the magnetic field H for three out -of-plane angles, (B) radial angle H for an applied field of 0.18T. The line correspond to the calculat ion of Eq.( S6) with minimization of the energy Eq.( S1). (C). Transverse voltage on Py(40nm)/Cu(40nm) as a function of the amplitude of the magnetic field for out -of-plane config uration at 180° . II-3. Heat power and magneto -voltaic signal In our experimen t, Joule heating is generated using AC current of pulsation , injected into a resistance through a second electrode deposited on the ferromagnetic layer (see Fig.1 of the main text). The h eat power flowing through the sample is proportional to the square of the current . As a consequence, the magneto -voltaic response to th e heat excitation is measured at the double frequency 2. We checked that the signal is proportional to the injected power as shown in Fig. 12. The extrapolation to zero shows that the hea t current JQ measured at the level of the electrode is simply proportional to the heat power injected by Joule effect: JQ x = c P Joul , where the constant c (such that 0<c<1) contain s all contribution of heat dissipation (including the coefficient ). It depends on the frequency (see Fig. S13) and va ries fro m one sample to the other . The change of the magneto -voltaic signal U observed for different values of the out -of-plane external field (here for H=-1T and H=1T ) is due to the anomalous Righi -Leduc ( or anomalous magnon -Hall) effect studied in this work. 23 Fig.S1 2: Measurement of the 2Magneto -voltaic signal as a function of the Joule power injecte d for different value s of the out -of-plane applied field. The typical frequency we used is 0 ,01Hz. Smalle r frequenc ies would lead to too long measurement time, while higher frequenc ies would give a too weak magneto -voltaic response. The amplitude U() of the magneto -voltaic signal as a function of the frequency of the heat excitation is presented in Fig. 13. This typical profile depends mainly on three characteristics , contained in the constant c , that are (i) the distance between the heat source and the electrode on which the magneto -voltaic signal is measured, (ii) the thermal conduct ivity of the substrate, and (iii) the electric contacts that thermally couple the sample to the voltmeters. We checked, using a vacuum cell , that the heat dissipated through the surfaces of the layer does not affect the signal (see Fig .S13). Fig.S13. Frequency dependence of the 2 magneto -voltaic signal. The typical profile is due to the thermal losses between the power injection and the measurement electrode. The local maximum at about 0.025 Hz is a good compromise between rapidity of the measure and amplitude of the signal . The curve has been measured in a va cuum cell (red points) in order to check that that dissipation throughout the surfaces is negligible. II-4. Measurement of the thermocouples 24 In order t o measure the Seebeck coefficient , we used a single line sample wired with aluminum wires and silver paint which is our reference material. One side of the sample was kept cool using an iced water bath and the other side was at room temperature. The voltage was measured over time. The maximum variation of U(t) – that corresponds to the maximum temperature difference T – gives a measure of the thermocouple S= U/T (V/K). It is shown that the contribution o f the Permalloy -Al interface is strong and negative ( -16.2 V/K) while the other contributions (Pt -Al, Pt -Ag, Pt -Au, etc…) are small and positive (< 1 V/K) . As a consequence, the thermocouple of NiFe dominates and the total thermocouple is always negative and of the order of -10 V/K. In contrast, the thermocouple for Pt electrodes depos ited on a YIG instead of NiFe (Pt/Al thermocouple ), has a positive thermocouple of the order of +1 V/K. This sign inversion of S explains the sign change observed between the magneto -voltaic signals of YIG ferromagnet and NiFe ferromagnet (see main text) . Fig.S14. Time dependent m easurement of the thermocouple generated by a contact wir e of Al with the electrode of (A) NiFe (Py), (B) Cu, and (C ) Pt. (D) sketch for the measure of the thermocouple. The temperature difference of T = 15.6°C is imposed at t =0, and the relaxation due to thermalization of the metallic line is measured as a function of time. The calculated line is the exponential relaxation. The temperature difference measured between the two extremit ies of the e lectrode is typically T = U/S = 0.002 K. The transport coefficient s related to anomalous Righi -Leduc effect (out - of-plane measurements) is found to be c1rRL = 0.16 K/W for NiFe and c2rRL = 0.13 K/W for YIG . The transport coefficient related to planar Righi-Leduc effect ( in-plane measurements) is found to be c1r = 0 .07 K/W for NiFe and c2r = 0.02K/W for YIG . The unknown parameter 0.1<ci<1 (i={1,2}) takes into account heat dissipation between the heater and the electrode (including shunt effect) and varies from one sample to the other . 25 Refer ence [31] McGuire , T. R. & Potter , R. I. Anisotropic Magnetoresistance in Ferromagnetic 3d Alloys, IEEE Trans actions On Magn etics 11, No4 1018 -1038 (1975)

2014-12-11

The Righi-Leduc effect refers to the thermal analogue of the Hall effect, for which the electric current is replaced by the heat current and the electric field by the temperature gradient. In both cases, the magnetic field generates a transverse force that deviates the carriers (electron, phonon, magnon) in the direction perpendicular to the current. In a ferromagnet, the magnetization plays the role of the magnetic field, and the corresponding effect is called anomalous Hall effect. Furthermore, a second transverse contribution due to the anisotropy, the planar Hall effect, is superimposed to the anomalous Hall effect. We report experimental evidence of the thermal counterpart of the Hall effects in ferromagnets, namely the magnon Hall effect (or equivalently the anomalous Righi-Leduc effect) and the planar Righi-Leduc effect, measured on ferromagnets that are either electrical conductor (NiFe) or insulator (YIG). The study shows the universal character of these new thermokinetic effects, related to the intrinsic chirality of the anisotropic ferromagnetic degrees of freedom.

Magnon Hall effect and anisotropic thermal transport in NiFe and YIG ferromagnets

1412.3723v1

Detection of spin pumping from YIG by spin-charge conversion in a Au jNi80Fe20 spin-valve structure N. Vlietstra and B. J. van Wees Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands F. K. Dejene Max Planck Institute for Microstructure Physics, Weinberg 2, 06120 Halle(Saale), Germany (Dated: June 20, 2021) Many experiments have shown the detection of spin-currents driven by radio-frequency spin pump- ing from yttrium iron garnet (YIG), by making use of the inverse spin-Hall eect, which is present in materials with strong spin-orbit coupling, such as Pt. Here we show that it is also possible to directly detect the resonance-driven spin-current using Au jpermalloy (Py, Ni 80Fe20) devices, where Py is used as a detector for the spins pumped across the YIG jAu interface. This detection mech- anism is equivalent to the spin-current detection in metallic non-local spin-valve devices. By nite element modeling we compare the pumped spin-current from a reference Pt strip with the detected signals from the Au jPy devices. We nd that for one series of Au jPy devices the calculated spin pumping signals mostly match the measurements, within 20%, whereas for a second series of devices additional signals are present which are up to a factor 10 higher than the calculated signals from spin pumping. We also identify contributions from thermoelectric eects caused by the resonant (spin-related) and non-resonant heating of the YIG. Thermocouples are used to investigate the pres- ence of these thermal eects and to quantify the magnitude of the Spin-(dependent-)Seebeck eect. Several additional features are observed, which are also discussed. PACS numbers: 72.25.-b, 75.78.-n, 76.50.+g, 85.75.-d I. INTRODUCTION Employing a ferro/ferrimagnetic insulating material (FMI) for spintronics research has attracted a lot of in- terest in the past years owing to the possibility of gen- erating pure spin-currents, without accompanying spu- rious charge-currents. Besides, in these materials, it is shown that spin information can be transported over large distances on the m-scale1or even mm-scale2,3, opening up new possibilities for spin-based data stor- age and transport. In these devices, Yttrium iron garnet (YIG), which is a room-temperature FMI with very low magnetic damping, is most often employed. Together with the (inverse) spin-Hall eect ((I)SHE) in Pt, it of- fers a platform for studying pure spin-current generation, transport and detection. An example of such an exper- iment is the electrical detection of spin pumping in a YIGjPt system, where the resonance of the YIG mag- netization leads to a spin-current pumped into the adja- cent Pt layer, which can electrically be detected via the ISHE.2,4{7 Pure spin-currents can also be generated and detected by making use of metallic magnetic jnon-magnetic nanos- tructures such as permalloy (Ni 80Fe20, Py) or cobalt.8 This method is mostly used in spin-valve structures, where a spin-current is generated by sending a charge- current through one magnet, which can be detected (ei- ther locally or non-locally) by a second magnetic strip, as a change in electric potential when switching between relative parallel and anti-parallel magnetic states of both magnets.9,10In this manuscript, we show that by combining a FMI (YIG) with a conducting magnetic material (Py) it is possible to electrically detect the magnetic resonance of the FMI, without the need of a high spin-orbit coupling material like Pt. Here the magnetic-resonance-induced dc spin-current pumped into an adjacent Au layer is de- tected as an electrical voltage by a Py detector connected to a Au spacer. This alternative method for detection of spin-currents from FMI-materials opens up new ways of investigating the origin of the spin-Seebeck eect11without the possi- ble presence of non-equilibrium proximity magnetization in the heavy metal Pt.12{14Besides, because of its anal- ogy to measuring a conventional spin-valve structure, this method also helps to determine the sign of the pumped spin-current from YIG into Pt,15and expands the pos- sibilities for designing devices, including spin transport through FMI materials. In the experiments we rst induce magnetic resonance in the YIG by sending RF currents through a microwave stripline, which is placed near the Au jPy devices that are connected in series to maximize the total signal. Part of the build-up potential we attribute to the spin- current generation by spin pumping from the YIG into the adjacent structure [schematically shown in Fig. 1(a)]. Hereby we compare spin pumping signals from a standard YIGjPt device structure with the signal from YIG jAujPy devices placed in series. Furthermore, we also identify signals that are related to heating and induction eects, which are rather small to explain the observed signals. It is found that we not only detect the resonance spin pumping from the magnetic YIG layer, but also observearXiv:1601.05605v1 [cond-mat.mes-hall] 21 Jan 20162 the Py resonance state. This self-detection of FMR by a Py strip has been observed before,16however, here we discuss that the mechanism is possibly dierent and re- lated to the interaction of spins at the YIG interface. II. SAMPLE CHARACTERISTICS The studied devices are fabricated on a 4 4 mm2 sized sample, which is cut from a wafer consisting of a 500- m-thick single crystal (111)Gd 3Ga5O12(GGG) substrate and a 210 nm thick layer of YIG, grown by liquid phase epitaxy (from the company Matesy GmbH). The YIG magnetization shows isotropic behavior of the magnetization in the lm plane, with a low coercive eld of less than 1 mT (measured by SQUID). Fig. 1(b) shows a schematic of one device from the studied series, fabricated by several steps of electron beam lithography. It consists of an 8-nm-thick Au layer deposited on YIG by dc sputtering, followed by a 20-nm- thick Py layer (30 2:5m2for area 1, and 60 10m2 for area 2), contacted with a top Ti jAu layer of 5j100 nm, both deposited by e-beam evaporation. To prevent short- ing between the top and bottom Au layers, when placing several devices in series, a 60-nm-thick Al 2O3layer was deposited over the edges of the Au jPy stack before the deposition of the top Ti jAu layer. Ar-ion milling has been used to clean the surfaces and etch the native oxide layer of Py before deposition of the Py and Ti jAu layers, respectively. Fig. 1(c) shows a microscope image of a full series of devices. With the employed meander struc- ture for the series of devices possible signals generated by the ISHE cancel out. In Fig. 1(c) also the reference Pt-strip (40030m2, 7-nm-thick, dc sputtered) can be seen, placed below and perpendicular to the 60 m wide TijAu microwave stripline (5 j100 nm thick) used to ex- cite the magnetization resonance in the magnetic layers. Between the Pt-strip and the microwave stripline a 60- nm-thick Al 2O3layer (dark brown) is added, to prevent electrical shorts. Fig. 1(d) and 1(e) show a close-up of the devices in area 1 and 2, respectively. For the spin pumping experiment using the YIG jPt system, the obtained signal scales linearly with the length of the Pt detection strip. For the YIG jAujPy devices the detected signal is not directly scalable by the size of the device, rather by the number of YIG jAujPy de- vices connected in series. As the expected signal for one YIGjAujPy device is below the signal-to-noise ratio of our measurement setup, we fabricated a structure where we increase the detected signal by placing many separate devices in series. Two sets of devices were investigated, having dierent surface area, as is shown in g. 1(c). For the presented experiments 96 (area 1) and 62 (area 2) YIGjAujPy devices in series were used. (c) (e) BVISHEVISHE YIG Ti/Au Py Al2O3 Au Ti/Au Au YIG MYIG Au V Py MPy (d) Stripline +- VPy -VPy+Pt VPy - VPy+ Area 1Area 2 (b) (a) Ti/Au Al 2O3 Py Au Area 1 Area 2Ti/Au Al 2O3 Py Au 100 µmFIG. 1. (a) Schematic representation of the spin pumping process and voltage detection in a YIG jAujPy device. (b) Schematic drawing of one YIG jAujPy device. Each device consists of Au (8 nm), Py (20 nm), Al 2O3(60 nm), and TijAu (5j100 nm) layers. (c) Microscope image of the nal device structures. In area 1 (area 2) 96 (62) YIG jAujPy devices are placed in series. The Pt strip is used as a reference for the measurements on the YIG jAujPy devices, and this strip is electrically insulated from the stripline by an Al 2O3layer (60 nm thick). During the measurements, an external magnetic eld is applied and three separate sets of voltage probes are connected as pointed out in the gure. (d) and (e) show close- ups of the devices placed in area 1 and 2, respectively. The dierent material layers are marked. III. MEASUREMENT METHODS The microwave signal is generated by a Rohde-Schwarz vector network analyzer (ZVA-40), connected to the waveguide on the sample via a picoprobe GS microwave probe. To be able to use the low-noise lock-in detection method, the power of the applied microwave signal is modulated between `on' and `o' using a triggering sig- nal which is synchronized with the trigger of the lock-in ampliers. Typically used `on' and `o' powers are 10 dBm and -30 dBm, respectively. The RF frequency is xed for each measurement (ranging from 1 GHz to 10 GHz), while sweeping the static in-plane magnetic eld. During this magnetic eld sweep, the voltage from the Pt strip and both series of Py devices are separately recorded by connecting them to three dierent lock-in ampliers, using the connections as is shown in Fig. 1(c). The ISHE voltage detected from the Pt strip is used as a reference for the data obtained from the Py devices. As the power is modulated during all measurements, the absorbed power cannot be measured simultaneously, and therefore no detailed information is obtained about the dependence of power absorption on applied RF frequency. The frequency dependence of the absorbed power was ob-3 (a) (b)-120-80-4004080120 -101234567 -300 -200 -100 0 100 200 300-0.4-0.3-0.2-0.10.00.10.2 VISHE [µV]Frequency [GHz] 1 5 9 2 6 10 3 7 4 8 P = 10mW Area 1VPy [µV] Area 2VPy [µV] Magnetic Field [mT]-0.50.00.51.01.5 0 20 40 60 80-0.2-0.10.00.1-120-80-400VPy [µV] Magnetic Field [mT] VISHE [µV] VPy [µV] (f)(e)(d) (c) FIG. 2. Magnetic eld sweeps for dierent applied frequen- cies, with an applied RF power of 10 mW for (a) the Pt strip, detecting the ISHE voltage and (b), (c) voltage generated by the series of Py-devices in area 1 and 2, respectively. (d)-(f) close-up of marked areas in (a)-(c). tained by separate measurements of the S 11parameter as in Ref.6. All measurements are performed at room tem- perature. IV. RESULTS AND DISCUSSION Results of the magnetic eld sweeps for dierent RF frequencies are shown in Fig. 2 where the ISHE voltage signal detected by the Pt strip [Fig. 2(a)] and the corre- sponding signals from the series of Py devices in area 1 [Fig. 2(b)] and area 2 [Fig. 2(c)] are shown. The detected voltage of the Pt strip shows the expected peaks for YIG resonance, changing sign when changing the magnetic eld direction, as also observed in for example Refs.4,6,17. The magnitude of the peaks is in the order of 100 V. Comparing these results to the data shown in Figs. 2(b) and 2(c); peaks at exactly the same position are observed for the Py devices. Here the detected peaks do not change sign by changing the magnetic eld direction, as the sign of the signal in these devices is determined by the relative orientation of the YIG and Py magnetization. Because of the low coercive elds ( <10 mT) of both the YIG and Py layers, their magnetizations always align parallel to each other when applying a small magnetic eld. At rst glance, it appears that when the YIG is excited into resonance conditions, the pumped spin-current into the Au layer is detected as an electrical voltage gener- ated by a conducting ferromagnet placed on top of the Au layer. Fig. 3 shows the frequency dependence of the magnitude of the observed peaks at YIG resonance for all three types of devices (Pt strip and Py devices area 1 and2). A few points can be made regarding these dependen- cies. First, where the dependence of the Pt shows some analogy with the dependence observed in area 2, the fre- quency dependence of the signals from area 1 (narrow Py strips) and area 2 (wider Py strips) largely diers. Besides, the signals from area 1 and 2 show about one order dierence in magnitude. These observations show that by changing the surface area of the YIG jAujPy de- vices, dierent physics phenomena can be present. In section IV B we calculate the contribution of spin pump- ing to the observed signals in both Py device areas and discuss these results. Secondly, from Fig. 3, we observe that the signals for positive and negative applied magnetic elds consis- tently dier. Interestingly, for area 1 the higher signals are observed for positive applied elds, whereas for area 2 the higher signals appear for negative applied elds. At rst glance, the device is fully symmetric and there- fore one would not expect any dependence on the sign of the applied magnetic eld. Further investigation leads to the existence of nonreciprocal magnetostatic surface spin- wave modes (MSSW), whose traveling direction (perpen- dicular to the stripline) is determined by the applied mag- netic eld direction.18,19As the spin pumping process in insensitive to the spin-wave mode, and the device areas 1 and 2 are placed on opposite sides of the stripline, the presence of these unidirectional MSSWs can lead to the observed dierence in peak-height for positive and nega- tive applied elds. Besides the peaks at YIG resonance, the Py devices show more peaks at lower applied magnetic elds (most clearly visible in area 1, Fig. 2(b)), these peaks are at- tributed to the ferromagnetic resonance of the magnetic (a) 0 2 4 6 8 100.00.10.20.30.40.5Peak Height Py [ µV]Area 2 -B +B Frequency [GHz](c) (b)0 2 4 6 8 10406080100120140 Frequency [GHz]Pt -B +B Peak height Pt [ µV] 0 2 4 6 8 10012345Peak Height Py [ µV] Area 1 -B +B Frequency [GHz] FIG. 3. Magnitude of the peaks at YIG resonance obtained from the measurements shown in Fig. 2, for positive and negative applied magnetic elds as a function of applied RF frequency. (a) For the Pt strip, (b) and (c) for the Py devices of area 1 and 2, respectively.4 Py layers, as will be shown in section IV A and further discussed in section IV E. A. Position of the resonance peaks To check the position of the observed peaks with respect to the predicted resonance conditions of the magnetic layers (YIG and Py), the Kittel equation is used:20,21 f= 2q (B+Nk0Ms)(B+N?0Ms); (1) where is the gyromagnetic ratio ( = 176 GHz/T), B is the applied magnetic eld, NkandN?are in-plane and out-of-plane demagnetization factors and 0Msis the saturation magnetization. Taking Nk0Ms= 10 mT andN?0Ms= 1:1 T (consistent with previously re- ported values for Py)16,22, we obtain the red solid curve shown in Fig. 4, and nd that the position of the inner peaks, detected only for the series of Py devices, matches the Py resonance conditions (the shown data is obtained from area 1). Therefore, the origin of the inner peaks is related to the Py layers being in ferromagnetic resonance. The position of the outer peaks, detected for both se- ries of Py devices, corresponds to the voltage peaks ap- pearing in the Pt strip, which are ascribed to the YIG magnetization being in resonance. To check the expected YIG resonance conditions, a more simplied form of the Kittel equation can be used, because of its isotropic in- plane magnetization behavior ( Nk= 0 andN?= 1): f= 2p (B(B+0Ms): (2) For the curve corresponding to the YIG resonance peaks as shown in Fig. 4, 0Ms= 176 mT is used, 0 2 4 6 8 10 12050100150200250300350 Magnetic Field [mT] Frequency [GHz]Kittel Equation: YIG Py Measured: YIG/Pt peak Py outer peak Py inner peak FIG. 4. The measured peak positions for both the Py devices area 1 (red symbols) as well as the Pt strip (black symbols), plotted together with the calculated resonance conditions by the Kittel equation [Eqs. (1) and (2)].which is the reported bulk saturation magnetization of YIG.2,6,23The close match of the calculated curve and the measured data proves that the outer peaks from the Py devices and the peaks from the Pt strip indeed origi- nate from the YIG magnetization being in resonance. B. Estimation of the spin pumping signal in YIG jAu jPy devices In this section we will only focus on the origin of the peaks at YIG resonance. The possible origin of the de- tected inner peaks for the Py devices will be discussed in section IV E. Using the data from the Pt strip as a ref- erence, we calculate the expected signal caused by spin pumping in the YIG jAujPy devices. Dierent steps in this calculation are: 1) calculate the pumped spin-current density from the ISHE voltage detected by the Pt strip, 2) obtain an estimate for the spin-mixing conductance of the YIGjAu interface, and 3) use a nite element spin trans- port model to nd the expected spin pumping signal in the YIGjAujPy devices, setting the results of 1) and 2) as boundary conditions and input parameters, respectively. For the calculations we assume that the spin accumula- tion in the layer adjacent to the YIG, dened as the ratio between the injected spin-current Jsand the real part of the spin-mixing conductance Gr, is constant when con- sidering dierent types of interfaces and devices.24 As the magnitude of the excited spin-waves decays with distance from the stripline, we fabricated another sample, where a 6-nm-thick Pt strip and a Au jPt strip (8j6 nm) were placed exactly on the location of the series of Py devices in the previous batch, as is shown in Fig. 5 (The dimensions of these strips are 334 30m). Using this sample, the average injected spin-current on the loca- tion of the Py devices can directly be calculated. Signals obtained from these new devices show similar behavior as the data shown in Fig. 2(a) and Fig. 3(a), only the magnitude of the signals diers: The Pt (Au jPt) strip on the new sample results in ISHE-voltages around 30 V (3 V), compared to 120 V for the Pt strip directly below the RF line. These signicantly lower signals prove the decay of spin-waves with distance from the stripline. The signal for the AujPt strip is even more suppressed com- pared to the Pt strip, as the Au layer short-circuits the structure because of its lower resistance compared to Pt, while the inverse spin-Hall voltage is mainly generated in the Pt layer. In the following calculation, the Au jPt strip is modeled using the 3D nite-element modeling software Comsol Multiphysics, taking into account these losses. Step 1: From the measured ISHE-voltage VISHE in the Pt strip, the injected spin-current density can be calcu- lated by Js=VISHEtPt l SH1 1 tanh(tPt 2); (3) wheretPt,l,,, andSHare the thickness (6 nm), length (330 m), resistivity (3 :5107 m), spin relax-5 B V+V- V+V-100 µm Pt Au/Pt Pt FIG. 5. Microscope image of the Pt and Au jPt strips used to estimate the injected spin-currentby spin pumping at the location of the Py-devices. ation length (1.2 nm) and the spin-Hall angle (0.08) of the Pt strip, respectively, and = [1+Gr,Ptcoth(tPt 2)]1is the back ow term,25whereGr,Ptis the spin-mixing con- ductance of the YIG jPt interface (4 :41014 1m2). The given material properties are taken from previously reported spin-Hall magnetoresistance measurements.26 Using Eq. (3), an injected spin-current density of 2 :0 107A/m2is found for a detected VISHE of 30 V. Step 2: The spin-mixing conductance of the YIG jPt interface diers from the YIG jAu interface, as present in the measured YIG jAujPy devices. Therefore we use the obtained signals from the combined Au jPt strip to nd an estimate of the spin-mixing conductance of the YIGjAu interface Gr,Au. Spin pumping into the Au jPt strip results in ISHE signals in the order of 3 V. Using Comsol Multiphysics, we model the YIG jAujPt device, including spin-diusion by the two-channel model and the spin-Hall eect as explained in Ref.27. To include the contribution of the spin-mixing conductance in this model, such that back ow is accounted for, a thin inter- face layer (t= 1 nm) is dened between the YIG and Au layers (resulting in a stack: YIG jinterfacejAujPt). The interface layer acts as an extra resistive channel for the in- jected spin-current, parallel to the spin-resistance of the device on top (in this case the Au jPt strip), such that there eectively are two spin-channels: one for back ow and one for injection into the Au layer. The conductivity of this interface layer is dened as int=Gr,Au
t. The input parameter at the interface jAu boundary is the dc spin-current Jsobtained in step 1 for the YIG jPt device. By scaling Jswith the ratio of Gr,AuandGr,Ptwe take into account that the injected spin-current is lower when the spin-mixing conductance is lower. We now tune the value ofGr,Auin the model such that the modeling re- sult matches the measured VISHE of the AujPt strip. By doing so we nd Gr,Au= (2:20:2)1014 1m2, in order to match the measured VISHE of the AujPt strip. This value is similar as reported by Heinrich et al. ,28who obtainedGr,Au-values up to 1 :91014 1m2. Used modeling parameters for the Au layer are Au= 6:8106 S/m andAu= 80 nm.29The modeling parameters for the Pt layer are as mentioned above. By replacing the Pt layer by the Au jPt strip, we nd that the back ow spin-current almost doubles (around75% increase). This increased back ow is mainly caused by the larger spin-diusion length in Au as compared to Pt, resulting in a higher spin-resistance for the injection of spins into the thin Au layer, as compared to the back- ow spin-channel (In other words: Pt is a better spin- sink). Furthermore, the initially injected spin-current is a factor 2 lower for the YIG jAujPt strip, compared to YIGjPt, caused by the lower spin-mixing conductance. These two cases result in intrinsically lower signals when placing Au on top of YIG as compared to Pt. Step 3: After having calculated the injected spin- current and the spin-mixing conductance of the YIG jAu interface, these parameters are used as input for the Com- sol Multiphysics model of one YIG jAujPy device. This model is again based on the two-channel model for spin transport, including an additional interface layer to add back ow to the model, as described above. Detailed in- formation about the modeling and the used equations can be found in Ref.27. The dierent sizes of devices present in area 1 and area 2 are both separately modeled. The properties of the Py layer added to the model are P= 0:3 (dening the spin polarization), Py= 2:9 106S/m andPy= 5 nm.29The spin-current density obtained from Eq. (3) is used as input, and is set as a boundary ux/source term at the bottom interface of the Au layer. As explained in step 2, also here a thin interface layer is placed below the Au layer, such that it acts as a spin-current channel parallel to the injection of spin- current into the device. The above estimated value for Gr,Auis used to dene the interface layer conductivity. Also from this model we nd a large back ow from the initially injected Js, which is mainly caused by the spins accumulating at the Au jPy interface, increasing the spin- resistance in the injection-channel with respect to the back ow spin-channel. To obtain the dependence on RF frequency of the ex- pected voltage signal, Jswas calculated from Eq. (3) for each frequency, using the measurements on the YIG jPt strip, and the YIG jAujPy model was run for each ob- tainedJs. The calculated voltage signal caused by spin pumping for one Py device was multiplied by 96 (62), the number of devices placed in series in area 1 (area 2), and the nal results of these calculations are shown as the red curve in Fig. 6(a) and 6(b), together with the absolute values of the measured peaks, shown in black, for area 1 and 2, respectively. For the Py-devices in area 1, the calculated spin pump- ing voltages are one order of magnitude smaller than the measured signals, from which we conclude that the major contribution of the measured signal is caused by another source than spin accumulation created by spin pump- ing. Additionally, the calculated signals generated by spin pumping clearly show a dierent dependence on fre- quency, as compared to the measured signals from the YIGjAujPy devices. This also indicates that besides spin pumping there are other phenomena present which in- duce voltage signals in our devices. Interestingly the peaks obtained from the devices in6 012345 0 100 200 3000.00.20.40.6 0 50 100 150 200 250 3000.00.10.20.3 Vpeak [µV] Spin Pumping from Model Vpeak [µV] Magnetic Field [mT]Area 1 Area 2(a) (b) FIG. 6. Calculated spin pumping voltage (red squares) ver- sus measured signal (black peaks) as a function of applied RF frequency and magnetic eld. The peaks from left to right correspond to applied RF frequencies from 1 to 10 GHz, re- spectively, directly copied from the measurements shown in Fig 2. (a) Results for the Py-devices in area 1. The inset shows a close-up of the calculated spin pumping voltage. (b) Results for the Py-devices in area 2. area 2 show a very good agreement with the calculated spin pumping signals. From these results we conclude that in this case the major contribution of the measured signals is caused by spin pumping. Comparing the results from area 1 and area 2, it is clear that in the experiments the exact device geometry largely in uences the signal and even results in a totally dierent dependence on applied RF frequency. From the calcula- tions of the spin pumping signals, such a big change in behavior cannot be reproduced and therefore additional phenomena must be present and becoming more promi- nent for more narrow devices, as present in area 1. As thermoelectric eects might also play a role in the performed experiments, next section discusses some further investigation of possible signals related to RF- heating. C. Thermal eects While an RF current ows through the stripline, heat is absorbed by the YIG layer causing the YIG temperature to rise. Together with Eddy currents that are induced in the Py devices, which can result in Joule heating, this power absorption leads to local heating. Besides heat- ing at the non-resonance conditions, especially at mag- netic resonance additional heat will be dissipated into the YIG due to the continuous damping of the YIG mag- netization precession.30The generation of temperature- (a) (b) 100 µmV+ V- B 0 2 4 6 8 100.00.20.40.60.8 VBackground [µV] Frequency [GHz](c)-300 -200 -100 0 100 200 3000.40.60.81.01.2 VThermocouple [µV] Magnetic Field [mT]F = 7 GHz P = 10 mW (d) 0 2 4 6 8 100.00.10.20.30.40.50.6 VPeak [µV] Frequency [GHz]05101520 ∆T [mK] +B -BFIG. 7. (a) Microscope image of the thermo-couples placed near the microstripline. Below all contact-leads a 70-nm-thick Al2O3layer is present (visible as the dark areas), to avoid spurious signals generated in these leads. Only the Au pad in the center of each thermo-couple is in direct contact to the YIG substrate. Each Au pad is contacted with a 40-nm-thick Pt lead on the left side and a 40-nm-thick NiCu lead on the right side. (b) Detected voltage from the thermocouple most close to the stripline for F= 7 GHz and P= 10 mW. (c) Frequency dependence of the detected peaks for positive and negative applied magnetic elds. The right axis shows the corresponding temperature change
T=
V=(SPtSNiCu). (d) Dependence of the thermo-couple background voltage on applied frequency. (c) and (d) are both for P= 10 mW and for the thermo-couple most close to the stripline. gradients caused by local heating gives rise to thermo- electric eects such as the Seebeck eect (caused by the dierence in Seebeck coecient of Au and Py), the spin- dependent-Seebeck eect (SdSE) (due to the spin depen- dency of the Seebeck coecient in Py, resulting in ther- mal spin injection at the Au jPy interface), and the spin- Seebeck eect (SSE) (spin pumping caused by thermally excited magnons in the YIG, leading to spin accumula- tion at the YIGjAu interface). In order to probe the RF induced heating, NiCu jPt thermocouples were placed near the stripline as is shown in Fig. 7(a). In this way, the temperature of the sub- strate can locally be measured by making use of the See- beck eect. From the measured thermo-voltage signals the increase in temperature at the NiCu jPt junction with respect to the reference temperature of the contact pads can be obtained using
V=(SPtSNiCu)
T, where SPt=5V/K andSNiCu =32V/K are the See- beck coecient of Pt and NiCu, respectively.30Besides a constant background voltage signal, indicating heating when the YIG magnetization is not in resonance, clear peaks are observed at the YIG resonance conditions, as is presented in Fig. 7(b) for an applied RF frequency of 7 GHz. The magnitude of the peaks at resonance is in the order of 40 nV ( F= 1 GHz,P= 10 mW, distance from microstrip 195 m) up to 0.6 V (F= 10 GHz, P= 10 mW, distance from microstrip 50 m); Higher signals were measured for higher frequencies and for de- vices closer to the RF line.7 Fig. 7(c) shows the extracted peak-height of the mea- surements for the thermocouple most close to the mi- crostrip (50 m), and the corresponding temperature in- crease is added on the right vertical axis. Additionally, Fig. 7(d) gives the evolution of the background voltage as a function of applied RF frequency. A maximum tem- perature increase at resonance conditions of 22 mK is ob- served in this thermocouple. Interestingly, all measure- ments show dierent behavior at the YIG resonance con- ditions for positive and negative applied magnetic elds; Consistently, the peak at negative applied magnetic elds is larger than the one for positive elds, increasing to a factor 2 in magnitude at F= 10 GHz. These observations are in agreement with the dif- ference in peak-height between positive and negative applied magnetic elds as observed for the Py-devices in area 2, which are placed on the same side of the stripline as the thermocouples. Also here the existence of nonreciprocal magnetostatic surface spin-waves (MSSW) might explain the observed behavior, as they will in u- ence the heating and lead to unidirectional heating of the substrate, as observed by An et al. ,31who used a measurement conguration very similar to the one we describe in this paper. The thermocouple measurements show non-negligible heating of the YIG surface, and therefore thermal ef- fects are likely to play a role in the observed voltage generation in the YIG jAujPy devices. To obtain the quantitative contribution of the Seebeck eect, SdSE, and SSE in the studied device geometry, it is needed to model the YIG jAujPy device, including its thermal properties. However, from the observed dependence on applied RF frequency of the thermocouple signals, even without knowing the expected quantitative contribution of the thermal eects, it can be concluded that these ef- fects cannot explain the large signals in the YIG jAujPy devices of area 1. As observed for spin pumping, the thermocouple signals saturate at higher RF frequencies, whereas the series of permalloy devices in area 1 shows a continuously increasing signal. This indicates that ad- ditional eects are present, which scale linearly with fre- quency, such as for example inductive coupling, where the RF current in the microstrip induces a current in the Py, causing Joule heating. D. Finite element simulation of the SdSE and SSE To determine the contribution of the SdSE (at the AujPy interface) and the SSE (at the YIG jAu interface) we performed a three-dimensional nite element (3D- FEM) simulation of our devices27where the charge- ( ~J) and heat- ( ~Q) current densities are related to the cor- responding voltage- ( V) and temperature- ( T) gradients using a 3D thermoelectric model. Details of this model- ing can be found in Refs.27,29,32{34. The input material parameters, such as the electrical conductivity, thermal conductivity, Seebeck coecients and Peltier coecientare adopted from Table I of Refs.32,33. 1. SdSE In the SdSE, the heat current owing across the Au jPy interface causes the injection of spins which are anti- aligned to the magnetization of the Py layer. In our modeling, we use the temperature values measured by the NiCujPt thermocouples, shown in Fig. 7(c), as a Dirichlet boundary condition in the 3D-FEM. Specically, for a given microwave power and fre- quency, by xing the temperature of the Au jYIG in- terface to the measured values and anchoring the leads (TijAu contacts) to the reference temperature, we can calculate the resulting temperature gradient rTFin the Py and hence the SdSE voltage. From this model, for a single AujPy interface, a total spin-coupled voltage drop of approximately 1 nV is obtained, which corresponds to96 nV for the series of Py devices in area 1. We can also compare this result with one obtained from a simple one-dimensional spin-diusion model using the following equation35
Vs=2sSSrTPyPRm; (4) wheres= 5 nm is the spin-diusion length, SS= 5V/K is the spin-dependent Seebeck coecient, P= 0:3 is the bulk spin-polarization, rTPy= 105K/m is the temperature gradient in the Py, obtained from a simple 1D heat diusion model across the interfaces, and Rm is a resistance mismatch term which is a value close to unity for such metallic interfaces considered here. The es- timated signal from this 1D-model is a factor two larger than that obtained from the 3D-FEM. Note that, in both modeling schemes, the distance dependence from the strip-line has not been taken into account, which would lead to an even lower signal. Therefore, we con- clude that the SdSE does not contribute signicantly to the measured signal. 2. SSE To estimate the maximum contribution of the SSE, caused by spin pumping due to thermal magnons, we need to obtain the temperature dierence between the magnons and electrons
Tmeat the YIGjAu interface. We again use the 3D-FEM, but this time, extended to include the coupled heat transport by phonons, elec- trons and magnons with the corresponding heat exchange lengths between each subsystem. The detailed descrip- tion of this model, which was used earlier to describe the interfacial spin-heat exchange at a Pt jYIG interface, can be found in Ref.33along with the used modeling param- eters. In our model, we set the bottom of the GGG substrate to the surrounding (phonon) temperature T0, the AujPy8 interface at the equilibrium temperatures of both elec- trons and phonons, i.e., Te=Tph=T0+ 20 mK while using a magnon heat conductivity m= 0:01 Wm1K1 and phonon-magnon heat exchange length m-ph = 1 nm. From our 3D-FEM model, we nd an interface temper- ature dierence of
Tme= 25 K between the magnon and electron subsystems. While
Tmeat the YIGjAu interface seem a rather small value, the comparison with earlier reports indicate equivalence between the ratio of
Tmeto the temperature increase of the YIG
Tph. In the SSE, the spin-current density Jspumped across the YIGjAu interface can be obtained using Js= LS
Tme, whereLS=Gr ~kB=2MsVa= 7:24109 Am2K1is the interface spin Seebeck coecient11,33 whereGr,Ms, andVaare the real part of the spin-mixing conductance per unit area, the YIG saturation magneti- zation, and the magnetic coherence volume (3pVa= 1:3 nm)25, respectively. Because the thickness t of the Au is much smaller than the spin diusion length in Au, we can assume a homogeneous spin accumulation
=Jst. The voltage drop at the Au jPy interface is thus
VSSE=P
=2, which gives the maximum SSE voltageVSSEdetected by the Py. Using
Tme= 25 K, we obtainVSSE= 54 pV for a single Py device and a total of 5 nV for 96 Py serially connected devices. From this discussion we conclude that the combined voltage contri- butions from the SSE and the SdSE cannot explain the observed enhancement at higher frequencies, suggesting that there are additional eects that need to be consid- ered here. E. Discussion Besides the measured signals at YIG resonance, a few other present features need attention. First, at low ap- plied frequency and magnetic eld, the voltages gener- ated by the Py-devices in series in area 1 show resonance behavior, as is clearly visible in Fig. 2(e). These res- onating signals decrease and nally disappear for higher frequencies. In area 2 also some small resonances are ob- served [see Fig. 2(f)], but these resonances are far less prominent. The origin of the resonances might be related to the fact that this system, like the YIG jPt system, is not only sensitive to the ferromagnetic resonance (FMR) mode, but to any spin-wave mode. This means that addi- tional signals can appear when multiple spin-wave modes exist, which might be more strongly present at lower fre- quencies, and for some reason more sensitively detected by the narrow strips of Py-devices in series in area 1 as compared to the wider Py-devices and the YIG jPt sys- tem. Furthermore, for the lower frequencies the YIG resonance conditions are very similar to those of the Py layer, which could lead to coupling between those states, resulting in a broader range of possible resonance mag- netic elds for a certain applied frequency. A second feature that needs attention is the back- ground signal for the YIG jAujPy devices. The magni-tude of the background signal increases with the applied RF power, similar in magnitude as the resonance-peaks, only having opposite sign. From the evolution of the background heating, as depicted in Fig. 7(d), it is ob- served that the background heating decreases by increas- ing the RF frequency. Therefore it is not possible to di- rectly attribute the measured background signals of the YIGjAujPy devices to heating of the substrate, and the origin of these signals is still unclear. Third are the peaks observed at Py resonance con- ditions. As a control experiment the same sequence of YIGjAujPy devices was fabricated on a Si jSiO2substrate, including the waveguide and stripline. In these samples no resonance peaks were present, neither at the YIG res- onance conditions nor at the Py resonance conditions. This experiment proves the need of the YIG substrate in order to detect Py resonance. A possible explanation of the origin of the detected peaks is as follows: By having the Py magnetization in resonance, a pure spin-current is pumped into the adja- cent layers. The polarization of this spin-current consists of both an ac- and a dc-component. The spin-current pumped into the upper contact will relax and does not give rise to any signal. The spin-current pumped into the thin Au layer below the Py strip will not relax before arriving at the YIG jAu interface. At this interface the component of the spin angular momentum perpendicular to the YIG-magnetization (here the ac-component of the pumped spins) will be absorbed and the parallel com- ponent (the dc-component of the pumped spins) will be re ected (as is the case for the spin-Hall magnetoresis- tance). This interaction with the YIG jAu interface re- sults in only the dc-component of the initially pumped spin-current being re ected. The re ected spins will dif- fuse back to the Py strip, where they accumulate, as their polarization direction is changed as compared to the spins being pumped. This spin-accumulation results in a build-up potential, which is measured. In the case YIG is replaced by SiO 2this mechanism does not work, as the absorption and re ection of spins at the SiO 2jAu interface is not spin-dependent. Finally, to prove the obtained signals are caused by spin pumping, and detected by the diusion of the gen- erated spin accumulation to the Py layers, as in typical non-local spin-valve devices, the observed peaks should change sign when the magnetization directions of the YIG and Py are placed anti-parallel, as is shown in Fig. 8(a). In the experiments, this situation turned out to be hard to accomplish, as the switching elds of both YIG and Py are relatively low: for YIG smaller than 1 mT, and for the 20-nm-thick Py strips on YIG a maximum switching eld of 10 mT was obtained for Py dimen- sions of 0:36m2. So when sweeping the magnetic eld, only elds between 1 and 10 mT will result in anti- parallel alignment of the magnetic layers. Besides this eld window being rather narrow, the resonance peaks in this regime are no clear single peaks [see Fig. 2(e)]. Nevetheless, one batch of samples was fabricated where9 -15 -10 -5 0 5 10 15-0.3-0.2-0.10.00.10.20.3 F = 1 GHz P = 10 mW Py [0.3 x 6 µm2]VPy [µV] Magnetic Field [mT]Py YIG -10 0 10 Magnetic Field [mT](b) (a) FIG. 8. (a) Theoretically expected resonance peaks, when in- dividual switching of the YIG and Py layers is obtained. The insets show the magnetization orientation of the YIG and Py layers for each resonance peak. (b) Measurement result for a batch of samples having smaller Py-strips (0 :36m2) as compared to the afore described devices, in order to observe the anti-parallel state of the YIG and Py magnetization direc- tions. The trace and retrace of the measurement are marked by the black and red data, respectively. the Py dimensions were set to 0 :36m2, and one result- ing measurement is shown in Fig. 8(b). While sweeping the magnetic eld from negative to positive values (black line), the resonance peak clearly changes sign. However, for the reverse eld sweep (red line) the expected switch- ing of the peak is not observed: There is a narrow positive peak, but less than halfway the expected peakwidth, it reverses sign. From this measurement it is not possible to unambiguously state the presence of the sign reversal for the anti-parallel state. To do so, a device is needed having a switching eld of the second magnetic layer in the order of 100 mT or higher (possibly accomplished by replacing Py for cobalt, which has a larger coercive eld), such that the anti-parallel magnetization state can also be obtained for slightly higher magnetic elds. V. SUMMARY In summary, we have observed the generation of volt- age signals in YIG jAujPy devices placed in series, caused by YIG magnetization resonance. Furthermore, the reso- nance of the magnetic Py layers, caused by direct excita- tion of the magnetization, or indirect dynamic coupling, is detected. By modeling our device structure, we nd that the signals of the wider Py structures (area 2) can very well be reproduced by the calculated spin pumping signals. For the narrow structures (area 1) additional signals are detected. The origin of these additionally observed signals and some other features, such as thedependence of the resonance peaks on applied RF fre- quency, the resonating peaks at low applied frequencies, and the increasing background voltage as a function of RF frequency, remain to be explained. Spin-dependent thermal eect are also quantied; The heating caused by the applied RF current was studied by placing thermocouples in close proximity to the stripline. Due to the temperature increase at the surface of the YIG substrate, especially at YIG resonance conditions, contributions of thermal eects to the generated voltage in the YIGjAujPy devices cannot be excluded. Never- theless, from nite element simulations we nd that the contribution of the SdSE and SSE are rather small and cannot explain the observed features. Additionally, the thermocouples showed that the heating of the substrate is dependent on the applied eld direction, indicating the possible presence of nonreciprocal magnetostatic surface spin-wave modes. Concluding, we have shown the possibility to electri- cally detect magnetization resonance from an electrical insulating material by a spin-valve-like structure, with- out making use of the ISHE. For the presented work, 96 and 62 AujPy devices were placed in series to increase the magnitude of the generated signal. By comparing the obtained data with signals from a reference Pt strip and using a nite element model of the devices, we nd that part of the detected signals can be ascribed to spin- current generation by spin pumping (for the 62 devices in area 2 an agreement between measurements and calcu- lated signals within 20% is found), however, especially for the 96 devices in area 1, additional signals are present, of which the origin remains to be explained. Once a better understanding of the origin of the full signals is obtained, the device geometry and injection eciency can be im- proved, such that the number of needed devices can be decreased, which opens up possibilities for new types of spintronic devices, where magnetic insulators can be in- tegrated. ACKNOWLEDGEMENTS We would like to acknowledge J. Flipse for sharing his ideas and M. de Roosz, H. Adema and J. G. Hol- stein for technical assistance. This work is supported by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners, by NanoLab NL and by the Zernike Institute for Advanced Materials (Dieptestrategie program). 1L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Nature Physics 11, 1022 (2015). 2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,262 (2010). 3A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010). 4K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Mat-10 suo, S. Maekawa, and E. Saitoh, Journal of Applied Physics 109, 103913 (2011). 5K. Harii, T. An, Y. Kajiwara, K. Ando, H. Nakayama, T. Yoshino, and E. Saitoh, Journal of Applied Physics 109, 116105 (2011). 6V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 86, 134419 (2012). 7C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013). 8F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature 410, 345 (2001). 9M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 10G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 11J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). 12S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012). 13Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). 14S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wil- helm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Applied Physics Letters 101, 262407 (2012). 15M. Schreier, G. E. W. Bauer, V. I. Vasyuchka, J. Flipse, K. ichi Uchida, J. Lotze, V. Lauer, A. V. Chumak, A. A. Serga, S. Daimon, T. Kikkawa, E. Saitoh, B. J. van Wees, B. Hillebrands, R. Gross, and S. T. B. Goennenwein, Jour- nal of Physics D: Applied Physics 48, 025001 (2015). 16M. V. Costache, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. B 78, 064423 (2008). 17V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 90, 214434 (2014). 18T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, and M. P. Kostylev, Phys. Rev. B 77, 214411 (2008). 19M. Agrawal, A. A. Serga, V. Lauer, E. T. Papaioannou, B. Hillebrands, and V. I. Vasyuchka, Applied Physics Let- ters105, 092404 (2014).20C. Kittel, Phys. Rev. 73, 155 (1948). 21C. Kittel, \Introduction to Solid State Physics," (Wiley, New York, 2009) Chap. 13, 8th ed. 22M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). 23H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Nature Mater. 10, 660 (2011). 24M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, Applied Physics Letters 103, 022411 (2013). 25M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B88, 094410 (2013). 26N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. van Wees, Applied Physics Letters 103, 032401 (2013). 27A. Slachter, F. L. Bakker, and B. J. van Wees, Phys. Rev. B84, 174408 (2011). 28C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Applied Physics Letters 100, 092403 (2012). 29F. K. Dejene, J. Flipse, G. E. W. Bauer, and B. J. van Wees, Nat Phys 9, 636 (2013). 30F. L. Bakker, J. Flipse, A. Slachter, D. Wagenaar, and B. J. van Wees, Phys. Rev. Lett. 108, 167602 (2012). 31T. An, V. I. Vasyuchka, K. Uchida, A. V. Chumak, K. Ya- maguchi, K. Harii, J. Ohe, M. B. Jung eisch, Y. Kajiwara, H. Adachi, B. Hillebrands, S. Maekawa, and E. Saitoh, Nature Materials 12, 549 (2013). 32F. L. Bakker, J. Flipse, and B. J. v. Wees, Journal of Applied Physics 111, 084306 (2012). 33J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). 34F. K. Dejene, J. Flipse, and B. J. van Wees, Phys. Rev. B86, 024436 (2012). 35A. Slachter, F. L. Bakker, J.-P. Adam, and B. J. van Wees, Nature Physics 6, 879 (2010).

2016-01-21

Many experiments have shown the detection of spin-currents driven by radio-frequency spin pumping from yttrium iron garnet (YIG), by making use of the inverse spin-Hall effect, which is present in materials with strong spin-orbit coupling, such as Pt. Here we show that it is also possible to directly detect the resonance-driven spin-current using Au/permalloy (Py, Ni$_{80}$Fe$_{20}$) devices, where Py is used as a detector for the spins pumped across the YIG/Au interface. This detection mechanism is equivalent to the spin-current detection in metallic non-local spin-valve devices. By finite element modeling we compare the pumped spin-current from a reference Pt strip with the detected signals from the Au/Py devices. We find that for one series of Au/Py devices the calculated spin pumping signals mostly match the measurements, within 20%, whereas for a second series of devices additional signals are present which are up to a factor 10 higher than the calculated signals from spin pumping. We also identify contributions from thermoelectric effects caused by the resonant (spin-related) and non-resonant heating of the YIG. Thermocouples are used to investigate the presence of these thermal effects and to quantify the magnitude of the Spin-(dependent-)Seebeck effect. Several additional features are observed, which are also discussed.

Detection of spin pumping from YIG by spin-charge conversion in a Au/Ni$_{80}$Fe$_{20}$ spin-valve structure

1601.05605v1

arXiv:1309.2965v1 [cond-mat.mes-hall] 11 Sep 2013NanomechanicalAC Susceptometry ofan IndividualMesoscop icFerrimagnet J.E.Losbya,b, Z.Diaoa,b, F. FaniSania,b,D.T.Grandmonta, M.Belovb,J.A.JBurgessa,b, W.K. Hieberta,b, M.R. Freemana,b aDepartment ofPhysics, University ofAlberta, Edmonton, Al berta, Canada T6G 2G7 bNational Institute for Nanotechnology, Edmonton, Alberta , Canada T6G 2M9 Abstract A novel method for simultaneous detection of both DC and time -dependent magnetic signatures in individual mesoscopic structures has emerged from early studies in spi n mechanics. Multifrequencynanomechanicaldetection of AC susceptibility and its harmonics highlights reversib le nonlinearities in the magnetization response of a single yttriumirongarnet(YIG)element,separatingthemfromhys tereticjumpsintheDC magnetization. Thismanuscriptwasacceptedforpublicationin SolidState Communications(SpecialIssue: SpinMechanics). Keywords: A.magneticallyorderedmaterials,A.nanostructures,B. n anofabrications,D.spin dynamics Nanomechanical torque magnetometry of quasi- static magnetization processes has sparked recent in- terest due to its exceptional sensitivity (with room- temperature magnetic moment resolution approaching 106µB) and ability to non-invasively measure single, mesoscopicelements1,2,3. Thestudyofindividualstruc- tures is a necessity to probe local e ffects from varia- tionin themagneticmicrostructuredueto grainbound- aries, vacancies, and other intrinsic or extrinsic inho- mogeneities. These are masked in measurementsof ar- rays of magnetic structures but can have a tremendous impact on the magnetization response of single ele- ments. Inthistechniqueamagnetostatictorque, τ,isex- erted on the magnetization, M, by an external field, H: τ=MV×µ0H,inwhich Visthesamplevolume. Ifthe magneticmaterialisa ffixedto atorsionalresonator,the induced magnetic torque is converted to a mechanical deflection proportional to its magnetization. For struc- tures with strong in-plane magnetic anisotropy, a small torquingAC (dither)field is appliedout-of-planeto the surfacewhile a DC field biases the in-planemagnetiza- tion. By ramping the DC field, the quasi-static magne- tizationevolutionwithappliedfieldcanberecorded. Techniques to probe time-dependent magnetic phe- nomena yield information complementary to the DC magnetization. In particular,the AC magneticresponse Emailaddress: mark.freeman@ualberta.ca (M.R.Freeman)aids in discriminatingreversible fromirreversiblemag- netization changes. In the small signal regime, an AC magnetization Mac=χHacisinducedbyanalternating field, whereχis a susceptibility tensor. In bulk mag- netic systems, low frequency susceptibility (including the use of higher order susceptibility4,5) measurements are used extensively to monitor myriad phenomena fromthe onsetof ferromagnetismat theCurie tempera- ture6,7,magnetizationreversalprocessesinthinfilms8,9 and patternednanostructures10, exchangeanisotropyin exchangebiasedsystems11, anddynamicsin spin-glass systems12. The advantages o ffered by nanomechanical transductionhaveyet to befullyexploitedformagnetic susceptibility measurements. It is highly desirable to develop an experimental technique for measuring both DCandACcomponentsofmagnetization,complemen- tarytoSQUIDmeasurements13,14byvirtueofoperating in a wider range of sample environments. Micro Hall magnetometerscan also be verysensitive15,16, but their low bandwidth will limit applicability to susceptibility measurements. Synchrotron-based XMCD-PEEM has alsobeenutilizedtorecordlow-frequencydynamicsus- ceptibilityin magneticthin-films17,18. Herewereportananomechanicalplatformforsimul- taneous DC and AC magnetic measurements through the introduction of an AC field component along the bias field direction, to probe the quasi-static longitudi- nal susceptibility. Two orthogonalAC fields, if applied Preprintsubmitted to Solid State Communications August14,2018at different frequenciessimultaneously, give rise to AC magnetictorquesatthesumanddi fferencefrequencies. These can be tuned to match the natural resonance fre- quency of the resonator, allowing it to function e ffec- tively as a signal mixer. Di fferent frequency compo- nentscanbeextractedfromthemechanicalsignalasthe DC field sweeps the magnetization texture to measure simultaneously the DC magnetization, AC susceptibil- ity,andhigherorderACsusceptibilityterms. Figure 1: Instrument schematic for frequency-mixed nanome chani- cal detection of AC susceptibility. The lock-in amplifier pr ovides the reference/drive frequencies, f1andf2, which induce orthogonal AC fields through aHelmholtz coil assembly ( Hac x) and single coil ( Hac z), both illustrated in the bottom-right inset). The lock-in ou tput signal is amplified by audio (AF) and radio (RF) frequency power ampl i- fiers, respectively. A HeNe laser is used to interferometric ally detect the mechanical motion of the torsional resonator (BS =beam splitter, PD=photodetector). The di fference and/or sum frequencies are de- modulated simultaneously at the lock-in. Top left inset: fa lse-colour scanning electron micrograph of thedevice (scale bar =1µm) Determination of the susceptibility requires some level of description of the spin-mechanical coupling, taking in to account the conversion of magnetic torque into mechanical torque. The full tensor analysis will include Einstein-de Haas terms, which we will neglect here to concentrate on the net mechanical torque aris- ing throughmagneticanisotropy. Conceptually,we can viewthe systemas two torsionspringsconnectedin se- ries. The geometrically-confinedmagnetizationtexture of the disk is the first spring, as is the one acted on di- rectly by the torquing fields. In turn, the net magnetic torquewill twist the torsionrod until an equivalentme- chanicalcounter-torquedevelops. Forthepresentanaly- sis,wedefinethee ffectivemagnetictorsionspringcon- stantsbyconvertingmagneticsusceptibilitiesintoangu- lar changesas if the magnetizationof the element werea single macrospin. Although far from the real case, theparallel axistheoremallowsforthe samenet torque from the same net magnetization independent of how themagnetizationisdistributed. The net magnetic anisotropy (shape plus magne- tocrystalline) of the mesoscale magnetization in this simplepicturemanifestsitselfthroughdi fferencesindi- agonal components of the magnetic susceptibility ten- sor. If no anisotropy exists, then the equilibrium mag- netization will always be parallel to the applied field, with a resultant torque of zero. In general, one will encounterstructureswith significantlydi fferentsuscep- tibilities parallel and perpendicular to the plane of the torsionpaddle,but whereneither componentis negligi- ble. This is the case for the specific example we con- sider below, a 3D vortex state in a short cylinder of YIG, where the low-field linear in-plane susceptibility isapproximatelytwicetheout-of-planesusceptibility19. Then,theresultanttorqueinsimultaneoussmall(forlin- earmagnetizationresponse)fields HxandHzis: −τy=µ0(MxHz−MzHx)V=µ0(χx−χz)HxHzV(1) This net torque will transfer to the lattice and twist the resonator. For a low frequency AC field, an AC torque ensuesinphasewiththefield. Ifthefrequencyisnottoo high, the magnetization remains in quasi-static equilib- rium with the field at all times. For the vortex structure in the absence of pinning (no slow thermal dynamics), this criterion is satisfied at frequenciesin the low MHz regime,convenientforresonantenhancementoftheme- chanical response in nanoscale torsional structures. In the presentexperiment,correctionsforthe appliedfield angles changing in the frame of reference of the pad- dlearenegligible: themechanicalspringismuchsti ffer (∼106×) than the magnetic spring, and the mechanical Q isonly2600. WhentwoorthogonalACfieldsactsimultaneouslyat differentfrequencies, f1andf2,ACmagnetictorquesat thesumanddifferencefrequenciesalsoarise,according to: −τy=µ0(χx−χz)Hac xcos(2πf1t)Hac zcos(2πf2t)V =µ0(χx−χz)Hac xHac z 2(cos(2π(f2−f1)t)+cos(2π(f2+f1)t))V (2) With the simultaneous application of a DC field to sweep the magnetization texture through a hysteresis cycle, this becomes the basis of the AC susceptome- trymeasurement,to complementtheDC magnetization data. For demonstration of the technique, a single-crystal, mesoscale YIG disk (radius =600nm,thickness =500 2Figure 2: DC torque magnetometry of an individual micromagn etic YIGdisk, (a) and the corresponding numerical derivative, ( b). nm)wasfocusedionbeam(FIB)-milledfromanepitax- ialthickfilmandnanomanipulated in-situontoaprefab- ricated torsional resonator19. A scanning electron mi- crograph of the completed device is shown in the inset of Fig 1. The FIB milling and ‘pick and place’ proce- dureswereemployedtoovercomethedi fficultyofform- ing monocrystalline mesoscale objects which are to be measured singularly. The pristine magnetic nature of the YIG disk is exhibited, within experimental resolu- tion,bythelackofBarkhausensignaturesinthechange of the magnetization with applied field, which are as- sociated with nanoscale imperfections such as grain boundaries in polycrystalline materials. In the present context YIG simplifies interpretation of the demonstra- tion susceptibility signals. Hysteretic minor loops as- sociated with Barkhausen transitions in the magnetiza- tion evolution2, can strongly influence and complicate theAC susceptibility. The YIG disk,on the otherhand, wasshowntohavenoobservableminorhysteresis. The resonatorwas drivenby the magnetictorque ex- erted on the YIG disk. The mechanical deflection was optically detected through the interferometric modula- tion of the light reflected from the device. The mag- netization was biased by a variable external field, ( Hdc x, CartesiancoordinatesindicatedinFig. 1)intheplaneof the resonator and perpendicularto the torsion rod. The AC torquing field was supplied by a small copper wire coilwhichproducedanout-of-planeditherfield, Hac z,to complete the field orientationsnecessary for the torque magnetometry measurement. The AC field was driven atthefundamentaltorsionfrequencytomaximizedetec- tionsensitivity. ThemagnetizationoftheYIGdiskwith appliedbias field is shown in Fig. 2, characterizedby a unipolarhysteresissimilartowhathasbeenobservedin two-dimensional disks where the sharp transitions areattributed to the nucleation and annihilation of a mag- netic vortex. However, the thickness of the YIG disk promotes additional three dimensional structure in the magnetizationtexture(detailedin Ref. 19). AresultfromthesimultaneousacquisitionoftheDC magnetization and AC susceptibility is shown in Fig 3. Featuresagreeingcloselyto the numericalderivativeof the DC magnetization curve, Fig. 2b, are evident with the differencesarisingfromthe susceptibilityscreening outirreversibleeventsinthefieldsweep. Here,thelock- in amplifier providedboth orthogonalac drive frequen- cies,f1(Hac x)andf2(Hac z),whilerecordingtheresponse of the resonator through simultaneous demodulation20 atf2andatf2−f1(seeFig. 1formeasurementscheme). It is interesting to note that a softening of the magne- tization texture (peak in the susceptibility) occurs just after the irreversible annihilation transition on the field sweep-up. Thehighestfieldsusceptibilityisat opposite phase in this measurementconsistent with the possibil- ityinherentinEqn. 2for χzcontributestoexceedthe χx contributionasthemagnetizationapproachessaturation inx. Thisfeatureisreplicatedagainonthesweepdown ataslightlylowerfield–ahystereticfeaturenotreadily apparentfromtheDCmagnetization. Figure 3: Simultaneous acquisition of the DC magnetometry t hrough the driven signal at f2=fres+f1, a), and low frequency AC suscepti- bility detected at the mixing (resonance) frequency, fres, b). Here, f1 was set to 500 Hz. The technique can easily be extended to detect the harmonics of AC susceptibility arising when the mag- netization response is nonlinear in field. For example, for a magnetization describable by a polynomial series infield, Mx=a1Hx+a2H2 x+a3H3 x+a4H4 x+a5H5 x+a6H6 x+..., (3) whereHx=Hdc x+Hac x,multipleharmonicsof f1arise and also mix with f2to generate unique torque terms. 3For this representation of nonlinear magnetization, Ta- ble 1 shows the amplitude coe fficients for the first six entriesin the Fouriersum forthe full nonlinearsuscep- tibility, χx(Hx)=∞/summationdisplay n=−∞χx n(Hx)e−i(2πnf1)t. (4) Table1: Harmonics of the AC susceptibility f2±nf1Amplitudecoefficients relatedto f2+0a1Hdc x+a2[(Hdc x)2+1 2]+ a3[(Hdc x)3+3 2Hdc x] +a4[(Hdc x)4+3Hdc x)2+3 8]+ a5[(Hdc x)5+5(Hdc x)3+15 8Hdc x] +a6[(Hdc x)6+15 2(Hdc x)4+ 45 8(Hdc x)2+5 16]χx 0(Hx) f2±f1a1+2a2Hdc x+a3[3(Hdc x)2+ 3 4]+a4[4(Hdc x)3+3Hdc x] +a5[5(Hdc x)4+15 2(Hdc x)2+5 8] +a6[6(Hdc x)5+15(Hdc x)3+ 15 4Hdc x]−dMz dHz|Hdcxχx 1(Hx) f2±2f11 2a2+3 2a3Hdc x+a4[3(Hdc x)2+ 1 2]+a5[5(Hdc x)3+5 2Hdc x] +a6[15 2(Hdc x)4+15 2(Hdc x)2+ 15 32]χx 2(Hx) f2±3f11 4a3+a4Hdc x+a5[5 2(Hdc x)2+ 5 16]+a6[5(Hdc x)3+15 8Hdc x]χx 3(Hx) f2±4f11 8a4+5 8a5Hdc x+a6[15 8(Hdc x)2+ 3 16]χx 4(Hx) f2±5f11 16a5+3 8a6Hdc x χx 5(Hx) f2±6f11 32a6 χx 6(Hx) A spectroscopic frequency-versus-field mapping of the higher order AC susceptibilities for the YIG disk is shown in Fig. 4 for the sweep-down portion of the hysteresis cycle (31 to 4 kA /m). Here the fre- quency response was recorded at each magnetic field step while keeping both f1(Hac x) andf2(Hac z) constant (500 Hz and 1.8095 MHz, respectively). The f2drive wasshifted fromthe resonancefrequency(1.808MHz) inordertoallowformoresusceptibilityharmonicstobe recorded within the mechanical resonance line-width. The brightest band is the demodulation of the f2drive, while the subsequent bands represent the harmonics of themagneticsusceptibility. Thefeaturesobservedinthe secondhalfofthefieldsweepinFig. 3barereproduced, includingthestrongsusceptibilitypeakjustbefore(very closetotheonsetof)thenucleationtransition. Through such strong nonlinearities in the magnetization curve, mixingfrequencies f2±nf1areobservedupto n=7inFig. 4. Figure 4: Spectroscopic mapping of higher harmonics of the A C sus- ceptibility. f1(Hac x) andf2(Hac z) were each driven simultaneously at constant frequency (500 Hz and 1.8095 MHz respectively) w hile the frequency response (100 Hzbandwidth) was measured with mag- netic field. The image shows the sweep-down portion of the fiel d sweep. The band of highest intensity is the demodulation of f2, with the successive bands representing theharmonics ofAC susce ptibility, f2±nf1. A line-scan thorough the high susceptibility peaks is show n in the bottom left, showing up to the n =7 harmonic. The original in- tensity scale was cropped to show the highest order signals. The multifrequency responses are all harmonics of the low AC frequency dithering the magnetiza- tion, mixed to fall within the mechanical resonance linewidth. High harmonics of susceptibility have been usedpreviouslytocharacterizehysteresisforbulkferro- magnetismandsuperconductivity. Inbulksystems, and in arrays of nanostructures, the strongest nonlinearities inmagnetizationcomefromhystereticfeatures. Typical ACfieldamplitudescanprobeminorhysteresisloops21, oreventhefullhysteresiscloseenoughtotheCurietem- perature4. By contrast, the single mesoscale structure we examine here exhibits no minor hysteresis, and its magnetization nonlinearities are observed without any diminution from array averaging. The harmonics char- acterizethenon-hysteretic,nonlinearresponse. In summary, nanomechanical detection of the low- frequency AC susceptibility (along with higher order harmonics)ofanindividualmicromagneticdiskwasac- complishedthroughfrequencymixingoforthogonalAC drivingfieldsmanifestingasamechanicaltorqueonthe resonator. In addition to providing further insight into quasi-static magnetization processes in geometrically- confined magnetic elements, this signal-mixing ap- proach, in principle, will enable very broadband mea- surementsoffrequency-dependentsusceptibility. 4Acknowledgements The authors would like to thank NSERC, CIFAR, Alberta Innovates and NINT for support. Partial de- vice fabrication was done at the University of Alberta NanoFab. FIBmillingandmanipulationwascarriedout byD.VickandS.R.ComptonattheNINTelectronmi- croscopyfacility. References [1] J. Moreland, A. Jander, J.A. Beall, P. Kabos and S.E. Russ ek, IEEET.Magn., 37,2770 (2001). [2] J.A.J. Burgess, A.E. Fraser, F. Fani Sani, D. Vick, B.D. H auer, J.P.Davis and M.R.Freeman, Science 339,1051 (2013). [3] J.E. Losby, J.A.J. Burgess, Z. Diao, D.C. Fortin, W.K. Hi ebert and M.R.Freeman, J.Appl. Phys. 111, 07D305 (2012). [4] C. R¨ udt, P J. Jensen, A. Scherz, J. Lindner, P. Poulopoul os and K.Baberschke, Phys.Rev. B 69,014419 (2004). [5] S.Pr¨ ufer and M.Ziese, Phys.Stat. Sol. 245, 1661 (2008). [6] B.Heinrich and A.S.Arrott, J.Appl. Phys. 53, 1991 (1982). [7] K.R.Sydney, D.H.Chaplin and G.V.H.Wilson,J.Magn.Mag n. Mater.2,345 (1976). [8] W. Kleemann, J. Rhensius, O. Petracic, J. F´ erre, J.P. Ja met and H.Bernas, Phys.Rev. Lett., 99,097203 (2007). [9] U.Stetter, M.Farle, K.Baberschke, and W.G.Clark, Phys .Rev. B45,503 (1992). [10] J. A. J. Burgess, D. C. Fortin, J. E. Losby, D. Grombacher , J. P. Davis and M.R.Freeman, Phys.Rev. B 82, 144403 (2010). [11] V. Str¨ om, B.J. J¨ onsson, K. V. Rao and D. Dahlberg, J. Ap pl. Phys.81,5003 (1997). [12] L. Lundgren, P. Svedlindh and O. Beckman, J. Magn. Magn. Mater.25, 33 (1981). [13] M.J.Mart´ ınez-P´ erez, J.Ses´ e,F.Luis,D.Drung and T .Schurig, Rev. Sci. Instrum. 81, 016108 (2010). [14] W. Wernsdorfer, B. Doudin, D. Mailly, K. Hasselbach, A. Benoit, J. Meier, J.-Ph. Ansermet, and B. Barbara, Phys. Rev . Lett.77, 1873 (1996). [15] K. S. Novoselov, A. K. Geim, S. V. Dubonos, E. W. Hill, and I. V.Grigorieva, Nature 426, 812 (2003). [16] S. Wirth, S. von Moln´ ar, M. Field, and D. D. Awschalom, J . Appl. Phys. 85,5249 (1999). [17] A.Aspelmeier, M.Tischer, M.Farle, M.Russo,K.Babers chke, and D.Arvanitis, J.Magn. Magn. Mater. 146, 256 (1995). [18] F.M.R¨ omer,F.Kronast,L.Heyne,C.Hassel,A.Banholz er, M. Kl¨ aui, R. Meckenstock, J. Lindner, and M. Farle, Appl. Phys . Lett.96, 122501 (2010). [19] J.E. Losby, Z. Diao, J.A.J. Burgess, S.R. Compton,T. Fi rdous, F. Fani Sani, M. Belov, D. Vick, W.K. Hiebert and M.R. Free- man,Nanomechanical characterization of a three-dimensional micromagnetic vortex state in a single-crystal yttrium iro n gar- net disk. Submitted: New J.Phys.(2013). [20] Zurich Instruments HF2LI with Multi-frequency (MF) op tion, Zurich, Switzerland. [21] C.P.Bean, Rev. Mod. Phys. 36, 31 (1964). 5

2013-09-11

A novel method for simultaneous detection of both DC and time-dependent magnetic signatures in individual mesoscopic structures has emerged from early studies in spin mechanics. Multifrequency nanomechanical detection of AC susceptibility and its harmonics highlights reversible nonlinearities in the magnetization response of a single yttrium iron garnet (YIG) element, separating them from hysteretic jumps in the DC magnetization.

Nanomechanical AC Susceptometry of an Individual Mesoscopic Ferrimagnet

1309.2965v1

1 Spin Hall magnetoresistance in Pt/YIG bilayers via varying magnon excitation Q. B. Liu1,2, K. K. Meng1*, S. Q. Zheng1, Y . C. Wu1, J. Miao1, X. G. Xu1 and Y . Jiang1** 1Beijing Advanced Innovation Center for Materials Genome Engineering, School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China 2Applied and Engineering Physics , Cornell University, Ithaca, NY 14853, USA Abstract: Spin Hall magnetoresistance (SMR) and magnon excitation magnetoresistance (M MR) that all generate via the spin Hall effect and inverse spin Hall effect in a nonmagnetic material are always related to each other. However, the influence of magnon excitation for SMR is often overlooked due to the negligible MMR. Here, we investigate the SMR in Pt/Y 3Fe5O12 (YIG) bilayers from 5 to 300K, in which the YIG are treated after Ar+-ion milling. The SMR in the treated device is smaller than in the non -treated. According to theoretical simulation, we attribute this phenomenon to the reduction o f the interfacial spin -mixing conductance at the treated Pt/YIG interface induced by the magnon suppression. Our experimental results point out that the SMR and the MMR are inter -connected, and the former could be modulated via magnon excitation. Our fin dings provide a new approach for separating and clarifying the underlying mechanisms. Key words: Spin-orbit coupling; Spin transport in metals ; Spin transport through interfaces *Authors to whom correspondence should be addressed: *kkmeng@ustb.edu.cn **yjiang@ustb.edu.cn 2 Ferromagnetic insulators (FMIs) has emerged as a promising and novel technology platform to generate, modulate and detect spin information over long dista nces [1,2]. The advantage of using FMIs against metallic ones is avoiding the flow of charge current, preventing ohmic losses and the emergence of undesired spurious effects [3]. Yttrium Iron Garnet (YIG) is one of the most prominent FMIs for the investiga tion of spintronics, magnonics [4,5], and spin caloritronics [6,7], due to its extremely low damping, soft ferrimagnetism and negligible in -plane magnetic anisotropy even in limit thickness. One can efficiently investigate and control the magnetization direction and spin waves propagation in FMIs through transport measurements based on the spin Hall effect (SHE) and the inverse spin Hall effect (ISHE) [2,8,9], by which the mutual conversion between magnon or spin current and charge current can be realized in a heavy metal (HM) with strong spin-orbit coupling (SOC) . Therefore, t he combin ation of S HE and ISHE can give rise to the spin Hall magnetoresistance (SMR) and magnon excitation magnetoresistance (MMR) [10,11], while the former can be ascribed to the spi n transfer torque at the HM/FMIs interface. Owing to the SHE, the spin current would accumulate at the HM/FMIs interface with spin polarization σ parallel to the surface. If σ is not collinear with the magnetization M of the FMIs, the accumulated spin current can exert a torque proportional to M×(M×σ) on the M of FMIs. It suggests that a finite spin current will be absorbed by FMIs if M and σ have a finite angle. The spin current absorption by FMIs represents the spin current reflection that is suppressed, and its resistance therefore will change with the magnetization orientation which is expected to be maximized (minimized) when M is perpendicula r (parallel) to σ [8,9]. On the other hand, the MMR is firstly observed in Pt/YIG heterostructures in which the two parallel Pt strips were separated by a distance. When the two Pt electrodes are closed enough, the spin accumulation at 3 one Pt strip will transmit into anot her strip by magnon, and the spin current will be converted into charge current via ISHE. Therefore, a large non -local charge current is supposed for M//σ, since the magnons beneath the first Pt strip can diffuse across the gap to the second Pt strip. In c ontrast, for M⊥σ, the non -local signals should be significantly reduced, since the spin transfer torque absorbs the magnon propagation. Therefore, the measured resistance or voltage signal is expected to be maximized (minimized) when M is parallel (perpend icular) to σ [12]. The MMR can thus be seen as a magnon -mediated, non -local counterpart of the S MR. In short , both the SMR and the MMR stem from spin accumulation and spin transport at the HM/FIMs interface s. Theoretically, the efficiency and strength of t he spin transfer across the interface depend on the magnitude of interfacial spin -mixing conductance (SMC) G↑↓=(G r+iGi), where Gr (Gi) denotes the real (imaginary) part [1 3,14]. Therefore, if the magnon transport can be suppressed or enhanc ed, it will cha nge the magnitude of Gr and the SMR is expected to change correspondingly . However, the influence of magnon excitation for SMR is often overlooked due to the negligible MMR, and few methods have been reported to separate and clarif y the underlying mechanis ms. In this work, we have employed the etching treatment to suppress the in -plane magnon transport and found that the SMR in Pt/YIG -based devices was dramatically altered when the part of YIG film out of the hall devices is treated with Ar+-ion milling etc hing. At room temperature, the SMR in the treated device was much smaller than that in the non -treated, while at low temperature the SMR for the two devices was similar. The anomalous SMR reduction has similar temperature dependence with magnon excitation. According to the theoretical simulation, we have attribute d this phenomenon to the reduction of the G↑↓ at the Pt/YIG interface induced by magnon suppression after etching . Our experimental results point ed out that the 4 SMR and the MMR were interconnected, while the former can be modulated via magnon excitation even though the value due to magnon excitatio ns was much smaller than the SMR . The epitaxial YIG film was grown on a [111] -oriented GGG substrate (lattice parameter a = 1.237 nm) by pulsed laser deposition technique with the substrate temperature TS = 780℃ and the oxygen pressure 10 Pa. Then the sam ples were annealed at 780℃ for 30 min at the oxygen pressure of 200 Pa . The base pressure of the PLD cavity was better than 2 ×10-6 Pa. Then, the Pt layer was deposited on YIG at room temperature by magnetron sputtering. In order to avoid the run -to-run err or, each large Pt/YIG sample was then cut into two small pieces. After the deposition, the electron beam lithography and Ar ion milling were used to pattern Hall bars, and a lift-off process was used to form contact electrodes. The size of all the Hall bar s is 20 μm×120 μm. For comparison , a part of YIG film out of the Hall bars was etched away by Ar+-ion milling which was defined as YIG+ as shown in Fig. 1(a). Fig. 1 (b) shows the XRD ω-2θ scan spectra of the 40-nm-thick YIG thin film, which was taken from represe ntative thin film of each type, and it shows predominant (444) diffraction peaks with no diffraction peaks occurring from impurity phases or other crystallographic orientation, indicating the single phase nature. According to the (444) diffraction peak pos ition and the reciprocal space maps of the (642) reflection of 30-nm-thick YIG films grown on GGG as shown in Fig. 1(c), we have found that the lattice constant of YIG layer is similar with the value of GGG substrate, indicating the high quality epitaxial growth without mismatch. Moreover, the saturated magnetization of the YIG layer measured by a vibrating sample magnetometer was determined to be 140 emu/cm3 as shown in Fig. 1(d), which is similar with the value of the bulk YIG. All the magnetotransport me asurements performed in the multilayers 5 were carried out using a Keithley 6221 sourcemeter and a Keithley 2182A nanovoltmeter. These measurements were performed at different temperatures from 5 and 300 K in a liquid -He cryostat that allows applying magneti c fields H up to 3 T and rotating the samples by 360 º. Using small and non -perturbative current densities (~ 106 A/cm2), we have investigated the angular -dependent magnetoresistance (ADMR) in Pt (5 nm)/YIG (40 nm) and Pt (5 nm)/YIG+ (40 nm) devices at roo m temperature. The measurement configuration, the definition of the axes, and the rotation angles (α, β, γ) are defined in the sketches as shown in Fig. 2(a). Figs. 2(b) and (c) show the longitudinal ADMR curves with applying magnetic field of 3T, and the ADMR was defined as ADMR =[ρ−ρ(0 deg)]/ρ(0 deg) [15]. The ADMR of the two devices show the expected behavior of the SMR, in agreement with the earlier report in Pt/YIG bilayers, and the values in the treated and non -treated devices are about 5.723 × 10−4 and 7.814 × 10−4, respectively. One can find that the SMR of Pt/YIG is 1.4 times larger than that in Pt/YIG+ device. In order to further investigate the anomalous SMR reduction in Pt/YIG+ device, we carried out the ADMR measurements at the temperature range from 5 to 300 K. The temperature dependent ADMR of Pt/YIG and Pt/YIG+ bilayers in the β scan with 3 T field are shown in Figs. 3 (a) and (b), which all can be well fitted by the SMR mechanism. The Fig. 3(c) displays the temperature dependent SMR of the two devices. It is obvious that the SMR changes non -monotonically with decreasing temperature , which is in debate since it might stem from the competition of two physical mechanisms: the spin Hall effect -induced magnetoresistance (SHE -MR) and the magnetic proximity effect -induced magnetoresistance (MPE -MR) [16]. The MPE -MR becomes evident a t relatively lower temperature due to the magnetization induced by the MPE , while at higher temperature, the thermal 6 fluctuations will dominate, disrupting the spontaneous Pt magnetization and eliminating the MPE -MR. More interestingly, the temperature dependence of SMR in Pt/YIG bilayers in the previous reports was weak, while the sharp drop of SMR below 100 K in our Pt/YIG bilayers would be discussed latter via systematic ADMR measurements with varying the thickness of Pt. Furthermore, we have defined the ratio ΔSMR as ΔSMR =[SMR (YIG)-SMR (YIG+)]/SMR (YIG), and the temperature dependence is shown in Fig. 3(d). It seems to be constant below 100 K and linearly increases from 100K to 300K. The anomalous temperature dependen ce behavior may be related to the magnon excit ations, which should also be suppressed by the MPE and increase with temperature. To further verify our speculation about the magnon excitation modulat ed SMR, we have carried out the Pt thickness (t) dependent measurements. The unusual ΔSMR fluctuation cur ve with increasing t was shown in Fig. 4 (a). We can find that the ΔSMR is irrelevant with the bulk spin Hall angle (SHA) and spin diffusion length (SDL) of Pt that should exhibit a peak value at t ~ 3 nm [1 7]. Therefore, the only possibility of the unusua l ΔSMR should originate from the change of the interface SMC due to the suppress ion of magnon transport after etching. To qualitatively analyze the experimental results, we employ a simulation within the spin drift -diffusion theoretical framework. Accordi ng to the SMR theory, the longitudinal resistivities of the Pt layer are given by ）（2 y 1 0 0 m-1 , where m(mx,my,mz)=M/Ms are the normalized projections of the magnetization of the YIG film to the three main axes, Ms the saturated magnetization of the YIG , ρ0 is the Drude resistivity , Δρ0 accounts for a number of corrections due to the SHE and Δρ1 is the main SMR term . The SMR is quantified by [10,1 8]: 7 )/(coth) 2/(1)2/( tanh/2 2 sd r sdsd sd SH λt Gρλλt tλθρΔρ (1) Where λsd, θSH, t, and Gr are the spin Hall angle , the spin diffusion length , the Pt layer thickness, and the real part of the SMC at the YIG/Pt interface, respectively. The thickness dependence of the longitudinal resistance is shown in Fig. 4(b), and the product of the longitudinal resistivity ρ and th e Pt film thickness t is found to change linearly with the film thickness as shown in the inset of Fig. 4(b). Considering the product is found to well obey the equation ρt = ρbt + ρs with the bulk resistivity ρb and the interfacial resistivity ρs, which are determined to be 8.0 μΩ.cm and 32.0 μΩ.cm2 based on the fitted lines, respectively. The bulk resistivity is close to the value of 10.0 μΩ.cm of the bulk Pt [ 19]. Here, we have used the SHA, the SDL and the Gr as 0.07, 1.5 nm and 5 ×1015 Ω-1 m-1 [16,18,20]. We can find a large discrepancy between the fitted and the measured results as shown in Fig. 4(c). However, the variation trend of the SMR could be fitted with giving SHA = 0.131 and SDL = 0.864 nm as shown in Fig. 4(d), and the large SHA should stem fro m the interfacial contribution [2 1]. Therefore, the sharp drop of SMR ratio below 100 K in our Pt/YIG bilayers could be derived from the large interfacial SHA. Notably, the Pt/YIG+ device should have similar SHA and SDL with Pt/YIG device. We can fit the Gr in Pt/YIG+ bilayers through giving SHA 0.131 and SDL 0.864 nm with Eq. (1). According to the fitted results as shown in Fig. 5(a), we can determine the Gr at the Pt/YIG+ interface is one order of magnitude smaller than Pt/YIG interface. Furthermore, the Hanle magnetoresistance (HMR) cloud modulate the resistance of the HM layer with H instead of M, exhibiting the similar angular dependent behavior with SMR: no resistance correction is observed for H parallel to σ, whereas a resistance increase is obtained for H perpendicular to σ [22,23]. Therefore, 8 the anomalous SMR reduction in our devices may also stem from the different HMR. Notably, the HMR is due to the spin precession around the external magnetic field H, leading to the spin relaxation , so we could distinguish SMR and HMR from the field dependent MR measurement. As shown in Fig. 5(b), we can find that the distinction of HMR is negligible as compared with SMR in Pt/YIG and Pt/YIG+ devices at H =3 T. Recently , Y . Dai, et al. have found that the simulation method by Eq. (1) exhibits discrepancy because the SHA is fluctuation with varying the thickness of Pt layer. They put forward the electron diffusion coefficient (EDC) and SDL that could be precisely estimate d through the ratio of HMR and SMR [2 1]. Therefore, we would use it to determine the EDC and SDL in the Pt/YIG devices. Then, the SMC at Pt/YIG+ interface could be calculated with the similar EDC and SDL in Pt/YIG. Notably, the HMR/SMR ratio is independent of the SHA and thus the HMR/SMR ratio reads [21]: DBigμλDBigμ λt Gρt/λ Gλρ λt//DBigμ λt DBgμiλSMR HMR B sdB sdr xxsd r sdxx sdB sd B sd 22222 2 1)1(coth 2 1) (coth 21 )2( tanh)21( tanh 11Re / (2) where g, μB, D, and B are the Landé factor, the Bohr magneton, the EDC, and the magnetic induction intensity, respectively . The SDL and the EDC are determined to be 2.02 nm and 4 × 10−6 m2s-1, respectively, where the longitudinal resistivity ρ = 25.5 μΩ.cm, Gr = 5 × 1015 Ω-1m-1, and the Landé factor g ≈ 2.0. As shown in Fig. 6, if one assigns the sd and the D with other values that deviate from 2.02 nm and 4 × 10−6 m2s-1, the fitted results cannot reproduce for all the samples. Based on the determined sd and D, the Gr for Pt/YIG+ device is five times smaller than Pt/YIG device. 9 However, there is still a problem which needs to be further clarified. From Fig. 6 (c), we can find that the fitted curves are insensitivity to the SMC which is used as the only free paramet er. It is difficult to point out which of the two fitted methods is more precise. Here, we put forward a simple model to further explain this result. According to the report by X. P. Zhang in Ref. [2 3], the Gr should be read as Gr = e2vF(1/τP-1/τT) = e2vF/τP + Gs, where e, vF, τP and τT are the elementary charge ( e > 0), the density of states per spin species at the Fermi level, the longitudinal and transverse spin relaxation times per unit length for the itinerant electron, respectively. We note that Gr represents the difference between the longitudinal spin relaxation with transverse spin relaxation and it does not have a physical meaning on its own. The Gs=-e2vF/τT originates entirely from spin -flip processes and associates with magnon emi ssion and absorption [2 4,25]. Therefore, magnon transport could affect the Gr. In the Pt/YIG+ device, the part of YIG film around the Hall bar is etched, which produces infinite high barriers at two sides of the Hall bar and suppresses the in -plane magnon transport. Obviously, it will increase the magnon accumulation and suppress the spin absorption at Pt/YIG+ interface, which will reduce the Gr and the corresponding SMR [2 6]. Q. Shao et al. also found that the measured SOT efficiency was significantly enha nced with increasing the FMIs (TmIG) thickness, which is completely different from the FMs based devices. Similarly, we also found that the SMR is significantly enhanced with increasing the YIG thickness as shown in Fig. 6(d). Therefore, we also ascribe the modulated SMR to the magnon excitations because the thicker YIG is benefit for magnon diffusion, reducing magnon accumulation [2 7]. In conclusion, we have found the SMR in Pt/YIG -based devices was dramatically altered when the part of YIG film out of the hall devices was etched by the Ar+ -ion milling. At room temperature, the SMR effect in the treated device was smaller than 10 in the non -treated one. According to theoretical simulation, we attributed this phenomenon to the reduction of the G↑↓ at the Pt/YIG interface induced by the suppressed magnon transport. Our experimental results pointed out that the SMR and the MMR that were inter -connected, and the SMR could be modulated via magnon excitation/suppression even though the magnitude of MMR from magnon ex citations was much smaller than the SMR . Our findings provide a new approach for modulating SMR for spintronic applications. Acknowledgements: This work was partially supported by the National Science Foundation of China (Grant Nos. 51971027 , 51731003, 51 671019, 51602022, 61674013, 51602025), and the Fundamental Research Funds for the Central Universities ( FRF-TP-19-001A3). 11 References [1] M. I. Dyakonov, and V . I. Perel, Phys. Lett. A 35, 459 (1971). [2] Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh. Nature 464, 262 (2010). [3] M.-Z.Wu and A. Hoffmann, Academic, New York, 2013. [4] Demokritov, S. O., Hillebrands, B. and Slavin, A. N. Phys. Rep. 348, 441 (2001). [5] A. A. Serga, A. V . Chumak, and B. Hillebrands, J. Phys. D 43, 264002 (2010). [6] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015). [7] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y . Kajiwara, H. Umezawa, H. Kaw ai, G. E. W. Bauer, S. Maekawa and E. Saitoh, Nat. Mater. 9, 894 (2010). [8] N. Vlietstra, J. Shan, V . Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). [9] C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). [10] H. Nakayama, M. Althammer, Y . T. Chen, K. Uchida, Y . Kajiwara, D. Kikuchi, T. Ohtani, S. Geprags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). [11] S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Appl. Phys. Lett 107, 172405 (2015). [12] S. S. -L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). [13] A. Brataas, Y . V . Nazarov, and G. E. W. Bauer, Phys. Re v. Lett. 84, 2481 (2000). 12 [14] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S.Altmannshofer, M. Weiler, H. Huebl, S. Geprä gs, M. Opel,R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y . -T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). [15] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E.W. Bauer, Phys. Rev. B 87, 144411 (2013). [16] Q. Shao, A. Grutter, Y . Liu, G. Yu, C. Y . Yang, D. A. Gilbert, E. Arenholz, P. Shafer, X. Che, C. Tang, M. Aldosary, A. Navabi, Q. L. He, B. J. Kirby, J. Shi, and K. L. Wang, Phys. Rev. B 99, 104401 (2019). [17] M. H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 116, 126601 (2016). [18] X. Jia, K. Liu, K. X ia, and G. E. W. Bauer, Europhys. Lett. 96, 17005 (2011). [19] S. Dutta, K. Sankaran, K. Moors, G. Pourtois, S. Van Elshocht, J. Bö mmels, W. Vandervorst, Z. Tӧei, and C. Adelmann, J. Appl. Phys. 122, 025107 (2017). [20] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ral ph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). [21] Y . Dai, S. J. Xu, S. W. Chen, X. L. Fan, D. Z. Yang, D. S. Xue, D. S. Song, J. Zhu, S. M. Zhou, and X. P. Qiu , Phys. Rev. B 100, 064404 (2019). [22] S. Vé lez, V . N. Golovach, A. Bedoya -Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett. 116, 016603 (2016). [23] H. Wu, X. Zhang, C. H. Wan, B. S. Tao, L. Huang, W. J. Kong, and X. F. Han, Phys. Rev. B 94, 174407 (2016). [24] X.-P. Zhang, F. S. Bergeret, and V. N. Golovach , Nano Lett. 19, 9, 6330 (2019). [25] F. K. Dejene, N. Vlietstra, D. Luc, X . Waintal, J. B en Youssef, and B. J. Van Wees, 13 Phys. Rev. B 91, 100404 (R) (2015) . [26] S. Vélez, A. Bedoya -Pinto, W. Yan, L. E. Hueso, and F. Casanova Phys. Rev. B 94, 174405 (2016). [27] Q. Shao, C . Tang, G . Yu, A . Navabi, H . Wu, C . He, J . Li, P. Upadhyaya, P . Zhang, S. A. Razavi, Q . L. He, Y . Liu, P . Yang , S. K. Kim, C . Zheng, Y . Liu, L . Pan, R . K. Lake, X . Han, Y . Tserkovnyak , J. Shi and K. L. Wang , Nat. Commun. 9, 3612 (2018). Figure captions Figure 1 (a) Sample and measurement configurations with the definition of the axes. (b) The XRD ω-2θ scans of the YIG film grown on GGG substrate. (c) High -resolution XRD reciprocal space maps of the YIG film grown on GGG substrate. (d) Fi eld dependence of out -of-plane magnetization (black) and in -plane magnetization (red) of YIG film. Figure 2 (a) Notations of different rotations of the angular α, β, and γ. (b) and (c) ADMR in Pt/YIG and Pt/YIG+ devices at 300 K with H = 3 T in the three rotation planes ( α, β and γ ). Figure 3 (a) and (b) ADMR curves measured in Pt/YIG and Pt/YIG+ devices with varying β at different temperature, and the applied magnetic field is 3 T. (c) Temperature dependence of the ADMR of Pt/YIG (black) and Pt/YIG+ (red) devices. (d) Temperature dependence of the ΔSMR . Figure 4 The ΔSMR (a) and longitudinal resistance in the two devices (b) at room temperature with varying Pt layer thickness (t). The inset of (b) shows the Pt layer 14 thickness dependence of the produc t of the longitudinal resistivity and thickness in the two devices . (c) and (d) The Pt layer thickness dependence of the experimental and fitted SMR in Pt/YIG with different parameters. Figure 5 (a) The experimental and fitted SMR in Pt/YIG+ with varying the Pt layer thickness, and Gr is parameter for fitting. (b) The MR versus the external magnetic field H along Y axis and Z axis. Figure 6 (a) and (b) The measured and fitted of HMR/SMR ( H) curves in Pt/YIG device . In panel (a), the red lines refer to the fitted results where the SDL and the EDC are free parameters. The blue and pink lines refer to the fitted results with fixed but different EDC. In panel (b), the red lines refer to the fitted results where the SDL and the EDC are free parameters. The blue and pink lines refer to the fitted results with fixed but different EDL. (c) The measured and fitted of HMR/SMR ( H) curves in Pt/YIG+ device. T he red lines refer to the fitted results where the SMC is free parameter. The blue and pink lines refer to the f itted results with fixed but different SMC. (d) Temperature dependence of the ADMR of Pt/YIG (20 nm) and Pt/YIG (60 nm) bilayers. 15 16 17

2020-02-28

Spin Hall magnetoresistance (SMR) and magnon excitation magnetoresistance (MMR) that all generate via the spin Hall effect and inverse spin Hall effect in a nonmagnetic material are always related to each other. However, the influence of magnon excitation for SMR is often overlooked due to the negligible MMR. Here, we investigate the SMR in Pt/Y3Fe5O12 (YIG) bilayers from 5 to 300K, in which the YIG are treated after Ar+-ion milling. The SMR in the treated device is smaller than in the non-treated. According to theoretical simulation, we attribute this phenomenon to the reduction of the interfacial spin-mixing conductance at the treated Pt/YIG interface induced by the magnon suppression. Our experimental results point out that the SMR and the MMR are inter-connected, and the former could be modulated via magnon excitation. Our findings provide a new approach for separating and clarifying the underlying mechanisms.

Spin Hall magnetoresistance in Pt/YIG bilayers via varying magnon excitation

2002.12550v1

Magnetic Nernst eect Sylvain D. Brechetand Jean-Philippe Ansermet Institute of Condensed Matter Physics, Station 3, Ecole Polytechnique F ed erale de Lausanne - EPFL, CH-1015 Lausanne, Switzerland The thermodynamics of irreversible processes in continuous media predicts the existence of a Magnetic Nernst eect that results from a magnetic analog to the Seebeck eect in a ferromagnet and magnetophoresis occurring in a paramagnetic electrode in contact with the ferromagnet. Thus, a voltage that has DC and AC components is expected across a Pt electrode as a response to the inhomogeneous magnetic induction eld generated by magnetostatic waves of an adjacent YIG slab subject to a temperature gradient. The voltage frequency and dependence on the orientation of the applied magnetic induction eld are quite distinct from that of spin pumping. I. INTRODUCTION In spincaloritronics, there has been recently quite some interest in the study of the propagation of spin waves across a ferromagnetic lm in the presence of a tempera- ture gradient1,2. For a conguration where the external magnetic induction eld is parallel to the temperature gradient, the propagation of magnetization waves induces a magnetic induction eld of magnitude proportional to the temperature gradient. Since the Seebeck eect refers to an electric eld induced by a temperature gradient, this eect demonstrated in a YIG slab3can be called the Magnetic Seebeck eect. The Magnetic Seebeck eect is a dynamical eect re- sulting from the precession of the magnetization. Thus, it should not be confused with the Spin Seebeck eect4{8 where the magnetization is at equilibrium. A theoreti- cal model of the Spin Seebeck eect was established by Adachi et al.9in a quantum framework, while Schreier et al.10attribute the eect to a dierence in the tempera- tures of the lattice and the magnetization. Here, we point out that the thermodynamics of irre- versible processes in a magnetic continuous medium11 predicts that the electrostatic potential depends on the gradient of the magnetic induction eld applied to it. This is a consequence of the magnetophoretic force ex- erted on a magnetized charge carrier. Magnetophoresis is commonly used in electrochemistry12and biophysics13. The Magnetic Seebeck eect implies the existence of a magnetic induction eld normal to the interface between the ferromagnet and the electrode. Thus, the magne- tophoretically induced electric eld, the thermally in- duced magnetic induction eld and the temperature gra- dient are orthogonal to one another as in the Nernst ef- fect. Hence, we call it the Magnetic Nernst eect14. Here, the YIG ferromagnet is an insulator and the Pt electrode is a paramagnetic conductor. The Magnetic Nernst ef- fect is thus a combination of the Magnetic Seebeck eect presented in reference3and magnetophoresis15. It is to be distinguished from the anomalous Nernst16eect and from the planar Nernst eect17. As shown below, the composition of these two eects results in a voltage that has a DC component and an AC component oscillating at twice the frequency of the mag-netization. The maximum amplitude occurs when the magnetic induction eld applied to carry the ferromag- netic resonance is oriented with a 45angle with respect to the orientation of the electrode and of the temperature gradient. II. MAGNETIC SEEBECK EFFECT As shown below, the composition of these two eects results in a voltage that has a DC component and an AC component oscillating at twice the frequency of the mag- netization. The maximum amplitude occurs when the magnetic induction eld applied to carry the ferromag- netic resonance is oriented with a 45angle with respect to the orientation of the electrode and of the temperature gradient. For the sake of clarity, we recall here in what sense a magnetic induction eld is induced by an out-of- equilibrium magnetization in a temperature gradient. The formalism presented in reference11implies the exis- tence of a magnetic counter-part to the well-known See- beck eect, where a magnetic induction eld BTis in- duced by a temperature gradient ryTin a YIG slab in the presence of an oscillating magnetic excitation eld and a constant external magnetic induction eld Bext in the slab plane (see axes ^x,^y,^zon Fig. 1). The existence of a magnetic induction eld BTcan be un- derstood as follows. In an insulator like YIG, there is no drift current. Thus, an induced magnetization force density MYryBTbalances the thermal force density nYkBryT, i.e. MYryBT=YnYkBryT; (1) where the index Yrefers to YIG, Y>0 is a phenomeno- logical dimensionless parameter, MYis the magnetiza- tion of YIG and nYis the Bohr magneton number density of YIG. The magnetization MYis the sum of the satu- ration magnetization and the magnetic linear response, i.e. MY=MSY+mYwhere mY
Bext= 0:(2) As detailed in reference11, the magnetic induction eld BTinduced by the temperature gradient ryTcan bearXiv:1509.04440v1 [cond-mat.mes-hall] 15 Sep 20152 written as, BT="MYryT; (3) where the phenomenological vector "MYis given by, "MY=YnYkB M2 SY r1 ymY
: (4) Hence, the thermally induced magnetic induction eld BTis oscillating at the frequency of mY. III. MAGNETOPHORESIS The continuity of the orthogonal component of the thermally induced magnetic induction eld BTacross the junction between the YIG and the Pt is ensured by Thomson's equation, i.e. r
BT= 0: (5) Therefore, the normal component of the magnetic induc- tion eld BTis acting also on the Pt electrode. Mag- netophoresis occurs in a Pt electrode that is suciently thick to be treated thermodynamically and suciently narrow for the temperature gradient to be neglected. The interaction between the magnetization of the con- duction electrons of the paramagnetic Pt electrode and the thermally induced magnetic induction eld in the fer- romagnetic YIG slab results in a magnetization force that leads to the diusion of the conduction electrons along the electrode, i.e. magnetophoresis. This generates in turn an electrostatic potential gradient rxVacross the Pt electrode (see Fig. 1) which can be thought of as a magnetophoretic electrochemical voltage. The existence of an the electrostatic potential gradient orthogonal to the temperature gradient depends on the orientation of the external magnetic induction eld, as we shall show. Concretely, in the Pt electrode, the thermodynamic formalism11yields linear phenomenological relations be- tween the electric current density and the magnetization and electrostatic forces densities respectively. By identi- fying the electric current in these relations, the electro- static force density qPrxVresulting from the drift of the conduction electrons is found to be proportional to the magnetization force density MPrxBTgenerating the drift, i.e. qPrxV=PMPrxBT; (6) where the index Prefers to Pt, qP<0 is the charge density of conduction electrons, MPis the magnetiza- tion of the conduction electrons in the Pt electrode and P>0 is a phenomenological dimensionless parameter. The magnetization MPis the sum of the paramagnetic contribution due to the constant external eld Bextand the linear response mPto the stray magnetic inductioneld generated by the propagating magnetization waves in the ferromagnet, i.e. MP=P 0Bext+mPwhere mP
Bext= 0;(7) 0is the magnetic permeability of vacuum, Pis the Pauli susceptibility of conduction electrons in Pt. As shown in reference18, the magnetization force density can be expressed in terms of the magnetization current den- sity, i.e. MPrxBT= (rxmP)BT: (8) Thus, the relations (6) and (8) imply that the electro- static potential gradient generated by transport of the conduction electrons is given by, rxV=P qP (rxmP)BT : (9) This eect is shown on Fig. 1 for a YIG slab with a Pt electrode. IV. MAGNETIC NERNST EFFECT The voltage dierence derived from rxVin equa- tion (9) is a Magnetic Seebeck eect detected electrically through the magnetophoresis of the conduction electrons in the Pt electrode. We show now explicitly this eect in the form of a Nernst eect. In a sense the Magnetic Nernst eect is a thermally induced magnetophoresis. In the Magnetic Seebeck eect, the temperature gradi- entryTimposed on the YIG slab induces a magnetic induction eld BTthat is oscillating in an orthogonal plane. Through magnetophoresis, this eld generates an electrostatic potential gradient rxVacross the Pt elec- trode. Using the Magnetic Seebeck eect (3) and the denition (4), the electrostatic potential gradient gener- ated by magnetophoresis (9) is recast explicitly as, rxV= PY(rxmP) r1 ymY
ryT ; (10) where PY=PYnYkB qPM2 SY: (11) Using the Jacobi identity for the cross product, the linear relation (10) yields the Magnetic Nernst eect, i.e. rxV=Nz mryT; (12) where the phenomenological vector Nz mis given by, Nz m= PY(rxmPz) r1 ymY z
(13) withmPz= (^z
mP)^zandmY z= (^z
mY)^zin order to satisfy the vectorial symmetries and contribute to the3 eect. The structure of equation (12) relating the gradi- entsryTandrxVis that of a Nernst eect. In place of a magnetic induction eld, there is a phenomenologi- cal vector Nz m, which depends on the out of equilibrium magnetization in the YIG slab. This eect is illustrated on Fig. 1 for a YIG slab with a Pt electrode, and a surface coil is presumed to excite the ferromagnetic resonance. Bext ˆyˆxˆzBT ∇yTPtYIGθ ∆Vω FIG. 1: YIG slab with a Pt electrode connected to a voltmeter and excited by a local probe. In order to determine the structure of the Magnetic Nernst vector Nz m, we perform a Fourier series expan- sion of the linear response elds mPzandmY z. In a stationary regime, the Fourier transform of the response elds are expressed in terms of real parameters as, mPz=X kPmkPzsin (kP
r!kPt+kP)^z;(14) mY z=X kYmkYzsin (kY
r!kYt+kY)^z;(15) wherekand'kare the dephasing angles and !kis the angular frequency of the eigenmodes k. Finally, the Fourier decompositions (14) and (15) imply that the re- lation (13) is recast in terms of the eigenmodes as, Nz m= PYX kP;kYkPxk1 YymkPzmkYz (16)
cos (kP
r!kPt+kP) cos (kY
r!kYt+kY)^z; wherekPx=^x
kPandk1 Yy=^y
k1 Y. The magnetic waves vectors in YIG and Pt are collinear to the external magnetic induction eld, i.e. kPBext=0andk1 Y Bext=0, which implies that kPx=kPsinandk1 Yy= k1 Ycoswhereis the orientation angle between the temperature gradient and the external magnetic eld in the plane of the YIG slab as shown on Fig. 1. Moreover, the specic mode k=kY=kPcorresponding to the excitation frequency !!kof the magnetization waves in YIG and Pt is determined by the quadratic dispersion relation of the magnetization waves , i.e. !k=Ak2whereAis the stiness. Thus, choosing the initial time to cancel the dephasing of the magnetization in YIG and using the trigonometric identity, cos (k
r!t+) cos (k
r!t) = 1 2 cos+ cos (2 k
r2!t+) ;(17) the Magnetic Nernst vector (16) is recast as, Nz m= PY 4mkPzmkYzsin (2)
 cos+ cos (2 k
r2!t+) ^z;(18) wherekPandkY= 0 . In the homogeneous elec- trode, the electrostatic potential varies linearly along the ^x-axis, i.e. ^x
rV=
V=`xwhere`xis the length of the electrode. Thus, the voltage across the electrode is given by,
V=`x^x
(Nz mryT): (19) The expressions (18) and (19) imply that the voltage
V along the electrode consists of DC and AC contributions. The DC contribution is proportional to cos , which im- plies that it is maximal in the absence dephasing between the magnetization in Pt and YIG. The AC contribution is oscillating with an angular frequency 2 !that corre- sponds to the double of the excitation angular frequency !. The Magnetic Nernst eect vanishes if the external magnetic eld Bextis collinear ( = 0) or orthogonal (==2) to the temperature gradient ryTand it is maximal if there is a =4 angle between these vectors in the YIG slab plane. It is important to mention that the Magnetic Nernst ef- fect is not equivalent to thermal spin pumping19. The an- gular dependence of these two eects are dierent. Ther- mal spin pumping is maximal for = 0 and minimal for ==2 whereas the Magnetic Nernst eect is maximal for==4 and minimal for = 0 and==2 . V. CONCLUSION In summary, a Nernst eect is predicted, which results from the interplay between the magnetization dynamics of a ferromagnet driven in a temperature gradient, and the linear response of the paramagnetic electrode to the inhomogeneous eld produced by the magnetization. First, in a magnetic insulating YIG slab, there is the Magnetic Seebeck eect, i.e. a magnetic induction eld induced by a temperature gradient. Second, the electrical detection of this eect in a paramagnetic Pt electrode contacted to the slab relies on the voltage induced by the inhomogeneous eld acting on the conductive electrode, causing a drift of charges that carry a magnetic dipole. The voltage has a DC component and an AC compo- nent that oscillates at twice the frequency of the eld driving the magnetization. The eect is zero when the4 applied eld is parallel or perpendicular to the tempera- ture gradient, and maximum at a 45angle in between. Hence, this eect is quite distinct from the angular de- pendence of spin pumping or the Spin Seebeck eect. Acknowledgments We would like to thank Fran cois Reuse and Klaus Maschke for useful comments and acknowledge the fol-lowing funding agencies : Polish-Swiss Research Program NANOSPIN PSRP-045 =2010; Deutsche Forschungsge- meinschaft SS1538 SPINCAT, no. AN762 =1. Electronic address: sylvain.brechet@ep .ch 1R. O. Cunha, E. Padr on-Hern andez, A. Azevedo, and S. M. Rezende, Phys. Rev. B 87, 184401 (2013). 2B. Obry, V. I. Vasyuchka, A. V. Chumak, A. A. Serga, and B. Hillebrands, Applied Physics Letters 101, 192406 (2012). 3S. D. Brechet, S. D., F. A. Vetro, F. A., E. Papa, S.-E. Barnes, and J.-P. Ansermet, Phys. Rev. Lett. 111, 087205 (2013). 4K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 5S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401 (2011). 6C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat Mater 9, 898 (2010). 7K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, et al., Nat Mater 9, 894 (2010). 8K.-I. Uchida, T. Nonaka, T. Ota, and E. Saitoh, Appl. Phys. Lett. 97, 262504 (2010). 9H. Adachi, K.-i. Uchida, E. Saitoh, and S. Maekawa, ArXiv 1209.6407 (2012). 10M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W.Bauer, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B 88, 094410 (2013). 11S. D. Brechet and J.-P. Ansermet, The European Physical Journal B 86, 1 (2013). 12N. Leventis and X. Gao, Analytical Chemistry 73, 3981 (2001). 13E. P. Furlani, Journal of Physics D: Applied Physics 40, 1313 (2007). 14H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd Edition (Wiley & Sons: New York, 1960). 15J. Lim, C. Lanni, E. R. Evarts, F. Lanni, R. D. Tilton, and S. A. Majetich, ACS Nano 5, 217 (2011). 16M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, et al., Phys. Rev. Lett. 108, 106602 (2012). 17M. Schmid, S. Srichandan, D. Meier, T. Kuschel, J.-M. Schmalhorst, M. Vogel, G. Reiss, C. Strunk, and C. H. Back, Phys. Rev. Lett. 111, 187201 (2013). 18D. J. Griths, Introduction to Electrodynamics (Prentice- Hall, Upper Saddle River, 1999), 3rd ed. 19K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kaji- wara, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Journal of Applied Physics 111, 103903 (2012).

2015-09-15

The thermodynamics of irreversible processes in continuous media predicts the existence of a Magnetic Nernst effect that results from a magnetic analog to the Seebeck effect in a ferromagnet and magnetophoresis occurring in a paramagnetic electrode in contact with the ferromagnet. Thus, a voltage that has DC and AC components is expected across a Pt electrode as a response to the inhomogeneous magnetic induction field generated by magnetostatic waves of an adjacent YIG slab subject to a temperature gradient. The voltage frequency and dependence on the orientation of the applied magnetic induction field are quite distinct from that of spin pumping.

Magnetic Nernst effect

1509.04440v1

1 Microwave to optical photon conversion by means of travelling-wave magnons in YIG films M.Kostylev1 and A.A. Stashkevich2 1School of Physics, the University of Western Australia, Crawley, 6009 WA, Australia 2LSPM (CNRS-UPR 3407), Université Paris 13, Sorbonne Paris Cité, 93430 Villetaneuse, France Abstract: In this work we study theoretically the efficiency of a travelling magnon based microwave to optical photon converter for applications in Quantum Information (QI). The converter employs an epitaxially grown yttrium iron garnet (YIG) film as the medium for propagation of travelling magnons (spin waves). The conversion is achieved through coupling of magnons to guided optical modes of the film. The total microwave to optical photon conversion efficiency is found to be larger than in a similar process employing a YIG sphere by at least 4 orders of magnitude. By creating an optical resonator of a large length from the film (such that the traveling magnon decays before forming a standing wave over the resonator length) one will be able to further increase the efficiency by several orders of magnitude, potentially reaching a value similar to achieved with opto-mechanical resonators. Also, as a spin-off result, it is shown that isolation of more that 20 dB with direct insertion losses about 5 dBm can be achieved with YIG film based microwave isolators for applications in Quantum Information. An important advantage of the suggested concept of the QI devices based on travelling spin waves is a perfectly planar geometry and a possibility of implementing the device as a hybrid opto-microwave chip. I. Introduction The magnon-based microwave-light converter is very attractive from the viewpoint of enlarging the potential of the superconducting qubits [1-5]. Such device whose first conversion stage is based on coherent coupling between a ferromagnetic magnon and a superconducting qubit is expected to have a bandwidth of around 1 MHz and thus operates faster than the lifetime of a superconducting qubit currently available (around 100 μs [1]). Moreover, the ferromagnetic magnon coherent coupling to a superconducting qubit has recently been confirmed experimentally [2,3]. The most natural and direct way of up-converting, at the second stage, thus excited microwave magnons (typical frequencies lying in the lower part of the GHz range) to the optical domain is via magneto-optical interactions, thus realizing coherent connection between distant superconducting qubits. In the basic configuration studied so far the microwave magnons, known as magnetostatic Kittel mode are excited in an uniformly 2 magnetized YIG sphere with a diameter of about 1 mm. This mode is spatially homogenuous and can be considered as the lowest ferromagnetic resonance (FMR) localized within a spherical ferromagnetic resonator. The FMR line-width Δ H is its major parameter determining the quality factor. To improve the efficiency of the magnon-photon magneto-optical (MO) coupling, the latter should be kept as large as possible. This implies a very narrow FMR line which in its turn corresponds to a narrow frequency bandwidth of effective MO interaction ΔfFMR. Thus, the value of FWHM 2Δ H=0.3 Oe typical for high-quality YIG specimen corresponds to a very modest bandwidth of 1 MHz. One has to note that the currently achieved quantum efficiency of the microwave to optical photon coupling via magnons in a YIG sphere is also quite modest 1010 [4]. This is much smaller than already achieved with other methods [6-10]. In particular, a 10% coupling was demonstrates through an opto-mechanical route [6], and electro-optics allows one to reach 0.1% [7]. Therefore, in this paper we investigate theoretically an alternative solution to the problem of magnon-to-light conversion. More specifically, we consider travelling magnons in the form of magnetostatic spin waves as a candidate for this role. A spin wave is a wave of electromagnetic nature which is propagating in a saturated ferrimagnetic film and which is strongly retarded (1000 - 10000 times with respect to the velocity of a normal electromagnetic wave) due to a strong resonance interaction with the matter. As a consequence of the slow propagation, the magnetic component of the wave is extremely strong. This is why magnetostatic spin waves can be exceptionally effective in MO modulating devices based on the Faraday Effect. Further, such a microwave to optical photon converter has an intrinsically planar geometry. Therefore, contrary to the YIG sphere in a microwave cavity concept [4], it potentially allows integration into a hybrid opto-microwave chip of reasonable sizes and fabrication of the microwave-photon to magnon transducers using optical lithography. First experiments on MO diffraction of light by magnetostatic surface spin waves (MSSW) date back to the early 1970s [11-12]. However, the interaction efficiency was low due to the inadequately chosen configuration; the light was impinging on a YIG film normally. Thus, the interaction length was equal to the film thickness. A major breakthrough was achieved a decade later through introduction of the so-called waveguide MO interaction [13] in which case both the MSSW and the optical wave propagate simultaneously in the YIG film as in a waveguiding structure. In this geometry the interaction length is determined by the MSSW free propagation path, which typically is on the order of several millimeters. As a result, the interaction intensity was increased several orders of magnitude. In the course of investigations that followed in the late 1980s, several basic SW-optical device geometries have been studied, pure YIG films typically having been used as a medium for magneto-optical interaction [14-16]. These experiments have stressed the significance of the improvement of such an important parameter as the interaction efficiency. Not surprisingly, theoretical effort was focused on the specificity of strong MSSW-Light interactions involving multiple-scattering mechanisms depicted mathematically by Feynman-type multiple-scattering diagrams [17]. 3 Further improvement of the interaction intensity (20 – 50 times) was due to the progress in the technology of highly bismuth-doped YIG films with higher values of the Faraday constant (5 – 10 times) [18-20]. Also, a more sophisticated optical geometries were suggested by theoreticians – the focus has been on using bi-layer ferromagnetic films and confining the optical modes in the film plane in order to improve the structure of optical modes and the light/spin wave interaction efficiency [21,22]. All these experiments have been performed at room temperatures and microwave and optical power levels well above the single photon/magnon one. No experiments have been carried out so far at cryogenic temperatures and at the single optical-photon / magnon power level. On the other hand, the first experiments probing microwave properties of magnons in YIG films at millikelvin temperatures have been already carried out [23,24]. Therefore, it is very timely to look at this problem from the theoretical prospective. The latest theoretical effort [21,22] in this field was focused on telecom applications, therefore its goal was to achieve a 100% conversion of incident light into scattered light by employing a high-power spin wave signal. The incident spin wave power was of no importance in that case. Furthermore, the efficiency of the conversion of the microwave signal into spin waves by microwave stripline transducers was not included in the theory. In the present work we focus on a qualitatively different situation. We assume that the microwave signal incident onto the device input port is at a single-photon level. Therefore, a 100% conversion of the incident light into an output optical signal is impossible. And we are not interested in achieving the 100% light-to light conversion. Instead, we are interested in a 100% conversion of the weak input microwave signal into the output optical one, expressed in a number of generated optical photons per one input microwave photon. The incident optical power can be arbitrary in this situation. We investigate this configuration theoretically and make predictions about the rate of scattering of the guided light from single traveling magnons. In the process of the calculation, the efficiency of microwave photon to magnon conversion is taken into account in full. In contrast to the case of an YIG sphere in a microwave cavity, theoretical description of coupling of traveling spin waves to microwave photons in microwave stripline transmission lines is much more involved (see e.g. [25]). Furthermore, it is difficult to obtain efficient microwave photon to magnon coupling experimentally, unless the coupling geometry is carefully optimized (which is feasible even in a mass-production setting). Therefore, an extended part of the present paper (Section IIA) is devoted to theoretical treatment of spin wave excitation by stripline antennas. Section IIB presents theoretical details of coupling of travelling-wave magnons to guided optical photons in the YIG film. Section IIIA reports on the results of numerical calculations of the microwave to optical photon conversion efficiency employing the formalism presented in Section II. Ways to further improve the conversion efficiency are discussed in Section IIIB. Conclusions are contained in Section IV. As a side result of this calculation, we also evaluate the efficiency of a microwave isolator based on travelling Damon-Eshbach magnetostatic surface spin waves [26] in YIG films. It has been recently shown that these devices are very important for isolation of qubits from the noise 4 in the microwave circuits to which they are connected [27]. Previous calculations and experiments showed that losses inserted by a travelling spin wave device in the forward direction can be small – about 5 dB theoretically and 10 dB experimentally (see Figs. 2-4 in Ref. [28]). In our calculation, we evaluate transmission of the device in both directions and find that a YIG-film based isolator employing asymmetric coplanar transducers has very good performance characteristics. II. Theory A. Excitation and propagation of spin waves in the YIG film waveguide. In this work we will be dealing with rather thick YIG films, that is why the Damon-Eshbach exchange-free approximation [26] is appropriate. However, if needed the exchange interaction can be easily included in the theory by using the approach from [26]. The geometry of the problem is shown in Fig. 1. It represents a YIG film of thickness L in the direction y. In this section, we consider the film being continuous in both in-plane directions x and z. The film is placed on top of a microwave stripline. The stripline runs along the z-axis and consists of a number of parallel conductors (lines). We will be interested in a general case of a backed coplanar line shown in Fig. 1. The line consists of three parallel conductors of widths w1, w, and w2 in the direction x. The central conductor of a width w is called the signal line. The two lines of widths w1 and w2 are ground lines. They are separated from the signal line by gaps of widths 1 and 2 respectively. If one of the ground lines is absent ( w2=2=0), one deals with a microwave transmission line called the asymmetric coplanar line. If both ground lines are absent ( w1=1=w2=2=0), but the ground plane at y=d is still present one deals with a microstrip line (We will use an asymmetric coplanar line in examples which will be considered in the discussion sections of the paper.) The stripline length in the direction z is assumed to be infinite at the first stage of the problem solution. Also, the YIG film will be considered as infinite in both in-plane directions x and z at the first stage. At the second stage, we will assign specific values to the stripline length and the film width in the direction z, once an expression for stripline’s linear impedance has been obtained. The thickness of the microstrip in the direction y is assumed to be zero. All these assumptions significantly simplify the calculations without any significant loss of generality [25,30,31]. The stripline is supported by a dielectric substrate of thickness d whose other surface (located at y=d) is metalized thus forming a backed coplanar line. The dielectric permittivity of the substrate is . A microwave current I flows through the line. The linear density of this current is j(x) may be non-uniform across the widths of both signal and ground lines and obeys the relation as follows: 5 Fig. 1. Vertical cross-section of the geometry of the problem. 1: Stripline ground plane. 2. Stripline substrate. 3 Stripline (ground, signal and ground lines of widths w1, w and w2 respectively). 4. Yttrium iron garnet (YIG) film. 5: Film substrate made of gadolinium gallium garnet. The static magnetic field H is applied along the z-axis. /2 /2( )w wj x dx I (1). A microwave Oersted field of the current drives magnetisation precession in the film. The total width of the stipline w1+1+w+2+w2 is assumed to be much smaller that the free propagation path of the magnetostatic surface spin waves in epitaxial YIG films (typically several mm for 4+ micron thick YIG films). Therefore the excited magnetization oscillation propagates as a plane wave from the stripline transducer in both directions along the x-axis. We will term this stripline “the input stripline transducer”. We also assume that a magnetic field zHH uis applied along the axis z thus forming conditions for propagation of a Damon-Eshbach surface wave (MSSW) [26] along the x- direction (here zuis the unit vector along z). Also, an optical guided mode can propagate in the YIG film along the x-direction with its evanescent field penetrating into the gadolinium gallium garnet (GGG) substrate of the film. We will deal with optical properties of this system in Section II B, but now let us focus on the formalism of excitation of the spin waves in the film by the input transducer and propagation of the excited spin waves in the film. The goal of this section is to express the efficiency of spin wave excitation in terms of a number of excited magnons per one microwave photon incident onto the input port of the input transducer. We will use a classical formalism for spin waves for the calculation; the quantum theoretical notions of the numbers of photons and magnons will appear at the very last step of this derivation (Eq.(21)). The reader not interested in the details of this calculation can skip directly to this equation. 6 The most comprehensive theory of the efficiency of spin wave excitation in this geometry was developed by Vugalter and Gilinski [31]. Our formalism will be similar to the one suggested by them, but because we are interested in maximising the efficiency of all involved interactions, we will be paying greater attention to detail. In Ref.[31] the focus was on obtaining analytical solutions. This was achieved by introducing a number of approximations valid in particular limiting cases, for instance, large values of d or small values of w with respect to spin wave wavelength. We will keep the theory as general as possible, keeping in mind that derived equations will be solved numerically at the last stage of the calculation, in order to obtain results which are as rigorous as possible. This is because the focus of the present paper is not on derivation of equations but on getting numbers which reflect efficiency of microwave to optical photon conversion via the travelling-magnon route. Our analysis will be similar to one we used in Refs. [31,26]. The translational invariance of the geometry in Fig. 1 in the z-direction enables calculation of a quantity called the complex impedance of stripline. This is achieved by using a quasi-static approach for the description of microwave transmission lines whose central point is an assumption that all the fields can be considered as uniform in the direction z. This allows decomposing the problem to a two dimensional one, where all dynamic variables depend only on x and y [25,30,31]. The magnetization dynamics in the films are described by the linearized Landau-Lifshitz equation ( )t mm H M h , (2) where the static magnetization within the film is given by zMM u where M is the saturation magnetization for the film. The dynamic magnetization vector m has only two components (perpendicular to M) and can be represented as mxex+myey. The dynamic dipole field of precessing magnetization h is attained from solving Maxwell’s equations in the magnetostatic approximation 0 h, (3) h m , (4) 0( )i e h m , (5) where e is the microwave electric field generated by the dynamic magnetization due to Faraday induction. In the framework of the quasi-static approach for the stripline description, from the symmetry of the problem it follows that e has only a z-component: e=ezuz. A solution for all dynamic variables entering the equation – m, h and e – is obtained representing it as a set of plane waves propagating along x: , , , , exp( )k k k i t ikx dk m h e m h e , (6) 7 where is the angular frequency of the microwave current flowing through the input transducer and k is the Fourier wave number. The analytical expressions for km, and kh are given in Appendix A. Once these expressions have been derived, one can calculate ( )xm, ( )xh and ( )xe by carrying out the inverse Fourier transformation (6). It this work, the transformation is carried out numerically. In order to enable this, magnetic losses H in the material are introduced into the expressions. We do this by replacing H with H+iH in the final expressions [32]. Also, a simple analytical solution exists for the far zone of the “spin wave antenna” which the input transducer actually represents. We will return to the far-zone solution below. Let us now turn to the expression for the complex linear impedance of the transducer rZ (often called the “Radiation impedance of a spin wave antenna” [31]). It may be obtained in the framework of the “Induced Electromotive force” method, as suggested in [30]. Following this method, /2 2 /21( ) ( , 0)w r z wZ j x e x y dxI , (7) where I is the microwave current through the antenna, j(x) is its linear density and ( , 0)ze x y is the z-component of the microwave electric field at the level of the antenna ( y=0) and the dash above j denotes complex conjugation. This electric field has two components – the self-field of the microstrip line and the electric field of the precessing magnetization in the film. Outside the spin wave frequency band the latter field vanishes, and ( , 0)ze x y reduces to the self-field of the stripline [31,33]. This property will be used in the following in order to calculate the characteristic impedance and the complex propagation constant for the transducer. Starting with this expression, a solution for rZ is obtained. We do not derive this solution here because it is a simplified version of a more general result from [29] and also because very similar solutions exist in the literature [25,30,31,33]. The final formula for the complex radiation impedance has the form as follows 22( 0)r k zkZ j e y dk I , (8) where the Fourier transform of the electric field at y=0 reads 0( 0) cosh( ) tanh( ) ( 0)zk k xie y k d j k d h yk . (9) In order to find the Fourier transform kj of the microwave current density in the transducer, we may use the fact that in the quasistatic approximation, the continuity equation for j(x) [31] reduces to 8 ( ) ( / ) ( )c j x i x , (10) where x is the linear charge density distribution across the transducer width for the current and c is the complex propagation constant for the stripline. This implies that kj scales as the Fourier transform of the charge density k ( / )k c kj i , where the concrete value of the constant ( / )ci K is of no importance for the following. The charge density can be obtained by solving the electrostatic problem for the transducer [29]. Because in the following we will be dealing with transducers of an unusual shape, in the present work we find x self- consistently. To formulate the self-consistent problem, we use the fact that in the electrostatic approximation, the electric potential u(x) should be uniform across the widths (in the direction x) of the transducer electrodes (strips). Requiring this and using Eq.(B11) from [29] which relates the Fourier transform of the potential uk to the Fourier transform of the charge density k one arrives at an integral equation for x This equation can be easily solved numerically to yield x for a stripline geometry of interest. x is then Fourier transformed numerically to yield k. Then k kj K. This method takes into account the dielectric properties of the ferromagnetic film and its substrate while calculating kj and k, but it does not take into account the magnetic properties of the film. However, in [28] it has been shown that the influence of the excited spin waves on the x-dependence of j is negligible for most of the spin wave vector range, except the upper edge of the range. In the present work, we will be interested in smaller spin wave wavenumbers, therefore this approximation is appropriate. Once we have obtained kj, the complex linear impedance rZ is easily computed numerically. (K cancels out in the course of this derivation, because of the static nature of the involved fields.) rZ is then transformed into the input impedance for the stripline by utilizing established formulas [33,29]. To maximize the efficiency of spin wave excitation, one has to maximize the microwave current through the stripline. This is obtained by shorting the end of the antenna. In order to incorporate this requirement in our model, now we assume that the stripline has a finite length al (in the direction z) and the width of the film in the direction z is also finite and equals the same al. The input impedance for a stripline obeys Telegrapher Equations [34]. For a stripline of a length al whose other end is shorted one has tanh( )in c c aZ Z l , (11) where cZ is the characteristic impedance for the stripline. Both cZ and c relate to the linear parallel capacitive conductance iC and in-series inductive ZL reactance of the stripline. The linear capacitance C is obtained as a by-product of the solution of the electrostatic problem for 9 x (see Eq.(B9) in [29]), and rZ plays the role of ZL in our case, as it reduces to ZL outside the spin wave band (see the comment after Eq.(7) above). Hence, /( )c rZ Z i C , (12) c ri CZ . (13) Then the complex reflection coefficient from the input transducer input port reads: 0 0( )/( )in inZ Z Z Z , (14) where 0Z=50 is the characteristic impedance of the microwave feeding line. Let us now assume that the microwave voltage inV incident onto the input port of the transducer has an amplitude of 1 Volt. The theory of transmission lines then allows us to express the current in the transducer as a function of the position 0 z la (z=0 coincides with the transducer input port and z=al with the shorted antenna end): 1( ) exp( ) exp( )1 exp( 2 )in s c sVI z z l zZ l . (15) As follows from Eq.(15), the amplitude of the driven precession of magnetization below the input transducer will be a function of the co-ordinate z along the transducer. So, each point z will be a point source of a travelling spin wave whose complex amplitude scales as I(z), and the wave front in the far field of antenna will be a result of interferences of these partial waves. Hence, strictly speaking, the problem of formation of the wave front of a travelling spin wave in the far zone of the antenna is 2-dimensional (see e.g. [31]). Solving the two-dimensional problem is beyond the scope of the current work, as it involves consideration of caustic beams which spin waves in in-plane magnetized films are prone to form (see e.g. [35]) if the films are not confined in the z direction. However, if they are confined, a guided spin wave mode will be formed. The m(z) distribution for the guided spin wave is not perfectly uniform [36] across the width of the waveguide, however we may neglect this non-uniformity as it is not of central importance for the present paper. Accordingly, below we will assume that the m(z) is uniform across the film width (the width coincides with the antenna width la) everywhere in the far zone of the antenna (the latter also implies that we assume that the guided mode is formed straight after the wave escapes the near antenna zone.) The energy conservation law tells us that the spin wave energy adjusted for energy losses due to the intrinsic magnetic damping in the film should be the same for any waveguide cross- section. Therefore, we may introduce an effective z-uniform current through the input antenna which produces the same spin wave energy carried through a waveguide cross-section Lxla. The magnitude of the effective current reads 10 2 01( )al eff aI I z dzl. (16) Note that this definition leaves the phase of the effective current undefined. The dynamic magnetisation of the travelling spin wave at any position x is given by Eq.(6), where mk is given by Eqs. (A1-A2) from Appendix A. In the closed form, the Fourier component of the dynamic magnetisation (for the input microwave signal of 1 Volt, see above) reads: ( ) ( ) ( )( )exp( ) ( )exp( ) 2( , )ˆdet( ( ))mx y mx y kx y k effAC k k y BC k k ym x y jI W k , (17a) where Cmx=1 and Cmy is given by Eq.(A3) in Appendix A. The inverse Fourier transformation of (Eq.(17a)) can be carried out analytically. The analytical solution of the integral in (6) is expressed in terms of exponential integral functions and two complex exponential functions [37]. The exponential integral term describes the near field of the transducer [38] and the complex exponential ones two travelling spin waves propagating in the two opposite directions from the transducer. A similar analytical solution can be obtained for the integral in Eq.(8). The near-field term of that solution yields the antenna linear reactance and the real-valued term represents the spin wave radiation resistance of the antenna. We will not evaluate analytically Zr in the present work. Instead we will use numerical integration in order to calculate Zr, which is just easier and more accurate. The near field of an MSSW transducer is localised in the closest vicinity of the transducer, as we previously demonstrated experimentally in Ref.[38]. Outside this area, only the travelling wave contributions to the integral exist. Evaluation of the integral in (6) in this far zone of the transducer using the Residues Theory is straightforward and results in a simple expression as follows 0( ) 3 ( ) 0 0 ( ) 0 0 0 0( , ) ( )exp( ) ( )exp( ) 2exp( )exp( ) ˆ(det( ( ))x y mx y mx y k effm x y AC k k y BC k k yj ik x vxd IW kdk (17b) where ˆdet( ( ))W k denotes the determinant of the matrix ˆ( )W k shown in Appendix A and A, B, mxC and myC are coefficients also given in Appendix A. k0 is the value of spin-wave wave number which satisfies the implicit dispersion relation for spin waves ˆdet( ( )) 0W k. In the limiting case d=∞ it reduces to the Damon-Eschbach dispersion relation [26] for spin waves in a film not sitting on top of a backed stripline: 2 2 2( )/ ( ) [1 exp( 2 )]4Mk H H M kL . 11 In the following we will refer to the case d=∞ as “a film in vacuum”. The quantity 0kj is obtained as the Fourier transform of j(x)=const(x), where the value of the constant is given by the condition 2 2/2 /2( )w eff wj x dx I . There are two solutions to the dispersion relation ˆdet( ( )) 0W k, one for a positive wave number k and one for a negative one. The magnitudes of the two wave numbers are different, as the presence of the metal ground plane of a stripline at y= d makes the spin wave spectrum non-reciprocal. The spatial decay factor for the wave is given by the expression as follows: 0 0ˆdet ( ) / ˆdet ( ) /d W k dk H d W k dH , Where H is the loss parameter for the film, and, as in Eq.(17b), the argument k0 means that the derivatives are evaluated for k= k0 . The analytical expression (17b) is in excellent agreement with the direct numerical evaluation of the integral in (17a) in the far zone of the transducer. This excellent agreement allows us to use the same analytical approach in order to evaluate the Poynting vector for spin waves [31]. An expression for the Poynting vector for a somewhat simpler geometry of a film in vacuum is shown in Appendix B. The derivation of an expression for the Poynting vector for the geometry of a film on top of a backed coplanar line is also straightforward (not shown here). Numerical evaluation of this expression demonstrates excellent agreement with the expression for the power irradiated by the transducer in the form of spin waves. The microwave power incident onto the input port of the transducer reads 2 02in inVPZ. (19) Then from the telegraphist equations it follows that the incident microwave power converted into the magnon power is given by 2 01Re (1 )(1 )2inVPZ . This power is redistributed between the two waves excited by the transducer in the two opposite propagation directions + k and –k. Excitation of the Damon-Eshbach spin waves is highly non-12 reciprocal (see e.g. [39]). The portion of the power nr carried by the spin wave propagating in the positive direction of the axis x (i.e. in the + k direction) is given by [30] /Re( )nr rR Z , where R+ is the component of the total radiation resistance associated with excitation of spin waves travelling in the + k direction: 2 02Re ( 0)k zk R j e y dk I . Hence, the power P+ carried by spin waves in the + k direction is given by 2 01( ) Re (1 )(1 ) exp( 2 )2in nrVP x xZ . (20) A result of numerical evaluation of this expression in the far zone of an asymmetric coplanar transducer ( x>w2+2+w3) is in excellent agreement with the value of the Poynting vector for spin waves, as has already been stated above. Eq.(20) also yields an expression for the efficiency of the microwave photon to magnon conversion ( 0) m inP xNP . (21) Since the incident microwave photons and the generated magnons have the same frequency (energy), the efficiency of the photon-magnon transduction Nm actually represents the number of generated magnons per one incident photon. B. Scattering of optical photons from magnons In this section we consider two guided optical modes propagating in the YIG film. One is incident, the other is generated due to a photon-magnon interaction. An incident optical mode scatters from a spin wave whose wave vector is collinear to light. This creates the second optical mode which we will call “scattered light”. The physics behind the process of the interaction of the guided light with the spin wave is light scattering from a diffraction grid formed by a spatial harmonic modulation of the material’s refractive index along the light propagation path. The modulation originates from the dynamic magnetisation of the spin wave – the spatial variation of the magnetisation vector leads to spatially periodic modulation of the strength of the Faraday and Cotton-Mouton Effects. The “diffraction grid” moves with the phase velocity of the spin wave which leads to important peculiarities of light scattering from it. They will be discussed in the very end of the Discussion section. 13 From the point of view of wave interactions, the magnon-optical photon interaction is a three- wave process. Classically, this process is described by a theory of coupled modes (see Eqs. (16-17) in [21]). The mode coupling coefficient scales as the overlap intergral for the three waves (which reflects spatial correlation of the interacting waves) 𝐼௩=∭𝐄(௦)(x,y,z)∙𝐦(x,y,z)∙𝐄()(x,y,z)dV. (22) Here functions E(i) (x,y,z), E(s) (x,y,z), m(x,y,z) describe the spatial distribution of the electric field in the incident and scattered optical waves and that of the magnetization in the scattering spin wave. Secondly, the efficiency of the MO interactions depends on the “correlation” of their polarizations (the vector factor). Mathematically, the symmetry of MO coupling in an optically isotropic media is described by the totally antisymmetric Levi-Civita tensor [40]. As a result, the vector factor is expressed via the mixed product of the interacting waves 𝐼௩=ቀ𝑒⃗(௦)∙൫𝐦×𝑒⃗()൯ቁ=∑ ∑ ∑ 𝛿𝑒(௦)𝑚 ୀ௫,௬,௭ ୀ௫,௬,௭ ୀ௫,௬,௭ 𝑒(). (23) Correspondingly, their polarizations are given by unit vectors 𝑒⃗(),𝑒⃗(௦),𝐦. An interested reader can find all details in an exhaustive theoretical analysis presented in Ref . [41]. A YIG film can be regarded as a planar dielectric waveguide. Due to its specific symmetry such a waveguide supports two types of electromagnetic waves, namely TE-modes 𝐄()்ா(𝑥,𝑦)=𝐸௭()்ா(𝑦)exp (−𝑖𝛽()்ா𝑥)𝑒⃗௭ and TM-modes 𝐄()்ெ(𝑥,𝑦)=ቀ𝐸௫()்ெ(𝑦)𝑒⃗௫+ 𝐸௬()்ெ(𝑦)𝑒⃗௬ቁ∙exp (−𝑖𝛽()்ெ𝑥) with 1<n<N. The value of N, for a given optical wavelength, depends on the film thickness. Here 𝛽()்ா and 𝛽()்ெ are waveguide wavenumbers of respectively n-th TE and n-th TM mode. The orthogonality of the modes in the sense of the cross-power allows expressing an arbitrary given field distribution as a superposition of waveguide modes. In this regard, the spin wave possesses the structure of a TE-mode with the dynamic magnetization in the “xy” plane. The selection rules relying directly on Eqs.(22,23) boil down to the following. First, due to Eq.(23), interactions between modes of the same type are impossible. Similarly, the overlap integrals in Eq.(22) forbid the interaction if n ≠ n’. In other words, there are only two permitted configurations, and these are TE n →TM n and TM n →TE n. The symmetry considerations in cubic ferrites allow for another MO effect, namely the Cotton-Mouton linear birefringence (quadratic in magnetization). As any three-wave process, a MO interaction can be efficient only under the condition that the interacting waves are in phase synchronism, referred to as the Bragg condition in optics. In the collinear geometry (see Fig. 2), which is better suited for our purposes, and in the TE →TM configuration it reads 𝛥𝛽=𝛽()்ெ−𝛽()்ா=𝑘. (24) Here k is a spin wave wave number. When this condition is satisfied, the integrand in (22) does not contain a rapidly oscillating function, which maximizes the overlap integral. If >0, the scattering MSSW propagates in the same direction as the optical modes. Consequently, the scattered TM-mode is frequency up-shifted 𝛺்ா=𝛺்ெ+𝜔, here 𝜔 is the frequency of the scattering MSW (anti-Stokes process). On the other hand, if <0, the scattering MSW 14 propagates in the opposite direction with respect to the optical modes, as a result the scattered TM-mode is frequency down-shifted 𝛺்ா=𝛺்ெ−𝜔 (Stokes process). Fig. 2. Geometry of the magnon-light interaction. (1,2,3) is the stripline used to excite magnons in the fiim. (1) stripline ground plane. (2) its dielectric substrate. (3) stripline itself (asymmetric coplanar). (4) YIG film. (5) film substrate. “TE” denotes the non-diffracted light beam and “TM” the diffracted beam (light scattered from magnons). Magnons excited by a microwave current in the stripline propagate in the film collinear with the optical photons. This results in light scattering from the magnons. In the TM →TE configuration , the Bragg condition reads 𝛥𝛽=𝛽()்ா−𝛽()்ெ=𝑘, (25) and the previous paragraph can be directly applied to the analysis of the scattering mechanism, except that the superscript TM should be replaced with TE and vice versa. Now consider the MO coupling coefficient, describing the efficiency of the MO interaction. As mentioned above, in garnets the magnetically induced perturbation of the dielectric permittivity at optical frequencies is comprised of two major contributions due to the Faraday (F) and Cotton-Mouton (CM) effects 442F CM ij ij ij F ijl l i j if M g M M , where Ff is the Faraday constant, g44 is a Cotton-Mouton constant, Mi is the component of the total vector zM M e m along the axis i, and summation over the dummy indices is assumed (Einstein convention). Here we have neglected the anisotropic part of the Cotton-Mouton contribution proportional to 11 12 442 g g g which is not essential. In our case, the total magnetization vector M is comprised of the following components ( ) ( ) exp( )exp( )z x x y yM m y m y i t ikx x M e e e (see (17)). The amplitude of the scattered optical mode, either TE or TM, i.e. the efficiency of the MO interaction follows from Eq.(22). One finds that it scales as a parameter vMO referred to as the index of phase modulation. Its magnitude is given by 15 0 , , 44 , , 2 ( ) 2 ( )2S F xz y k yz x kk xz x k y ez y ffk i f I m I m g M I m I mN (26) for the Stokes process and 0 , , 44 , , 2 ( ) 2 ( )2AS F xz y k yz x k x effz x k yz y k f I m iI m g M I m iI mN (27) for the anti-Stokes one. In Eqs.(26,27), 02 / is a light wavenumber in vacuum, and Neff is an effective refraction index of the “incident” waveguide mode; ( ) 0/TE n TEN N for the TE→TM optical mode conversion and ( ) 0/TM TM effnN N for TM →TE. The quantities ,x kkm and ,y km are thickness averages of the respective vector components of the amplitude mk (Eq.(17a)), calculated for k=. Stripline transducers excite spin waves with kL<1, therefore the y-dependencies of mkx(y) and mky(y) are close to uniform on the scale of the film thickness. For the purpose of our calculation, it is appropriate to assume that they are perfectly uniform and equal to the thickness averages of these quantities. (The expressions for the latter are given by (A16-A17) in Appendix B.) Under this assumption, Eq.(23) yields 0xzI and 1zyI. The latter relations illustrate symmetry of the optical waveguide modes. More specifically, the transverse components ( )( )n TE zE y and ( )( )n TM yE y have practically identical spatial distribution across the waveguide and 1zyI, while the longitudinal component ( )( )n TM xE y is characterized by a symmetry that is opposite to that of the transverse components, hence 0xzI. Thus, both effects, Faraday and Cotton-Mouton, couple only the transverse components of the optical fields, i.e. ( )( )n TE zE y and ( )( )n TM yE y . Taking all this into account, we finally arrive at 0 442S F kx ky effi f m g MmN , (28) 0 442 )AS F kx effky i f m g MmN . (29) Here we denoted ,x k kxm m and ,y k kym m , in order to simplify notations . The amplitude of the scattered optical mode is proportional to that of the incident one and the coupling coefficient /2MO. Hence the power of the scattered light Is is given by 2 4MO s LP P, (30) 16 where PL is the power of the incident light and ,MO S AS , depending on whether one deals with a Stokes or an anti-Stokes scattering process. The microwave-photon to optical-photon (microwave-to-light) conversion efficiency is given by Eq.(6) in [7]. s inP P , (31) where Pin is given by Eq.(19) and is the frequency of the light. III. Discussion A. Calculation results In this section, we show results of calculations by employing the theory above. For these calculations, we use the parameters as follows. The thickness of the YIG film L=20 micron (Fig. 1). The input transducer is an asymmetric backed coplanar line, having a 400 micron wide signal line and a 200 micron wide ground line ( w2=400 micron, w3=200 micron). The gap between the two 2=25 micron. The coplanar line is supported by a 0.5mm-thick dielectric substrate ( d=0.5mm) with =11 whose other surface is metalized and grounded. The length of the transducer la= 5 mm; it coincides with the width of the film. The applied field H=1000 Oe. We use magnetic parameters for the film extracted from a recent microwave experiment at 16 mK [23]: saturation magnetization 4 M=2360 G and gyromagnetic ratio =2.83 MHz/Oe. As the resonance linewidth could not be measured in that experiment (because traveling waves were excited in [23]) we use the fact that no difference between the resonance linewidths at the same temperature and at room temperature was found in [42]. Therefore we use the same magnetic loss parameter as typical for bismuth-substituted YIG films at room temperature: FWHM 2 H=0.8 Oe. Bismuth-dopped YIG is the best candidate for the magneto-optical interaction. Unfortunately, its optical constants at 16 mK are not known. However, in [4] it was found that the Verdet constant for a YIG sphere at 16 mK does not differ significantly from the literature value for the room temperature. Therefore, in our calculation we use the literature value for the bismuth- substituted YIG: fF=103/(4M) (measured in G1) obtained for room temperature [43]. The constant g44 can then be estimated based on fF and the Stokes/anti-Stokes peak asymmetry from [18]; one obtains 2 g44=0.58 103/(4M)2 (measured in G2). Optical wavelength is 1.15 µm which corresponds to β0 = 5.46∙106 rad/m and optical frequency =22.6 1014 Hz. N =2, and LP=15 mW as in the recent experiment with a YIG sphere [4]. Let us first calculate the efficiency of the YIG film with asymmetric coplanar transducer as a non-reciprocal device called a microwave isolator. The power transmitted by the device is obtained by calculating the electric field of the spin waves at the position of the output transducer. Then Telegraphist Equations which include a distributed source of a microwave 17 electric voltage are employed, in order to convert the electric field into the output voltage of the output transducer [31]. A result of this calculation is shown in Fig. 3. The distance between the transducers is 4 mm. One sees that transmission characteristics for the + k (from Port 1 to Port 2) and – k (from Port 2 to Port 1) directions differ by at least 20 dB. Also, the transmission losses in the + k directin are just 5 dBm. This characterizes the spin wave device as a very efficient microwave isolator. Note that the magnitude of losses in the + k direction is consistent with previous calculations and experiments [28,44]. Fig. 3. Non-reciprocity of the spin wave device. Solid line: loss of power transmitted from Port 1 to Port 2 (+ k direction), dotted line: loss microwave power for transmission from Port 2 to Port 1. Parameters of calculation: L=20 micron, la=5 mm, w2=400 micron, w3=200 micron, =25 micron and the distance between the Port 1 and Port 2 transducers is 4 mm. Applied field is 1000 Oe. Let us now proceed to discussion of the microwave photon to optical photon conversion efficiency . Central to the discussion is Eq.(17a), as it enters the expression for (31). One sees that this expression has a resonant denominator – for some k=k0 the real part of the denominator vanishes and the denominator value becomes equal to ˆIm[det( ( ))]i W k . Hence, ˆRe[det( ( ))] 0W corresponds to resonant interaction of light with a spin wave with eigenfrequency given by the condition ˆRe[det( ( ))] 0W k . In the vicinity of 0 k k and for small magnetic losses ( H H ), the denominator can be expanded into Taylor series 18 0 0 0 0ˆdet( ( , )) ˆ ˆ ˆRe[det( ( , )] ( ) Re[det( ( , ))]/ Re[det( ( , ))]/W k H i H W k H k k W k H k i H W k H H This expression can be re-written spin-wave frequency resolved: 0 0 0 0 0ˆdet( ( , )) ( )ˆ ˆ ˆRe[det( ( , )] Re[det( ( , ))]/ Re[det( ( , ))]/( )gW k H H W k H W k H k i H W k H HV k where 0 0( , )k H and ( , )k H is the dispersion law for spin waves given by ˆdet ( ) 0W k. The expression for the spin wave group velocity gV reads ˆ Re det / ˆ Re det /gd W dk V d W d . Substituting this expression into the expression above one finds that the bandwidth of the optical guided mode conversion is given by ˆRe[det( )]/ ˆRe[det( )]/WH W H evaluated at 0( )k,H. The first term of this expression is close to the gyromagnetic ratio (Eq.(2)). Therefore, the bandwidth of the optical-photon/magnon interaction is given by 2H, which is the same as in the case of a YIG sphere. However, for the sphere, the microwave photon to magnon conversion bandwidth is given by the same H, but for the travelling spin waves the latter bandwidth is much larger – several hundreds of MHz, as one sees from Fig. 3. This implies that multi-channel regime of magnon- optical photon conversion can be implemented, with each individual channel corresponding to a specific pair of optical modes “n” with a specific value of TE n-TMn birefringence n. (Recall that the k= condition should be satisfied, therefore the central microwave frequency 𝜔௧(𝑘=𝛥𝛽) for each independent MO channel will correspond to a particular point in the Damon-Eshbach MSW dispersion curve.) The bandwith of each channel, scaling as the inverse of the effective length of the MO interaction leff, typically varies from 10-1 to 10 MHz. Thus, it is considerably smaller than the above-mentioned frequency band of MSSW excitation by an MSSW transducer. The important point of the frequency characteristics of a MO channel steming from the Bragg condition based phase synchronism for the optical guided modes and MSSW will be addressed in the last section “B. Ways to further improve the efficiency”. In the following, we assume that the phase synchronism Bragg condition k=has been satisfied at any microwave frequency by properly choosing a respective pair of guided optical modes. Therefore, in all figures below we show the efficiencies of the microwave to optical photon conversion calculated for the maximum of the conversion bandwidth k= In other words, we assume that the condition k= is fulfilled for each frequency shown in the graphs. Mathematically 19 this means that the real part of the denominator of Eq.(17a) vanishes for each frequency and k for each frequency is chosen such that it makes ˆRe[det( ( , )]W k H H vanish. Figure 4 shows the result of our calculation for the asymmetric coplanar geometry. One sees that the efficiency of the anti-Stokes process is larger than that of the Stokes one in our geometry. Therefore, below we will focus on the anti-Stokes process. Fig. 4. MOP conversion efficiency for the parameters of Fig. 3 (except the second coplanar transducer is absent, because not needed). Solid line: anti-Stokes process; dotted line: Stokes process. The length of the area where magnons interact with guided optical photons is 3 lf=20 mm. In this figure, we actually demonstrate the efficiency of the magnon to optical photon (MOP) conversion. To carry out this calculation, we use the same Eq.(31) but equate Pin in it to the power P+ of spin waves radiated by the stripline transducer in the direction of optical mode propagation. The latter is given by Eq.(21). From the figure, one sees that the light scattering process is characterised by a pronounced maximum. One also sees that the efficiency of MOP conversion is quite high – on the order of 105, or tens of parts per million (PPM). In order to understand formation of the maximum, in Fig. 5 we compare this result to the MOP efficiency in the assumption that a traveling spin wave only exists in a medium. In this case, Eq.(29) can be cast in the form given by Eq.(A18) in Appendix B. This form of the equation for MOis convenient for calculation of MOP efficiency of eigen-waves. Let us use the case of a film in vacuum ( d=∞) as a reference. In Appendix B, we show a simple analytic expression (A20) for the Poynting vector eig for the 20 eigen-waves in this geometry. By using (A19) and equating Pin in (31) to | eig| one obtains the dotted line in Fig. 5. Fig. 5. MOP conversion efficiency for the anti-Stokes process. Dotted line: eigen-waves in a film in vacuum ( d=∞). Solid line: Film on top of the backed stripline, but only the traveling spin wave (i.e. far-field) contribution is accounted for. Dashed line: total MOP conversion efficiency for the backed stripline (contributions of both far and near spin wave field of the asymmetric coplanar transducer are taken into account). The solid line in this figure is obtained by calculating the amplitude of the traveling wave component of the total spin wave field excited by the transducer (in real space, Eq.(17b)). This component and P+ (Eq.(20)) are calculated for x=0 and then Eq.(A18) is applied to compute the MOP conversion efficiency. The latter is obtained by equating Pin to P+(x=0) in Eq.(31). One sees that the two curves agree perfectly for frequencies larger than 5.4 GHz. Below this frequency, the efficiency of MOP conversion for the real coplanar device goes down. This is explained by the effect of the metal of the stripline ground plane located at y=d. The effect of the ground plane is present for kd<1. The latter wave number range corresponds to the frequency range below 5.4 GHz. When d is increased in the numerical simulation, the maximum in the solid line shifts to lower frequencies. This confirms that the maximum is formed due to the effect of the ground plane. One also sees that the total MOP conversion efficiency (dashed line) is two times bigger than the traveling spin wave contribution. This evidences that the contribution of the near field of the transducer to the total MOP conversion efficiency may be very significant. Again, the decrease in the efficiency below 5.4 GHz is due to the effect of the metal ground plane. The respective total microwave photon to optical photon (MPOP) efficiency (Eq.(31)) and the efficiency of microwave photon to magnon conversion Nm (Eq.(21)) is shown in Fig. 6. One sees that the MPOP efficiency is of the same order of magnitude as the MOP one. This is 21 because the efficiency of the microwave photon to magnon conversion Nm by the asymmetric coplanar transducer is close to 1. Fig. 6. Total (MPOP) conversion efficiency (left-hand axis) for the asymmetric coplanar line geometry from Fig. 5. Solid line: anti-Stokes process; dotted line (Stokes process). Dashed line: microwave photon to magnon conversion efficiency by the asymmetric coplanar transducer (right-hand axis). B. Ways to further improve the efficiency Even though MPOP efficiency achievable with travelling magnons is significantly larger than one achievable with a standing wave oscillation [4], it is still much smaller than for optomechanical converters (10%). Therefore, below we discuss ways to further improve the conversion efficiency. However, one has to keep in mind that all these measures will decrease the available frequency bandwidth. The first obvious way to increase efficiency is by decreasing the area of cross-section for light scattering from magnons. Figure 7 demonstrates the results of calculations of MOP conversion efficiency for three different film cross-sections. The first one is the same as in Fig. 4 – L=20 micron, la=5 mm. The second one is for the film thickness L=4 micron, which is the same thickness as in the experiments [18,19], but keeping the film width the same la=5 mm. The third calculation is for a microscopic film cross-section L=4 micron, la=50 micron. One sees 22 that the decrease in the film thickness alone does not change the efficiency; this is because, ultimately, the efficiency scales as kL, as does the frequency of the Damon-Eshbach waves for kL<<1. In other words, k-vector resolved, one gains in efficiency by decreasing L. However, frequency-resolved, the net gain is zero, because the spin wave frequency for a given k decreases proportionally, such that for the same frequency the efficiency remains the same. This is illustrated in Fig. 7 by showing k() dependences for both film thicknesses. On the other hand, the decrease in the film width has an impact on the efficiency, as the same figure shows. Here we have to note that for the 4x50 square micron cross-section, L and la are comparable, therefore, strictly speaking, our theory developed under assumption L<< la is not fully applicable in this case. Notwithstanding this, it demonstrates that the reduction of the film width is a valid way to significantly boost the efficiency of the MOP conversion. However, the decrease in la decreases coupling of the microwave field of the transducer to the precessing spins. This happens because the number of spins in the volume where the microwave field is present shrinks with the decrease in la. In our case of a stripline shorted at its end this is seen as a decrease in the input antenna impedance for la =50 micron. It is possible to compensate the drop in the impedance to a significant extent by playing around with the geometry of the coplanar line, as shown in Fig. 8. To this end, one has to use a coplanar line with a microscopic cross-section. As this calculation shows, in this way it is possible to achieve an impressive total conversion efficiency of 0.3 percent. Importantly, the difference between the MPOP and MOP efficiencies is about 3 times in this case, which is not a huge number, and the input impedance of the coplanar transducer is large enough – about 8 Ohm. Hence, it should be technically easy to increase MPOP to 0.9% by inserting an impedance matching circuit between the feeding line and the coplanar transducer. The conversion efficiency of almost 1% is a very good result. However, the possibility of its further improvement through additional reduction of the YIG film width should be taken with caution, as our theory is not fully applicable for la comparable to L. One more way to improve the efficiency is by further increasing the time of spin wave interaction with the optical modes. This can be done by confining the optical field inside a length equal to lf by forming an optical resonator of this length. Then the time of spin wave interaction with the optical modes will increase by Q times, where Q is the quality factor for the optical resonator. To be specific, let us begin with a modest and easily realizable quality factor of Q = 100. Naturally, this will increase the length of magnon-light interaction in a 4 µm thick YIG film from 0.6 cm to 0.6 m which will increase the efficiency accordingly from 102 to 1 (or 100%), as the efficiency scales as lf (see Eq.(A18)). At the same time, this increase in the interaction length will inevitably lead to restrictions imposed by the Bragg selectivity in the reciprocal space. As a result, even a slight deviation from the phase-synchronism Bragg condition will lead to a drop in the efficiency of the MO interaction according to 2sinc ( ) where is the phase mismatch due to the above-mentioned deviation. To be specific, let us consider the TE→TM configuration with Δβ > 0 (see Eq.(24)), in which case the magnon propagating in the same direction as the “incident” TE optical mode will contribute to the anti-Stokes process 23 (the up-shifted scattered TM mode). There are two mechanisms, both dispersion related, contributing to this mismatch: magnetic (MSSW mode) and optical (optical waveguide mode). Fig. 7. Thick solid line: conversion efficiency for L=20 micron, la=5 mm; dotted line: the same, but L=4 micron, la=5 mm; dashed line: the same, but L=4 micron, la=50 micron. Thin solid line: spin wave wave number for L=4 micron (right-hand axis); dash-dotted line: the same, but for L=20 micron. Fig. 8. Solid line: MOP conversion efficiency or the microscopic device (left-hand axis). Dotted line: total (MPOP) conversion efficiency (left-hand axis). Dashed line (right-hand axis): microwave photon to magnon conversion efficiency by the asymmetric coplanar transducer (w1==0). Film thickness: L=4 micron, film width: 50 micron. Width of the signal line of the asymmetric coplanar transducer: w2=4 micron, width of the ground line: w3=20 micron, gap between the two: =250 micron. Distance to the ground plane d=500 micron. Dielectrip permittivity of the substrate of the coplanar line: 11. 24 First, numerical estimation of the value of the wave number of the scattered optical mode should take into account the Doppler shift imposed by the moving magnon, i.e. ( ) ( ) /TM TM opt gV , where opt gV is the group velocity of the optical mode. The latter approximation is perfectly justified since ω is really very small with respect to Ω and typically ω/Ω ~10-5. Thus, the Doppler related additional phase shift reads /2 /(2 )opt eff eff gl l V , here leff is the effective interaction length, while ( ) is the perturbation of the wavenumber of the scattered optical TM mode due to the Doppler frequency shift and, consequently, (0) 0. The optical wave number frequency dependence ( ) makes the total phase mismatch also frequency dependent ( ) . Suppose that in the absence of the Doppler shift the Bragg condition is fully satisfied and the phase synchronism is perfect, i.e. (0) 0 . Now let us estimate the consequences of the Doppler effect, namely the deviation from the phase synchronism ( ) ( ) (0) for a MSSW frequency of 4 GHz and two characteristic values of the effective interaction length mentioned above leff = 0.6 m (with the optical resonator) and leff = 6 mm (without the resonator). In the conventional no-resonance case one obtains (4GHz,6mm) 0.5rad which is not enough to influence appreciably the mechanism of the MO interaction that is why this effect was neglected in the earlier papers of the 1980s – 1990s. It follows from the same analysis that the effective bandwidth of the MO interaction defined as full width at half maximum (FWHM) and adapted for the Sinc function is equal to (6mm) 30GHzMSSWf . In the resonator case one obtains an impressive figure of 2( ) 0.5 10 rad and, as a result, (0.6m) 300 HzMSSWf M . Second, the MSSW dispersion also appears in the Bragg condition. Moreover, the phase synchronism is even more sensitive to MSSW frequency variations through this mechanism which is due to the direct presence of the wavenumber of extremely slow MSSW modes in the Bragg condition. As in the previous case, the coefficient weighing this contribution is the inverse group velocity, but this time that of the MSSW that is about 104 times slower. Let us suppose that the 4µm thick YIG film is magnetized to saturation by a 800 Oe magnetic field. In this case a MSSW with a frequency of 4 GHz will propagate with a group velocity gV of approximately 4∙104 m/s. Correspondingly, in the “without-resonator” and with-resonator configurations one obtains (6mm) 8MHzMSSWf and (0.6m) 80KHzMSSWf respectively. The former figure was confirmed in the experiments in the 1980s-1990s. In any case, it is the second mechanism that bottlenecks the frequency properties, its bandwidth in the first place, of the MO interaction. Thus, in this paragraph we provide useful quantitative data on the « bandwidth – efficiency » limitations, classical in photonics, which stem from the general laws of three-wave interactions requiring phase synchronism between the interacting waves. This information is indispensable for the design of specific devices specialized in coherently connecting distant superconducting qubits via light. On the other hand, our analysis emphasizes the importance of the role played by the Doppler shift in the scattering of light by MSSW in the case where the effective interaction length leff 25 exceeds a critical value of several centermeters in the lower GHz band addressed in this paper. While it is of less importance in the case of Brillouin light scattering by thermally excited incoherent magnons [5], it should not be overlooked when one considers coherent magnon- qubit up conversion to optical frequencies [45]. It is especially important for MO interactions involving the homogeneous Kittel mode in which case Δβ→0, especially if the optical resonator quality factor is as high as 105 [46]. In other words, the actual pertinent criterion of the smallness of the Doppler shift [45] cannot be formulated solely in terms of the ratio of the frequencies of the interacting waves ఠ ఆ≪ 1, even if it is as small as 10-5. It must follow directly from the Bragg phase synchronism and be expressed in terms of the maximum tolerable phase mismatch, thus reading ∆𝜑=ఠ ௩ ଶ< 1. It is should emphasized that boosting the efficiency of the MO interaction through a radical increase of the interaction length cannot be implemented in the absence of a reliable mechanism of fine-tuning to the Bragg condition. In this regard, the configuration relying on travelling spin wave has an advantage of an additional flexible degree of freedom, namely the tunable spatial periodicity in the form of the spin wave wavenumber k. Another important aspect of the Doppler frequency shift is its asymmetry with respect to the inversion of the direction of the incident optical mode which can be exploited in order to create non-reciprocal MO devices. Thus, if both interacting waves, the spin wave and the incident optical mode, propagate collinearly this shift will be positive ( +ω), whereas anti-collinear propagation will produce a negative shift ( ω). This, in its turn, means that if the Bragg condition is perfectly satisfied in the collinear geometry ∆𝜑(+𝜔) = 0, reversal of propagation direction of the optical incident wave will lead to a double phase asynchronism ∆𝜑(−𝜔) = 2ఠ ௩ ଶ . As a result, the amplitude of the “new-born” scattered optical wave will reduce accordingly. In other words, such a MO element can be regarded as an optical isolator with a ratio of nonreciprocity (which is defined as the ratio of amplitudes of counter propagating optical modes) equal to 𝑆𝑖𝑛𝑐(∆𝜑(−𝜔)). IV. Conclusions In this work we evaluated theoretically the efficiencies of a travelling magnon based microwave to optical photon converter for applications in Quantum Information. The microwave to optical photon conversion efficiency was found to be larger than in a similar process employing a YIG sphere by at least 4 orders of magnitude. By employing an optical resonator of a large length (such that the traveling magnon decays before forming a standing wave over the resonator length) it will be possible to further increase the efficiency by several orders of magnitude, potentially reaching a magnitude similar to one achieved with opto- mechanical resonators. However, this measure will decrease the frequency bandwidth of conversion. 26 Also, as a spin-off result, it has been shown that microwave isolation of more that 20 dB with direct insertion loss of about 5 dBm can be achieved with YIG film based isolators. These devices are needed to isolate qubits from noise in a microwave circuit to which they are connected. An important advantage of the concept of the travelling spin wave based Quantum Information devices is a perfectly planar geometry and a possibility of implementing a device as a hybrid opto-microwave chip. Acknowledgement Research Colaboration Award from the University of Western Australia is acknowledged. The authors also thank M. Goryachev, M. Tobar and V.N. Malyshev for fruitful discussions. Appendix A: Solution of the system of equations (2-5). The solution is obtained in the Fourier space (Eq.(6)). exp( ) exp( )kxm A k y B k y , (A1) ( )exp( ) ( )exp( )ky my mym AC k k y BC k k y (A2) where 2 2 2 2 2( )( )( )M my H M Hqk q kC q iq q k . (A3) Similarly, ( )exp( ) ( ( ))exp( )kx hx hxh AC k k y B C k k y , (A4) ( )exp( ) ( ( ))exp( )ky hy hyh AC k k y B C k k y , (A5) with 2 2 2( )( )( )H hx H M Hk q kC qq q k , (A6) and 2 2 2( )( )( )H hy H M Hq q kC q iq q k , (A7) 27 where HH and MM (or 4MM in Gaussian units ). Application of the electro-dynamic boundary conditions results in a vector-matrix equation sign( )ˆ 0k A i k jWB , (A8) where (| |) ( | |)ˆ (| |) ( | |)k kWk k , (A9) ( ) ( ) ( ) coth(| | ) sign( ) ( )my hy hxq C q C q q d i q C q , (A10) and ( ) ( ) ( ) sign( ) ( ) exp( )my hy hxq C q C q i q C q qL . (A11) Solving (A8) with respect to the vector on its left-hand side yields 1sign( )ˆ 0k A i k jWB , (A12) where 1 ( | |) ( | |)1ˆ ˆ(| |) (| |) det( )k kWk k W , (A13) and ˆdet( )W denotes the determinant of the matrix ˆW. Accordingly, ˆ sign( ) (| |) /det( )k A i k k j W , (A14) ˆ sign( ) ( | |) /det( )k B i k k j W . (A15) This concludes the solution of the problem of calculation of the spin wave amplitude mk. The closed form of the expression for mk is given by Eq.(17(a)). Appendix B: Calculation of quantities entering expressions for magnon to optical photon conversion efficiency The thickness-averaged spin wave amplitude is obtained from Eqs. (A1) and (A2) and reads: 28 ( ) ( )kxm AF k BF k , (A16) ( ) ( ) ( ) ( )ky my mym AC k F k BC k F k , (A17) where ( ) sign( )(exp( ) 1)/( )F k k kL kL . For eigen-waves we may set A=1. Then (A14) and A(15) yield ( | |)/ (| |)B k k . Substituting these A and B into (A16) and (A17) and the result into (28) and (29) we obtain the MOP conversion efficiency. In doing this we need to specify the length of the MO interaction area. It is found introducing the spin wave propagation path as /f gl H V , where His the magnetic loss parameter and /gV k is the group velocity of spin waves. Then the wave decays exponentially during its propagation 0exp( / )exp( )k k f x l ikx m m , where 0 km is its initial amplitude (at x=0). Substituting this expression into (17b) and it into the overlap integral (22) we arrive at the expression for the light-magnon coupling coefficient for the eigen-waves. 0 0 0 0 0 44 ( ) 2 ( )2f AS F xz y yz x xz x yz ylf I m iI m g M I m iI mN . (A18) In order to calculate the MOP conversion efficiency for the eigenwaves we also need the Poynting vector for them. It is obtained from (A2), (A4), (A5) and (5). For a film in vacuum (no metal ground plane at y= d) it reads: 20 1 3 ( ) Re( )exp( 2 / )2s flx x lk , (A19) where 2 1 1/(2 )A k , 2 3 3exp( 2 )/(2 )B k L k , 1( ) ( )hx hxA AC k BC k , 3( )exp(2 ) ( )hx hxB AC k k L BC k , and 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 exp(2 ) /(2 ) ( ) ( ) ( ) 1 exp( 2 ) /(2 )hy my hy hy my hy hy my hy hy my hyABL C k C k C k ABL C k C k C k A C k C k C k k L k B C k C k C k k L k . The MOP conversion efficiency is obtained by substituting (A19) into (31) and using ( 0)x (Eq.(A19)) as Pin. 29 References 1. M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013). 2. Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R.Yamazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015). 3. Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K.Usami, and Y. Nakamura, arXiv:1508.05290. 4. R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. B 93, 174427 (2016). 5. S. Sharma, Y. M. Blanter, and G. E. W. Bauer, Phys. Rev. B 96, 094412 (2017). 6. R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, Nature Phys . 10, 321 (2014). 7. A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, Optica 3, 596 (2016). 8. X. Fernandez-Gonzalvo, Y.-H. Chen, C. Yin, S. Rogge, and J. J. Longdel, Phys. Rev. A 92, 062313 (2015). 9. B. T. Gard, K. Jacobs, R. McDermott, M. Saffman, Phys. Rev. A 96, 013833 (2017). 10. S. Blum, C. O'Brien, N. Lauk, P. Bushev, M. Fleischhauer, and G. Morigi, Phys. Rev. A 91, 033834 (2015). 11. H. L. Hu and F. R. Morgenthaller, Appl.Phys.Lett . 18, 307 (1971). 12. B. Desormiere and H. Le Gall, IEEE Trans. Magn. MAG-8, 379 (1972). 13. A. D. Fisher, J. N. Lee, E. S. Gaynor, and A. B. Tveten, Appl. Phys. Lett. 41, 779 (1982). 14. C.S. Tsai, D. Young, W. Chen, L. Adkins, C. C. Lee and H. Glass, Appl. Phys. Lett. 47, 651 (1985). 15. A. A. Stashkevich, B. A. Kalinikos, N. G.Kovshikov, O. G. Rutkin, A. N. Sigaev, and A. N. Ageev, Sov. Tech. Phys. Lett . 13, 20 (1987). 16. A.N. Sigaev and A.A. Stashkevich, Sov. Tech. Phys. Lett. 14, 209 (1988). 17. A. A. Stashkevich, M. G. Cottam, and A. N. Slavin, Phys. Rev. B 54, 1072 (1996). 18. H. Tamada, M. Kaneko, and T. Okamoto, J. Appl. Phys . 64, 554 (1988). 19. V. V. Matyushev, A. A. Stashkevich and J. M. Desvignes, J. Appl. Phys . 69, 5972 (1991). 20. V. V. Matyushev, M. P. Kostylev, A. A. Stashkevich and J. M. Desvignes, J. Appl. Phys . 77, 2087 (1995). 21. R. Bhandari and Y. Miyazaki, IEICE Trans. Electron . Ecc-C, 255 (2000). 22. A. Noguchi, N. Goto, and Y. Miyazaki, Electrical Engineering in Japan 162, 240 (2008). 23. M. Goryachev, S. Watt, M. Kostylev, and M. E. Tobar, “Strong coupling of 3D cavity photons to travelling magnons at low temperatures”, unpublished. 24. S. Kosen, R. G. E. Morris, A. F. van Loo, and A. D. Karenowska, arXiv:1711.00958v1 (2017). 25. P. R. Emtage, J. Appl. Phys . 49, 4475 (1978). 26. R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids , 19 308 (1961). 30 27. A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, Phys. Rev. X 7, 011007 (2017). 28. V. F. Dmitriev, Sov. J. Commun. Technol. Electron . 36, 34 (1991). 29. S. Balaji and M. Kostylev, J. Appl. Phys . 121, 123906 (2017). 30. V. F. Dmitriev and B. A. Kalinikos, Sov. Phys. J. 31, 875 (1988). 31. G. A. Vugal’ter and I. A. Gilinski, Radiophysics and Quantum Electronics 32, 869 (1989). 32. A. G. Gurevich and G. A. Melkov, “ Magnetization Oscillations and Waves ”, CRC Press, Florida: 1996. 33. M. Kostylev, J. Appl. Phys. 119, 013901 (2016). 34. A. K. Ganguly and D. C. Webb, IEEE Trans. MTT MTT-23, 998 (1975). 35. T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich and A. N. Slavin, Phys. Rev. Lett .104, 197203 (2010). 36. M. P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T. Ono, and R. L. Stamps, Phys. Rev. B 76, 054422 (2007). 37. M. P. Kostylev A. A. Serga, T. Schneider, T. Neumann, B. Leven, B. Hillebrands, and R. L. Stamps, Phys. Rev. B 76, 184419 (2007). 38. C. S. Chang, M. Kostylev, E. Ivanov, J. Ding and A. O. Adeyeye, Appl. Phys. Lett. 104, 032408 (2014). 39. T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, and M. P. Kostylev, Phys. Rev. B 77, 214411 (2008). 40. M. Cottam and D. Lockwood, Light Scattering in Magnetic Solids , Wiley- Interscience, New York: 1986. 41. A. A. Stashkevich, " Chapter 9. Diffraction of Guided Light by Spin Waves in Ferromagnetic Films " in "High-Frequency Processes in Magnetic Materials" Eds.: G. Srinivasan and A. N. Slavin, World Scientific /Singapore, New Jersey, London, Hong-Kong: 1995. 42. M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, Phys. Rev. Appl . 2, 054002 (2014). 43. A. Zvezdin and V. Kotov, “Modern Magnetooptics and Magnetooptical Materials”, IOP Publishing, Bristol: 1997. 44. G. A. Vugalter, B. N. Gusev, A. G. Gurevich, and O. A. Chivileva, Sov. Phys. Tech. Phys. 31, 87 (1986). 45. P. A. Pantazopoulos, N. Stefanou, E. Almpanis, and N. Papanikolaou, Phys. Rev. B 96, 104425 (2017). 46. A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Phys. Rev. Lett . 116, 223601 (2016).

2017-12-12

In this work we study theoretically the efficiency of a travelling magnon based microwave to optical photon converter for applications in Quantum Information (QI). The converter employs an epitaxially grown yttrium iron garnet (YIG) film as the medium for propagation of travelling magnons (spin waves). The conversion is achieved through coupling of magnons to guided optical modes of the film. The total microwave to optical photon conversion efficiency is found to be larger than in a similar process employing a YIG sphere by at least 4 orders of magnitude. By creating an optical resonator of a large length from the film (such that the traveling magnon decays before forming a standing wave over the resonator length) one will be able to further increase the efficiency by several orders of magnitude, potentially reaching a value similar to achieved with opto-mechanical resonators. Also, as a spin-off result, it is shown that isolation of more that 20 dB with direct insertion losses about 5 dBm can be achieved with YIG film based microwave isolators for applications in Quantum Information. An important advantage of the suggested concept of the QI devices based on travelling spin waves is a perfectly planar geometry and a possibility of implementing a the device as a hybrid opto-microwave chip.

Microwave to optical photon conversion by means of travelling-wave magnons in YIG films

1712.04304v2

Macroscopic distant magnon modes entanglement via a squeezed reservoir Kamran Ullah,1,∗M. Tahir Naseem,2,†and ¨Ozg¨ur E. M ¨ustecaplıo ˘glu1, 3,‡ 1Department of Physics, Ko c ¸University, 34450 Sarıyer, Istanbul, T ¨urkiye 2Faculty of Engineering Science, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Topi 23640, Khyber Pakhtunkhwa, Pakistan 3T¨UB˙ITAK Research Institute for Fundamental Sciences, 41470 Gebze, T ¨urkiye (Dated: December 19, 2023) The generation of robust entanglement in quantum system arrays is a crucial aspect of the realization of efficient quantum information processing. Recently, the field of quantum magnonics has garnered significant attention as a promising platform for advancing in this direction. In our proposed scheme, we utilize a one-dimensional array of coupled cavities, with each cavity housing a single yttrium iron garnet (YIG) sphere coupled to the cavity mode through magnetic dipole interaction. To induce entanglement between YIGs, we employ a local squeezed reservoir, which provides the necessary nonlinearity for entanglement generation. Our results demonstrate the successful generation of bipartite and tripartite entanglement between distant magnon modes, all achieved through a single quantum reservoir. Furthermore, the steady-state entanglement between magnon modes is robust against magnon dissipation rates and environment temperature. Our results may lead to applications of cavity-magnon arrays in quantum information processing and quantum communication systems. I. INTRODUCTION Quantum entanglement plays a pivotal role in various ap- plications of quantum information processing [ 1], encompass- ing quantum cryptography [ 2], quantum teleportation [ 3], and quantum metrology [ 4]. It is a crucial resource for enhanc- ing the performance of quantum devices and technologies [5]. However, preparing long-lived entangled states, especially at the macroscopic scale, becomes challenging due to inevitable interactions between quantum systems and their environments. Consequently, the quest for generating macroscopic entangled states using various physical setups has gained significant at- tention. In this context, steady-state entanglement between atomic ensembles has been successfully demonstrated [ 6–8]. Additionally, entangled states involving macroscopic systems have been reported in different setups, such as entanglement between a single cavity mode and a mechanical resonator in an optomechanical configuration [ 9]. Recently, experimental achievements include entanglement between two macroscopic resonators coupled to a common cavity mode through optome- chanical interaction [10]. Another central challenge in harnessing entanglement to improve various quantum tasks lies in the ability to create and distribute entanglement across large arrays of quantum systems [11–17]. An intriguing approach to address this challenge is reservoir engineering, where external control drives are used to engineer desirable dissipative dynamics, leading the quan- tum system to relax in the desired quantum state [ 14,18–25]. However, most reservoir engineering schemes require multiple external control fields to be applied over different elements of the quantum system array [26, 27]. In recent years, hybrid quantum systems, combining differ- ent physical subsystems, have received significant attention ∗kullah19@ku.edu.tr †mnaseem16@ku.edu.tr ‡omustecap@ku.edu.tr[28–30]. Among these systems, cavity magnon setups offer a unique platform to investigate light-matter interactions [ 31,32]. In cavity quantum electrodynamics, the strong and ultrastrong coupling between magnons and photons has been increasingly studied [ 31,33–37], with experimental demonstrations of these regimes [ 38–40]. This strong coupling opens possibilities for exploring novel phenomena, such as the magnon Kerr effect [41], cavity-magnon polaritons [ 42], magnon-induced trans- parency [ 43], magnomechanically induced slow-light [ 44], bistability [ 45], exceptional points [ 42,46], magnon block- ade [47], and nonreciprocity [48]. Recent proposals aim to realize genuine quantum effects in macroscopic magnon-cavity systems, including magnon blockade [ 49], magnon squeezed states [ 50], Schr ¨odinger cat states [ 51,52], Bell states [ 53], and nontrivial bipartite and multipartite entangled states [ 54,55]. Of particular interest is the generation of entanglement between distant magnon modes [ 56], which holds promise for quantum information processing applications. However, to the best of our knowledge, existing studies have primarily focused on entangling only two magnon modes, such as entangling two YIG spheres in a single cavity via Kerr nonlinearity [ 57], magnetostrictive interaction [ 58], or applying a vacuum-squeezed drive [ 59, 60]. Further, stationary quantum entanglement between two massive magnetic spheres can be induced by subjecting each sphere to two-tone Floquet fields, which effectively generates parametric interaction between magnon modes [ 61]. In the case of two cavities, each containing a single YIG, entanglement between the YIGs has been generated through optomechanical- like coupling [ 62], vacuum squeezed drive [ 63], or reservoir engineering [ 56,64,65]. Recently, the bipartite entanglement between magnon modes is investigated in a one-dimensional array of cavities. The entanglement is generated by exploiting the coupling of the magnon modes with a central giant atom via virtual photons [66]. Hybrid quantum magnonic systems hold great promise for efficient quantum networks [ 67]. Therefore, a crucial task is to establish and control entanglement between multiple magnon modes in arrays based on quantum magnonic systems. In thisarXiv:2308.13586v3 [physics.optics] 17 Dec 20232 Figure 1. (a) Schematic description of magnon-magnon entanglement generation scheme, consisting of a one-dimensional 2N+ 1array of identical lossy cavities coupled via hopping interaction J. Each cavity contains a yttrium iron garnet (YIG) sphere with volume V. The cavities’ decay and dissipation rates are κaandκj, respectively. The central YIG is coupled to a quantum-squeezed reservoir, enabling our entanglement generation scheme. (b) Describes the effective model, which consists of only magnon modes obtained after removing the cavities via employing Schrieffer-Wolf transformation. (c) shows the behavior of the dispersion relation of the cavity mode in momentum space. regard, we propose a scheme to generate entanglement between multiple yttrium iron garnet (YIG) spheres using a reservoir engineering approach with a single squeezed bath. Specifically, our setup consists of an array of cavities, each housing a macro- scopic YIG coupled to the cavity through beam splitter-like interaction. However, entanglement cannot be generated with this interaction alone; thus, we introduce the necessary nonlin- earity by driving one of the magnon modes with a squeezed reservoir. Previous studies have demonstrated that a single and two-mode squeezed drive can create entanglement between magnon modes [ 59,60,62,63]. In our work, we show that when one of the magnon modes is subjected to a squeezed reservoir, it is sufficient to generate steady-state bipartite and tripartite entanglement between distant magnon modes. We would like to highlight that, in principle, the local squeezed reservoir on the central magnon mode ( m2) can be realized by coupling an auxiliary quantum system. Genera- tion of the squeezed input field can be accomplished using a method similar to the proposal discussed in [ 68]. For example, the input squeezed reservoir can be engineered by coupling an auxiliary microwave cavity to the central magnon mode through beam-splitter-type interaction. The weak squeezed vacuum field, generated via a flux-driven Josephson parametric amplifier, acts as the driving force for the auxiliary cavity. Con- sequently, the auxiliary cavity serves as a quantum squeezed reservoir for the central magnon mode [ 50,69]. Similarly, the squeezed reservoir can also be achieved by coupling the central magnon mode to a superconducting qubit [ 70]. More specif- ically, the approach proposed in [ 71] can be implemented by replacing the coplanar waveguide resonator with a YIG placed inside a microwave cavity. This scheme enables the coupling of a superconducting qubit with an adjustable en- ergy gap to a magnon mode characterized by the frequency ω2. By modulating the energy gap of the qubit with a bichro- matic field, the squeezed reservoir for the magnon mode can be engineered [ 50]. Alternatively, the input squeezed radia- tion can be generated by applying a two-tone microwave field drive to the central YIG, where the magnon mode is coupledto its mechanical vibrational mode through magnetostrictive interaction [72, 73]. The paper is structured into the following sections: In Sec. II, we introduce the model; and in Sec. III derivation of effective Hamiltonian is presented. The results for bipartite and tripartite entanglement between the magnon modes are discussed in Secs. IV A and IV B, respectively. Finally, in Sec. V, we offer concluding remarks and summarize the key findings. II. THE MODEL We investigate a cavity-magnon system comprising a one- dimensional array of 2N+ 1cavities, as illustrated in Fig. 1. These cavities are interconnected through photon-hopping in- teractions. Inside each cavity, a single YIG sphere is present, coupled to the cavity mode via magnetic dipole interaction. In this study, we consider a single magnon mode, which repre- sents quasi-particles with collective spin excitations, associated with each YIG in the cavities. The Hamiltonian governing the field for the cavity array is as follows: Ha/ℏ=ωc/summationdisplay jˆa† jˆaj−J/summationdisplay j(ˆajˆa† j+1+ ˆa† jˆaj+1).(1) The first term represents the free energy of the cavities, while the second term accounts for the exchange energy of the field. ˆajandˆa† jare the bosonic annihilation and creation operators of thej-th cavity and the corresponding resonance frequency ωc. Herej(j=−N,.., 0,...,N ) describes the index of the cavity in a one-dimensional array, with each cavity interconnected through a hopping coupling called J. We consider only a single photon process in each cavity. The ferromagnetic sample (YIG) holds within it scattering spin waves, with the assumption that only the spatially uniform Kittel mode [ 74] exhibits a pronounced interaction with pho- tons within the cavity. The free Hamiltonian for these magnon3 modes is given by Hm/ℏ=/summationdisplay nωnˆm† nˆmn, (2) here ˆmn(ˆm† n) is the annihilation (creation) operator of the magnon mode, and ωnis the associated frequency of the mode. In addition, the index nin the summation is an integer number such thatn∈[−N,N ]. In general, some cavities can be empty while some can contain YIG spheres. For example, in Sec. IV, we assume that three of the cavities are occupied while the rest remain empty. In this scenario, Eq. (2) includes only three terms for magnon modes, however, 2N+ 1cavities are present. The magnon frequencies can be calculated based on the applied magnetic fields Hn, and given by ωn=γHn, whereγ/2π= 28 GHz/T represents the gyromagnetic ratio. The interaction between the magnon and cavity modes is given by HI/ℏ=/summationdisplay ngn(ˆanˆm† n+ ˆa† nˆmn), (3) here,gnis the coupling strength between the magnon and associated cavity modes and is given by gn= ζγ/2/radicalbig 5ℏωnµ0N/V . The volume of the cavity is given by V, Nrepresents the total number of spins in the YIG, µ0is the permeability of free space, and ζdescribes the spatial over- lap between the magnon and photon modes. We note that the interaction term in Eq. (3) is obtained after performing the Holstein-Primakoff transformation in which collective spin operators are written in the form of bosonic magnon operators ˆm(ˆm†) [57]. In addition, the counter-rotating terms ˆmˆa,ˆm†ˆa† are ignored assuming the validity of rotating wave approxima- tion [59] III. THE SCHRIEFFER–WOLFF APPROXIMATION AND EFFECTIVE HAMILTONIAN To generate entanglement between two magnon modes, a nonlinear interaction such as effective parametric-type nonlin- ear coupling ( ˆm† 1ˆm† 2+ ˆm1ˆm2)between the modes is required. This interaction can be achieved by introducing strong Kerr nonlinearity [ 57] or nonlinear magnomechanical interaction [58]. It is typically easier to engineer an effective beamsplitter- type coupling ( ˆm† 1ˆm2+ ˆm1ˆm† 2)between the magnon modes. For instance, in a system with two YIGs spheres placed in- side a microwave cavity, it is possible to generate the de- sired coupling ( ˆm† 1ˆm2+ ˆm1ˆm† 2)by carefully selecting the system parameters that allow adiabatic elimination of the cav- ity mode [ 75]. Another approach involves an array of three cavities, where the central cavity contains a qubit, and each end cavity houses a YIG sample [ 65]. By employing the dispersive regime and using the Schrieffer-Wolff (Frohlich-Nakajima) approximation [ 76–78], the cavity modes can be eliminated, resulting in an effective beamsplitter-type coupling between the magnon modes. In our case, we adopt the latter method to create an array of YIGs with an effective coherent coupling( ˆm† nˆmn+1+ ˆmnˆm† n+1). Our strategy is to utilize these easier- to-engineer beam-splitter-type magnon-magnon interactions but introduce the required nonlinearity for generating entan- glement between distant magnon modes through a squeezed thermal bath [68, 79–82]. The Hamiltonian Ha(Eq. (1)) associated with a one- dimensional array of coupled cavities represents the tight- binding bosonic model. To diagonalize it, we introduce new operators in the momentum space ( k-space). The resulting diagonal Hamiltonian is given by Hdiag/ℏ=/summationdisplay kωkˆa† kˆak, (4) here, we have introduced ˆak=1√ 2N+ 1/summationdisplay jˆajeikj, (5) ˆa† k=1√ 2N+ 1/summationdisplay jˆa† je−ikj. (6) We have assumed the periodic boundary conditions such that k:=km= 2πm/(2N+ 1) withm∈[−N,N ], and con- sidering a large cavity array ( N≫1) results ink∈[−π,π]. Moreover,ωk=ωc−2Jcoskis the dispersion relation of the cavity mode. The interaction Hamiltonian, as presented in Eq. (3), modifies to HI/ℏ=1√ 2N+ 1/summationdisplay k,n/bracketleftig gn(ˆakˆm† neikln+ ˆa† kˆmne−ikln)/bracketrightig . (7) We note that ln=n, and it is associated with the position of the YIG in the array (see Fig. 1). To derive an effective Hamiltonian involving only magnon modes, the elimination of cavity modes is necessary. This can be accomplished by invoking the Schrieffer-Wolf (Frohlich-Nakajima) transforma- tion [76, 77, 83, 84]. [S,H 0] =−HI. (8) WhereH0is the free Hamiltonian consisting of cavity ( Hdiag), and magnon ( Hm) modes. In addition, HIis the interaction Hamiltonian between the cavity and magnon modes, as given in Eq. (7).Sis called a generator and it is anti-Hermitian in nature i.e.,S†=−S, and it is given by S=/summationdisplay k,n′/bracketleftig αn′ kˆakˆm† n′−αn′ k⋆ˆa† kˆmn′/bracketrightig , (9) hereαn′ k=gn′/(√ 2N+ 1(ωn′−ωk))e−ikln′. The effec- tive Hamiltonian of the system can be determined by uni- tary transformation eSHe−S. In the dispersive regime, such thatωn′−ωk≫gn′/√ 2N+ 1, we can ignore higher or- der terms in the expansion of the unitary transformation, i.e., H0+1/2[HI,S], and keep terms up to the second order in αn′ k. Consequently, the approximate effective Hamiltonian, compris- ing solely of magnon modes, can be explicitly expressed as4 follows [65, 66, 85, 86]: Heff/ℏ=/summationdisplay nω′ nˆm† nˆmn+/summationdisplay n,n′Gnn′( ˆm† nˆmn′+ ˆmnˆm† n′). (10) Where we have replaced discrete modes with a continuous distribution 1 2N+ 1/summationdisplay k=1 2π/integraldisplayπ −πdk, (11) in addition, the following integral identity is employed in the derivation of Eq. (10) /integraldisplayπ −πdk 2πe−ilx U+Vcosx= (−1)|l|/radicalbigg 1 U2−V2e−|l|arccosh (U/V). (12) The effective frequency ω′ nof then-th magnon mode is given by ω′ n=ωn+g2 n/radicalbig ∆2n+ 4J∆n,and (13) Gnn′=gngn′(−1)|lnn ′| 2/parenleftbigge−|lnn ′|arccosh (1+∆n′/2J) /radicalbig ∆2 n′+ 4J∆n′+ e−|lnn ′|arccosh (1+∆ n/2J) /radicalbig ∆2n+ 4J∆n/parenrightbigg , (14) is spatially dependent effective coupling strength between the magnon modes. Further, ∆n′= (ωn′−δa)2+ 4J(ωn′−δa), ∆n= (ωn−δa)2+ 4J(ωn−δa), andδa=ωa−2Jis the lower bound frequency of the cavity mode. In addition, lnn′=ln−ln′is the distance between n-th andn′-th YIG placed inside the cavity array. In our numerical simulations, we consider identical magnon modes: Since all the YIGs oscillate inside the cavity with the same frequency, therefore, resonance condition ∆n′=∆n= ∆ , andδn′=ωn′−ωaandδn=ωn− ωa=δ. The effective Hamiltonian in Eq. (10) takes on the form of a beam splitter for magnon modes, which can be used for state transfer between distant magnon modes, as discussed in the previous studies [ 65,66]. We note that, in our case, the interaction between magnon modes is mediated by virtual photons; and when contrasted with actual photons, virtual photons possess the advantages of being non-propagating and non-radiative. Consequently, energy exchange facilitated by virtual photons results in entirely coherent dynamics devoid of dissipation [65]. IV . RESULTS To illustrate the working principles of our scheme, we ini- tially focus on a simple scenario where only three cavities are occupied, each containing one YIG. The remaining cavities in the array are unoccupied. It is worth noting that the place- ment of YIGs is not restricted to neighboring cavities; they can be situated in any of the three cavities within the array. Inaddition, we consider a squeezed thermal reservoir coupled to the central YIG. It is worth noting that this entanglement generation scheme remains applicable for arrays with arbitrary cavities lengths [68]. We employ the quantum Langevin equations to describe the system’s dynamics. By working in the interaction picture through the unitary evolution operator ˆU(t) = exp[−it(/summationtext nω′ nˆm† nˆmn)], the equations of motion take the following form (for convenience, we will now omit the hat symbol from operators) ˙m1=−κ1m1−iG12m2−iG13m3+/radicalbig 2κ1min 1, ˙m2=−κ2m2−iG12m1−iG23m3+√2κ2min 2, ˙m3=−κ3m3−iG13m1−iG23m2+/radicalbig 2κ3min 3.(15) Here,κnis the dissipation rate of the n-th magnon modes, and min nrepresents the corresponding input noise operator. The in- put noise operator min 2accounts for driving the central magnon mode through a squeezed vacuum field. It is characterized by zero mean and following correlation functions [87] ⟨min 2(t)min† 2(t′)⟩=(N+ 1)δ(t−t′), ⟨min† 2(t)min 2(t′)⟩=Nδ(t−t′), ⟨min 2(t)min 2(t′)⟩=Mδ(t−t′), ⟨min† 2(t)min† 2(t′)⟩=M∗δ(t−t′). (16) HereN= sinh2r+ ¯n2(sinh2r+ cosh2r)andM= eiθsinhrcoshr(1 + 2¯n2)withθandrbeing the phase and squeezing parameter of the input squeezed reservoir, respec- tively. In addition, NandMaccount for the number of exci- tations and correlations, respectively. We observe that due to the Schrieffer-Wolf transformation as described in Eq. (9), ad- ditional terms are introduced into the bath correlation function. Nonetheless, the alterations induced by this transformation in the correlation function are deemed negligible. Among these parameters,Mplays the most influential role in entanglement generation. The equilibrium mean thermal magnon number can be determined by ¯n2(ω2) = [exp( ℏω2/KBT2)−1]−1. The input noise operators for the other two magnon modes are characterized by the following correlation functions ⟨min α(t)min† α(t′)⟩=(¯nα+ 1)δ(t−t′), ⟨min† α(t)min α(t′)⟩=¯nαδ(t−t′), (17) hereα= 1,3, and ¯nα= [exp( ℏωα/KBTα)−1]−1. In our scheme, we showcase the possibility of generating entanglement between all potential bipartitions of the magnon modes by driving the central magnon mode with an input quantum-squeezed field. The quadratures for both the magnon modes and the input noise operators are defined as follows xn= (mn+m† n)/√ 2,yn= (mn−m† n)/√ 2i, andxin n= (min n+min† n)/√ 2,yn= (min n−min† n)/√ 2i, respectively. In terms of quadrature fluctuations, the quantum Langevin equation (15) can be rewritten as ˙F(t) =F(t)A+N(t), (18)5 0 0.1 0.2 0.300.050.10.15 T(K)ℰ Figure 2. Magnon-magnon bipartite entanglement as a function of the environment temperature T. The blue solid line represents the logarithmic negativity E1,2, betweenm1andm2modes, while the black dashed line indicates the logarithmic negativity E2,3between m2andm3modes. Further, the red solid line shows the logarithmic negativity E1,3betweenm1andm3modes. Parameters: ωn/2π= 10 GHz,gj/2π= 10 MHz,J/2π= 12 MHz,κn/2π=5MHz, and r= 1. whereF(t)=[x1(t),y1(t),x2(t),y2(t),x3(t),y3(t)]T, and N(t)=[xin 1(t),yin 1(t),xin 2(t),yin 2(t),xin 3(t),yin 3(t)]Tdenote the quadrature fluctuation vectors of magnon and input noise operators, respectively. The drift matrix Fis given by F= −κ10 0G12 0G13 0−κ1−G12 0−G13 0 0G12−κ20 0G23 −G12 0 0−κ2−G23 0 0G13 0G23−κ30 −G13 0−G23 0 0−κ3 . (19) As the effective Hamiltonian of the magnon modes, as given in Eq. (10), is quadratic, and the input quantum noise is Gaus- sian, as a result, the state of the system also remains Gaussian. The reduced state, consisting of three magnon modes, forms a continuous variable three-mode Gaussian state. This state can be entirely characterized by a 6×6covariance matrix V, which is expressed as Vij=1 2⟨Fi(t)Fj(t) +Fj(t)Fi(t)⟩i,j= 1,2,..., 6. (20) The steady-state solution can be obtained by solving the Lya- punov equation FV+VFT=−D (21) whereDis the diffusion matrix and can be derived from the noise correlation matrix; ⟨Ni(t)Nj(t) +Nj(t)Ni(t)⟩/2 = Dijδ((t−t′). It can be written as a direct sum of D=D1⊕ D2⊕D3, withDα=diag[κα(2¯nα+ 1),κα(2¯nα+ 1)] , and D2=/parenleftbigg κ2U1iκ2(M∗−M ) iκ2(M∗−M )κ2U2/parenrightbigg . (22) 0 0.2 0.4 0.6 0.8 1 1.200.050.10.150.2 rℰFigure 3. Magnon-magnon bipartite entanglement Eas a function of squeezing parameter r. The solid blue line illustrates the logarithmic negativity E1,2characterizing the entanglement between m1andm2 magnon modes. The red dashed curve showcases the logarithmic negativity E2,3representing the entanglement between the m2andm3 modes. Furthermore, the blue dot-dashed line shows the logarithmic negativity E1,3betweenm1andm3considering the same parameters as those specified in Fig. 2. whereU1= (2N+ 1 +M+M∗), andU2= (2N+ 1− M−M∗). A. Bipartite entanglement To investigate the entanglement between the magnon modes, we compute the logarithmic negativity EN. This quantity has previously been proposed as a measure of entanglement and helps establish the conditions under which the two modes are entangled [ 88]. In the continuous variable case, the logarithmic negativity is defined as [89] E=max[0,−ln 2V−], (23) whereV−=/radicalbig 1/2(Σ−(Σ2−4 detV′)1/2with Σ = detA+ detB−2 detC.V′is a4×4matrix obtained from steady-state covariance matrix Vby removing the two rows and associated columns related to the traced-out magnon mode. The reduced covariance matrix V′is given by V′=/parenleftbiggA C CTB/parenrightbigg . We compute the bipartite entanglement among all possible pairs of magnon modes by employing the logarithmic neg- ativityEN. The analytical solution of equation (21) is too cumbersome and we don’t report it here. Instead, we exten- sively examine the logarithmic negativity across various system parameters. For numerical evaluations, we employ experimen- tally attainable parameters reported in recent studies [ 55,90– 93]. We consider the magnon density at low temperature with ground state spin s= 5/2of theFe+ 3ion in the YIG sphere. The total number of spins N=ρVwithρ= 4.22×1027/m36 characterizing the density of each YIG and Vis the volume of each sphere with diameter 250 µ-meter; this results in the total number of spins N= 3.5×106in each YIG. In Fig. 2, we present a plot illustrating bipartite entan- glement, quantified using the logarithmic negativity, among all possible pairs of magnon modes within an effective sys- tem comprising three magnon modes. These results are de- rived using a set of experimentally feasible parameters [ 43]: ωn/2π= 10 GHz,gj/2π= 10 MHz,J/2π= 12 MHz, ωc/2π= 10 GHz,κ1/2π=κ3/2π=5MHz,r= 1, and the phase angle θ= 0. The entanglement is robust against tem- perature and survives up to about T= 0.2K (solid blue line and dashed black line), and T= 0.15K (red solid line). The couplingG13between YIG1 and YIG3 is relatively weak as compared toG12orG23due to their farthest location and as a re- sult, YIG1 and YIG3 are relatively weakly entangled described by red line in Fig. 2. However, the more pronounced squeezing parameterrwould result in a more robust entanglement against the environment temperature. Here, we consider a reasonable value of the squeezing parameter r= 1. The logarithmic neg- ativityE1,2between YIG1 and YIG2 and E2,3between YIG2 and YIG3 are the same due to the same couplings; G12=G23 represented by solid blue and black dashed lines. The quan- tum squeezed reservoir coupled to the central magnon mode is responsible for entanglement generation between the magnon modes. The degree of entanglement decreases by reducing the squeezing parameter rand dies out in the absence of the squeezed reservoir. We observe that the difference in the order of magnon decays at resonance frequencies can produce sig- nificant changes in the amount of steady-state entanglement between the two magnon modes. Each magnon-pair has a state-swap interaction and the squeezing can be transferred from a squeezed magnon state to another magnon state. As a result, each magnon pair is to be entangled due to swap-state type interaction via squeezing. This implies that squeezing can affect the bipartite entanglement generated in between a pair of the magnon modes. To observe the impact of squeezing on entanglement gen- eration, we plot the bipartite entanglement, quantified using logarithmic negativity ( E), as a function of the squeezing pa- rameter (r). This can be observed in Fig. 3. The results indi- cate that without the presence of the squeezed thermal bath, entanglement generation within our scheme is unattainable. Therefore, it is evident that squeezing plays a pivotal role in the generation of entanglement, with logarithmic negativity exhibiting a notable increase as the squeezing parameter ( r) is raised. The entanglement between the magnon modes m1-m2 andm2-m3is identical, owing to the equal coupling strength between these magnon pairs. However, the magnon modes m1-m3are relatively weakly coupled, primarily because of the greater distance between the associated YIGs. Consequently, the entanglementEm1,m3between modes m1-m3is lower in comparison to the other two pairs. Figure 4. Two-dimensional plot for tripartite entanglement, quantified by minimal residual contangle Ras a function of bath temperature T, and squeezing parameter r. The rest of the system parameters are the same as given in Fig. 2. B. Tripartite entanglement We further investigate the possibility of the generation of the distant magnon-magnon tripartite entanglement in the array as shown in Fig. 1. To this end, we employ the minimal residual contangle given by [94, 95] Ri|jk=Ci|jk−Ci|j−Ci|k. (24) Here, the contangle of subsystems xandyis denoted by Cx|y, andymay represent more than one mode. Cx|yis a proper entanglement monotone, and it is determined by the square of the logarithmic negativity between the respective modes, which is given by Ei|jk=max[0,−ln 2Vi|jk], (25) whereVi|jk=min/vextendsingle/vextendsingleeigiΩ3˜V/vextendsingle/vextendsingleis the smallest symplectic eigenvalues. Ω3is the symplectic matrix Ω3=⊕3 j=1iσy, withσybeing they-Pauli matrix and ⊕symbol describes the direct sum of the σymatrices. The 6×6covariance matrix ˜Vis obtained by inverting the momentum quadra- ture of one of the magnon modes. The transformed covari- ance matrix ˜Vis determined by ˜V=Pi|jkVPi|jkwhere P1|23=diag[1,−1,1,1,1,1],P2|13=diag[1,1,1,−1,1,1], andP3|12=diag[1,1,1,1,1,−1]are partial transposition di- agonal matrices. The steady state of the magnon modes can be fully characterized by the 6×6covariance matrix because of its Gaussian nature. The tripartite entanglement for Gaus- sian states can be determined by minimum residual cotangle [94, 95] Rmin=min[R1|23,R2|13,R3|12]. (26) This ensures that the tripartite entanglement remains un- changed regardless of how the modes are permuted. The7 density plot of magnon-magnon tripartite entanglement de- termined via minimum residual cotangle Rminis shown in Fig. 4. It is evident from the results that considerable tripartite entanglement can be generated between magnon modes when the central magnon mode is coupled to a squeezed reservoir. Fig. 4 shows that the degree of entanglement is significantly enhanced with the increase in the squeezing parameter r. V . CONCLUSION In summary, we have shown that steady-state bipartite and tripartite entanglement can be generated between distant magnon modes when only one of these magnon modes is cou- pled to a quantum-squeezed reservoir. In particular, we con- sidered an array of Ncavities each of which houses a single YIG sphere, and the central YIG is coupled to a single-mode squeezed vacuum bath. We have demonstrated that steady- state bipartite and tripartite entanglement can be generated among magnon modes hosted by YIGs when some of the cav- ities, up to five, are occupied. In contrast to prior proposals for magnon-magnon entanglement generation [ 56–60,62–65], both bipartite and tripartite entanglement are possible in our scheme. Further, in principle, our scheme can be extended to an arbitrary number of YIGs within the cavity array [ 68,81]. However, achieving entanglement across the entire array may necessitate negligible or extremely small magnon decay rates. To address this challenge, we propose the use of an external squeezed drive on each cavity in the array, enabling strong long-range interactions between YIGs positioned at greater dis- tances within the array [ 86]. In our scheme, the entanglement generation does not require nonlinearity, instead, entanglement is generated via a squeezed reservoir and its strength dependson the squeezing parameter r. The generated entanglement is robust against the environment temperature provided the magnons dissipation rates are sufficiently low. It may be in- teresting to look for multipartite entangled states of magnons, which is left for future works. Our work may be useful in designing quantum networks based on cavity-magnon systems [67]. Appendix A: Bipartite entanglement with five YIGs Here, we extend our results for five YIGs placed inside the cavities array, and the squeezed reservoir is coupled to the central YIG. The Langevin equations of motion in this case are given by ˙m1=−κ1m1−i5/summationdisplay n′=2G1n′mn′+√2κ1m1in, ˙m2=−κ2m2−iG12m1−i5/summationdisplay n′=3G2n′mn′+√2κ2m2in, ˙m3=−κ3m3−i2/summationdisplay n=1Gn3m3−i5/summationdisplay n′=4G3n′mn′+√2κ3m3in, ˙m4=−κ4m4−i3/summationdisplay n=1Gn4mn−iG45m5+√2κ4m4in, ˙m5=−κ5m5−i4/summationdisplay n=1Gn5m5+√2κ5m5in. (A1) Again we write the Langevin equations in matrix form as equation (18). The A matrix is now a 10×10matrix which can be described as A= −κ10 0G12 0G13 0G14 0G15 0−κ1−G12 0−G13 0−G14 0−G15 0 0G12−κ20 0G23 0G24 0G25 −G12 0 0−κ2−G23 0−G24 0−G25 0 0G13 0G23−κ30 0G34 0G35 −G13 0−G23 0 0−κ3−G34 0−G35 0 0G14 0G24 0G34−κ40 0G45 −G14 0−G24 0−G34 0 0−κ4−G45 0 0G15 0G25 0G35 0G45−κ50 −G15 0−G25 0−G35 0−G45 0 0−κ5 . (A2) WithF(t)=[(xi(t),yi(t)]TandN(t)=[xiin(t),yiin(t)]T i=1,2,3,. . . , 10. Since the squeezed reservoir for a 5 YIG sam- ple is taken on YIG3, this shifts the frequency of each magnon mode by the amount of the frequency i.e., ∆′ j=ω′ j−ω0after applying the rotating wave approximation corresponding to each magnon mode. We can write equation (A2) for N magnon modes in general m×nmatrix form in the interaction picture can be expressed asQnn′=/parenleftbigg KnGnn′ Gnn′Kn/parenrightbigg where,Kn=/parenleftbigg −κn0 0−κn/parenrightbigg , andGij=/parenleftbigg 0Gnn′ −Gnn′0/parenrightbigg for n=1 . . . N, and n′= 2,. . . are the sub-block 2×2matrices of8 0 0.1 0.20.000.050.100.15 T(K)ℰ (a) 0 0.1 0.20.0000.0050.0100.0150.0200.025 T(K)ℰ (b) Figure 5. (Color online). In the given panel, the pairs of bipartite entanglement for a 5-YIG sample are shown. (a) The black solid, blue dashed, and red dotdashed lines indicate the entanglement between m1andm4modes,m2andm4, andm3andm4, respectively. (b) Describes the bipartite entanglement between m1andm5(solid black curve), m2andm5(red dashed curve), m3andm5blue dotdashed Curve. The parameters remain the same as used in Fig. 3 n-th YIG decays with corresponding zero detuning frequency in the interaction picture, and the effective spatially dependent couplingGnn′as discussed earlier.Our model can be extended to 2N+ 1cavities with a squeezed reservoir coupled to the central YIG and can be manifested as a quantum network. [1]F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, Reports on Progress in Physics 82, 016001 (2018). [2]T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, Quantum cryptography with entangled photons, Phys. Rev. Lett. 84, 4729 (2000). [3]Y .-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y . Lu, A. Zeilinger, and J.-W. Pan, Quantum teleportation in high dimensions, Phys. Rev. Lett. 123, 070505 (2019). [4]R. Demkowicz-Dobrza ´nski and L. Maccone, Using entangle- ment against noise in quantum metrology, Phys. Rev. Lett. 113, 250801 (2014). [5]S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513 (2005). [6]B. Julsgaard, A. Kozhekin, and E. S. Polzik, Experimental long- lived entanglement of two macroscopic objects, Nature 413, 400 (2001). [7]C. W. Chou, H. de Riedmatten, D. Felinto, S. V . Polyakov, S. J. van Enk, and H. J. Kimble, Measurement-induced entanglement for excitation stored in remote atomic ensembles, Nature 438, 828 (2005). [8]R. McConnell, H. Zhang, J. Hu, S. ´Cuk, and V . Vuleti ´c, Entan- glement with negative wigner function of almost 3,000 atoms heralded by one photon, Nature 519, 439 (2015). [9]T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehn- ert, Entangling mechanical motion with microwave fields, Sci- ence 342, 710 (2013). [10] C. F. Ockeloen-Korppi, E. Damsk ¨agg, J.-M. Pirkkalainen, M. As- jad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanp ¨a¨a, Stabilized entanglement of massive mechanical oscillators, Na- ture556, 478 (2018).[11] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. B ¨uchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nat. Phys. 4, 878 (2008). [12] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Quantum com- putation and quantum-state engineering driven by dissipation, Nat. Phys. 5, 633 (2009). [13] H. Weimer, M. M ¨uller, I. Lesanovsky, P. Zoller, and H. P. B¨uchler, A rydberg quantum simulator, Nat. Phys. 6, 382 (2010). [14] J. T. Barreiro, M. M ¨uller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, An open-system quantum simulator with trapped ions, Nature 470, 486 (2011). [15] J. Cho, S. Bose, and M. S. Kim, Optical pumping into many- body entanglement, Phys. Rev. Lett. 106, 020504 (2011). [16] G. Morigi, J. Eschner, C. Cormick, Y . Lin, D. Leibfried, and D. J. Wineland, Dissipative quantum control of a spin chain, Phys. Rev. Lett. 115, 200502 (2015). [17] F. Reiter, D. Reeb, and A. S. Sørensen, Scalable dissipative preparation of many-body entanglement, Phys. Rev. Lett. 117, 040501 (2016). [18] J. F. Poyatos, J. I. Cirac, and P. Zoller, Quantum reservoir engi- neering with laser cooled trapped ions, Phys. Rev. Lett. 77, 4728 (1996). [19] A. R. R. Carvalho, P. Milman, R. L. de Matos Filho, and L. Davi- dovich, Decoherence, pointer engineering, and quantum state protection, Phys. Rev. Lett. 86, 4988 (2001). [20] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. Petersen, J. I. Cirac, and E. S. Polzik, Entanglement generated by dissipation and steady state entanglement of two macroscopic objects, Phys. Rev. Lett. 107, 080503 (2011). [21] S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. V ool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H.9 Devoret, Autonomously stabilized entanglement between two superconducting quantum bits, Nature 504, 419 (2013). [22] D. Kienzler, H.-Y . Lo, B. Keitch, L. de Clercq, F. Leupold, F. Lindenfelser, M. Marinelli, V . Negnevitsky, and J. P. Home, Quantum harmonic oscillator state synthesis by reservoir engi- neering, Science 347, 53 (2015). [23] J.-M. Pirkkalainen, E. Damsk ¨agg, M. Brandt, F. Massel, and M. A. Sillanp ¨a¨a, Squeezing of quantum noise of motion in a micromechanical resonator, Phys. Rev. Lett. 115, 243601 (2015). [24] M. T. Naseem and ¨O. E. M ¨ustecaplıo ˘glu, Ground-state cooling of mechanical resonators by quantum reservoir engineering, Commun. Phys. 4, 95 (2021). [25] M. T. Naseem and ¨Ozg¨ur E M ¨ustecaplıo ˘glu, Engineering entan- glement between resonators by hot environment, Quantum Sci. Technol. 7, 045012 (2022). [26] J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller, “dark” squeezed states of the motion of a trapped ion, Phys. Rev. Lett. 70, 556 (1993). [27] A. Kronwald, F. Marquardt, and A. A. Clerk, Arbitrarily large steady-state bosonic squeezing via dissipation, Phys. Rev. A 88, 063833 (2013). [28] Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Quantum magnonics: The magnon meets the superconducting qubit, Comptes Rendus Physique 17, 729 (2016), quantum microwaves / Micro-ondes quantiques. [29] D. Lachance-Quirion, Y . Tabuchi, A. Gloppe, K. Usami, and Y . Nakamura, Hybrid quantum systems based on magnonics, Appl. Phys. Express 12, 070101 (2019). [30] H. Yuan, Y . Cao, A. Kamra, R. A. Duine, and P. Yan, Quantum magnonics: When magnon spintronics meets quantum infor- mation science, Phys. Rep. 965, 1 (2022), quantum magnonics: When magnon spintronics meets quantum information science. [31] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons, Phys. Rev. Lett. 113, 156401 (2014). [32] L. Bai, M. Harder, Y . P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Spin pumping in electrodynamically coupled magnon-photon systems, Phys. Rev. Lett. 114, 227201 (2015). [33] J. Bourhill, N. Kostylev, M. Goryachev, D. L. Creedon, and M. E. Tobar, Ultrahigh cooperativity interactions between magnons and resonant photons in a yig sphere, Phys. Rev. B 93, 144420 (2016). [34] N. Kostylev, M. Goryachev, and M. E. Tobar, Superstrong cou- pling of a microwave cavity to yttrium iron garnet magnons, Ap- plied Physics Letters 108, 10.1063/1.4941730 (2016), 062402. [35] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Optomagnonic whispering gallery microresonators, Phys. Rev. Lett. 117, 123605 (2016). [36] J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Fergu- son, Triple-resonant brillouin light scattering in magneto-optical cavities, Phys. Rev. Lett. 117, 133602 (2016). [37] A. Osada, R. Hisatomi, A. Noguchi, Y . Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y . Nakamura, Cavity optomagnonics with spin-orbit coupled photons, Phys. Rev. Lett. 116, 223601 (2016). [38] J. T. Hou and L. Liu, Strong coupling between microwave pho- tons and nanomagnet magnons, Phys. Rev. Lett. 123, 107702 (2019). [39] Y . Li, T. Polakovic, Y .-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. Xiao, V . Novosad, A. Hoffmann, and W. Zhang, Strong coupling between magnons and microwave photons in on-chip ferromagnet-superconductor thin-film devices, Phys. Rev. Lett. 123, 107701 (2019).[40] I. A. Golovchanskiy, N. N. Abramov, V . S. Stolyarov, M. Wei- des, V . V . Ryazanov, A. A. Golubov, A. V . Ustinov, and M. Y . Kupriyanov, Ultrastrong photon-to-magnon coupling in multi- layered heterostructures involving superconducting coherence via ferromagnetic layers, Science Advances 7, eabe8638 (2021). [41] Y .-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. Q. You, Magnon kerr effect in a strongly coupled cavity-magnon system, Phys. Rev. B 94, 224410 (2016). [42] D. Zhang, X.-Q. Luo, Y .-P. Wang, T.-F. Li, and J. Q. You, Obser- vation of the exceptional point in cavity magnon-polaritons, Nat. Commun. 8, 1368 (2017). [43] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity mag- nomechanics, Science Advances 2, e1501286 (2016). [44] K. Ullah, M. T. Naseem, and O. E. M ¨ustecaplıo ˘glu, Tunable multiwindow magnomechanically induced transparency, fano resonances, and slow-to-fast light conversion, Phys. Rev. A 102, 033721 (2020). [45] Y .-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, Bistability of cavity magnon polaritons, Phys. Rev. Lett. 120, 057202 (2018). [46] G.-Q. Zhang and J. Q. You, Higher-order exceptional point in a cavity magnonics system, Phys. Rev. B 99, 054404 (2019). [47] J.-k. Xie, S.-l. Ma, and F.-l. Li, Quantum-interference-enhanced magnon blockade in an yttrium-iron-garnet sphere coupled to superconducting circuits, Phys. Rev. A 101, 042331 (2020). [48] C. Kong, H. Xiong, and Y . Wu, Magnon-induced nonreciprocity based on the magnon kerr effect, Phys. Rev. Appl. 12, 034001 (2019). [49] Z.-X. Liu, H. Xiong, and Y . Wu, Magnon blockade in a hybrid ferromagnet-superconductor quantum system, Phys. Rev. B 100, 134421 (2019). [50] J. Li, S.-Y . Zhu, and G. S. Agarwal, Squeezed states of magnons and phonons in cavity magnomechanics, Phys. Rev. A 99, 021801 (2019). [51] S. Sharma, V . A. S. V . Bittencourt, A. D. Karenowska, and S. V . Kusminskiy, Spin cat states in ferromagnetic insulators, Phys. Rev. B 103, L100403 (2021). [52] F.-X. Sun, S.-S. Zheng, Y . Xiao, Q. Gong, Q. He, and K. Xia, Remote generation of magnon schr ¨odinger cat state via magnon- photon entanglement, Phys. Rev. Lett. 127, 087203 (2021). [53] H. Y . Yuan, P. Yan, S. Zheng, Q. Y . He, K. Xia, and M.-H. Yung, Steady bell state generation via magnon-photon coupling, Phys. Rev. Lett. 124, 053602 (2020). [54] Z.-B. Yang, H. Jin, J.-W. Jin, J.-Y . Liu, H.-Y . Liu, and R.-C. Yang, Bistability of squeezing and entanglement in cavity magnonics, Phys. Rev. Res. 3, 023126 (2021). [55] J. Li, S.-Y . Zhu, and G. S. Agarwal, Magnon-photon-phonon entanglement in cavity magnomechanics, Phys. Rev. Lett. 121, 203601 (2018). [56] D.-W. Luo, X.-F. Qian, and T. Yu, Nonlocal magnon entangle- ment generation in coupled hybrid cavity systems, Opt. Lett. 46, 1073 (2021). [57] Z. Zhang, M. O. Scully, and G. S. Agarwal, Quantum entangle- ment between two magnon modes via kerr nonlinearity driven far from equilibrium, Phys. Rev. Res. 1, 023021 (2019). [58] J. Li and S.-Y . Zhu, Entangling two magnon modes via magne- tostrictive interaction, New J. Phys. 21, 085001 (2019). [59] J. M. P. Nair and G. S. Agarwal, Deterministic quantum entan- glement between macroscopic ferrite samples, Applied Physics Letters 117, 10.1063/5.0015195 (2020). [60] Q. Zheng, W. Zhong, G. Cheng, and A. Chen, Genuine magnon– photon–magnon tripartite entanglement in a cavity electro- magnonical system based on squeezed-reservoir engineering,10 Quantum Information Processing 22, 140 (2023). [61] J. Xie, H. Yuan, S. Ma, S. Gao, F. Li, and R. A. Duine, Sta- tionary quantum entanglement and steering between two distant macromagnets, Quantum Science and Technology 8, 035022 (2023). [62] D. Zhao, W. Zhong, G. Cheng, and A. Chen, Controllable magnon–magnon entanglement and one-way epr steering with two cascaded cavities, Quantum Inf. Processing 21, 384 (2022). [63] M. Yu, S.-Y . Zhu, and J. Li, Macroscopic entanglement of two magnon modes via quantum correlated microwave fields, J. Phys. B: At. Mol. Opt. Phys. 53, 065402 (2020). [64] W.-J. Wu, Y .-P. Wang, J.-Z. Wu, J. Li, and J. Q. You, Re- mote magnon entanglement between two massive ferrimagnetic spheres via cavity optomagnonics, Phys. Rev. A 104, 023711 (2021). [65] Y .-l. Ren, J.-k. Xie, X.-k. Li, S.-l. Ma, and F.-l. Li, Long-range generation of a magnon-magnon entangled state, Phys. Rev. B 105, 094422 (2022). [66] J.-Y . Liu, J.-W. Jin, X.-M. Zhang, Z.-G. Zheng, H.-Y . Liu, Y . Ming, and R.-C. Yang, Distant entanglement generation and controllable information transfer via magnon–waveguide sys- tems, Results in Physics 52, 106854 (2023). [67] J. Li, Y .-P. Wang, W.-J. Wu, S.-Y . Zhu, and J. You, Quantum network with magnonic and mechanical nodes, PRX Quantum 2, 040344 (2021). [68] S. Zippilli, J. Li, and D. Vitali, Steady-state nested entangle- ment structures in harmonic chains with single-site squeezing manipulation, Phys. Rev. A 92, 032319 (2015). [69] A. Bienfait, P. Campagne-Ibarcq, A. H. Kiilerich, X. Zhou, S. Probst, J. J. Pla, T. Schenkel, D. Vion, D. Esteve, J. J. L. Morton, K. Moelmer, and P. Bertet, Magnetic resonance with squeezed microwaves, Phys. Rev. X 7, 041011 (2017). [70] Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Coherent coupling between a fer- romagnetic magnon and a superconducting qubit, Science 349, 405 (2015). [71] D. Porras and J. J. Garc ´ıa-Ripoll, Shaping an itinerant quantum field into a multimode squeezed vacuum by dissipation, Phys. Rev. Lett. 108, 043602 (2012). [72] W. Zhang, D.-Y . Wang, C.-H. Bai, T. Wang, S. Zhang, and H.-F. Wang, Generation and transfer of squeezed states in a cavity magnomechanical system by two-tone microwave fields, Opt. Express 29, 11773 (2021). [73] J. Li, Y .-P. Wang, J.-Q. You, and S.-Y . Zhu, Squeez- ing microwaves by magnetostriction, National Science Re- view 10, nwac247 (2022), https://academic.oup.com/nsr/article- pdf/10/5/nwac247/50426152/nwac247.pdf. [74] C. Kittel, On the theory of ferromagnetic resonance absorption, Phys. Rev. 73, 155 (1948). [75] J. Zhao, Y . Liu, L. Wu, C.-K. Duan, Y .-x. Liu, and J. Du, Ob- servation of anti- PT-symmetry phase transition in the magnon- cavity-magnon coupled system, Phys. Rev. Appl. 13, 014053 (2020). [76] H. Fr ¨ohlich, Theory of the superconducting state. i. the ground state at the absolute zero of temperature, Phys. Rev. 79, 845(1950). [77] S. Nakajima, Perturbation theory in statistical mechanics, Ad- vances in Physics 4, 363 (1955). [78] S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer–wolff trans- formation for quantum many-body systems, Annals of Physics 326, 2793 (2011). [79] S. Ma, M. J. Woolley, I. R. Petersen, and N. Yamamoto, Pure gaussian states from quantum harmonic oscillator chains with a single local dissipative process, J. Phys. A: Math. Theor. 50, 135301 (2017). [80] Y . Yanay and A. A. Clerk, Reservoir engineering with localized dissipation: Dynamics and prethermalization, Phys. Rev. Res. 2, 023177 (2020). [81] S. Zippilli and D. Vitali, Dissipative engineering of gaussian entangled states in harmonic lattices with a single-site squeezed reservoir, Phys. Rev. Lett. 126, 020402 (2021). [82] J. Angeletti, S. Zippilli, and D. Vitali, Dissipative stabilization of entangled qubit pairs in quantum arrays with a single localized dissipative channel, Quantum Sci. Technol. 8, 035020 (2023). [83] J. M. Luttinger and W. Kohn, Motion of electrons and holes in perturbed periodic fields, Phys. Rev. 97, 869 (1955). [84] J. R. Schrieffer and P. A. Wolff, Relation between the anderson and kondo hamiltonians, Phys. Rev. 149, 491 (1966). [85] C. Aron, M. Kulkarni, and H. E. T ¨ureci, Photon-mediated inter- actions: A scalable tool to create and sustain entangled states of natoms, Phys. Rev. X 6, 011032 (2016). [86] Y .-l. Ren, S.-l. Ma, S. Zippilli, D. Vitali, and F.-l. Li, Enhancing strength and range of atom-atom interaction in a coupled-cavity array via parametric drives, Phys. Rev. A 108, 033717 (2023). [87] C. W. Gardiner, Inhibition of atomic phase decays by squeezed light: A direct effect of squeezing, Phys. Rev. Lett. 56, 1917 (1986). [88] G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002). [89] G. Adesso, A. Serafini, and F. Illuminati, Extremal entanglement and mixedness in continuous variable systems, Phys. Rev. A 70, 022318 (2004). [90] Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Hybridizing ferromagnetic magnons and mi- crowave photons in the quantum limit, Phys. Rev. Lett. 113, 083603 (2014). [91] M. E. Kimchi-Schwartz, L. Martin, E. Flurin, C. Aron, M. Kulka- rni, H. E. Tureci, and I. Siddiqi, Stabilizing entanglement via symmetry-selective bath engineering in superconducting qubits, Phys. Rev. Lett. 116, 240503 (2016). [92] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity mag- nomechanics, Science advances 2, e1501286 (2016). [93] R.-C. Shen, J. Li, Z.-Y . Fan, Y .-P. Wang, and J. Q. You, Mechan- ical bistability in kerr-modified cavity magnomechanics, Phys. Rev. Lett. 129, 123601 (2022). [94] G. Adesso and F. Illuminati, Continuous variable tangle, monogamy inequality, and entanglement sharing in gaussian states of continuous variable systems, New J. Phys. 8, 15 (2006). [95] G. Adesso and F. Illuminati, Entanglement in continuous- variable systems: recent advances and current perspectives, J. Phys. A: Math. Theor. 40, 7821 (2007).

2023-08-25

The generation of robust entanglement in quantum system arrays is a crucial aspect of the realization of efficient quantum information processing. Recently, the field of quantum magnonics has garnered significant attention as a promising platform for advancing in this direction. In our proposed scheme, we utilize a one-dimensional array of coupled cavities, with each cavity housing a single yttrium iron garnet (YIG) sphere coupled to the cavity mode through magnetic dipole interaction. To induce entanglement between YIGs, we employ a local squeezed reservoir, which provides the necessary nonlinearity for entanglement generation. Our results demonstrate the successful generation of bipartite and tripartite entanglement between distant magnon modes, all achieved through a single quantum reservoir. Furthermore, the steady-state entanglement between magnon modes is robust against magnon dissipation rates and environment temperature. Our results may lead to applications of cavity-magnon arrays in quantum information processing and quantum communication systems.

Macroscopic distant magnon modes entanglement via a squeezed reservoir

2308.13586v3

Probing magnon-magnon coupling in exchange coupled Y3Fe5O12/Permalloy bilayers with magneto-optical eects Yuzan Xiong,1, 2Yi Li,3,a)Mouhamad Hammami,1Rao Bidthanapally,1Joseph Sklenar,4 Xufeng Zhang,5Hongwei Qu,2Gopalan Srinivasan,1John Pearson,3Axel Homann,6, 3 Valentine Novosad,3and Wei Zhang1, 3,b) 1)Department of Physics, Oakland University, Rochester, MI 48309, USA 2)Department of Electronic and Computer Engineering, Oakland University, Rochester, MI 48309, USA 3)Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA 4)Department of Physics and Astronomy, Wayne State University, Detroit, MI 48201, USA 5)Center for Nanoscale Materials, Argonne National Laboratory, Argonne, IL 60439, USA 6)Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA (Dated: 13 July 2020) We demonstrate the magnetically-induced transparency (MIT) eect in Y3Fe5O12(YIG)/Permalloy(Py) coupled bilayers. The measurement is achieved via a heterodyne detection of the coupled magnetization dynamics using a single wavelength that probes the magneto-optical Kerr and Faraday eects of Py and YIG, respectively. Clear features of the MIT eect are evident from the deeply modulated ferromagnetic resonance of Py due to the perpendicular-standing-spin- wave of YIG. We develop a phenomenological model that nicely reproduces the experimental results including the induced amplitude and phase evolution caused by the magnon-magnon coupling. Our work oers a new route towards studying phase-resolved spin dynamics and hybrid magnonic systems. a)Electronic mail: yili@anl.gov b)Electronic mail: weizhang@oakland.edu 1arXiv:1912.13407v2 [cond-mat.mtrl-sci] 10 Jul 2020I. INTRODUCTION Hybrid magnonic systems are becoming rising contenders for coherent information processing1{3, owing to their capability of connecting distinct physical platforms in quantum systems as well as the rich emerging physics for new functionalities4{21. Magnons have been demonstrated to eciently couple to cavity quantum electrodynamics systems including superconducting resonators and qubits4{8; magnonic systems are therefore well-positioned for the next advances in quantum information. In addition, recent studies also revealed the potential of magnonic systems for microwave-optical transduction22{28, which are promising for combining quantum information, sensing, and transduction. To fully leverage the hybrid coupling phenomena with magnons, strong and tunable cou- plings between two magnonic systems have attracted considerable interests recently29{32. They can be considered as hosting hybrid magnonic modes in a \magnonic cavity" as op- posed to microwave photonic cavity in cavity-magnon polaritons (CMPs)1{3, which allows excitations of forbidden modes and high group velocity of spin waves owing to the state- of-the-art magnon bandgap engineering capabilities30,33. The detuning of the two magnonic systems can be easily engineered by the thickness of the thin lms, which set the wavenum- bers and the corresponding exchange eld. Furthermore, in such strongly coupled magnetic heterostructures, both magneto-optical Kerr and Faraday eects can be utilized for light modulation, in terms of light re ection by metals and/or transmission in insulators, re- spectively. In this architecture, the freedom of lateral dimensions is maintained for device fabrication and large-scale, on-chip integration. To date, both magnon-photon and magnon-magnon couplings are predominantly investi- gated by the cavity ferromagnetic resonance (FMR) spectroscopy, i.e. microwave transmis- sion and/or re ection measurements, typically involving a vector-network analyzer (VNA) or a microwave diode4,6{12,29{32,34. Strong magnon-magnon couplings have been observed in yttrium iron garnet (Y 3Fe5O12, YIG) coupled with ferromagnetic (FM) metals, where exchange spin waves were excited by a combined action of exchange, dampinglike, and/or eldlike torques that are localized at the interfaces29{32. In this work, we investigate the magnon-magnon coupling in YIG/Permalloy(Py) bilayers by a phase-resolved, heterodyne optical detection method. We reveal the coupled magnon modes in the regime exhibiting the magnetically-induced transparency (MIT) eect, i.e. the 2GGS HYIG Faraday effect Py Kerr effect(1)(2) (3)(4) hrf(a) (b)VO(µV)Py/YIGPy/SiO2/YIG 40 (c)Re[VO] Im[VO] |VO|FIG. 1. (a) Schematic illustration of the experimental setup. Modulated and linearly-polarized 1550-nm light enter the sample at a polarization angle (1); dynamic Faraday eect of the YIG causes the polarization to rotate (2); dynamic Kerr eect of the Py causes polarization to further rotate (3); the re ected light, upon the returning path, picks up again the Faraday eect and causes the polarization to further rotate (4), before entering light detection and analysis. The applied dc magnetic eld is parallel to the ground-signal-ground (G-S-G) lines of the CPW. (b) Example signal trace for YIG/Py (solid) and YIG/SiO 2/Py (dashed) measured at 5.85 GHz, showing the in-phase X(top) and quadrature Y(middle), and the total amplitude,p X2+Y2(bottom). (c) Plotting and the tting of the observed PSSW modes versus the resonance elds. magnetic analogy of electromagnetically induced transparency (EIT)10,35{39, akin to a spin- wave induced suppression of FMR. In the hybrid magnon-photon systems, the MIT eect arises when the coupling strength, g=2is larger than the photon dissipation rate p=2 but smaller than the magnon dissipation rate m=210. Under such a condition, the mode hybridization leads to an abrupt suppression of the microwave transmission at a certain frequency range. A transparency window, whose linewidth is determined by the low-loss mode, can be observed in the broad resonance of the other lossy mode. Such resonant transparency is controlled by an external magnetic eld. Our measurement is achieved via detecting the coupled magnetization dynamics of the insulating and metallic FMs using a single 1550-nm telecommunication wavelength. Unlike the ultrafast optical pump-probes26, the method herein is a continue-wave (cw), heterodyne technique in which the 1550-nm 3laser light is modulated at the FMR frequencies (in GHz range) simultaneously with the sample's excitation. This feature makes the method eectively an optical \lock-in" type measurement, akin to the electrical lock-in detection40{42. The phase information between the Py and YIG FMRs, as well as the YIG perpendicular standing spin waves (PSSWs) can be obtained by simultaneously analyzing both the Kerr and Faraday responses. II. SAMPLES AND MEASUREMENTS The commercial YIG lms (from MTI Corporation) used in this work are 3- m thick, single-sided grown on double-side-polished Gd 3Ga5O12(GGG) substrates via liquid phase epitaxy (LPE). The Py lms (t Py= 10 nm and 30 nm) were subsequently deposited on the YIG lms using magnetron sputtering following earlier recipes32. To ensure the strong coupling, we used in situ Ar gas rf-bias cleaning for 3 minutes, to clean the YIG surface before depositing the Py layer. Reference samples of GGG/YIG/SiO 2(3-nm)/Py(10-nm) GGG/YIG/Cu(3-nm)/Py(10-nm) were also prepared at the same growth condition. Figure 1(a) illustrates the measurement conguration. The modulated and linearly- polarized 1550-nm light passes through the transparent GGG substrates and detects the dynamic Faraday and Kerr signals upon their FMR excitation. As the light travels through the YIG bulk, the dynamic Faraday rotation due to the YIG FMR is picked up. Similarly, the dynamic Kerr rotation caused by the Py FMR is then picked up, when the light reaches the Py layer. The Py layer also serves as a mirror and re ects the laser light. Upon re ec- tion, the dynamic Faraday eect from the YIG is picked up again, making the eective YIG thickness 6- m, i.e. twice the lm thickness. It should be noted that the Faraday rotations for the incoming and returning light add up as opposed to cancel, due to the inversion of both the chirality of the Faraday rotation and the projection of the perpendicular magne- tization of YIG along the wavenumber direction, whose mechanism is akin to a commercial \Faraday rotator" often encountered in ber optics. The YIG/Py samples are chip- ipped on a coplanar waveguide (CPW) for microwave excitation and optical detection, as depicted in Fig. 1(a). An in-plane magnetic eld, H, along they-direction saturates both the YIG and Py magnetizations. We scan the frequency (from 4 to 8 GHz) and the magnetic eld, and then measure the optical responses using a lock-in amplier's in-phase X(Re[VO]) and quadrature Y(Im[VO]) channels as well as the 4microwave transmission using a microwave diode. A detailed description of the measurement setup is in the Supplemental Materials (SM), Figure S1. III. RESULTS AND DISCUSSIONS Figure 1(b) compares the optical rectication signals between the 10-nm-Py sample and the YIG/SiO 2/Py reference sample measured at 5.85 GHz. The 10-nm-Py sample (solid line) shows the representative features of the detected FMR and hybridized PSSW modes. The complete ne-scan including the FMR diode dataset are summarized in the SM, Fig. S2. The optical signals with the phase information are obtained by the lock-in's in-phase X(Re[VO], top panel) and quadrature Y(Im[VO], middle panel), which are further used to calculate the total amplitude,p X2+Y2, (bottom panel). The technical details of the measured signal versus the optical and electrical phases are summarized in the SM. The YIG FMR signal at 1.3 kOe is accumulated from the Faraday eect corresponding to the spatially uniform precession of the YIG magnetization. The FMR dispersion is de- scribed by the Kittel formula: !2= 2=HFMR(HFMR+Ms), where!is the mode frequency, =2= (ge=2)28 GHz/T is the gyromagnetic ratio, geis the g-factor, HFMRis the reso- nance eld, and Msis the magnetization. The excitation of the YIG PSSW modes introduces an additional exchange eld Hexto the Kittel equation, as 0Hex= (2Aex=Ms)(n=d YIG)2, which denes the mode splitting between the PSSW modes and the uniform mode. Here Aexis the exchange stiness, and dYIGis the YIG lm thickness. A total of more than 30 PSSW modes can be identied for the 10-nm-Py sample. In Fig. 1(c), the quadratic increase of Hexwith the mode number nconrms the observation of the PSSWs. Fittings to the Kittel equation and the exchange eld expression yield Ms= 1.97 kOe and Aex= 3.76 pJ/m, which are in good agreement with the previously reported values29{32. In Fig. 1(b), the Py FMR at 0.6 kOe is strongly modulated by the YIG PSSWs, exhibiting the MIT eect, due to the formation of hybrid magnon modes. Besides, the YIG PSSW signals near the Py FMR regime ( n>25) are much stronger than the o-resonance regime (n < 25), which indicates the important role of the Py/YIG coupling in exciting the relevant PSSW modes and resonantly enhancing the magnetization dynamics. As a comparison, no apparent PSSW modes are observed for the Py/SiO 2/YIG reference sample, in Fig.1(b) (dashed line), indicating that only the Py but not the YIG PSSWs couples to 5the microwave drive in the MIT regime. The Py resonance linewidth also is much narrower. (e) (a)(b)(c) PyYIG (d) (f)PyYIG FIG. 2. (a) Theoretical signal trace of the MIT eect of the YIG/Py bilayer. 7 hybrid PSSW modes are shown as an example. (b) Example tting of the complex optical singal at 6 GHz for the 30-nm-Py sample. (c) Full scan of the signal as a function of the magnetic eld and frequency. (d) Theoretical calculated dispersion using the tting parameters, reproducing the experimental data in (c). (e) and (f) are the ne-scans at smaller eld and frequency steps corresponding to the boxes in (c) and (d) (5.7 - 6.3 GHz). Our experimental conguration, similar to previously reported29{32, is relevant to the Schl omann excitation mechanism of spin wave43with a dynamic pinning at the interface44. The interfaces of two distinct, coupled magnetic layers have been recently recognized as an interesting source of spin dynamics generation and manipulation45,46. In particular, the 6critical role of the microwave susceptibilities of the distinct magnetic layers has been theoret- ically laid out47,48that are directly relevant to the magnon-magnon coupled experiments29{32. Here, we introduce a phenomenological model by considering a series of YIG harmonic os- cillators coupled with the Py oscillator, and using practical experimental tting parameters, in which the measured complex optical signal, VO, can be expressed as: VO=Aei(Lm) i(HPy FMRH)
HPy+ jg2 i(HYIG PSSW ;jH)
HYIG;j(1) whereAis the total signal amplitude, HPy FMR andHYIG PSSW is the resonance eld of Py and YIG-PSSWs, respectively,
HYIG(Py) is the half-width-half-maximum linewidth, and gis the eldlike coupling strength from the interfacial exchange. This model disregards the linear frequency-dependent phase (exists in the Re[ VO] and Im[VO] signals) but directly analyzes the total optical signal. Figure 2(a) plots the theoretically predicated MIT eect according to Eq.1, showing the lineshape of the Py FMR mode that is coupled to the YIG PSSW modes. The center curve with a zero resonance detuning denotes the MIT eect. The magnon-magnon coupling induces a set of sharp dips in the spectra. Such dips in the optical re ection means a peak in their transmission, which is referred to as a \transparency window" in quantum optics, resembling the EIT phenomenon in photonics37and optomechanics38,39. Using a singlegvalue, the amplitude and phase of the hybrid modes display a clearly evolution with respect to the dierent Py-YIG resonance detuning. Away from the Py FMR, the lineshape appears to be more antisymmetric, whilst around the Py FMR, the hybrid mode appears to be more symmetric. Such a phase evolution is contained in the Eq.1 and is not a t parameter. Fig.2(b) is an example tting result of a signal trace at 6 GHz. The ttings nicely reproduce the complex lineshapes arising from the coupled YIG PSSW modes and the Py FMR, including the phase evolution across the involved PSSW modes ( n). Our model allows extracting the YIG and Py magnon dissipation rates,
HYIG= 1.8 Oe,
HPy= 43.3 Oe, and the coupling strength: g= 18.7 Oe. In the frequency domain, these values correspond to g=2= 90 MHz, YIG=2= 6 MHz, and Py=2= 308 MHz. The numbers satisfy the condition for the MIT eect:
HYIG<g<
HPy. The YIG/Py interfacial exchange coupling can be also found from the HPy FMRshift com- paring to the YIG/SiO 2/Py reference sample. As shown in Fig. 1(b), the Py resonance occurs at a higher eld when Py is in direct contact with YIG due to the interfacial ex- 7change coupling. The increase of HPy FMR suggests that the YIG/Py interface induces a negative eective eld onto Py, which agrees well with the antiferromagnetic coupling in the previous reports29,32. We nd a resonance oset of HPy FMR,ofst = 0:17 kOe from Fig. 1(b), which further yields a eldlike coupling strength from the interfacial exchange32, g=HPy FMR,ofst0:9p MPytPy=MYIGtYIG. TakingMYIG= 1:97 kOe,MPy= 10 kOe,tPy= 10 nm andtYIG= 3m, we estimate g= 19:8 Oe. This value is in good agreement with that obtained earlier from the MIT eect. Figure 2 (c and d) compare the experimental and theoretical spin-wave dispersion using the total amplitude signal (p X2+Y2), whilst the individual channels, XandY, as well as the corresponding theoretical plots, are included in the SM, Fig. S3. To better analyze the hybrid PSSW modes, we show the zoom-in scan between 5.7 and 6.3 GHz and 0.2 to 0.9 kOe in Fig.2 (e and f), which covers the Kittel dispersion of the Py. We clearly identify the distinct PSSW modes strongly \chopping" the Py FMR line. In particular, the Py resonance is attenuated to nearly the background level (non-absorption condition) at the PSSW resonance dips. The same measurements are also performed for the reference samples, YIG/SiO 2/Py and YIG/Cu/Py, as summarized in the SM, Fig. S4. Despite the observation of the YIG and Py FMR modes, we do not measure any signal of the hybrid PSSW modes, reconrming that the excitation of the PSSW modes are primarily via the interfacial exchange coupling, rather than the dipolar interactions. To further examine the detuning range and its characteristics, we separate the Py reso- nance envelope with the YIG PSSW modes. Such analysis can be made via tting either the raw Re[ VO] or Im[VO] data. Fig. 3(a) shows the raw Re[ VO] signal of the hybrid modes at a representative frequency window (5.85 - 6.1 GHz) for the 10-nm-Py sample. After subtracting the Py resonance prole (details are in the SM, Fig. S5), we can t each PSSW series (labeled n= 3243) to a phase-shifted Lorentzian function yielding the resonance and linewidth for each PSSWs, as shown in Fig.3(b) (where the highlighted section shows an example series at n= 39). We dene a \resonance distance",
Hres= [HYIG PSSWHPy FMR], which represents their frequency detuning and the coupling eciency. Figure 3(c) shows the resonance eld HYIG PSSW of the PSSW series (thin lines, from n= 3243) comparing to the HPy FMR (single thick line). The shaded area indicates the Py linewidth, which is centered at the HPy FMRand is also much enhanced as compared to the case 8(a)(b) YIG433239 (d) (c)39394332 43326.10 GHz 5.85 GHzVO(arb. units) YIG+PyYIGPyFIG. 3. (a) The Re[ VO] signal at a representative frequency window (5.85 - 6.1 GHz) for the 10- nm-Py sample. (b) YIG PSSW lineshape (12 mode series near the Py resonance are labeled and analyzed, n= 32 - 43) after subtracting the Py resonance prole. The highlighted section is an example series at n= 39. (c) Resonance eld, HYIG PSSW of the PSSW series and the HPy FMRenvelope. The shaded area re ects the Py linewidth. (d) The extracted YIG PSSW linewidth
HYIGversus the
Hresat each frequency and for all the PSSW series. without the mode coupling (in the YIG/SiO 2/Py sample). Next, we plot the
Hresat each frequency and for all the PSSW series with the corresponding YIG PSSW linewidth,
HYIG, in Fig. 3(d). We clearly observe a modulation eect of the YIG linewidth,
HYIG, from 2 Oe to10 Oe, spanning across the magnon-magnon coupling regime. This observation provides strong evidence that the MIT linewidth is broadened due to the additional energy dissipation by coupling the YIG PSSW modes to the Py FMR mode, also known as the Purcell regime2,3,10. From the theoretical model in Eq.1, we obtain a relationship between 9the YIG linewidth broadening due to a nite gand the overlapped resonance:
HMIT YIG;j=
HYIG;j+g2
HPy (
Hres)2+ (
HPy)2(2) where the derivation is included in the SM. Since the MIT regime is within a relatively narrow frequency window from 5.7 - 6.6 GHz, we take the average of the Py linewidth within this frequency window, as
HPy= 608 Oe from Fig.3(c). For YIG, we take the same linewidth as in Fig.2(b),
HYIG;j= 1:8 Oe, therefore leaving gas the only tting parameter. The best t yields g= 17:71:2 Oe, with a tting curve indicated in Fig.3(d), dashed line. This value agrees with the lineshape tting results of 18.7 Oe as discussed above. Finally, the observed multiple PSSW modes and their coupling to a \magnonic cavity" are similar to the multi-mode coupling in the magnon-photon system49, in which the proles and properties of each PSSW mode are greatly modied as compared to the free-space conditions. We envisage that a stronger coupling condition may be fullled by combining an appropriately designed optical cavity such as the whispering gallery modes23, or replacing the Py with a low damping ferromagnetic material with reduced dissipation rate29. IV. CONCLUSION In summary, we report the observation of the magnetically-induced transparency in YIG/Py bilayers exhibiting magnon-magnon coupling. The use of the thin-lm YIG system shows great potential in practical applications. The series of standing waves in YIG may allow to build an evenly distributed resonance array in a single YIG device, which may lead to relevant applications such as memory and comb generation. In addition, compared with the so-far widely used hybrid magnonic systems that utilize the ferromagnetic resonances, our results pave the way towards building more complex hybrid systems with spin-waves. Our measurement is achieved via a simultaneous and stroboscopic detection of the coupled magnetization dynamics using a single wavelength, therefore avoids the possible artifacts due to multiple probes. Our work, performed in a planar structure as opposed to 3D cavities, also paves the way towards solving strong magnon-magnon couplings by the state-of-the-art spin-orbitronic toolkits53,54, involving emerging materials such as antiferromagnets55{57, 2D monolayers58{61, and topological insulators62,63. 10Acknowledgements W.Z. acknowledges useful discussions with V. Tyberkevych and A. Slavin. This work, in- cluding apparatus buildup, experimental measurements, and data analysis, was supported by AFOSR under Grant No. FA9550-19-1-0254, National Science Foundation under Grants No. DMR-1808892 and ECCS-1933301 . Work at Argonne, including sample preparation, was supported by U.S. DOE, Oce of Science, Materials Sciences and Engineering Division. M.H. acknowledges the Michigan Space Grant Consortium Student Fellowship for nancial support. Author Contributions Y.L. and W.Z. conceived the idea. Y.X., M.H., R.B., W.Z. performed the experiment. J.S., X.Z., and Y.L. developed the theoretical model. J.P., A.H., and V.N. participated in the thin-lm sample fabrication. G.S. and H.Q. participated in the microwave measurement. Y.L., X.Z., J.S., and W.Z. prepared the gures and manuscript draft. All authors discussed the results and participated in the writing and nalizing of the manuscript. Competing Interests Te authors declare no competing interests. Additional Information Supplementary information is available for this paper at https://doi.org/XX.XXXX. Correspondence and requests for materials should be addressed to Y.L. and W.Z. REFERENCES 1D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, \Hybrid quan- tum systems based on magnonics", Applied Physics Express 12, 070101 (2019). 2M. Harder and C. -M. Hu, \Cavity Spintronics: An Early Review of Recent Progress in the Study of Magnon-Photon Level Repulsion, Solid State Physics, 70, 47 - 121 (2018). R. Stamps and R. Camley (Ed.), Academic Press. 3B. Bhoi and S. -K. Kim, \Photon-magnon coupling: Historical perspective, status, and 11future directions, Solid State Physics, 69, 1 - 77 (2019). R. Stamps and H. Schultheiss (Ed.), Academic Press. 4Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Naka- mura, \Coherent coupling between a ferromagnetic magnon and a superconducting qubit", Science 349, 405 (2015). 5L. McKenzie-Sell, J. Xie, C.-M. Lee, J. W. A. Robinson, C. Ciccarelli, and J. A. Haigh, \Low-impedance superconducting microwave resonators for strong coupling to small mag- netic mode volumes", Phys. Rev. B 99, 140414 (2019). 6Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W. K. Kwok, Z. Xiao, V. Novosad, A. Homann, and W. Zhang, \Strong Coupling between Magnons and Microwave Photons in On-Chip Ferromagnet-Superconductor Thin-Film Devices", Phys. Rev. Lett. 123, 107701 (2019). 7J. T. Hou and L. Liu, \Strong coupling between microwave photons and nanomagnet magnons", Phys. Rev. Lett. 123, 107702 (2019). 8H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, \High Cooperativity in Coupled Microwave Resonator Ferrimagnetic Insulator Hybrids", Phys. Rev. Lett. 111, 127003 (2013). 9L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C. M. Hu, \Spin Pumping in Electrodynamically Coupled Magnon-Photon Systems", Phys. Rev. Lett. 114, 227201 (2015). 10X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, \Strongly Coupled Magnons and Cavity Microwave Photons", Phys. Rev. Lett. 113, 156401 (2014). 11X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, \Magnon dark modes and gradient memory", Nat. Commun. 6, 8914 (2015). 12Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and G. E. W. Bauer, \Exchange magnon- polaritons in microwave cavities", Phys. Rev. B 91, 094423 (2015). 13M. Harder, Y. Yang, B. M. Yao, C.H. Yu, J.W. Rao, Y.S. Gui, R.L. Stamps, and C.-M. Hu, \Level Attraction Due to Dissipative Magnon-Photon Coupling", Phys. Rev. Lett. 121, 137203 (2018). 14B. Bhoi, B. Kim, S.-H. Jang, J. Kim, J. Yang, Y.-J. Cho, and S.-K. Kim, \Abnormal anticrossing eect in photon-magnon coupling", Phys. Rev. B 99, 134426 (2019). 15M. N. Winchester, M. A. Norcia, J. R. K. Cline, and J. K. Thompson, Phys. Rev. Lett. 12118, 263601 (2017). 16D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. Q. You, \Observation of the exceptional point in cavity magnon-polaritons", Nat. Commun. 8, 1368 (2017). 17Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, \Hybridiz- ing Ferromagnetic Magnons and Microwave Photons in the Quantum Limit", Phys. Rev. Lett. 113, 083603 (2014). 18T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. Uchida, Z. Qiu, G. E.W. Bauer, and E. Saitoh, \Magnon Polarons in the Spin Seebeck Eects", Phys. Rev. Lett. 117, 207203 (2016). 19X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, \Cavity magnomechanics", Science Ad- vances, 2, e1501286 (2016). 20L. Bai, M. Harder, P. Hyde, Z. Zhang and C. M. Hu, \Cavity Mediated Manipulation of Distant Spin Currents Using a Cavity-Magnon-Polariton", Phys. Rev. Lett. 118, 217201 (2017). 21P. Andrich, C. F. de las Casas, X. Liu, H. L. Bretscher, J. R. Berman, F. J. Heremans, P. F. Nealey, and D. D. Awschalom, \Long-range spin wave mediated control of defect qubits in nanodiamonds", npj Quantum Inf. 3, 28 (2017). 22A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, \Cavity Optomagnonics with Spin-Orbit Coupled Photons", Phys. Rev. Lett. 116, 223601 (2016). 23X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, \Optomagnonic Whispering Gallery Mi- croresonators", Phys. Rev. Lett. 117, 123605 (2016). 24R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, \Bidirectional conversion between microwave and light via ferromagnetic magnons", Phys. Rev. B 93, 174427 (2016). 25J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, \Triple-Resonant Bril- louin Light Scattering in Magneto-Optical Cavities", Phys. Rev. Lett. 117, 133602 (2016). 26A. Kirilyuk, A. V. Kimel, and Th. Rasing, \Ultrafast optical manipulation of magnetic order", Rev. Mod. Phys. 82, 2731 (2010). 27J. Li, S. Y. Zhu, and G. S. Agarwal, \Magnon-poton-phonon entanglement in cavity mag- nomechanics", Phys. Rev. Lett. 121, 203601 (2018). 28D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura, 13\Entanglement-based single-shot detection of a single magnon with a superconducting qubit", Science 367, 425 (2020). 29S. Klingler, V. Amin, S. Geprags, K. Ganzhorn, H. Maier-Flaig, M. Althammer, H. Huebl, R. Gross, R. D. McMichael, M. D. Stiles, S. T. B. Goennenwein, and M. Weiler, \Spin- Torque Excitation of Perpendicular Standing Spin Waves in Coupled YIG/Co Heterostruc- tures", Phys. Rev. Lett. 120, 127201 (2018). 30J. Chen, C. Liu, T. Liu, Y. Xiao, K. Xia, G. E. W. Bauer, M. Wu, and H. Yu, \Strong In- terlayer Magnon-Magnon Coupling in Magnetic Metal-Insulator Hybrid Nanostructures", Phys. Rev. Lett. 120, 217202 (2018). 31H. Qin, S. J. Hamalainen, and S. van Dijken, \Exchange-torque-induced excitation of perpendicular standing spin waves in nanometer-thick YIG lms", Sci. Rep. 8, 5755 (2018). 32Y. Li, W. Cao, V. P. Amin, Z. Zhang, J. Gibbons, J. Sklenar, J. Pearson, P. M. Haney, M. D. Stiles, W. E. Bailey, V. Novosad, A. Homann, and W. Zhang, \Coherent spin pumping in a strongly coupled magnon-magnon hybrid system", Phys. Rev. Lett. 124, 117202 (2020). 33V. V. Kruglyak, S. O. Demokritov, and D. Grundler, \Magnonics", J. Phys. D: Appl. Phys. 43, 264001 (2010). 34K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurdequint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, \Coherent long-range transfer of angular momentum between magnon Kittel modes by phonons", Phys. Rev. B 101, 060407(R) (2020). 35A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, \Fano resonance in nanoscale struc- tures", Rev. Mod. Phys. 82, 2257 (2010). 36M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, \Fano resonances in photonics", Nature Photonics 11, 543 (2017). 37B. Peng, S. K. Ozdemir, W. Chen, F. Nori, and L. Yang, \What is and what is not electro- magnetically induced transparency in whispering-gallery microcavities", Nature Commun. 55082 (2014). 38S. Weis, R. Rivire, S. Delglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, \Optomechanically Induced Transparency", Science 330, 1520 (2010). 39A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eicheneld, M. Winger, Q. Lin, J. 14T. Hill, D. E. Chang, and O. Painter, \Electromagnetically induced transparency and slow light with optomechanics", 472, 69 (2011). 40S. Yoon, J. Liu, and R. D. McMichael, \Phase-resolved ferromagnetic resonance using a heterodyne detection method", Phys. Rev. B 93, 144423 (2016). 41Y. Li, H. Saglam, Z. Zhang, R. Bidthanapally, Y. Xiong, J. E. Pearson, V. Novosad, H. Qu, G. Srinivasan, A. Homann, and W. Zhang, \Simultaneous Optical and Electrical Spin- Torque Magnetometry with Phase-Sensitive Detection of Spin Precession", Phys. Rev. Applied 11, 034047 (2019). 42Y. Li, F. Zeng, H. Saglam, J. Sklenar, J. E. Pearson, T. Sebastian, Y. Wu, A. Ho- mann, and W. Zhang, \Optical Detection of Phase-Resolved Ferromagnetic Resonance in Epitaxial FeCo Thin Films", IEEE Trans. Magn. 55, 6100605 (2019). 43E. Schl omann, \Generation of spin waves in nonuniform magnetic eld. I. Conversion of electromagnetic power into spinwave power and vice versa" J. Appl. Phys. 35, 159 (1964). 44P. E. Wigen, C. F. Kooi, M. R. Shanabarger, and T. D. Rossing, \Dynamic Pinning in Thin-Film Spin-Wave Resonance", Phys. Rev. Lett. 9, 206 (1962). 45V. V. Kruglyak, C. S. Davies, V. S. Tkachenko, O. Y. Gorobets, Y. I. Gorobets, and A. N. Kuchko, \Formation of the band spectrum of spin waves in 1D magnonic crystals with dierent types of interfacial boundary conditions", J. Phys. D: Appl. Phys. 50, 094003 (2017). 46R. Verba, V. Tiberkevich, and A. Slavin, \Spin-wave transmission through an internal boundary: Beyond the scalar approximation", Phys. Rev. B 101, 144430 (2020). 47V. D. Poimanov, A. N. Kuchko, and V. V. Kruglyak, \Emission of coherent spin waves from a magnetic layer excited by a uniform microwave magnetic eld", J. Phys. D: Appl. Phys. 52, 135001 (2019). 48V. D. Poimanov, A. N. Kuchko, and V. V. Kruglyak, \Magnetic interfaces as sources of coherent spin waves", Phys. Rev. B 98, 104418 (2018). 49X. Zhang, C. Zou, L. Jiang, and H. X. Tang, \Superstrong coupling of thin lm magneto- static waves with microwave cavity", J. Appl. Phys. 119, 023905 (2016). 50N. M. Sundaresan, Y. Liu, D. Sadri, L. J. Szocs, D. L. Underwood, M. Malekakhlagh, H. E. Tureci, and A. A. Houck, \Beyond Strong Coupling in a Multimode Cavity", Phys. Rev. X 5, 021035 (2015). 51B. A. Moores, L. R. Sletten, J. J. Viennot, and K. W. Lehnert, \Cavity Quantum Acoustic 15Device in the Multimode Strong Coupling Regime", Phys. Rev. Lett. 120, 227701 (2018). 52N. Kostylev, M. Goryachev, and M. E. Tobar, \Superstrong coupling of a microwave cavity to yttrium iron garnet magnons", Appl. Phys. Lett. 108, 062402 (2016). 53W. Zhang, M. B. Jung eisch, W. Jiang, J. Sklenar, F. Y. Fradin, J. E. Pearson, J. B. Ketterson, and A. Homann, \Spin pumping and inverse spin Hall eectsInsights for future spin-orbitronics", J. Appl. Phys. 117, 172610 (2015). 54R. Ramaswamy, J. M. Lee, K. Cai, and H. Yang, \Recent advances in spin-orbit torques: Moving towards device applications", Applied Physics Reviews 5, 031107 (2018). 55V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, \Antiferro- magnetic spintronics", Rev. Mod. Phys. 90, 015005 (2018). 56W. Zhang and K. M. Krishnan, \Epitaxial exchange-bias systems: From fundamentals to future spin-orbitronics", Materials Science and Engineering: R: Reports 105, 1-20 (2016). 57L. Liensberger, A. Kamra, H. Maier-Flaig, S. Geprags, A. Erb, S. T. B. Goennenwein, R. Gross, W. Belzig, H. Huebl, and M. Weiler, \Exchange-Enhanced Ultrastrong Magnon- Magnon Coupling in a Compensated Ferrimagnet", Phys. Rev. Lett. 123, 117204 (2019). 58D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, and D. C. Ralph, \Control of spinorbit torques through crystal symmetry in WTe2/ferromagnet bilayers", Nat. Phys. 13, 300 (2017). 59W. Zhang, J. Sklenar, B. Hsu, W. Jiang, M. B. Jung eisch, K. Sarkar, F. Y. Fradin, Y. Liu, J. E. Pearson, J. B. Ketterson, Z. Yang, and A. Homann, \Spin transfer torques in permalloy on monolayer MoS2", APL Mater. 4, 032302 (2016). 60Y. Li, F. Ye, J. Xu, W. Zhang, P. X. -L. Feng, and X. Zhang, \Gate-Tuned Temperature in a Hexagonal Boron Nitride-Encapsulated 2-D Semiconductor Device", IEEE Trans. Electron Devices 65, 4068 (2018). 61X. Zhang, \Characterization of Layer Number of Two-Dimensional Transition Metal Dis- elenide Semiconducting Devices Using Si-Peak Analysis", Advances in Materials Science and Engineering, 2019 , 7865698 (2019). 62A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, \Spin-transfer torque generated by a topological insulator", Nature 511, 449 (2014). 63P. Li et al. , \Magnetization switching using topological surface states", Science Advances 5, eaaw3415 (2019). 16

2019-12-31

We demonstrate the magnetically-induced transparency (MIT) effect in Y$_3$Fe$_5$O$_{12}$(YIG)/Permalloy(Py) coupled bilayers. The measurement is achieved via a heterodyne detection of the coupled magnetization dynamics using a single wavelength that probes the magneto-optical Kerr and Faraday effects of Py and YIG, respectively. Clear features of the MIT effect are evident from the deeply modulated ferromagnetic resonance of Py due to the perpendicular-standing-spin-wave of YIG. We develop a phenomenological model that nicely reproduces the experimental results including the induced amplitude and phase evolution caused by the magnon-magnon coupling. Our work offers a new route towards studying phase-resolved spin dynamics and hybrid magnonic systems.

Probing magnon-magnon coupling in exchange coupled Y$_3$Fe$_5$O$_{12}$/Permalloy bilayers with magneto-optical effects

1912.13407v2

1 Spin Wave Interference Detect ion via Inverse Spin Hall Effect Michael Balynsk iy, Howard Chiang, David Gutierrez, and Alexander Khitun* Department of Electrical and Computer Engineering, University of California -Riverside, Riverside, California, USA 92521 Abstract In this letter, we present experimental data demonstrating spin wave interference detection using spin Hall effect (ISHE). Two coherent spin waves are excited in a yttrium -iron garnet (YIG) waveguide by continuous microwave signals. The initial phase difference between the spin waves is controlled by the external phase shifter. The ISHE voltage is de tected at a distance of 2 mm and 4 mm away from the spin wave generating antennae by an attached Pt layer. Experimental data show ISHE voltage oscillation as a function of the phase difference between the two interfering spin waves . This experiment demons trates an intriguing possibility of using ISHE in spin wave logic circuit converting spin wave phase into an electric signal. Keywords: spin waves, inverse spin Hall effect, wave interference Corresponding Author: akhitun@engr.ucr.edu There is a big im petus in the development of spin -based logic devices aimed to benefit spin in addition to charge [1]. Spin wave (SW) devices are one of the promising approaches exploiting collective spin oscillation s [2]. SW can propagate on much larger distances (e.g., 1 cm at room temperature) compared to the spin -diffusion length in metals [3]. It makes it possible to exploit th e phase of SW signal for information transfer [4]. There were several prototypes demonstrated during the past decade including SW majo rity logic gate [5], SW holographic memory [6], and devices for special type data processing [7, 8] . In all of the above cited works, SW output was detected by micro -antenna which implies certain restriction on the size/position of the input -output ports due to the stray fi eld coupling. ISHE is one of the possible 2 alternati ves which may be more scalable and less sensitive to the direct input -output coupling. SW detection using ISHE was presented in a number of works [9-12]. For instance, it was experimentally demonstrated magnon spin tra nsport between the spatially separated inductive pulse spin -wave source and the ISHE detector [12]. The role of the travelling SW was revealed by a delay in the detection of the ISHE voltage. In this letter, we extend this approach to two -SW inte rference. The primarily objective of this work is to demonstrate the correlation between the phase difference of the interfering spin waves and the produced ISHE voltage. The test structure is schematically shown in Fig. 1. It comprises a 9.5 μm thick YIG waveguide ( 10 mm × 2 mm) with a 10 nm thick (0.2 × 2 mm2, 374 Ohm) Pt strip deposited on the top. The YIG waveguide is magnetized along its long axis by an external bias magnetic field providing conditions for the propagation of a backward volume magnetostatic wave (BVMSW). There are two 50 μm -wide Au microstrip antennae fabricated on the edges of waveguide. These antennae are used for continuous SW signal generation. The antennas are i ntentionally placed at the different distances from the Pt detector. The distance from the left antenna to the center of the detector is about 2 mm while the distance from the antenna on the right to the detector is 4 mm. The asymmetry in th e excitation p ort position helps to unmask the effect of direct coupling on ISHE voltage . The antennae are connected to a programmable network analyzer (PNA) Keysight N5241A. PNA serves as a co mmon input for the both antennae . It also allows us to detect SW signal inde pendently from ISHE measurements. There is a p hase shifter included in the antennae -PNA circuit. This shifter is to control the initial phase d ifference between the excited spin waves . The DC ISHE voltage is detected via Lock -In Amplifier SR830 DSP which i s synchronized with sweeping frequency of PNA operating in power sweep mode. The test structure is placed inside an electromagnet (GMW model 3472 – 70) with the pole cap of 50 mm (2 inch) diameter tapered. The system provided a uniform bias magnetic field H/H<10-4 per 1 mm in the range from −2000 Oe to +2000 Oe. Based on the power source specification (KEPCO), the magnetic field instability was estimated about 0.15 Oe. All measurements are accomplished at room temperature. 3 The first set of experiments is aimed at confirming the spin wave generation by the antennas via the inductive voltage measurements, and finding the optimum operational frequency 𝑓 and at the bias magnetic field 𝐻0. Based on our prior studies of SW propagation in YIG films [13, 14] , we searched for the maximum output signal in the frequency range from 4 GHz to 6 GHz. In Fig.2., there are shown experimental data for 𝑆21 parameter in the loop PNA – antenna 1 – antenna2 – PNA. The maximum signal is observed for f = 4.972 GHz and bias magnetic field 𝐻0=1100 Oe. This combination of frequency and bias magnetic field fits well the BVMS W dispersion given by [15]: 𝑓𝐵𝑉𝑀𝑆𝑊=√𝑓𝐻(𝑓𝐻+𝑓𝑀1−exp(−𝑘𝑑) 𝑘𝑑) , (1) where 𝑓𝐻=𝛾𝐻0, 𝑓𝑀=4𝜋𝛾𝑀0,𝛾=2.8 𝑀𝐻𝑧/𝑂𝑒, 𝑘~ 30 𝑐𝑚−1 is the SW wavenumber, 𝑑 is the film thickness. The data show prominent spin wave propagation over the 10 mm distance at room temperature. Next, ISHE voltage produced by spin waves was detected across the Pt strip. Spin waves are excited by the two antenna e biased by PNA with the fixed frequency f = 4.972 GHz . The att enuators (e.g. A1 and A2 show n in Fig.1) were used to equalize the amplitudes o f SW signals coming to Pt detector . ISHE voltage was measured at different bias magnetic field s. The experimental data are shown in Fig. 3. In Fig.3(a) there is shown the dependence for magnetic field directed along the axes of YIG waveguide as shown in Fig.1 . The maximum of the ISHE voltage −5.5 𝜇𝑉 appears at 𝐻0=1100 Oe which corresponds to the maximum of SW signal. In order to further verify the origin of the detected voltage, the direction of the bias magnetic field has been reversed. The electric field induced by the ISHE 𝐸𝐼𝑆𝐻𝐸 can be written as follows [16]: 𝐸⃗ 𝐼𝑆𝐻𝐸∝𝐽 𝑠×𝜎 , (2) where 𝐽𝑠 is the spin current injected from YIG in Pt and 𝜎 is the spin polarization vector of the spin current defined by the bias magnetic field 𝐻0. As expected from Eq. (2), the reverse of the bias magnetic field resulted in the ISHE voltage polarity change as can be seen in Fig. 3(b). The maximum of the detec ted voltage 4.2 μV appears at 𝐻0= −1100 Oe. A difference in the peak ISHE voltage seen in Fig.3(a) and Fig.3(b) can be 4 attributed to the weak non-reciprocity of BVMSW and to an asymmetrical orientation of our device in magnetic field . The collection of experimental data in Figs.2 and 3 confirms the origin of the ISHE voltage as a result of spin wave conversion into an electron carried spin current. Finally, ISHE voltage was measured as a function of the phase difference between the propagating spin waves. Spin waves were generated by antenna 1 and antenna 2 at constant f = 4.972 GHz . The bia s magnetic field was fixed to 𝐻0=1100 Oe. The initial phase difference between the antennae is the only parameter which has been changed. The obtained experimental data are shown in Fig.4. It is clearly seen the oscillation of the ISHE voltage depending on the phase difference of interfering spin waves. The maximum voltage is −5.5 𝜇𝑉 which is the same as in Fig.3(a). The minimum voltage is about −0.6 𝜇𝑉. This minimum voltage is attribu ted to the destructive SW interference corres ponding to the phase difference ∆𝜙=𝜋. All other phases in Fig.4 are defined to this phase difference. The oscillation of the ISHE voltage can be well approximated by the classical formula: 𝑉𝐼𝑆𝐻𝐸=√𝑉12+ 𝑉22+2𝑉1𝑉2cos(∆𝜙) (3) where 𝑉1 and 𝑉2 are the voltage s produced separately by each SW . There may be several reasons for 𝑉𝐼𝑆𝐻𝐸 not coming to zero for the destr uctive spin wave interference (i.e., ∆𝜙=𝜋). For instance, the wave front of the SW signals may be disturbed by the reflection from the waveguide boundar ies. There are several observations we want to make based on the obtained experimental data. (i) ISHE voltage is produced by the interfering spin waves. This conclusion is supported by the experimental data shown in Fig.2 and Fig.3. ISHE voltage attains its maximum for magnetic field allowing BVMSW propagation which is in good agreement with the results of PNA measurements (e.g., more than 10 dB increase of 𝑆21 parameter). The combination of frequency and bias magnetic field is well fitted by the BVMSW dispersion (i.e., Eq. 1). (ii) There is an ISHE voltage dependence on the phase difference bet ween the interfering spin waves. The voltage oscillates with the phase 5 difference following the classical interference formula Eq. 3 . There is a prominent difference between the maximu m and minimum voltages which is attributed to the cases of constructive and destructive SW interference, respectively. A careful comparison of the ISHE voltage measured across Pt strip and SW intensity measured via PNA shows that the maximum of the DC signal occurs for constructive spin wave interference. In conclusion, we presented experimental data demonstrating the detection of two interfering spin waves via ISHE voltage measurements. The amplitude of the voltage oscillates depending on the phase difference between the waves. ISHE detectors may be pote ntially utilized for connect ing phase -based SW logic devices with output electronic devices [6]. It may be also possible to remotely control pure spin currents using spin waves interference . Acknowledgement This was supported in part by the National Science Foundation (NSF) under Award # 2006290. It was also supported in part by the Spins and Heat in Nanoscale Electronic Systems (SHINES), an Energy Frontier Research Center fun ded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (BES) under Award # SC0012670. The data that support the findings of this study are ava ilable from the corresponding author upon reasonable request. Figure Captions Figure 1: Schematic illustration of the experimental setup: two coherent spin waves are excited in the YIG waveguide using two microstrip antennae. The antennae are connected to PNA via the set of attenuators (A1 and A2) and a phase shifter. There is Pt stripe on to p of the waveguide for ISHE voltage detection. The distances from the strip to the antennae are 2 mm and 4 mm. 6 Figure 2: Experimental data showing the coupling (i.e., S21 parameter) between the antennae depending on the bias magnetic field 𝐻0. The coupl ing is maximum at 𝐻0=1100 Oe for excitation frequency f = 4.972 GHz , which is attributed to BVMSW s propagating in the waveguide. Figure 3: Experimental data showing the variation of the ISHE voltage depending on the bias magnetic field. (a) Magnetic fi eld is directed along the axes of YIG waveguide . b)The direction of magnetic field is reversed. Figure 4: Experimental data showing ISHE voltage oscillation depending on the phase difference between the interfering spin waves. References [1] S. A. Wolf, A. Y. Chtchelkanova, and D. M. Treger, "Spintronics - A retrospective and perspective," Ibm Journal of Research and Development, vol. 50, pp. 101 - 110, Jan 2006 [2] A. Khitun and K. L. Wang, "Nano scale computational archi tectures with Spin Wave Bus," Superlattices and Microstructures, vol. 38, pp. 184 -200, Sep 2005. DOI: 10.1016/j.spmi.2005.07.001 [3] J. Bass and W. P. Pratt, "Spin -diffusion lengths in metals and alloys, and spin - flipping at metal/metal interfaces: an ex perimentalist's critical review," Journal of Physics: Condensed Matter, vol. 19, 2007 [4] A. Khitun, "Spin wave phase logic," Cmos and Beyond: Logic Switches for Terascale Integrated Circuits, pp. 359 -378, 2015 [5] Y. Wu, M. Bao, A. Khitun, J. -Y. Kim, A. Hong, and K.L. Wang, " A Three -Terminal Spin-Wave Device for Logic Applications," Journal of Nanoelectronics and Optoelectronics, vol. 4, pp. 394 -7, 2009 [6] F. Gertz, A. Kozhevnikov, Y. Filimonov, and A. Khitun, "Magnonic Holographic Memory," Ieee Trans actions on Magnetics, vol. 51, Apr 2015. DOI: 10.1109/tmag.2014.2362723 [7] A. Kozhevnikov, F. Gertz, G. Dudko, Y. Filimonov, and A. Khitun, "Pattern recognition with magnonic holographic memory device," Applied Physics Letters, vol. 106, Apr 2015. DOI: 10.1063/1.4917507 [8] Y. Khivintsev, M. Ranjbar, D. Gutierrez, H. Chiang, A. Kozhevnikov, Y. Filimonov, and A. Khitun, "Prime factorization using magnonic holographic devices," Journal of Applied Physics, vol. 120, Sep 2016. DOI: 10.1063/1.4962740 [9] K. Ando, J. Ieda, K. Sasage, S. Takahashi, S. Maekawa, and E. Saitoh, "Electric detection of spin wave resonance using inverse spin -Hall effect," Appl. Phys. Lett., vol. 94, 2009. DOI: https://doi.org/10.1 063/1.3167826 7 [10] L. Feiler, K. Sentker, M. Brinker, N. Kuhlmann, F. -U. Stein, and G. Meier, "Inverse spin -Hall effect voltage generation by nonlinear spin -wave excitation," Phys. Rev. B, vol. 93, 2016. DOI: https://doi.org/10.1103/PhysRevB.93.064408 [11] T. Brächer, M. Fabre, T. Meyer, T. Fischer, S. Auffret, O. Boulle, U. Ebels, P. Pirro, and G. Gaudin, "Detection of Short -Waved Spin Waves in Individual Microscopic Spin -Wave Waveguides Using the Inverse Spin Hall Effect," Nano Letters, vol. 17, pp. 7234 –7241, 2017. DOI: https://doi.org/10.1021/acs.nanolett.7b02458 [12] A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, "Direct detection of magnon spin transport by the inverse spin Hall effect," Applied Physics Letters, vol. 100, 2012. DOI: https://doi.org/10.1063/1.3689787 [13] A. Kozhevnikov, F. Gertz, G. Dudko, Y. Filimonov, and A. Khitu n, "Pattern recognition with magnonic holographic memory device," Applied Physics Letters, vol. 106, p. 142409, Apr 6 2015. DOI: 142409 10.1063/1.4917507 [14] M. Balynsky, A. Kozhevnikov, Y. Khivintsev, T. Bhowmick, D. Gutierrez, H. Chiang, G. Dudko, Y. Filimonov, G. X. Liu, C. L. Jiang, A. A. Balandin, R. Lake, and A. Khitun, "Magnonic interferometric switch for multi -valued logic circuits," Journal of Applied Physics, vol. 121, p. 024504, Jan 2017. DOI: 10.1063/1.4973115 [15] A. A. Serga, A. V. Chumak , and B. Hillebrands, "YIG magnonics," Journal of Physics D -Applied Physics, vol. 43, Jul 7 2010. DOI: 264002 10.1088/0022 - 3727/43/26/264002 [16] V. E. Demidov, M. P. Kostylev, K. Rott, P. Krzysteczko, G. Reiss, and S. O. Demokritov, "Excitation of micro waveguide modes by a stripe antenna," Appl. Phys. Lett., vol. 95, 2009. DOI: https://doi.org/10.1063/1.3231875 8 Figure 1 9 Figure 2 10 Figure 3 11 Figure 4

2021-05-06

In this letter, we present experimental data demonstrating spin wave interference detection using spin Hall effect (ISHE). Two coherent spin waves are excited in a yttrium-iron garnet (YIG) waveguide by continuous microwave signals. The initial phase difference between the spin waves is controlled by the external phase shifter. The ISHE voltage is detected at a distance of 2 mm and 4 mm away from the spin wave generating antennae by an attached Pt layer. Experimental data show ISHE voltage oscillation as a function of the phase difference between the two interfering spin waves. This experiment demonstrates an intriguing possibility of using ISHE in spin wave logic circuit converting spin wave phase into an electric signal

Spin Wave Interference Detection via Inverse Spin Hall Effect

2105.02979v1

Enhancement of YIG jPt spin conductance by local Joule annealing R. Kohno,1N. Thiery,1K. An,1P. Noel,1L. Vila,1V. V. Naletov,1, 2N. Beaulieu,3, 4J. Ben Youssef,4G. de Loubens,3and O. Klein1,a) 1)Universit e Grenoble Alpes, CEA, CNRS, Spintec, 38054 Grenoble, France 2)Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 3)SPEC, CEA-Saclay, CNRS, Universit e Paris-Saclay, 91191 Gif-sur-Yvette, France 4)LabSTICC, CNRS, Universit e de Bretagne Occidentale, 29238 Brest, France (Dated: 8 September 2020) We report that Joule heating can be used to enhance the interfacial spin conductivity between a metal and an oxide. We observe that local annealing of the interface at about 550 K by injecting large current densities (>1012A/m2) into a pristine 7 nm thick Pt nanostrip evaporated on top of yttrium iron garnet (YIG), can improve the spin transmission up to a factor 3: a result of particular interest for interfacing ultra thin garnet lms where strong chemical etching of the surface has to be avoided. The eect is conrmed by dierent methods: spin Hall magnetoresistance, spin pumping and non-local spin transport. We use it to study the in uence of the YIG jPt coupling on the non-linear spin transport properties. We nd that the cross-over current from a linear to a non-linear spin transport regime is independent of this coupling, suggesting that the behavior of pure spin currents circulating in the dielectric are mostly governed by the physical properties of the bare YIG lm beside the Pt nanostrip. The transport of pure spin information through local- ized magnetic moments is at the heart of a new research topic called insulatronic (forinsula -tor spin- tronic )1{4. Interest stems here from the recognition that magnetic insulators are superior spin conductors than metals or semi-conductors. Among magnetic insulators, garnets, and in particular yttrium iron garnet (YIG), have the lowest known magnetic damping5. One can exploit here the spin Hall eect (SHE) to interconvert pure spin cur- rents circulating inside the dielectric into charge currents, which can then be probed electrically. This is usually achieved by depositing a heavy metal electrode, advan- tageously in Pt6,7, on top of the YIG surface. The en- suing ow of spin escaping through the metal-oxide in- terface can be measured through spin pumping (SP)8, spin Hall magnetoresistance (SMR)9{11, spin Seebeck ef- fect (SSE)12or spin orbit torque (SOT) using non-local transport devices1,13. The eciency of the process is determined by the spin transparency of the interface and parametrized by the so-called spin mixing conductance, G"#. To optimize its strength, the YIG surface is usually treated by strong process such as O+/Ar+plasma14, by annealing15and piranha etching16,17prior to the metal deposition in order to achieve good chemical and structural YIG jPt interface. However these treatments are performed on m thick YIG samples and are dicult to implement once the thickness of the lm is in the nanometer range as a modulation of surface roughness signicantly disturbs its magnetic properties. Additionally an enhancement of SMR by global annealing at very high temperature18has been reported, yet this process is not necessarily compatible at the device level. In addition a)Corresponding author: oklein@cea.frto the cleaning of the YIG surface, the use of sputtering technique to deposit the metal is known to lead to betterG"#than the evaporation technique19. The later however usually leads to lower resistivity, which should be favored when one wants to inject large current densi- ties. Considering that evaporation is also advantageous to have better lift-o during nanofabrication, certainly a process allowing to improve interfacial quality of evaporated Pt is required. In this paper, we investigate the impact of local Joule heating at 550 K on the spin transport between thin YIG lm and evaporated Pt. We use SMR and spin pumping measurements to show a clear 3 times post-deposition enhancement of the interfacial spin transmission, which is irreversible. We have exploited this feature to study the in uence of the interfacial spin conductivity on non- linear spin transport properties in lateral devices. We nd that the later are mostly governed by the physical properties of the bare YIG lm not covered by Pt. Any enhancement of the coupling to the electrodes seems to play a negligible role in determining the non-linear char- acteristics of pure spin transport. We use a tYIG = 56 nm thick YIG lm grown by liquid phase epitaxy on a 500 m Gd 3Ga5O12(GGG) substrate9,20. Ferromagnetic resonance experiments have shown a damping parameter of YIG= 2:2104re- vealing an excellent crystal quality of the YIG lm21. Two similar Pt nanostrips, respectively Pt 1and Pt 2, are patterned by e-beam lithography to have a width of 300 nm and length of 30 m. A 7 nm thick Pt layer is then deposited by e-beam evaporation on the YIG lm. The nanostrips are connected to Ti jAu (5 nmj50 nm) electrical contacts. The sample is mounted on a rota- tional stage and exposed to an in-plane magnetic eld of 0H0= 200 mT to fully magnetize the YIG lm. All the magneto-transport experiments are performed at roomarXiv:2009.02785v1 [cond-mat.mtrl-sci] 6 Sep 20202 FIG. 1. a) Schematic of the YIG jPt structure with two nanos- trips oriented along the y-direction. The rst one (Pt 1) is connected to a current source and it is used to measure the SMR ratio. The second electrode (Pt 2) is used to measure the non-local spin transport properties. b) Resistance of the Pt 1 nanostrip,RI=V1=I, as a function of the injected current Iinside. c) Angular dependence of the SMR ratio when an applied magnetic 0H0= 200 mT is rotated in the xyplane. The data shows the result before (blue dots) and after (red dots) local Joule annealing at 550 K. Solid lines are t with cos2'. temperature, T0. The schematic of the sample geometry is shown in Fig.1a). Transport measurements are per- formed by injecting 10 ms current pulses with a duty cycle of 10% into the Pt 1nanostrip via a 6221 Keithley current source which is synchronized to a 2182A Keithley nano- voltmeter. We rst investigate the conguration where the nanovoltmeter is connected to the same nanostrip to extract the magnetoresistive response through V122. Fig.1b) represents the evolution of the Pt 1resistance, RI, as a function of the electrical current, I, showing a quadratic dependence due to Joule heating. The Pt elec- trode can also be used as a temperature sensor. The tem- perature rise of the Pt is simply inferred from the change of Pt resistance with TT0=Pt(RI-R0)/R0, where R0= 2:6 k is the Pt resistance at I= 0 andPt= 478 K is a thermal coecient specic to our Pt nanostrip. In our structure, the local temperature reaches about 550 K when the current density is Jmax= 1:21012A/m2. The spin transparency of the interface can be evalu- ated from the SMR ratio dened as ( R'Rk)=R0, which measures the relative change of the Pt resistance as a function of ', the azimuthal angle between H0(and thus the magnetization) and the x-axis. In this notation Rkis the resistance when H0is applied parallel to the nanos- trip direction ( '=90ory-axis). Fig.1c) presents the SMR signal measured with a bias current of I= 100A. The data in blue dots show the values obtained directly after the nanofabrication process. The angular depen-dence follows a cos2'behavior (see solid line t), and the maximum SMR deviation is observed when H0is ap- plied perpendicular to the nanostrip direction ( '= 0 orx-axis) with a value of ( R?Rk)=R0=2:9105 extracted from the t (blue line). From the theory of SMR11,23, the amplitude of the SMR ratio is expressed as : R?Rk R0=2 SHE2(2 sf=tPt)g"#tanh2tPt 2sf 1 + 2sfg"#cothtPt sf;(1) where= 19:5 cm is the Pt resistivity with tPtits thickness,sfis the spin diusion length, and SHEis the SHE angle inside the Pt layer. In our notation, g"#is an eective spin mixing conductance of the YIG jPt interface (see discussion below). Using SMR measurement, we then investigate the eect of 550 K local Joule annealing on the eective spin mixing conductance of YIG jPt. The electrical annealing is provided by applying current density pulses ofJmax= 1:21012A/m2for about 60 minutes. The red dots shows the SMR ratio measured after. The amplitude is increased to ( R?Rk)=R0=8:9105 (red line t in Fig.1c), which is 3 times larger than the value before annealing (blue line). This enhanced SMR is irreversible and increasing the annealing time above an hour leads to negligible gain in the SMR value. We also observe that the Pt 1resistance is changed by less than 1%, indicating no major structural changes in the Pt, which suggests that SHEandsfremain the same throughout this treatment24,25. Knowing the product SHEsfto be 0.18 nm26, we conclude that the spin mix- ing conductance is improved from g"#= 0:641018m2 to 1.901018m2. The enhanced value is comparable to the ones obtained after Ar+-ion milling process14 and "piranha" etch16,17, which means that the observed increase of g"#is more a catching up of the decit of spin conductance probably associated with the use of evaporation rather than an overall improvement of the result from what is obtained by sputtering. Next, we focus on spin pumping measurements using the same sample batch. The experiment is performed at a xed frequency of 9.65 GHz in a X-band cavity while applying a static in-plane magnetic eld perpendicularly to the Pt nanostrip ( x-axis). Conversely the rf magnetic eldhrfis applied along the Pt nanostrip ( y-axis). The rf power is xed at 5 mW ( 0hrf= 2T) to maximize the SP signal while minimizing non-linear eects such as distorsion of the lineshape (foldover eect)27,28. At the ferromagnetic resonance condition, a ow of angular momentum from the YIG relaxes into the Pt29. The normalized spin-pumping spectra ispare obtained by dividing the generated inverse spin Hall eect (ISHE) voltage by both h2 rfand the Pt resistance R0,i.e.isp= VISHE=(R0h2 rf). The shape of the spectral line does not follow a simple Lorentzian, which is attributed to inho-3 FIG. 2. Spin pumping spectra at dierent annealing step by injecting a current density Jmax= 1:21012A/m2into the Pt nanostrip for three dierent annealing time. The inset shows the normalized spin pumping spectra at each annealing time. mogeneous broadening. The main peak of the spectrum is identied as the Kittel mode (uniform precession) and we have measured its amplitude to estimate the eciency of spin transmission through the interface. After char- acterizing the pristine YIG jPt interface, we performed a local annealing by applying pulse current density of Jmax=1.21012A/m2for various durations (10, 30 and 60 minutes) following the same procedure as before. The eect of annealing on the spectra are shown in Fig.2. Similar to the SMR measurement, Joule annealing in- creases the spin pumping signal. From the amplitude of the main peak of the spectra and using the model in ref.29, one can estimate the enhancement of g"#from 0.601018m2to 1.231018m2, compatible with our previous SMR estimation. The interesting feature of this experiment lies in the measurement of the full width at half maximum (FWHM) of the main peak. As the amplitude of the peak become larger with the annealing time, FWHM re- mains constant over the whole annealing process (see the inset of Fig.2). The modulation of FWHM can be in general attributed to the extra damping induced by the coupling between YIG and Pt30,31. The constant FWHM reveals that the linewidth is mostly controled by the bare YIG lm beside the nanometric Pt nanostrip, rather than the sole relaxation of precession dynamics at the YIG jPt interface. This suggests that the additional relaxation channel provided by the adjacent metallic nanostrip is a weak perturbation of the overall relaxation of the ex- tended YIG thin lm underneath. Since this nding dif- fers from what is observed when the adjacent Pt covers the whole YIG lm21, we attribute the dierence to - nite size eects. For nanostructured Pt electrodes, the additional relaxation channel of the magnons in the YIG provided by the adjacent metal becomes weak when the lateral size of the Pt is smaller than the magnon wave- length. We emphasize that such scenario would then mostly concern the long wavelength magnons, such as FIG. 3. Non-local spin signals  I(left) and
I(right) as a function of injected current for dierent annealing steps. The inset of the left panel displays the current variation of 0=I, where0is the initial slope of the spin current increase (see gray dashed line). The intercept with the 0.75 level is a land- mark that denes Jc13. The (a) panel represents non-local spin signals measured directly after the nanofabrication. The (b) panel shows the result when the injector is annealed. The (c) panel shows the result when both injector and detector nanostrip are annealed. those excited by the cavity. Finally we study the impact of local Joule annealing on the non-local spin transport properties. In these sam- ples, the second Pt nanostrip (Pt 2), placed 2 m away from the rst nanostrip (Pt 1), is used as a detector of the spin current (see Fig.1a). When a charge current is sent to the Pt 1nanostrip (injector), a spin accumu- lation is generated at the YIG jPt interface due to the SHE and its angular momentum is transferred from Pt to YIG32,33. The angular momentum is then carried in YIG by magnons , which are then detected via the ISHE volt- age at the Pt 2nanostrip (detector). Again, the non-local spin transport is probed by applying pulses of electrical current in the injector while simultaneously monitoring the non-local voltage V'on the detector as a function of the angle '. Similar to the SMR measurement, we4 use the background-subtracted voltage V'=V'Vk to extract the spin contribution13. The background is again measured by applying the external eld H0par- allel to the nanostrip direction ( y-axis). We distinguish SSE from SOT by dening two quantities based on the yzmirror symmetry13: ';I;
';I(V';IV';I)=2, where'='. Previously we have reported13that in 18 nm thick YIG lm, injecting electrical current above a cross-over threshold of the order of Jc=6.01011A/m2is sucient to excite low energy magnetostatic magnons (in the GHz range). This phenomenon is characterized by the emer- gence a non-linear spin conduction as a function of the applied current. In the following, we want to use this incident change of the interface transmission to investi- gate the in uence of non-equilibrium spin accumulation at the YIGjPt interface on the non-linear properties. Let us rst consider in Fig.3a) the pristine state where neither interface of YIG jPt at the injector nor the de- tector has been treated by Joule annealing. In the top panel a) of Fig.3, we display the non-local  '=0;Iand
'=0;Ias a function of the applied current in the injec- tor. In this conguration Pt 1is connected to the current source while we record V2the voltage drop across Pt 2(cf. Fig.1a). It can be seen that the
Ishows a quadratic rise due to its thermal origin while the  Iseems to evolve quasi-linearly with current Ion the range explore. The non-linear properties are analyzed by plotting in the in- set of Fig.3a) the current dependence of 0=I, where 0= (@I=@I)jI=0is the slope of a linear regression through the  Idata measured at I <0:5 mA (see gray dashed lines). We dene Jcas the intercept with the 75% decrease1334. We report on the main graph the estimate ofIc, which is here not precise due to the low spin con- ductivity of the device. Next, we perform Joule annealing of the injector (Pt 1) for 60 minutes with the same proce- dure as before. The impact on the non-local signals can be seen in the second panel b). We observe that the  I is now 3 times larger, indicating that the excited magnon density in YIG is enhanced due to the higher spin trans- mission at the injector. Remarkably, now we are able to distinguish clearly on the gure at Jc= 71011A/m2, the cross-over threshold from a linear to a non-linear spin transport regime, which is the signature of the participa- tion of low energy magnetostatic magnons to spin trans- port. Such magnons are in principle solely excited by SOT exerted at the interface between YIG and the injec- tor. We also note that the
I-signal produced by ther- mally generated magnons remains unchanged. This is expected because the spin conversion of SSE signal oc- curs only at the detector Pt 2nanostrip (which is not an- nealed for this moment) whereas the injector nanostrip only plays the role of a heater. The last step, shown in panel c), both injector and detector are annealed. The
Iis enhanced by a factor of 3 due to the higher spin conversion of the probing interface. As can be seen there too is that the  Iis now 9 times larger than the ref- erence (panel a). It follows the fact that injection anddetection of magnons at each YIG jPt interface are now 3 times more eective, leading to a factor of 9 increase of the SOT signals. Nonetheless, the crossover current den- sityJc13is not aected by the annealing and it occurs at the same value for both cases (b) and (c). It supports the observation in SP experiments, where local annealing hasn't induced additional broadening the linewidth (in- set of Fig.2). It also highlights the striking dierence of out-of-equilibrium behavior between closed (nano-pillar) and open (extended lms) magnetic geometries. A possible explanation compatible with the observed behavior could be an improved wettability of the Pt on YIG after local Joule annealing35. The eect could be parametrized by introducing an additional transmission coecient 0 <T61 to the spin transparency of the interface. This coecient represents for example the eective contact area ratio between the YIG and the Pt. This transmission alters the eective spin conductance g"#=TG"#, while retaining constant G"#the intrinsic spin mixing conductance inferred from the interfacial spin pumping contribution to the linewidth measured in YIG lms completely covered by Pt. Since the enhancement of a factor of 3 is consistently observed in all the devices, we believe that the change of contact area must have a geometrical origin probably linked to the nanofabrication process. Because the temperature does not exceed 550 K during the heating pulses, it is unlikely that the chemical and structural quality of the YIG surface is aected by this treatment36. Thus we expect the local Joule annealing to aect only the interface between Pt nanostrip and the YIG layer37and could result in a larger coverage of Pt onto the YIG surface. This improves the number of spin transmission channels at the interface which posses similar spin mixing conductance. Through this mechanism the emission, re ection or absorption of the spin current are enhanced. In summary, we consistently observed enhancement of the spin-induced voltage at the YIG jPt interface after local Joule annealing of evaporated Pt nanostrips. This enhancement possibly occurred through a higher cover- age of Pt on the YIG surface, increasing the number of spin transmission channels available. This results can also explain the large dierence in G"#,SHE andsf in works performed via dierent deposition and mea- surement methods38{40. Additionally the spin pumping measurements and non-local magnon transport measure- ments showed that an enhancement of the spin transmis- sion does not involve necessarily an increase or reduction of the cross-over threshold current to excite subthermal magnons, pointing out the important role of the bare YIG lm away from the Pt nanostrip in the relaxation of pure spin current transport. 1L. Cornelissen, J. Liu, R. Duine, J. B. Youssef, and B. Van Wees, Nature Physics 11, 1022 (2015). 2S. T. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn,5 M. Althammer, R. Gross, and H. Huebl, Applied Physics Letters 107, 172405 (2015). 3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, and K. Takanashi, Nature 464, 262 (2010). 4A. Brataas, B. van Wees, O. Klein, G. de Loubens, and M. Viret, Phys. Reports (2020), https://doi.org/10.1016/j.physrep.2020.08.006. 5V. Cherepanov, I. Kolokolov, and V. L'vov, Physics reports 229, 81 (1993). 6S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 7J. R. S anchez, L. Vila, G. Desfonds, S. Gambarelli, J. Attan e, J. De Teresa, C. Mag en, and A. Fert, Nature communications 4, 2944 (2013). 8B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur- ban, and G. E. W. Bauer, Physical Review Letters 90, 187601 (2003). 9C. Hahn, G. De Loubens, O. Klein, M. Viret, V. V. Naletov, and J. B. Youssef, Physical Review B 87, 174417 (2013). 10H. Wang, C. Du, P. C. Hammel, and F. Yang, Applied Physics Letters 110, 062402 (2017). 11H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kaji- wara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, et al. , Physical review letters 110, 206601 (2013). 12K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Applied Physics Letters 97, 172505 (2010). 13N. Thiery, A. Draveny, V. Naletov, L. Vila, J. Attan e, C. Beign e, G. de Loubens, M. Viret, N. Beaulieu, J. B. Youssef, et al. , Phys- ical Review B 97, 060409 (2018). 14S. V elez, A. Bedoya-Pinto, W. Yan, L. E. Hueso, and F. Casanova, Physical Review B 94, 174405 (2016). 15Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Applied Physics Letters 103, 092404 (2013). 16M. Jung eisch, V. Lauer, R. Neb, A. Chumak, and B. Hille- brands, Applied Physics Letters 103, 022411 (2013). 17S. P utter, S. Gepr ags, R. Schlitz, M. Althammer, A. Erb, R. Gross, and S. T. Goennenwein, Applied Physics Letters 110, 012403 (2017). 18Y. Saiga, K. Mizunuma, Y. Kono, J. C. Ryu, H. Ono, M. Kohda, and E. Okuno, Applied Physics Express 7, 093001 (2014). 19N. Vlietstra, J. Shan, V. Castel, B. Van Wees, and J. B. Youssef, Physical Review B 87, 184421 (2013). 20V. Castel, N. Vlietstra, B. Van Wees, and J. B. Youssef, Physical Review B 86, 134419 (2012). 21N. Beaulieu, N. Kervarec, N. Thiery, O. Klein, V. Naletov, H. Hurdequint, G. de Loubens, J. B. Youssef, and N. Vukadi- novic, IEEE Magnetics Letters 9, 1 (2018). 22N. Thiery, V. Naletov, L. Vila, A. Marty, A. Brenac, J.-F. Jacquot, G. de Loubens, M. Viret, A. Anane, V. Cros, et al. , Physical Review B 97, 064422 (2018). 23Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. Goennenwein, E. Saitoh, and G. E. Bauer, Physical Review B 87, 144411 (2013). 24E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Physical Review B 94, 060412 (2016). 25P. Laczkowski, Y. Fu, H. Yang, J.-C. Rojas-S anchez, P. Noel, V. Pham, G. Zahnd, C. Deranlot, S. Collin, C. Bouard, et al. , Physical Review B 96, 140405 (2017). 26J.-C. Rojas-S anchez, N. Reyren, P. Laczkowski, W. Savero, J.- P. Attan e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and H. Jar es, Phys. Rev. Lett. 112, 106602 (2014). 27Y. K. Fetisov, C. E. Patton, and V. T. Synogach, IEEE trans- actions on magnetics 35, 4511 (1999). 28Y. Gui, A. Wirthmann, and C.-M. Hu, Physical Review B 80, 184422 (2009). 29Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Physical Review B66, 224403 (2002). 30B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y.Song, Y. Sun, and M. Wu, Physical Review Letters 107(2011), 10.1103/physrevlett.107.066604. 31O. Mosendz, V. Vlaminck, J. Pearson, F. Fradin, G. Bauer, S. Bader, and A. Homann, Physical Review B 82, 214403 (2010). 32S. S.-L. Zhang and S. Zhang, Physical review letters 109, 096603 (2012). 33S. S.-L. Zhang and S. Zhang, Physical Review B 86, 214424 (2012). 34In a single mode model, 0=Ifollows a parabola, which in- tercepts the abscissa at the threshold current density for damp- ing compensation, Jth. In this picture, the 75% landmark corre- sponds to half this value Jc=Jth=2. 35For the sake of completeness, we have also annealed the whole sample in an oven at the same temperature during an hour. No signicant changes of the interfacial spin conductance were ob- served, indicating the importance of performing local Joule an- nealing. 36Y.-H. Rao, H.-W. Zhang, Q.-H. Yang, D.-N. Zhang, L.-C. Jin, B. Ma, and Y.-J. Wu, Chinese Physics B 27, 086701 (2018). 37E. T. Papaioannou, P. Fuhrmann, M. B. Jung eisch, T. Br acher, P. Pirro, V. Lauer, J. L osch, and B. Hillebrands, Applied Physics Letters 103, 162401 (2013). 38M. Jung eisch, A. Chumak, V. Vasyuchka, A. Serga, B. Obry, H. Schultheiss, P. Beck, A. Karenowska, E. Saitoh, and B. Hille- brands, Applied Physics Letters 99, 182512 (2011). 39N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. Bauer, and B. Van Wees, Applied Physics Letters 103, 032401 (2013). 40L. Liu, R. Buhrman, and D. Ralph, arXiv preprint arXiv:1111.3702 (2011).

2020-09-06

We report that Joule heating can be used to enhance the interfacial spin conductivity between a metal and an oxide. We observe that local annealing of the interface at about 550\,K by injecting large current densities ($>10^{12}\text{A/m}^{2}$) into a pristine 7\,nm thick Pt nanostrip evaporated on top of yttrium iron garnet (YIG), can improve the spin transmission up to a factor 3: a result of particular interest for interfacing ultra thin garnet films where strong chemical etching of the surface has to be avoided. The effect is confirmed by different methods: spin Hall magnetoresistance, spin pumping and non-local spin transport. We use it to study the influence of the YIG$|$Pt coupling on the non-linear spin transport properties. We find that the cross-over current from a linear to a non-linear spin transport regime is independent of this coupling, suggesting that the behavior of pure spin currents circulating in the dielectric are mostly governed by the physical properties of the bare YIG film beside the Pt nanostrip.

Enhancement of YIG$|$Pt spin conductance by local Joule annealing

2009.02785v1

arXiv:2106.09542v1 [physics.app-ph] 15 Jun 2021Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity Mei Wang,1,∗Cui Kong,2Zhao-Yu Sun,3Duo Zhang,3Yu-Ying Wu,3and Li-Li Zheng4,† 1School of Electrical Engineering, Wuhan Polytechnic Unive rsity, Wuhan, 430040, the People’s Republic of China 2College of Physics and Electronic Science, Hubei Normal Uni versity, Huangshi, 435002, the People’s Republic of China 3Department of mathematics and Physics, Wuhan Polytechnic U niversity, Wuhan, 430040, the People’s Republic of China 4Key Laboratory of Optoelectronic Chemical Materials and De vices of Ministry of Education, Jianghan University, Wuhan, 430074, People’s Republic of C hina (Dated: June 18, 2021) We propose an effective approach for creating robust nonreci procity of high-order sidebands, including the first-, second- and third-order sidebands, at microwave frequencies. This approach relies onmagnon Kerrnonlinearity in acavity magnonics sys tem composed of twomicrowave cavities and one yttrium iron garnet (YIG) sphere. By manipulating th e driving power applied on YIG and the frequency detuning between the magnon mode in YIG and the driving field, the effective Kerr nonlinearity can be strengthened, thereby inducing st rong transmission non-reciprocity. More interestingly, we find the higher the sideband order, the str onger the transmission nonreciprocity marked by the higher isolation ratio in the optimal detuning regime. Such a series of equally-spaced high-order sidebands have potential applications in frequ ency comb-like precision measurement, besides structuring high-performance on-chip nonrecipro cal devices. PACS numbers: 42.25.Bs, 03.65.Ta I. INTRODUCTION Recently years, cavity magnonics system, benefited from its unique features, has shown potential applica- tion prospects in information processing. Especially, the vast spin excitation in YIG sphere called magnon, which plays a vital role in the system. Benefited from the high spin density, strong and ultra-strong magnon- microwave photon coupling [ 1–6], observation of ex- ceptional point and precision magnetometry [ 7,8] have been demonstrated in experiment. Furthermore, thanks to the high tunability, extensibility and low dissipation of the magnon, complex information interaction plat- forms [9] can be structured via establishing the inter- action of magnon to an (artificial) atom [ 10], acoustic phonons [ 11–15], optical photons [ 16,17], or supercon- ducting coplanar microwave resonator [ 18]. In addition to above merits, another advantage of the system is the Kerr nonlinearity of magnon originated from the magne- tocrystalline anisotropy in YIG sphere. Based on Kerr nonlinearity, a series of studies [ 19–22] have been carried out, and one of the most notable is the nonreciprocity. Nonreciprocity , associatedwiththebreakingofLorentz reciprocity [ 23,24], is manifested as asymmetric propa- gation of signal in two opposite directions. Nonrecipro- cal physics has vast and valuable applications in com- munication technology and information processing. For examples, one-way signal communication and manufac- turing chip-scale nonreciprocal devices: isolator, circu- lator, diode. Stimulated by the bright prospect, con- ∗Electronic address: wangmei 2@vip.163.com †Electronic address: zhenglili@jhun.edu.cnsiderable efforts has been made in the study of nonre- ciprocal physics. One of the most watched is the non- reciprocity of electromagnetic fields [ 25,26], which is mainly concerned with the nonreciprocal propagation of electromagnetic fields in the microwave and optical re- gions. Microwave and optical nonreciprocity are both typicallyachievedfrom the magneto-opticcrystal[ 27–31] relying on Faraday-rotation [ 32–34] to break the Lorentz reciprocity. However, the tricky thing is the bulky and heavy structure of the magneto-optic crystal, that poses a serious hurdle to flexibly control and chip-scale inte- gration. Over the past few decades, alternative mecha- nisms towardsrealizing nonreciprocal propagation at mi- crowaveand opticalfrequencieshavebeen proposed. The most promising ones are based on nonlinearity [ 35–44], reservoir engineering [ 45,46], parametric time modula- tion [47,48]. Inspired by previous studies, based on nonlinearity mechanism, we propose a flexible scheme to achieve ro- bust nonreciprocity of the first-, second-, and third-order sidebands at microwave frequencies. This approach is implemented in a three-mode cavity magnonics system, which consists of two coupled microwave cavities and a YIG (or another suitable magnonic material) sphere. Due to the magnon Kerr nonlinearity in YIG sphere, strong nonreciprocity of high-order sidebands can be ob- tained by flexibly manipulating the driving power in- jected on YIG sphere and the frequency detuning be- tween magnon mode (in YIG) and the driving field. On one hand, the driving power applied on YIG affects exci- tation of magnon that determines the effective strength of the Kerr nonlinearity. It therefor induces varying de- greesasymmetric responsesofthe system in two opposite directions. On the other hand, frequency detuning be- tween the driving field and the magnon mode also affects2 magnon excitation in YIG sphere. Thus the effective Kerr nonlinearity will be influenced, as well as degree of the asymmetric responses in two directions. More inter- estingly, in the optimal parameters regime, the greater the order of the sideband, the stronger transmission non- reciprocity marked by the enhanced isolation ratio. Such aseriesofequally-spacedhigh-ordersidebandscanbe ap- plied in frequency comb-like precision measurement, Be- sides structuring nonreciprocal devices. In addition, fre- quenciesofthesidebandsavoidthemaincavityfrequency and this will strongly inhibit dissipation. The system pa- rameters are referenced from recent experiments [ 19,49], specially, Kerr nonlinear coefficient ( Kreaches 102nHz refereingto[ 50]) in the systemis highlydependent onthe specific experimental implementation. This study broad- ens applications of the cavity magnonics system in infor- mation processing. This paper is organized as follows: In Sec. II, we introduce theoretical scheme of the three-mode cavity magnonics system and its effective Hamiltonian with ex- ternal pump fields. Then we derive transmission co- efficients and isolation ratios of the first-, second- and third-order sidebands in two cases. In Sec. III, we study transmission nonreciprocity of the high-order sidebands throughmanipulatingthe drivingfieldonthe YIGsphere and the frequency detuning between driving field and the magnon mode in YIG sphere. Conclusions are finally drawn in Sec. IV. (b) ǻp Qǻ pYIG Cavities stokes field Ȧ (3) (1) (2) high-order sidebands (n) Ȧm /ȦdȦc Ȧp(a) acavity xya saJ cavity b sbYIG m bHz ǻc į ǻp ǻp FIG. 1: (Color online) (a) Schematic diagram of a three-mode cavity magnonics system consisting of two coupled microwav e cavities and a YIG sphere. (b) Frequency spectrogram of the cavity magnonics system. Here the driving field (with amplitude ε) applied on YIG sphere and the control fields (withamplitudes εa,εb)injectedfrommicrowavecavitieshave the same frequency ωd. Frequencies of the control fields and the probe fields (with amplitudes sa,sb) are respectively de- tuned from the cavities’ frequency by ∆ candδ. And the frequency detuning between the probe field and the control field is ∆ p. There are higher-order sidebands at frequency ωd+n∆p, where nis an integer representing the sideband order.II. THE MODEL AND FORMULA DERIVATION We consider a three-mode cavity magnonics sys- tem [19] as shown in Fig. 1(a). It is composed of two three-dimensional rectangular microwave cavities ( aand b, frequency ωc) and one YIG sphere, where collective spin excitation is quantified as magnon ( m, frequency ωm). The cavities are linearly coupled to each other with coupling rate J; and each of them is driven and detected by a bichromatic microwave source, i.e., a strong control field with amplitude εa(εb) at frequency ωdand a weak probe field sa(sb) at frequency ωp. The YIG sphere, located in one cavity, is exposed to a uniformly DC mag- netic bias field ( Hz) in the zdirection. Meanwhile, it interacts with one cavity with strength g, and is pumped by a microwave driving field with amplitude εand fre- quencyωd. Note that amplitudes of the input microwave fields can be converted into corresponding powers, i.e., Pj=/planckover2pi1ωdε2 j,P=/planckover2pi1ωdε2andPsj=/planckover2pi1ωps2 j(j=a,b) with control power Pj, driving power Pand probe power Psj. In the rotating frame, including all external microwave sources, the system Hamiltonian (referingto [ 19,49]) can be written as ˆHtot//planckover2pi1= ∆c(ˆa†ˆa+ˆb†ˆb)+∆mˆm†ˆm+Kˆm†ˆmˆm†ˆm +J(ˆa†ˆb+ˆaˆb†)+g(ˆb†ˆm+ˆbˆm†) +i√ηaκaεa(ˆa†−ˆa)+i√ηbκbεb(ˆb†−ˆb) +i√ηaκasa(ˆa†e−i∆pt−ˆaei∆pt) +i√ηbκbsb(ˆb†e−i∆pt−ˆbei∆pt) +i√ηmκmε(ˆm†−ˆm), (1) here ∆ m=ωm−ωd, ∆p=ωp−ωcand ∆ c=ωc−ωd is constant. In addition, the frequency detunings satisfy relationship ∆ p= ∆c+δ, whereδ=ωp−ωcrefereing to Fig.1(b). The third term in Hamiltonian is magnon Kerr nonlinear term originating from the magnetocrys- talline anisotropyin the YIG sphere. The Kerr nonlinear coefficient K=µ0Kanγ2/(M2Vm), where µ0is the mag- netic permeability of free space, Kanis the first-order anisotropy constant, γis the gyromagnetic ratio, Mis the saturation magnetization, and Vmis the volume of the YIG sphere. κj(j=a,b,m) represents the total dis- sipation of any microwave cavity or magnon mode with cavity (magnon) coupling parameter ηj. In this work, we employ analytic solution method to solve Heisenberg-Langevin equations. It not only de- scribes the dynamical evolution of the cavity magnon- ics system, but also responses the expectation values of all operators, namely, /angbracketleftˆo/angbracketright=o(ˆois a normal annihilation operator). Takingaccountdampingprogressesinthesys- tem, Heisenberg-Langevinequationsofthe systemcanbe3 expressed as ˙a=−(i∆c+κa 2)a−iJb+√ηaκaεa +√ηaκasae−i∆pt, (2a) ˙b=−(i∆c+κb 2)b−iJa−igm+√ηbκbεb +√ηbκbsbe−i∆pt, (2b) ˙m=−(i∆m+κm 2)m−igb−i(2K|m|2+K)m +√ηmκmε. (2c) Equations ( 2a) and (2b) describe the dynamics of cav- itiesaandb. Equation ( 2c) describes the dynamics of the magnon mode. The cubic term −2iK|m|2min Equation ( 2c) presents the nonlinearity of the system. It is derived from the Kerr nonlinearity term in Hamil- tonian (1) with factorizing averages approximation, i.e., /angbracketleftbc/angbracketright=/angbracketleftb/angbracketright/angbracketleftc/angbracketright. The input quantum and thermal noise terms are ignored here because their mean values are zero. Forexistenceofthenonlinearterm, equations( 2a)-(2c) are inherently nonlinear and cannot be solved directly and precisely. However, due to the strength of the con- trol fields (5 mW) are far stronger than the probe fields (1nW) in the system, we can take perturbation method tosolveequations( 2). Thatisdividingeachoperatorinto FIG. 2: (Color online) The steady-state microwave photon numbers |α|2,|β|2and magnon number |̟|2versus with the resonant driving power Pon the magnon mode in two cases. (a) case 1: Pa= 5mW,Psa= 1nW,Pb=Psb= 0; (b) case 2:Pb= 5mW,Psb= 1nW,Pa=Psa= 0. Other pa- rameters are given as ωc/2π= 10.1 GHz,ωm/2π=ωd/2π= 10.06 GHz, ∆ c/2π= 40 MHz, ∆ m= 0,κa/2π= 3.8 MHz, κb/2π= 5 MHz, κm/2π= 20 MHz, δ/2π=−12 MHz, K/2π= 3.9×10−7Hz,J/2π= 12 MHz, g/2π= 8 MHz.two parts: a steady-state mean value and a perturbation term, i.e., a=α+δa,b=β+δb,m=̟+δm. It means that the steady-state mean values of the operators are determined by the strong control fields, and the pertur- bation terms are caused by the weak probe fields. When the probe fields are absent, with dα/dt= 0,dβ/dt= 0 andd̟/dt= 0, the mean values of the operators from the steady-state equations can be derived as α=√ηaκaεa−iJβ i∆c+κa 2, (3) β=g(∆c−iκa 2)̟−iJ√ηaκaεa+(i∆c+κa 2)√ηbκbεb (i∆c+κa 2)(i∆c+κb 2)+J2, (4) ̟=−gJ√ηaκaεa+g(∆c−iκa 2)√ηbκbεb/bracketleftbig (i∆c+κa 2)(i∆c+κb 2)+J2/bracketrightbig Θ+√ηmκmε Θ, (5) where we have defined Θ =g2(i∆c+κa 2) (i∆c+κa 2)(i∆c+κb 2)+J2+κm 2 +i(∆m+2K|̟|2+K). (6) In Fig.2, evolutions of the steady-state microwave- photon numbers and the magnon number versus the res- onant driving power on magnon mode are shown in two different cases. Figure 2(a) corresponds to case 1, only cavityaisdrivenbythebichromaticmicrowavefieldsand cavitybis free of driven. In case 2, the same bichromatic microwavefields are only applied on cavity bin Fig.2(b). As we can see the steady-state microwave photon num- bers and the magnon number in case 1 are larger than those in case 2 even with the same driving power P. In addition, in both cases the microwave-photon numbers in cavity aare smaller than that in cavity b, though the dissipation of cavity ais lower than that of cavity band the magnon mode, i.e., κa< κb,κm(on the basis ofg J>1 3andK≥1nHz). The result illustrates that the counterintuitive asymmetric responds of the steady- state mean values are affected by the nonlinearity term in equations ( 2). Meanwhile, it also plays an important role in evolutions of the perturbation terms. Expandingequations( 2a)-(2c) with the ansatzofoper- ators and removingthe steady-state terms, the perturba- tion terms canbe expressedby the linearizedHeisenberg- Langevin equations δ˙a=−(i∆c+κa 2)δa−iJδb+√ηaκasae−i∆pt,(7a) δ˙b=−(i∆c+κb 2)δb−iJδa−igδm +√ηbκbsbe−i∆pt, (7b) δ˙m=−[i(∆m+4K|̟|2+K)+κm 2]δm−igδb −i2K(̟2δm∗+̟∗δm2+2̟δm∗δm) −i2Kδm∗δm2, (7c)4 the quadratic terms −i2K̟∗δm2,−i4K̟δm∗δmin equation ( 7c) denote the system nonlinearity and can not be sneezed at. They inevitably cause asymmetric re- sponses of the perturbation terms, just as the responses of the steady-state mean values in Fig. 2. However, the cubicterm −i2Kδm∗δm2inequations( 7)issosmallthat it will be ignored in the following calculation. To study asymmetric responses of the perturbations from a new perspective, we explore the nonreciprocal transmission of the output fields, mainly including the first-, second- and third-order sidebands, in case 1 and case 2. To get analytical solutions of these high-order sidebands, we define the perturbation terms have the fol- lowing forms δa=A− 1e−i∆pt+A+ 1ei∆pt+A− 2e−2i∆pt+A+ 2e2i∆pt +A− 3e−3i∆pt+A+ 3e3i∆pt, (8a) δb=B− 1e−i∆pt+B+ 1ei∆pt+B− 2e−2i∆pt+B+ 2e2i∆pt +B− 3e−3i∆pt+B+ 3e3i∆pt, (8b) δm=M− 1e−i∆pt+M+ 1ei∆pt+M− 2e−2i∆pt+M+ 2e2i∆pt +M− 3e−3i∆pt+M+ 3e3i∆pt, (8c) this ansatz is rooted from the internal mechanism of the system, where a series of high-order sidebands [ 51] with frequency ωd+n∆p(n= 1,2,3...) are generated. n represents the sideband order as shown in Fig. 1(b). The negativeandpositiveexponentsofthepowersareinpairs and respectively denote the upper and lower sidebands. For examples, the term A− 1e−i∆ptdenotes the first-order upper sideband with coefficient A− 1, andA+ 1ei∆ptcor- responds to the first-order lower sideband (the Stokes field [52] in Fig. 1(b)) with coefficient A+ 1. For the sake of simplicity, in what follows we only take the high-order upper sidebands into consideration, because of similar characteristics in the lower sidebands. Of course the cor- responding lower sidebands can also be studied by utiliz- ing the same method. With the ansatz we can analytically solve equa- tions (7), the coefficients of the high-order sidebands can be derived, respectively. Firstly, the coefficients of the first-order upper sideband are derived as B− 1=i√ηbκbsb(∆p−∆c+iκa 2)(iC −∆p+2̟2KL) g2(∆p−∆c+iκa 2)+(iC −∆p+2̟2KL)S(−) +iJ√ηaκasa(iC −∆p+2̟2KL) g2(∆p−∆c+iκa 2)+(iC −∆p+2̟2KL)S(−), (9a) A− 1=JB− 1+i√ηaκasa ∆p−∆c+iκa 2, (9b) M− 1=−gB− 1 iC −∆p+2K̟2L, (9c) M+ 1=L∗M−∗ 1, (9d)with C=−i(∆m+K+4K|̟|2)−κm 2, (10) S(±)= (∆p±∆c+iκa 2)(∆p±∆c+iκb 2)−J2,(11) L=2KS(+)̟2∗ g2(∆p+∆c+iκa 2)−(∆p−iC)S(+).(12) For the second-order upper sideband, the coefficients are given as B− 2=g(2∆p−∆c+iκa 2)M− 2 G, (13a) A− 2=JB− 2 2∆p−∆c+iκa 2, (13b) M− 2=−2KG(̟2E∗+̟∗M−2 1+2̟M− 1M+∗ 1) (iC −2∆p+2K̟2D∗)G+g2(2∆p−∆c+iκa 2), (13c) M+ 2=DM−∗ 2+E, (13d) where D=2̟2KF g2(2∆p+∆c−iκa 2)−F(2∆p+iC),(14) E=2̟∗KFM+2 1+4̟KFM+ 1M−∗ 1 g2(2∆p+∆c−iκa 2)−F(2∆p+iC),(15) F= (2∆ p+∆c−iκb 2)(2∆p+∆c−iκa 2)−J2,(16) G= (2∆ p−∆c+iκb 2)(2∆p−∆c+iκa 2)−J2.(17) To the third-order upper sideband, the coefficients have the following forms B− 3=g(3∆p−∆c+iκa 2)M− 3 (3∆p−∆c+iκa 2)(3∆p−∆c+iκb 2)−J2,(18a) A− 3=JB− 3 3∆p−∆c+iκa 2, (18b) M− 3=2K̟2H∗ ++H−(3∆c+gT∗−iC∗) (3i∆p+C −igQ)(−3∆p−gT∗+iC∗)−4iK2|̟|4, (18c) M+ 3=2̟2KM−∗ 3+iH+ −3∆p−gB3−iC, (18d) in which Q=g(3∆p−∆c+iκa 2) (3∆p−∆c+iκa 2)(3∆p−∆c+iκb 2)−J2,(19) T=−g(3∆p+∆c−iκa 2) (3∆p+∆c−iκa 2)(3∆p+∆c−iκb 2)−J2,(20) H+=−4i̟∗KM+ 1M+ 2−4i̟K(M+ 1M−∗ 2+M+ 2M−∗ 1), (21) H−=−4i̟∗KM− 1M− 2−4i̟K(M− 2M+∗ 1+M− 1M+∗ 2). (22)5 It can be seen that the coefficients A− 1,B− 1,A− 2,B− 2, A− 3andB− 3of the high-order sidebands are closely de- pendent on the magnon mode including the steady-state amplitude and the perturbation terms. With above analytical solutions, we calculate the out- put fields of each sideband in two cases. In case 1, the control and probe fields are injected from cavity aand the output fields from cavity bare explored. With input- output relation, the output fields of high-order sidebands from cavity bcan be expressed as saout=√ηbκbB− 1e−i(ωd+∆p)t+√ηbκbB− 2e−i(ωd+2∆p)t +√ηbκbB− 3e−i(ωd+3∆p)t. (23) In the direction from cavity atob, the transmission co- efficients of the first-, second- and third-order sidebands are respectively given as tba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB− 1 sa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24a) τba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB− 2 sa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (24b) ℓba=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηbκbB− 3 sa/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (24c) In case 2, the same control and probe fields are ap- plied on cavity band the output fields from cavity aare researched. In order to avoid confusion, we use A− j,B− j andM− j(j= 1,2,3) to distinguish from the coefficients in case1. Combiningthis condition andthe input-output relation, output fields of the high-order upper sidebands are given as follows sbout=√ηaκaA− 1e−i(ωd+∆p)t+√ηaκaA− 2e−i(ωd+2∆p)t +√ηaκaA− 3e−i(ωd+3∆p)t. (25) Similarly, along the direction from cavity btoa, the transmission coefficients of the first-, second- and third- order sidebands are defined as tab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA− 1 sb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (26a) τab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA− 2 sb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (26b) ℓab=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle√ηaκaA− 3 sb/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (26c) In order to more clearly describe nonreciprocal trans- missions of the high-order sidebands in two cases, we introduce definition of the nonreciprocal isolation ratio, which is expressed as Io=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglelog10|√ηbκbB− j|2 |√ηaκaA− j|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (27)hereostands for either t,τorℓcorresponding to j= 1,2,3.It,IτandIℓrespectively denote isolation ratio of the first-, second- and third-order sidebands. Then we use isolation ratio Ioto quantitatively describe de- gree of the nonreciprocal transmissions in two cases. It is decided by the value of Io: ifIo= 0, equivalently, |√ηbκbB− j|2/|√ηaκaA− j|2= 1, it means transmissions of thehigh-ordersidebandsarereciprocal(symmetry)along two opposite directions. For other situations, the higher the value of Io, the stronger is the nonreciprocity of the transmissions in two cases. For example, Io≥1, which illustrates transmission coefficient in one direction is at least 10 times that of the other direction. It also shows that transmittances along two opposite directions have a strong asymmetry. This situation can be approximated as one-way transmission of high-order sidebands. III. NONRECIPROCAL TRANSMISSIONS OF THE HIGH-ORDER SIDEBANDS IN A CAVITY MAGNONICS SYSTEM Since the transmission coefficients in equations ( 24) and (26) and the isolation ratio in equation ( 27) are mainly dominated by coefficients of each sideband, which are closely associated with the steady-state amplitude and the perturbation terms of magnon mode. Under this situation, we analyze influence of the driving power, ap- plied on the magnon mode, and the frequency detuning between the driving field and the magnon mode on the nonreciprocal transmission of each sideband. 0 1 2 3 4051015 FIG. 3: (Color online) The logarithmic amplitudes lo g10S(ω) of the transmitted high-order sidebands vs frequency ω/∆p in case 1 and case 2. Here P= 15mW, ∆ p/∆c= 0.7, other parameters are same as in Fig. 2. For the same purpose, but utilizing a different method, we first exhibit an overview comparison of the high-order sideband spectrums between case 1 and case 2. Here the magnon mode is resonantly driven by a 15 mW driv- ing field. As shown in Fig. 3, spectrums of the adjacent sidebands are equally spaced apart from each other at6 frequency ∆ pin the rotating frame. Where ω/∆p= 0 denotes the control field, ω/∆p=n(n= 1,2,3) cor- responds to the nth-order sideband. It can be seen that there exists amplitude gap between the sidebands with the same order in case 1 and case 2. Scilicet, the transmissions of the high-order sidebands are non- reciprocal in two cases. What’s more, amplitudes of the high-order sidebands decrease rapidly as the side- band order increases. Specifically, S(∆p)/S(0)<10−3, S(2∆p)/S(0)<10−7andS(3∆p)/S(0)<10−12in both cases. This proves that it is valid to regard high-order sidebands as perturbation comparedwith the strongcon- trol field. Then, we analyze in detail the transmission nonreciprocity of each sideband in the following content. A. Dependence of nonreciprocal transmissions for high-order sidebands on the resonant driving power P Figure4shows transmission coefficients |tba|2and |tab|2vary with frequency detuning δ/∆cunder differ- ent resonant driving powers P. In Fig. 4(a), we choose driving power P= 0. Under this situation, |tba|2and |tab|2aresmallerthan0 .2andevolutionsofthem aresyn- chronous. It means the transmissions of the first-order FIG. 4: (Color online) Transmission coefficients |tba|2and |tab|2of the first-order sideband vs frequency detuning δ/∆c. To observe transmission coefficients clearly, |tab|2in (b) and |tba|2in (c) have been increased by one order of magnitude. The driving powers P= 0, 20 mW, 30mW in (a), (b) and (c), respectively. Other parameters are same as that in Fig. 2.sidebandintwocasesarereciprocalwhenthedrivingfield on magnon mode is absent. Increasing driving power to 20mW in Fig. 4(b), intuitively, only a peak with value 150 of|tba|2is seen at δ/∆c=−0.3; and a peak value 5.6 of|tab|2is shown at δ/∆c=−0.25. However, if one zooms in locally, a local peak of |tba|2and a tiny one of|tab|2can be found at the position of the blue-dotted line. This illustrates that the first-order sideband (probe field) is magnified hundreds of times compared with the original input probe field and mainly transmitted at δ/∆c=−0.3 in case 1. Whereas, in case 2, magnifica- tion and transmission of the first-order sideband mainly occurs at δ/∆c=−0.25. From another point of view, the transmissions of the first-order sideband in two cases are robustly nonreciprocal near δ/∆c=−0.3. Continue to enhance the driving power to 30 mW in Fig. 4(c), we find|tba|2with two peaks is always smaller than 2; while |tab|2reaches its maximum value 15 .5 atδ/∆c=−0.31, with the other invisible peak at the blue-dotted line. The results illustrate that the first-order sideband is magni- fied less than twice in case 1, but more than ten times in case 2; and the strong transmission nonreciprocity ex- ists near δ/∆c=−0.3. From above numerical analysis, we find the transmission nonreciprocity of the first-order sideband is effectively modulated by the driving power acted on the magnon mode. This stems from the fact thatthe increased driving power excites more magnon polarons and simultaneously enhances the effective Kerr FIG. 5: (Color online) Transmission coefficients τba,τaband ℓba,ℓabof the second- and third-order sidebands vs frequency detuning δ/∆c. In order to clearly observe the transmission coefficients, the values of τabin (b) and ℓabin (e) have been respectively magnified by three and five orders of magnitude. HereP= 0 in panels (a) and (d), P= 20mW in panels (b) and (e), and P= 30mW in panels (c) and (f). Other parameters are same as that in Fig. 2.7 nonlinearity of the magnon ,which eventually strengthens the nonlinear asymmetric responses of the system in two cases. In addition, it needs to be emphasized that peaks on each curve should have been resonant with the effec- tivesupermodesmadeupoftwomicrowavecavitymodes. Frequencies of the supermodes shift ±/radicalBig J2−(κb−κa 2)2 (equal to ±0.3∆c) from the original cavity frequency ωc. This is originated from the strong cavity-cavity coupling rate, i.e., J > κ a,κbin the system. Practically, the ac- tural resonant frequency exhibit shifts due to the joint interference of the strong cavity-magnon coherent inter- action (g > J) and the magnon Kerr nonlinearity [ 20]. Figure5shows transmission coefficients τba,τaband ℓba,ℓabof the second- and the third-order sidebands versus frequency detuning δ/∆cwith different resonant driving powers. When the driving field is absent in Figs.5(a) and 5(d), although the transmission coeffi- cients are very tiny, the transmission nonreciprocity is very obvious in the regime δ/∆c∈[−0.5−0.2]. Once thedrivingfieldisconsidered,boththetransmissioncoef- ficients and nonreciprocity will be further enhanced. For instance, P= 20mW in Figs. 5(b) and 5(e),τbaand ℓbareach respective unique maximum 0 .7 and 1.5×10−3 atδ/∆c=−0.3. While τabandℓabkeep much smaller maximums. Continue to increase the driving power to 30mW in Fig. 5(c) and 5(f). Both τbaandℓbahave very tiny maximums. While τabandℓabhave soared to their peaks 4 .8×10−3and 4.6×10−7at the point δ/∆c=−0.31. This illustrates that the resonant driving field can not only effectively manipulate the transmis- sion nonreciprocity of the first-order sideband, but also of the second- and the third-order sidebands. What’s more, the transmission nonreciprocity of the high-order sidebands have a cooperative and consistent characteris- tic when facing different driving powers. Besides being applied in high performance nonreciprocal directional- switching isolator[ 33,53] and diode [ 54–56], such a series -0.4 -0.2 0 0.2 0.401020304050 012345 FIG. 6: (Color online) The isolation ratio Itof the first-order sideband vs the resonant driving power Pand frequency de- tuningδ/∆c. Parameters are the same as that in Fig. 2.of equally spaced high-order sidebands also have poten- tial applications in frequency comb-like precision mea- surement through simply operating the driving power. Then, we quantitatively describe the degree of trans- mission nonreciprocity for each sideband by isolation ra- tio. Figure 6intuitively displays isolation ratio Itvaries with the resonant driving power Pand frequency detun- ingδ/∆c. As the driving power increases, one bright re- gion, composed of an upper and a lowerparts, appearsat the center of δ/∆c=−0.3. In this region the optimal It is 5.6 dB obtained with P= 21.8mW. This can also be explained by the result in Fig. 4, i.e., the prominent large differencesoftransmissioncoefficients |tba|2and|tab|2be- tween the left resonant peaks near δ/∆c=−0.3. For the second- and third-order sidebands, the cor- responding isolation ratios IτandIℓversus the driv- ing power Pand frequency detuning δ/∆care given in Fig.7(a) and7(b). In Fig. 7(a), the value of Iτin most areas is lower than 3 .5 dB, except two bright areas cen- tered at δ/∆c=−0.3, where Iτreaches its maximum 9.9 dB. While the distribution of Iℓis quite different. As presented in Fig. 7(b), except a small dark area, the values of Iℓin other areas are higher than 10 dB. Es- pecially in the yellow area centered at δ/∆c=−0.4 withP∈[0 10]mW and the other one centered at δ/∆c=−0.3 withP∈[22 28]mW,Iℓcan be higher than 30 dB. By an overallobserving, what IℓandIτhave in common is that the optimal value is distributed near δ/∆c=−0.3, this is consistent with the result in Fig. 5. TakentogetherFig. 6andFig. 7, wefind that the isola- tion ratios It,IτandIℓshow an overall increasing trend with the increase of the sideband order. Specifically, op- timalIt,IτandIℓare enhanced in turn from 5 .6 dB, to 9.9 dB and lastly 30 dB near detuning δ/∆c=−0.3. The reasonis rooted fromthe strengthened effective Kerr nonlinearity in each sideband. Besides the steady-state amplitude of the magnon mode, the first-order sideband is also associated with the first-order perturbation terms (in equations ( 9)) of the magnon mode; the second-order sideband is simultaneously related to the first- and the second-orderperturbation terms of the magnon mode (in -0.5 -0.4 -0.3 -0.2 -0.101020304050 02468 -0.5 -0.4 -0.3 -0.2 -0.101020304050 51015202530 FIG. 7: (Color online) isolation ratios (a) Iτof the second- order sideband and (b) Iℓof the third-order sideband vs the resonant driving power Pand frequency detuning δ/∆c. Pa- rameters are same as that in Fig. 2.8 equations. ( 13)); and the third-order sideband is totally related to the first-, second-and third-orderperturbation terms (in equations ( 18)) of the magnon mode. Except for the steady-state amplitude of the magnon, the strong transmission nonreciprocity of each sideband originates from the aggregatenonlinear effects in each perturbation term. Therefor, the higher the order of the sideband, the stronger nonlinear asymmetry of the transmission, it lastly leads to much higher isolation ratio. FIG. 8: (Color online) The isolation ratios (a) It, (b)Iτ and (c) Iℓvsδ/∆cwith different detunings ∆ mbetween the magnon mode and the correspong driving field. Here P= 25mW, and the other parameters are the same as in Fig.2. B. Dependence of the nonreciprocal transmissions for high-order sidebands on frequency detuning ∆m In this section, we research influence of frequency de- tuning ∆ monthe nonreciprocaltransmissionofthe high- ordersidebands. InFig. 8,Isolationratios It, IτandIℓvs δ/∆cwith different detunings ∆ mare intuitively given. Whenmagnonmodeisresonantwiththedrivingfieldi.e., ∆m= 0, isolation ratios It, IτandIℓincrease in turn. Especially in the optimal detuning regime (centered at the resonant frequency of one effective supermode of the microwave cavities [ 20]),It,IτandIℓreach their max- imum values. The same evolution trend is also suitable for non-resonant situations, i.e., ∆ m/2π=−19.2 MHz, 20 MHz, except for the reduced optimal isolation ratios and the changed optimal detuning regimes. It is because excitation of the magnetic polarons is suppressed in thenon-resonant situations and results in much weaker effec- tive Kerr nonlinearity . This lastly induces much weaker transmission nonreciprocity and shifts the optimal de- tuning regime. For actual operation to obtain strong nonreciprocity of the high-order sidebands, the bandwidth of the op- timal detuning regime is often more concerned. In the resonant situation, we centralize It,IηandIℓin Fig.9. It can be obviously seen It,IηandIℓincrease in turn FIG. 9: (Color online) The isolation ratios It,Iτand (c)Iℓ vsδ/∆cwhen the magnon mode is resonant with the driving field, i.e., ∆ m= 0. Here P= 25mW and the other parame- ters are the same as in Fig.2. and reach their optimal values in the gray shaded area. This means an overall controlling of the strong nonre- ciprocity for high-order sidebands can be realized in the optimal detuning regime ranging from δ/∆c=−0.301 to δ/∆c=−0.273. The bandwidth of the optimal detun- ing regime is seven megahertz with ∆ c= 2π×40 MHz in the system. It is achievablefor experimentaloperationto capture strong transmission nonreciprocity of the high- order sidebands simultaneously in such a bandwidth. IV. CONCLUSION In summary, we have theoretically provided a method torealizeanoverallcontrollingofthe strongtransmission nonreciprocity of the first-, second- and third-order side- bands in a cavity magnonics system. It is consist of two coupledmicrowavecavitiesandoneYIG (oranothersuit- able magnonic material) sphere. Our approach utilizes the self-Kerr nonlinearity of magnon in the YIG sphere. We have shown that strong transmission nonreciprocity of the high-order sidebands can be achieved by respec- tively regulating the driving power on the magnon mode and the frequency detuning between the magnon mode and the driving field. We also showed that the higher the order of the sideband, the stronger is the nonreciprocity markedby the enhancedisolationratioin the optimalde- tuning regime. We have illustrated that the bandwidth9 of the optimal detuning regime can reach several mega- hertz when magnon mode is resonant drived. This im- plies that it is experimentally feasible to capture robust transmission nonreciprocity of the high-order sidebands simultaneously. This study providesa promisingroute to realize strong nonreciprocity of the high-order sidebands at microwave frequencies and has potential applications in frequency comb-like precision measurement.Acknowledgments We acknowledge professor Xin-You L¨ u for his valuable suggestions on our work. And this work was supported by the National Natural Science Foundation of China (NSFC) under Grants12005078,11675124and11704295. [1] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev and M. E. Tobar, High-Cooperativity Cavity QED with Magnons at Microwave Frequencies, Phys. Rev. Applied. 2, 054002 (2014). [2] S. Sharma, B. Z. Rameshti, Y. M. Blanter and G. E. W. Bauer, Optimal mode matchingin cavityoptomagnonics, Phys. Rev. B 99, 214423(2019). [3] C. Braggio, G. Carugno, M. Guarise, A.Ortolan and G. Ruoso, Optical Manipulation of a Magnon-Photon Hy- brid System, Phys. Rev. Lett. 118, 107205 (2017). [4] X. Zhang, C.-L. Zou, L. Jiang and H. X. Tang, Strongly Coupled Magnons and Cavity Microwave Photons, Phys. Rev. Lett. 113, 156401 (2014). [5] J. T. Hou, L. Liu, Strong coupling between microwave photons and nanomagnet magnons, Phys. Rev. Lett. 123, 107702 (2019). [6]¨O. O. Soykal and M. E. Flatt´ e, Strong Field Interactions between a Nanomagnet and a Photonic Cavity, Phys. Rev. Lett. 104, 077202 (2010). [7] N. Crescini, C. Braggio, G. Carugno, A. Ortolan and G. Ruoso, Cavity magnon polariton based precision magne- tometry, arXiv:2008.03062. [8] Y. Cao and P. Yan, Exceptional magnetic sensitivity of PT-symmetric cavity magnon polaritons, Phys. Rev. B 99, 214415 (2019). [9] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Us- ami, Y. Nakamura, Hybrid quantum systems based on magnonics, Appl. Phys. Express 12, 070101 (2019). [10] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, Y. Nakamura, Coherent coupling be- tween a ferromagnetic magnon and a superconducting qubit, Science 349, 6246 (2015). [11] J. Li, S.-Y. Zhu and G. S. Agarwal, Squeezed states of magnons and phonons in cavity magnomechanics, Phys. Rev. A99, 021801 (2019). [12] R. F. Sabiryanovand S. S. Jaswa, Magnons and Magnon- Phonon Interactions in Iron, Phys. Rev. Lett. 83, 10 (1999). [13] X. Zhang, C.-L. Zou, L. Jiang, H. X. Tang, Cavity mag- nomechanics, Sci. Adv. 2, e1501286 (2016). [14] Y.-P. Gao, C. Cao, T.-J. Wang, Y. Zhang and C. Wang, Cavity-mediated coupling of phonons and magnons, Phys. Rev. A 96, 023826 (2017) [15] L. Wang, Z.-X. Yang, Y.-M. Liu, C.-H. Bai, D.-Y. Wang, S. Zhang and H.-F. Wang, Magnon blockade in a PT-symmetric-like cavity magnomechanical system, arXiv:2007.14645v1. [16] S. V. Kusminskiy, H. X. Tang and F. Marquardt, Cou- pled spin-light dynamics in cavity optomagnonics, Phys. Rev. A94, 033821 (2016). [17] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baum-berg, A. J. Ramsay, A. Nunnenkamp and A. J. Fer- guson, Magneto-optical coupling in whispering-gallery- mode resonators, Phys. Rev. A 92, 063845 (2015). [18] H.Huebl, C.W.Zollitsch, J. Lotze, F.Hocke, M.Greifen - stein, A. Marx, R. Gross, and S. T. B. Goennenwein, High cooperativity in coupled microwave resonator ferri- magnetic insulator hybrids, Phys. Rev. Lett. 111, 127003 (2013). [19] Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, J. Q.You, Bistability ofCavityMagnon Polaritons, Phys. Rev. Lett. 120, 057202 (2018). [20] L. V. Abdurakhimov, Yu. M. Bunkov and D. Konstanti- nov, Normal-Mode Splitting in the Coupled System of Hybridized Nuclear Magnons and Microwave Photons, Phys. Rev. Lett. 114, 226402 (2015). [21] X. Zhang, A. Galda, X. Han, D. Jin and V. M. Vinokur, Broadband Nonreciprocity Enabled by Strong Coupling of Magnons and Microwave Photons, Phys. Rev. Applied 13, 044039 (2020). [22] C. Kong, H. Xiong and Y. Wu, Magnon-Induced Nonre- ciprocity Based on the Magnon Kerr Effect, Phys. Rev. Applied. 12, 034001 (2019). [23] R. J. Potton, Reciprocity in optics, Rep. Prog. Phys. 67, 717-754 (2004). [24] C. Caloz, A. Al` u, S. Tretyakov, D. Sounas, K. Achouri and Z.-L. Deck-L´ eger, Electromagnetic Nonreciprocity, Phys. Rev. Applied 10, 047001 (2018). [25] C. Caloz, A. Al` u, S. Tretyakov, D. Sounas, K. Achouri, and Z. Deck-L´ eger, Electromagnetic Nonreciprocity, Phys. Rev. Applied 10, 047001 (2018). [26] V. S. Asadchy, M. S. Mirmoosa, A. ´Daz-Rubio, S. Fan and S. A. Tretyakov, Tutorial on electromagnetic nonre- ciprocity and its origin, arXiv:2001.04848v2. [27] J. Fujita, M. Levy, R. M. Osgood, L. Wilkens and H. D¨ otsch, Waveguide optical isolator based on Mach- Zehnder interferometer, Appl. Phys. Lett. 76, 2158 (2000). [28] T. R. Zaman, X. Guo and R. J. Ram, Faraday rotation in an InP waveguide, Appl. Phys. Lett. 90, 023514 (2007). [29] F. D. M. Haldane and S. Raghu, Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 100, 013904 (2008). [30] Y. Shoji, T. Mizumoto, H. Yokoi, I. Hsieh and R. M. Osgood, Magneto-optical isolator with silicon waveg- uides fabricated by direct bonding, Appl. Phys. Lett. 92, 071117 (2008). [31] Z. Wang, Y. Chong, J. D. Joannopoulos and M. Soljaˇ ci´ c, Observation of unidirectional backscatterin g- immune topological electromagnetic states, Nature (Lon- don)461, 772 (2009).10 [32] M. Soljacic, C. Luo, J. D. Joannopoulos and S. Fan, Non- linear photonic crystal microdevices for optical integra- tion, Opt. Lett. 28, 637 (2003). [33] A. Alberucci and G. Assanto, All-optical isolation by d i- rectional coupling, Opt. Lett. 33, 1641 (2008). [34] B. Anand, R. Podila, K. Lingam, S. R. Krishnan, S. S. S. Sai, R. Philip and A. M. Rao, Optical diode action from axially asymmetric nonlinearity in an all-carbon solid- state device, Nano Lett. 13, 5771 (2013). [35] M. Hafezi and P. Rabl, Optomechanically induced non- reciprocity in microring resonators, Opt. Express 20, 007672 (2012); F. Ruesink, M.-A. Miri, A. Al` u, E. Ver- hagen, Nonreciprocity and magnetic-free isolation based on optomechanical interactions, Nat. Commun. 7, 13662 (2016); S. Manipatruni, J. T. Robinson and M. Lip- son, Optical Nonreciprocity in Optomechanical Struc- tures, Phys. Rev. Lett. 102, 213903 (2009); Z. Shen, et al. Experimental realization of optomechanically in- duced non-reciprocity, 10, 657-661 (2016); A. Seif and M. Hafezi, Broadband optomechanical non-reciprocity, Nat. Photonics 12, 60-61 (2018); B. Peng, et al., Parity- time-symmetric whispering-gallery microcavities, Natur e Phys.10, 2927 (2014). [36] L. Du, Y.-T. Chen, J.-H. Wu and Y. Li, Nonrecipro- cal quantum interference and coherent photon routing in a three-port optomechanical system, Opt. Express 28, 3647-3659 (2020); R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori and H. Jing, Nonreciprocal Photon Blockade, Phys. Rev. Lett. 121, 153601 (2018). [37] B. Megyeri, G. Harvie, A. Lampis, and J. Goldwin, Di- rectional Bistability and Nonreciprocal Lasing with Cold Atoms in a Ring Cavity. Phys. Rev. Lett. 121, 163603 (2018); L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao and P. Del’Haye, Microresonator iso- lators and circulators based on the intrinsic nonreciproc- ity of the Kerr effect, Optica 5, 000279 (2018). [38] A. E. Miroshnichenko, E. Brasselet and Y. S. Kivshar, Reversible nonreciprocity in photonic structures infil- trated with liquid crystals, Appl. Phys. Lett. 96, 063302 (2010); [39] J. Mund, D. R. Yakovlev, A. N. Poddubny, R. M. Dubrovin, M. Bayer and R. V. Pisarev, Magneto- toroidal nonreciprocity of second harmonic generation, arXiv:2005.05393v1. [40] K. Wang, Q. Wu, Y. Yu, Z. Zhang, Nonreciprocal pho- ton blockade in a two-mode cavity with a second-order nonlinearity, Phys. Rev. A 100, 053832 (2019); [41] S.Toyoda, M. Fiebig, T.-h. Arima, Y. Tokura, N. Ogawa, Nonreciprocal second harmonic generation in a magneto- electric material, arXiv:2006.01728. [42] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel and J. Aumentado, Non-reciprocal Microwave Signal Processing with a Field- Programmable Josephson Amplifier, Phys. Rev. Applied 7, 024028 (2017). [43] K. G. Fedorov, S. V. Shitov, H. Rotzinger and A. V. Ustinov, Nonreciprocal microwave transmission through a long Josephson junction, Phys. Rev. B 85, 184512 (2012). [44] N.R. Bernier, L.D. T´ oth, A. Koottandavida, M. A. Ioan- nou, D. Malz, A. Nunnenkamp, A.K. Feofanov, T. J. Kippenberg, Nonreciprocal reconfigurable microwave op- tomechanical circuit, Nat. Commun. 8, 604 (2018). [45] K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Mar- quardt, A. A. Clerk and O. Painter, Generalized non- reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering, Nat. Phys. 13, 465 (2017). [46] A. Metelmann and A. A. Clerk, Nonreciprocal Photon Transmission and Amplification via Reservoir Engineer- ing, Phys. Rev. X 5, 021025 (2015). [47] D. L. Sounas and A. Al` u, Non-reciprocal photonics base d on time modulation, Nat. Photonics 11, 774 (2017). [48] L. Ranzani and J. Aumentado, A geometric description of nonreciprocity in coupled two-mode systems, New J. Phys.16, 103027 (2014). [49] Y.-P. Wang et al. Magnon Kerr effect in a strongly cou- pled cavity-magnon system, Phys. Rev. B 94, 224410 (2016). [50] G.-Q. Zhang, Y.-P. Wang, J. Q. You, Theory of the magnon Kerreffectincavitymagnonics, Sci.China-Phys. Mech. Astron. 62, 987511 (2019). [51] H. Xiong, L.-G. Si, X.-Y. L¨ u, X. Yang and Ying Wu, Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system, Opt. Lett. 38, 353-355 (2013). [52] Q. Bin, X.-Y. L¨ u, F. P. Laussy, F. Nori and Y. Wu, N- Phonon Bundle Emission via the Stokes Process, Phys. Rev. Lett. 124, 053601 (2020). [53] Y. Kawaguchi, S. Guddala, K. Chen, A. Al` e, V. Menon, A. B. Khanikaev, All-optical nonreciprocity due to valley polarization in transition metal dichalcogenides, arXiv:2007.14934. [54] C. Lu, X. Hu, Y. Zhang, Z. Li, X. Xu, H. Yang, Q. Gong, Ultralow power all-optical diode in photonic crys- tal heterostructures with broken spatial inversion sym- metry, Appl. Phys. Lett. 99, 051107 (2011). [55] D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Ev- ers, S.-Y. Zhu, Optical diode made from a moving pho- tonic crystal, Phys. Rev. Lett. 110, 093901 (2013). [56] L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, M. Qi, An All-Silicon Passive Op- tical Diode, Science 335, 6067 (2012).

2021-06-15

We propose an effective approach for creating robust nonreciprocity of high-order sidebands, including the first-, second- and third-order sidebands, at microwave frequencies. This approach relies on magnon Kerr nonlinearity in a cavity magnonics system composed of two microwave cavities and one yttrium iron garnet (YIG) sphere. By manipulating the driving power applied on YIG and the frequency detuning between the magnon mode in YIG and the driving field, the effective Kerr nonlinearity can be strengthened, thereby inducing strong transmission non-reciprocity. More interestingly, we find the higher the sideband order, the stronger the transmission nonreciprocity marked by the higher isolation ratio in the optimal detuning regime. Such a series of equally-spaced high-order sidebands have potential applications in frequency comb-like precision measurement, besides structuring high-performance on-chip nonreciprocal devices.

Nonreciprocal high-order sidebands induced by magnon Kerr nonlinearity

2106.09542v1

Study of strong photon–magnon coupling in a YIG–fil m split–ring resonant system B. Bhoi 2*, T. Cliff 1* , I. S. Maksymov 1, M. Kostylev 1†, R. Aiyar 2, N. Venkataramani 3, S. Prasad 2, and R. L. Stamps 1,4 1School of Physics M013, University of Western Austr alia, Crawley 6009, Australia 2Department of Physics, Indian Institute of Technolo gy Bombay, Powai, Mumbai 400076, India 3Department of Metallurgical Engineering and Materia ls Science, Indian Institute of Technology Bombay, Mumbai 400076, India 4SUPA, University of Glasgow, Glasgow, G12 8QQ, Unit ed Kingdom Abstract: By using the stripline Microwave Vector–Network Ana lyser Ferromagnetic Resonance and Pulsed Inductive Microwave Magnetometry spectro scopy techniques, we study a strong coupling regime of magnons to microwave photons in the planar geometry of a lithographically formed split–ring resonator (SRR) loaded by a singl e–crystal epitaxial yttrium–iron–garnet (YIG) film. Strong anti-crossing of the photon modes of S RR and of the magnon modes of the YIG film is observed in the applied-magnetic-field resolved mea surements. The coupling strength extracted from the experimental data reaches 9% at 3 GHz. Theoretically, we propose an equivalent circuit mo del of the SRR loaded by a magnetic film. This model follows from the results of our nu merical simulations of the microwave field structure of the SRR and of the magnetisation dynam ics in the YIG film driven by the microwave currents in the SRR. The equivalent-circuit model i s in good agreement with the experiment. It provides simple physical explanation of the process of mode anti-crossing. Our findings are important for future applications in microwave quantum photonic devices as well as in nonlinear and magnetically tuneable m etamaterials exploiting the strong coupling of magnons to microwave photons. 1. Introduction In order to be useful for quantum application, a p roposed technology has to be able to exchange information with preserved coherence [ 1-3]. To this end, a system which consists of two sub–systems has to operate in a regime called 'stro ng coupling'. The strong coupling regime is characterised by strength of coupling between the s ubsystems which is larger than the mean energy loss in both of them. A straightforward way to decr ease losses is to make use of resonant systems [4–6]. Systems that exploit plasmonic resonances make u se of strong coupling between electric dipoles and optical fields localised at the sub-wav elength scale (i.e. on a spatial scale smaller than the wavelength of light). This makes possible the c reation of solid–state sources of quantum states of light (single photons, indistinguishable or enta ngled photons) such as those based on semiconductor quantum dots (QDs) or nitrogen–vacanc y centres in diamonds (NVs), embedded in optical microcavities [ 7–10 ] or precisely placed in close proximity to an opti cal nanoantenna [ 11– 13 ]. * Both authors contributed equally to this paper. † Corresponding author On the other hand, strong light–matter coupling is also achievable by using magnetic dipoles [14 ]. These schemes make possible the processing of qu antum information in various microwave resonator systems comprising ultra–cold atomic clou ds [ 15 ], molecules [ 16 ], NVs [ 17, 18 ] and ion– doped crystals [ 19 ]. Recently, strong coupling of microwave photons to magnons was demonstrated for a system representing a microwave cavity loaded by a single– crystal yttrium–iron–garnet (YIG) sphere [ 1, 20–22 ]. Furthermore, a more planar geometry of a microwa ve stripline resonator loaded by a single–crystal YIG film was shown to produce streng th of coupling of 150 MHz [ 23 ]. Nowadays, a microstrip line has largely replaced a traditional microwave cavity as a source of driving microwave magnetic field in the ferromag netic resonance (FMR) experiment [ 24, 25 ]. On the other hand, the stripline arrangement is use d to investigate planar microwave metamaterials [26 ]. Consequently, it makes this technique very suita ble for the investigation of coupling of microwave photons to magnons. Additionally, the us e of microstrip lines opens up avenues for the development of magnetically tuneable metamaterials based on arrays of highly–conductive meta– molecules (resonant elements that comprise metamate rials) coupled to magnetically–active materials of different sizes and shapes [ 27, 28 ]. In a recent work – Ref [ 23 ], the interaction between a non–magnetic split–rin g resonator (SRR) and a thin film of YIG was investigated with the Microwave Vector–Network Analyser (VNA) Ferromagnetic Resonance (FMR) method. Strongl y hybridised resonances were observed. The YIG film was grown on a single–crystal yttrium aluminium garnet substrate by pulsed laser deposition using an excimer laser. Although the SRR supported an anti–symmetric (low–frequency) and a symmetric (high–frequency) mode, only strong coupling of the YIG static magnetic field– dependent mode to the symmetric mode was studied, a nd a coupling strength of 150 MHz was observed. In the present work, we study the interaction of m agnetic resonances in a YIG film with microwave photon resonances in SRRs. We use both th e VNA–FMR (frequency domain) and Pulsed Inductive Microwave Magnetometry ('PIMM', ti me domain) spectroscopy techniques to investigate the coupling between photons and magnon s. We observe a strong coupling between the YIG mode and both low–frequency anti–symmetric and high–frequency symmetric modes of the SRR. In the VNA–FMR experiment, this interaction ma nifests itself as a strong anti–crossing between the photon and magnon mode. Naturally, the same result is confirmed by Fourier– transforming time–domain PIMM traces into the frequ ency domain. Additionally, the inspection of time–domain traces reveals the presence of a strong beat effect. The presence of the beat signal in time–domain traces is a signature of the entangleme nt of qubits [ 29, 30 ]. We conduct numerical simulations of the microwave field structure of the split rings and of the magnetisation dynamics driven by the microwave currents in the ring. We also suggest an equivalent circuit of an SRR loaded by a magnetic f ilm. These calculations are in very good agreement with the experiment and deliver a clear p hysical picture of the process of anti–crossing of the SRR and magnon modes. 2. Experimental arrangement The investigated SRR (Fig. 1) represents a small m etallic loop with a slit in it. It is a kind of LC resonant contour in which the loop acts as an indu ctance and the slit represents a lumped capacitance. 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 represents a microwave resonant system which suppor ts FMR. FMR is the collective precession of spins about the equilibrium spin direction in the m aterial. In a ferro– or ferrimagnetic materials the spins are coupled by the exchange and dipole–dipole interactions. The energies of the (inhomogeneous) exchange and dipole–dipole interact ions determine the FMR frequency. For the present work it is important that the FMR frequency also depends on the static magnetic field [ 31 ] in which the material is placed. The dependence on the applied field is through the Zeeman energy contribution to the energy of magnons which represe nt quanta of FMR [ 32 ]. Fig. 1. Sketch of the split ring resonator structur e. The split ring is inductively coupled to a microstrip feeding line. In experiment, the input a nd output of the microstrip line are connected to a VNA and the static applied magnetic field H is created by an electromagnet (not shown). For th e measurements, a YIG film (not shown) with the dimen sions 10 mm × 15 mm × 25 µm is placed on top of the split ring. The measured dimensions of t he split ring and the microstrip line are: a = 8.5 mm, b = 7.5 mm, g = 0.06 mm (the distance between the microstrip lin e and the SRR is also g), w = 3 mm. Also not shown: the total thickness of the SRR plus the dielectric substrate (grey area, ε = 2.3) and the back–side metallisation is h = 0.84 mm. The thickness of the copper lines of th e SRR and the microstrip is 0.01 mm [ 33 ]. Usually, FMR in a sample is excited by placing the sample in a uniform (or quasi–uniform) microwave magnetic field. An onset of FMR is easily seen as applied–field dependent resonance absorption of the microwave power by the magnetic m aterial. A convenient way to excite FMR in ferro– and ferrimagnetic films is by placing them on top of a microstrip or copla nar microwave stripline transmission line [ 24, 25, 34 ] or forming a microstrip line directly on top of t he film [ 35 ]. A microwave current flowing through the stripline i nduces a microwave Oersted field in the space above the stripline. This field drives spin precess ion in the material. A split–ring resonator can be formed lithographica lly as a loop of a microstrip line on top of a microwave substrate. One can use the Oersted fiel d of a microwave current flowing through such an 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 is the central point of our paper. The geometry of the device under test is shown in Fig. 1. It represents an SRR inductively coupled to a microwave transmission line. A single– crystal yttrium–iron–garnet film grown on a 1 mm–thick gadolinium–gallium garnet (GGG) substrat e sits on top of the resonator with the YIG layer facing the SRR. The film is 25 µm thick and was grown with liquid–phase epitaxy (LP E). A static magnetic field H is applied in the plane of the film in the directi on perpendicular to the microstrip line (Fig. 1). The measurements have been taken at room temperatu re. In order to take the measurements in the frequency domain, the input and the output o f the microstrip line have been connected to the ports of a VNA and the transmission characteristic of the microstrip line (S21=Re(S21)+ iIm(S21)) has been measured as a function of microwave freque ncy f and the strength H of the applied field. In the alternative time–domain (PIMM) arrangement, the input port of the microstrip line is connected to the output port of a generator of shor t pulses. We excite our system with a pulse of a rectangular shape which is 1 V in amplitude and 10 ns in duration. The nominal pulse rise time at the output port of the pulse generator is 55 ps. 3. Experimental results A set of representative |S21| vs. microwave freque ncy f dependencies taken with VNA– FMR for a number of values of the applied field is shown in Fig. 2. In each of the panels (except one for 300 Oe) one observes two peaks. One peak is very strongly dependent on the applied field. Essentially it moves across the displayed frequency range with an increase in the frequency. For H=300 Oe this peak has an almost vanishing amplitude and is located at about 2.25 GHz. Its relative amplitude becomes much larger when it move s closer to second – higher–frequency – peak (the peaks at 2.7 GHz and 3.4 GHz respectively in t he panel or H=430Oe). For H=450 Oe the lower–frequency peak ceases moving across, but the higher–frequency peak starts to move quickly with H. Its amplitude drops with an increase in the frequ ency separation from the lower–frequency peak. The respective frequency vs. field dependence of the peak positions are shown in Fig. 3(a). One clearly sees strong anti–crossing of the two li nes which suggests strong coupling of the modes. In order to identify the resonances these measureme nts have been repeated when the YIG film covered not only the SRR but also a section of the feeding microstrip line. These measurements have been taken on a different SRR str ucture whose geometry was able to accommodate the film in this position. The shape of the SRR for this structure was the same as in Fig. 1, but its sizes were slightly different, ther efore the frequencies of the resonances in Fig. 3(b ) do not coincide with the ones in Fig. 3(a) (and in Fig. 4 which we will discuss later on). From Fig. 3(b) one sees that for this YIG film placement one observes a third mode located in between two peaks of the type shown in Fig. 2. The frequenc y vs. applied field dependence for this extra mode is well fitted by the Kittel formula [ 36 ] 0 s ( ) 2f H H M γµπ= + (1) for the FMR frequency of an in–plane magnetised fil m with a saturation magnetisation value µ0Ms=0.2 T=2 kOe. (Here γ/(2 π) is gyromagnetic ratio; typically 2.8 MHz/Oe for Y IG.) This value is very close to the standard value of µ0Ms for YIG: 0.175 T. 300 Oe |S21| (dB) -4 -2 02 400 Oe |S21| (dB) -4 -2 02 430 Oe |S21| (dB) -4 -2 02 470 Oe |S21| (dB) -4 -2 0 500 Oe Frequency (GHz) 2.0 2.5 3.0 3.5 4.0 |S21| (dB) -4 -2 02450 Oe |S21| (dB) -4 -2 0 Fig. 2. Representative |S21| vs. microwave frequenc y traces taken with the vector network analyser. The numbers in the panels are the strengths of the applied static magnetic fields applied to the YIG film to take the measurements. This suggests that this extra mode is the ferromagn etic resonance mode ('magnon mode') of the YIG film excited directly by the Oersted field of the feeding microstrip line. Because of this origin, it is decoupled from the other two modes. F or this reason its frequency vs. applied field dependence is a smooth monotonic function. From Fig . 3(b) one also sees that the four other modes converge 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 magnon mode and a mode located at the higher–field one. Importantly, far away from the “anti– crossing” with the dispersion line the frequencies are almost the same within each pair and the line slope is practically vanishing. This identifies the horizontal sections of these four lines as uncoupl ed SRR resonances ('photon modes'). The sections of th ese lines with significant slopes (close to the anti–crossing area) are SRR resonances coupled to m agnon modes of the YIG film. (a) Applied field (Oe) 0 200 400 600 800 Frequency (GHz) 2.0 2.5 3.0 3.5 4.0 (b) Applied field (kOe) 1 2 3 4 5Frequency (GHz) 46810 12 14 16 Fig. 3. (a) Frequencies of the peaks from Fig. 1 as functions of the applied field. Dots – experiment , dashed lines – fits with Eq. (2). (b) Results of a measurement when a YIG film covers both an SRR and a section of the microstrip line. Dots – experi ment, dashed line – fit with the Kittel formula (1) for FMR frequency of an in–plane magnetised film. T he solid lines in both panels are the guides for the eye. The strong anti–crossing between the photon and mag non modes seen in Fig. 3 suggests a strong coupling between them. Figure 4 displays the results we obtained on the sample from Fig. 1 in a broad range of frequencies and applied fields. Two SRR modes are visible in it: at 3.2 GHz and 6.6 GHz. [the lower one is the same as in Fig. 3(a) ]. Both modes strongly interact with the YIG magnon mode. The dashed lines in Fig. 3(a) are the fits of the experimental data for the lower–frequency section of Fig. 4 with the model of two coupled res onators (see, e.g., Ref. [ 22 ]) 20 0 0 0 2 1 2 1 2 1(2) 2 2 f f f f f + − = ± + ∆ , (2) where 1f and 2fare the frequencies of the coupled resonances, 0 1f and 0 2f are the respective resonance frequencies in the absence of coupling an d ∆ is the coupling strength (measured in frequency units). By assuming that the frequency 0 2f is given by the Kittel formula Eq. (1) (depends on the applied field) and 0 1fis independent on the frequency (uncoupled SRR mode ), from the fit we obtain ∆ = 270 MHz or 0 1/ 9% f∆ = . Similarly, for the higher–frequency anti–crossing (located between 5 and 7 GHz) the best fit with Eq. (1) is obtained for ∆ = 450MHz or 6.8%. This latter value is significantly larger than the one previously observed for a pulse–laser deposited YIG film (1.3%) but weaker than the one f or a bulk YIG crystal (approximately 20%) [23 ]. The SRR geometries and sizes in Fig. 1 and in Re f. [23 ] are essentially different, therefore it is difficult to estimate the contribution of the SRR d esign to the performance of our device–prototype. However, from the comparison of the three results i t becomes certain that the usage of a much thicker YIG film in the present work (25 µm) than in Ref.[ 23 ] (2.4 µm), combined with potentially smaller intrinsic magnetic losses for the film in o ur case delivers a significant contribution to the improvement of the device performance observed in t he present work. Indeed, it is known that coupling of magnon dynamics in ferromagnetic films to microwave fields of striplines scales with the film thickness [ 37 ] and that the amplitude of any resonance driven by an external source is inversely proportional to the intrinsic losses in t he resonating medium. Although the magnetic loss parameter for the YIG film from [ 23 ] is not specified in the paper, it is natural to s uppose that the film from [ 23 ] is characterised by an at least half an order of magnitude larger loss parameter than our YIG film, because the former was grown with pul se laser deposition and the latter with liquid– phase epitaxy (LPE). It is known that LPE is the on ly method able to produce extremely low–loss films [ 38 ]. The strength of coupling of 20% observed in [ 23 ] for the case of the bulk YIG crystal is explained in a similar way. It is due to the very l arge thickness of this material potentially combine d with very low magnetic losses, if the used YIG slab represents a single–crystal material. In Ref. [ 23 ] a strong positive peak of |S21| was observed betw een the two usual negative peaks of resonance absorption for the maximum of an ti–crossing of the two resonances (middle line in Fig. 3 in Ref. [ 23 ]). It was claimed that this peak could be explaine d as due to a negative refraction index for the YIG+SRR meta–material in t hese conditions. Interestingly, we do not observe this behaviour in our Fig. 2. Instead, for H=450 Oe we see strong reduction in the peak amplitudes. It reaches 3dB with respect to the peak shown in the panel for 300 Oe. (Note that our simulations in Sect. 3 do not reproduce this extra peak either.) The result of the time–domain measurements is shown in Fig. 5(a). The main observation in this figure is the damped high–frequency oscillatio ns on top of the rectangular pulse. The shape and the amplitude of this oscillatory pattern strongly depend on the applied field. The results of the Fourier analysis of these time sequences are consis tent with the frequency–resolved measurements in Fig. 4 and Fig. 2. In particular, for 430 Oe one observes strong reduction in the amplitude of the high–frequency oscillation and noticeable change in the regularity of oscillations. Also, for 900 Oe and 1300 Oe the dominant period of the temporal seq uence is practically the same and practically equal to the one for 0 Oe. This fact can be explain ed as the dominant contribution to the total oscillation amplitude originating from the lowest S RR resonance (at 3 GHz for this field). This is expected 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, F(3.0 GHz)/ F(6.6 GHz) = 15. Hence, the oscillations at 6.6 GHz may be excited by a rectangular pulse less efficiently than at 3.0 GHz by an order of magnitude or so. For all these reasons, the change in the oscillation pattern between 0 Oe and 430 Oe is very noticeable and may be explained as the beat of two damped resonance modes from the respective panel in Fig. 2. On the contrary, the respective beat pattern is practically invisible in the time trace for 1400 Oe. This is because the dominating contribution to this pattern is delivere d by the SRR resonance at 3 GHz. However, the Fourier analysis in Fig. 5(b) reveals the presence of this beat pattern. Fig. 4. Grey–scale plot of the linear magnitude of the real part of S21 (Re(S21)) as a function of the applied field and microwave frequency obtained usin g VNA–FMR. 3. Numerical results and discussion We have conducted rigorous numerical three–dimensi onal finite–difference time–domain (FDTD) simulations in order to understand electrody namic properties of the SRR. The FDTD method is a very well–known time–domain Maxwell's e quations solver, which allows simulating of open–space problems using so–called absorbing bound ary conditions [ 39 ]. Conceptually, this numerical method is a counterpart of the time–domai n PIMM technique. In brief, we simulate the propagation of a short pulse of microwave current t hrough the microstrip line coupled to the SRR. For the sake of clarity, in the first approximation we consider the SRR without the YIG film. This simulation is repeated for an isolated microstrip l ine (the SRR is absent). Similar to PIMM, the simulated time–domain traces are Fourier–transforme d to obtain transmission characteristics. The transmission 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 line, which are unavoidable in our numerical model. The numerical results are shown in Fig. 6. We see that the FDTD simulation qualitatively reproduces the experimental picture. The low– and t he high–frequency resonances of the SRR are easily identified [Fig. 6(a)]. The inspection of th e magnetic field profiles of these resonances [Fig. 6(b–e)] reveals that the microwave magnetic f ield of SRR is strongly localised in close proximity to the split ring; it drops sharply with distance from the ring edges. From this point of view the double–SRR structure from Ref. [ 23 ] may be more advantageous since it ensures more complete filling of the area inside the SRR with th e microwave magnetic field. Figure 6(f) also confirms the result in Ref. [ 23 ] that the profile of the microwave magnetic field for the low– frequency mode is anti–symmetric but the one for th e high–frequency mode is symmetric with respect to an imaginary horizontal axis running thr ough the middle of the ring gap in Fig. 1. In the following we will call these SRR modes as 'symmetri c' and 'anti–symmetric', respectively. Fig. 5 (a) Time–domain traces for the selected valu es of the applied field. For the sake of clarity, each trace has been offset vertically by –0.2 dB. ( b) Grey–scale plot of the linear magnitude of Fourier–transformed PIMM traces as a function of th e applied field and microwave frequency. Note a different grey–scale bar as compared with Fig. 4. We used the 'Window filtering procedure' to remove noise from the time traces and thus improve the quality of the Fourier–space data. In our VNA–FMR experiment we noticed that the reso nant frequencies shift when we remove the YIG film from the SRR. The YIG film is k ept on top of a significantly thicker GGG substrate. Due to computation constraints, in our s imulations the GGG substrate cannot be as thick as it is in the experiment. Nevertheless, simulatio ns with a significantly thinner GGG substrate (not shown) qualitatively reproduce the experimental res ults: the resonance frequencies of the SRR are shifted and an additional very strong peak arises a t ~6 GHz [ 40 ]. This peak is probably due to the microwave resonance of the GGG substrate, and it se ems to be realistic as confirmed by the data in Figs. (4) and 5(a). In these figures one sees a wea k resonance at about 5.2 GHz which does not interact with the magnon mode at all. The unrealist ically large amplitude of this peak in our simulation seems to be an artefact, possibly due to constraints of the numerical model (which assumes much smaller GGG thickness, etc.). A separate numerical simulation was run to qualita tively explain the origin of the mode anti–crossing. It was based on a completely differe nt approach. As has been mentioned in Ref. [ 23 ], the magnetic dynamics excited by microwave magnetic field of the SRR is actually not a genuine ferromagnetic resonance, i.e. the dynamics does not take the form of magnetisation precession whose amplitude is (quasi) uniform and the phase is constant over the film area covered by the SRR. The excitations driven by the microwave field of the SRR are actually travelling spin waves. Two types of spin waves are relevant for our geomet ry. Both exist in in–plane magnetised ferromagnetic films [ 41 ]. The Damon–Eshbach (DE) wave propagates in the fi lm plane at a right angle to the applied magnetic field H. It is characterised by a positive dispersion (gro up velocity) – dω/d k > 0, where k is the wave number. The backward volume magnetostat ic spin wave (BVMSW) is characterised by a negative dispersion d ω/d k < 0. Its wave vector is parallel to the applied field . The frequency (energy) gaps ω(k=0) for both types of waves are the same and are gi ven by the Kittel formula Eq. (1). Microwave Oersted fields of microstrip lines are a ble to efficiently excite spin waves in the wave number range −2π/s<k<2 π/ s, (3) where s is the width of the microstrip (in Fig. 1 s=( a – b)/2 = 0.5 mm). The excited wave propagates perpendicular to the longitudinal axis of the micro strip. Hence, given the direction of the applied field (along x in Fig. 1), the two SRR sides parallel to the x–axis excite DE waves and the sides along the z–axis excite BVMSW. Hereafter, we will focus on the fundamental SRR resonance (at f10=3.106 GHz in the experiment and at 3.7 GHz in the simulation data in Fig. 6 obtained for the SSR without the YIG film). An important peculiarity of BVMSW excitation is that this type of spin wave is excited by the out–of–plane component ( hy) of the microwave magnetic field of the microstrip (see, e.g., Ref. [ 42 ]). From Figs. 6(a) and 6(g) one sees that for the fundamental mode this field is practically absent for the SRR side c ontaining the gap; it is present only at the opposi te SRR side. Furthermore, the BVMSW excitation is sign ificantly less efficient than that of the DE wave [ 42 ], because the latter is excited by both in–plane ( hz) and out–of–plane components of the microstrip field. The in–plane microwave field comp onent gives a significantly larger contribution to the total dynamic magnetisation amplitude. The hz–contribution is larger because of elliptical polarisation of spin waves (i.e. ellipticity of mag netisation precession) in in–plane magnetised films [42 ]. For those reasons, to keep our theory simple, we will neglect the contribution of BVMSW excitation to the total SRR response, and concentra te on the waves which deliver the main contribution to the magnon mode – the DE waves. (If necessary, the BVMSW contribution can be included by using the theory from [ 42 ]). A parameter which is often used to describe the ef ficiency of spin wave excitation by microstrip transducers is the radiation impedance ( see, e.g., Ref. [ 43 ]). We use the theory from Ref. [ 44 ] to calculate the radiation resistance of spin wav es excited by the microwave current in the split–ring resonator. Figure 7(a) shows the calcula ted dependence of the radiation impedance for a transducer which represents two parallel microstrip lines at a distance 8 mm from each other. The length of the lines is assumed to be equal to a in Fig. 1 and the microwave currents in the two parallel microstrips were assumed to be in phase. T his arrangement mimics the two SRR sides which are parallel to the x–axis. The in–phase currents ensure that the hz fields for the two sides are in anti–phase as seen in Fig. 6(f) for the mode at 3.7 GHz. The applied field H = 485 Oe was chosen such that the maximum of Re( Zr) is at the same frequency as the experimental unco upled SRR resonance – 3.106 GHz, i.e. f20(H) = f10 in the notations of Eq. (1). As a proxy to the unc oupled resonance we consider the case H = 0. For this field value, DE waves exist in the fr equency range 0 to 2.8 GHz, but should be excited in a much smaller frequency range, due to the very large value of s [see Eq. (3)]. Hence, no magnetisation dynamics wi th noticeable amplitude are expected for H = 0 at 3.106 GHz. As seen from Fig. 7(a), the radiation impedance ha s non–negligible values in a narrow range of frequencies δf = 500MHz or so. Analysis of this frequency range de monstrates that it corresponds to the DE wave–number range given by Eq . (3). For s = 0.5 mm, 2 π/s =kmax = 125 cm –1. On one hand, this very small value of kmax shows that the system operates in the regime which is close to FMR conditions (FMR corresponds to k = 0). On the other hand δf >> (γ ∆ Η)/(2π ), where ∆Η= 0.5 Oe, is the standard magnetic loss parameter for LPE YI G films. The latter parameter determines the intrinsic resonance linewidth for FM R. (Basically, (γ ∆ Η)/(2π ) is the frequency– resolved linewidth and ∆Η is the field–resolved one). The fulfilment of the condition δf >> (γ ∆ Η)/(2π ) formally implies that the travelling–spin–wave con tribution to the FMR linewidth is significant [ 45 ]. This happens because s is significantly smaller that the free propagation path for spin–waves in YIG films. (For our 25 µm–thick LPE-grown film the latter amounts to severa l millimetres.) Indeed, in Fig. 8(a) one clearly sees travelling s pin waves propagating away from the two microstrip lines, placed at a distance a from each other. The existence of a travelling spi n wave is evidenced by a linear spatial dependence of the amp litude of dynamic magnetisation seen in this figure for | z| > 5 mm or so, i.e. outside the SRR perimeter. Thi s linear dependence on the logarithmic scale corresponds to an exponential dec ay of the dynamic magnetisation amplitude my on the linear scale. The exponential decay of ampli tude is characteristic to a travelling wave in any medium with non–negligible losses. One notices a large difference in the wave amplitud es propagating in the positive and the negative direction of the z–axis: my(z<−5mm) > my(z>+ 5mm). This is due to the strong non– reciprocity of the DE wave excitation by microstrip transducers (see, e.g., Ref. [ 42 ]). Because the free propagation path of spin waves in YIG is very large, the partial wave excited by the right–hand–side microstrip in Fig. 8 (a) reaches the left–hand–side microstrip. This leads to interference of the two partial waves exci ted by the two parallel sides of the SRR. The interference forms an oscillatory amplitude pattern characteristic to a partial standing wave in the space 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 Fig. 8(a) for the radiation impedance and is formed because the DE wave number varies with frequency according to the dispersion law for the D E wave [ 41 ]. As a result, the phase φ= ka accumulated by the wave after crossing the distance a between the two sides of the SRR is a function of frequency. The maxima and the minima of the 'interference pattern' in Fig. 7(a) correspond to φ equal to either an even or odd multiple of 2 π. Figure 8(b) demonstrates an equivalent circuit for the fundamental mode of SRR resonance loaded by the magnon mode in YIG. Because of the mu ch smaller length of the SRR perimeter with the respect to the wavelength of the electromagneti c wave in the SRR substrate, one can consider the SRR as a series LC contour with lumped L and C. Physically, the capacitance C is concentrated in the gap. Although the inductance L is actually distributed along the perimeter of the SRR we can also consider it as a lumped one for the same reaso n of the smallness of the perimeter with respect to the microwave wavelength at 3 GHz. The losses in this contour are due to ohmic losses in the SRR’s metal; in our model they are accounted for by an equivalent resistance R0. Zr is an equivalent lumped element whose complex impedance is given by the plots in Fig. 7(a). The dashed line in Fig. 7(b) shows the reactance X0 of this contour for H = 0. As stated above, for this field Zr = 0, hence X0 = −1/( ωC)+ ωL, (4) where ω= 2πf. The resonance frequency f10 of the unloaded SRR follows from the condition X0 = 0. It is given by the well–known formula 0 11/ f LC = . (5). X0 = 0 corresponds to a maximum of the real part of t he complex conductance Y 0 0 0 0 1/ ( ) Y R iX = + . (6). In Fig. 7(b) one clearly sees a sharp resonance pea k of Re( Y0) at f10 = 3.06GHz. 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 two frequencies for which Im( Zr) = − X 0. The existence of the two compensation points is p ossible because the SRR resonance represents an 'anti–reson ance' (given by a maximum of the complex conductance ) [ 46 ], but the spin wave excitation represents a 'reson ance' (given by a maximum of the complex impedance ). The resonance condition for the whole sequence of t he equivalent elements connected in series follows from X0+Im( Zr) = 0 or, equivalently, it is given by the frequenc y position of the maximum of the real part of the total complex condu ctance 0 0 1/ ( ) r Y R iX Z = + + . (7) Accordingly, the presence of the two frequencies fo r which Im( Zr) = −X0 results in two resonances for the coupled system, seen as the anti–crossing o f f1 and f2 [in the notations of Eq. (1)] in the field and 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 calculation we assume that the value of C is given. Then the value of L is obtained from Eq. (5) by setting f10 to the frequency corresponding to the experimental SRR frequency for H = 0 (Step 1). The extracted value of L allows us to determine R0 as R0 = ωL/Q, (8) where Q is the experimental quality factor for the SRR res onance for H = 0 (Step 2). The last step of the calculation (Step 3) is to determine the two frequencies for which X0+Im( Zr) = 0. An additional step of the calculation is converting th e results for the 'internal' dynamics of the SRR in to the signal from the output of the feeding microstri p line. To include the coupling of the loaded SRR to the feeding microstrip line into our model, we i ntroduce an equivalent lumped mutual inductance M into the equivalent circuit in Fig. 8(b). It can b e then shown that S21 = 1 −KY, (9) where K is a coefficient – 'SRR to microstrip coupling coe fficient' – which depends on M. To obtain the traces of Re( Y) shown in Fig. 7(b) we fit the experimental data w ith Eqs. (4–8) using C as a single fit parameter. To this end, we iterati vely perform Steps 1–3 for different values of C until the frequency difference between the positio ns of the two peaks, shown by the solid line in Fig. 7(b), becomes equal to the experimental one from the panel for 470 Oe in Fig. 2: 592 MHz. The Zr(f) profile is kept the same throughout the fitting p rocedure; we use the one shown in Fig. 7(a). We choose the particular value of the ex perimental field – 470 Oe – because from Fig 3(a) one sees that for this field value the frequency se paration of the two coupled resonances is a minimum. According to Eq. (1) this implies that for this field f20(H) = f10 = 3.106 GHz and hence the results of the calculation in Fig. 7(a) corresp ond to this particular experimental field value. As an initial (guess) value for C we employ a value which we extracted from the FDTD simulation of the microwave electric field inside t he gap: 4.9 pF. Once the value of C has been extracted from the fit, the value of K may be found by equating Re(S21) in Eq. (9) to the maximum of the negative peak of the experimentally measured Re(S21) for H = 0 [Fig. 7(d)]. The best fit is obtained for C=2.0 pF which is of the same order of magnitude as the above mentioned guess value. The corresponding values of L and R0 are 1.01 nH and 0.71 Ohm respectively. We believe that both quantities are q uite reasonable. The respective plots of Re(S21) are shown in Fig. 7(c). These data correspond to K = 0.23. The main observation from Fig. 7(c) is in good qual itative agreement with the experimental data in Fig. 7(d). Furthermore, in both Fig. 7(c) a nd 7(d) one sees a fine structure (small amplitude oscillations) on top of the higher–frequency peak. Comparison of Figs. 7(a) and 7(c) reveals that this fine structure is due to the spin–wave interfe rence (see above). One more observation is that the linewidth of the l ower–frequency coupled resonance is smaller than that of the uncoupled one, both in exp eriment and theory. The decrease in the peak linewidths, due to coupling to the magnon system, m ay be explained based on the observation that the slope of the Re( Zr) vs. f dependence, near the lower–frequency compensation point, is of the same sign as of X0(f). Therefore Im{ Zr(f)}+ X0(f) is a steeper function of f than X0(f). This leads to quicker detuning from the resonance with f than for the uncoupled resonance and hence a quick er drop in the amplitude with the frequency. This idea is in agreement with the much larger expe rimental and theoretical resonance linewidths for the higher–frequency coupled resonan ce [Figs. 7(c) and 7(d)]. Indeed, if one neglects the fast oscillations of Im( Zr) [Fig. 7(a)] and follows the local mean value of I m{ Zr(f)}, then one finds that at the frequencies slightly below this r esonance Im{ Zr(f)} first becomes flatter and then the slope changes from positive to negative. This c hange in the slope results in a long tail of the higher–frequency coupled resonance peak, spanning o ver the range of 400 MHz towards smaller frequencies. This tail is very visible in Figs. 7(c ) and 7(d); it significantly broadens the upper– frequency peak with respect to the lower–frequency one. Two small discrepancies between the theory and the experiment become clear from the comparison of Fig. 7(c) and 7(d). The first one is a noticeable shift upwards in frequency of the theoretical resonance pair for H = 485 Oe with respect to H = 0. Indeed, the experimental pair is located more symmetrically with respect to the H = 0 data than the theoretical one. We believe that this may be due to the fact that in our calculation we neglected the BVMSW contribution to Zr. Inclusion of the excitation of BVMSW would modify t he lower frequency slope of the peak of Re( Zr) by making it less steep. Accordingly, the lower–f requency slope of Im( Zr) would become less steep and non–vanishing values of Im( Zr) would extend to smaller frequencies. This would potentially move the lower–frequency peak for H = 485 Oe in Fig. 7(c) to lower frequencies, thus making the location of the two peaks more symmetric with respect to the H = 0 peak. The other noticeable discrepancy is the peak amplit udes. The experimental data in Fig. 7(d) demonstrate a larger reduction in the peak amplitud es for H = 470 Oe with respect to the H = 0 case than the theoretical ones. This difference may sugg est that K is dependent on H. Indeed, the YIG film covers the SRR but does not cover the feeding microstrip line. Therefore one may expect that the mutual inductance M is a function of H. With the approach of the magnon–mode resonance, t he microwave magnetic permeability of the YIG film [ 47 ] grows. This affects the microwave fields of the SRR. However, the microwave magnetic permeabili ty in the vicinity of the feeding microstrip remains the same, since this part is not covered by the film. This applied–field dependent jump in the spatial profile of the permeability may make th e strength inductive coupling of the microstrip to the SRR magnetic–field dependent. 4. Conclusions By using both the frequency–domain VNA–FMR and time –domain PIMM spectroscopy techniques, we have demonstrated a strong coupling regime of magnons to microwave photons in the planar geometry of a lithographically formed sp lit–ring resonator loaded by a single–crystal epitaxial YIG film. Whereas in the VNA–FMR experime nt this interaction manifests itself as a strong anti–crossing between the photon and magnon mode, the time–domain PIMM traces exhibit a signature of a strong beat effect. We have conducted numerical simulations of the mic rowave field structure of the SRR and of the magnetisation dynamics driven by the microwa ve currents in the SRR and suggested an equivalent circuit of an SRR loaded by a magnetic f ilm. These calculations are in very good agreement with the experiment and reveal the physic al origins of the effect of anti–crossing. Our results are important for the progress of micro wave quantum photonic devices such as, e.g., generators of entangled microwave radiation [ 48] required for quantum teleportation, quantum communication, or quantum radar with continuous var iables at microwave frequencies [49]. Moreover, 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 microwave photons. Fig. 6 Results of 3D FDTD simulations. The YIG film and its GGG substrate are not taken into account. (a) Spectrum of the SRR excited by the mic rostrip line. (b–e) Intensity profiles of the out– of–plane ( hout = |hy|) and in–plane [ hin = (| hx|2 + |hz|2)1/2 ] fields. The false colour is slightly oversaturated for the sake of illustration. (f, g) Re( hz)–field and Re( hx)–field profiles across the z– direction and x–direction, respectively, for the resonance frequen cies 3.7 GHz (solid line) and 7.2 GHz (dashed line). These results, which are com plementary to those in panels (b–e), show the phase relationship for the in–plane field component s. Frequency (GHz) Reactance (Ohm) -50 -40 -30 -20 -10 010 20 30 Resistance (Ohm) 0246810 12 14 16 18 20 (b) Frequency (GHz) Conductance Re( Y) (Sm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (a) Re(S21) (dB) -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 (c) (d) Frequency (GHz) 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 Re(S21) (dB) -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 Fig. 7. (a) Complex radiation impedance Zr of two 0.5 mm–wide and 8.5 mm long parallel microstrip lines (solid lines). Left–hand axis: Im( Zr); right–hand axis: Re( Zr). The distance between the longitudinal axes of the microstrips is 8 mm. T he static magnetic field is applied along the microstrips and is 485 Oe. Film saturation magnetis ation 4 πMs = 2000 G. This geometry mimics excitation of the Damon–Eshbach spin waves by the t wo sides of SRR which are parallel to the x– axis in Fig. 1. Dashed line: reactance for the unco upled SRR (Im( X0)). (b) Calculated real part of conductance Re( Y). Solid line: H = 485 Oe; dashed line: H = 0. (In the latter case it is assumed that Zr(H=0) = 0). (c) Simulated signal from the output of t he feeding microstrip line Re(S21). Solid line: H=485 Oe; dashed line: H = 0. (d) Experimental data for Re(S21). Solid line: H = 470 Oe; dashed line: H = 0. (a) Co-ordinate z across SRR (mm) -15 -10 -5 0 5 10 15 Magnetization amplitude (dB) -14 -12 -10 -8 -6 -4 -2 0 Fig. 8. (a) Calculated profile of the dynamic magne tisation amplitude in the z direction. The SRR sides are located at z = ±4 mm. Frequency is 3.242 GHz. The other parameters of calculation are the same as for Fig. 7. One notices a linear decay of t he magnetisation amplitude with the distance from the SRR outside the ring. One also notices a standi ng–wave pattern inside the ring. Note that the slight waviness on top of the linear slopes is an a rtefact of the calculation. It originates from fini te accuracy of the numerical inverse Fourier transform used to produce these data. (b) Equivalent circuit for the fundamental mode of SRR resonance l oaded by the magnon mode in YIG. Acknowledgements Support by Department of Science and Technology of India, Australia–India Strategic Research Fund, and the Faculty of Science of the University of Western Australia is acknowledged. ISM has been supported by the UWA UPRF scheme. References [1] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan , M. Kostylev, and M. E. Tobar, ArXiv 1408.2905 (2014). [2] E. Jaynes and F. Cummings, Proc. IEEE 51 , 89 (1973). [3] M. Tavis and F. W. Cummings, Phys. Rev. 170 , 379 (1968). [4] C. Ciuti and I. Carusotto, Phys. Rev. A 74 , 033811 (2006). [5] T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia–Ripoll, D. Zueco, T. Hummer, E. Solano, A. Marx, and R. Gro ss, Nat. Phys. 6, 772 (2010). [6] J. Plumridge, E. Clarke, R. Murray, and C. Phil lips, Solid State Comm. 146 , 406 (2008). [7] A. J. Shields, Nat. Photon. 1, 215 (2007). [8] C. Santori, D. Fattal, J. Vuckovic, G. S. Solom on, and Y. Yamamoto, Nature (London) 419 , 594 (2002). [9] R. M. Stevenson, R. J. Young, P. Atkinson, K. C ooper, D. A. Ritchie, and A. J. Shields, Nature (London) 439 , 179 (2006). [10] T. M. Babinec, B. J. M. Hausmann, M. Khan, Y. Zhang, J. R. Maze, P. R. Hemmer, and M. Lon čar, Nat. Nanotechn. 5, 195 (2010). [11] A. G. Curto, G. Volpe, T. H. Taminiau, M. P. K reuzer, R. Quidant, and N. F. van Hulst, Science 20 , 930 (2010). [12] I. S. Maksymov, A. E. Miroshnichenko, and Yu. S. Kivshar, Phys. Rev. A 86 , 011801(R) (2012). [13] S. D’Agostino, F. Alpeggiani, and L. C. Andrea ni, Opt. Express 21 , 27602 (2013). [14] A. Imamoglu, Phys. Rev. Lett. 102 , 083602 (2009). [15] J. Verdú, H. Zoubi, C. Koller, J. Majer, H. Ri tsch, and J. Schmiedmayer, Phys. Rev. Lett. 103 , 043603 (2009). [16] A.W. Eddins, C. C. Beedle, D. N. Hendrickson, and J. R. Friedman, Phys. Rev. Lett. 112 , 120501 (2014). [17] D. Marcos, M. Wubs, J. M. Taylor, R. Aguado, M . D. Lukin, and A. S. Sø rensen, Phys. Rev. Lett. 105 , 210501 (2010). [18] X. Zhu, S. Saito, A. Kemp, K. Kakuyanagi, S. K arimoto, H. Nakano, W. J. Munro, Y. Tokura, M. S. Everitt, K. Nemoto, M. Kasu, N. Mizuochi, and K. Semba, Nature 478 , 221 (2011). [19] D. I. Schuster, A. P. Sears, E. Ginossar, L. D iCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J . Schoelkopf, Phys. Rev. Lett. 105 , 140501 (2010). [20] H. X. Tang, X. Zhang, X. Han, and M. Balinskiy , Proc. SPIE 8373 , 83730D (2012). [21] X. Zhang, C.–L. Zou, L. Jiang, and H.X. Tang, ArXiv: 1405.7062 (2014). [22] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazak i, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 113 , 083603 (2014) [23] G. B. G. Stenning, G. J. Bowden, L. C. Maple, S. A. Gregory, A. Sposito, R. E. Eason, N. I. Zheludev, and P. A. J. de Groot, Opt. Express 21 , 1456 (2013). [24] T. J. Silva, C. S. Lee, T. M. Crawford, C. T. Rogers, J. Appl. Phys. 85 7849 (1999). [25] I. Maksymov and M. Kostylev, to appear in Phys ica E. [26] F. Falcone, F. Martín, J. Bonache, R. Marqués, and M. Sorolla, Microw. Opt. Technol. Lett. 40 , 3 (2004). [27] L. Kang, Q. Zhao, H. Zhao, and J. Zhou, Opt. Express 16 , 8825 (2008). [28] J. N. Gollub, J. Y. Chin, T. J. Cui, and D. R. Smith, Opt. Express 17 , 2122 (2009). [29] A. Sato, J. Ishi–Hayase, F. Minami, and M. Sas aki, J. Luminescence 119–120 , 508 (2006). [30] J. Majer, J. M. Chow, J. M. Gambetta, Jens Koc h, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Nature 449 , 443 (2007). [31] 7. A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves, CRC, Boca Raton, FL, 1996. [32]. More precisely, magnons are quantised spin wa ves [31]. A spin wave with a zero wave number is conventionally referred to as FMR. [33] In simulations, the electrical conductivity of the SRR and microstrip line is taken to be σ = 5.96×10 7 S/m, which corresponds to the conductivity of bulk copper. [34] R. Magaraggia, M. Hambe, M. Kostylev, R. L. St amps, and V. Nagarajan, Phys. Rev. B 84 , 104441 (2011). [35] C. S. Wolfe, V. P. Bhallamudi, H. L. Wang, C. H. Du, S. Manuilov, R. M. Teeling–Smith, A. J. Berger, R. Adur, F. Y. Yang, and P. C. Hammel, Phys. Rev. B 89 , 180406(R) (2014). [36] D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications . Springer, Berlin, 2009. [37] P. R. Emtage, J.Appl.Phys . 49 4475 (1978). [38] Y. Sun and M. Wu , Solid State Physics 64 , 157 (2013). [39] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite–Differenc e Time– Domain Method , 3rd ed., Artech House Publishers, Boston, 2005. [40] We neglect ferrimagnetism of YIG by setting its ma gnetic permeability to 1 but include the high dielectric permittivity of GGG and YIG – 15. [41] R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). [42] T. Schneider, A. A. Serga, T. Neumann, B. Hill ebrands, and M. P. Kostylev, Phys. Rev. B 77 , 214411 (2008). [43] B. A. Kalinikos, Sov. J. Phys. 24 , 718 (1981). [44] C. S. Chang, M. Kostylev, E. Ivanov, J. Ding, and A. O. Adeyeye, Appl. Phys. Lett. 104 , 032408 (2014). [45] G. Counil, J.–V. Kim, T. Devolder, C. Chappert , K. Shigeto, and Y. Otani, J. Appl. Phys. 95 , 5646 (2004). [46] A. F. McKinley, T. P. White, I. S. Maksymov, a nd K. R. Catchpole, J. Appl. Phys. 112 , 094911 (2012). [47] N. S. Almeida and D. L. Mills, Phys. Rev. B 53 , 12232 (1996). [48] E. Flurin, N. Roch, F. Mallet, M. H. Devoret, and B. Huard, Phys. Rev. Lett. 109 , 183901 (2012). [49] E. P. Menzel, R. Di Candia, F. Deppe, P. Eder, L. Zhong, M. Ihmig, M. Haeberlein, A. Baust, E. Hoffmann, D. Ballester, K. Inomata, T. Yamamoto, Y. Nakamura, E. Solano, A. Marx, and R. Gross, Phys. Rev. Lett. 109 , 250502 (2012). [50] M. Lapine, I. V. Shadrivov, and Yu. S. Kivshar, Rev. Mod. Phys. 86 , 1093 (2014).

2014-09-19

By using the stripline Microwave Vector Network Analyzer Ferromagnetic Resonance and Pulsed Inductive Microwave Magnetometry spectroscopy techniques, we study a strong coupling regime of magnons to microwave photons in the planar geometry of a lithographically formed split-ring resonator (SRR) loaded by a single-crystal epitaxial yttrium-iron garnet (YIG) film. Strong anti-crossing of the photon modes of SRR and of the magnon modes of the YIG film is observed in the applied-magnetic-field resolved measurements. The coupling strength extracted from the experimental data reaches 9 percent at 3 GHz. Theoretically, we propose an equivalent circuit model of an SRR loaded by a magnetic film. This model follows from the results of our numerical simulations of the microwave field structure of the SRR and of the magnetization dynamics in the YIG film driven by the microwave currents in the SRR. The equivalent circuit model is in good agreement with the experiment. It provides a simple physical explanation of the process of mode anti-crossing. Our findings are important for future applications in microwave quantum photonic devices as well as in magnetically tunable metamaterials exploiting the strong coupling of magnons to microwave photons.

Study of strong photon-magnon coupling in a YIG-film split-ring resonant system

1409.5499v1

Breaking surface plasmon excitation constraint via surface spin waves H. Y. Yuan1,2and Yaroslav Blanter1 1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands and 2Institute for Advanced Study in Physics, Zhejiang University, 310027 Hangzhou, China (Dated: February 8, 2024) Surface plasmons in two-dimensional (2D) electron systems have attracted great attention for their promising light-matter applications. However, the excitation of a surface plasmon, in particular, transverse-electric (TE) surface plasmon, remains an outstanding challenge due to the difficulty to conserve energy and momentum simultaneously in the normal 2D materials. Here we show that the TE surface plasmons ranging from gigahertz to terahertz regime can be effectively excited and manipulated in a hybrid dielectric, 2D material and magnet structure. The essential physics is that the surface spin wave supplements an additional freedom of surface plasmon excitation and thus greatly enhances the electric field in the 2D medium. Based on widely-used magnetic materials like yttrium iron garnet (YIG) and manganese difluoride (MnF 2), we further show that the plasmon excitation manifests itself as a measurable dip in the reflection spectrum of the hybrid system while the dip position and the dip depth can be well controlled by the electric gating on the 2D layer and an external magnetic field. Our findings should bridge the fields of low-dimensional physics, plasmonics and spintronics and open a novel route to integrate plasmonic and spintronic devices. Introduction.— Plasmons are collective excitations of electronic charge density in metallic structures. In three- dimensional (3D) systems, one has to overcome a gap of several electronvolts to excite the bulk plasma oscilla- tions, which makes it challenging to be manipulated. The situation in two-dimensional (2D) systems is very differ- ent since the plasmon frequency is usually proportional to√qwith qbeing the propagating wavevector of plasmons [1], implying that the excitation energy can be desirably tuned far below the optical regime. Another benefit of the a 2D configuration is the electrical tunability of the Fermi energy and thus of the charge carrier density [2, 3]. As a result, surface plasmons in 2D materials, for exam- ple, graphene, have attracted significant attention with the well-developed fabrication technology of 2D materials [2–7]. In particular, transverse magnetic (TM) plasmons are broadly studied while transverse electric (TE) plas- mons are seldom studied for its restrictive excitation con- dition. For usual 2D systems with the parabolic electron dispersion, it is widely believed that the TE plasmons are not present for their positive imaginary component of conductivity, which is well described by the Drude model [8]. For graphene, it was theoretically proposed that the sign of the imaginary part of the conductivity may reverse near the spectral onset of intraband scatter- ing to unlock the TE modes [9]. However, the resulting TE plasmons locating in infra and terahertz regime are yet to be verified. On the other hand, spin waves – collective excitations of spins in ordered magnets – can carry information even in magnetic insulators, which largely reduces the Joule heating problem during information processing [10, 11]. Spintronic systems are also easily integrated with other physical systems, for example, photonic platforms, qubits and phonons, to form hybrid systems for multifunctional information processing [12, 13]. The frequency of spin FIG. 1. Schematic of the modified Otto configuration com- posing of a prism, a dielectric layer, a 2DEG layer and a fer- romagnetic layer. An incident electromagnetic wave induces an evanescent wave in the dielectric layer above a critical in- cident angle. The evanescent wave propagates toward + ez direction and excites a surface plasmon in the 2DEG layer as well as surface spin waves in the magnetic layer. waves ranges from gigahertz (GHz) in ferromagnets to terahertz (THz) in antiferromagnets [14]. This makes it possible to couple them to surface plasmons in 2D mate- rials that have a continuous spectrum [15–17]. Hybrid 2D materials which include magnetic films, which attracted a lot of interest recently, [18–22] also provide an accessi- ble platform to investigate the hybrid magnon-plasmon excitation. In this work, we show how the conventional constraint on a TE surface plasmon can be overcome by the inter- play of surface plasmons and spin waves. In particular, we investigate the wave propagation in a hybrid dielectric (DE), 2D electron gas (2DEG) and magnetic insulator structure as shown in Fig. 1. An incident electromag- netic wave first induces a surface wave at the interfacearXiv:2402.04626v1 [cond-mat.mes-hall] 7 Feb 20242 of media 4-3 and an evanescent wave inside medium 3. The evanescent wave propagates towards the 2DEG layer and excites a TE surface plasmon and surface spin waves simultaneously free from the constraint of 2DEG conduc- tivity. The frequency of this surface plasmon-magnon po- lariton can be well tuned by an external magnetic field and falls into the GHz regime for ferromagnets (FM) and THz for antiferromagnets (AFM). For comparison, plas- mons are barely generated when a magnet is replaced by a (non-magnetic) dielectric material. Furthermore, the excitation of the surface plasmon manifests as a sharp dip in the reflection spectrum of the layered structure. The depth and position of the dip are tunable by elec- tric gating and by external magnetic field. These find- ings give a state-of-art demonstration of surface-plasmon excitations in hybrid 2D material-magnet structures and they should provide a feasible platform to study the inter- play of magnon spintronics and plasmonics. The reported GHz and THz plasmons may find promising applications in designing novel plasmonic devices. Physical model and the excitation spectrum. —Let us first look at the excitation spectrum of the hybrid system DE/2DEG/FM(DE) shown in Fig. 1, where the DE and FM layers are semi-infinite. The electromagnetic proper- ties of the hybrid structure should satisfy the Maxwell’s equations ∇ ×E=−∂tB,∇ ×H=∂tD, (1) where EandHare respectively electric and magnetic fields, while D=ϵ0ϵEandB=µ0(H+M) are respec- tively the electric displacement and the magnetic induc- tance with ϵ0,µ0andϵbeing the vacuum permittivity, vacuum permeability and material permittivity, respec- tively. After eliminating the electric components, Eqs. (1) can be combined to (∇2+k2)H− ∇(∇ ·H) +k2M= 0, (2) where k2=ϵµ0ω2,M=Msmwith Msbeing the satura- tion magnetization and mthe normalized magnetization vector of the FM layer. On the other hand, the magnetization dynamics in the FM layer is governed by the Landau-Lifshitz-Gilbert (LLG) equation [23–25] ∂tm=−γm×Heff+αm×∂tm. (3) The first and second terms on the right-hand side of Eq. (3) describe the precessional and damped motion of the magnetization toward the effective field Heffwith γand αbeing the gyromagnetic ratio and the Gilbert damp- ing parameter, respectively. In general, Heffis a sum of the external field He, the dipolar field H, the crys- talline anisotropy field, and the exchange field. It is as- sumed that the external field is applied along the x-axis He=Hexand is strong enough to generate a uniformequilibrium state M0=Msex. Then the spin-wave ex- citation above this ground state can be represented as M=M0+Myey+Mzezwith My,z≪Msand the dynamics of ( My, Mz) is derived by linearizing the LLG equation (3) around M0as  My Mz = κ−iν iν κ Hy Hz , (4) where κ= (ωh−iαω)ωm/((ωh−iαω)2−ω2),ν= ωmω/((ωh−iαω)2−ω2) with ωh=γH,ωm=γMs. Without loss of generality, we have neglected the ex- change field, because it does not contribute significantly to the low-energy excitation of spin-waves in the soft magnets like yttrium iron garnet (YIG). By substituting Eq. (4) into the Maxwell’s equations (2), we can derive self-contained equations of Hyand Hz. We consider an incident wave with momentum k(i)= (0, k4cosθ, k4sinθ). Then the spins mainly os- cillate in the yandzdirections, and the combined LLG and Maxwell equations in medium 2 read  ∂zz+k2 2(1 +κ)−∂yz−ik2 2ν −∂yz+ik2 2ν ∂ yy+k2 2(1 +κ) Hy Hz = 0.(5) By solving Eqs (5), we derive the surface spin-wave mode with H2= (0 , H(−) 2,y, H(−) 2,z)eik2,yy−κ2z,E2= (E(−) 2,x,0,0)eik2,yy−κ2z,H(−) 2,y=iκ2E(−) 2,x/(µ0ω) and k2,y, κ2, ωare related to each other by the determinantal equation. Unless stated otherwise, we always label the wavevector and decay exponent in medium ibyki,yand κi, and they satisfy k2 i,y−κ2 i=ω2/c2ϵi(i= 3,4). In the dielectric medium 3, M= 0, the TE wave solution to the Maxwell’s equations reads H3= (0, H(+) 3,y, H(+) 3,z)eik3,yy+κ3z,E3= (E(−) 3,x,0,0)eik3,yy+κ3z with the relation H(+) 3,y=−iκ3E(+) 3,x/(µ0ω). At the in- terface of media 3 and 2, the tangential components of electric field should be continuous while the tangential components of magnetic field are connected by the sur- face electric currents jx=σE2,xcorresponding to surface plasmons excitations in the 2DEG layer, i.e. H(+) 3,y−H(−) 2,y=σE(−) 2,x, E(+) 3,x=E(−) 2,x, (6) where we shifted the z= 0 plane to the interface of 2DEG for simplicity and imposed the requirement of in-plane momentum conservation k3,y=k2,y≡q. Nontrivial so- lutions of Eqs. (6) appear provided r q2−ω2ϵ3 c2+ϵ2ω2 c2δp−iµ0ωσ= 0, (7) where δp(p∈ {FM,DE}) depends on the nature of medium 2 such that δFM=−i k2,yk2 2,z−k2 2(1 +κ) k2,yk2,z−ivk2 2−k2,z! , (8a) δDE=k2 2/κ2. (8b)3 (a) (b) Light coneH=0.3T H=0.4T 0 2 4 6 805101520 q(mm-1)ωr/2π(GHz ) ωh+ωm/2 DE/GRA /DEEF=0.1eV EF=0.2eV EF=0.8eV 0.0 0.1 0.2 0.3 0.4 0.505101520 Field (T)ωr/2π(GHz ) FIG. 2. (a) Dispersion relation of the surface plasmon- magnon polariton. The light cone is bounded by ω=cq/√ϵ4. ϵ4= 14 , ϵ3= 2, EF= 0.8 eV. The parameters of YIG are used with ϵ3= 10.8,Ms= 0.175 T [28]. (b) Resonant frequency of the surface plasmon-magnon polariton as a func- tion of external field at different values of the Fermi energy in the hybrid structure shown in Fig. 1. θ= 1.1θc. The black line at ω= 0 is the solution to resonant condition (7) in DE/GRA/DE structure. This is the first key result of our work. When medium 2 is a dielectric, the resonance condition is reduced to the familiar form in literature by inserting δDEinto Eq. (7) [9]. This condition only has real solutions when σis purely imaginary and otherwise has a negative imaginary component. Therefore, it cannot be fulfilled for conven- tional 2DEG whose conductivity is well described by the Drude model [9, 26, 27] i.e. σ=σ0EF/(πΓ−iπℏω) with σ0=e2/4ℏ,EFbeing the Fermi energy, and Γ being the relaxation rate of carriers. This implies that the TE surface plasmons cannot be resonantly excited using the conventional Otto setup. The situation changes dramatically when medium 2 is a ferromagnet. By plugging δFMinto Eq. (7), we find that the resonant frequency should satisfy the equation r q2−ω2ϵ3 c2+q2−k2 2ν q(1 +κ+ν)+µ0σ0EF πℏ= 0.(9) Firstly, we recover the frequency of surface magnon mode in the magnetostatic limit ( ω≪cq) when EF= 0 as ωr=ωh+ωm/2 (blue and red dashed lines in Fig. 2(a)) [29, 30]. As we go beyond this limit, the spectrum can be obtained by numerically solving Eq. (9), with the result shown in Fig. 2(a). Clearly, there is a crossing be- tween the light cone and surface plasmon-magnon disper- sion, suggesting the possibility to match both momentum and frequency between the incident photons and hybrid plasmon-magnon modes and thus enabling the plasmon excitations. It is noteworthy that the resonant frequency can be well tuned in the GHz regime by the external field, as shown in Fig. 2(b). Reflection rate. –Now we proceed to demonstrate that the surface plasmons and magnons can be simultane- ously excited by shining a proper wave on the hy- brid system. The excited surface plasmon will carry away electromagnetic energy and reduce the reflec-tion rate of the system, which provides a feasible way to detect the excitation of surface plasmons in our setup. Here we consider a s-polarized incident wave with electric field perpendicular to the incident plane k(i)= (0 , ky, kz) in medium 4, where ky=k4sinθ and kz=k4cosθ, as shown in Fig. 1. To sat- isfy the Maxwell’s equations, the magnetic and electric fields should read H(i/r) 4 = (0, H(i/r) 4,y, H(i/r) 4,z)ei(kyy±kzz) andE(i/r) 4 = (E(i/r) 4,x,0,0)ei(kyy±kzz)with H(i/r) 4,y = ±E(i/r) 4,xkz/(µ0ω), H(i/r) 4,z=−E(i/r) 4,xky/(µ0ω), where i, r label the incident and reflected waves, respectively. The finite thickness of medium 3 allows for the coexistence of exponential increase and decay modes, i.e. H3= (0, H(−) 3,y, H(−) 3,z)eik3,yy−κ3z + (0, H(+) 3,y, H(+) 3,z)eik3,yy+κ3z, E3= (E(−) 3,x,0,0)eik3,yy−κ3z+ (E(+) 3,x,0,0)eik3,yy+κ3z, (10) where H(±) 3,y =∓iE(±) 3,xκ3/(µ0ω) and H(±) 3,x = −iE(±) 3,xk3,y/(µ0ω). Now the boundary conditions require the continuity of the tangential components of both electric and magnetic fields at interfaces of media 4-3 and 3-2, i.e. H(i) 4,y+H(r) 4,y=H(+) 3,y+H(−) 3,y, (11a) E(i) 4,x+E(r) 4,x=E(+) 3,x+E(−) 3,x, (11b) H(+) 3,ye(κ2+κ3)d+H(−) 3,ye(κ2−κ3)d−σE(−) 2,x=H(−) 2,y,(11c) E(+) 3,xe(κ2+κ3)d+E(−) 3,xe(κ2−κ3)d=E(−) 2,x. (11d) By expressing all the magnetic fields by their elec- tric fields counterparts and solving the resulting lin- ear equations, we can derive the reflection coefficient R≡E(r) 4,x/E(i) 4,xas [31] R=k2 2(kzsinh(κ3d)−iκ3cosh( κ3d)) +δpc+ k2 2(kzsinh(κ3d) +iκ3cosh( κ3d)) +δpc−,(12) where c±= (∓µ0σω+kz)κ3cosh( κ3d)∓i(κ2 3± kzµ0σω) sinh( κ3d). For very thin dielectric medium 3 (κ3d→0), the reflection coefficient is simplified as R=−ik2 2+δp(kz−µ0σω) ik2 2+δp(kz+µ0σω). (13) This is the second key result of our work. Figure 3(a) shows the reflection rate |R|2as a function of the fre- quency of incident wave when θ= 1.1θcwith the critical angle θc= arcsinp ϵ2/ϵ4(ϵ4> ϵ3,2). A sharp dip in the reflection rate appears at the resonant frequency (vertical dashed line), implying a resonant excitation of the sur- face plasmon-magnon polariton. As a comparison, the reflection rate is approximately one when the magnetic layer is replaced by a normal dielectric with the same4 (b) (c) (d) Fano -likeLorentz -like 0.20.30.40.50.60.70.80.050.100.150.20 EF(eV)Γ(meV )ρ 0246810(a) H=0.3T H=0.4T H=0.5T 0.05 0.10 0.15 0.200.0.10.20.3 EF(eV)|Rm2 Γ=0.01 meV Fano -like Γ=0.2meV Lorentz -like 6 810 12 14 160.50.60.70.80.91.0 ω/2π(GHz )|R2 DE/GRA /FM DE/GRA /DE Re(E/E(i))FM Re(E/E(i))DE 3 my2+mz2 8 10 12 14 160.00.20.40.60.81.0 ω/2π(GHz )|R2 FIG. 3. (a) Reflection rate of the hybrid system, electric field strength at the FM surface, spin-wave excitation amplitude as a function of the incident wave frequency. d= 2.5µm, H= 0.3 T, α= 10−4,Γ = 0 .01 meV , θ= 1.1θc, EF= 0.3 eV. (b) Fano-like and Lorentz-like reflection spectrum at small relax- ation rate and large relaxation rate of carriers, respectively. The dashed lines are the results of analytical formula Eq. (14). (c) Density plot of the lineshape index ρin the EF−Γ plane. The lineshape is Fano-like for ρ≪1 and Lorentz-like forρ≫1. The black dashed line is ρ= 1. (d) The minimum reflection rate as a function of the Fermi energy at different external fields. Γ = 0 .01 meV. permittivity ϵ2(blue line), indicating very weak plasmon excitations. This comparison explicitly confirms that the magnetic layer releases the constraint to excite the TE surface plasmon. To understand the essential physics, we further plot the electric field in the 2DEG layer as well as the spin-wave amplitude as a function of the wave frequency in Fig. 3(a). When medium 2 is a magnetic layer, the spin-wave is maximally excited at the resonant frequency, which also significantly enhances the electric field in the 2DEG layer and thus strongly excites the sur- face plasmon mode. However, there is no enhancement of electric fields when medium 2 is a dielectric. Now it seems safe to conclude that the surface spin waves boost the surface plasmon excitations, which carry away signifi- cant amount of electromagnetic energy and thus generate a considerable dip in the reflection spectrum. Lineshape of the reflection spectrum. — We further no- tice that the lineshape of the reflection spectrum near the resonance is asymmetric. Physically, this may be inter- preted as an interference effect between the background continuum spectrum and a discrete mode. Here the con- tinuous mode is the flat reflection spectrum without con- sidering the magnetic properties of medium 2 (blue line in Fig. 3(a)) while the discrete mode is the hybrid sur- face plasmon-magnon mode. Specifically, we can expand the reflection rate (13) around the resonance frequencyand derive that [31] |R|2=A0(ω−ω0+λβ)2+η2 (ω−ω0)2+β2, (14) where λis the well-known Fano parameter, ω0=ωr−∆ω is the modified resonance frequency, βis the effective linewidth, and ηis the strength of the Lorentz contri- bution. In general, near the resonance position, we may characterize the relative weight of the Fano and Lorentz lineshapes by defining a lineshape index ρ≡η/(∆ω+λβ) as [31] ρ=πkzΓ2−µ0σ0ωEFΓ +πkz(ℏω)2 µ0σ0EFℏω2. (15) When the relaxation rate of carriers Γ in 2DEG is very small, the ratio ρ≈πkzℏ/µ0σ0EFis much smaller than one for higher Fermi energy, then the reflection spectrum is Fano-like [32], as shown in Fig. 3(b) (red line). When the relaxation rate Γ is very high, the ratio becomes ρ≈πkzΓ2/(µ0σ0EFℏω2). In this regime, the Lorentz contribution can be comparable and even dominate the Fano contribution (orange line in Fig. 3(b)). The com- plete phase diagram of ρin the EF−Γ plane is shown in Fig. 3(c). It is noteworthy that the Fermi energy of the 2DEG can be tuned by electric gating [33], which makes it possible to tune the lineshape and thus the minimum reflection rate of the hybrid system. Figure 3(d) shows that the minimum reflection rate |Rm|2can reach zero if the Fermi energy and external fields are appropriately tuned. In this situation, all the incident wave energy is converted to excite surface plasmons. Extension to antiferromagnet. — The essential physics presented above can be extended to AFMs. As an exam- ple, we consider a two-sublattice AFM insulator with the easy axis and external field both aligned along the x-axis. Following the theoretical approach presented above, we derive a similar form of reflection coefficient (13) with κ andνreplaced by their AFM counterparts [31]. Figure 4 shows the reflection rate as a function of the incident wave frequency. Unlike the ferromagnetic case, two dis- tinguished dips appear in the sub-THz regime (red and blue lines) depending on the direction of in-plane mo- mentum ( q) of incident wave. This difference is because there are two surface spin-wave modes in an AFM prop- agating in ±eydirections respectively [36]. The incident wave with q > 0 (q < 0) only excites the surface spin wave and plasmon propagating in the + ey(−ey) direc- tions. Therefore one may generate nonreciprocal surface plasmons by properly choosing the wave frequency. Discussions and conclusions. —In conclusion, we have shown that surface spin waves in both ferromagnet and antiferromagnet can boost the excitation of TE surface plasmons ranging from GHz to THz regime in 2D materi- als. The excitation condition does not require the purely negative imaginary conductivity and is thus applicable5 ωh/ωspq>0 q<0ωr/ωspDE/GRA/AFM(q<0)100 my2+mz2DE/GRA/AFM(q>0) 265 270 275 280 285 2900.00.20.40.60.8 ω/2π(GHz )|R20 0.602 FIG. 4. (a) Reflection rate and spin-wave excitation am- plitude of an AFM as a function of incident frequency in an DE/GRA/AFM structure. Here, mrefers to the mag- netization order of an AFM. The inset shows the external field dependence of the frequency of plasmon-magnon mode. ωsp=γp Han(2Hex+Han). Parameters of MnF 2are used with the exchange field Hex= 55 .6 T, anisotropy Han= 0.88 T, Ms= 0.059 T , ϵ2= 7.645 [34, 35], Γ = 0 .8 meV. Other parameters are the same as Fig. 3(a). to a wide class of 2D systems. The excitation of surface plasmons carries away electromagnetic energy and gen- erates a local minimum in the reflection spectrum of the system. The position of minimal reflection is tunable by both external magnetic field and electric gating on the 2D systems, providing an accessible way to probe the plas- mon excitation in the experiments. Our findings should open a novel and feasible hybrid platform to study the surface plasmons and further promote its application in designing plasmonic devices down to GHz regime. In the future, it would be interesting to extend our formalism to the quantum regime to study the entanglement between magnons and plasmons and their potential applications in quantum information science. Acknowledgments. — The work was supported by the Dutch Research Council (NWO). H.Y.Y acknowledges the helpful discussions with Mathias Kl¨ aui, Rembert Duine, Alexander Mook, Pieter Gunnink, Artem Bon- darenko, Mikhail Cherkasskii and Zhaoju Yang. [1] A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, Phys. Rep. 408, 131 (2005), ISSN 0370- 1573, URL https://www.sciencedirect.com/science/ article/pii/S0370157304004600 . [2] A. Rodin, M. Trushin, A. Carvalho, and A. H. Cas- tro Neto, Nat. Rev. Phys. 2, 524 (2020), ISSN 2522-5820, URL https://doi.org/10.1038/s42254-020-0214-4 . [3] M. S. Ukhtary and R. Saito, Carbon 167, 455 (2020), ISSN 0008-6223, URL https://www.sciencedirect. com/science/article/pii/S0008622320304607 .[4] D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. G. de Abajo, V. Pruneri, and H. Altug, Science 349, 165 (2015). [5] D. A. Iranzo, S. Nanot, E. J. C. Dias, I. Epstein, C. Peng, D. K. Efetov, M. B. Lundeberg, R. Parret, J. Osmond, J.-Y. Hong, et al., Science 360, 291 (2018). [6] I. Epstein, D. Alcaraz, Z. Huang, V.-V. Pusapati, J.-P. Hugonin, A. Kumar, X. M. Deputy, T. Khodkov, T. G. Rappoport, J.-Y. Hong, et al., Science 368, 1219 (2020). [7] W. Zhao, S. Wang, S. Chen, Z. Zhang, K. Watan- abe, T. Taniguchi, A. Zettl, and F. Wang, Nature 614, 688 (2023), ISSN 1476-4687, URL https://doi.org/10. 1038/s41586-022-05619-8 . [8] M. Jablan, H. Buljan, and M. Soljaˇ ci´ c, Phys. Rev. B 80, 245435 (2009), URL https://link.aps.org/doi/ 10.1103/PhysRevB.80.245435 . [9] S. A. Mikhailov and K. Ziegler, Phys. Rev. Lett. 99, 016803 (2007), URL https://link.aps.org/doi/10. 1103/PhysRevLett.99.016803 . [10] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015), ISSN 1745- 2481, URL https://doi.org/10.1038/nphys3347 . [11] P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Rev. Mater. 6, 1114 (2021), ISSN 2058-8437, URL https://doi.org/10.1038/ s41578-021-00332-w . [12] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Phys. Rep. 965, 1 (2022), ISSN 0370- 1573, URL https://www.sciencedirect.com/science/ article/pii/S0370157322000977 . [13] B. Zare Rameshti, S. Viola Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.- M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Phys. Rep. 979, 1 (2022), ISSN 0370- 1573, URL https://www.sciencedirect.com/science/ article/pii/S0370157322002460 . [14] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018), URL https://link.aps.org/doi/10. 1103/RevModPhys.90.015005 . [15] A. Dyrda l, A. Qaiumzadeh, A. Brataas, and J. Barna´ s, Phys. Rev. B 108, 045414 (2023), URL https://link. aps.org/doi/10.1103/PhysRevB.108.045414 . [16] S. Ghosh, G. Menichetti, M. I. Katsnelson, and M. Polini, Phys. Rev. B 107, 195302 (2023), URL https://link. aps.org/doi/10.1103/PhysRevB.107.195302 . [17] A. T. Costa, M. I. Vasilevskiy, J. Fernandez-Rossier, and N. M. R. Peres, Nano Lett. 23, 4510 (2023), ISSN 1530- 6984, URL https://doi.org/10.1021/acs.nanolett. 3c00907 . [18] J. B. S. Mendes, O. Alves Santos, L. M. Meireles, R. G. Lacerda, L. H. Vilela-Le˜ ao, F. L. A. Machado, R. L. Rodr´ ıguez-Su´ arez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 115, 226601 (2015), URL https://link.aps. org/doi/10.1103/PhysRevLett.115.226601 . [19] T. Song, X. Cai, M. W.-Y. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, et al., Science 360(2018), ISSN 0036-8075. [20] K. Takiguchi, L. D. Anh, T. Chiba, T. Koyama, D. Chiba, and M. Tanaka, Nat. Phys. 15, 1134 (2019), ISSN 1745-2481, URL https://doi.org/10. 1038/s41567-019-0621-6 . [21] T. S. Ghiasi, A. A. Kaverzin, A. H. Dismukes, D. K. de Wal, X. Roy, and B. J. van Wees, Nat. Nanotech. 16,6 788 (2021), ISSN 1748-3395, URL https://doi.org/10. 1038/s41565-021-00887-3 . [22] J. Hu, J. Luo, Y. Zheng, J. Chen, G. J. Omar, A. T. S. Wee, and A. Ariando, J. Alloys Compd. 911, 164830 (2022), ISSN 0925-8388, URL https://www.sciencedirect.com/science/article/ pii/S092583882201221X . [23] L. D. Landau, in Collected Papers of L.D. Landau , edited by D. ter Haar (Pergamon, 1965), pp. 101–114, ISBN 978-0-08-010586-4, URL https://www.sciencedirect. com/science/article/pii/B9780080105864500237 . [24] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [25] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005), URL https: //link.aps.org/doi/10.1103/RevModPhys.77.1375 . [26] J. Jackson, Classical Electrodynamics (Wiley, 1998), ISBN 9780471309321, URL https://books.google.nl/ books?id=FOBBEAAAQBAJ . [27] Y.-C. Chang, C.-H. Liu, C.-H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, Nat. Commun. 7, 10568 (2016), ISSN 2041-1723, URL https://doi.org/10.1038/ncomms10568 . [28] E. A. Kuznetsov, A. B. Rinkevich, and D. V. Perov, Tech. Phys. 64, 629 (2019), ISSN 1090-6525, URL https:// doi.org/10.1134/S1063784219050128 . [29] J. R. Eshbach and R. W. Damon, Phys. Rev.118, 1208 (1960), URL https://link.aps.org/doi/10. 1103/PhysRev.118.1208 . [30] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010), URL https://dx. doi.org/10.1088/0022-3727/43/26/264002 . [31] See the Supplementary Material for the detailed deriva- tion of the reflection rate and spin-wave amplitude in the ferromagnetic case, the reflection spectrum as a su- perposition of Fano and Lorentz shape together with an analytical form of the lineshape paramters λ, β, η and the reflection spectrum in the antiferromagnetic case. [32] U. Fano, Phys. Rev. 124, 1866 (1961), URL https:// link.aps.org/doi/10.1103/PhysRev.124.1866 . [33] M. Craciun, S. Russo, M. Yamamoto, and S. Tarucha, Nano Today 6, 42 (2011), ISSN 1748-0132, URL https://www.sciencedirect.com/science/article/ pii/S1748013210001623 . [34] F. M. Johnson and A. H. Nethercot, Phys. Rev. 114, 705 (1959), URL https://link.aps.org/doi/10.1103/ PhysRev.114.705 . [35] M. S. Seehra and R. E. Helmick, J. Appl. Phys. 55, 2330 (1984), ISSN 0021-8979, URL https://doi.org/ 10.1063/1.333652 . [36] R. E. Camley, Phys. Rev. Lett. 45, 283 (1980), URL https://link.aps.org/doi/10.1103/PhysRevLett.45. 283.

2024-02-07

Surface plasmons in two-dimensional (2D) electron systems have attracted great attention for their promising light-matter applications. However, the excitation of a surface plasmon, in particular, transverse-electric (TE) surface plasmon, remains an outstanding challenge due to the difficulty to conserve energy and momentum simultaneously in the normal 2D materials. Here we show that the TE surface plasmons ranging from gigahertz to terahertz regime can be effectively excited and manipulated in a hybrid dielectric, 2D material and magnet structure. The essential physics is that the surface spin wave supplements an additional freedom of surface plasmon excitation and thus greatly enhances the electric field in the 2D medium. Based on widely-used magnetic materials like yttrium iron garnet (YIG) and manganese difluoride ($\mathrm{MnF}_2$), we further show that the plasmon excitation manifests itself as a measurable dip in the reflection spectrum of the hybrid system while the dip position and the dip depth can be well controlled by the electric gating on the 2D layer and an external magnetic field. Our findings should bridge the fields of low-dimensional physics, plasmonics and spintronics and open a novel route to integrate plasmonic and spintronic devices.

Breaking surface plasmon excitation constraint via surface spin waves

2402.04626v1

Anisotropy-assisted magnon condensation in ferromagnetic thin films Therese Frostad,1Philipp Pirro,2Alexander A. Serga,2 Burkard Hillebrands,2Arne Brataas,1and Alireza Qaiumzadeh1 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 2Fachbereich Physik and Landesforschungszentrum OPTIMAS, Rheinland-Pf¨ alzische Technische Universit¨ at Kaiserslautern-Landau, 67663 Kaiserslautern, Germany We theoretically demonstrate that adding an easy-axis magnetic anisotropy facilitates magnon condensation in thin yttrium iron garnet (YIG) films. Dipolar interactions in a quasi-equilibrium state stabilize room-temperature magnon condensation in YIG. Even though the out-of-plane easy- axis anisotropy generally competes with the dipolar interactions, we show that adding such magnetic anisotropy may even assist the generation of the magnon condensate electrically via the spin transfer torque mechanism. We use analytical calculations and micromagnetic simulations to illustrate this effect. Our results may explain the recent experiment on Bi-doped YIG and open a pathway toward applying current-driven magnon condensation in quantum spintronics. Introduction—. Magnon condensate with nonzero momentum at room temperature is a fascinating phe- nomenon first observed in 2006 [1]. Condensed magnons were observed at two degenerate magnon band minima of high-quality yttrium iron garnet (YIG), an easy-plane ferrimagnetic insulator with a very low magnetic dissi- pation [2–4], as the spontaneous formation of a quasi- equilibrium and coherent magnetization dynamics in mo- mentum space [5–8]. To generate magnon condensate, nonequilibrium magnons must be pumped into the sys- tem by an incoherent stimulus such as parametric pump- ing [1, 9–18], rapid cooling of thermal magnons [19– 21], and spin-transfer torque [22–29]. Above a critical nonequilibrium magnon density, magnons may finally (quasi)thermalize to form a quasi-equilibrium magnon condensate at the bottom of magnon bands. The study of magnon condensation is not only inter- esting from an academic point of view but also of great importance in various areas of emerging quantum tech- nology and applied spintronics [15, 30–33]. Therefore, it is crucial to clarify the intricate microscopic mechanisms at play and to present theoretical proposals to electrically control the generation of magnon condensate. At low magnon densities, the interaction between magnons is weak, and they behave as free quasiparti- cles. But when the magnon population increases, as in magnon condensation experiments, the interactions between magnons become stronger and more crucial. Moreover, nonlinear magnon interactions facilitate the quasi-thermalization process of injected nonequilibrium magnons. A stable and steady quasi-equilibrium magnon condensation requires an effective repulsive interaction between injected magnon quasiparticles. It is known that in a system mainly influenced by Heisenberg exchange interactions, interaction between magnons is attractive. However, it was shown that dipolar interactions may as- sist in the generation of a metastable double-degenerate magnon condensate in YIG [12–14, 34–42]. Recently, it was theoretically shown that the quasi- thermalization time of magnon condensation is reducedin confined nanoscopic systems [43]. It was also demon- strated that the lateral confinement in YIG thin films enhances the dipolar interaction along the propagation direction of magnons and causes a deeper band depth, i.e., the difference between the ferromagnetic resonance (FMR) and magnon band minima. Increasing the life- time of the magnon condensate was attributed to this enhancement of the band depth [43]. In another recent achievement in magnon condensa- tion experiments, Divinsky et al. [22] found evidence of condensation of magnons by spin-transfer torque mech- anism. They introduced a small perpendicular magne- tocrystalline anisotropy (PMA) through bismuth doping in the thin film of YIG, while the magnetic ground state still resides within the plane. This discovery opens a route toward electrical control of magnon condensation. However, the interplay between the dipolar interac- tions, which was previously shown to be essential for the stability and thermalization of magnon condensation, and the counteracting out-of-plane easy-axis magnetic anisotropy is so far uncharted. In this Letter, we ana- lyze the stability of condensate magnons in the presence of a PMA in YIG. We present simulations within the Landau-Lifshitz-Gilbert framework [44–46] that support our analytical calculations. Model—. We consider a thin ferromagnetic film in the y−zplane to model YIG. The magnetic moments are di- rected along the zdirection by an in-plane external mag- netic field of strength H0. The magnetic potential energy of the film contains contributions from the isotropic ex- change interaction Hex, Zeeman interaction HZ, dipolar interaction Hdip, and additionally a PMA energy Hanin thexdirection, normal to the film plane. YIG has a weak in-plane easy-axis that can be neglected compared to the other energy scales in the system. The total spin Hamiltonian of the system reads, H=Hex+HZ+Hdip+Han. (1)arXiv:2309.05982v3 [cond-mat.mes-hall] 10 Feb 20242 The PMA energy is given by, Han=−KanX j(Sj·ˆx)2, (2) where Kan>0 is the easy-axis energy, ℏSjis the vector of spin operator at site j, and ℏis the reduced Planck constant. Details of the Hamiltonian can be found in the Supplemental Material (SM) [47]. The Holstein-Primakoff spin-boson transformation [48] allows us to express the spin Hamiltonian in terms of magnon creation and annihilation operators. The ampli- tude of the effective spin per unit cell in YIG at room temperature is large S≈14.3≫1, [39, 49, 50], and thus we can expand the spin Hamiltonian in the inverse pow- ers of the spin S, which is equivalent to the semiclassical regime. Up to the lowest order in nonlinear terms, the magnon Hamiltonian Hof a YIG thin film can be ex- pressed as the sum of two components: H2andH4. The former represents a noninteracting magnon gas compris- ing quadratic magnon operators. The latter, on the other hand, constitutes nonlinear magnon interactions charac- terized by quartic magnon operators; see SM for details [47]. Note that three-magnon interactions are forbidden in our geometry by the conservation laws [51] Magnon dispersion of YIG with a finite PMA—. The magnon dispersion in YIG is well known and has been studied extensively in both experimental and theoreti- cal works [2, 52, 53]. Magnons traveling in the direc- tion of the external magnetic field have the lowest en- ergy. These so-called backward volume magnetostatic (BVM) magnons have a dispersion with double degener- ate minima at finite wavevectors qz=±Q. When pump- ing magnons into the thin film, the magnons may ther- malize and eventually form a condensate state in these two degenerate minima with opposite wavevectors. The noninteracting magnon Hamiltonian and the dis- persion of BVM magnons, along the zdirection, in the presence of a finite PMA reads, H2=X qzℏωqzˆc† qzˆcqz, (3a) ℏωqz=q A2qz−B2qz, (3b) where ˆ c† qz(ˆcqz) are magnon creation (annihilation) oper- ators, which are Bogoliubov bosons [47], and Aqz=Dexq2 z+γ(H0+ 2πMSfq)−KanS, (4a) Bqz= 2πMSfq−KanS. (4b) Here, Dex=JexSa2is the exchange constant, Jexis the Heisenberg exchange coupling and MS=γℏS/a3is the saturation magnetization, where γ= 1.2×10−5eV Oe−1 is the gyromagnetic ratio, and a= 12 .376˚A is the lattice constant of YIG. The form factor fq= (1 − e−|qz|Lx)/(|qz|Lx) stems from dipolar interactions in a thin magnetic film with thickness Lx[54, 55]. FIG. 1. The analytical dispersion of noninteracting BVM magnons in a YIG thin film for various PMA strengths, Eq. (3b). The inset shows the depth of the magnon band minima as a function of the PMA strength. We set the thickness Lx= 50 nm and the magnetic field in the z direction H0= 1 kOe. Figure 1 shows the effect of PMA on the magnon dispersion of YIG. PMA decreases the FMR frequency ωqz=0, in addition to a more significant decrease in the magnon band minima ωqz=±Q. Therefore the band depth ∆ω=ωqz=0−ωqz=±Qis increased. The position of the band minima at qz=±Qis also shifted to larger momenta. In addition, the curvature of the minima in- creases as a function of the anisotropy strength. Above a critical PMA, Kc2an, the magnetic ground state is desta- bilized and the in-plane magnetic state becomes out- of-plane. We are interested in the regime in which the magnetic ground state remains in the plane, and thus the effective saturation magnetization is positive Meff=MS−2Kan/(µ0MS)>0. The effect of PMA on magnon dispersion resembles the effect of confinement in the magnon spectra of YIG. In Ref. 43, it was shown that transverse confinement in a YIG thin film leads to an increase of the FMR frequency, the band depth, as well as shifting the band minima to higher momenta while the magnon band gap at the band minima is also increased. It was shown that this change of the spectrum in confined systems increases the magnon condensate lifetime. Therefore, we expect PMA to in- crease the magnon condensate lifetime and assist in the generation of magnon condensation. Nonlinear magnon interactions in the presence of PMA—. To check the stability of condensate magnons at low energy, we should show that the interactions do not destabilize and destroy magnon condensation. To achieve this goal, we turn on the magnon interaction between the condensate magnons at qz=±Q. This magnon in- teraction consists of intra- and inter-band contributions,3 H4=Hintra 4+Hinter 4, where Hintra 4=A(ˆc† Qˆc† QˆcQˆcQ+ ˆc† −Qˆc† −Qˆc−Qˆc−Q), (5a) Hinter 4= 2B(ˆc† Qˆc† −QˆcQˆc−Q) +C(ˆc† Qˆc−QˆcQˆc−Q ˆc† −Qˆc−Qˆc−QˆcQ+ H.c.) + D(ˆc† Qˆc† Qˆc† −Qˆc† −Q+ H.c.) .(5b) The intraband magnon interaction, parametrized by A, preserves magnon number. However, the interband magnon interaction includes both a magnon conserving contribution, parametrized by B, and nonconserving con- tributions, parametrized by CandD, see SM [47]. The interaction amplitudes are given by A=−γπM S SN (α1+α3)fQ−2α2(1−f2Q) −DexQ2 2SN(α1−4α2) +Kan 2N(α1+α3), (6a) B=γ2πMS SN (α1−α2)(1−f2Q)−(α1−α3)fQ) +DexQ2 2SN(α1−2α2) +Kan N(α1+α3), (6b) C=γπM S 2SN (3α1+ 3α2+ 4α3)fQ−8 3α3(1−f2Q) +DexQ2 3SNα3+Kan 4N(3α1+ 3α2+ 4α3), (6c) D=γπM S 2SN (3α1+ 3α2+ 4α3)fQ−2α2(1−f2Q) +DexQ2 2SNα2+Kan 2N(3α2+α3). (6d) Here, Nis the total number of spin sites. The dimen- sionless parameters α1,α2, and α3are related to the Bogoliubov transformation coefficients, listed in SM [47]. An off-diagonal long-rage order characterizes the con- densation state. The condensate state is a macroscopic occupation of the ground state and can be represented by a classical complex field. Therefore, to analyze the sta- bility of the magnon condensate, we perform Madelung’s transform ˆ c±Q→p N±Qeiϕ±Q, in which the macroscopic condensate magnon state is described with a coherent phase ϕ±Qand a population number N±Q[39, 40]. The total number of condensed magnons is Nc=N+Q+N−Q, while the distribution difference is δ=N+Q−N−Q.Ncis set in the system by an external magnon pumping mech- anism and is a constant. We also define the total phase as Φ = ϕ+Q+ϕ−Q. Finally, the macroscopic four-magnon interaction energy of condensed magnons is expressed as , V4(δ,Φ) =N2 c 2 A+B+ 2Ccos Φs 1−δ2 N2c +Dcos 2Φ − B−A+Dcos 2Φ
δ2 N2c .(7) This expression is similar to the one recently obtained without PMA [56], but the interaction amplitudes, Eq. FIG. 2. The analytical nonlinear interaction energy of magnon condensate state, Eq. (7), as a function of the PMA strength. NandNcare the total spins and condensate magnons, respectively. Kc1anrepresents the critical value of the PMA at which the sign of nonlinear interaction energy is changed. On the other hand, Kc2ancorresponds to the critical value of PMA at which the in-plane magnetic ground state becomes unstable. We set Lx= 50 nm and H0= 1 kOe. Ksim an= 0.5µeV denotes the PMA used in our micromagnetic simulations. (6), depend on the PMA through the Bogoliubov coeffi- cients, see SM [47] . We now look at the total interaction energy and am- plitudes of condensate magnons in more detail. Figure 2 shows the effective interaction potential of condensate magnons as a function of the PMA. In a critical PMA strength, Kc1an, the sign of the interaction changes. This means that below Kc1an, the interaction reduces the total energy of condensate magnons while above Kc1anthe inter- action increases its energy. This critical anisotropy is well below the critical magnetic anisotropy strength Kc2anthat destabilizes the in-plane magnetic ground state. In the following, we consider a PMA strength below the critical anisotropy Kan< Kc1an. The interacting potential energy of the condensate magnons, Eq. (7), has five extrema, ∂V4(δi,Φi) = 0, at, δ1= 0,Φ1= 0; (8a) δ2= 0,Φ2=π; (8b) δ3= 0,Φ3= cos−1(−C D); (8c) δ4=Nc 1−(C B−A+D)21 2,Φ4= 0; (8d) δ5=δ4,Φ5=π. (8e) δi= 0 indicates condensate states with symmetric magnon populations in the two magnon band minima while δi̸= 0 represents states with nonsymmetrical4 TABLE I: The material parameters used in the micromagnetic simulations. Parameter Symbol Value Saturation magnetization 4 πMS1.75 kOe Effective spin S 14.3 Exchange constant Dex 0.64 eV ˚A2 Gilbert damping parameter α 10−3 (a) (b) FIG. 3. The theoretical phase diagram of the condensate magnons in the absence (a) and presence (b) of PMA. We plot the magnon interaction energy V4/N2 c, Eq. (7), as a function of the film thickness Lxand the magnetic field strength H0, applied along the zdirection. Different states are labeled based on different extrema listed in Eq. (8). The dashed black lines indicate the boundaries between the different condensate phases, Eq. (8). We set Kan= 0.5µeV in (b). magnon populations. Whether any of these extrema rep- resents the actual minimum of the interacting potential energy, i.e., ∂2V4(δi,Φi)>0, depends on the system pa- rameters. Finding these minima allows us to construct the phase diagram for magnon condensate. Phase diagram for magnon condensate—. Now, we theoretically explore the (meta)stability of the magnoncondensate as a function of the film thickness Lxand the magnetic field strength H0, applied along the zdirection in the plane, using the YIG spin parameters, see Ta- ble I. We characterize a (meta)stable state of a magnon condensate in the phase diagram as one that minimizes the interaction potential V4while satisfying the condi- tionV4<0, ensuring a reduction in the total magnon condensate energy at qz=±Qthrough interactions. First, we present the phase diagram for magnon con- densation in YIG, in the absence of PMA, in Fig. 3a. The thinner films are expected to have a symmetric dis- tribution of magnons between the two magnon band min- ima, the state with δ1= 0, and only thicker films with larger applied magnetic fields tend to have nonsymmetric magnon populations, the state with δ4̸= 0. This phase diagram is in agreement with previous studies [39, 56]. Next, we add a PMA, with strength Kan= 0.5µeV, and plot the phase diagram of the magnon condensate in Fig. 3b for different thicknesses. Compared to the case without PMA, we see that the condensate magnons can only be stabilized for thinner films, and within our mate- rial parameters, we do not have any metastable conden- sation above 90nm since the sign of the total interaction energy becomes positive. In addition, we have a richer phase diagram in the presence of PMA. PMA tends to push the magnon condensate within our material param- eters toward a more nonsymmetric population distribu- tion between the two magnon band minima, states with δ4̸= 0 and δ5̸= 0. Since both minima are degener- ate, there is an oscillation of magnon population between these two minima. In very thin films, less than 30 nm, we may have a symmetric condensate magnon state, δ1= 0, in our system. This phase diagram shows that in the presence of a PMA, magnon condensate can be still survived as a metastable state. In addition, as we discussed earlier, a PMA increases the band depth and reduces the cur- vature of noninteracting magnon dispersion, see Fig. 1, leading to an enhancement of the condensate magnon lifetime. Thus, we expect that introducing a small PMA into a thin film of YIG facilitates the magnon conden- sation process. It is worth mentioning that the stabi- lizing condensated magnons in thinner films with finite PMA, is not a real problematic issue since the injection of magnons into the system by electrical means is more efficient in thin films. Micromagnetic simulation of magnon condensate—. To validate our theoretical predictions and illustrate the facilitation of magnon condensate formation by incor- porating a PMA, we conducted a series of micromag- netic simulations. Simulations were performed using MuMax3[57], which solves the semiclassical Landau- Lifshitz-Gilbert (LLG) equation that describes magneti- zation’s precessional motion; see SM [47]. In the limit of large S, it is important to note that we enter the semiclassical regime and hence the LLG equation may effectively capture and accurately describe the spin dy- namics of the system. In our ferromagnetic thin film5 (a) (b) (c) (d) FIG. 4. Micromagnetic simulation of nonequilibrium magnon distribution injected by spin torque mechanism for a YIG thin film with a thickness of Lx= 50 nm and lat- eral sizes of Ly=Lz= 5µm,in the presence of an external magnetic field along the zdirection H0= 1 kOe. (a) and (b) show magnon distributions of the initial nonequilibrium injected magnons and final quasi-equilibrium magnon con- densate steady state, respectively, when Kan= 0. (c) and (d) show magnon distributions of initial nonequilibrium ex- cited magnons and final quasi-equilibrium magnon conden- sate steady state, respectively, when Kan= 0.5µeV. The dotted line indicates the analytical dispersion relation of non- interacting magnons, Eq. 3b. Because of magnon-magnon interactions, the simulated magnon dispersion has a nonlin- ear spectral shift compared to the analytical noninteracting magnon dispersion. Although the duration of magnon pump- ing by spin-transfer torque is the same in the absence or pres- ence of the PMA, the critical torque amplitude is lower in the presence of PMA. simulation, magnons are excited via spin-transfer torque at zero temperature, eliminating thermal magnons. The nonequilibrium magnons in the film are the result of in- jection of spin current across the sample surface [22]. Optimal spin torque strength ensures that the magnon population reaches the critical density required for form- ing condensed magnons; see SM for simulation details [47]. By introducing spin torque into the system, we excite magnons with varying wavevectors and frequen- cies, as illustrated in Figs. 4(a) and 4(c). A portion of these nonequilibrium magnons undergoes thermalization through nonlinear magnon-magnon interactions, leading to the establishment of a stable and quasi-equilibriumstate of condensed magnons located at the minima of the magnon band spectra, ±Q, as depicted in Figs. 4(b) and 4(d). This sharp peak in the number of magnons at the band minima is a signature of magnon condensate. The numerical simulations confirm the supportive role of PMA in the condensation process. First, there is a reduction in the threshold of spin-transfer torque neces- sary to inject the critical magnon density into the sys- tem, enabling the system to attain said critical magnon density even at lower torque amplitudes. Second, the fi- nal condensate magnons in the presence of the PMA are more localized around the band minima than in the ab- sence of PMA. Simulations also indicate that PMA shifts the population of condensate magnons from a symmetric distribution between two band minima to a nonsymmet- ric distribution, Fig. 4. This agrees with the analytical phase diagram in Fig. 3(b). Summary and concluding remarks—. Dipolar inter- actions are assumed to be relevant to stabilizing the magnon condensate within YIG. The presence of a PMA is expected to counteract dipolar interactions. In this Letter, we show that even at intermediate strengths of the PMA field, a magnon condensate state can exist as a metastable state. We note that the anisotropy increases the band depth and curvature of the magnon disper- sion. These adjustments to the magnon spectrum are ex- pected to facilitate magnon condensate formation. From the calculations of effective magnon-magnon interactions and minimizing the interaction potential at the band minima, we find the magnon condensate phase diagram. We demonstrate that the inclusion of PMA results in a magnon condensate with a more intricate phase diagram compared to when PMA is absent. A finite PMA has the tendency to drive the magnon condensate towards a nonsymmetric magnon population at band minima in thinner films and lower magnetic fields, as compared to the absence of PMA. Micromagnetic simulations within the LLG framework confirm our analytical results and analyses. ACKNOWLEDGEMENTS The authors thank Anne Louise Kristoffersen for help- ful discussions. We acknowledge financial support from the Research Council of Norway through its Centers of Excellence funding scheme, project number 262633, ”QuSpin”. A.Q. was supported by the Norwegian Fi- nancial Mechanism Project No. 2019/34/H/ST/00515, ”2Dtronics”. P.P., A.A.S., and B.H. acknowledge fi- nancial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) TRR 173 Grant No. 268565370 Spin+X (Projects B01 and B04). [1] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N.Slavin, Bose–Einstein condensation of quasi-equilibrium6 magnons at room temperature under pumping, Nature 443, 430 (2006). [2] V. Cherepanov, I. Kolokolov, and V. L’vov, The saga of YIG: Spectra, thermodynamics, interaction and relax- ation of magnons in a complex magnet, Phys. Rep. 229, 81 (1993). [3] L. Soumah, N. Beaulieu, L. Qassym, C. Carr´ et´ ero, E. Jacquet, R. Lebourgeois, J. Ben Youssef, P. Bortolotti, V. Cros, and A. Anane, Ultra-low damping insulating magnetic thin films get perpendicular, Nat. commun. 9, 3355 (2018). [4] C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schnei- der, K. Lenz, J. Grenzer, R. H¨ ubner, and E. Wendler, Low damping and microstructural perfection of sub- 40nm-thin yttrium iron garnet films grown by liquid phase epitaxy, Phys. Rev. Mater. 4, 024416 (2020). [5] S. Demokritov, Comment on “Bose–Einstein Condensa- tion and Spin Superfluidity of Magnons in a Perpendicu- larly Magnetized Yttrium Iron Garnet Film”, JETP Lett. 115, 691 (2022). [6] T. B. Noack, V. I. Vasyuchka, A. Pomyalov, V. S. L’vov, A. A. Serga, and B. Hillebrands, Evolution of room-temperature magnon gas: Toward a coherent Bose- Einstein condensate, Phys. Rev. B 104, L100410 (2021). [7] D. Snoke, Coherent questions, Nature 443, 403 (2006). [8] R. E. Troncoso and A. S. N´ u˜ nez, Dynamics and sponta- neous coherence of magnons in ferromagnetic thin films, J. Phys. Cond. Mat. 24, 036006 (2011). [9] P. Anderson and H. Suhl, Instability in the motion of ferromagnets at high microwave power levels, Phys. Rev. 100, 1788 (1955). [10] H. Suhl, The non-linear behaviour of ferrites at high sig- nal level, Proceedings of the IRE 44, 1270 (1956). [11] V. E. Demidov, O. Dzyapko, M. Buchmeier, T. Stock- hoff, G. Schmitz, G. A. Melkov, and S. O. Demokritov, Magnon kinetics and Bose-Einstein condensation studied in phase space, Phys. Rev. Lett. 101, 257201 (2008). [12] S. M. Rezende, Theory of coherence in Bose-Einstein con- densation phenomena in a microwave-driven interacting magnon gas, Phys. Rev. B 79, 174411 (2009). [13] P. Nowik-Boltyk, O. Dzyapko, V. Demidov, N. Berloff, and S. Demokritov, Spatially non-uniform ground state and quantized vortices in a two-component Bose-Einstein condensate of magnons, Sci. Rep. 2, 482 (2012). [14] A. A. Serga, V. S. Tiberkevich, C. W. Sandweg, V. I. Vasyuchka, D. A. Bozhko, A. V. Chumak, T. Neumann, B. Obry, G. A. Melkov, A. N. Slavin, et al. , Bose–Einstein condensation in an ultra-hot gas of pumped magnons, Nat. Commun. 5, 3452 (2014). [15] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, F. Heussner, G. A. Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Supercurrent in a room-temperature Bose–Einstein magnon condensate, Nat. Phys. 12, 1057 (2016). [16] C. Sun, T. Nattermann, and V. L. Pokrovsky, Unconven- tional superfluidity in yttrium iron garnet films, Phys. Rev. Lett. 116, 257205 (2016). [17] O. Dzyapko, P. Nowik-Boltyk, B. Koene, V. E. Demidov, J. Jersch, A. Kirilyuk, T. Rasing, and S. O. Demokritov, High-resolution magneto-optical Kerr-effect spectroscopy of magnon Bose–Einstein condensate, IEEE Magn. Lett. 7, 1 (2016). [18] I. Borisenko, B. Divinskiy, V. Demidov, G. Li, T. Nat- termann, V. Pokrovsky, and S. Demokritov, Direct evi-dence of spatial stability of Bose-Einstein condensate of magnons, Nat. Commun. 11, 1691 (2020). [19] M. Schneider, T. Br¨ acher, and e. a. Breitbach, D., Bose–Einstein condensation of quasiparticles by rapid cooling, Nat. Nanotechnol. 15, 457 (2020). [20] M. Schneider, D. Breitbach, R. O. Serha, Q. Wang, M. Mohseni, A. A. Serga, A. N. Slavin, V. S. Tiberkevich, B. Heinz, T. Br¨ acher, B. L¨ agel, C. Dubs, S. Knauer, O. V. Dobrovolskiy, P. Pirro, B. Hillebrands, and A. V. Chu- mak, Stabilization of a nonlinear magnonic bullet coex- isting with a Bose-Einstein condensate in a rapidly cooled magnonic system driven by spin-orbit torque, Phys. Rev. B104, L140405 (2021). [21] C. Safranski, I. Barsukov, and e. a. Lee, H. K., Spin caloritronic nano-oscillator, Nat. Commun. 8, 117 (2017). [22] B. Divinskiy, H. Merbouche, V. E. Demidov, K. Niko- laev, L. Soumah, D. Gou´ er´ e, R. Lebrun, V. Cros, J. B. Youssef, P. Bortolotti, A. Anane, and S. O. Demokritov, Evidence for spin current driven Bose-Einstein conden- sation of magnons, Nat. Commun. 12, 6541 (2021). [23] M. Schneider, D. Breitbach, R. O. Serha, Q. Wang, A. A. Serga, A. N. Slavin, V. S. Tiberkevich, B. Heinz, B. L¨ agel, T. Br¨ acher, C. Dubs, S. Knauer, O. V. Do- brovolskiy, P. Pirro, B. Hillebrands, and A. V. Chumak, Control of the Bose-Einstein Condensation of Magnons by the Spin Hall Effect, Phys. Rev. Lett. 127, 237203 (2021). [24] D. Breitbach, M. Schneider, B. Heinz, F. Kohl, J. Maskill, L. Scheuer, R. O. Serha, T. Br¨ acher, B. L¨ agel, C. Dubs, V. S. Tiberkevich, A. N. Slavin, A. A. Serga, B. Hillebrands, A. V. Chumak, and P. Pirro, Stimulated Amplification of Propagating Spin Waves, Phys. Rev. Lett. 131, 156701 (2023). [25] V. E. Demidov, S. Urazhdin, E. R. J. Edwards, M. D. Stiles, R. D. McMichael, and S. O. Demokritov, Control of magnetic fluctuations by spin current, Phys. Rev. Lett. 107, 107204 (2011). [26] V. Demidov, S. Urazhdin, G. De Loubens, O. Klein, V. Cros, A. Anane, and S. Demokritov, Magnetization oscillations and waves driven by pure spin currents, Phys. Rep.673, 1 (2017). [27] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Electronic pumping of quasiequilibrium Bose-Einstein- condensed magnons, Phys. Rev. Lett. 108, 246601 (2012). [28] S. A. Bender, R. A. Duine, A. Brataas, and Y. Tserkovnyak, Dynamic phase diagram of dc-pumped magnon condensates, Phy. Rev. B 90, 094409 (2014). [29] Y. Tserkovnyak, S. A. Bender, R. A. Duine, and B. Fle- bus, Bose-Einstein condensation of magnons pumped by the bulk spin Seebeck effect, Phys. Rev. B 93, 100402(R) (2016). [30] K. Nakata, K. A. van Hoogdalem, P. Simon, and D. Loss, Josephson and persistent spin currents in bose-einstein condensates of magnons, Phys. Rev. B 90, 144419 (2014). [31] S. N. Andrianov and S. A. Moiseev, Magnon qubit and quantum computing on magnon Bose-Einstein conden- sates, Phys. Rev. A 90, 042303 (2014). [32] M. Mohseni, V. I. Vasyuchka, V. S. L’vov, A. A. Serga, and B. Hillebrands, Classical analog of qubit logic based on a magnon Bose–Einstein condensate, Commun. Phys. 5, 196 (2022). [33] Y. M. Bunkov, A. N. Kuzmichev, T. R. Safin, P. M. Vetoshko, V. I. Belotelov, and M. S. Tagirov, Quantum7 paradigm of the foldover magnetic resonance, Sci. Rep. 11, 7673 (2021). [34] O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk, V. E. Demi- dov, S. O. Demokritov, B. Koene, A. Kirilyuk, T. Rasing, V. Tiberkevich, and A. Slavin, Magnon-magnon inter- actions in a room-temperature magnonic Bose-Einstein condensate, Phys. Rev. B 96, 064438 (2017). [35] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, and A. N. Slavin, Quantum coherence due to Bose–Einstein condensation of parametrically driven magnons, New J. Phys. 10, 045029 (2008). [36] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, Observation of spontaneous coherence in Bose-Einstein condensate of magnons, Phys. Rev. Lett. 100, 047205 (2008). [37] S. M. Rezende, Theory of microwave superradiance from a Bose-Einstein condensate of magnons, Phys. Rev. B 79, 060410 (2009). [38] I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin, Stability of bose-einstein condensates of hot magnons in yttrium iron garnet films, Phys. Rev. Lett. 100, 257202 (2008). [39] F. Li, W. M. Saslow, and V. L. Pokrovsky, Phase diagram for magnon condensate in yttrium iron garnet film, Sci. Rep.3, 1372 (2013). [40] H. Salman, N. G. Berloff, and S. O. Demokritov, Micro- scopic theory of Bose-Einstein condensation of magnons at room temperature, in Universal Themes of Bose- Einstein Condensation , edited by N. P. Proukakis, D. W. Snoke, and P. B. Littlewood (Cambridge University Press, Cambridge, 2017) Chap. 25, p. 493. [41] J. Hick, F. Sauli, A. Kreisel, and P. Kopietz, Bose- Einstein condensation at finite momentum and magnon condensation in thin film ferromagnets, Eur. Phys. J. B 78, 429 (2010). [42] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A. Melkov, and A. N. Slavin, Thermalization of a paramet- rically driven magnon gas leading to Bose-Einstein con- densation, Phys. Rev. Lett. 99, 037205 (2007). [43] M. Mohseni, A. Qaiumzadeh, A. A. Serga, A. Brataas, B. Hillebrands, and P. Pirro, Bose–Einstein condensation of nonequilibrium magnons in confined systems, New J. Phys. 22, 083080 (2020). [44] M. Lakshmanan, The fascinating world of the Landau– Lifshitz–Gilbert equation: an overview, Philos. Trans. R. Soc. A 369, 1280 (2011). [45] L. Landau and E. Lifshitz, 3 - On the theory of the dis- persion of magnetic permeability in ferromagnetic bod- ies, in Perspectives in Theoretical Physics , edited by L. P. Pitaevski (Pergamon, Amsterdam, 1992) pp. 51–65. [46] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40, 3443 (2004). [47] See Supplementary Material for more details on the magnon Hamiltonian and simulation. [48] T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940). [49] S. Streib, N. Vidal-Silva, K. Shen, and G. E. W. Bauer, Magnon-phonon interactions in magnetic insula- tors, Phys. Rev. B 99, 184442 (2019). [50] H. Maier-Flaig, S. Klingler, C. Dubs, O. Surzhenko, R. Gross, M. Weiler, H. Huebl, and S. T. B. Goen- nenwein, Temperature-dependent magnetic damping of yttrium iron garnet spheres, Phys. Rev. B 95, 214423(2017). [51] A. D. Boardman and S. A. Nikitov, Three-and four- magnon decay of nonlinear surface magnetostatic waves in thin ferromagnetic films, Phys. Rev. B 38, 11444 (1988). [52] A. J. Princep, R. A. Ewings, S. Ward, S. T´ oth, C. Dubs, D. Prabhakaran, and A. T. Boothroyd, The full magnon spectrum of yttrium iron garnet, npj Quantum Mater. 2, 63 (2017). [53] A. A. Serga, A. V. Chumak, and B. Hillebrands, YIG magnonics, J. Phys. D: Appl. Phys. 43, 264002 (2010). [54] F. J. Buijnsters, L. J. Van Tilburg, A. Fasolino, and M. I. Katsnelson, Two-dimensional dispersion of mag- netostatic volume spin waves, J. Phys. Condens. Matter. 30, 255803 (2018). [55] H. Zabel and M. Farle, Magnetic Nanostructures: Spin Dynamics and Spin Transport , Vol. 246 (Springer, New York, 2012). [56] C. Sun, T. Nattermann, and V. L. Pokrovsky, Bose– Einstein condensation and superfluidity of magnons in yttrium iron garnet films, J. Phys. D: Appl. Phys. 50, 143002 (2017). [57] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. 4, 107133 (2014). [58] A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Mi- croscopic spin-wave theory for yttrium-iron garnet films, Eur. Phys. J. B 71, 59 (2009). [59] T. Frostad, H. L. Skarsv˚ ag, A. Qaiumzadeh, and A. Brataas, Spin-transfer-assisted parametric pumping of magnons in yttrium iron garnet, Phys. Rev. B 106, 024423 (2022).8 Appendix A: Diagonalization of Magnon Hamiltonian The total spin Hamiltonian of a thin film, in the y−zplane with a small perpendicular anisotropy along the ˆ x direction reads, H=Hex+HZ+Hdip+Han. (A1) The exchange energy between neighboring spins reads Hex=−1 2JexX i,jSi·Sj, (A2) where Jex>0 is the ferromagnetic exchange constant and ais the lattice constant. The Zeeman energy due to an inplane external magnetic field of strength H0along the ˆ zdirection reads, HZ=−gµBH0X jSz j, (A3) where µBis the Bohr magneton and gis the the effective Land´ e g-factor. The dipolar field is expressed as [58], Hdip=−1 2X i,jX α,βDα,β i,jSα iSβ j, (A4a) Dα,β i,j= (gµB)2(1−δi,j)∂2 ∂rα ij∂rβ ij1 |rij|, (A4b) where α, βdenote the spatial components x, yandz; and rijis the distance vector between the spin sites iandj. Finally, the PMA anisotropy is given by, Han=−KanX j(Sj·ˆx)2. (A5) The Holstein-Primakoff transformation allows us to express the spin operators in terms of bosonic creation and annihilation operators ˆ a†and ˆarespectively. Using the large- Sapproximation, we have, S+≈ℏ√ 2S(ˆa−ˆa†ˆaˆa/(4S)) ,S−≈ℏ√ 2S(ˆa†−ˆa†ˆa†ˆa/(4S)), and Sz=ℏ(S−ˆa†ˆa) . The corresponding noniteracting boson Hamiltonian in the Fourier space reads, H2=X qAqˆa† qˆaq+1 2Bqˆaqˆa−q+1 2B∗ qˆa† qˆa† −q, (A6) where ˆ a† j=1 NP qeik·rjˆa† qandAq(Bq) is presented in Eq. (4) in the main text. We utilize the following Bo- goliubov transformation to diagonalize this bosonic Hamiltonian and find the corresponding noninteracting magnon Hamiltonian, ˆaq=uqˆcq+vqˆc† −q, (A7a) ˆa† −q=v† qˆcq+uqˆc† −q, (A7b) where uq=u−qandvq=v−qare the Bogoliubov coefficients, given by, uq=Aq+ 2ℏωq 2ℏωq
1 2, (A8a) vq= sgn( Bq)Aq−2ℏωq 2ℏωq
1 2. (A8b) The Bogoliubov coefficients depend on the easy-axis magnetic anisotropy Kanvia both magnon dispersion ωqand Aq. It can be shown that the off-diagonal terms ( α̸=β) of the dipolar interaction vanish in the uniform mode approximation for a thin film of infinite lateral lengths [41, 58]. In this case, the dipolar interaction contains no three-magnon operator terms. We omit any renormalization correction to the noninteracting magnon Hamiltonian, as9 FIG. S1. The interaction parameters, Eq. (6) in the main text, as a function of PMA. AandBare, respectively, the magnon-conserving intraband and interband interaction parameters while CandDare the magnon-non-conserving interband interaction parameters. We set Lx= 50 nm and H0= 1 kOe. they are of the order of 1 /Sand small. We define the following parameters in the 4-magnon interaction amplitudes, introduced in the main text Eq. (6), [39, 40]. α1=u4 Q+v4 Q+ 4u2 Qv2 Q, (A9a) α2= 2u2 Qv2 Q, (A9b) α3= 3uQvQ(u2 Q+v2 Q). (A9c) These parameters depend on Kanvia Bogoliubov coefficients. Appendix B: The nonlinear interaction amplitudes In Fig. S1, we plot the different intraband and interband interaction apmlitudes, see Eq. (6) in the main text, as a function of PMA.10 TABLE II: Simulation parameters Parameter Value Excitation time, first interval 100 ns Excitation time, second interval 200 ns Excitation strength, first interval −1.2×1010A m−2 Excitation strength, second interval −0.35×1010A m−2 (Kan= 0) Excitation strength, second interval −0.2×1010A m−2 (Kan= 0.5µeV) Appendix C: Micromagnetic Simulations We solve the following LLG equation in the continuum model for the magnetization direction m(r, t) =S/S, using micromagnetic simulation code MuMax3 [57], to study spin dynamics in the system, ∂m ∂t=γ 1 +α2 m×Beff+αm×(m×Beff)
+τSTT, (C1) where γis the gyromagnetic ration, αis the Gilbert damping parameter, Beff=−M−1 SdH/dmis the effective magnetic field,τSTT=βm×(mp×m) is the Slonczewski spin-transfer torque, with βis depends on material parameters and applied charge current density and mpis the polarization of the spin current. In the continuum model, the exchange stiffness in the SI unit is related to the Heisenberg exchange interaction via Aex= 104MSDex/(2γ). We perform simulations of magnon creation by spin-transfer torque with and without out-of-plane anisotropy. We define an initial state in which the spins, on average, point along the ˆ zdirection. We introduce a random noise in the spin direction to mimic the thermal noise as the initial condition of our simulation. Next, we excite magnons in the ferromagnetic thin film by applying a spin torque, with mp∥ˆ z, to the entire film surface. The film is discretized in the lateral directions and uniform in the ˆ xdirection. The lateral dimensions of the film are large compared to the film thickness. In this way, the film is effectively a 2D magnetic system. The LLG equation is solved for each successive time step, using the open-source software MuMax3 [57]. We refer to Ref. [59] for more numerical details. The spin accumulation determines the strength of the spin torque, see Ref. [59]. We start the simulations by exciting magnons with a strong spin torque (interval I1), before lowering the torque strength to keep the magnetization dynamics in a semi-stable state where the total number of magnons does not change dramatically over time (interval I2). In this semistable state, the two magnon populations may interact with each other, and the density of magnons in both minima may oscillate in time. However, the total number of magnons remains relatively unchanged. The current strength and time duration of the two intervals are listed in Table I in the SM. The magnetization data in Fig. 4 in the main text are from the last 50 ns of the intervals. The nonequilibrium magnon density in the film is proportional to the deviation of total magnetization for the ground state, η= 1− ⟨mz⟩[59]. The torque strength determines the number of excited magnons. A finite PMA lowers the magnon spectrum minima, meaning that one can use a weaker spin torque to generate the critical magnon density needed for the generation of condensate magnons. In Fig. 4 in the main text, we choose a spin-transfer torque strength for the generation of condensate magnons, resulting in a magnon density of approximately η≈0.01. This density is small enough to only take into account two-magnon scattering possesses and not higher order processes. The distribution of magnons in the magnon spectrum ξcan be found by performing a Fourier transform of the transverse magnetization components, ξ(qy, qz;ω) =|F[mx(y, z;t)]|2+|F[my(y, z;t)]|2[59]. To reduce the consequences of the finite-size effect in our results, we analyze the magnetization data in the middle region of the thin film, ( y, z)∈[L/8,7L/8], where L=Ly=Lz= 5µm are the lateral dimensions of the square thin film.

2023-09-12

We theoretically demonstrate that adding an easy-axis magnetic anisotropy facilitates magnon condensation in thin yttrium iron garnet (YIG) films. Dipolar interactions in a quasi-equilibrium state stabilize room-temperature magnon condensation in YIG. Even though the out-of-plane easy-axis anisotropy generally competes with the dipolar interactions, we show that adding such magnetic anisotropy may even assist the generation of the magnon condensate electrically via the spin transfer torque mechanism. We use analytical calculations and micromagnetic simulations to illustrate this effect. Our results may explain the recent experiment on Bi-doped YIG and open a pathway toward applying current-driven magnon condensation in quantum spintronics.

Anisotropy-assisted magnon condensation in ferromagnetic thin films

2309.05982v3

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. 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) along 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 2nmYIGPtRuMgOTaOX Y Pt Ru Mg2nm505152YIG(444)Intensity (a.u.)2q (degree)GGG(444)(a)(b) (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]. 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&-+,($), and the IOHE of Ru (𝐼&-+,./), respectively. The ISHE of Ru (𝐼&*+,./) is much weaker compared to the other contributions, thus not shown in this figure. The solid red line represents the fitting curves 𝐼0-012 with all four contributions mentioned above. In consideration of the large orbital Hall conductivity of Ru (𝜎-+./ = 9100 (ħ/e) (Ω cm)-1) that was (a)(b) (c)(d)-20-1001020-2-1012ISSE (nA)Hy (Oe)YIG/Pt(1.5)/Ru(4)YIG/Pt(1.5) spincurrentISHE GGGYIGPtRuorbitalcurrentIOHE with L-Sconversion xzHy>0V xzHyGGGYIGPtobserved in recent works [33-35], we propose a physical mechanism as shown in Fig. 2 (c). The temperature difference drives spin current 𝑆($ 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./ is 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&*+,($ . 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. 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) (c)(d)1.21.62.02.4-40-2002040ISP (μV)H (Oe) qH = 90°qH = 270°6 GHz7 GHz8 GHz -500500306090120150ISP (nA)H-HR (Oe) YIG/Pt(1.5)/Ru(4) YIG/Pt(1.5)024120306090120150ISP(nA)tRu (nm)V 𝑆"#$%&'(GGGYIGPt𝑆)%𝐿)%𝑆+,𝐿+,RuxyHshow a significant increase, which contradicts the theory of ISHE. According to ISHE, the SSE and SP signals generated in Ta and W should cancel out the Pt signal due to the opposite signs of the spin Hall conductivity (𝜎*+) compared to Pt. However, the 𝜎-+ of Ta, W, and Pt have the same positive sign. Consequently, the IOHE overcomes the cancellation effect of ISHE in Ta and W, resulting in an increased signal. Furthermore, the sample of Cu, which is a light metal and a poor spin sink material (spin diffusion length over 200 nm measured at room temperature [45]), exhibits a substantial increase, supporting the notion that the spin sink effect is not prominent in our experiments. FIG. 4. (a) SP measurements and (b) SSE measurements of YIG/Pt/NM samples with different NM (NM = Ta, W, Cu, and Ru). TABLE I. The sign of the 𝐼&*+, and 𝐼&-+, from the NM layer in YIG/Pt(Gd)/NM heterostructures. NM1 NM2 NM3 𝜎*+78/𝜎-+78 +/+ -/+ +/- 𝜂2*($>0 𝐼&*+,/𝐼&-+, +/+ -/+ +/- 𝜂2*'9 <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) (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) H (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. Rev. Mater. 6, 095001 (2022). [6] L. Onsager, Phys. Rev. 37, 405 (1931). [7] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). [8] V. T. P h a m et al., Nat. Electron. 3, 309 (2020). [9] Z. Guo, J. Yin, Y . Bai, D. Zhu, K. Shi, G. Wang, K. Cao, and W. Zhao, Proc. IEEE 109, 1398 (2021). [10] K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602 (2015). [11] M. Dc, J.-Y. C h e n , T. P e t e r son, P. Sahu, B. Ma, N. Mousavi, R. Harjani, and J.-P. Wang, Nano Lett. 19, 4836 (2019). [12] W. Lin, K. Chen, S. Zhang, and C. L. Chien, Phys. Rev. Lett. 116, 186601 (2016). [13] D. Yue, W. Lin, J. Li, X. Jin, and C. L. Chien, Phys. Rev. Lett. 121, 037201 (2018). [14] Y. X u , F. Z h a n g , Y. L i u , R . X u , Y. J i a n g , H . C h e n g , A . F e r t , a n d W. Z h a o , a r X i v : 2 2 0 8 . 0 1 8 6 6 (2022). [15] T. S. Seifert, D. Go, H. Hayashi, R. Rouzegar, F. Freimuth, K. Ando, Y. Mokrousov, and T. Kampfrath, arXiv:2301.00747 (2023). [16] G. Torosyan, S. Keller, L. Scheuer, R. Beigang, and E. T. Papaioannou, Sci Rep 8, 1311 (2018). [17] M. Hennecke, I. Radu, R. Abrudan, T. Kachel, K. Holldack, R. Mitzner, A. Tsukamoto, and S. Eisebitt, Phys. Rev. Lett. 122, 157202 (2019). [18] N. Bergeard, V . López-Flores, V . Halte, M. Hehn, C. Stamm, N. Pontius, E. Beaurepaire, and C. Boeglin, Nat. Commun. 5, 3466 (2014). [19] C. Boeglin, E. Beaurepaire, V . Halté, V . López-Flores, C. Stamm, N. Pontius, H. A. Dürr, and J. Y . Bigot, Nature 465, 458 (2010). [20] H. Cheng et al., Sci. China Phys. Mech. Astron. 65, 287511 (2022). [21] See Supplemental Material at UTL for the details about the thin film deposition, the SSE signal as a function of the heating power, thickness dependent SSE measurements in YIG(40)Pt(1.5)Ru(tRu) heterostructures, the schematic of the spin sink and orbital sink effect, modeling, and thickness dependent SP measurements in YIG(40)Pt(1.5)Ru(tRu) heterostructures, which includes Refs. [2,5,24,26,27,36,37,42,43]. [22] N. Sokolov et al., J. Appl. Phys. 119, 023903 (2016). [23] T. Liu et al., Phys. Rev. Lett. 125, 017204 (2020). [24] D. Tian, Y . Li, D. Qu, X. Jin, and C. Chien, Appl. Phys. Lett. 106, 212407 (2015). [25] H. Li, Y . Kurokawa, T. Niimura, T. Yamauchi, and H. Yuasa, Jpn. J. Appl. Phys. 59, 073001 (2020). [26] H. Kontani, T. Tanaka, D. S. Hirashima, K. Yamada, and J. Inoue, Phys. Rev. Lett. 102, 016601 (2009). [27] H. Yuasa, F. Nakata, R. Nakamura, and Y . Kurokawa, J. Magn. Magn. Mater. 51, 134002 (2018). [28] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Phys. Rev. Lett. 95, 066601 (2005). [29] R. Fukunaga, S. Haku, H. Hayashi, and K. Ando, Phys. Rev. Research 5, 023054 (2023). [30] Y.-G. Choi et al., Nature 619, 52 (2023). [31] D. Go et al., Phys. Rev. Research 2, 033401 (2020). [32] J. Kim and Y . Otani, J. Magn. Magn. Mater. 563, 169974 (2022). [33] L. Salemi and P. M. Oppeneer, Phys. Rev. Mater. 6, 095001 (2022). [34] L. Liao et al., Phys. Rev. B 105, 104434 (2022). [35] A. Bose, F. Kammerbauer, R. Gupta, D. Go, Y . Mokrousov, G. Jakob, and M. Kläui, Phys. Rev. B 107, 134423 (2023). [36] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 87, 174411 (2013). [37] K.-W. Kim, Phys. Rev. B 99, 224415 (2019). [38] M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 116, 126601 (2016). [39] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D. Bader, and A. Hoffmann, Appl. Phys. Lett. 103, 242414 (2013). [40] S. Yakata, Y . Ando, T. Miyazaki, and S. Mizukami, Jpn. J. Appl. Phys. 45, 3892 (2006). [41] V. C a s t e l , N . V l i e t s t r a , J . B e n Yo u s s e f , a n d B . J . v a n We e s , A p p l . P h y s . L e t t . 101 (2012). [42] L. Qin et al., J. Magn. Magn. Mater. 560, 169600 (2022). [43] X. Tao et al., Sci. Adv. 4, 1670 (2018). [44] L. Liao et al., Phys. Rev. B 105, 104434 (2022). [45] T. Kimura, T. Sato, and Y. Otani, Phys. Rev. Lett. 100, 066602 (2008). [46] S. 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).

2023-08-25

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.

Giant orbit-to-charge conversion induced via the inverse orbital Hall effect

2308.13144v1

arXiv:1810.00380v1 [physics.app-ph] 30 Sep 2018Magnon Valves Based on YIG/NiO/YIG All-Insulating Magnon J unctions C. Y. Guo,∗C. H. Wan,∗X. Wang, C. Fang, P. Tang, W. J. Kong, M. K. Zhao, L. N. Jiang, B. S. Tao, G. Q. Yu, and X. F. Han† Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Scie nces, Chinese Academy of Sciences, Beijing 100190, China. (Dated: October 2, 2018) As an alternative angular momentum carrier, magnons or spin waves can be utilized to encode information and breed magnon-based circuits with ultralow power consumption and non-Boolean data processing capability. In order to construct such a cir cuit, it is indispensable to design some electronic components with both long magnon decay and coher ence length and effective control over magnon transport. Here we show that an all-insulating m agnon junctions composed by a magnetic insulator (MI 1)/antiferromagnetic insulator (AFI)/magnetic insulator (MI2) sandwich (Y3Fe5O12/NiO/Y 3Fe5O12) can completely turn a thermogradient-induced magnon curr ent on or off as the two Y 3Fe5O12layers are aligned parallel or anti-parallel. The magnon de cay length in NiO is about 3.5 ∼4.5 nm between 100 K and 200 K for thermally activated magnons . The insulating magnon valve (magnon junction), as a basic building block, p ossibly shed light on the naissance of efficient magnon-based circuits, including non-Boolean log ic, memory, diode, transistors, magnon waveguide and switches with sizable on-off ratios. PACS numbers: 72.25.Rb, 72.25.Ba, 73.50.Bk, 73.40.Rw INTRODUCTION Data processing and transmission in sophisticated mi- croelectronics rely strongly on electric current, which in- evitably wastes a large amount of energy due to Joule heating. Magnons represent the collective excitations in magnetic systems. Though charge neutral, they possess angular momenta and can also transfer the momenta as information carrierfree from Joule heating [1]. The main dissipation channel of magnons in magnetic insulators is spin-lattice coupling which is much weaker than Joule heating. Moreover, the wave nature of magnons pro- vides additional merits, (1) long propagation distance up to millimeters [2, 3] and (2) a new degree of free- dom (magnon phase) owing to which non-Boolean logic processing [1, 4–6] are anticipated. These characteristics make magnons the ideal information carriers based on which some electronic components for future magnonic circuits are being developed recently [7–11]. Benderet al.[7] theoretically proposed a spin valve with magnetic insulator (MI)/nonmagnetic metal/MI structure where magnetization switching induced by thermally driven spin torques was expected. Wu et al.[8] proposed a concept of magnon valve with heterostruc- ture of MI/spacer/MI and experimentally realized the heterostructure in a YIG/Au/YIG sandwich whose spin Seebeck effect (SSE) depends on the relative orientation of the top and bottom YIG layers. In this structure, magnon transport occurs in two YIG layers while spin transport in Au remains limited to the electrons. Angu- lar momentum transfer through the structure thus relies on mutual conversion between magnon spin current and electron spin current. Cramer et al.[9] prepared another type ofhybridspin valvesin YIG/CoO/Co. In this valve, inverse spin Hall voltage (ISHE) in Co induced by spinpumping effect is determined by spin configurations be- tween YIG and Co. The ferromagnetic Co electrode can supportboth electronand magnoncurrents, andthus the output signalhaveboth electronspin and magnoncontri- butions [3, 9]. In the above cases, additional conversions between magnon current and spin current or vice visa could first reduce effective decay length of angular mo- mentum. Second, more noticeably, magnon phase would be lost in the above conversions since phase information cannot be encoded by ordinary spin current. Thus, it is very desirable to construct pure magnon valves in an all-insulating structure such that the spin information propagation is uniquely limited to magnons. Very recently, a sandwichconsisting oftwo ferromagnetic insulators and an antiferromagnetic spacer was proposed by Cheng et al.[11] where both giant spin Seebeck effect and magnon transfer torques were predicted. Here we design and further experimentally realize a typical MI/antiferromagneticinsulator (AFMI)/ MI het- erostructure using YIG/NiO/YIG sandwiches. We en- title such a MI/AFMI/MI heterostructure as insulating magnon junction (IMJ) for short. Output magnon cur- rent of an IMJ generated by SSE can be regulated by its parallel (P) or antiparallel (AP) states. Especially, the output magnon current can be totally shut down in the AP state while superposed in the P state near room tem- perature,contributingtoalargeon-offratio. Demonstra- tion of the pure magnon junction based on all-insulators may further help to develop magnon-based circuits with fast speed and ultralow energy dissipation in the coming future.2 FIG. 1 Microstructure of the GGG//YIG(100)/NiO(15)/YIG(60 n m) IMJ. (a) The cross-sectional TEM im- age of the sample. HRTEM images of (b) the GGG//YIG(100 nm) and ( c) YIG(100)/NiO(15)/YIG(60 nm) interfaces, respectively. Fourier transformation of HRTEM for ( d) the bottom YIG and (e) the top YIG only. (f) Larger area HRTEM for the YIG/NiO(15 nm)/YIG IMJ and ( g) inverse Fourier Transformation of the yellow ring in the inset diffraction pattern which is obtained by Fo urier transformation of Fig.1(f). EXPERIMENTS IMJs stacks YIG(100)/NiO( t)/YIG(60 nm) ( t= 4, 6, 8, 10, 15, 20, 30, 60, all thickness number in nanometers)were deposited on Gd 3Ga5O12(GGG) (111) substrates3 in a sputtering system (ULVAC-MPS-4000-HC7 model) with base vacuum of 1 ×10−6Pa. After deposition, high temperature annealing in an oxygenatmosphere was car- ried out to further improve the crystalline quality of the YIG layers. The stacks with t= 6, 8, 15, 20, 30 and 60 nm were fabricated in the same round. Then a 10 nm Pt stripe with 100 µm×1000µm lateral dimensions for SSE measurement were fabricated by standard photolithog- raphy combined with an argon-ion dry etching process. Finally, an insulating SiO 2layer of 100 nm and a Pt/Au stripe were successively deposited on top of the Pt stripe for on-chip heating. Before platinum deposition, vibrat- ing sample magnetometer (VSM, EZ-9 from MicroSense) was used to characterize magnetic properties of the YIG layers. After microfabrication process, SSE of the IMJs were measured in a physical property measurement sys- tem (PPMS-9T from Quantum Design). Keithley 2400 provided a heating current ( I) to the Pt/Au stripe while Keithley 2182 picked up a voltage along the Pt stripe induced by SSE and ISHE. Magnetic field was applied along the transverse direction of the Pt stripe. Control samples YIG(100)/Pt(10nm), YIG(100)/NiO(15)/Pt(10 nm) and NiO(15)/YIG(60)/Pt(10 nm) were also pre- pared on GGG substrates and measured for comparison. We have also confirmed insulating nature of oxide parts in the YIG/NiO/YIG and the control stacks by electrical transport measurements. RESULTS AND DISCUSSION (1) STRUCTURE CHARACTERIZATION Fig.1 shows crystalline structure of an IMJ stack GGG//YIG(100)/NiO(15)/YIG(60 nm). The NiO spacerhas uniform thickness without pinholes (Fig.1(a)). The bottom YIG (B-YIG) is epitaxially grown on GGG substrate with atomically sharp interface (Fig.1(b)). Fourier transformation of the high-resolution transmis- sion electron microscope (HRTEM) in the inset only shows one diffraction pattern, further confirming the epitaxial relation. However, the interfaces of the NiO spacer with adjacent YIG layers become rougher than the GGG//YIG interface, especially for the interface with the top YIG (T-YIG) (Fig.1(c)). It is worth not- ing the two YIG layers are both single-crystalline but with different orientations (Fig.1(d) and (e)). The B- YIG grows along [111] direction of the substrate while theT-YIGdoesnot, probablybecausethepolycrystalline NiO spacer broke epitaxial relation (Fig.1(c)). Fig.1(f) shows a HRTEM image acquired across the bottom- YIG/NiO/top-YIG interfaces. Two sets of diffraction patterns corresponding to Fig.1(d) and Fig.1(e) can be identified. From the patterns and lattice parameters of YIG, we can accurately calibrate camera length of the TEM. Then, besides of the already-known patterns ow- ingtoYIGs, diffractionpatternsowingtoNiOcanbealsoidentified as highlighted by the yellow ring and the two red circles in Fig.1(g) inset. The yellow ring corresponds to (200) plane of NiO while the red circles correspond to (111) plane of NiO. After inversely Fourier transforming the yellow ring pattern, NiO polycrystals can be clearly observed as shown in Fig.1(g). (2) SPIN SEEBECK EFFECT MEASUREMENT Fig.2(a) schematically shows setup to measure SSE of an IMJ. A Pt/Au electrode on top of a 100 nm SiO 2 insulating layer is heated by a current and then a tem- perature gradient ∇Talong the stack normal (+ zaxis) is built. ∇Tintroduces inhomogeneous distribution of magnons inside a MI and produce a magnon current along∇T[12, 13]. The magnon current can be fur- ther transformed as a spin current penetrating into an adjacent heavy metal and then generate a sizable volt- age by ISHE [14–17], which is so-called longitudinal SSE. Here we use a Pt stripe to measure the voltage ( VSSE) induced by ISHE and monitor the magnitude and direc- tion of spin current exuded from the top YIG. Fig.2(b) shows angle scanning of VSSEof an IMJ with tNiO=8 nm. The Pt stripe is along the yaxis. Therefore SSE can be observed only if magnetization has component in thexaxis. This requirement gives the angle dependence showninFig.2(b). Fig.2(c)showsspinSeebeckvoltageas a function of applied field measured at different heating currents. Saturated VSSEparabolicly depends on current (Fig.2(d)). This parabolic dependence confirms the ther- mopower essence of the measured voltage signals while the angle dependences confirm the voltage signals are in- duced by spin Seebeck effect. We have also estimated temperature rise (∆ T) of spin detector (Pt stripe) as el- evating heating current by calibrating resistivity of the Pt stripe. Heating current of 10 mA, 15 mA and 20 mA would lead to ∆ Tof 2.7 K, 6.0 K and 10 K, respec- tively, as background temperature within (50 K, 325 K). In order to enhance signal-to-noise ratio and minimize influence of heating current on IMJs, we have selected 15 mA as heating current to conduct the following SSE measurement in Fig.4 and Fig.5. (3) FIELD DEPENDENCE OF VSSEOF AN INSULATING MAGNON JUNCTION Fig.3(a) shows a typical hysteresis of an IMJ YIG(100)/NiO(8)/YIG(60 nm). Two magnetization re- versals have been identified. Epitaxial growth and larger thickness endow the bottom YIG with lower coercivity (HC) and larger magnetization than the top YIG on the NiO spacer. Thus, the reversal with HC≈3 Oe and ∆M≈1.3(MS,B+MS,T) is attributed to the switching ofthe bottom YIG while the other reversalwith HC≈16 Oe and ∆ M≈0.7(MS,B+MS,T) is attributed to the top YIG.MS,BandMS,Tare saturated magnetization of the4 FIG. 2 (a) Schematic diagram of spin Seebeck ef- fect and its measurement setup for an IMJ. Dur- ing measurement, a field in the x-axis is applied. (b) Angle scanning of VSSEof an IMJ with tNiO=8 nm. The Pt stripe is along the yaxis. (c) Spin Seebeck voltage as a function of applied field mea- sured at elevated heating currents. (d) Parabolic fitting of the field dependence of saturated VSSE. bottom and top YIG layers, respectively. The obtained MS,B/MS,Tis 13/7, close to the ratio 5/3 in the nomi- nal thickness. Due to the difference in coercivity, parallel and antiparallel spin configurations can be formed, as il- lustrated in the figure. Fig.3(b) shows field-dependence of VSSE. Besides of a large ∆VSSE≈1.6VSSEmaxoccurring at 13 Oe, VSSEalso sharply changes by about 0.4 VSSEmaxat 2 Oe. VSSEmax is the saturation value (2.6 µV) in Fig.3(b). d VSSE/dH and dM/dH(Fig.3(c)) are used to show correspondence between SSE and VSM results. A peak in Fig.3(c) rep- resents a sharp reversal of a YIG layer. There are four peaks in both field dependences. The middle two la- beled as (P1-) and (P1+) originate from the reversal of the bottom YIG while the outer ones marked as (P2-) and (P2+) are caused by the reversal of the top YIG. SSE of the control samples YIG/NiO/Pt, NiO/YIG/Pt and YIG/Pt have also been measured (Fig.3(d)). Ex- cept YIG/NiO/Pt, the other samples show comparable VSSEat the same heating current, which may be owing to higher spin mixing conductance of YIG/Pt interface than that of NiO/Pt interface in our case. YIG/NiO/Pt and YIG/Pt are indeed much softer than NiO/YIG/Pt. Furthermore, only onemagnetizationreversalis observed for the control samples. If the NiO spacer is replaced by an MgO spacer, VSSEsignal due to the reversal of the bottom layer disappears as shown in Ref [8]. The above observation indicates that the magnon current from the bottom layer can flow through the NiO spacer and thetop YIG and finally penetrate into Pt. Magnon decay length in epitaxial and polycrystalline YIG is about 10 µm [2] and several tens of nanometers [18, 19], respec- tively. Wang et al. [20] reported spin relaxation length of 9.8 nm in NiO. Our IMJs have comparable dimensions with those reported values, indicating magnon current from the bottom YIG layer capable of flowing into Pt. Remarkably, though solidly confirmed in experiments, longitudinal spin Seebeck effect is regarded to be po- tentially caused by (1) difference in electron tempera- ture and magnon temperature across an interface be- tween heavy metal and magnetic insulator [12, 13] or by (2) inhomogeneous magnon distribution inside bulk re- gionof amagnetic insulatorand as-inducedpure magnon flow [21, 22]. These two mechanisms are hard to tell in classic magnetic insulator/heavy metal bilayer systems. Though not ruled out possibility of the 1stmechanism, nevertheless, our experiment strongly proved rationality of the 2ndmechanism since the bottom YIG layer could only deliver magnon current into Pt via the bulk effect. FIG. 3 (a) Hysteresis loop and (b) field depen- dence of VSSEfor an IMJ with t= 8 nm at 300 K. (c) Corresponding field dependences of d VSSE/dH and dM/dH. (d) VSSEof the control samples. (4)TANDtNiODEPENDENCE OF SSE OF INSULATING MAGNON JUNCTIONS We have measured field dependence of VSSEat 15 mA with elevating Tfor different IMJs as shown in Fig.4. In order to distinguish switching fields from different YIG layers, their d VSSE/dHare shown in Fig.5. First, all IMJs show a significant exchange bias below blocking temperature of about 100 K (Fig.4, Fig.5 and Fig.6(b)), which evidences the appearance of NiO antiferromag- netism. Similarblockingtemperaturesforallthe samples indicate interfacial nature of exchange bias effect whose5 magnitude is dominantly determined by exchange cou- pling strength between interfaciallayersofNiO and YIG. Second, only two peaks are unambiguously identifiable for the IMJ with 30 nm and 60 nm NiO at all temper- atures (Fig.5(g,h)). These peaks belong to (P2+) and (P2-) because their positions are identical with those of YIG(100)/NiO(15)/YIG(60 nm) determined by VSM andHCof NiO/YIG/Pt as shown in Fig.6(a). Due to too thick NiO spacer and its blocking effect on magnon current, the Pt stripe can only detect magnon current from the top YIG. Thus it is nature that the peaks in Fig.5(g,h) share the same positions with P2+ and P2-. Fort=6∼20 nm, temperature evolution of d VSSE/dH vs.Hcurves seem nontrivial. Within a certain tempera- ture region, 4 peaks corresponding to 4 switching events of two YIG layers can be clearly resolved. For the t= 15 nm IMJ, for example, four peaks can be clearly re- solved between 50 K and 275 K. The relative intensity of (P1+/P2+) or (P1-/P2-) decreases gradually with in- creasing T(Fig.5(e)). The trend has also been repro- duced in the IMJ with t= 8 nm (Fig.5(c)). At T <175 K, the pair of peaks (P1+) and (P1-) are dominant. At intermedium temperature from175K to325K, the other pair of peaks (P2+) and (P2-) emerge and are enhanced with increasing T. AtT >325 K, (P1+) and (P1-) even- tually fade away but (P2+) and (P2-) remain (Fig.5(c)). This trend indicates the layer dominating SSE of an IMJ can be changed between the two YIG layers though spin detector is only directly connected to the top layer. The IMJ with 4 nm NiO is unique, in which only P1+ and P1- are observed at all temperatures, which is also indicated in Fig.6(a). Though P2+ and P2- are absent here, two broad shoulders outsides of P1+ and P1- can be identified above 275 K, which probably still originate from switching of the top YIG. For thin enough NiO spacer, magnon current generated in the bottom layer can still survive after a weak decay in the NiO spacer at high temperatures. Thus VSSEdue to the bottom layer is still observable or even dominated in this case. (5) MAGNON DECAY LENGTH OF NIO In order to further check the correlation between the peaks at different temperatures in Fig.5 and switch- ing fields of the YIG layers, we have plotted them (open symbols) together with HCof YIG/NiO/Pt and NiO/YIG/Pt (solid triangles) determined by SSE and HCof YIG/NiO(15)/YIG determined by VSM (solid hexagons) in Fig.6(a). Remarkably, nearly all the peaks lie on 4 branches defined by HCof the control samples andHCfrom the VSM results, which unambiguously demonstrates origin of the four peaks in the entire tem- perature range, i.e., the outer peaks from the top and the innerpeaksfromthe bottom YIG. Thus, wecanconclude the magnon current that overwhelmingly contributes to SSE comes from the bottom YIG at low temperatureswhile the magnon current from the top YIG becomes significant at high temperatures. A plausible explana- tion of the relative contributions of the magnon current from the two YIG layers is as follows. The bottom YIG grownon GGG has much better quality and thus a larger SSE coefficient. At low temperature and for a small NiO thickness, the magnon current from the bottom layer is able to propagate through both NiO and the top YIG without noticeable decay. When NiO becomes thicker, magnon current from the bottom YIG decays significant. On the other hand, the magnon current of the top YIG does not suffer such decay since it is in direct contact with Pt. Thus the bottom and top YIG contribute more dominantly at low and high temperatures, respectively. Fig.6(c) summarizes magnon valve ratio ηmvof the IMJs as a function of T. Here ηmvis defined as VSSE,AP/VSSE,P.VSSEinduced by ISHE is propor- tional to the injected magnon current Jmflowing to- ward Pt from the top YIG layer. According to the SSE theory [12, 13, 21, 22], Jm,T/Bis proportional to ST/B(∇T)T/B. Here Jm,Tis the spin Seebeck coeffi- cient of the top/bottom YIG and ( ∇T)T/Bis the in- duced temperature gradient across the top/bottom YIG and proportional to I2. ThusVSSE∝I2, as confirmed in Fig.2(d). The Jm,P/AP=Jm,T±aNiOaYIGJm,Bin the P and AP states. Here, Jm,TandJm,Bare the generated magnon currents by SSE in the top and bottom YIG, respectively, while aNiOandaYIGare the decay ratio of the magnon current from the bottom YIG in NiO and the top YIG, respectively. The magnon valve ratio ηmv can be thus rewritten as ηmv=Jm,AP Jm,P=Jm,T−aNiOaYIGJm,B Jm,T+aNiOaYIGJm,B(1) ηmvsign reflects the relative magnitude of the magnon currents from the top and the bottom layers. The nega- tive values in Fig.6(c) indicates the magnon current from the bottom YIG is larger at low temperature. The posi- tiveηmvseen in the 6 and 8 nm NiO IMJs of Fig.6(c) at high temperatures means a larger magnon current from the top YIG. The most interesting case is ηmv= 0 where the net magnon current at the YIG/Pt interface becomes zero, i.e., the exact cancellation of the two magnon cur- rents generated by two YIG layers (inset of Fig.6(c)). Such cancellation only occurs at the AP state. The fig- urealsoshowsatrendthatthecriticaltemperaturewhere ηmv=0 increases with decreasing tNiO. Next, weestimatethemagnondecaylength λNiOinthe NiO spacers. Magnon decay ratio in NiO is aNiO. From Eq.(1), one can easily see aNiOis proportional to δ= (1- ηmv)/(1+ηmv). We then plot ln δas a function of tNiObe- tween 100 K to 200 K (Fig.6(d)). In this temperature re- gion, the switching fields of both YIGs are well separated for the IMJs. Worth noting, data from the IMJs with 4 nm and 10 nm NiO spacer are not used here. For the IMJ with 4 nm NiO, only two peaks are clearly observed (Fig.5(a)). Thus it is hard to obtain reliable ηmvfor the6 FIG. 4 Field dependence of VSSEat elevated temperatures for IMJs with (a-h) t=4 nm, 6 nm, 8 nm, 10 nm, 15 nm, 20 nm, 30 nm and 60 nm, respectively. device. For the IMJs with 10 nm and 4 nm NiO, the stacks were deposited and annealed in different rounds with the others. The other stacks were fabricated in the sameroundandtheirdataweresystematicandthusmore comparable. Fig.6(d) shows a good linear dependence of lnδontNiO, which suggests that magnons generated in the bottom YIG can pass through NiO in a diffusive way [11]. The inset in Fig.6(d) shows T-dependence of the derived λNiOwhich increases slightly with T.λNiO is about 3.5 nm ∼4.5 nm between 100 K and 200 K. Though λNiOslightly increases with T. However, VSSE from the top YIG gradually dominates at higher temper- atures. Itindicatesnotonlymagnontransportproperties of an IMJ but also T-dependent spin Seebeck coefficients of YIG layers and thermoconductivity of YIG and NiO layers will eventually affect the magnon valve effect. The influence of these parameters is already out of scope of this article. Therefore, we only use magnon valve ratio and its thickness dependence at a fixed temperature to measuremagnondecaylengthasshowninthemainpanel of Fig.6(d). Spin decay length in antiferromagnetic materials such as NiO, CoO, Cr 2O3and IrMn have been measured by spin pumping or spin Hall magnetoresistance tech- niques [20, 23–29]. For NiO thin films, λNiOis reported about 10 nm [20, 23]. Our value has the same orderof magnitude with theirs. Qiu et al[27, 28] reported an enhanced spin pumping effect near N´ eel temperature TNof antiferromagnetic materials. For 6 nm CoO and 1.5 nm NiO, they observed the most significant enhance- ment at about 200 K and 285 K, respectively [27]. In our case, NiO spacers have thickness of 4 nm ∼60 nm. They probably have even higher TN. Then the increase inλNiOwithin (100 K, 200 K), we think, is also due to similar enhancement in magnon transport efficiency as T approaching TN. (6) EXCHANGE BIAS DEPENDENCE OF SSE OF IMJS We have also changed exchange bias direction of the same IMJ with 8 nm NiO by field cooling technique. In this case, we first elevated temperature to 400 K (the highest temperature of our PPMS system) and then ap- plied 5 T field along different directions (along the Pt strip or in-plane vertical to the Pt stripe or normal to stacks) and then cooled the device down to 10 K with the field of 5 T maintained. Finally, the high field was reduced to 0 in oscillating mode to minimize remanence field of magnet. After all the above procedures, we be- gan SSE measurement with the Pt stripe along the yaxis7 FIG. 5 Field dependence of d VSSE/dHat elevated temperatures for IMJs with (a- h)t=4 nm, 6 nm, 8 nm, 10 nm, 15 nm, 20 nm, 30 nm and 60 nm, respectively. and field applied along the xaxis. At 10 K, only the case with cooling field in-plane vertical to the stripe shows re- markable exchange bias effect while the other two cases show small or even negligible exchange bias effect. This means we have successfully changed the exchange bias directions by the above procedures. However, as shown in Fig.7, all the three cases show nearly the same sat- urationVSSEat all the temperatures. Furthermore, we can see from the data between 150 K and 300 K that magnon valve ratio was also independent on exchange bias directions. It indicates different exchange coupling directions at the interfaces would not deterioratetransfer efficiency of magnon current across the NiO/YIG inter- faces, which is luckily benefit for applications. Indepen- denceofspintorquetransferondirectionofexchangebias was very recently reported in metallic system [29]. Our data show this independence was also solidly reproduced for magnon transfer. CONCLUSION In conclusion, fully electric-insulating and merely magnon-conductive magnon valves have been demon- strated by magnetic insulator YIG/antiferromagnetic in-sulator NiO/magnetic insulator YIG IMJs in which out- put spin current in Pt detector can be regulated with an high on-off ratio between P and AP states of the two YIG layers near room temperatures. The magnon current is dominated by the bottom (top) YIG layer at low (high) temperature regions. The transition temper- ature depends on tNiO. The magnon decay length in NiO is about several nanometers. Magnon transfer effi- ciency is independent on exchange bias directions. Most importantly, similar to the fundamental role played by magnetic tunnel junction (MTJ) in spintronics, the IMJ can also provide a basic building block for magnonics and oxide spintronics. Pure magnonic devices/circuits based on IMJs can be constructed in an ideally insu- lating system with magnon being the only angular mo- mentum carrier. In these devices/circuits, information processing and transport can be accomplished only by magnons without mobile electrons and ultralow energy consumption and more versatile functions such as non- Boolean logics utilizing magnon phase coherence, mag- netic memory based on insulators, magnon diode, tran- sistors, waveguide and switches with large on-off ratios can be expected.8 FIG. 6 The dependence of (a) switching fields and (b) exchange bias of all the IMJs and the control samples on temperatures. (c) VSSE,AP/VSSE,Pratios for the IMJs. Inset shows the field dependence of VSSEfor the IMJ with 6 nm NiO spacer and sizeable on-off ratio at 260 K. (d) Thickness dependence of ln δat medium tem- peratures. Insets show derived λNiOas function of T. ACKNOWLEDGEMENTS We gratefully thank Prof. S. Zhang in Univer- sity of Arizona for enlightening discussions. This work was supported by the National Key Research and Development Program of China [MOST, Grants No. 2017YFA0206200], the National Natural Sci- ence Foundation of China [NSFC, Grants No.11434014, No.51620105004, and No.11674373], and partially sup- ported by the Strategic Priority Research Program (B) [Grant No. XDB07030200], the International Partner- ship Program (Grant No.112111KYSB20170090), and the Key Research Program of Frontier Sciences (Grant No. QYZDJ-SSWSLH016) of the Chinese Academy of Sciences (CAS). ∗These two authors contributed equally to this work. FIG. 7 Independence of magnon transfer on di- rections of exchange bias. Three field cooling ge- ometries are also shown as insets. Colors of the VSSEvs.Hcurves are the same with the cooling fields in the left insets. FC, IP and OOP denote cooling field, in-plane and out-of-plane, respectively. †Corresponding Author: xfhan@iphy.ac.cn [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). [2] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015). [3] K. Wright, Physics 11(2018). [4] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D43, 264002 (2010). [5] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D 43, 264001 (2010). [6] B. Lenk, H. Ulrichs, F. Garbs, and M. Muenzenberg, Phys. Rep. 507, 107 (2011). [7] S. A. Bender and Y. Tserkovnyak, Phys. Rev. B 93, 064418 (2016). [8] H. Wu, L. Huang, C. Fang, B. S. Yang, C. H. Wan, G. Q. Yu, J. F. Feng, H. X. Wei, and X. F. Han, Phys. Rev. Lett.120, 097205 (2018). [9] J. Cramer, F. Fuhrmann, U. Ritzmann, V. Gall, T. Ni- izeki, R. Ramos, Z. Qiu, D. Hou, T. Kikkawa, J. Sinova, U. Nowak, E. Saitoh, and M. Klaeui, Nat. Commun. 9, 1089 (2018). [10] L. J. Cornelissen, J. Liu, B. J. van Wees, and R. A. Duine, Phys. Rev. Lett. 120, 097702 (2018). [11] Y. Cheng, K. Chen, and S. Zhang, Appl. Phys. Lett. 112, 052405 (2018). [12] J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B. 81, 214418 (2010). [13] H. Adachi, K.-i. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013). [14] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). [15] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X. F. Jin, and E. Saitoh, Phys.9 Rev. Lett. 110, 067207 (2013). [16] H. Wu, C. H. Wan, Z. H. Yuan, X. Zhang, J. Jiang, Q. T. Zhang, Z. C. Wen, and X. F. Han, Phys. Rev. B. 92, 054404 (2015). [17] H.Wu,X.Wang, L.Huang, J.Y.Qin, C.Fang, X.Zhang, C. H. Wan, and X. F. Han, J Magn Magn Mater 441, 149 (2017). [18] H. Wu, C. H. Wan, X. Zhang, Z. H. Yuan, Q. T. Zhang, J. Y. Qin, H. X. Wei, X. F. Han, and S. Zhang, Phys. Rev. B.93, 060403(R) (2016). [19] J. Li, Y. Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake, and J. Shi, Nat. Commun. 7, 10858 (2016). [20] H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 113, 097202 (2014). [21] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine, Phys. Rev. Lett. 116, 117201 (2016). [22] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B. 94, 014412 (2016).[23] H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. B.91, 220410(R) (2015). [24] C. Hahn, G. De Loubens, V. V. Naletov, J. Ben Youssef, O. Klein, and M. Viret, EPL 108, 57005 (2014). [25] L. Frangou, S. Oyarzun, S. Auffret, L. Vila, S. Gam- barelli, and V. Baltz, Phys. Rev. Lett. 116, 077203 (2016). [26] T. Shang, Q. F. Zhan, H. L. Yang, Z. H. Zuo, Y. L. Xie, L. P. Liu, S. L. Zhang, Y. Zhang, H. H. Li, B. M. Wang, Y. H. Wu, S. Zhang, and R.-W. Li, Appl. Phys. Lett. 109, 032410 (2016). [27] Z.Qiu, J.Li, D.Hou, E.Arenholz, A.T.N’Diaye, A.Tan, K.-i. Uchida, K.Sato, S.Okamoto, Y. Tserkovnyak, Z.Q. Qiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016). [28] Z. Qiu, D. Hou, J. Barker, K. Yamamoto, O. Gomonay, and E. Saitoh, Nature Mater. 17, 577 (2018). [29] H. Saglam, J. C. Rojas-Sanchez, S. Petit, M. Hehn, W. Zhang, J. E. Pearson, S. Mangin, and A. Hoffmann, Phys. Rev. B. 98, 094407 (2018).

2018-09-30

As an alternative angular momentum carrier, magnons or spin waves can be utilized to encode information and breed magnon-based circuits with ultralow power consumption and non-Boolean data processing capability. In order to construct such a circuit, it is indispensable to design some electronic components with both long magnon decay and coherence length and effective control over magnon transport. Here we show that an all-insulating magnon junctions composed by a magnetic insulator (MI1)/antiferromagnetic insulator (AFI)/magnetic insulator (MI2) sandwich (Y3Fe5O12/NiO/Y3Fe5O12) can completely turn a thermogradient-induced magnon current on or off as the two Y3Fe5O12 layers are aligned parallel or anti-parallel. The magnon decay length in NiO is about 3.5~4.5 nm between 100 K and 200 K for thermally activated magnons. The insulating magnon valve (magnon junction), as a basic building block, possibly shed light on the naissance of efficient magnon-based circuits, including non-Boolean logic, memory, diode, transistors, magnon waveguide and switches with sizable on-off ratios.

Magnon Valves Based on YIG/NiO/YIG All-Insulating Magnon Junctions

1810.00380v1

NRSWB High wave vector non-reciprocal spin wave beams L. Temdie,1,a)V. Castel,1,a)C. Dubs,2G. Pradhan,3J. Solano,3H. Majjad,3R. Bernard,3Y. Henry,3M. Bailleul,3and V. Vlaminck1,a) 1)IMT- Atlantique, Dpt. MO, Technopole Brest-Iroise CS83818, 29238 Brest Cedex 03 France 2)INNOVENT e.V. Technologieentwicklung, Pruessingstrasse 27B, 07745 Jena, Germany 3)IPCMS - UMR 7504 CNRS Institut de Physique et Chimie des Matériaux de Strasbourg France (*Electronic mail: vincent.vlaminck@imt-atlantique.fr) (Dated: 13 January 2023) We report unidirectional transmission of micron-wide spin waves beams in a 55 nm thin YIG. We downscaled a chiral coupling technique implementing Ni 80Fe20nanowires arrays with different widths and lattice spacing to study the non-reciprocal transmission of exchange spin waves down to l80 nm. A full spin wave spectroscopy analysis of these high wavevector coupled-modes shows some difficulties to characterize their propagation properties, due to both the non-monotonous field dependence of the coupling efficiency, and also the inhomogeneous stray field from the nanowires. I. INTRODUCTION Non-reciprocity is an essential properties of today’s information systems1. The ability to inhibit signal flow in one direction while allowing it in the reverse direction is crucial to either protect devices from reflection, isolate transmitters and receivers in radar architecture, or even shield qubits from its environment in quantum computers. Most of the current non-reciprocal functionalities rely on the gyrotropic nature of the magnetization dynamics in field-based ferrimagnetic systems (namely Yttrium Iron Garnet - YIG), which tend to be large, and costly to assemble2. Future progress in communication systems, and most critically in quantum technologies, rely heavily on the possibility to miniaturize and integrate these non-reciprocal devices3. The field of magnonics, which implements magnetic excita- tion called spin waves -or their quanta magnons-, is actively involved in the search of non-reciprocal scalable solutions4. Extensive research in the last decade revealed the possibility to engineer both amplitude and frequency non-reciprocity of spin waves in many different ways5. Firstly, the well-known Damon-Eshbach (DE) configuration, where the equilibrium magnetization of a thin film lies in the plane of the film and perpendicular to the wavevector6, displays non-reciprocal dynamic amplitude across the thickness for oppositely travel- ing waves, which couple chirally to an excitation antenna7. Moreover, this non-reciprocity was recently proven to be strongly enhanced in the presence of magnetic nanostructures, where unidirectional transmission of spin waves was achieved using the chiral coupling between the FMR of Co nanowires and exchange spin waves in a thin YIG film8,9. Secondly, small frequency non-reciprocity (namely f(k) 6=f(-k)) was demonstrated more recently in various systems, either with asymmetrical surface anisotropies between top and bottom surfaces10, or with the coupled dynamics of ferromagnetic multilayers11, or also with the interfacial Dzyaloshinskii- Moriya interaction (DMI) of a ferromagnetic layer coupled a)Also at Lab-STICC - UMR 6285 CNRS, Technopole Brest-Iroise CS83818, 29238 Brest Cedex 03 Franceto a high spin-orbit material12,14 ?. Additionally, asymmetric spin-wave dispersion was recently predicted in non-planar geometries due to a topographically induced dynamic dipolar effect15. Nevertheless, all these frequency non-reciprocity effects only becomes significant when the magnons wave- length reaches sub micrometric sizes. Lastly, non-reciprocal functionalities have also been predicted very recently using an innovative inverse design approach16,17. In this communication, we further miniaturized the method of Chen et al.8and demonstrate the possibility of shaping non-reciprocal spin wave beams in a continuous thin YIG film. This paper is organized as follows: in Sec. II, we present the sample design and the experimental protocol used to measure broadband multi-modes spin waves transmission. In Sec. III, we present the non-reciprocal spin wave beams spectra, and address through a spin wave spectroscopy study the peculiarities encountered in shaping narrow spin wave beams via this method. II. EXPERIMENTAL SET-UP A. Sample design and fabrication The chiral excitation of propagating spin waves is achieved by coupling the ferromagnetic resonnance (FMR) of mag- netic nanowires array (NWA) to a low damping continuous film such as YIG. As illustrated in Fig.1-(a), the phase pro- file of a propagating spin wave excited by the dynamic dipo- lar field of nanowires, all precessing in phase, only matches for a single propagation direction. The degree of chirality de- pends strongly on the ellipticity of the dynamic dipolar field of the NWA18. Typically, the elliptical polarization of the Kit- tel mode in flat rectangular nanowires breaks the perfect chi- rality. For this reason we made the nanowires 60 nm thick to come as close as possible to an aspect ratio t=w=1 which pro- duces circularly polarized dipolar field. Moreover, the Kittel mode frequency of the nanowires can be tuned up with de- creasing their width w, therefore the NWA can excite prop- agating modes ( kNW) with much higher wavevector than thearXiv:2211.05514v2 [physics.app-ph] 12 Jan 2023NRSWB 2 (a) 𝒎 YIG film𝒉𝒅𝑘𝑁𝑊 𝑴𝒆𝒒𝒕𝒘𝒂 n=4n=6 𝑘𝐶𝑃𝑊 𝑘𝑁𝑊 𝑴𝒆𝒒 (b) FIG. 1. (a) Sketch of the NWA-FMR mediated chiral cou- pling mechanism. (b) SEM image of device I, w=300 nm width and a=500 nm lattice constant of Ni 80Fe20NWA grown on top of 55 nm thickYIG film, 80 nm Au-antennas grown on top of YIG|Ni 80Fe20. Red and Yellow color represent respectively antenna ( kCPW) and Ni80Fe20NWA( kNW) excite mode . one directly coupled by the microwave field of the antenna (kCPW). In the case of an extended array of NWA, these high vector ( kNW) correspond simply to integer values of2p aac- cordingly with the periodic boundary conditions of the phase precession19,20, minus some possible extinction related to the ratio of w=a. In order to study the dependence of the NWA ar- ray density on the excitation efficiency of the spin wave beam, we designed two devices with different width wand lattice constant a. Namely, for device I, w=300 nm and a=500 nm; and for device II, w=200 nm and a=400 nm. The NWA length of 10 mm was kept the same for both devices. Such a localized distribution of excitation field produces a focused emission of spin waves in a similar fashion to the spin wave beam excited from a constricted CPW21?. In the present geometry, the dis- tance of propagation is well within the near-field region de- fined as the ratio L2=lNW, with Lbeing the antenna length and lNWthe magnon wavelength, so that the near-field diffraction pattern from such a localized excitation follows very closely its shape. For this reason, the emission of kNWmagnons re- mains confined within the length of the NW, thus forming a spin wave beam of width closely equal to 10 µm as sketched in Fig. 1-(b). Fig.1-(b) shows a SEM image of device I. The sample fabri- cation required two steps of e-beam lithography followed bye-beam evaporation lift-off process on a 55 nm thick liquid phase epitaxy YIG film23. To circumvent the insulating na- ture of the substrate, we resorted to an extra layer of conduc- tive resist AR Electra 9224on top of PMMA. In the first step, we structured the 60 nm thick Ni 80Fe20(Permalloy) using a 3 nm thin Ti adhesion layer, and a 8 nm Au capping layer via e-beam evaporation. In the second step, we aligned for both devices the same 20 mm long coplanar wave guides (CPW) with the following lateral dimensions: 2 mm wide central line and 1 mm wide ground lines each spaced by 1 mm with the central line. These pairs of spin wave antenna made of 4 nm Ti + 80 nm Au also via ebeam evaporation. The center-to-center distance D between the two CPW are respectively 20 mm for device I, and 15 mm for device II. B. Broadband Spin Wave Spectroscopy We incorporated an home-made confined electromagnet onto a PM8 probe station to perform VNA spin wave spectroscopy using a Rohde & Schwarz ZNA43GHz Vector Network Analyzer, calibrated via a SOLT procedure with 150mm pitch picoprobes. The sample is carefully placed within the 11 mm gap of the electromagnet, whose poles are 15 mm in diameter, and produce an homogeneous in-plane field along the poles axis (namely less than 0.3% variation at most) up to 500 mT at 3 A. Due to the small hysteresis of the electromagnet, we always initialize a measurement with a high current of 5A in order to be consistent with the calibration of our magnet. We performed broadband frequency sweep in the [0.5;20] GHz range at constant applied field acquiring 3201 points with a resolution bandwidth of 100 Hz in order to resolve widely spread-out multi-modes transmission spectra. We measure 2-ports S-parameters for several field values, which we translate into the corresponding Zi jimpedance matrices. Furthermore, due to the small area of NWA (namely 10*5 mm2), the typical range of Si jsignal amplitudes are of the order of 104in linear scale (or -80 dB in log scale), while the noise floor lays at 106. We retrieve a flat base line by subtracting the measurement at given field with a reference measurement at another field value Hre ffar enough that there are no dynamic feature in the frequency span. Finally, as the coupling of the spin wave to an antenna is of inductive nature, we chose to represent our relative measurements in units of inductance defined as21,25,26: DLi j(f;H) =1 i2pf(Zi j(f;H)Zi j(f;Hre f)) (1) where the subscripts (i;j)=1 or 2 denote either a transmission measurement between two antennas ( i6=j), or a reflection measurement done on the same port ( i=j).NRSWB 3 (c) (d) 𝒌𝑪𝑷𝑾 𝒌𝑵𝑾𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟓𝒎𝑻 𝜇0𝐻𝑒𝑥𝑡(𝑚𝑇) 15 269Δ𝐿11 (a)Im(∆𝐿11) Im(∆𝐿21) Im(∆𝐿12)(b) (e) (f) (g) 𝒌𝑪𝑷𝑾 𝒌𝑵𝑾0.00.20.40.60.81.0 0.00.20.40.60.81.0 𝑘𝑎𝐼𝑘𝑏𝐼 𝑘𝑎𝐼𝑘𝑏𝐼∆𝑘 FIG. 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 m0Hext=+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- ing from 50 mT to 269 mT. III. EXPERIMENTAL RESULTS AND DISCUSSION A. Perfect non-reciprocal transmission We present in Fig.2 a mapping in the (f,H) plane of the spec- traDL11,DL21, andDL12acquired with device I for field grad- ually changing from +46 mT to -46 mT. We observe a myriad of modes that we can separated into two main regions. The first region at lower frequencies corresponds to all the kCPW modes coupled directly by the microwave field of the antenna, and which range typically from k 0 (namely the section of the CPW around the picroprobes) to k 15 rad. mm1(the 8th satellites peak of the antenna). As shown in Fig.2-(e) with the spectra measured at m0Hext=+15 mT, some partial non- reciprocity occurs between DL21andDL12due to the elliptic- ity of the microwave field of the antenna as mentioned above. The second region above 8 GHz corresponds to the higher kNWmodes mediated by the FMR of the permalloy NWA. In this frequency region, Fig.2-(b) and 2-(c) show a perfect 100% non-reciprocal transmission of spin waves. Namely, at positive field values, DL21shows large oscillations, while no transmission occurs from port 2 to port 1. Conversely at neg- ative fields, the non-reciprocity is reversed, and no transmis- sion occurs from port 1 to port 2. We emphasize that the dy- namic range employed for our measurements (iBW=100Hz, P=-10dBm) gives a noise floor of about 5 fH, while typical transmission amplitude are of the order of 100 fH. Besides, one notices between 0 and -15 mT that the transmission for DL12is somewhat dispersed, indicating that the magnetiza- tion of the permalloy NWA do not conserve an anti-parallel orientation with the YIG magnetization, and that a gradual switching of the Py NWA is likely occurring. Furthermore, as shown with the spectra at +15 mT of Fig.2-(f), a closer lookin this region shows an additional perfectly non-reciprocal mode around 8.4 GHz on top of the more pronounced mode at 9.4 GHz. Nevertheless, the frequency spacing between these two modes is much smaller than expected, as the wavevec- tor difference between two adjacent modes, kn+1kn=2p awould result in a frequency difference of about 3 GHz, as can be deduced from Fig.2-(g). We investigate further the nature of these kNWmultimodes in the next section. B. High-k spin wave beam characterization We make use of several distinct peaks to characterize the propagation of high k-vector magnons on both devices I and II. To begin with, we use the Kittel mode to fit the lower branch of the reflection spectra in Fig.2-(a), which corre- sponds to the quasi k=0 FMR resonance of the YIG happen- ing in the larger portion of the CPW close by the picoprobe and far from the NWA. From this analysis, we extract a value of the gyromagnetic ratio of g=2p=27.70:2 GHz.T1and an effective magnetization m0Me f f=1851 mT, which is identi- cal to the saturation magnetization M s23, suggesting no in- plane anisotropy. Then, we track the small reflection peak visible in between the two regions (e.g. the peak at 7.4 GHz in Fig.2-(d)), which corresponds to the first perpendicular stand- ing spin wave (PSSW) mode, and we fit the difference of the square of the frequencies between the PSSW and the Kittel modes27: f2 PSSWf2 FMR= (g 2pm0)2[2MsL2p2 t2Hext +M2 sL2p2 t2(1+L2p2 t2)](2)NRSWB 4 𝑘(𝑟𝑎𝑑.µ𝑚−1) 𝑘(𝑟𝑎𝑑.µ𝑚−1) Device I 𝝁𝟎𝑯𝒆𝒙𝒕=𝟑𝟕𝒎𝑻(a) 𝑘𝑏𝐼 𝑘𝑐𝐼 𝑘𝑎𝐼 𝑘𝑏𝐼𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟔𝟑𝒎𝑻 Device II(c) 𝑘𝑎𝐼𝐼 𝑘𝑎𝐼𝐼 𝑘𝑎𝐼𝐼𝝁𝟎𝑯𝒆𝒙𝒕=𝟑𝟕𝒎𝑻 𝝁𝟎𝑯𝒆𝒙𝒕=𝟏𝟔𝟑𝒎𝑻 (b)(d) 𝜇0𝐻𝑒𝑥𝑡(𝑚𝑇) 23 106 269 𝑘𝑎𝐼,𝐶𝑃𝑊=8 𝑘𝑐𝐼,𝐶𝑃𝑊=15.2𝑘𝑏𝐼,𝐶𝑃𝑊=11.2(e) (f) (g)𝑘(𝑟𝑎𝑑.µ𝑚−1) 𝑘𝑎𝐼 𝑘𝑎𝐼𝐼𝑓𝑜𝑠𝑐𝑘𝑐𝐼 FIG. 3. Transmission spectra at 37 mT and 163 mT for (a) device I and (c) device II, blue and red line represent respectively transmission spectra DL21andDL12. Field dependence of the frequency for the several mode of (b) device I and (d) device II. (e) Field dependence of all the mode amplitudes. (f) and (g) Comparison of the measured group velocity with theoretical expression. where L=q 2Aex m0M2sis the exchange length. In doing so, we ob- tain the following exchange constant A exch=3.850:1 pJ.m1. At last, we study the field dependence of the several kNW modes features. For this purpose, we use a simple Gaus- sian function multiplied by a cosine to fit the oscillations of each transmission peak (see colored fit in Fig.3-(a) and 3-(c)), which gives us the peak position, its amplitude, and the pe- riod of oscillation. This leaves us fitting the field dependence of the frequency of each mode with only the wavevector as fitting parameter. We show in the Fig.3 the results of this methodology. For device I with a lattice constant a=500 nm, we fol- lowed three modes which correspond to kI a=61.3 rad. mm1, kI b=66.7 rad. mm1, and kI c=74 rad. mm1; and for device II with a=400 nm, we tracked two modes kII a=72 rad. mm1, and kII b=77 rad. mm1. It may seem curious at first that none of these wavevec- tors corresponds to an integer value of2p a. However, the Fourier transform of the NWA dynamic dipolar field distri- bution, which gives the spectral efficiency of the excitation25, is rather complex to depict as it consists of the modulation of the NWA periodicity with the field distribution of the CPW. We therefore perceive these kNWmodes to be neighboring rip- ples of a convoluted spectral distribution that depends on the lattice periodicity, the width of the nanowire, and the antenna field distribution. We also reported the field dependence of the mode amplitudes in Fig.3-(e), which shows non-monotonous behavior. As onecan see in the spectra of Fig.3-(a) and Fig.3-(c), the efficiency of the coupling to a particular mode will be all the stronger that its frequency matches with the one of the NWA FMR. As the gyromagnetic ratio of YIG is smaller than the one of permalloy ( gPy=2p=29.5 GHz.T1), the NWA dispersion will gradually cross the kNWmultimodes dispersion across the field range. Unfortunately, this non-monotonous field depen- dence of the excitation efficiency makes it arduous to charac- terize the attenuation of these high k-vector with just one kind of device. A series of devices with different distance D would be more suited for determining the attenuation length of these high-k SWB. Finally, from the period of oscillation f oscof the transmis- sion spectra, we estimate the group velocity according to vg=fosc*D21. We compare our measurement of v gwith the theoretical expression in Fig.3-(f) and Fig.3-(g). Although the agreement at lower wavevector is fair, we observe some sig- nificant discrepancies in the group velocity among the kNW modes. We foresee that an additional phase delay must oc- cur through the propagation path. Namely, considering that the length of NWA is comparable to the propagation distance D, one can expect the static stray field of the wires to cause some inhomogeneities in the static field distribution between the two antennas.NRSWB 5 IV. CONCLUSION We carried out a spin wave spectroscopy study on two dif- ferent 50 mm2Ni80Fe20nanowire arrays resonantly coupled to a continuous 55 nm thin YIG film. We demonstrated uni- directional transmission of 10 mm wide spin wave beams up to 77 rad. mm1in the [8;20] GHz frequency range. An at- tempt to characterize the propagation properties of these high k-vector spin wave beams reveals several peculiarities regard- ing the modes selection, their coupling efficiency, and pos- sible additional phase lag due to inhomogeneous stray field from the nanowires. These findings serve as a guideline for future miniaturization of nonreciprocal magnonic devices. ACKNOWLEDGMENTS The authors acknowledge the financial support from the French National research agency (ANR) under the project MagFunc , the Region Bretagne with the CPER-Hypermag project, and the Département du Finistère through the project SOSMAG . We also want to thanks Bernard Abiven for the implementation of the electromagnet and Guillaume Bourcin for fruitful discussions. CD acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 271741898. DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request. 1W. Palmer, et al., IEEE Micr. Magazine 20, 36 (2019). 2V . G. Harris, Modern microwave ferrites, IEEE Trans. Magn. 48, 3 (2012). 3M. Devoret, et al., Superconducting circuits for quantum information, Sci- ence 339, 1169 (2013). 4A. V . Chumak et al. , IEEE TRANSACTIONS ON MAGNETICS 58, 6 (2022).5A. Barman et al, J. Phys.: Condens. Matter 33, 413001 (2021). 6R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 7R. Camley, Surf. Sci. Rep. 7,103 (1987). 8J. Chen, T. Yu, C. Liu, T. Liu, M. Madami, K. Shen, J. Zhang, S. Tu, M. S. Alam, K. Xia, M. Wu, G. Gubbiotti, Y . M. Blanter, G. E. W. Bauer, and H. Yu, Phys. Rev. B 100, 104427 (2019). 9H. Wang, et al., Nano Res. 14, 2133–2138 (2021). 10O. Gladii, M. Haidar,Y . Henry, M. Kostylev, and M. Bailleul, Phys. Rev. B 93, 054430 (2016). 11M. Grassi, M. Geilen, D. Louis, M. Mohseni, T. Brächer, M. Hehn, D. Stoeffler, M. Bailleul, P. Pirro, and Y . Henry, Phys. Rev.Appl. 14, 024047 (2020). 12J.-H. Moon, S.-M. Seo, K.-J. Lee, K.-W. Kim, J. Ryu, H.-W. Lee, R. D. McMichael, and M. D. Stiles, Spin-wave propagation in the presence of interfacial Dzyaloshinskii-Moriya interaction, Phys. Rev. B 88, 184404 (2013). 13T. Brächer, O. Boulle, G. Gaudin, and P. Pirro, Creation of unidirectional spin-wave emitters by utilizing interfacial Dzyaloshinskii-Moriya interac- tion, Phys. Rev. B 95, 064429 (2017) 14K. Di, V . L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuo, J. Yu, J. Yoon, X. Qiu, and H. Yang, Phys. Rev. Lett. 114, 047201 (2015). 15J. A. Otalora, M. Yan, H. Schultheiss, R. Hertel, A. Kakay, Curvature- induced asymmetric spin-wave dispersion, Phys. Rev. Lett. 117, 227203 (2016). 16Q. Wang , A. V . Chumak, and P. Pirro, Nature Commun. 12, 2636 (2021). 17A. Papp, W. Porod, and G. Csaba, Nature Commun. 12, 6422 (2021). 18T. Yu, and G. E. W. Bauer, "Chiral Coupling to Magnetodipolar Radia- tion",Topics in Applied Physics Springer book series, vol. 138, (2021). 19C. Liu et al, Nature Comm. 9, 738 (2018). 20J. Chen, C. Liu, T. Liu, Y . Xiao, K. Xia, G. E. W. Bauer, M. Wu, and H. Yu, Phys. Rev. Lett. 120, 217202 (2018). 21N. Loayza, M. B. Jungfleisch, A. Hoffmann, M. Bailleul, and V . Vlaminck, Phys. Rev. B 98, 144430 (2018). 22H. S. Körner, J. Stigloher, C. H. Back, Phys. Rev. B 96, 100401(R) (2017). 23C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. Hübner, and E. Wendler, Phys. Rev. Materials 4, 024416 (2020). 24Protective Coating AR-PC 5091.02 (Electra 92), https://www.allresist.com/portfolio-item/ protective-coating-ar-pc-5091-02-electra-92/ . 25V . Vlaminck and M. Bailleul, Phys. Rev. B 81, 14425 (2010). 26O. Gladii, M. Collet, K. Garcia-Hernandez, C. Cheng, S. Xavier, P. Bor- tolotti, V . Cros, Y . Henry, J.-V . Kim, A. Anane, and M. Bailleul, Appl. Phys. Lett. 108, 202407 (2016). 27B. A. Kalinikos, and A. N. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986).

2022-11-10

We report unidirectional transmission of micron-wide spin waves beams in a 55 nm thin YIG. We downscaled a chiral coupling technique implementing Ni80Fe20 nanowires arrays with different widths and lattice spacing to study the non-reciprocal transmission of exchange spin waves down to lambda = 80 nm. A full spin wave spectroscopy analysis of these high wavevector coupled-modes shows some difficulties to characterize their propagation properties, due to both the non-monotonous field dependence of the coupling efficiency, and also the inhomogeneous stray field from the nanowires.

High wave vector non-reciprocal spin wave beams

2211.05514v2

Magnetic-eld-induced suppression of spin Peltier eect in Pt/Y3Fe5O12system at room temperature Ryuichi Itoh,1Ryo Iguchi,1, 2,Shunsuke Daimon,1, 3Koichi Oyanagi,1Ken-ichi Uchida,2, 4, 5and Eiji Saitoh1, 3, 5, 6 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2National Institute for Materials Science, Tsukuba 305-0047, Japan 3WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 5Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan Abstract We report the observation of magnetic-eld-induced suppression of the spin Peltier eect (SPE) in a junction of a paramagnetic metal Pt and a ferrimagnetic insulator Y 3Fe5O12(YIG) at room temperature. For driving the SPE, spin currents are generated via the spin Hall eect from applied charge currents in the Pt layer, and injected into the adjacent thick YIG lm. The resultant temperature modulation is detected by a commonly-used thermocouple attached to the Pt/YIG junction. The output of the thermocouple shows sign reversal when the magnetization is reversed and linearly increases with the applied current, demonstrating the detection of the SPE signal. We found that the SPE signal decreases with the magnetic eld. The observed suppression rate was found to be comparable to that of the spin Seebeck eect (SSE), suggesting the dominant and similar contribution of the low-energy magnons in the SPE as in the SSE. IGUCHI.Ryo@nims.go.jp 1arXiv:1709.08997v1 [cond-mat.mtrl-sci] 26 Sep 2017I. INTRODUCTION Thermoelectric conversion is one of the promising technologies for smart energy utilization [1]. Owing to the progress of spintronics in this decade, the spin-based thermoelectric conversion is now added to the scope of the thermoelectric technology [2{7]. In particular, the thermoelectric generation mediated by ow of spins, or spin current, has attracted much attention because of the advantageous scalability, simple fabrication processes, and exible design of the gure of merit [4, 8{12]. This is realized by combining the spin Seebeck eect (SSE) [13] and spin-to-charge conversion eects [14{16], where a spin current is generated by an applied thermal gradient and is converted into electricity owing to spin{orbit coupling. The SSE has a reciprocal eect called the spin Peltier eect (SPE), discovered by Flipse et al. in 2014 in a Pt/yttrium iron garnet (Y 3Fe5O12: YIG) junction [5, 17]. In the SPE, a spin current across a normal conductor (N)/ferromagnet (F) junction induces a heat current, which can change the temperature distribution around the junction system. To reveal the mechanism of the SPE, systematic experiments have been conducted [17{ 19]. Since the SPE is driven by magnetic uctuations (magnons) in the F layer, detailed study on the magnetic-eld and temperature dependence is indispensable for clarifying the microscopic relation between the SPE and magnon excitation and the reciprocity between the SPE and SSE [20{29]. A high magnetic eld is expected to aect the magnitude of the SPE signal via the modulation of spectral properties of magnons. In fact, the SSE thermopower in a Pt/YIG system was shown to be suppressed by high magnetic elds even at room temperature against the conventional theoretical expectation based on the equal contribution over the magnon spectrum [22]. This anomalously-large suppression highlights the dominant contribution of sub-thermal magnons, which possess lower energy and longer propagation length than thermal magnons [23, 25, 28, 30]. Thus, the experimental examination of the eld dependence of the SPE is an important task for understanding the SPE. Although the SPE has recently been measured in various systems by using the lock-in thermography (LIT) [17], it is dicult to be used at high elds and/or low temperatures. For investigating the high-magnetic-eld response of the SPE, an alternative method is required. In this paper, we investigate the magnetic eld dependence of the SPE up to 9 T at 300 K by using a commonly-used thermocouple (TC) wire. As revealed by the LIT experiments [17, 19], the temperature modulation induced by the SPE is localized in the vicinity of N/F 2interfaces. This is the reason why the magnitude of the SPE signals is very small in the rst experiment by Flipse et al. [5], where a thermopile sensor is put on the bare YIG surface, not on the Pt/YIG junction. Here, we show that the SPE can be detected with better sensitivity simply by attaching a common TC wire on a N/F junction. This simple SPE detection method enables systematic measurements of the magnetic eld dependence of the SPE, since it is easily integrated to conventional measurement systems. In the following, we describe the details of the electric detection of the SPE signal using a TC, the results of the magnetic eld dependence of the SPE signal in a high-magnetic-eld range, and its comparison to that of the SSE thermopower. II. EXPERIMENTAL The spin current for driving the SPE is generated via the spin Hall eect (SHE) from a charge current applied to N [15, 16]. The SHE-induced spin current then forms spin accumulation at the N/F interface, whose spin vector representation is given by s/SHEjcn; (1) whereSHEis the spin Hall angle of N, jcthe charge-current-density vector, and nthe unit vector normal to the interface directing from F to N. sat the interface exerts spin-transfer torque to magnons in F via the interfacial exchange coupling at nite temperatures, when sis parallel or anti-parallel to the equilibrium magnetization ( m) [31, 32]. The torque increases or decreases the number of the magnons depending on the polarization of the torque ( skmorskm), and eventually changes the system temperature by energy transfer, concomitant with the spin-current injection [5, 17, 29]. The energy transfer induces observable temperature modulation in isolated systems, which satises the following relation
TSPE/s
m/(jcn)
m: (2) A schematic of the sample system and measurement geometry is shown in Fig. 1(a). The sample system is a Pt strip on a single-crystal YIG. The YIG layer is 112- m-thick and grown by a liquid phase epitaxy method on a 500- m-thick Gd 3Ga5O12substrate with the lateral dimension 10 ×10 mm2, where small amount of Bi is substituted for the Y-site of the YIG to compensate the lattice mismatching to the substrate. The Pt strip, connected 3(a) Vres Current Source Voltmeter syncVTC Jc H (b) Thermocouple Varnish Al 2O3 YIG Substrate Pt JcμsThermo Couple 0Jc time0VTC ∆Jc -∆Jc Joule heating ( ∝Jc2)spin Peltier effect (∝Jc)T signal acquisition tdelay ∆T -∆TTYIG Pt VTC + VTC -convert Jq 2∆VTC mThermal anchoringFIG. 1. (a) Schematic of the Pt/YIG sample and measurement system with a thermocouple wire. Jc,s,m,H, and Jqdenote the applied current, the spin accumulation, the unit vector of the equilibrium magnetization, the magnetic eld, and the heat current concomitant with the spin-current injection. In an isolated system, Jqinduces a temperature gradient by accumulating heat, which can be detected by the thermocouple. The resistance of the Pt strip is obtained by measuringVres. The measurements were carried out by using the physical property measurement system, Quantum Design. (b) Expected responses due to the SPE and Joule heating when Jcis periodically changed from
Jcto
Jcwith the zero oset current J0 c= 0. Corresponding voltage signalsV+ TCandV TCare obtained after the time delay tdelay. The SPE signal is extracted from the dierence
VTC= V+ TCV TC
=2 . to four electrodes, is 5-nm-thick, 0.5-mm-wide, fabricated by a sputtering method, and patterned with a metal mask. Then, the whole surface of the sample except the electrodes is covered by a highly-resistive Al 2O3lm with a thickness of 100 nm by means of an atomic layer deposition method. We attached a TC wire to the top of the Pt/YIG junction, where the wire is electrically insulated from but thermally connected to the Pt layer owing to the Al 2O3layer. We used a type-E TC with a diameter of 0.013 mm (Omega engineering CHCO-0005), and xed its junction part on the middle of the Pt strip using varnish. The rest of the TC wires were xed on the sample surface for thermal-anchoring and avoiding thermal leakage from the top of the Pt/YIG junction [see the cross sectional view in Fig. 1(a)]. The expected thickness of the varnish between the TC and the sample surface is in 4the order of 10 m [33]. Note that the thickness of the varnish layer does not aect the magnitude of the signal while it aects the temporal response of the TC [34]. The ends of the Pt strip are connected to a current source and the other two electrodes are used for measuring resistance based on the four-terminal method. The magnetic eld His applied in the lm plane and perpendicular to the strip, thus is along s, satisfying the symmetry of the SPE [Eq. (2)]. The TC is connected to electrodes on a heat bath, which acts as a thermal anchor, and further connected to a voltmeter via conductive wires. The measurements were carried out at 300 K and 103Pa. For the electric detection of the SPE, we measured the amplitude dierence (
VTC) of the TC voltage VTCin response to a change (
Jc) in the current Jc(so-called Delta-mode of the nanovoltmeter Keithley 2182A). After the current is set to Jc=J0 c
Jc, time delay (tdelay) is inserted before measuring the corresponding voltages V TC. Then,
VTCis obtained as
VTC= V+ TCV TC
=2. The time delay tdelay is necessary because the temperature modulation occurred at the Pt/YIG junction takes certain time to reach and stabilize the TC. The appropriate value of tdelaycan be determined from the delay-dependence of the SPE signal, which will be shown in Sec. III. For the SPE measurements, we set no oset current (J0 c= 0) so that
VTCis free from Joule heating ( /J2 c), and the SPE (/Jc) is expected to dominate the
VTCsignal [see Fig. 1(a)]. By using the measured values of
VTC, the temperature modulation (
T) is estimated via the relation
T=STC
VTC, whereSTC is the Seebeck coecient of the TC. For the low-eld measurements ( 0H < 0:1 T), the reference value of STC= 61V=K at 300 K is used, while, for the high-eld measurements, the eld dependence of STC, determined by the method shown in Appendix A, is used. III. RESULTS AND DISCUSSION First, we demonstrated the electric detection of the SPE at low elds. Figure 2(a) shows
VTCas a function of the eld magnitude Hat
Jc= 10 mA and tdelay= 50 ms.
VTC clearly changes its sign when the eld direction is reversed and the appearance of the hystere- sis demonstrates that it re ects the magnetization curve of the YIG, showing the symmetry expected from Eq. (2) [5, 17]. The small oset of
VTCmay be attributed to the tempera- ture modulation by the Peltier eect appearing around the current electrodes, Joule heating due to small uncanceled current osets, and possible electrical leakage of the applied current 5(a) (b) (c) 1.0 0.9 0.8 0.7normalized ∆ TSPE 1 100 tdelay (ms)10 1000 ∆J2 1 0∆TSPE (mK) 15 10 5 0 c (mA)310 305 300TPt (K) -1 01∆T (mK) -20 -10 0 10 20 µ0H (mT)0.1 0.0 -0.1∆VTC (uV) 2∆ TSPEFIG. 2. (a) Field magnitude Hdependence of
VTC(the right axis) and
Twithout oset (the left axis), measured at
Jc= 10 mA and tdelay = 50 ms. (b) Current Jcdependence of
TSPEat0jHj= 0:1 T, where 0represents the permeability of vacuum. (c) Delay-time tdelay dependence of
TSPE. Plotted data are estimated from the results measured at
Jc= 10 mA and 5 mT<  0jHj<20 mT and normalized based on the tting with B[1exp (tdelay=)], where Bdenotes the proportional constant and = 0:4 ms denotes the characteristic time scale. The acquisition time of 20 ms is used for measurements. from the sample to the TC. Since the Peltier and resistance eects are of even functions of the magnetization cosines though the SPE is of an odd function, the SPE-induced tem- perature modulation
TSPEcan be extracted by subtracting the symmetric response to the magnetization:
TSPE= [
T(+H)
T(H)]=2 [5]. Figure 2(b) shows the
Jcdepen- dence of
TSPEand the temperature ( TPt) of the Pt strip, estimated from the resistance of the strip. While TPtincreases parabolically with the magnitude of
Jcfor Joule heating,
TSPEincreases linearly as is expected from the characteristic of the SPE. This distinct de- pendencies show negligibly small contribution to
TSPEfrom the Joule heating in this study [35]. The magnitude of the SPE signal is estimated to be
TSPE=
jc= 3:41013Km2=A, where
jcis the dierence in jc. This value is almost same as the value obtained in the thermographic experiments [17, 19]; since in Ref. [19] the sine-wave amplitude Aof
TSPEis divided by the rectangular-wave amplitude of
jc, a correction factor of =4 is necessary, i.e. 6(a) (b) (d) 2 1 0∆R/R0 (x10 -4 ) -10 -5 0 510 µ0H (T)2 1 0∆TSPE (mK) 15 10 5 0 Jc (mA) 0.1 T 8.0 T -2 -1 012∆TSPE (mK) -10 -5 0 5 10 µ0H (T)-1.0-0.50.00.51.0normalized SSSE -10 -5 0 5 10 µ0H (T) 1 mA 5 mA 10 mA 15 mA(c) H JqV+ -FIG. 3. (a) Field dependence of
TSPEincluding the high-magnetic-eld range measured at various
Jcvalues. (b) Jcdependence of
TSPEat0jHj= 0:1 T and 8:0 T. The solid lines represent the linear tting. (c) Field dependence of the resistance change
R=R(H)R0, whereRdenotes the resistance and R0that at0H= 0:1 T. (d) Field dependence of the spin Seebeck voltage VSSE normalized at 0H= 0:1 T, where VSSEis obtained by measuring the voltage under heater power of 100 mW and subtracting the component symmetric to H.
TSPE=
jc=A=(4
jc) = 3:71013Km2=A in the previous study. We note that, in the above and following measurements, tdelay= 50 ms is chosen based on the tdelaydependence of
TSPE[Fig. 2(c)], where
TSPEis almost saturated at tdelay>10 ms. Such nite but small thermal-stabilization time can be explained by the thermal diusion from the junction to the TC and rapid thermal stabilization of the SPE-induced temperature modulation [17]. Next, we measured the eld dependence of the SPE at higher elds up to 0H= 9:0 T. Figure 3(a) shows
TSPEas a function of H, where
Jcis changed from 1 to 15 mA. The suppression of the
TSPEsignal at higher elds is clearly observed for all the
Jcvalues. As shown in Fig. 3(b), the signal shows a linear variation with Jcboth at0H= 0:1 T and 8.0 T, demonstrating a constant suppression rate. Since the resistance of the sample varies only 1 % at most [Fig. 3(c)], the junction temperature keeps constant during the eld scan. The eld dependence of the thermal conductivity of YIG is also irrelevant to the
TSPEsuppression as it is known to be negligibly small at room temperature [36]. Thus we can conclude that the suppression is attributed to the nature of the SPE. By calculating the suppression magnitude as SPE= 1
TSPE(0H= 8:0 T)
TSPE(0H= 0:1 T); 7we obtained SPE= 0:26. To compare SPEto the eld-induced suppression of the SSE, we performed SSE mea- surements in a longitudinal conguration using a Pt/YIG junction system fabricated at the same time as the SPE sample. The SSE sample has the lateral dimension 2 :06:0 mm2and the same vertical conguration as the SPE sample except for the absence of the Al 2O3layer. The detailed method of the SSE measurement is available elsewhere [27, 37, 38]. Figure 3(d) shows the eld dependence of the SSE thermopower in the Pt/YIG junction. The clear suppression of the SSE thermopower is observed. Importantly, the high-eld response of the SSE is quite similar to that of the SPE in the Pt/YIG system. The suppression magnitude of the SSESSE, dened in the same manner as the SPE, is estimated to be 0:22, consistent with the previously reported values [22, 23, 25]. The observed remarkable eld-induced suppression of the SPE at room temperature shows that the SPE is likely dominated by low-energy magnons because the energy scale of the ap- plied eld is less than 10 K and thus much lower than the thermal energy of 300 K [22]. The origin of the strong contribution of the low-energy magnons in the SPE can be (i) stronger coupling of the spin torque to the low-energy (sub-thermal) magnons and (ii) greater propa- gation length of the low-energy magnons than those of high-energy (thermal) magnons [30]. While (i) is not well experimentally investigated, the existence of the m-range length scale in the SPE [19] and the similarity between SPEandSSEsuggest the dominant contribution from (ii) as in the case of the SSE [25, 30]. In fact, recently, it has been demonstrated that the high magnetic elds reduce the propagation length of magnons contributing to the SSE [30]. This length-scale scenario can qualitatively explain the suppression in the SPE. Recall- ing that a heat current density ( jq) existing over a distance ( l) generates the temperature dierence
T=1jqlin an isolated system,
Tshould decrease when ldecreases, where is the thermal conductivity of the system. In the SPE, lcorresponds to the magnon prop- agation length [39], and a ow of magnons accompanies a heat current [36]. Consequently, when the high magnetic eld is applied and the magnons with longer propagation length are suppressed by the Zeeman gap, the averaged magnon propagation length decreases and thus results in the reduced
T. To further investigate the microscopic mechanism of the SPE, consideration of the spectral non-uniformity may be vital both in experiments and theories. 8IV. SUMMARY In this study, we showed the magnetic eld dependence of the spin Peltier eect (SPE) up to 9.0 T at 300 K in a Pt/YIG junction system. We established a simple but sensitive detection method of the SPE using a commonly-available thermocouple wire. The SPE signals were observed to be suppressed at high magnetic elds, highlighting the stronger contribution of low-energy magnons in the SPE. The similar suppression rate of the SPE- induced temperature modulation to that of the SSE-induced thermopower suggests that the suppression originates the decrease in the magnon propagation length as in the case of the SSE. We anticipate that the experimental results and the method reported here will be useful for systematic investigation of the SPE. ACKNOWLEDGMENTS The authors thank T. Kikkawa for the aid in measuring the SSE and G. E. W. Bauer and Y. Ohnuma for the valuable discussion. This work was supported by PRESTO \Phase Interfaces for Highly Ecient Energy Utilization" (Grant No. JPMJPR12C1) and ERATO \Spin Quantum Rectication Project" (Grant No. JPMJER1402) from JST, Japan, Grant- in-Aid for Scientic Research (A) (Grant No. JP15H02012), and Grant-in-Aid for Scientic Research on Innovative Area \Nano Spin Conversion Science" (Grant No. JP26103005) from JSPS KAKENHI, Japan, NEC Corporation, the Noguchi Institute, and E-IMR, Tohoku University. S.D. was supported by JSPS through a research fellowship for young scientists (Grant No. JP16J02422). K.O. acknowledges support from GP-Spin at Tohoku University. 9(a) (b)0Jc time0VTC Jc+∆Jc Joule heating VTC + VTC - 2∆V TC Jc-∆Jc00 2.26 2.24 2.22∆VJoule (µV) -10 -5 0 5 10 µ0H (T)FIG. 4. (a) Expected responses due to the Joule heating at the nite oset current J0 c6= 0. (b) Field dependence of
VJoule at
Jc= 0:5 mA and J0 c= 5 mA. The solid curve represents the calibration line determined by the tting. Appendix A: Calibration of Thermo Couple at High Magnetic Fields To measure the eld dependence of STC, we used the Joule-heating-induced signal as a reference. By adding a non-zero oset ( J0 c) to the applied current, we obtained the temperature modulation induced by the Joule heating, of which the power Pchanges from P(H) =R(H) (J0 c
Jc)2toP(H) =R(H) (J0 c+
Jc)2, whereRdenotes the resistance of the strip [Fig. 4(a)]. Figure 4(b) shows the magnetic eld dependence of the component of
VTCsymmetric to the eld (
VJoule= [
TTC(+H) +
TTC(H)]=2). As the change in R, due to the ordinary, spin Hall, and Hanle magnetoresistance eects [40, 41], is in the order of 0.02 % [Fig.3(c)], its contribution to Pcan be neglected. Similarly, the eld dependence of the thermal conductivity of YIG is negligibly small [36], ensuring the constant temperature change. Accordingly, the eld dependence of
VJouledirectly re ects STC(H). It increases by a factor of1 % when the eld magnitude increases up to 9.0 T. We approximated the eld dependence of STC(H) asSTC(H) = 61 1 + 4:54103j0Hj0:453
V=K by determining the relative change from the measurement results (
VJoule(H)=
VJoulejH=0) and the absolute value from the reference value. [1] X. Zhang and L.-D. Zhao, J. Materiomics 1, 92 (2015). 10[2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). [3] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). [4] K. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S. Maekawa, and E. Saitoh, Proc. IEEE 104, 1946 (2016). [5] J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). [6] J. Flipse, F. L. Bakker, A. Slachter, F. K. Dejene, and B. J. van Wees, Nat. Nanotech. 7, 166 (2012). [7] A. Slachter, F. L. Bakker, J. P. Adam, and B. J. van Wees, Nat. Phys. 6, 879 (2010). [8] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [9] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010). [10] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, Nat. Mater. 11, 686 (2012). [11] R. Ramos, T. Kikkawa, M. H. Aguirre, I. Lucas, A. Anad on, T. Oyake, K. Uchida, H. Adachi, J. Shiomi, P. A. Algarabel, L. Morell on, S. Maekawa, E. Saitoh, and M. R. Ibarra, Phys. Rev. B92, 220407 (2015). [12] H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013). [13] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [14] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 2509 (2006). [15] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). [16] A. Homann, IEEE Trans. Magn. 49, 5172 (2013). [17] S. Daimon, R. Iguchi, T. Hioki, E. Saitoh, and K. Uchida, Nat. Commun. 7, 13754 (2016). [18] K. Uchida, R. Iguchi, S. Daimon, R. Ramos, A. Anad n, I. Lucas, P. A. Algarabel, L. Morelln, M. H. Aguirre, M. R. Ibarra, and E. Saitoh, Phys. Rev. B 95, 184437 (2017). [19] S. Daimon, K. Uchida, R. Iguchi, T. Hioki, and E. Saitoh, Phys. Rev. B 96, 024424 (2017). [20] K. Uchida, T. Kikkawa, A. Miura, J. Shiomi, and E. Saitoh, Phys. Rev. X 4, 041023 (2014). 11[21] S. M. Rezende, R. L. Rodriguez-Suarez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. F. Guerra, J. C. L. Ortiz, and A. Azevedo, Phys. Rev. B 89, 014416 (2014). [22] T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 064413 (2015). [23] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015). [24] J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117, 217201 (2016). [25] E.-J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D. A. MacLaren, G. Jakob, and M. Kl aui, Phys. Rev. X 6, 031012 (2016). [26] V. Basso, E. Ferraro, A. Magni, A. Sola, M. Kuepferling, and M. Pasquale, Phys. Rev. B 93, 184421 (2016). [27] R. Iguchi, K. Uchida, S. Daimon, and E. Saitoh, Phys. Rev. B 95, 174401 (2017). [28] A. Miura, T. Kikkawa, R. Iguchi, K. Uchida, E. Saitoh, and J. Shiomi, Phys. Rev. Materials 1, 014601 (2017). [29] Y. Ohnuma, M. Matsuo, and S. Maekawa, to be published in Phys. Rev. B (2017). [30] T. Hioki, R. Iguchi, Z. Qiu, D. Hou, K. Uchida, and E. Saitoh, Appl. Phys. Express 10, 073002 (2017). [31] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [32] S. S. L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012). [33] This is estimated from the thickness of the varnish layer sandwiched between glass substrates pressed with the same pressure applied to the sample; we pressed the sample and the TC wire with an additional glass cover. [34] As the radiation to the outer environment at the surface is negligibly small, the vertical heat current in the varnish layer is zero at the steady-state condition. The eect of the lateral heat currents, expected at the edges of the Pt strip, is also small as the total thickness from the top of the Pt strip to the surface ( 30m) is smaller than the width (500 m). [35] The SPE signal at 305 and 310 K (nominal) was observed to show the same magnitude as that at 300 K. Thus the temperature increase due to the Joule heating does not aect the measured SPE value. [36] S. R. Boona and J. P. Heremans, Phys. Rev. B 90, 064421 (2014). 12[37] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa, and E. Saitoh, J. Appl. Phys. 111, 103903 (2012). [38] A. Sola, P. Bougiatioti, M. Kuepferling, D. Meier, G. Reiss, M. Pasquale, T. Kuschel, and V. Basso, Sci. Rep. 7, 46752 (2017). [39] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). [40] H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). [41] S. V elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. Lett. 116, 016603 (2016). 13

2017-09-26

We report the observation of magnetic-field-induced suppression of the spin Peltier effect (SPE) in a junction of a paramagnetic metal Pt and a ferrimagnetic insulator ${\rm Y_{3}Fe_{5}O_{12}}$ (YIG) at room temperature. For driving the SPE, spin currents are generated via the spin Hall effect from applied charge currents in the Pt layer, and injected into the adjacent thick YIG film. The resultant temperature modulation is detected by a commonly-used thermocouple attached to the Pt/YIG junction. The output of the thermocouple shows sign reversal when the magnetization is reversed and linearly increases with the applied current, demonstrating the detection of the SPE signal. We found that the SPE signal decreases with the magnetic field. The observed suppression rate was found to be comparable to that of the spin Seebeck effect (SSE), suggesting the dominant and similar contribution of the low-energy magnons in the SPE as in the SSE.

Magnetic-field-induced suppression of spin Peltier effect in Pt/${\rm Y_{3}Fe_{5}O_{12}}$ system at room temperature

1709.08997v1

Fresnel diraction of spin waves N. Loayza,1M. B. Jung eisch,2, 3A. Homann,2M. Bailleul,4and V.Vlaminck1 1Colegio de Ciencias e Ingeniera, Universidad San Francisco de Quito, Quito, Ecuador 2Material Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA 3Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA 4Institut de Physique et Chimie des Mat eriaux de Strasbourg, UMR 7504 CNRS, Universit e de Strasbourg, 23 rue du Loess, BP 43, 67034 Strasbourg Cedex 2, France (Dated: November 8, 2021) Abstract The propagation of magnetostatic forward volume waves excited by a constricted coplanar waveg- uide is studied via inductive spectroscopy techniques. A series of devices consisting of pairs of sub-micrometer size antennae is used to perform a discrete mapping of the spin wave amplitude in the plane of a 30-nm thin YIG lm. We found that the spin wave propagation remains well focused in a beam shape of width comparable to the constriction length and that the amplitude within the constriction displays oscillations, two features which are explained in terms of near-eld Fresnel diraction theory. 1arXiv:1807.11754v1 [cond-mat.mes-hall] 31 Jul 2018The emerging eld of magnonics [1, 2] has sparked a renewed interest in non-uniform magnetization dynamics. Spin-waves are now considered as a very promising information carrier for performing basic logic operations [3{7], or implementing novel computation archi- tectures [8, 9]. A key advantage of spin-waves is that their dispersion can be easily tailored in a wide band of the microwave spectrum, particularly in the so-called magnetostatic wave regime for which the magnetic dipolar interaction plays the dominant role. Recently, it has been demonstrated that the propagation of spin waves in ferromagnetic thin lm could be shaped using several concepts borrowed from optics [10]. A special attention has been set on understanding their refraction and re ection eects [11{14], and also on generating and manipulating spin-wave beams. The latter is of particular importance in order to exploit the potential of multi-beam interference. So far, three dierent mechanisms have been investigated to shape spin-wave beams: (i) the so-called caustic eect [15{17, 21] associated with the very strong anisotropy of mag- netostatic wave dispersions; (ii) the connement by the strongly inhomogeneous internal magnetic elds existing at strip edges [10], in magnetic domain-walls [18], or close to nano-contact spin torque nanoscillators [19, 20]; (iii) the coupling to specially designed constricted microwave antennae providing a suitable non-uniform magnetic pumping eld prole [22, 23]. The last method appears the most versatile, being able to produce a co- herent spin wave beam in a homogeneous magnetic layer without any special requirement on its magnetic conguration. It was rst proposed theoretically by Gruszecki et al. via micromagnetic simulation [22], and was recently veried experimentally by K orner et al. via time resolved magneto-optical imaging of magnetostatic surface wave beams generated in a relatively thick NiFe lm [23]. In this letter, we show experimentally that the spin wave beam generated by a constricted coplanar waveguide (CPW) follows closely a near-eld diraction pattern. To this objective, we resort to all-electrical measurements performed in a conguration providing isotropic spin-wave propagation (thin Yttrium Iron Garnet lm magnetized out-of-plane) and analyze them using elementary Fresnel diraction modeling. The spin wave antennae are designed in such a way that the constricted region of the CPW reproduces as closely as possible the case of an isolated rectangular slit. We rst focus on the geometry-A of spin-wave antennae shown in Fig. 1-(a),(b). It consists of a pair of identical shorted CPWs, whose constriction is shaped symmetrically with a gradual bend 2in order to have two narrow sections of CPW facing each other. The constricted region of the CPW consists of a central track of width wS= 400nmand two ground tracks of widthwG= 200nmwith a gap of 200 nm. The generated spinwaves have a wavelength of the order of the distance between the center of the ground tracks, i.e. '1m, which remains much smaller than the constriction length. We adopted a much sharper constric- tion than in the geometries study by Gruszecki et al. [22], and K orner et al. [23], with a factor of ten between the widths of the constricted section I and the non-constricted section II. This allows us to fully separate the peaks associated with the excitation of spin waves in the two sections, as illustrated in Fig. 1-(c) which shows the corresponding Fourier transforms of the current density (assumed to be uniform in each CPW track). For section I, one distinguishes a main peak centered at kI= 5:92rad:m1with a full width at half maximum
kI= 4:15rad:m1, and for section II a main peak at kII= 0:59rad:m1with
kII= 0:41rad:m1. We fabricated ve spin-wave transducers of geometryAwith separation distance D=f4;6;8;10;12gm and constriction length lexc= 5m, which we used for preliminary characterization and val- idation of the spin-wave transduction in continuous layer. The antennae were fabricated by e-beam lithography and lift-o of 5 nmTi= 80nmAu directly on top of 30 nmthin sputtered YIG (Y 3Fe5O12) lms deposited on gadolinium gallium garnet by magnetron sputtering and post-annealed [24{26]. The fabricated device is then placed in the center of the lower pole of an electromagnet tting in a home-made probe station, and we proceed to the propagative spin-wave spectroscopy measurement [27] while applying an external magnetic eld Hlarge enough to magnetize the lm out of the plane. This corresponds to the so-called magnetostatic forward volume wave (MSFVW) conguration, for which the isotropic dispersion relation does not favor any propagation direction. This is in strong contrast with the situation of in-plane magnetized lms, for which the spin-wave dispersion is strongly anisotropic with a maximal group velocity in the so-called magnetostatic surface wave conguration. For practical reasons, most studies of nanomagnonics, including recent ones in YIG have been done in this last conguration [28, 29]. In the present case, we simplify the analogy with optics by employing the isotropic MSFVW conguration, and take directly advantage of the low damping and low magnetization of the YIG lms. The microwave spectra were acquired using a vector network analyzer ( AgilentE 8342B, 10MHz - 50GHz ) at low input power ( 20dBm ), 100Hzbandwidth, and in a single sweep 3kIkII k=0D = 4 /uni03BCm D = 8 /uni03BCm D = 12 /uni03BCmfosc kIkII(b) 5/uni03BCm kIIkI(a) lexc 2/uni03BCmD λIλII (c)(d)FIG. 1. (a) Geometry-A spin-wave antennae with a separation distance D= 12m. (b) Scanning electron microscope image of the same device zoomed in the region of the constriction. (c) Fourier transform of the current distribution for the two sections I and II, and MSFVW dispersion relation for0Hext= 308mT. (c) Self- and mutual inductance spectra obtained at 0Hext= 308mTfor three devices of geometry-A with separation distance D= 4m,D= 8m, andD= 12m. mode in order to limit the possible temperature drift of the electromagnet. We always per- form two measurements: a rst one at a resonant eld ( Hres), followed by a second one at a reference eld ( Href) for which no resonance occurs within the frequency range swept. In this manner, we retrieve the variation of inductance
Lij=Lij(Hres)Lij(Href) due to spin wave excitation [29];
L11the self-inductance measured on antenna 1, and
L21the 4mutual inductance characterizing the transduction of spin wave excited by antennae 1 and detected by antennae 2. Figure 1-(d) shows typical spectra obtained with identical antennae ofgeometry-A atHext= 308mT, for three dierent separation distances (4, 8, and 12 m). We can identify from the re ection spectra three main peaks which are attributed to the dierent parts of the CPW. Namely, the lowest frequency peak corresponds to the quasi- uniform resonance ( k= 0) of the wide section of the CPW where the 150 mpitch coplanar probe are contacted. The second peak corresponds to the non-constricted region II of the CPW, and the last peak to the constricted region I. For the mutual inductance spectra, we observe oscillations only underneath the last peak conrming that only the constricted region of the CPW contributes to the spin wave transduction between antennae. These oscillations are attributed to the phase delay kDaccumulated by the spin-waves during its propagation between the two antennas, and therefore are more numerous the longer the separation distance between antennae. The level of amplitude of the mutual inductance spectra is comparable to the one found when performing simulation of MSFVW transduc- tion [27, 30] on a stripe of width equal to the length of the constricted region. This suggests already that the excitation of the spin wave from this type of constriction should remain fairly focused. To validate our spin wave transduction technique when applied to a continuous magnetic layer, we rst analyze the microwave spectra measured for dierent elds and dierent dis- tances between the antennas. In particular, we can take advantage of the three section of our waveguides to perform k-resolved ferromagnetic resonance (FMR). As shown in Fig. 2-(a), we track the peak position in function of applied eld respectively for k= 0,kII, andkI. Fitting it to the MSFVW dispersion relation [31] for ve dierent devices, we obtain an average value for the gyromagnetic ration =2= 28:260:07GHz:T1and the eective magnetization 0Meff= 1362mT. Furthermore, we can estimate independently the saturation magneti- zationMsby plotting the eld-dependence of the dierence f2 res(kI)-f2 res(kII) = 22 0(Hext- Meff)Ms(kIkII)t=2 (wheretis the YIG lm thickness) as shown in Fig. 2-(b), from which we nd a nice linear dependence and the average value 0Ms= 1968mT. Next, we use the observed decay of the amplitude of the mutual-inductance as a function of the distance as shown in Fig. 2-(c) to extract the characteristic attenuation length of the spin wave. For each applied eld, we observe a clear linear dependence of ln(j
L21j) on the antenna separation D, which is consistent with an exponential decay jL21j/eD=L att). This constitutes another 5(a) (b) γμ0MS(kI-kII)t/4π(c) 1/vg-1/L att (d)2πα eff (g)(e) (f)FIG. 2. (a) Field dependence of the resonance peaks k= 0,kII, andkI. (b) Dierence of the square of the resonance frequencies f2(kI) -f2(kII). Separation distance dependence of (c) ln(
L21), and (d) the inverse of the oscillation period of
L21. Frequency dependence of: (e) the ratio of the group velocity to the attenuation length, (f) the attenuation length, and (g) the group velocity. evidence for a proper focusing of the spin wave excitation. Obviously, a diused emission of opening angle would reduce the amplitude by an additional factor ldet=(D), which is not observed here. Then, from the period of oscillation foscof the mutual inductance spectra, we can estimate the group velocity vgaccording to vg=foscD[27]. Fig. 2-(d) shows clear linear dependence of 1 =foscwithDfor the dierent applied elds. Finally, we perform a linear t of the frequency dependence of the ratio vg=Latt[32],and identify the slope to 2eff[cf. Fig. 2-(e)], which gives us a value of the eective damping eff= 7:50:2 104in good agreement with previous measurements on similar lms [33, 34]. We obtain fairly good agreements with the theoretical group velocity and attenuation length estimated from the MSFVW dispersion relation [dotted lines in Fig. 2(f,g)], which validate the implementation of the spin wave transduction technique to continuous layers for this geometry of CPW. We now turn to the main result of this work, which is the evolution of the ampli- tude
L21between several pairs of antennae at various separation distances D, and with various shift swith respect to their axis in order to map in a discrete manner the spin- 6wave emission from a constriction. We fabricated two series of pairs of non-identical spin-wave antennae with a long excitation antenna ( lexc= 10m), and a shorter detection antenna of ( ldet= 2m) in order to rene the spatial resolution of the mapping. The rst series consists of the symmetrical geometry-A as shown in Fig. 3-(a), for which we fabri- cated six devices having the same separation distance D= 5m, and only one-sided shift s=f0;1;2;3;4;6gm. For the second series, geometry-B shown in Fig. 3-(b), which con- sists of an asymmetrical constriction short-circuited right at its end and also with a steeper bend, we fabricated eighteen devices covering two separation distances D=f8;12gm, and nine shifts=f8;6;4;2;0;2;4;6;8gm. Fig. 3-(c) shows the shift dependence of the peak amplitudej
L21jmaxforgeometry-A at various applied eld (see typical examples of mutual-inductance spectra in the supplementary materials [35]). We observe an oscillation of the amplitude within the width of the constriction and a clear drop of amplitude for the devices= 6m, which lays just entirely outside of the constriction. Similar observations are made with geometry-B shown in Fig. 3-(d) although the drop of amplitude outside the constriction is slower for negative shifts due to the non-symmetrical shape of the antennae. Indeed, the shorted ends of the constrictions, which come close to each other for positive s [see Fig. 3(b)], radiate much less spin-wave power out of the constriction than the broader convex CPW access, which come close to each other for negative s. To describe these features of spin wave emission from a constricted CPW, we propose to implement the common equations of optics used in the case of the Fresnel diraction from a rectangular slit [36]. This choice is particularly relevant for the range of wavelength considered, for which the Fresnel radius RFremains much smaller than the length of the constriction: RF=p D << l exc. We simplify the problem by considering that each track of the CPW [ j=fG;S;G +g, see sketch in Fig. 3-(e)] acts a single rectangular source of coherent, circular, and monochromatic waves, of wavelength =2 kI. We also account for the spin wave attenuation with an exponential decay factor ( er=L att), whereLattis eld- (or frequency-) dependent with a value given in Fig. 2-(f). The normalized spin-wave amplitude ~mFresnel (D;s) at a distance Dand a shift semitted by the track jof the CPW is written as: ~mj(D;s) =Rl=2 l=2dyRwj=2 wj=2dx1prjerj=Latteikrj (1) 7(c) (d) 5 10 15 | | | G- G+ S λ+(x,y) + +1015 1510 D [μm]s [μm] mFresnel(e) 2 μmD s 2 μms D(a) (b) (x,y)0FIG. 3. (a) Geometry-A device for the mapping of spin wave with a separation distance D= 5m and a shifts=5m. (b) geometry-B with a separation distance D= 12mand a shifts= +8m. Evolution of the measured mutual inductance amplitude with the antennae shift sfor: (c) geometry- Amapping devices, and (d) geometry-B antennae separated by D= 8m, andD= 12m. The dotted lines are the calculated spin-wave amplitude from the Fresnel diraction model with the corresponding Latt. The symbols are the measured ampitude for the dierent devices. (e) Color mapping of the wave amplitude for Latt= 10mtaking into account the probe size ldet= 2m. The vertical dotted lines indicate sections of amplitude <~mCPW (s;D)>atD= 5;8;12m, 8Whererj=p (Dxj)2+ (sy)2is the distance between an element of surface dxdy of the source centered at ( xj;y) and a detection point of coordinates ( D;s);lis the antennae length and wjthe width of the CPW track. Now, the complete amplitude of the Fresnel diracted spin-wave ~ mCPW results from the linear combination of the three branches of the CPW: ~mCPW (D;s) =~mG(D+ 2;s) + ~mS(D;s)~mG+(D 2;s) (2) Where the negative signs accounts for the opposite phase of the excitation in the ground lines with respect to the central line. Fig. 3-(e) shows the color mapping of the spin-wave am- plitude in the ( D;s) plane calculated from Eq. (2) with an attenuation length Latt=10m. In order to compare our measurement with this Fresnel diraction model, we took into account the non-punctual aspect of the detection antenna by averaging the amplitude over the probe antenna extension ( ldet=2m). This near-eld diraction patterns reproduces the main features of our measurement, which are on one hand an emission that remains focused in a beam shape of width similar to the CPW length, and on the other hand, some oscillations of the amplitude within the beam width that depend mostly on the distance D. Finally, we compare the measured amplitudes with the calculated ones <~mCPW(s;D)> [dotted lines in Fig. 3-(c),(d)] for the specic distances D, and with the corresponding values of attenuation length obtained in Fig. 2-(f). We nd a remarkable agreement between this Fresnel diraction model of spin waves emitted from an antenna of nite extension and our measurements in the two dierent geometries of waveguide, which constitutes a direct demonstration of the focused nature of spin wave beams in constricted CPW. In summary, we rst demonstrated the possibility of performing spin-wave spectroscopy in thin magnetic lms without the need to structure a spin-wave guide, only by using su- ciently sharp constrictions in CPWs. We rstly showed that the signal amplitudes measured for pairs of identical antennae shifted gradually along the beam direction follow precisely an exponential decay, which suggests that the emission remains well-focused. Secondly, via a series of devices consisting of pairs of non-identical antennae covering dierent location of the 2D-plane, we performed a discrete mapping of the spin-wave amplitude for two dierent geometries conceived in such a way to reproduce the case of an optical rectangular slit. We found that the spin wave amplitude oscillates within the constriction zone, while it decays 9rapidly outside of it, which is notably well-explained with a Fresnel diraction model of circular waves. These ndings draw a deeper parallel between the excitation of spin-waves from sub-micrometric antennae and the basic concepts of optics, and therefore pave the way for future studies of spin wave beam interference, which could nd applications for spin wave logic devices. We thank Olga Gladii, Hicham Majjad, Romain Bernard, and Alain Carvalho for sup- port with the nanofabrication in the STnano platform, and Guy Schmerber for X-ray measurements. This work was supported by the French Agence National de la Recherche grant ANR-11-LABX-0058 NIE, and the USFQs PolyGrant] 431 program. The synthesis of the YIG lms at Argonne was supported by the U.S. Department of Energy, Oce of Science, Materials Science and Engineering Division. [1] S. O. Demokritov, A. N. Slavin, Magnonics, From Fundamentals to Applications, Springer (2012). [2] A.V. Chumak, el al., Nature Physics 11, 453-461 (2015). [3] T. Schneider, et al., Appl. Phys. Lett. 92, 022505 (2008). [4] A. Chumak, et al., Nat. Commun. 5, 4700 (2014). [5] K. Vogt, et al., Nat. Commun. 5, 3727 (2014). [6] S. Klingler, et al., Appl. Phys. Lett. 106, 212406 (2015). [7] S. Louis, et al., AIP Advances 6, 065103 (2016). [8] A. Kozhevnikov, et al., Appl. Phys. Lett. 106, 142409 (2015). [9] A. Papp, et al., Sci. Rep. 7, 9245 (2017). [10] V. Demidov et al., Appl. Phys. Lett. 92, 232503 (2008). [11] P. Gruszecki, et al., Appl. Phys. Lett. 105, 242406 (2014). [12] J. Stigloher, et al., Phys. Rev. Lett. 117, 037204 (2016). [13] P. Gruszecki, et al., Phys. Rev. B 95, 014421 (2017). [14] J. Gr afe et al., arXiv:1707.03664. [15] V. Demidov, et al., Phys. Rev. B 80, 014429 (2009). 10[16] T. Sebastian, et. al, Phys. Rev. Lett. 110, 067201 (2013). [17] J-V. Kim, et. al, Phys. Rev. Lett. 117, 197204 (2016). [18] K. Wagner, et al., Nature Nanotechnology 11, 432 (2016). [19] A. Houshang, et al., Nature Nanotechnology 11, 280 (2016). [20] V. Demidov et al., Nature Comm. 7, 10446 (2016). [21] T. Schneider, et al., Phys. Rev. Lett. 104, 197203 (2010). [22] P. Gruszecki, et al., Sci. Rep. 6, 22367 (2016). [23] H. S. K orner, et al., Phys. Rev. B 96, 100401(R) (2017). [24] S. Li et al, Nanoscale 8, 388 (2016). [25] M. B. Joung eisch, et al. Nano Lett. 17, pp 814 (2017). [26] Supplementary online materials (S-1). [27] V. Vlaminck, et al., Phys. Rev. B 81, 014425 (2010). [28] S. Maendl, et. al., Appl. Phys. Lett. 111, 012403 (2017). [29] M. Collet, et al., Appl. Phys. Lett. 110, 092408 (2017). [30] Supplementary online materials (S-2). [31] A. G. Gurevich, G. A. Melkov, Magnetization Oscillations and Waves, CRC (1996). [32] O. Gladii et al. Appl. Phys. Lett. 108, 202407 (2016) [33] T. Liu, et al., J. Appl. Phys. 115, 17A501 (2014). [34] M. B. Jung eisch, et. al, J. Appl. Phys. 117, 17D128 (2015). [35] Supplementary online materials (S-3). [36] E. Hecht, Optics, Addison Wesley (2002). 11

2018-07-31

The propagation of magnetostatic forward volume waves excited by a constricted coplanar waveguide is studied via inductive spectroscopy techniques. A series of devices consisting of pairs of sub-micrometer size antennae is used to perform a discrete mapping of the spin wave amplitude in the plane of a 30-nm thin YIG film. We found that the spin wave propagation remains well focused in a beam shape of width comparable to the constriction length and that the amplitude within the constriction displays oscillations, two features which are explained in terms of near-field Fresnel diffraction theory.

Fresnel diffraction of spin waves

1807.11754v1

Microwave-to-Optical Quantum Transduction Utilizing the Topological Faraday Effect of Topological Insulator Heterostructures Akihiko Sekine,1,∗Mari Ohfuchi,1and Yoshiyasu Doi1 1Fujitsu Research, Fujitsu Limited, Kawasaki 211-8588, Japan (Dated: November 14, 2023) The quantum transduction between microwave and optical photons is essential for realizing scalable quantum computers with superconducting qubits. Due to the large frequency difference between microwave and optical ranges, the transduction needs to be done via intermediate bosonic modes or nonlinear processes. So far, the transduction efficiency ηvia the magneto-optic Faraday effect (i.e., the light-magnon interaction) in the ferromagnet YIG has been demonstrated to be small asη∼10−8−10−15due to the sample size limitation inside the cavity. Here, we take advantage of the fact that three-dimensional topological insulator thin films exhibit a topological Faraday effect that is independent of the sample thickness. This leads to a large Faraday rotation angle and therefore enhanced light-magnon interaction in the thin film limit. We show theoretically that the transduction efficiency can be greatly improved to η∼10−4by utilizing the heterostructures consisting of topological insulator thin films such as Bi 2Se3and ferromagnetic insulator thin films such as YIG. Introduction.— The quantum transduction, or quan- tum frequency conversion, is an important quantum tech- nology which enables the interconnects between quan- tum devices such as quantum processors and quantum memories. In particular, the quantum transduction be- tween microwave and optical photons has so far gath- ered attention in pursuit of large-scale quantum com- puters with superconducting qubits [1–3]. Due to the large frequency difference between microwave and opti- cal ranges, the transduction needs to be done via inter- mediate interaction processes with bosonic modes or via nonlinear processes, such as the optomechanical effect [4– 13] electro-optic effect [14–21], and magneto-optic effect [22–26]. To date, the transduction efficiency η, whose maximum value is 1 by definition, has recorded the high- est value η∼10−1with a bandwidth ∼10−2MHz [6] (η∼10−2with∼1 MHz [15]) among the transductions utilizing the optomechanical effect (electro-optic effect). The focus of this Letter is the microwave-to-optical quantum transduction via the magneto-optic Faraday ef- fect, i.e., the light-magnon interaction. Such a quan- tum transduction mediated by ferromagnetic magnons can have a wide bandwidth ∼1 MHz and can be oper- ated even at room temperature [1–3]. Also, the coherent coupling between a ferromagnetic magnon and a super- conducting qubit has been realized [27, 28]. However, the current major bottleneck when using the ferromagnetic insulators (FIs) such as YIG is the low transduction ef- ficiency η∼10−8−10−15[22–26, 29] due to the sample size limitation inside the microwave cavity, as can be un- derstood from the relation η∝dFIwith dFIthe sample thickness [22]. The purpose of this study is to challenge this issue by utilizing topological materials, which are a new class of materials that are expected to exhibit un- usual materials properties due to their topological nature. In this Letter, we take advantage of the fact that three-dimensional (3D) topological insulator (TI) thin films ex- hibit a topological Faraday effect that is independent of the sample thickness, thus leading to a large Faraday ro- tation angle in the thin film limit. To this end, we partic- ularly consider the heterostructures consisting of TI thin films such as Bi 2Se3and FI thin films such as YIG. We find that the transduction efficiency ηis inversely pro- portional to the thickness of the FI layers ( η∝1/dFI), which is in sharp contrast to the above case of conven- tional FIs. We show theoretically that the transduction efficiency can be greatly improved to η∼10−4in a het- erostructure of a few dozen of layers of nanometer-thick TI and FI thin films. Quantum transduction.— Let us start with a generic description of our setup depicted in Fig. 1. We consider the interaction Hamiltonian Hint=Hκ+Hg+Hζ[22], where Hκ=−iℏ√κcZ∞ −∞dω 2πh ˆa†ˆain(ω)−a† in(ω)ˆai (1) describes the coupling between the microwave cavity pho- ton ˆaand an itinerant microwave photon ˆ ain(ω), Hg=ℏg ˆa†ˆm+ ˆm†ˆa
(2) describes the coupling between the microwave cavity pho- ton and the magnon (in the ferromagnetic resonance FIG. 1. Schematic illustration of our setup in terms of the operators, coupling strengths, and losses.arXiv:2311.07293v1 [cond-mat.mes-hall] 13 Nov 20232 state) ˆ m, and Hζ=−iℏp ζ ×Z∞ −∞dΩ 2π ˆm+ ˆm†
h ˆbin(Ω)eiΩ0t−ˆb† in(Ω)e−iΩ0ti (3) describes the coupling between the magnon and an itiner- ant optical photon ˆbin(Ω), which is indeed the sum of the beam-splitter-type and parametric-amplification-type in- teractions [30]. Ω 0is the input light frequency. In order to relate the incoming and outgoing itinerant photons, we can employ the standard input-output formalism, to obtain ˆ aout= ˆain+√κcˆaandˆbout=ˆbin+√ζˆm. We solve the equations of motion for the cavity and magnon modes in the presence of intrinsic losses κandγ: ˙ˆa=i ℏ[Htotal,ˆa]−κc+κ 2ˆa−√κcˆain, (4) ˙ˆm=i ℏ[Htotal,ˆm]−γ 2ˆm−p ζˆbin, (5) where Htotal=H0+Hintis the total Hamiltonian of the system with H0being the noninteracting Hamiltonian for the cavity and magnon modes. The microwave-to-optical quantum transduction effi- ciency, which is defined by the ratio between the outgo- ing and incoming photon numbers η(ω) =⟨ˆaout(ω)⟩ ⟨ˆbin(Ω)⟩2 = ⟨ˆbout(Ω)⟩ ⟨ˆain(ω)⟩2 , is obtained as [22] η(ω) =4Cκc κc+κζ γ C+ 1−4∆c κc+κ∆m γ2 + 4 ∆c κc+κ+∆m γ2,(6) where C=4g2 (κc+κ)γis the cooperativity, ∆ c=ω−ωc is the detuning from the microwave cavity frequency ωc, and ∆ m=ω−ωmis the detuning from the ferromagnetic resonance frequency ωm. Topological insulator heterostructures.— It has been shown that the magnitude of the magnon-mediated transduction efficiency (6) is essentially determined by the light-magnon coupling strength ζ, i.e., η∝ζ∝ ϕ2 F/Ns[22], where ϕFandNsare respectively the Fara- day rotation angle and total number of spins of the ferro- magnet. From this relation we see that the transduction efficiency can be improved in materials which exhibit a large Faraday rotation angle even with a small sample size. We take advantage of the fact that 3D TI thin films exhibit a topological Faraday effect arising from the sur- face anomalous Hall effect, whose rotation angle ϕF,TI is independent of the material thickness [31–34]. Here, the bandgap 2∆ of the surface Dirac bands, generated by the exchange coupling between the surface electrons and the proximitized magnetic moments having the com- ponents perpendicular to the surface, is essential for the FIG. 2. (a) Heterostructure consisting of a magnetically doped TI and a nonmagnetic insulator. (b) Heterostructure consisting of a (nonmagnetic) TI and a FI. occurrence of the surface anomalous Hall effect. In other words, ϕF,TI= 0 in the absence of proximitized mag- netic moments. The applicable range of the input light frequency Ω 0is limited by the cutoff energy εcof the surface Dirac bands (given typically by the half of the TI bulk bandgap), such that ℏΩ0< εc[31]. In particular, ϕF,TItakes a universal value in the low-frequency limit ℏΩ0≪εcand when the Fermi level µFis in the bandgap 2∆ [31–33] ϕF,TI= tan−1α≈α, (7) where α=e2/ℏc≈1/137 is the fine-structure con- stant. This universal behavior has been experimentally observed [34–37]. We propose to utilize two types of TI heterostructures [38–40], as shown in Fig. 2. One is the heterostructures consisting of magnetically doped TIs and nonmagnetic insulators [38, 40]. The other is the heterostructures consisting of (nonmagnetic) TIs and FIs [39–43]. In what follows, we focus on the latter because the surface anoma- lous Hall effect (and thereby the quantum transduction) can occur at a higher temperature ∼100 K than the for- mer [41, 42]. Light-magnon interaction in the TI heterostructure.— Suppose that a linearly polarized light is propagating along the zdirection. Microscopically, the Hamiltonian for the Faraday effect in a ferromagnet is described by the coupling between the z-component of the magnetiza- tion density and the z-component of the Stokes operator of the light [22, 30]. We extend this Hamiltonian to the heterostructure of NLTI layers and NLFI layers [see Figs. 2(b) and 3] as HF=ℏANLX i=1Zti+τ tidt G i(t)mi,z(t)Sz(t), (8) where idenotes the i-th FI layer, Gi(t) is the coupling constant, Ais the cross section of the light beam, and τ=dFI/c(with dFIthe thickness of each FI layer and cthe speed of light in the material) is the interaction time. We assume that the coupling constant describing the topological Faraday effect [Eq. (7)] takes a δ-function form, since it is a surface effect. Taking also into account3 FIG. 3. Enlarged view of a heterostructure consisting of TIs and FIs. δm⊥(t) is the small precessing component around the direction of the effective field Beff. The applied magnetic field needs to be tilted from the zaxis in order to induce a finite angle θbetween the zaxis and Beff. the conventional contribution to the Faraday effect, cG0, which takes a constant value across the sample [22], we obtain Gi(t) =cG0+1 2GTIδ(t−ti) +1 2GTIδ(t−ti−τ).(9) Thezcomponent of the magnetization density ˆ mi,z(t) is given by [44] mi,z(t) =δm⊥(t) sinθ=√Ns 2Vsinθh ˆmi(t) + ˆm† i(t)i , (10) where V(Ns) is the volume (total number of spins) of each FI layer, and ˆ mi(t) is the magnon annihilation op- erator satisfying [ ˆ mi(t),ˆm† j(t)] = δij. Note that a fi- nite angle θbetween the zaxis and the effective field Beff=−∂F/∂mi(with Fthe free energy of each FI layer), which can be realized by a tilt of the applied mag- netic field from the zaxis, is required. Here, let us define a collective magnon operator ˆ m(t)≡1√NLPNL i=1ˆmi(t) satisfying [ ˆ m(t),ˆm†(t)] = 1, in a similar way as spin en- sembles [45, 46]. Then, Eq. (8) is simplified to be HF=ℏAp NLZτ 0dt cG0+1 2GTIδ(t) +1 2GTIδ(t−τ) ×mz(t)Sz(t), (11) where mz(t) =√Ns 2Vsinθ[ ˆm(t)+ ˆm†(t)]. The zcomponent of the Stokes operator for the polarization of light Sz(t) is given by [30] Sz(t) =1 2Ah ˆb† R(t)ˆbR(t)−ˆb† L(t)ˆbL(t)i , (12) where ˆbR(t) [ˆbL(t)] is the annihilation operator of the mode of the right-circular (left-circular) polarized light propagating in the zdirection. For a strong x-polarized light we have ˆbR,L(t) =1√ 2(ˆbx±iˆby)≃1√ 2(⟨ˆbx⟩ ±iˆby)[30], where ⟨ˆbx⟩=q P0 ℏΩ0e−iΩ0twith P0(Ω0) the power (angular frequency) of the input light. Because the interaction time τ=dFI/c∼10−8m/(3× 108m/s)∼10−17s is much shorter than the time scale of the magnon dynamics (i.e., the ferromagnetic resonance frequency) 1 /ωm∼10−10s, the operators in Eq. (11) can be regarded as constant during the interaction. Then, setting ˆby≡ˆbin, we arrive at Eq. (3). The light-magnon coupling strength ζis obtained as ζ=ϕ2 FNL Nssin2θP0 ℏΩ0, (13) where the Faraday rotation angle ϕF= (G0dFI+ GTI)ns/4 with ns=Ns/Vthe spin density. As expected, this expression for ϕFproperly describes the physical sit- uation: the conventional contribution is proportional to the thickness dFI, while the topological contribution is independent of the thickness, i.e., is a surface contribu- tion. Coupling between magnon and microwave-cavity photon.— Next, we calculate the coupling strength gin the heterostructure. The total Hamiltonian describing the coupling between magnon and microwave-cavity pho- ton is given by the sum of the contribution from each FI layer, Hg=PNL i=1ℏgi(ˆa†ˆmi+ ˆm† iˆa), where gi=g0√Ns with g0the single-spin coupling strength [47, 48]. As we have done in the case of the light-magnon coupling, we may introduce a collective magnon operator ˆ m(t)≡ 1√NLPNL i=1ˆmi(t) satisfying [ ˆ m(t),ˆm†(t)] = 1. Then, we obtain Eq. (2) with the coupling strength g=g0p NLNs. (14) Magnetization dynamics in the FI layer.— So far, we have treated the electronic response of the TI surface state, i.e., the topological Faraday effect [Eq. (7)] as the magnonic response of the FI layer. Indeed, these two pic- tures are equivalent because the effective spin model is derived by integrating out the electronic degree of free- dom in the surface Dirac Hamiltonian coupled to the FI layer via the exchange interaction [49, 50]. The derived spin model takes the form of the exchange interaction and the easy-axis anisotropy when the chemical potential µF lies in the mass gap of the surface Dirac fermions, i.e., µF<|∆|, while it takes the form of the Dzyaloshinskii– Moriya interaction when µF>|∆|[49]. It has been shown that the rotation angle of the topo- logical Faraday effect decreases as the carrier density (i.e., the value of µF) becomes larger [31, 50]. We point out that such a behavior is consistent with the above- mentioned effective spin model analysis. Generally, the Faraday rotation angle is proportional to the spin density asϕF∝ns. On the other hand, as represented by the skyrmion lattice, a canted spin structure is favored due to the Dzyaloshinskii–Moriya interaction, which leads to4 FIG. 4. The input light frequency Ω 0dependence of the transduction efficiency η. We set θ= 30◦,εc= 150 meV, and ∆ = 15 meV. The input photon number is fixed here toP0 ℏΩ0= 1.5×1019Hz, which can be obtained, for example, with P0= 10 mW and Ω 0/2π= 1 THz. the decrease of the magnetization density, i.e., the spin density ns. Therefore, the decrease of the value of ϕF,TI from the case of µF<|∆|to the case of µF>|∆|can also be explained from the effective spin model analysis. Transduction efficiency.— We are now in a position to obtain the transduction efficiency ηin the TI het- erostructures. We use the typical (possible) values of YIG for the FI layer and those of microwave cavity: ns= 2.1×1019µBmm−3,g0/2π= 40 mHz, κ/2π= 1 MHz, κc/2π= 3 MHz, γ/2π= 1 MHz [22, 47]. We assume the sample area size of 0 .5 mm×0.5 mm, and treat NL anddFIas key variables. We also assume the topological transport regime of µF<|∆|. In the following, we focus on the thin film regime for the FI layer where the conven- tional contribution to the Faraday rotation angle can be neglected as ϕF,0=G0nsdFI/4≪1/137, as well as the thin film regime for the TI layer where the thickness of the TI thin film can be neglected as dTI≪λ= 2πc/Ω0. As can be seen from Eq. (6), the transduction effi- ciency ηis proportional to the light-magnon interaction strength ζ. Thus, we firstly consider the dimensionless prefactor η/(ζ/γ). The cooperativity Cis obtained as C= 1.6×10−15×NLNs. Assuming NL∼101and d∼1 nm, we find that C ∼10−1. Accordingly, it turns out that the prefactor η/(ζ/γ) takes a maximum value ≈0.5 when ∆ c= ∆ m= 0, i.e., ω=ωc=ωm[51]. From this result we see that the magnitude of ηin our case is also essentially determined by ζ. Next, we show the dependence of the transduction ef- ficiency on the input light frequency Ω 0in Fig. 4. Here, note that the input photon numberP0 ℏΩ0is fixed in Fig. 4. In other words, the Ω 0dependence in Fig. 4 originates from the topological Faraday rotation angle [31, 50]. Im- portantly, the Faraday rotation angle needs not be the universal value ϕF,TI≈1/137 (in the low frequency limit) and the transduction efficiency can be enhanced more than an order of magnitude near the sharp peak at the interband absorption threshold 2∆ [31] by tuning Ω 0. As- FIG. 5. Schematic Illustration of the microwave-to-optical quantum transduction utilizing a TI heterostructure. suming the cutoff energy εc= 150 meV and the surface mass gap ∆ = 15 meV [38–40], we find that the frequency at which ηtakes the maximum value is Ω 0/2π= 7.3 THz. Note that the maximum value of ηis mainly determined byNLanddFI, not by εcor ∆. The transduction efficiency η∼10−3−10−4obtained in Fig. 4 is greatly improved [52] compared to that of a spherical YIG of 0 .75 mm diameter, η∼10−10, obtained withP0= 15 mW and Ω 0/2π= 200 THz [22, 53]. Here, it should be noted that the Verdet constant V(=G0ns/4) in YIG in the terahertz range ( ∼1 THz) is about an order of magnitude smaller than that in the telecom frequency range ( ≈200 THz) [54]. This means that the light-magnon interaction strength ζdoes not change largely even in the terahertz range due to the relation ζ∝ϕ2 F,0/Ω0where ϕF,0=VdFI. Then, it follows that the transduction efficiency using YIG would be as small asO(10−10) even in the terahertz range. There is a fundamental difference in the sample size dependence of the transduction efficiency ηbetween pre- vious studies and our study, while the expression η∝ ϕ2 F/Nsis the same. In conventional ferromagnets such as YIG, one obtains η∝dFIsince both ϕFandNsare proportional to dFI. This indicates that the value of η becomes very small in the thin film limit. On the other hand, in TI heterostructures, one obtains η∝1/dFIsince ϕFis constant whereas Nsis proportional to dFI. This is the mechanism for the enhancement of ηin the thin film limit. Finally, we show in Fig. 5 a schematic illustration of the microwave cavity setup in our microwave-to-optical quantum transduction. Here, a linearly polarized light in the terahertz range is applied perpendicular to the heterostructure plane. A finite tilt angle between the light propagation direction and the ground state direc- tion of the ferromagnetic moments is required by apply- ing a static magnetic field. Discussion.— We briefly discuss a possible application of our finding. While optical fibers in the telecom fre-5 quency range are currently used widely, we would like to stress that optical fibers in the terahertz range are also in principle able to interconnect quantum devices. Actu- ally, terahertz optical fibers are under active research and development [55]. Thus, we expect that in the future TI heterostructures might be used as a quantum transducer for superconducting quantum computers interconnected via terahertz optical fibers. One possible way for bringing the light frequency closer to the telecom frequency range is to find TIs with a large bulk bandgap ( ≈2εc), as well as combinations of FIs and such TI surface states that allow a strong exchange interaction between them and therefore enable a large surface bandgap 2∆. If a TI heterostructure with ∆ ≈ 100 meV is discovered, then the maximum transduction efficiency will be obtained at Ω 0/2π= 2∆ ≈48 THz. In this case, infrared optical fibers can be used. Summary.— To summarize, we have shown that the transduction efficiency of microwave-to-optical quantum transduction mediated by ferromagnetic magnon can be greatly improved by utilizing the topological Faraday ef- fect in 3D TI thin films. By virtue of the topological contribution to the Faraday rotation angle which is inde- pendent of the thickness of the FI layer, the transduction efficiency is strongly enhanced in the thin film limit. Our study opens up a way for possible applications of topo- logical materials in future quantum interconnects. Acknowledgements.— We would like to thank Shintaro Sato, Ryo Murakami, Kenichi Kawaguchi, and Shoichi Shiba for their advice and support. ∗akihiko.sekine@fujitsu.com [1] N. Lauk, N. Sinclair, S. Barzanjeh, J. P. Covey, M. Saffman, M. Spiropulu, and C. Simon, Perspectives on Quantum Transduction, Quantum Sci. Technol. 5, 020501 (2020). [2] N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L. Schwefel, Coherent Conversion Between Microwave and Optical Photons–An Overview of Physical Implementa- tions, Adv. Quantum Technol. 3, 1900077 (2020). [3] X. Han, W. Fu, C.-L. Zou, L. Jiang, and H. X. Tang, Microwave-Optical Quantum Frequency Conversion, Op- tica8, 1050 (2021). [4] R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Ci- cak, R. W. Simmonds, C. A. Regal, and K. W. Lehn- ert, Bidirectional and Efficient Conversion between Mi- crowave and Optical Light, Nat. Phys. 10, 321 (2014). [5] A. Vainsencher, K. J. Satzinger, G. A. Peairs, and A. N. Cleland, Bi-Directional Conversion between Microwave and Optical Frequencies in a Piezoelectric Optomechan- ical Device, Appl. Phys. Lett. 109, (2016). [6] A. P. Higginbotham, P. S. Burns, M. D. Urmey, R. W. Peterson, N. S. Kampel, B. M. Brubaker, G. Smith, K. W. Lehnert, and C. A. Regal, Harnessing Electro-Optic Correlations in an Efficient Mechanical Converter, Nat. Phys. 14, 1038 (2018).[7] M. Forsch, R. Stockill, A. Wallucks, I. Marinkovi´ c, C. G¨ artner, R. A. Norte, F. van Otten, A. Fiore, K. Srini- vasan, and S. Gr¨ oblacher, Microwave-to-Optics Con- version Using a Mechanical Oscillator in Its Quantum Ground State, Nat. Phys. 16, 69 (2020). [8] M. Mirhosseini, A. Sipahigil, M. Kalaee, and O. Painter, Superconducting Qubit to Optical Photon Transduction, Nature 588, 599 (2020). [9] W. Jiang, C. J. Sarabalis, Y. D. Dahmani, R. N. Patel, F. M. Mayor, T. P. McKenna, R. Van Laer, and A. H. Safavi-Naeini, Efficient Bidirectional Piezo- Optomechanical Transduction between Microwave and Optical Frequency, Nat. Commun. 11, 1166 (2020). [10] X. Han, W. Fu, C. Zhong, C.-L. Zou, Y. Xu, A. Al Sayem, M. Xu, S. Wang, R. Cheng, L. Jiang, and H. X. Tang, Cavity Piezo-Mechanics for Superconducting- Nanophotonic Quantum Interface, Nat. Commun. 11, 3237 (2020). [11] G. Arnold, M. Wulf, S. Barzanjeh, E. S. Redchenko, A. Rueda, W. J. Hease, F. Hassani, and J. M. Fink, Con- verting Microwave and Telecom Photons with a Silicon Photonic Nanomechanical Interface, Nat. Commun. 11, 4460 (2020). [12] S. H¨ onl, Y. Popoff, D. Caimi, A. Beccari, T. J. Kippen- berg, and P. Seidler, Microwave-to-Optical Conversion with a Gallium Phosphide Photonic Crystal Cavity, Nat. Commun. 13, 2065 (2022). [13] S. Barzanjeh, A. Xuereb, S. Gr¨ oblacher, M. Paternostro, C. A. Regal, and E. M. Weig, Optomechanics for Quan- tum Technologies, Nat. Phys. 18, 15 (2022). [14] A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, G. Leuchs, and H. G. L. Schwefel, Efficient Microwave to Optical Photon Conversion: An Electro- Optical Realization, Optica 3, 597 (2016). [15] L. Fan, C.-L. Zou, R. Cheng, X. Guo, X. Han, Z. Gong, S. Wang, and H. X. Tang, Superconducting Cavity Electro- Optics: A Platform for Coherent Photon Conversion be- tween Superconducting and Photonic Circuits, Sci. Adv. 4, eaar4994 (2018). [16] W. Hease, A. Rueda, R. Sahu, M. Wulf, G. Arnold, H. G. L. Schwefel, and J. M. Fink, Bidirectional Electro-Optic Wavelength Conversion in the Quantum Ground State, PRX Quantum 1, 020315 (2020). [17] J. Holzgrafe, N. Sinclair, D. Zhu, A. Shams-Ansari, M. Colangelo, Y. Hu, M. Zhang, K. K. Berggren, and M. Lonˇ car, Cavity Electro-Optics in Thin-Film Lithium Niobate for Efficient Microwave-to-Optical Transduction, Optica 7, 1714 (2020). [18] T. P. McKenna, J. D. Witmer, R. N. Patel, W. Jiang, R. Van Laer, P. Arrangoiz-Arriola, E. A. Wollack, J. F. Her- rmann, and A. H. Safavi-Naeini, Cryogenic Microwave- to-Optical Conversion Using a Triply Resonant Lithium- Niobate-on-Sapphire Transducer, Optica 7, 1737 (2020). [19] Y. Xu, A. Al Sayem, L. Fan, C.-L. Zou, S. Wang, R. Cheng, W. Fu, L. Yang, M. Xu, and H. X. Tang, Bidirec- tional Interconversion of Microwave and Light with Thin- Film Lithium Niobate, Nat. Commun. 12, 4453 (2021). [20] A. Youssefi, I. Shomroni, Y. J. Joshi, N. R. Bernier, A. Lukashchuk, P. Uhrich, L. Qiu, and T. J. Kippenberg, A Cryogenic Electro-Optic Interconnect for Superconduct- ing Devices, Nat. Electron. 4, 326 (2021). [21] R. Sahu, W. Hease, A. Rueda, G. Arnold, L. Qiu, and J. M. Fink, Quantum-Enabled Operation of a Microwave-6 Optical Interface, Nat. Commun. 13, 1276 (2022). [22] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Bidirectional Conversion between Microwave and Light via Ferromagnetic Magnons, Phys. Rev. B 93, 174427 (2016). [23] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya- mazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Cavity Optomagnonics with Spin- Orbit Coupled Photons, Phys. Rev. Lett. 116, 223601 (2016). [24] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Op- tomagnonic Whispering Gallery Microresonators, Phys. Rev. Lett. 117, 123605 (2016). [25] J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, Triple-Resonant Brillouin Light Scattering in Magneto-Optical Cavities, Phys. Rev. Lett. 117, 133602 (2016). [26] N. Zhu, X. Zhang, X. Han, C.-L. Zou, C. Zhong, C.-H. Wang, L. Jiang, and H. X. Tang, Waveguide Cavity Op- tomagnonics for Microwave-to-Optics Conversion, Optica 7, 1291 (2020). [27] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent Coupling between a Ferromagnetic Magnon and a Superconducting Qubit, Science 349, 405 (2015). [28] J. J. Viennot, M. C. Dartiailh, A. Cottet, and T. Kontos, Coherent Coupling of a Single Spin to Microwave Cavity Photons, Science 349, 408 (2015). [29] Theoretically, it has been suggested that η= 0.5 is the minimum value required for the quantum state transfer. See, for example, M. M. Wolf, D. P´ erez-Garc´ ıa, and G. Giedke, Quantum Capacities of Bosonic Channels, Phys. Rev. Lett. 98, 130501 (2007). [30] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Quantum Interface between Light and Atomic Ensembles, Rev. Mod. Phys. 82, 1041 (2010). [31] W.-K. Tse and A. H. MacDonald, Giant Magneto- Optical Kerr Effect and Universal Faraday Effect in Thin-Film Topological Insulators, Phys. Rev. Lett. 105, 057401 (2010). [32] J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, Topological Quantization in Units of the Fine Structure Constant, Phys. Rev. Lett. 105, 166803 (2010). [33] W.-K. Tse and A. H. MacDonald, Magneto-Optical Fara- day and Kerr Effects in Topological Insulator Films and in Other Layered Quantized Hall Systems, Phys. Rev. B 84, 205327 (2011). [34] A. Sekine and K. Nomura, Axion Electrodynamics in Topological Materials, J. Appl. Phys. 129, 141101 (2021). [35] L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. P. Armitage, Quantized Faraday and Kerr Rotation and Axion Electrodynamics of a 3D Topological Insulator, Science 354, 1124 (2016). [36] K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki, and Y. Tokura, Terahertz Spectroscopy on Faraday and Kerr Rotations in a Quantum Anomalous Hall State, Nat. Commun. 7, 12245 (2016). [37] V. Dziom, A. Shuvaev, A. Pimenov, G. V. Astakhov, C. Ames, K. Bendias, J. B¨ ottcher, G. Tkachov, E. M. Han- kiewicz, C. Br¨ une, H. Buhmann, and L. W. Molenkamp, Observation of the Universal Magnetoelectric Effect ina 3D Topological Insulator, Nat. Commun. 8, 15197 (2017). [38] Y. Tokura, K. Yasuda, and A. Tsukazaki, Magnetic Topological Insulators, Nat. Rev. Phys. 1, 126 (2019). [39] S. Bhattacharyya, G. Akhgar, M. Gebert, J. Karel, M. T. Edmonds, and M. S. Fuhrer, Recent Progress in Prox- imity Coupling of Magnetism to Topological Insulators, Adv. Mater. 33, 2170262 (2021). [40] J. Liu and T. Hesjedal, Magnetic Topological Insulator Heterostructures: A Review, Adv. Mater. 35, 2102427 (2023). [41] C. Tang, C.-Z. Chang, G. Zhao, Y. Liu, Z. Jiang, C.- X. Liu, M. R. McCartney, D. J. Smith, T. Chen, J. S. Moodera, and J. Shi, Above 400-K Robust Perpendicular Ferromagnetic Phase in a Topological Insulator, Sci. Adv. 3, e1700307 (2017). [42] Y. T. Fanchiang, K. H. M. Chen, C. C. Tseng, C. C. Chen, C. K. Cheng, S. R. Yang, C. N. Wu, S. F. Lee, M. Hong, and J. Kwo, Strongly Exchange-Coupled and Surface-State-Modulated Magnetization Dynamics in Bi 2Se3/Yttrium Iron Garnet Heterostructures, Nat. Commun. 9, 223 (2018). [43] R. Watanabe, R. Yoshimi, M. Kawamura, M. Mogi, A. Tsukazaki, X. Z. Yu, K. Nakajima, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Quantum Anomalous Hall Effect Driven by Magnetic Proximity Coupling in All- Telluride Based Heterostructure, Appl. Phys. Lett. 115, 102403 (2019). [44] B. Zare Rameshti, S. Viola Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. W. Bauer, and Y. M. Blanter, Cavity Magnonics, Phys. Rep. 979, 1 (2022). [45] J. H. Wesenberg, A. Ardavan, G. A. D. Briggs, J. J. L. Morton, R. J. Schoelkopf, D. I. Schuster, and K. Mølmer, Quantum Computing with an Electron Spin Ensemble, Phys. Rev. Lett. 103, 070502 (2009). [46] A. Imamo˘ glu, Cavity QED Based on Collective Magnetic Dipole Coupling: Spin Ensembles as Hybrid Two-Level Systems, Phys. Rev. Lett. 102, 083602 (2009). [47] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Hybridizing Ferromagnetic Magnons and Microwave Photons in the Quantum Limit, Phys. Rev. Lett. 113, 083603 (2014). [48] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly Coupled Magnons and Cavity Microwave Photons, Phys. Rev. Lett. 113, 156401 (2014). [49] R. Wakatsuki, M. Ezawa, and N. Nagaosa, Domain Wall of a Ferromagnet on a Three-Dimensional Topological Insulator, Sci. Rep. 5, 13638 (2015). [50] See Supplemental Material for details. [51] Note that even when the cooperativity Cis as large as O(102), the maximum value of the prefactor η/(ζ/γ) is ≈0.7 at the nonzero detunings ∆ c̸= 0 and ∆ m̸= 0. [52] If a heterostructure of NL= 500 can be realized (where the total thickness of the heterostructure 2 NL(dTI+ dFI)≈10µm is still shorter than the wavelength λ= 2πc/Ω0), the maximum value of ηin Fig. 4 can be fur- ther improved to η∼10−2. [53] To make a more precise comparison, the input photon number should be the same as in Fig. 4. The value in Ref. [22] (P0 ℏΩ0= 1.1×1017Hz obtained with P0= 15 mW and Ω 0/2π= 200 THz) is two orders of magnitude smaller than that in Fig. 4. IfP0 ℏΩ0= 1.5×1019Hz is7 employed in the setup in Ref. [22], then the transduction efficiency would be η∼10−8. [54] Y. Li, T. Li, Q. Wen, F. Fan, Q. Yang, and S. Chang, Terahertz magneto-optical effect of wafer-scale La: yt-trium iron garnet single-crystal film with low loss and high permittivity, Opt. Express 28, 21062-21071 (2020). [55] M. S. Islam, C. M. B. Cordeiro, M. A. R. Franco, J. Sultana, A. L. S. Cruz, and D. Abbott, Terahertz Optical Fibers [Invited], Opt. Express 28, 16089 (2020). Supplemental Material for “Microwave-to-Optical Quantum Transduction Utilizing the Universal Faraday Effect of Topological Insulator Heterostructures” MAGNETIZATION DYNAMICS OF THE FERROMAGNETIC INSULATOR LAYER We consider the surface Hamiltonian of the topological insulator (TI) layer exchange-coupled to the ferromagnetic insulator (FI) layer, which is given by H=P kψ† kH(k)ψkwith H(k) =ℏvF(kxσx+kyσy) +Jm·σ ≡ℏvF(kxσx+kyσy) + ∆ σz+X αJαδmασα, (S1) where vFis the Fermi velocity, σiare the Pauli matrices describing electrons’ spin on the TI surface, Jis the strength of the exchange coupling, and m=m0+δm(with m0the ground state direction) is the magnetization density vector of the FI layer. The effective action for the FI layer can be derived by integrating out the electronic degree of freedom: Z=Z D[ψ,¯ψ]eiS≡eiSeff[δm]= exp" Tr lnG−1 0
−∞X n=11 nTr (−G0V)n# , (S2) where G0(k, iωn) = [iωn−H(k)−µF]−1is the unperturbed Green’s function and V=P αJαδmασαis a perturbation. At the one-loop level and in the low-frequency limit [49], the effective action for δmis written in terms of the static susceptibility χαβ(q,0) as Seff[δm] =1 2iTr (G0V)2=1 2X qX α,βJαJβδmα(q)χαβ(q,0)δmβ(−q), (S3) which takes the form of the exchange interaction and the easy-axis anisotropy when the chemical potential µFlies in the mass gap of the surface Dirac fermions, i.e., µF<|∆|, while it takes the form of the Dzyaloshinskii–Moriya interaction when µF>|∆|[49]. This indicates that the magnetization dynamics of the FI layer is modified by the presence of the TI surface state, depending on the value of the chemical potential. TOPOLOGICAL FARADAY EFFECT IN TOPOLOGICAL INSULATOR THIN FILMS General expression for the Faraday rotation angle We consider the electronic response of the TI surface state which is described by the effective Hamiltonian of the form, H(k) =ℏvF(kxσx+kyσy) + ∆ σz, (S4) where vFis the Fermi velocity, σiare the Pauli matrices describing electrons’ spin on the TI surface, and ∆ is the mass gap induced by the exchange coupling between the proximitized ferromagnetic moments. The optical conductivity of the system described by Eq. (S4) can be obtained by solving a quantum kinetic equation [31]. In the following, the conductivities are given in units of e2/ℏ=cα, where we set c= 1. The longitudinal conductivity8 σxx(Ω0) =σR xx(Ω0) +iσI xx(Ω0) and the transverse conductivity σxy(Ω0) =σR xy(Ω0) +iσI xy(Ω0) are written explicitly as [31] σR xx(Ω0) ="∆ 2Ω02 +1 16# θ[Ω0−2 max( µF,∆)] +µ2 F−∆2 4µFδ(Ω0)θ(µF−∆), (S5) σI xx(Ω0) =1 8π( 1 2" 4∆ Ω02 + 1# F(Ω0)−2∆2 Ω01 εc−1 max( µF,∆)) +1 4πµ2 F−∆2 µFΩ0θ(µF−∆), (S6) and σR xy(Ω0) =−∆ 4πΩ0F(Ω0), (S7) σI xy(Ω0) =∆ 4Ω0θ[Ω0−2 max( µF,∆)], (S8) where F(Ω0) = lnΩ0+ 2εc Ω0−2εc−lnΩ0+ 2 max( µF,∆) Ω0−2 max( µF,∆), (S9) εcis the cutoff energy of the surface Dirac bands ±q ℏ2v2 F(k2x+k2y) + ∆2which is given typically by the half of the TI bulk bandgap, and we have considered the case of µF>0 and ∆ >0 without loss of generality. In what follows, we derive a general expression for the Faraday rotation angle at a single interface. Suppose that an electromagnetic wave is propagating along the zdirection from medium ito medium jwith dielectric constant ϵiandϵjand magnetic permeability µiandµj, respectively. The boundary conditions for the electric and magnetic fields are [33] Ei=Ejand−iτy(Bj−Bi) =µ0Js, (S10) where the first equation follows from the continuity of the electric field by Faraday’s law and the second equation follows from Amp` ere’s law integrated over the zdirectionR dz∇ ×B=R dz µ 0j.τyis the y-component of the Pauli matrices. Here, we have assumed that the bulk of the media is insulating, i.e., the electric current Js=σEiflows only at the boundary. Note also that Ez=Bz= 0 because we are considering an transverse wave. LetE0,Er, andEtbe incident, reflected, and transmitted electric fields, respectively. Then, the electric fields in media iandjare given by Ei=eikizEti+e−ikizEri, Ej=eikjzEtj. (S11) At the boundary, the incoming filedsEtiErjTand the outgoing fieldsEriEtjTare related by the scattering matrix S=r t′ t r′ with r=rxxrxy −rxyryy andt=h txxtxy −txytyyi (and similarly for r′andt′) [33]. We therefore have  Eri Etj = r t′ t r′E0 0 =rE0 tE0 . (S12) The explicit forms of randtcan be obtained by solving the boundary condition equations (S10) with the use of Faraday’s law in media iandj,∇ ×E=−(1/c)∂B/∂t. The Faraday rotation angle is calculated from the arguments of the transmitted electric field (in medium j): ϕF(Ω0) = arg(Et +)−arg(Et −) /2, (S13) where Et ±=Et x±iEt yare the left-handed (+) and right-handed ( −) circularly polarized components of the transmitted electric field Et. The explicit forms are given by arg(Et ±) = tan−1" 4πασI xx(Ω0)±4πασR xy(Ω0)p ϵi/µi+p ϵj/µj+ 4πασRxx(Ω0)−4πασIxy(Ω0)# . (S14)9 FIG. S1. (a) The input light frequency Ω 0dependence and (b) the chemical potential µFdependence of the Faraday rotation angle 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, respectively. Faraday rotation angle in the thin film regime Here, we calculate the Faraday rotation angle in a TI thin film. In the thin film regime where dTI≪λ= 2πc/Ω0, the thickness of the TI thin film can be neglected. In other words, we may regard the system as a 2D system consisting only of the top and bottom surfaces. Then, we can simply setp ϵi/µi=p ϵj/µj→1, as well as σxx(Ω0)→2σxx(Ω0) andσxy(Ω0)→2σxy(Ω0), which accounts for the contributions from both the top and bottom surfaces. Figure S1(a) shows the input light frequency Ω 0dependence of the Faraday rotation angle ϕF, which reproduces the result in Ref. [31]. The value of ϕFin the low-frequency limit Ω 0/εc≪1 is almost universal such that ϕF= tan−1[α(1−∆/εc)]≃tan−1α. Figure S1(b) shows the chemical potential µFdependence of ϕF. The value of ϕFis constant as long as the chemical potential is in the surface gap (i.e., µ/∆<1), reflecting the constant anomalous Hall conductivity σR xy= (α/4π)(1−∆/εc) in the topological transport regime. On the other hand, we can see that the value of ϕFbegins to decrease as the carrier density, i,e., the value of µ/∆(>1), becomes larger.

2023-11-13

The quantum transduction between microwave and optical photons is essential for realizing scalable quantum computers with superconducting qubits. Due to the large frequency difference between microwave and optical ranges, the transduction needs to be done via intermediate bosonic modes or nonlinear processes. So far, the transduction efficiency $\eta$ via the magneto-optic Faraday effect (i.e., the light-magnon interaction) in the ferromagnet YIG has been demonstrated to be small as $\eta\sim 10^{-8} \mathrm{-} 10^{-15}$ due to the sample size limitation inside the cavity. Here, we take advantage of the fact that three-dimensional topological insulator thin films exhibit a topological Faraday effect that is independent of the sample thickness. This leads to a large Faraday rotation angle and therefore enhanced light-magnon interaction in the thin film limit. We show theoretically that the transduction efficiency can be greatly improved to $\eta\sim10^{-4}$ by utilizing the heterostructures consisting of topological insulator thin films such as Bi$_2$Se$_3$ and ferromagnetic insulator thin films such as YIG.

Microwave-to-Optical Quantum Transduction Utilizing the Topological Faraday Effect of Topological Insulator Heterostructures

2311.07293v1

Positive-Negative Birefringence in Multiferroic Layered Metasurfaces R. Khomeriki Institut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, D-06099 Halle/Saale, Germany Physics Department, Tbilisi State University, 3 Chavchavadze, 0128 Tbilisi, Georgia L. Chotorlishvili Institut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, D-06099 Halle/Saale, Germany I. Tralle Faculty of Mathematis and Natural Sciences, University of Rzeszow, Pigonia 1, 35-310 Rzeszow, Poland J. Berakdar Institut f ur Physik, Martin-Luther-Universit at, Halle-Wittenberg, 06099 Halle/Saale, Germany We uncover and identify the regime for a magnetically and ferroelectrically controllable nega- tive refraction of light traversing multiferroic, oxide-based metastructure consisting of alternating nanoscopic ferroelectric (SrTiO 2) and ferromagnetic (Y 3Fe2(FeO 4)3, YIG) layers. We perform an- alytical and numerical simulations based on discretized, coupled equations for the self-consistent Maxwell/ferroelectric/ferromagnetic dynamics and obtain a biquadratic relation for the refractive index. Various scenarios of ordinary and negative refraction in dierent frequency ranges are ana- lyzed and quantied by simple analytical formula that are conrmed by full- edge numerical simula- tions. Electromagnetic-waves injected at the edges of the sample are propagated exactly numerically. We discovered that for particular GHz frequencies, waves with dierent polarizations are charac- terized by dierent signs of the refractive index giving rise to novel types of phenomena such as a positive-negative birefringence eect, and magnetically controlled light trapping and accelerations. INTRODUCTION According to Veselago's predictions [1, 2] that were later conrmed experimentally [3{5], negative refraction phenomena occur in metamaterials where both the elec- tric permittivity and the magnetic permeability are negative. The Poynting and the wave vectors are then antiparallel resulting in a phase decrease during the prop- agation process. A clear example of this phenomena is based on metallic heterostructures [6] which are inher- ently absorptive at relevant frequencies [7]. To avoid losses, insulating multiferroics may oer a solution [8, 9], but also new possibilities for external control and ex- ploitations of functional materials. Multiferroics are one- phase or composite, synthesized structures exhibiting si- multaneously multiple orderings such as ferromagnetic, ferro and/or piezoelectric order and respond thus to a multitude of conjugate elds. This class of materials plays a key role for addressing fundamental issues regard- ing the interplay between electronic correlation, symme- try, magnetism, and polarization. Potential applications are diverse, ranging from sensorics and magnetoelectric spintronics to environmentally friendly devices with ul- tralow energy consumption [10, 11]. Here, we demonstrate how multiferroics properties lead to exotic electromagnetic wave propagation features. In particular, we demonstrate the existence of a negative refraction in ferroelectric (FE)/ferromagnetic(FM) mul- tilayers. The large (FE and FM) resonance frequency mismatch between ferroelectric and ferromagnetic mediais usually an obstacle. Indeed, the paradigm ferroelectric BaTiO 3has a resonance frequency in THz range [12{15], while the insulating ferromagnet, rhodium-substituted "- RhxFe2xO3with largest known coercivity has charac- teristic frequencies in 200GHz range [16]. On the other hand, ferroelectric SrTiO 2(STO) [17] does possess over- lapping resonances with the well-investigated insulating ferromagnet Y 3Fe2(FeO 4)3(also called YIG) [18]. A number of further insulating FE, and FM insulating ox- ides are also possible, for concreteness we present and dis- cuss here the results for STO/YIG/STO/


structures. In the pilot numerical simulations below we choose the FE layer to be 10 nmand the FM layer to be 1 m. As we will be working in reduced units, meaning the ef- fects are scalable; for materials composites responding at higher frequencies, smaller structures are appropriate. In earlier studies the negative refraction eect was ob- served in specic systems embracing two dierent sub- parts with  <0, > 0, and >0, < 0 respectively [8, 9]. We note that in the same medium propagating pos- itively refracting mode can exist as well [9]. In the present paper, we study the FE/FM composite (cf. the schemat- ics setup in Fig. 1a) with an eye to nd the frequency domain where both negatively and positively refracting waves with dierent polarizations simultaneously coex- ist for the same excitation frequency. By this we predict that in the suggested system unpolarized electromagnetic wave undergoes both positive and negative refractions, manifesting a novel positive-negative birefringence eect.arXiv:1710.05995v1 [physics.optics] 16 Oct 20172 𝑷𝟐 𝑷𝟒 𝑷𝟔 𝑺𝟓 x y z 𝑺𝟑 𝑺𝟏 (a) (c) (b) FIG. 1: (a) Schematics for the pho- tonic/ferroelectric/ferromagnetic heterostructure. The electric polarization Pjin the layer jis aligned along xaxis. The incident light wave propagating along zaxis.EandH denote the electric and magnetic eld components. The mag- netization Siin the layer ipoints also along the zaxis. (b) and (c) graphs show real (dashed blue curve) and imaginary (dashed-dotted green curve) parts of the refraction index nversus the mode frequency according to the biquadratic equation (5). Red solid curve is the zcomponent of the time averaged dimensionless Poynting vector hWzi=P0S0as calculated by means of the formula (6). Graphs (b) and (c) correspond to the negatively and positively refracting waves, respectively for the same excitation frequency 30GHz indicated by vertical dotted lines in the insets. Insets display enlarged views of the frequencies in interest. The FE/FM specic materials are detailed in the text. MODEL AND PARAMETERS Let both the thickness of the YIG and the STO layers be equala, for clarity. We denote the positions of STO and YIG layers by even (2 m) and odd (2 m1) inte- ger numbers respectively, where m= 1;2;:::; N= 2. For the description of light-induced FE/FM dynamics and its backaction on the light propagation properties we will utilized a discretized Maxwell materials equation self- consistently coupled to the FE dynamics as described by the Ginzburg-Landau-Devonshire (GLD) method, and a classical Heisenberg model for the magnetization preces- sion. This low-energy eective treatment is well-justied due to the choice of the appropriate frequency and the (low to moderate) intensity of the incident light wave. The discretized FE polarization P2m(initially along x axis) and FM magnetization ~S2m1(initially along zaxis) (cf. Fig. 1a) are thus described by the energy func- tional H=HP+HS;HP=N=2X m=10 2dP2m dt2 1 2(P2m)2+2 4(P2m)4P2mEx 2m ; (1) HS=N=2X m=1h H0Sz+D Sz 2m1
2+~H2m1~S2m1i ; where0stands for the kinetic constant, 1and2are potential coecients of the FE part, and Dis a uniaxial anisotropy constant in FM layers. H0is a static external magnetic eld applied along the zaxis which will prove useful for tuning the functional properties of the setup, e.g. for switching between the ordinary and the negative refraction regimes (see below). We assume that the multilayer structure is rst driven to saturation by appropriately strong elds. The rem- nant FE and FM polarizations are then denoted by P0 andS0. Let us introduce dimensionless photonic eld ~h~H=S 0, and~E~E=P 0and denote small deviations around the ordering directions by p2mP0P2m, and ~ s2m1~S0~S2m1. The thickness of the layers (along the propagation direction) should be small enough such that no domains are formed along the zaxis (the gen- eral case of large ais captured also with this model by adding pinning sites and appropriate energy contribu- tions to eq.(1), but this is expected to be subsidiary to the eects discussed here). Our propagating electromag- netic wave cannot create domains since the wavelength far exceeds a. The discretized form of Maxwell's equa- tions for the electromagnetic eld vectors read 1 cd dt hx 2m+1+ 4sx 2m+1
=1 2a Ey 2m+2Ey 2m
1 cd dt hy 2m+1+ 4sy 2m+1
=1 2a Ex 2m+2Ex 2m
1 cd dt(Ex 2m+ 4p2m) =1 2a hy 2m+1hy 2m1
1 cd dtEy 2m=1 2a hx 2m+1hx 2m1
:(2) These equations need to be propagated simultaneously with the dynamics governed by eqs.(1). Nonlinear cor- rections are irrelevant when the relative values of the po- larization and the magnetization of eigenmodes are much smaller than unity. Thus, the validity of the linear ap- proximation can be checked directly by monitoring the relative eigenmodes. After these simplications, from (1) we infer the linearized, coupled photonic-matter evolu-3 tion equations d2p2m dt2=!2 Pp2m+!2 P 4Ex 2m (3) @sx 2m+1 @t=!0sy 2m+1+!M 4hy 2m+1 @sy 2m+1 @t=!0sx 2m+1!M 4hx 2m+1: The FE resonance frequency !Pof STO is around !P= 10 GHz. The GDL potential curvature at equi- librium= 21is related to the electric susceptibil- ity at the zero mode frequency as (0) = 1=4. For the large permittivity observed in Ref. [17] the potential curvature is of the order of 104. For YIG [18] the Larmor frequency is !0= H0+2D S 0(in zero external eld!0= 20GHz) and !M= 4 S 0= 30GHz ( is the gyromagnetic ratio for electrons). For a linearly polarized electromagnetic waves in the FE/FM multilayers the refractive index follows from the matter equations (3). The expressions for the permittiv- ity= 1+!2 P= !2 P!2and the permeability = 1+!0!M !2 0!2for the linearly polarized wave components Ex,hyindicate that n2 1== 1 +!2 P= !2 P!2 1 +!0!M !2 0!2 :(4) In spite of the fact that a linearly polarized wave is not an eigenmode of the FE-FM system, the eigenmode (4) has a certain merit. The asymptotic solution corresponds to the large susceptibility limit (see below). In order to precisely calculate the refractive index, one needs to solve a complete set of coupled Maxwell (2) and matter (3) equations. Thus looking for a general solution we proceed further and adopt an ansatz presenting eld and matter wave components as follows: Ex m=Exei(!tkam). Herea is a FE/FM lattice constant !andkare the eigenmode frequency and wavenumber, respectively. Analyzing the linear algebraic equations (see Supporting Information for the details) we arrive at the following biquadratic equation for the refractive index nck=! . (!2 0!2)(!2 P!2)n4 (5) (!2 0+!M!0!2) 2(!2 P!2) +!2 P n2+ + (!2 P!2) +!2 P (!0+!M)2!2 = 0; and a set of amplitudes for the eld and matter wave com- ponents. Then it is straightforward to calculate the time averaged dimensionless Poynting vector W=hWzi=P0S0 as W=hExHyEyHxi=P0S0= Exh yEyh x +c:c: =n( (!2!2 P) (!2!2 0)(n21) +!0!M2 (!2!2 P!2 P=)!2!2 M+ 1) :(6) FIG. 2: Graphs (a) and (c) represent negative refraction in- jecting the signal with a polarization Ex;y= 0:19 + 1:81i. while Graphs (b) and (d) display positive refraction of the wave with polarization Ex;y= 0:015+1:985icorresponding to a positive refraction. In graphs (c) and (d) blue (solid), green (dashed-dotted) and red (dashed) curves correspond to time oscillations of the FE/FM layers with increasing site numbers. It is straightforward to derive from (5) the two obvious limiting cases of pure ferromagnet or ferroelectrics. For zero magnetization, we set !M= 0 and from (5) obtain two linearly polarized eigenmodes with the refractive in- dexesn1= 1 andn2=r 1 +!2 P= !2 P!2corresponding to the polarizations along yandxaxis, respectively. In the case of a vanishing polarization in the system we set !1 and nd two circularly polarized eigenmodes character- ized by dierent refractive indexes n1=q 1 +!M !0!and n2=q 1 +!M !0+!. Obviously all of these modes in both limiting cases are characterized by ordinary refraction properties for low excitation frequencies !, while for large !one of re- fractive indexes becomes purely imaginary corresponding to the non-propagating regime, while other remains real and corresponds to an ordinary refraction. It should be emphasized that in the above mentioned cases with the linear polarized wave at the input always obtain linearly polarized wave at the output: In purely ferromagnetic case we nd a Faraday rotation and for pure ferroelectrics just a phase shift (that however can be retrieved by in- terference measurements). The situation changes drastically in the presence of both ferroelectric and ferromagnetic layers. An injected linear polarized electromagnetic wave emerges after traversing the FE/FM heterostructure with an elliptical polariza- tion, as detailed below. We can further simplify the analytic solutions (5) and4 (6) in the general case (both ferroelectric and ferromag- netic layers are present in the system) assuming !0 which is just the case of a large susceptibility in FE [17]. Then one infers two roots of (5). The rst one matches exactly the asymptotic result (4), and the second one reads n2 2=(!0+!M)2!2 !0(!0+!M)!2: (7) Now it is evident that in the same limit of !0, for the rst mode (4) the Poynting vector has the val- uesW1n, meaning that a negative refraction takes place. For the second mode (7) the Poynting vector is W2ncorresponding to a positive refraction case. To identify the positive-negative birefringence regime both propagating modes should be present, i.e. n2 1>0 and n2 2>0. From these relations we deduce the restrictions on the mode frequency Maxf!P;!0g<!<p !0(!0+!M): (8) These relations set a reliable range in which positive- negative birefringence eect to be found. RESULTS The exact solutions of (5) are illustrated by graphs (b) and (c) in Fig. 1. The frequency dependencies of the two roots with positive real parts n0>0 of the refractive in- dexn=n0+n00are displayed as to pinpoint the wavevec- tor direction and compare it with the sign of the Poynting vector as calculated according to (6). Apparently in Fig. 1 (b), the sign of the Poynting vector is negative in the frequency range close to != 30GHz, i.e. the Poynting vector is antiparallel to the wave vector direction and therefore we are in the negative refraction regime, while in graph (c) another root with a positive n0is presented. The Poynting vector in this case is positive, that means we have a second mode with an ordinary (positive) re- fraction for the same wave frequency != 30GHz. The clear evidence of the coexistence of a positive and nega- tive refractions for an unpolarized electromagnetic wave is fully compatible with the approximate conditions (8). To substantiate the analytical estimation we per- formed full numerical experiment considering two wave modes with dierent polarizations being injected into the FE/FM composite metastructure. An oscillating electric eld with the characteristic frequency != 30GHz oper- ating on the left edge of the dipolar/spin chain is due to the action of the light source on the sample. The wave propagation proceeds self-consistently as governed by the set of equations (2) and (3). The results shown in Fig. 2 conrm evidently the existence of the birefringence eect. In numerical simulations we act on the left end of the system with an electric eld having the polarizationEx;y= 0:19 + 1:81iand corresponding to the negative refraction regime. Obviously a Gaussian pulse prop- agates with a positive group velocity see Fig. 2 (a), while according to Fig. 2 (c) the phase velocity is neg- ative (blue, green and red curves correspond to time os- cillations of subsequent sites with increasing site num- ber). Fig. 2 (b) displays the wave propagation process (again with positive group velocity) with the polarization Ex;y= 0:015 + 1:985i. However, the corresponding phase velocity is now positive and ordinary (positive) refraction scenario holds, see Fig. 2 (d). In the above considerations the damping eects were neglected which allows obtaining analytical expressions. In practice, ferromagnetic layers can be engineered such that the damping [20] is very small (in s range) and hence can neglected on our relevant ns time scale. In fer- roelectrics damping is much stronger and its eect should be considered. FE damping impacts the rst mode only (4). In the limit !0 (i.e. for large susceptibility) the modied expression of the refraction index of the rst mode reads n2 1= 1 +!2 P= !2 P!2+i! 1 +!0!M !2 0!2 ;(9) (here is the damping parameter) while the second mode (7) is left unchanged. The experimentally observed peaks in the susceptibility correspond to the frequency !p= 10GHz see Ref [17]. In our case != 30GHz, !P!. Hence, we conclude that also FE damping has no signicant eect on the rst mode (9) clarifying so the role of losses for the predicted phenomena. Until now we considered FM and FE layers of equal thickness. Absorption eects are rather sensitive to the thickness of FE layers and much less sensitive to the thickness of FM layers. A recipe for minimizing losses is thus straightfor- ward by fabricating thinner FE layers. In the numerical simulations the thickness enters through the values of  in (3). For example, taking = 102instead of= 104 in (3), one can make the average polarization 100 times smaller (which aects the electromagnetic waves). If we take the FE layer 100 times thinner than the FM layer, then the wave spends much less time in FE layer and the absorption eects are reduced drastically. On the other hand, tuning leaves the main qualitative characteristics of the considered eect unchanged. The frequency range for which the negative refraction takes place depends on the external magnetic eld (see Fig. 3). The external magnetic eld induces a shift in the Larmor frequency !0=!a+ H0, while the anisotropy frequency is xed to !a= 20GHz. By means of an exter- nal magnetic eld the frequency !0is tunable within an interval 10GHz to 50GHz. The results for the negative refraction are shown in Fig. 3 (left graph). These results conrm that the frequency range for which a negative refraction takes place can be controlled by an external magnetic eld. In the experiment one may switch so neg-5 0 50 100−15−10−505 ω [GHz]Wz / P0S0 −0.100.1−0.0200.020.040.060.08 H0 [ T ] vg [units of c] ω0=50GHzω0=40GHzω0=20GHzω0=10GHz bomgr ω0=30GHz FIG. 3: Left graph: The Poynting vector magnitude ver- sus the mode frequency for dierent external magnetic elds: !0= 10;20;30;40;50 GHz correspond to blue (b), orange (o), magenta (m), green (g), and red (r) lines. Solid curves in- dicate the modes with a negative refraction regime and dashed lines describe the positively refractive ones. The frequency ranges with a vanishing Poynting vector correspond to non- propagating evanescent modes. Right graph: The mode group velocity dependence on the static magnetic eld for a xed ex- citation frequency != 24GHz. Blue (dashed) and red (solid) curves describe the group velocities (in units of the speed of light) of positively and negatively refracting modes, respec- tively. ative refraction media to ordinary media and vice versa simply by turning the magnetic eld o and on. In the right graph of Fig. 3 for negatively and pos- itively refracting waves we plot the dependence of the group velocity on a static magnetic eld. It is evident that by a ramped static magnetic eld we can achieve ac- celeration or even trapping of the electromagnetic waves. Finally we note that eects related to photonic-magneto- elastic and/or piezoelectric couplings are straightfor- wardly incorporated in the above formulism by including the respective energy term in eqs.(1), along the lines as done in Refs.[21, 22]. Furthermore, to access a higher frequency regime (cf. eq.7)) it would be advantages to utilize FE/aniferromagnetic/FE/.... layer structures. For example, recently an antiferromagnetic resonance fre- quency of 22 THz were observed for KNiF 3[23]. SUMMARY Summarizing, in the present work we illustrated theoretically an insulating multiferroic metamate- rial featuring simultaneous positive and negative re- fraction (positive-negative birefringes) and provided concrete predictions for a realization on the ba- sis ofSrTiO 2=YIG multilayers. In addition to full- edge numerical simulations for the coupled Maxwell/ferroelectric/ferromagnetic dynamics, we wereable to derive credible analytical solutions and concrete frequency regimes in which the predicted eects are to be expected. The theory and predictions are of a general nature and are applicable to a wide range of material classes. The ndings point to new exciting applications of insulating nanostructured oxides in photonics. This research is funded by the German Science founda- tion under SFB 762 "Functionality of Oxide Interfaces". We thank J. Schilling for comments on the experimen- tal aspects. R. Kh. acknowledges nancial support from Georgian SRNSF (grant No FR/25/6-100/14) and travel grants from Georgian SRNSF and CNR, Italy (grant No 04/24) and CNRS, France (grant No 04/01). APPENDIX To obtain wave solutions of the set Eqs.(2,3) in the main text we express the eld and the polariza- tion/magnetization components as follows: Ex m=Exei(!tkam)+c:c;Ey m=Eyei(!tkam)+c:c; hx m=hxei(!tkam)+c:c; hy m=hyei(!tkam)+c:c; sx m=Sxei(!tkam)+c:c; sy m=Syei(!tkam)+c:c; pm=Pei(!tkam)+c:c: (10) Substituting this into Eqs.(2,3) of the main text and con- sidering the large wavelength limit k!0 we nd i!(hx+ 4Sx) =ikEy;i!(hy+ 4Sy) =ikEx i!(Ex+ 4P) =ikhy; i!Ey=ikhx i!Sx+!0Sy!M 4hy= 0i!Sy!0Sx+!M 4hx= 0  !2!2 P
P+!2 P 4Ex= 0: (11) After some algebra one can reduce this equation to the three coupled equations i!(n21)hx+ !0(n21)!M hy!0nP= 0;  !0(n21)!M hx+i!(n21)hyi!nP= 0; !2 Pnhy+ (!2!2 P)!2 P P= 0; (12) wherenck=! is a refractive index. Thus nally we arrive at the matrix 0 @i!(n21)!0(n21)!M!0n !M!0(n21)i!(n21)i!n 0 !2 Pn  (!2!2 P)!2 P1 A the determinant of which should be equal to zero which leads to the relation  (!2!2 P)!2 Pn !0(n21)!M2!2(n21)2o +!2 Pn2 (!2 0!2)(n21)!M!0 = 06 This could be straightforwardly simplied to a bi- quadratic equation for the refractive index n (!2 0!2)(!2 P!2)n4 (!2 0+!M!0!2) 2(!2 P!2) +!2 P n2+ + (!2 P!2) +!2 P (!0+!M)2!2 = 0 which is exactly the same relation (5) as in the main text. The above matrix together with the relations (11) of this supplementary materials gives the eigenmodes of the system and denes the time averaged Poynting vector value as hWzi=hExHyEyHxi=P0S0[Ex(hy)Ey(hx)]+c:c: Dening now the dimensionless version of the Poynting vector asW=hWzi=P0S0we obtain eq.(6) of the main text W=n( (!2!2 P) (!2!2 0)(n21) +!0!M2 (!2!2 P!2 P=)!2!2 M+ 1) which in the limit !0 gives the following expressions for the negatively refracting n=n1and positively re- fractingn=n2modes W1=n(!2 0+!0!M!2)2!2 P (!2!2 P)!2!2 M; W2=n 1(!2!2 P)!2!2 M (!2 0+!0!M!2)2!2 P and as it could be easily seen W1nfor! >!Pand W2nfor!0. Electronic address: Jamal.Berakdar@physik.uni-halle.de [1] V.G. Veselago, Properties of materials having simulta- neously negative values of the dielectric and magnetic susceptibilities Sov. Phys. Solid State 8, 2854 (1967). [2] V.G. Veselago, The electrodynamics of substances with simultaneously negative values of andSov. Phys. Usp. 10, 509 (1968). [3] J.B. Pendry, A.J. Holden, D.J. Robbins and W. J. Stew- art, Magnetism from conductors and enhanced nonlin- ear phenomena IEEE Trans. Microwave Theory Tech. 47 2075 (1999). [4] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, Composite Medium with Simultaneously Negative Permeability and Permittivity Phys. Rev. Lett. 84, 4184 (2000). [5] R.A. Shelby, D.R. Smith and S. Schultz, Experimental Verication of a Negative Index of Refraction Science 292, 7 (2001). [6] J. B. Pendry, Negative refraction Contemp. Phys. 45, 191 (2004). [7] M. I. Stockman, Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality Phys. Rev. Lett. 98, 177404 (2007).[8] D.W. Ward, K.A. Nelson, K.J. Webb, On the physical origins of the negative index of refraction New J. Phys. 7, 213 (2005). [9] D.R. Fredkin, A. Ron, Eectively left-handed (negative index) composite material App. Phys. Lett. 81, 1753 (2002). [10] M. Bibes and A. Barthe le  my, Multiferroics: Towards a magnetoelectric memory Nat. Mater. 7, 425 (2008). [11] D. Pantel, S. Goetze, D. Hesse, and M. Alexe, Reversible electrical switching of spin polarization in multiferroic tunnel junctions Nat. Mater. 11, 289 (2012). [12]Physics of Ferroelectrics , edited by K. Rabe, Ch. H. Ahn, and J.-M. Triscone (Springer, Berlin, 2007). [13] A. Picinin, M.H. Lente, J.A. Eiras, and J.P. Rino, The- oretical and experimental investigations of polarization switching in ferroelectric materials Phys. Rev. B 69, 064117 (2004). [14] J.J. Wang, P.P. Wu, X.Q. Ma, and L.Q. Chen, Temperature-pressure phase diagram and ferroelectric properties of BaTiO3 single crystal based on a modied Landau potential J. Appl. Phys. 108, 114105 (2010). [15] O.E. Fesenko and V.S. Popov, Phase T,E-diagram of bar- ium titanate Ferroelectrics 37, 729 (1981). [16] A. Namai, A. Namai, M. Yoshikiyo, K. Yamada, S. Saku- rai, T. Goto, T. Yoshida, T. Miyazaki, M. Nakajima, T. Suemoto, H. Tokoro, and S.-i. Ohkoshi Hard mag- netic ferrite with a gigantic coercivity and high frequency millimetre wave rotation Nat. Communications, 3, 1035 (2012). [17] J. H. Haeni, P. Irvin, W. Chang, R. Uecker, P. Reiche, Y. L. Li, S. Choudhury, W. Tian, M. E. Hawley, B. Craigo, A. K. Tagantsev, X. Q. Pan, S. K. Streier, L. Q. Chen, S. W. Kirchoefer, J. Levy, and D. G. Schlom et al. , Room- temperature ferroelectricity in strained SrTiO 3Nature, 430, 758 (2004). [18] J. Xiao, G.E.W. Bauer, K.-C. Uchida, E. Saitoh, and S. Maekawa, Theory of magnon-driven spin Seebeck eect Phys. Rev. B, 81, 214418 (2010). [19] C. Xiong, W. H. P. Pernice, J. H. Ngai, J. W. Reiner, D. Kumah, F. J. Walker, C. H. Ahn, and H. X. Tang, Active Silicon Integrated Nanophotonics: Ferroelectric BaTiO3 Devices Nano Lett. 14, 1419 (2014). [20] C. Hauser, T. Richter, N. Homonnay, Ch. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. G. Ebbing- haus, and G. Schmidt Yttrium Iron Garnet Thin Films with Very Low Damping Obtained by Recrystallization of Amorphous Material Sci. Rep., 620827 (2016). [21] C.-L. Jia , N. Zhang, A. Sukhov, and J. Berakdar, Ultra- fast transient dynamics in composite multiferroics New J. Phys. 18, 023002 (2016) [22] C.-L. Jia, A. Sukhov, P. P. Horley, and J. Berakdar, Piezoelectric control of the magnetic anisotropy via inter- face strain coupling in a composite multiferroic structure Europhys. Lett. 99, 17004 (2012) [23] D. Bossini, S. Dal Conte, Y. Hashimoto, A. Secchi, R. V. Pisarev, Th. Rasing, G. Cerullo, and A. V. Kimel Macrospin dynamics in antiferromagnets triggered by sub-20 femtosecond injection of nanomagnons. Nat. Com- mun. 7:10645 doi: 10.1038/ncomms10645 (2016).

2017-10-16

We uncover and identify the regime for a magnetically and ferroelectrically controllable negative refraction of light traversing multiferroic, oxide-based metastructure consisting of alternating nanoscopic ferroelectric (SrTiO$_2$) and ferromagnetic (Y$_3$Fe$_2$(FeO$_4$)$_3$, YIG) layers. We perform analytical and numerical simulations based on discretized, coupled equations for the self-consistent Maxwell/ferroelectric/ferromagnetic dynamics and obtain a biquadratic relation for the refractive index. Various scenarios of ordinary and negative refraction in different frequency ranges are analyzed and quantified by simple analytical formula that are confirmed by full-fledge numerical simulations. Electromagnetic-waves injected at the edges of the sample are propagated exactly numerically. We discovered that for particular GHz frequencies, waves with different polarizations are characterized by different signs of the refractive index giving rise to novel types of phenomena such as a positive-negative birefringence effect, and magnetically controlled light trapping and accelerations.

Positive-Negative Birefringence in Multiferroic Layered Metasurfaces

1710.05995v1

PSWS at millikelvin temperatures Propagating spin-wave spectroscopy in nanometer-thick YIG films at millikelvin temperatures Sebastian Knauer,1Kristýna Davídková,2David Schmoll,1, 3Rostyslav O. Serha,1, 3Andrey Voronov,1, 3Qi Wang,1Roman Verba,4Oleksandr V. Dobrovolskiy,1Morris Lindner,5Timmy Reimann,5Carsten Dubs,5Michal Urbánek,2and Andrii V. Chumak1 1)University of Vienna, Faculty of Physics, A-1090 Vienna, Austria 2)CEITEC BUT, Brno University of Technology, 612 00 Brno, Czech Republic 3)University of Vienna, Vienna Doctoral School in Physics, A-1090 Vienna, Austria 4)Institute of Magnetism, Kyiv 03142, Ukraine 5)INNOVENT e.V. Technologieentwicklung, Prüssingstraße 27B, Jena, Germany (*Electronic mail: knauer.seb@gmail.com) (Dated: 24 January 2023) Performing propagating spin-wave spectroscopy of thin films at millikelvin temperatures is the next step towards the realisation of large-scale integrated magnonic circuits for quantum applications. Here we demonstrate spin-wave prop- agation in a 100nm-thick yttrium-iron-garnet film at the temperatures down to 45mK, using stripline nanoantennas deposited on YIG surface for the electrical excitation and detection. The clear transmission characteristics over the distance of 10 mm are measured and the subtracted spin-wave group velocity and the YIG saturation magnetisation agree well with the theoretical values. We show that the gadolinium-gallium-garnet substrate influences the spin-wave propagation characteristics only for the applied magnetic fields beyond 75mT, originating from a GGG magnetisation up to 47kA =m at 45mK. Our results show that the developed fabrication and measurement methodologies enable the realisation of integrated magnonic quantum nanotechnologies at millikelvin temperatures. I. INTRODUCTION Yttrium-Iron-Garnet (YIG, Y 3Fe5O12) is the ideal choice of material to build and develop classical and novel quan- tum technologies1,2by coupling spin waves, and their single quanta magnons, to phonons3, fluxons4, or to microwave and optical photons5–8. These technologies may be realised by the coupling to bulk or spherical YIG samples (e.g. Ref.9,10), or by the fabrication of integrated structures in thin YIG films11,12. Such nanometer-thick films can be grown using liquid phase epitaxy (LPE)13,14, exhibiting long spin wave propagation lengths, narrow linewidths and low damping con- stants13–16. Significant progress was made in realising YIG nano-waveguides with lateral dimensions down to 50nm11, in understanding the spin-wave properties in these waveg- uides17, and in using them for room-temperature data process- ing12. To create, propagate and read out spin waves at single magnon level, millikelvin temperatures are required to sup- press thermal magnons according to the Bose-Einstein statis- tics2. The established technique of ferromagnetic resonance (FMR) spectroscopy was used to characterise YIG films of micrometer18and nanometer thicknesses19–21at kelvin tem- peratures. At millikelvin temperatures FMR measurements were performed on micrometer22and nanometer-thick19–21 YIG films. Another method, propagating spin-wave spec- troscopy (PSWS), is often used to characterise magnon trans- port between spatially-separated sources and detectors. This technique was successfully used in thin films at room23,24and near room temperature25, and at millikelvin temperatures for micrometer-thick YIG slabs26,27and micrometer-scaled hy- brid magnon-superconducting systems28. The ability to process information in sub-100nm sized magnonic structures is one of the key advantages of magnon-ics, which translates also to the fields of hybrid opto-magnonic quantum systems and quantum magnonics. To couple PSW to these nanostructures efficiently at millikelvin temperatures, integrated nanoantennas24,29are required. Here we demon- strate PSWS at millikelvin temperatures, with base tempera- tures reaching 45mK in a 100nm-thick YIG film, using inte- grated nanoantennas separated by 10 micrometers for excita- tion and detection. The analysis is focused on magnetostatic surface spin waves (MSSWs, also called “Damon-Eshbach” mode) that propagate perpendicular to an in-plane magnetic field k?B. We find that magnon transport at the nanome- ter structure scale can be measured also down to millikelvin temperatures. Although the propagation signal is measurable across a wide field and temperature range, we observe that the transmitted signal is distorted for applied magnetic fields above 75mT. This effect is largely caused by the magneti- sation of the gadolinium-gallium-garnet (GGG) substrate, on which the YIG film is grown. It reaches 47kA =m for 75mT of applied external magnetic field at 45mK temperature. In general, our findings agree with the increase in the damping of YIG grown on GGG at low-temperatures reported in the literature19–22. First, we explain the sample preparation and experimental techniques, before we continue to pre-characterise the sam- ple at room and base temperature, using standard FMR tech- niques. Then we discuss the first PSWS experiment, in which a fixed external magnetic field is applied and the temperature is swept from base to room temperature. We continue to anal- yse the propagation characteristics in more detail by compar- ing the low-temperature measurements to room-temperature results and extract the spin-wave group velocities. Finally, we perform PSWS at higher external magnetic fields, to inves- tigate the propagation characteristics between the room and base temperature. The magnetisation of GGG is measured byarXiv:2212.02257v3 [physics.app-ph] 22 Jan 2023PSWS at millikelvin temperatures 2 (a) (b) (c) 1μm380μm e- (I) (II) (III) (IV) (V) (VI) YIG GGG PMMA Ti/Au Antenna 1 Antenna 2S21,21Stripline nanoantennas with CPWReference CPW GGGYIGPort 2Port 1Antenna 1 Antenna 2 FIG. 1. Overview of the electron-beam lithographed stripline nanoantennas on the yttrium-iron-garnet film. (a) Sketch of the sample used in these measurements. Stripline nanoantennas coupled to coplanar waveguide (CPW) and the reference CPW are fabricated atop a 100nm-thick yttrium-iron-garnet film on a 500 mm-thick gadolinium-gallium-garnet substrate. (b) The coplanar-waveguide coupled nanoantennas are fabricated with electron-beam lithography. These nanoantennas are made of Ti(5nm)/Au(55nm) (more details in main text). (c) Optical and secondary-electron images of the CWP nanoantennas used in the manuscript. These nanoantennas are 10 mm spaced apart and have a width of 330nm and length of 120 mm. The propagating spin waves (PSW) are excited and detected by the stripline nanoantennas 1 and 2 respectively. The transmission is measured through the S-parameters, acquired by a vector network analyser. vibrating sample magnetometry (VSM) of a GGG-only sub- strate at low-temperatures. II. SAMPLE AND EXPERIMENTAL SETUPS In our experiments we use an LPE-grown 100nm-thick (111)-orientated YIG film on a 500 mm-thick GGG substrate, as sketched in Fig. 1 (a). Atop the YIG film we fabricate nanoantennas connected to CPWs, using an electron-beam lithography process Fig. 1 (b). First, a single layer of PMMA is spin-coated and baked. After, we use electron-beam lithog- raphy to write the antenna structures, develop the sample and deposit a layer of Ti(5nm)/Au(55nm), using electron beam physical vapour deposition, followed by lift-off. Figure 1(c) shows an optical (top) and a secondary-electron image (bot- tom) with the coplanar waveguides and stripline nanoanten- nas used in this work. Here the nanoantennas have a spac- ing of 10 mm. The stripline nanoantennas possess a width of 330nm and a length of 120 mm. Additionally, we fabricate a reference coplanar waveguide, to measure the FMR signals only (see Fig. 1 (a)). After fabrication, the sample is glued and then wire-bonded, with a 75 mm diameter gold wire, to a high-frequency printed circuit board and mounted into the dilution refrigerator. Our setup is based on a cryogenic-free dilution refrigerator system (BlueFors-LD250), which reaches base temperatures below 10mK at the mixing chamber stage. The sample space possesses a base temperature of about 16mK. During oper- ation, the sample space heats up to about 45mK. At these temperatures, the thermal excitations of gigahertz-frequency magnons and phonons are still suppressed. The input signal is transmitted and collected from the sample (ports 1 and 2 Fig. 1(a)), using high-frequency copper and superconducting wiring each attenuated by 7dB to reduce thermal noise. The signals are collected with a 70GHz vector network analyser (Anritsu VectorStar MS4647B).The room-temperature measurements are carried out on a home-built setup. The setup consists of a VNA (Anritsu MS4642B) connected to an H-frame electromagnet GMW 3473-70 with an 8cm air gap for various measurement config- urations and magnet poles of 15cm diameter to induce a suf- ficiently uniform biasing magnetic field. The electromagnet is powered by a bipolar power supply BPS-85-70EC (ICEO), allowing it to generate up to 0 :9T at 8cm air gap. The in- put powers are adjusted, to obtain the same power levels at the sample as in the cryogenic measurements, to account for cable losses and the previously mentioned attenuators. The precise microwave powers for each individual experiment are stated later. III. RESULTS AND DISCUSSION First, we use the reference CPW to pre-characterise the sample and to estimate the Gilbert damping as. We plot our FMR and PSWS data according to S0 21(f) =S21;sig(f)S21;ref(f) S21;ref(f); (1) where fdenotes the set of frequency points of the com- plex transmission signal S21;sig(f)and reference S21;ref(f) values30. The reference signal is obtained by detuning the external magnetic field by +50mT. For the FMR refer- ence measurements, we find the Gilbert damping parameter as= (5:980:3)104for room temperature, and as= (31:5)103at 45mK respectively. The large error in the low-temperature case originates from the fit uncertainty in the slope of FMR linewidth versus FMR frequency. The methodology developed in Ref.30, which accounts for asym- metry and phase offset in the distorted FMR signal, was used. The order of magnitude in the Gilbert damping at 45mK is in good agreement with previously reported values for thin YIG films at Kelvin temperatures19.PSWS at millikelvin temperatures 3 0.5K 0.045K 1.0K 1.5K2.0K2.5K FIG. 2. Linear magnitude, real and imaginary part of the S0 21 parameters for propagating spin waves (PSW) in the Damon- Eshbach mode, using 50mT of external magnetic field and dif- ferent temperatures. The applied microwave power was set to 28dBm (at the sample) with an average sampling of 50 for 45mK- 1K and 100 for 1 :5K-2 :5K. The FMR point ( k=0) is constant at 3:36GHz (189kA =m) for all measured PSW. We perform the first PSWS experiment at a fixed exter- nal magnetic field of 50mT, using the stripline nanoanten- nas shown in Fig. 1. Figure 2 displays the linear magni- tude (black), real (blue) and imaginary (red) part of the trans- mission data (cw-mode), together with a temperature sweep from the base temperature of 45mK up to 2 :5K, i.e. about the Curie-Weiss temperature of GGG31,32. The spin waves are excited with a power of 28dBm at the sample, with the external magnetic field applied perpendicular to the propaga- tion direction. In Fig. 2 we verify the ability to measure the transmission across the entire temperature range and observe a propagation signal with a fixed FMR point ( k=0) of about 3:36GHz, corresponding to an effective saturation magnetisa- tion of about 189kA =m. The signal amplitude increases by about 30% from 45mK to 2 :5K. We continue to investigate the spin-wave propagation in more detail and compare the results to room-temperature mea- surements. Figure 3 (first column) depicts the imaginary part of the S0 21parameters for PSWs between the two nanoanten- nas at three different selected temperatures: (a) 297K, (b) 500mK, and (c) 45mK, at a fixed external magnetic field of 50mT. The second column in Fig. 3 shows the correspond- ing calculated dispersion relations for MSSWs (black), us- ing the Kalinikos-Salvin model33. The maximum excitation efficiency J(green line Fig. 3) is governed by the 330nm stripline nanoantennas23. The third column in Fig. 3 shows (a) (b) (c)297K 0.5K 0.045K FIG. 3. Imaginary part of the S0 21parameter, calculated disper- sion relation, antenna excitation efficiency and group velocity for PSW (Damon-Eshbach mode), using 50mT external field at dif- ferent temperatures. The theoretical group velocity is calculated as the derivation of the dispersion relation and measured as vg=df
D, with the periodicity of the transmission in the Im(S0 21) parameters dfand the gap between the nanoantennas D(see Ref.23). The pa- rameters measured and used for the calculation are the following: (a) 297K, Ms=142kA =m, (b) 500mK, Ms=189kA =m, (c) 45mK, Ms=189kA =m. The effective saturation magnetisation increases and thus group velocity increases by about 50% at millikelvin tem- peratures. the theoretical group velocities as the derivation of the dis- persion relation (black curve) and the measured values given byvg=df
D(red dots), where dfis the periodicity of the oscillations in the Im (S0 21)parameters and Dthe gap be- tween the nanoantennas23. The errors in the calculated group velocities are estimated from the error propagation of the frequency reading. We observe a reduction in propagation amplitude by about 50% between the room and both cryo- genic temperatures caused by the increase in Gilbert damping. We find values for the effective saturation magnetisation of Ms=142kA =m at room temperature and Ms=189kA =m for 45mK and 500mK. The constant effective saturation mag- netisation at millikelvin temperatures is in good agreement with literature34,35, with a value close to the observed ones in micrometer-thick YIG samples26. In accordance with the increase in effective saturation magnetisation, we observe an increase of the group velocity by about 50%. The measured values are in good agreement with the theoretically calculated group velocities. We continue our investigations by comparing the spin- wave propagation for higher external magnetic fields than inPSWS at millikelvin temperatures 4 (a) (b)50mT297K175mT0.045K25mT 200mT 75mT FIG. 4. Linear magnitude, real and imaginary part of the S0 21parameters for PSWS in the Damon-Eshbach mode at different external fields. The applied microwave power was set to 28dBm (at the sample) with an averaging of 10 (for 297K) and 25 (for 0 :045K). (a) Room temperature (297K): The spin-wave propagation can be measured over a wide magnetic field range. (b) Base temperature (45mK): The spin-wave propagation for magnetic fields in the range from about 25mT to 75mT is trackable, while above 75mT the magnitude and its propagation characteristics start to be distorted. This effect is a result of the increased magnetisation of the GGG substrate (see Fig. 5). the previous measurements, at 297K (Fig. 4 (a)) and 45mK (Fig. 4 (b)). Figure 4 shows the linear magnitude (black), real (blue) and imaginary (red) part for PSW in the Damon- Eshbach mode at selected magnetic fields. At room temper- ature, we measure the spin-wave signal over a wide external magnetic field range up to about 900mT. Examples for low fields are given in Fig. 4 (a). However, at 45mK the propa- gation characteristics are changing (Fig. 4 (b)). After about 75mT the magnitude of the spin-wave signal is reduced sig- nificantly and only a signature in the oscillation behaviour can be observed. Moreover, the fixed phase relation between the imaginary and real parts disappears, causing challenges in plotting the linear magnitude of the propagation signal. Exam- ples for the reduced spin-waves signals are given for 175mT and 200mT. This opposing behaviour between the room and base temperature is a clear indication, that beyond an external field of about 75mT the GGG substrate magnetises enough to influence the propagation characteristics of the spin waves. Thus, future millikelvin measurements at high magnetic fields may rely on suspended YIG membranes or triangular nanos- tructures, which have already been demonstrated in other ma- terial systems (e.g. Ref.36). To estimate the influence of the paramagnetic GGG sub- strate on the spin-wave propagation in YIG, we conclude our investigations by measuring the GGG magnetisation MGGG of a 440:5mm GGG-only substrate, using a vibrating sam- ple magnetometer (VSM) in the temperature range from 2K to 300K in the presence of fields up to 9T. The results at 2K for our magnetic fields of interest are shown in Fig. 5 (dark- blue dots). As the VSM is limited to kelvin temperatures, we extrapolate magnetisation values for GGG at 45mK (blue dashed line), using the 2K data. For example at 75mT (Fig. 5 black dots) we find, that GGG possesses a magnetisation value of 28 :5kA =m at 2K, which increases to about 47kA =m at 45mK. Thus, the temperature and magnetic field dependant GGG magnetisation may explain the observed reduction in the 47kA/m 28.5kA/mFIG. 5. Magnetisation of the GGG substrate versus the applied magnetic field. A GGG-only sample is measured using a vibrating sample magnetometer (VSM) at 2K (dark-blue dots), leading for ex- ample to an effective magnetisation of 28 :5kA =m. From the data the magnetisation values for 45mK are extrapolated (blue dashed line). At 75mT the magnetisation increases to about 47kA =m. PSW amplitudes and the propagation distortions above exter- nal magnetic fields of 75mT. However, the role of the paramagnetic GGG substrate on spin waves in YIG is the subject of separate systematic stud- ies. Our PSWS measurements, supported by the FMR and VSM studies, suggest that the magnetic moment induced in GGG at millikelvin temperatures by the application of rela- tively large magnetic fields is at least partly responsible for the increase in spin-wave damping. The increase in the Gilbert damping constant acan only be approximately quantified, as this requires plotting the FMR linewidth DBagainst the FMR frequency fFMR over a wide range of applied fields.PSWS at millikelvin temperatures 5 However, since the FMR linewidth depends on the degree of the magnetisation of the GGG (given by the temperature and the applied field - see Fig. 5), the dependence DB(fFMR) becomes nonlinear and the parameter aloses its original physical meaning. Moreover, the measurement of FMR on nanometer-thick samples requires the careful subtraction of the reference microwave transmission signal (see Eq. 1) at a 50mT detuned magnetic field. Since this reference signal also depends significantly on the GGG magnetisation at low- temperatures, the measurement uncertainties increase. Nev- ertheless, we can qualitatively conclude that the increase in spin-wave damping in the nanometer-thick YIG films on GGG corresponds to the previously reported increase in damping in the micrometer-thick films on GGG18,19,22. Other phenomena that could contribute to the distortion of the PSWS experi- ments at the nanoscale at fields above 75mT are the possible dependence of the magnetocrystalline anisotropy caused by the dependence of the YIG/GGG lattice mismatch on temper- ature and the absorption/distortion of the microwave signal in the CPW transmission lines (see Fig. 1(a)) by the magnetised GGG substrate. IV. CONCLUSIONS In conclusion, we have shown for the first time that propa- gating spin-wave spectroscopy in 100nm-thin YIG films can be performed in a wide temperature range, from millikelvin to room temperature, without changing the propagation charac- teristics. At a fixed external magnetic field of 50mT we con- firm that the propagating spin waves maintain a constant fer- romagnetic resonance frequency below temperatures of about 2:5K. However, the signal amplitude increases by 30% be- tween 45mK and 2 :5K, and further by about 50% when the temperature is raised to room temperature. In contrast to previous work we demonstrate, that only beyond an ex- ternal field of about 75mT the GGG substrate magnetises up to 47kA =m influence the spin-wave propagation at low- temperatures. With our experiments, we illustrate that al- though the GGG substrate influences the spin-wave propaga- tion characteristics at millikelvin temperatures, future large- scale integrated YIG nanocircuits can be realised and mea- sured. ACKNOWLEDGMENTS The authors thank Vincent Vlaminck for useful discussions and feedback. SK acknowledges the support by the H2020- MSCA-IF under the grant number 101025758 (OMNI). KD was supported by the Erasmus+ program of the European Union. The authors acknowledge CzechNanoLab Research Infrastructure supported by MEYS CR (LM2018110). The work of CD was supported by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) under grant 271741898. The work of ML was supported by the German Bundesministerium für Wirtschaft und Energie (BMWi) under grant 49MF180119. CD thanks O. Surzhenko and R. Meyer(INNOVENT) for their support. The authors thank Oleksandr Dobrovolskiy for his support in the initial configuration of the dilution refrigerator. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Authors Contributions SK and MU conceived the experiment in discussion with AC. SK and KD performed the experiments under the guid- ance of MU and AC. SK and KD analysed and interpreted the data with support from AC. RS and OD performed the VSM measurements at kelvin temperatures, and A V interpo- lated the data for millikelvin temperatures. ML and TR pre- pared the LPE sample. CD conceived and supervised the LPE film growth. QW and RV supported the measurements with theoretical expertise. DS and SK set up the cryogenic system. RS supported the measurements and analysis of the measure- ments. SK wrote the manuscript with support from all co- authors. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye, M. Krawczyk, J. Grafe, C. Adelmann, S. Cotofana, A. Naeemi, V . I. Vasyuchka, B. Hillebrands, S. A. Nikitov, H. Yu, D. Grundler, A. V . Sadovnikov, A. A. Grachev, S. E. Sheshukova, J. Y . Duquesne, M. Marangolo, G. Csaba, W. Porod, V . E. Demidov, S. Urazhdin, S. O. Demokritov, E. Albisetti, D. Petti, R. Bertacco, H. Schultheiss, V . V . Kruglyak, V . D. Poimanov, S. Sahoo, J. Sinha, H. Yang, M. Münzenberg, T. Moriyama, S. Mizukami, P. Lan- deros, R. A. Gallardo, G. Carlotti, J. V . Kim, R. L. Stamps, R. E. Cam- ley, B. Rana, Y . Otani, W. Yu, T. Yu, G. E. Bauer, C. Back, G. S. Uhrig, O. V . Dobrovolskiy, B. Budinska, H. Qin, S. Van Dijken, A. V . Chumak, A. Khitun, D. E. Nikonov, I. A. Young, B. W. Zingsem, and M. Win- klhofer, “The 2021 Magnonics Roadmap,” Journal of Physics Condensed Matter 33(2021). 2A. V . Chumak, P. Kabos, M. Wu, C. Abert, C. Adelmann, A. O. Adey- eye, J. Akerman, F. G. Aliev, A. Anane, A. Awad, C. H. Back, A. Bar- man, G. E. W. Bauer, M. Becherer, E. N. Beginin, V . A. S. V . Bitten- court, Y . M. Blanter, P. Bortolotti, I. Boventer, D. A. Bozhko, S. A. Bun- yaev, J. J. Carmiggelt, R. R. Cheenikundil, F. Ciubotaru, S. Cotofana, G. Csaba, O. V . Dobrovolskiy, C. Dubs, M. Elyasi, K. G. Fripp, H. Fu- lara, I. A. Golovchanskiy, C. Gonzalez-Ballestero, P. Graczyk, D. Grundler, P. Gruszecki, G. Gubbiotti, K. Guslienko, A. Haldar, S. Hamdioui, R. Her- tel, B. Hillebrands, T. Hioki, A. Houshang, C.-M. Hu, H. Huebl, M. Huth, E. Iacocca, M. B. Jungfleisch, G. N. Kakazei, A. Khitun, R. Khymyn, T. Kikkawa, M. Klaui, O. Klein, J. W. Klos, S. Knauer, S. Koraltan, M. Kostylev, M. Krawczyk, I. N. Krivorotov, V . V . Kruglyak, D. Lachance- Quirion, S. Ladak, R. Lebrun, Y . Li, M. Lindner, R. Macedo, S. Mayr,PSWS at millikelvin temperatures 6 G. A. Melkov, S. Mieszczak, Y . Nakamura, H. T. Nembach, A. A. Nikitin, S. A. Nikitov, V . Novosad, J. A. Otalora, Y . Otani, A. Papp, B. Pigeau, P. Pirro, W. Porod, F. Porrati, H. Qin, B. Rana, T. Reimann, F. Riente, O. Romero-Isart, A. Ross, A. V . Sadovnikov, A. R. Safin, E. Saitoh, G. Schmidt, H. Schultheiss, K. Schultheiss, A. A. Serga, S. Sharma, J. M. Shaw, D. Suess, O. Surzhenko, K. Szulc, T. Taniguchi, M. Urbanek, K. Us- ami, A. B. Ustinov, T. van der Sar, S. van Dijken, V . I. Vasyuchka, R. Verba, S. V . Kusminskiy, Q. Wang, M. Weides, M. Weiler, S. Wintz, S. P. Wolski, and X. Zhang, “Advances in Magnetics Roadmap on Spin-Wave Comput- ing,” IEEE Transactions on Magnetics 58, 1–72 (2022). 3Y . Li, W. Zhang, V . Tyberkevych, W. K. Kwok, A. Hoffmann, and V . Novosad, “Hybrid magnonics: Physics, circuits, and applications for co- herent information processing,” Journal of Applied Physics 128(2020). 4O. V . Dobrovolskiy, R. Sachser, T. Brächer, T. Böttcher, V . V . Kruglyak, R. V . V ovk, V . A. Shklovskij, M. Huth, B. Hillebrands, and A. V . Chumak, “Magnon–fluxon interaction in a ferromagnet/superconductor heterostruc- ture,” Nature Physics 15, 477–482 (2019). 5Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, “Quantum magnonics: The magnon meets the supercon- ducting qubit,” Comptes Rendus Physique 17, 729–739 (2016). 6I. Boventer, M. Pfirrmann, J. Krause, Y . Schön, M. Kläui, and M. Weides, “Complex temperature dependence of coupling and dissipation of cavity magnon polaritons from millikelvin to room temperature,” Physical Review B97, 1–9 (2018). 7V . A. Bittencourt, V . Feulner, and S. V . Kusminskiy, “Magnon heralding in cavity optomagnonics,” Physical Review A 100, 1–15 (2019). 8D. Lachance-Quirion, Y . Tabuchi, A. Gloppe, K. Usami, and Y . Nakamura, “Hybrid quantum systems based on magnonics,” Applied Physics Express 12(2019). 9D. Lachance-Quirion, S. P. Wolski, Y . Tabuchi, S. Kono, K. Usami, and Y . Nakamura, “Entanglement-based single-shot detection of a single magnon with a superconducting qubit,” Science 367, 425–428 (2020). 10R. G. Morris, A. F. Van Loo, S. Kosen, and A. D. Karenowska, “Strong coupling of magnons in a YIG sphere to photons in a planar superconduct- ing resonator in the quantum limit,” Scientific Reports 7, 1–6 (2017). 11B. Heinz, T. Brächer, M. Schneider, Q. Wang, B. Lägel, A. M. Friedel, D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, and A. V . Chumak, “Propagation of Spin-Wave Packets in Individual Nanosized Yttrium Iron Garnet Magnonic Conduits,” 20, 4220–4227 (2020). 12Q. Wang, M. Kewenig, M. Schneider, R. Verba, F. Kohl, B. Heinz, M. Geilen, M. Mohseni, B. Lägel, F. Ciubotaru, C. Adelmann, C. Dubs, S. D. Cotofana, O. V . Dobrovolskiy, T. Brächer, P. Pirro, and A. V . Chumak, “A magnonic directional coupler for integrated magnonic half-adders,” Na- ture Electronics 3, 765–774 (2020). 13C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Brückner, and J. Del- lith, “Sub-micrometer yttrium iron garnet LPE films with low ferromagnetic resonance losses,” Journal of Physics D: Applied Physics 50(2017). 14C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. Hübner, and E. Wendler, “Low damping and microstruc- tural perfection of sub-40nm-thin yttrium iron garnet films grown by liquid phase epitaxy,” Phys. Rev. Materials 4(2020). 15R. D. Henry, P. J. Besser, D. M. Heinz, and J. E. Mee, “Ferromagnetic Resonance Properties of LPE YIG Films,” IEEE Transactions on Magnetics 9, 535–537 (1973). 16W. Hongxu and W. Wenshu, “The ghowth of LPE YIG films with narrow FMR linewidth,” IEEE Transactions on Magnetics 20, 1222–1223 (1984). 17Q. Wang, B. Heinz, R. Verba, M. Kewenig, P. Pirro, M. Schneider, T. Meyer, B. Lägel, C. Dubs, T. Brächer, and A. V . Chumak, “Spin Pinning and Spin-Wave Dispersion in Nanoscopic Ferromagnetic Waveg- uides,” Physical Review Letters 122, 247202 (2019). 18L. Mihalceanu, V . I. Vasyuchka, D. A. Bozhko, T. Langner, A. Y . Nechiporuk, V . F. Romanyuk, B. Hillebrands, and A. A. Serga, “Temperature-dependent relaxation of dipole-exchange magnons in yttrium iron garnet films,” Physical Review B 97, 1–9 (2018).19C. L. Jermain, S. V . Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brang- ham, M. R. Page, P. C. Hammel, F. Y . Yang, and D. C. Ralph, “Increased low-temperature damping in yttrium iron garnet thin films,” Physical Re- view B 95, 1–5 (2017). 20S. Guo, B. Mccullian, P. C. Hammel, and F. Yang, “Low damping at few-K temperatures in Y 3Fe5O12epitaxial films isolated from Gd 3Ga5O12sub- strate using a diamagnetic Y 3Sc2:5Al2:5O12spacer,” Journal of Magnetism and Magnetic Materials 562, 169795 (2022). 21I. A. Golovchanskiy, N. N. Abramov, M. Pfirrmann, T. Piskor, J. N. V oss, D. S. Baranov, R. A. Hovhannisyan, V . S. Stolyarov, C. Dubs, A. A. Gol- ubov, V . V . Ryazanov, A. V . Ustinov, and M. Weides, “Interplay of Mag- netization Dynamics with a Microwave Waveguide at Cryogenic Tempera- tures,” Physical Review Applied 11, 1 (2019). 22S. Kosen, A. F. Van Loo, D. A. Bozhko, L. Mihalceanu, and A. D. Karenowska, “Microwave magnon damping in YIG films at millikelvin temperatures,” APL Materials 7(2019). 23V . Vlaminck and M. Bailleul, “Spin-wave transduction at the submicrome- ter scale: Experiment and modeling,” Physical Review B - Condensed Mat- ter and Materials Physics 81, 1–13 (2010). 24M. Va ˇnatka, K. Szulc, O. Wojewoda, C. Dubs, A. V . Chumak, M. Krawczyk, O. V . Dobrovolskiy, J. W. Kłos, and M. Urbánek, “Spin- Wave Dispersion Measurement by Variable-Gap Propagating Spin-Wave Spectroscopy,” Physical Review Applied 16, 054033 (2021). 25M. S. Alam, C. Wang, J. Chen, J. Zhang, C. Liu, J. Xiao, Y . Wu, L. Bi, and H. Yu, “Temperature control of spin wave propagation over 100 mm distance in 100nm-thick YIG film,” Physics Letters, Section A: General, Atomic and Solid State Physics 383, 366–368 (2019). 26A. F. Van Loo, R. G. Morris, and A. D. Karenowska, “Time-Resolved Measurements of Surface Spin-Wave Pulses at Millikelvin Temperatures,” Physical Review Applied 10, 1 (2018). 27D. Schmoll, Enabling technology for high-frequency quantum magnonics , Master’s thesis, University of Vienna, Wien (2022). 28P. G. Baity, D. A. Bozhko, R. Macêdo, W. Smith, R. C. Holland, S. Danilin, V . Seferai, J. Barbosa, R. R. Peroor, S. Goldman, U. Nasti, J. Paul, R. H. Hadfield, S. McVitie, and M. Weides, “Strong magnon-photon coupling with chip-integrated YIG in the zero-temperature limit,” Applied Physics Letters 119(2021). 29H. Yu, O. D’Allivy Kelly, V . Cros, R. Bernard, P. Bortolotti, A. Anane, F. Brandl, R. Huber, I. Stasinopoulos, and D. Grundler, “Magnetic thin-film insulator with ultra-low spin wave damping for coherent nanomagnonics,” Scientific Reports 4, 2–6 (2014). 30S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Ka- bos, T. J. Silva, and J. P. Nibarger, “Ferromagnetic resonance linewidth in metallic thin films: Comparison of measurement methods,” Journal of Applied Physics 99(2006). 31O. A. Petrenko, D. M. K. Paul, C. Ritter, T. Zeiske, and M. Yethiraj, “Mag- netic frustration and order in gadolinium gallium garnet,” Physica B: Con- densed Matter 266, 41–48 (1999). 32M. Sabbaghi, G. W. Hanson, M. Weinert, F. Shi, and C. Cen, “Terahertz response of gadolinium gallium garnet (GGG) and gadolinium scandium gallium garnet (SGGG),” Journal of Applied Physics 127(2020). 33B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary condi- tions,” Journal of Physics C: Solid State Physics 19, 7013–7033 (1986). 34P. Hansen, P. Röschmann, and W. Tolksdorf, “Saturation magnetization of gallium-substituted yttrium iron garnet,” Journal of Applied Physics 45, 2728–2732 (1974). 35I. V . Zavislyak and M. A. Popov, “Microwave Properties and Applications of Yttrium Iron Garnet (YIG) Films: Current State of Art and Perspectives. In Yttrium: Compounds, Production and Applications,” (Nova, 2009) pp. 87–125. 36M. J. Burek, Y . Chu, M. S. Liddy, P. Patel, J. Rochman, S. Meesala, W. Hong, Q. Quan, M. D. Lukin, and M. Loncar, “High quality-factor op- tical nanocavities in bulk single-crystal diamond,” Nature Communications 5, 1–7 (2014).

2022-12-05

Performing propagating spin-wave spectroscopy of thin films at millikelvin temperatures is the next step towards the realisation of large-scale integrated magnonic circuits for quantum applications. Here we demonstrate spin-wave propagation in a $100\,\mathrm{nm}$-thick yttrium-iron-garnet film at the temperatures down to $45 \,\mathrm{mK}$, using stripline nanoantennas deposited on YIG surface for the electrical excitation and detection. The clear transmission characteristics over the distance of $10\,\mu \mathrm{m}$ are measured and the subtracted spin-wave group velocity and the YIG saturation magnetisation agree well with the theoretical values. We show that the gadolinium-gallium-garnet substrate influences the spin-wave propagation characteristics only for the applied magnetic fields beyond $75\,\mathrm{mT}$, originating from a GGG magnetisation up to $47 \,\mathrm{kA/m}$ at $45 \,\mathrm{mK}$. Our results show that the developed fabrication and measurement methodologies enable the realisation of integrated magnonic quantum nanotechnologies at millikelvin temperatures.

Propagating spin-wave spectroscopy in nanometer-thick YIG films at millikelvin temperatures

2212.02257v3

Crystallization dynamics of amorphous yttrium iron garnet thin films Sebastian Sailler,1Gregor Skobjin,1Heike Schlörb,2Benny Boehm,3Olav Hellwig,3, 4Andy Thomas,2, 5Sebastian T. B. Goennenwein,1and Michaela Lammel1 1)Department of Physics, University of Konstanz, 78457 Konstanz, Germany 2)Leibniz Institute of Solid State and Materials Science, 01069 Dresden, Germany 3)Institute of Physics, Technische Universität Chemnitz, 09126 Chemnitz, Germany 4)Center for Materials Architectures and Integration of Nanomembranes (MAIN), Technische Universität Chemnitz, 09107 Chemnitz, Germany 5)Institut für Festkörper- und Materialphysik (IFMP), TUD Dresden University of Technology, 01069 Dresden, Germany (*Electronic mail: michaela.lammel@uni-konstanz.de) (*Electronic mail: sebastian.sailler@uni-konstanz.de) (Dated: 19th April 2024) Yttrium iron garnet (YIG) is a prototypical material in spintronics due to its exceptional magnetic properties. To exploit these properties high quality thin films need to be manufactured. Deposition techniques like sputter deposition or pulsed laser deposition at ambient temperature produce amorphous films, which need a post annealing step to induce crystallization. However, not much is known about the exact dynamics of the formation of crystalline YIG out of the amorphous phase. Here, we conduct extensive time and temperature series to study the crystallization behavior of YIG on various substrates and extract the crystallization velocities as well as the activation energies needed to promote crystallization. We find that the type of crystallization as well as the crystallization velocity depend on the lattice mismatch to the substrate. We compare the crystallization parameters found in literature with our results and find an excellent agreement with our model. Our results allow us to determine the time needed for the formation of a fully crystalline film of arbitrary thickness for any temperature. I. INTRODUCTION Yttrium iron garnet (Y 3Fe5O12, YIG) is an electrically in- sulating ferrimagnet, crystallizing in a cubic crystal lattice with Ia ¯3d symmetry.1,2Its electric and magnetic properties include a long spin diffusion length, which makes YIG an ideal material for spin transport experiments with pure spin currents.3–5Additionally, YIG shows an exceptionally low Gilbert damping and a low coercive field, which allows in- vestigations of magnon dynamics via e.g. ferromagnetic res- onance experiments.6–10These exceptional properties caused YIG to be intensively studied and made it the prototypical ma- terial in the field of spintronics, which almost exclusively re- lies on devices in thin film geometry. Several deposition techniques are known to produce high quality YIG thin films, including pulsed laser deposition (PLD),11–18liquid phase epitaxy (LPE)10,19–23and radio- frequency (RF) magnetron sputtering.24–41Some deposition techniques like magnetron sputtering give the opportunity to deposit both, amorphous and crystalline thin films, depend- ing on the process temperatures during deposition.25,42Here, room temperature magnetron sputtering processes yield amor- phous films.24–33,35–42For the deposition of YIG onto gadolin- ium gallium garnet (Gd 3Ga5O12, GGG) substrates, which fea- ture a lattice constant very similar to the one of YIG, direct epitaxial growth was observed for process temperatures of 700◦C.25,42On quartz a post annealing step is needed to en- able the formation of polycrystalline YIG.43 The annealing process is usually performed in air26,36or reduced oxygen atmosphere28,40,44,45to counteract potential oxygen vacancies in the YIG lattice. For amorphous PLD films annealing in inert argon atmosphere has been reported tohave no deteriorating influence.15Annealing crystalline, sput- tered YIG films in vacuum, however, showed a reduction in typical characteristic properties like the spin Hall magnetore- sistance in YIG/Pt.44 Furthermore, the annealing process itself can lead to an in- terdiffusion at the substrate interface,36,46often leading to the formation of a magnetic dead layer,23,36,46as well as an in- crease of the ferromagnetic resonance linewidth, especially at low temperatures.13,47On the one hand, YIG grown on GGG by LPE requires no post annealing, which allows for the sup- pression of the gadolinium interdiffusion, leading to an ex- tremely sharp interface.23On the other hand, scaling the LPE process is not straightforward. Sputter deposition31or solu- tion based methods45,48allow for wafer scale processes, but the mandatory post annealing step should be optimized to al- low fast processing, which then could simultaneously reduce the interdiffusion of yttrium and gadolinium. To achieve this, the annealing time required to yield fully crystalline YIG films needs to be kept as low as possible. However, the dynamics describing the crystallization of YIG thin films during the post annealing step are only selec- tively reported in the literature. Typically, only the tempera- ture and a time proven to yield a completely crystalline thin film with the desired properties are reported. Here, we present an extended picture of the crystalliza- tion dynamics of YIG at different temperatures and annealing times, which allows us to define different crystallization win- dows depending on the substrate material. Our results pro- vide a general crystallographic description of the crystalliza- tion process for YIG thin films and summarize the crystalliza- tion parameters found in the literature.arXiv:2308.00412v2 [cond-mat.mtrl-sci] 18 Apr 20242 II. METHODS Ahead of the deposition, all substrates were cleaned for five minutes in aceton and isopropanol, and one minute in de-ionized water in an ultrasonic bath. YIG thin films were then deposited at room temperature onto different substrate materials using RF sputtering from a YIG sinter target at 2.7·10−3mbar argon pressure and 80 W power, at a rate of 0.0135 nm /s. The nominal thickness upon deposition was 33 nm. The post-annealing steps were carried out in a tube zone furnace under air. As substrates yttrium aluminum garnet (Y 3Al5O12, YAG, CrysTec ) and gadolinium gallium garnet (Gd 3Ga5O12, GGG, SurfaceNet ) with a <111> crystal orientation along the sur- face normal were used. Additionally, silicon wafers cut along the <100> crystal direction with a 500 nm thick thermal ox- ide layer (Si/SiO x,MicroChemicals ) were used. Since GGG and YAG crystallize in the same space group Ia ¯3d as YIG and their lattice parameters are 1 .2376 nm49and 1 .2009 nm,50re- spectively, they are considered lattice matched in regards to the 1 .2380 nm for YIG.51The lattice mismatch εcan be cal- culated with Eq. (1) ε=aYIG−asubstrate asubstrate·100% (1) and translates to 0 .03 % for GGG and 3 .09 % for YAG.52Due to the amorphous SiO xlayer the Si/SiO xsubstrates do not pro- vide any preferential direction for crystallization. But even considering the underlying Si layer, we do not expect it to influence the crystallization direction in any way, as it fea- tures a fundamentally different space group (Fd ¯3m) and lat- tice constant.53Therefore, Si/SiO xis considered non lattice matched and fulfills the function as an arbitrary substrate. For the structural characterization X-ray diffraction mea- surements (XRD) were performed using a Rigaku Smart Lab Diffractometer with Cu Kαradiation. Scanning electron mi- croscopy as well as electron backscatter diffraction (EBSD) measurements were conducted using a Zeiss Gemini Scan- ning Electron Microscope (SEM). The magnetic properties were characterized via magneto-optical Kerr effect measure- ments in longitudinal geometry (L-MOKE) in a commercial Kerr microscope from Evico Magnetics. III. RESULTS AND DISCUSSION The crystallization mechanism of a thin film crucially de- pends on the substrate: for substrates where the lattices of film and substrate are sufficiently similar, the thin film layer crystallizes epitaxially, whereas for a substrate where the two lattices do not match, nucleation is needed. Figure 1 shows the different crystallization mechanisms and the resulting YIG micro structure depending on the cho- sen substrate. As depicted in Figure 1(a), a lattice matched substrate acts as a seed on which the film can grow epitaxi- ally. Therefore, a single crystalline front is expected to move from the substrate towards the upper boundary of the film,54,55which is commonly referred to as solid phase epitaxy (SPE) in the literature. For a substrate with a sufficiently large lattice mismatch or no crystalline order, no such interface is given, see Fig. 1(b). Here, a nucleus needs to be formed first from which further crystallization takes place. The formation of the initial seeds by nucleation is expected to yield random crys- tal orientations. The polycrystalline seeds grow until reaching another grain or one of the sample’s boundaries. For any of these processes, SPE or nucleation, to take place, the system needs to be at a temperature characteristic for this specific thin film/substrate system.56 To distinguish between amorphous, partly and fully crys- talline films we apply several characterization methods, prob- ing the structural and magnetic properties of the YIG thin films. The typical fingerprints of amorphous versus crystalline YIG on different substrates as determined by X-ray diffrac- tion (XRD), the longitudinal magneto-optical Kerr effect (L- MOKE) and electron back scatter diffraction (EBSD) are de- picted in Figure 2. From top to bottom we gain an increased spacial resolution, probing increasingly smaller areas of the sample. With XRD, the structural properties of YIG on YAG and GGG can be evaluated. For the amorphous films, the XRD measurements in Fig. 2 (a-c) show a signal stemming only from the substrate (cp. gray dashed lines). Upon annealing, YIG is visible in the form of Laue-oscillations on GGG (pur- ple) and as a peak on YAG (red). In stark contrast to that no signal, which could be attributed to YIG, can be found on SiO x, even when annealing at 800◦C for 48 h. The sharp peak in Fig. 2(c) at 32 .96◦can be attributed to a detour ex- citation of the substrate, as it is visible in the as deposited state and fits the forbidden Si (200) peak.57In the literature, YIG on SiO xhas been reported to be polycrystalline at lower annealing temperatures than in the exemplary data shown in Fig. 2(c).26,28,43These films show peaks in the XRD, however they were at least one order of magnitude thicker. We there- fore do not expect the YIG layer on Si/SiO xto be amorphous, which will be confirmed in the following. By probing the magnetic properties of the thin films with Fig. 1. Expected crystallization of an amorphous, as-deposited (a.d.) YIG thin film on lattice matched substrates (a) and non lat- tice matched substrates (b). In the first case of solid phase epitaxy, a homogeneous crystal front forms at the substrate and propagates towards the upper thin film border. For the latter, nucleation is nec- essary and crystallites form in various orientations. This results in a single crystalline (sc) film for the epitaxy and a polycrystalline (pc) film when nucleation occurs.3 Fig. 2. (a)-(c) XRD analysis of YIG thin films pre and post annealing on different substrates as given above in the respective column. The nominal positions of the substrate and the thin film are marked by the grey and black dashed lines, respectively. The additional peak marked with Si(200) in (c) is a detour reflex from the substrate. (d)-(f) Background corrected Kerr microscopy data in L-MOKE configuration for the same samples before and after the annealing procedure. The change in the measured gray value corresponds to a change in the magnetization of 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 to the surface normal as extracted from the Kikuchi-patterns determined by EBSD. The as-deposited films showed no Kikuchi-Patterns and are therefore not shown here. L-MOKE (cp. Fig. 2(d-f)), a clear distinction between amor- phous and crystalline YIG can be made. While the film shows a linear L-MOKE signal in the as-deposited state, it changes to a hysteresis for all three samples upon anneal- ing. In general, the sharpest hysteresis is visible for YIG on GGG, which becomes broader for an increasing structural misfit. Naïvely polycrystalline samples are expected to con- sist of multiple domains pointing towards different directions, which lead to an increase of the coercive field. This is con- sistent with our results and also with the magnetic properties found in literature.14,28,35,58These coercive fields are below0.1 mT for YIG on GGG14,35and between 2.2-3 mT for YIG on Si/SiO x.28,58The L-MOKE measurements therefore indi- cate the spontaneous formation of a phase with magnetic or- dering on all three substrates. For additional characterization of the magnetic properties of the films via ferromagnetic resonance and SQUID magne- tometry please refer to the supplemental information.59The corresponding data show the same dependence on the type of substrate, that is also apparent in the L-MOKE measurements. Once the YIG is fully crystallized, however, we do not find a dependence of the magnetic parameters of our thin films on4 the annealing parameters. While L-MOKE correlates the magnetic properties with amorphous and crystalline films, it lacks the ability to quan- tify the amount of crystalline YIG. The hysteretic response for the annealed YIG on SiO xstrongly supports the forma- tion of crystalline YIG, however, we cannot correlate this to a percentage of crystalline material. Therefore, a structural characterization with higher spacial resolution than XRD is needed. To that end electron back scatter diffraction (EBSD) mea- surements were performed. With this technique Kikuchi pat- terns, which are correlated to the crystal structure, are detected and later evaluated. The results are shown for crystalline sam- ples only, as the amorphous film showed no Kikuchi patterns. This confirms, that the detected patterns stem from the YIG thin film itself and not from the crystallographically simi- lar substrates of YAG or GGG. This is consistent with the EBSD signal depth given in the literature of 10 to 40 nm.60 The extracted crystal orientations along the surface normal can be seen in Fig. 2 (g-i). On YAG and GGG a single color corresponding to the <111> direction is visible in the mapping, which is consistent with the XRD data and corrob- orates SPE from the substrate in the <111> direction. On SiO x, however, various crystal directions are present, con- firming the polycrystalline nature of the YIG. The crystallo- graphic data from our EBSD measurements show random nu- cleation. The cross shape of the individual crystalline areas point towards an anisotropic crystallization with a preferen- tial direction along <110> or higher indexed directions like <112>, which is consistent with earlier studies on YIG and other rare earth garnets24,61–63as well as PLD grown bismuth iron garnet.11 The use of EBSD enables the quantification of the amount of crystalline material in a YIG thin film on SiO xor any ar- bitrary substrate. Combining the magnetic and structural data from L-MOKE and EBSD, respectively, allows for an unam- biguous identification of the formation of polycrystalline YIG on SiO x. We presume that the absence of any XRD peaks in the sym- metric θ−2θscan results from the small volume of the in- dividual crystallites of YIG on SiO x.34,37,48We approximate the volume of a single polycrystalline grain, i.e. one cross from the EBSD data (cp. Fig. 2(i)) to be 0 .5µm3, stemming from an area of about 15 µm and a film thickness of 32 nm. This is also the size of individually contributing grains to the diffraction within the XRD. Assuming a single crystalline thin film, where the whole irradiated area contributes additively, the contributing area amounts to 7 ·105µm3, which is six orders of magnitude larger than that of an individual grain. Therefore, the contributions of the individual grains of the YIG layer on SiO xto the XRD intensity are too small to result in a finite peak for a 30 nm thick film. These results provide the basis for the investigation of the crystallization behavior and reveal how different techniques enable us to distinguish between amorphous, partly and fully crystalline films. We utilize the structural information to ana- lyze the crystallization dynamics on the different substrates. The percentage of crystalline YIG was quantified differ- Fig. 3. Evolution of the crystallinity as a function of the annealing temperature for a constant annealing time of 4 h (a,c,e) and for dif- ferent times at a constant temperature of 600◦C on GGG, YAG (b,d) or 800◦C on SiOx(f). The dotted lines act as a guide to the eye. ently for the three different substrates. For YIG on YAG the amount of crystalline YIG correlates to the intensity of the Bragg peak. A certain film thickness corresponds to a max- imum area under the peak, to which the intensity is normal- ized. For YIG on GGG, the percentage of crystalline YIG is extracted from the Laue oscillations (cp. Fig 2(a)). The frequency of the oscillation corresponds to the number of in- terfering lattice planes, enabling the calculation of the thick- ness of the crystalline layer. Using X-ray reflectivity, the ab- solute film thickness was measured for each film. For these measurements and evaluation, please refer to the supplemen- tal information.59Comparing the thickness of the crystalline layer with the film thickness then enables to monitor the crys- tallization of YIG on GGG. For the non lattice matched sub- strates, EBSD mappings were taken to extract the amount of crystalline YIG. Further evaluation of partly crystalline YIG on SiO xcan be found in the supplemental information.59For each of the YIG thin films, a percentage of crystalline YIG at a given time and temperature is extracted, which allows an evaluation of the crystallization process for this specific tem- perature. First, we find the onset temperature for the crystalliza- tion of YIG on each substrate. As crystallization is ther- mally activated, it depends exponentially on the annealing temperature,64which leads to a very narrow temperature win- dow of incomplete crystallization. To extract this window, multiple samples were annealed for four hours at different temperatures. Figure 3 shows the results for YIG on GGG (a), YIG on YAG (c) and YIG on SiO x(e). At substrate de-5 pendent temperatures of 550◦C, 575◦C and 700◦C for YIG on GGG (a), YAG (c) and SiO x(e), respectively, a steep in- crease in the crystallinity can be seen. Towards higher tem- peratures the extracted value stays the same or is only slightly reduced, which suggests, that the YIG film is fully crystal- lized and no further changes are expected. A crystalline YIG film on YAG and GGG can therefore be obtained at a temper- ature range around 600◦C, whereas on SiO x, temperatures of approximately 700◦C are necessary. For our samples, the heating up and cooling down is in- cluded in the annealing time. An in-situ study on a represen- tative sample with dYIG=100nm yielded data in good agree- ment with the crystallization behavior in the one zone furnace. It should be noted that the use of different equipment led to a small variation in the absolute temperature, see supplemental information.59 The lower extracted crystallinities for YIG on YAG and GGG at 800◦C and above (cp. Fig. 3(a)+(c)) hint towards the occurrence of competing crystallization processes. We at- tribute the reduction in crystallinity at annealing temperatures above 800◦C to additional formation of polycrystals enabled by the elevated temperatures, which competes with the solid phase epitaxy and by that reduces the crystal quality of the thin film. Analyzing the rocking curves of these samples (see supplemental information) confirms an increased full width at half maximum value at higher temperatures.59This can be correlated with a lower crystal quality, which supports an ad- ditional crystallization process. To study the crystallization dynamics, the time evolution of the normalized crystallinity for a given temperature is eval- uated, shown in Fig. 3 for YIG on GGG(b), YAG(d) and SiO x(f). Here, a sample was subjected to the same temper- ature for multiple repeats until the extracted value and there- with the crystalline amount did saturate. This saturation can be seen on all substrates and represents a fully crystallized thin film, where no further changes are expected. To describe the crystallization at an arbitrary temperature, we find a general crystallographic description for each of the substrates. A phase transition in a solid like crystallization can generally be described by the Avrami equation:64–67 θc=1−e−k·tn(2) where θcis the crystallinity normalized to one, with respect to a complete crystallization, kthe rate constant and tthe anneal- ing time. The exponent nis often referred to as the Avrami ex- ponent and describes how the crystallization takes place.66It can take values between 1 and 4, where one contribution stems from the nucleation and takes values of 0 for controlled and 1 for random nucleation, while the other contributions origi- nate from the type of crystallization in the three spacial direc- tions. For the rate constant k we use an exponential Arrhenius dependency:54,68 k=k0·e−EA kB·T (3) where both the pre-factor k0and the activation energy EAare unique for each combination of film and substrate material.The Avrami equation (cp. Eq.(2)) lets us describe the crys- tallization on all three substrates. To that end we fit the nor- malized crystallinity values of YIG with the Avrami equation (cp. Eq.(2)), where we fix the Avrami exponent n between 1 and 4 (cp. Fig. 4(a)+(b)). The rate constants kthen describe the crystallization velocities on the respective substrate in h−1. The crystallization behavior of YIG on GGG and YAG at an annealing temperature of 600◦C is shown in Fig. 4(a). On GGG at 600◦C (cp. Fig. 4(a)), YIG immediately starts to crystallize with a rate constant of 1 .96 h−1and an Avrami exponent of 1. This means, that the crystallization takes place without nucleation and in one spacial direction, which is con- sistent with the monotonously moving crystallization front ex- pected for SPE. The rate constant translates to a initial veloc- ity of 0 .98 nm /min for the 30 nm films. Towards longer an- nealing times, the curve flattens, meaning that the crystalline material reaches the sample’s surface. The crystallization of YIG on YAG shows an initial time de- lay, despite the comparably small lattice mismatch of 3 .09 % Fig. 4. (a) and (b) show the time evolution of the YIG crystallization on the three substrates after normalizing the data with the maximum value to 1. The dots represent the crystallinity values from XRD (YAG/GGG) and EBSD (SiOx), while the solid lines show the fit of the data using Eq.(2). Because of the inherently different crystalliza- tion processes, the time scales and the temperatures differ. Conduct- ing these time evolutions at different temperatures for each substrate result in a rate constant k(T)for this temperature. A logarithmic rep- resentation of the k(T)values over the inverse temperature is given by the symbols in (c). For each substrate a linear expression was fitted, where the slope represents the activation Energy EAand the intercept of the y-axis the pre-factor k0for YIG on each substrate.6 Fig. 5. (a) Annealing parameters to obtain a fully crystalline YIG film on the respective substrates. We expect every point in the colored area to yield a fully crystalline sample. We use Eq.(4) with the values obtained in Fig. 4(c) to determine the boundary separating crystalline YIG (shaded areas, sc = single crystalline, pc = poly crystalline) from amorphous YIG (white areas). The open circles represent the samples from Fig. 4(c) which are used for the fit. Further studied, fully crystalline samples are marked by the full circles. There are different regions where the YIG is fully crystalline depending on the substrate. Panel (b) gives a comparison of our crystallization diagram with other studies.24–33,35–41 Note that, while we here consider only the crystallization of sputtered thin films by post annealing, the crystallization diagram also fits for comparable samples obtained by PLD (not shown here).11–17 (cp. Fig. 4(a)). The fitting of the data at 600◦C leads to a rate constant of 0 .10 h−1with n=3.8. This means, that the crys- tallization does not follow a typical SPE behavior and nucle- ation processes in the thin film cannot be excluded. However, also for the crystallization on YAG, single crystalline YIG is obtained (cp. Fig. 2(b)+(h)). This deviation from YIG on GGG is most likely due to the larger lattice mismatch which causes an energetically costly strain in the film.69The crys- tallization velocity along the surface normal direction is ob- tained by the n-th root out of the rate constant and translates to 0.27 nm /min. The crystallization of YIG on SiO xis fundamentally differ- ent (cp. Fig. 4(b)). Here, polycrystalline grains were found at temperatures of 675◦C and above. The time evolution of the crystallinity is depicted in Figure 4(b), where fitting the data by the Avrami equation (Eq. (2)) yields n=4 and a rate constant of 9 .9·10−5h−1. This confirms our initial hypothesis of nucleation and subsequent crystallization in three dimen- sions. Higher temperatures compared to the garnet substrates are needed to provide enough energy for nucleation, which causes the crystallization process to be visible at 675◦C and above. An approximation of the crystallization velocity can be ex- tracted from the EBSD data. Here, we assume that the crys-tallization starts in the middle of a cross shape structure (cp. Fig. 2(i)) and stops when reaching a boundary given by neigh- boring crystallites. The distance covered depends on the num- ber of nuclei formed and is highly dependent on the crystal- lographic direction. To ensure comparability with the two lat- tice matched substrates, we consider grains growing in plane along the <111> direction. At 700◦C, the YIG crystallites on SiO xmeasured up to 10 µm in length after at least 12 h of annealing. This translates into a propagation velocity of 16.7 nm /min at 700◦C on an arbitrary substrate along the <111> direction. To compare the three crystallization velocities, the temper- ature dependence of the rate constants kneeds to be taken into consideration. Using the Arrhenius equation (Eq. (3)) we are able to extrapolate the crystallization rate at any temperature. To that end, the logarithm of each rate constant is plotted over the inverse temperature. The linear dependency of Eq. (3) in the logarithmic plot allows us to extract the activation energy and the pre-factor k0for YIG on each substrate. The resulting values are plotted in Tab. I. While at first glance the crys- tallization velocity for YIG on SiO xseems faster, the differ- ent annealing temperatures of 600◦C for the garnet substrates and 700◦C for SiO xneed to be taken into account (cp. Fig.4). Extrapolating the crystallization velocity for YIG on GGG at7 700◦C reveals that here YIG would crystallize approximately 30 times faster than on SiO x. Our activation energy of 3 .98 eV for YIG on GGG is in good agreement with the literature. For the formation of bulk YIG from oxide powders, a value of 5 .08 eV was reported.70 Further, for the crystallization of bulk polycrystalline YAG, which is expected to behave similarly as it has the same crys- tal structure, an activation energy of 4 .5 eV was found.71The lower value of 3 .98 eV for YIG on GGG highlights the re- duced energy needed, due to the SPE from the lattice matched GGG. The activation energies for YIG on YAG as well as on SiO x are much higher than the value on GGG. As the general crys- tallization windows and times needed for a fully crystalline film stay the same, we ascribe this behavior to a kinetic block- ing, originating from the lattice mismatch and the nucleation. Understanding the exact mechanism however, would need fur- ther study. These results allow to establish a diagram to underline which annealing parameters will lead to a fully crystalline YIG thin film on the three substrates (cp. Fig. 5(a)). For a mathematical description, we combine the Avrami equation Eq. (2) with the Arrhenius equation Eq. (3) to be able to express the crystallinity in terms of annealing time and tem- perature. t= −ln(1−θc) k0 ·eEA kBT1 n (4) We use a crystallinity θcof 0.999 to avoid the divergence of the logarithm and the respective n,k0andEAfound in Tab. I. Figure 5(a) outlines the temperature and time combination where crystalline YIG (shaded areas) can be obtained. Re- gions where the YIG thin film remains amorphous are left in white. The boundary between non crystalline and crystalline for each substrate is given by Eq. (4). Each of the circles seen in Fig. 5(a) represents one fully crystalline sample obtained as described for Fig. 3(b). The filled circles represent fully crys- talline samples, where no time dependence of the crystallinity was measured. As already anticipated, YIG exhibits different crystallization behavior depending on the substrate. Note, that the formation of polycrystalline YIG on SiO xor any arbitrary substrate needs notably higher temperatures than SPE, where an annealing at 660◦C for 100 h would be necessary to result in a fully crystalline film. The different temperatures and times necessary to induce crystallization stem from the different types of substrates. For Tab. I. Extracted activation energies EAand pre-factors k0for YIG on each substrate EA(eV) k0(1/h) n YIG on GGG 3 .98±0.32 2 .0·10231 YIG on YAG 15 .70±1.59 2 .6·10893.8 YIG on SiOx 16.37±0.85 8 .4·10804YIG on GGG and YAG the seed for the crystallization is given by the lattice of the substrate. Therefore, we ascribe the dis- crepancy between YAG and GGG to the different lattice mis- match compared to YIG. In the YIG thin films on YAG a higher strain is expected to exist in the film, which leads to the formation of energetically costly dislocations. This in turn results in the slightly higher temperature needed for YIG to crystallize on YAG. On SiO x, however, a significantly higher temperature than for the lattice matched substrates is needed for crystalline YIG to form. Here, as no initial seed is given by the substrate, nucleation is required, which is a thermally activated process that needs additional energy, i.e. higher tem- peratures. This random formation of seeds leads to a polycrys- talline YIG thin film on SiO x A comparison with the literature shows, that parame- ters which have been previously reported to result in a fully crystalline YIG layer, fit into our extracted area, (cp. Fig. 5(b)).24–33,35–41Additionally to the sputtered films, also amorphous films obtained from PLD with subsequent anneal- ing fit in the observed regions.11–17The extracted diagram in Fig. 5 therefore acts as a general description for the crystal- lization of YIG thin films out of the amorphous phase. IV. CONCLUSION Extensive time and temperature series were used to ana- lyze the crystallization kinetics of sputtered amorphous YIG thin films on different substrates. We find the formation of single crystalline YIG thin films on garnet substrates where the crystallization on gadolinium gallium garnet can be co- herently described in a solid phase epitaxy picture, whereas a more complicated crystallization scheme is found on yttrium aluminum garnet. On SiO xa polycrystalline YIG thin film develops, with slower crystallization dynamics than for the garnet substrates. A fully crystalline YIG film on GGG was found for tem- peratures as low as 537◦C and annealing times of 110 h. On silicon oxide (representing any type of amorphous or non lat- tice matched substrate), the nucleation of the YIG crystals is not expected for reasonable time scales below 660◦C. The results summarized in Tab. I allow for the determination of the crystallization velocity of YIG on those substrates for any temperature. Thus, we provide a complete description of the crystalliza- tion process from the amorphous phase for YIG on GGG, YAG and arbitrary substrates such as SiO x, which allows us to define the range in which crystalline YIG thin films can be obtained. V. ACKNOWLEDGMENTS This work was funded by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) – Project-ID 446571927 and via the SFB 1432 - Project-ID 425217212. We cordially thank F. Michaud and J. Ben Youssef from the Uni- versité de Bretagne Occidentale in Brest (France) for fruitful8 discussions and for letting us use their in-situ X-ray diffrac- tometer. We also gratefully acknowledge technical support and advice by the nano.lab facility of the University Konstanz. VI. REFERENCES 1F. Bertaut and F. Forrat, C. R. Acad. Sci. 242, 382 (1956). 2S. Geller and M. Gilleo, J. Phys. Chem. Solids 3, 30 (1957). 3Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 4M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprägs, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y .-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 5L. Cornelissen, J. Liu, R. Duine, J. B. Youssef, and B. Van Wees, Nat. Phys. 11, 1022 (2015). 6J. F. Dillon, Phys. Rev. 105, 759 (1957). 7V . Cherepanov, I. Kolokolov, and V . L’vov, Phys. Rep. 229, 81 (1993). 8H. L. Wang, C. H. Du, Y . Pu, R. Adur, P. C. Hammel, and F. Y . Yang, Phys. Rev. B 88, 100406(R) (2013). 9H. Sakimura, T. Tashiro, and K. Ando, Nat. Commun. 5, 5730 (2014). 10C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. Hübner, and E. Wendler, Phys. Rev. Materials 4, 024416 (2020). 11A. Heinrich, S. Leitenmeier, T. Körner, R. Lux, M. Herbort, and B. Stritzker, J. Magn. Soc. Jpn. 30, 584 (2006). 12Y . Krockenberger, H. Matsui, T. Hasegawa, M. Kawasaki, and Y . Tokura, Appl. Phys. Lett. 93, 092505 (2008). 13M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas, S. Khartsev, and J. Åk- erman, J. Appl. Phys. 117, 17D119 (2015). 14C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. Ebbinghaus, and G. Schmidt, Sci. Rep. 6, 20827 (2016). 15C. Hauser, C. Eisenschmidt, T. Richter, A. Müller, H. Deniz, and G. Schmidt, J. Appl. Phys. 122, 083908 (2017). 16F. Heyroth, C. Hauser, P. Trempler, P. Geyer, F. Syrowatka, R. Dreyer, S. G. Ebbinghaus, G. Woltersdorf, and G. Schmidt, Phys. Rev. Applied 12, 054031 (2019). 17G. Gurjar, V . Sharma, S. Patnaik, and B. K. Kuanr, Mater. Res. Express 8, 066401 (2021). 18O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carrétéro, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebour- geois, J.-C. Mage, G. De Loubens, O. Klein, V . Cros, and A. Fert, Appl. Phys. Lett. 103, 082408 (2013). 19M. Shone, Circuits, Syst. Signal Process. 4, 89 (1985). 20P. Görnert, R. Hergt, E. Sinn, M. Wendt, B. Keszei, and J. Vandlik, J. Cryst. Growth 87, 331 (1988). 21J.ˇCermák, A. Abrahám, T. Fabián, P. Kaboš, and P. Hyben, J. Magn. Magn. Mater. 83, 427 (1990). 22N. Beaulieu, N. Kervarec, N. Thiery, O. Klein, V . Naletov, H. Hurdequint, G. De Loubens, J. B. Youssef, and N. Vukadinovic, IEEE Magn. Lett. 9, 1 (2018). 23A. R. Will-Cole, J. L. Hart, V . Lauter, A. Grutter, C. Dubs, M. Lindner, T. Reimann, N. R. Valdez, C. J. Pearce, T. C. Monson, J. J. Cha, D. Heiman, and N. X. Sun, Phys. Rev. Materials 7, 054411 (2023). 24M.-B. Park and N.-H. Cho, J. Magn. Magn. Mater. 231, 253 (2001). 25P. W. Jang and J. Y . Kim, IEEE Trans. Magn. 37, 2438 (2001). 26T. Boudiar, B. Payet-Gervy, M.-F. Blanc-Mignon, J.-J. Rousseau, M. Le Berre, and H. Joisten, J. Magn. Magn. Mater. 284, 77 (2004). 27S. Yamamoto, H. Kuniki, H. Kurisu, M. Matsuura, and P. Jang, Phys. Stat. Sol. (a) 201, 1810 (2004). 28Y .-M. Kang, S.-H. Wee, S.-I. Baik, S.-G. Min, S.-C. Yu, S.-H. Moon, Y .-W. Kim, and S.-I. Yoo, J. Appl. Phys. 97, 10A319 (2005). 29A. D. Block, P. Dulal, B. J. H. Stadler, and N. C. A. Seaton, IEEE Photonics J.6, 1 (2014).30T. Liu, H. Chang, V . Vlaminck, Y . Sun, M. Kabatek, A. Hoffmann, L. Deng, and M. Wu, J. Appl. Phys. 115, 17A501 (2014). 31H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann, L. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014). 32J. Lustikova, Y . Shiomi, Z. Qiu, T. Kikkawa, R. Iguchi, K. Uchida, and E. Saitoh, J. Appl. Phys. 116, 153902 (2014). 33M. B. Jungfleisch, W. Zhang, W. Jiang, H. Chang, J. Sklenar, S. M. Wu, J. E. Pearson, A. Bhattacharya, J. B. Ketterson, M. Wu, and A. Hoffmann, J. Appl. Phys. 117, 17D128 (2015). 34Y . Zhang, J. Xie, L. Deng, and L. Bi, IEEE Trans. Magn. 51, 1 (2015). 35S. Li, W. Zhang, J. Ding, J. E. Pearson, V . Novosad, and A. Hoffmann, Nanoscale 8, 388 (2016). 36J. F. K. Cooper, C. J. Kinane, S. Langridge, M. Ali, B. J. Hickey, T. Niizeki, K. Uchida, E. Saitoh, H. Ambaye, and A. Glavic, Phys. Rev. B 96, 104404 (2017). 37J. Lian, Y . Chen, Z. Liu, M. Zhu, G. Wang, W. Zhang, and X. Dong, Ceram. Int.43, 7477 (2017). 38A. Talalaevskij, M. Decker, J. Stigloher, A. Mitra, H. S. Körner, O. Ces- pedes, C. H. Back, and B. J. Hickey, Phys. Rev. B 95, 064409 (2017). 39N. Zhu, H. Chang, A. Franson, T. Liu, X. Zhang, E. Johnston-Halperin, M. Wu, and H. X. Tang, Appl. Phys. Lett. 110, 252401 (2017). 40J. Ding, C. Liu, Y . Zhang, U. Erugu, Z. Quan, R. Yu, E. McCollum, S. Mo, S. Yang, H. Ding, X. Xu, J. Tang, X. Yang, and M. Wu, Phys. Rev. Appl. 14, 014017 (2020). 41J. Ding, T. Liu, H. Chang, and M. Wu, IEEE Magn. Lett. 11, 1 (2020). 42P. Jang, S. Yamamoto, and H. Kuniki, Phys. Stat. Sol. (a) 201, 1851 (2004). 43M. Roumie, B. A. Samad, M. Tabbal, M. Abi-Akl, M.-F. Blanc-Mignon, and B. Nsouli, Mater. Chem. Phys. 124, 188 (2010). 44H. Bai, X. Z. Zhan, G. Li, J. Su, Z. Z. Zhu, Y . Zhang, T. Zhu, and J. W. Cai, Appl. Phys. Lett. 115, 182401 (2019). 45J.-H. Seol, J.-H. An, G.-W. Park, T. Nguyen Thi, D. Duong Viet, B.-G. Park, P. C. Van, and J.-R. Jeong, Thin Solid Films 774, 139846 (2023). 46A. Mitra, O. Cespedes, Q. Ramasse, M. Ali, S. Marmion, M. Ward, R. M. D. Brydson, C. J. Kinane, J. F. K. Cooper, S. Langridge, and B. J. Hickey, Sci. Rep. 7, 11774 (2017). 47C. L. Jermain, S. V . Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brang- ham, M. R. Page, P. C. Hammel, F. Y . Yang, and D. C. Ralph, Phys. Rev. B95, 174411 (2017). 48P. Cao Van, T. T. Nguyen, V . D. Duong, M. H. Nguyen, J.-H. Seol, G.-W. Park, G.-H. Kim, D.-H. Kim, and J.-R. Jeong, Curr. Appl. Phys. 42, 80 (2022). 49S. Gates-Rector and T. Blanton, Powder Diffr. (PDF: 00-013-0493) 34, 352 (2019). 50S. Gates-Rector and T. Blanton, Powder Diffr. (PDF: 00-033-0040) 34, 352 (2019). 51S. Gates-Rector and T. Blanton, Powder Diffr. (PDF: 00-043-0507) 34, 352 (2019). 52R. Gross and A. Marx, Festkörperphysik , 2nd ed. (De Gruyter, Berlin ; Boston, 2014). 53S. Gates-Rector and T. Blanton, Powder Diffr. (PDF: 00-005-0565) 34, 352 (2019). 54L. Csepregi, J. Mayer, and T. Sigmon, Phys. Lett., A 54, 157 (1975). 55L. Csepregi, J. W. Mayer, and T. W. Sigmon, Appl. Phys. Lett. 29, 92 (1976). 56Y . Chen, M. H. Yusuf, Y . Guan, R. Jacobson, M. G. Lagally, S. E. Babcock, T. F. Kuech, and P. G. Evans, ACS Appl. Mater. Interfaces 9, 41034 (2017). 57M. Renninger, Z. Phys. 106, 141 (1937). 58H. Zheng, J. Zhou, J. Deng, P. Zheng, L. Zheng, M. Han, Y . Yang, L. Deng, and H. Qin, Mater. Lett. 123, 181 (2014). 59See supplemental information at [URL] for further structural and mag- netic characterization of the YIG thin films. For the evaluation of the mag- netic properties ferromagnetic resonance (FMR) measurements, SQUID and MOKE magnetometry are presented. X-ray based techniques are uti- lized to access the film thicknesses and the crystalline quality. Additionally, we confirm the time evolution of the crystallization process by in-situ X-ray measurements. Further information regarding the surface morphology and the characteristics of partly crystalline YIG thin films on SiOxare obtained using SEM based techniques. 60D. Dingley, J. Microsc. 213, 214 (2004). 61J. W. Nielsen and E. F. Dearborn, J. Phys. Chem. Solids 5, 202 (1958).9 62W. Tolksdorf and I. Bartels, J. Cryst. Growth 54, 417 (1981). 63P. Bennema, E. A. Giess, and J. E. Weidenborner, J. Cryst. Growth 62, 41 (1983). 64M. Avrami, J. Chem. Phys. 7, 1103 (1939). 65J. William and R. Mehl, Trans. Metall. Soc. AIME 135, 416 (1939). 66M. Avrami, J. Chem. Phys. 8, 212 (1940). 67M. Avrami, J. Chem. Phys. 9, 177 (1941).68W. D. Callister, Materials science and engineering: an introduction , 7th ed. (John Wiley & Sons, New York, 2007). 69H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. B 89, 134404 (2014). 70W. F. F. Wan Ali, M. Othman, M. F. Ain, N. S. Abdullah, and Z. A. Ahmad, J. Am. Ceram. Soc. 99, 315 (2016). 71B. R. Johnson and W. M. Kriven, J. Mater. Res. 16, 1795 (2001).

2023-08-01

Yttrium iron garnet (YIG) is a prototypical material in spintronics due to its exceptional magnetic properties. To exploit these properties high quality thin films need to be manufactured. Deposition techniques like sputter deposition or pulsed laser deposition at ambient temperature produce amorphous films, which need a post annealing step to induce crystallization. However, not much is known about the exact dynamics of the formation of crystalline YIG out of the amorphous phase. Here, we conduct extensive time and temperature series to study the crystallization behavior of YIG on various substrates and extract the crystallization velocities as well as the activation energies needed to promote crystallization. We find that the type of crystallization as well as the crystallization velocity depend on the lattice mismatch to the substrate. We compare the crystallization parameters found in literature with our results and find an excellent agreement with our model. Our results allow us to determine the time needed for the formation of a fully crystalline film of arbitrary thickness for any temperature.

Crystallization Dynamics of Amorphous Yttrium Iron Garnet Thin Films

2308.00412v2

Spin Hall magnetoresistance in heterostructures consisting of noncrystalline paramagnetic YIG and Pt Michaela Lammel,1, 2,a)Richard Schlitz,3Kevin Geishendorf,1, 2Denys Makarov,4Tobias Kosub,4Savio Fabretti,3 Helena Reichlova,3Rene Huebner,4Kornelius Nielsch,1, 2, 5Andy Thomas,1and Sebastian T.B. Goennenwein3,b) 1)Institute for Metallic Materials, Leibnitz Institute of Solid State and Materials Science, 01069 Dresden, Germany 2)Technische Universit at Dresden, Institute of Applied Physics, 01062 Dresden, Germany 3)Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden, 01062 Dresden, Germany 4)Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany 5)Technische Universit at Dresden, Institute of Materials Science, 01062 Dresden, Germany (Dated: 30 January 2019) The spin Hall magnetoresistance (SMR) eect arises from spin-transfer processes across the interface be- tween a spin Hall active metal and an insulating magnet. While the SMR response of ferrimagnetic and antiferromagnetic insulators has been studied extensively, the SMR of a paramagnetic spin ensemble is not well established. Thus, we investigate herein the magnetoresistive response of as-deposited yttrium iron gar- net/platinum thin lm bilayers as a function of the orientation and the amplitude of an externally applied magnetic eld. Structural and magnetic characterization show no evidence for crystalline order or sponta- neous magnetization in the yttrium iron garnet layer. Nevertheless, we observe a clear magnetoresistance response with a dependence on the magnetic eld orientation characteristic for the SMR. We propose two models for the origin of the SMR response in paramagnetic insulator/Pt heterostructures. The rst model de- scribes the SMR of an ensemble of non-interacting paramagnetic moments, while the second model describes the magnetoresistance arising by considering the total net moment. Interestingly, our experimental data are consistently described by the net moment picture, in contrast to the situation in compensated ferrimagnets or antiferromagnets. Spin Hall magnetoresistance (SMR)1{3is commonly observed in ferrimagnetic insulator (FMI)/normal metal (NM) heterostructures when the metal exhibits a large spin-orbit coupling. The SMR arises due to the interplay of the spin-transfer torque, the spin Hall eect (SHE) and the inverse spin Hall eect at the FMI/NM interface.4{6 While the SMR eect is usually discussed in terms of the total (net) magnetization,1recent experimental work showed that the SMR does not only probe the net magne- tization of FMIs, but is also sensitive to the contributions of the dierent magnetic sublattices.7,8This observation is key to understand the SMR response of more com- plex magnetic systems, such as canted ferrimagnets7{9, antiferromagnets10{15, spin spirals16or helical phases.17 To date, SMR measurements have been performed ex- tensively in samples with dierent long-range (sponta- neous) magnetic ordering.2,7,10,16{18In contrast, param- agnetic materials have not been in the focus of prior work done for SMR measurements. However, the magnetore- sistive response of paramagnetic materials is an interest- ing topic. For example, magnetoresistance measurements were recently performed in a gated paramagnetic ionic liquid.19The presence of SMR has been reported by two a)Electronic mail: m.lammel@ifw-dresden.de b)Electronic mail: sebastian.goennenwein@tu-dresden.degroups in dierent magnetically ordered materials, in the paramagnetic phase above the ordering temperature.16,18 Since the SMR is primarily studied in the magnetically ordered phase in those works, the authors do not provide a microscopic picture for the SMR in a randomly ordered spin ensemble. Therefore, in this work, we systematically study the SMR in a paramagnetic insulator (PMI)/spin Hall metal bilayer and critically compare the experimen- tal results to the SMR expected from two dierent micro- scopic models: one model assumes an ensemble of nonin- teracting moments, while the other model considers the (induced) net magnetization. More specically, we in- vestigate bilayers fabricated by sputtering of Y 3Fe5O12 (YIG) and Pt at room temperature. These heterostruc- tures do not show a crystalline order of the YIG layer or spontaneous magnetization, such that we take the YIG layer to be paramagnetic, but they nevertheless exhibit a clear SMR-like magnetoresistive response. The YIG/Pt bilayers were fabricated via sputtering at room temperature from 2 inch YIG and Pt tar- gets on commercially available (111)-oriented single- crystalline yttrium aluminum garnet (Y 3Al5O12, YAG) substrates.2,7,9To rule out crystallization of the de- posited YIG layer on the YAG substrate due to the low lattice mismatch, reference samples were fabricated in the same manner on (100)-oriented Si wafers ter- minated by a thermal oxide layer of 1 µm. The sub- strates were immersed in isopropanol and ethanol andarXiv:1901.09986v1 [cond-mat.mes-hall] 28 Jan 20192 a) b) t jnc) d)YIGPt sub. FIG. 1. a) Schematic of the YIG/Pt bilayer sample. b) X- ray diraction -2scan of a typical noncrystalline YIG/Pt bilayer. The colored vertical lines give the expected positions for dierent crystalline diraction peaks. c) Normalized mag- netization of a noncrystalline YIG/Pt bilayer and two YAG substrates as a function of temperature. The light blue shad- ing around the data indicates the scatter of two subsequent measurement runs. The observed moment is negative due to the diamagnetic substrate. d) Electrical contacting scheme of the patterned sample. cleaned in an ultrasonic bath prior to the deposition. The YIG layer was deposited via RF sputtering at 80 W for 6000 s. Subsequently Pt was deposited using DC sput- tering for 73 s at 30 W without breaking the vacuum. A schematic of a typical stack is given in Fig.1a. The above-mentioned sputtering parameters resulted in layer thicknesses of d YIG= (301) nm for the YIG layer and dPt= (2:50:5) nm for the Pt layer, as conrmed by X-ray re ectometry. X-ray diraction measurements were performed using a Bruker D8 Advanced diractometer equipped with a cobalt anode. As shown in Fig.1b, we do not observe diraction peaks that could be linked to YIG. We take this as evidence that the YIG does not grow as small crys- tallites, but rather as an unordered \amorphous" layer. Therefore such YIG layers will be referred to as \non- crystalline" in the following. In contrast, the Pt layer is textured inh111i-direction which has been reported to be the preferred orientation direction of Pt deposited at room temperature.20The sharp step at 2 = 60 deg re- sults from the iron lter that is used to suppress the Co K radiation. TEM studies on the YIG/Pt bilayers addition- ally conrm the noncrystallinity of the YIG layer while energy-dispersive X-ray spectroscopy analyses show the YIG layer to be stoichometrically identical to the YAG substrate. For the magnetic characterization, a Quan- tum Design MPMS-XL7 SQUID magnetometer with re- ciprocating sample option was used. Figure 1c shows themagnetization as a function of temperature measured at 500 mT after cooling the sample in zero magnetic eld. As a reference, two substrates were measured by the iden- tical procedure. To ensure comparability and to account for dierences in sample size, the data were normalized to the magnetization at 300 K. Comparing the normalized M(T) data from the YIG/Pt bilayer to the bare YAG substrates, we conclude that within the measurement error, no (spontaneous) magnetization of the lm can be detected in our samples. Moreover, the samples ex- hibit only the negative magnetization expected for a dia- magnetic substrate. The low temperature paramagnetic- like behavior is likely caused by paramagnetic dopants that were consistently observed in the commercial YAG substrates.21 For magnetotransport measurements, Hall bars with a contact separation of l= 400 µm along the direction of current ow and a width of w= 50 µm were dened by using optical lithography and consecutive Ar ion etching. Subsequently, the samples were mounted into a chip car- rier and contacted by aluminum wire bonding. The elec- tric contacting scheme as well as the used coordinate sys- tem is given in Fig.1d. A current I= 90 µA was applied along the Hall bar (along jdirection) utilizing a Keith- ley 2450 sourcemeter. To decrease the noise level and to enhance the sensitivity, a current reversal technique was used.22The longitudinal voltage V, i.e. the voltage drop along the direction of current ow, was measured by a Keithley 2182 nanovoltmeter. Field orientation dependent magnetoresistance mea- surements at dierent temperatures and in three orthog- onal rotation planes were performed in a 3D vector mag- netic eld cryostat. The in-plane rotation of a constant external magnetic eld Haround the surface normal nis herein denoted as ip (angle ), the out-of-plane rotation around the current direction jas oopj (angle ) and the out-of-plane rotation around the tdirection as oopt (an- gle ), as is shown above Fig.2a, b and c, respectively. In a model FMI/NM system with one single magnetic sublattice pointing along the magnetization unit vector m=M M, the SMR can be described by:1 =0+
(1mt2) =0+
[1sin2(;)](1) where
 >0 gives the change of resistance as a func- tion of the projection of the magnetization unit vector mon the tdirection m t, as dened above. Thus, follow- ing Eq.1, the resistance in a typical SMR measurement is minimal for mjjtand maximal for m?t. Figure 2 shows the dependence of the magnetoresistance on the angles,and , at 200 K for dierent amplitudes of the external magnetic eld. For each eld amplitude, the voltageVis recorded for clockwise and anticlockwise ro- tation of the magnetic eld direction, and the data are averaged before normalization to account for slow tem- perature drifts. For the ip and the oopj rotation, the data (open symbols) can be well described by a sin2(;)3 t jn t jn t jn a) b) c)ip oopj oopt FIG. 2. Magnetoresistance measurements as a function of magnetic eld orientation, recorded at 200 K in three or- thogonal rotation planes. The rotation geometries are dis- played above the corresponding panel. Measurements were performed at dierent, xed magnetic eld strengths 0H= 0:5 T, 1 T, 1:5 T and 2 T, which are given by the open blue rhombi, yellow triangles, red squares and gray circles, respec- tively. A sin2(;; ) t to the data is given by the solid line in the associated color (cf. Eq.1). dependence (solid line), whereas no modulation is vis- ible in the oopt rotation. This angular dependence is characteristic for both the SMR1,2(cf. Eq.1) and the Hanle magnetoresistance (HMR).23The HMR is micro- scopically ascribed to the dephasing of the spin accumu- lation at the Pt interface, also exists in pure Pt without the adjacent magnetic layer, and scales with the exter- nal magnetic eld H. In contrast, the SMR depends on the magnetization orientation m, which is expected to be eld dependent in a paramagnetic material. However, we do not expect the HMR to contribute signicantly to our results, since the magnitude of the HMR for the external magnetic elds used in our measurements ( 0H2 T) is reported to be
HMR=02:5106and there- fore roughly one order of magnitude smaller than the MR ratio observed here23
=0=3105. There- with,
=0of the noncrystalline YIG/Pt bilayers on YAG is roughly one order of magnitude smaller than in comparable samples featuring a crystalline, ferrimagnetic YIG layer.2,3Similar measurements performed on refer- ence non-crystalline YIG/Pt bilayers, in particular also on the ones on Si/SiO 2substrates, show the same de- pendencies, with a magnetoresistance in the same order of magnitude, further supporting the fact that it is in- deed the non-crystalline YIG layer that is responsible for the SMR. The angular dependence additionally dis- putes long-range antiferromagnetic ordering in our bi- layers, since a shift of the extrema by 90 deg compared to the SMR introduced in Eq.1 would be expected for an AFM.10{12,18,24The existence of a magnetoresistance in PMI/Pt heterostructures has already been reported above the Curie temperature for CoCr 2O4/Pt bilayers a) b) c)FIG. 3. Field-dependent magnetoresistance measurements at dierent temperatures for Hjjj(panel a), Hjjt(panel b) and Hjjn(panel c). Measurements were performed at dierent temperatures T = 10 K, 100 K, 200 K and 300 K, which are given by the open gray rhombi, blue triangles, violet squares and red circles, respectively. with a MR ratio of
=0<2106by Aqeel et al.16 and above the Neel temperature for Cr 2O3/Pt bilayer structures by Schlitz et al.18with
=0>1104. It has also been reported that no MR was detectable in paramagnetic Gd 3Ga5O12(GGG)/Pt heterostructures at room temperature. In contrast to the other systems, the magnetic moment of GGG has its origin in the 4 felec- trons (vs. 3 delectrons), which have been suggested not to couple well to the spin accumulation of the Pt layer due to their strong localization.18Therewith, the reported values of the MR ratio in paramagnetic phases vary be- tween zero and nearly the MR of crystalline YIG/Pt bilayers.16,18Our data fall well within this range. We, however, address a real paramagnet and not the param- agnetic phase of a system that orders at lower tempera- tures. Field-dependent measurements were performed on the same sample and in the same experimental setup as de- scribed before. The resistivity ratio =H=01 forHjjj (Hjjt,Hjjn) is given in Fig.3a (b, c). Note that the re- sults are normalized with respect to the value at zero magnetic eld to ensure comparability of the data ac- quired for dierent temperatures. An increase in the external magnetic eld magnitude along the jandndi- rection leads to an increase in resistivity. For an ex- ternal magnetic eld applied along tdirection, how- ever, no substantial modulation of the resistivity is ob- served. For these results a signicant contribution of the HMR is again excluded since the reported resistivity ratio
=2:5106at 2 T is one order of magnitude lower than the values for the same eld magnitude in Fig.3.23 For low temperatures, an increase in the external mag- netic eld does lead to a parabolic increase in the tand ndirection that we ascribe to a Kohler's rule type ordi- nary magnetoresistance in the Pt layer.25Additionally, weak antilocalization has been reported to occur in Pt thin lms on various substrates for temperatures below 50 K which might give an additional contribution to the eld dependency.23,26,27No saturation of the resistance can be observed in our samples, even for the highest mag- netic elds that can be applied in our setup ( 2 T in the4 a) b) c) Hsat -Hsat0d) e) f) Hsat -Hsat0 Hsat -Hsat0jt n FIG. 4. Assuming that all magnetic moments in a param- agnet contribute individually to the SMR (i.e. hmt2i), one obtains the evolution of the resistivity sketched as a blue line forHjjj(a),Hjjt(b),Hjjn(c). Presuming that the SMR is dependent on the net magnetization (i.e. hmti2), the expected magnetoresistive response is shown as a red line in panel d, e and f, for Hjjj,HjjtandHjjn, respectively. The external magnetic eld applied in each direction is indicated by the arrows below the panels. The alignment of the magnetic mo- ments for dierent external eld amplitudes is given by the arrows on above and below the resistance curves. jandtdirection and6 T in the ndirection) and the lowest temperatures accessible (10 K). We now compare the SMR expected from a model that considers an ensemble of non-interacting moments (Fig.4a-c) with the SMR arising in a model that addresses the total net moment (Fig.4d-f). To that end, we con- sider external magnetic elds applied along j,tandndi- rection as schematically shown above the panels in Fig.4. In a paramagnet, the moments at zero magnetic eld are unordered and point in random directions, as schemati- cally sketched in Fig.4. For a suciently large magnetic eld H! Hsat, though, the majority of the magnetic moments are aligned collinear to the external eld. Since the SMR is sensitive to m t2, one would expect the small- est resistivity (i.e. 0) when all moments are parallel to thetdirection and the largest resistivity (i.e. 0+
) when all moments are perpendicular to the tdirection (c.f. Eq.1). Thus, 0is the value expected for open boundary conditions (i.e. no magnetic layer), while the introduction of a magnetic layer to the system can only lead to an increase in the resistivity depending on its magnetization orientation.1 Presuming non-interacting moments in the PMI, the contribution of each moment to the SMR is considered separately. Thus, m t2in Eq.1 has to be understood as hmt2i. Applying suciently large magnetic elds leads to a saturation of the resistivity at a minimum value 0 forHjjt(Fig.4b) and maximum value 0+
forHjjj;n (Fig.4a, c). In turn, this implies that for zero magnetic eld an intermediate resistance (H = 0) is expected with0<(H = 0)<0+
, since some but not all magnetic moments are collinear to the tdirection (panel a, b and c of Fig.4 at H = 0). Assuming that the SMR depends on the total net mag- netization in the paramagnetic layer instead, m t2in Eq.1 has to be understood as hmti2. Hence, no magnetic mo- ment is expected for zero external magnetic eld, which leads to a vanishing SMR and therefore to the minimum resistivity value 0(Fig.4d-f). Applying suciently large external magnetic elds along jandnleads to an increase in resistivity up to the saturation value of 0+
as is shown in Fig.4d and e. In contrast, no change in the resistivity is expected for elds applied along tdirection (Fig.4e). Comparing the expected behavior from Fig.4 with the eld-dependent measurements in Fig.3, shows that the measurements do not agree with the SMR stemming from an ensemble of non-interacting moments (see Fig.4a-c). Instead, the measurements corroborate that the SMR in the paramagnetic YIG/Pt bilayers is determined by the total net magnetization of the system (cf. Fig.4d-e). This result contradicts previous ndings in compensated garnets and antiferromagnets where the results were ex- plained by taking into consideration the magnetizations of the dierent sublattices separately.7,8,10,11To conrm the saturation of the eect and to corroborate our ob- servations, further experimental work on dierent PMI materials, as well as at higher external magnetic elds and/or lower temperatures, is necessary. In summary, we have studied the magnetoresistive response in noncrystalline, paramagnetic YIG/Pt het- erostructures. Upon rotating the external magnetic eld at a xed magnitude in dierent planes, we observe a magnetoresistance with the characteristics of the spin Hall magnetoresistance with a magnitude of j
=j= 3105at0H = 2 T and 200 K. Field-dependent mea- surements show an increase of resistivity for an increasing magnetic eld along jandndirection, whereas no change of the resistance is observed for elds applied along tdi- rection. No saturation is detected for the maximum mag- netic elds accessible in our setup. Furthermore, we pro- pose two possible models for the origin of the SMR in a simple paramagnetic insulator/Pt heterostructure taking into consideration either an ensemble of non-interacting moments (i.e.hmt2i), or the total net magnetization (i.e. hmti2). Comparing the experimentally observed signa- ture with those models, we nd that the data are better described in terms of the total moment picture. Thus, we conclude that in a paramagnetic insulator, the mo- ments do not contribute individually, but in a collective net fashion to the SMR. We thank K. Nenkov and B. Weise for technical support. We acknowledge nancial support by the Deutsche Forschungsgemeinschaft via SPP 1538 (project GO 944/4). 1Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Physical Review B87, 144411 (2013).5 2M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Alt- mannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Physical Review B 87, 224401 (2013). 3H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kaji- wara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Physical Review Letters 110, 206601 (2013). 4D. Ralph and M. Stiles, Journal of Magnetism and Magnetic Materials 320, 1190 (2008). 5M. Dyakonov and V. Perel, Physics Letters A 35, 459 (1971). 6J. E. Hirsch, Physical Review Letters 83, 1834 (1999). 7K. Ganzhorn, J. Barker, R. Schlitz, B. A. Piot, K. Ollefs, F. Guillou, F. Wilhelm, A. Rogalev, M. Opel, M. Althammer, S. Gepr ags, H. Huebl, R. Gross, G. E. W. Bauer, and S. T. B. Goennenwein, Physical Review B 94, 094401 (2016). 8B.-W. Dong, J. Cramer, K. Ganzhorn, H. Y. Yuan, E.-J. Guo, S. T. B. Goennenwein, and M. Klui, Journal of Physics: Con- densed Matter 30, 035802 (2018). 9S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Applied Physics Letters 101, 262407 (2012). 10J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vlietstra, M. Althammer, H. Huebl, M. Opel, R. Gross, S. T. B. Goennen- wein, and S. Gepr ags, Physical Review B 97, 014417 (2018). 11G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Palstra, and B. J. van Wees, Applied Physics Letters 111, 052409 (2017). 12E. V. Gomonay and V. M. Loktev, Low Temperature Physics 40, 17 (2014). 13D. Hou, Z. Qiu, J. Barker, K. Sato, K. Yamamoto, S. V elez, J. M. Gomez-Perez, L. E. Hueso, F. Casanova, and E. Saitoh, Physical Review Letters 118, 147202 (2017). 14J. H. Han, C. Song, F. Li, Y. Y. Wang, G. Y. Wang, Q. H. Yang, and F. Pan, Physical Review B 90, 144431 (2014). 15Y. 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 (2017). 16A. Aqeel, N. Vlietstra, J. A. Heuver, G. E. W. Bauer, B. Noheda, B. J. van Wees, and T. T. M. Palstra, Physical Review B 92, 224410 (2015). 17A. Aqeel, M. Mostovoy, B. J. van Wees, and T. T. M. Palstra, Journal of Physics D: Applied Physics 50, 174006 (2017). 18R. Schlitz, T. Kosub, A. Thomas, S. Fabretti, K. Nielsch, D. Makarov, and S. T. B. Goennenwein, Applied Physics Letters 112, 132401 (2018). 19L. Liang, J. Shan, Q. H. Chen, J. M. Lu, G. R. Blake, T. T. M. Palstra, G. E. W. Bauer, B. J. van Wees, and J. T. Ye, Physical Review B 98, 134402 (2018). 20J. Narayan, P. Tiwari, K. Jagannadham, and O. W. Holland, Applied Physics Letters 64, 2093 (1994). 21The measured moment of the substrates corresponds to a con- centration of paramagnetic dopants of 1017cm1. Besides, one would expect a magnetic moment of 108Am2at 10 K as- suming that all Fe moments of YIG are independently contribut- ing whereas a magnetic moment of 1014Am2is expected for the YAG substrate with assuming a susceptibilty of YAG of YAG = 105. 22S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Applied Physics Letters 107, 172405 (2015). 23S. V elez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Physical Review Letters 116, 016603 (2016). 24L. Baldrati, A. Ross, T. Niizeki, R. Ramos, J. Cramer, O. Gomonay, E. Saitoh, J. Sinova, and M. Kl aui, arXiv:1709.00910 (2017). 25M. Kohler, Annalen der Physik 424, 211 (1938). 26H. Homann, F. Hofmann, and W. Schoepe, Physical Review B 25, 5563 (1982). 27Y. Niimi, D. Wei, H. Idzuchi, T. Wakamura, T. Kato, and Y. Otani, Phys. Rev. Lett. 110, 016805 (2013).

2019-01-28

The spin Hall magnetoresistance (SMR) effect arises from spin-transfer processes across the interface between a spin Hall active metal and an insulating magnet. While the SMR response of ferrimagnetic and antiferromagnetic insulators has been studied extensively, the SMR of a paramagnetic spin ensemble is not well established. Thus, we investigate herein the magnetoresistive response of as-deposited yttrium iron garnet/platinum thin film bilayers as a function of the orientation and the amplitude of an externally applied magnetic field. Structural and magnetic characterization show no evidence for crystalline order or spontaneous magnetization in the yttrium iron garnet layer. Nevertheless, we observe a clear magnetoresistance response with a dependence on the magnetic field orientation characteristic for the SMR. We propose two models for the origin of the SMR response in paramagnetic insulator/Pt heterostructures. The first model describes the SMR of an ensemble of non-interacting paramagnetic moments, while the second model describes the magnetoresistance arising by considering the total net moment. Interestingly, our experimental data are consistently described by the net moment picture, in contrast to the situation in compensated ferrimagnets or antiferromagnets.

Spin Hall magnetoresistance in heterostructures consisting of noncrystalline paramagnetic YIG and Pt

1901.09986v1

arXiv:1701.05320v1 [cond-mat.mes-hall] 19 Jan 2017Separation ofinversespinHalleffect and anomalousNernst effect inferromagnetic metals Hao Wu, Xiao Wang, Li Huang, Jianying Qin, Chi Fang, Xuan Zhan g, Caihua Wan, and Xiufeng Han∗ Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China Inverse spinHalleffect (ISHE)inferromagnetic metals(FM ) canalsobe used todetect the spin current gen- eratedbylongitudinal spinSeebeck effect ina ferromagnet ic insulator YIG. However, anomalous Nernst effect (ANE)in FMitself always mixes inthe thermal voltage. Inthi s work, the exchange bias structure (NiFe/IrMn) isemployed toseparate these twoeffects. The exchange bias structure provides a shift fieldtoNiFe,which can separate the magnetization of NiFe from that of YIG in M-Hloops. As a result, the ISHE related to magneti- zationof YIG and the ANE relatedtothe magnetization of NiFe can be separated as well. Bycomparison with Pt, a relative spin Hall angle of NiFe (0.87) is obtained, whi ch results from the partially filled 3 dorbits and the ferromagnetic order. This work puts forward a practical method to use the ISHE in ferromagnetic metals towards future spintronic applications. Spincaloritronicsfocusesoncouplingheat,spinandcharg e in magnetic materials. [1] Spin Seebeck effect (SSE) ori- gins from the excitation of spin wave in magnetic materials by a temperature gradient, which can pump a spin current into a contact metal. [2–5] In the past several years, SSE has been achieved in magnetic metals[2], semiconductors[6 ] and insulators. [3, 4] Especially SSE in magnetic insulator s draws many attentions since a pure spin current without any chargeflowisoneofthemostdesirablepropertiesfordevice s with dramatically reduced power consumption. Transverse and longitudinal spin Seebeck effect are divided by differe nt experimental configurations, while the detected spin curre nt can be perpendicular or parallel to the temperature gradien t. Especially longitudinal spin Seebeck effect (LSSE) in ferr o- magneticinsulatorsiswidelyusedtopumpaspincurrentint o theneighboringmaterials. InversespinHalleffect(ISHE)canconvertthespincurrent Jsintothechargecurrent Je,whichcanbedetectedbyavolt- age signal: EISHE= (θSHρ)Js×σ, whereEISHEis the ISHE electric field, θSHis the spin Hall angle, ρis the resistivity, andσis the unit vector of the spin. [7–9] It is generally be- lieved that the magnitude of spin Hall angle θSHdepends on the strength of spin orbit coupling(SOC), and the strength o f SOC is proportionalto Z4, whileZis the atomic number, so heavymetals(HM)withlarge ZhavearelativelargespinHall angle. [10] Similar to spin Hall effect (SHE) in non-magnetic metals, [11–13]anomalousHalleffect(AHE)inferromagneticmetal s (FM) comes fromthe spin dependentscattering of the charge current. [14]Duetothespinpolarizationofthechargecurr ent in FM, the spin accumulationaccompaniedwith a charge ac- cumulationcanbegeneratedinthetransversedirection. Wh en a pure spin current is injected to FM, as the inverse effect of AHE, ISHE in FM provides another potential application in detectingthespincurrentbychargesignals. Recent works draw attention on using the ISHE in FM to detect the spin currentgeneratedby LSSE in a ferromagnetic insulatorY 3Fe5O12(YIG).[15–18] However,thetemperature gradient will also introduce additional anomalous Nernst e f- ∗Email: xfhan@iphy.ac.cnfect (ANE)in FM: EANE∝ ∇Tz×M, whereEANEisthe ANE electricfield, ∇Tzisthetemperaturegradientalongthethick- nessdirection,and MisthemagneticmomentofFM.[19,20] Therefore,the separationof ANE and ISHE in FM is in great demand. Several works have used two magnetic materials with different coercivity to separate the ISHE (related to t he magnetization of YIG) and ANE (related to the magnetiza- tion of the ferromagnetic detector).[16, 17] However, ISHE andANEarestillmixedwitheachother,whichpreventsusto directlydetectthespin currentbyFM. In this work, we designed the exchange bias structure (NiFe/IrMn) to detect the spin current generated by LSSE in YIG. A Cu layer with a negligible spin Hall angle is inserted between NiFe and YIG to reduce the magnetic coupling be- tweenYIGandNiFe,andthespincurrentcanalsopasswith- out too much loss at the same time. The exchangebias struc- ture provides a bias field for NiFe, which can separate the magnetizationswitching processof NiFe fromthat of YIG in M-Hloops. [21, 22] As a result, the ISHE related to magne- tization of YIG and the ANE related to the magnetization of NiFecanbeseparatedaswell. Moreimportantly,wecaneven observe the only ISHE contribution in a field range which is smaller than the exchange bias field that only the magneti- zation of YIG switches, while the magnetization of NiFe is fixed. TheexchangebiasstructureCu(5)/NiFe(5)/IrMn(12)/Ta(5 ) (thickness in nanometers) was fabricated on polished 3.5 µm YIG films on GGG substrates by a magnetron sputtering system. In order to introduce the exchange bias effect in FM/AFM, an in-plane magnetic field was applied during the depositionprocess. ThespinSeebeckvoltagewasmeasureby ananovoltmeter(Keithley2182A)inaSeebeckmeasurement system with a Helmholtz coil, and the longitudinal temper- ature difference along the thickness direction was measure d between the bottom of the GGG substrate and the top of the film. All datawasperformedatroomtemperature. Fig. 1(a)showstheschematicdiagramofthemeasurement method and the physical process. In longitudinal spin See- beck measurement, the temperature gradient ( ∇T) is applied along the out-of-plane zdirection, and the magnetic field is scanned along xdirection (also the direction of the exchange bias field). According to EISHE= (θSHρ)Js×σ, the thermal voltageshouldbe measuredalong ydirection. Firstly, aspre-2 FIG. 1. Under the longitudinal temperature gradient, only t he inverse spin Hall effect (ISHE) exists in YIG/Pt sample, w hile both ISHE and the anomalous Nernst effect (ANE)exist inYIG/Cu/NiFe/IrM n/Tasample. Once an insulating layer MgO is insertedbetwee n NiFe and YIG, the spincurrent willbe blocked, soISHEvanishes whileANE s tillexists. FIG.2. Highresolutiontransmissionelectronmicroscopy( HRTEM)resultsoftheYIG/Cu/NiFe/IrMn/Tasample(a)andse lectedareaelectron diffraction (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 the magnetic fieldis appliedalong the directionof exchange bias field. viousworks,we use a Pt layer whichhas a relative largespin Hallangleabout0.1[23–25]tomeasurethepurespinSeebeck induced ISHE signal. Then, we changed the Pt with the ex- changebiasstructureCu/NiFe/IrMn/Ta. ApartfromtheISHE signal, ANE from NiFe itself will also contribute to the ther - mal voltage. Once we inserted an insulating layer MgO to block the spin current injected from YIG to NiFe, ISHE sig- nalshouldvanishwhereonlyANEfromNiFe exists. Fig. 2(a) shows the cross-sectional transmission electron microscopy (HRTEM) results of the YIG/Cu/NiFe/IrMn/Ta multilayers,andtheinterfacebetweenCuandYIGisveryflat and clear. The selected area electron diffraction(SAED) pa t- ternisshowninFig. 2(b),demonstratingthattheepitaxial di- rectionofYIGfilmcrystalisalongthe(111)directionandth elattice parameter is 12.4 ˚A. A Si-SiO 2/Cu/NiFe/IrMn/Ta ref- erencesampleisusedtochecktheexchangebiaseffect,wher e a 220 Oe exchange bias field is obtained from the M-Hloop (alongxdirection) [Fig. 2(c)]. Then we measured the mag- netic propertiesof YIG/Cu/NiFe/IrMn/Ta sample [Fig. 2(d) ], andthesaturationmagnetization MsofYIGis120emu/ccand thesaturationfieldofYIGislessthan10Oe. Fromthezoom- infigurein Fig. 2(d),we cansee the magnetizationswitching rangeofNiFe/IrMnexchangebiasstructureisfrom150Oeto 250Oe,whichisfarfromthe rangeofYIG. Then we measured the magnetic field dependence of the thermal voltage, and the longitudinal temperature differe nce ∆Tkeeps 13 K during the measurement in Fig. 3(a)-(c). Firstly, as conventional LSSE measurement, a heavy metal3 FIG. 3. (a)-(c) shows the spin dependent thermal volt- age measurement of YIG/Pt, YIG/Cu/NiFe/IrMn/Ta, and YIG/MgO/NiFe/IrMn/Ta samples, where the magnetic field is applied along xdirection (also the direction of the exchange bias field). Pt is used to measure the spin current generated by SSE in YIG, anda 0.4 µV ISHE voltagewhichis relatedto the mag- netization of YIG is observed, as shown in Fig. 3(a). While in YIG/Cu/NiFe/IrMn/Ta structure, due to the exchange bias effect, ISHE (related to the magnetization of YIG) and ANE (related to the magnetization of NiFe) exist in different fie ld ranges. The ISHE signal is around 0 Oe and the ANE signal is around 150 Oe, and the exchange bias field here is a little smaller than that from the M-Hloop because the exchange bias decreaseswith the increasedtemperature,as seen in Fi g. 3(b). It is worth noting that the polarityof ISHE and ANE in NiFe is the same, which is contrary to the result for CoFeB. [16] In order to prove that the ISHE voltage related to the magnetizationofYIG indeedcomesfromthe spin currentin- jection from YIG to NiFe, we insert an insulating layer MgO toblockthisspincurrent. Asexpected,theISHEsignaldisa p- pearswhiletheANEstill exists,asseeninFig. 3(c). ForSSE inheavymetal/ferromagnetstructures,themixingofmagne tic proximity effect [26, 27] (MPE) and ANE has a debate for a long time, which means that magnetized Pt shows some fer- romagneticpropertiesintransportmeasurementsuchasANE andAHE.Inourwork,wedirectlyuseaFMtodetectthespin current generatedby SSE, and our results show that although both ISHE and ANE take place in the thermal voltage, how- ever,byusingtheexchangebiaseffect,ISHEandANEcanbe separated in different field ranges. These results demonstr ate that SSE and ANE share different physical origins, and ANE isnottheessential conditionofSSE. Then, we changed the temperature differences ∆Tfrom 2.5 K to 13K, and the field dependent ISHE voltage for dif- FIG.4. (a)and(b)showthespindependentthermalvoltageme asure- ment in YIG/Pt, YIG/Cu/NiFe/IrMn/Ta samples respectively under the varied longitudinal temperature difference from 2.5 K t o 13 K, andtheoffsetvoltagehasbeenremovedtoobtainthefielddep endent ISHE contribution. The magnetic field is applied along xdirection (also the direction of the exchange bias field), and the field r ange is smallerthan the exchange bias field. ferent∆Tis shown in Fig. 4(a) (YIG/Pt) and Fig. 4(b) (YIG/Cu/NiFe/IrMn/Ta). The ISHE voltages gradually in- crease with increasing the temperature gradient in both sam - ples, which are in accordance with the spin Seebeck mecha- nism. Under a ±80 Oe field range which is smaller than the exchangebiasfield, onlythe pureISHE signalswithout ANE shows that the comparableutility of FM (NiFe) with conven- tionalheavymetals(Pt)indetectingthespincurrent. Beca use inthiscaseonlythemagnetizationofYIGreverses,whileth e magnetization of NiFe keeps fixed. And the optimization of FMwithlargespinHallanglewillbeanessentialsteptoward s futureapplications. In order to compare the spin Hall angle in Pt and NiFe, the ISHE voltage is normalized by the resistance of the de- tecting electrode R, and the VISHE/R-∆Tcurve is fitted by the linear shape, as seen in Fig. 5. VISHE/R=βθSH∆T, where βrepresents the efficiency from thermal current to the de- tected spin current. If we assume the same βin YIG/Pt and YIG/Cu/NiFe/IrMn/Ta samples, we can calculate the relativ e spin Hall angle of NiFe: θSH(NiFe)/θSH(Pt)≈0.87, which is closetoourpreviousresult(0.98)bytransverseSSEmeasur e- ment. [28] InconventionalunderstandingofSOC,thestrengthofSOC follows a Z4dependence. While NiFe is composed of light atoms, so SOC in NiFe should be small in this mechanism. However,previousworkshave also shownthat SOC not only dependson the atomic number Zbut also dependson the fill- ing ofd-orbit, both Ni and Fe have partially filled 3 dorbits,4 FIG. 5. The VISHE/R−∆Tcurves and the linear fitting curves mea- sured in YIG/Pt and YIG/Cu/NiFe/IrMn/Ta samples, and the IS HE voltage has been normalized by the resistance of the detecti ng elec- trode. soSOCfromthe d-orbitfillingcouldtakeanimportantrolein NiFe. [29, 30] Moreover,ferromagneticorderinducedintri n- sic spin dependent scattering which is solely determined by the electronic band structure can also contribute to the ISH E in FM, because the ISHE in FM is independentof its magne-tization. [31] In conclusion, we have designed the exchange bias struc- ture (NiFe/IrMn) to separate the ISHE and ANE in FM. As expected, the ISHE related to magnetization of YIG and the ANE related to the magnetization of NiFe can be separated in different ranges of magnetic field. By linear fitting the VISHE/R-∆Tcurves of NiFe and Pt, we calculated the relative spin Hall angle θSH(NiFe)/θSH(Pt)=0.87, and the partial filling of 3dorbits and the ferromagneticorder play important roles in thislargespin Hall angle ofNiFe. Thisworkdemonstrates that ferromagnetic metals can also be used to detect the spin currentinspintronicsdevices. ACKNOWLEDGMENTS This work was supported by the 863 Plan Project of Ministry of Science and Technology (MOST) [Grant No. 2014AA032904], the National Key Research and Develop- ment Program of China [Grant No. 2016YFA0300802], the NationalNaturalScienceFoundationofChina(NSFC)[Grant Nos. 11434014, 11404382], and the Strategic Priority Re- search Program (B) of the Chinese Academy of Sciences (CAS)[GrantNo. XDB07030200]. [1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391(2012). [2] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K.Ando, S.Maekawa, andE.Saitoh, Nature 455, 778 (2008). [3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ied a, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S.Maekawa, and E.Saitoh,Nat.Mater. 9, 894 (2010). [4] K.Uchida, H.Adachi,T.Ota,H.Nakayama, S.Maekawa, and E.Saitoh,Appl. Phys.Lett. 97, 172505 (2010). [5] J.Xiao,G.E.W.Bauer,K.Uchida,E.Saitoh, andS.Maekaw a, Phys.Rev. B 81, 214418 (2010). [6] C. Jaworski, J. Yang, S. Mack, D. Awschalom, J. Heremans, andR.Myers, Nat.Mater. 9,898 (2010). [7] S.O.Valenzuela and M. Tinkham, Nature 442, 176(2006). [8] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa , Phys.Rev. Lett. 98, 156601 (2007). [9] R.Karplus and J.Luttinger, Phys.Rev. 95, 1154 (1954). [10] H.L.Wang,C.H.Du,Y.Pu,R.Adur,P.C.Hammel, andF.Y. Yang,Phys. Rev. Lett. 112, 197201 (2014). [11] J.E.Hirsch, Phys.Rev. Lett. 83, 1834 (1999). [12] Y. Kato, R. Myers, A. Gossard, and D. Awschalom, Science 306, 1910 (2004). [13] O. Mosendz, J. Pearson, F. Fradin, G. E. W. Bauer, S. Bade r, andA. Hoffmann, Phys.Rev. Lett. 104, 046601 (2010). [14] N. Nagaosa, J. Sinova, S. Onoda, A. MacDonald, and N. Ong , Rev. Mod. Phys. 82, 1539 (2010). [15] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett.111, 066602 (2013). [16] S. M. Wu, J. Hoffman, J. E. Pearson, and A. Bhattacharya, Appl.Phys. Lett. 105, 092409 (2014). [17] D.Tian,Y.F.Li,D.Qu,X.F.Jin, andC.L.Chien,Appl.Ph ys.Lett.106, 212407 (2015). [18] T. Seki, K. Uchida, T. Kikkawa, Z. Qiu, E. Saitoh, and K. Takanashi, Appl. Phys.Lett. 107, 092401 (2015). [19] A.Slachter,F.L.Bakker, andB.J.vanWees,Phys.Rev. B 84, 020412 (2011). [20] M. Mizuguchi, S. Ohata, K. Uchida, E. Saitoh, and K. Takanashi, Appl. Phys.Express 5,093002 (2012). [21] N. Koon, Phys.Rev. Lett. 78, 4865 (1997). [22] A. Berkowitz and K. Takano, J. Magn. Magn. Mater. 200, 552 (1999). [23] H. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602 (2013). [24] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D. Bade r, and A.Hoffmann, Appl. Phys.Lett. 103, 242414 (2013). [25] M. Obstbaum, M. H¨ artinger, H. Bauer, T. Meier, F. Swien tek, C. Back, and G.Woltersdorf, Phys.Rev. B 89, 060407 (2014). [26] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012). [27] Y. M. Lu, Y. Choi, C.M. Ortega,X. M. Cheng, J. W.Cai, S.Y. Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). [28] H.Wu,C.H.Wan,Z.H.Yuan,X.Zhang,J.Jiang,Q.T.Zhang , Z.C. Wen, andX. F.Han,Phys. Rev. B 92, 054404 (2015). [29] C. H. Du, H. L. Wang, F. Y. Yang, and P. C. Hammel, Phys. Rev. B90, 140407 (2014). [30] M. Morota, Y. Niimi, K. Ohnishi, D. Wei, T. Tanaka, H. Kon - tani,T.Kimura, andY.Otani,Phys.Rev.B 83,174405(2011). [31] D.Tian,Y.F.Li,D.Qu,S.Y.Huang,X.F.Jin, andC.L.Chi en, Phys. Rev. B 94, 020403 (2016).

2017-01-19

Inverse spin Hall effect (ISHE) in ferromagnetic metals (FM) can also be used to detect the spin current generated by longitudinal spin Seebeck effect in a ferromagnetic insulator YIG. However, anomalous Nernst effect(ANE) in FM itself always mixes in the thermal voltage. In this work, the exchange bias structure (NiFe/IrMn)is employed to separate these two effects. The exchange bias structure provides a shift field to NiFe, which can separate the magnetization of NiFe from that of YIG in M-H loops. As a result, the ISHE related to magnetization of YIG and the ANE related to the magnetization of NiFe can be separated as well. By comparison with Pt, a relative spin Hall angle of NiFe (0.87) is obtained, which results from the partially filled 3d orbits and the ferromagnetic order. This work puts forward a practical method to use the ISHE in ferromagnetic metals towards future spintronic applications.

Separation of inverse spin Hall effect and anomalous Nernst effect in ferromagnetic metals

1701.05320v1

Concomitant enhancement of longitudinal spin Seebeck eect with thermal conductivity Ryo Iguchi,1,Ken-ichi Uchida,1, 2, 3, 4Shunsuke Daimon,1, 5and Eiji Saitoh1, 4, 5, 6 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2National Institute for Materials Science, Tsukuba 305-0047, Japan 3PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 5WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 6Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan Abstract We report a simultaneous measurement of a longitudinal spin Seebeck eect (LSSE) and thermal conductivity in a Pt/Y 3Fe5O12(YIG)/Pt system in a temperature range from 10 to 300 K. By directly monitoring the temperature dierence in the system, we excluded thermal artifacts in the LSSE measurements. It is found that both the LSSE signal and the thermal conductivity of YIG exhibit sharp peaks at the same temperature, dierently from previous reports. The maximum LSSE coecient is found to be SLSSE>10V=K, one-order-of magnitude greater than the previously reported values. The concomitant enhancement of the LSSE and thermal conductivity of YIG suggests the strong correlation between magnon and phonon transport in the LSSE. iguchi@imr.tohoku.ac.jp 1arXiv:1612.08142v1 [cond-mat.mtrl-sci] 24 Dec 2016A spin counterpart of the Seebeck eect, the spin Seebeck eect (SSE) [1], has attracted much attention from the viewpoints of fundamental spintronic physics [2{4] and future thermoelectric applications [5{7]. The SSE converts temperature dierence into a spin current in a magnetic material, which can generate electrical power by attaching a conductor with spin{orbit interaction [1, 5]. The SSE originates from thermally-excited magnons, and it appears even in magnetic insulators. In fact, after the pioneer work by Xiao et al. [8], the SSE has been discussed in terms of the thermal non-equilibrium between magnons in a magnetic material and electrons in an attached conductor. Recent experimental and theoretical works have been focused on the transport and excitation of magnons contributing to the SSE in the magnetic material [9{17], whose importance can be recognized in the temperature dependence, magnetic-eld-induced suppression, and thickness dependence of SSEs [18{ 23]. Most of the SSE experiments have been performed by using a junction comprising a ferrimagnetic insulator Y 3Fe5O12(YIG) and a paramagnetic metal Pt since YIG/Pt enables pure driving and ecient electric detection of spin-current eects; a YIG/Pt junction is now recognized as a model system for the SSE studies. Figure 1(a) shows a schematic illustration of the SSE in an YIG/Pt-based system in a longitudinal conguration, which is a typical conguration used for measuring the SSE. In the longitudinal SSE (LSSE) conguration, when a temperature gradient rTis applied along thezdirection, it generates a spin current across the YIG/Pt interface [5, 8, 24, 25]. This thermally-induced spin current is converted into an electric eld ( EISHE) by the inverse spin Hall eect (ISHE) in Pt according to the relation [26, 27] EISHE/Js; (1) where Jsis the spatial direction of the spin current and is the spin-polarization vector of Js, which is parallel to the magnetization Mof YIG [see Fig. 1(a)]. When Mis along the xdirection, the LSSE is detected as a voltage, VLSSE =R EISHEdy, between the ends of the Pt layer along the ydirection. In the LSSE research, temperature dependence of the voltage generation has been essen- tial for investigating its mechanisms, such as spectral non-uniformity of magnon contribu- tions [21{23] and phonon-mediated eects [18, 19, 28]. The recent studies demonstrated that the LSSE voltage in a single-crystalline YIG slab exhibits a peak at a low temperature, and the peak temperature is dierent from that of the thermal conductivity of YIG [20{22, 29]. 2MYIG slab Pt film (bottom)Pt film (top) x zy ∇T TT JsAu electrode EISHE (a) (b) (c) thermal grease (Apiezon N)Au electrode Hsapphire plate heat bathheat bath Pt/YIG/Pt sample sapphire plateIn ∆T T 5 K (fixed)TH TLTave∇TTHset TLset H Au wire zFIG. 1. (a) A schematic illustration of the Pt/YIG/Pt sample. rT,H,M,EISHE, and Js denote the temperature gradient, magnetic eld (with the magnitude H), magnetization vector, electric eld induced by the ISHE, and spatial direction of the thermally generated spin current, respectively. The electric voltage VH(VL) and resistance RH(RL) between the ends of top (bottom) Pt layers were measured using a multimeter. (b) Experimental conguration for applying rT. The thickness and width of the sapphire plates are 0.33 and 2.0 mm, respectively. (c) A schematic plot of temperature prole along the zdirection. The dierence in the peak temperatures was the basis of the recently-proposed scenario that the LSSE is due purely to the thermal magnon excitation, rather than the phonon-mediated magnon excitation [9, 12, 15]. In this paper, we report temperature dependence of LSSE in an YIG/Pt-based system free from thermal artifacts and its strong correlation with the thermal conductivity of YIG. The LSSE signal and thermal conductivity were simultaneously measured without changing the experimental conguration. The intrinsic temperature dependence of the LSSE shows signicant dierence from the prior results, while the thermal conductivity shows good agreement with the previous studies; the peak temperatures are found to be exceedingly close to each other. These data will be useful for comparing experiments and theories quantitatively and for developing comprehensive theoretical models for the SSE. The quantitative measurements of the thermoelectric properties are realized by directly monitoring the temperature dierence between the top and bottom surfaces of a single- 3crystalline YIG slab. To do this, we extended a method proposed in Ref. 23, which is based on the resistance measurements of Pt layers covering the top and bottom surfaces of the slab [Fig. 1(a)]. The lengths of the YIG slab along the x,y, andzdirections ( w,l, andt) are 1.9 mm, 6.0 mm, and 1.0 mm, respectively. After polishing the x-ysurfaces [(111) plane] of the YIG slab, the 10-nm-thick Pt lms were sputtered on the whole of the surfaces. The Pt lms are electrically insulated from each other [30]. Au electrodes were formed on the edges of the Pt layers, of which the gap length l0is 5.0 mm. The thermoelectric voltage and resistance inside the gap were measured by a multimeter. The sample is put between heat baths with two sapphire plates with a length of l0for electrical insulation. For thermal connection between them, thermal grease is used [Fig. 1(b)]. During the LSSE measurements, we set Tset H=Tset L+5 K with Tset H(L)being the temperature of the top (bottom) heat bath (hereafter, we use the subscripts H and L to represent the corresponding quantities of the top and bottom Pt lms, respectively). In this condition, we monitored the temperature dierence between the top and bottom of the sample
T=THTLby using the Pt lms not only as spin-current detectors but also as temperature sensors [Fig. 1(c)] [23][31] . A magnetic eld with the magnitude His applied in the xdirection. Importantly, the above method allows us to estimate the thermal conductivity YIGof the YIG slab at the same time as the LSSE measurements. This is realized simply by recording the heater power PHeater in addition to
T;YIGcan be calculated as YIG=t wlPHeater f(l0)
T: (2) in a similar manner to the steady heat- ow method, where f(l0) = 0:88 denotes a form factor for adjusting the measured
Tto the temperature dierence averaged over the sample length [32]. Note that the in uence of the thermal resistances of the Pt layers [33] and YIG/Pt interfaces [34] are negligibly small for the YIGestimation. The simultaneous measurements enable quantitative comparison of the LSSE and YIG. The inset to Fig. 2(a) shows the estimated values of
Tbased on the resistance measure- ments. We found that the
Tvalue is smaller than the temperature dierence applied to the heat baths ( Tset HTset L= 5 K in this study) at each temperature and strongly decreases with decreasing the temperature. This behavior can be explained by dominant consumption of the applied temperature dierence by the thermal grease layers; typical thermal resis- tance of 10- m-thick thermal grease is comparable to that of the YIG slab at 300 K and 4(a) (b)THset-TLset TH-TL 0.1110 ∆T (K) 300 200 1000 T (K) 2VLSSE -5 05VH (µV) -0.5 0.0 0.5 µ0H (T)THset=305K100 50 0κYIG (Wm -1 K-1 ) 300 250 200 150 100 50 0 T (K)50 25 0VLSSE /∆T (µV/K) VH/∆ T VL/∆ TVH/( THset-THset)FIG. 2. Temperature ( T) dependence of the thermal conductivity YIGestimated from
Tand the heater power PHeater (a) , andVLSSE=
T(b). The inset to (a) shows
Testimated from RHand RLunder the temperature gradient. The inset to (b) shows Hdependence of VHatTset H(L)= 305 (300) K. The peak temperatures are determined by parabolic tting of ve points around the maximums. much greater than that at low temperatures as YIGincreases at low temperatures [29][35]. Consequently, the actual temperature dierence (
T), and resultant VLSSE measured with xingTset HTset L, strongly decrease. This result indicates that the conventional method, which monitors only Tset HTset L, cannot reach the intrinsic temperature dependence of the LSSE. Figure 2(a) shows YIGas a function of Tave= (TH+TL)=2, estimated from Eq. (2). The temperature dependence and magnitude of YIGare well consistent with the previous studies [29], supporting the validity of our estimation. The YIGvalue exhibits a peak at around 27 K and reaches 1 :3102Wm1K1at the peak temperature; this temperature dependence can be related to phonon transport, i.e., the competition between the increase of the phonon life time due to the suppression of Umklapp scattering and the decrease of the phonon number with decreasing the temperature [29]. The inset to Fig. 2(b) shows the Hdependence of VHin the Pt/YIG/Pt sample at 5Tset H(L)= 305 K (300 K). The clear LSSE voltage was observed; the voltage shows a sign reversal in response to the reversal of the magnetization direction of the YIG slab [5, 24]. We extracted the LSSE voltage VLSSE from the averaged values of VL(VH) in the region of 0:2 T<j0Hj<0:5 T, where 0denotes the vacuum permittivity (note that the eld- induced suppression of the LSSE is negligibly small in this Hrange [20{22]). Figure 2(b) shows the LSSE voltage normalized by the estimated temperature dierence VLSSE/
Tin the top (bottom) Pt layers of the Pt/YIG/Pt sample as a function of TH(TL). With decreasing the temperature, the VLSSE=
Tvalue rst increases and, after taking a peak around 31 K, monotonically decreases toward zero. Although this behavior is qualitatively consistent with the previous reports [20{22, 29], the observed peak structure is very steep and the peak temperature signicantly diers from the previous results, where the peak temperature was reported to be 70 K. We found that, if VLSSE is normalized by the temperature dierence between the heat baths Tset HTset L, our data also exhibits a peak 74 K as previously reported [20{22, 29]. The overestimated peak temperature is attributed to the misestimation of
Tshown in the inset to Fig. 2(a), showing that the previous results may not represent intrinsic temperature dependence of the LSSE because of the thermal artifacts, and the direct
Tmonitoring is necessary for the quantitative LSSE measurements. We also note that the magnitude of the intrinsic LSSE thermopower is in fact much greater than that estimated from the previous experiments. The dierence in the signal magnitude is more visible at low temperatures; the LSSE voltage in our Pt/YIG/Pt sample at the peak temperature reaches to VLSSE=
T55V/K. In Fig. 3, we show the complete temperature dependence of the LSSE thermopower SLSSE=tVLSSE=(l0
T), thermal conductivity YIG, and saturation magnetization Msof the YIG, which includes the data in the high temperature range reported in Ref. 23. In the fol- lowing, we discuss the origin of the temperature dependence of the LSSE. The temperature dependence of the LSSE originates from the spin mixing conductance [36, 37] at Pt/YIG interfaces, spin diusion length, spin Hall angle [4], and resistance of Pt, and dynamics of thermally-excited magnons in YIG. The spin mixing conductance, which is proportional to the LSSE voltage, may depend on TandMs. The predicted relation, /M2 s[38], cannot explain the LSSE enhancement at low temperatures because the maximum possible enhance- ment is calculated as a factor of 1.9. Similarly, the spin diusion length or spin Hall angle of Pt cannot explain the enhancement of the LSSE voltage at low temperatures as they 6100 50 0κYIG (Wm-1K-1) 600 550 500 450 400 350 300 250 200 150 100 50 0 T (K)10 5 0SLSSE (µV/K)0.2 0.1µ0Ms (T) SLSSE κYIG ∝a+bT-0.85±0.02 ∝a+bT-1∝bT3∝bT0.9±0.6 ∝(T-Tc)3Tc=553 K (iii) (iv) (i) (ii) 10-810-710-610-5SLSSE (µV/K) 102 4681002 46 T (K)10100κYIG (Wm-1K-1) SLSSE KYIG FIG. 3. The saturation magnetization Msof the YIG slab as a function of T, the LSSE thermo- electric coecient SLSSE of the top Pt layer as a function of TH, and the thermal conductivity YIGas a function of the averaged temperature Tave.Mswas measured using a vibrating sample magnetometer. The data at temperatures higher than 300 K is drawn from Ref. 23, which shares the same YIG slab and Pt thickness. The inset shows a Log-Log plot. The solid lines represent tting curves discussed in the main text. are almost temperature-independent below 300 K [39, 40]. The temperature dependence of the resistance of the Pt is also irrelevant because it decreases with decreasing the temper- ature. Therefore, the LSSE enhancement at the peak likely comes from the properties of thermally-excited magnons. Here, two scenarios have been proposed for magnon excitation in the SSE: one is the pure thermal magnon excitation [9, 12, 15, 20, 22] and the other is the phonon-mediated magnon excitation [28, 41, 42]. Importantly, the peak temperature of the observed LSSE signal in our Pt/YIG/Pt sample is almost the same as that of YIG, indicating the strong correlation between VLSSEandYIG at low temperatures. This behavior is consistent with the scenario of the phonon-mediated SSE, where SSE voltage is expected to be proportional to the phonon life time in YIG (note again that the peak in the T-YIGcurve re ects the phonon transport) [28]. Although the recent studies proposed the scenario that the LSSE is due purely to the magnon-driven contribution, it is based on the dierence in the peak temperatures between the LSSE and thermal conductivity, which now turned out to be relevant to the thermal artifacts [see Fig. 2(b)]. The magnon- and phonon-driven contributions cannot be separated completely by the present experiments. However, at least, the presence of the phonon-mediated process 7(a) (b)5 0PLSSE (x10-4 Wm-1K-2) 300 250 200 150 100 50 0 T (K)4 2 0ZLSSET (x10-4 )FIG. 4.Tdependence of the power factor PLSSE (a) and the gure of merit ZLSSETof our LSSE device (b). cannot be excluded because of the similar peak temperatures between SLSSE andYIG. According to the previous studies on the transverse SSE [18], the observed temperature dependence of the LSSE can be separated into the following four regions: (i) the low temper- ature region from 10 to 30 K, (ii) from 30 K to room temperature, (iii) the high temperature region from room temperature to the Curie temperature Tcof YIG, and (iv) above Tc. In the region (i), both SLSSE andYIGincrease with the temperature Tand then reach their maximums. This tendency can be expressed by bT with = 0:90:6 forSLSSE up to 15 K, whereband are tting parameters. YIGin this temperature range is reproduced by setting = 3 [29] . The dierence in the exponents between SLSSEandYIGcan be due to dierence in the Tdependence of the heat capacitance and group velocity between magnons and phonons [22, 29, 41]. In the region (ii), SLSSEandYIGstart to decrease with increasing T. WhileYIGshowsa+bT1dependence originating from Umklapp scattering of phonons, SLSSE shows thea+bT0:850:02dependence. This dierence suggests the coexistence of the magnon- and phonon-induced processes in the LSSE, re ecting dierent temperature depen- dence of the magnon and phonon life times [29, 43]. In the region (iii), SLSSE shows strong correlation to Msrather than YIG, dierently from (i). Here, SLSSE andMsare described by (TcT) with = 3 and 0.5, respectively [23], while YIGgradually decreases. Finally, in the region (iv), SLSSE andMsvanish. The microscopic and quantitative explanation of the above behavior remains to be achieved. Since the experimental method demonstrated here enables simultaneous measurements of 8the LSSE thermopower SLSSE, thermal conductivity of YIG YIG, and electrical conductivity of PtPt, we can also obtain the quantitative temperature dependence of the thermoelectric performance of our Pt/YIG/Pt sample. Figure 4(a) shows the power factor PLSSE=S2 LSSEPt as a function of T. Owing to the strong enhancement of SLSSE at low temperatures, the power factor exhibits a sharp peak at 32 K; the PLSSE value at the peak temperature is 6104Wm1K2,1000 times greater than that at room temperature. In contrast, the gure of merit ZLSSET= (S2 LSSEPt=YIG)T[5] exhibits a maximum at 67 K due to the competition between SLSSE andYIG[Fig. 4(b)]. The low-temperature enhancement of the power factor and gure of merit is in sharp contrast to the typical behavior of the conventional Seebeck devices [44], the thermoelectric performance of which decreases with decreasing the temperature, indicating potential thermoelectric applications of LSSE at low temperatures. In conclusion, we systematically investigated the longitudinal spin Seebeck eect (LSSE) in an Y 3Fe5O12(YIG) slab sandwiched by two Pt lms in the low temperature range from 10 K to room temperature. The direct temperature monitoring based on the resistance measurements of the Pt layers successfully reveals the intrinsic LSSE behavior, unreachable by the conventional method. We found that the magnitude of the LSSE in the Pt/YIG/Pt sample rapidly increases with decreasing temperature and takes a maximum at a temper- ature very close to the peak temperature of the thermal conductivity of YIG. The strong correlation between the LSSE and thermal conductivity shed light again on the importance of the phonon-mediated processes in the SSE. Although more detailed experimental and theoretical investigations are required, we anticipate that the nding of the intrinsic tem- perature dependence of the LSSE will be helpful for obtaining the full understanding of its mechanism. ACKNOWLEDGMENTS The authors thank J. Shiomi, A. Miura, T. Oyake, H. Adachi, T. Kikkawa, T. Ota, R. Ramos, and G. E. W. Bauer for valuable discussions. This work was supported by PRESTO \Phase Interfaces for Highly Ecient Energy Utilization" and ERATO \Spin Quantum Rectication" from JST, Japan, Grant-in-Aid for Scientic Research (A) (JP15H02012), Grant-in-Aid for Scientic Research on Innovative Area \Nano Spin Conversion Science" 9(JP26103005) from JSPS KAKENHI, Japan, NEC Corporation, the Noguchi Institute, and E-IMR, Tohoku University. S.D. is supported by JSPS through a research fellowship for young scientists (No. 16J02422). [1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). [3] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E. Saitoh, J. Phys. Soc. Jpn. 82, 102002 (2013). [4] S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura, Spin Current , Vol. 17 (Oxford University Press, 2012). [5] K. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S. Maekawa, and E. Saitoh, Proc. IEEE 104, 1946 (2016). [6] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, Nat. Mater. 11, 686 (2012). [7] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). [8] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). [9] S. M. Rezende, R. L. Rodriguez-Suarez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. F. Guerra, J. C. L. Ortiz, and A. Azevedo, Phys. Rev. B 89, 014416 (2014). [10] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015). [11] A. Kehlberger, U. Ritzmann, D. Hinzke, E.-J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jung eisch, B. Hillebrands, U. Nowak, and M. Kl aui, Phys. Rev. Lett.115, 096602 (2015). [12] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). [13] T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. Uchida, Z. Qiu, G. E. W. Bauer, and E. Saitoh, Phys. Rev. Lett. 117, 207203 (2016). [14] J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117, 217201 (2016). 10[15] V. Basso, E. Ferraro, and M. Piazzi, Phys. Rev. B 94, 144422 (2016). [16] H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013). [17] S. Homan, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [18] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, R. C. Myers, and J. P. Heremans, Phys. Rev. Lett. 106, 186601 (2011). [19] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa, and E. Saitoh, J. Appl. Phys. 111, 103903 (2012). [20] E.-J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D. A. MacLaren, G. Jakob, and M. Kl aui, Phys. Rev. X 6, 031012 (2016). [21] T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 064413 (2015). [22] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Heremans, Phys. Rev. B 92, 054436 (2015). [23] K. Uchida, T. Kikkawa, A. Miura, J. Shiomi, and E. Saitoh, Phys. Rev. X 4, 041023 (2014). [24] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [25] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). [26] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 2509 (2006). [27] A. Azevedo, L. H. Vilela Le~ ao, R. L. Rodriguez-Suarez, A. B. Oliveira, and S. M. Rezende, J. Appl. Phys. 97, 10C715 (2005). [28] H. Adachi, K. Uchida, E. Saitoh, J. Ohe, S. Takahashi, and S. Maekawa, Appl. Phys. Lett. 97, 252506 (2010). [29] S. R. Boona and J. P. Heremans, Phys. Rev. B 90, 064421 (2014). [30] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). [31] See Supplemental Material attached for details of the temperature estimation based on the resistance measurements. [32] See Supplemental Material attached for details of the YIGmeasurements. [33] L. Lu, W. Yi, and D. L. Zhang, Rev. Sci. Instrum. 72, 2996 (2001). [34] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goen- nenwein, Phys. Rev. B 88, 094410 (2013). 11[35] For thermal grease, thermal conductivity of 0.2 Wm1K1is assumed, of which the thermal resistance becomes 5.3 W/K in our experiment. This is comparable to that of the YIG slab of 13 W/K at 300 K and is much greater than that of 0.7 W/K at 27 K. [36] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [37] S. S. L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012). [38] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev. B 89, 174417 (2014). [39] L. Vila, T. Kimura, and Y. Otani, Phys. Rev. Lett. 99, 226604 (2007). [40] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Phys. Rev. B 94, 060412 (2016). [41] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat. Mater. 9, 898 (2010). [42] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, Nat. Mater. 10, 737 (2011). [43] S. M. Rezende, R. L. Rodriguez-Suarez, J. C. L. Ortiz, and A. Azevedo, Phys. Rev. B 89, 134406 (2014). [44] X. Zhang and L.-D. Zhao, J. Materiomics 1, 92 (2015). 12SUPPLEMENTAL MATERIALS S. 1. TEMPERATURE ESTIMATION To estimate the temperature dierence of the sample, we monitored the temperatures THandTLof the Pt layers on the top and bottom surfaces of the YIG slab based on the resistance measurements as in Ref. [23]. In this experiment, the series resistance of the Pt layer, 0.5-mm-long Au electrodes, and Au wires were measured using a four probe method. The Au wires with a diameter of 50 m were rigidly connected to the Au electrodes on the Pt/YIG/Pt sample via indium soldering. Since the electrodes and wires have negligibly small resistance, the measured values and temperatures can be attributed to those in the region l0=2<y<l0=2 of the Pt layer, where the gap length l0is 5 mm in this study and y= 0 is set to the center of the YIG slab. This does not aect to the LSSE estimation, because the output voltage only appears in the Pt layers in the area without the Au electrodes, while it should be considered in the thermal-conductivity estimation (as discussed in the next section). The temperature of the Pt layers are determined by comparing the resistances of the Pt lmsRHandRLunder the temperature dierence with the isothermal RHandRLat various temperatures. The LSSE measurements started from 8 K to 300 K. The resistance mea- surements were performed three times at each temperature: (1) at the isothermal condition before the LSSE measurements, (2) at the steady-state condition with the heater output on (Tset HTset L= 5:0 K), and (3) again at the isothermal condition after the heater is turned o. TheRH(L)values recorded in the process (2) were transformed into the TH(L)by comparing them with the isothermal RH(L)values. Immediately after the process (2), the magnetic eld Hdependence of the voltages VHandVL, the LSSE voltages, was measured. After the LSSE measurement, the process (3) was performed to check the reproducibility of the resistance. The current amplitude for sensing RHandRLis 1 mA, which induces suciently small heat [0.1 % of typical applied heater output ( PHeater )], and does not aect the original sample temperatures. Figure S1 shows the temperature ( T) dependence of RHandRLof the Pt lms measured under the isothermal conditions and their dierentials. As the RH(L)Tcurve is monotonic above 10 K, we conned the measurements in the range from 10 to 300 K, where the sucient 1380 60 40 R (Ω) 300 200 100 0 T (K)0.12 0.10 0.08 0.06 0.04 0.02 0.00dR/d T (Ω/K) RH RL dRH/d T dRL/d TFIG. S1. Temperature Tdependence of RH,RL, and their dierentials. temperature sensitivity is ensured with our measurement accuracy. The dierence in the resistance between before and after the LSSE measurements was <1103 below 100 K, corresponding to the error of <0:01 K except 10{13 K. The standard deviation of the typical resistance measurements is about 1 104 . The dierence values and the standard deviation of the resistance values were treated as the error of the temperature estimation, which contributes to the error bars in Fig. 2{4. S. 2. THERMAL CONDUCTIVITY ESTIMATION The thermal conductivity of the system is calculated based on the thermal diusion equation in the steady state condition, r2T= 0. We consider a model system shown in Fig. S2(a). By considering the temperature distribution symmetric with respect to the z axis, we obtain a thermal diusion equation for the averaged temperature ~T(z) over they direction in the Pt/YIG/Pt system, @2 z~T(z) = 0: (S1) In this condition, can be obtained as =t+ 2tPt wlP
~T(S2) with
~T=~T(t=2 +tPt)~T(t=2tPt) under the boundary conditions: @z~T([t=2 +tPt]) =Jq lw; (S3) wheret(tPt) denotes the thickness of the YIG slab (the Pt layer) wthe width of the sample, andJq=PHeater the applied heat current. Here, z= 0 is the center of the sample. This 14FIG. S2. (a) Model system for temperature distribution calculation. YIG slab with a length ( l) of 6.0 mm and a thickness ( t) of 1.0 mm is sandwiched by two 10-nm-thick Pt layers and two sapphire plates with the gap length l0= 5:0 mm. The heat current Jq ows from the top sapphire plate. (b) Calculated temperature distribution in the system. Because of the size dierence of the sample and the input heat ow, the heat distribution becomes non uniform. The averaged temperature dierence
~Toverldiers from the measured averaged temperature
T, which is averaged over l0=2<y <l0=2. (c) Temperature prole along the thickness direction (the zaxis) at the center of thexyplane. equation is valid for nonuniform heat ows when the entire heat goes through the sample and sapphire plates. Since our estimated
T, which is the averaged temperature dierence overl0=2< y < l0=2, slightly diers from
~T, we need a form factor, f(l0) =
~T=
T, calculated by a numerical calculation of the heat distribution using the COMSOL software. The dierence is due to the existence of the conductive Au electrodes and the nonuniform heat- ow due to the size dierence between the input heat current and the sample dimension [ See Fig. S2(a)]. Figure S2(b) shows the calculated 2D temperature prole in the sample, where we assumed that the thermal conductivity of YIG and Pt is 1.3 102Wm1K1(the maximum value at 27 K in our experiment) and 20 10 Wm1K1[33], respectively, and the interfacial thermal conductance of the YIG/Pt is 2.79 108Wm2K1[34]. The calculated form factor becomes f(l0) = 0:88 and is temperature independent. In our system, can be regarded as that of the YIG slab because of the small values oftPtand the interfacial thermal resistance of the YIG/Pt interface. Fig. S2(c) shows the temperature distribution along the zaxis assuming T= 28 K, at which the thermal resistance of the YIG slab is minimized and thus the eect of the lm and interface may be maximized. As expected, the temperature drop in the Pt layers is about 7 104% of that in the YIG slab, and the interfacial temperature drop is about 5 102%, negligibly small 15FIG. S3. Form factor f(l0) for various l0values. Here, the other dimensions are the same as that used for the calculation of Fig. S2. contributions. For experiments with a sample having another dimension, we calculate the form factor f(l0) for various l0values, which is shown in Fig. S3. 16

2016-12-24

We report a simultaneous measurement of a longitudinal spin Seebeck effect (LSSE) and thermal conductivity in a Pt/${\rm Y_{3}Fe_{5}O_{12}}$ (YIG)/Pt system in a temperature range from 10 to 300 K. By directly monitoring the temperature difference in the system, we excluded thermal artifacts in the LSSE measurements. It is found that both the LSSE signal and the thermal conductivity of YIG exhibit sharp peaks at the same temperature, differently from previous reports. The maximum LSSE coefficient is found to be $S{\rm_{LSSE}}>10\ \mu{\rm V/K}$, one-order-of magnitude greater than the previously reported values. The concomitant enhancement of the LSSE and thermal conductivity of YIG suggests the strong correlation between magnon and phonon transport in the LSSE.

Concomitant enhancement of longitudinal spin Seebeck effect with thermal conductivity

1612.08142v1

arXiv:2010.12954v1 [quant-ph] 24 Oct 2020Enhanced sensing of weak anharmonicities through coherenc es in dissipatively coupled anti-PT symmetric systems Jayakrishnan M. P. Nair,1,∗Debsuvra Mukhopadhyay,1,†and G. S. Agarwal1, 2,‡ 1Institute for Quantum Science and Engineering, Department of Physics and Astronomy, Texas A &M University, College Station, TX 77843, USA 2Department of Biological and Agricultural Engineering, Texas A &M University, College Station, TX 77843, USA (Dated: October 27, 2020) In the last few years, the great utility of PT-symmetric syst ems in sensing small perturbations has been rec- ognized. Here, we propose an alternate method relevant to di ssipative systems, especially those coupled to the vacuum of the electromagnetic fields. In such systems, which typically show anti-PT symmetry and do not require the incorporation of gain, vacuum induces coherenc e between two modes. Owing to this coherence, the linear response acquires a pole on the real axis. We demonstr ate how this coherence can be exploited for the en- hanced sensing of very weak anhamonicities at low pumping ra tes. Higher drive powers ( ∼0.1 W), on the other hand, generate new domains of coherences. Our results are ap plicable to a wide class of systems, and we specif- ically illustrate the remarkable sensing capabilities in t he context of a weakly anharmonic Yttrium Iron Garnet (YIG) sphere interacting with a cavity via a tapered fiber wav eguide. A small change in the anharmonicity leads to a substantial change in the induced spin current. In the modern world with proliferating technological ad- vances, sensing is of fundamental importance, with far- reaching applications [1–5] across various scientific disc i- plines, with adoptions as particle sensors, motion sensors and more. Both semiclassical and quantum phenomena provide us with a wide range of techniques to attain remarkable e ffi- cacy in sensing operations. Over the past decade, parity-ti me (PT) symmetric [6–9] systems with a balanced loss and gain have been revealed to possess an enormous potential in boost - ing sensitivity. Thus, the non-Hermitian degeneracies kno wn as exceptional points (EPs) in PT-symmetric systems [10–14 ] have rendered a new avenue to engineer augmented response in an open quantum system [15–24]. Some recent experiments include the demonstration of enhanced sensitivity in micro - cavities near EPs [15] and the observation of higher-order E Ps in a coupled-cavity arrangement [16]. While this is a truly remarkable development based on a balanced gain-loss distr i- bution, one would like to examine the possibilities of newer sensing methodologies, where one can avoid the use of gain and yet, attain extremely high sensitivity. All of this woul d be without using quantum resources such as entangled photons, the innate potential of which would further boost the sensit iv- ity. In this letter, we demonstrate a new physical basis for en- hanced sensing without utilizing a commensurate gain-loss profile. We consider dissipatively coupled systems where th e coupling is produced via interaction with the vacuum of the electromagnetic field [25]. Such systems have the novel prop - erty that vacuum induces coherence between two modes. The phenomenon of vacuum induced coherence (VIC) has been the subject of intense activity [26–40] with applications r ang- ing from heat engines [30], nuclear gamma ray transmission [36] to photosynthesis [37] and molecular isomerization in vi- sion [40]. In a system with strong VIC, one of the eigen- values characterizing its dynamics moves to the real axis. W e demonstrate the great utility of this key property to the sen singof extremely weak nonlinearities which are, otherwise, di ffi- cult to detect. This new paradigm is applicable generally to a wide class of systems encountered across various scientifi c disciplines. Examples include quantum dots coupled to plas - monic excitations in a nanowire [41], superconducting tran s- mon qubits [42], quantum emitters coupled to meta materials [26–28, 43], optomechanical systems [44], hybrid magnon- photon systems [45] and more. We show explicit results on enhanced sensing by employ- ing the VIC paradigm in an anti-PT symmetric configuration to the detection of very weak magnetic nonlinearities in a YI G sphere coupled to a cavity [45–57]. This system is specifical ly chosen in view of the ongoing experimental activities. YIGs are endowed with high spin density and the collective mo- tion of these spins are embodied in the form of quasiparticle s named magnons. Dissipative coupling using YIGs has been observed in a multitude of settings, involving, for instanc e, a Fabry-perot cavity [45] or a coplanar waveguide [46]. Note that under most circumstances, weak nonlinearities of the o r- der of nHZ would require immense drive power to be detected in experiments. However, a dissipatively coupled system af - fords a prodigious response in the magnetization of the YIG which goes up spectacularly with the weakening strength of nonlinearity. That our system does not require the administ ra- tion of gain to draw out such a divergent response underpins its utility in sensing applications. Upon ramping up the dri ve power, anharmonic e ffects are strongly reinforced and new do- mains of VIC are brought to light, with peaks in the response function corresponding to strong linewidth suppression. We start offby considering the general model for a two- mode anharmonic system, which is pertinent to a wide range of physical systems. This is characterized by a Hamiltonian H//planckover2pi1=ωaa†a+ωbb†b+g(ab†+a†b) +U(b†2b2)+iΩ(b†e−iωdt−beiωdt),(1) whereωaandωbdenote the respective resonance frequencies2 FIG. 1: Schematic of a general two-mode system dissi- patively coupled through a waveguide. γa(b)andΓde- scribe decay into the surrounding (local heat bath) and cou- pling to the fiber (shared bath) respectively. Optimal ef- fect of VIC can be realized in the limit of γa(b)≪Γ. of the uncoupled modes aandb, and gconstitutes the co- herent hermitian coupling between them. The parameter U is a measure of the strength of anharmonicity intrinsic to th e mode b, which is driven externally by a laser at frequency ωd. The quantityΩrepresents the Rabi frequency. In addition, these modes could be interfacing with a dissipative environ - ment. Dissipative environments in an open quantum system fall roughly under two classifications - one, where the modes are coupled independently to their local heat baths, and an- other, where a common reservoir interacts with both, as de- picted in figure (1). A complete description of the two-mode system, in terms of its density matrixρ, is provided by the master equation [25] dρ dt=−i /planckover2pi1[H,ρ]+γaL(a)ρ+γbL(b)ρ+2ΓL(c)ρ, (2) whereγaandγbare, respectively, the intrinsic damping rates of the modes, induced by coupling with their independent heat baths. The parameter Γintroduces coherences, and the Liouvillian operator Lis defined by its action L(σ)ρ= 2σρσ†−σ†σρ−ρσ†σ. Assuming symmetrical couplings of the modes to the common reservoir we have the relation c=1√ 2(a+b). The mean value equations for aandbare obtained to be /parenleftBigg˙a ˙b/parenrightBigg =−iH/parenleftBigga b/parenrightBigg −2iU(b†b)R/parenleftBigga b/parenrightBigg +Ωe−iωdt/parenleftBigg0 1/parenrightBigg , (3) where H=/parenleftBiggωa−i(γa+Γ) g−iΓ g−iΓωb−i(γb+Γ)/parenrightBigg ,R=/parenleftBigg0 0 0 1/parenrightBigg , and the notation∝angbracketleft.∝angbracketrighthas been dropped for conciseness. In dealing with the nonlinear term, we have taken recourse to the mean-field approximation ∝angbracketleftX1X2∝angbracketright=∝angbracketleftX1∝angbracketright∝angbracketleftX2∝angbracketrightfor any two operators X1andX2. A canonical transformation of the form ( a,b)→e−iωdt(a,b) stamps out the time dependence on the final term in (3) and translates HtoH−ωd1without tampering with the nonlinear term, where 1is the 2×2 iden- tity matrix. This transformation takes us to the frame of the applied laser frequency. FIG. 2: a), b) Eigenfrequencies and linewidths for an anti- PT symmetric system, plotted against a variable ∆, viewed in units ofγ0= Γ . While EPs emerge at ∆ =±γ0, the VIC-induced linewidth suppression corresponds to ∆ = 0. c), d) Analogous plots for the PT symmetric sys- tem, against the coupling strength g, in units ofγab, at ∆ = 0. EPs are found at g=±γab. In all of these cases, the eigenvalues refer to the Hamiltonian H−ωd1. Before proceeding with a generalized treatment, let us first deconstruct the linear dynamics, i.e. when Uis dropped. Defining the detuning parameters ∆a=ωa−ωd,∆b=ωb−ωd, we have the eigenvalues of Hgiven byλ±=ωd+∆ 0− i(γ0+Γ)±/radicalbig (∆−iγab)2+(g−iΓ)2], with∆0=(∆a+∆ b)/2, ∆=(∆a−∆b)/2,γ0=(γa+γb)/2 andγab=(γa−γb)/2. Con- tingent on the stability condition Im( λ±)<0, which averts exponential amplification, the steady-state solutions for the mean values O(0)=∝angbracketleftO∝angbracketright(O(0)=A,B;O=a,b) would un- fold as A=−g+iΓ (ωd−λ+)(ωd−λ−)Ω B=∆b−i(γb+Γ) (ωd−λ+)(ωd−λ−)Ω (4) It makes for a straightforward inference that the resonant r e- sponses of these steady-state amplitudes pertain, in princ iple, to the eigenfrequencies of H. In the most generic setting, since the eigenmodes have finite linewidths, it is not possib le to chart a real parameter trajectory through these roots. Ho w- ever, there exist leeways for certain classes of systems, wh ich we make clear in the following discussion. The generic 2×2 matrix Hencompasses two distinct sub- types: (i) coherently coupled systems with Γ=0, and (ii) dis- sipatively coupled systems with g=0. Two special symme- tries can be realized within these folds, namely PT-symmetr y (allowed by (i)), which obey ( ˆPT)H(ˆPT)=H, and anti-PT3 symmetry (allowed by (ii)), with ( ˆPT)H(ˆPT)=−H. An anti-PT symmetric realization of mode hybridization can be pinned down by the parameter choices ∆0=γab=g=0 andΓ/nequal0. Clearly, while the modes are oppositely detuned in character, both of them retain their lossy nature. Here, it makes sense to switch to the rotating frame of the laser, where His substituted by H−ωd1. The shifted eigenval- ues would be obtained as −i(γ0+Γ)±√ ∆2−Γ2for|∆|>Γ and−i(γ0+Γ)±i√ Γ2−∆2(broken anti-PT) for |∆|<Γ. The behavior of the real and imaginary parts of these eigenvalue s is provided in figure 2 (a), (b). As long as the stability cri- terion is fulfilled, the responses in (4) are inversely relat ed to (ωd−λ+)(ωd−λ−)=−γ0(2Γ+γ0)−∆2. The broken anti-PT phase brings in real singularities at ωd=1 2(ωa+ωb) in the limitγ0→0, which is evidenced by the resonant inhibition in the imaginary part of λ+, as marked by the point X in fig- ure 2 (b). The extreme condition γ0=0 holds when none of the modes suffers spontaneous losses to its independent sur- rounding, all the while interacting with the mediating rese r- voir. Therefore, by harnessing dissipative coupling betwe en two modes and optimizing the e ffect of VIC, we observe a pre- cipitous divergence in the linear response under steady-st ate conditions. The PT-symmetric configuration for coherently coupled systems is conformable with the parameter structure ∆=γ0= Γ= 0. The constraintγb=−γaimplies that a loss in mode amust be offset by a commensurate gain in mode b. With the corresponding eigenvalues given as ∆0±i/radicalBig γ2 ab−g2for |γab|>gand as∆0±/radicalBig g2−γ2 abfor|γab|<g, it follows that the transition point |γab|=g, which defines the EP, introduces real singularities and strongly accentuated resonances in the mode amplitudes at ωa=ωb=ωd. Figure 2 (c), (d) depicts the eigenvalue structure in the PT-symmetric case. Next, we illustrate the importance of the condition Im(λ+)→0 in the context of the nonlinear response observed in the system. The nonlinear behavior depends on the intrin- sic symmetry properties of the matrix H. Specifically, the extraordinary response achievable in anti-PT symmetric mo d- els yields a convenient protocol for the fine-grained estima tion of weak anharmonicity. We restrict our focus to the anti-PT symmetric framework because it circumvents the possible di f- ficulties of gain fabrication. We now consider a full treatme nt of Eq. (3) by factoring in the e ffect of U. In the rotating frame, upon choosing∆a=−∆b=δ/2, andγa=γb=γ0, Eq. (3) leads to the modified steady-state relations: −(iδ/2+γ0+Γ)a−Γb=0 −(−iδ/2+γ0+Γ)b−2iU|b|2b−Γb+Ω= 0 (5) Definingγ=γ0+Γand eliminating a, the intensity x=|b|2is found to satisfy a cubic relation β2 γ2+(δ/2)2x−2Uβδ γ2+(δ/2)2x2+4U2x3=I, (6)whereβ= Γ2−γ2−(δ/2)2and I= Ω2. Eq. (6) can entail a bistable response under the condition Uδ < 0 and δ2>12γ2. However, throughout this manuscript, we operate at adequately low drive powers to ward o ffbistable signature. Now, in the limitγ0→0 andδ→0,βbecomes vanishingly small, and the first two terms in Eq. (6) recede in importance, for a given Rabi frequency Ω. Consequently, in the neighbor- hood ofδ=0, the response becomes highly sensitive to vari- ations in U. To be more precise, for su fficiently low values of the detuning, the response mimics the functional dependenc e x≈(I/4U2)1/3. A tenfold decrease in U, therefore, scales up the peak intensity of bby a factor of 4.64. In this con- text, it is useful to strike a correspondence with the sensit ivity in eigenmode splitting around an EP which is typically em- ployed in PT symmetric sensing protocols [15, 16, 19]. For two mode systems, where the EP is characterized by a square root singularity, this splitting δωscales as the square root of the perturbation parameter ǫimplying a sensitivity that goes as/vextendsingle/vextendsingle/vextendsingleδω δǫ/vextendsingle/vextendsingle/vextendsingle∝|ǫ|−1/2. However, in our setup, the sensitivity to Uin the response is encoded as/vextendsingle/vextendsingle/vextendsingleδx δU/vextendsingle/vextendsingle/vextendsingle∝|U|−5/3. The importance of the above result in the context of sens- ing is hereby legitimized for dissipatively coupled system s. Guided by the recent experiments on dissipatively coupled h y- brid magnon-photon systems [45–50], we apply these ideas to the specific example of Kerr nonlinearity in a YIG sam- ple [57]. However, the bulk of these works have restricted their investigations to the linear domain. Here, we transce nd this restriction and study the nonlinear response to an exte r- nal drive. We consider an integrated apparatus comprising an optical cavity and a YIG sphere, both interfacing with a one-dimensional waveguide. The direct coupling between th e cavity and the magnon modes can be neglected. However, the interaction with the waveguide would engender an indi- rect coupling between them. In order to excite the weak Kerr nonlinearity of the YIG sphere, a microwave laser is used to drive the spatially uniform Kittel mode. The full Hamiltoni an in presence of the external drive can be cast exactly in the form of Eq. (1), with bsuperseded by the magnonic operator m[58], Heff//planckover2pi1=ωaa†a+/bracketleftBig ωmm†m+U(m†2m2)/bracketrightBig + iΩ/parenleftBig m†e−iωdt−meiωdt/parenrightBig .(7) As discussed earlier, the mediating e ffect of the waveguide is reflected as a dissipative coupling between the two modes, which instills VIC into the system. With the anti-PT sym- metric choices∆a=−∆m=δ/2,γa=γb=γ0, and the redefinitionγ0+Γ =γ, we recover Eq. (6) in the steady state, with the obvious substitution b→mandx=|b|2de- noting the spin current response. We now expound the utility of engineering a lossless system in sensing weak Kerr non- linearity. To that end, we zero in on the parameter subspace Γ=γ=2π×10 MHz. Sinceβ=δ2/4, the contributions from the first two terms in Eq. (6) taper o ffas resonance is approached. As outlined earlier, we find that for all practi- cal purposes, the nonlinear response can be approximated as4 FIG. 3: a) The spin current plotted against δat two dif- ferent nonlinearities; b) spin currents away from the VIC condition, compared against the lossless scenario, at di ffer- ent drive powers and at U/2π=7.8 nHz - for ease of comparison, the blue and maroon curves have been scaled up by 10; c) contrasting responses observed at a drive power of 1 mW for two different strengths of nonlinearity. x≈(I/4U2)1/3in the regionδ/2π< 1 MHz, which demon- strates its stark sensitivity to U. A lower nonlinearity begets a higher response, as manifested in figure 3 (a), where plots of x againstδare studied at differing strengths of the nonlinearity. Even at Dp=1µW, we observe a significant enhancement in the induced spin current of the YIG around δ=0. The result is a natural upshot of the VIC-induced divergent response in an anti-PT symmetric system in the linear regime. Quite con- veniently, the inclusion of nonlinearity dispels the seemi ngly absurd problem of a real singularity noticed in the linear ca se. IfΓ<γ, a strong quenching in the response is observed, as depicted in figure 3 (b). The sensitivity to variations in Ualso incurs deleterious consequences. Nevertheless, we can cou n- teract this decline by boosting the drive power. A drive powe r close to 1 mWcan bring back the augmented response and the pronounced sensitivity to U(figure 3 (c)). This mechanism can serve as an efficient tool to sense small anharmonicities present in a system. The fact that even a minute pump power in the range of 1µWto 1mW generates substantial enhance- ment in the spin current makes it all the more robust. As evident from the preceding discussion, it is imperative that the waveguide-mediated coupling overshadows the ef- fect of spontaneous emissions. Moreover, the protocol hing es on the anti-PT symmetric character and eigenmodes of H, which largely control the dynamics at low drive powers. At larger drive powers ( ∼0.1W), the nonlinear correction in (3) becomes important and activates new coherences. A theoreti - cal explanation of this phenomenon can be spelled out by lin- earizing the dynamics of the mode mabout its steady-state FIG. 4: Nonlinearity induced coherences and higher dimen- sionality: (a) Imaginary parts of the eigenmodes of an ef- fectively four-dimensional system at a drive power of 0 .1 W- aroundδ/γ=−3.2, an extreme linewidth narrow- ing is observed; b) the spin current response, with a peak near toδ/γ=−2.9, highly skewed due to a stronger drive. value ensuing from (6). This linearization yields a higher- dimensional eigensystem, portrayed in figure 4 (a). The new coherences are closely correlated with the extreme linewid th narrowing manifested in the higher-dimensional model. Fig - ure 4 (b) exemplifies new VIC-induced peaks that emerge even when spontaneous decays become comparable to the dis- sipative coupling. Details on this calculation are provide d in the supplementary material [58]. In summary, we have proposed an optical test bed that shows enhanced sensitivity to Kerr nonlinearity in the mode responses to an external field, hence qualifying it as a proto - typical agency to gauge the strength of anharmonic perturba - tions under optimal conditions. The physical origin of this peculiar behavior lies in an e ffective coupling induced be- tween the cavity and the magnon modes in the presence of a shared ancillary reservoir. Such a coupling is purely an ar - tifact of third-party mediation, and is dissipative in natu re as the two modes synergistically drive up the energy being chan - neled into the interposing reservoir. Optimal results vis- ` a-vis the estimation of nonlinearity are obtained when VIC strong ly dominates, i.e when spontaneous emissions from the modes to the surrounding environments become negligible in com- parison to the waveguide-mediated coupling. Since dissipa - tively coupled systems do not require the synthetic introdu c- tion of gain for efficient sensing, our setup o ffers a clear edge over PT-symmetric systems which rely on a balanced trade- offbetween gain and loss. At higher drive powers, we observe skewed VIC peaks as a testimony to strongly anharmonic re- sponses, even when decays into the environment become sig- nificant. These nonlinearity-induced VICs could bear rele- vance in other contexts and merits further investigation. A l- though, to provide numerical estimates, our analysis has be en tailored to demonstrate the sensitivity in the context of YI G oscillations, the essence of our assessment would be applic a- ble to any two-mode nonlinear system.5 ACKNOWLEDGMENTS The authors gratefully acknowledge the support of The Air Force Office of Scientific Research [AFOSR award no FA9550-20-1-0366], The Robert A. Welch Foundation [grant no A-1243] and the Herman F. Heep and Minnie Belle Heep Texas A&M University endowed fund. GSA thanks Dr. C. M. Hu for discussions on the dissipative coupling and for shari ng his data with us. ∗jayakrishnan00213@tamu.edu †debsosu16@tamu.edu ‡girish.agarwal@tamu.edu [1] E. Gil-Santos, D. Ramos, J. Mart´ ınez, M. Fern´ andez-Re g´ ulez, R. Garc´ ıa, ´A. San Paulo, M. Calleja, and J. Tamayo, Nature nanotechnology 5, 641 (2010). [2] J. Zhu, S. K. Ozdemir, Y .-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, Nature photonics 4, 46 (2010). [3] L. He, S ¸. K. ¨Ozdemir, J. Zhu, W. Kim, and L. Yang, Nature nanotechnology 6, 428 (2011). [4] F. V ollmer and L. Yang, Nanophotonics 1, 267 (2012). [5] S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. S waim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein - Dunlop, Phys. Rev. Lett. 108, 120801 (2012). [6] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). [7] C. M. Bender, M. Berry, and A. Mandilara, Journal of Physi cs A: Mathematical and General 35, L467 (2002). [8] C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002). [9] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z . H. Musslimani, Opt. Lett. 32, 2632 (2007). [10] L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li , G. Wang, and M. Xiao, Nature photonics 8, 524 (2014). [11] I. I. Arkhipov, A. Miranowicz, O. Di Stefano, R. Stassi, S. Savasta, F. Nori, and i. m. c. K. ¨Ozdemir, Phys. Rev. A 99, 053806 (2019). [12] W. Heiss, Journal of Physics A: Mathematical and Theore tical 45, 444016 (2012). [13] H. Xu, D. Mason, L. Jiang, and J. Harris, Nature 537, 80 (2016). [14] H. Cao and J. Wiersig, Rev. Mod. Phys. 87, 61 (2015). [15] W. Chen, S ¸. K. ¨Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017). [16] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El- Ganainy, D. N. Christodoulides, and M. Khajavikhan, Nature 548, 187 (2017). [17] Z. Xiao, H. Li, T. Kottos, and A. Al` u, Phys. Rev. Lett. 123, 213901 (2019). [18] Z. Lin, A. Pick, M. Lonˇ car, and A. W. Rodriguez, Physica l review letters 117, 107402 (2016). [19] J. Wiersig, Phys. Rev. Lett. 112, 203901 (2014). [20] P.-Y . Chen and J. Jung, Phys. Rev. Applied 5, 064018 (2016). [21] M.-A. Miri and A. Alu, Science 363(2019). [22] X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Phys. Rev. Lett. 123, 237202 (2019). [23] C. Zeng, Y . Sun, G. Li, Y . Li, H. Jiang, Y . Yang, and H. Chen , Optics express 27, 27562 (2019). [24] J. Wiersig, Phys. Rev. A 93, 033809 (2016).[25] G. S. Agarwal, in Quantum Optics (Springer, 1974) pp. 1–128. [26] G. Agarwal, Physical Review Letters 84, 5500 (2000). [27] P. K. Jha, X. Ni, C. Wu, Y . Wang, and X. Zhang, Physical review letters 115, 025501 (2015). [28] E. Lassalle, P. Lalanne, S. Aljunid, P. Genevet, B. Stou t, T. Durt, and D. Wilkowski, Physical Review A 101, 013837 (2020). [29] D. Kornovan, M. Petrov, and I. Iorsh, Physical Review A 100, 033840 (2019). [30] M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, and A. Svidzinsky, Proceedings of the National Academy of Sci- ences 108, 15097 (2011). [31] M. Kiffner, M. Macovei, J. Evers, and C. Keitel, in Progress in Optics , V ol. 55 (Elsevier, 2010) pp. 85–197. [32] C. H. Keitel, Physical review letters 83, 1307 (1999). [33] P. Zhou and S. Swain, Physical Review A 56, 3011 (1997). [34] E. Paspalakis and P. Knight, Physical review letters 81, 293 (1998). [35] E. Paspalakis, C. H. Keitel, and P. L. Knight, Physical R eview A58, 4868 (1998). [36] K. P. Heeg, H.-C. Wille, K. Schlage, T. Guryeva, D. Schu- macher, I. Uschmann, K. S. Schulze, B. Marx, T. K¨ ampfer, G. G. Paulus, et al. , Physical review letters 111, 073601 (2013). [37] K. E. Dorfman, D. V . V oronine, S. Mukamel, and M. O. Scull y, Proceedings of the National Academy of Sciences 110, 2746 (2013). [38] M. O. Scully, Phys. Rev. Lett. 104, 207701 (2010). [39] A. A. Svidzinsky, K. E. Dorfman, and M. O. Scully, Phys. Rev. A 84, 053818 (2011). [40] A. Dodin and P. Brumer, The Journal of chemical physics 150, 184304 (2019). [41] H. Wei, Z. Li, X. Tian, Z. Wang, F. Cong, N. Liu, S. Zhang, P. Nordlander, N. J. Halas, and H. Xu, Nano letters 11, 471 (2011). [42] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster , J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007). [43] I. Thanopulos, V . Yannopapas, and E. Paspalakis, Phys. Rev. B 95, 075412 (2017). [44] N. R. Bernier, L. D. T´ oth, A. K. Feofanov, and T. J. Kippe n- berg, Phys. Rev. A 98, 023841 (2018). [45] M. Harder, Y . Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y . S. Gui, R. L. Stamps, and C.-M. Hu, Phys. Rev. Lett. 121, 137203 (2018). [46] B. Bhoi, B. Kim, S.-H. Jang, J. Kim, J. Yang, Y .-J. Cho, an d S.-K. Kim, Phys. Rev. B 99, 134426 (2019). [47] Y .-P. Wang, J. W. Rao, Y . Yang, P.-C. Xu, Y . S. Gui, B. M. Ya o, J. Q. You, and C.-M. Hu, Phys. Rev. Lett. 123, 127202 (2019). [48] J. W. Rao, Y . P. Wang, Y . Yang, T. Yu, Y . S. Gui, X. L. Fan, D. S. Xue, and C.-M. Hu, Phys. Rev. B 101, 064404 (2020). [49] B. Yao, T. Yu, Y . Gui, J. Rao, Y . Zhao, W. Lu, and C.-M. Hu, Communications Physics 2, 1 (2019). [50] Y .-P. Wang and C.-M. Hu, Journal of Applied Physics 127, 130901 (2020). [51] A. Metelmann and A. A. Clerk, Phys. Rev. X 5, 021025 (2015). [52] W. Yu, J. Wang, H. Y . Yuan, and J. Xiao, Phys. Rev. Lett. 123, 227201 (2019). [53] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [54] Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usam i, and Y . Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [55] Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamaz aki, K. Usami, and Y . Nakamura, Science 349, 405 (2015). [56] S. P. Wolski, D. Lachance-Quirion, Y . Tabuchi, S. Kono, A. Noguchi, K. Usami, and Y . Nakamura,6 Phys. Rev. Lett. 125, 117701 (2020). [57] Y .-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. Q. You,Phys. Rev. B 94, 224410 (2016). [58] Refer to supplementary material.

2020-10-24

In the last few years, the great utility of PT-symmetric systems in sensing small perturbations has been recognized. Here, we propose an alternate method relevant to dissipative systems, especially those coupled to the vacuum of the electromagnetic fields. In such systems, which typically show anti-PT symmetry and do not require the incorporation of gain, vacuum induces coherence between two modes. Owing to this coherence, the linear response acquires a pole on the real axis. We demonstrate how this coherence can be exploited for the enhanced sensing of very weak anhamonicities at low pumping rates. Higher drive powers ($\sim 0.1$ W), on the other hand, generate new domains of coherences. Our results are applicable to a wide class of systems, and we specifically illustrate the remarkable sensing capabilities in the context of a weakly anharmonic Yttrium Iron Garnet (YIG) sphere interacting with a cavity via a tapered fiber waveguide. A small change in the anharmonicity leads to a substantial change in the induced spin current.

Enhanced sensing of weak anharmonicities through coherences in dissipatively coupled anti-PT symmetric systems

2010.12954v1

arXiv:1309.4841v2 [cond-mat.mtrl-sci] 10 Oct 2013Tuning magnetotransport in PdPt/Y 3Fe5O12: Effects of magnetic proximity and spin orbital coupling X. Zhou, L. Ma, Z. Shi, and S. M. Zhou∗ Shanghai Key Laboratory of Special Artificial Microstructu re and Pohl Institute of Solid State Physics and School of Physics Science and Engi neering, Tongji University, Shanghai 200092, China (Dated: June 17, 2018) Abstract Anisotropic magnetoresistance (AMR) ratio and anomalous H all conductivity (AHC) in PdPt/Y 3Fe5O12(YIG) system are tuned significantly by spin orbital couplin g strength ξthrough varying the Pt concentration. For both Pt/YIG and Pd /YIG, the maximal AMR ratio is located at temperatures for the maximal suscept ibility of paramagnetic Pt and Pd metals. The AHC and ordinary Hall effect both change the s ign with temperature for Pt-rich system and vice versa for Pd-rich system. The pre sent results ambiguously evidence the spin polarization of Pt and Pd atoms in contact w ith YIG layers. The global curvature near the Fermi surface is suggested to change with the Pt concentration and temperature. PACS numbers: 72.25.Mk, 72.25.Ba, 75.47.-m, 75.70.-i ∗Correspondence author. Electronic mail: shiming@tongji.edu.cn 1Generation, manipulation, and detection of pure spin current are p opular topic in the community of spintronics because of its prominent advantage of negligible Joule heat in spintronic devices1–5. Pure spin current can be generated by spin Hall effect, spin Seebeck effect (SSE), and etc. By spin Hall effect, the pure spin current can be achieved in semiconductors due to strong spin orbital couplin g (SOC). In the SSE approach, the spin current is produced in ferromagnetic m aterials with a temperature gradient and injected into another nonmagnetic lay er through the interface. In general, the pure spin current cannot be probed by conventional electric approach. Instead, it is detected by inverse spin Hall effec t6,7. With strong SOC in Pt layers and long spin diffusion length in Y 3Fe5O12(YIG) insulator layers, the Pt/YIG systems are particularly suitable for d esign and fabrication of spintronic devices8–17. In studies of the SSE phenomena of Pt/YIG system, the SEE and the anomalous Nernst effect were argued to b e entangled9, where the latter comes from the spin polarization due to the magnet ic proximity effect (MPE) of the nearly ferromagnetic Pt layers. Many attempt s have been made to study the MPE in Pt/YIG system. Since the atomic magnetic momen t of Pt is too small to be measured by magnetometry, anisotropic magnetor esistance (AMR) and anomalous Hall effect (AHE) have been investigated intensively a s a function of the Pt layer thickness and sampling temperature ( T)10–16. Up to date, however, magnetotransport results are controversial. The AMR ratio of Pt /YIG system exhibits an angular dependence different from the conventional AM R in magnetic films, and it changes nonmonotonically with the Pt layer thickness. Alt hough these phenomena were attributed to spin Hall magnetoresistance (SMR)14,15instead of the conventional AMR, the nonmonotonic variation of the AMR with Tcannot be understood in the SMR model13. Moreover, the ferromagnetic ordering in Pt layers was proved by the anomalous Hall effect (AHE) in Pt/YIG system9. In particular, the mechanism of the observed sign change of the AHE with Tis still unclear. Very recently, the x-ray magnetic circular dichroism measurements hav e been performed by different groups and experimental results are still controvers ial possibly due to either small atomic magnetic moments of Pt or antiparallel alignmen t of spins between neighboring Pt atoms11,13,18. Therefore, an alternative ideal experimental approach must be taken to reveal the MPE in Pt/YIG system. 2In this work, we will study the SOC effect on the AMR and AHE by using Pd1−xPtx(PdPt)/YIG systems. Here, Pd and Pt atoms are isoelectric elemen ts with different atomic order numbers such that the effective SOC str ength can be significantly adjusted by modifying x19. It is surprising that the AHE and AMR can be tuned significantly for xfrom 0 to 1.0. Meanwhile, the nonmonotonic variation of the AMR with Tis revealed to be caused by the unique Tdependence of the spin polarization of Pt and Pd metals. The sign change of AHE with Tis found in Pt-rich samples and vice versa in Pd-rich systems. The intriguing ph enomena provide strong evidence for the MPE. Meanwhile, the Ttuning effects on the curvature near Fermi surface of polarized Pt and Pd layers are illus trated. A series of PdPt/YIG bilayers were fabricated by pulse laser deposit ion and subsequent magnetron sputtering in ultrahigh vacuum on (111)-o riented, single crystalline Gd 3Ga5O12(GGG) substrates. The 70 nm thick YIG thin films were epitaxially grown via pulsed laser deposition from a stoichiometric polyc rystalline target using a KrF excimer laser. Secondly, PdPt layers were depos ited on YIG thin films by magnetron sputtering. The thickness of the YIG and Pd Pt layers was determined by the X-ray reflection (XRR) as shown in Fig. 1(a). Figu re 1(b) shows that the x-ray diffraction (XRD) peaks at 2 θ= 51 degrees for (444) orientations in GGG substrate and YIG films overlap each other. The epitaxial gro wth of the YIG films was confirmed by Φ and Ψ scan with fixed 2 θfor the (008) reflection of GGG substrates and YIG films, as shown in Fig. 1(c). In-plane mag netization hysteresis loops of the YIG films were measured at room temperatu re by vibrating sample magnetometer in Fig. 1(d). The measured magnetization of 1 34 emu/cm3 is almost equal to the theoretical value, and the coercivity is as sma ll as 6.0 Oe. In experiments, half width at half height of the ferromagnetic resona nce absorption peak is about 3 Oe17. Therefore, high quality epitaxial YIG films are achieved in the present work. Before measurements, the films were patterned into normal Hall b ar, and then AMR and AHE were measured from 10 to 300 K. Figure 2(a) shows the longitudinal resistivity ρxxversus the external magnetic field Hat room temperature. At the saturation state, at the angle between the magnetization and the sensing current φH= 0 theρxxis larger than that of ρxxatφH= 90 degrees, similar to the conven- 3tional AMR in thick magnetic metallic films such as permalloy13. Figure 2(b) shows the in-plane angular dependence of the AMR at room temperature c an be fitted by a linear function of cos2φH, exhibiting a similar attribute in permalloy films. The AMR ratio depends on both the sampling Tandx, as shown in Fig. 2(c). For both Pt/YIG13and Pd/YIG systems, the ∆ ρxx/ρxxshows nonmonotonic variations with T; whereas for most ferromagnetic materials it changes monotonica lly. The max- imal value is located near 120 K and 60 K for Pt/YIG and Pd/YIG, resp ectively. Remarkably, the susceptibility of paramagnetic Pt and Pd was early o bserved to have broad peaks almost at the same temperatures20,21. For nonmagnetic transition metals, the enhanced susceptibility χ=χ0/(1−IN(EF)), where χ0,I, andN(EF) refer to the susceptibility without the presence of Coulomb interac tion, the Stoner parameter, and the density of states (DOS) near Fermi level, res pectively. Both the Stoner parameter and the MPE induced magnetic moment are als o expected to change nonmonotonically22,23. Therefore, the nonmonotonic dependence of the AMR in Pt/YIG and Pd/YIG systems should stem from their unique Tdependence of the induced magnetic moments of both Pt and Pd layers. The mono tonic change of the AMR ratio for intermediate xmay be due to the contribution of the impurity scattering, as proved below by the large ρxxat intermediate x. It is noted that the present AMR ratio in Pd/YIG is much larger than the values reported by Linet al24, possibly due to the weak spin polarization of Pd atoms. In experiments, the Hall resistivity ρxywas measured as a function of Hin the out-of-plane geometry. The anomalous Hall resistivity ρAHwas extrapolated from the linear dependence of ρxyat large H. Figure 3 shows the Hall loops for Pt/YIG and Pd/YIG at 10 K and 300 K. For Pt/YIG, both ρAHand the ordinary Hall coefficient R0are negative at room temperature but positive at 10 K. In contrast, they are negative at both 10 K and 300 K for Pd/ YIG samples24. Figure 4(a) shows the σAHas a function of Tfor all samples, where σAH=ρAH/(ρ2 xx+ρAH∗ρxx)≃ρAH/ρ2 xxsinceρAH≪ρxx. TheσAHchanges from the positive to negative for Pt-rich samples10; whereas it is always negative in the measured Tregion for small x. Intriguingly, Figure 4(b) shows that the R0also changes the sign for large xwhereas no sign change occurs for small x. Apparently, the Tdependence of σAHis correlated with that of R0as a function of 4x. Figure 4(c) shows at high T,ρxxof all samples increases approximately linearly withTand deviates from the linear dependence at low T. The residual resistivity changes nonmonotonically as a function xwith a maximum near x= 0.6, verifying almost random location of Pt and Pd atoms25. For spherical Fermi surface, the R0sign is directly determined by the numbers of electrons and holes. For nonspherical Fermi surface, howeve r, it is also strongly related to the curvature near the Fermi surface. As the integra tion of the Berry curvature over the Brillouin zone, the intrinsic AHC of magnetic tran sition metals is naturally determined by the curvature near the Fermi surface. For paramagnetic Pt, the DOS near the Fermi surface changes sharply with the ener gy23and theR0 changes the sign nearthe Fermi level26. Due to the exchange splitting and SOC in polarized Pt, not only the numbers of electrons and holes but also th e curvature near Fermi surface are significantly different from those of param agnetic ones27. Therefore, the R0(at lowT) is positive for polarized Pt (in Fig. 4(b)), opposite to that of paramagnetic one26,28,29. With weak exchange splitting and SOC at high T, theR0in polarized Pt is negative, like paramagnetic Pt, as shown in Fig. 4(b). With the prominent Teffect on the Berry curvature near the Fermi surface, the intrinsic contribution to the σAHin polarized Pt is expected to change the sign withT. As well known, the σAHconsists of the skew scattering, side-jump, and intrinsic terms30. Since the magnitude of the skew scattering term (proportional t o σxx) changes slightly with T, theσAHfor Pt-rich systems is also expected to change the sign, as shown in Fig. 4(a). The co-occurrent sign changes of both R0andσAH strongly verify the globally varying curvature near the Fermi surf ace. In contrast, for Fe and Mn 5Ge3films, the R0changes from the negative to positive near 80 K whereas the σAHis always positive below room temperature, and the sign change is attributed to the change of the conductivity ratio of dandsbands instead of the global curvature change near the Fermi surface31–33. For NiPt thin films, the σAHrather than R0changes the sign with T, which is attributed to other reasons rather than the global change of the curvature near the Fermi s urface34. Due to weak SOC in polarized Pd, the global curvature near the Fermi surf ace changes less prominently, compared with that of paramagnetic one and ther eforeR0in the measuring Tregion is always negative, like the paramagnetic one. Meanwhile, 5neitherR0norσAHchanges the sign with Tas shown in Figs. 4(a)& 4(b). Similarly, for pure Ni films neither R0norσAHchanges the sign below room temperature35. The present correlation between the σAHandR0with the Pt concentration verifies that they are largely determined by the curvature near the Fermi surface. The T tuning effect on the electronic band structure is also demonstrate d in Pt/YIG. Significant SOC effect on the magnetotransport properties in PdPt /YIG is illustrated. Figures 5(a)& 5(b) show the ∆ ρxx/ρxxandσAHat 10 K as a function of x, respectively. The AMR ratio is 8 ×10−4and 1×10−4for Pt/YIG and Pd/YIG bilayers, respectively. It is enhanced in magnitude by a factor of ab out one order fromx= 0 to 1.0. In principle, the AMR in ferromagnetic materials arises from thes-dscattering, and it is theoretically predicted to be proportional to t he square of the SOC strength ξ2if the resistivity ratio of spin-up and spin-down channels is fixed according to the perturbation theory36. Since the ξof Pt is about 3 times that of Pd19, it is experimentally proved that the ratio of the AMR between Pt/YI G and Pd/YIG is close to that of ξ2. For intermediate x, the AMR ratio deviates from the quadratic dependence due to large contribution of the impurity scattering as shown in Fig. 5(c). With increasing x, theσAHat 10 K changes from the negative to positive. For Pt/YIG and Pd/YIG, it is about 3.0 and -1.0 (S/cm), r espectively, and the magnitude ratio is close to that of ξbetween two elements19,33. Therefore, the sign and magnitude of σAHin PdPt/YIG system are tuned by changing ξwith variousx. In summary, the AMR ratio in PdPt/YIG system can be enhanced by a factor of about one order from x= 0 tox= 1. It changes nonmonotonically with T due to similar Tdependence of the atomic magnetic moment. At 10 K, the AHC magnitude of x= 1 is about 3 times that of x= 0. For Pt-rich samples both theR0and AHC change their signs with Tand vice versa for Pd-rich system, due to the global change of the curvature near the Fermi surfac e withx. The SOC tuning effects on the magnetotransport properties can be un derstood based on the perturbation theory. All present phenomena directly evide nce the MPE in the PdPt/YIG. The Ttuning effect on the electronic band structure is also demonstrated in polarized PdPt layers. The present work will also be helpful for optimizing the spintronics devices. 6Acknowledgments This work was supported by the National Science Founda- tion of China Grant Nos.11374227, 51331004, 51171129, and 5120 1114, the State Key Project of Fundamental Research Grant No.2009CB929201, and Shanghai Nanotechnology Program Center (No. 0252nm004). 71M. I. D’Yakonov and V. I. Perel, Phys. Lett. 35A, 459(1971) 2J. E. Hirsch, Phys. Rev. Lett. 83, 1834(1999) 3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Sci ence306, 1910(2004) 4F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Baselmans, a nd B. J. van Wees, Nature (London) 416, 713(2002) 5S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176(2006) 6K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, Phys. Rev. B 78, 014413(2008) 7E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Le tt.88, 182509(2006) 8F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althamm er, I. M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601(2011) 9S. Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L. Chien, Phys. Rev. Lett. 107, 216604 (2011) 10S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T.Y. Che n, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012) 11D. Qu, S. Y. Huang, J. Hu, R. Q. Wu, and C. L. Chien, Phys. Rev. Le tt.110, 067206(2013) 12T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Na kayama, X. F. Jin, and E. Saitoh, Phys. Rev. Lett. 110, 067207 (2013) 13Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huan g, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207(2013) 14H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara , D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel, S. Takahashi, R. Gross, G. E.W. Bauer, S. T. B. Goen- nenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601(2013) 15N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. B. Yous sef, Phys. Rev. B 87, 184421(2013) 16Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien , Phys. Rev. 8B87, 220409(R)(2013) 17Y. Y. Sun, H. C. Chang, M. Kabatek, Y. Y. Song, Z. H. Wang, M. Jan tz, W. Schneider, M. Z. Wu, E. Montoya, B. Kardasz, B. Heinrich, S. G. E. te Velth uis, H. Schultheiss, and A. Hoffmann, Phys. Rev. Lett. 111, 106601(2013) 18S. Gepr¨ ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm , A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407(2012) 19N. E. Christensen, J. Phys. F: Metal Phys. 8, L51(1978) 20N. Inoue and T. Sugawara, J. Phys. Soc. Jpn. 44, 440(1978) 21W. Gerhardt, F. Razavi, J. S. Schiling, D. H¨ user, and J. A. My dosh, Phys. Rev. B 24, 6744(1981) 22B.Zellermann,A.Paintner, andJ.Voitl¨ ander, J.Phys.: Co ndens.Matter 16, 919(2004) 23J. Kubler, Theory of Itinerant Electron Magnetism (Oxford U niversity Press, Wiltshire UK, 2009) 24T. Lin, C. Tang, and J. Shi, Appl. Phys. Lett. 103, 132407 (2013) 25P. Blood and D. Greig, J. Phys. F: Metal Phys. 2, 79(1972) 26W. W. Schulz, P. B. Allen, and N. Trivedi, Phys. Rev. B 45, 10886(1992) 27I. A. Campbell, Phys. Rev. Lett. 24, 269(1970) 28T. Dosdale and D. Livesey, J. Phys. F: Metal Phys. 4, 68(1974) 29D. Greig and D. Livesey, J. Phys. F: Met. Phys. 2, 699(1972) 30N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong , Rev. Mod. Phys. 82, 1539(2010) 31R. W. Klaffy and R. V. Coleman, Phys. Rev. B 10, 2915(1974) 32M. G. Cottam and R. B. Stikchombe, J. Phys. C: Solid State Phys .1, 1052(1968) 33C. Zeng, Y. Yao, Q. Niu, and H. H. Weitering, Phys. Rev. Lett. 96, 037204(2006) 34T. Golod, A. Rydh, P. Svedlindh, and V. M. Krasnov, Phys. Rev. B87, 104407(2013) 35S. P. McAlister and C. M. Hurd, J. Appl. Phys. 50, 7526(1979) 36A. P. Malozemoff, Phys. Rev. B 34, 1853(1986) 9FIGURE CAPTIONS Figure 1 (color online): For typical Pt/YIG films, x-ray refle ctivity at small angles (a) and XRD diffraction at large angles (b), Φ and Ψ scan with fixed 2 θfor the (008) reflection of GGG substrate and YIG film (c). In (d) is shown the room temperature in-plane magnetization hysteresis loop of the YIG layer. In (a) black and red lines correspond to YIG and Pt layers, respectively. Figure 2 (color online): For Pt (1 nm)/YIG films, AMR curves at φH= 0 and 90 degrees (a) and angular dependent AMR at H= 10 kOe (b). For PdPt (1 nm)/YIG films, the AMR ratio versus Tfor various x(c). Figure 3 (color online): For Pt (1 nm)/YIG (a, b) and Pd (1 nm)/ YIG (c, d) films, ρxy versusHat 10 K (a, c) and 300 K(b, d). Figure 4 (color online): For PdPt (1 nm)/YIG films, σAH(a),R0(b), and ρxx(c) versus Tfor various x. Figure 5 (color online): For PdPt (1 nm)/YIG films, AMR (a), σAH(b), and ρxx(c) at 10 K versus x. Solid lines serve a guide to the eye. 101 2 3 4 5 40455055600 30 60 90 120 150180210240270300330 40 45 50 55 6049 50 51 52 53 (d)(c)(b)Intensity (a.u) 2θ (deg)(a) 75-2 YIG (444) 2θ (deg) GGG (444) -5 -50 0 50-101M (102 emu/cm3) H (Oe) FIG. 1: 11-90 0 90 180 270389.8390.0390.2 0 100 200 300110-600 -400 -200 0 200 400 600389.9390.0390.1390.2 (b) φH (deg) Rxx (Ω) (c) x=1 0.8 0.7 0.5 0.25 0.13 0104 Δρ/ρ T (K)Rxx(Ω) H (Oe)(a) FIG. 2: 12-202 -0.50.00.5 -20 0 20-1.0-0.50.00.51.0 -20 -10 0 10 20-0.50.00.5102 ρxy (µΩ⋅cm)(a) (d)(c) (b)102 ρxy (µΩ⋅cm) H (kOe) H (kOe) FIG. 3: 13-50510-2024 0 100 200 300253035106 R0 (µΩ cm/Oe)x=1.0 0.8 0.7 0.5 0.25 0.13 0 (b)(a)σAH (S/cm) T (K)ρxx (µΩ cm)(c) FIG. 4: 14-10123 0.0 0.2 0.4 0.6 0.8 1.02530048 σAH(S/cm)(b) x (c) ρxx (µΩ cm)10 K(a) 104 Δρ/ρ FIG. 5: 15

2013-09-19

Anisotropic magnetoresistance (AMR) ratio and anomalous Hall conductivity (AHC) in PdPt/Y$_3$Fe$_5$O$_{12}$ (YIG) system are tuned significantly by spin orbital coupling strength $\xi$ through varying the Pt concentration. For both Pt/YIG and Pd/YIG, the maximal AMR ratio is located at temperatures for the maximal susceptibility of paramagnetic Pt and Pd metals. The AHC and ordinary Hall effect both change the sign with temperature for Pt-rich system and vice versa for Pd-rich system. The present results ambiguously evidence the spin polarization of Pt and Pd atoms in contact with YIG layers. The global curvature near the Fermi surface is suggested to change with the Pt concentration and temperature.

Tuning Magnetotransport in PdPt/Y3Fe5O12: Effects of magnetic proximity and spin orbital coupling

1309.4841v2

arXiv:2311.01711v1 [cond-mat.mes-hall] 3 Nov 2023Cryogenic spin Peltier effect detected by a RuO 2-AlOxon-chip microthermometer Takashi Kikkawa,1,∗Haruka Kiguchi,2,1Alexey A. Kaverzin,1,3Ryo Takahashi,2and Eiji Saitoh1,3,4 1Department of Applied Physics, The University of Tokyo, Tok yo 113-8656, Japan 2Department of Physics, Ochanomizu University, Tokyo 112-8 610, Japan 3Institute for AI and Beyond, The University of Tokyo, Tokyo 1 13-8656, Japan 4WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan (Dated: November 6, 2023) We report electric detection of the spin Peltier effect (SPE) in a bilayer consisting of a Pt film and a Y 3Fe5O12(YIG) single crystal at the cryogenic temperature Tas low as 2 K based on a RuO 2−AlOxon-chip thermometer film. By means of a reactive co-sputteri ng technique, we successfully fabricated RuO 2−AlOxfilms having a large temperature coefficient of resistance (TC R) of∼100% K−1at around 2 K. By using the RuO 2−AlOxfilm as an on-chip temperature sensor for a Pt/YIG device, we observe a SPE-induced temperature ch ange on the order of sub- µK, the sign of which is reversed with respect to the external magnet ic fieldBdirection. We found that the SPE signal gradually decreases and converges to zero by i ncreasing Bup to 10 T. The result is attributed to the suppression of magnon excitations due to t he Zeeman-gap opening in the magnon dispersion of YIG, whose energy much exceeds the thermal ene rgy at 2 K. I. INTRODUCTION One of the important features in spintronics is that various phenomena have been found at room tempera- ture in simple stacked structures, leading to their prac- tical device applications1–4. Meanwhile, exploring the spintronic phenomena at low temperatures often resulted in a discovery of new functional properties with both fundamental and practical prospects5–11. A typical ex- ample is the spin Seebeck effect (SSE), which refers to the generation of a spin current as a result of a tem- perature gradient in a magnetic material, and has been observed at room temperature in a variety of magnetic materials, including garnet- and spinel-ferrites with high magnetic ordering temperatures12–14. When SSEs are measured at low temperatures in certain systems how- ever, intriguing physics comes to the surface. Majorfind- ings include the signal anomalies induced by hybridized magnon-phonon excitations14–19, unconventional sign re- versal due to competing magnon modes having oppo- site spin polarizations20, observationofa spin-superfluid- mediated nonlocal SSE signal21, and SSEs driven by paramagnetic spins22,23and exotic elementary excita- tions in quantum spin systems24–26. Furthermore, re- cently, a nuclear SSE has been observed in an antiferro- magnet having strong hyperfine coupling14,27. The sig- nal increases down to ultralow temperatures on the order of 100 mK, which is distinct from conventional thermo- electric effects in electronic (spin) systems14,27, and may offer an opportunity for exploring thermoelectric science and technologies at ultralow temperatures, an important environment in quantum information science. In contrast to the intense research on SSEs, the spin Peltier effect28–39, the reciprocal of the SSE, remains to be explored at low temperatures below 100 K because of its experimental difficulty. The SPE modulates the temperature of a junction consisting of a metallic film and a magnet in response to a spin current29, and has been detected usually by means of lock-in thermogra-phy (LIT)29,30,32,36and thermocouples28,31,34. The LIT measures the infrared intensity emitted from the sample surface based on a combination of the lock-in with tem- perature imaging technique, whose intensity is in pro- portion to the fourth power of the absolute tempera- tureT(the Stefan–Boltzmann law32,38). This results in a typical resolution of 0.1 mK at room temperature38, which is sufficient to measure a SPE in a prototypical Pt/Y3Fe5O12(YIG) system at higher temperatures ( ∼ room temperature and above)29,30. However, the LIT may not be applicable for detecting the low-temperature SPE, because its sensitivity is dramatically reduced with decreasing temperature32,38. Furthermore, a thermocou- ple micro-sensor with a high resolution of ∼5µK was used to measure a SPE down to 100 K in Ref. 34. How- ever, it was found to be difficult to conduct the measure- ments below 100K as the sensitivity of the thermocouple decreases with decreasing T. It is therefore important to establish an alternative experimental method for detect- ing cryogenic SPEs4. An ultimate goal in this direction would be to find cryogenic SPEs driven by nuclear and quantum spins that can be activated even at ultralow temperatures, toward future possible cooling- and heat- pump technologies in such an environment. In this study, we have explored the SPE at a cryo- genic temperature below the liquid-4He temperature in a prototypical Pt/YIG system. There are three cru- cial requirements for practical realization of such mea- surement that are (1) the high temperature-resolution of∼sub-µK-order or better at low temperatures, (2) ability to detect a temperature change of a metallic (Pt) thin film (which implies for contact-mode mea- surements sufficient thermal coupling and low heat ca- pacity), and (3) reliability under a high magnetic-field environment. To realize the thermometry that meets these requirements, we adopted a RuO 2-based micro- thermometer40–49(RuO2−AlOxcomposite film in our case). In general, RuO 2-based resistors show a high temperature-sensitivity due to their large negative tem-2 perature coefficient of resistance. Besides, they show reasonably small magnetoresistance and can be made in a thin-film form. Owing to these advantages, in fact, RuO2-based chip resistors have widely been used as tem- peraturesensorsatcryogenictemperatures40,41. We have fabricatedRuO 2−AlOxfilms bymeans ofa co-sputtering technique and found the optimal fabricationcondition by characterizing their electric transport properties. By us- ing a RuO 2−AlOxfilm as an on-chip temperature sensor for a Pt-film/YIG-slab system, we successfully measured a SPE-induced temperature change on the order of sub- µK atT= 2 K. Our results provide an important step toward a complete physical picture of the SPE and es- tablishment of cryogenic spin(calori)tronics37. II. EXPERIMENTAL PROCEDURE A. Fabrication of RuO 2−AlOxfilms We have fabricated RuO 2−AlOxcomposite films as a micro-thermometer by means of d.c. co-sputtering technique from RuO 2(99.9%, 2-inch diameter) and Al (99.999%, 2-inch diameter) targets under Ar and O 2at- mosphere. To obtain the most suitable thermometer film for the SPE at low temperatures, a series of co-sputtered RuO2−AlOxfilms on thermally-oxidized Si substrates was first prepared at several d.c. power values for the RuO2target (PRuO2= 25, 26, 27, 28, and 30 W) and the fixedd.c. powerfortheAltarget( PAlOx= 25W)undera sputtering gasofAr + 7.83vol.%O 2at a pressureof 0.13 Pa at room temperature. Here, the values of Ar −O2gas amount and the d.c. power of PAlOx= 25 W were cho- sen such that highly-insulating AlO xfilms are obtained with a reasonable deposition rate ( ∼1 nm/min) when AlOxis sputtered solely from the Al target. We note that, if the O 2gas amount exceeds an onset value, the d.c. sputtering rate suddenly decreases due to the sur- face oxidization of the Al target50,51, whereas if the O 2 gas amount is insufficient, the resultant AlO xfilm may show finite electrical conduction. We found that the in- troduction ofO 2by itselfdoes not playan important role in the temperature variation of the resistance for pure RuO2films (for details, see APPENDIX A). To keep the sputtering conditions and resultant films’ quality as con- sistent as possible through repeated deposition cycles, we introduced common pre-sputtering processes just before actual depositions. To remove a possible oxidized top layer of the Al target, it was pre-sputtered at a relatively high power of PAlOx= 30 W for 600 s without introduc- ing O2gas, and then the RuO 2and Al targets were pre- sputtered for 60 s under the actual deposition conditions (i.e., Ar + 7.83 vol.% O 2)52. For electric transport mea- surements of the RuO 2−AlOxfilms, they were patterned intoaHall-barshapehavingthelength, width, andthick- ness of 1 .0 mm, 0.5 mm, and ∼100 nm, respectively, by co-sputtering RuO 2−AlOxthrough a metal mask. The RuO2content in the RuO 2−AlOxfilms under the differ-ent RuO 2sputtering power PRuO2wasevaluated through scanning electron microscopy with energy dispersive X- rayanalysis(SEM-EDX)andthesurfaceroughnessofthe films was characterized through atomic force microscopy (AFM). B. Fabrication of SPE device To investigate the SPE below the liquid-4He tem- perature, we have prepared devices consisting of a Pt-film/YIG-slab bilayer, where a 100-nm-thick RuO2−AlOxfilm with Au/Ti electrodes is attached on the top surface of the Pt film to detect its SPE-induced temperature change ∆ T[see the schematic illustrations and the optical microscope image of a typical SPE device showninFigs. 1(a)–1(c)]. Threephotolithographysteps were employed to make the SPE devices, where all the film depositions were performed at room temperature. First, a 5-nm-thick Pt wire with the width of 200 µm was formed on the (111) surface of a single-crystalline YIG slabwiththesizeof5 ×5×1mm3byd.c. magnetronsput- tering in a 0 .1 Pa Ar atmosphere under the d.c. power of 20 W. In the next photolithography step, a 70-nm- thick insulating AlO xlayer was formed at the area of 230×350µm2[300×500µm2forthe deviceshownin Fig. 1(c)] on top of the Pt/YIG layer to electrically isolate the RuO 2−AlOxfilm from the Pt layer. Here, the AlO x deposition was done by r.f. magnetron sputtering from an Al2O3target (99.99%, 2-inch diameter) under the r.f. power of 150 W and a sputtering gas of Ar + 1.0 vol.% O253at a pressure of 0.6 Pa. We later confirmed that the AlO xfilm shows a high electric resistance on the or- der of 1−10GΩ along the out-of-plane direction at room temperature. Subsequently, a100-nm-thickRuO 2−AlOx thermometer film was deposited on top of the AlO xlayer at the area of 230 ×350µm2[300×500µm2for the de- vice shown in Fig. 1(c)] through the co-sputtering un- der the d.c. sputtering power of PRuO2= 28 W and PAlOx= 25 W. Here, the dimensions and sputtering power for the RuO 2−AlOxfilm were chosen such that the resistance Rof the resulting film is several tens of kΩ at2Kanditssensitivitymonotonicallyincreaseswithde- creasing Tdown to 2 K54[as shown in Figs. 3and4(d) and discussed in Sec. IIIA]. We then proceeded with the final photolithography step for Au(150 nm)/Ti(20 nm) electrodes, where the numbers in parentheses represent the thicknesses of the deposited films. Each Au/Ti elec- trode wire on the RuO 2−AlOxfilm has the 30- µm width and is placed at 50- µm intervals. To reduce the contact resistance between the RuO 2−AlOxand Ti films, Ar- ion milling was performed directly before depositing the Au/Ti film. Both the Ti and Au layers were formed by r.f. magnetron sputtering in succession without breaking vacuum. The first lithography process for the Pt layer was done using a single-layer photoresist (AZ5214E) fol- lowed by a lift-off process, whereas the second and third processes for the AlO x/RuO2−AlOxand Au/Ti layers3 B(d) Input: (e) Output:RTM Rhigh RlowRoffsetJc/g39Jc /g16/g39 Jc0 Time tTime t data acquisition /g87delay0Resistance change /g39RTM /g39RTM Rhigh RlowRoffset 0/g39RTM /g39TRTM Temperature T (K) (f) /g39RTM /g3/g111/g3 /g39T conversion via RTM -T curve YIGPt xzy~105 times repeated for each B valueI+ I-V+ V-Ti/Au RuO 2-AlOx thermometer insulating AlOx layer (a) RuO 2-AlOx (100 nm) AlOx (70 nm) Pt (5 nm) YIG slab (1 mm)Ti(20 nm)/Au(150 nm) (b) xzy (c) Ti/Au Pt YIG300 μmRuO 2-AlOx on insulating AlO xSquare-wave charge current Jc Jc FIG. 1: (a) A schematic illustration of the SPE device con- sisting of a Pt-film/YIG-slab bilayer, on top of which a RuO2−AlOxthermometer (TM) film is attached for the de- tection of the SPE-induced temperature change ∆ Tin the Pt film. Besides, in this device, Au/Ti electrodes are formed on the RuO 2−AlOxfilm for the 4 terminal resistance measure- ments and an AlO xfilm is inserted between the RuO 2−AlOx and Pt films for the electrical insulation between them. (b) A schematic side-view image of the SPE device, where the num- bers in parentheses represent the thickness. (c) An optical microscope image of a typical SPE device. (d) Input signal: A square-wave charge current Jcwith amplitude ∆ Jcapplied to the Pt film. (e) Output signal: A resistance RTMin the RuO2−AlOxfilm that responds to the change in the Jcpolar- ity, ∆RTM(≡Rhigh−Rlow) originating from the SPE-induced ∆Tof the Pt film ( ∝∆Jc)31,34. Here, the Joule-heating- induced temperature change ( ∝∆J2 c) is constant in time, and does not overlap with ∆ RTM. (f) A schematic illustra- tion of the temperature Tdependence of RTM, from which the ∆RTMvalue can be converted to the temperature change ∆T. were done using a double-layered photoresist (LOR-3A and AZ5214E) to provide an undercut structure for a better success rate of the lift off process. C. SPE and SSE measurements Figure1(a) shows a schematic illustration of the SPE device and the experimental setup in the present study. The SPE appears as a result of the interfacial spin and energy transfer between magnons in YIG and electron spins in Pt28–31. Suppose that the magnetization Mofthe YIG layer is oriented along the + ˆ zdirection by the external magnetic field B||+ˆ z, as shown in Fig. 1(a). With the application of a charge current Jc=Jcˆ yto the Ptfilm, thespinHalleffect(SHE)55,56inducesanonequi- librium spin, or magnetic moment, accumulation at the Pt/YIG interface28–31,34. ForJc||+ˆ y(Jc|| −ˆ y), the accumulated magnetic moment δmsat the interfacial Pt orientsalongthe −ˆ z(+ˆ z)direction30,57, whichisantipar- allel (parallel) to the Mdirection in Fig. 1(a). Through the interfacial spin-flip scattering, δmscreates or annihi- lates a magnon in YIG; the number of magnons in YIG increases (decreases) when δms|| −M(δms||M)4,37. Because of energy conservation, this process is accompa- nied by a heat flow Jqbetween the electron in Pt and the magnon in YIG4,37. The temperature of Pt (YIG) thus decreases(increases) when δms||−MunderJc||+ˆ yand B||+ˆ z[Fig.1(a)], whereas the temperature of Pt (YIG) increases (decreases) when δms||Mby reversing either JcorBin Fig.1(a)29–31,34. The SPE-induced tempera- ture change ∆ Tsatisfies the following relationship29–31 ∆T∝δms·M∝(Jc×M)·ˆ x. (1) For the electric SPE detection based on the on-chip thermometer (TM), we utilized the highly-accurate re- sistance measurement scheme called the Delta mode, a combination of low-noise current source and nanovolt- meter(KeithleyModel6221and2182A31,34). Weapplied a square-wave charge current Jcwith amplitude ∆ Jcto the Pt film [Figs. 1(a) and1(d)] and measured the 4 ter- minal RuO 2−AlOxresistance RTMthat responds to the change in the Jcpolarity, ∆ RTM≡Rhigh−Rlow, where Rhigh(Rlow) represents the RTMvalue for Jc= +∆Jc (−∆Jc) and was measured under the sensing current of 100 nA applied to the RuO 2−AlOxfilm [see Figs. 1(a) and1(e)]31. Here, the ∆ RTMvalue isfree fromthe Joule- heating-induced resistance change ( ∝∆J2 c) that is inde- pendent of time, which thereby only contributes to the offset resistance Roffsetof the RuO 2−AlOxfilm shown in Fig.1(e)58. During the SPE measurement, the magnetic fieldB(with magnitude B) was applied in the film plane and perpendicular to the Pt wire, i.e., B||ˆ zin Fig.1(a), except for the control experiment shown in Fig. 4(b), whereB||ˆ x. The resistance Rhigh,lowwas recorded af- ter the time delay τdelayof 50 ms [except for the τdelay dependence shown in Fig. 4(e)] during the data acqui- sition time τsensof 20 ms59, and then was accumulated by repeating the process of the Jc-polarity change ∼105 times for each Bpoint [see Fig. 1(d)] to improve the signal-to-noise ratio. ∆ RTMcan be converted into the corresponding temperature change ∆ T(= ∆RTM/S) by using the sensitivity S≡ |dRTM/dT|of the RuO 2−AlOx film [see Figs. 1(f) and4(d)]. To compare the Bdependence of the SPE signal with that of the SSE, we also measured the SSE at T= 2 K using the same device, for which all SPE results pre- sented in this paper were obtained, but in a different experimental run from the SPE measurement. Here,4 the SSE measurement was done by means of a lock- in detection technique19,22,27and the RuO 2−AlOxlayer was used as a resistive heater; an a.c. charge current Ic=√ 2Irmssin(ωt) with the amplitude of Irms= 5.48µA and the frequency of ω/2π= 13.423 Hz was applied to the RuO 2-AlOxfilm, and the second harmonic voltage in the Pt layer induced by a spin current (driven by a heat current due to the Joule heating of the RuO 2-AlOx filmPheater=RTMI2 rms) was detected. During the SSE measurement, the externalfield Bwasapplied in the film plane and perpendicular to the Pt wire, i.e., B||ˆ zin Fig. 1(a). III. RESULTS AND DISCUSSION A. Electrical conduction in RuO 2−AlOxfilms We first characterize the electrical conduction of the RuO2−AlOxfilms on thermally-oxidized Si substrates. Figure2(a) shows the Tdependence of the resistivity ρ for the films grown under the several (fixed) sputtering power values for the RuO 2(Al) target PRuO2(PAlOx). For all the films, ρincreases with decreasing Tin the entire temperature range, showing a negative tempera- ture coefficient of resistance (TCR). Both ρand its slope |dρ/dT|increase significantly at low temperatures and monotonically by decreasing the RuO 2sputtering power PRuO2. The overall ρ–Tcurve shifts toward the upper right by decreasing PRuO2. The result shows that the ρversusTcharacteristics of the RuO 2−AlOxfilms can be controlled simply by changing the sputtering power PRuO2. SEM-EDX analysis reveals that the RuO 2/AlOx ratio decreases by decreasing PRuO2[Fig.2(c)], which leads to the ρincrease in the electric transport. We also characterized the RuO 2−AlOxfilms by means of AFM andfoundthat atypicalroot-mean-squaredsurface roughnessis Rrms∼1nm, muchsmallerthan their thick- ness∼100 nm [see the AFM image of the RuO 2−AlOx films grown under PRuO2= 28 W and PAlOx= 25 W (the RuO 2content of 41%) shown in Fig. 2(d)]. The electrical conduction at sufficiently low tem- peratures for RuO 2-based thermometers has often been analyzed by the variable-range hopping (VRH) model for three dimensional (3D) systems proposed by Mott41–46,60,61, ρ=ρ0exp/parenleftbiggT0 T/parenrightbigg1/4 , (2) whereρ0is the resistivity coefficient and T0is the char- acteristic temperature related to the electron localization lengtha. To discuss our result in light of the VRH, we plot lnρversusT−1/4for the RuO 2−AlOxfilms in Fig. 2(b). We found that ln ρscales linearly with T−1/4at low-Tranges, and the ln ρ–T−1/4data is well fitted by Eq. (2) [see the black solid lines in Fig. 2(b)], sug- gesting that the low- Telectrical conduction is indeed (a) (b) 0.6 0.4 0.2 0 20 40 60 80 100 T (K)ρ (Ωcm) 300 50 25 10 5 2T (K) T-1/4 (K-1/4 )ln[ ρ (Ωcm)] 0.2 0.4 0.6 0.802 -4-2 -6PRuO 2 = 25 W = 26 W = 27 W = 28 W = 30 WPAlO x = 25 W (fixed) RuO 2 37% RuO 2 38% RuO 2 39% RuO 2 41% RuO 2 43%3D VRH fittingRuO 2 content (%) RuO2 sputtering power PRuO 2 (W) (c) 25 26 27 28 30 PAlO x = 25 W (fixed) 36 34 32 42 40 38 44 (d) Surface image of RuO 2-AlO x 500 nm 7.8 0 (nm) FIG. 2: (a) Tdependence of the resistivity ρfor the RuO2−AlOxfilms fabricated on thermally-oxidized Si sub- strates under the several d.c. sputtering power for the RuO 2 target (PRuO2) and the fixed d.c. power for the Al target (PAlOx). (b) ln ρversusT−1/4for the RuO 2−AlOxfilms. The black solid lines are obtained by fitting Eq. ( 2) (the 3D Mott VRH model) to the experimental data. (c) Rela- tionship between the RuO 2content in the RuO 2−AlOxfilms and the RuO 2sputtering power PRuO2determined by SEM- EDX. Using this correspondence, the figure legends in (b) and also Fig. 3are described in terms of the RuO 2content. (d) A typical AFM image of the RuO 2−AlOxfilm grown un- derPRuO2= 28 W and PAlOx= 25 W (the RuO 2content of 41%), where the root-mean-squared surface roughness is Rrms= 1.2 nm. The white scale bar represents 500 nm. governed by the VRH. From the fitting, the T0val- ues are obtained as 2 .58×105, 1.41×105, 8.95×104, 5.73×103, and 2.60×102K for the RuO 2−AlOxfilms grown under PRuO2= 25, 26, 27, 28, and 30 W, respec- tively. We note that, at all the Tranges adopted for the VRH fitting, the average hopping distance ( Rhop) is largerthan the electron localizationlength ( a) that is the requirement for the VRH model to be valid: Rhop/a= (3/8)(T0/T)1/4>162–66. Besides, the Mott hopping en- ergyEhop= (1/4)kBT(T0/T)1/4(kB: the Boltzmann constant) obtained for the present films is larger than (or comparable to) the thermal energy kBT, allowing for the electron hopping62–66. The above argument further con- firms the validity of the 3D Mott VRH model to describe the conduction mechanism in the RuO 2−AlOxfilms. We here discuss the T-dependent thermometer char- acteristics of the RuO 2−AlOxfilms. Figure 3shows the Tdependence of (a) the resistance R, (b) the sensitivity S≡ |dR/dT|,(c)thetemperaturecoefficientofresistance (TCR)ST≡ |(1/R)dR/dT|, and (d) the dimensionless sensitivity SD≡ |(T/R)dR/dT|=|d(lnR)/d(lnT)|for5 1 10 100 T (K) 1 10 100 T (K)1 10 100 T (K)1 10 100 T (K) 10 -310 -210 -110 010 310 410 510 6 10 310 410 510 6 10 010 110 2 10 -110 1 10 0R (Ω) |dR / dT | (Ω K-1) |(1/ R) dR / dT | (K -1) |( T/R) dR / dT | T-coefficient of resistance (TCR) ST Dimensionless sensitivity SDSignal sensitivity S Resistance R (a) (b) (c) (d)RuO 2 37% RuO 2 38% RuO 2 39% RuO 2 41% RuO 2 43%RuO 2 37% RuO 2 38% RuO 2 39% RuO 2 41% RuO 2 43% RuO 2 37% RuO 2 38% RuO 2 39% RuO 2 41% RuO 2 43% RuO 2 41% (RuO 2-AlO x film on AlO x/Pt/YIG) RuO 2 41% (RuO 2-AlO x film on AlO x/Pt/YIG) RuO 2 37% RuO 2 38% RuO 2 39% RuO 2 41% RuO 2 43% FIG. 3: Tdependence of (a) the resistance R, (b) the sen- sitivityS≡ |dR/dT|, (c) the temperature coefficient of re- sistance (TCR) ST≡ |(1/R)dR/dT|, and (d) the dimension- less sensitivity SD≡ |(T/R)dR/dT|=|d(lnR)/d(lnT)|for the RuO 2−AlOxfilms with different RuO 2content fabricated on thermally-oxidized Si substrates. The films are patterne d into a Hall-bar shape having the length, width, and thick- ness of 1 .0 mm, 0 .5 mm, and ∼100 nm, respectively, by co-sputtering RuO 2−AlOxthrough a metal mask. In (c) and (d), the ST(T) andSD(T) results for the RuO 2−AlOxther- mometer film on the Pt/YIG sample are coplotted (red star marks). the RuO 2−AlOxfilms. Here, the sensitivity Sis an es- sential quantity when the thermometer is used as an ac- tual temperature-sensor device in its original form. The TCRSTis the normalized sensitivity Sby the measured resistance R, given that Sis geometry dependent (i.e., dR/dTscales with R)40. The dimensionless sensitivity SDis a measure often used to compare the performance of the thermometers made of different materials, regard- less of their size40,67–69. For the present RuO 2−AlOx films with low RuO 2content ( <40%), the sensitivity S takes a high value on the order of 104−106Ω/K below ∼10 K. For such a low- Trange, however, their resis- tanceRvalues are highly enhanced, and exceed 1 MΩ at 2 K, which is too high to use such films as thermometers in their originaldimensions below the liquid-4He temper- ature. Besides, their TCR values start to show a satu- ration behavior by decreasing Tin such a low- Tenvi- ronment. By contrast, the RuO 2−AlOxfilm with the RuO2content of 41% (fabricated under PRuO2= 28 W andPAlOx= 25 W) shows a moderate R(S) value of 104−105Ω (104−105Ω/K) and the best TCR char- acteristic of ∼100% K−1around 2 K. We therefore adopt its growth condition for our SPE device. Overall, theS, TCR, and SDvalues of the present RuO 2−AlOx films are comparable to those of commercially avail-able Cernox ™zirconium oxy-nitride sensors40,69, carbon composites41,70,71, and AuGe films68commonly used at a similar Trange. B. Observation of SPE based on RuO 2−AlOx on-chip thermometer We are now in a position to demonstrate a cryogenic SPEinthePt/YIGsamplebasedontheRuO 2−AlOxon- chip thermometer. Figure 4(a) shows the Bdependence of the RuO 2−AlOxresistance change ∆ RTMmeasured atT= 2 K and a low- Brange of |B| ≤0.2 T. With the application of the charge current ∆ Jc(= 0.15 mA) to the Pt film, a clear ∆ RTMsignal appears with a magnitude saturated at ∼30 mΩ72and its sign changes depending on theB(||±ˆ z) direction. The signal disappears either when ∆Jcis essentially zero [gray diamonds in Fig. 4(a)] or when Bis applied perpendicular to the Pt/YIG in- terface (B|| ±ˆ x) [Fig.4(b)]. We also confirmed that theBdependence of ∆ RTMis consistent with that of the SSE in the identical Pt/YIG device [see Fig. 4(c)]. These are the representative features of the SPE29–38. Furthermore, the sign of ∆ RTMagrees with the SPE- induced temperature change29,30. As shown in Fig. 4(a), the measured ∆ RTMvalue is positive for B >0, mean- ing that the resistance RTMincreases (decreases) when Jc||+ˆ y(Jc||−ˆ y), for which the orientation of the SHE- induced magnetic moment at the interfacial Pt layer is δms|| −ˆ z(δms||+ˆ z) in Fig. 1(a). According to the negative TCR of the RuO 2−AlOxfilm, this implies that the temperatureofthe Ptfilm decreases(increases)when δms|| −ˆ z(δms||+ˆ z) underM||B||+ˆ z. This cor- respondence between the sign of the temperature change ∆Tand the relative orientation of δmswith respect to Mis consistent with the scenarioof the SPE described in Sec.IIC. We thus conclude that we succeeded in mea- suring a cryogenic SPE using the RuO 2−AlOxon-chip thermometer film. To convertthe ∆ RTMvalue to the temperature change ∆T, we measured the RTM–Tcurve for the RuO 2−AlOx film. As shown in Fig. 4(d), similar to the results de- scribed in Sec. IIIA, its resistance RTMincreases dra- matically with decreasing Tat low temperatures, and the sensitivity S=|dRTM/dT|is as large as 55 .3 kΩ/K at 2 K [The TCR STand dimensionless sensitivity SD for the film are plotted in Figs. 3(c) and3(d), respec- tively, and ρand|dρ/dT|are plotted in Figs. 7(a) and 7(b) in APPENDIX B, respectively, together with the results for the RuO 2−AlOxfilms grown on thermally- oxidized Si substrates]. In Fig. 4(c), we replot the Bde- pendence of the SPE in units of the temperature change ∆T(= ∆RTM/S) using the above Svalue. We evaluate the magnitude of the SPE-induced temperature change to be ∆TSPE= 482±39 nK, by averaging the ∆ Tval- ues for 0 .08 T≤B≤0.2 T, at which the magnetization Mof the YIG slab fully orients along the Bdirection74 [see the dashed line in Fig. 4(c)]. The standard devia-6 100 0 200 300 T (K)40 30 20 RTM (kΩ) (d) 10 2 3 4 5 T (K)Sensitivity |dR TM / dT | (kΩ K-1) 60 40 20 0-0.2 0 0.2 B (T) 40 20 /g39RTM (mΩ) -40 -20 T = 2 K, B x<(b) /g39Jc = 150 μA 0 -0.2 -0.1 0 0.1 0.2 B (T) T = 2 K, B z< 1.0 0.5 0/g39T (μK) -1.0-0.530 ms 50 ms 80 ms/g87delay :(e)-0.2 -0.1 0 0.1 0.2 B (T) 40 20 0/g39RTM (mΩ) -40 -20 T = 2 K, B z 0.15 mA 0 mA <(a) /g39Jc : 1.0 0.5 0/g39T (μK) -1.0-0.5 -0.2 -0.1 0 0.1 0.2 B (T) T = 2 K, B z< SPE SSE(c) 10 0 -10 VSSE /Pheater (mV W -1)/g39TSPE 1.0 0.5 0/g39TSPE (μK) (f) 50 100 /g87delay (ms) FIG. 4: (a) Bdependence of the SPE-induced ∆ RTMat T= 2 K and B≤0.2 T (B||ˆ z) under ∆ Jc= 0.15 and 0.00 mA and τdelay= 50 ms. The dashed lines connect adja- cent plots. (b) Bdependence of ∆ RTMforB||ˆ x(B≤0.3 T) under ∆Jc= 0.15 mA. Note that the applied Bis larger than the out-of-plane ( B||ˆ x) saturation field for bulk YIG, which is∼0.2 T73. (c) Comparison between the Bdependence of the SPE-induced temperature change ∆ T(blue filled circles) and the SSE-induced voltage normalized by heating power VSSE/Pheater(orange solid curve) at T= 2 K and B||ˆ z. The SPE data shown here is the same as that plotted in (a), but the left vertical axis is converted from ∆ RTMto ∆Tvia the RTM–Tcalibration curve plotted in (d). For details of the SSE measurement, see Sec. IIC. (d)Tdependence of RTM (main) and |dRTM/dT|(inset) for the RuO 2−AlOxfilm on the Pt/YIG sample. (e) Bdependence of the SPE-induced ∆TatT= 2 K under ∆ Jc= 0.15 mA and several τdelay values. (f) τdelaydependence of the magnitude of the SPE- induced temperature change ∆ TSPE, where ∆ TSPEis evalu- ated by averaging the ∆ Tvalues for 0 .08 T≤B≤0.2 T [see also (c)]. The dashed line represents the averaged value. Al l the ∆RTMand ∆Tdata were anti-symmetrized with respect to the magnetic field B. tion of 39 nK shows that our measurement scheme based on the RuO 2−AlOxon-chip thermometer can resolve an extremely small ∆ Ton the order of several tens of nK (which is a value achieved by repeating the process of theJc-polarity change of 7 ×104times at each B). The ∆Tresolution is much higher than that reported in the previous SPE measurements based on lock-in thermog- raphy, lock-in thermoreflectance, and thermocouples, forwhich the typical resolution is 100, 10 −100, and 5 µK, respectively34,38,39. We found that the magnitude of ∆TSPEnormalized by the charge-currentdensity ∆ jcap- plied to the Pt wire is ∆ TSPE/∆jc= 3.2×10−15Km2/A, which is two orders of magnitude smaller than the cor- responding value for Pt/YIG systems measured at room temperature30,31. The low- Tsignal reduction of the SPE is consistent with that found in the SSE17,75–77, and is attributed mainly to the reduction of the thermally acti- vated magnons contributing to these phenomena at cryo- genic temperatures. Besides, there can be a finite tem- perature gradient across the insulating AlO xfilm, be- tweenthe Pt andRuO 2−AlOxlayers,resultingin further decreaseofthedetected∆ Tsignal. Wealsomeasuredthe delay time τdelaydependence of the SPE and found that the ∆TSPEtakes almost the same value in the present τdelayrange (30 ms ≤τdelay≤80 ms) [see Figs. 4(e) and 4(f)], showing that all the data were obtained under the steady-state condition31,35. We also explored the high magnetic field response of the SPE signal. Figure 5(a) displays the ∆ TversusB data measured at T= 2 K and B≤10 T (B||ˆ z). We found that ∆ Texhibits a maximum at a low B(/lessorsimilar0.2 T) and, by increasing B, gradually decreases and is eventu- ally suppressed. The Bdependence of the SPE agrees well with that of the SSE measured with the identical device [see Fig. 5(a)]. We note that the magnetoresis- tance (MR) ratio of the RuO 2−AlOxfilm is as small as ∼3.7% forB≤10 T at T= 2 K, so that the device can be used reliably under the high- Brange. The ob- served ∆ T(B) feature is explained in terms of the sup- pression of magnon excitations by the Zeeman effect, as established in the previous SSE research17,27,75–77[see Fig.5(b)]. By increasing B, the magnon dispersion shifts toward high frequencies due to the Zeeman inter- action (∝γB). AtB= 10 T, the Zeeman energy /planckover2pi1γB is∼13.5 K in units of temperature, which is greater than the thermal energy kBTat 2 K [see Fig. 5(b)], re- sulting in an insignificant value of the Boltzmann factor: exp(−/planckover2pi1γB/kBT)∼10−3≪1, where γand/planckover2pi1represent the gyromagnetic ratio and Dirac constant, respectively. Therefore, the thermal magnons that can contribute to the SPE at a low Bare gradually suppressed with the increase of Band, atB∼10 T, are hardly excited by the strong Zeeman gap in the magnon spectrum [Fig. 5(b)], which leads to the suppression of the SPE in the low-Tand high- Benvironment. We also compared the experimental result with a calculation for the interfacial heat current induced by the SPE JSPE qand spin current induced by the SSE JSSE s, which are expressed as JSPE q∝/integraldisplayd3k (2π)3ω2∂nBE ∂ω, JSSE s∝ −/integraldisplayd3k (2π)3ωT∂nBE ∂T,(3) respectively75,78–81. Here, ω=Dexk2+γBis the parabolic magnon dispersion for YIG with the stiff-7 10 0 -10 VSSE /Pheater (mV W -1) -10 -5 0 5 10 B (T)0.2 0/g39T (μK) -0.2(a) T = 2 K, B z< SPE SSE 0.2 0ω/2 /g83 (THz) (b) Magnon dispersion 0.3 0.1 k (10-8 m -1)1 2kBT at 2 KZeeman gap ( /g118/g3 γB ) hγB /kB ~ 13.5 K at 10 T B = 10 T = 0 T magnon freeze-out magnon Experiment Calculation (arb. units) FIG. 5: (a) Comparison between the high magnetic field B response of the SPE-induced temperature change ∆ T(blue filled circles) and the SSE-induced voltage normalized by heating power VSSE/Pheater(blue solid curve) at T= 2 K andB≤10 T (B||ˆ z). The SPE data was obtained un- der ∆Jc= 0.15 mA and τdelay= 50 ms. For details of the SSE measurement, see Sec. IIC. The orange dashed curve shows the numerically calculated result based on Eq. ( 3) for T= 2 K. (b) Magnon dispersion relations for YIG15atB= 0 and 10 T, at which the magnon-excitation gap values are ∼0 and 13.5 K in units of temperature, respectively, where krep- resents the wavenumber. The thermal energy ( kBT) level of 2 K is also plotted with a green dashed line, above which thermal excitation is exponentially suppressed. ness constant of Dex= 7.7×10−6m2/s15andnBE= [exp(/planckover2pi1ω/kBT)−1]−1is the Bose −Einstein distribu- tion function. Note that the relation ω∂nBE/∂ω= −T∂nBE/∂Tensures the Onsager reciprocity between the SSE and SPE81, which makes the above expressions to be of the same form in terms of the Bdependence. As shown by the orange dashed curve in Fig. 5(a), the calculated result based on Eq. ( 3) well reproduces the experiment. This result further supports the origin of the measured ∆ Tsignal and provides additional clues for further understanding of the physics of the SPE. IV. CONCLUSIONS In this study, we have fabricated RuO 2−AlOxfilms by means of a d.c. co-sputtering technique and charac- terized their electrical conduction and sensitivity at low temperatures. Thesensitivitywasfoundtobetuned sim- ply bythe relativesputtering powerapplied forthe RuO 2 and Al targets, and the TCR value reaches ∼100% K−1 for the RuO 2−AlOxfilms with the moderate RuO 2con- tent (/greaterorsimilar41%). By using the RuO 2−AlOxfilm as an on- chip micro-thermometer, we successfully measured the SPE-induced temperature change ∆ Tin a Pt-film/YIG- slab system at the low temperature of 2 K based on the so-called Delta method, which can resolve an extremely small ∆Tvalue of several tens of nK. We also measured the high Bresponse of the SPE at T= 2 K up to B= 10 T, and found that, by increasing B, the SPE signal gradually decreases and is eventually suppressed.100 200 3000.51.01.5 0Ar only Ar + 33.3 vol.% O 2 T (K) R / R (300 K) R-T characteristics of RuO 2 film FIG. 6: Tdependenceof R/R(T= 300 K)for thepure RuO 2 films grown under only Ar gas flow and also under a large amount of O 2gas flow (Ar + 33.3 vol.% O 2), for which the resistivity ρvalues at T= 300 K are evaluated as 3 .21×10−4 and 3.96×10−4Ωcm, respectively. TheBdependence can be interpreted in terms of the field-induced freeze-out of magnons due to the Zeeman- gap opening in the magnon spectrum of YIG. We an- ticipate that our experimental methods based on an on- chip thin-film thermometer will be useful for exploring low-Tthermoelectric heating/cooling effects in various types of micro devices, including a system based on two- dimensional van der Waals materials82–84. Besides, with an appropriate optimization of the resistance and sensi- tivity of the RuO 2−AlOxfilms by controllingthe content of RuO 2, our results can be extended toward even lower temperature ranges below 1 K, where they can be used to detect unexplored cryogenic spin caloritronic effects driven by nuclear and quantum spins. ACKNOWLEDGMENTS We thank S. Daimon, R. Yahiro, J. Numata, K. K. Meng, H. Arisawa, T. Makiuchi, and T. Hioki for valu- able discussions. This work was supported by JST- CREST (JPMJCR20C1 and JPMJCR20T2), Grant-in- Aid for Scientific Research (JP19H05600, JP20H02599, and JP22K18686) and Grant-in-Aid for Transformative Research Areas (JP22H05114) from JSPS KAKENHI, MEXT Initiative to Establish Next-generation Novel In- tegratedCircuits Centers (X-NICS) (JPJ011438),Japan, Murata Science Foundation, Daikin Industries, Ltd, and Institute for AI and Beyond of the University of Tokyo. APPENDIX A: ELECTRICAL CONDUCTION IN PURE RUO 2FILMS To check the effect of O 2gas introduction during sput- teringontheRuO 2film, wealsofabricatedpristine(poly- crystalline) RuO 2films under only Ar gas flow and also under a large amount of O 2gas flow (Ar + 33.3 vol.%8 1 10 100 T (K)1 10 100 T (K)(on AlOx/Pt/YIG) ρ (Ωcm) 10 -310 -210 -110 010 1Resistivity slope Resistivity ρ (a) (b) 10 -310 -210 -110 0|dρ / dT | (Ωcm K-1) 10 -510 -4 PRuO 2 = 28 WPRuO 2 = 25 W = 26 W = 27 W = 28 W = 30 W (on AlO x/Pt/YIG) PRuO 2 = 28 W(on SiO 2/Si) PRuO 2 = 25 W = 26 W = 27 W = 28 W = 30 W (on SiO 2/Si) FIG. 7: Tdependence of (a) the resistivity ρand (b) its slope|dρ/dT|for theRuO 2−AlOxfilms onthermally-oxidized Si substrates (filled circles) and on the Pt/YIG sample (red star marks) grown under the several d.c. sputtering power for the RuO 2target (PRuO2) and the fixed d.c. power for the Al target ( PAlOx= 25 W). Note that ST≡ |(1/R)dR/dT|= |(1/ρ)dρ/dT|andSD≡ |(T/R)dR/dT|=|(T/ρ)dρ/dT|, the Tdependences of which are shown in Figs. 3(c) and 3(d), respectively. O2, which is ∼5 times greater than that used for the RuO2−AlOxdeposition)andmeasuredtheir R–Tcurves. Here, the RuO 2films were patterned into a Hall-bar shape having the length, width, and thickness of 2 .0 mm, 0.3 mm, and ∼10 nm, respectively, by sputtering RuO 2 through a metal mask. Figure 6shows the Tdependence ofRnormalized by the value at 300 K for each RuO 2film. For both the films, the R–Tcurve shows almost the same characteristics; Rgradually increases with de- creasing Tand the R(T)/R(300 K) value at T= 2 K (the temperature of interest) deviates only ∼2% with each other. This result shows that the effect of oxygen on the RuO 2deposition does not play an essential role in theRversusTcharacteristics of RuO 2. APPENDIX B: COMPARISON OF ρ−TCURVES BETWEEN RUO 2−ALOxFILMS ON SIO 2/SI SUBSTRATES AND ON PT/YIG DEVICE Figures7(a) and7(b) show the double logarithmicplot of (a) the resistivity ρand (b) its slope |dρ/dT|versus temperature Tfor the RuO 2−AlOxfilms on thermally- oxidized Si substrates (filled circles) and on the Pt/YIG sample (red star marks) grown under the several d.c. sputtering power for the RuO 2target (PRuO2) and the fixed d.c. power for the Al target ( PAlOx= 25 W). Al- though a small deviation of the ρand|dρ/dT|values is observed even under the same growth condition depend- ing on the substrate layer (i.e., SiO 2/Si or Pt/YIG), the overallTdependent feature agrees well with each other. Note that the substrate-dependent difference in the ρ and|dρ/dT|values does not have a significant impact on the observation of the cryogenic SPE, if the sensitivity is large enough for its detection. ∗Electronic address: t.kikkawa@ap.t.u-tokyo.ac.jp 1H. Ohno, M. D. Stiles, and B. Dieny, Spintronics, Proc. IEEE104, 1782 (2016). 2A. V. Chumak, P. Kabos, M. Wu, C. Abert, C. Adelmann, A. O. Adeyeye, J. ˚Akerman, F. G. Aliev, A. Anane, A. Awadet al., Advances in magnetics roadmap on spin-wave computing, IEEE Trans. Magn. 58, 0800172 (2022). 3H. Yang, S. O. Valenzuela, M. Chshiev, S. Couet, B. Dieny, B. Dlubak, A. Fert, K. Garello, M. Jamet, D.-E. Jeong, K. Lee, T. Lee, M.-B. Martin, G.S.Kar, P.S´ en´ eor, H.-J.Shin, and S. Roche, Two-dimensional materials prospects for non-volatile spintronic memories, Nature 606, 663 (2022). 4S. Maekawa, T. Kikkawa, H. Chudo, J. Ieda, and E. Saitoh, Spin and spin current–From fundamentals to re- cent progress, J. Appl. Phys. 133, 020902 (2023). 5Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z. Wang, J. Tang, L. He, L.-T. Chang, M. Montazeri, G. Yu, W. Jiang, T. Nie, R. N. Schwartz, Y. Tserkovnyak, and K. L. Wang, Magnetization switching through giant spin- orbit torque in a magnetically doped topological insulator heterostructure, Nat. Mater. 13, 699 (2014). 6J. Linder and J. W. A. Robinson, Superconducting spin- tronics, Nat. Phys. 11, 307 (2015). 7M. Umeda, Y. Shiomi, T. Kikkawa, T. Niizeki, J. Lustikova, S. Takahashi, and E. Saitoh, Spin-current co- herence peak in superconductor/magnet junctions, Appl. Phys. Lett. 112, 232601 (2018). 8Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan, Y. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han,Probe of spin dynamics in superconducting NbN thin films via spin pumping, Phys. Rev. B 97, 224414 (2018). 9Y. Shiomi, J. Lustikova, S. Watanabe, D. Hirobe, S. Taka- hashi, and E. Saitoh, Spin pumping from nuclear spin waves, Nat. Phys. 15, 22 (2019). 10K.-R. Jeon, J.-C. Jeon, X. Zhou, A. Migliorini, J. Yoon, and S. S. P. Parkin, Giant Transition-State Quasiparticle Spin-Hall Effect in an Exchange-Spin-Split Superconduc- tor Detected by Nonlocal Magnon Spin Transport, ACS Nano14, 15874 (2020). 11C. L. Tschirhart, E. Redekop, L. Li, T. Li, S. Jiang, T. Arp, O. Sheekey, T. Taniguchi, K. Watanabe, M. E. Hu- ber, K. F. Mak, J. Shan, and A. F. Young, Intrinsic spin Hall torque in a moir´ e Chern magnet, Nat. Phys. 19, 807 (2023). 12K. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S.Maekawa, andE. Saitoh, Thermoelectric gen- eration based on spin Seebeck effects, Proc. IEEE 104, 1946 (2016)., ibid.104, 1499 (2016). 13S. M. Rezende, Fundamentals of Magnonics (Springer Na- ture Switzerland AG, Switzerland, 2020). 14T. Kikkawa and E. Saitoh, Spin Seebeck Effect: Sensitive Probe for Elementary Excitation, Spin Correlation, Trans- port, Magnetic Order, and Domains in Solids, Annu. Rev. Condens. Matter Phys. 14, 129 (2023). 15T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. Uchida, Z. Qiu, G. E. W. Bauer, and E. Saitoh, Magnon Polarons in the Spin Seebeck Effect, Phys. Rev. Lett. 117, 207203 (2016).9 16L. J. Cornelissen, K. Oyanagi, T. Kikkawa, Z. Qiu, T. Kuschel, G. E. W. Bauer, B. J. van Wees, and E. Saitoh, Nonlocal magnon-polaron transport in yttrium iron gar- net, Phys. Rev. B 96, 104441 (2017). 17K. Oyanagi, T. Kikkawa, and E. Saitoh, Magnetic field de- pendence of the nonlocal spin Seebeck effect in Pt/YIG/Pt systems at low temperatures, AIP Adv. 10, 015031 (2020). 18J. Li, H. T. Simensen, D. Reitz, Q. Sun, W. Yuan, C. Li, Y. Tserkovnyak, A. Brataas, and J. Shi, Observation of Magnon Polarons in a Uniaxial Antiferromagnetic Insula- tor, Phys. Rev. Lett. 125, 217201 (2020). 19T. Kikkawa, K. Oyanagi, T. Hioki, M. Ishida, Z. Qiu, R. Ramos, Y. Hashimoto, and E. Saitoh, Composition- tunable magnon-polaron anomalies in spin Seebeck effects in epitaxial Bi xY3−xFe5O12films, Phys. Rev. Materials 6, 104402 (2022). 20S. Gepr¨ ags, A. Kehlberger, F. D. Coletta, Z. Qiu, E.-J. Guo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Al- thammer, H. Huebl, G. Jakob, Y. Ohnuma, H. Adachi, J. Barker, S. Maekawa, Gerrit E. W. Bauer, E. Saitoh, R. Gross, S. T. B. Goennenwein, and M. Kl¨ aui, Origin of the spin Seebeck effect in compensated ferrimagnets, Nat. Commun. 7, 10452 (2016). 21W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen, Y. Ma, X. Lin, J. Shi, R. Shindou, X. C. Xie, and W. Han, Experimental signatures of spin superfluid ground state in canted antiferromagnet Cr 2O3via nonlocalspin transport, Sci. Adv. 4, eaat1098 (2018). 22S. M. Wu, J. E. Pearson, and A. Bhattacharya, Paramag- netic Spin Seebeck Effect, Phys. Rev. Lett. 114, 186602 (2015). 23K. Oyanagi, S. Takahashi, T. Kikkawa, and E. Saitoh, Mechanism of paramagnetic spin Seebeck effect, Phys. Rev. B107, 014423 (2023). 24D. Hirobe, M. Sato, T. Kawamata, Y. Shiomi, K. Uchida, R. Iguchi, Y. Koike, S. Maekawa, and E. Saitoh, One- dimensional spinonspincurrents, Nat.Phys. 13, 30(2017). 25Y. Chen, M. Sato, Y. Tang, Y. Shiomi, K. Oyanagi, T. Masuda, Y. Nambu, M. Fujita, E. Saitoh, Triplon current generation in solids, Nat. Commun. 12, 5199 (2021). 26W. Xing, R. Cai, K. Moriyama, K. Nara, Y. Yao, W. Qiao, K. Yoshimura, and W. Han, Spin Seebeck effect in quantum magnet Pb 2V3O9, Appl. Phys. Lett. 120, 042402 (2022). 27T. Kikkawa, D. Reitz, H. Ito, T. Makiuchi, T. Sugimoto, K.Tsunekawa, S.Daimon, K.Oyanagi, R.Ramos, S.Taka- hashi, Y. Shiomi, Y. Tserkovnyak, and E. Saitoh, Obser- vation of nuclear-spin Seebeck effect, Nat. Commun. 12, 4356 (2021). 28J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Observation of the Spin Peltier Effect for Magnetic Insulators, Phys. Rev. Lett. 113, 027601 (2014). 29S. Daimon, R. Iguchi, T. Hioki, E. Saitoh, and K. Uchida, Thermal imaging of spin Peltier effect, Nat. Commun. 7, 13754 (2016). 30S. Daimon, K. Uchida, R. Iguchi, T. Hioki, and E. Saitoh, Thermographic measurements of the spin Peltier effect in metal/yttrium-iron-garnet junction systems, Phys. Rev. B 96, 024424 (2017). 31R. Itoh, R. Iguchi, S. Daimon, K. Oyanagi, K. Uchida, and E. Saitoh, Magnetic-field-induced decrease of the spin Peltier effectinPt/Y 3Fe5O12systemat room temperature, Phys. Rev. B 96, 184422 (2017).32A. Yagmur, R. Iguchi, S. Gepr¨ ags, A. Erb, S. Daimon, E. Saitoh, R. Gross, and K. Uchida, Lock-in thermography measurements of the spin Peltier effect in a compensated ferrimagnet and its comparison to the spin Seebeck effect, J. Phys. D: Appl. Phys. 51, 194002 (2018). 33A. Sola, V. Basso, M. Kuepferling, C. Dubs, and M. Pasquale, Experimental proof of the reciprocal relation between spin Peltier and spin Seebeck effects in a bulk YIG/Pt bilayer, Sci. Rep. 9, 2047 (2019). 34R. Yahiro, T. Kikkawa, R. Ramos, K. Oyanagi, T. Hioki, S. Daimon, and E. Saitoh, Magnon polarons in the spin Peltier effect, Phys. Rev. B 101, 024407 (2020). 35T. Yamazaki, R. Iguchi, T. Ohkubo, H. Nagano, and K. Uchida, Transient response of the spin Peltier effect re- vealed by lock-in thermoreflectance measurements, Phys. Rev. B101, 020415(R) (2020). 36S. Daimon, K. Uchida, N. Ujiie, Y. Hattori, R. Tsuboi, and E. Saitoh, Thickness dependence of spin Peltier effect visu- alized by thermal imaging technique, Appl. Phys. Express 13, 103001 (2020). 37K. Uchida, Transport phenomena in spin caloritronics, Proc. Jpn. Acad., Ser. B 97, 69 (2021). 38K. Uchida and R. Iguchi, Spintronic Thermal Manage- ment, J. Phys. Soc. Japan 90, 122001 (2021). 39A. Takahagi, R. Iguchi, H. Nagano, and K. Uchida, Highly sensitive lock-in thermoreflectance temperature measure- mentusingthermochromicliquidcrystal, Appl.Phys.Lett. 122, 172401 (2023). 40Cernox Sensors Catalog, Cryogenic temperature sen- sor characteristics, Cernox vs. Rox Sensor Perfor- mance, in Temperature Sensor Information , Lake Shore https://www.lakeshore.com/resources/sensors 41F. Pobell, Chapter 12: Low-Temperature Thermometry in Matter and Methods at Low Temperatures (Springer-Verlag Berlin Heidelberg, Heidelberg, 2007). 42W. A.Bosch, F. Mathu, H. C. Meijer, andR.W. Willekers, Behaviour of thick film resistors (Philips type RC-01) as low temperature thermometers in magnetic fields up to 5 T, Cryogenics 26, 3 (1986). 43Q. Li, C. H. Watson, R. G. Goodrich, D. G. Haase, and H. Lukefahr, Thick film chip resistors for use as low temper- ature thermometers, Cryogenics 26, 467 (1986). 44I. Bat’ko, K. Flachbart, M. Somora, and D. Vanick´ y, De- sign of RuO 2-based thermometers for the millikelvin tem- perature range, Cryogenics 35, 105 (1995). 45B. Neppert and P. Esquinazi, Temperature and magnetic field dependence of thick-film resistor thermometers (Dale type RC550), Cryogenics 36, 231 (1996). 46M. Affronte, M. Campani, S. Piccinini, M. Tamborin, B. Morten, M. Prudenziati, and O. Laborde, Low Tempera- ture Electronic Transport in RuO 2-Based Cermet Resis- tors, J. Low Temp. Phys. 109, 461 (1997). 47Y.-Y. Chen, Low-temperature Ru-Sapphire Film Ther- mometer and Its Application in Heat Capacity Measure- ments, AIP Conf. Proc. 684, 387 (2003). 48Y. Y. Chen, P. C. Chen, C. B. Tsai, K. I. Suga, and K. Kindo, Low-Magnetoresistance RuO 2−Al2O3Thin-Film Thermometer and its Application, Int. J. Thermophys. 30, 316 (2009). 49J. Nelson and A. M. Goldman, Thin film cryogenic ther- mometers defined with optical lithography for thermomag- neticmeasurements onfilms, Rev.Sci. Instrum. 86, 053902 (2015). 50S. Maniv and W. D. Westwood, Oxidation of an aluminum10 magnetron sputtering target in Ar/O 2mixtures, J. Appl. Phys.51, 718 (1980). 51E. Wallin and U. Helmersson, Hysteresis-free reactive high power impulse magnetron sputtering, Thin Solid Films 516, 6398 (2008). 52Except for this procedure, we do not need to be extremely careful about the maintenance of the RuO 2and Al tar- gets. As a side note, during the pre-sputtering process, RuO2thin flakes that are easily peeled off may form on the shutter in the sputtering chamber, so we need to clean itregularly topreventtheseflakesfrom fallingontheRuO 2 target. 53M. Voigtand M. Sokolowski, Electrical properties of thinrf sputtered aluminum oxide films, Mater. Sci. Eng.: B 109, 99 (2004). 54Since the SPE measurement in this study was intended to be performed at 2 K, the conditions for the thermometer fabrication were chosen to realize high sensitivity at 2 K. Our method can be extended to different temperatures by making thermometer films with high sensitivity at the tar- geted temperature range. However, as the temperature is lowered toward zero kelvin, the SPE-induced temperature difference is expected to decrease because the population of thermal magnons in YIG is suppressed. 55A Hoffmann, Spin Hall Effects in Metals, IEEE Trans. Magn.49, 5172 (2013). 56J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015). 57M. Schreier, G. E. W. Bauer, V. I. Vasyuchka, J. Flipse, K. Uchida, J. Lotze, V. Lauer, A. V. Chumak, A. A. Serga, S. Daimon, T. Kikkawa, E. Saitoh, B. J. van Wees, B. Hillebrands, R. Gross, and S. T. B. Goennenwein, Sign of inverse spin Hall voltages generated by ferromagnetic resonance and temperature gradients in yttrium iron gar- net platinum bilayers, J. Phys. D: Appl. Phys. 48, 025001 (2015). 58The Joule-heating-induced temperature increase under a typical ∆ Jcvalue of 0.15 mA is less than10 mK, which was confirmed bymeasuring the RTMvalue of the RuO 2−AlOx film under the d.c. current of 0.15 mA applied to the Pt wire atT= 2 K and by comparing it with the RTM–T calibration curve shown in Fig. 4(d). The current value of 0.15 mA was adopted such that the sample’s temperature increase is less than 1% compared to the base temperature of 2 K and also that the SPE signal can be measured with a sufficient signal-to-noise ratio. 59For the delay time of τdelay= 50 ms and the data acqui- sition time τsens= 20 ms, the frequency of the applied square-wave current is estimated to be f= 1/[2(τdelay+ τsens)]∼7 Hz. 60N. F. Mott, Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands, Philos. Mag. 19, 835 (1969). 61V. Ambegaokar, B. I. Halperin, and J. S. Langer, Hopping Conductivity in Disordered Systems, Phys. Rev. B 4, 2612 (1971). 62R. Rosenbaum, Crossover from Mott to Efros-Shklovskii variable-range-hopping conductivity in In xOyfilms, Phys. Rev. B44, 3599 (1991). 63S. Lafuerza, J. Garc´ ıa, G. Sub´ ıas, J. Blasco, K. Con- der, and E. Pomjakushina, Intrinsic electrical properties of LuFe 2O4, Phys. Rev. B 88, 085130 (2013).64C. Lu, A. Quindeau, H. Deniz, D. Preziosi, D. Hesse, and M. Alexe, Crossover of conduction mechanism in Sr 2IrO4 epitaxial thin films, Appl. Phys. Lett. 105, 082407 (2014). 65X. Chen, B. Wang, Y. Chen, H. Wei, and B. Cao, Tuning Jahn −Teller distortion and electron localization of LaMnO 3epitaxial films via substrate temperature, J. Phys. D: Appl. Phys. 54, 235302 (2021). 66Y. Li, M. You, X. Li, B. Yang, Z. Lin, and J. Liu, Tunable sensitivity of zirconium oxynitride thin-film temperature sensor modulated by film thickness, J. Mater. Sci.: Mater. Electron. 33, 20940 (2022). 67C. T. Harris and T.-M. Lu, A PtNiGe resistance ther- mometer for cryogenic applications, Rev. Sci. Instrum. 92, 054904 (2021). 68E. A. Scott, C. M. Smyth, M. K. Singh, T.-M. Lu, P. Sharma, D. Pete, J. Watt, and C. T. Harris, Optimization of gold germanium (Au 0.17Ge0.83) thin films for high sensi- tivity resistance thermometry, J. Appl. Phys. 132, 065103 (2022). 69S. S. Courts and P. R. Swinehart, Review of Cer- nox™(Zirconium Oxy-Nitride) Thin-Film Resistance Tem- perature Sensors, AIP Conf. Proc. 684, 393 (2003). 70J. R. Clement and E. H. Quinnell, The Low Tempera- ture Characteristics of Carbon-Composition Thermome- ters, Rev. Sci. Instrum. 23, 213 (1952). 71W. N. Lawless, Thermometric Properties of Carbon- Impregnated Porous Glass at Low Temperatures, Rev. Sci. Instrum. 43, 1743 (1972). 72A typical error at each field is evaluated to be 9 .8 mΩ for the data under ∆ Jc= 0.15 mA, so the slight undulation behavior of ∆ RTMaboveB= 0.05 T is attributed to the uncertainty of the measurement. 73T. Kikkawa, M. Suzuki, J. Okabayashi, K. Uchida, D. Kikuchi, Z. Qiu, and E. Saitoh, Detection of induced para- magnetic moments in Pt on Y 3Fe5O12via x-ray magnetic circular dichroism, Phys. Rev. B 95, 214416 (2017). 74K. Uchida, J. Ohe, T. Kikkawa, S. Daimon, D. Hou, Z. Qiu, and E. Saitoh, Intrinsic surface magnetic anisotropy in Y3Fe5O12as the origin of low-magnetic-field behavior of the spin Seebeck effect, Phys. Rev. B 92, 014415 (2015). 75T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Critical suppression of spin Seebeck effect by magnetic fields, Phys. Rev. B 92, 064413 (2015). 76H. Jin, S. R. Boona, Z. Yang, R. C. Myers, and J. P. Here- mans, Effect of the magnon dispersion on the longitudinal spin Seebeck effect in yttrium iron garnets, Phys. Rev. B 92, 054436 (2015). 77T. Kikkawa, K. Uchida, S. Daimon, and E. Saitoh, Com- plete Suppression of Longitudinal Spin Seebeck Effect by Frozen Magnetization Dynamics in Y 3Fe5O12, J. Phys. Soc. Japan 85, 065003 (2016). 78H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Theory of the spin Seebeck effect, Rep. Prog. Phys. 76, 036501 (2013). 79S. A. Bender andY. Tserkovnyak, Interfacial spinand heat transfer between metals and magnetic insulators, Phys. Rev. B96, 134412 (2015). 80L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016). 81Y. Ohnuma, M. Matsuo, and S. Maekawa, Theory of the spin Peltier effect, Phys. Rev. B 96, 134412 (2017). 82K. Kanahashi, J. Pu, and T. Takenobu, 2D Materials for11 Large-Area Flexible Thermoelectric Devices, Adv. Energy Mater.10, 1902842 (2020). 83D. Li, Y. Gong, Y. Chen, J. Lin, Q. Khan, Y. Zhang, Y. Li, H. Zhang, and H. Xie, Recent Progress of Two- Dimensional Thermoelectric Materials, Nano-Micro Lett. 12, 36 (2020).84Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes, A. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q.-C. Sun, E. J. Telfordet al., The Magnetic Genome of Two-Dimensional van der Waals Materials, ACS Nano 16, 6960 (2022).

2023-11-03

We report electric detection of the spin Peltier effect (SPE) in a bilayer consisting of a Pt film and a Y$_{3}$Fe$_5$O$_{12}$ (YIG) single crystal at the cryogenic temperature $T$ as low as 2 K based on a RuO$_2$$-$AlO$_x$ on-chip thermometer film. By means of a reactive co-sputtering technique, we successfully fabricated RuO$_2$$-$AlO$_x$ films having a large temperature coefficient of resistance (TCR) of $\sim 100\% ~\textrm{K}^{-1}$ at around $2~\textrm{K}$. By using the RuO$_2$$-$AlO$_x$ film as an on-chip temperature sensor for a Pt/YIG device, we observe a SPE-induced temperature change on the order of sub-$\mu \textrm{K}$, the sign of which is reversed with respect to the external magnetic field $B$ direction. We found that the SPE signal gradually decreases and converges to zero by increasing $B$ up to $10~\textrm{T}$. The result is attributed to the suppression of magnon excitations due to the Zeeman-gap opening in the magnon dispersion of YIG, whose energy much exceeds the thermal energy at 2 K.

Cryogenic spin Peltier effect detected by a RuO$_2$-AlO$_x$ on-chip microthermometer

2311.01711v1

arXiv:1408.2972v2 [cond-mat.mtrl-sci] 30 Sep 2014Quantitative Temperature Dependence of Longitudinal Spin Seebeck Effect at High Temperatures Ken-ichi Uchida,1,2,∗Takashi Kikkawa,1Asuka Miura,3Junichiro Shiomi,2,3and Eiji Saitoh1,4,5,6 1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan 3Department of Mechanical Engineering, The University of To kyo, Tokyo 113-8656, Japan 4WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan 5CREST, Japan Science and Technology Agency, Tokyo 102-0076 , Japan 6Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan (Dated: August 16, 2018) This article reports temperature-dependent measurements of longitudinal spin Seebeck effects (LSSEs) in Pt/Y 3Fe5O12(YIG)/Pt systems in a high temperature range from room tempe rature to above the Curie temperature of YIG. The experimental resu lts show that the magnitude of the LSSE voltage in the Pt/YIG/Pt systems rapidly decreases with increasing the temperature and disappears above the Curie temperature. The critical ex ponent of the LSSE voltage in the Pt/YIG/Pt systems at the Curie temperature was estimated to be 3, which is much greater than that for the magnetization curve of YIG. This difference high lights the fact that the mechanism of the LSSE cannot be explained in terms of simple static magnet ic properties in YIG. PACS numbers: 85.75.-d, 72.25.-b, 72.15.Jf, 73.50.Lw I. INTRODUCTION The Seebeck effect converts a temperature difference into electric voltage in conductors [1]. Since the discov- ery of the Seebeck effect nearly 200 years ago, it has been studiedintensivelytorealizesimpleandenvironmentally- friendly energy-conversion technologies [2]. The See- beck effect has been measured using various materials in a wide temperature range to investigate thermoelec- tricconversionperformanceandthermoelectrictransport properties. Temperature-dependent measurements in a high temperature range are especially important in the investigation of the Seebeck effect, since thermoelectric devices are often used above room temperature [3, 4]. In the field of spintronics [5–7], a spin counterpart of the Seebeck effect —the spin Seebeck effect (SSE)— was recently discovered. The SSE converts a temperature difference into spin voltage in ferromagnetic or ferrimag- netic materials. When a conductor is attached to a mag- net under a temperature gradient, the thermally gener- ated spin voltage in the magnet injects a spin current [8–10] into the conductor. Since the SSE occurs not only in metals [11–14] and semiconductors [15, 16] but also in insulators [13, 17–32], it enables the construction of “insulator-based thermoelectric generators” [21] in com- bination with the inverse spin Hall effect (ISHE) [33–37], which was impossible if only the conventional Seebeck effect was used. The observation of the SSE in insulators has been re- ported mainly in a longitudinal configuration [13, 18– 32]. The sample system for measuring the longi- tudinal SSE (LSSE) is a simple paramagnetic metal (PM)/ferrimagnetic insulator (FI) junction system. In many cases, Pt and Y 3Fe5O12(YIG) are used as PMand FI, respectively. When a temperature gradient ∇T is applied to the PM/FI system perpendicular to the in- terface, the spin voltage is thermally generated and in- jects a spin current into the PM along the ∇Tdirection owing to thermal spin-pumping mechanism [38–46]. This thermally induced spin current is converted into an elec- tric field EISHEby the ISHE in the PM according to the relation EISHE∝Js×σ, (1) whereJsis the spatial direction of the thermally induced spin current and σis the spin-polarization vector of elec- trons in the PM, which is parallel to the magnetization Mof FI [see Fig. 1(a)]. By measuring EISHEin the PM, one can detect the LSSE electrically. In the experimental research on the SSE, temperature- dependent measurements also have been used for inves- tigating various thermo-spin transport properties, such as phonon-mediated effects [15, 16, 39, 47], correlation between the SSE and magnon excitation [29, 48], and ef- fects of metal-insulator phase transition [13]. However, all the experiments on the SSE to date have been per- formedaroundandbelowroomtemperature. Inthisarti- cle, we report quantitative temperature-dependent mea- surements of the LSSE in Pt/YIG systems in the high temperature range from room temperature to above the Curie temperature of YIG. II. EXPERIMENTAL PROCEDURE The sample system used in this study consists of a single-crystalline YIG slab covered with Pt films. One difference from conventional samples is that the Pt films2 VL VH YIG slab Pt film (Pt_H) Pt film (Pt_L) (a) (b) x z y (c) (d) set temperature of heat baths P1 P2 P3 estimate THPt and TLPt under ∇T by combining P1 and P2 data n + 1 → n n <_ Nyes no measure RH and RL of Pt films measure RH and RL of Pt films set THset = THn, TLset = TLn ( THn > TLn)set THset = TLset = TLn measure LSSE ( H dep. of VH and VL) n = 1 n = 2 n = 3TL1 time THset TLsetTL2 TH1 TL3 TH2 TL4 TH3 P1 P2 P3 HEISHE Js ∇TJs (step) Msapphire plate heat bath ( TLset)heat bath ( THset) Pt/YIG/Pt sampleVH VLthermal grease Au/Ti contacts∇T H Pt wiresmetal paste FIG. 1: (a) A schematic illustration of the Pt/YIG/Pt sample .∇T,H,M,EISHE, andJsdenote the temperature gradient, magnetic field (with the magnitude H), magnetization vector, electric field induced by the ISHE, and spatial direction of the thermally generated spin current, respectively. The el ectric voltage VH(VL) and resistance RH(RL) between the ends of PtH (PtL) were measured using a nanovolt/micro-ohm meter (Agilent 34420A) in the ‘DC-voltage’ and ‘2-wire-resistance’ modes, respectively. The DC-voltage (2-wire-resistance) mode corresponds to the switch-off (switch-on) state in this schematic illustration. (b) Experimental configuration for measurin g the LSSE used in the present study. (c) A flow chart of the measurement processes. (d) A schematic graph of the set temp eratures, THnandTLn, of the heat baths as a function of time or the measurement step number n. The process P1 was performed under the isothermal conditio n, while the processes P2 and P3 were under the temperature gradient. Before starting the processes P1 and P2, we waited for ∼30 minutes at each nto stabilize the Tset HandTset Lvalues. are put on both the top and bottom surfaces of the YIG slab [Fig. 1(a)], while only the top surface of YIG is covered with a Pt film in conventional samples [18, 23, 27, 32]. The lengths of the YIG slab along the x,y, andzdirections are 3 mm, 7 mm, and 1 mm, re- spectively. 10-nm-thick Pt films were sputtered on the whole of the 3 ×7-mm2(111) surfaces of the YIG. The top and bottom Pt films are electrically insulated from each other because YIG is a very good insulator. Since YIG has the largechargegap of2.7eV [49], thermal exci- tation of charge carriers in YIG is vanishingly small even at the high temperatures. To attach electrodes to both the Pt films symmetri- cally and to generate a uniform temperature gradient, we made the configuration shown in Fig. 1(b). Here, the Pt/YIG/Pt sample was sandwiched between two 0.5-mm-thick sapphire plates of which the surface is covered with two separated Au/Ti contacts. The distance be- tween the two Au/Ti contacts is ∼6 mm. To extract voltage signals in the Pt films, both the ends of the Pt films are connected to the Au/Ti contacts via sintering metal paste, which can be used up to 900◦C, and thin Pt wires with the diameter of 0.1 mm were attached to the end of the contacts [see Fig. 1(b)]. The sapphire plates are thermally connected to heat baths of which the tem- peratures are controlled with the accuracy of <0.6 K by usingPID(proportional-integral-derivative)temperature controllers. We attached thermal paste to both the sur- faces of the sapphire plates, except for the regions of the Au/Ti contacts, to improve the thermal contact [see Fig. 1(b)]. During the measurements of the LSSE, the tem- perature of the upper heat bath, Tset H, is set to be higher3 than that of the lower one, Tset L. According to the direc- tion of the temperature gradient, hereafter, the top and bottom Pt films of the Pt/YIG/Pt sample are referred to as ‘PtH’ and ‘Pt L’, respectively. In this setup, we can measure the electric voltage VH(VL) and resistance RH (RL) between the ends of Pt H (PtL) without changing electrodes, wiring, and measurement equipment [see also the caption of Fig. 1]. An external magnetic field H (with the magnitude H) was applied along the xdirec- tion. To investigate the temperature dependence of the LSSE quantitatively, it is important to estimate the tem- perature difference between the top and bottom of the YIG sample. However, in the conventional experiments, the temperatures of the heat baths, not the sample it- self, were usually monitored [20]. Therefore, the mea- sured temperature difference includes the contributions from the interfacial thermal resistance between the sam- ple and heat baths and from small temperature gradients in the sample holders. To avoid this problem, in this study, we used the Pt films not only as spin-current de- tectors but also as temperature sensors [28, 31]; we can know the temperatures of the Pt films from the temper- ature dependence of the resistance of the films, enabling the estimation of the temperature difference between the top and bottom of the sample during the LSSE measure- ments. Figure 1(c) shows the flow chart of the measure- ment processes. The measurement comprises the fol- lowing three processes P1-P3, and the processes are re- peatedNtimes. Hereafter, THnandTLndenote the set temperatures for each measurement step number n(= 1,2,..., N) (see Table I, where the values of THn andTLnfor each nare shown). The process P1 is the measurement of the resistance of Pt H and Pt L,RH TABLE I: Set temperatures of the heat baths for each mea- surementstep number n(≤N). Here, the THnandTLnvalues are increased by 10 K with every nincrease, while THn−TLn is fixed at 8 K. Step number n T Ln(K)THn(K) 1 290 298 2 300 308 3 310 318 4 320 328 5 330 338 · · · · · · · · · 31 590 598 32 600 608 33 610 618 34 (=N) 620 628andRL, in the isothermal condition, where the tempera- tures of the heat baths are set to the same temperature: Tset H=Tset L=TLn. In this isothermalcondition, the tem- perature of the Pt/YIG/Pt sample is uniform and very close to the set temperature of the heat baths irrespec- tive of the presence of the interfacial thermal resistance. Next, weapplyatemperaturegradienttothePt/YIG/Pt sample by increasing the temperature of the upper heat bath, where Tset H=THnandTset L=TLnwithTHn> TLn (see Table I). After waiting until the temperatures are stabilized, we measure the resistance of the Pt films un- derthe temperature gradient: this is the processP2. The processes P1 and P2 are performed without applying H. Immediately after the process P2, we proceed to the pro- cess P3; the LSSE, i.e. the Hdependence of VHandVL, is measured with keeping the magnitude of Tset H−Tset L constant. After finishing the LSSE measurements, we go on to the next step and increase nby 1, where the values ofTHnandTLnare increased as shown in Table I. These measurement processes are summarized in Fig. 1(d). The calibration method of the sample temperatures is as follows. From the process P1, we obtain the tem- perature ( TLn) dependence of RHandRLunder the isothermal condition. By comparing the resistance un- der the temperature gradient, obtained from the process P2, with the isothermal RH,L-TLncurves, we can cali- brate the temperature of the Pt films, TPt HandTPt L, un- der the temperature gradient, allowingus to estimate the averagetemperature Tav(= (TPt H+TPt L)/2) and the tem- perature difference ∆ T(=TPt H−TPt L) in the YIG slab during the LSSE measurements. The Tavand ∆Tvalues are free from the contributions from the interfacial ther- mal resistance between the sample and heat baths and from the temperature gradients in the sapphire plates, enabling the quantitative evaluation of the LSSE at var- ious temperatures. Although the experimental protocol proposed here cannot estimate the contributions of the interfacial thermal resistance at the Pt/YIG interfaces and temperature gradients in the Pt films, they are neg- ligible compared with the temperature difference applied to the YIG slab [26, 32]. III. RESULTS AND DISCUSSION Figures 2(a) and 2(b) respectively show the Hdepen- dence of VHandVLin the Pt/YIG/Pt sample for each stepnumber n, measuredwhen THn−TLn= 8K. Around room temperature, we observed clear voltage signals in both the Pt films; the signs of VHandVLare reversed in response to the reversal of the magnetization direction of the YIG slab. Since the contribution of anomalous Nernst effects induced by proximity ferromagnetism in Pt [50] is negligibly small in Pt/YIG systems, the volt- age signals observed here are due purely to the LSSE [23, 27, 32]. The sign of the LSSE voltage in Pt H was4 020406080100 510 15 300400500600300 400 500 600 TLn (K) Tav (K) ∆T (K) 10 20 30 0 nRH,L ( Ω) -1 0 1 H (kOe) Pt_H Pt_L n = 1 n = 29 n = 25 n = 21 n = 17 n = 13 n = 9n = 5 n = 33 Step -1 0 1 H (kOe) Pt_H Pt_L 10.0 10.0 (c) (d)VH ( µV) (a) 10 20 30 nRH.L ( Ω) 020 60 40 80 Pt_H Pt_L VLSSE VLSSE VL ( µV) (b) THset = TLset = TLn THset = THn TLset = TLnn = 1 n = 29 n = 25 n = 21 n = 17 n = 13 n = 9n = 5 n = 33 Step 200 400 VBG ( µV) 10 20 30 0 n(e) Background Pt_H Pt_L FIG. 2: (a) Hdependence of VHfor PtH in the Pt/YIG/Pt sample for various values of n. (b)Hdependence of VLfor PtL. The LSSE voltage VLSSEfor PtH (PtL) is defined as VH(VL) atH= 1 kOe. (c) TLndependence of RH(RL) for Pt H (PtL) in the isothermal condition: Tset H=Tset L=TLn. (d) The average temperature Tavand temperature difference ∆ Tof the Pt/YIG/Pt sample during the LSSE measurements as a function ofn. The inset to (d) shows RH(RL) for Pt H (PtL) as a function of nunder the temperature gradient: Tset H=THnandTset L=TLnwithTHn> TLn. (e) The background voltage VBG for PtH and Pt L as a function of nunder the temperature gradient. observed to be the same as that in Pt L, a situation con- sistent with the scenario of the SSE [38, 40, 44, 51]. Here we note that, although the large backgroundvoltage VBG appears due to unavoidable thermopower differences in the wires with increasing n[Fig. 2(e)], the noise level in theVHandVLsignals does not increase and the drift ofVBGis small [Figs. 2(a) and 2(b)]. Therefore, we can extract the LSSE voltage simply by measuring the Hdependence of VHandVL. We also found that the magnitude of the LSSE voltage in the Pt/YIG/Pt sam- ple monotonically decreases with increasing the temper- ature. To quantitatively discuss the temperature dependence of the LSSE voltage in the Pt/YIG/Pt sample, we esti- mateTavand ∆Tat each step number nby using the method explained in Sec. II. Figure 2(c) shows the TLn dependence of RHandRLfor the Pt films measured un- der the isothermal condition. By combining the isother- malRH,L-TLncurves with the RHandRLdata under the temperature gradient [the inset to Fig. 2(d)], we obtain theTavand∆Tvaluesateach n[Fig. 2(d)]. Importantly, the calibrated values of ∆ Tare dependent on nand al- ways smaller than the temperature difference applied to the heat baths due to the interfacial thermal resistance and temperature gradients in the sapphire plates (notethatTHn−TLn= 8 K for all the measurements as shown in Table I). Figure 3(c) shows the LSSE voltage normalized by the calibrated temperature difference applied to the Pt/YIG/Pt sample, VLSSE/∆T, as a function of Tav. Here,VLSSEdenotesVH(VL) for Pt H (PtL) atH= 1 kOe. We confirm again that the magnitude of VLSSE/∆Tmonotonically decreases with increasing the temperatureanddisappearsabovetheCurietemperature Tcof YIG, where Tcof our YIG slab was experimentally estimated to be 553 K (see Appendix A). This behav- ior was observed not only in one sample but also in our different samples as exemplified in Fig. 3(c). Interest- ingly, the temperature dependence of VLSSE/∆Tis sig- nificantly different from the magnetization (4 πM) curve of the YIG slab [compare Figs. 3(a) and 3(c)]; the mag- nitude of VLSSE/∆Trapidly decreases with a concave- up shape, while the magnetization curve of YIG exhibits a standard concave-down shape. We also checked that the strong temperature dependence of the LSSE voltage in the Pt/YIG/Pt sample cannot be explained by the weak temperature dependence of the thermal conductiv- ity of YIG (see Appendix B). Similar difference in the temperature-dependent data between the ISHE voltage and magnetization wasobservedalso in Pt/GaMnAs sys-5 Tc of YIG 4πMs4πM at H = 2 kOe Fitting with Eq. (2) 300 400 500 600 T (K) 300 400 500 600 Tav (K)Tc of YIG Sample A Sample B Pt_H Pt_L Pt_H Pt_L Fitting with Eq. (3) 12 04πM (kG) 3(a) 0123VLSSE /∆T ( µV/K) (c)10 110 210 3 10 110 210 3 Tc − Tav (K)10 -1 10 04πMs (kG) 10 1 (b) 10 -2 10 -1 10 010 1VLSSE /∆T ( µV/K) (d)Tc − T (K) ∝ ( Tc − T)3∝ ( Tc − T)0.5 FIG. 3: (a) Tdependence of the bulk magnetization 4 πMof the YIG slab. The green curve shows the 4 πMdata atH= 2 kOe, measured with a vibrating sample magnetometer (VSM). The gr ay circles show the saturation magnetization 4 πMsof the YIG slab. The gray curve was obtained by fitting the 4 πMsdata with Eq. (2). The values of 4 πMsand Curie temperature Tc(= 553 K) of the YIG slab were estimated by using the Arrott plo t method (see Appendix A). (b) Double logarithmic plot of theTc−Tdependence of 4 πMsof the YIG slab. (c) Tavdependence of VLSSE/∆Tfor two different Pt/YIG/Pt samples A (circles) and B (triangles). The experimental results sho wn in Fig. 2 were measured using the Pt/YIG/Pt sample A. The orange curve was obtained by fitting the VLSSE/∆Tdata with Eq. (3). (d) Double logarithmic plot of the Tc−Tavdependence ofVLSSE/∆Tfor the Pt/YIG/Pt samples A and B. tems in the measurement of the transverse SSEs [15, 16]. The behavior of physical quantities near continuous phase transitions can be described by critical exponents in general. Here, we compare the critical exponents for the observed temperature dependences of the LSSE volt- age in the Pt/YIG/Pt sample and the magnetization of YIG. First, we checked that the magnetization curve of YIG is well reproduced by a standard mean-field model [52]: 4πMs=A(Tc−T)0.5, (2) where the critical exponent is fixed at 0.5 and Ais an adjustable parameter [see Figs. 3(a) and 3(b)]. The crit- ical exponent γfor the LSSE was estimated by fitting the experimental data in Fig. 3(c) with the following equation: VISHE ∆T=S(Tc−T)γ, (3) where both Sandγare adjustable parameters and Tav is regarded as Tfor the LSSE data. We found that the observed temperature dependence of VLSSE/∆Tfor the Pt/YIG/Pt sample is well fitted by Eq. (3) with γ= 3,which is much greater than the critical exponent for the magnetization curve [see also the double logarithmic plot in Fig. 3(d)]. This big difference in the critical exponents betweentheLSSEandmagnetizationemphasizesthefact that the LSSE is not attributed solely to static magnetic properties in YIG. Here, we qualitatively discuss the origin of the temper- ature dependence of the LSSE voltage in the Pt/YIG/Pt sample. According to the thermal spin-pumping mech- anism [38, 44] and phenomenological calculation of the ISHE combined with short-circuit effects [53], the mag- nitude of the LSSE voltage is determined mainly by the following factors: the spin-mixing conductance [54–57] at the Pt/YIG interfaces, spin-diffusion length and spin- Hall angle of Pt, and difference between an effective magnon temperature in YIG and an effective electron temperature in Pt. Since the LSSE voltage is propor- tional to the spin-mixing conductance [38], it can con- tribute directly to the observed temperature dependence of the LSSE voltage. Recently, Ohnuma et al.formu- lated the relation between the spin-mixing conductance and interface s-dinteraction at paramagnet/ferromagnet interfaces, and predicted that the spin-mixing conduc- tance is proportional to (4 πMs)2of the ferromagnet [58].6 By combining this prediction with Eq. (2), the spin- mixing conductance is proportional to Tc−T, of which the critical exponent (= 1) is greater than that for the magnetization curve. Although the temperature depen- dence of the spin-mixing conductance can explain the factsthat the LSSEvoltagemonotonicallydecreaseswith increasingthe temperatureanddisappearsat Tc, itisstill much weaker than the observed ( Tc−T)3dependence of the LSSE voltage. Furthermore, if the spin-diffusion length of Pt decreases with increasing the temperature [59], it can also contribute to reducing the LSSE volt- age at high temperatures, while the spin-Hall angle of Pt was shown to exhibit weak temperature dependence [60] (note that the magnitude of the ISHE voltage monoton- ically decreases with decreasing the spin-diffusion length when the spin-Hall angle is constant [53]). The effec- tive magnon-electron temperature difference could also be an important factor, but there is no clear framework to determine its temperature dependence at the present stage. Therefore, more elaborate investigations are nec- essaryforthe completeunderstandingofthetemperature dependence of the LSSE voltage. IV. CONCLUSION In this study, we reported the longitudinal spin See- beck effects (LSSEs) in Y 3Fe5O12(YIG) slabs sand- wiched by two Pt films in the high temperature range from room temperature to above the Curie temperature Tcof YIG. To investigate the temperature dependence of the LSSE quantitatively, we used the Pt films not only asspin-currentdetectorsbut alsoastemperaturesensors. The measurement processes used here enabled accurate estimation of the average temperature and temperature difference of the sample, being free from thermal arti- facts. We found that the magnitude of the LSSE in the Pt/YIG/Pt sample rapidly decreases with increasing the temperature and disappears above Tcof YIG; the ob- served LSSE voltage exhibits the ( Tc−T)3dependence of which the critical exponent (= 3) is much greater than thatofthemagnetizationofYIG(= 0 .5). Althoughmore detailed experimental and theoretical investigations are required to clarify the microscopic origin of this discrep- ancy, we anticipate that the quantitative temperature- dependent LSSE data at high temperatures will be help- ful for obtaining full understanding of the mechanism of the LSSE. ACKNOWLEDGMENTS The authors thank S. Maekawa, H. Adachi, Y. Ohnuma, N. Yokoi, K. Sato, and J. Ohe for valuable dis- cussionsandY.Zhangforhisassistanceinmagnetometry measurements. This work was supported by PRESTO-JST “Phase Interfaces for Highly Efficient Energy Uti- lization”, CREST-JST “Creation of Nanosystems with Novel Functions through Process Integration”, Grant- in-Aid for Young Scientists (A) (25707029), Grant-in- Aid for Challenging Exploratory Research (26600067), Grant-in-Aid for Scientific Research(A) (24244051)from MEXT, Japan, LC-IMR of Tohoku University, the Sum- itomo Foundation, the Tanikawa Fund Promotion of Thermal Technology, the Casio Science Promotion Foun- dation, and the Iwatani Naoji Foundation. APPENDIX A: ESTIMATION OF CURIE TEMPERATURE OF YIG The Curie temperature Tcof the YIG slab used in the present study was estimated from vibrating sample mag- netometry and Arrott-plot analysis [52, 61]. The inset to Fig. 4(a) shows the Hdependence of the magnetiza- tion 4πMof the YIG slab for various values of T. From this result, we obtained the Arrott plots, i.e. H/4πM dependence of (4 πM)2, of the YIG slab [see Fig. 4(a)]. The saturation magnetization 4 πMsof the YIG at each temperature was extracted by extrapolating the (4 πM)2 datain the high-magnetic-fieldrangeto zerofield [see red dotted lines in Fig. 4(a)]. As shown in Fig. 4(b), the T dependence of (4 πMs)2of the YIG slab is well fitted by a linear function; the horizontal intercept of the linear fit line corresponds to Tc. The fitting result shows that the Curie temperature of our YIG slab is Tc= 553 K, which is consistent with literature values [62, 63]. APPENDIX B: TEMPERATURE DEPENDENCE OF THERMAL CONDUCTIVITY OF YIG Figure 5(a) shows the thermal conductivity κof the YIG slab used in the present study as a function of T. Theκvalues were obtained by the combination of ther- maldiffusivitymeasuredbyalaser-flashmethodandspe- cific heat Cmeasured by a differential scanning calorime- try. Here, we measured the thermal diffusivity along the [111] direction of the single-crystallineYIG slab, which is paralleltothe ∇Tdirectionin theLSSEsetup. Asshown in Fig. 5(b), the measured Cvalues are consistent with theDulong-Petit(DP) law[1]; thedifferenceofthe Cval- uesfrom the DPspecific heatofYIG, CDP= 0.676J/gK, is less than 10 % of CDPforT >350 K. The observed Tdependence of κis well fitted by κ∝T−1, indicating that the thermal conductivity of the YIG is dominated by phonons in this temperature range [see also the inset to Fig. 5(a)]. The κvalue at 300 K is consistent with literature values [64–66]. We also confirmed that the T dependence of κis much weaker than that of the LSSE voltage in the Pt/YIG/Pt sample [compare Figs. 3(c) and 5(a)] and shows no anomaly around Tc.7 Tc = 553 K 0 2 4 6 823 300 400 500 600 T (K)(4 πMs)2 (kG 2) 013 2H/ 4πM (Oe/G) (4 πM)2 (kG 2) 1T = 295.3 K 359.6 K 400.4 K 450.5 K 501.7 K 550.7 K (a) -1 0 1-2-10124πM (kG) H (kOe)601.5 K 295.3 K 550.7 K (b) HYIG slab xz y FIG. 4: (a) Arrott plots of the YIG slab for various values of T. The inset to (a) shows the Hdependence of 4 πMof the YIG slab for various values of T, measured with VSM. The lengths along the x,y, andzdirections of the YIG slab used for the magnetometry measurements are 3 mm, 7 mm, and 1 mm, respectively. Hwas applied along the xdirection. (b) Tdependence of (4 πMs)2of the YIG slab. ∗Electronic address: kuchida@imr.tohoku.ac.jp [1] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976). [2] S. B. Riffat and X. Ma, Thermoelectrics: a Review of Present and Potential Applications , Appl. Therm. Eng. 23, 913 (2003). [3] J. P. Heremans, V. Jovovic, E. S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, and G. J. Snyder, Enhancement of Thermoelectric Efficiency in PbTeby Distortion of the Electronic Density of States , Science321, 554 (2008). [4] L.-D. Zhao, S.-H. Lo, Y. Zhang, H. Sun, G. Tan, C. Uher, C. Wolverton, V. P. Dravid, and M. G. Kanatzidis, Ultralow Thermal Conductivity and High Thermoelec- tric Figure of Merit in SnSeCrystals, Nature 508, 373 (2014). [5] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Spintronics: A Spin- Based Electronics Vision for the Future , Science 294, 1488 (2001). [6] I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Spintronics: Fun- damentals and Applications , Rev. Mod. Phys. 76, 323 (2004).300 400 500 600 T (K)0515 10 (W/mK) κ10 210 3 10 010 110 2 T (K)(W/mK) κ∝ T -1 ∝ T -1 00.20.40.60.81.0 300 400 500 600 T (K)(a) (b) C (J/gK)CDP = 0.676 J/gK FIG. 5: (a) Tdependence of the thermal conductivity κof the YIG slab. The black circles show the measured thermal conductivity and the black curve shows the fitting of the ex- perimental data with κ=KT−1, where Kis an adjustable parameter. The inset shows thedouble logarithmic plot ofth e Tdependence of κ. (b)Tdependence of the specific heat C of the YIG slab. The gray dotted line shows the DP specific heat of YIG. [7] S. Maekawa, ed., Concepts in Spin Electronics (Oxford University Press, Oxford, 2006). [8] J. C. Slonczewski, Current-Driven Excitation of Mag- netic Multilayers , J. Magn. Magn. Mater. 159, L1(1996). [9] S. Maekawa, E. Saitoh, S. O. Valenzuela, and T. Kimura, eds.,Spin Current (Oxford University Press, Oxford, 2012). [10] S. Maekawa, H. Adachi, K. Uchida, J. Ieda, and E. Saitoh,Spin Current: Experimental and Theoretical As- pects, J. Phys. Soc. Jpn. 82, 102002 (2013). [11] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Observation of the Spin Seebeck Effect , Nature 455, 778 (2008). [12] S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E. Saitoh, and K. Takanashi, Spin Seebeck Effect in Thin Films of the Heusler Compound Co2MnSi, Phys. Rev. B 83, 224401 (2011). [13] R. Ramos, T. Kikkawa, K. Uchida, H. Adachi, I. Lucas, M. H. Aguirre, P. Algarabel, L. Morell´ on, S. Maekawa, E. Saitoh, andM. R.Ibarra, Observation of the Spin Seebeck Effect in Epitaxial Fe3O4Thin Films , Appl. Phys. Lett. 102, 072413 (2013). [14] S. H. Wang, L. K. Zou, J. W. Cai, B. G. Shen, and J. R. Sun, Transverse Thermoelectric Effects in Platinum Strips on Permalloy Films , Phys. Rev. B 88, 214304 (2013).8 [15] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Observation of the Spin- Seebeck Effect in a Ferromagnetic Semiconductor , Nat. Mater.9, 898 (2010). [16] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, R. C. Myers, and J. P. Heremans, Spin-Seebeck Effect: A Phonon Driven Spin Distribution , Phys. Rev. Lett. 106, 186601 (2011). [17] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck Insulator , Nat. Mater. 9, 894 (2010). [18] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Observation of Longitudinal Spin-Seebeck Effect in Magnetic Insulators , Appl. Phys. Lett.97, 172505 (2010). [19] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Local Charge and Spin Currents in Magnetothermal Landscapes , Phys. Rev. Lett. 108, 106602 (2012). [20] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Thermal Spin Pumping and Magnon-Phonon-Mediated Spin-Seebeck Effect , Journal of Applied Physics 111, 103903 (2012). [21] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, Spin- Current-Driven Thermoelectric Coating , Nat. Mater. 11, 686 (2012). [22] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, In- trinsic Spin Seebeck Effect in Au/YIG, Phys. Rev. Lett. 110, 067206 (2013). [23] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Longitudi- nal Spin Seebeck Effect Free from the Proximity Nernst Effect, Phys. Rev. Lett. 110, 067207 (2013). [24] D. Meier, T. Kuschel, L. Shen, A. Gupta, T. Kikkawa, K. Uchida, E. Saitoh, J.-M. Schmalhorst, and G. Reiss, Thermally Driven Spin and Charge Currents in Thin NiFe2O4/PtFilms, Phys. Rev. B 87, 054421 (2013). [25] K. Uchida, T. Nonaka, T. Kikkawa, Y. Kajiwara, and E. Saitoh, Longitudinal Spin Seebeck Effect in Various Garnet Ferrites , Phys. Rev. B 87, 104412 (2013). [26] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Magnon, Phonon, and Electron Temperature Profiles and the Spin Seebeck Effect in Magnetic Insulator/normal Metal Hy- brid Structures , Phys. Rev. B 88, 094410 (2013). [27] T. Kikkawa, K. Uchida, S. Daimon, Y. Shiomi, H. Adachi, Z. Qiu, D. Hou, X.-F. Jin, S. Maekawa, and E. Saitoh, Separation of Longitudinal Spin See- beck Effect from Anomalous Nernst Effect: Determina- tion of the Origin of Transverse Thermoelectric Voltage in Metal/insulator Junctions , Phys. Rev. B 88, 214403 (2013). [28] M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Current Heating Induced Spin Seebeck Effect , Appl. Phys. Lett. 103, 242404 (2013). [29] S. M. Rezende, R. L. Rodr´ ıguez-Su´ arez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J.C. LopezOrtiz, andA.Azevedo, Magnon Spin-Current Theory for the Longitudinal Spin-Seebeck Effect , Phys.Rev. B89, 014416 (2014). [30] N. Roschewsky, M. Schreier, A. Kamra, F. Schade, K. Ganzhorn, S. Meyer, H.Huebl, S. Gepr¨ ags, R. Gross, and S. T. B. Goennenwein, Time Resolved Spin Seebeck Effect Experiments , Appl. Phys. Lett. 104, 202410 (2014). [31] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. Kirihara, P. Pirro, T. Langner, M. B. Jungfleisch, A. V. Chumak, E. Th. Papaioannou, and B. Hillebrands, Role of Bulk- Magnon Transport in the Temporal Evolution of the Lon- gitudinal Spin-Seebeck Effect , Phys. Rev. B 89, 224414 (2014). [32] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Mu- rakami, and E. Saitoh, Longitudinal Spin Seebeck Effect: from Fundamentals to Applications , J. Phys.: Condens. Matter26, 343202 (2014). [33] A. Azevedo, L. H. Vilela Le˜ ao, R. L. Rodriguez-Suarez, A. B. Oliveira, andS.M. Rezende, dc Effect in Ferromag- netic Resonance: Evidence of the Spin-Pumping Effect? , J. Appl. Phys. 97, 10C715 (2005). [34] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Con- version of Spin Current into Charge Current at Room Temperature: Inverse Spin-Hall Effect , Appl. Phys. Lett. 88, 182509 (2006). [35] S. O. Valenzuela and M. Tinkham, Direct Electronic Measurement of the Spin Hall Effect , Nature 442, 176 (2006). [36] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Electrical Detection of Spin Pumping due to the Precessing Magnetization of a Single Ferromagnet , Phys. Rev. Lett. 97, 216603 (2006). [37] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Room-Temperature Reversible Spin Hall Ef- fect, Phys. Rev. Lett. 98, 156601 (2007). [38] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Theory of Magnon-Driven Spin Seebeck Effect , Phys. Rev. B 81, 214418 (2010). [39] H. Adachi, K. Uchida, E. Saitoh, J. Ohe, S. Takahashi, and S. Maekawa, Gigantic Enhancement of Spin Seebeck Effect by Phonon Drag , Appl. Phys. Lett. 97, 252506 (2010). [40] H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, Linear-Response Theory of Spin Seebeck Effect in Fer- romagnetic Insulators , Phys. Rev. B 83, 094410 (2011). [41] J. Ohe, H. Adachi, S. Takahashi, and S. Maekawa, Nu- merical Study on the Spin Seebeck Effect , Phys. Rev. B 83, 115118 (2011). [42] S. S.-L. Zhang and S. Zhang, Spin Convertance at Mag- netic Interfaces , Phys. Rev. B 86, 214424 (2012). [43] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Spin Seebeck Effect in Antiferromagnets and Compen- sated Ferrimagnets , Phys. Rev. B 87, 014423 (2013). [44] H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, The- ory of the Spin Seebeck Effect , Rep. Prog. Phys. 76, 036501 (2013). [45] H. Adachi and S. Maekawa, Linear-Response Theory of the Longitudinal Spin Seebeck Effect , J. Korean Phys. Soc.62, 1753 (2013). [46] S. Hoffman, K. Sato, and Y. Tserkovnyak, Landau- Lifshitz Theory of the Longitudinal Spin Seebeck Effect , Phys. Rev. B 88, 064408 (2013). [47] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hille- brands, S. Maekawa, and E. Saitoh, Long-Range Spin Seebeck Effect and Acoustic Spin Pumping , Nat. Mater. 10, 737 (2011).9 [48] S. R. Boona and J. P. Heremans, Magnon Thermal Mean Free Path in Yttrium Iron Garnet , Phys. Rev. B 90, 064421 (2014). [49] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Transmission of Electrical Signals by Spin-Wave Interconversion in a Magnetic Insulator , Nature 464, 262 (2010). [50] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Transport Magnetic Proximity Effects in Platinum , Phys.Rev.Lett. 109, 107204 (2012). [51] Note that the sign of the SSE-induced spin voltage is reversed between the higher- and lower- temperature ends of the ferromagnet [38, 40, 44]. Therefore, in the Pt/YIG/Pt samples used in the present study, the spin voltage in YIG injects a spin current into Pt on one side, while it ejects a spin current from Pt on the other side; the direction of Jsin PtH is the same as that in Pt L in the longitudinal configuration. [52] S. Chikazumi, Physics of Ferromagnetism (Oxford Uni- versity, Oxford, 1997), 2nd ed. [53] H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka- hashi, Y. Kajiwara, K. Uchida, Y. Fujikawa, and E. Saitoh,Geometry Dependence on Inverse Spin Hall Effect Induced by Spin Pumping in Ni81Fe19/PtFilms, Phys. Rev. B85, 144408 (2012). [54] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal Magnetization Dynamics in Fer- romagnetic Heterostructures , Rev. Mod. Phys. 77, 1375 (2005). [55] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Spin Trans- fer Torque on Magnetic Insulators , Europhys. Lett. 96, 17005 (2011). [56] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra,J. Xiao, Y.-T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein, Experimental Test of the Spin Mixing Interface Conductivity Concept , Phys. Rev. Lett. 111, 176601 (2013). [57] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Taka- hashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Spin Mixing Conductance at a Well-Controlled Plat- inum/Yttrium Iron Garnet Interface , Appl. Phys. Lett. 103, 092404 (2013). [58] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, En- hanced dc Spin Pumping into a Fluctuating Ferromagnet nearTc, Phys. Rev. B 89, 174417 (2014). [59] S. R. Marmion, M. Ali, M. McLaren, D. A. Williams, and B. J. Hickey, Temperature Dependence of Spin Hall Magnetoresistance in Thin YIG/Pt Films, Phys. Rev. B 89, 220404(R) (2014). [60] L. Vila, T. Kimura, and Y. Otani, Evolution of the Spin Hall Effect in PtNanowires: Size and Temperature Ef- fects, Phys. Rev. Lett. 99, 226604 (2007). [61] A. Arrott, Criterion for Ferromagnetism from Obser- vations of Magnetic Isotherms , Phys. Rev. 108, 1394 (1957). [62] M. A. Gilleo and S. Geller, Magnetic and Crystallo- graphic Properties of Substituted Yttrium-Iron Garnet, 3Y2O3·xM2O3·(5−x)Fe2O3, Phys. Rev. 110, 73 (1958). [63] E. E. Anderson, Molecular Field Model and the Magne- tization of YIG, Phys. Rev. 134, A1581 (1964). [64] G. A. Slack and D. W. Oliver, Thermal Conductivity of Garnets and Phonon Scattering by Rare-Earth Ions , Phys. Rev. B 4, 592 (1971). [65] N. P. Padture and P. G. Klemens, Low Thermal Conduc- tivity in Garnets , J. Am. Ceram. Soc. 80, 1018 (1997). [66] A.M. Hofmeister, Thermal Diffusivity of Garnets at High Temperature , Phys. Chem. Minerals 33, 45 (2006).

2014-08-13

This article reports temperature-dependent measurements of longitudinal spin Seebeck effects (LSSEs) in Pt/Y$_3$Fe$_5$O$_{12}$ (YIG)/Pt systems in a high temperature range from room temperature to above the Curie temperature of YIG. The experimental results show that the magnitude of the LSSE voltage in the Pt/YIG/Pt systems rapidly decreases with increasing the temperature and disappears above the Curie temperature. The critical exponent of the LSSE voltage in the Pt/YIG/Pt systems at the Curie temperature was estimated to be 3, which is much greater than that for the magnetization curve of YIG. This difference highlights the fact that the mechanism of the LSSE cannot be explained in terms of simple static magnetic properties in YIG.

Quantitative Temperature Dependence of Longitudinal Spin Seebeck Effect at High Temperatures

1408.2972v2

1 Anomalous Hall -like Transvers e Magnetoresistance in Au thin films on Y3Fe5O12 Tobias Kosub1, Saül Vélez2,†, Juan M. Gomez -Perez2, Luis E. Hueso2,3, Jürgen Faßbender1, Fèlix Casanova2,3, Denys Makarov1 1 Helmholtz -Zentrum Dresden -Rossendorf e.V ., Institute of Ion Beam Physics and Materials Research , 01328 Dresden, Germany 2 CIC nanoGUNE, 20018 Donostia -San Sebastian, Basque Country, Spain †Present address: Department of Materials, ETH Z ürich, 8093 Zürich, Switzerland 3 IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain Anomalous Hall -like signals in platinum in contact with magnetic insulators are common observations that could be explained by either proximity magnetization or spin Hall magnetoresistance. In this work, longitudinal and transverse magnetoresistance s are measured in a pure g old thin film on the ferrimagnetic insulator Y3Fe5O12 (Yttrium Iron Garnet, YIG) . We show that both the longitudinal and transverse magnetoresistance s have quantitatively consistent scaling in YIG/Au and in a YIG/Pt reference system when applying the Spin Hall magnetoresistance framework . No contribution of an anomalous Hall effect due to the magnetic proximity effect is evident . Throughout the last few years, systems of magnetic insulators and nonmagnetic metals (MI/NM) have seen significant attention from the spintronics community [1–10]. Such systems general ly display Spin Hall magnetoresistance (SMR) [9–19] enabled by the eponymous Spin H all effect [20,21] of the metal which also offers elegant charge -spin-interconversion . One of the most popular metals in such systems is platinum (Pt) due to its strong spin orbit coupling, large spin Hall angle and its benign chemical properties. Still, a larger variety of well -studied metals is urgently needed in the field of insulator spintronics for two reasons: Firstly, some metals – Pt being a prot otypical example – are close to the Stoner criterion for ferromagnetism and can thus show a strong magnetic proximity effect [22]. When Pt becomes a magnetic conductor in this way, its strong SHE becomes an anomalous Hall effect (AHE) [20] that would mirror the magnetization of a nearby MI [23–26]. In contrast, the same phenomenon could be attributed solely due to the transverse SMR [27–29] or the nonlocal AHE [30] creating an ambiguity about the physics of the anomalous Hall -like signal in MI/NM systems . Secondly, studying more diverse systems can probe the applicability limits of the SMR theory [31]. The longitudinal and transverse SMR magnitudes are typically explained in terms of the real and imaginary components of the interfacial spin -mixing conductivity 𝐺↑↓=𝐺𝑟+𝑖𝐺𝑖 [32,33] , but higher order effects are already known [34]. Here, we report longitudinal as well as transverse signatures of the spin Hall magnetoresistance for a gold (Au) layer on YIG and compare this to a YIG/Pt reference system . Au shares or exceeds the excellent chemical and electrical properties of Pt. At the same time, static proximity magnetization is not expected for Au [35] and we take care to avoid dynamic proximity magnetization due to thermal spin pumping [Suppl. Information , 1]. As a result, we exclude possible influences from proximity magnetization and its associated anomalous Hall effect. Using empirical data, we quantitatively predict the longitudinal and transverse spin Hall magnetoresistance in YIG/Au and confirm their respective magnitudes by measurements. Hence, the key finding of this work is the experimental observation of the transverse SMR in Au. We prepar ed a YIG/Au(10nm) and a YIG/Pt(2nm) system using DC sputtering under identical conditions on top of 3.5 -µm-thick liquid phase epitaxy YIG /GGG (gallium gadolinium garnet) substrates from Innovent e.V., Jena, Germany . The larger thickness for the Au metal fi lm was chosen to guarantee the continuity of the film [Suppl. Information, 2] . Hall bars 100 µm wide and 800 µm long were patterned by e -beam lithography and Ar ion milling for b oth systems . 2 Figure 1: Magnetoresistance measurement geometries: (a) Longitudinal resistivity measurement (𝜌𝑥𝑥) while rotating the magnetic field along the angles 𝛼 (green), 𝛽 (blue) and 𝛾 (brown) and while current is flowing in 𝑥-direction. (b) Transverse resistivity m easurement (𝜌T) while sweeping the magnetic field along the 𝑧-direction and current alternatingly in the 𝑥- and 𝑦- directions (Zero -Offset Hall measurement) . (c,e) Longitudinal ADMR for YIG/Pt and YIG/Au, respectively , at a magnetic field strength of 1 T. (d,f) Zero-Offset Hall measurement after subtracting the normal Hall effect for YIG/Pt and YIG/Au, respectively. Grey 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 normal Hall signal. The longitudinal resistance in 𝑥-direction 𝜌𝑥𝑥 and its magnetic field angle dependent magnetoresistance (ADMR) were measured using a conventional Kelvin contact layout [ Figure 1(a)] in a l-He cryostat with 9 T field and 360° sample rotation capabilities. The transvers e resistance and its magnetoresistance were obtained by out -of-plane field Zero-Offset Hall measurements [ Figure 1(b)] using an integrated measurement device from HZDR innovation s GmbH [more details in Suppl. Information, 2] . In the context of SMR , the resistivit y tensor of the NM layer in lab coordinates is given by [12]: 𝜌𝑥𝑥=𝜌0−∆𝜌1 𝑚𝑦2 𝜌𝑦𝑦=𝜌0−∆𝜌1 𝑚𝑥2 𝜌𝑥𝑦=∆𝜌1𝑚𝑥 𝑚𝑦−∆𝜌2 𝑚𝑧 (1) where 𝐦(𝑚x,𝑚y,𝑚z)=𝐌/𝑀s are the normalized projections of the magnetization of the YIG film the three main axes and 𝑀s is the saturation magnetization of the MI layer. 𝜌0 is the Drude resistivity and 3 ∆𝜌1 and ∆𝜌2 are the characteristic amplitudes of the SMR. The first term in the expression for 𝜌𝑥𝑦 is a planar Hall effect resulting from the anisotropy of the longitudinal resistivity ( 𝜌PHE =𝜌𝑥𝑥−𝜌𝑦𝑦). In the following, w e do n either discuss nor measure planar Hall effects, which are universally rejected by the Zero-Offset Hall measurement technique [36] due to their origin in the longitudinal tensor components [Suppl. Information, 3]. In our measurement, the transvers e signature of the SMR thus simplifies to 𝜌T=−∆𝜌2 𝑚𝑧 (2) The longitudinal and transvers e amplitudes of the SMR, ∆𝜌L≡∆𝜌1 and ∆𝜌T≡∆𝜌2, are related to the spintronic properties of the MI/NM bilayer via: ∆𝜌L≡∆𝜌1=2 𝜌0 𝜃SH2 𝜆 𝑡 𝑅𝑒[𝜆 𝐺↑↓ tanh2 (𝑡 2𝜆) 1 𝜌0+2 𝜆 𝐺↑↓ coth (𝑡 𝜆)] (3) ∆𝜌T≡∆𝜌2=−2 𝜌0 𝜃SH2 𝜆 𝑡 𝐼𝑚[𝜆 𝐺↑↓ tanh2 (𝑡 2𝜆) 1 𝜌0+2 𝜆 𝐺↑↓ coth (𝑡 𝜆)] (4) where 𝜆 and 𝜃 are the spin diffusion length and the spin Hall angle of the NM layer , and 𝐺↑↓ is the spi n mixing conductiv ity of the MI/NM interface. Taking into account that 𝐺r𝐺i⁄ ≫1 [11,33] , we can combine Eqs. (3) and (4) to obtain 𝐺r 𝐺i≈∆𝜌L ∆𝜌T 1 1+2 𝜌0 𝜆 𝐺rcoth (𝑡 𝜆) (5) The SMR amplitude ∆𝜌L was evaluated in YIG/Pt and YIG/Au as shown in Figure 1(a,c,e ) from longitudinal ADMR measurements at a field of 1 T to keep YIG saturated . ∆𝜌T was evaluated in the same samples from out -of-plane field Zero -Offset Hall measurements by driving the YIG film into magnetic saturation along the 𝑧-direction as shown in Figure 1(b,d,f ). For YIG/Pt , the longitudinal ADMR reveals the behavior expected from the SMR [Eq. (1)] , namely a resistivity change of ∆𝜌L in the 𝛼 and 𝛽 angles and no modulation in the 𝛾 angle [ Figure 1(c)]. The slight effect seen in 𝛾 is consistent with a sample misalignment of less than 0.1°. From the measured resistivity 𝜌0=516 nΩm in our Pt thin film, the values 𝜆Pt=(1.2±0.2) nm and 𝜃Pt=0.09±0.01 can be inferred from the empirical relationships fo und for Pt thin films [37]. Together with the measured ∆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 is in good agreement with previous reports in the same system [9,11,13,15,18,19] . When applying a magnetic field along the 𝑧-direction, the YIG film is gradually brought into saturation, while a proportional transverse magnetoresistance develops [ Figure 1(d)]. The value ∆𝜌T corresponds to the saturation value of the resistivity change. Eq. ( 5) then yields (𝐺r𝐺i⁄)YIG /Pt=22±3 for the YIG/Pt reference sample [ Table 1]. This ratio is in good agreement with the theoretical calculations of 𝐺r𝐺i⁄ ≈ 20 [33], as well as experimental values of 𝐺r𝐺i⁄ =16±4 [18] and 𝐺r𝐺i⁄ =33 [11]. The experimental values of ∆𝜌L,Au=(1.05±0.12) pΩm and ∆𝜌T,Au=(11.2±1.7) fΩm of YIG/Au are obtained in the same manner as for the YIG/Pt reference sample [see Figure 1(e,f)]. To actually measure a transverse magnetoresistance at this level of approximately 1 μΩ, special care must be taken to isolate the anomalous Hall signal from the much larger background contributions [ inset in Figure 1(f)], which we accomplish by eliminating the longitudinal resistance by Zero -Offset Hall [36] and by accounting for the nonlinearity of the normal Hall effect itself [Suppl. Information , 4]. 4 Figure 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, and fits in context of the Elliott -Yafet relation (a) and resistivity dependence on the intrinsic spin Hall effect (b) shown as red lines with 1𝜎 confidence bands. The data points are empirical data taken f rom: Kimura [38] (solid diamond ), Brangham [39] (hollow squares ), Vlaminck & Obstbaum [40,41] (hollow cir cle), Isasa [42] (hollow triangle), Niimi [43] (solid triangle), Mosendz & Obstbaum [44,45] (solid square) , Laczkowski [46] (hollow diamond) and Qu [47] (solid circle) . 𝜃 values from Refs. [42,43] have been multiplied by 2 for a proper comparison. The solid blue line denotes the resistivity of the Au film in the present YIG/Au system; dashed blue lines show the obtained spintronic quantities 𝜆 and 𝜃 for our Au film . In the following, we will provide an independent calculation of the SMR magnitud es for YIG/Au. First, we have to estimate the spintronic quantities 𝜆Au and 𝜃Au of our Au film. Concerning 𝜆, it is well established that the spin relaxation in metals is dominated by the Elliott -Yafet mechanism ( 𝜆∝ 𝜌−1) [37,48 –53]. Figure 2(a) illustrates how we obtain 𝜆Au=(25−8+12) nm based on the measured resistivity of the Au layer and empirical data [38–44,46,47] . This corresponds to a 𝜌- 𝜆-product of (2.1−0.7+1.0) 10−15 Ωm2. Regarding 𝜃, different mechanism have been suggested to contribute in Au [39,42,43] , but the origin of its SHE is not well established, yet. In order to estimate a reasonable value, we consider for simplicity that the intrinsic scattering contribution dominates the spin Hall angle . In this case, 𝜃=𝜎SHint×𝜌 holds , where the intrin sic spin Hall conductivity 𝜎SHint depends on the band structure and is thus constant for a given metal [21]. By taking reported data on Au [39–43,45 –47], we estimate 𝜎SHint=(2.0−1.0+2.0) [ℏ 2𝑒] 105 Ω−1m−1, touching the upper end of theoretical predictions ranging from (0.22⋯0.9) [ℏ 2𝑒] 105 Ω−1m−1 [47,54,55] . The fitted 𝜎SHint value leads to 𝜃Au=0.017−0.008+0.016 for our Au film as shown in Figure 2(b). In addition, we assume identical interface spin mixing conductivities in our reference Pt system and the Au system 𝐺YIG /Pt≡𝐺YIG /Au, owing to the identical fabrication conditions, similar chemical qualities of the metals and similar Fermi energies and Sha rvin conductivities of the metals [12]. Given these values, we can estimate the SMR magnitudes ∆𝜌L,Au=(0.7−0.4+2.0) pΩm and ∆𝜌T,Au=(7−5+25) fΩm for our YIG/Au sample . The calculated values are quantitatively consistent with the measured values. The uncertainty of the calculation will decrease in the future when the Elliott -Yafet scaling constant and intrinsic spin Hall conductivity are better understood for Au, as well as for other metals. Table 1: Overview of the obtained quantities for YIG/ Pt and YIG/Au studied here: Resistivity 𝜌0, the re lative longitudinal and transverse amplitudes of the spin Hall magnetoresistance ∆𝜌L and ∆𝜌T, spin diffusion length 𝜆, spin Hall angle 𝜃 and real and imaginary spin mixing conductivities 𝐺r, 𝐺i. * 𝜆 and 𝜃 are derived from empirically observed scaling. * * Spin mixing conductivit ies are calculated for YIG/Pt and assumed to be identical for YIG/Au. Pt(2nm) Au(10nm) 𝜌0 (nΩm) 516 84.4 ∆𝜌L/𝜌0 6.8×10−4 1.3×10−5 ∆𝜌T/𝜌0 2.2×10−5 1.3×10−7 𝜆 (nm) 1.2±0.2 * 25−8+12 * 𝜃 0.09±0.01 * 0.017−0.008+0.016 * 𝐺r (1014 Ω−1m−2) 3.8±1.0 3.8 ** 𝐺r𝐺i⁄ 22±3 22 ** 5 We conclude that the observed magnitudes of the longitudinal and transvers e magnetoresistance in our two systems, YIG/Au and YIG/Pt, are consistent with the same physical picture. Namely, the transvers e magnetoresistance in these M I/NM systems can be fully understood as emergent from the transvers e part of the spin Hall magnetoresistance due to th e imaginary component of the spin -mixing conductivity. No evidence points to a contribut ion of proximity magnetization via the anomalous Hall effect. Au is prototypical for materials with a low resistivity and intermediate spin Hall angle, which indicates that the conventional SMR theory applies in this regime. In addition, the possibility of measuring both the longitudinal and the transverse spin Hall magnetoresistance amplitudes for a wide range of MI/NM interfaces provide s an elegant way to study the spin -mixing conduct ivity and its fundamental dependencies . Supplementary Material Supplementary material contains further details on the transport measurements including the approach to compensate for the normal Hall effect a nd remark on the absence of the planar Hall effect in Zero -Offset Hall measurements. Electron microscopy imaging of the samples in cross section is reported as well. Additional measurements are reported to assess the contribution of the anomalous Hall effe ct due to the thermal spin pumping. The data includes hysteresis loops measured at different current densities as well as transport data taken at higher harmonics. Acknowledgements Support by the Structural Characterization Facilities Rossendorf at the Ion Beam Center (IBC) at the HZDR is greatly appreciated. The work was financially supported in part via the German Research Foundation (DFG) Grant MA 5144/9 -1, the BMBF project GUC -LSE (federal research fun ding of Germany FKZ: 01DK17007), the BMWi project WiTenso (03THW12G01), by the Spanish MINECO under the Maria de Maeztu Units of Excellence Programme (MDM -2016 -0618) and under Project No. MAT2015 -65159 -R and by the Regional Council o f Gipuzkoa (Project No. 100/16). J.M.G. -P. thanks the Spanish MINECO for a Ph.D. fellowship (Grant No. BES -2016 -077301). References [1] T. Kosub, M. Kopte, R. Hühne, P. Appel, B. Shields, P. Maletinsky, R. Hübner, M. O. Liedke, J. Fassbender, O. G. Schmidt, and D. Makarov, Purely Antiferromagnetic Magnetoelectric Random Access Memory , Nat. Commun. 7 13985 (2017). [2] S. Meyer, Y. -T. Chen, S. Wimmer, M. Althammer, T. Wimmer, R. Schlitz, S. Geprägs, H. Huebl, D. Ködderitzsch, H. Ebert, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Observation of the spin Nernst effect , Nat. Mater. 16 977 EP (2017). [3] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck insulator , Nat. Mater. 9 894 EP (2010). [4] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Long -dista nce transport of magnon spin information in a magnetic insulator at room temperature , Nat. Phys. 11 1022 EP (2015). [5] D. Ellsworth, L. Lu, J. Lan, H. Chang, P. Li, Z. Wang, J. Hu, B. Johnson, Y. Bian, J. Xiao, R. Wu, and M. Wu, Photo -spin-voltaic effect , Nat. Phys. 12 861 EP (2016). [6] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Transmission of electrical signals by spin - wave interconversion in a magnetic in sulator , Nature 464 262 EP (2010). [7] E. Villamor, M. Isasa, S. Vélez, A. Bedoya -Pinto, P. Vavassori, L. E. Hueso, F. S. Bergeret, and F. Casanova, Modulation of pure spin currents with a ferromagnetic insulator , Phys. Rev. B 91 020403 (2015). [8] J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, Observation of the Spin Peltier Effect for Magnetic Insulators , Phys. Rev. Lett. 113 027601 (2014). [9] H. Nakayama, M. Althammer, Y. -T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel, S. Takahashi, and others, Spin Hall Magnetoresistance Induced by a Nonequilibrium Proximity Effect , Phys. Rev. Lett. 110 206601 (2013). 6 [10] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y. -T. Chen, H. Jiao, G. E. W. Bauer, and S. T. B. Goennenwein, Experimental Test of the Spin Mixing Interface Conductivity Concept , Phys. Rev. Lett. 111 176601 (2013). [11] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Hueb l, S. Geprägs, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. -M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y. -T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Quantitative study of the spin Hall magnetoresistance in ferromagnetic insula tor/normal metal hybrids , Phys. Rev. B 87 224401 (2013). [12] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Theory of spin Hall magnetoresistance , Phys. Rev. B 87 144411 (2013). [13] C. Hahn, G. d e Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Comparative measurements of inverse spin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta , Phys. Rev. B 87 174417 (2013). [14] M. Isasa, A. Bedoya -Pinto, S. Vélez, F. Golmar, F. Sánch ez, L. E. Hueso, J. Fontcuberta, and F. Casanova, Spin Hall magnetoresistance at Pt/CoFe 2O4 interfaces and texture effects , Appl. Phys. Lett. 105 142402 (2014). [15] S. R. Marmion, M. Ali, M. McLaren, D. A. Williams, and B. J. Hickey, Temperature dependence of spin Hall magnetoresistance in thin YIG/Pt films , Phys. Rev. B 89 220404 (2014). [16] S. Meyer, R. Schlitz, S. Geprägs, M. Opel, H. Huebl, R. Gross, and S. T. B. Goennenwein, Anomalous Hall effect in YIG|Pt bilayers , Appl. Phys. Lett. 106 132402 (2015). [17] S. Vélez, A. Bedoya -Pinto, W. Yan, L. E. Hueso, and F. Casanova, Competing effects at Pt/YIG interfaces: Spin Hall magnetoresistance, magnon excitations, and magnetic frustration , Phys. Rev. B 94 174405 (2016). [18] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, and B. J. van Wees, Exchange magnetic field torques in YIG/Pt bilayers observed by the spin -Hall magnetoresistance , Appl. Phys. Lett . 103 032401 (2013). [19] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Spin-Hall magnetoresistance in platinum on yttrium iron garnet: Dependence on platinum thickness and in-plane/out -of-plane magnetization , Phys. Rev. B 87 184421 (2013). [20] J. E. Hirsch, Spin Hall Effect , Phys. Rev. Lett. 83 1834 (1999). [21] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects , Rev. Mod. Phys. 87 1213 (2015). [22] S. Rüegg, G. Schütz, P. Fischer, R. Wienke , W. B. Zeper, and H. Ebert, Spin-dependent x -ray absorption in Co/Pt multilayers , J. Appl. Phys. 69 5655 (1991). [23] S.-Y. Huang, X. Fan, D. Qu, Y. Chen, W. Wang, J. Wu, T. Chen, J. Xiao, and C. Chien, Transport magnetic proximity effects in platinum , Phys. Rev. Lett. 109 107204 (2012). [24] Y. Lu, Y. Choi, C. Ortega, X. Cheng, J. Cai, S. Huang, L. Sun, and C. Chien, Pt Magnetic Polarization on Y 3Fe5O12 and Magnetotransport Characteristics , Phys. Rev. Lett. 110 147207 (2013). [25] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Physical Origins of the New Magnetoresistance in Pt/YIG , Phys. Rev. Lett. 112 236601 (2014). [26] T. Shang, Q. F. Zhan, L. Ma, H. L. Yang, Z. H. Zuo, Y. L. Xie, H. H. Li, L. P. Liu, B. M. Wang, Y. H. Wu, S. Zhang, and R. -W. Li, Pure spin -Hall magnetoresistance in Rh/Y 3Fe5O12 hybrid , Sci. Rep. 5 17734 EP (2015). [27] S. Geprägs, S. Meyer, S. Altmannshof er, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Investigation of induced Pt magnetic polarization in Pt/Y 3Fe5O12 bilayers , Appl. Phys. Lett. 101 262407 (2012). [28] S. Vélez, V. N. Golovach, A. Bedoya -Pinto, M. Isasa, E. Sagasta, M . Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Hanle Magnetoresistance in Thin Metal Films with Strong Spin -Orbit Coupling , Phys Rev Lett 116 016603 (2016). [29] M. Valvidares, N. Dix, M. Isasa, K. Ollefs, F. Wilhelm, A. Rogalev, F. Sán chez, E. Pellegrin, A. Bedoya -Pinto, P. Gargiani, L. E. Hueso, F. Casanova, and J. Fontcuberta, Absence of magnetic proximity effects in magnetoresistive Pt/CoFe 2O4 hybrid interfaces , Phys. Rev. B 93 214415 (2016). [30] S. S. -L. Zhang and G. Vignale, Nonlocal Anomalous Hall Effect , Phys. Rev. Lett. 116 136601 (2016). [31] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Theory of spin Hall magnetoresistance (SMR) an d related phenomena , J. Phys.: Cond. Mat. 28 103004 (2016). [32] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Finite -Element Theory of Transport in Ferromagnet –Normal Metal Systems , Phys Rev Lett 84 2481 (2000). 7 [33] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Spin transfer torque on magnetic insulators , Europhys. Lett. 96 17005 (2011). [34] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F. Alvarado, and P. Gambardella, Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers , Nat. Phys. 11 570 EP (2015). [35] D. Hou, Z. Qiu, R. Iguchi, K. Sato, E. K. Vehstedt, K. Uchida, G. E. W. Bauer, and E. Saitoh, Observation of temperature -gradient -induced magnetization , Nat. Commun. 7 12265 EP (2016). [36] T. Kosub, M. Kopte, F. Radu, O. G. Schmidt, and D. Makarov, All-electric access to the magnetic -field-invariant magnetization of antiferromagnets , Phys. Rev. Lett. 115 097201 (2015). [37] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Tuning the spin Hall effect of Pt from the moderately dirty to the superclean regime , Phys. Rev. B 94 060412 (2016). [38] T. Kimura, J. Hamrle, and Y. Otani, Estimation of spin -diffusion length from the magnitude of spin-current absorption: Multiterminal ferromag netic/nonferromagnetic hybrid structures , Phys. Rev. B 72 014461 (2005). [39] J. T. Brangham, K. -Y. Meng, A. S. Yang, J. C. Gallagher, B. D. Esser, S. P. White, S. Yu, D. W. McComb, P. C. Hammel, and F. Yang, Thickness dependence of spin Hall angle of Au g rown on Y 3Fe5O12 epitaxial films , Phys. Rev. B 94 054418 (2016). [40] M. Obstbaum, M. Härtinger, H. G. Bauer, T. Meier, F. Swientek, C. H. Back, and G. Woltersdorf, Inverse spin Hall effect in Ni 81Fe19/normal -metal bilayers , Phys. Rev. B 89 060407 (2014). [41] V. Vlaminck, J. E. Pearson, S. D. Bader, and A. Hoffmann, Depende nce of spin -pumping spin Hall effect measurements on layer thicknesses and stacking order , Phys. Rev. B 88 064414 (2013). [42] M. Isasa, E. Villamor, L. E. Hueso, M. Gradhand, and F. Casanova, Temperature dependence of spin diffusion length and spin Hall a ngle in Au and Pt , Phys. Rev. B 91 024402 (2015). [43] Y. Niimi, H. Suzuki, Y. Kawanishi, Y. Omori, T. Valet, A. Fert, and Y. Otani, Extrinsic spin Hall effects measured with lateral spin valve structures , Phys. Rev. B 89 054401 (2014). [44] O. Mosendz, G. Woltersdorf, B. Kardasz, B. Heinrich, and C. H. Back, Magnetization dynamics in the presence of pure spin currents in magnetic single and double layers in spin ballistic and diffusive regimes , Phys. Rev. B 79 224412 (2009). [45] M. Obstbaum, M. Decker, A. K. Greitner, M. Haertinger, T. N. G. Meier, M. Kronseder, K. Chadova, S. Wimmer, D. Ködderitzsch, H. Ebert, and C. H. Back, Tuning Spin Hall Angles by Alloying , Phys. Rev. Lett. 117 167204 (2016). [46] P. Laczkowski, L. Vila, V. -D. Nguyen, A. Marty, J. -P. Attané, H. Jaffrès, J. -M. George, and A. Fert, Enhancement of the spin signal in permalloy/gold multiterminal nanodevices by lateral confinement , Phys. Rev. B 85 220404 (2012). [47] D. Qu, S. Y. Huang, G. Y. Guo, and C. L. Chien, Inverse spin Hall effect in Au xTa1-x alloy films , Phys. Rev. B 97 024402 (2018). [48] J. Bass and W. P. Pratt Jr, Spin-diffusion lengths in metals and alloys, and spin -flipping at metal/metal interfaces: an e xperimentalist’s critical review , J. Phys.: Cond. Mat. 19 183201 (2007). [49] A. J. Berger, E. R. J. Edwards, H. T. Nembach, O. Karis, M. Weiler, and T. J. Silva, Determination of the spin Hall effect and the spin diffusion length of Pt from self -consisten t fitting of damping enhancement and inverse spin -orbit torque measurements , Phys. Rev. B 98 024402 (2018). [50] A. Kiss, L. Szolnoki, and F. Simon, The Elliott -Yafet theory of spin relaxation generalized for large spin -orbit coupling , Sci. Rep. 6 22706 EP (2016). [51] M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Spin Torque Study of the Spin Hall Conductivity and Spin Diffusion Length in Platinum Thin Films with Varying Resistivity , Phys. Rev. Lett. 116 126601 (2016). [52] K. Roy, Estimating the spin diff usion length and the spin Hall angle from spin pumping induced inverse spin Hall voltages , Phys. Rev. B 96 174432 (2017). [53] E. Sagasta, Y. Omori, M. Isasa, Y. Otani, L. E. Hueso, and F. Casanova, Spin diffusion length of Permalloy using spin absorption in lateral spin valves , Appl. Phys. Lett. 111 082407 (2017). [54] K. Chadova, D. V. Fedorov, C. Herschbach, M. Gradhand, I. Mertig, D. Ködderitzsch, and H. Ebert, Separation of the individual contributions to the spin Hall effect in dilute alloys within the first-principles Kubo -Stifmmode relse rfieda approach , Phys. Rev. B 92 045120 (2015). [55] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, Intrinsic spin Hall effect and orbital Hall effect in 4d and 5d transition metals , Phys. Rev. B 77 165117 (2008). 8 Supplementary Information 1. Thermal spin pumping induced anomalous Hall effect It has been reported that thermal gradients can lead to thermal spin pumping, which in turn can lead to proximity magnetization of Au thin films on YIG depending on the strength of the thermal gradient [1]. Such proximity magnetization will also lead to an anomalous Hall effect but would be not due to the spin Hall magnetoresistance. We argue that the contribution of thermal spin pumping induced proximity magnetization is negligible in our present study. Several references and statements within D. Hou et al. [1] confirm tha t Au thin films are not expected to reveal any static magnetic proximity effect. This is also exemplified by the experiments in that work, which were performed without a thermal gradient resulting in no detected AHE signal. The authors argue that thermal s pin pumping leads to magnetization in Au. As our Au film is Joule heated due to transport experiments, we expect a thermal gradient from Au towards YIG also in our samples. D. Hou et al. used an inverted notation of Hall voltage which is evident from their positive slope of the normal Hall effect in their Figure 2(b). Therefore, the positive slope anomalous Hall effect (AHE) curves in that work are identically signed as our negative slope AHE curves. Taking into account our twice lower Au thickness, we woul d be able to explain our AHE signal magnitude with a purported thermal gradient of about 3.5 K/mm. If this thermal gradient is indeed present and is the origin of our Hall signal, the expectation from the cited work would be an increase in the magnitude of AHE curves when larger currents are used to probe the Au film. Figure S1: Variation of the transverse magnetoresistance upon employing different probe current amplitudes 𝐼amp. We did measure our sample at various levels of current. Due to our narrow st ripe pattern these currents lead to varying extents of Joule heating and thus thermal gradients orthogonal to the film plan. We did not observe an increasing trend of the AHE magnitude with increasing probe current. In fact, despite a more than 7 -fold vari ation of the Joule heating dissipation, we did not observe any significant change in the magnitude of the AHE effect [Figure S1]. The slight (but non -significant) variation of the AHE magnitude is due to the processing of the nonlinear background, which is slightly different scan to scan due to long term nature of the scans. Even if the variation of the AHE magnitude with the current amplitude is taken to be real, it is opposite to that expected from thermal spin pumping described by D. Hou et al. Therefore , we conclude that the thermal spin pumping induced AHE is negligible for the total magnitude of our observed transversal magnetoresistance. Furthermore, it is also possible to separate the contributions from the transverse spin Hall magnetoresistance and from thermal spin pumping AHE to the total transverse magnetoresistance through harmonic measurements. The signal in Figure S1 follows from the transverse voltage at the excitation current frequency (first harmonic) and can contain co ntributions from both effects. In contrast, the voltage at the third harmonic frequency should contain only contributions from thermal 9 spin pumping: When we supply a low frequency ( 𝑓) AC current to probe our Au films, we expect to heat the Au film not onl y to an equilibrium value – which we discussed above – but the Au film would also exhibit temperature variations with a 2𝑓 periodicity. The temperature would peak every time the probe current sine wave is at its tips and temperature would dip below the av erage value at the zero - crossings of the sine wave. According to D. Hou et al. such a temperature behavior would result in more transient magnetization being present in Au while the probe current sine wave is near its tips. Therefore, the resulting AHE vol tage would be larger than a pure sine wave near the tips of the sine wave probe current resulting in an additional third harmonic Hall signal. This third harmonic Hall signal should exhibit the same behavior as the magnetization of YIG, i.e. an antisymmetr ical curve saturating at about 0.15 T . Figure S2: Third harmonic Hall signal using 𝐼amp =5.61 mA. We indeed also probed this third harmonic signal and found no significant YIG -like variation of it as we swept the field [Figure S2]. Fitting a YIG like si gnal to the third harmonic Hall signal of Figure S2 leads to a thermal spin pumping AHE of about 2 fΩm. This is a) much lower than the first harmonic transverse magnetoresistance magnitude we observe and b) is questionable to exist at all in light of the d ata in Figure S2 which could be also just a linear trend due to the normal Hall effect and slightly non-sinusoidal excitation. Judging from the signal levels reported by D. Hou et al., we should have definitely observed these trends using our very sensitiv e methodology if the thermal gradient was sufficient and the effect as strong as reported. We hypothesize that the absence of the thermal gradient AHE signals in our study could be due to either a lower thermal gradient or the different preparation conditi ons. Regarding the latter, a lthough D. Hou et al. did not provide structural information on their YIG/Au interface, we believe that it is of more coherent quality due to their etching and in -situ-prenannealing procedure prior to Au deposition. Instead, to observe the SMR of Au/YIG, no interface optimization was found necessary. As a result, we deposited Au on untreated YIG substrates, yielding dense continuous films, but no epitaxy probably because of amorphous surface termination of the YIG surface due to adsorbates. These adsorbates could also render the thermal spin pumping inefficient in our samples. 2. Sample preparation and measurement details The reason we chose 10 nm Au as our main sample is because we wanted to avoid discontinuous metal films. Such discontinuities appear due to the bad wetting of the noble metals on the untreated YIG, especially for metals with low melting temperatures, such as Cu, Ag, Au [Figure S3]. Discontinuous layers can give rise to spurious effects like wrong resistivity values when assuming nominal thicknesses which would be problematic for our determination of the spin Hall angle and the spin diffusion length. 10 Figure S3: Cross -sectional STEM image of 10 nm film of Au on an untreated liquid phase epitaxy YIG -on-GGG sub strate. The Au film is fully continuous and has a homogenous thickness. On the other hand, films thicker than 10 nm would produce so much shunting of the interfacial SMR effect (or magnetic proximity effect), that the detection of the transversal signal would be no longer possible. We want to stress that the presented measurements have been very challenging even in this 10 nm Au sample. Even though w e used a precisely patterned Hall cross with only about 0.3 % contact skew, the residual longitudinal resistance background amounted to roughly 11 mΩ, which is roughly 105 over the minimum necessary precision to detect the AHE curves of about 100 nΩ and places high demands for continuous dynamic measurement range. In addition, to achieve this precision at a noise density of about 2 µΩ√Hz⁄ (at 2 mA current amplitude) required approximately 400 seconds of integration for each of the 100 field bins, amounting to an integration time of approximately half a day. Therefore, although our purpose made measurement device unites great continuous dyn amic range and extremely low noise, we had to employ sophisticated drift compensation methods to achieve a low enough corner frequency to actually resolve the AHE effect in our samples. 3. Planar Hall effect When measuring the transverse voltage drop develop ed by a slab of anisotropically conducting material, a planar Hall effect is generally observed. In context of polycrystalline metal films used in spintronics, such anisotropy commonly appears as a result of e.g. the anisotropic magnetoresistance [2,3] or spin Hall magnetoresistance [4]. When a voltage is applied to this anisotropically conducting thin film, the current will in general not flow collinearly w ith the voltage gradient, but experience a transverse deflection towards the high conductivity axis, which results in an equilibrium transverse voltage drop. In a stationary measurement layout, this planar Hall effect only vanishes for a) perfectly isotrop ic conduction or b) conduction exactly along the high or low conductivity axes. However, despite its name, the planar Hall effect is fundamentally different from other Hall effects such as the normal and an omalous Hall effects (real Hall effects) . When averaging all in -plane current directions, the planar Hall effect assumes both positive and negative values as current experience alternatingly left -handed and right -handed deflection. In contrast , the real Hall effects persist as they experience the same handedness of deflection for any in -plane current direction . Therefore, planar Hall effects disappear in Zero -Offset Hall measurements [5], while the real Hall effects are maximally preserved. 4. Normal Hall effect In order to study tiny anomalous Hall-like signals, it is mandatory to dynamically reject the influence of the longitudinal resistance, which – even for carefully patterned – Hall cross structures can be much larger than the Hall signal of interest [5]. While this is elegantly achieved using the Zero -Offset Hall measuremen t scheme, the normal Hall effect cannot be rejected using this approach. D. Hou et al. reported an approach to reject the normal Hall effect via a lock -in method [1], but this is possible only when the origin of the anomal ous Hall effect can be switched on and off, which is not the case in our sample. 11 We remove the normal Hall effect after the measurement by subtracting it from the measured transverse resistance. The observed normal Hall effect is not completely linear in the magnetic field for two reasons: a) our magnetic field reading is not fully linear over the actual magnetic field mainly due to nonlinearities of the Si hall probe. b) the normal H all effect of the thin metal ( Pt or Au) films can be slightly non -linear itself. These influences cause a total nonlinearity of the normal Hall effect in the range of 10−3 to 10−2, which is non -negligible in our study. The function we use to model the norm al Hall effect is composed of three contributions: 𝜌NHE =𝐴1𝐻𝑧+𝐴2(1−sech (𝐻𝑧 𝐻NL))+𝐴3tanh (𝐻𝑧 𝐻NL) (1) where 𝐻𝑧 is the magnetic field applied out -of-plane, 𝐻NL is a characteristic field determining the shape of the nonlinearity and 𝐴1,2,3 are scaling constants. The first contribution 𝐴1𝐻𝑧 is the largest by far, while the sech and tanh function s capture smaller even and odd nonlinearities, respectively . One motivation for this model ch oice is that it cannot accidently introduce YIG -like signals, when 𝐻NL is sufficiently large, namely 𝐻NL≳2𝐻sat,YIG with 𝜇0𝐻sat,YIG≈0.16 T. When this is fulfilled, the tanh contribution is essentially linear in the relevant field range. For the present ed measurements, we used 𝜇0𝐻NL≈1.1 T to model and subtract the normal Hall contribution to 𝜌T [Figure 1(d,f) of the main text]. References [1] D. Hou, Z. Qiu, R. Iguchi, K. Sato, E. K. Vehstedt, K. Uchida, G. E. W. Bauer, and E. Saitoh, Observation of temperature -gradient -induced magnetization , Nat. Commun. 7 12265 EP (2016). [2] K. M. Seemann, F. Freimuth, H. Zhang, S. Blügel, Y. Mokrousov, D. E . Bürgler, and C. M. Schneider, Origin of the Planar Hall Effect in Nanocrystalline Co 60Fe20B20., Phys. Rev. Lett. 107 086603 (2011). [3] P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R. P. Campion, V. Novák, K. Olejník, F. Maccherozzi, S. S. Dh esi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kuneš, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth, Electrical switching of an antiferromagnet , Science 351 587 (2016). [4] H. Nakayama, M. Althammer, Y. -T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel, S. Takahashi, and others, Spin Hall Magnetoresistance Induced by a Nonequilibrium Proximity Effect , Phys. Rev. Lett. 110 206601 (2013). [5] T. Kosub, M. Kopte, F. Radu, O. G. Schmidt, and D. Makarov, All-electric access to the magnetic - field-invariant magnetization of antiferromagnets , Phys. Rev. Lett. 115 097201 (2015).

2018-11-14

Anomalous Hall-like signals in platinum in contact with magnetic insulators are common observations that could be explained by either proximity magnetization or spin Hall magnetoresistance. In this work, longitudinal and transverse magnetoresistances are measured in a pure gold thin film on the ferrimagnetic insulator Y$_3$Fe$_5$O$_{12}$ (Yttrium Iron Garnet, YIG). We show that both the longitudinal and transverse magnetoresistances have quantitatively consistent scaling in YIG/Au and in a YIG/Pt reference system when applying the Spin Hall magnetoresistance framework. No contribution of an anomalous Hall effect due to the magnetic proximity effect is evident.

Anomalous Hall-like transverse magnetoresistance in Au thin films on Y$_3$Fe$_5$O$_{12}$

1811.05848v1

A spinwave Ising machine Artem Litvinenko1, Roman Khymyn1, Victor H. Gonz ´alez1, Ahmad A. Awad1, Vasyl Tyberkevych2, Andrei Slavin2and Johan ˚Akerman1 1Department of Physics, University of Gothenburg, Fysikgr ¨and 3, 412 96 Gothenburg, Sweden 2Department of Physics, Oakland University, 48309, Rochester, Michigan, USA, To whom correspondence should be addressed; E-mails: artem.litvinenko@physics.gu.se, johan.akerman@physics.gu.se We demonstrate a spin-wave-based time-multiplexed Ising Machine (SWIM), implemented using a 5 m thick Yttrium Iron Garnet (YIG) film and off-the- shelf microwave components. The artificial Ising spins consist of 34–68 ns long 3.125 GHz spinwave RF pulses with their phase binarized using a phase- sensitive microwave amplifier. Thanks to the very low spinwave group veloc- ity, the 7 mm long YIG waveguide can host an 8-spin MAX-CUT problem and solve it in less than 4 s while consuming only 7 J. Using a real-time oscillo- scope, we follow the temporal evolution of each spin as the SWIM minimizes its energy and find both uniform and domain-propagation-like switching of the spin state. The SWIM has the potential for substantial further miniatur- ization, scalability, speed, and reduced power consumption, and may become a versatile platform for commercially feasible optimization problem solvers with high performance. 1arXiv:2209.04291v1 [cond-mat.mes-hall] 9 Sep 2022Introduction As Moore’s law comes to an end due to physical limitations, while the amount processing data is constantly growing, novel analog and digital computing paradigm are being investigated. Large part of data processing tasks consist of combinatorial optimization problems and require special purpose accelerators for power efficient and fast computing. Combinatorial optimiza- tion is essential to select the optimal path from an enormous range of choices and appears in various social and industrial fields such as optimizing financial transactions ( 1), logistics, in- cluding travel ( 2) and channel assignment for wireless communications ( 3), genetic engineering and molecular design for drug discovery ( 4). Conventional computers based on V on Neumann architectures are inefficient in solving hard combinatorial optimization problems due to the fac- torial growth of all possible combinations to be evaluated using brute force. Fortunately, Ising machines have emerged as a promising non-von-Neumann computing scheme that can acceler- ate computation of NP-hard optimization problems. An Ising machine maps an NP-problem onto the Ising Hamiltonian of an array of Nbina- rized and interacting physical entities, commonly referred to as spins si=1. H(s1;:::;s N) =X i<jJijsisjX i=1hisi (1) The coding of a specific problem is performed by adjusting the pairwise coupling Jijterms such that the lowest energy spin glass configuration represents the solution to the NP-problem. This is possible because spin glass solutions belong to the NP-hard complexity class ( 5), and many NP-complete and NP-hard problems can be reformulated as Ising Hamiltonians ( 6). Then, the energy minimization of the entire array can be employed as a solution search algorithm in a process called annealing ( 7). Various types of Ising machines ( 8) have been proposed in the recent decades exploiting novel physical building blocks such as spintronic devices ( 9–12 ), memristor crossbars ( 13), 2quantum superconducting junctions ( 14, 15 ), metal-insulator relaxation oscillators ( 16), and degenerate optical parametric oscillators ( 17–19 ) as well as employing conventional CMOS technology with analog electric oscillators ( 12, 20 ), FPGAs ( 21, 22 ). In particular, optical Co- herent Ising Machines have attracted great attention due to their high computational speed, time-multiplexing method that provides all-to-all Ising spin connections and the largest amount of supported Ising spins amongst all other implementations. Optical CIMs use time-multiplexed degenerate optical parametric oscillators (DOPOs) that propagate in a ring cavity in the form of short light-wave pulses. All-to-all coupling between DOPOs can be implemented either physically, via external optical delay lines ( 23–26 ), or elec- tronically, using FPGA blocks ( 18, 19, 24, 27 ). In contrast, spatially-resolved IMs based on physical oscillators are inherently limited to low density connectivity as the number of all-to-all interconnections grows as O(n2)and hence represent an unsolvable problem for 2.5-D mass electronics production technology. DOPOs phase degeneracy allows one to exploit the bistable DOPOs phase as equivalent two-state Ising spins. CIM belongs to a annealing scheme class known as dynamical system solvers as their collective steady-state oscillation minimizes the individual phases described by the Kuramoto model. _i=!0+KX jJijsin( ij) +Khisin( i!0t) (2) WhereKis a global coupling factor. See that eq.2 reduces to eq.1 for sin( ij) =1, promoting binarization which can be done by the parametric pumping at double frequency which will result in an additional term in eq.2. In the rotating frame, the steady-state solution (_i= 0) necessitates ij= 0;andi!0t= 0;forNmutually coupled oscillators. This state will have the properties of a spin glass solution and thus the architecture can be used as scalable NP-solvers, as shown in a progressive evolution from 100 to 100,000 optical spins in ring-based CIMs. Nevertheless, the commercial feasibility of optical CIMs remains elusive 3as the technology requires optical tables, kilowatts of power, and kilometers of optical fibers, which altogether blocks its further development from a proof-of-principle demonstration to a miniaturized commercially viable device. In this work, we present a spinwave Ising machine (SWIM) with artificial spin states im- plemented via the phase of spinwave RF pulses propagating in an Yttrium iron garnet (YIG) thin film. Spinwaves are being actively used for implementation of signal processing ( 28–35 ), frequency synthesis ( 36–38 ), logic ( 39–45 ) and computing ( 46–50 ) applications due to their inherent non-linearities and exceptionally slow propagation velocities. In comparison to op- tical pulses, the use of spinwaves allows for 5–7 orders of magnitude reduction in delay line length. Another advantage comes from the simplicity and efficiency of microwave current-to- spinwave transducers and the off-the-shelf availability of power-efficient microwave electronic components and circuits for the signal processing required in a time-multiplexing architecture of Ising machines. In particular, it allows the use of electronic phase-sensitive amplification blocks consuming only milliwatts of power in contrast to optical kilowatt-power parametric phase-sensitive amplifiers. Altogether, the use of propagating spinwaves for the design of a time-multiplexed Ising machine brings the concept from a laboratory demonstrator to a com- mercially feasible technology with potential for CMOS integration. Design and characteristics of the spinwave Ising Machine The SWIM employs a time-multiplexing approach similar to optical CIMs ( 17–19, 23–27 ) but in the microwave frequency domain, which leads to certain signal processing modifications and substantial performance improvements. As shown in Fig. 1, it consists of two blocks: a) an elec- tronic part for linear and parametric amplification of the RF pulses, RF pulse interconnection, and RF pulse measurement, and b) a physical YIG spinwave waveguide where all the spinwave RF pulses are excited by a first transducer (antenna), allowed to propagate along the film length, 4and finally transformed back to a microwave RF pulses by a second transducer. In contrast to optical CIMs, where optical pulses propagate and are amplified in an optical ring system, never leaving the ring, the SWIM is a multi-physical system where amplification of the propagating RF pulses and their further signal processing is performed outside of the YIG waveguide in a power-efficient electronic system. This is possible because the carrier frequency of the prop- agating spinwaves is around 3 GHz and, therefore, can be easily handled by inexpensive and readily available commercial RF components. Spin waves are a fundamental type of excitation in magnetic systems ( 51–55 ), representing collective precession of elementary magnetic moments coupled via exchange and dipole-dipole interactions. The spin-wave excitation frequency depends on the strength and orientation of the static magnetic field applied to the film and on the magnetic properties of its material, such as the saturation magnetization ( MS), the exchange stiffness ( A), and gyromagnetic ratio ( ). In this work, the static magnetic field is applied at angle =53w.r.t. to the film plane, with its in-pane component parallel to the antennas, i.e.perpendicular to the spinwave propagation along the waveguide. In this configuration, the propagating waves are of a mixed type between magnetostatic surface spin waves (MSSW) and forward volume spin waves (FVMSW). The frequency in this configuration is described by the dispersion relation ( 56, 57 ): !0= 0p (Hint+Msl2 exk2)(Hint+Msl2 exk2+MsF0); (3) F0=P0+ cos2int[1P0+MsP0(1P0)= Hint+Msl2 exk2
 ; (4) wherekis the full wavevector of the spin wave, P0= 1(1exp(kd))andd=5m being the thickness of the film, lex=p 2A=(0M2 s)'1:7
108m for YIG, and Hintandintdefine the value and out of plane angle of the internal field; these can be found from the strength and angle of the applied field ( Hand) using the solution of the magnetostatic problem ( 56, 57 ). In our case H=0.04 T and the choice of = 53is motivated by an empirically found high 5excitation efficiency and low total losses in the YIG delay line when using simple thin copper wire antennas. Moreover, spinwaves in such a configuration are strongly unidirectional and will be excited most efficiently in only one direction, which prohibits multi-transit spinwave echo signals and, therefore, improves the frequency stability of the SWIM. Our time-multiplexed SWIM can be considered as a ring oscillator circuit. According to the circuit design theory, a circuit oscillates when the Barkhausen stability criteria are satisfied: A= 1; (5) 6A= 2n; n2f0;1;2:::g (6) where Ais the total amplification in the loop, is the total loss, and 6Ais the phase accu- mulated along the loop. The Barkhausen criteria are further narrowed by adding a parametric phase-sensitive amplifier (PSA), which limits stable oscillations only at either phase 0 or rel- ative to a pumping reference signal at twice the oscillating frequency and consequently phase binarize the system. The PSA also induces second-harmonic frequency locking ( 58, 59 ) to the external reference signal, !0t!reft=2 =n; n2f0;1;2:::g (7) !0=!ref=2 (8) which further improves the frequency stability. These phase-binarized oscillation conditions are valid for continuous oscillations in the loop. In order to define separate time-multiplexed and phase-binarized aritificial Ising spins, we introduce an RF switch ( 1) that is triggered by a square wave signal with variable frequency to control the spinwave RF pulse length, p. The frequency of the switching is determined by the total delay in the ring and the required number of supported artificial spins: fsw=N= delay (9) 6The RF switch also prevents propagating spinwave RF pulses from spreading due to spinwave dispersion and non-constant delay time over the occupied frequency range. Figure 1: The spinwave Ising machine (SWIM). (a) Schematic of the SWIM with the in- plane magnetized YIG waveguide in the center together with peripheral microwave components, described in the text, and a real-time oscilloscope for direct studies of the time-evoluation of all spins; couplings between spins are implemented using three delay cables of different lengths. (b) Measurement of S21of the YIG delay line showing the spinwave spectrum. BW sw=60 MHz is the spinwave spectral bandwidth measured at –3 dB. ( c) The frequency-dependent delay time of the YIG delay line. delay =271 ns is the average delay time over BW swwith a total deviation of
delay =30 ns. ( d) Amplification of the parametric phase-sensitive amplifier as a function of the relative phase between the propagating RF pulses and the reference signal !ref=6.25 GHz, with a phase sensitivity of APSA=6.1 dB. One major advantage of using spinwaves is their 5–7 orders of magnitude slower propaga- 7tion than the vacuum speed of light, which allows us to reduce the length of the YIG waveguide to millimeters compared to the kilometers-long optical fibers of CIM. As our artificial Ising spins consist of spinwave RF pulses of length p, the maximum number of supported spins is N=delay=p, where the waveguide delay time delay =lYIG=vg;swis the ratio of the waveg- uide length ( lYIG) and the spinwave group velocity ( vg;sw). An approximate theoretical value of the minimal pulse length is given by convolution of i) the 3-dB bandwidth BW swof the spin- wave spectrum, which controls the slew rate of the spinwave RF pulses, and ii) the delay time deviation
delay withinBW sw, which broadens the pulses as they propagate between the two transducers. Fig. 1(b) shows the spinwave spectrum of the YIG spinwave waveguide in the form of itsS21-parameter. The bandwidth of the spinwave generation spectrum is BW sw=60 MHz measured at the –3 dB level. Fig. 1(c) shows the corresponding frequency-dependent delay time, with an average value across BW swofdelay=271 ns and a deviation of 30.2 ns.The limit of minimal pulse width derived from bandwidth is 16.8 ns. However, the delay time deviation causes a significant broadening of propagating spinwave RF pulses limiting the total minimal pulse width by 34 ns. Defining and solving MAX-CUT problems In this section, we evaluate the computational performance of the SWIM on simple 4-spin and 8- spin MAX-CUT optimization problems. While we have successfully populated the waveguide with up to 10 spins, in agreement with the theoretical estimate above, we here limit the problem size to eight spins to avoid any undesired direct interaction between spinwave RF pulses as they begin to partially overlap when more closely packed. The first experiment demonstrates the principle of operation and routes for obtaining a so- lution to the simple 4-spin MAX-CUT problem with antiferromagnetic nearest-neighbor cou- 8plings described by the following Ising matrix: Figure 2: Experimental demonstration of 4-spin SWIM solving a MAX-CUT problem. (a) Transformation of the initial non-optimal state to an optimal solution for 4-spin MAX-CUT optimization problem. The number of cuts in each case is defined by the number of couplings which cross the line which separates spins if they were regrouped according to their spin values. Time traces of SWIM control and computational signals in the initial non-optimal state ( b) and the final solution ( c). Top panel: a control signal of switch 1with a repetition frequency of 14.8 MHz and duty cycle of 50 %that forms oscillations in the SWIM ring oscillator into separate propagating RF pulses. Middle panel: an amplified signal of RF propagating pulses Vampl. Bottom panel: a calculated instantaneous phase signal. The time traces in middle panels are colored according to a value of the calculated instantaneous signal to visually show the pulse instantaneous phase. 9Jij=0 BB@01 0 0 0 01 0 0 0 01 1 0 0 01 CCA(10) This matrix can be realized using a single coupling delay cable ( 3) with a delay of 68 ns, i.e.of exactly one period of the pulse repetition time. Each microwave RF pulse that passes through Coupler 2generates both a spinwave RF pulse in the YIG delay line and a coupling microwave RF pulse propagating through the coupling delay cable 1. After 68 ns, the coupling microwave RF pulse reaches Coupler 3and combines with a different RF microwave RF pulse converted from a propagating spinwave RF pulse, effectively implementing the coupling between these two spins. The negative sign of the coupling coefficients Jijis realized using an additional 180 degree of phase shift implemented with a variable phase shifter. For a demonstration of the computational dynamics, the SWIM is first placed in a randomly chosen steady state s1 j=f+1111gwith the coupling cable 3disconnected. The value +1 corresponds to 180of phase difference between the RF pulse and the reference signal fref, while the value -1 corresponds to 0phase difference. The time traces of the signal of propagating RF pulses and their instantaneous phase are shown in Fig. 2(b), where the color of the time trace corresponds to the instantaneous phase. As can be seen, the propagating spinwave RF pulses are well-defined and separated from each other; their individual phases are similarly well-defined and uniform in time. The initial s1 jrepresents a non-optimal MAX-CUT solution with the number of cuts equal to 2 (see Fig. 2(a)&(b)). At t= 0we connect the coupling cable 3and let the SWIM evolve for 3.76 s or 12 circulation periods, during which it reaches a stable state and finds the optimal number of cuts. Fig. 2(c) shows the resulting spin state during the 12thcirculation period, where the third spin has now switched its phase to under the influence of the coupling matrix. The SWIM has hence changed its state to s12 j=f+11 + 11g, which represents an optimal solution with the number of cuts equal to 4 (see Fig. 2(a)). 10Similarly, we confirmed the SWIM’s capability to solve 8-spin MAX-CUT problems. The frequencyfswof RF switch 1was hence increased to 29.6 MHz according to eq.9 to increase the spin capacity to eight. As the corresponding repetition period decreased to 34 ns, a shorter coupling delay cable with 34 ns delay was used for this experiment. As can be seen in Fig. 2(b) the spinwave RF Figure 3: Experimental demonstration of 8-spin MAX-CUT problem computation. (a) An optimal solution for 8-spin MAX-CUT optimization problem with 8 cuts. ( b) Time traces of SWIM computational signals in the final optimal solution state. Top panel: an amplified signal of RF propagating pulses Vampl. The total duration of the time trace signals corresponds to the delay time in YIG delay line. Bottom panel: a calculated instantaneous phase signal pulses start to show some minor overlap, which could nevertheless be neglected, as the SWIM 11had reached the correct solution when interrogated after 12 cycles. Figure 4: Spin switching scenarios for 4- and 8-spin MAX-CUT problem computations. (a,b) Evolution of the 3rd artificial spin from an initial non-optimal state to an optimal solu- tion for 4-spin and 8-spin MAX-CUT optimization problem. Top panels: an amplified signal Vampl within time intervals which correspond to the 3rd propagating RF pulse. Bottom panels: a calculated instantaneous phase signal. ( c,d) Envelopes (left panels) and instantaneous phase signals (right panels) of the 3rd propagating RF pulse signal within 12 circulation periods plot- ted in the form of overlapping traces in a relative scale. The color of the time traces corresponds to the circulation number. To further visualize the routes to a solution in greater detail, time traces and the instanta- neous phases for the 3rdspin for both 4- and 8-spin MAX-CUT problems were captured at twelve consecutive circulation steps and compiled into Fig. 4(a,b) with scale brakes. The signal envelopes and instantaneous phases are also compiled in Fig. 4(c,d) in the form of overlap- ping traces with different colors corresponding to the number of circulations. Interestingly, the 12SWIM demonstrates two different scenarios of switching between c1 3=1andc12 3= +1 . The first scenario is observed for the 4-spin MAX-CUT problem where the pulses are longer (40 ns). During the first 4 circulations, the spinwave RF pulse gradually separates into two distinct domains of approximately equal width, where only the trailing domain reverses its instanta- neous phase. Withing the next 4-5 circulations the amplitude of the switching domain starts to decrease. The corresponding domain wall can also be clearly seen in the instantaneous phase. From this point on, the domain wall starts to propagate through the pulse until a fully switched single domain pulse is established at about the 10thcycle. The exact origin of this behavior is yet to be identified. A different switching mechanism can be observed for the 8-spin MAX-CUT problem with the 34 ns long coupling delay cable 3. In this case, the evolution of the instantaneous phase occurs via a gradual and uniform shift from 0 to . The difference between the two mechanisms is particularly clear when the evolutions of the envelope and instantaneous phase are plotted in Fig.4c&d. It is possible that the shorter pulse length in the 8-spin case promotes a more uniform response to the coupling. Regardless of its origin, this gradual uniform switching mechanism might open up the possibility to use the SWIM concept as a potential platform to implement XY-Ising machines ( 60). A key parameter for benchmarking the performance of an Ising machine is its time-to- solution. In order to achieve optimal solutions in the simple cases of 4- and 8-spin MAX-CUT problems, a single system run appears sufficient. However, for larger combinatorial problems with dense coupling graphs, there may exist a number of local minima in the Ising Hamiltonian. Therefore, a larger SWIM will require classical annealing schemes with multiple system runs and randomly chosen or idle initial conditions in order to perform statistical analysis on the probability of different stable solutions. The shorter the time-to-solution, the more such runs can be carried out in a given time. 13In our experiments, we periodically switch off the amplification in the loop for 2 s to ensure that the SWIM returns to an idle state, after which we again switch on the loop amplification. Fig. 4(a) shows the temporal evolution of the SWIM prior to and after shutdown. It first takes 1.9 s for the SWIM to develop circulating RF pulses to saturation level. It then takes an additional 1.5s to reach a stable state that represents the optimal solution. The rapid development of the amplitude of the circulating pulses is explained by a large small-signal over-amplification in the loop of the ring-based SWIM circuit ( 6 dB) and leads to system evolution into a non-optimal solution before the signal reaches amplitude saturation. Reducing this over-amplification will lead to slower development of oscillation and it might improve the time-to-solution for large size optimization problems as at low amplitude of the RF pulses the SNR is lower and the system would more easily switch between its states. Another approach that could be attempted, similar to CIM ( 26), is to gradually increase the amplification ratio until the system starts oscillating in the lowest energy state that corresponds to the minimum state of the Ising Hamiltonian. The SWIM design allows for the control of the overall coupling strength between spins by attenuating the signal coming from the coupling delay lines with a variable attenuator (see Fig. 1(a)). To benchmark the SWIM dynamic range, a series of ten consecutive time-to-solution measurements was conducted at different coupling strengths for the 4-spin MAX-CUT problem. A statistical analysis is presented in Fig. 5(b). For this very simple problem, the SWIM demon- strates almost constant time-to-solution values for coupling strengths from 0 to –12 dB with about 20 circulations or 5 s. The very weak trend of a longer time-to-solution at weaker cou- pling strength is then more pronounced at –15 dB. The large SWIM dynamical range of 15 dB should allow the mapping of a wide range of optimization problems. 14Figure 5: Measurement of the time-to-solution parameter for 4-spin MAX-CUT problem computation. (a) SWIM time traces for the measurement of time-to-saturation and time-to- solution parameters. Top panel: a control signal Vctrl ampl for the linear amplifier. Middle panel: an amplified signal of RF propagating pulses Vampl. Bottom panel: phase values in the center of each propagating RF pulse sampled at each circulation period. The linear amplifier switches down at the moment 1swith blackout period of 2sto ensure signal suppression. (b) Circulation number and time-to-solution parameters for 4-spin MAX-CUT problem as a function of the coupling strength. Discussion and outlook The SWIM concept has a high potential for further scaling in terms of spin capacity and physical size and the improvement of power consumption and speed performance parameters. Exploita- tion of non-linear spinwave solitons ( 61–64 ) and proper choice of magnetization angle could compensate spinwave dispersion and flatten delay time over a wide range of frequencies allow- ing to use shorter spinwave RF pulses effectively increasing the number of supported Ising spins with the same length of YIG waveguide. The use of exchange-dominated spinwaves with ex- ceptionally slow propagation velocity ( 51, 65–67 ) in nanoscale YIG films could help to further miniaturize the YIG waveguide. 15The use of miniaturized microwave signal processing components opens a possibility of SWIM scaling by parallelizing the system using multiple ring circuits comprising identical phase sensitive amplifiers, YIG delay lines, etc. Parallelizing SWIM would improve the time- to-solution parameter by the number of the parallel rings as it decreases circulation time. Such a solution would require an FPGA ( 19) as a highly parallel control system for the interconnection of Ising spins. Conclusion We have demonstrated a spinwave time-multiplexed Ising machine (SWIM) and characterized its computational performance and functionality. We have shown how the SWIM can solve 4- and 8-spin NP-hard MAX-CUT problems in about 3.5 s, which is comparable to D-wave’s performance and faster than CIM. The multiphysics design allows the use of power-efficient and conventional low-power microwave components consuming 2 W of power, which amounts to an energy consumption of only 7 J. This outperforms both D-wave’s Advantage and CIM by 3–5 orders of magnitude. Thanks to the exceptionally slow spin wave propagation speed, the 7 mm long YIG waveguide can host up to 11 spins and further scaling is envisioned using both slower spinwave modes and patterned YIG waveguides. The vast variety of nonlinear spinwave modes and phenomena, and the direct availability of more optimized microwave signal processing components, bodes well for the SWIM scaling potential in terms of number of supported Ising spins, device size, and power consumption, making SWIM a commercially feasible platform for solving a wide range of optimization problems. References 1. K. Tatsumura, R. Hidaka, M. Yamasaki, Y . Sakai, H. Goto, 2020 IEEE International Sym- posium on Circuits and Systems (ISCAS) (IEEE, 2020), pp. 1–5. 162. T. Zhang, Q. Tao, J. Han, 2021 18th International SoC Design Conference (ISOCC) (IEEE, 2021), pp. 288–289. 3. M. Hasegawa, H. Ito, H. Takesue, K. Aihara, IEICE Transactions on Communications 104, 210 (2021). 4. R. Liu, X. Li, K. S. Lam, Current opinion in chemical biology 38, 117 (2017). 5. F. Barahona, Journal of Physics A: Mathematical and General 15, 3241 (1982). 6. A. Lucas, Frontiers in Physics 2(2014). 7. N. Mohseni, P. L. McMahon, T. Byrnes, Nature Reviews Physics 4, 363 (2022). 8. N. Mohseni, P. L. McMahon, T. Byrnes, Nature Reviews Physics 4, 363 (2022). 9. D. I. Albertsson, et al. ,Applied Physics Letters 118, 112404 (2021). 10. A. Houshang, et al. ,Physical Review Applied 17, 014003 (2022). 11. B. C. McGoldrick, J. Z. Sun, L. Liu, Physical Review Applied 17, 014006 (2022). 12. B. Sutton, K. Y . Camsari, B. Behin-Aein, S. Datta, Scientific Reports 7, 44370 (2017). 13. F. Cai, et al. ,Nature Electronics 3, 409 (2020). 14. M. W. Johnson, et al. ,Nature 473, 194 (2011). 15. P. I. Bunyk, et al. ,IEEE Transactions on Applied Superconductivity 24, 1 (2014). 16. S. Dutta, et al. ,2019 IEEE International Electron Devices Meeting (IEDM) (2019), pp. 37.8.1–37.8.4. ISSN: 2156-017X. 17. T. Honjo, et al. ,Science Advances 7, eabh0952 (2021). 1718. T. Inagaki, et al. ,Science 354, 603 (2016). 19. P. L. McMahon, et al. ,Science 354, 614 (2016). 20. T. Wang, L. Wu, J. Roychowdhury, Proceedings of the 56th Annual Design Automation Conference 2019 , DAC ’19 (Association for Computing Machinery, New York, NY , USA, 2019). 21. S. Patel, L. Chen, P. Canoza, S. Salahuddin, arXiv preprint arXiv:2008.04436 (2020). 22. K. Tatsumura, M. Yamasaki, H. Goto, Nature Electronics 4, 208 (2021). 23. K. Takata, et al. ,Scientific reports 6, 1 (2016). 24. Y . Yamamoto, et al. ,npj Quantum Information 3, 1 (2017). 25. Y . Haribara, S. Utsunomiya, Y . Yamamoto, A Coherent Ising Machine for MAX-CUT Prob- lems: Performance Evaluation against Semidefinite Programming and Simulated Anneal- ing(Springer Japan, Tokyo, 2016), pp. 251–262. 26. A. Marandi, Z. Wang, K. Takata, R. L. Byer, Y . Yamamoto, Nature Photonics 8, 937 (2014). 27. S. Kako, et al. ,Advanced Quantum Technologies 3, 2000045 (2020). 28. ´A. Papp, W. Porod, ´A. I. Csurgay, G. Csaba, Scientific reports 7, 1 (2017). 29. A. B. Ustinov, A. V . Drozdovskii, B. A. Kalinikos, Applied physics letters 96, 142513 (2010). 30. A. B. Ustinov, B. A. Kalinikos, Applied Physics Letters 93, 102504 (2008). 31. A. B. Ustinov, B. A. Kalinikos, E. L ¨ahderanta, Journal of Applied Physics 113, 113904 (2013). 1832. A. Sadovnikov, V . Gubanov, S. Sheshukova, Y . P. Sharaevskii, S. Nikitov, Physical Review Applied 9, 051002 (2018). 33. A. A. Grachev, A. V . Sadovnikov, S. A. Nikitov, Nanomaterials 12, 1520 (2022). 34. I. Ustinova, et al. ,Technical Physics Letters 42, 891 (2016). 35. Y . L. Etko, A. Ustinov, Technical Physics Letters 37, 1015 (2011). 36. A. Litvinenko, S. Grishin, Y . P. Sharaevskii, V . Tikhonov, S. Nikitov, Technical Physics Letters 44, 263 (2018). 37. A. Litvinenko, et al. ,Physical Review Applied 15, 034057 (2021). 38. V . V . Vitko, A. A. Nikitin, A. B. Ustinov, B. A. Kalinikos, IEEE Magnetics Letters 9, 1 (2018). 39. A. B. Ustinov, E. L ¨ahderanta, M. Inoue, B. A. Kalinikos, IEEE Magnetics Letters 10, 1 (2019). 40. A. A. Nikitin, et al. ,Applied Physics Letters 106, 102405 (2015). 41. Q. Wang, et al. ,Nature Electronics 3, 765 (2020). 42. A. Khitun, M. Bao, K. L. Wang, Journal of Physics D: Applied Physics 43, 264005 (2010). 43. A. Khitun, K. L. Wang, Journal of Applied Physics 110, 034306 (2011). 44. A. Khitun, M. Bao, K. L. Wang, IEEE Transactions on Magnetics 44, 2141 (2008). 45. C. Davies, et al. ,IEEE Transactions on Magnetics 51, 1 (2015). 46. S. Watt, M. Kostylev, A. B. Ustinov, Journal of Applied Physics 129, 044902 (2021). 1947. S. Watt, M. Kostylev, A. B. Ustinov, B. A. Kalinikos, Physical Review Applied 15, 064060 (2021). 48. M. Balinskiy, A. Khitun, AIP Advances 12, 045307 (2022). 49. Y . Khivintsev, et al. ,Journal of Magnetism and Magnetic Materials 545, 168754 (2022). 50. Z. Guo, et al. ,Proceedings of the IEEE (2021). 51. A. Serga, A. Chumak, B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010). 52. V . Cherepanov, I. Kolokolov, V . L’vov, Physics Reports 229, 81 (1993). 53. H. Glass, Proceedings of the IEEE 76, 151 (1988). 54. M. Haidar, et al. ,Journal of Applied Physics 117, 17D119 (2015). 55. A. Navabi, et al. ,Physical review applied 11, 034046 (2019). 56. B. Kalinikos, A. Slavin, Journal of Physics C: Solid State Physics 19, 7013 (1986). 57. S. Muralidhar, et al. ,Physical Review Letters 126, 037204 (2021). 58. R. Lebrun, et al. ,Phys. Rev. Lett. 115, 017201 (2015). 59. A. Litvinenko, et al. ,Physical Review Applied 16, 024048 (2021). 60. K. P. Kalinin, A. Amo, J. Bloch, N. G. Berloff, Nanophotonics 9, 4127 (2020). 61. A. B. Ustinov, N. Y . Grigor’eva, B. A. Kalinikos, JETP letters 88, 31 (2008). 62. A. B. Ustinov, B. A. Kalinikos, V . E. Demidov, S. O. Demokritov, Physical Review B 80, 052405 (2009). 2063. P. A. Kolodin, et al. ,Phys. Rev. Lett. 80, 1976 (1998). 64. M. Kostylev, B. Kalinikos, Technical Physics 45, 277 (2000). 65. A. Chumak, A. Serga, B. Hillebrands, Journal of Physics D: Applied Physics 50, 244001 (2017). 66. A. Mahmoud, et al. ,Journal of Applied Physics 128, 161101 (2020). 67. V . Tikhonov, A. Litvinenko, Applied Physics Letters 115, 072410 (2019). Author Contributions A.L. conceived the concept and designed the circuit; A.L., V .G., and A.A. performed the mea- surements and analyzed the data; A.L, R.K. performed theoretical calculations with help from A.S. and V .T.; J. ˚A. managed the project; all co-authors contributed to the manuscript, the dis- cussion, and analysis of the results. 21

2022-09-09

We demonstrate a spin-wave-based time-multiplexed Ising Machine (SWIM), implemented using a 5 $\mu$m thick Yttrium Iron Garnet (YIG) film and off-the-shelf microwave components. The artificial Ising spins consist of 34--68 ns long 3.125 GHz spinwave RF pulses with their phase binarized using a phase-sensitive microwave amplifier. Thanks to the very low spinwave group velocity, the 7 mm long YIG waveguide can host an 8-spin MAX-CUT problem and solve it in less than 4 $\mu$s while consuming only 7 $\mu$J. Using a real-time oscilloscope, we follow the temporal evolution of each spin as the SWIM minimizes its energy and find both uniform and domain-propagation-like switching of the spin state. The SWIM has the potential for substantial further miniaturization, scalability, speed, and reduced power consumption, and may become a versatile platform for commercially feasible optimization problem solvers with high performance.

A spinwave Ising machine

2209.04291v1

arXiv:0711.1574v1 [cond-mat.other] 10 Nov 2007Stability of Bose Einstein condensates of hot magnons in YIG I. S. Tupitsyn,1P. C. E. Stamp,1and A. L. Burin2 1Pacific Institute of Theoretical Physics, University of Bri tish Columbia, 6224 Agricultural Road, Vancouver, BC Canada, V6T 1Z1 2Department of Chemistry, Tulane University, New Orleans, L A 70118, USA (Dated: November 5, 2018) We investigate the stability of the recently discovered roo m temperature Bose-Einstein condensate (BEC) of magnons in Ytrrium Iron Garnet (YIG) films. We show th at magnon-magnon interactions dependstronglyontheexternalfieldorientation, andthatt heBECincurrentexperimentsisactually metastable - it only survives because of finite size effects, a nd because the BEC density is very low. On the other hand a strong field applied perpendicular to the s ample plane leads to a repulsive magnon-magnon interaction; we predict that a high-density magnon BEC can then be formed in this perpendicular field geometry. PACS numbers: In aremarkableveryrecentdiscovery[1], a Room Tem- perature Bose-Einstein condensate (BEC) of magnon ex- citations was stabilized for a period of roughly 1 µsin a thin slab of the well-known insulating magnet Yttrium IronGarnet(YIG).Thedensityofmagnonswasquitelow (n∼10−4perlatticesite), andthe density nooftheBEC was apparently unknown, but no/n≪1. This result can be understood naively in terms of a weakly-interacting dilute gas of bosons, provided that (i) one assumes that the number of magnons is conserved, so their chemical potential may be non-zero, and (ii) the interactions be- tween them are repulsive (attractive interactions favour depletion of the BEC, causing a negative compressibility and instability of the BEC [2]). In the experiment, the magnon dispersion was controlled both by the sample ge- ometry and external magnetic field, in such a way that magnon-magnon collisions conserved magnon number - the decayofthe BECwasthen attributed tospin-phonon couplings [1]. The magnonsin the experiment had rather long wavelengths, of order a µm - hitherto such excita- tions have been treated entirely classically [3]. This experiment raises a number of important ques- tions, not least of which concern the kind of superfluid properties possessed by a BEC of such unusual objects. However a much more basic question about the stability of the system must first be answered. In fact we find the rather startling result that under the conditions of the experiments reported so far, these interactions were ac- tuallyattractive , ie., the BEC ought to be unstable! We shall see that because of the particular geometry used, the BEC is actually metastable to thermally activated or tunneling decay, and that it only survives because its density is low - above a critical density, given below, it is absolutely unstable. However we also find that by chang- ingthe fieldconfigurationinthe systemonecanmakethe interactions repulsive, and the BEC should then stabilize at a much higherdensity - opening the wayto much more interesting experiments. YIG is one of the best characterized of all insulat-ing magnets [4]. It is cubic, with lattice constant ao= 12.376˚A, ordering ferrimagnetically below Tc= 560K. At room temperature the long-wavelengthproperties can be understood using a Hamiltonian with ferromagnetic exchange interactions between effective ’block spins’ Sj, one per unit cell, whose magnitude Sj=|Sj|=a3 oMs/γ, withγ=geµB, is defined by the experimental saturated magnetisation density Ms(Ms≈140Gat room temper- ature; with ge≈2, onehas Sj≈14.3),alongwithdipolar couplings between these; the resulting lattice Hamilto- nian takes the form: /hatwideH=−γ/summationdisplay iSi·Ho−Jo/summationdisplay i,δSiSi+δ +Ud/summationdisplay i/negationslash=jSiSj−3(Si·nij)(Sj·nij) |rij|3,(1) where the sums i,jare taken over lattice sites at po- sitionsRi, etc.,δdenotes nearest-neighbor spins, rij= (Ri−Rj)/ao, andnij=rij/|rij|. The nearest-neighbour dipolar interaction Ud=γ2/a3 o≈1.3×10−3K. The isotropic exchange Jois determined experimentally from J= 2SJoa2 o≈0.83×10−28erg cm2at room tem- perature. One then has Jo≈1.37K, andUd/Jo≈ 0.95×10−3. In what follows we set up a theoretical description of the BEC, taking into account the external field, dipolar and exchange interactions, and boundary conditions in the finite geometry. We evaluate the interactions and the BEC stability for 2 different field configurations; the general picture then becomes clear. (i) In-plane field : All experiments so far have had Hoin the slab plane. The combination of exchange, dipolar, and Zeeman couplings then leads to a magnon spectrum ωqshown in Fig.1, in which the competition between dipolar and exchange interactions leads to a finite-qminimum in ωqat a wave-vector Q∼1/d, where dis the slab thickness. To completely specify ωqand the inter-magnon interactions one needs boundary con-2 ditions, which can involve partial pinning of the surface spins [5, 6, 7]. Demokritov et al [1] assume free surface spins, implying that (i) |(∂M(r)/∂r)·ns|= 0 when ris at the surface; here nsis the normal to the surface, and (ii) that the allowed momenta along ˆ z(see Fig.1, inset) areq⊥=n⊥π/d, leading to different magnon branches labelled by n⊥. For now we assumea continuous in-plane momentum, and later discuss the effect of in-plane quan- tization; andweassume n⊥= 0, takingthelowest-energy magnon branch. 1x10 31x10 5110 Magnon spectrum (GHz) q (cm-1)d=2 mµ 5 mµ 10 mµX ZY dHo||X 3 5 FIG. 1: (color online). The magnon spectrum, Eq.2, for d= 2,5 and 10 µmatHo= 700Gauss. The inset shows the sample geometry. The above assumptions yield an ωqwhich agrees with experiment [1], and which for Ho∝bardblˆxtakes the form [7] /planckover2pi1ωq=/bracketleftbig (γHi+Jq2)(γHi+Jq2+/planckover2pi1ωMFq)/bracketrightbig1/2,(2) hereHi=Ho−4πNxMsis the internal field, Nxthe demagnetization factor (for a slab in the xy-plane,Nx= Ny= 0,Nz= 1),/planckover2pi1ωM= 4πγMs(for YIG, /planckover2pi1ωM≈ 0.236Kat room temperature), and /planckover2pi1ωH=γHi. The form of the dimensionless function Fqdepends on the direction of q. In the important case when qis parallel toHo, ie., along ˆ x, one has Fq→/parenleftbig 1−e−qxd/parenrightbig /(qxd), (3) In the experiment [1] magnons are argued to condense at the minima qx=Q; whend= 5µm, andHo= 700Gauss, one has |Q| ≈5.5×104cm−1. We now set up a theoretical description of the BEC, including all 4-magnon scattering processes (3-magnon scattering is excluded by the kinematics), using a gener- alized Bogoliubov quasi-average technique [8] to incor- porate the BEC. Defining magnon operators bq,b† q, a magnon BEC at q=Q, withNocondensed magnons, has quasi-averages < b±Q>=< b† ±Q>=/radicalbig N0/2, (4)corresponding to a condensate wave-function Ψ Q(y)∝ cos(Qy). More generally Ψ Q(y) is multiplied by a phase factoreiφ(r,t)//planckover2pi1, which is crucial to the BEC dynamics, but not necessary for a stability analysis of the BEC. We make a Holstein-Primakoff magnon expansion [9] up to 4th-order in magnon operators, including contribu- tions from both (2 in - 2 out) and (3 in - 1 out) magnon scattering processes [10] (we ignore multiple-scattering contributions here, which are ∼O(Ud/Jo)∼10−3rela- tive to the leading terms). Then, taking quasi-averages, we can write the Hamiltonian in the form H=/planckover2pi1/summationdisplay qωq(b† qbq+1 2)+/hatwideVp int+/hatwideV−p int (5) where the interaction term /hatwideVp inttakes the form /hatwideVp int=n0(Γ0+ΓS 4)/summationdisplay p<Q/bracketleftbig b† Q+pbQ+p+b† −Q−pb−Q−p/bracketrightbig +ΓSn0 4/summationdisplay p<Q/bracketleftbig b† Q+pb−Q+p+b† −Q+pbQ+p+ +bQ+pb−Q−p+b† Q+pb† −Q−p/bracketrightbig +Γ0n0 2/summationdisplay p<Q/bracketleftbig bQ+pbQ−p+b−Q−pb−Q+p+ +b† Q+pb† Q−p+b† −Q−pb† −Q+p/bracketrightbig .(6) Here Γ 0and Γ Sare the four-magnon scattering ampli- tudes between states ( Q,Q)→(Q,Q) and (Q,−Q)→ (Q,−Q) respectively. For the sample geometry in Fig.1, withHo∝bardblˆx, these scattering amplitudes are found to be Γ0=−/planckover2pi1ωM 8S/bracketleftbig (α1−α3)FQ−2α2(1−F2Q)/bracketrightbig −JQ2 4S/bracketleftbig α1−4α2/bracketrightbig ; (7) ΓS=/planckover2pi1ωM 2S/bracketleftbig (α1−α2)(1−F2Q)−(α1−α3)FQ/bracketrightbig +JQ2 S/bracketleftbig α1−2α2/bracketrightbig ,(8) withα1=u4 Q+ 4u2 Qv2 Q+v4 Q,α2= 2u2 Qv2 Qandα3= 3uQvQ(u2 Q+v2 Q), where {uq,vq}=/bracketleftbig (Aq±/planckover2pi1ωq)/2/planckover2pi1ωq/bracketrightbig1/2 andAqandBqare given by Aq=γHi+Jq2+2πγMsFq andBq=−πγMsFqrespectively. Higher-ordermultiple- scattering contributions to Γ 0,ΓSare∼O(Ud/Jo) rela- tive to the leading terms given here. We plot the combinations 2Γ 0±ΓSin Figures 2 and 3. Both these amplitudes are sensitive to the external field and the film thickness. The first amplitude, 2Γ 0+ ΓS, becomes negative in the entire region of fields at d < dc≈2(J//planckover2pi1ωM)1/2≈0.032µm. The second amplitude is positive at d < dc. Near the energy minimum (when |p|<<|Q|), one has ωQ+p≈ωQ−p, and the Bogoliuibov transformation is3 straightforward because His symmetric when p↔ −p andQ↔ −Q. The spectrum thus has 4 branches, with excited quasiparticle energies ǫpη=/radicalBig ΩQ(p)[ΩQ(p)+no(2Γ0+ηΓS)] (9) whereη=±1, and Ω Q(p) =/planckover2pi1(ωQ+p−ωQ). 2Γ + Γ 0 5 m d=0.01 m 0.05 m 10 m 2 m 0.00 0.02 0.04 0.06 0.08 0.10 Ho (in Tesla)-0.01 0.00 0.01 0.02 µµµµµS FIG. 2: (color online). The effective amplitude 2Γ 0+ΓS(in Kelvins) as a function of field Hoat different values of the film thickness d. Dashed lines: d= 2 and 10 µm. Solid lines: d= 0.01,0.05 and 5 µm. The inset in Fig.3 shows the ”phase diagram” of the quasi-two dimensional YIG for different values of dand Hx o. Oneimmediately seesaparadox: the amplitudesare never both positive, so bulk BEC should not exist - yet BEC has been observed [1] in samples with an in-plane fieldHx o∼0.07T, in which d∼5µm. The paradox is resolved by noting that in a finite ge- ometry, the energy gap from the condensate to excited modes can be larger than the scattering amplitude, lead- ingtoapotentialbarriertodecay. Forweakly-interacting Bosegasesthis yields [11] an upper critical number ncr oof condensate particles, above which the barrier disappears; one hasαoncr o/lo=k, whereαois the s-wave scattering length, and lothe characteristic size of the BEC wave function. The constant k∼O(1), and depends on the sample geometry. In the present case we can write the critical density ncr oas|2Γ0−ΓS|ncr o∼ǫmin p, whereǫmin pis the minimum quasiparticle energy in the presence of the BEC; below thiscriticaldensitytheBECismetastabletotunnelingor thermal activation. If the BEC were to spread through the entire slab, then ncr o|2Γ0−ΓS|= ΩQ(π/L), where the length Ldepends on the direction of the field rela- tive to the slab axes. Taking this result literally for the experiment [1], with a slab measuring 20 ×2mm2in the plane, one finds 10−8< ncr o<10−6(for fields along thelong and short sides of the slab respectively). However this result is certainly too low, since it assumes a per- fectly uniform BEC - in reality disorder and edge effects will smear the magnon spectrum and restrict the size of the BEC, and a more realistic estimate for Lis then Leff= (LxLyLz)1/3∼0.6mm. This yields ncr o∼10−5 for the experiment. Thus we conclude that for in-plane fields, it will be impossible to raise noabove this value; this could be checked experimentally (eg., by increasing the pumping rate). 2Γ − Γ 0 5 m d=0.01 m 0.05 m 10 m 2 m 0.00 0.02 0.04 0.06 0.08 0.10 Ho (in Tesla)-0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 µ µ µ µ µS 0 0.04 0.08 0123Ho (in Tesla) d (in µm)dc2Γ0+ΓS 2Γ0−ΓS FIG. 3: (color online). The effective amplitude 2Γ 0−ΓS(in Kelvins) as a function of external field Hoat different values of the film thickness d. Dashed lines: d= 2 and 10 µm. Solid line:d= 0.01,0.05 and 5 µm. The inset shows the ”phase diagram”ofYIGinthe( Ho,d)plane. Theamplitude2Γ 0+ΓS is positive everywhere to the right of the solid line (the reg ion ⊕). The amplitude 2Γ 0−ΓSis negative everywhere to the right of the dashed line (the region ⊖). (ii) Perpendicular field : Now the results are very different. Consider first an infinite thin slab, which is simple to analyze. The competition between the external fieldHo= ˆzHoand the demagnetization field (which favours in-plane magnetisation) gradually pulls the spins out of the plane; below a critical field Hc=/planckover2pi1ωM/γ≈ 1760G, the Free Energy is degenerate with respect to rotation around ˆ zand so the magnons are gapless, but atHcthey align with Hoand a gap /planckover2pi1ωo≡/planckover2pi1ωq=0= γHo−/planckover2pi1ωMopens up. The minimum in the magnon spectrum is always at q= 0 (Fig.4). The inter-magnon scattering amplitude Γ is now al- ways positive; neglecting a very small exchange contri- bution one finds Γ(q) = (/planckover2pi1ωM/4S)[1−(1−Fq)/2] →(/planckover2pi1ωM/4S)[1−q||d/4 +O(q2 /bardbl)] (10) whereq/bardblis the momentum in the xyplane, and Fqtakes the form (3) but with qx→q/bardbl. This radically changes4 the situation - now a BEC is stable with no restriction on the condensate density. There are however restric- tions on Ho; when 2200 G < Hz o<3500Gthe system has a “kinetic instability” [12], in which the pumping of the magnons at one frequency destabilizes the magnon distribution, along with strong microwave emission. 0 1000 2000 00.5 11.5 Magnon spectrum (GHz) q (cm-1)2000 G 1760 GX Z, Ho|| Z Y dd=5 mµ FIG. 4: (color online). The magnon spectrum for an infinite slab of YIG, assuming free surface spin boundary conditions , with magnetisation polarised perpendicular to the plane (s ee inset). The spectra are shown for Hz o=Hc≈1760Gwhere the spectrum is gapless, and for Hz o= 2000G. In a real finite sample things are much more compli- cated. Evenwithoutsurfaceanisotropythespinsnearthe slab edge are put out of alignment with the bulk spins by edge demagnetisation fields; and surface anisotropy does the same to spins on the slab faces. However in the cen- tral region of the sample, at distances further from the surface than the exchange length, the spectrum returns to the infinite plane form. For fields well above Hc, eg., forHz o∼2000G, all of the spins will be aligned along ˆ z, and (10) will then be valid everywhere. In this case we have the striking result that a BEC of pumped magnons should be possible with densities much higher than present. To give an upper bound is complicated since the problem then becomes essentially non-perturbative (similar to, eg., liquid4He), beyond the range of higher-order magnon expansions. However there appears to be no obstacle in principle to raising no/n∼O(1). At present the highest achievable density is probably limited by experimental pumping strengths rather than any fundamental restrictions. Such a high- density BEC existing at room temperature would be ex- tremely interesting, and certainly possess unusual mag- netic properties. Remarks : The 2 cases studied above are actually lim- iting cases of a more general situation in which one can manipulate the inter-magnon interactions by varying the field direction and strength, and vary the upper criti- cal density for BEC formation by changing the samplegeometry. Thus the analysis can be easily generalized to long ’magnetic wires’ or whiskers, and we also ex- pect that BEC will be stabilized there when the external field is perpendicular to the sample axis, but unstable or metastable when the field is parallel to the sample axis. Further details of the various possible cases will be pub- lished elsewhere. Acknowledgements : We acknowledge support by the Pacific Institute of Theoretical Physics, the Na- tional Science and Engineering Council of Canada, the CanadianInstitute forAdvancedResearch,the Louisiana Board of Regents (Contract No. LEQSF (2005-08)-RD- A-29), the Tulane University Researchand Enhancement Fund, and the US Air Force Office of Scientific Research (Award no. FA 9550-06-1-0110). We would also like to thank B. Heinrich and D. Uskov for very useful conver- sations. [1] S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A.A.Serga, B. Hillebrands, A.N.Slavin, Nature 443, 7110 (2006); O. Dzyapko, V. E. Demidov, S. O. Demokritov, G. A. Melkov, A. N. Slavin, New J. of Phys. 9, 64 (2007). [2] K Huang, Statistical Mechanics , Wiley, 1963; L. D. Lan- dau, E. M. Lifshitz, Statistical Physics , Butterworth- Heinemann, 1980. [3] L.R. Walker, in Magnetism, vol. I , ed G Rado, H Suhl, Academic, 1963; S.O. Demokritov et al., Phys. Reports 348, 441 (2001). [4] The parameters used here are taken from R. Pauthenet, Ann. Phys. (Paris) 3, 424 (1958), from M.A. Gilleo, S. Geller, Phys. Rev. 110, 73 (1958), and from M. Sparks, Ferromagnetic-relaxation theory , McGraw-Hill, New York, 1964. [5] G. Rado, J Weertman, J. Phys. Chem. Sol. 11, 315 (1959); M. Sparks, Phys. Rev. B 1, 3831, 3856, 3869 (1970). [6] M.J. Hurben, C.J. Patton, J. Mag, Mag. Mat 139, 263 (1995);ibid163, 39 (1996). [7] B.A. Kalinikos and A.N. Slavin, J. Phys. C 19, 7013 (1986); and J Phys CM2, 9861 (1990). [8] S. T. Belyaev, Sov. Phys. JETP 7, 289 (1958). [9] T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940). [10] M. Sparks, R. Loudon, C. Kittel, Phys. Rev. 122, 791 (1961); P. Pincus, M. Sparks, RC LeCraw, Phys. Rev. 124, 1015 (1961). [11] P.A. Ruprecht, M.J. Holland, K. Burnett, M. Edwards, Phys. Rev. A51, 4704 (1995); Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. Lett. 79, 2604 (1997); M. Ueda, A.J. Leggett, Phys. Rev. Lett. 80, 1576 (1998). [12] G.A. Melkov and S.V. Sholom, Sov. Phys. JETP 72(2), 341 (1991); see also G. A. Melkov, V. I. Safonov, A. Yu. Taranenko, S. V. Sholom, J. Magn. Magn. Mater. 132, 180 (1994).

2007-11-10

We investigate the stability of the recently discovered room temperature Bose-Einstein condensate (BEC) of magnons in Ytrrium Iron Garnet (YIG) films. We show that magnon-magnon interactions depend strongly on the external field orientation, and that the BEC in current experiments is actually metastable - it only survives because of finite size effects, and because the BEC density is very low. On the other hand a strong field applied perpendicular to the sample plane leads to a repulsive magnon-magnon interaction; we predict that a high-density magnon BEC can then be formed in this perpendicular field geometry.

Stability of Bose Einstein condensates of hot magnons in YIG

0711.1574v1

Bose -Einstein Condensation of Quasi-Particles by Rapid Cooling Authors: M. Schneider1, T. Br ächer1, D. Breitbach1, V. Lauer1, P. Pirro1, D. A. Bozhko2, H. Yu. Musiienko- Shmarova1, B. Heinz1,3, Q. Wang1, T. Meyer1,4, F. Heussner1, S. Keller1, E. Th. Papaioannou1, B. L ägel5, T. Löber5, C. Dubs6, A. N. Slavin7, V. S. Tiberkevich7, A. A. Serga1, B. Hillebrands1, and A. V. Chumak1,8,* The fundamental phenomenon of Bose -Einstein Condensation (BEC) has been observed in different systems of real and quasi -particles. The condensation of real particles is achieved through a major reduction in temperature while for quasi -particles a mechanism of external injection of bosons by irradiation is required. Here, we present a novel and universal approach to enable B EC of quasi -particles and to corroborate it experimentally by using magnons as the Bose -particle model system. The critical point to this approach is the introduction of a disequilibrium of magnons with the phonon bath. After heating to an elevated tempera ture, a sudden decrease in the temperature of the phonons, which is approximately instant on the time scales of the magnon system, results in a large excess of incoherent magnons. The consequent spectral redistribution of these magnons triggers the Bose -Einstein condensation. State of the art of Bose -Einstein c ondensation. Bosons are particles of integer spin that allow for the fundamental quantum effect of Bose –Einstein Condensation (BEC), which manifests itself in the formation of a macroscopic coherent state in an otherwise incoherent, thermalized many - particle system. The phenomenon of BEC was originally predict ed for an ideal gas by Albert Einstein in 1924 based on the theory developed by Satyendra Nath Bose. Nowadays, Bose -Einstein condensates are investigated experimentally in a variety of different systems , which includes real particles such as ultra- cold gas es 1,2, as well as quasi -particles with the likes of exciton – polaritons3,4, photons5,6, or magnons in quantum magnets7-9, liquid helium 3H10,11, and ferrimagnets12-15. The phenomenon can be reached by a major decrease in the system temperature or by an increase in the particle density. In order to condensate atomic gases, extremely low temperatures on the order of mK are required since the density of such gases must be very low to prevent their cohesion. In contrast, cooling of a quasi -particle system is accompanied by a decrease in its population and prevents BEC. Thus, an injection of bosons is required to reach the threshold for BEC in such systems. Since quasi -particle systems allow for high densities of bosons, BEC by quasi -particle injection is possible even at room temperature. Prominent examples are the injection of exciton -polaritons by a laser 3,4 and of magnons by either microwave parametric pumping12–16 or due to the spin Seebeck effect induced spin currents17. Concept of BEC by rapid c ooling. In this letter, we propose and demonstrate experimentally a different and universal way to achieve BEC of quasi -particles. In a solid body, each quasi -particle system interacts with the phonon bath and these two systems stay in equilibrium in a quasi -stationary state. An instant reduction in the phonon temperature results in a disequilibrium between these systems and in a large excess of quasi -particles when compared to the equilibrium state at the new temperature. In contrast to all previous studies3-6,12-15, this kind of “ injection ” is a-priori incoherent , populates the entire energy spectrum rather than a narrow spectral range, and, consequently, excludes the necessit y of an initial thermaliz ation of the injected quasi -particles. The consequent redistribution of the quasi -particles due to multi- particle and particle -phonon sc attering processes results in the increase in the chemical potential µ required for the BEC. 2 The energy -distribution n (hf, µ, T) of the density of bosons depends on temperature T and is described by the Bose -Einstein distribution function multiplied with their energy -dependent density of states D (hf): ()() ( )() B,µ , µ( )exp 1()D hfn hf t T t hf t kTt=−− , (1) where hf is the energy of the quasi -particle, f is its frequency, t is the elapsed time, T(t) is the phonon or lattice temperature, µ( t) is the chemical potential, h is Planck’s constant, and Bk is Boltzmann’s constant. In the work presente d, BEC is studied experimentally using a magnon system. For their description, the approximated density of states for ferromagnetic exchange magnons is used: ()min D hf f f∝− , where minf is the frequency at the bottom of the magnon spectrum. The dashed blue lines in all panels of Fig. 1b show the steady -state density distributions when the magnon and the phonon systems are in equilibrium at room temperature T = TRoom for µ = 0. In order to achieve a BEC of magnons , the chemical potential µ has to be increased up to the value of the minimum magnon energy hf min. In this case, Equation (1) diverges, reflecting the condensation of the quasi particles into the same quantum state at the smalles t energy of the system. The distribution of magnon density in the particular case of µ = hfmin at room temperature T = TRoom is shown in Fig. 1b by the dashed green lines. Fig. 1 | Theoretical modelling of the BEC by rapid cooling and experimental approach. a, and b, show schematic depiction of the condensation process. a, shows the time evolution of the temperature of the phonon system (top panel), the quasi -particle chemical potential (middle panel) and the quasi - particle population at the lowest energy state (bottom panel). The time of the instant cooling is marked by tOff. The inset shows the simulated magnon spectrum (white -to-red linear scale) for the YIG structure depicted in c, and magnetized along its long axis by an external field of µ0𝐻𝐻ext=188 mT. b, shows the quasi -particle density as a function of frequency. The red lines in the different panels show the densities for the different times marked in a . Dashed blue lines: Steady -state room - temperature distributions. Dashed green lines: Room -temperature distribution with µ = hfmin. c, Structure under investigation and sketch of the experimental setup. 3 Our approach to provide conditions for the BEC is as follows. First, the temperature of the phonon system is raised to a certain critical value. This rise in temperature increases the number of phonons and magnons simultaneously (see point “1” in Fig. 1a ). The magnon density in the heated equilibrium state is given by Eq. 1 at the increased temperature and µ = 0 is depicted in the Panel “1” in Fig. 1b by the solid red line. The proposed mechanism for BEC is based on the rapid cooling of the phonon system as it is shown in Fig. 1a . As the lattice temperature drops to Tmin, a disequilibrium between the magnon and phonon systems results in an excess of magnons over the whole spectrum with respect to the equilibrated state. The consequent magnon distribution (see Panel “2” in Fig. 1 b) is calculated following the dynamic rate equations given in the Methods section . Two mechanisms are of highest i mportance for this magno n redistribution. The first one, is the phonon- magnon coupling that transfers energy from the magnon to the phonon system. With the assumption of the simplest case of viscous Gilbert damping18, the rate of decay of the magnons is proportional to their energy. At the same time, mag non-magnon scattering processes18, i.e., the interactions within the magnon system, allow for a redistribution of magnons towards low energies. Consequently, the density of low -energy magnons increases while the high- energy magnons dissipate. This leads to a decrease in the temperature and to a simultaneous increase in the chemical potential µ of magnons over time – see middle panel in Fig. 1 a. If the critical number of magnons in the heated state is reached, the chemical potential will rise up to the minimum magnon energy hfmin and BEC of magnons occurs (see Panels “2” and “3” in Fig. 1b) . This is manifested in the form of a sharp increase in the magnon density at the lowest energy state shown in the bottom panel of Fig. 1 a. The simulated magnon spectrum (see Methods ) is shown in the inset of Fig. 1a and the lowest magnon energy state is indicated. Ultimately, the magnon system returns to room -temperature equilibrium (see point “4” in Figs. 1a and Panel “4” in Fig. 1b) due to scattering into the phonon system, which is responsible for the finite magnon lifetime. For simplicity, our model only considers the case of quasi -particle conserving four magnon scattering processes (see Methods) but, other mechanisms like Cherenkov radiation will also contribute to the magnon redistribution towards low energies18,19 and can, thus, significantly enhance the condensation efficiency. The proposed mechanism can be used universally for all bosonic quasi -particles showing internal inelastic scattering mechanisms and that are interacting with the phonon system in a way that the relaxation rate increases with the increase in the quasi -particle energy. The key experimental challenge, which has inhibited its realization so far, i s the achievement of a sufficiently high cooling rate. Recent progress in the creation of (magnetic) nano -structures enabled quasi -instant cooling. In these structures, the phonon transport from a heated nano- structure to the quasi -bulk surroundings ensure s a cooling time shorter than the characteristic magnon scattering time and the characteristic interaction time of the quasi- particles with the phonon bath. Experimental o bservation and proof of BEC by rapid c ooling. In our study, as model system, we investigate magnons at room temperature in an Yttrium Iron Garnet (YIG)20 – Platinum nano- strip (width 500 nm, length 5 µm, YIG thickness 70 nm, Pt thickness 7 nm (see Fig. 1c). A biasing magnetic field sufficiently large to magnetize the strip along the field direction is applied either in-plane parallel ( B||x) or perpendicular ( B||y) to the long axis of the strip – for the exact geometry, see Fig. 1 c. Ti/Au leads have been deposited on top of the strip to apply electric current pulses to the Pt layer, which heat up the YIG/Pt structure due to Joule heating. Switching off the pulse results in a fast cooling with rates on the order of tens of Kelvin per nanosecond. 4 Fig. 2 | BLS Measurements of the BEC by rapid cooling. a, Depicts the mea sured BLS spectrum measured as a function of time for B ||x. The BLS signal (color -coded, log scale) is proportional to the density of magnons. The vertical dashed lines indicate the start and end of the pulse. Switching off the pulse results in the formation of a pronounced magnon signal at the bottom of the magnon band. b, Normalized BLS intensity integrated from 4.95 GHz to 8.1 GHz as a function of time (black line), frequency of the fundamental mode (blue dots) and measured chemical potential (red dots). The BEC of magnons manifests itself as the peak in the BL S intensity that is proportional to the magnon density. Moreover, it is clearly visible that the chemical potential reaches the frequency of the fundamental mode. The inset shows the sample geometry and the coordinate system used. C, The magnon intensity a t the bottom of the spectrum (analogue to panel b but with and B ||y). The pink dashed line shows the magnon population at the bottom of the spectrum calculated using the dynamic rate equations discussed in the Methods Section with τFall = 1 ns. D, The same experiment with increased fall times of 50 ns (blue line). Here, the BEC is strongly suppressed. The pink dashed line shows the calculated magnon density for τ Fall = 50 ns. For this slow cooling, the chemical potential reaches a maximal value of only 0.58× fmin. e, The same experiment with increased fall times of 100 ns (red line) and theory for τFall = 100 ns (pink line). See the inset for the time profiles of the pulses. 5 This fast cooling is provided by the phonon transport to the quasi -bulk Gadolinium Gallium Garnet (GGG) substrate and to the Ti/Au leads. In order to measure the time evolution of the magnon density in the YIG strip, micro -focused time -resolved Brillouin Light Scattering (BLS) spectroscopy21 is used (see Met hods ). Figure 2a shows the measured magnon spectrum as a function of time when the voltage pulse with a duration of P120 nsτ= and a rise and fall time of Rise Fall 1ns τ= τ= is applied. The beginning and the end of the applied pulse are indicated by the vertical dashed lines. The amplitude of the voltage pulse was U = 0.9 V, corresponding to an estimated current density of 7.8×1011 A/m2 in the Pt overlayer. The in- plane biasing magnetic field of µ0𝐻𝐻ext=188 mT is applied along the strip. The thermal population of two magnon modes is visible before the current pulse is applied. The magnon mode at f FM = 7.5 GHz has the smallest frequency and is the fundamental mode. The magnon mode of frequency f 1st mode = 14.9 GHz is the first standing thickness mode18. Once the current pulse is switched on at the time t = 0 ns, the frequencies of both modes decrease with time due to heating and the consequent decrease in the saturation magnetization M s and the exchange stiffness Dex of YIG22. Another observable effect is the decrease of the BLS intensity during the current pulse that is due to a temperature- dependent decrease in the sensitivity of the BLS23. One can clearly see in Fig. 2 a that switching off the current pulse at the time Pt=τ results in the formation of a pronounced magnon signal at the frequency of the fundamental mode , which corresponds to the bottom of the magnon spectrum. This is the characteristic fingerprint of the process of BEC of ma gnons described by the proposed theoretical model. The signal maximum is reached approximately 10 ns after the current pulse is switched off. Afterwards, the BLS intensity decreases exponentially, corresponding to a magnon lifetime of 21 ± 4 ns. In addition, the recovery of the frequency of both modes is observed. The black line in Fig. 2b presents the time evolution of the BLS intensity integrated over the frequency range at the bottom of the magnon spectrum. A clear peak in the magnon density is in agreem ent with the result of the simplified theoretical model in the bottom panel of Fig. 1a . The criterion of a BEC is that the chemical potential reaches the minimal energy of the quasi - particles . Our experiment allows for the direct measurement of the temporal evolution of the chemical potential µ/ℎ since the BLS intensity is proportional to the magnon density and the density of the first standing mode can be used as a reference (see Methods ). The red dots in Fig. 2 b show s the measured chemical potential in GHz units . One can clearly see that the potential strongly increases after the current pulse is switched off and reaches the frequency of the fundamental mode defin ing the minimal magnon energy. This experimental finding confirms the BEC of magnons directly. It is also visible that the thermal equilibrium between the magnon and phonon bathes in the nanostructured YIG film is recovered on a time scale of about 100 –150 ns, when the chemical potenti al of the magnon system recovers to its zero value. This time scale is consistent with the BEC process since it is sufficiently long for the BEC to be formed. It is remarkable that the temporal behaviors of the experimentally determined chemical potential agrees both qualitatively and quantitatively with the results of our theoretical calculations presented in the Extended Data Fig. 1. Please note that a weak increase in the chemical potential is also observed during the current pulse. It can be attributed to the spin Seebeck effect (SSE), as reported recently 17,24,25. From the simulated temperature gradient ( to 3.5 × 108 K/m, see Supplement) and with the approximation that the spin -diffusion length is determined by the thickness of the YIG layer, the calculated26 initial increase in the chemical potential caused by the spin Seebeck effect is of µ/h = 0.7f min = 6 5.25 GHz. As the YIG sample heats up from room temperature to the threshold temperature of 440 K (see, Extended Data Fig. 5), this value, w hich is still not enough to reach the BEC in our experiment, rapidly decreases due to the tenfold reduction in the SSE magnitude (see Fig. 3c in Ref. [ 27]). As a result, no increase in the low -energy magnon density is observed during the application of the heating electric current even in the continuous regime. Thus, we can state that in contrast to the SSE -induced magnon BEC measured at low temperatures17, where the maximal temperature of the YIG -layer was near room temperature, the thermally induced spin injection from the Pt layer plays only a minor role in our experiments. This statement is directly confirmed by our observations of magnon accumulation at f min in an experiment with test structures containing an Al/Au heater instead of the Pt layer (see Su pplement). In addition, the same experiment was performed when the field was applied perpendicular to the long axis of the strip ( B||y, see Supplement ) in different directions . No qualitative difference between the results was observed excluding the spin Hall e ffect induced s pin transfer torque28-30 as the mechanism responsible for the experimental findings . To verify the crucial role of the fast cooling for the BEC, additional measurements in which the current pulses were switched off with longer fall times of 50 ns and 100 ns were performed . For these measurements, the pulse duration was increased to P300 nsτ= . As can be seen from Fig. 2 c, the increased pulse duration does not inhibit the BEC. The influence of the fall time is shown in Fig s. 2d, and 2e . It is evident that the condensate disappears for long fall times, i.e ., slow cooling rates. The maximal temperature approaches a value of max sim 491K T= at the end of the current pulse according to simulations performed with COMSOL Multiphysics (see Supplement ). The maximal cooling rates of the phonon system ()max/Tt∂∂ obtained from these simulations are also indicated in the Figures. The values clearly demonstrate that the cooling rate of 2 K/ns is already too slow to trigger the magnon BEC, while a rate of 20.5 K/ns is still fast enough. Moreover, the simulations show that the cooling rate is not constant and decreases with time. In order to model this, the simple approach of an exponential decay ()Fall exp /Tt∝ −τ of the temperature was used in the theoretical model instead of the step function discussed in F ig. 1. The theoretical results are presented in Fig. 2 c-e and support the experimental findings – the BEC is observed only in the first case of Fall 1nsτ= . Please note that the theoretical calculations are normalized to the intensity level of the thermal waves in equilibrium. Thus, also the generated total number of magnons is in good agreement with the predictions from the model. Fig. 3 | Threshold behaviour of the BEC. a, Measured intensity of magnons at the bottom of the spectrum (color -coded) as a function of time and applied voltage ( B||y). The applied current pulse is marked by the vertical dashed lines. b, Maximal magnon intensity as a function of the applied voltage extracted from a. A clear threshold is visible at 1.2 V, independent of the current polarity. 7 The BEC of quasi -particles is a threshold -like phenomenon that takes place when the chemical potential μ reaches the minimal energy level hf min. This implies that in this experiment, a certain threshold temperature needs to be exceeded which corresponds to a certain critical magnon density in the system. Therefore, for a given pulse duration Pτ, a certain voltage needs to be applied to reach this critical density. The clear threshold -like behaviour is visible in Fig. 3a , in which the frequency -integrated BLS intensity (color -coded) is plotted as a function of time and voltage for P40 nsτ= . The extracted maximum magnon intensity is depicted in Fig. 3 b as a function of the applied voltage. It can be clearly seen that the application of voltage pulses with amplitudes smaller than ± 1.2 V does not result in BEC, since the critical magnon density is not reached. The threshold magnon density is exceeded and the magnon condensate is formed only for larger voltages. Further details on the threshold nature can be found in the Supplement . In general, the threshold behavior in Fig. 3 is one of the main three indicators of the BEC presented in our work. T ogether with the observation of spontaneous condensation of magnons at the lowest energy state (Fig. 2a) and with the experimentally determined increase in the chemical potential up to the energy of this state (Fig. 2b) , it allows us to confidently declare the creation of the magnon Bose -Einstein condensate in our experiment. Conclusions . In conclusion, a novel way to create a magnon Bose -Einstein condensate by rapid cooling in an individual magnetic nano- structure has been proposed and verified experimentally. BEC of magnons has been achieved in a typical spintronic structure by the application of mere current pulses with powers of a few mW rather than by the usage of (intricate) high -power microwave pumping. This paves the way for the usage of macroscopic quantum magnon states in conventional spintronics and to on- chip solid- state quantum computing. We would like to stress that the observed way to BEC is genuine to any solid- state quasi -particles in exchange with the phonon bath. The injection mechanism is originally incoherent , which is in direct opposition to laser or microwave irradiation and can be applied to other bosonic systems such as exciton-polarito ns and photons in cavities. 8 Methods Theoretical model of the magnon spectral redistribution process . The spectral redistribution of a magnon gas in response to rapid changes of the lattice temperature can be modelled by the following model. For simplicity, we assume that the magnon distribution remains isotropic and depends only on the magnon frequency f , i.e. it is described by the distribution function n (f,t). In the quasi -equilibrium state with lattice temperature T L (t) and chemical potential µ (t), the distribution function is given by Eq. (1) in the manuscript. The main reason we may approximate the real quantized spectrum as a continuous one is that, in our experiments and simulations, the magnon subsystem is heated and cooled by thermal contact with lattice and this process involves magnons in the whole thermal range – up to several THz. The frequency bandwidth of such magnons (which are primarily responsible for the nonlinear thermalization in the magnon subsystem) is much larger than the frequency distance between quantized magnon modes and, therefore, the magnon spectrum can be safely approximat ed as a continuum. Numerical calculations were performed on a uniform frequency grid: min kf f fk= +∆, (2) where k is the step number and Δf is the grid cell size. Now, instead of the continuous distribution function n (f,t) we obtain a discrete set of magnon population numbers: ()(),.kkn t nft= The dynamics of the population numbers n k (t) can be described by the following set of equations: ()()0,k k kk k kdnD n nt Ndt+Γ − = (3) where Dk = D (fk)is the density of states for the k -th magnon level ( Dknk is the density of magnons in the frequency range f k ± Δf/2). The density of states is determined by the dispersion law of quasi - particles and the dimensionality of space. In our case, for a quadratic dispersion in the 3D case31: ()min kk kDf ff f= ∆− (4) For the lowest state k = 0, D0 = 1/V was taken, where V is the sample’s volume. Γk = Γ(fk) in Eq. (3) is the spin- lattice relaxation time, for which one can use the simple “Gilbert” approximation of viscous damping32 𝛤𝛤(𝑓𝑓)=2𝛼𝛼𝐺𝐺𝑓𝑓. (5) nk0 (t) = n0 (fk,t) in Eq. (3) is the instantaneous equilibrium magnon population, which is determined by the instantaneous lattice temperature T L (t) () ()( )0 eq L , , ,µ 0 . n ft n fT t= = (6) The right -hand side of Eq. ( 3) Nk describes magnon transitions due to nonlinear four -magnon processes18. We consider the simplest model of nonlinear magnon scattering, in which only the scattering of magnons with similar energies was taken into account. Namely, we considered only the scattering processes of the form 11 .kk k kff f f+−+↔ + (7) 9 The net rate of such processes is ()()()2 2 1 1 111 1 1,k kkk k kk kF Cnn n nn n+ − +−= + +− + (8) where 3 min mink kkffC Cf ff−= − ∆. Here the constant C is a fitting parameter, which defines the efficiency of the four magnon scattering processes. The four magnon term N k in Eq. ( 3) has the form 112.k k kkN F FF+−=−+ (9) This constitutes a closed system of equations for the determination of the dynamics of nk (t) for a given lattice temperature temporal profile T L (t). The dynamics of the magnon population number n k (t) is obtained by solving Eq. ( 3) numerically with the parameters shown in Extended Data Table 1, t he results of these numerical calculations are shown in Extended Data Fig. 1. The lattice temperature temporal profile TL (t) is assumed to be a step -like function, which changes from an elevated temperature T1 to room temperature T 0 in the moment of time tOff = 0 as is shown in the upper panel in Extended Data Fig. 1a. The simulated magnon densities D knk (t) are exemplarily plotted for four characteristic times in Extended Data Fig. 1b. The calculated magnon distributions n k (t), are fitted with the Bose -Einstein distribution function in order to obtain the corresponding chemical potential µ( t). The proced ure of this fitting is as follows. First, we linearize the Bose -Einstein distribution Eq. (1) : () ()() ()B eff B effµ 1ln 1 .,t hf n f t kT t kT t+= − (10) Consequently, the right -hand -side of Eq. ( 10) is a linear function of energy (frequency). We fit the low-energy area (below 50 GHz) of the magnon distribution. The fit converges at every time point, indicating that the low -energy magnons are in equilibrium at every instance of time. From fitting the simulated distributions at each moment of time, we obtain the time dependence of the chemical potential µ( t). The resulting dependence is shown in the middle panel in Extended Data Fig. 1a . In order to compare to the experimental data (see Fig. 2 in the main manuscript), we perform an integration of the magnon density Dknk (t) over the same frequency range as in the experiment. The corresponding plot is shown in Extended Data Fig. 1a . Fabrication of the YIG/Pt nano -structures under investigation. The investigated YIG/Pt strips are 500 nm (Fig.1, Fig. 2a -b and Fig. 3) and 1000 nm (Fig. 2c -e) wide and 5 µm long. The distance between the leads is 4 μm. The structures were fabricated using a YIG film of 70 nm thickness grown by means of Liquid Phase Epitaxy (LPE) on a (111) oriented Gadolinium Gallium Garnet (GGG) substrate20. Standard micr owave- based ferromagnetic resonance (FMR) measurements yield a Gilbert damping parameter of α YIG = 1.8×10−4 for the bare out -of-plane magnetized YIG film with an inhomogeneous broadening of μ 0H0 = 0.1 mT, and an effective saturation magnetization of M S = 123 kA/m. Next, plasma -assisted cleaning was used to remove potential contaminations before the sample was transferred into a Molecular Beam Epitaxy (MBE) facility, where the sample was heated up to 200°C for 2 hours at a pressure of p = 3.7 × 10-9 mbar bef ore a 7 nm thick Pt layer was deposited on top of the YIG film33. The deposited Pt layer was found to 10 increase the Gilbert damping to α YIG/Pt = 1.2 × 10−3. The subsequent structuring of the YIG/Pt strips was achieved by Electron -Beam Lithography and Argon- Ion Milling34 to a depth of approximately 40 nm. Afterwards the 150 nm thick gold contacts were structured by Electron -Beam Lithography and Electron -Beam evaporation, with a 10 nm thick Titanium seed layer. Then a focused Ion- Beam was used to remove the YI G adjacent to the structures and to polish the edges, resulting in a material removal to a depth of 300 nm. The resistance of the investigated devices is typically in the range of 600 Ohm for a width of 500 nm and in the range of 400 Ohm for a width of 1000 nm. Indicators of magnon condensate formation. F ormation of a spontaneous coherency caused by a thermodynamic phase transition in a bosonic system is considered13,35 to be the most prominent property of a BEC of quasiparticles. It relates to the fact that “experimentally, with an energy distribution alone, it is difficult to tell whether particles really are in the same quantum state, or just close to it.” 34 Unfortuna tely, in our experiment, the coherency of a magnon BEC cannot be proven directly due to the limited frequency resolution of the used BLS setup. Even the increased frequency resolution in a special BLS experiment in Ref. 12 was not high enough to prove the coherency directly. Nevertheless , using the micro -focused space -resolved BLS spectroscopy , the coherency of a magnon BEC has been proven by the demonstration of a stable standing wave pattern 36. As well, the direct measurement of the coherency of a magnon BEC has been done for the SSE -mediated BEC using specialized microwave technique17. In addition, such phenomena as quantized vorticity36 or Bogoliubov waves37, being canonical features of both atomic and quasiparticle quantum condensates, can be recognized as indicators38,39 of the magnon BEC formation. All these observations evidence the ability for the room -temperature magnon BEC in magnetic films12,36 ,37 and nanowires17. At the same time, the proof of a BEC can be performed using its other general featur es. In our studies, we decided for the set of three indicators for the BEC, which directly follow ed from the Bose -Einstein distribution function given by Eq. (1). These are the following: (i) a threshold behavior, (ii) the spontaneous population of the low est energy level, and (iii) the increase of the chemical potential up to the energy of this state. We show that all these three signs of condensate formation are present in our case. Clear threshold behaviours of BEC is shown in Fig. 3. Similar to Ref. 12, where the magnon BEC has been reported for the first time, here, a magnon accumulation at the lowest energy state within the frequency resolution of our setup is observed (Fig. 2a). Additionally, the extracted chemical potential does not only rea ch the minimum energy of the system (Fig. 2b) but fits perfectly to the presented theoretical predictions and a clear threshold behaviour is observed (Fig. 3). Measurements of the magnon densi ty by time -resolved BLS spectroscopy . In order to apply curren t pulses to the Pt nano- strip, a pulse generator ( Keysight 81160 A ) providing transition times down to 1 ns is connected to the sample using RF probes ( picoprobes ). For the BLS measurements 21, a laser beam with a wavelength of 457 nm and a power of 1.5 mW is focused onto the center of the structures. To be able to probe the ultra -thin YIG film covered by the reflective metal layer (Pt), the p robing light is directed to the YIG film from the opposite sample side through the transparent Gallium Gadolinium Gar net (GGG) substrate. The laser -spot size is approximately 400 nm in diameter. The experiment was performed with a repetition rate of 1 µs. The intensity of the frequency -shifted and backscattered light is proportional to the magnon density21. The maximal magnon wave vector, which can be detected is given by the laser wavelength and the numerical aperture of the objective N A = 0.85. In our setup, magnons with an in -plane wave vector up to 11 23.3 rad/µm can be detected. The frequency shifted light passes a thr ee-pass tandem Fabry -Pérot interferometer and is detected frequency selectively by a single photon counting module with a resolution of Δ fBLS = 150 MHz. So broad frequency band does not allow direct measurement of the BEC coherency35 like it was done in Re f. 17 by means of electrical detection of the SSE -induced BEC but is very suitable for the observations of transient magnon dynamics reported here. The output of the optical detector is connected to a fast data acquisition module , which is synchronized with the applied pulses. The time resolution is limited by the time the scattered photons stay in the interferometer, which is determined by the finesse of the interferometer and the distance between the mirrors (d = 5 mm in the experiment), resulting in a time resolution of approximately Δ t = 2 ns. An in -plane biasing field of B ext = 188 mT is applied, which is sufficiently large to magnetize the strips in the direction of the applied field. The orientation of the external field with respect to the strip is changed by turning the sample by 90 degrees. Measurements of magnon chemical potential by BLS spectroscopy . The condition for the quasi -particle BEC is that the chemical potential is approaching the minimal energy of these quasi- particles. To subtract the chemical potential directly from our experiment, the following procedure was used. For simplicity, the chemical potential µ(𝑡𝑡) ℎ in units of frequency is calculated from Eq. (1) in the approach 𝑘𝑘B𝑇𝑇≫ℎ𝑓𝑓−µ (corresponding to the Rayleigh -Jeans limit). Hence, the number of magnons 𝑛𝑛𝑖𝑖 occupying a certain mode of the frequency 𝑓𝑓𝑖𝑖 is given by 𝑛𝑛𝑖𝑖=𝐷𝐷𝑘𝑘B𝑇𝑇 ℎ𝑓𝑓𝑖𝑖−µ. Since both, the first standing thickness mode and the fundamental mode obey the Bose -Einstein di stribution, the chemical potential can be expressed as µ(𝑡𝑡) ℎ=𝑛𝑛1st mode (𝑡𝑡)𝑓𝑓1st mode (𝑡𝑡)−𝑛𝑛FM(𝑡𝑡)𝑓𝑓FM(𝑡𝑡) 𝑛𝑛1st mode (𝑡𝑡)−𝑛𝑛FM(𝑡𝑡). (11) Thereby µ(𝑡𝑡) ℎ becomes a function of the number of magnons in both modes 𝑛𝑛1st mode , 𝑛𝑛FM and of the corresponding frequencies 𝑓𝑓1st mode , 𝑓𝑓FM. For the case of thermal equilibrium, the chemical potential is zero and therefore, the number of magnons in one mode is given by 𝑛𝑛𝑖𝑖= 𝐷𝐷𝑘𝑘B𝑇𝑇 ℎ𝑓𝑓𝑖𝑖=𝑏𝑏𝑖𝑖𝐼𝐼𝑖𝑖. Here, 𝐼𝐼𝑖𝑖 are the measured integrated BLS intensities and the factors 𝑏𝑏 𝑖𝑖 expre ss the BLS sensitivities for different modes. Thus, the factors 𝑏𝑏𝑖𝑖 can be calculated from the thermal BLS intensities ( 𝑡𝑡>300 ns in Fig. 2 a, 2b) corresponding to a magnon system in equilibrium. Both, the measured BLS frequency of the fundamental mode 𝑓𝑓FM and the chemical potential obtained experimental ly with help of Eq. 11 are shown in Fig. 2b. In addition, the black line in Fig. 2 b presents the time evolution of the BLS intensity integrated over the frequency range at the bottom of the magnon spectrum . As can be seen from Fig. 2 b, a good agreement with the simplified theoretical model in the middle panel of Fig. 1 a is visible. The BLS intensity at the bottom of the magnon spectrum increases 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 chemical potential and the minimum energy of the system – is fulfilled. Please note that an increase in the chemical potential is also observed during the pulse. This increase can be attribut ed to the s pin Seebeck effect 17,24,25. As it is reported by I. Barsukov 12 and I. N. Krivorotov at a variety of international conferences, the spin Seebeck effect serv s as the main mechanism of magnon BEC in their experiments17. However, in our experiments, t he thermally induced spin injection from the Pt layer is not sufficient to reach the BEC condition. Even though the experiment was performed at various voltages and pulse durations, we observed the BEC always after the pulse, i.e., at times at which the gr adient has already disappeared, and never during the pulse. The same experiment in the continuous regime also did not show any magnon generation in the lowest or any other observable magnon state. This can be related Therefore, in our experiment, the s pin Seebeck effect can be seen just as a supportive mechanism for the BEC while the key mechanism is rapid cooling. 13 References : 1. Anderson, M. H. , Ensher, J. R., Matthews, M. R., Wieman, C. E. & Cornell, E. A. Observation of Bose –Einstein condensation in a dilute atomic vapor. Science 269, 198– 201 (1995). 2. Santra, B. et al. Measuring finite -range phase coherence in an optical lattice using Talbot interferometry. Nat. Commun. 8, 15601 (2017). 3. Kasprzak, J. et al. Bose –Einstein condensation of exciton polaritons. Nature 443, 409–414 (2006). 4. Lerario , G. et al. Room -temperature superfluidity in a polariton condensate. Nat. Phys. 13, 837–841 (2017). 5. Klaers, J., Schmitt, J., Vewinger, F. & Weitz, M. Bose –Einstein condensation of photons in an optical microcavity . Nature 468, 545–548 (2010). 6. Damm, T., et al. Calorimetry of a Bose –Einstein -condensed photon gas. Nat. Commun. 7, 11340 (2016). 7. Nikuni, T., Oshikawa M., Oosawa, A. & Tanaka, H. Bose –Einstein condensation of dilute magnons in TlCuCl 3. Phys. Rev. Lett. 84, 5868–5871 (2000). 8. Yin, L., Xia, J. S., Zapf, V. S., Sullivan, N. S. & Paduan -Filho, A. Direct measurement of the Bose -Einstein condensation universality class in NiCl 2-4SC(NH 2)2 at ultralow temperatures . Phys. Rev. Let. 101, 187205 (2008). 9. Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose -Einstein condensation in magnetic insulators . Nature Phys . 4, 198 (2008). 10. Borovik- Romanov, A. S. , Bun'kov, Yu. M., Dmitriev, V. V. & Mukharskiǐ, Yu. M. Long - lived induction signal in superfluid 3He-B. JETP Lett . 40, 1033–1037 (1984). 11. Bunkov, Yu. M. & Volovik, G. E. Magnon condensation into a Q ball in 3He-B. Phys. Rev. Lett. 98, 265302 (2007). 12. S. O. Demokritov, et al. Bose –Einstein condensation of quasi -equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006). 13. Rezende, S. M. Theory of coherence in Bose -Einstein condensation phenomena in a microwave -driven interacting magnon gas. Phys. Rev. B 79, 174411 (2009). 14. Serga A. A. et al. Bose –Einstein condensation in an ultra -hot gas of pumped magnons. Nat. Commun. 5, 3452 (2014). 15. Bozhko, D. A et al. Supercurrent in a room temperature Bose -Einstein magnon condensate. Nature Phys . 12, 1057–1062 (2016). 16. Brächer, T., Pirro & P., Hillebrands, B. Parallel pumping for magnon spintronics: Amplification and manipulation of magnon spin currents on the micron- scale. Phys. Rep. 699, 1–34 (2017). 17. Safranski C. et al. Spin caloritronic nano- oscillator. Nat. Commun. 8, 117 (2017). 18. Gurevich, A. G. & Melkov, G. A. Magnetization oscillations and waves (CRC, 1996). 19. Hüser, J. Kinetic theory of ma gnon Bose -Einstein condensation, PhD thesis, Westfälische Wilhelms -Universität Münster, Münster, Germany (2016). 20. Dubs, C. et al. Sub- micrometer yttrium iron garnet LPE films with low ferromagnetic resonance losses . J. Phys. D: Appl. Phys. 50, 204005 (2017). 14 21. Sebastian T., Schultheiss, K., Obry, B., Hillebrands, B. & Schultheiss, H. Micro -focused Brillouin light scattering: ima ging spin waves at the nanoscale. Front. Phys. 3, 35 (2015). 22. Cherepanov, V., Kolokolov, I. & L'vov, V. The saga of YIG. Phys. Rep. 229, 81–144 (1993). 23. Olsson, K. S. et al. Temperature- dependent Brillouin light scattering spectra of magnons in yttrium iron garnet and permalloy. Phys. Rev. B 96, 024448 (2017). 24. Bauer, G. E. W., Saitoh, E. & van Wees, B. J. Spin caloritronics. Nature Mater. 11, 391– 399 (2012). 25. Bender, S. A. & Tserkovnyak, Y. Thermally driven spin torques in layered magnetic insulators. Phys. Rev. B 93, 064418 (2016). 26. Tserkovnyak, Y., Bender, S. A., Duine, R. A. & Flebus, B. Bose -Einstein condensation of magnons pumped by the bulk spin Seebeck effect . Phys. Rev. B 93, 100402 (2016) . 27. Uchida, K., Kikkawa, T., Miura, A., Shiomi, J. & Saitoh, E. Quantitative temperature dependence of longitudinal spin Seebeck effect at high temperatures. Phys. Rev. X 4, 041023 (2014). 28. Demidov, V. E. et al. Magnetization oscillations and waves driven by pure spin currents. Phys. Rep. 673, 1 (2017). 29. Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453 (2015). 30. Cornelissen, L. J., Liu, J., Duine, R. A., Ben Youssef, J. & van Wees, B. J. Long -distance transport of magnon spin information in a magnetic insulator at room temperature, Nat. Phys. 11, 1022 (2015). 31. Kittel, C. Introduction to Solid State Physics (Wiley, 2005). 32. Stancil, D. D. & Prabhakar, A. Spin Waves: Theory and applications (Springer, 2009). 33. Jungfleisch, M. B., Lauer, Neb, V. R., Chumak, A. V. & Hillebrands, B. Improvement of the yttrium iron garnet/platinum interface for spin pumping -based applications. Appl. Phys. Lett. 103, 022411 (2013). 34. Pirro , P. et al. Spin- wave excitation and propagation in microstructured waveguides of yttrium iron garnet /Pt bilayers . Appl . Phys . Lett . 104, 012402 (2014). 35. Snoke, D. Coherent questions. Nature 443, 403–404 (2006). 36. Nowik -Boltyk, P., Dzyapko, O., Demidov, V. E., Berloff, N. G. & Demokritov, S. O. Spatially non- uniform ground state and quantized vortices in a two- component Bose - Einstein condensate of magnons. Sci. Rep. 2, 482 (2012). 37. Bozhko, D. A. et al. Bogoliubov waves and distant transport of magnon condensate at room temperatur e. Nat. Commun. 10, 2460 (2019). 38. Snoke, D. Polariton condensates: A feature rather than a bug. Nat. Phys. 4, 673 –673 (2008). 39. Eisenstein, J. P. & MacDonald, A. H. Bose –Einstein condensation of excitons in bilayer electron systems. Nature 432, 691–694 (2004). 15 Acknowledgements: This research has been supported by ERC Starting Grant 678309 MagnonCircuits, ERC Advanced Grant 694709 Super -Magnonics, by the DFG in the framework of the Research Unit TRR 173 “Spin+X” (Projects B01 and B07) and Project DU 1427/2 -1, by the grants Nos. EFMA -1641989 and ECCS -1708982 from the National Science Foundation of the USA, and by the DARPA M3IC grant under the contract W911- 17-C-0031. Author contributions: M.S. and D.B. performed the measurements and analyzed the experimental results. T.B. and A.V.C. supervised the measurements. T.B., P.P., A.A.S., A.V.C. planned the experiment. M.S., Bj.H., T.M. developed the experimental set -up. V.L. performed FMR characterizations and preliminary experiments. C.D. has grown the LPE YIG film. S.K. and E.Th.P. deposited the Pt overlayer. M.S., Bj.H., B.L., T.L., D.B. fabricated the structures under investigation. V.S.T. developed the quasi -analyti cal model of the magnon spectral redistribution . D.A.B., H.Yu.M.- S., V.S.T., A.N.S. performed the theoretical calculations. M.S. and F.H. performed the COMSOL simulations. Q.W. and P.P. performed the MuMax3 simulations. B.H. and A.V.C. led the project. All authors discussed the results and wrote the manuscript. Competing interests: None declared. Affiliations: 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, D -67663 Kaiserslautern, Germany. 2Department of P hysics and Energy Science, University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USA 3Graduate School Materials Science in Mainz, Staudingerweg 9, D -55128 Mainz, Germany. 4THATec Innovation GmbH, Bautzner Landstraße 400, D -01328 Dresden, Germany. 5Nano Structuring Center, Technische Universität Kaiserslautern, D -67663 Kaiserslautern, Germany. 6INNOVENT e.V. Technologieentwicklung, Prüssingstraße 27B, D -07745 Jena, Germany. 7Department of Physics, Oakland University, Rochester, MI 48309, USA. 8Faculty of Physics, University of Vienna, Boltzmanngasse 5, AT -1090 Wien, Austria. *Corresponding author. Email: chumak@physik.uni -kl.de 16 Supplementary Information Extended Data Fig. 1 | Detailed results of the numerical modelling of the BEC by rapid cooling. a, Time evolution of the temperature of the phonon system (top panel), the magnon chemical 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 Off and the vertical dashed line. b, Frequency distributions of the quasi -particles density for different times. The red lines in panels 1–4 show the snapshots of the magnon densities (for the times marked by the pink dots in panel a calculated using the dynamics equations. Dashed blue lines show the steady -state room - temperature distribution calculated using Eq. (1) in the manuscript. The green dashed lines denote the distribution when the chemical potential µ is equal to the minimal magnon energy hf min. Panel 1: Stationary heated case before the instant cooling. Panel 2: Distribution just after the instant cooling. Panel 3: Snapshot of magnon density when the chemical potential µ reaches the minimal energy hf min, resulting in BEC. Panel 4: Final room temp erature equilibrium state. c, Time evolution 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 pinning of the chemical pot ential µ at f min is clear ly visible for the case of the stronger heating. 17 Parameter Value Width of the strip 500 nm 1000 nm Current pulse duration τ p 120 ns 300 ns Applied voltage U 0.9 V 1.05 V Resistance R 600 Ohm 403 Ohm Room temperature 0T 288 K 288 K Elevated temperature 1T 440 K 490 K Gilbert damping constant Gα 1.5×10-3 1.5×10-3 Maximal frequency f max 600 GHz 600 GHz Frequency step f∆ 250 MHz 250 MHz Four -magnon scattering efficiency C 0.7 0.7 Extended Data Table 1 | Parameters used for the calculations of the magnon density . The table shows the parameters according to the developed quasi -analytical theoretical model for two different experimentally investigated strips. Spin- wave dispersion in the YIG nano -structures . The dispersion relations were obtained by means of micromagnetic simulations, which were performed using MuMax3 1 – see Extended Data Fig. 2 . The following parameters were used for the micromagnetic simulations. The size of the waveguide is 20 µm × 500 nm × 70 nm (length × width × thickness). The mesh was set to 10 nm × 10 nm × 70 nm. The following parameters for YIG were used: Saturation magnetization Ms = 123 kA/m, exchange constant A = 8.5 pJ/m, and Gilbert damping α = 2 × 10-4. In the simulations, the damping at the ends of the waveguides was set to increase exponentially to 0.5 in order to eliminate spin -wave reflections. The external field is 188 mT for bot h, the Backward Volume and the Damon -Eshbach geometries18. In order to excite spin- wave dynamics in the waveguide, a sinc -function- shaped field pulse was applied to a 50 nm wide area in the center of the waveguide. To gain access to both, odd and even spin- wave- width modes, this area was slightly shifted away from the center along the short axis of the waveguide. The sinc field is given by bz = b0 sinc (2πfct), with an oscillation field b 0 = 1 mT and a cutoff frequency f c = 20 GHz. The out -of- plane component Mz (x, y, t ) of each cell was collected over a period of t = 50 ns and stored in t s = 12.5 ps intervals, which results in a frequency resolution of Δ f = 1/t = 0.02 GHz, whereas the highest resolvable frequency was f max = 1/(2ts) = 40 GHz. The fluctuations in m z (x, y, t ) were calculated for all 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 1 Vansteenkiste, A., et al. The design and verification of MuMax3, AIP Adv. 4, 107133 (2014). 18 state. To obtain the spin- wave dispersion curves, a two- dimensional fast Fourier transformation in space and time has been performed2. To visualize the dispersion curve, Extended Data Fig. 2 shows a 3D grey-scale map of the spin- wave intensity, which is proportional to mz2 (kx, f ), in logarithmic scale as a function of f and kx. In both configurations, several width modes are visible, of which the first three a re marked in the graphs in Extended Data Fig. 2. Since the first three width modes are at similar frequencies, they 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 with Brillouin -Light Scattering (BLS) spectroscopy ( k x = 13 rad/µm for B ||y and kx = 0 rad/µm for B||x). The frequency of the minimum of the magnon spectrum coincides with the ferromagnetic resonance frequency ( k = 0) in the case of the Damon -Eshbach B||y geometry and differs by a value of 380 MHz in the case of the Backward Volume geometry B ||x. This difference is on the ord er of magnitude 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 the increase in temperature within the YIG nano -structure. 2 Venkat G. , et al. Proposal of a standard micromagnetic problem: Spin wave dispersion in a magnonic waveguide, IEEE Trans. Magn. 49, 524 – 529 (2013). Extended Data Fig. 2 | Simulated magnon dispersions. a, Simulated spin -wave intensity shown in a logarithmic black -to-white scale as a function of the frequency f and the wavenumber kx along the waveguide for B ||x. b, Simulated spin -wave intensity as a function of the frequency f and the wavenumber k x along th e waveguide for B ||y. 19 Simulation of time evolution of YIG/Pt structure by COMSOL Multiphysics . The numerically calculated time evolution of the temperature of the YIG/Pt strips was determined by solving a 3D heat -transfer model of the experimental set -up with the COMSOL Multiphysics Software using the heat transfer module and the electric currents m odule. Hereby, the conventional heat conduction differential equation and the differential equations for current conservation are solved taking into consideration the boundary conditions applied to the model as well as the material parameters of the mater ials used. The model comprises a 9 µm × 4.5µm × 6.5 µm large volume which includes half of the strip, exploiting the symmetry of the system with respect to its long axis. The simulated geometry includes the YIG/Pt strip, a part of the Au leads, which have a width of 1000 nm where they meet the strip and a width of 8 µm at a distance of 1 µm to the strip. In addition, the removal of the material by focused Ion -Beam with a width of 4 µm at the sides of the strip with a depth of 300 nm is taken into account. T he used material parameters can be found in Table Extended Data Table 2. The electrical conductivity and the corresponding temperature coefficient of Pt were measured experimentally, whereas the other parameters are taken from the referenced literature or from the COMSOL library. For the heat -transfer model, the boundary on the edges were set to the symmetry boundary condition for the according edges and to open boundary for the others. For the electric- currents model they were set to electric insulation, e xcept for the two faces of the leads where the electric potential is applied. The Joule heating is modelled for all metallic layers in the system. The simulation differs from the experiment by the fact that the convective cooling due to the surrounding air as well as the laser heating are not implemented. Extended Data Fig. 3 shows the obtained temperature profile in the center of the Pt overlayer. The profile was obtained for a 300 ns long pulse with an amplitude of U Sim = 1.872 V (corresponding to a set v oltage of U = 1.05 V at the pulse generator for a 50 Ohms load resistance in the experiment) and transition times of 1 ns at the edges. It can be seen that the temperature increases rapidly just after the current pulse is applied to Pt but tends to saturat e. After a time of approximately 300 ns the temperature still grows but the heating has already strongly decelerated allowing for the formation of the quasi -equilibrium between the magnon and phonon temperatures3 considered in the theoretical model. The maximal temperature reached at the end of the applied current pulse in this particular case is 491 K. 3 Agrawal, M. , et al. Direct measurement of magnon temperature: New insight into magnon - phonon coupling in magnetic insulators. Phys. Rev. Lett. 111, 107204 (2013). 20 Parameter Material Value/ Source Density YIG 5170 kg m-3 Clark, A. E. & Strakna , R. E. Elastic constants of single - crystal YIG, J. Appl. Phys. 32, 1172 (1961) Heat conductivity YIG 6 W m-1K-1 Hofmeister, A. M. Thermal diffusivity of garnets at high temperature, Phys. Chem. Minerals 33, 45 -62 (2006) Heat capacity YIG 570 J kg-1 K-1 Guillot, M., Tchéou, F., Marchand, A., Feldmann, P. & Lagnier, R., Specific heat in Erbium and Yttrium Iron garnet crystals, Z. Phys. B – Condensed Matter 44, 53- 57 (1981) Density GGG 7080 kg m-3 Hofmeister A. M. Thermal diffusivity of garnets at high temperature, Phys. Chem. Minerals 33, 45 -62 (2006) Heat conductivity GGG 7080 kg m-3 Hofmeister A. M. Thermal diffusivity of garnets at high temperature, Phys. Chem. Minerals 33, 45 -62 (2006) Heat capacity GGG 7080 kg m-3 Hofmeister A. M. Thermal diffusivity of garnets at high temperature, Phys. Chem. Minerals 33, 45 -62 (2006) Density Pt 21450 kg m-3 Lide, D. CRC Handbook of Chemistry and Physics, 89th ed. (Taylor & Francis, London, 2008) Electrical conductivity Pt 1.41 × 106 S m-1 (Extended Data Fig. 3), 1.90 × 106 S m-1 (Extended Data Fig. 4), 1.79 × 106 S m-1 (Extended Data Fig. 5a) Measured Resistance- Temperature Coefficient Pt 7.135 × 10-4 K-1 Measured Thermal Conductivity Pt 22 W m-1 K-1 Yoneoka, S., et al. Electrical and thermal conduction in atomic layer deposition nanobridges down to 7 nm thickness, Nano Lett. 12, 683- 686 (2012) Heat capacity Pt 130 J kg-1 K-1 Furukawa, G. T., Reilly, M. L., Gallagher, J. S. Critical Analysis of Heat Capacity data and evaluation of thermodynamic properties of Ruthenium, Rhodium, Palladium, Iridium, and Platinum from 0 to 300K. A survey of the literature data on Osmium, J . Phys. Chem. Ref. Data 3, 163 (1974) Density Au 1900 kg m-3 COMSOL Library 21 Electrical conductivity Au 4.1 × 107 S m-1 COMSOL Library Thermal Conductivity Au 190 W m-1 K-1 Langer, G., Hartmann, J. & Reichling M. Thermal conductivity of thin metallic films mea sured by photothermal profile analysis , Rev. Sci. Instrum. 68, 3 (1997) Heat capacity Au 130 J kg-1 K-1 Geballe, T. H., & Giauque, W. F. The heat capacity and entropy of Gold from 15 to 300°K, J. Am. Chem. Soc. 1952, 74) Density Ti 4500 kg m-3 COMSOL Library Electrical conductivity Ti 2 × 106 S m-1 COMSOL Library Thermal Conductivity Ti 22 W m-1 K-1 Ho, C. Y., Powell, R. W. & Liley, P. E. Thermal conductivity of the elements, J. Phys. Chem. Ref. Data 1, 2 (1972) Heat capacity Ti 521.4 J kg-1 K-1 Kothen, C. W. & Johnston, H. L. Low Temperature Heat Capacities of Inorganic Solids. XVII. Heat Capacity of Titanium from 15 to 305°K. J. Am. Chem. Soc. 75, 3101 (1953) Heat capacity Air 1047.6366 - 0.3726 T + 9.4530 × 10-4 T2 - 6.0241 × 10-7 T3+1.2859 × 10-10 T4 J kg-1 K-1 COMSOL Library Thermal Conductivity Air -0.0023 + 1.1548 × 10-4 T -7.9025 × 10-8 T2 + 4.1170 × 10-11 T3-7.4386 × 10-15 T4 W m-1 K-1 COMSOL Library Density Air 8.53 T-1 kg m-3 COMSOL Library Extended Data Table 2 | Parameters used for the C OMSOL Simulations. 22 As it is seen in Extended Data Fig. 3a , switching off the current pulse results in a fast decrease in temperature of the YIG/Pt nano -structure due to the thermal diffusion in the quasi - bulk surroundings. The cooling rate i s not constant and decreases with time. Moreover, our simulations further revealed (not shown) that the cooling rate depends on the duration of the applied current pulse which is associated with the increase in temperature of the surroundings of the nano -structure for long current pulses. To prove the crucial role of the fast cooling for the BEC, additional measurements were performed where the current pulses were switched off with fall times of 50 ns and 100 ns – see Fig. 2 in the main text of the manuscr ipt. It is evident that the condensate disappears for long fall times, i.e. slow cooling rates. For reference, Extended Data Fig. 3 shows the corresponding simulated temperature evolutions in the phonon system. The maximal cooling rates indicated in Extend ed Data Fig. 3b clearly show that a cooling rate of 2 K/ns is already too slow to trigger the magnon BEC, while a rate of 20.5 K/ns is still fast enough. Extended Data Fig. 3 | Results of the COMSOL simulations for different fall times. a, Simulated temperature as a function of time in the middle of the Pt overlayer for a 300 ns long pulse with a voltage of 1.05 V applied to a 1 µm wide waveguide. The fall time of the pulse was 1 ns. The dashed blue lines indicate the time when the pulse is present. The room temperature is marked with the dashed pink line. b, Simulated temperature as a function of time for different fall times τ Fall = 1 ns (green line), τ Fall = 50 ns (blue line), τ Fall = 100 ns (red line). The maximal time derivatives are ma rked at their respective time of occurrence. To support the COMSOL simulations an additional experiment is performed, measuring the temperature of the Platinum layer time -resolved by means of electrical measurements. First, τP = 300 ns long DC -pulses with an amplitude of U = 0.85 V are applied to a 1 µm wide YIG/Pt waveguide. The corresponding BLS -spectrum is shown in Extended Data Fig. 4b. The frequency shift during the pulse indicates the heating of the magnetic system, whereas the magnon ΒEC is observed after the pulse is switched off. To get access to the time -resolved resistivity (which depends on the temperature) during the applied pulse instead of the DC -pulse, a RF -pulse with the same duration and a frequency of 12.4 GHz is applied to heat up the sample. The RF -power was adjusted in such a way, that Joule heating results in the same steady state temperature as for a continuous applied voltage of U = 0.85 V. Thus, the resistance of the Pt layer was found to increase from 319.5 Ohm to 372.1 Ohm when either a continuous DC -voltage of 0.85V or a continuous RF signal with a power of 18.84 dBm and a frequency of 12.4 GHz is applied. Hence, the RF -power of 18.84 dBm results in the same Joule heating of the Platinum layer as a DC -voltage of 0.85 V. Then, 23 a continuous DC -voltage of 30 mV, which is used to measure the resistance change and a 300 ns long RF -pulse are applied in parallel using a DC -block and a RF -circulator. From the changed voltage drop on the sample, which is measured with an oscilloscope, the temperature is calculated. The resulting temperature profile is shown in Extended Data Fig. 4a (black curve). In addition, we have performed the same COMSOL simulation as for all other shown simulated temperature profiles. Only the cor responding conductivity of the Pt -layer was matched to the measured resistivity of 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 COMSOL model can precisely describe the temperature dynamics of the investigated structures. Further, from the BLS measurement the temperature is derived by considering the frequency shift determined by BLS and the Kittel equation. The re sulting temperature profile is shown in Extended Data Fig. 4 a (red points) and deviates from the simulated and electrically measured 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 during the pulse is not only given by the number of magnons, but also by the flattening of the dispersion, 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 frequencies. At the time the BEC occurs, a too high temperature is measured by BLS, which indicates ones more, that magnons condense to the lowest energy state. Extended Data Fig. 4 | Temperature profiles derived by electrical measurements, COMSOL and BLS | a, Temperature as a function of time measured electrically (black curve) by means of resistivity change for an applied RF -pulse with a frequency of f = 12.4 GHz, a power of PRF = 18.84 dBm and a duration of τ P = 300 ns, obtained by COMSOL simulations (blue curve) for an applied voltage of U = 0.85 V and a pulse duration of τ P = 300 ns, derived from the shift of the BLS -frequency (red data points) b, Time resol ved BLS spectrum for the temperature profiles shown in a, for the case when a pulse of an amplitude of U = 0.85 V and a duration of τP = 300 ns is applied. Thermal dynamics and the spin Seebeck effect. Another possible contribution to the observed BEC c an be the spin- orbit -mediated spin Seebeck effect as it was shown in Ref. 17. To clarify the role of the SSE, we extracted the temperature profile across the YIG/Pt structure and simulated the temporal evolution of the temperature and temperature gradient in the YIG film – see Extended 24 Data Fig. 5. The simulation was performed for t he experiment shown in Fig. 2b in the main manuscript. The current pulse duration was set to τP = 120 ns. The pulse voltage was of U Sim = 1.662 V, which corresponded to a set voltage of U = 0.9 V at the pulse generator for a 50 Ohms load resistance in the experiment. A maximum temperature of T = 443 K was reached at the end of the pulse. One can see that the gradient can reach values up to 3.5 × 108 K/m. However, the temporal evolution of the gradient shows that it is formed within few nanoseconds after the current pulse is applied, stays practically constant during the pulse and, finally, disappears within few nanoseconds after the pulse is switched off. Thus, at the moment of BEC in our experiment, there is no thermal gradient present as well as a possible contribution from the SSE. At the same time, we can assume that the SSE contributes to the increase in the magnon chemical potential during the current pulse is applied17. However, in course of time, such a contribution should decrease with increase in the YIG temperature due to a strong rapid temperature -dependent reduction in the SSE magnitude (see Fig. 3c in Ref. 27). Extended Data Fig. 5 | Temperature profile and time evolution of temperature gradient. Simulated temperature (red curve, left axis) and temperature gradient (black curve, right axis) as a function of time. The profiles were simulated with COMSOL for the parameters correspondi ng to the experiment shown in Fig. 2b in the main manuscript. Dependence of the BEC on the applied magnetic field . The described magnon BEC is driven by the change of the temperature of the phonon system. In order to prove this, a set of measurements using different structure sizes of width of 500 nm and 1000 nm, of different scanning laser powers in the range from 0.75 mW to 3.3 mW, and different scanning points on the structure were performed. The experimental finding of the BEC, which manifests itself as a strong magnon peak at the bottom of the spectrum, could always be confirmed. In particular, the threshold for BEC is found to be independent of the applied external field. Measurements of the threshold for a pulse duration of τP = 18 ns are performed in the range from 110 mT to 270 mT. The maximal integrated BLS intensities of the fun damental mode can be seen in Extended Data Fig. 6. 25 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 depend on the external field, whereas the inset shows that t he magnon lifetime is decreased for higher fields, as expected. The measured lifetime is 21 ns and is small compared to the lifetime of magnons in YIG films of µm thicknesses, used in previous experiments 12,14,15 ,36,37. It is known, that for YIG films with thicknesses in the nm -range, the spin -wave damping is larger, and consequently, the lifetime is smaller. Additionally, due to the spin pumping effect to the Pt layer and due to distortions induced in YIG sample by the structuring process the effective mag netic damping is enhanced. Furthermore, the fact that there is no dependence of the phenomenon on the direction of the external field was confirmed by changing the direction of the applied field. Extended Data Fig. 7 shows 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 of the strip for similar pulse durations of τ P = 120 ns and τ P = 150 ns. Further, the integrated intensity of the fundamental mode is shown for both cases in Extended Data Fig. 7b (B||y) and 7d (B||x). Extended Data Fig. 6 | Dependency of the threshold on the external field. Threshold curves 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 strip. The inset shows exemplary time traces for a supercritical voltage of U = 1.5V and for three different fields. The threshold is found to be approximately constant over the measured field range, whereas the magnon lifetime decreases for higher values of the applied magnetic field . 26 The frequencies of the fundamental modes are slightly la rger if the magnetic field is applied parallel to the long axis of the strip. This is associated with the demagnetization fields that result in a smaller internal field when the strip is magnetized along its short axis. In addition, the inhomogeneity 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 (see. Fig. Extended Data Fig. 7a ) as well as in the simulations (see Extended Data Fig. 2b). The magnon BEC is observed in both cases at the bottom of the band of the waveguide modes, i.e., the lowest frequency of the fundamental mode. Even though the edge mode is the total energy minimum at room temperature equilibrium, it cannot be separated in the experiment from the fundamental mode during the BEC since, in contrast to the other modes, its frequency is not significantly decreased during the pulse. The reason for this is that the frequency of the edge mode strongly depends on the strength of t he demagnetization fields. These fields are reduced as well when the saturation magnetization is decreased due to Joule heating. This behaviour was confirmed by additional MuMax3 simulations (not shown). The BEC for both field directions excludes a contrib ution from the Spin Hall Effect to the observed condensation, since it would only lead to an efficient injection of spin current when the magnetization is perpendicular to the direction of the electric current 26,28,29. Moreover, since the phenomenon does neither depend on Extended Data Fig. 7 | BEC by rapid cooling for different geometries of the external field. a, BLS spectrum as a function of time. The BLS signal (color -coded, log scale) is proportional to the density of magnons. FM indicates the fundamental mode, EM – the edge mode, and 1st M – the first thickness mode. The vertical dashed lines indicate the star t and the end of the pulse (τP = 150 ns , U = 0.9 V). The external field was parallel to the short axis of the strip ( B||y). b, BLS spectrum as a function of time for the case when the external magnetic field is parallel to short axis of the strip ( B||x), (τP = 120 ns , U = 0.9 V). c, d, Normalized magnon intensity integrated from 4.95 GHz to 8.1 GHz as a function of time for the cases in a and b. The insets show the sample and measurement geometry. 27 the direction nor on the value of the applied magnetic field (and thus also not on the frequency of the BEC), we can conclude that the Oersted fields generated by the eclectic current in the Pt overla yer do not play any sizable role in the experiments. Dependence of the BEC threshold on the current pulse duration. The threshold voltages are determined experimentally for different pulse durations 18 ns ≤ τP ≤ 175 ns. For each pulse duration, BLS measurements were performed for different voltages. The maximum of the BLS intensity integrated 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 Fig. 3 b in the main manuscript. The corresponding threshold voltage is determined by fitting the slope linearly and calculating the intercept with the averaged subcritical (thermal) BLS intensity. The extracted threshold voltages are shown in Extended Data Fig. 8. As can be seen, the threshold voltage increases for the shortest pulse durations, whereas it saturates 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 P > 70ns (blue points in Extended Data Fig. 8). Extended Data Fig. 8 | Dependency of the threshold on the pulse duration. Threshold voltage as a function of the pulse duration (black squares) and corresponding simulated temperatures at the end of the pulses (blue squares). The slight increase in the threshold temperature for the shorter pulses is likely related to incomplete thermalization of low energy magnons for such short times. Note, that in our experiment the spin -lattice relaxation time for the low energy magnons is about 21 ± 4 ns. The solid 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 such a case, the chemical potential induced by the rapid cooling might not reach the bottom of the magnon spectrum hf min and no BEC occurs. Nevertheless, as it is supported by the experimental findings shown in Extended Data Fig. 8, the incomplete equilibration of magnon and phonon systems before t he beginning of the fast 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 maximal sample temperature is increased to 490 K. It is worth mentioning, that the “rapid heating” of the magnons' environment just after the application 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 28 thermalization time. This effect is clearly visible in the Extended Data Fig. 9 but is not observe d in our experiments. The reason could be the following. In general, the spin Seebeck effect leads to an increase of magnon chemical potential and the magnon dynamics during the current pulse is determined by the interplay of two counter -acting processes: “rapid heating” and SSE. Proper theoretical description of the magnon dynamics in this interval would require development of a completely new theory that would treat these effects on equal footing and is outside the scope of the present work. However, simple estimation of the SSE -induced effects just after the current pulse is switched on shows, that temperature gradient might lead to magnon chemical potential µ ~ 5.25 GHz and should rapidly decrease afterwards due to the increase in the YIG temperature27. Thus, 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. Extended Data Fig. 9 | Time evolution of the chemical potential for a short heating pulse. The pulse duration τ P = 20 ns. The initial 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 increases the threshold temperature of the BEC formation from 440 K to 490 K. Dependence 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 P = 50 ns long current pulses at a 1 µm wide YIG/Pt structure. The fall times of the applied pulses varied in the range of 1 ns ≤ τFall ≤ 100 ns. Extended Data Figure 10 shows the integrated BLS intensity as a function of the fall time. The strong increase for shorter fall times is clearly visible (above -threshold regime), whereas for fall times larger than the spin wave lifetime the accumulation at the bottom of the spectrum vanishes indicating that the BEC threshold is not reached. This supports the conclusion from the main manuscript, that a rapid cooling of the phonon system is crucial t o achieve the magnon BEC and the required cooling rate is mainly determined by the lifetime of the low energy magnons. 29 Extended Data Fig. 10 | Dependence of the magnon BEC intensity on the fall time of the applied pulses. Integrated BLS intensity of the fundamental mode as a function of the fall time of the applied current pulses. Only for fall times shorter than the lifetime of magnons t m = 21 ± 4 ns a clear accumulation at the bottom of the spectrum is observed. Influence of thermally induced spin currents on the magnon BEC by rapid cooling. An additional experiment using an Al (7 nm)/Au (5 nm) heating layer on top of a 2 µm wide and 34 nm thick 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 thermally 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 applied in- plane perpendicular to the waveguide long axis. Extended Data Figure 11 shows the accumulation 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 the rapid cooling rather than spin- orbit mediated Spin Seebeck effect 17 is responsible for the reported BEC of magnons. Extended Data Fig. 11| Rapid cooling BEC generated in an Au/Al/YIG structure. a, Time resolved BLS spectrum for the case when a 50 ns long pulse is applied b, Integrated BLS intensity over the frequency range shown in a.

2016-12-21

The fundamental phenomenon of Bose-Einstein Condensation (BEC) has been observed in different systems of real and quasi-particles. The condensation of real particles is achieved through a major reduction in temperature while for quasi-particles a mechanism of external injection of bosons by irradiation is required. Here, we present a novel and universal approach to enable BEC of quasi-particles and to corroborate it experimentally by using magnons as the Bose-particle model system. The critical point to this approach is the introduction of a disequilibrium of magnons with the phonon bath. After heating to an elevated temperature, a sudden decrease in the temperature of the phonons, which is approximately instant on the time scales of the magnon system, results in a large excess of incoherent magnons. The consequent spectral redistribution of these magnons triggers the Bose-Einstein condensation.

Bose-Einstein Condensation of Quasi-Particles by Rapid Cooling

1612.07305v3

1 Spin Transport in Antiferromagnetic Insulators Mediated by Magnetic Correlations Hailong Wang†, Chunhui Du†, P. Chris Hammel* and Fengyuan Yang* Department of Physics, The Ohio State University, Columbus, OH, 43210, USA †These authors made equal contributions to this work *Emails: hammel@physics.osu.edu; fyyang@physics.osu.edu We report a systematic study of spin transport in antiferromagnetic (AF) insulators having a wide range of ordering temperatures . Spin current is dynamically injected from Y3Fe5O12 (YIG) into various AF insulators in Pt/insulator/ YIG trilayers. Robust , long -distance spin transport in the AF insulators is observed , which shows strong correlation with the AF ordering temperatures . We find a striking linear relationship between the spin decay length in the AFs and the damping enhancement in YIG, suggesting the critical role of magnetic correlations in the AF insulators as well as at the AF/YIG interface s for spin transport in magnetic insul ators . PACS: 75.50.Ee , 75.70.Cn , 76.50.+g, 81.15.Cd 2 Spin current s carried by mobile charges in metallic and semiconducting ferromagnetic (FM) and nonmagnetic (NM) materials ha ve been the central focus of spintronic s for the past two decades [1]. However, s pin transport in AF insulators has been essentially unexplored due to the difficulty in generating magnetic excitations in these insulators . Ferromagnetic resonance (FMR) and thermally driven spin pumping [2-15] have attracted intense interest in magnon - mediated spin current s, which can propagate in both conducting and insulating FM s and AF s. We recently reported observation of high ly efficien t spin transport in AF insulator NiO with long spin decay length [16]. In this letter, we probe the mechanism s responsible for spin transport in AF insulators by investigati ng several series of Pt/insulator/YIG trilayers ; this study is enabled by the large inverse spin Hall effect (ISHE) signals in our YIG -based structures [9-15]. Epitaxial YIG (epi-YIG) films are grown on (111) -oriented Gd 3Ga5O12 (GGG) substrates by sputtering [9-17]. X-ray diffraction and a tomic force microscopy measurements reveal high crystalline quality and smooth surfaces of the YIG films [18 ]. Figure 1(a) shows a n in-plane magnetic hysteresis loop for a 20 -nm YIG film which exhibits a small coercivity ( Hc) of 0.40 Oe and sharp magnetic reversal, indicating high magnetic uniformity. Figure 1 (b) presents a FMR derivative absorption spectrum for a 20 -nm YIG film taken in a cavity at radio -frequency (rf) f = 9.65 GHz and microwave power Prf = 0.2 mW with an in -plane magnetic field, which gives a narrow linewidth ( H) of 7.7 Oe. All of these measurements are carried out at room temperature. In order to probe spin transport in insulators of various magnetic structures , we select six materials, including: 1) amorphous SrTiO 3, a diamagnet , 2) epitaxial Gd 3Ga5O12, a paramagnet with a large magnetic susceptibility ( ), and four antiferromagnets , 3) Cr2O3 [19], 4) amorphous YIG ( a-YIG) [20], 5) amorphous NiFe 2O4 (a-NFO) [21], and 6) NiO [19]. All insulator layers 3 are deposited by off -axis sputtering. Lattice matched, s train-free Gd3Ga5O12 films are epitaxially grown on YIG at high temperature; the remaining five insulators are grown at room temperature to avoid strain ing the epi -YIG films which can significantly alter the magnetic resonance in YIG . Electrical transport measurements confirm the highly insulating nature of all these films. Figur es 1(a) and 1(b) indicate that the a-YIG film has negligible magnetization and FMR absorption (a- NFO films exhibit similar behavior ). Thus, the six insulators include a diamagnet , a paramagnet , and four AFs with a wide range of ordering temperatures , allowing us to probe magnetic excitation s and spin propagation in insulators both above and below the AF ordering temperatures , hence illuminating the roles of both static and dynamic magnetic correlations . Bulk Cr 2O3 and NiO have Néel temperature s TN = 318 and 525 K, respectively [19]. Both YIG and NiFe 2O4 are ferrimagnets when in crystalline form; however, amorphous YIG and NiFe 2O4 become AFs due to the lack of crystalline ordering required for ferrimagnetism [20, 21]. The temperature ( T) dependence of exchange bias in FM/AF bilayers allows direct measure ment of the blocking temperature, Tb, of the AFs. Our YIG(20 nm)/NiO(20 nm) bilayer exhibits a clear exchange bias field, HE = 13.5 Oe and a n enhanced coercivity Hc = 19.2 Oe ( Hc = 0.40 O e for a single YIG film) , demonstrat ing exchange coupling between YIG and NiO [18]. However, the very large paramagnetic background of GGG substrates prohibit s the measurement of exchange bias at low temperatures needed for Cr 2O3, a-YIG, and a-NFO . To determine Tb for each AF stud ied here , we use Ni81Fe19 (Py) as the FM and measure exchange bias in Py(5 nm)/AF(20 nm) bilayers grown on Si . Figure 2 (a) shows the hysteresis loops of four Py/AF bilayers at T = 5 K after field cooling from above Tb. All four samples exhibit substantial exchange bias: HE = 646, 1403, 568, and 97 Oe for Py/NiO, Py/ a-NFO, Py/ a-YIG, and Py/Cr 2O3, respectively. Figures 2 (b) and 2 (c) show the temperature dependenc ies of HE for the four 4 bilayers, from which we determine Tb = 20, 45, 70, and 330 K for Cr 2O3, a-YIG, a-NFO, and NiO, respectively. Spin currents in insulators propagat e via precessional spin wave modes , e.g., magnons in ordered FMs and AFs . However, it is challenging to excite AF magnons which, for example, requires THz frequency in NiO [22]. Furthermore, the AF ordering temperatures in thin films decrease at lower thicknesses and conventional magnons cannot be sustained above the ordering temperatures . Here, we leverage the established technique of FMR spi n pumping in YIG -based structures to excite the AF insulators via exchange coupling to the precessing YIG magnetization and to probe spin transfer in these insulators . For each of the six insulators, we grow a series of Pt(5 nm) /insulator( t)/epi-YIG(20 nm) trilayers with various insulator thickness es t on YIG films cut from the same YIG/GGG wafer to ensure consistency of the YIG quality. Since Pt is the only conduct or in the trilayers , the voltage signals detected are exclusively from the ISHE (VISHE), which proportionally reflects the spin current s pumped into Pt across the insulat ors. Room -temperature s pin pumping measurements [18] are conducted on all trilayers (~1 mm wide and ~5 mm long) in an FMR cavity at f = 9.65 GHz and Prf = 200 mW in an in-plane DC field ( H), as illustrated in Fig. 3(a). The mV-level ISHE voltage s provide a dynamic range of more than three orders of magnitude for detecting the decay of spin current across the insulat ors. The rates at which VISHE decays with increasing insulator thickness (t) differ dramatically among the six spacers. A 0.5-nm SrTiO 3 [18] already suppresses VISHE by a factor of 17 from the corresponding Pt/YIG bilayer [10]. As we change the insulator from SrTiO 3 Gd3Ga5O12 Cr2O3 a-YIG a-NFO NiO, the spin current s exhibit substantially increasing propagat ion lengths . Figure 3(b) summarizes the t-dependencies of the normalized peak VISHE at YIG resonance , Hres, for all six series . From the linear relationship in the semi -log 5 plots, we extract the spin decay length s 𝜆 in the insulat ors by fitting to 𝑉ISHE(𝑡)/𝑉ISHE(0)= 𝑒−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- YIG, a-NFO, and NiO, respectively (Table I) . More surprisingly, Fig. 3(c) shows that VISHE initially increases by a factor of 2.1 and 1.6 when a 1 - or 2-nm NiO and a-NFO, respectively, is inserted between YIG and Pt (the point for t = 0 is excluded from the exponential fit for NiO and NFO) . This dramatic variation in the spin current propagation length -scale most likely arise s from different magnetic characteristics of the six insulators. For dynamically generated spin current to transmit across insulating spacers beyond the tunneling range (~1 nm), magn etic excit ations in the insulat ors are expected to play a major role. Except SrTiO 3, all other five insulators have strong magnetic character , including paramagnetic Gd3Ga5O12 and four AFs with various ordering temperatures . For the same AF material, Tb can vary significantly depending on the film thickness [ 23]. Among the four AFs, NiO is the most robust AF with Tb = 330 K for our 20 -nm NiO film, while for very thin NiO layers (<5 nm) , Tb is expected to be well below 300 K [ 24]. For a-NFO , a-YIG and Cr2O3, the AF ordering temperatures are well below room temperature (Table I) . It is interesting to note that Gd3Ga5O12 also exhibits magnetic order at very low temperatures [ 25]. Thus, magnetic correlation s amongst thermally fluctuating AF moment s are critically important for the observed spin transport in insulators . These results suggest that , at resonance, the precessing YIG magnetization generates magnetic excitations in the adjacent insulat or (either with static AF ordering or fluctuati ng correlated moments ) via interfacial exchange coupling , which in turn e nhances magnetic damping of the YIG. We measure the Gilbert damping constant [26] from the frequency dependencies of FMR linewidth s H for six insulator(20 nm)/YIG(20 nm) bilayers and a single 6 epi-YIG film using a microstrip transmission line . Figure 4(a) show s the linear frequency dependence of Δ𝐻 given by Δ𝐻=Δ𝐻0+4𝜋𝛼𝑓 √3𝛾 for the seven samples , where Δ𝐻0 is the y- intercept and 𝛾 is the gyromagnetic ratio . From t he slope s of least -squares fits , we obtain = 8.1 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 YIG, SrTiO 3/YIG, Gd3Ga5O12/YIG, Cr2O3/YIG, a-YIG/YIG, a-NFO/YIG, NiO/ YIG, and Pt/YIG, respectively (Table I) . The diamagnetic SrTiO 3 does not enhance the damping of YIG within experimental uncertainty while its spin current decay s over an atomic length scale (𝜆 = 0.18 nm) due to quantum tunneling [10]. The large paramagnetic moment s in Gd3Ga5O12 can absorb angular momentum via exchange coupling to YIG and conduct spin current , resulting in a longer 𝜆 = 0.69 nm. The four AFs show much longer spin d ecay lengths together with enhanced damping of YIG due to strong magnetic correlations [23]. NiO more than triples the damping of YIG and its spin d ecay length is almost 10 nm , while clear spin current is detected over a NiO thickness of 100 nm . The AF resonance frequency of NiO is about 1 THz [22] which is much higher than the 9.65 GHz used in our FMR excitation of YIG. Despite the difference in the dispersion relation s of YIG and the AFs, our result clearly demonstrate s highly efficient spin transport across the AFs. Considering that strong AF spin correlation s have been o bserved well above TN for NiO [27], we believe the excitation s responsible for spin transport in AFs must be m agnon s in ordered AFs and AF fluctuations in insulators with low blocking temperatures. In either case, the strongly correlated AF spins are excited via exchange coupling to the precessing YIG magnetization (either the net or staggered ferrimagnetic moments) at the AF/YIG interface and transfer the spin current across the insulator to the interface with Pt, where it is converted to a 7 spin-polarized electr on current in Pt. This is analogous to the predicted magnon current s in FM insulators [28]. The independent ly measure d spin d ecay length 𝜆 and damping enhancement ∆𝛼 both increase monotonically following SrTiO 3 Gd3Ga5O12 Cr2O3 a-YIG a-NFO NiO (Table I) . Figure 4(b) further show s that 𝜆 and ∆𝛼 exhibit a nearly perfect linear relationship for all insulators excluding SrTiO 3. The excellent linear relationship between 𝜆 and ∆𝛼 of five significantly different insulat ors indicates that spin transfer across the YIG/AF interfaces (measured by ∆𝛼) and spin propagation inside the AF insulators (characterized by 𝜆) are tightly related. T he exchange coupling between YIG magnetization and AF spins at the interfaces and the exchange interaction between adjacent AF spins within the AFs play a dominant role in spin transport in insulators. Lastly , the strength of magnetic correlation s depend s on the AF ordering temperature s. Our results shown in Figs. 2 to 4 indicate that the correlation strength increases following the order SrTiO 3 (diamagnet) Gd3Ga5O12 (paramagnet ) Cr2O3 (AF, Tb = 20 K) a-YIG (AF, Tb = 45 K) a-NFO (AF, Tb = 70 K) NiO (AF, Tb = 330 K). As magnetic correlation increases, exchange interaction becomes stro nger, which, 1) facilitates the propagation of spin current s carried by magnetic excitation s in the insulators, and 2) enhances the magnetic damping of the underlying YIG films . The surprising enhancement of ISHE signals for the trilayers with 1- or 2-nm NiO and a-NFO [Fig. 3(c)] indicates that the Pt/NiO/YIG and Pt/a-NFO/YIG trilayer structures are highly efficient in spin transfer , while the underlying mechanism remains to be understood . In summary, we observ e clear spin current s in AF insulat ors mediated by AF magnetic correlation s be they static or fluctuating . This result brings a large family of insulators, in 8 particular, AF insulators, into the exploration of spintronic applications utilizing pure spin currents. This work was primarily supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, under Grants No. DEFG02 -03ER46054 (FMR and spin pumping characterization) and No. DE -SC0001304 (sample synthesis and magnetic characterization). This work was supported in part by the Center for Emergent Materials, an NSF-funded MRSEC, under Grant No. DMR -1420451 (structural characterization). Partial support was provided by Lake Shore Cryogenics, Inc., and the NanoSystems Labor atory at the Ohio State University . 9 Reference s: 1. I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 3. Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe , K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 4. E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 5. O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader, and A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2010). 6. A. Hoffmann , IEEE Trans. Magn. 49, 5172 (2013). 7. B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y. -Y. Song, Y. Y. Sun, and M. Z. Wu, Phys. Rev. Lett . 107, 066604 (2011) . 8. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa , and E. Saitoh, Nature 455, 778 (2008) . 9. H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 88, 100406(R) (2013). 10. C. H. Du, R. Adur , H. L. Wang, A. J. Hauser, F. Y. Yang, and P. C. Hammel, Phys. Rev. Lett. 110, 147204 (2013). 11. H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. B 89, 134404 (2014). 12. C. S. Wolfe, V. P. Bhallamudi, H. L. Wang, C. H. Du, S. Manuilov, A. J. Berge r, R. Adur, F. Y. Yang, and P. C. Hammel, Phys. Rev. B 89, 180406(R) (2014). 13. H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 14. C. H. Du, H. L. Wang, F. Y. Yang, and P. C. Hammel, Phys. Rev. Applied 1, 044004 (2014). 10 15. H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Appl. Phys. Lett. 104, 202405 (2014). 16. H. L. Wang, C. H. Du, P. C. Hammel and F. Y. Yang, Phys. Rev. Lett. 113, 097202 (2014). 17. C. H. Du, R. Adur, H. L. Wang, A. J. Hauser, F. Y. Yang, and P. C. Hammel, Phys. Rev. Lett. 110, 147204 (2013) . 18. See Supplementary Materials for details in XRD, AFM, exchange bias, and FMR spin pumping results. 19. D. R. Lide, (eds) Handbook of Chemistry and Physics, 86th Ed. (Taylor & Francis, New York, 2005). 20. E. M. Gyorgy , K. Nassau, K. Nassau, M. Eibschutz, J. V. Waszczak, C. A. Wang, and J. C. Shelton, J. Appl. Phys . 50, 2883 (1979) . 21. V. Korenivski, R. B. van Dover, Y. Suzuki, E. M. Gyorgy, J. M. Phillips, and R. J. Felder, J. Appl. Phys . 79, 5926 (1996) . 22. R. Cheng and Q. Niu, Phys. Rev. B 89, 081105(R) (2014). 23. J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999) . 24. A. Baruth and S. Adenwalla, Phys. Rev. B 78, 174407 (2008). 25. O. A. Petrenko , D. McK Paul , C. Ritter, T. Zeiske, and M. Yethiraj , Physica B 266, 41 (1999). 26. S. S. Kalarickal, P. Krivosik, M. Z. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006). 27. T. Chatterji, G. J. McIntyre, and P. A. Lindgard , Phys. Rev. B 79, 172403 (2009). 28. S.-L. Zhang and S. F. Zhang , Phys. Rev. B 86, 214424 (2012) . 11 Table I . Type of magnetism , blocking temperatures (for AFs only), and spin d ecay length (𝜆) for the s ix insulator s as well as the Gilbert damping constant ( ) for the six insulator s (20-nm) on YIG. The parameters for a single epitaxial YIG film are also included for comparison . Layer Magnetism Tb (K) 𝜆 (nm) epi-YIG ferrimagnet (8.1 ± 0.6) 10-4 SrTiO 3 diamagnet 0.18 ± 0.01 (8.6 ± 1.0) 10-4 Gd 3Ga5O12 paramagnet 0.69 ± 0.02 (11 ± 1) 10-4 Cr2O3 antiferromagnet 20 1.6 ± 0.1 (12 ± 1) 10-4 a-YIG antiferromagnet 45 3.9 ± 0.2 (14 ± 1) 10-4 a-NFO antiferromagnet 70 6.3 ± 0.3 (17 ± 2) 10-4 NiO antiferromagnet 330 9.8 ± 0.8 (26 ± 3) 10-4 12 Figure captions: Figure 1. (a) Room temperature in -plane magnetic hysteresis loop s of a 20-nm epitaxial YIG film (blue) and a 20-nm amorphous YIG film (red) grown on GGG , where the paramagnetic background is from the GGG substrate. Inset: low -field hysteresis loop of the epitaxial YIG film showing a coercivity of 0. 40 Oe. (b) FMR derivative absorption spectra of an epitaxial (blue) and an amorphous (red) 20-nm YIG film on GGG . Figure 2. (a) Magnetic hysteresis loops of four Py( 5 nm)/AF(20 nm) bilayers at 5 K after field cooling, all demonstrating clear exchange bias. Temperature dependence of HE for the four Py(5 nm)/AF(20 nm) bilayers in (b) linear and (c) log y-scale give the AF blocking temperature Tb = 20, 45, 70, and 330 K for Cr 2O3, a-YIG, a-NFO, and NiO, respectively. Figure 3. (a) Schematic of the ISHE measurement on various Pt/Insulator/YIG structures. (b) Semi -log plots of VISHE as a function of the insulator thickness for the six series normalized to the values for the corresponding Pt/YIG bilayers , where the straight lines are exponential fits to each series, from which the spin d ecay length s are determined . (c) Details of behavior shown in (b) for insulators below 10 nm. Figure 4. (a) Frequency dependencies of FMR linewidth s of a bare epitaxial YIG film, SrTiO 3(20 nm)/YIG, Gd3Ga5O12(20 nm) /YIG, Cr2O3(20 nm) /YIG, a-YIG/(20 nm) /YIG, a- NFO(20 nm)/YIG, and NiO(20 nm) /YIG bilayers. (b) Excellent linear correlation between spin decay length 𝜆 and Gilbert damping enhancement ∆𝛼=𝛼Insulator /YIG−𝛼YIG. The line is a least-squares linear fit to all data points excluding SrTiO 3. 13 Figure 1 roughness: 0.105 nm-1-0.500.51 -40 -20 0 20 40 M (memu) H (Oe)(a)epi-YIG/GGG a-YIG/GGG -4-2024 2600 2700dIFMR / dH (a.u.) H (Oe)epi-YIG/GGG a-YIG/GGG (b)-0.200.2 -1 01(b)14 Figure 2 05001000 HE (Oe)Py/a-NFO Py/Cr2O3Py/a-YIGPy/NiO (b)-101 -3 -2 -1 0 1 2M/Ms H (kOe)(a) T = 5 K Py/a-NFO Py/Cr2O3Py/a-YIGPy/NiO 10-1100101102103 0 100 200 300 T (K)Py/a-NFO Py/Cr2O3 Py/a-YIGPy/NiO (c)15 Figure 3 10-1100 0 5 10 Insulator thickness t (nm)SrTiO3 GGGCr2O3a-YIGa-NFONiOVISHE(t) / VISHE(0) (c) 10-310-210-1100 0 10 20 30 40 50VISHE(t) / VISHE(0) Insulator thickness t (nm)(b) SrTiO3: = 0.18 nmGd3Ga5O12: = 0.69 nm Cr2O3: = 1.6 nma-YIG: = 3.9 nma-NFO: = 6.3 nmNiO: = 9.8 nm (a)16 Figure 4 0102030 0 5 10 15 20 H (Oe) f (GHz)NiO/YIG a-NFO/YIG a-YIG/YIG Cr2O3/YIG Gd3Ga5O12/YIG single YIG filmSrTiO3/YIG(a) 0510 0 2 4 6 8 10 (nm) (10-4 ) Gd3Ga5O12 Cr2O3 a-YIG a-NFO NiO SrTiO3(b)

2015-09-14

We report a systematic study of spin transport in antiferromagnetic (AF) insulators having a wide range of ordering temperatures. Spin current is dynamically injected from Y3Fe5O12 (YIG) into various AF insulators in Pt/insulator/YIG trilayers. Robust, long-distance spin transport in the AF insulators is observed, which shows strong correlation with the AF ordering temperatures. We find a striking linear relationship between the spin decay length in the AFs and the damping enhancement in YIG, suggesting the critical role of magnetic correlations in the AF insulators as well as at the AF/YIG interfaces for spin transport in magnetic insulators.

Spin Transport in Antiferromagnetic Insulators Mediated by Magnetic Correlations

1509.04336v1

Non-local magnetoresistance in YIG/Pt nanostructures Sebastian T. B. Goennenwein,1, 2, 3, Richard Schlitz,1, 3Matthias Pernpeintner,1, 2, 3Matthias Althammer,1Rudolf Gross,1, 2, 3and Hans Huebl1, 2, 3 1Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany 2Nanosystems Initiative Munich (NIM), Schellingstrae 4, M unchen, Germany 3Physik-Department, Technische Universit at M unchen, Garching, Germany (Dated: September 3, 2018) We study the local and non-local magnetoresistance of thin Pt strips deposited onto yttrium iron garnet. The local magnetoresistive response, inferred from the voltage drop measured along one given Pt strip upon current-biasing it, shows the characteristic magnetization orientation dependence of the spin Hall magnetoresistance. We simultaneously also record the non-local voltage appearing along a second, electrically isolated, Pt strip, separated from the current carrying one by a gap of a few 100 nm. The corresponding non-local magnetoresistance exhibits the symmetry expected for a magnon spin accumulation-driven process, conrming the results recently put forward by Cornelissen et al.[1]. Our magnetotransport data, taken at a series of dierent temperatures as a function of magnetic eld orientation, rotating the externally applied eld in three mutually orthogonal planes, show that the mechanisms behind the spin Hall and the non-local magnetoresistance are qualitatively dierent. In particular, the non-local magnetoresistance vanishes at liquid Helium temperatures, while the spin Hall magnetoresistance prevails. Magneto-resistive phenomena are powerful probes for the magnetic properties. The anisotropic magnetoresis- tance in ferromagnetic metals [2], or the giant magne- toresistance [3] and the tunneling magnetoresistance [4] observed in thin lm heterostructures based on mag- netic metals are widely used in sensing and data stor- age applications [5]. Heterostructures consisting of an insulating magnetic layer (such as yttrium iron garnet Y3Fe5O12(YIG)) and a heavy metal (such as platinum (Pt)) also exhibit a magnetoresistance [6{9]. This so- called spin Hall magnetoresistance (SMR) is due to spin torque transfer across the magnetic insulator/metal in- terface [10]. Owing to the spin Hall eect [11, 12], a spin accumulation arises in the metal, at the interface to the magnet (cf. Fig. 1). Given that is not collinear with the magnetization Min the magnet, the spin accumulation can exert a torque proportional to M(M) onM. In other words, a nite spin current ow across the interface is possible if Mandenclose a nite angle. Since the spin current ow across the interface represents a dissipa- tion channel for the charge transport in the metal layer, its resistance therefore will change with the magnetiza- tion orientation as
R/M(M) [6{9]. In order to experimentally resolve this SMR ngerprint, magneto- resistance measurements as a function of magnetization orientation in at least three dierent planes are manda- tory [7]. Recently, Cornelissen et al. [1] discovered a non-local magneto-resistance eect in YIG/Pt heterostructures and attributed it to magnon accumulation and transport. We will refer to this eect as magnon-mediated magneto- resistance (MMR) in the following. The MMR is ob- served in two parallel Pt strips separated by a distance ddeposited onto YIG, as sketched in Fig. 1. Driving a charge current through the left Pt strip will generate a YIG (magnetic insulator)Pt(metal)JCJS JCJSσd w wFIG. 1. Sketch of the magnon-mediated magnetoresistance (MMR) following Cornelissen et al. [1]. Driving a charge cur- rentJcthrough the left strip of platinum (Pt) results in an or- thogonal spin current with propagation direction Jsand spin polarization . The spin accumulation in the metal induces a magnon accumulation (wiggly red arrows) in the adjacent magnetic insulator yttrium iron garnet (YIG). This magnon accumulation decays with increasing distance to the current- carrying Pt injector strip (shaded region). If a second Pt strip, electrically isolated from the rst, is within the range of magnon accumulation, a spin current will ow back from the magnetic insulator into the second Pt strip and give rise to an inverse spin Hall charge current there. The two Pt strips of widthware separated by an edge-to-edge distance d. spin accumulation in Pt at the interfaces. As shown in Fig. 1,is perpendicular to the direction of charge cur- rent ow Jcand orthogonal to the spin current ow Js across the interface. This spin accumulation in particular also induces a magnon (spin) accumulation in YIG [1] { an eect which usually is assumed small and ignored in the treatment of SMR [10]. The non-equilibrium magnon accumulation diuses out into the magnetic insulator, as indicated by the shading in Fig. 1. Given that the sec- ond Pt electrode is close enough such that the diusing magnons can reach it before decaying, the magnon accu-arXiv:1508.06130v1 [cond-mat.mtrl-sci] 25 Aug 20152 γH jtn Hjtn α βH jtn(b) ip(a) (c) oopj (d)ooptI+ -+ -Vloc+ -Vnl 10µm FIG. 2. (a) Optical micrograph of a typical YIG/Pt nanos- tructure. The two bright thin vertical lines in the center of the gure are the two Pt strips under investigation, the YIG lm beneath appears black. A current source attached to the left Pt strip supplies a constant current I. The voltage drop Vloc along the same Pt strip, as well as the non-local voltage drop Vnlalong the second Pt strip, are simultaneously recorded. Panels (b), (c), and (d) show the dierent magnetic eld ro- tation planes and the corresponding magnetic eld orientation angles,and , respectively. mulation will drive a spin current back into the second Pt electrode. In turn, this spin current then generates an inverse spin Hall charge current in the second Pt elec- trode. A large non-local charge current is expected for Mjj, since the magnons (the spin angular momenta) be- neath the rst Pt strip then can diuse across the gap to the second Pt strip. For M?, in contrast, the non- local signal should be signicantly reduced, since now spin torque transfer suppresses the magnon accumula- tion and/or the magnon propagation. This picture of a non-local magnon-based magnetoresistance, put forward by Cornelissen et al. in Ref. [1], to date only has been tested against non-local inverse spin Hall voltage data taken as a function of magnetic eld orientation for the magnetic eld in the plane of the YIG lm. Neither a di- rect comparison of the non-local magnetoresistance with the SMR, nor a study of the evolution of the non-local voltage as a function of out-of-plane magnetization orien- tation, have been put forward. Note also that the MMR is dierent from the non-local electrical detection of spin pumping [13, 14] or magnetoresistance experiments in metallic spin valves [15, 16], since the YIG (the mag- netically ordered material) only passively acts as a 'spin transport' medium, contacted with conventional metallic nano-electrodes. In this letter, we systematically compare the magnetization-orientation dependent evolution of the (non-local) MMR and the (local) SMR in YIG/Pt nanos-tructures. We have simultaneously measured the MMR and SMR as sketched in Fig. 2, rotating the externally applied magnetic eld of xed magnitude in three mu- tually orthogonal planes. Our data taken close to room temperature corroborate the picture that the MMR is mediated by magnon diusion, and reveal a qualitatively dierent evolution of SMR and MMR as a function of temperature. The YIG/Pt bilayers investigated were obtained start- ing from a commercially available, 3 m thick YIG lm grown onto GGG via liquid phase epitaxy. The as- purchased YIG lms were cleaned in a so-called Piranha etch solution (3 volumes H 2SO4mixed with 1 volume H2O2) for 120 seconds and annealed in 50 bar oxygen for 40 minutes at 500C. Without breaking the vacuum, the samples were then transferred to an electron beam evaporation chamber, where we deposited a 9 :6 nm thick Pt lm. After removing the sample from the vacuum chamber, the Pt strips were dened using a combina- tion of electron beam lithography and Argon ion beam milling. The Pt strips studied here are 100 m long and have a lateral width of w= 1m. We focus on a de- vice with a strip separation (edge to edge, see Fig. 2) of d= 200 nm in the following, but have also studied devices withd= 500 nm and d= 1000 nm. For the magneto- transport experiments, the YIG/Pt nanostructures were wire-bonded to a chip carrier and inserted into the vari- able temperature insert of a superconducting 3D vector magnet cryostat, allowing to rotate the externally ap- plied magnetic eld 0H2 T in any desired plane with respect to the sample. The magnetotransport data were taken by current-biasing one Pt strip with I= 100A using a Keithley 2400 sourcemeter, while simultaneously recording the local voltage drop Vloc(along the strip car- rying the current) as well as the non-local voltage Vnl appearing along the second, nearby Pt strip using Keith- ley 2182 nanovoltmeters as sketched in Fig. 2(a). To enhance sensitivity, we use the current reversal (delta mode) method [17]. We here discuss transport data taken as a function of magnetic eld orientation, for xed eld magnitude H. To ensure full saturation of the YIG mag- netization along the externally applied magnetic eld, we took all data using the maximum available magnetic eld j0Hj= 2 T. We rotated the eld in three mutually or- thogonal planes, as sketched in Fig. 2(b),(c),(d). The rotation of Haround the direction nnormal to the lm plane, such that the magnetic eld always resides within the lm plane, is referred to as ip (Fig. 2(b)). In the oopj rotation depicted in Fig. 2(c), His rotated around the direction j(along which the charge current ows), while in the oopt rotation depicted in Fig. 2(d), His rotated around the direction t, which is orthogonal to jandn. Figure 3 exemplarily shows magneto-transport data taken on the sample with d= 200 nm, with the vari- able temperature insert thermalized to T= 300 K. Since all data discussed in the following were taken with3 370.7370.8 -90 0 90 180 270-200-1000Vloc,0 (b)(a) oopt oopjVloc(mV) ip∆VSMR 300K,2T ∆VnlVnl(nV) α,β,γ(deg)oopt oopjip FIG. 3. The local voltage Vloc(panel (a)) and the non-local voltageVnl(panel (b)), recorded in two Pt strips separated by d= 200 nm as a function of the orientation (ip),(oopj), (oopt) of the externally applied magnetic eld H(see Fig. 2). The data were taken at T= 300 K and 0H= 2 T.Vlocis essentially constant in the oopt rotation plane (blue triangles), and varies in a sin2-type fashion in the ip (black rectangles) and oopj (red circles) rotation planes, respectively. The sin2 modulation with amplitude
VSMR<0 (gray vertical arrow) is superimposed on a constant voltage of magnitude Vloc;0 (horizontal dashed arrow). Vnlis qualitatively similar to Vloc. However,Vnlalways is negative, and does not show a constant oset voltage but only a sin2-type modulation with amplitude
Vnl. j0Hj= 2 T which exceeds the anisotropy elds in YIG by at least one order of magnitude, we assume MjjH and use the magnetic eld orientations ,and (see Fig. 2(b),(c),(d)) synonymously for MandH. The lo- cal voltage Vlocdepicted in Fig. 3(a) exhibits the depen- dence on magnetization orientation characteristic of the SMR. Upon rotating the external magnetic eld in the plane of the YIG lm (ip, see Fig. 2(b)), or in the plane perpendicular to j(oopj, Fig. 2(c)), a sin2-like modu- lation ofVlocwith amplitude
VSMR<0 on top of a constant level Vloc;0is observed. Vloc;0hereby is the voltage level observed when the YIG magnetization is along the tdirection (e.g., = 90in ip or= 90 in oopj). The magnitude of the SMR j
VSMRj=Vloc;0 4:5104agrees reasonably well with the SMR ampli- tude6104observed in YIG/Pt heterostructures in which the10 nm thick Pt lms were deposited in-situ, 0501001502002503000-50-100-150-200-2500-50-100-150∆VSMR(µV) 160200240280320360400(a) ∆VSMR Vloc,0(mV) Vloc,0 (b)∆Vnl(nV) Temperature(K)FIG. 4. Evolution of (a) the oset voltage Vloc;0(open squares) and the SMR modulation voltage
VSMR (full squares) recorded in the local geometry, and of (b) the non- local voltage change
Vnl(full circles) as a function of tem- perature for the d= 200 nm sample. directly after the YIG growth process [7]. The evolu- tion of
VSMR andVloc;0with temperature is shown in Fig. 4(a).j
VSMR(T)jmonotonically decreases by about a factor of 3 from T= 300 K to T= 5 K, very simi- larly to the behaviour observed in other YIG/Pt samples fabricated at the Walther-Meissner-Institut [18]. Vloc;0 only decreases by about 15% in the same temperature interval, showing that defect or surface scattering is very strong in the thin Pt lm. The non-local voltage Vnlrecorded in the same experi- ment is shown in Fig. 3(b). Vnlis qualitatively very simi- lar toVloc, showing a sin2-like modulation with amplitude
Vnl. We would like to stress, however, that on the one hand, there is no nite constant oset in Vnl, such that Vnl= 0 (to within the experimental noise) for the oopt rotation. On the other hand, Vnlinvariably assumes neg- ative values (Vnl0). According to our wiring scheme (Fig. 2(a)), a negative non-local voltage implies that the non-local inverse spin Hall charge current arising in the second Pt strip (due to a diusion of the magnon accu- mulation generated beneath the rst Pt strip) must ow in the same direction as the charge current in the rst Pt strip. Since we detect Vnlusing open circuit bound- ary conditions, this non-local ISHE charge current is ex- actly balanced by an electric potential of opposite (that is negative) sign. The negative sign of
Vnlin our ex- periments thus is consistent with the positive non-local
R > 0 reported by Cornelissen et al. [1], since these4 authors use an inverted sign convention for the non-local voltage signal. The data shown in Fig. 3 furthermore are consistent with the notion that magnon accumulation is at the origin of Vnl, since one would expect maximum magnon diusion signal (maximum Vnl<0 in our exper- iment) for Mjjjjtand minimal magnon diusion signal (Vnl= 0) for M?, which translates to Vnl= 0 for Mjjj andMjjn. The non-local voltage observed in our exper- iment in all three rotation planes indeed conrms this expectation. Note also that magneto-thermal (spin See- beck) voltages cannot account for Vnl, since these have a qualitatively dierent dependence on magnetization ori- entation [1, 19]. The magnitudej
Vnlj250 nV of the magnetization- orientation dependent modulation in the non-local volt- age atT= 300 K is about 1000 times smaller than the localj
VSMRj  150V. Fitting the
Vnlob- served for pairs of strips with separation d= 200 nm, d= 500 nm and d= 1m, respectively, using
Vnl= (C=) exp(d=)=(1exp(2d=)) derived as Eq. (7) in Ref. [1] for 1D spin diusion, we obtain 700 nm. This value ofis about one order of magnitude smaller than the value reported by Cornelissen et al. [1] for their sam- ples. The discrepancy might be evidence for enhanced magnon scattering owing to YIG surface damage caused by our fabrication process. In addition, 700 nm is smaller than the YIG lm thickness of 3 m in our case, suggesting that diusion in more than one dimension could be important. To conclusively resolve this point, multiple samples with dierent YIG lm thicknesses and a series of dierent Pt strip separations dmust be sys- tematically compared, which is beyond the scope of this work. Interestingly, the temperature dependencies of
Vnl and
VSMRare very dierent. As evident from Fig. 4(b), the magnitude of
VSMR at lowTis only about a factor of 3 smaller than at room temperature, while
Vnl= 0 forT10 K. The strong decrease in
Vnl(T) can be rationalized considering an increase of the magnon prop- agation length with decreasing T. In a simple picture, the non-equilibrium magnons generated at the YIG/Pt interface spread across a volume Vmag/3, such that the magnon accumulation (viz. the non-equilibrium magnon density) scaling with 1 =Vmagdecreases with T. In the limit of innite , the magnon accumulation and thus also
Vnlvanishes. More sophisticated theoretical anal- yses corroborate this intuitive picture [20, 21]. Note also that the nite SMR signal at low Tis direct evidence that the spin Hall eect is only weakly temperature de- pendent [18], such that the decrease of the MMR (viz. of
Vnl) withTcannot be simply attributed to spin Hall physics. In conclusion, we have simultaneously measured the local and the non-local magnetoresistive response of two parallel Pt strips separated by a gap of a few 100 nm, deposited onto yttrium iron garnet. The local mag-netoresistance (current-biasing one Pt strip and mea- suring the magnetization-orientation dependent voltage drop along this same Pt strip) shows the characteristic ngerprint of spin Hall magnetoresistance, as expected for a YIG/Pt heterostructure. We furthermore observe a non-local voltage Vnlalong the second, electrically iso- lated Pt strip upon current biasing the rst one. Our data taken at room temperature conrm the results put forward by Cornelissen et al. [1]. In addition, we have measuredVnlas a function of magnetization orientation in three mutually orthogonal rotation planes, and studied the evolution of both the local and the non-local magne- toresistance from room temperature down to 5 K. All our experimental data can be consistently understood assum- ing that the non-local magnetoresistance is mediated via magnon accumulation. We gratefully acknowledge discussions with G. E. W. Bauer and S. Klingler, and funding via the priority programme spin-caloric transport (spinCAT) of Deutsche Forschungsgemeinschaft, project GO 944/4. goennenwein@wmi.badw.de [1] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, arXiv:1505.06325 (2015). [2] T. R. McGuire and R. I. Potter, IEEE Trans. Mag. MAG-11 , 1018 (1975). [3] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petro, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). [4] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser- vey, Phys. Rev. Lett. 74, 3273 (1995). [5] R. C. O'Handley, Modern Magnetic Materials (John Wi- ley, New York, 2000). [6] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). [7] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). [8] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). [9] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Nale- tov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). [10] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). [11] M. I. D'akonov and V. I. Perel, JETP Lett. 13, 467 (1971). [12] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [13] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). [14] E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and5 M. Shiraishi, Phys. Rev. Lett. 110, 127201 (2013). [15] F. Jedema, A. Filip, and B. van Wees, Nature 410, 345 (2001). [16] H. X. Tang, F. G. Monzon, F. J. Jedema, A. T. Filip, B. J. van Wees, and M. L. Roukes, in Semiconduc- tor Spintronics and Quantum Computation , NanoScience and Technology, edited by D. D. Awschalom, D. Loss, and N. Samarth (Springer, Berlin, 2002) Chap. 2, pp. 31{92. [17] D. R uer, Non-local Phenomena in Metallic Nanostruc- tures , Diploma thesis, Walther-Meiner-Institut, Tech-nische Universit at M unchen (2009). [18] S. Meyer, M. Althammer, S. Gepr ags, M. Opel, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 104, 242411 (2014). [19] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [20] J. Xiao and G. E. W. Bauer, arXiv:1508.02486 (2015). [21] S. A. Bender and Y. Tserkovnyak, Phys. Rev. B 91, 140402 (2015).

2015-08-25

We study the local and non-local magnetoresistance of thin Pt strips deposited onto yttrium iron garnet. The local magnetoresistive response, inferred from the voltage drop measured along one given Pt strip upon current-biasing it, shows the characteristic magnetization orientation dependence of the spin Hall magnetoresistance. We simultaneously also record the non-local voltage appearing along a second, electrically isolated, Pt strip, separated from the current carrying one by a gap of a few 100 nm. The corresponding non-local magnetoresistance exhibits the symmetry expected for a magnon spin accumulation-driven process, confirming the results recently put forward by Cornelissen et al. [1]. Our magnetotransport data, taken at a series of different temperatures as a function of magnetic field orientation, rotating the externally applied field in three mutually orthogonal planes, show that the mechanisms behind the spin Hall and the non-local magnetoresistance are qualitatively different. In particular, the non-local magnetoresistance vanishes at liquid Helium temperatures, while the spin Hall magnetoresistance prevails.

Non-local magnetoresistance in YIG/Pt nanostructures

1508.06130v1

arXiv:2005.14133v2 [cond-mat.mtrl-sci] 2 Aug 2020A First Principle Study on Magneto-Optical Effects in Ferrom agnetic Semiconductors Y3Fe5O12and Bi 3Fe5O12 Wei-Kuo Li1and Guang-Yu Guo1,2,∗ 1Department of Physics and Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan 2Physics Division, National Center for Theoretical Science s, Hsinchu 30013, Taiwan (Dated: August 4, 2020) The magneto-optical (MO) effects not only are a powerful prob e of magnetism and electronic structure of magnetic solids but also have valuable applica tions in high-density data-storage tech- nology. Yttrium iron garnet (Y 3Fe5O12) (YIG) and bismuth iron garnet (Bi 3Fe5O12) (BIG) are two widely used magnetic semiconductors with significant magne to-optical effects. In particular, YIG has been routinely used as a spin current injector. In this pa per, we present a thorough theoretical investigation on magnetism, electronic, optical and MO pro perties of YIG and BIG, based on the density functional theory with the generalized gradient ap proximation plus onsite Coulomb repul- sion. We find that YIG exhibits significant MO Kerr and Faraday effects in UV frequency range that are comparable to ferromagnetic iron. Strikingly, BIG shows gigantic MO effects in visible frequency region that are several times larger than YIG. We fi nd that these distinctly different MO properties of YIG and BIG result from the fact that the magnit ude of the calculated MO conduc- tivity (σxy) of BIG is one order of magnitude larger than that of YIG. Inte restingly, the calculated band structures reveal that both valence and conduction ban ds across the semiconducting band gap in BIG are purely spin-down states, i.e., BIG is a single spin semiconductor. They also show that in YIG, Ysdorbitals mix mainly with the high lying conduction bands, le aving Fe dorbital dominated lower conduction bands almost unaffected by the SOC on the Y at om. In contrast, Bi porbitals in BIG hybridize significantly with Fe dorbitals in the lower conduction bands, leading to large SOC-induced band splitting in the bands. Consequently, the MO transitions between the upper valence bands and lower conduction bands are greatly enhanc ed when Y is replaced by heavier Bi. This finding suggests a guideline in search for materials wit h desired MO effects. Our calculated Kerr and Faraday rotation angles of YIG agree well with the av ailable experimental values. Our calculated Faraday rotation angles for BIG are in nearly per fect agreement with the measured ones. Thus, we hope that our predicted giant MO Kerr effect in BIG wil l stimulate further MOKE ex- periments on high quality BIG crystals. Our interesting find ings show that the iron garnets not only offer an useful platform for exploring the interplay of m icrowave, spin current, magnetism, and optics degrees of freedom, but also have promising applicat ions in high density MO data-storage and low-power consumption spintronic nanodevices. I. INTRODUCTION Yttrium iron garnet (Y 3Fe5O12, YIG) is a ferrimag- netic semiconductor with excellent magnetic properties such as high curie temperature Tc[1], low Gilbert damp- ingα∼6.7×10−5[2–4] and long spin wave propragating length [5]. Various applications such as spin pumping re- quire a non-metallic magnet. YIG is thus routinely used for spin pumping purposes [4]. It is also widely used as a magnetic insulating substrate for purposes such as intro- ducingmagneticproximityeffect whileavoidingelectrical short-cut. [6] YIG has high Curie temperature, which is good for applications across a wide temperature range. The low Gilbert damping of YIG also makes it a good microwave material. YIG thus becomes a famous mate- rial in the field of spintronics, where coupling between magnetism, microwave and spin current becomes possi- ble. Magneto-optical (MO) effects are important examples of light-matter interactions in magnetic phases. [7, 8] ∗gyguo@phys.ntu.edu.twWhen a linearly polarized light beam is shined onto a magnetic material, the reflected and transmitted light becomes elliptically polarized. The principal axis is ro- tated with respect to the polarization direction of inci- dentlight beam. The formerandlattereffects aretermed MO Kerr (MOKE) and MO Faraday (MOFE) effects, respectively. MOKE allowes us to detect the magnetiza- tion locally with a high spatial and temporal resolution in a non-invasive fashion. Furthermore, magnetic ma- terials with large MOKE would find valuable MO stor- age and sensor applications [9, 10]. Thus it has been widely used to probe the electronic and magnetic proper- ties of solids, surface, thin films and 2D magnets [8]. On the other hand, MOFE can be used as a time-reversal symmetry-breaking element in optics [11], and its ap- plications such as optical isolators are consequenses of time-reversal symmetry-breaking [12]. Magnetic materi- als with large Kerr or Faraday rotation angles have tech- nological applications. YIG is also known to be MO active [13]. Various ex- periments have been carried out to study the MOKE and MOFE of iron garnets in the visible and near-UV regime [14, 15]. Substituting yttrium with bismuth results in2 bismuth iron garnet (Bi 3Fe5O12) (BIG). BIG has ap- proximately 7 times larger Faraday rotation angles than that of YIG. The effect of doping bismuth into YIG on the MOFE spectrum was studied [16, 17]. The large ra- dius of bismuth atoms seems to make bulk BIG unstable. Thus high quality BIG film is difficult to synthesize [18]. Though numerous experimental studies have been done on these systems, first-principle calculations are scarce. This is probably due to the complexity of the structures ofBIG and YIG. As shown in Fig. 1(a), they havea total of 80 atoms in the primitive cell. Although the electronic structures of YIG and BIG have been theoretically stud- ied [19, 20], no first principle calculation on the MOKE or MOFE spectra of YIG and BIG have been reported. Therefore, here we carry out a systematic first-principle density functional study on the optical and MO proper- ties of YIG and BIG. The rest of this paper is organized as follows. A brief description of the crystal structures of YIG and BIG as well as the theoretical methods used is given in Sec. II. In Sec. III, the calculated magnetic mo- ments, electronic structure, optical conductivities, MO Kerr and Faraday effects are presented. Finally, the con- clusions drawn from this work are given in section IV. II. CRYSTAL STRUCTURE AND COMPUTATIONAL METHODS YIG and BIG crystalize in the cubic structure with space group Ia3d[21, 22], as illustrated in Fig. 1(a). In each unit cell, there are 48 oxygen atoms at the Wyck- off 96h positions, 8 octahedrally coordinated iron atoms (FeO) at the 16a positions, and 12 tetrahedrally coor- dinated iron atoms (FeT) at the 24d positions in the primitive cell. In other words, there are two FeOions and three FeTions per formula unit (f.u.). The ex- perimental lattice constant a= 12.376˚A, and the ex- perimental Wyckoff parameters for oxygen atoms are (x,y,z) = (0.9726,0.0572,0.1492). [21] The experimen- tal lattice constant for BIG a= 12.6469˚A. [22] Accurate oxygen position measurement for BIG is still on demand and under debate [18]. Therefore we use the experimen- tal lattice constant for BIG with the atomic positions determined theoretically (see Table I), as described next. We use the experimental lattice constant and atomic po- sitions for all YIG calculations, Our first principle calculations are based on the den- sity functional theory with the generalized gradient ap- proximation (GGA) of the Perdew-Burke-Ernzerhof for- mula [23] to the electron exchange-correlation potential. Furthermore, we use the GGA + Umethod to have a better description for on-site interaction for Fe delec- trons. [24] Here we set U= 4.0 eV, which was found to be rather appropriate for iron oxides [25]. Indeed, as we will show below, the optical and MO spectra calculated using this Uvalue agree rather well with the available experimental spectra. All the calculations are carried out by using the accurate projector-augmented wave [26]TABLE I. Structuralparameters of Y 3Fe5O12and Bi 3Fe5O12. For YIG, experimental lattice constant a= 12.376˚A and oxygen positions [21] are used. For BIG, experimental latti ce constant a= 12.6469˚A [22] is used while the oxygen positions are determined theoretically. Y3Fe5O12Wyckoff position x y z FeO16a 0.0000 0.0000 0.0000 FeT24d 0.3750 0.0000 0.2500 Y 24c 0.1250 0.0000 0.2500 O 96h 0.9726a0.0572a0.1492a Bi3Fe5O12Wyckoff position x y z FeO16a 0.0000 0.0000 0.0000 FeT24d 0.3750 0.0000 0.2500 Bi 24c 0.1250 0.0000 0.2500 O 96h 0.0540 0.0300 0.1485 aRef. 21. method, as implemented in Vienna ab initio Simulation Package (VASP). [27, 28] A large energy cutoff of 450 eV for the plane-wavebasis is used. A 6 ×6×6k-point mesh is used for both systems in the self-consistent chargeden- sitycalculations. The densityofstates(DOS) calculation is performed with a denser k-point mesh of 10 ×10×10. Wefirstcalculatethe opticalconductivitytensorwhich determine the MOKE and MOFE. We let the magnetiza- tion of our systems be along (001) ( z) direction. In this case, our systems have the four-fold rotational symmetry along the zaxis and thus the optical conductivity tensor can be written in the following form [29]: σ= σxxσxy0 −σxyσxx0 0 0 σzz . (1) The optical conductivity tensor can be formulated within the linear response theory. Here the real part of the di- agonal elements and imaginary part of the off-diagonal elements are given by [29–31]: σ1 aa(ω) =πe2 /planckover2pi1ωm2/summationdisplay i,j/integraldisplay BZdk (2π)3|pa ij|2δ(ǫkj−ǫki−/planckover2pi1ω), (2) σ2 xy(ω) =πe2 /planckover2pi1ωm2/summationdisplay i,j/integraldisplay BZdk (2π)3Im[px ijpy ji]δ(ǫkj−ǫki−/planckover2pi1ω), (3) where/planckover2pi1ωis the photon energy, and ǫki(j)are the en- ergy eigenvalues of occupied (unoccupied) states. The transition matrix elements pa ij=/angbracketleftkj|ˆpa|ki/angbracketrightwhere|ki(j)/angbracketright are thei(j)th occupied(unoccupied) states at k-pointk, and ˆpais the Cartesian component aof the momentum operator. The imaginary part of the diagonal elements and the real part of the off-diagonal elements are then obtained from σ1 aa(ω) andσ2 xy(ω), respectively, via the Kramers-Kronig transformations as follows: σ2 aa(ω) =−2ω πP/integraldisplay∞ 0σ1 aa(ω′) ω′2−ω2dω′,(4)3 FIG. 1. (a) 1/8 of BIG conventional unit cell. Oxygen atoms are shown as red balls; bismuth atoms are shown as purple balls; FeTatoms are shown as yellow balls; FeOatoms are shown as blue balls. (b) Brillouin zone of both YIG and BIG. The red lines denote the high symmetry lines where the calculated energy bands will be plotted. σ1 xy(ω) =2 πP/integraldisplay∞ 0ω′σ2 xy(ω′) ω′2−ω2dω′, (5) wherePdenotes the principal value of the integration. We can see that Eq. (2) and Eq. (3) neglect transitions acrossdifferent k-points since the momentum of the opti- cal photon is negligibly small compared with the electron crystal momentum and thus only the direct interband transitions need to be considered. In our calculations pa ijare obtained in the PAW formalism [32]. We use a 10×10×10k-point mesh and the Brillouin zone inte- gration is carried out with the linear tetrahedron method (see [33] and references therein), which leads to well con- verged results. To ensure that the σ2 aa(ω) andσ1 xy(ω) in the optical frequency range (e.g., /planckover2pi1ω <8 eV) obtained via Eqs. (4) and (5) are converged, we include the unoc- cupied states at least 21 eV above the Fermi energy, i.e., a total of 1200 (1300) bands are used in the YIG (BIG) calculations. For a bulk magnetic material, the complex polar Kerr rotation angle is given by [34, 35], θK+iǫK=−σxy σxx/radicalbig 1+i(4π/ω)σxx. (6)Similarly, the complex Faraday rotation angle for a thin film can be written as [39] θF+iǫF=ωd 2c(n+−n−), (7) wheren+andn−represent the refractive indices for left- and right-handed polarized lights, respectively, and are related to the corresponding dielectric function (or opti- cal conductivity via expressions n2 ±=ε±= 1+4πi ωσ±= 1 +4πi ω(σxx±iσxy). Here the real parts of the optical conductivity σ±can be written as σ1 ±(ω) =πe2 /planckover2pi1ωm2/summationdisplay i,j/integraldisplay BZdk (2π)3|Π± ij|2δ(ǫkj−ǫki−/planckover2pi1ω), (8) whereΠ± ij=/angbracketleftkj|1√ 2(ˆpx±iˆpy)|ki/angbracketright. Clearly, σxy=1 2i(σ+− σ−), and this shows that σxywould be nonzeroonly if σ+ andσ−are different. In other words, magnetic circular dichroism is the fundamental cause of the nonzero σxy and hence the MO effects. III. RESULTS AND DISCUSSION A. Magnetic moments Here we first present calculated total and atom- decomposed magnetic moments in Table I. As expected, Y3Fe5O12is a ferrimagnet in which Fe ions of the same type couple ferromagnetically while Fe ions of different types couple antiferromagnetically. Since there are two FeOions and three FeTions in a unit cell, Y 3Fe5O12is ferrimagnetic with a total magnetic moment per f.u. be- ing∼5.0µB(see Table I). The calculated spin magnetic moments of Fe ions of both types are ∼4.0µB, being consistent with the high spin state of Fe+2(d5↑t1↓ 2g) ions in either octahedral or tetrahedral crystal field. We note that the orbital magnetic moments of Fe are parallel to their spin magnetic moments. Nonetheless, the calcu- lated orbital magnetic moments of Fe are small, because of strong crystal field quenching. Interestingly, there is a significantspin magnetic moment on eachO ion, and this together with the spin magnetic moment of one net Fe ion per f. u. leads to the total spin magnetic moment per f.u. of∼5.0µB. The calculatedFe magneticmomentsfor both symmetry sites agree rather well with the measured ones of∼4.0µB. [36] The calculated total magnetization of∼5.0µB/f.u. is also in excellent agreement with the experiment. [36] Bi3Fe5O12is also predicted to be ferrimagnetic, al- though the calculated magnetic moments of both FeO and FeTions are slightly smaller than the corresponding ones in Y 3Fe5O12(see Table I). The total magnetization and local magnetic moments of the other ions in BIG are almost identical to that in YIG. However, the experimen- talmtotfor BIG is only 4 .4µB, [37] being significantly smaller than the calculated value. As mentioned before,4 TABLE II. Total spin magnetic moment ( mt s), atomic spin magnetic moments ( mFe s,mO s,mY(Bi) s), atomic Fe orbital magnetic moments ( mFe o) and band gap ( Eg) of ferrimagnetic Y 3Fe5O12and Bi 3Fe5O12from the full-relativistic electronic structure calculations. For comparison, the available measured opti calEgand total magnetization mt expare also listed. structure mt(mt exp)mFe(16a) s(mFe(16a) o)mFe(24d) s(mFe(24d) o) mO s mY(Bi) s Eg(Eexp g) (µB/f.u.) ( µB/atom) ( µB/atom) ( µB/atom) ( µB/atom) (eV) Y3Fe5O124.999 (5.0a) -4.177 (-0.016) 4.075 (0.018) 0.067 0.005 1.81 (2.4b) Bi3Fe5O124.996 (4.4c) -4.161 (-0.018) 4.068 (0.019) 0.066 0.005 1.82 (2.1d) aRef. 36.bRef. 14.cRef. 37.dRef. 17. stablehigh qualityBIGcrystalsarehardto grow. Conse- quently, this notable discrepancy in total magnetization between the calculation and the previous experiment [37] could be due to the poor quality of the samples used in the experiment. B. Electronic structure Here we present the calculated scalar-relativistic band structures of YIG and BIG in Fig. 2(a) and Fig. 3(a), respectively. The calculated band structures show that YIG and BIG are both direct band-gap semiconductors, where the conduction band minimum (CBM) and va- lence band maximum (VBM) are both located at the Γ point. For BIG, both CBM and VBM are purely spin-up bands. This means that BIG is a single-spin semiconduc- tor, which may find applications for spintronic and spin photovoltaic devices. The origin of the MO effects is the magnetic circular dichorism [see Eq. (8)], as mentioned above, which cannot occur without the presence of the spin-orbit coupling (SOC). Therefore, it is useful to ex- amine how the SOC influence the band structures. The fully relativistic band structures for YIG and BIG are presented in Fig. 2(b) and Fig. 3(b), respectively. First, we notice that with the inclusion of the SOC, YIG and BIG are still direct band-gap semiconductors, where the CBM and VBM are both located at the Γ point. Second, Fig. 3(b) indicates that when the SOC is considered, the BIG band structure changes significantly, while the YIG band structure hardly changes [see Fig. 2(b)]. For exam- ple, the band gap for BIG decreases from 2.0 to 1.8 eV after the SOC is included. Also, the gap, which was at 3.4 to 3.7 eV above the Fermi energy [see Fig. 3(a)], now becomes from 3.9 to 4.5 eV above the Fermi energy [see Fig. 3(b)]. Interestingly, the substitution of yittrium by bismuth not only enhances the SOC but also changes the electronic band structure significantly, as can be seen by comparing Figs. 2 and 3. We also calculate total as well as site-, orbital-, and spin-projected densities of states (DOS) for YIG and BIG, as displayed in Fig. (4) and (5), respectively. First, Figs. (4) and (5) show that in both YIG and BIG, the upper valence bands ranging from -4.0 to 0.0 eV, are dominated by O p-orbitals with minor contributions from Fed-orbitals as well as Y d-orbitals in YIG and Bi FIG. 2. (a) Scalar-relativistic spin-polarized band struc ture and (b) fully relativistic band structure of Y 3Fe5O12.5 FIG. 3. (a) Scalar-relativistic spin-polarized band struc ture and (b) fully relativistic band structure of Bi 3Fe5O12. sp-orbitals in BIG. Second, the lower conduction band manifold, ranging from 1.8 to ∼3.9 eV in YIG (Fig. 4) and from 2.0 to 3.4 eV in BIG (Fig. 5), stems predom- inately from Fe d-orbitals with small contributions from Op-orbitals. Therefore, the semiconducting band gaps in YIG and BIG are mainly of the charge transfer type. Furthermore, on the FeTsites, the d-DOS in this conduc- tion band is almost fully spin-down [see Figs. 4(e) and 5(e)]. On the FeOsites, on the other hand, the d-DOS in this conduction band is almost purely spin-up [see Figs. 4(d) and 5(d)]. Here, the DOS peak marked amostly consists of t2gorbital while that marked babove peak a, is made up of mainly egorbital. The gap between peaks FIG. 4. Spin-polarized density of states (DOS) of Y 3Fe5O12 from the scalar-relativistic calculation. FIG. 5. Spin-polarized density of states (DOS) of Bi 3Fe5O12 from the scalar-relativistic calculation.6 FIG. 6. Calculated optical conductivity of Y 3Fe5O12. (a) Real part and (b) imaginary part of the diagonal element; (c) imaginary part and (d) real part of the off-diagonal ele- ment. All the spectra have been convoluted with a Lorentzian of 0.3 eV to simulate the finite quasiparticle lifetime effect s. Red lines are the optical conductivity derived from the expe r- imental dielectric constant. [14] aandbis thus caused by the crystal field splitting. Figure 4 indicates that in YIG, the upper conduction bands from 4.4 to 6.0 eV are mainly of Y dorbital char- acter with some contribution from O porbitals. In BIG, on the other hand, the upper conduction bands from 3.6 to 6.0 eV are mainly the Bi and O porbital hybridized bands (seeFig. 5). Notably, there is sizableBi spDOS in the lower conduction band region from 2.0 to 3.4 eV (see Fig. 5(c)], indicating that the lower conduction bands in BIG are significantly mixed with Bi sporbitals, as no- ticed already by Oikawa et al.[20], Since the SOC of the Biporbitals are very strong, this explains why the band width of the lower conduction bands in BIG increases from∼1.4 to 2.1 eV when the SOC is included (see Fig. 3). In contrast, the band width of the lower conduction bands in YIG remains unaffected by the SOC (see Fig. 2). This also explains why the MO effects in BIG are much stronger than in YIG, as reported in Sec. III.D. below. C. Optical Conductivity Here we present the optical and magneto-optical con- ductivities for YIG and BIG which are ingredients for calculating the Kerrand Faradayrotation angles [see Eq. (6) and Eq. (7)]. In particular, the MO conductivity FIG. 7. Calculated optical conductivity of Bi 3Fe5O12. (a) Real part and (b) imaginary part of the diagonal element; (c) imaginary part and (d) real part of the off-diagonal ele- ment. All the spectra have been convoluted with a Lorentzian of 0.3 eV to simulate the finite quasiparticle lifetime effect s. Red lines are the optical conductivity derived from the expe r- imental dielectric constant. [17] (i.e., the off-diagonal element of the conductivity tensor σxy) is crucial, as shown by Eq. (8). Calculated optical conductivity spectra of YIG and BIG are plotted as a function of photon energy in Fig. 6 and Fig. 7, respec- tively. For YIG, the real part of the diagonal element of the conductivity tensor ( σ1 xx) starts to increase rapidly from the absorption edge ( ∼2.3 eV) to ∼4.0 eV, and then further increases with a smaller slope up to ∼5.6 eV [see Fig. 6(a)]. It then decreases slightly until 6.6 eV and finally increases again with a much steeper slope up to∼8.0 eV. Similarly, in BIG, σ1 xxincreases steeply from the absorption edge ( ∼2.0 eV) to ∼4.0 eV, and then further increases with a smaller slope up to ∼6.0 eV [see Fig. 7(a)]. It then decrease steadily from ∼6.0 eV to∼8.0 eV. The behaviors of the imaginary part of the diagonal element ( σ2 xx) of YIG and BIG are rather similar in the energy range up to 5.0 eV [see Figs. 6(b) and 7(b)]. The σ2 xxspectrum has a broad valley at ∼3.5 eV (∼3.0 eV) in the case of YIG (BIG). However, the σ2 xxspectra of YIG and BIG differ from each other for energy>5.0 eV. There is a sign change in σ2 xxoccuring at∼5.8 eV for BIG, while there is no such a sign change inσ2 xxof YIG up to 8.0 eV. The striking difference in the off-diagonal element of theconductivity( σxy)(i.e., magneto-opticalconductivity or magnetic circular dichroism) between YIG and BIG is thatσxyof BIG is almost ten times larger than that of7 YIG (see Figs. 6 and 7). Nonetheless, the line shapes of the off-diagonal element of YIG and BIG are rather similar except that their signs seem to be opposite and their peaks appear at quite different energy positions. In particular,inthelowenergyrangeupto ∼4.4eV,theline shape of the imaginary part of the off-diagonal element (σ2 xy) of BIG looks like a ”W” [see Fig. 7(c)], while that of YIG in the energy region up to ∼7.0 eV seems to have the inverted ”W” shape [see Fig. 6(c)], The main difference is that the σ2 xyof BIG decreases oscillatorily from 4.4 to 8.0 eV. On the other hand, the line shape of the real part of the off-diagonal element ( σ1 xy) of BIG looks like a ”sine wave” between 2.0 and 4.7 eV [see Fig. 7(d)], while that of YIG appears to be an inverted ”sine wave”between 2.6and6.4eV[seeFig. 6(d)]. The largest magnitude of σ2 xyof YIG is ∼1.6×1013s−1at∼4.3 eV, while that of BIG is ∼1.9×1014s−1at∼3.1 eV. The largest magnitude of σ1 xyof YIG is ∼1.2×1013s−1at∼ 4.8 eV, while that of BIG is ∼1.9×1014s−1at∼2.6 eV. In order to compare with the available experimental data, we also plot the experimental optical conductivity spectra [14, 17] in Figs. 6 and 7. The theoretical spectra of the diagonalelement of the optical conductivity tensor for both YIG and BIG match well with that of the exper- imental onesin the measuredenergyrange[see Figs. 6(a) and 6(b) as well as Figs. 7(a) and 7(b)]. Interestingly, we note that the relativistic GGA+U calculations give rise to the band gaps of YIG and BIG that are smaller than the experimental ones (see Table II), and yet the cal- culated and measured optical spectra agree rather well with each other. This apparently contradiction can be resolved as follows. In YIG, for example, the lowest con- duction bands at E= 1.8∼2.4 eV above the VBM are highly dispersive (see Fig. 2) and thus have very low DOS (see Fig. 4). This results in very low optical tran- sition. Therefore, the main absoption edge that appears in the optical spectrum ( σ1 xx) is∼2.2 eV, which is close to the experimental absorption edge of 2.5 eV, instead of 1.8 eV as determined by the calculated band struc- ture (see Table II). In contrast, no such highly dispersive bands appear at the CBM in BIG, Thus the calculated band gap agrees better with the measured band gap [17] (Table II). Figures 7(c) and 7(d) showthat the calculated σ1 xyand σ2 xyof BIG agree almost perfectly with the experimental data [17]. The peak positions, peak heights and overall trend of the theoretical spectra are nearly identical to that of the experimental ones [17]. On the other hand, the calculated σ1 xyandσ2 xyfor YIG do not agree so well with the experimental data [14] [Figs. 6(c) and 6(d]. For example, there is a sharp peak at ∼4.8 eV in the ex- perimental σ1 xyspectrum, which seems to be shifted to a higher energy at 5.6 eV with much reduced magnitude in the theoretical σ1 xyspectrum [seeFig. 6(c)]. Also, for σ2 xy spectrum, there is a sharp peak at ∼4.5 eV in the exper- imentalσ2 xyspectrum, which appears at ∼4.8 eV with considerably reduced height [see Fig. 6(d)]. Nonetheless, the overall trend of the theoretical σxyspectra of YIGis in rather good agreement with that of the measured ones [14]. Equations(2), (3), and (8) indicate that the absorptive parts of the optical conductivity elements ( σ1 xx,σ1 zz,σ2 xy andσ1 ±) are directly related to the dipole allowed in- terband transitions. Thus, we analyze the origin of the main features in the magneto-optical conductivity ( σ2 xy) spectrum by determining the symmetries of the involved band states and the dipole selection rules (see the Ap- pendix for details). The absorptive optical spectra are usually dominated by the interband transitions at the high symmetry points where the energy bands are gener- ally flat (see, e.g., Figs. 2 and 3), thus resulting in large joint density of states. As an example, here we consider the interband optical transitions at the Γ point where the band extrema often occur. Based on the determined band state symmetries and dipole selection rules (see Ta- ble III in the Appendix) as well as calculated transition matrix elements [Im( px ijpy ji)], we assign the main features inσ2 xy[labelled in Figs. 6(c) and 7(c)] to the main inter- band transitionsat the Γ point asshown in Figs. 8 and 9. The details of these assignments, related interband tran- sitions and transition matrix elements for YIG and BIG are presented in Tables IV and V in the Appendix, re- spectively. Since there are too many possible transitions to list, we present only those transitions whose transition matrix elements |Im(px ijpy ji)|>0.010 a.u. in YIG (Table IV) and |Im(px ijpy ji)|>0.012 a.u. in BIG (Table V). Figure 8 shows that nearly all the main optical tran- sitions in YIG are from the upper valence bands to the upper conduction bands, and only one main transition (P3) to the lower conduction bands. Consequently, these transitions contribute to the main features in σ2 xyat pho- ton energy >4.0 eV [see Fig. 6(c)]. In contrast, in BIG, a large number of the main transitions (e.g., P1-5, P7, N1-4, N5-8) are from the upper valence bands to lower conduction bands (see Fig. 9). This gives rise to the main features in σ2 xyfor photon energy <4.0 eV [see Fig. 7(c)], whose magnitudes are generally one order of magnitude larger than that of σ2 xyin YIG, as men- tioned above. The largely enhanced MO activity in BIG stems from the significant hybridization of Bi p-orbitals with Fed-orbitalsin the lowerconductionbands, asmen- tioned above. Since heavy Bi has a strongspin-orbit cou- pling, this hybridization greatly increases the dichroic in- terband transitions from the upper valence bands to the lower conduction bands in BIG. As mentioned above, Y sdorbitals contribute significantly only to the upper con- duction bands in YIG, and this results in the pronounced magneto-optical transitions only from the upper valence bands to the upper conduction bands (Fig. 8). Further- more, Y is lighter than Bi and thus has a weaker SOC than Bi. The discussion in the proceeding paragraph clearly indicates that the significant hybridization of heavy Bi porbitals with Fe dorbitals in the lower conduction bands just above the band gap is the main reason for the large MO effect in BIG. The magnetism in BIG is8 FIG. 8. Relativistic band structures of Y 3Fe5O12. Horizontal dashed lines denote the top of valance band. The principal interband transitions at the Γ point and the corresponding peaks in the σxyin Fig. 6 (c) are indicated by red and blue arrows. mainly caused by the iron dorbitals which have a rather weak SOC. However, through the hybridization between Biporbitals and Fe dorbitals, the strong SOC effect is also transfered to the lower conduction bands. Large exchange splitting and strong spin-orbit coupling in the valence and conduction bands below and above the band gaparecrucialforstrongmagneticcirculardichroismand hence large MO effects. Therefore, in search of materials with strongMO effects, one shouldlook formagneticsys- tems that contain heavy elements such as Bi and Pt [38]. D. Magneto-optical Kerr and Faraday effect Finally, let us study the polar Kerrand Faradayeffects inYIG andBIG.ThecomplexKerrandFaradayrotation angles for YIG and BIG are plotted as a function of pho- ton energy in Figs. 8 and 9, respectively. First of all, we notice that the Kerr rotation angles of BIG [Fig. 10(c)] are many times larger than that of YIG [Fig. 10(a)]. For example, the positiveKerrrotationmaximumof0.10◦in YIG occurs at ∼3.6 eV, while that (0.80◦) for BIG ap- pears at ∼3.5 eV. The negative Kerr rotation maximum (-0.12◦) of YIG occurs at ∼4.8 eV, while that (-1.21 ◦) for BIG appears at ∼2.4 eV. This may be expected because Kerr rotation angle is proportional to the MO conductivity ( σ1 xy) [Eq. (6)], which in BIG is nearly ten times larger than in YIG, as mentioned in the proceed- ing subsection. Similarly, the Kerr ellipticity maximum (0.16◦) of YIG occurs at ∼4.1 eV [Fig. 10(b)], whereas FIG. 9. Relativistic band structures of Bi 3Fe5O12. Horizontal dashed lines denote the top of valance band. The principal interband transitions at the Γ point and the corresponding peaks in the σxyin Fig. 7 (c) are indicated by red and blue arrows. that (0.54◦) of BIG [Fig. 10(d)] appears at ∼1.9 eV. The negative Kerr ellipticity maximum (-0.07◦) of YIG occurs at ∼5.7 eV while that (-1.16◦) of BIG is located at∼2.9 eV. Let us now compare our calculated Kerr rotation an- gles with some known MO materials such as 3 dtran- sition metal alloys and compound semiconductors. [8] For magnetic metals, ferromagnetic 3 dtransition metals and their alloys are an important family. Among them, manganese-basedpnictides areknownto havestrongMO effects. In particular, MnBi thin films were reported to have a large Kerr rotation angle of 2.3◦. [39, 40] Plat- inum alloys such as FePt, Co 2Pt [38] and PtMnSb [41] also possess large Kerr rotation angles. It was shown that the strong SOC on heavy Pt in these systems is the main cause of the strong MOKE. [38] Among semicon- ductor MO materials, diluted magnetic semiconductors Ga1−xMnxAs were reported to show Kerr rotations an- gle as large as 0.4◦at 1.80 eV. [42] Therefore, the strong MOKE effect in YIG and BIG could have promising ap- plicationsinhighdensityMOdata-storagedevicesorMO nanosensors with high spatial resolution. Figure 9 shows that as for the Kerr rotation angles, the Faraday rotation angles of BIG are generally up to ten times larger than that of YIG. The Faraday rotation maximum (7.2◦/µm) of YIG occurs at ∼3.9 eV, while that (51.2◦/µm) of BIG is located at ∼3.7 eV. The Fara- day ellipticity maximum (7.9◦/µm) for YIG appears at ∼4.4 eV, whereas that (54.1◦/µm) of BIG occurs at ∼2.3 eV. On the other hand, the negative Faraday rota-9 FIG. 10. Calculated complex Kerr rotation angles (blue curves). (a) Kerr rotation ( θK) and (b) Kerr ellipticity ( εK) spectra of Y 3Fe5O12; (c) Kerr rotation ( θK) and (d) Kerr el- lipticity ( εK) spectra of Bi 3Fe5O12. Red circles in (a) and (b) denote the experimental values from Ref. [15]. tion maximum (-5.7◦/µm) occurs at ∼5.4 eV, while that (-74.6◦/µm) for BIG appears at ∼2.7 eV. The negative Faraday ellipticity maximum (-3.6◦/µm) of YIG occurs at∼6.6 eV, while that (-70.2◦/µm) for BIG is located at∼3.2 eV. For comparision, we notice that MnBi films are known to possess large Faraday rotation angles of ∼80◦/µm at 1.8 eV. [39, 40] Finally, we compare our predicted MOKE and MOFE spectrawiththeavailableexperimentsinFigs. 10and11. AllthepredictedMOKEandMOFEspectraareinrather goodagreementwiththeexperimentalonesintheexperi- mental photon energyrange[14, 15, 17, 43]. Nonetheless, our theoretical predictions would have a better agree- ment with the experiments if all the calculated spectra are blue-shifted slightly by ∼0.3 eV, thus suggesting that the theoretical band gaps are slightly too small. IV. CONCLUSION Tosummarize,wehavesystematicallystudiedthe elec- tronic structure, magnetic, optical and MO properties of cubic iron garnets YIG and BIG by performing GGA+U calculations. We find that YIG exhibits significant MO Kerr and Faraday effects in UV frequency range that are comparable to cubic ferromagnetic iron. Strikingly, we find that BIG shows gigantic MO effects in the visible frequency region that are several times larger than YIG. In particular, the Kerr rotation angle of BIG becomes FIG. 11. Calculated complex Faraday rotation angles (blue curves). (a) Faraday rotation ( θF) and (b) Faraday ellipticity (εF) spectra of Y 3Fe5O12; (c) Kerr rotation ( θF) and (d) Kerr ellipticity ( εF) spectra of Bi 3Fe5O12. Red dashed line in (a) denotes the measured values from Ref. [15]. Black circles in (a) and (b) are the experimental values from Ref. [14]. Red (green) circles in (c) and (d) are the experimental values fr om Ref. [43] ([17]) as large as -1.2◦at photon energy 2.4 eV, and the Fara- day rotation angle for the BIG film reaches -75◦/µm at 2.7 eV. Calculated MO conductivity ( σ2 xy) spectra re- veal that these distinctly different MO properties of YIG and BIG result from the fact that the magnitude of σ2 xy of BIG is nearly ten times larger than that of YIG. Our calculatedKerrandFaradayrotationanglesofYIG agree well with the available experimental values. Our calcu- latedFaradayrotationanglesofBIG areinnearlyperfect agreement with the measured ones. Thus, we hope that our predicted giant MO Kerr effect in BIG will stimulate further MOKE experiments on high quality BIG crys- tals.‘ Principal features in the optical and MO spectra are analyzed in terms of the calculated band structures espe- cially the symmetry of the band states and optical tran- sition matrix elements at the Γ point of the BZ. We find that in YIG, Y sdorbitals mix mainly with the upper conduction bands that are ∼4.5 eV abovethe VBM, and thus leave the Fe dorbital dominated lower conduction bands from 1.8 to 3.8 eV above the VBM almost unaf- fected by the SOC on the Y atom. In contrast, Bi por- bitals in BIG hybridize significantly with Fe dorbitals in the lower conduction bands and this leads to large SOC- induced band splitting andmuch increasedband width of the lowerconduction bands. Consequently, the MO tran-10 sitions between the upper valence bands and lower con- duction bands are greatly enhanced when Y is replaced by heavier Bi. This finding thus provides a guideline in search for materials with desired MO effects, i.e., one should look for magnetic materials with heavy elements such as Bi whose orbitals hybridize significantly with the MO active conduction or valence bands. Finally, our findings of strong MO effects in these iron garnetsand alsosingle-spinsemiconductivityin BIGsug- gest that cubic iron garnets are an useful playground of exploring the interplay of microwave, spin current, mag- netism, and optics degrees of freedom, and also have promising applications in high density semiconductor MO data-storage and low-power consumption spintronic nanodevices. ACKNOWLEDGMENTS The authors thank Ming-Chun Jiang for many valu- able discussions throughout this work. The authors ac- knowledge the support from the Ministry of Science and Technology and the National Center for Theoretical Sci- ences (NCTS) of The R.O.C. The authors are also grate- ful to the National Center for High-performance Com- puting (NCHC) for the computing time. G.-Y. Guo also thanks the support from the Far Eastern Y. Z. Hsu Sci-ence and Technology Memorial Foundation in Taiwan. APPENDIX: DIPOLE SELECTION RULES AND SYMMETRIES OF BAND STATES AT Γ In this Appendix, to help identify the origins of the mainfeaturesinthe magneto-opticalconductivity σxy(ω) spectra of YIG and BIG, we provide the dipole selection rules and the symmetries of the band states at the Γ as well as the main optical transitions between them. BothYIG andBIGhavethe Ia¯3dspacegroupandthus they have the C4h(4/mm′m′) point group at the Γ point in the Brillouin zone. Based on the character table of the C4hpoint group [44], we determine the dipole selection rules for the optical transitions between the band states at the Γ point, as listed in Table III. We calculate the eigenvalues for all symmetry elements of each eigenstate of the Γ point using the Irvspprogram [45] and then determine the irreducible representation and hence the symmetry of the state. Based on the obtained symme- triesof the band states and alsocalculatedoptical matrix elements [Im( px ijpy ji)] [see Eq. (3)], we assign the peaks in theσxy(ω) spectra of YIG [see Fig. 6(c)] and BIG [see Fig. 7(c)] to the main optical transitions at the Γ point (see Fig. 8 and 9, respectively), as listed in Tables IV and V, respectively. [1] V. Cherepanov, I. Kolokolov, and V. Lvov, The saga of YIG: spectra, thermodynamics, interaction and relax- ation of magnons in a complex magnet, Phys. Rep. 229, 81 (1993). [2] S. Mizukami, Y. Ando, and T. Miyazaki, Effect of spin diffusion on Gilbert damping for a very thin permal- loy layer in Cu/permalloy/Cu/Pt films, Phys. Rev. B 66, 104413 (2002). [3] S. Chikazumi, Physics of Ferromagnetism , 2nd ed. (Ox- ford University Press, Oxford, 1997) [4] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Transmission of electrical signals by spin-wave interconversion in a mag- netic insulator, Nature (London) 464, 262 (2010). [5] T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Realization of spin-wave logic gates, Appl. Phys. Lett. 92(2), 022505 (2008). [6] Y. Sun, H. Chang, M. Kabatek, Y.-Y. Song, Z. Wang, M. Jantz, W. Schneider, M. Wu, E. Montoya, B. Kardasz, B. Heinrich, S. G. E. te Velthuis, H. Schultheiss, and A. Hoffmann, Damping in Yttrium Iron Garnet Nanoscale Films CappedbyPlatinum, Phys.Rev.Lett. 111, 106601 (2013). [7] P. M. Oppeneer, Chapter 1 Magneto-optical Kerr Spec- tra, pp. 229-422, in Handbook of Magnetic Materials , edited by K. H. J. Buschow. Elsevier, Amsterdam, (2001). [8] V. Antonov, B. Harmon, and A. Yaresko. Elec- tronic structure and magneto-optical properties of solids .TABLE III. Dipole selection rules for the C4hpoint group at the Γ point in the Brillouin zone of YIG and BIG. polarization Γ+ 6Γ+ 5Γ+ 8Γ+ 7Γ− 6Γ− 5Γ− 8Γ− 7 z Γ− 6Γ− 5Γ− 8Γ− 7Γ+ 6Γ+ 5Γ+ 8Γ+ 7 x+iy Γ− 5Γ− 8Γ− 7Γ− 6Γ+ 5Γ+ 8Γ+ 7Γ+ 6 x−iy Γ− 7Γ− 6Γ− 5Γ− 8Γ+ 7Γ+ 6Γ+ 5Γ+ 8 Springer Science & Business Media, (2004). [9] J. P. Castera, in Magneto-optical Devices , Vol.9ofEncy- clopedia of Applied Physics , edited byG. L. Trigg (Wiley- VCH, New York, 1996), p. 133. [10] M. Mansuripur, The Principles of Magneto-Optical Recording (Cambridge Univ. Press, Cambridge, 1995). [11] F. D. M. Haldane and S. Raghu, Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 100, 013904 (2008). [12] L. J. Aplet and J. W. Carson, A Faraday effect optical isolator, Appl. Opt. 3, 544 (1964). [13] J. F. Dillon, Optical properties of several ferrimagne tic garnets, J. Appl. Phys. 29, 539 (1958). [14] S. Wittekoek, T. J. A. Popma, J. M. Robertson, P. F. Bongers, Magneto-optic spectra and the dielectric ten- sor elements of bismuth-substituted iron garnets at pho- ton energies between 2.2-5.2 eV, Phys. Rev. B 12, 2777 (1975).11 TABLE IV. Main optical transitions between the states at the Γ point of the Brillouin zone of YIG. Symbols in the first column denote the assigned peaks in the magneto-optica l conductivity ( σ2 xy) spectrum (Figs. 6 and 8). iandjdenote the initial and final states, respectively. Im( px ijpy ji) denote the calculated transition matrixelement (inatomic units)[se e Eq. (3)].EiandEjrepresent the initial and final state energies (in eV), respectively. ∆ Eij=Ej−Eiis the transition energy. Peak state istatejIm(px ijpy ji) ∆EijEjEi P8 545 (Γ+ 6) 802 (Γ− 5) 0.0103 6.928 4.358 -2.569 P3 556 (Γ− 5) 761 (Γ+ 8) 0.0115 5.438 3.010 -2.428 P6 609 (Γ− 7) 803 (Γ+ 6) 0.0102 6.207 4.442 -1.765 N4 632 (Γ− 5) 803 (Γ+ 6) -0.0273 5.825 4.442 -1.384 P5 635 (Γ− 7) 803 (Γ+ 6) 0.0251 5.818 4.442 -1.376 N3 649 (Γ− 5) 803 (Γ+ 6) -0.0146 5.567 4.442 -1.125 P4 651 (Γ− 7) 803 (Γ+ 6) 0.0195 5.564 4.442 -1.123 P7 668 (Γ+ 6) 844 (Γ− 5) 0.0136 6.744 5.994 -0.750 N5 670 (Γ+ 8) 844 (Γ− 5) -0.0119 6.739 5.994 -0.745 P2 690 (Γ− 6) 801 (Γ+ 5) 0.0116 4.537 4.280 -0.257 N2 692 (Γ− 8) 801 (Γ+ 5) -0.0107 4.535 4.280 -0.254 P1 698 (Γ− 6) 801 (Γ+ 5) 0.0431 4.292 4.280 -0.012 N1 700 (Γ− 8) 801 (Γ+ 5) -0.0452 4.280 4.280 0.000 TABLE V. Main optical transitions between the states at the Γ point of the Brillouin zone of BIG. Symbols in the first col- umn denote the assigned peaks in the magneto-optical con- ductivity ( σ2 xy) spectrum (Figs. 7 and 9). iandjdenote the initial and final states, respectively. Im( px ijpy ji) denote the cal- culated transition matrix element (in atomic units) [see Eq . (3)].EiandEjrepresent the initial and final state energies (in eV), respectively. ∆ Eij=Ej−Eiis the transition energy. Peak state istatejIm(px ijpy ji) ∆EijEjEi N8 578 (Γ+ 8) 783 (Γ− 5) -0.0130 5.234 2.304 -2.930 P7 578 (Γ+ 8) 789 (Γ− 7) 0.0123 5.331 2.401 -2.930 N7 579 (Γ+ 6) 782 (Γ− 7) -0.0139 5.228 2.298 -2.930 N9 661 (Γ− 6) 856 (Γ+ 7) -0.0136 5.333 3.452 -1.882 P4 684 (Γ− 7) 846 (Γ+ 6) 0.0127 4.685 3.242 -1.443 P5 709 (Γ+ 8) 869 (Γ− 7) 0.0139 4.839 3.825 -1.014 P9 710 (Γ+ 6) 873 (Γ− 5) 0.0149 5.453 4.467 -0.986 N5 712 (Γ+ 8) 842 (Γ− 5) -0.0121 4.153 3.175 -0.979 P3 715 (Γ+ 7) 854 (Γ− 6) 0.0135 4.271 3.336 -0.935 P8 723 (Γ+ 7) 876 (Γ− 6) 0.0142 5.374 4.571 -0.803 N6 727 (Γ+ 8) 873 (Γ− 5) -0.0145 5.206 4.467 -0.738 P2 741 (Γ− 6) 872 (Γ+ 5) 0.0122 4.254 3.917 -0.337 P6 743 (Γ− 5) 878 (Γ+ 8) 0.0127 5.305 4.991 -0.314 N3 743 (Γ− 5) 749 (Γ+ 6) -0.0184 2.113 1.799 -0.314 N2 744 (Γ− 8) 755 (Γ+ 5) -0.0160 2.037 1.973 -0.063 N1 745 (Γ− 7) 753 (Γ+ 8) -0.0142 1.980 1.917 -0.062 N4 747 (Γ− 7) 871 (Γ+ 8) -0.0138 3.891 3.891 0.000 P1 715 (Γ+ 7) 760 (Γ− 6) 0.0125 3.032 2.097 -0.935 [15] F. J. Kahn, P. S. Pershan, and J. P. Remeika, Ultravio- let Magneto-Optical Properties of Single-Crystal Ortho- ferrites, Garnets, and Other Ferric Oxide Compounds, Phys. Rev. 186, 891 (1969). [16] M. -Y. Chern, F. -Y. Lo, D. -R. Liu, K. Yang, and J. -S. Liaw, Red shift of Faraday rotation in thin films of completely bismuth-substituted iron garnet Bi 3Fe5O12,Jpn. J. App. Phys., Part 1 38, 6687 (1999). [17] E. Jesenska, T. Yoshida, K. Shinozaki, T. Ishibashi, L. Beran, M. Zahradnik, R. Antos, M. Kuˇ cera, and M. Veis, Optical and magneto-optical properties of Bi substituted yttrium iron garnets prepared by metal organic decom- position, Opt. Mater. Express 6(6), 1986-1997 (2016). [18] B. Vertruyen, R. Cloots, J. S. Abell, T. J. Jackson, R. C. da Silva, E. Popova, and N. Keller, Curie temperature, exchange integrals, andmagneto-optical properties inoff- stoichiometric bismuth iron garnet epitaxial films, Phys. Rev. B78, 094429 (2008). [19] Y. -N. Xu, Z. -Q. Gu, and W. Y. Ching, First-principles calculation ofthe electronic structureof yttriumiron gar - net (Y 3Fe5O12), J. Appl. Phys. 87, 4867 (2000). [20] T. Oikawa, S. Suzuki, and K. Nakao, First-principles study of spin-orbit interactions in bismuth iron garnet, J. Phys. Soc. Jpn. 74, 401 (2005). [21] F. Bertaut, F. Forrat, A. Herpin, and P. M´ eriel, ´Etude par diffraction de neutrons du grenat ferrimagn´ etique Y3Fe5O12, Compt. rend. 243, 898 (1956). [22] H. Toraya and T. Okuda, Crystal structure analysis of polycrystalline Bi 3Fe5O12thin film by using asymmet- ric and symmetric diffraction techniques, J. Phys. Chem. Solids56, 1317 (1995). [23] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996). [24] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, Electron-energy-loss spec- tra and the structural stability of nickel oxide: An LSDA+ U study, Phys. Rev. B 57, 1505 (1998). [25] H.-T. Jeng, G. Y. Guo and D. J. Huang, Charge-orbital ordering and Verwey transition in magnetite, Phys. Rev. Lett.93, 156403 (2004). [26] G. Kresse and D. Joubert, From ultrasoft pseudopoten- tials to the projector augmented-wave method, Phys. Rev. B59, 1758 (1999). [27] G. Kresse and J. Furthm¨ uller, Efficient iterative schem es forab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996). [28] G. Kresse and J. Furthm¨ uller, Efficiency of ab-initio to tal energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mat. Sci 6, 15 (1996). [29] W. Feng, G.-Y. Guo, J. Zhou, Y. Yao, and Q. Niu, Large magneto-optical Kerr effect in noncollinear antiferromag- nets Mn 3X(X= Rh, Ir, Pt) Phys. Rev. B 92, 144426 (2015). [30] C. S. Wang and J. Callaway, Band structure of nickel: Spin-orbit coupling, the Fermi surface, and the optical conductivity, Phys. Rev. B 9, 4897 (1974). [31] P. M. Oppeneer, T. Maurer, J. Sticht, and J. Kbler, Ab initiocalculated magneto-optical Kerreffect offerromag- netic metals: Fe and Ni, Phys. Rev. B 45, 10924 (1992). [32] B. Adolph, J. Furthmller, and F. Bechstedt, Optical properties of semiconductors using projector-augmented waves, Phys. Rev. B 63, 125108 (2001). [33] W. M. Temmerman, P. A. Sterne, G. Y. Guo, and Z. Szotek, Electronic StructureCalculations of High T cMa- terials, Mol. Simul. 63, 153 (1989). [34] G.-Y. Guo and H. Ebert, Theoretical investigation of th e orientation dependence of the magneto-optical Kerr ef- fect in Co, Phys. Rev. B 50, 10377 (1994). [35] G.-Y. Guo and H. Ebert, Band-theoretical investigatio n12 of the magneto-optical Kerr effect in Fe and Co multilay- ers, Phys. Rev. B 51, 12633 (1995). [36] D.Rodic, M.Mitric, R.Tellgren, H.Rundlof, andA.Kre- menovic, True magnetic structure of the ferrimagnetic garnet Y 3Fe5O12and magnetic moments of iron ions, J. Magn. Magn. Mater. 191, 137 (1999). [37] N. Adachi, T. Okuda, V. P. Denysenkov, A. Jalali- Roudsar, and A. M. Grishin, Magnetic proper- ties of single crystal film Bi 3Fe5O12prepared onto Sm3(Sc,Ga) 5O12(1 1 1), J. Magn. Magn. Mater. 242-245, 775 (2002). [38] G. Y. Guo and H. Ebert, On the origins of the enhanced magneto-optical Kerr effect in ultrathin Fe and Co mul- tilayers, J. Magn. Magn. Mater. 156, 173 (1996). [39] P. Ravindran, A. Delin, P. James, B. Johansson, J. Wills , R. Ahuja, and O. Eriksson, Magnetic, optical, and magneto-optical properties of MnX (X=As, Sb, or Bi) from full-potential calculations, Phys. Rev. B 59, 15680(1999). [40] G. Q. Di and S. Uchiyama, Optical and magneto-optical properties of MnBi film, Phys. Rev. B 53, 3327 (1996). [41] P. Van Engen, K. Buschow, R. Jongebreur, and M. Er- man, PtMnSb, amaterial with veryhigh magneto-optical Kerr effect, Appl. Phys. Lett. 42, 202–204 (1983). [42] R. Lang, A. Winter, H. Pascher, H. Krenn, X. Liu, and J. K. Furdyna, Polar Kerr effect studies of Ga 1−xMnxAs epitaxial films, Phys. Rev. B 72, 024430 (2005). [43] M. Deb, E. Popova, A. Fouchet, and N. Keller, Magneto- optical Faraday spectroscopy of completely bismuth- substituted Bi 3Fe5O12garnet thin films, J. Phys. D 45, 455001 (2012). [44] G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of the thirty-twopoint groups (Vol. 24). MIT press (1963). [45] J.-C. Gao, Q.-S. Wu, C. Persson, and Z.-J. Wang, Irvsp: to obtain irreducible representations of electronic state s in the VASP, arXiv preprint arXiv:2002.04032 (2020).

2020-05-28

The magneto-optical (MO) effects not only are a powerful probe of magnetism and electronic structure of magnetic solids but also have valuable applications in high-density data-storage technology. Yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$) (YIG) and bismuth iron garnet (Bi$_3$Fe$_5$O$_{12}$) (BIG) are two widely used magnetic semiconductors with strong magneto-optical effects and have also attracted the attention for fundamental physics studies. In particular, YIG has been routinely used as a spin current injector. In this paper, we present a thorough theoretical investigation on magnetism, electronic, optical and MO properties of YIG and BIG, based on the density functional theory with the generalized gradient approximation plus onsite Coulomb repulsion. We find that both semiconductors exhibit large MO effects with their Kerr and Faraday rotation angles being comparable to that of best-known MO materials such as MnBi. Especially, the MO Kerr rotation angle for bulk BIG reaches -1.2$ ^{\circ}$ at photon energy $\sim2.4$ eV, and the MO Faraday rotation angle for BIG film reaches -74.6 $ ^{\circ}/\mu m$ at photon energy $\sim2.7$ eV. Furthermore, we also find that both valence and conduction bands across the MO band gap in BIG are purely spin-down states, i.e., BIG is a single spin semiconductor. These interesting findings suggest that the iron garnets will find valuable applications in semiconductor MO and spintronic nanodevices. The calculated optical conductivity spectra, MO Kerr and Faraday rotation angles agree well with the available experimental data. The main features in the optical and MO spectra of both systems are analyzed in terms of the calculated band structures especially by determining the band state symmetries and the main optical transitions at the $\Gamma$ point in the Brillouin zone.

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$_{12}$

2005.14133v2

Magnetization dynamics aected by phonon pumping Richard Schlitz,1, 2,Luise Siegl,3, 2Takuma Sato,4Weichao Yu,5, 6, 4Gerrit E. W. Bauer,4, 7, 8Hans Huebl,9, 10, 11and Sebastian T. B. Goennenwein3, 2 1Department of Materials, ETH Z urich, 8093 Z urich, Switzerland 2Institut f ur Festk orper- und Materialphysik, Technische Universit at Dresden and W urzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany 3Department of Physics, University of Konstanz, 78457 Konstanz, Germany 4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 5State Key Laboratory of Surface Physics and Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, China 6Zhangjiang Fudan International Innovation Center, Fudan University, Shanghai 201210, China 7AIMR and CSRN, Tohoku University, Sendai 980-8577, Japan 8Zernike Institute for Advanced Materials, Groningen University, Groningen, The Netherlands 9Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 10Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 11Munich Center for Quantum Science and Technology (MCQST), 80799 M unchen, Germany (Dated: February 8, 2022) \Pumping" of phonons by a dynamic magnetization promises to extend the range and functionality of magnonic devices. We explore the impact of phonon pumping on room-temperature ferromagnetic resonance (FMR) spectra of bilayers of thin yttrium iron garnet (YIG) lms on thick gadolinium gallium garnet substrates over a wide frequency range. At low frequencies the Kittel mode hybridizes coherently with standing ultrasound waves of a bulk acoustic resonator to form magnon polarons that induce rapid oscillations of the magnetic susceptibility, as reported before. At higher frequencies, the phonon resonances overlap, merging into a conventional FMR line, but with an increased line width. The frequency dependence of the increased line broadening follows the predictions from phonon pumping theory in the thick substrate limit. In addition, we nd substantial magnon-phonon coupling of a perpendicular standing spin wave (PSSW) mode. This evidences the importance of the mode overlap between the acoustic and magnetic modes, and provides a route towards engineering the magnetoelastic mode coupling. Magnons and phonons are, respectively, the elemen- tary excitations of the magnetic and atomic order in condensed matter. They are coupled by weak magne- toelastic and magnetorotational interactions, which can often simply be disregarded. However, recent experimen- tal and theoretical research reveals that the magnon- phonon interaction may cause spectacular eects in (i) ferromagnets close to a structural phases transition such as Galfenol [1, 2] or (ii) magnets with exceptionally high magnetic and acoustic quality such as yttrium iron gar- net [3{9]. Magnons are promising carriers for future low-power information and communication technologies [10, 11]. The magnon-phonon interaction can benet the function- ality of magnonic devices by helping to control and en- hance magnon propagation when coherently coupled into magnon polarons [4, 9]. On the other hand, magnon non- conserving magnon-phonon scattering is the main source of magnon dissipation at room temperature [12, 13]. The study of magnon-phonon interactions in high- quality magnets has a long history [14{21]. The arrival of crystal growth techniques, strongly improved microwave technology, and discovery of new phenomena such as the spin Seebeck eect, led to a revival of the subject in the past few years, with emphasis on ultrathin lms and het- erostructures [4, 6, 22{30].High-quality yttrium iron garnet (YIG) is an excellent material to study magnons and phonons. Thin lms grow best on single-crystal substrates of gadolinium gallium garnet (GGG), a paramagnetic insulator that is mag- netically inert at elevated temperatures. However, the acoustic parameters of GGG are almost identical to YIG such that phonons are not localized to the magnet and thus the substrate cannot be simply disregarded. Streib et al. [3] pointed out that magnetic energy can leak into the substrate by magnon-phonon coupling by a process called \phonon pumping" and predicted that it should cause an increased magnetization damping with a char- acteristic non-monotonous dependence on frequency. Phonon pumping has been experimentally observed in the ferromagnetic resonance of YIG lms on GGG sub- strates [4, 29, 30]. These experiments revealed coher- ent hybridization of the (uniform) Kittel magnon with standing sound waves extended over the whole sample. In YIG/GGG/YIG trilayers phonon exchange couples magnons dynamically over mm distances [4, 9]. However, the predicted increased damping due to phonon pump- ing and the coupling of other than the macro-spin Kittel magnon remained elusive. The direct detection of the increased damping is challenging due to the presence of inhomogeneous FMR line broadening and the resulting changes of the resonance line shape, in particular in thearXiv:2202.03331v1 [cond-mat.mes-hall] 7 Feb 20222 low frequency regime for thin lms [31] or due to the pres- ence of several modes in the resonance for thicker YIG lms [32{35]. In this Letter, we report FMR spectra of YIG/GGG bilayers over a large frequency range, demonstrating the coupling of magnons and phonons from the high coop- erativity to the incoherent regime. We reproduce the magnon polaron ne structure at low frequencies [4, 29], and evidence the presence of the acoustic spin pumping eect on the magnetic dissipation predicted in Ref. [3]at higher frequencies. The excellent agreement with an analytical model allows us to extract the parameters for the phonon pumping by the (even) Kittel mode in the strong coupling regime, and provides insights into the strong-weak coupling regime at higher frequencies. In addition, we observe that the magnon-phonon coupling strength also is characteristically modulated for an (odd) perpendicular standing spin wave mode. This shows that the overlap integral between magnon and phonon modes governs the coupling strength, thus opening a pathway for controlling it. (a) hrf CPWVNA P1 P2 z || H0x y low high (g)3(e) (f)GGG GGG FIG. 1. (a) A thin YIG lm on a thick paramagnetic GGG substrate (gray square) is placed face down on a coplanar waveguide. The latter is connected to a vector network analyzer to obtain the transmission parameter S21as a function of frequency!. The external static magnetic eld H0is applied normal to the surface. (b-d) High resolution maps of jS21jfor dierent!andH0. Shear waves with sound velocity ctand wavelength pform standing waves across the full layer stack. The three panels correspond to tYIGp=2;pand 3p=2, respectively. The fundamental (Kittel) mode and the rst perpendicular standing spin wave (PSSW) are marked with orange and blue dashed lines and arrows, respectively. (e,f) The thickness prole of the magnetic excitation ml(z) is shown for the Kittel mode (e) and the rst PSSW (f) in the YIG lm for p= 0:5;together with the eigenmodes of the acoustic strain @(z)=@zcorresponding to panels (b-d). The positive and negative contributions to the magnetoelastic exchange integral are shaded in red and gray, respectively. The magnetic excitation and thus the mode overlap vanishes in the GGG layer, whereas the phonons extend across full YIG/GGG sample stack. (g) The magnetoelastic mode coupling gmeis proportional to the overlap of the phonon and magnon modes and shows characteristic oscillations. Our sample consists of a 630 nm Y 3Fe5O12lm on a 560µm thick Gd 3Ga5O12substrate glued onto a copla- nar waveguide (CPW) with a center conductor width w= 110 µm. It is inserted into the air gap of an electro- magnet with surface normal parallel to the magnetic eld [cf. Fig. 1(a)]. We improve the magnetic eld resolution to the 1 µT range by an additional Helmholtz coil pair in the pole gap of the electromagnet that is biased with a separate power supply. We measure the complex mi- crowave transmission spectra S21(!) by a vector network analyzer for a series of xed magnetic eld strengths over a large frequency interval at room temperature. We rst address S21(!) in the strong-couplingregime [4] in the form of jS21jas a function of magnetic eld and frequency, see Fig. 1(b). The FMR reduces the transmission, emphasized by blue color and centered at the dashed orange line. Periodic perturbations in the FMR at xed frequencies with period of 3:2 MHz (dashed white lines) correspond to the acoustic free spec- tral range of the sample
!p 2ct 2tGGG3:2 MHz; (1) wherect= 3570 m s1is the transverse sound veloc- ity of GGG [36]. These are the anticrossings of the FMR dispersion with eld-independent standing acous-3 tic shear wave modes across the full YIG/GGG layer stack [4, 29, 30, 37, 38]. In Fig. 1(b) we addition- ally observe a resonance corresponding to the PSSW (dashed blue line) shifted to a lower magnetic elds by exchange splitting 0
H=D2=t2 YIG1:4 mT, where D= 51017Tm2is the exchange stiness of YIG [32{ 35], but without visible coupling to the phonons. At 6:15 GHz [cf. Fig. 1(c)] the periodic anticrossings vanish for the Kittel mode resonance, which implies a strongly suppressed magnon-phonon coupling. In con- trast, the PSSW now exhibits clear anticrossings similar to that of the Kittel mode in panel (b). Increasing the frequency further [cf. Fig. 1(d)] to around 9 :63 GHz, the periodic oscillations in the PSSW vanish again, but the anticrossings of the Kittel mode do not recover. We interpret the suppression of the magnon polaron signal at higher frequencies around 9 GHz in terms of a transition from the (underdamped) high cooperativity [4] to the (overdamped) weak coupling regime. In the latter, the dierent phonon modes overlap, leading to a constant contribution of the phonon pumping to the magnon line width. As a consequence, the periodic magnon polaron signatures vanish in favor of a slowly varying additional broadening of the FMR line that was predicted theoret- ically in the limit of thick GGG substrates [2, 3]. The coupling between the elastic and the magnetic sub- systems in a conned magnet scales with the overlap in- tegral of the phonon and magnon modes [3, 29]. The prole of a PSSW with index lcan be modelled by ml(z) =psin ([l+ 1][z+tYIG]=tYIG) + (1p) cos (l[z+tYIG]=tYIG); (2) wherez2[tYIG;0] and 0p1 interpolates be- tween free ( p= 0) and pinned ( p= 1) surface dynamics. Assuming free elastic boundary conditions, a shear wave across the full layer stack with amplitude and frequency !creates a strain prole (disregarding the standing wave formation and thus the nite free spectral range) in the YIG lm that is given by @(z) @z=! ectsin!(tYIG+z) ect; (3) where@(z) is the local displacement and ect= 3843 m s1 is the transverse sound velocity of YIG. Note that the ladder of modes is disregarded here for simplicity. The overlap integral of the fundamental (Kittel) mode with l= 0 (Fig. 1(e)) and the rst PSSW with l= 1 (Fig. 1(f)) enters the interaction magnetoelastic coupling gmeas [29] gme;l=s 2b2 !M stGGGtYIGZ0 tYIGml(z)@(z) @zdz;(4) where the parameters for YIG at room temperature are the magnetoelastic coupling constant b= 7105J=m3,the mass density = 5:1 g=cm3, the gyromagnetic ratio =2= 28:5 GHz T1and the saturation magnetization Ms= 143 kA m1[4].gme;0andgme;1in Fig. 1(g) for p= 0:5 (yellow and blue lines, respectively) reveal dif- ferences in the magnetoelastic coupling of the dierent magnetic modes. In both cases the coupling oscillates as a function of frequency, but the maxima for l= 0 and l= 1 are shifted by l
ect=2tYIG3 GHz. Note that this is true also for the higher standing spin wave modes, so that even at high frequencies, strong magnon-phonon in- teractions can be realized. In other materials the results may depend on the details of the interface and surface boundary conditions [23]. (a) photon phonon magnon FIG. 2. (a) The phonons and magnons in YIG/GGG bilay- ers form a two-partite system that can be modeled as coupled harmonic oscillators that are driven by microwave photons [4]. The parameters are the resonance frequencies !m;!p, damp- ing constants m; p;and coupling strength gme. The coplanar waveguide with transmission amplitude a, phaseand elec- tric length, interacts with the magnon system parametrized by the coupling strength . (b)jS21jas a function of the frequency for 0H259 mT. A Gaussian t (red line) deter- mines the FMR frequency !m(right dashed line). (c) Zoom-in on the phonon line with !m!p= 22 MHz (marked by the left dashed line in panel a). We obtain the line width pand the amplitude hpof the phonon resonance by a Lorentzian t. A phonon and a magnon mode with discrete frequen- cies!pand!m[= 0(HMe) for the Kittel mode] and amplitudes ApandAm;respectively, behave as two harmonic oscillators coupled by the interaction gme[4]: Am( m+i(!m!))iApgme=2 += 0 (5) Ap( p+i(!p!))iAmgme=2 = 0;(6) where mand pare the decay rates (in angular fre- quency units). parametrizes the coupling of the mag- netic order to the external microwaves at frequency !, see Fig. 2(a). The solution for the magnetic amplitude is Am=gme 22 1 p+i(!p!)+ ( m+i(!m!))1 : (7)4 This resonator couples to a CPW according to [39] S21(!) =aexp(i) exp(i!) [1Am]: (8) in which the rst part in the square brackets rep- resents the external circuit with frequency-dependent amplitude and phase shift aand;respectively, and is an electronic delay time. We can t the un- known parameters andgmeto the observed spec- tra in Fig. 2(b,c)]. can be extracted from the amplitude of the ferromagnetic resonance by solving hm=jS21(!=!m)=0jjS21(!=!m)gme=0j f(). Similarly, the data for a phonon resonance hp= jS21(!=!p)gme=0jjS21(!=!p)j g(gme) can be solved forgme.hpandhmcan be extracted from ts to the experimental data. We t the Kittel mode lines in jS21(!)jat dierent xed magnetic elds by a Gaussian to distill the reso- nance frequency !m, the amplitude hmand width m (cf. Fig. 2b). A good t by a Gaussian line shape in- dicates inhomogeneous broadening of the FMR, see be- low. Next, we select an acoustic resonance at a frequency !p;0with (!m!p;0)=2 >2 m=222 MHz, which is only weakly perturbed by the magnon-phonon coupling, but still has a signicant oscillator strength. For a bet- ter statistics, we independently t a total of six phonon resonances with frequencies below !p;0by Lorentzians [cf. Fig. 2(c)] to extract their average height hpand broadening p. 1012m/2 (MHz) G=1.7×104 m,0/2=9.3 MHz (a) 0.250.500.75p/2 (MHz) /2=5.2×106 GHz1 p,0/2=144 kHz (b) 0 2 4 6 8 10 12 /2 (GHz) 02gme/2 (MHz) (c) FIG. 3. (a) Half width at half maximum (HWHM) obtained from Gaussian ts to the FMR lines. (b) HWHM from the Lorentzian ts to the acoustic resonances. (c) Magnetoelastic mode coupling obtained from the harmonic oscillator model using the parameters from the ts shown in (a), (b) and in the SM [40]. The maximum coupling strength is 2:2 MHz. The resulting t parameters are summarized in Fig. 3 and in the supporting material [40]. The line width of theFMR m= m;0+G!shown in panel (a) is dominated by inhomogeneous broadening m;0=2= 9:3 MHz, which we associate to variations of the local (eective) magne- tization over the 6 mm long sample and across the thick- ness prole. The low Gilbert damping G1:7104 is evidence for an intrinsically high quality of the YIG lm. We associate the parabolic increase of the sound attenuation p=!2+ p;0[cf. Fig. 3(b)] to thermal phonon scattering in GGG [41{43]. The inhomogeneous phonon line width p;0=2= 144 kHz may be caused by a small angle1°between the bottom and top surfaces of our sample [44], where the estimate is based on the phonon mean-free-path ct= p4 mm [4]. We do not observe a larger scale disorder in the substrate thickness that would contribute a term !to the attenuation [42]. In the lower frequency regime !=2.10 GHz the phonon mean-free path  > 1 mm is larger than twice the thickness of the bilayer. At frequencies above 10 GHz, however, we enter the cross-over regime between high co- operativity and weak coupling in which the phononic free spectral range approaches its attenuation (
!p2 p). The tting with individual phonon lines becomes increas- ingly inaccurate, as the baseline of the FMR signal with- out contributions due to phonons cannot be established from the data. In this regime, the strongly overlap- ping phonon lines thus lead to an average increase of the FMR line width in addition to the rapidly oscillat- ing contributions. If the frequency is increased further, the mode overlap further increases and the thickness of the stack becomes irrelevant so that we can take it to be innite. In this limit, and for nite magnetoelastic cou- pling, phonons just give rise to an average broadening of the FMR line. While indirect, our observations thus conrm the predicted damping enhancement by phonon pumping in the incoherent limit [2, 3]. The oscillations observed in magnetoelastic mode cou- plinggmein panel 3(c) agree well with the model Eq. (4) (red line) for a YIG lm with a thickness of tYIG = 630 nm and a pinning parameter p= 0:5 (from Fig. 1(g)) for!=2.7 GHz. An alternative assessment based on a full t of the experiments by the coupled equations for the complex scattering parameter leads to a similar gme=2= 1:6 MHz at!=22:2 GHz (see SM [40]). The model likely overestimates the coupling strength, since the inhomogeneous contributions to the line broadening are not considered independently here. In summary, our high resolution FMR data taken over a broad frequency range conrm that magnon-phonon coupling in conned systems depends not only on the material parameters, but also qualitatively changes with the mode overlap. This provides the option of tun- ing the magnon-phonon coupling strength by the fre- quency, magnetic eld variations and sample geometry. We analyzed the magnon-phonon mode coupling over a broad frequency range by a simple harmonic oscilla- tor model, revealing the oscillating nature of the acous-5 tic spin pumping eciency as predicted theoretically [3]. Broadband phonon pumping experiments in heterostruc- tures as presented here can thus be used as experimental platform to study the in uence of the magnetic phase diagram on the acoustic properties also in an adjacent magnetic substrate, e.g. in the frustrated magnetic phase at very low temperatures in GGG [45]. We would like to thank O. Klein and A. Kamra for fruitful discussions. We acknowledge nancial support by the Deutsche Forschungsgemeinschaft via SFB 1432 (project no. B06), SFB 1143 (project no. C08), the W urzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 39085490), and the Cluster of Excellence \Mu- nich Center for Quantum Science and Technology" (EXC 2111, project-id 390814868). richard.schlitz@mat.ethz.ch [1] F. Godejohann, A. V. Scherbakov, S. M. Kukhtaruk, A. N. Poddubny, D. D. Yaremkevich, M. Wang, A. Nadzeyka, D. R. Yakovlev, A. W. Rushforth, A. V. Akimov, and M. Bayer, Phys. Rev. B 102, 144438 (2020). [2] T. Sato, W. Yu, S. Streib, and G. E. W. Bauer, Physical Review B 104, 014403 (2021). [3] S. Streib, H. Keshtgar, and G. E. W. Bauer, Physical Review Letters 121, 027202 (2018). [4] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hur- dequint, N. Beaulieu, J. B. Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Physical Review B 101, 060407(R) (2020). [5] A. R uckriegel and R. A. Duine, Physical Review Letters 124, 117201 (2020). [6] J. Holanda, D. S. Maior, O. A. Santos, A. Azevedo, and S. M. Rezende, Applied Physics Letters 118, 022409 (2021). [7] S. M. Rezende, D. S. Maior, O. A. Santos, and J. Holanda, Physical Review B 103, 144430 (2021). [8] J. Graf, S. Sharma, H. Huebl, and S. V. Kusminskiy, Physical Review Research 3, 013277 (2021). [9] K. An, R. Kohno, A. N. Litvinenko, R. L. Seeger, V. V. Naletov, L. Vila, G. de Loubens, J. B. Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, arXiv 2108.13272 (2021), arXiv:2108.13272 [cond-mat.mtrl-sci]. [10] A. V. Chumak and H. Schultheiss, Journal of Physics D: Applied Physics 50, 300201 (2017). [11] A. V. Chumak, P. Kabos, M. Wu, C. Abert, C. Adel- mann, A. Adeyeye, J. Akerman, F. G. Aliev, A. Anane, A. Awad, C. H. Back, A. Barman, G. E. W. Bauer, M. Becherer, E. N. Beginin, V. A. S. V. Bittencourt, Y. M. Blanter, P. Bortolotti, I. Boventer, D. A. Bozhko, S. A. Bunyaev, J. J. Carmiggelt, R. R. Cheenikundil, F. Ciubotaru, S. Cotofana, G. Csaba, O. V. Dobro- volskiy, C. Dubs, M. Elyasi, K. G. Fripp, H. Fulara, I. A. Golovchanskiy, C. Gonzalez-Ballestero, P. Graczyk, D. Grundler, P. Gruszecki, G. Gubbiotti, K. Guslienko,A. Haldar, S. Hamdioui, R. Hertel, B. Hillebrands, T. Hioki, A. Houshang, C. M. Hu, H. Huebl, M. Huth, E. Iacocca, M. B. Jung eisch, G. N. Kakazei, A. Khi- tun, R. Khymyn, T. Kikkawa, M. Kl aui, O. Klein, J. W. K los, S. Knauer, S. Koraltan, M. Kostylev, M. Krawczyk, I. N. Krivorotov, V. V. Kruglyak, D. Lachance-Quirion, S. Ladak, R. Lebrun, Y. Li, M. Lindner, R. Mac^ edo, S. Mayr, G. A. Melkov, S. Mieszczak, Y. Nakamura, H. T. Nembach, A. A. Nikitin, S. A. Nikitov, V. Novosad, J. A. Otalora, Y. Otani, A. Papp, B. Pigeau, P. Pirro, W. Porod, F. Porrati, H. Qin, B. Rana, T. Reimann, F. Riente, O. Romero-Isart, A. Ross, A. V. Sadovnikov, A. R. San, E. Saitoh, G. Schmidt, H. Schultheiss, K. Schultheiss, A. A. Serga, S. Sharma, J. M. Shaw, D. Suess, O. Surzhenko, K. Szulc, T. Taniguchi, M. Urb anek, K. Usami, A. B. Ustinov, T. van der Sar, S. van Dijken, V. I. Vasyuchka, R. Verba, S. V. Kus- minskiy, Q. Wang, M. Weides, M. Weiler, S. Wintz, S. P. Wolski, and X. Zhang, arXiv 2111.00365 (2021), arXiv:2111.00365 [physics.app-ph]. [12] V. Cherepanov, I. Kolokolov, and V. L 'vov, Physics Re- ports 229, 81 (1993). [13] T. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004). [14] M. Pomerantz, Physical Review Letters 7, 312 (1961). [15] H. Matthews and R. C. LeCraw, Physical Review Letters 8, 397 (1962). [16] C. F. Kooi, Physical Review 131, 1070 (1963). [17] P. M. Rowell, British Journal of Applied Physics 14, 60 (1963). [18] C. F. Kooi, P. E. Wigen, M. R. Shanabarger, and J. V. Kerrigan, Journal of Applied Physics 35, 791 (1964). [19] P. E. Wigen, W. I. Dobrov, and M. R. Shanabarger, Physical Review 140, A1827 (1965). [20] T. Kobayashi, R. C. Barker, and A. Yelon, Physical Review B 7, 3286 (1973). [21] Y. Sunakawa, S. Maekawa, and M. Takahashi, Journal of Magnetism and Magnetic Materials 46, 131 (1984). [22] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Materials 9, 894 (2010). [23] M. Bombeck, A. S. Salasyuk, B. A. Glavin, A. V. Scherbakov, C. Br uggemann, D. R. Yakovlev, V. F. Sapega, X. Liu, J. K. Furdyna, A. V. Akimov, and M. Bayer, Physical Review B 85, 195324 (2012). [24] T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. ichi Uchida, Z. Qiu, G. E. Bauer, and E. Saitoh, Physical Review Letters 117, 207203 (2016). [25] M. Goryachev, S. Galliou, and M. E. Tobar, Physical Review B 100, 174108 (2019). [26] K. Harii, Y.-J. Seo, Y. Tsutsumi, H. Chudo, K. Oy- anagi, M. Matsuo, Y. Shiomi, T. Ono, S. Maekawa, and E. Saitoh, Nature Communications 10, 2616 (2019). [27] M. Goryachev, S. Galliou, and M. E. Tobar, Physical Review Research 2, 023035 (2020). [28] N. K. P. Babu, A. Trzaskowska, P. Graczyk, G. Centa la, S. Mieszczak, H. G lowi nski, M. Zdunek, S. Mielcarek, and J. W. K los, Nano Letters 21, 946 (2020). [29] A. Litvinenko, R. Khymyn, V. Tyberkevych, V. Tikhonov, A. Slavin, and S. Nikitov, Physical Review Applied 15, 034057 (2021). [30] S. N. Polulyakh, V. N. Berzhanskii, E. Y. Semuk, V. I. Belotelov, P. M. Vetoshko, V. V. Popov, A. N. Shaposh-6 nikov, A. G. Shumilov, and A. I. Chernov, Journal of Experimental and Theoretical Physics 132, 257 (2021). [31] C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schnei- der, K. Lenz, J. Grenzer, R. H ubner, and E. Wendler, Physical Review Materials 4, 024416 (2020). [32] C. Kittel, Physical Review 110, 1295 (1958). [33] J. T. Yu, R. A. Turk, and P. E. Wigen, Physical Review B11, 420 (1975). [34] F. Schreiber and Z. Frait, Physical Review B 54, 6473 (1996). [35] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, Journal of Physics D: Applied Physics 48, 015001 (2014). [36] M. Ye and H. D otsch, Physical Review B 44, 9458 (1991). [37] R. L. Comstock and R. C. LeCraw, Journal of Applied Physics 34, 3022 (1963).[38] M. Ye, A. Brockmeyer, P. E. Wigen, and H. D otsch, Le Journal de Physique Colloques 49, C8 (1988). [39] S. Probst, F. B. Song, P. A. Bushev, A. V. Ustinov, and M. Weides, Review of Scientic Instruments 86, 024706 (2015). [40] See Supplemental Material at [URL will be inserted by publisher] for an alternative approach to extracting the coupling strength and a discussion of the full set of t parameters. [41] M. Dutoit and D. Bellavance, in 1972 Ultrasonics Sym- posium (IEEE, 1972). [42] M. Dutoit, Journal of Applied Physics 45, 2836 (1974). [43] B. C. Daly, K. Kang, Y. Wang, and D. G. Cahill, Phys- ical Review B 80, 174112 (2009). [44] M. Krzesi nska and T. Szuta-Buchacz, Physica Status So- lidi (a) 82, 421 (1984). [45] P. P. Deen, O. Florea, E. Lhotel, and H. Jacobsen, Phys- ical Review B 91, 014419 (2015).

2022-02-07

"Pumping" of phonons by a dynamic magnetization promises to extend the range and functionality of magnonic devices. We explore the impact of phonon pumping on room-temperature ferromagnetic resonance (FMR) spectra of bilayers of thin yttrium iron garnet (YIG) films on thick gadolinium gallium garnet substrates over a wide frequency range. At low frequencies the Kittel mode hybridizes coherently with standing ultrasound waves of a bulk acoustic resonator to form magnon polarons that induce rapid oscillations of the magnetic susceptibility, as reported before. At higher frequencies, the phonon resonances overlap, merging into a conventional FMR line, but with an increased line width. The frequency dependence of the increased line broadening follows the predictions from phonon pumping theory in the thick substrate limit. In addition, we find substantial magnon-phonon coupling of a perpendicular standing spin wave (PSSW) mode. This evidences the importance of the mode overlap between the acoustic and magnetic modes, and provides a route towards engineering the magnetoelastic mode coupling.

Magnetization dynamics affected by phonon pumping

2202.03331v1

arXiv:1906.01560v3 [cond-mat.mtrl-sci] 3 Sep 2020Thickness dependence of spin Peltier effect visualized by th ermal imaging technique Shunsuke Daimon,1, 2, 3,∗Ken-ichi Uchida,1, 4, 5, 6, †Naomi Ujiie,7Yasuyuki Hattori,7Rei Tsuboi,1and Eiji Saitoh1, 2, 3, 6, 8 1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3Department of Applied Physics, The University of Tokyo, Tok yo 113-8656, Japan 4National Institute for Materials Science, Tsukuba 305-004 7, Japan 5Department of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan 6Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 7ALPS ALPINE CO., LTD., Tokyo 145-8501, Japan 8Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan Abstract Magnon propagation length in a ferrimagnetic insulator ytt rium iron garnet (YIG) has been investigated by measuring and analyzing the YIG-thickness tYIGdependence of the spin Peltier effect (SPE) in a Pt/YIG junction system. By means of the lock-in thermography techn ique, we measured the spatial distribution of the SPE-induced temperature modulation in the Pt/YIG syste m with the tYIGgradation, allowing us to obtain the accurate tYIGdependence of SPE with high tYIGresolution. Based on the tYIGdependence of SPE, we verified the applicability of several phenomenological mod els to estimate the magnon diffusion length in YIG. 1Interconversion between spin and heat currents has been ext ensively studied in the field of spin caloritronics [1, 2]. One of the spin-caloritronic phenome na is the spin Seebeck effect (SSE) [3], which generates a spin current as a result of a heat current in metal/magnetic-insulator junction systems. The Onsager reciprocal of SSE is the spin Peltier eff ect (SPE) [4–11]. A typical system used for studying SPE and SSE is paramagnetic metal Pt/ferri magnetic insulator yttrium iron garnet (YIG) junction systems [3–5, 7, 12–15], where the spin and he at currents are carried by electron spins in Pt and magnons in YIG [16–19]. SPE and SSE are characterized by their length scale includin g the magnon-spin diffusion length lmand the magnon-phonon thermalization length lmpin YIG [18, 20, 21]. These length parameters have been investigated by measuring discrete YIG-thicknes stYIGdependence of SPE and SSE in several Pt/YIG junction systems with different tYIG[7, 13, 14, 21, 22]. However, different junctions may have different magnetic properties, surface roughness, crystallinity, and interface condition. These variations make it hard to analyze the fine tYIGdependence and obtain correct values of the length parameters. In this letter, we measured the tYIGdependence of SPE by using a single Pt/YIG system with atYIGgradient ( ∇tYIG). Since SPE induces temperature change reflecting the local tYIGvalue, we can extract the tYIGdependence of SPE from a temperature distribution. The spat ial distribution of the SPE-induced temperature change was visualized by means of the lock-in thermography (LIT) [5]. The LIT method allows us to obtain the accurate tYIGdependence of SPE with high tYIG resolution. By means of the thermoelectric imaging techniq ue based on laser heating [23], we also measured the tYIGdependence of SSE in a single Pt/YIG system, which shows the s ame behavior as that of SPE. By analyzing the measured tYIGdependence of SPE and using phenomenological models, we estimated lmand determined the upper limit of lmpfor YIG. The sample used for measuring SPE consists of a Pt film and an YI G film with ∇tYIG[Fig. 1(a),(d)]. ∇tYIGwas introduced by obliquely polishing a single-crystallin e YIG (111) film grown by a liquid phase epitaxy method on a single-crystalline Gd 3Ga5O12(GGG) (111) substrate. The obtained ∇tYIGis almost uniform in the measurement range, which was observ ed to be 9.2µm per a 1 mm lateral length by a cross-sectional scanning elect ron microscopy [Fig. 1(f)]. After the polishing, a U-shaped Pt film with a thickness of 5 nm and width of 0.5 mm was sputtered on the surface of the YIG film. The longer lines of the U-shaped Pt film were along the ∇tYIGdirection. In the microscope image of the sample in Fig. 1(d), the yellow (gray) area above (below) the white dotted line corresponds to the YIG film with ∇tYIG(GGG substrate). 2FIG. 1: (a) Schematic of the SPE measurement using a Pt/YIG/G GG sample by means of the lock-in thermography method. A charge current Jcis applied to the U-shaped Pt film fabricated on the YIG film with a thickness gradient ∇tYIG. (b),(c) Time tprofile of an input a.c. charge current Jcand output temperature change ∆Tfor the (b) SPE and (c) Joule heating configurations. (d) An op tical microscope image of the sample. The yellow (gray) area above (below) the white dotted line corresponds to the YIG film with ∇tYIG(GGG substrate). His an applied magnetic field. (e) An infrared image of the samp le. (f) AtYIGprofile and cross-sectional image of the sample obtained wit h a scanning electron microscope. SPE induces temperature modulation in the Pt/YIG/GGG sampl e in response to a charge current in the Pt film. When we apply a charge current Jcto the Pt film as shown in Fig. 1(a), a spin current is generated by the spin Hall effect in Pt [24, 25]. The spin current induces a heat current across the Pt/YIG interface via SPE. The heat current result s in a temperature change ∆Twhich satisfies the following relation [4, 5]: ∆T∝(Jc×M)·n, where Mandnare the magnetization vector of YIG and the normal vector of the Pt/YIG interface pl ane, respectively. Significantly, the SPE-induced temperature change reflects local tYIGinformation because the temperature change induced by SPE is localized owing to the formation of dipolar heat sources [5, 7]. Based on the ∇tYIGvalue and the spatial resolution of our LIT system, we obtain ed the high tYIGresolution of 92 nm. The procedure of the LIT-based SPE measurements are as follo ws [5, 26–31]. To excite SPE, we applied a rectangular a.c. charge current Jcwith the amplitude J0, frequency f, and zero offset to the Pt film [Fig. 1(b)]. By extracting the first harmonic res ponse of a temperature change ∆T1f 3FIG. 2: (a) Lock-in image of an infrared light emission ∆I1finduced by SPE. (b) Lock-in amplitude of the infrared emission Ainduced by the Joule heating. (c),(d) YIG-thickness tYIGdependence of (c) ∆I1fand the emissivityǫand (d) the temperature change ∆T1finduced by SPE. in this condition (SPE configuration), we can detect the pure SPE signal free from a Joule heating contribution [5, 7]. Here, ∆T1fis defined as the temperature change oscillating in the same p hase asJcbecause∆Tgenerated by SPE follows the Jcoscillation [Fig. 1(b)] and the out-of-phase signal is negligibly small [5, 7, 32, 33]. In the LIT measurem ent, we detect the first harmonic component of the infrared light emission ∆I1fcaused by∆T1f, where∆I1f=Acosφwith Aand φrespectively being the lock-in amplitude and phase. We conv erted∆I1finto∆T1fby considering spatial distribution of an infrared emissivity ǫof the sample [5, 7]. All measurements of SPE were performed at room temperature and atmospheric pressure und er a magnetic field with a magnitude of 20 mT, where Maligns along the field direction. Figure 2(a) shows the ∆I1fimage from the Pt/YIG/GGG sample in the SPE configuration wit h J0=8 mA and f=5 Hz. Clear signals appear only on the Pt film with finite tYIGbut disappears on the Pt/GGG interface with tYIG=0 [compare Figs. 1(d) and 2(a)]. The sign of ∆I1fis reversed when Jcis reversed, which confirms that the observed infrared signa l comes from SPE [4, 5]. The spatial profile of ∆I1falong the ydirection is plotted in Fig. 2(c), where the ∆I1fvalues are averaged along the xdirection in the area surrounded by the dotted line in Fig. 2( a).∆I1fgradually increases with small oscillation with increasing tYIG. The oscillation originates from the oscillation 4ofǫof the sample due to multiple reflection and interference of t he infrared light in the YIG film [see the infrared light image of the sample in Fig. 1(e)] [7]. To calibrate theǫoscillation in the tYIG dependence of SPE, the infrared emission induced by the Joul e heating was measured [Fig. 2(b)], where Jcwith the amplitude ∆Jc=0.5 mA, frequency f=5 Hz, and d.c. offset Jdc=8.0 mA was used as an input for the LIT measurement [Fig. 1(c)] [5]. Sinc e the temperature change induced by the Joule heating is uniform on the Pt film, the lock-in ampl itude of the infrared emission A on the Pt film depends only on the ǫdistribution. We found that the obtained tYIGdependence ofǫ(∝Adue to the Joule heating) and the ∆I1fsignal due to SPE exhibit the similar oscillating behavior [Fig. 2(c)]. By calibrating ∆I1fbyǫ, we obtained the tYIGdependence of∆T1finduced by SPE [Fig. 2(d)]. ∆T1fmonotonically increases with increasing tYIG. The saturated∆T1fvalue fortYIG>6µm was determined by using the same Pt/YIG/GGG sample coated w ith a black-ink infrared emission layer with high emissivity larger than 0. 95. The accurate tYIGdependence of SPE with high tYIGresolution allows us to verify the appli- cability of several phenomenological models used for discu ssing the behaviors of SPE. First of all, we found the obtained tYIGdependence of the SPE-induced temperature modulation cann ot be explained by a simple exponential approximation [7, 13, 14, 22]. Based on the assumption that the magnon diffuses in YIG with the magnon diffusion length lm, the simple exponential decay has been used for the analysis of the tYIGdependence: ∆T∝1−exp(−tYIG/lm). (1) However, in general, this expression cannot be used for the s mall thickness region since the exponential function should be modulated by the boundary co nditions for the spin and heat currents. In fact, the fitting result using Eq. (1) significantly deviat es from the experimental data when tYIG<4µm (see the green curve in Fig. 3). The observed continuous tYIGdependence of SPE thus requires advanced understanding of the spin-heat conv ersion phenomena. Next we focus on two phenomenological models proposed in Ref s. 19 and 34. The model in Ref. 19 is based on the linear Boltzmann’s theory for magnons and the tYIGdependence of SPE is described as ∆T∝1 Scoth(tYIG/lm)+1, (2) where Sis atYIG-independent constant used as a fitting parameter in our anal ysis. The red curve in Fig. 3 shows the fitting result based on Eq. (2). We found that E q. (2) shows the best agreement with the experimental result and lmis estimated to be 3 .9µm. We also analyzed the experimental 5FIG. 3: Experimental results of the tYIGdependence of∆T1finduced by SPE for the Pt/YIG/GGG sample and fitting curves using Eqs. (1)-(3). results by the model based on the non-equilibrium thermodyn amics [34]: ∆T∝cosh(tYIG/lm)−1 sinh(tYIG/lm)+rcosh(tYIG/lm), (3) where ris atYIG-independent constant used as a fitting parameter in our anal ysis. In contrast to Eq. (2), Eq. (3) is less consistent with the experimental res ults in the whole thickness range (see the yellow curve in Fig. 3) and gives a shorter magnon diffusio n length of 0.6µm. From the fitting results using Eqs. (1-3), we conclude that Eq. (2) is the best phenomenological model to explain the experimental results on SPE. To check the reciprocity between SPE and SSE [35], we also mea sured the tYIGdependence of SSE by using a single Pt/YIG/GGG sample with ∇tYIG. The YIG film used in the SSE measurement was obtained from the same YIG/GGG substrate and the SSE samp le was prepared by the same method as that for the SPE sample. The ∇tYIGvalue of the SSE sample was determined to be 12.2µm per a 1 mm lateral length. To obtain the tYIGdependence of SSE, the SSE signal was measured in the Pt/YIG/GGG sample by means of the thermoelec tric imaging technique based on laser heating [23, 36–38]. As shown in Fig. 4(a), the sample s urface was irradiated by laser light with the wavelength of 1 .3µm and the laser spot diameter of 5 .2µm to generate a temperature gradient across the Pt/YIG interface. The local temperatur e gradient induces a spin current across the interface due to SSE. The spin current is then converted i nto a charge current via the inverse spin Hall effect in Pt [25]. To avoid the reduction of the spati al resolution of the SSE signal due to the thermal diffusion, we adopted a lock-in technique in th e laser SSE measurement, where 6FIG. 4: (a) Schematic of the SSE measurement using a laser hea ting method. JSSE cis a charge current due to the inverse spin Hall effect induced by SSE. The thickness of t he Pt film is 50 nm, which is thick enough to prevent light transmission; the Pt layer is heated by the las er light. (b) Infrared light image of the sample. (c) The SSE signal SSSEimage induced by the laser heating. (d) tYIGdependence of SSSEand comparison with that of the temperature change induced by SPE. Here, we used a n arbitrary unit for the SSE signal because the temperature gradient induced by the laser heating canno t be estimated experimentally. Nevertheless, the relative change of SSSEis reliable because the heating of the Pt layer is uniform ove r the sample. the laser intensity was modulated in a periodic square wavef orm with the frequency f=5 kHz and the thermopower signal S1foscillating with the same frequency as that of the input lase r was measured. The measurements at the high lock-in frequency re alized high spatial resolution for the SSE signal because temperature broadening due to the hea t diffusion is suppressed. Here, we defined the SSE signal SSSEas/bracketleftbig S1f(+50 mT)−S1f(−50 mT)/bracketrightbig /2 to remove magnetic-field- independent background. By scanning the position of the las er spot on the sample, we visualized the spatial distribution of the SSE signal with high tYIGresolution of 64 nm. Figure 4(c) shows the spatial distribution of SSSEfor the Pt/YIG/GGG sample. In response to the laser heating, the clear signal was observed to appear in the Pt film. The tYIGdependence of the SSE signal is plotted in Fig. 4(d), where the SSSEvalues were averaged along the xdirection 7in the area surrounded by the dotted line in Fig. 4(c). The SSE signal monotonically increases with increasing tYIG. Significantly, the tYIGdependence of SSSEshows the same behavior as that of the SPE-induced temperature modulation [Fig. 4(d)]. This r esult supports the reciprocity between SPE and SSE and strengthens our conclusion in the SPE measure ment. In the recent study on SSE in Ref. 21, Parakash et al. reported non-monotonical increase of the SSE signal with tYIG. Since the SSE signal takes a local maximum at tYIG∼lmp, they estimated lmpas 250 nm from the maximum point. However, in our Pt/YIG sampl es, the SSE and SPE signals monotonically increase with increasing tYIG. These results suggest that lmpis shorter than thetYIGresolution, 64 nm, in our experiments. The conclusion is con sistent with the theoretical expectation of lmp∼1 nm [18]. In conclusion, we have discussed the length scale of the spin and heat transport by magnons in YIG by measuring the tYIGdependence of SPE in the Pt/YIG sample. This measurement was realized by using the YIG film with the tYIGgradient and the LIT method, which allow us to obtain the continuous tYIGdependence of SPE in the single Pt/YIG sample. The experimen tal result is well reproduced by the phenomenological model bas ed on the linear Boltzmann’s theory for magnons referenced in Ref. 19 and lmis estimated to be 3 .9µm for our YIG sample. We also measured the tYIGdependence of SSE. The SPE and SSE signals show the same behav ior in the tYIGdependence and monotonically increase as tYIGincreases. The monotonic increase implies that lmpis shorter than 64 nm for our YIG sample. These results give cr ucial information to understand the microscopic origin of the spin-heat conve rsion phenomena. The authors thank R. Iguchi, T. Kikkawa, M. Matsuo, Y. Ohnuma , and G. E. W. Bauer for valuable discussions. This work was supported by CREST “Cre ation of Innovative Core Tech- nologies for Nano-enabled Thermal Management” (JPMJCR17I 1), PRESTO “Phase Interfaces for Highly Efficient Energy Utilization” (JPMJPR12C1), and ERAT O “Spin Quantum Rectification Project” (JPMJER1402) from JST, Japan, Grant-in-Aid for Sc ientific Research (A) (JP15H02012) and Grant-in-Aid for Scientific Research on Innovative Area “Nano Spin Conversion Science” (JP26103005) from JSPS KAKENHI, Japan, the Inter-Universi ty Cooperative Research Program of the Institute for Materials Research (17K0005), Tohoku U niversity, and NEC Corporation. ∗Electronic address: daimon@ap.t.u-tokyo.ac.jp †Electronic address: UCHIDA.Kenichi@nims.go.jp 8[1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). [2] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Enviro n. Sci. 7, 885 (2014). [3] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [4] J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. Ben Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). [5] S. Daimon, R. Iguchi, T. Hioki, E. Saitoh, and K. Uchida, N at. Commun. 7, 13754 (2016). [6] K. Uchida, R. Iguchi, S. Daimon, R. Ramos, A. Anadón, I. Lu cas, P. A. Algarabel, L. Morellón, M. H. Aguirre, M. R. Ibarra, and E. Saitoh, Phys. Rev. B 95, 184437 (2017). [7] S. Daimon, K. Uchida, R. Iguchi, T. Hioki, and E. Saitoh, P hys. Rev. B 96, 024424 (2017). [8] Y. Ohnuma, M. Matsuo, and S. Maekawa, Phys. Rev. B 96, 134412 (2017). [9] R. Iguchi, A. Yagmur, Y.-C. Lau, S. Daimon, E. Saitoh, M. H ayashi, and K. Uchida, Phys. Rev. B 98, 014402 (2018). [10] K. Uchida, M. Sasaki, Y. Sakuraba, R. Iguchi, S. Daimon, E. Saitoh, and M. Goto, Sci. Rep. 8, 16067 (2018). [11] T. Yamazaki, R. Iguchi, T. Ohkubo, H. Nagano, and K. Uchi da, Phys. Rev. B 101, 020415(R) (2020). [12] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev. Lett. 110, 067207 (2013). [13] T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E . Saitoh, Phys. Rev. B 92, 064413 (2015). [14] A. Kehlberger, U. Ritzmann, D. Hinzke, E.-J. Guo, J. Cra mer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jungfleisch, B. Hillebrands, U. Nowak, and M. Kläui, Phys. Rev. Lett. 115, 096602 (2015). [15] R. Itoh, R. Iguchi, S. Daimon, K. Oyanagi, K. Uchida, and E. Saitoh, Phys. Rev. B 96, 184422 (2017). [16] S. S.-L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012). [17] S. M. Rezende, R. L. Rodríguez-Suárez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Phys. Rev. B89, 014416 (2014). [18] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Du ine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). [19] S. S. Costa and L. C. Sampaio, J. Phys. D: Appl. Phys. 53, 355001 (2020). [20] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine, Phys. Rev. Lett. 116, 117201 (2016). [21] A. Prakash, B. Flebus, J. Brangham, F. Yang, Y. Tserkovn yak, and J. P. Heremans, Phys. Rev. B 97, 9020408(R) (2018). [22] E.-J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D. A . MacLaren, G. Jakob, and M. Kläui, Phys. Rev. X 6, 031012 (2016). [23] R. Iguchi, S. Kasai, K. Koshikawa, N. Chinone, S. Suzuki , and K. Uchida, Sci. Rep. 9, 18443 (2019). [24] A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013). [25] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, a nd T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). [26] H. Straube, J.-M. Wagner, and O. Breitenstein, Appl. Ph ys. Lett. 95, 052107 (2009). [27] O. Breitenstein, W. Warta, and M. Langenkamp, Lock-in Thermography: Basics and Use for Evaluating Electronic Devices and Materials (Springer, Belrin/Heidelberg, 2010). [28] O. Wid, J. Bauer, A. Müller, O. Breitenstein, S. S. P. Par kin, and G. Schmidt, Sci. Rep. 6, 28233 (2016). [29] O. Wid, J. Bauer, A. Müller, O. Breitenstein, S. S. P. Par kin, and G. Schmidt, J. Phys. D: Appl. Phys. 50, 134001 (2017). [30] T. Seki, R. Iguchi, K. Takanashi, and K. Uchida, Appl. Ph ys. Lett. 112, 152403 (2018). [31] K. Uchida, S. Daimon, R. Iguchi, and E. Saitoh, Nature 558, 95-99 (2018). [32] R. Iguchi and K. Uchida, Jpn. J. Appl. Phys. 57, 0902B6 (2018). [33] A. Yagmur, R. Iguchi, S. Geprägs, A. Erb, S. Daimon, E. Sa itoh, R. Gross, and K. Uchida, J. Phys. D 51, 194002 (2018). [34] V. Basso, E. Ferraro, A. Magni, A. Sola, M. Kuepferling, and M. Pasquale, Phys. Rev. B 93, 184421 (2016). [35] A. Sola, V. Basso, M. Kuepferling, C. Dubs, and M. Pasqua le, Sci. Rep. 9, 2047 (2019). [36] M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. W agner, M. Opel, I. M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett .108, 106602 (2012). [37] N. Roschewsky, M. Schreier, A. Kamra, F. Schade, K. Ganz horn, S. Meyer, H. Huebl, S. Geprägs, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 104, 202410 (2014). [38] J. M. Bartell, C. L. Jermain, S. V. Aradhya, J. T. Brangha m, F. Yang, D. C. Ralph, and G. D. Fuchs, Phys. Rev. Appl. 7, 044004 (2017). 10

2019-06-04

Magnon propagation length in a ferrimagnetic insulator yttrium iron garnet (YIG) has been investigated by measuring and analyzing the YIG-thickness t_YIG dependence of the spin Peltier effect (SPE) in a Pt/YIG junction system. By means of the lock-in thermography technique, we measured the spatial distribution of the SPE-induced temperature modulation in the Pt/YIG system with the t_YIG gradation, allowing us to obtain the accurate t_YIG dependence of SPE with high t_YIG resolution. Based on the t_YIG dependence of SPE, we verified the applicability of several phenomenological models to estimate the magnon diffusion length in YIG.

Thickness dependence of spin Peltier effect visualized by thermal imaging technique

1906.01560v3

arXiv:2101.09931v1 [quant-ph] 25 Jan 2021Nonreciprocal Transmission and Entanglement in a cavity-m agnomechanical system Zhi-Bo Yang1, Jin-Song Liu1, Ai-Dong Zhu1, Hong-Yu Liu1,∗and Rong-Can Yang2,3† 1Department of Physics, College of Science, Yanbian Univers ity, Yanji, Jilin 133002, China 2Fujian Provincial Key Laboratory of Quantum Manipulation a nd New Energy Materials, and College of Physics and Energy, Fujian Normal University , Fuzhou 350117, China and 3Fujian Provincial Collaborative Innovation Center for Opt oelectronic Semiconductors and Efficient Devices, Xiamen 361005, China (Dated: January 26, 2021) Quantum entanglement, a key element for quanum information is generated with a cavity mag- nomechanical system. It comprises of two microwave cavitie s, a magnon mode and a vibrational mode, and the last two elements come from a YIG sphere trapped in the second cavity. The two microwave cavities are connected by a superconducting t ransmission line, resulting in a linear coupling between them. The magnon mode is driven by a strong m icrowave field and coupled to cavity photons via magnetic dipole interaction, and at the s ame time interacts with phonons via magnetostrictive interaction. By breaking symmetry of the configuration, we realize nonreciprocal photon transmission and one-way bipartite quantum entangl ement. By using current experimental parameters for numerical simulation, it is hoped that our re sults may reveal a new strategy to built quantum resources for the realization of noise-tolerant qu antum processors, chiral networks, and so on. I. INTRODUCTION Although reciprocity is ubiquitous in nature, nonre- ciprocity promptes diverse applications such as chiral engineering, invisible sensing, and backaction-immune information processing [ 1–3]. So far, electromagnetic nonreciprocal transmission [ 4,5] has been demonstrated with various systems ranging from microwave [ 6–9], ter- ahertz [10], optical [ 11,12] photons to x-rays [13]. As a special type of nonreciprocity which only allows one- way transmission, unidirectional transmission of light is also revealed with some hybrid systems, such as atomic gases [14], nonlinear devices [ 11,15–17], moving me- dia [18] and synthetic materials [ 19]. While in quantum regime, a recent scheme has been proposed, where nonre- ciprocal entangled states is prepared by using an optical diode with a high isolation rate [ 20], making it possi- ble to swift a single device between classical isolator and quantum diode, or to protect quantum resources from backscattering losses. Cavity spintronics [ 21–35] is an emerging and rapidlydevelopinginterdisciplinarythatstudiesmagnons strongly couple with microwave photons via magnetic dipole interaction. Such a hybrid system shows great ap- plication prospect in the field of quantum information processing, especially for quantum transducer [ 36–42] and quantum memory [ 43]. The reason is that magnetic materials have several distinguishing advantages such as long lifetime, high spin density and easy tunability. Fur- thermore, a collective excitation of spins in these mate- rials(called magnon mode or Kittel mode )in magnetic materialscan easilybe coupled to a varietyofothertypes of systems. Thus, cavity spintronics seems to be a po- ∗liuhongyu@ybu.edu.cn †rcyang@fjnu.edu.cntentialcandidatetostudymultiple quantumcorrelations, etc. By using cavity spintronics, in this manuscript, we propose a scheme to realize an optical diode with both quantum and classicalcharacteristics, i.e., the implemen- tation of both nonreciprocal microwavetransmission and nonreciprocal bipartite quantum entanglement. In this manuscript, we present a proposal to carry out nonreciprocal microwave transmission and one-way bi- partite entanglement by using an additional microwave cavity coupled to a cavity-magnon system to break sym- metryofspatialinversion[ 9]. Thetwomicrowavecavities are linked by superconducting transmission lines [ 44] and one of them interacts with magnon mode of a ferrimag- netic yttrium iron garnet (YIG)sphere [21,24,45–49]. Simultaneously, the magnon mode is coupled to phonons due to vibration of YIG sphere induced by the mag- netostrictive force [ 26,50,51]. In addition, magnons are driven by a strong microwave field at the first red sideband with respect to phonons because entanglement mainly survives with small thermal phonon occupancy. The(anti-Stokes )process not only realizes phonon cool- ing, a prerequisite for observing quantum effects in the system [52], but also also enhances the effective magnon- phonon coupling in orderfor the generationof magnome- chanical entanglement. If both microwave cavities are resonant with the red sideband, then entanglement be- tween the other two subsystems will be generatedthough some entanglement is very small. For the same driving power, different driving direction induces different effec- tive magnetostrictive interaction, leading the generated subsystem entanglement to exhibit rare nonreciprocity and unidirectional invisibility. Furthermore, most of bi- partite entanglement is robust against ambient tempera- ture. The manuscript is organized as follows. At first, we presenta generalmodel ofthe scheme, and then solvethe system dynamics by means of the standard Langevin for-2 malism with linearization treatment. Next, we illustrate nonreciprocity of microwave transmission and bipartite entanglement in the stationary state. Finally, we show how to measure generated entanglement and analyze the validity of our model. II. MODEL AND EQUATION OF MOTION As schematically shown in Fig. 1(a), we study a cavity magnomechanical system which consists of two coupled microwave cavities and one of them coupled to a YIG sphere. The sphere is uniformly magnetized to satura- tion by a bias magnetic field with B0=B0ez, whereB0 andezrepresent magnetic amplitude and the unit vector a Kittel mode #side view top view #Ea Ec left cavity right cavity Emsuperconducting transmission linea cout out YIG sphere vibrations mode ωa ωd ωm ωc ωd + ωb ωd - ωb-ΔaΔm-Δc κcκaκm ω b z xy FIG. 1. (a) A schematic diagram of two-cavity magnome- chanic system, consisting of two microwave cavities and a YIG sphere, which is placed in the right cavity. The YIG sphere, which is magnetized to saturation by a bias magnetic fieldB0aligned along the z-direction, is mounted near the right cavity wall, where the magnetic field of the cavity mode is the strongest and polarized along x-direction to excite the magnon mode in YIG. The magnon mode is driven by a mi- crowave field along the y-direction. Three magnetic fields are mutuallyperpendicular ar thesite ofthe YIGsphere. In addi - tion, thesystemwill exhibitvaryingdegrees ofmagnomecha n- ical interaction when the left or right cavity is driven alon e. (b) Mode frequencies and linewidths. The magnon mode (two MW cavity modes )with frequency ωm(ωaandωc)is driven by a strong MW field at frequency ωd, and the mechanical motion of frequency ωbscatters the driving photons onto two sidebands at ωd±ωb. If the magnon mode is resonant with the blue (anti-Stokes )sideband, and two cavity modes are resonant with the red (Stokes)sideband, all subsystems are prepared in entangled state. See text for more details.inzdirection, respectively [ 23]. At the same time, the YIG sphere is also directly driven by a microwave field with driving strength Emand frequency ωd. In addition, either the first cavity or the second one is driven by a mi- crowave light beam with driving strength EaorEcwith the same frequency ωd. The two cavities are linked by a superconducting transmission line [ 44] and the magnon modecouplestothesecondcavitymodeandavibrational mode [21] via the collective magnetic-dipole interaction and magnetostrictive interaction, respectively. The mag- netostatic mode with finite wave number has a distinct frequency different from the Kittel mode so that the se- lective excitation may be implemented through the driv- ing field wavelength and cavity mode selection. When all of the driving fields are included, the total Hamiltonian of the system can be written as follows H//planckover2pi1=ωaˆa†ˆa+ωcˆc†ˆc+ωmˆm†ˆm+ωbˆb†ˆb+gac(ˆa†ˆc+ˆc†ˆa) +gcm(ˆc†ˆm+ ˆm†ˆc)+gmbˆm†ˆm(ˆb†+ˆb) +iEa(ˆa†e−iωdt−ˆaeiωdt)+iEc(ˆc†e−iωdt−ˆceiωdt) +iEm(ˆm†e−iωdt−ˆmeiωdt). (1) Here, ˆa, ˆc, ˆmandˆb(ˆa†, ˆc†, ˆm†andˆb†)are individu- ally the annihilation (creation)operators of correspond- ing cavities, magnons and phonons with resonance fre- quencyωa,ωc,ωmandωb. The magnon frequency is determined by the external bias magnetic field B0, i.e., ωm=γB0withγ/2π= 2.8 MHz/Oe being the gyromag- netic ratio. gacis the photon coupling rate [ 53] andgcm represents the linear photon-magnon coupling strength which can be estimated by measuring the reflection spec- trum of YIG sphere in right cavity and can be adjusted by varying the direction of the bias field or the position of the YIG sphere inside the right cavity [ 50]. The single- magnon magnomechanical coupling rate gmbis typically small, but the magnomechanical interaction can be en- hanced if magnons are strongly driven [ 21,22,54,55]. The amplitude for each driving fields is Ei=√κiεiwith theeffectivestrength εi=/radicalbig Pi//planckover2pi1ωdwiththecorrespond- ing driving power and driving frequency being Piand ωd, respectively [ 25].κa/candκm/bseparately repre- sent the total loss rate of the first/second cavity and the magnon/phonon mode. In the frame rotating at the driving frequency ωd, the quantum Langevin equations (QLEs)describing the sys- tem are given by ˙ˆa=−(i∆a+κa)ˆa−igacˆc+Ea+√2κaˆain, ˙ˆc=−(i∆c+κc)ˆc−igacˆa−igcmˆm+Ec+√2κcˆcin, ˙ˆm=−(i∆m+κm)ˆm−igcmˆc−igmbˆm(ˆb†+ˆb)+Em +√2κmˆmin, ˙ˆb=−(iωb+κb)ˆb−igmbˆm†ˆm+√ 2κbˆbin, (2) where ∆ j=ωj−ωdand ˆain, ˆcin, ˆmin,ˆbinare input noise operators with zero mean value acting on the cavi- ties, magnon and mechanical modes, respectively, which are characterized by the following correlation functions:3 /angbracketleftˆain(t)ˆain†(t′)/angbracketright=/angbracketleftˆcin(t)ˆcin†(t′)/angbracketright=/angbracketleftˆmin(t)ˆmin†(t′)/angbracketright= δ(t−t′),/angbracketleftˆbin(t)ˆbin†(t′)/angbracketright= (nb+ 1)δ(t−t′), and /angbracketleftˆbin†(t)ˆbin(t′)/angbracketright=nbδ(t−t′) where the equilibrium mean thermal phonon numbers nb= [exp(/planckover2pi1ωb kBT)−1]−1withkB the Boltzmann constant and Tthe ambient temperature. Because the magnon mode and microwavephotons are strongly driven, they all have large amplitude, i.e., |/angbracketlefta/angbracketright|, |/angbracketleftc/angbracketright|,|/angbracketleftm/angbracketright|≫1, whichallowsustolinearizethedynamics of the system around the steady-state values by writing any a operator as ˆ o=/angbracketlefto/angbracketright+δo(o=a,c,m,b)and ne- glecting second order fluctuation terms. Then, we obtain a set of differential equations for the mean values: ˙/angbracketlefta/angbracketright=−(i∆a+κa)/angbracketlefta/angbracketright−igac/angbracketleftc/angbracketright+Ea, ˙/angbracketleftc/angbracketright=−(i∆c+κc)/angbracketleftc/angbracketright−igac/angbracketlefta/angbracketright−igcm/angbracketleftm/angbracketright+Ec, ˙/angbracketleftm/angbracketright=−(i˜∆m+κm)/angbracketleftm/angbracketright−igcm/angbracketleftc/angbracketright+Em, ˙/angbracketleftb/angbracketright=−(iωb+κb)/angbracketleftb/angbracketright−igmb|/angbracketleftm/angbracketright|2, (3) and the linearized QLEs for the quantum fluctuations: δ˙a=−(i∆a+κa)δa−igacδc+√2κaˆain, δ˙c=−(i∆c+κc)δc−igacδa−igcmδm+√2κcˆcin, δ˙m=−(i˜∆m+κm)δm−igcmδc−1 2Gmb(δb†+δb) +√2κmˆmin, δ˙b=−(iωb+κb)δb−1 2Gmb(δm†−δm)+√2κbˆbin, (4) where˜∆m= ∆m+gmb(/angbracketleftb/angbracketright+/angbracketleftb/angbracketright∗)istheeffectivedetuning of the magnon mode including the frequency shift caused by the magnetostrictive interaction. Gmb=i2gmb/angbracketleftm/angbracketright is the effective magnomechanical coupling rate. If we consider the time to be t→ ∞and the detunings to satisfy|∆a|,|∆c|,|˜∆m| ≫κa,κc,κm, then/angbracketlefta/angbracketright,/angbracketleftc/angbracketright,/angbracketleftm/angbracketright and/angbracketleftb/angbracketrightcan be given by /angbracketlefta/angbracketright ≃iEmgacgcm−Eag2 cm−Ecgac˜∆m+Ea∆c˜∆m g2cm∆a+g2ac˜∆m−∆a∆c˜∆m, /angbracketleftc/angbracketright ≃ −iEmgcm∆a+Eagac˜∆m−Ec∆a˜∆m g2cm∆a+g2ac˜∆m−∆a∆c˜∆m, /angbracketleftm/angbracketright ≃ −iEmg2 ac−Eagacgcm+Ecgcm∆a−Em∆a∆c g2cm∆a+g2ac˜∆m−∆a∆c˜∆m, /angbracketleftb/angbracketright=−igmb|/angbracketleftm/angbracketright|2/(iωb+κb). (5) Inwhatfollows,wefirstshowthattheeffectoftherelated parameters on the nonreciprocal transmission of a mi- crowave field with a forward and backward driving-field input by solving the equations numerically, and then we show that the classical nonreciprocity can also be used to prepare nonreciprocal subsystem entangled states.III. NONRECIPROCAL TRANSMISSION In order to study the nonreciprocity of cavity output field, we let the first cavity mode driven (Ea/negationslash= 0 & Ec= 0)or the second cavity mode driven (Ea= 0 &Ec/negationslash= 0)by a classical field, and measure the output field of the other cavity [ 25]. For the sake of concision, we denote the first (second)case as the driving-field in- put from the forward (backward )direction in the follow- ing passages. Thus, the transmission coefficients for the two cases are separately defined as T12=|cout/εa|= |√κc/angbracketleftc/angbracketright/εa|andT21=|aout/εc|=|√κa/angbracketlefta/angbracketright/εc|with the average number of the first (second)cavity|/angbracketlefta(c)/angbracketright| calculated with Eq. ( 5). In addition, in order to de- pict the nonreciprocality, we define the isolation Tiso= 20×log10|T12/T21|which is given in decibels (dB)[9,25]. For the sake of simplicity, we set the power of the mi- crowavesource focused on the first or second cavity to be the same, i.e. Pa=Pc=P, and the two cavities to be identical, i.e. κa=κc=κ. The transmission coefficient T12/21and the isolation Tisoas a function of the driving powerPareplotted in Fig. 2(a)and (c) with gac/ωb= 1, and (b) and (d) with gac/ωb= 0.32. The other param- eters are chosen as ωa/2π= 10 GHz, ωb/2π= 10 MHz, ˜∆m= 0.9ωb, ∆c= ∆a=−ωb,κ/2π=κm/2π= 1 MHz, andgcm/2π= 3.2 MHz [23,54]. From Eq. ( 5), we find that when gac=ωbthe same transmissioncoefficientisobservedforthe differentdirec- tions(forward or backward injection )of the driving-field input. We regard gac=ωbasaimpedance-matchingcon- dition [53] for the same microwave transmission. From Fig.2(a)withgac/ωb= 1, wecanclearlyseethatthetwo transmission coefficients (T12andT21)are always equal. However, if we decrease the coupling rate gacbetween the two cavities, then the two coefficients (T12andT21) will be distinctly different when Pis small. For example, T12≃0.08,0 andT21≃0,0.02 when P≃10,110 mW (see Fig.2(b)). While the increaseof Pwill lead T12and T21to be closer and then to keep some difference. There- fore, if the impedance-matching condition is broken, the transmission for both two directions will be distinct. This is a typical nonreciprocal phenomenon, which cor- responds to the classical nonreciprocal microwave trans- mission [ 9]. In addition, the degree of the nonreciprocal transmission of a microwavefield can be described by the isolation as follows: Tiso= 20×log10|T12/T21|given in decibels(dB). The calculation results for Tiso, plotted as a function of driving power, is shown in Fig. 2(c) and (d). Fig. 2(c) shows that when the impedance-matching condition is met, the microwave transmission coefficients in the two directions are almost the same, i.e., Tiso= 0. However, Fig. 2(d) gives us a clearer perspective for the differences between the transmission coefficients T12and T21. At this time, the impedance-matching condition is broken, and the microwavetransmission in the two direc- tions has a larger transmission isolation ratio Tiso>70 dB. It is worthnoting that when P= 0,Tiso/negationslash= 0 in Fig. 2 (d), thisisbecausethemagnonsmodeisalwaysdrivenby4 FIG. 2. (a) (b) Transmission coefficients T12andT21and (c) (d) transmission isolation ratio Tisoversus driving power Pfor (a) (c) gac/ωb= 1 and (b) (d) gac/ωb= 0.32, where Pm= 94.5 mW. Other parameters are ωa/2π= 10 GHz, ωb/2π= 10 MHz, ˜∆m= 0.9ωb, ∆c= ∆a=−ωb,κ/2π= κm/2π= 1 MHz, and gcm/2π= 3.2 MHz which are mostly based on the latest experimental parameters [ 23,54]. a microwave source with driving power P= 94.5 mW to maintain a certain magnitude of magnomechanical cou- pling. IV. DEFINITION OF ENTANGLEMENT We thereby obtain the linearized QLEs for the quadratures δXo,δYodefined as δXo= (δo+δo†)/√ 2,δYo= (δo−δo†)/i√ 2, can be written as ˙ σ(t) =Aσ(t) +n(t) with σ(t) = [δXa(t),δYa(t),δXc(t),δYc(t),δXm(t),δYm(t),δXb(t), δYb(t)]Tandn(t) = [√2κaδXin a(t),√2κaδYin a(t),√2κc δXin c(t),√2κcδYin c(t),√2κmδXin m(t),√2κmδYin m(t),√2κbδXin b(t),√2κbδYin b(t)]Tbeing the vectors for quan- tum fluctuations and noise, respectively. In addition, the drift matrix Areads A= −κa∆a0gac0 0 0 0 −∆a−κa−gac0 0 0 0 0 0gac−κc∆c0gcm0 0 −gac0−∆c−κc−gcm0 0 0 0 0 0 gcm−κm˜∆m−Gmb0 0 0 −gcm0−˜∆m−κm0 0 0 0 0 0 0 0 −κbωb 0 0 0 0 0 Gmbωb−κb . (6) In this case, when the forwardinput direction ofthe driv- ing field is taken into account (i.e.,Ea/negationslash= 0 and Ec= 0),magnomechanical coupling can be written as follows Gmb,12=2gmb(Emg2 ac−Eagacgcm−Em∆a∆c) g2cm∆a+g2ac˜∆m−∆a∆c˜∆m.(7) However, when the backward input direction of the driv- ing field is taken into account (i.e.,Ea= 0 and Ec/negationslash= 0), magnomechanical coupling changes to Gmb,21=2gmb(Emg2 ac+Ecgcm∆a−Em∆a∆c) g2cm∆a+g2ac˜∆m−∆a∆c˜∆m.(8) Since we are using linearized QLEs, the Gaussian nature of the input states will be preserved during the time evo- lution for the system. So, the quantum fluctuation is in a continuous four-mode Gaussian state which can be com- pletelycharacterizedbya8 ×8covariancematrix (CM)V in the phase space 2 Vij(t,t′) =/angbracketleftσi(t)σj(t′)+σj(t′)σi(t)/angbracketright, (i,j= 1,···,8). Then the vector can be obtained straightforwardly by solving the Lyapunov equation AV+VAT=−D (9) withD= diag [κa,κa,κc,κc,κm,κm,(2nb+1)κb,(2nb+ 1)κb] defind through 2 Dijδ(t−t′) =/angbracketleftni(t)nj(t′) + nj(t′)ni(t)/angbracketright. In this manuscript, we use logarithmic neg- ativity [58,59] to quantify the degree of the quantum entanglement for the four-mode Gaussian state, which is defined as EN≡max[0,−2lnν−], where ν−= min eig | ⊕2 j=1(−σy)PVP|withσyandP=σz⊕1 are respec- tivelyy-Pauli matrix and the matrix that realizes partial transposition at the CM level [ 58,60]. FIG. 3. (a) Eac, (b)Ecm, (c)Emband (d)Eabversus detun- ings ∆ aand ∆ c. We take Gmb/2π= 2.5 MHz,κb/2π= 100 Hz,T= 20 mK and the other parameters are same as Fig. 2.5 V. NONRECIPROCAL ENTANGLEMENT The foremost taskof studying entanglementproperties betweenanytwosubsystemsinsuchahybridsystemis to find the optimal effective interaction among modes, i.e., tofindoptimalfrequencydetuningthatcangeneratesub- system entanglement [ 21]. In Fig. 3, we show four types of subsystem entanglement (Eac,Ecm,EmbandEab) as the function of cavity detunings ∆ aand ∆ c, where Eac,Ecm,EmbandEabdenote the cavity-cavity entan- glement, cavity-magnon entanglement, magnon-phonon entanglement, and cavity-phonon distant entanglement, respectively. In addition, we choose Gmb/2π= 2.5 MHz, κa=κc=κm,κb/2π= 100 Hz, T= 20 mK and the other parameters are chosen as the same as that in Fig.2. Furthermore, we also set ˜∆m≃ωbwhich im- ply that magnon mode is in the blue sideband with re- spect to the first cavity mode, which corresponds to the anti-Stokes process, i.e., significantly cooling the phonon mode. Thus, the elimination of the main obstacle for ob- serving entanglement is obtained [ 21]. It is noted that all results are satisfying with the condition of the steady state guaranteed by the negative eigenvalues (real parts ) of the drift matrix A. From Fig. 3, it is shown that a parameter regime exists, i.e., ∆ a= ∆c=−ωb, where the entanglement within any two subsystems occurs (see Fig.1(b)). This is similar to the realization of entangle- ment between magnon modes in a magnomechanic sys- tem with two YIG spheres proposed in Ref. [ 61]. In or- der to obtain the entanglement between any two subsys- tems and keep the system stable at the same time, the three coupling rates gac,gcmandGmbshould be on the same order of magnitude and chosen as a separate and moderate value. Based on it, we choose gacgcmGmb≪ |∆a∆c˜∆m| ≃ω3 bandGmb= 2.5 MHz which corresponds toEm≃Gmbω3 b/2gmb(ω2 b−g2 ac)≃4.2×1013Hz and Pm≃1.85 mW, when Ea=Ec= 0 in Fig. 3. All entan- glement in the system comes from the magnetostrictive interaction between the magnon and the phonon [ 21]. Next, we will show the realization of nonreciprocal sub- system entanglement by controlling the classical nonre- ciprocal transmission, i.e., controlling the effective mag- nomechanical interaction. For the sake of brevity, we mainly focus on the en- tanglement EabandEmbdue to they being larger than the other types of entanglement. In Fig. 4(a) (b), we showEmbandEabas the function of driving power Pon the cavity aorc, whereEij,12(Eij,21)represents the en- tanglement between mode iand mode jwhen the direc- tion of driving microwave source is forward (backward ). Additionally, gmb/2π= 0.3 Hz,κa=κc=κmand T= 20 mK are set and the other parameters are cho- sen as the same as that in Fig. 2. Due to the existence of magnon-phonon nonlinear coupling (magnetostrictive force), the entanglement of the magnomechanical sub- system is generated. And because there is a direct or indirect linear state-swap interaction between the two microwave cavities and the magnon mode, the resultingmagnomechanical entanglement is distributed to other subsystems. Therefore, controlling the optimal effective magnomechanical interaction will be an effective opera- tion to control whether entanglement exists in each sub- system. From Fig. 4(a) we can see that the magnome- chanical subsystem entanglement in the two directions shows very different trends with the increase of the driv- ing power. Therefore, entanglement of other subsystems will be affected accordingly (see Fig.4(d)). The results in Fig.4(a) (b) demonstrate that the entanglement of multiple subsystems in a hybrid system can be prepared in a highly asymmetric way. This is in good agreement with our previous expectations. This is a clear signature of quantum nonreciprocity, which is fundamentally dif- ferent from that in classical devices [ 62,63] showing only nonreciprocal transmission rates. Next, the difference between subsystem entanglement with forward driving direction and backward driving direction is extracted in the decibel scale (defined as 20×log10|Eij,12/Eij,21|, similar to Tiso[3]), and we take its value as the entanglement isolation ratio Eij,iso(units of dB). For the case of reciprocal subsystem entangle- ment, we have Eij,12/Eij,21= 1 and Eij,iso= 0. A nonzeroEij,isopresents nonreciprocal entanglement and the greater the value of Eij,iso, the higher is the de- gree of the nonreciprocal entanglement. The calculation results for Eij,iso, plotted as a function of the driving powerP, are shown in Fig. 4(c) (d), which gives us a clearer perspective for the differences between the entan- glements Eij,12andEij,21. Notice the cutoffin Fig. 4(d). The position of the cutoff corresponds to the position of Eab,21= 0 in Fig. 4(b), which shows the unidirectional invisibility of subsystem entanglement. In the multilayer microwave integrated quantum circuit, we can further FIG. 4. (a) Emb, (b)Eab, (c)Emb,iso, and (d) Eab,isoversus driving power P. We take gmb/2π= 0.3 Hz,κb/2π= 100 Hz, T= 20 mK and the other parameters are same as Fig. 2.6 design superconducting transmission lines and intercon- nects based on these existing technologies to provide the large range of necessary couplings and to minimize any parasitic losses [ 44]. In fact, we can find from Eqs. ( 7–8) that when gac=ωb, the system reaches the impedance matching condition [ 25,53], i.e.,Gmb,12=Gmb,21. This realizes the switch from nonreciprocal to reciprocal of subsystem entangled state and shows the potential ad- vantage of our solution as a tunable quantum diode. Fig.5,EmbandEabas a function of ambient tem- perature Tfor driving power P= 1 W, shows that the generated subsystem entanglements is robust to ambient temperature and survives up to ∼100 mK, below which the average phonon number is always smaller than 1, showing that mechanical cooling is, thus, a precondition for observing quantum entanglement in the system [ 21]. Compared with the scheme that a strong squeezed vac- uum field proposed in Ref. [ 64] is used to generate en- tanglement between magnon modes, in our scheme due to the inherent low frequency of phonon modes, this ro- bustness is generally weak. VI. DISCUSSION AND CONCLUSION Lastly, we discuss how to detect the entanglement and verify the effectiveness of two-cavitymagnomechanicsys- tem. The generated subsystem entanglements can be detected by measuring the CM of two cavity output fields [21]. Such measurement in the microwave domain has been realized in the experiments [ 65,66]. In addi- tion, for a 0.5-mm-diameter YIG sphere, the number of spinsN≃2.8×1017, andPm= 189 mW corresponds toEm≃3×1014Hz, and |/angbracketleftm/angbracketright| ≃5.9×106, leading to FIG. 5. (a) Emband (b)Eabversus ambient temperature T. We take P= 0.5 W,gmb/2π= 0.3 Hz,κb/2π= 100 Hz and the other parameters are same as Fig. 2./angbracketleftm†m/angbracketright ≃3.5×1013≪5N= 1.4×1018which is well fulfilled. It is worth noting that two cavity modes is res- onant with the red-sideband (∆a= ∆c=−ωb)results in a higher magnon excitation number in the stable state in the presence of EaorEc, where the magnon number has a simpler form /angbracketleftm†m/angbracketright ≃[Em(ω2 b−g2 ac)+Eagacgcm]2/ω6 b or/angbracketleftm†m/angbracketright ≃[Em(ω2 b−g2 ac) +Ecωbgcm]2/ω6 b. Therefore, under the premise that the magnon mode is continu- ously driven by the classical microwave field with driv- ing power Pm= 94.5 mW,P= 1 W corresponds to /angbracketleftm†m/angbracketright12≃1.4×1015≪5N(Ea/negationslash= 0,Ec= 0)and /angbracketleftm†m/angbracketright21≃3.47×1015≪5N(Ec/negationslash= 0,Ea= 0), re- spectively. In order to keep the Kerr effect negligible, K|/angbracketleftm/angbracketright|3≪/summationtext i=a,c,mEimust hold [ 21]. Kerr coefficient Kis inversely proportional to the volume of the sphere. In this manuscript, we use a 5-mm-diameter YIG sphere, K/2π≃8×10−10Hz which corresponds to K|/angbracketleftm/angbracketright|3≃ 4.2×1013Hz≪Em+Ea≃1.3×1015(Pm= 189 mW, P= 2 W, and Ec= 0)andK|/angbracketleftm/angbracketright|3≃1.64×1014Hz ≪Em+Ec≃1.3×1015(Pm= 189 mW, P= 2 W, and Ea= 0). This implying that the nonlinear effects are negligible and the linearization treatment of the model is a good approximation. In summary, we show how to use an asymmetric cav- ity magnomechanic system to produce classical nonre- ciprocal transmission, and extend this nonreciprocity to quantum states, thereby generating nonreciprocal sub- system entangled states, rather than by means of nonre- ciprocal devices [ 62]. The introduction of an additional cavity mode successfully breaks the symmetry of spa- tial inversion. Our work opens up a range of exciting opportunities for quantum information processing, net- working and metrology by exploiting the power of quan- tum nonreciprocity. The ability to manipulate quan- tum states or nonclassical correlations in a nonrecipro- cal way sheds new lights on chiral quantum engineering and can stimulate more works on achieving and operat- ing quantum nonreciprocal devices, such as directional quantum squeezing [ 67,68], backaction-immune quan- tum sensing [ 69,70], and quantum chiral coupling of cav- ity optomechanics devices to superconducting qubits or atomic spins [ 21,71,72]. VII. ACKNOWLEDGMENTS This work is supported by the Science and Technol- ogy project of Jilin Provincial Education Department of China during the 13th Five-Year Plan Period (Grant No. JJKH20200510KJ )and the Fujian NaturalScienceFoun- dation(Grant No. 2018J01661 and No. 2019J01431 ).7 [1] Y. Shoji and T. Mizumoto, Sci. Technol. Adv. Mater. 15, 014602 (2014). [2] A. Li and W. Bogaerts, OPTICA 7, 7 (2020). [3] Z. B. Yang, J. S. Liu, A. D. Zhu, H. Y. Liu, and R. C. Yang, Ann. Phys. (Berlin) 532, 2000196 (2020). [4] C. Caloz, A. Al` u, S. Tretyakov, D. Sounas, K. Achouri, and Z. L. Deck-L´ eger, Phys. Rev. Applied 10, 047001 (2018). [5] N. T. Otterstrom, E. A. Kittlaus, S. Gertler, R. O. Be- hunin, A. L. Lentine, and P. T. Rakich, OPTICA 6, 1117 (2019). [6] N.R.Bernier, L.D.T´ oth, A.Koottandavida, M. A.Ioan- nou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, Nat. Commun. 8, 604 (2017). [7] F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Aumentado, J. D. Teufel, and J. Aumentado, Phys. Rev. Applied 7, 024028 (2017). [8] G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Phys. Rev. X 7, 031001 (2017). [9] Y. P. Wang, J. W. Rao, Y. Yang, P. C. Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C. M. Hu, Phys. Rev. Lett. 123, 127202 (2019). [10] M. Shalaby, M. Peccianti, Y. Ozturk, and R. Morandotti, Nat. Commun. 4, 1558 (2013). [11] Z. Shen, Y. L. Zhang, Y. Chen, C. L. Zou, Y. F. Xiao, X. B. Zou, F. W. Sun, G. C. Guo, and C. H. Dong, Nat. Photonics 10, 657 (2016). [12] M. A. Miri, F. Ruesink, E. Verhagen, and A. Al` u, Phys. Rev. Applied 7, 064014 (2017). [13] J. Goulon, A. Rogalev, C. Goulon-Ginet, G. Benayoun, L. Paolasini, C. Brouder, C. Malgrange, and P. A. Met- calf, Phys. Rev. Lett. 85, 4385 (2000). [14] H. Ramezani, P. K. Jha, Y. Wang, and X. Zhang, Phys. Rev. Lett. 120, 043901 (2018). [15] S. Manipatruni, J. T. Robinson, and M. Lipson, Phys. Rev. Lett. 102, 213903 (2009). [16] C. H. Dong, Z. Shen, C. L. Zou, Y.-L. Zhang, W. Fu, and G. C. Guo, Nat. Commun. 6, 6193 (2015). [17] L. D. Bino, J. M. Silver, M. T. M. Woodley, S. L. Steb- bings, X. Zhao, and P. Del’Haye, OPTICA 5, 279 (2018). [18] D. W. Wang, H. T. Zhou, M. J. Guo, J. X. Zhang, J. Ev- ers, and S. Y. Zhu, Phys. Rev. Lett. 110, 093901 (2013). [19] Q. Zhong, S. Nelson, S. K. ¨Ozdemir, and R. El-Ganainy, Opt. Lett. 44, 5242 (2019). [20] Y. F. Jiao, S. D. Zhang, Y. L. Zhang, A. Miranowicz, L. M. Kuang, and H. Jing, Phys. Rev. Lett. 125, 143605 (2020). [21] J. Li, S. Y. Zhu, and G. S. Agarwal, Phys. Rev. Lett. 121, 203601 (2018). [22] Y. P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, Phys. Rev. B 94, 224410 (2016). [23] G. Q. Zhang, Y. P. Wang, and J. Q. You, Sci. China- Phys. Mech. Astron. 62, 987511 (2019). [24] Z. B. Yang, J. S. Liu, H. Jin, Q. H. Zhu, A. D. Zhu, H. Y. Liu, Y. Ming, and R. C. Yang, Opt. Express 28, 31862 (2020). [25] C. Kong, H. Xiong, and Y. Wu, Phys. Rev. Applied 12, 034001 (2019).[26] J. Li, S. Y. Zhu, and G. S. Agarwal, Phys. Rev. A. 99, 021801(R) (2019). [27] Z. Zhang, M. O. Scully, and G. S. Agarwal, Phys. Rev. Research 1, 023021 (2019). [28] H. Y. Yuan and X. R. Wang, Appl. Phys. Lett. 110, 082403 (2017). [29] H. Y. Yuan, S. Zhang, Z. Ficek, Q. Y. He, and M. H. Yung, Phys. Rev. B 101, 014419 (2020). [30] S. S. Zheng, F. X. Sun, H. Y. Yuan, Z. Ficek, Q. H. Gong, and Q. Y. He, Sci. China-Phys. Mech. Astron. 64, 210311 (2020). [31] H. Y. Yuan, P. Yan, S. S. Zheng, Q. Y. He, K. Xia, and M. H. Yung, Phys. Rev. Lett. 124, 053602 (2020). [32] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [33] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Us- ami, and Y. Nakamura, Appl. Phys. Express 12, 070101 (2019). [34] M. Harder and C. M. Hu, in Solid State Physics 69 , edited by R. E. Camley and R. L. Stamp (Academic, Cambridge, 2018), pp. 47–121. [35] D. Zhang, X. Q. Luo, Y. P. Wang, T. F. Li, J. Q. You, Nat. Commun. 8, 1368 (2017). [36] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015). [37] D. Lachance-Quirion, Y. Tabuchi, S. Ishino, A. Noguchi , T. Ishikawa, R. Yamazaki, and Y. Nakamura, Sci. Adv. 3, e1603150 (2017). [38] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. No- mura,and Y. Nakamura, Phys. Rev. Lett. 116, 223601 (2016). [39] X. Zhang, N. Zhu, C. L. Zou, and H. X. Tang, Phys. Rev. Lett.117, 123605 (2016). [40] J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, Phys. Rev. Lett. 117, 133602 (2016). [41] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. B 93, 174427 (2016). [42] C. Braggio, G. Carugno, M. Guarise, A. Ortolan, and G. Ruoso, Phys. Rev. Lett. 118, 107205 (2017). [43] X. Zhang, C. L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Nat. Commun. 6, 8914 (2015). [44] T. Brecht, W. Pfaff, C. Wang, Y. Chu, L. Frunzio, M. H Devoret, and R. J Schoelkopf, npj QuantumInf. 2, 16002 (2016). [45] C. Kittel, Phys. Rev. 73, 155 (1948). [46] H.Huebl, C.W.Zollitsch, J. Lotze, F.Hocke, M.Greifen - stein, A. Marx, R. Gross, and S. T. B. Goennensein, Phys. Rev. Lett. 111, 127003 (2013). [47] X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [48] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C. M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [49] D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, and J. Q. You, npj Quantum Inf. 1, 15014 (2015). [50] X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, Sci. Adv. 2, e1501286 (2016).8 [51] C. Kong, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, Opt. Express 27, 5544 (2019). [52] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014). [53] X. W. Xu, L. N. Song, Q. Zheng, Z. H. Wang, and Y. Li, Phys. Rev. A 98, 063845 (2018). [54] Y. P. Wang, G. Q. Zhang, D. Zhang, T. F. Li, C. M. Hu, and J. Q. You, Phys. Rev. Lett. 120, 057202 (2018). [55] For a 250- µm diameter YIG sphere, the corresponding gmb/2π≃0.2 Hz [21]. In this manuscript, however, in order to avoid undesired Kerr nonlinearity, we choose a YIG sphere with a larger volume (5-mm diameter), which leads to a smaller single-magnon magnomechani- cal coupling gmb. Nevertheless, when any cavity field and magnon mode are strongly driven at the same time, we can still obtain a strong Gmbunder the premise that the Kerr effect is negligible. [56] The external loss rate of the cavity field comes from the connection of the superconducting transmission line and the input and output ports on the cavity. For the port- induced decay rates κex aandκex c, their magnitudes can be tuned by adjusting the lengths of the port pins in- side the cavity and when the YIG sphere is inserted into the cavity, both the cavity frequency and intrinsic loss rate shift slightly at different displacements of the YIG sphere [35]. However, the loss rate of the magnon comes from the surface roughness as well as the impurities and defects in the YIG sphere [ 32,49]. [57] C.W.GardinerandP.Zoller, QuantumNoise (Springer, New York, 2004). [58] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).[59] M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005). [60] R. Simon, Phys. Rev. Lett. 84, 2726 (2000). [61] J. Li and S. Y. Zhu, New J. Phys. 21, 085001 (2019). [62] S. Maayani, R. Dahan, Y. Kligerman, E. Moses, A. U. Hassan, H. Jing, F. Nori, D. N. Christodoulides, and T. Carmon, Nature (London) 558, 569 (2018). [63] N. Liu, J. Zhao, L. Du, C. Niu, C. Sun, X. Kong, Z. Wang, and X. Li, Opt. Lett. 45, 5917 (2020). [64] J. M. P. Nair and G. S. Agarwal, Appl. Phys. Lett. 117, 084001 (2020). [65] S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, Nature (London) 570, 480 (2019). [66] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Science 342, 710 (2013). [67] E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A.Clerk, andK. C. Schwab, Science349, 952 (2015). [68] J. M. Pirkkalainen, E. Damsk¨ agg, M. Brandt, F. Mas- sel, and M. A. Sillanp¨ a¨a, Phys. Rev. Lett. 115, 243601 (2015). [69] F. Wolfgramm, C. Vitelli, F. A. Beduini, N. Godbout, and M. W. Mitchell, Nat. Photonics 7, 28 (2013). [70] Y. Ma, H. Miao, B. H. Pang, M. Evans, C. Zhao, J. Harms, R. Schnabel, and Y. Chen, Nat. Phys. 13, 776 (2017). [71] L. DiCarlo, M. D. Reed, L. Sun, B. R. Johnson, J. M. Chow, J. M. Gambetta, L. Frunzio, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, Nature (London) 467, 574 (2010). [72] W. Qin, A. Miranowicz, P. B. Li, X. Y. L¨ u, J. Q. You, and F. Nori, Phys. Rev. Lett. 120, 093601 (2018).

2021-01-25

Quantum entanglement, a key element for quantum information is generated with a cavity-magnomechanical system. It comprises of two microwave cavities, a magnon mode and a vibrational mode, and the last two elements come from a YIG sphere trapped in the second cavity. The two microwave cavities are connected by a superconducting transmission line, resulting in a linear coupling between them. The magnon mode is driven by a strong microwave field and coupled to cavity photons via magnetic dipole interaction, and at the same time interacts with phonons via magnetostrictive interaction. By breaking symmetry of the configuration, we realize nonreciprocal photon transmission and one-way bipartite quantum entanglement. By using current experimental parameters for numerical simulation, it is hoped that our results may reveal a new strategy to built quantum resources for the realization of noise-tolerant quantum processors, chiral networks, and so on.

Nonreciprocal Transmission and Entanglement in a cavity-magnomechanical system

2101.09931v1

1 Approaching quantum anomalous Hall effect in proximity -coupled YIG/ graphene/h -BN sandwich structure Chi Tang1, Bin Cheng1, Mohammed Aldosary1, Zhiyong Wang1, Zilong Jiang1, K. Watanabe3, T. Taniguchi3, Marc Bockrath1, 2, and Jing Shi1, a 1Department of Physics and Astronomy, University of California, Riverside, C A 92521, USA 2Department of Physics, The Ohio State University, Columbus, O H 43210, USA 3Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305- 0044, Japan Quantum anomalous Hall state is expected to emerge in Dirac electron systems such as graphene under both sufficiently strong exchange and spin -orbit interactions. In pristine graphene, neither interaction exists; however, both interactions can be acquired by coupling graphene to a magnetic insulator (MI) as revealed by the anomalous Hall effect. Here, we show enhanced magnetic proximity coupling by sandwiching graphene between a ferrimagnetic insulator yttrium iron garnet (YIG) and hexagonal -boron nitride (h-BN) which also serves as a top gate dielectric. By sweeping the top -gate voltage, we observe Fermi level- dependent anomalous Hall conductance . As the Dirac p oint is approached f rom both electron and hole sides, the anomalous Hall conductance reaches ¼ of the quantum anomalous Hall conductance 2e2/h. The exchange coupling strength is determined to be as high as 27 meV from the transition temperature of the induced magnetic phase. YIG/graphene/h -BN is an excellent heterostructure for demonstrating proximity -induced interactions in two -dimensional electron systems . 2 Long-range fer romagnetic order in two -dimensional electron systems (2DES) has long been sought to involve the spin degree of freedom in spatially confined quantum systems1-5; however, the transition metal doping approach failed to deliver high-temperature ferromagnetism . Recently it became possible w ith the advent of 2D layered materials such as graphene6,7 and other van der Waals (vdW) materials8-11. In the former, ferromagnetism is introduced by proximity coupling in a heterostructure comprising of graphene and magnetic insulator; while in the later, the material itself is a spontaneously ferromagnetically order ed 2D vdW crystal, either a 2D H eisenberg or an Ising ferromagnet. Such 2D ferromagnetic systems provide unprecedented opportunities to create novel heter ostructures for exploiting spin- dependent phenomena in spatial ly confined electron systems . A particularly interesting quantum phenomenon in graphene was theoretically proposed by Qiao et al.12,13, namely the quantum anomalous Hall effect (QAHE) , which emerges in the presence of finite exchange interaction and spin- orbit coupling (SOC) . Under these two interactions, a topological band gap or an exchange gap is opened up at the Dirac point which consequently gives rise to a quantized anomalous Hall conductance of ± 2e2/h. However, neither interaction is present in pristine graphene. F erromagnetic order has recently been demonstrated via proximity coupling in both yttrium iron garnet(YIG)/graphene6 and EuS/graphene7, where YIG or EuS breaks the time reversal and inversion symmetries and therefore serves as the source of the se interaction s. To ultimately realize the QAHE in graphene, besides sufficiently strong exchange and SOC, graphene needs to have relatively weak disorder so that the energy scale associated with the disorder is smaller than the exchange gap . Therefore, a main challenge is to maximize the exchange gap and simultaneously minimize the disorder energy scale. Here, we show an enhanced proximity exchange effect in a YIG/graphene/h -BN sandwich structure revealed by the anomalous Hall effect (AHE) . In this study, t he h-BN layer replaces the poly - methyl methacrylate (PMMA) layer used in previous devices6, which not only makes an efficient top gate but also protects graphene underneath to preserve its high mobility14. As a result, the AHE conductance shows a clear gate voltage dependence as the Fermi level is tuned towards the Dirac point, which is expected from a gapped Dirac spectrum. Moreover, the AHE conduc tance reaches ¼ of the QAHE conductance 2e2/h. The induced magnetic phase transition temperature in the heterostructure is as high as room temperature, with an exchange coupling strength of ~27 meV. 3 We first epitaxially gro w ~ 20 nm thick YIG films using pulsed laser deposition on (111) -oriented gadolinium gallium garnet substrate15. The surface topography of the YIG films is characteriz ed by atomic force microscope (AFM) with a root -mean -square (rms) roughness of ~1 - 2 Ǻ over a 2 × 2 μm2 scan area, as sh own in Fig. 1(b). Due to the interfacial nature of the proximity coupling, t he atomically flat surface of YIG films is critical to create a strong exchange interaction . Standard Hall bar patterned graphene devices are fabricated on 290 nm - thick Si/SiO 2 substrates. Single -layer graphene flakes are loc ated under an optical microscope and confirmed by Raman spectroscopy . We use a 20 – 30 nm thick h- BN layer via a micro mechanical transfer process16 to cover the entire graphene region, which simultaneou sly protects graphene from chemical solvent or resist contaminations and serves as an effective top gate dielectric la yer. After top gate electrodes are fabricated and tested, we transfer the entire functional graphene/h- BN devices from Si/SiO 2 substrates to YIG utilizing a previously developed transfer technique6,17. The optical images of graphene/h -BN devi ces before and after transfer are illustrated in Fig. 1(c), which show a successful transfer. Transport measureme nts are performed on the same graphene/h -BN devices before and after transfer to track the magneto - transport response s on different substrates . The soft magnetic hysteresis of a bare YIG film with the magnetic field applied along the out-of-plane direction shown in Fig. 2 (a) confirms the in -plane magnetic anisotropy arising primarily from the shape anisotropy15. Additional growth- or interface strain -induced in-plane magnetic anisotropy contributes to a higher saturation magnetic field (usually between 2000 and 3000 Oe ) than the demagnetizing field 4πMs18-20. When graphene is placed on SiO 2, the Hall resistance is linear in external magnetic field , which originates from the ordinary Hall effect (OHE). A fter the linear background is subtracted, no net signal is left as shown in Fig. 2 (a) . In the same device transferred to YIG, a clear nonlinear Hall curve stands out after subtraction of a linear OHE background also shown in Fig. 2 (a) . Below the Hall curves, the magnetic hysteresis of the YIG film measured with the out-of-plane magnetic fields is displayed. It is clear that the nonlinear Hall curve resembles the magnetization of the underlying YIG layer. The observed nonlinear Hall behavior can arise from the following possible mechanisms : (i) the AHE response of graphene in which the carriers are spin polarized due to the interfacial exchange coupling with YIG; (ii) the Lorentz force induced nonlinear Hall effect either from the coexistence of two types of carriers in graphene21 or the stray magnetic field produced by YIG . As will be discussed, t he 4 gate dependence measurements exclude the latter possibility, which favors the first scenario, i.e. induc ed ferromagn etism in graphene due to the exchange coupling with the YIG layer. Here we ascribe the nonlinear Hall response to the AHE. To quantitatively characterize the exchange coupling strength in YIG/graphene/h -BN, the AHE response is studied over a wide range of temperatures from 13 K to 300 K . Fig. 2(b) shows the data taken at several selected temperatures with a top gate voltage of 0.9 V. The AHE resistance progressively decreases as the temperature increases , but persists up to 300 K. Note that the AHE resista nce is proportional to the spontaneous magnetization and also has power -law dependence on the longitudinal resistance22. Since t he latter is a smooth function of temperature, near the ferromagnetic transition temperature T c, the AHE resistance can be expressed as RAH ~ (T – Tc) β with the critical component β = 0.5. By fitting the temperature dependence data, we extract the transition temperature T c ~ 308 K, or kBTc ~ 27 meV in exchange energy, which is higher than what was previously reported6. As mentioned earlier, a trivial cause of the nonlinear Hall signal in YIG/graphene/h -BN is the Lorentz force, either from the coexistence of two types of carriers or from the stray magnetic field pro duced by the YIG substrate . Such a nonlinear Hall response should reverse its sign when the carrier s change from electron - to hole -type or vice versa, whereas the AHE sign does not have to change as the carrier type switche s. Thus, we measure the nonlinear Hall response at each fixed gate voltage and repeat for a range of top gate voltages which set different carrier densities on both hole and electron sides . Fig. 3(a) shows the sheet resistance of YIG/graphene/ h-BN as a function of the top gate voltage Vtg measured at 13 K. The Dirac point VDP of the graphene device is at 0.5 V, very close to zero. T he Hall mobility is as high as 18,400 cm2/Vs, about the same or der of magnitude as in SiO 2/graphene /h-BN, indicating no negative effect on carrier mobi lity due to the transfer process. The mobility is at least 3 times as high as that in graphene on SiO 2 without h- BN. Due to the thinner h- BN sheet, only much smaller applied gate voltage s (decreased by a factor of 15) are needed to tune the carrier density over a wide range. To avoid the region where electron s and hole s coexist in the vicinity of the Dirac point which can also produce a strong and complex nonlinear OHE , we measure the Hall voltages in YIG/graphene/h -BN only with 𝑉𝑡𝑔≤0 𝑉 and 𝑉𝑡𝑔≥0.9 𝑉, as indicated in Fig. 3(b). T he sign of the anomalous Hall resistance is independent of the carrier type. If the Hall response were 5 generated by the Lorentz force, the sign of nonlinear Hall resistance would switch once the carrier type changes . The observed same nonlinear Hall sign clearly excludes the Lorentz force related mechanism , and therefore unambiguously demonstrate s the anomalous Hall origin due to the SOC in the proximity -induced ferromagnetic phase of graphe ne. As expected for QAHE insulator s12,13,22,23, the intrinsic AHE from the Berry curvature indeed has the same sign on both electron and hole sides in unquantized regions where the Fermi level is outside the exchange gap . We calculate the anomalous Hal conductance 𝜎𝐴𝐻 as a function of the top gate voltage and plot it in Fig. 4. At large gate voltages on both sides, 𝜎𝐴𝐻 is relatively small. As the gate voltage decreases to approach the Dirac point from both sides, 𝜎𝐴𝐻 gradually increases and follows the trend predicted by Qiao et al .12,13 Here w e intentionally stay away from the two - carrier dominated region near the Dirac point. Over the measured gate voltage range, 𝜎𝐴𝐻 is clearly not quantized. The maximum 𝜎𝐴𝐻 in our best device reaches ¼ of the quantum anomalous Hall co nductance , 2e2/h. Both the magnitude and the clear gate voltage dependence of 𝜎𝐴𝐻 indicate much improved proximity -induced exchange and SOC strengths compared to the previous YIG/graphene/PMMA devices, thanks to the h- BN that preserves the high quality of graphene sheet and reduces the disorder strength (~ 13 meV) . However, the absence of the QAHE plateau suggests a small exchange gap which is smeared out by thermal fluctuation s and disorder . The exchange gap is determined by the small interaction of the exchange and SOC. Since we have achieved relatively large exchange interaction, the small exchange gap is mainly due to relatively weak SOC. To ultimately demonstrate the QAHE in graphene, greatly enhanced SOC is clearly required. Recent experiments indic ate that it is possible to drastically enhance the Rashba SOC via proximity coupling with transition metal dichalcogenide materials such as WSe 224,25. In summary, the observed AHE dem onstrates the existence of long -range ferromagnetic order in graphene proximity coupled with a magnetic insulator. The maximu m anomalous Hall conductance reaches ¼ of the QAHE conductance 2e2/h. The exchang e coupling strength is determined to be as high as 27 meV , indicating an effective role the h- BN played in promoting both exchange and SOC and reducing the disorder strength. Further exploration of incorporating a transition metal dichalcogenide layer into the sandwich structure to enlarge SOC in graphene and utilizing thulium iron garnet with robust perpendicular magnetic anisotropy26 is promising to realize the QAHE at high temperatures and zero magnetic field . 6 YIG film growth and ch aracterizations, YIG/graphene heterostructur e device fabrication, transport measurements, and data analysis were supported in part by DOE BES Award No. DE - FG02 -07ER46351. Construction of the transfer microscope and device characterizations were supported b y NSF -ECCS under Awards No. 1202559 and NSF -ECCS and No. 1610447. Exfoliation and transfer of h- BN were supported by DOE ER 46940- DE-SC0010597. 7 Fig. 1. (a) Schematic view of YIG/graphene/ h-BN device; (b) AFM image of a typical YIG film grown by pulsed laser deposition with roughness ~ 0.15 nm across a 2 × 2 μm2 scan area; (c) Top: exfoliated graphene on SiO 2 covered with h- BN. Bottom: transferred graphene/ h-BN devices on YIG substrate. 8 Fig. 2. (a) Top: A nomalous Hall resistance of graphene/h- BN on YIG (orange) and SiO 2 (black) at 300 K after the OHE background is subtracted . No nonlinear response is left in graphene on SiO 2 whereas there is a clear anomalous Hall signal in graphene transferr ed onto YIG. Bottom: Magnetic hysteresis b ehavior of YIG film (green) when an external magnetic field is applied perpendicular to graphene . The nonlinear Hall resistance in YIG/graphene/h -BN follows the magnetization of the YIG film. (b) A nomalous Hall resistance curves of YIG/graphene/h- BN at the top gate voltage of 0.9 V measured from 13 to 300 K; (c) T emperature dependence of anomalous Hall resistance of YIG/graphene/h -BN. The magnetic phase transition temperature T c of the induced ferromagne tism in graphene is extracted to be 308 K by using R AH ~ (T – Tc) β with the critical component β = 0.5. 9 Fig. 3. (a) T op gate voltage dependence of the sheet resistance of graphene sandwiched between YIG and h -BN measured at 13 K with a mobility of 18,400 cm2/Vs and the Dirac point near 0.5 V. Sever al top gate voltages are selected to show the anomalous Hall resistance at differ ent carrier densities on both electron and hole dominated regions. (b) A nomalous Hall resistance in YIG/graphene/h -BN at different top gate voltages . The anomalous Hall resistance sign remains the same for both electron and hole carrier types . 10 Fig. 4. T op gate voltage and carrier density dependence of anomalous H all conductance measured at 13 K. R ed squares are experimental data and dashed green curve is drawn for th e purpose of eye guidance. The green area marks the electron or hole -dominated region where clear AHE is observed. The largest anomalous Hall conductance in YIG/graphene/h -BN reaches ¼ of the quantum anomalous Hall conductance. The white region is t he e- h coexisting region where the F ermi level is too close to the Dirac point and additional oscillatory features are observed due to multi- carriers in this region. 11 Reference: 1 Kikkawa, J. M., Baumberg, J. J., Awschalom, D. D., Leonard, D. & Petroff, P. M. Optical Studies of Locally Implanted Magnetic Ions in Gaas. Phys. Rev. B 50, 2003- 2006 (1994). 2 Smorchkova, I. P., Samarth, N., Kikkawa, J. M. & Awschalom, D. D. Spin transpor t and localization in a magnetic two -dimensional electron gas. Phys. Rev. Lett. 78, 3571- 3574 (1997). 3 Haury, A. et al. Observation of a ferromagnetic transition induced by two- dimensional hole gas in modulation- doped CdMnTe quantum wells. Phys. Rev. Lett . 79, 511- 514 (1997). 4 Kikkawa, J. M., Smorchkova, I. P., Samarth, N. & Awschalom, D. D. Room -temperature spin memory in two -dimensional electron gases. Science 277, 1284 -1287 (1997). 5 Smorchkova, I. P., Samarth, N., Kikkawa, J. M. & Awschalom, D. D. Gia nt magnetoresistance and quantum phase transitions in strongly localized magnetic two - dimensional electron gases. Phys. Rev. B 58, R4238- R4241 (1998). 6 Wang, Z. Y., Tang, C., Sachs, R., Barlas, Y. & Shi, J. Proximity -Induced Ferromagnetism in Graphene Rev ealed by the Anomalous Hall Effect. Phys. Rev. Lett. 114, 016603 (2015). 7 Wei, P. et al. Strong interfacial exchange field in the graphene/EuS heterostructure. Nat. Mater. 15, 711 (2016). 8 Huang, B. et al. Layer -dependent ferromagnetism in a van der Waal s crystal down to the monolayer limit. Nature 546, 270- 273 (2017). 9 Gong, C. et al. Discovery of intrinsic ferromagnetism in two -dimensional van der Waals crystals. Nature 546, 265 (2017). 10 Tian, Y., Gray, M. J., Ji, H. W., Cava, R. J. & Burch, K. S. Ma gneto -elastic coupling in a potential ferromagnetic 2D atomic crystal. 2d Mater 3, 025035 (2016). 11 Xing, W. Y. et al. Electric field effect in multilayer Cr2Ge2Te6: a ferromagnetic 2D material. 2d Mater 4, 024009 (2017). 12 Qiao, Z. H. et al. Quantum Anomalous Hall Effect in Graphene Proximity Coupled to an Antiferromagnetic Insulator. Phys. Rev. Lett. 112, 116404 (2014). 13 Qiao, Z. H. et al. Quantum anomalous Hall effect in graphene from Rashba and exchange effects. Phys. Rev. B 82, 161414 ( 2010). 14 Dean, C. R. et al. Boron nitride substrates for high- quality graphene electronics. Nat. Nanotechnol. 5, 722- 726 (2010). 15 Tang, C. et al. Exquisite growth control and magnetic properties of yttrium iron garnet thin films. Appl. Phys. Lett. 108, 102403 (2016). 16 Wang, L. et al. One-Dimensional Electrical Contact to a Two -Dimensional Material. Science 342, 614 -617 (2013). 17 Sachs, R., Lin, Z. S., Odenthal, P., Kawakami, R. & Shi, J. Direct comparison of graphene devices before and after transfer to different substrates. Appl. Phys. Lett. 104, 033103 (2014). 18 Wang, H. L., Du, C. H., Hammel, P. C. & Yang, F. Y. Strain- tunable magnetocrystalline anisotropy in epitaxial Y3Fe5O12 thin films. Phys. Rev. B 89, 134404 (2014). 19 Krockenberger, Y. et al. Layer -by-layer growth and magnetic properties of Y3Fe5O12 thin films on Gd3Ga5O12. J. Appl. Phys. 106, 123911 (2009). 12 20 Sellappan, P., Tang, C., Shi, J. & Garay, J. E. An integrated approach to doped thin films with strain -tunable magnetic anisotropy: po wder synthesis, target preparation and pulsed laser deposition of Bi:YIG. Materials Research Letters 5, 41- 47 (2017). 21 Bansal, N., Kim, Y. S., Brahlek, M., Edrey, E. & Oh, S. Thickness -Independent Transport Channels in Topological Insulator Bi 2Se3 Thin F ilms. Phys. Rev. Lett. 109, 116804 (2012). 22 Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82 , 1539- 1592 (2010). 23 Chang, C. Z. et al. Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator. Science 340, 167 -170 (2013). 24 Wang, Z. et al. Strong interface -induced spin- orbit interaction in graphene on WS 2. Nat. Commun. 6, 8339 (2015). 25 Yang, B. W. et al. Tunable spin- orbit coupling and symmetry -protected edge states in graphene/WS 2. 2d Mater 3, 031012 (2016). 26 Tang, C. et al. Above 400- K robust perpendicular ferromagnetic phase in a topological insulator. Sci Adv 3, e1700307 (2017).

2017-10-11

Quantum anomalous Hall state is expected to emerge in Dirac electron systems such as graphene under both sufficiently strong exchange and spin-orbit interactions. In pristine graphene, neither interaction exists; however, both interactions can be acquired by coupling graphene to a magnetic insulator (MI) as revealed by the anomalous Hall effect. Here, we show enhanced magnetic proximity coupling by sandwiching graphene between a ferrimagnetic insulator yttrium iron garnet (YIG) and hexagonal-boron nitride (h-BN) which also serves as a top gate dielectric. By sweeping the top-gate voltage, we observe Fermi level-dependent anomalous Hall conductance. As the Dirac point is approached from both electron and hole sides, the anomalous Hall conductance reaches 1/4 of the quantum anomalous Hall conductance 2e2/h. The exchange coupling strength is determined to be as high as 27 meV from the transition temperature of the induced magnetic phase. YIG/graphene/h-BN is an excellent heterostructure for demonstrating proximity-induced interactions in two-dimensional electron systems.

Approaching quantum anomalous Hall effect in proximity-coupled YIG/graphene/h-BN sandwich structure

1710.04179v1

Quantum information diode based on a magnonic crystal Rohit K. Shukla Department of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi - 221005, India Levan Chotorlishvili E-mail: levan.chotorlishvili@gmail.com Department of Physics and Medical Engineering, Rzeszow University of Technology, 35-959 Rzeszow Poland Vipin Vijayan Department of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi - 221005, India Harshit Verma Centre for Engineered Quantum Systems (EQUS), School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia Arthur Ernst Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany Institute of Theoretical Physics, Johannes Kepler University Alterger Strasse 69, 4040 Linz, Austria E-mail: Arthur.Ernst@jku.at Stuart S. P. Parkin Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany Sunil K. Mishra E-mail: sunilkm.app@iitbhu.ac.in Department of Physics, Indian Institute of Technology (Banaras Hindu University) Varanasi - 221005, India Abstract. Exploiting the effect of nonreciprocal magnons in a system with no inversion symmetry, we propose a concept of a quantum information diode, i.e., a device rectifying the amount of quantum information transmitted in the opposite directions. We control the asymmetric left and right quantum information currents through an applied external electric field and quantify it through the left and right out-of-time-ordered correlation (OTOC). To enhance the efficiency of the quantum information diode, we utilize a magnonic crystal. We 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 Nonreciprocal magnons propagating in opposite directions have different dispersion relations. Magnons propagating in one direction match resonant conditions and scatter on gate magnons. Therefore, magnon flux in one direction is damped in the magnonic crystal leading to an asymmetric transport of quantum information in the quantum information diode. A quantum information diode can be fabricated from an yttrium iron garnet (YIG) film. This is an experimentally feasible concept and implies certain conditions: low temperature and small deviation from the equilibrium to exclude effects of phonons and magnon interactions. We show that rectification of the flaw of quantum information can be controlled efficiently by an external electric field and magnetoelectric effects.Quantum information diode based on a magnonic crystal 3 1. Introduction A diode is a device designated to support asymmetric transport. Nowadays, household electric appliances or advanced experimental scientific equipment are all inconceivable without extensive use of diodes. Diodes with a perfect rectification effect permit electrical current to flow in one direction only. The progress in nanotechnology and material science passes new demands to a new generation of diodes; futuristic nano-devices that can rectify either acoustic (sound waves), thermal phononic, or magnonic spin current transport. Nevertheless, we note that at the nano-scale, the rectification effect is never perfect, i.e., backflow is permitted, but amplitudes of the front and backflows are different [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. In the present work, we propose an entirely new type of diode designed to rectify the quantum information current. We do believe that in the foreseeable future the quantum information diode (QID) has a perspective to become a benchmark of quantum information technologies. The functionality of a QID relies on the use of magnonic crystals, i. e. , artificial media with a characteristic periodic lateral variation of magnetic properties. Similar to photonic crystals, magnonic crystals possess a band gap in the magnonic excitation spectrum. Therefore, spin waves with frequencies matching the band gap are not allowed to propagate through the magnonic crystals [15, 16, 17, 18, 19, 20, 21]. This effect has been utilized earlier to demonstrate a magnonic transistor in a YIG strip [15, 16]. The essence of a magnonic transistor is a YIG strip with a periodic modulation of its thickness (magnonic crystal). The transistor is complemented by a source, a drain, and gate antennas. A gate antenna injects magnonic crystal magnons with a frequency ωGmatching the magnonic crystal band gap. In the process, the gate magnons cannot leave the crystal and may reach a high density. Magnons emitted from a source with a wave vector ksflowing towards the drain run into the magnonic crystal. The interaction between the source magnons and the magnonic crystal magnons is a four-magnon scattering process. The magnonic current emitted from the source attenuates in the magnonic crystal, and the weak signal reaches the drain due to the scattering. The relaxation process is swift if the following condition holds [16, 22] ks=m0π a0, (1) where m0is the integer, and a0is the crystal lattice constant. The magnons with wave vectors satisfying the Bragg conditions Eq. (1) will be resonantly scattered back, resulting in the generation of rejection bands in a spin-wave spectrum over which magnon propagation is entirely prohibited. Experimental verification of this effect is given in Ref. [16]. 2. Results 2.1. Proposed set-up for QID A pictorial representation of a QID is shown in Fig. 1. A magnonic crystal can be fabricated from a YIG film. Grooves can be deposited using a lithography procedure in a few nanometer steps, and, for our purpose, we consider parallel lines in width of 1 µm spaced with 10 µm from each other. Therefore, the lattice constant, approximately a0=11µm,i. e., is muchQuantum information diode based on a magnonic crystal 4 Figure 1. Illustration of a quantum information diode: A plane of a YIG film with grooves orthogonal to the direction of the propagation of quantum information. In the middle of the QID, we pump extra magnons to excite the system. A quantum excitation propagates toward the left, and the right ends asymmetrically. To describe the propagation process of quantum information, we introduce the left and right OTOC CL(t)andCR(t). Because the left-right inversion is equivalent to D→− D, meaning Ey→− Ey, we can invert the left and right OTOCs by switching the applied external electric field. larger than the unit cell size a=10nm used in our coarse-graining approach. Due to the capacity of our analytical calculations, we consider quantum spin chains of length about N=1000 spins and the maximal distance between the spins ri j=d(in the units of a), d=i−j=40. In what follows, we take k(ω)a≪1. The mechanism of the QID is based on the effect of direction dependence of nonreciprocal magnons [23, 24, 25]. In the chiral spin systems, the absence of inversion symmetry causes a difference in dispersion relations of the left and right propagating magnons, i. e., ωs,L(k)̸=ωs,R(−k). Due to the Dzyaloshinskii–Moriya interaction (DMI), magnons of the same frequency ωspropagating in opposite directions have different wave vectors [26]: a(k+ s−k− s) =D/J, where Jis the exchange constant, and Dis the DMI constant. Therefore, if the condition Eq.(1) holds for the left propagating magnons, it is violated for the right propagating magnons and vice versa. These magnons propagating in different directions decay differently in the magnonic crystal. Without loss of generality, we assume that the right propagating magnons with k+ ssatisfy the condition Eq.(1), and the current attenuates due to the scattering of source magnons by the gate magnons. The left propagating magnons k− sviolate the condition Eq.(1), and the current flows without scattering. Thus, reversing the source and drain antennas’ positions rectifies the current. Following ref. [16], we introduce a suppression rate of the source to drain the magnonic current ξ(D) =1−n+ D/n− D, where n+ D<n− Dare densities of the drain magnons with and without scattering. The parameter ξ(D)is experimentally accessible, and it depends on a particular setup. Therefore, in this manuscript, we take ξ(D)as a free theory parameter. Multiferroic (MF) materials are considered as a good example of a system with broken inversion symmetry, (see Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35]) and references therein. MF properties of YIG are studied in ref. [36]. Moreover, in accordance with theQuantum information diode based on a magnonic crystal 5 scanning tunneling microscopy experiments, a change in the spin direction at one edge of a chiral chain was experimentally probed by tens of nanometers away from the second edge [35]. 2.2. Model We consider a 2D square-lattice spin system with nearest-neighbor J1and the next nearest- neighbor J2coupling constants: ˆH=J1∑ ⟨n,m⟩ˆσnˆσm+J2∑ ⟨⟨n,m⟩⟩ˆσnˆσm−P·E, (2) where ⟨n,m⟩and⟨⟨n,m⟩⟩indicates all the pairs with nearest-neighbor and next nearest- neighbor interactions, respectively. The last term in Eq. (2) describes a coupling of the ferroelectric polarization with unit vector ex i,i+1,P=gMEex i,i+1×(ˆσi׈σi+1)with an applied external electric field and mimics an effective Dzyaloshinskii–Moriya interaction term D= EygMEbreaking the left-right symmetry, where −P·E=D∑ n(ˆσn׈σn+1)z. (3) Here we consider only the nearest neighbor DMI and only in one direction. As a consequence, the left-right inversion is equivalent to D→ − D, orEy→ − Ey. The broken left-right inversion symmetry can be exploited in rectifying the information current by an electric field. More importantly, the procedure is experimentally feasible. We can diagonlize the Hamiltonian in Eq. (2) by using the Holstein-Primakoff transformation [37, 38, 39, 40][See Appendix A for detailed derivation] as: ˆH=∑ ⃗kω(±D,k)ˆa† ⃗kˆa⃗k,ω(±D,k) = ω(⃗k)±ωDM(⃗k)
,ωDM(⃗k) =Dsin(kxa), ω(⃗k) =2J1(1−γ1,k)+2J2(1−γ2,k),gamma 1,k=1 2(coskxa+coskya), γ2,k=1 2[cos(kx+ky)a+cos(kx−ky)a]. (4) Here±Dcorresponds to the magnons propagating in opposite directions and the sign change is equivalent to the electric field direction change. We note that a 1D character of the DM term is ensured by the magnetoelectric effect [27] and to the electric field applied along the y axis. The speed limit of information propagation is usually given in terms of Lieb-Robinson (LR) bound, defined for the Hamiltonians that are locally bounded and short-range interacting [41, 42, 43]. Since the Hamiltonian in Eq. (2) satisfies both conditions, the LR bound can be defined for the spin model. However, when we transform the Hamiltonian using Holstein- Primakoff bosons, we have to take extra care as the bosons are not locally bounded. To define LR bound, we take only a few noninteracting magnons and exclude the magnon-magnon interaction to truncate terms beyond quadratic operators. In a realistic experimental setting, a low density of propagating magnons in YIG can easily be achieved by properly controlling the microwave antenna. In the case of low magnon density, the role of the magnon-magnonQuantum information diode based on a magnonic crystal 6 interaction between propagating magnons in YIG is negligible. Therefore, for YIG, we have a quadratic Hamiltonian, which is a precise approach in a low magnon density limit. Our discussion is valid for the experimental physical system [16], where magnons of YIG do not interact with each other, implying that there is no term in the Hamiltonian beyond quadratic. We can estimate LR bounds [44] defining the maximum group velocities of the left-right propagating magnons v± g(⃗k) =∂(ω(⃗k)±ωDM(⃗k)) ∂k. Taking into account the explicit form of the dispersion relations, we see that the maximal asymmetry is approximately equal to the DM constant i. e., v+ g(0)−v− g(0)≈2D. We note that the effect of nonreciprocal magnons is already observed experimentally [45, 46, 47, 48, 49] but up to date, never discussed in the context of the quantum information theory. We formulate the central interest question as follows: At t=0, we act upon the spin ˆσnto see how swiftly changes in the spin direction can be probed tens of sites away d=n−m≫1, and whether the forward and backward processes (i.e., probing for ˆσmthe outcome of the measurement done on ˆσn) are asymmetric or not. Due to the left-right asymmetry, the chiral spin channel may sustain a diode rectification effect when transferring the quantum information from left to right and in the opposite direction. We note that our discussion about the left-right asymmetry of the quantum information flow is valid until the current reaches boundaries. Thus the upper limit of the time reads tmax=Na/v± g(⃗k), where Nis the size of the system. 2.3. Out-of-time-order correlator Larkin and Ovchinnikov [50] introduced the concept of the out-of-time-ordered correlator (OTOC), and since then, OTOC has been seen as a diagnostic tool of quantum chaos. The concern of delocalizations in the quantum information theory (i.e., the scrambling of quantum entanglement) was renewed only recently, see Refs. [51, 52, 53, 54, 55, 56, 57, 58, 59, 60] and references therein. OTOC is also used for describing the static and dynamical phase transitions [61, 62, 63]. Dynamics of the semi-classical, quantum, and spin systems can be discussed by using OTOC [64, 51, 65, 66, 67]. We utilize OTOC to characterize the left-right asymmetry of the quantum information flow and thus infer the rectification effect of a diode. Let us consider two unitary operators ˆVand ˆWdescribing local perturbations to the chiral spin system Eq. (2), and the unitary time evolution of one of the operators ˆW(t) =exp(iˆHt)ˆW(0)exp(−iˆHt). Then the OTOC is defined as C(t) =1 2DˆW(t),ˆV(0)†ˆW(t),ˆV(0)E , (5) where parentheses ⟨···⟩ denotes a quantum mechanical average over the propagated quantum state. Following the definition the OTOC at the initial moment is zero C(0) =0, provided that [ˆW(0),ˆV(0)] = 0. In particular, for the local unitary and Hermitian operators of our choice ˆW† m(t)≡ˆσz m(t) =exp(iˆHt)ˆηmexp(−iˆHt), and ˆV† n=ˆσz n=ˆηn, where ˆηn=ˆ2a† nˆan−1. The bosonic operators are related to the spin operators via σ− n=2a† n,σ+ n=2an,σz n=2a† nan−1.Quantum information diode based on a magnonic crystal 7 0 20 4000.511.52 0 20 4000.20.40.60.8 0 20 4000.20.40.60.8 Figure 2. (a) Left-OTOC and (b)Right-OTOC with time t(in the units of 1 /J) for different distances r1,2=10a, 20aand 30 a.(c)Right-OTOC with time for r1,2=10aand suppression rates of the magnon current ζ=0.8, 0.6 and 0 .4. Parameters are N=1000, D=J1=2J2=1. Periodic boundary conditions are considered. The values of the parameters: m0=1 to N, a=10−3anda0=1. In terms of the occupation number operators, the OTOC is given as C(t) =1 2 ⟨ηnηm(t)ηm(t)ηn⟩+⟨ηm(t)ηnηnηm(t)⟩ − ⟨ηm(t)ηnηm(t)ηn⟩−⟨ηnηm(t)ηnηm(t) . (6) Indeed, the OTOC can be interpreted as the overlap of two wave functions, which are time evolved in two different ways for the same initial state |ψ(0)⟩. The first wave function is obtained by perturbing the initial state at t=0 with a local unitary operator ˆV, then evolved further under the unitary evolution operator ˆU=exp(−iˆHt)until time t. It is then perturbed at time twith a local unitary operator ˆW, and evolved backwards from ttot=0 underˆU†. Hence, the time-evolved wave function is |ψ(t)⟩=ˆU†ˆWˆUˆV|ψ(0)⟩=ˆW(t)ˆV|ψ(0)⟩. To get the second wave function, the order of the applied perturbations is permuted, i. e., first ˆW attand then ˆVatt=0. Therefore, the second wave function is |φ(t)⟩=ˆVˆU†ˆWˆU|ψ(0)⟩= ˆVˆW(t)|ψ(0)⟩and their overlap is equivalent to F(t) =⟨φ(t)|ψ(t)⟩. The OTOC is calculated from this overlap using C(t) =1−ℜ[F(t)]. What breaks the time inversion symmetry for the OTOC is the permuted sequence of operators ˆWand ˆV. However, in spin-lattice models with a preserved spatial inversion symmetry ∧PˆH=ˆH, the spatial inversion ∧Pd(ˆW,ˆV) = −d(ˆW,ˆV) =d(ˆV,ˆW)can restore the permuted order between ˆVand ˆW, where d(ˆW,ˆV) denotes the distance between observables ˆWand ˆV. Permuting just a single wave function, one finds C(t) =1−ℜ(⟨φ(t)|∧P∧T|ψ(t)⟩) =C(0). Thus, a scrambled quantum entanglement formally can be unscrambled by a spatial inversion. However, in chiral systems ∧PˆH̸=ˆHand the unscrambling procedure fails. Taking into account Eq. (4), we analyze quantum information scrambling along the x axis i. e.,ω(±D,k) =ω(±D,kx,0)and along the yaxis, ω(0,k) =ω(0,0,ky). It is easy to see that the quantum information flow along the yaxis is symmetric, while along the xaxis it is asymmetric and depends on the sign of the DM constant, i.e., the flow along the xis different from −x. Let us assume that Eq. (1) holds for right-moving magnons and is violated for left-moving magnons. Excited magnons with the same frequency and propagating intoQuantum information diode based on a magnonic crystal 8 different directions have different wave vectors ωs(D,k+ s) =ωs(−D,k− s)where: ωs ±D,k± s
=2J1(1−1/2cos k± xa)+2J2(1−cosk± xa)±Dsink± xa, (7) k+ m0x=m0π a0,m0=Nandk− m0xwe find from the condition ωs(D,k+ s) =ωs(−D,k− s)leading tok− m0x=k+ m0x+2 atan−1 D J1+2J2 . Here we use shortened notations ωm0=ωs(D,k+ s) = ωs(−D,k− s)and set dimensionless units J1=2J2≡J=1. We excite in the diode magnons of different frequencies m0= [1,N]. Considering Eq. (6), Eq. (7) and following Ref. [39], we obtain expressions for the left and right OTOCs CL(t)andCR(t)as: CL(t) =8 N2ΩL 1ΩL 2−8 N4ΩL 1ΩL 2ΩL 1ΩL 2, CR(t) =ζ4(D)8 N2ΩR 1ΩR 2−8 N4ΩR 1ΩR 2ΩR 1ΩR 2 , (8) where frequencies ΩL/R 1/2and details of derivations are presented in Appendix B. Parameter ξenters into the right OTOC CR(t)expression because the right propagating magnons are scattered on the gate magnons. This is due to the non-reciprocal magnon dispersion relations associated with the DMI term. Since the value of ξdepends on the experimental setup, we consider experimentally feasible values in our calculations. It should be noted that in the calculation of OTOC, we consider expectation value over the one magnon excitation state ˆ a† n|φ⟩, where |φ⟩is the vacuum state. Such a state shows the presence of the quantum blockade effect in a magnonic crystal. The calculation of equal time second-order correlation function showing the quantum blockade effect is given in Appendix C. Fig. 2(a) and Fig. 2(b) are the variation of CL(t)andCR(t)for|n+−m|and|n−−m| distant spins, respectively. Both show similar behavior with increasing separation between the spins. However, the amplitude of CR(t)is less than CL(t)because the decay amplitude of theCR(t)varies due to the suppression coefficient ζ. In the case of the dominant attenuation by the gate magnons, the OTOC decreases significantly. The difference in CL(t)andCR(t) originated due to the asymmetry arising from the DMI term. The time required to deviate the OTOC from zero increases as the separation between the spins increases. This observation indicates that quantum information flow has a finite "butterfly velocity." On the contrary, the amplitude of OTOC decreases as the separation between the spins increases because the initial amount of quantum information spreads among more spins. At the large time, OTOC again becomes zero because it spreads over the whole system. Fig. 2(c) is the behavior of CR(t) with decreasing suppression coefficient ζand fixed value of distance between the spins r1,2. As suppression coefficient ζdecreases, the amplitude of CR(t)decreases which is an indicator of increasing rectification. A detailed discussion of rectification is in the next subsection. A high density of magnons can invalidate the assumption of a pure state or spin-wave approximation that works only for a low density of magnons. However, the key point in our case is that one has to distinguish between two sorts of magnons, gate magnons and propagating nonreciprocal magnons. The density of the propagating magnons can be regulated in the experiment through a microwave antenna, and one can always ensure that their density is low enough. It is easy to regulate the density of the gate magnons, and anQuantum information diode based on a magnonic crystal 9 0 0.5 10.40.60.81 Figure 3. Rectification coefficient Ris plotted against DMI coefficient ( D) for suppression rateζ(D)≈e−D/5. The parameters are J1=2J2=1,N=1000, r12=10a,a0=1 and m0=1 toN. experimentally accessible method is discussed in Ref. [16]. In the magnonic systems, the Kerr nonlinearity may lead to interesting effects, for example, the magnon-magnon entanglement and frequency shift [68, 69]. On the other hand, we note that DMI term and strong magneto- electric coupling may be responsible for nonlinear coupling terms similar to the magnon Kerr effect. This effect is studied in Ref. [70]. 2.4. Rectification The efficiency of the quantum information diode is given by the rectification coefficient i.e., the ratio between the left and right propagating magnons that is calculated by left and right OTOC. DMI term and non-reciprocal magnon dispersion relations influence the rectification coefficient in two ways. a) directly meaning that in the left and right OTOCs appear different left and right dispersion relations and b) non-directly meaning that magnonic crystal due to the scattering on the gate magnons bans propagation of the drain magnons in one direction only (damping of the OTOC current). This non-reciprocal damping effect is experimentally observed in magnonic crystals [16]. The non-reciprocal damping enhances the rectification effect and it was not studied in the context of quantum information and OTOC before. Let us calculate the total amount of correlations transferred in opposite directions followed by the rectification coefficient, a function of the external electric field as R=∞R 0CR(t)dt ∞R 0CL(t)dt. We interpolate the suppression rate as a function of the DMI coefficient in the form ζ(D)≈ e−D/5. The coefficient ζ(D)mimics a scattering process of the drain magnons on the gate magnons [16]. In Fig. 3 we see the variation of the rectification coefficient as a function ofD. The electric field has a direct and important role in rectification. In particular, DMI constant Ddepends on the electric field EyasD=EygME, where gMEis the magneto-electric coupling constant. In the case of zero electric field, Dwill be zero, implying the absence of rectification effect R=1. As the electric field increases, Dalso increases linearly, and rectification decreases exponentially. A detailed study of the role of the electric field in DM has been done in Ref. [36].Quantum information diode based on a magnonic crystal 10 3. Discussions We studied a quantum information flow in a spin quantum system. In particular, we proposed a quantum magnon diode based on YIG and magnonic crystal properties. The flow of magnons with wavelengths satisfying the Bragg conditions k=m0π/aois reflected from the gate magnons. Due to the absence of inversion symmetry in the system, left and right-propagating magnons have different dispersion relations and wave vectors. While for the right propagating magnons, the Bragg conditions hold, left magnons violate them, leading to an asymmetric flow of the quantum information. We found that the strength of quantum correlations depends on the distance between spins and time. The OTOC for the spins separated by longer distance shows an inevitable delay in time, meaning that the quantum information flow has a finite "butterfly velocity." On the other hand, the OTOC amplitude becomes smaller at longer distances between spins. The reason is that the initial amount of quantum information spreads among more spins. After the quantum information spreads over the whole system, which is pretty large ( N=1000 sites), the OTOC again becomes zero. We proposed a novel theoretical concept that can be directly realized with the experimentally feasible setup and particular material. There are several experimentally feasible protocols for measuring OTOC in the spin systems [71, 72]. According to these protocols, one needs to initialize the system into the fully polarized state, then apply quench and measure the expectation value of the first spin. All these steps are directly applicable to our setup from YIG. The fully polarized initial state can be obtained by switching on and off a strong magnetic field at a time moment t = 0. Quench, in our case, is performed by a microwave antenna which is an experimentally accessible device. Polarization of the initial spin can be measured through the STM tip. Overall our setup is the experimentally feasible setup studied in Ref. [16]. Data Availability Statement The data that support the findings of this study are available within the article. Acknowledgments SKM acknowledges the Science and Engineering Research Board, Department of Science and Technology, India for support under Core Research Grant CRG/2021/007095. A.E. acknowledges the funding by the Fonds zur Förderung der Wissenschaftlichen Forschung (FWF) under Grant No. I 5384.Quantum information diode based on a magnonic crystal 11 Appendix A. Diagonalization of Hamiltonian Eq. (2) 2D square-lattice spin system with nearest-neighbor J1and the next nearest-neighbor J2 coupling constants (taking ℏ=1): ˆH=J1∑ ⟨n,m⟩ˆσnˆσm+J2∑ ⟨⟨n,m⟩⟩ˆσnˆσm−P·E, =J1∑ ⟨n,m⟩ˆσnˆσm+J2∑ ⟨⟨n,m⟩⟩ˆσnˆσm−D∑ n(ˆσn׈σn+1)z, =4h J1∑ ⟨n,m⟩ˆSnˆSm+J2∑ ⟨⟨n,m⟩⟩ˆSnˆSm+D i∑ n(ˆS+ nˆS− n+1−ˆS− nˆS+ n+1)i , =4h J1∑ ⟨n,m⟩1 2n ˆS− nˆS+ m+ˆS+ nˆS− m +ˆSz nˆSz mo +J2∑ ⟨⟨n,m⟩⟩1 2n ˆS− nˆS+ m+ˆS+ nˆS− m +ˆSz nˆSz mo +D i∑ n(ˆS+ nˆS− n+1−ˆS− nˆS+ n+1)i . (A.1) Spin-half systems have two permitted states on each site, i.e.,| ↑⟩and| ↓⟩. Operation of spin operators on these state are given as ˆS+| ↓⟩=| ↑⟩,ˆS+| ↑⟩=0, (A.2) ˆS−| ↑⟩=| ↓⟩,ˆS−| ↓⟩=0, (A.3) ˆSz| ↑⟩=1 2| ↑⟩,ˆSz| ↓⟩=−1 2| ↓⟩, (A.4) Transformation of the spin operators in hard-core bosonic creation and annihilation operators are given as ˆS+ m,n=ˆam,n, ˆS− m,n=ˆa† m,n, ˆSz m,n=1/2−ˆa† m,nˆam,n (A.5) Hamiltonian in the bosonic representation is given as ˆH=2h J1∑ ⟨n,m⟩ ˆa† nˆam+ˆanˆa† m−ˆa† nˆan−ˆa† mˆam+1 2+ˆa† nˆanˆa† mˆam +J2∑ ⟨⟨n,m⟩⟩ ˆa† nˆam+ˆanˆa† m−ˆa† nˆan−ˆa† mˆam+1 2+ˆa† nˆanˆa† mˆam +D i∑ n ˆanˆa† n+1−ˆa† nˆan+1i . (A.6) Fourier transform of ˆ a† n(ˆan)is ˆa† ⃗k(ˆa⃗k). ˆa† ⃗k=1√ N∑ nei⃗k⃗rna† n, ˆa⃗k=1√ N∑ ne−i⃗k⃗rnan. (A.7)Quantum information diode based on a magnonic crystal 12 Inverse Fourier transform is given as ˆa† n=1√ N∑ nei⃗k⃗rna† ⃗k, ˆan=1√ N∑ ne−i⃗k⃗rna⃗k. (A.8) After summing over nwe get Hamiltonian (Eq. A.1) in ⃗kspace as ˆH=∑ ⃗kω⃗kˆa† ⃗kˆa⃗k−D∑ ⃗ksin(⃗ka)ˆa† ⃗kˆa⃗k, =∑ ⃗kω(±D,k)ˆa† ⃗kˆa⃗k(A.9) where, ω(±D,k) = ω(⃗k)±ωDM(⃗k)
,ωDM(⃗k) =Dsin(kxa), ω(⃗k) =2J1(1−γ1,k)+2J2(1−γ2,k),γ1,k=1 2(coskxa+coskya), γ2,k=1 2[cos(kx+ky)a+cos(kx−ky)a]. (A.10) Appendix B. Calculation of left and right out-of-time ordered correlation functions We will calculate OTOC exactly for one magnon excitation state given in Eq. (7) as C(t) =1 2 ⟨ˆηnˆηm(t)ˆηm(t)ˆηn⟩+⟨ˆηm(t)ˆηnˆηnˆηm(t)⟩ −⟨ˆηm(t)ˆηnˆηm(t)ˆηn⟩−⟨ ˆηnˆηm(t)ˆηnˆηm(t) .(B.1) Here, ˆηm/n=ˆσz m/nis Hermitian and unitary, therefore, Eq. (B.1) transforms in the form given as C(t) =1−⟨ˆηm(t)ˆηnˆηm(t)ˆηn⟩=1−F(t), (B.2) where F(t)is given as F(t) =⟨φ|ˆanˆηm(t)ˆηnˆηm(t)ˆηna† n|φ⟩. (B.3) In the above equation, the expectation value is taken over one magnon excitation state ˆa† n|φ⟩, where |φ⟩is the vacuum state, equivalent to a polarized state. First of all we calculate the product of four observables in F(t)(Eq. (B.3))in bosonic representation as ˆηm(t)ˆηnˆηm(t)ˆηn= [1−2 ˆa† mˆam(t)][1−2 ˆa† nˆan][1−2 ˆa† mˆam(t)][1−2 ˆa† nˆan], =h 1−2 ˆa† mˆam(t)−2 ˆa† nˆan+4 ˆa† mˆam(t)ˆa† nˆani ×h 1−2 ˆa† mˆam(t)−2 ˆa† nˆan+4 ˆa† mˆam(t)ˆa† nˆani , =1−4 ˆa† mˆam(t)−4 ˆa† nˆan+4 ˆa† mˆam(t)ˆa† nˆan+4 ˆa† nˆanˆa† mˆam(t) +4 ˆa† mˆamˆa† mˆam(t)+4 ˆa† nˆanˆa† nˆan+4 ˆa† mˆamˆa† nˆan+4 ˆa† nˆanˆa† mˆamQuantum information diode based on a magnonic crystal 13 −8 ˆa† mˆamˆa† mˆam(t)ˆa† nˆan−8 ˆa† nˆanˆa† mˆam(t)ˆa† nˆan −8 ˆa† mˆam(t)ˆa† nˆanˆa† mˆam(t)−8 ˆa† mˆam(t)ˆa† nˆanˆa† nˆan +16 ˆa† mˆam(t)ˆa† nˆanˆa† mˆam(t)ˆa† nˆan. (B.4) Further, we calculate the expectation value of the last term of Eq. (B.4) over one magnon excitation state i. e.,⟨φ|ˆanˆa† mˆam(t)ˆa† nˆanˆanˆa† mˆam(t)ˆa† nˆanˆa† n|φ⟩,using the properties of bosonic operators [ˆai,ˆa† j] =δi j,(ˆai)2=0, and (ˆa† i)2=0. We get ⟨φ|ˆanˆa† mˆam(t)ˆa† nˆanˆanˆa† mˆam(t)ˆa† nˆanˆa† n|φ⟩=⟨φ|ˆaneiˆHtˆa† mˆame−iˆHtˆa† nˆaneiˆHtˆa† mˆame−iˆHtˆa† n|φ⟩, =⟨Ψ(t)|Ψ(t)⟩, (B.5) where |Ψ(t)⟩=ˆaneiˆHtˆa† mˆame−iˆHtˆa† n|φ⟩.Fourier transformation of the |Ψ(t)⟩and diagonalized Hamiltonian will provide |Ψ(t)⟩=1 N∑ kei(−k(m−n)+ωkt/ℏ)1 N∑ k′ei(k′(m−n)−ωk′t/ℏ)|φ⟩ =1 N2Ω1Ω2|φ⟩. Hence, ⟨Ψ(t)|Ψ(t)⟩=1 N4Ω1Ω2Ω1Ω2. (B.6) Similarly, ⟨φ|ˆanˆamˆam(t)ˆa† n|φ⟩=1 N2Ω1Ω2 (B.7) After doing simple bosonic algebra, time-dependent terms of Eq. (B.4) are converted either in the form of Eq. (B.5) or Eq. (B.7). By using Eq. (B.6) and Eq. (B.7), we calculate F(t)as F(t) =1−4 N2Ω1Ω2−4+4 N2Ω1Ω2+4 N2Ω1Ω2+4 N2Ω1Ω2+4 N2Ω1Ω2+4 N2Ω1Ω2+4 −8 N2Ω1Ω2−8 N2Ω1Ω2−8 N4Ω1Ω2Ω1Ω2−8 N2Ω1Ω2+16 N4Ω1Ω2Ω1Ω2 =1−8 N2Ω1Ω2+8 N4Ω1Ω2Ω1Ω2 (B.8) Then, we get the left and right OTOCs’ analytical expressions as CL(t) =8 N2ΩL 1ΩL 2−8 N4ΩL 1ΩL 2ΩL 1ΩL 2, CR(t) =ζ4(D)8 N2ΩR 1ΩR 2−8 N4ΩR 1ΩR 2ΩR 1ΩR 2 , (B.9) where frequencies ΩL/R 1/2are given as ΩR 1=ΩR∗ 2=∑ m0exp −im0πr1,2 a0 expiωm0t ℏ , and ΩL 1=ΩL∗ 2=∑ m0exp(−ik− sr1,2)expiωm0t ℏ . (B.10)Quantum information diode based on a magnonic crystal 14 Appendix C. Quantum blockade effects To analyze the magnon blockade effect, we calculate the equal time second-order correlation function defined as [73, 74, 75, 76, 77, 78] g2 a(0) =Tr(ˆρˆa†2 mˆa2 m) h Tr(ˆρˆa†mˆam)i2=⟨ˆa†2 mˆa2 m⟩ ⟨a†mˆam⟩2, (C.1) where am(a† m)are the annihilation (creation) operators of the magnon excitation. The magnon blockade is inferred from the condition g2 a(0)→0 meaning that magnons can be excited individually, and two or more magnons cannot be excited together. We note that ˆ a2 mˆa† m|φ⟩=0 leading to g2 a(0) =0. Therefore, the quantum blockade effect occurs in this case. References [1] Liang B, Yuan B and Cheng J c 2009 Phys. Rev. Lett. 103(10) 104301 [2] Liang B, Guo X, Tu J, Zhang D and Cheng J 2010 Nature materials 9989–992 [3] Li X F, Ni X, Feng L, Lu M H, He C and Chen Y F 2011 Phys. Rev. Lett. 106(8) 084301 [4] Maldovan M 2013 Nature 503209–217 [5] Ren J 2013 Phys. Rev. B 88(22) 220406 [6] Lepri S, Livi R and Politi A 1997 Phys. Rev. Lett. 78(10) 1896–1899 [7] Komatsu T S and Ito N 2011 Phys. Rev. E 83(1) 012104 [8] Kim P, Shi L, Majumdar A and McEuen P L 2001 Phys. Rev. Lett. 87(21) 215502 [9] Kobayashi W, Teraoka Y and Terasaki I 2009 Applied Physics Lett. 95171905 [10] Li B, Lan J and Wang L 2005 Phys. Rev. Lett. 95104302 [11] Terraneo M, Peyrard M and Casati G 2002 Phys. Rev. Lett. 88094302 [12] Li B, Wang L and Casati G 2004 Phys. Rev. Lett. 93184301 [13] Li N, Ren J, Wang L, Zhang G, Hänggi P and Li B 2012 Rev. Mod. Phys. 84(3) 1045–1066 [14] Chotorlishvili L, Etesami S, Berakdar J, Khomeriki R and Ren J 2015 Phys. Rev. B 92134424 [15] Chumak A, Serga A, Hillebrands B and Kostylev M 2008 Applied Physics Lett. 93022508 [16] Chumak A V , Serga A A and Hillebrands B 2014 Nature communications 51–8 [17] Nikitov S, Tailhades P and Tsai C 2001 Journal of Magnetism and Magnetic Materials 236320–330 [18] Kruglyak V and Kuchko A 2004 Journal of magnetism and magnetic materials 272302–303 [19] Wang Z, Zhang V , Lim H, Ng S, Kuok M, Jain S and Adeyeye A 2009 Applied Physics Lett. 94083112 [20] Gubbiotti G, Tacchi S, Madami M, Carlotti G, Adeyeye A and Kostylev M 2010 Journal of Physics D: Applied Physics 43264003 [21] Ustinov A B, Drozdovskii A V and Kalinikos B A 2010 Applied physics Lett. 96142513 [22] Gurevich A G and Melkov G A 1996 Magnetization oscillations and waves (CRC press) [23] Takashima R, Shiomi Y and Motome Y 2018 Phys. Rev. B 98020401 [24] Matsumoto T and Hayami S 2020 Phys. Rev. B 101224419 [25] Shiomi Y , Takashima R, Okuyama D, Gitgeatpong G, Piyawongwatthana P, Matan K, Sato T and Saitoh E 2017 Phys. Rev. B 96180414 [26] Wang X g, Chotorlishvili L, Guo G h and Berakdar J 2018 Journal of Applied Physics 124073903 [27] Katsura H, Nagaosa N and Balatsky A V 2005 Phys. Rev. Lett. 95057205 [28] Mostovoy M 2006 Phys. Rev. Lett. 96067601 [29] Ramesh R and Spaldin N A 2010 Nanoscience And Technology: A Collection of Reviews from Nature Journals 20–28 [30] Bibes M and Barthélémy A 2008 Nature materials 7425–426Quantum information diode based on a magnonic crystal 15 [31] Fiebig M 2005 Journal of physics D: applied physics 38R123R152 [32] Hemberger J, Schrettle F, Pimenov A, Lunkenheimer P, Ivanov V Y , Mukhin A, Balbashov A and Loidl A 2007 Phys. Rev. B 75035118 [33] Meyerheim H, Klimenta F, Ernst A, Mohseni K, Ostanin S, Fechner M, Parihar S, Maznichenko I, Mertig I and Kirschner J 2011 Phys. Rev. Lett. 106087203 [34] Cheong S W and Mostovoy M 2007 Nature materials 613–20 [35] Menzel M, Mokrousov Y , Wieser R, Bickel J E, Vedmedenko E, Blügel S, Heinze S, von Bergmann K, Kubetzka A and Wiesendanger R 2012 Phys. Rev. Lett. 108197204 [36] Liu T and Vignale G 2011 Phys. Rev. Lett. 106247203 [37] Maruyama K, Iitaka T and Nori F 2007 Phys. Rev. A 75(1) 012325 [38] Udvardi L and Szunyogh L 2009 Phys. Rev. Lett. 102(20) 207204 [39] Guo Z X, Hu X D, Su Q Q and Li Z 2021 arXiv preprint arXiv:2109.11371 [40] Stagraczy ´nski S, Chotorlishvili L, Schüler M, Mierzejewski M and Berakdar J 2017 Phys. Rev. B 96(5) 054440 [41] Kuwahara T and Saito K 2021 Physical review letters 127070403 [42] Hastings M B and Koma T 2006 Communications in mathematical physics 265781 [43] Nachtergaele B, Ogata Y and Sims R 2006 Journal of statistical physics 1241 [44] Lieb E H and Robinson D W 1972 The finite group velocity of quantum spin systems Statistical mechanics (Springer) pp 425–431 [45] Zakeri K, Zhang Y , Prokop J, Chuang T H, Sakr N, Tang W X and Kirschner J 2010 Phys. Rev. Lett. 104(13) 137203 [46] Iguchi Y , Uemura S, Ueno K and Onose Y 2015 Phys. Rev. B 92(18) 184419 [47] Seki S, Okamura Y , Kondou K, Shibata K, Kubota M, Takagi R, Kagawa F, Kawasaki M, Tatara G, Otani Y and Tokura Y 2016 Phys. Rev. B 93(23) 235131 [48] Sato T J, Okuyama D, Hong T, Kikkawa A, Taguchi Y , Arima T h and Tokura Y 2016 Phys. Rev. B 94(14) 144420 [49] Gitgeatpong G, Zhao Y , Piyawongwatthana P, Qiu Y , Harriger L W, Butch N P, Sato T J and Matan K 2017 Phys. Rev. Lett. 119(4) 047201 [50] Larkin A I and Ovchinnikov Y N 1969 Soviet Journal of Experimental and Theoretical Physics 281200 [51] Maldacena J, Shenker S H and Stanford D 2016 Journal of High Energy Physics 2016 106 [52] Roberts D A, Stanford D and Susskind L 2015 Journal of High Energy Physics 2015 51 [53] Iyoda E and Sagawa T 2018 Phys. Rev. A 97(4) 042330 [54] Chapman A and Miyake A 2018 Phys. Rev. A 98(1) 012309 [55] Swingle B and Chowdhury D 2017 Phys. Rev. B 95(6) 060201 [56] Klug M J, Scheurer M S and Schmalian J 2018 Phys. Rev. B 98(4) 045102 [57] del Campo A, Molina-Vilaplana J and Sonner J 2017 Phys. Rev. D 95(12) 126008 [58] Campisi M and Goold J 2017 Phys. Rev. E 95(6) 062127 [59] Hosur P, Qi X l, Roberts D A and Yoshida B 2016 Journal of High Energy Physics 2016 4 ISSN 1029-8479 [60] Yunger Halpern N 2017 Phys. Rev. A 95(1) 012120 [61] Heyl M, Pollmann F and Dóra B 2018 Phys. Rev. Lett. 121016801 [62] Chen B, Hou X, Zhou F, Qian P, Shen H and Xu N 2020 Applied Phys. Lett. 116194002 [63] Shukla R K, Naik G K and Mishra S K 2021 EPL13247003 [64] Rozenbaum E B, Ganeshan S and Galitski V 2017 Phys. Rev. Lett. 118086801 [65] Shukla R K, Lakshminarayan A and Mishra S K 2022 Physical Review B 105224307 [66] Shukla R K and Mishra S K 2022 Physical Review A 106022403 [67] Kukuljan I, Grozdanov S and Prosen T 2017 Phys. Rev. B 96060301 [68] Zhang Z, Scully M O and Agarwal G S 2019 Physical Review Research 1023021 [69] Moslehi M, Baghshahi H R, Faghihi M J and Mirafzali S Y 2022 The European Physical Journal Plus 137 1–7 [70] Toklikishvili Z, Chotorlishvili L, Khomeriki R, Jandieri V and Berakdar J 2023 Physical Review B 107 115126Quantum information diode based on a magnonic crystal 16 [71] Nie X, Wei B B, Chen X, Zhang Z, Zhao X, Qiu C, Tian Y , Ji Y , Xin T, Lu D and Li J 2020 Phys. Rev. Lett. 124250601 [72] Li J, Fan R, Wang H, Ye B, Zeng B, Zhai H, Peng X and Du J 2017 Phys. Rev. X 7031011 [73] Liu Z X, Xiong H and Wu Y 2019 Physical Review B 100134421 [74] Xie J k, Ma S l and Li F l 2020 Physical Review A 101042331 [75] Wu K, Zhong W x, Cheng G l and Chen A x 2021 Physical Review A 103052411 [76] Zhao C, Li X, Chao S, Peng R, Li C and Zhou L 2020 Physical Review A 101063838 [77] Wang F, Gou C, Xu J and Gong C 2022 Physical Review A 106013705 [78] Wang L, Yang Z X, Liu Y M, Bai C H, Wang D Y , Zhang S and Wang H F 2020 Annalen der Physik 532 2000028

2023-07-12

Exploiting the effect of nonreciprocal magnons in a system with no inversion symmetry, we propose a concept of a quantum information diode, {\it i.e.}, a device rectifying the amount of quantum information transmitted in the opposite directions. We control the asymmetric left and right quantum information currents through an applied external electric field and quantify it through the left and right out-of-time-ordered correlation (OTOC). To enhance the efficiency of the quantum information diode, we utilize a magnonic crystal. We excite magnons of different frequencies and let them propagate in opposite directions. Nonreciprocal magnons propagating in opposite directions have different dispersion relations. Magnons propagating in one direction match resonant conditions and scatter on gate magnons. Therefore, magnon flux in one direction is damped in the magnonic crystal leading to an asymmetric transport of quantum information in the quantum information diode. A quantum information diode can be fabricated from an yttrium iron garnet (YIG) film. This is an experimentally feasible concept and implies certain conditions: low temperature and small deviation from the equilibrium to exclude effects of phonons and magnon interactions. We show that rectification of the flaw of quantum information can be controlled efficiently by an external electric field and magnetoelectric effects.

Quantum information diode based on a magnonic crystal

2307.06047v1

Sign of inverse spin Hall voltages generated by ferromagnetic resonance and temperature gradients in yttrium iron garnet jplatinum bilayers Michael Schreier,1, 2,Gerrit E. W. Bauer,3, 4, 5Vitaliy I. Vasyuchka,6Joost Flipse,7Ken-ichi Uchida,3, 8Johannes Lotze,1, 2Viktor Lauer,6Andrii V. Chumak,6Alexander A. Serga,6Shunsuke Daimon,3Takashi Kikkawa,3Eiji Saitoh,3, 4, 9, 10Bart J. van Wees,7Burkard Hillebrands,6Rudolf Gross,1, 11, 2and Sebastian T. B. Goennenwein1, 11 1Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany 2Physik-Department, Technische Universit at M unchen, Garching, Germany 3Institute for Materials Research, Tohoku University, Sendai, Japan 4WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan 5Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands 6Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit at Kaiserslautern, Kaiserslautern, Germany 7Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands 8PRESTO, Japan Science and Technology Agency, Saitama, Japan 9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan 10CREST, Japan Science and Technology Agency, Tokyo, Japan 11Nanosystems Initiative Munich, Munich, Germany We carried out a concerted eort to determine the absolute sign of the inverse spin Hall eect voltage generated by spin currents injected into a normal metal. We focus on yttrium iron garnet (YIG)jplatinum bilayers at room temperature, generating spin currents by microwaves and temper- ature gradients. We nd consistent results for dierent samples and measurement setups that agree with theory. We suggest a right-hand-rule to dene a positive spin Hall angle corresponding to the voltage expected for the simple case of scattering of free electrons from repulsive Coulomb charges. The bon mot that the sign is the most dicult con- cept in physics since there are no approximate methods to determine it has been ascribed to Wolfgang Pauli. In- deed, the struggle to obtain correct signs permeates all of physics. While the negative sign of the electron charge is just a convention, that of derived properties, such as the (conventional) Hall voltage, has real physical mean- ing. Often it is much easier and sucient to determine sign dierences between related quantities. However, a complete understanding requires not only the relative but also the absolute sign. Here we address the sign of the (inverse) spin Hall eect [(I)SHE] [1{10] and related phe- nomena. The characteristic parameter is the spin Hall angle, dened as the ratio SH/Js=Jcof the transverse spin current Jscaused by an applied charge current Jc(a more precise denition is given below). The sign of SH may dier for dierent materials. Since the spin Hall an- gle for Pt is generally taken to be positive, SHof Mo [11], Ta [12], and W [13] must be negative. The sign of SHgoverns the direction of the spin trans- fer torque on a magnetic contact relative to that caused by the Oersted magnetic eld induced by the same cur- rentJc[12, 13]. It also determines the sign of the induced transverse voltage in experiments in which the ISHE is used to detect spin currents [7]. This technique is now widely used to study spin current injection by a mag- netic contact, through \spin pumping" induced by ferro- magnetic resonance (FMR) [11, 14{18] or by temperature dierences [19{23] (\spin Seebeck eect", SSE). michael.schreier@wmi.badw.deHowever, the pitfalls that can aect the determination of the sign of the SH, such as the sign of the spin cur- rents [24] and magnetic eld direction are often glossed over in experimental and theoretical papers. Moreover, a mechanism for a sign reversal of the longitudinal spin Seebeck eect has recently been proposed [25]. A careful analysis of experimental results with respect to the signs of of FMR and thermal spin pumping voltages generated by the inverse spin Hall eect is therefore overdue. In this letter we present the results of a concerted ac- tion to resolve the sign issue by comparing experiments on microwave-induced spin pumping and spin Seebeck ef- fect for a bilayer of the magnetic insulator yttrium iron garnet (YIG) and platinum (Pt) at room temperature. Samples grown by dierent techniques have been used in four experimental setups at the Institute for Materials Research, Tohoku University (IMR), Technische Univer- sit at Kaiserslautern (UniKL) Zernike Institute for Ad- vanced Materials, University of Groningen (RUG) and the Walther-Meiner-Institut in Garching (WMI). Con- sidering the dierent sample properties the variations in the magnitude of the observed voltages is not surpris- ing. However, all groups nd identical signs for the ISHE Hall voltages that agree with the standard theory for spin pumping by FMR [26] and spin Seebeck eect [27{29]. A positive spin Hall angle can be associated with scatter- ing at negatively charged Coulomb centers in the weakly relativistic electron gas. Let us dene the the electron charge as e <0:We recall that the thumb of the right hand points along the angular momentum L=rpof a circulating parti- cle with mass m, position rand momentum p=mvarXiv:1404.3490v2 [cond-mat.mtrl-sci] 11 Feb 20152 Figure 1. (a) The transverse de ection of polarized electrons generated by a xed point charge Q < 0 that we associate with a positive spin Hall angle. (b) A magnetic eld is positive when aligned with the Earth's magnetic eld or as the mag- netic eld generated by a negatively charged particle current owing through a coil when congured as sketched. [(c),(d)] Typical setups for spin pumping and spin Seebeck experi- ments, respectively. [(c)] An rf-microwave eld excites mag- netization precession that relaxes by emitting a spin current into the adjacent Pt layer. [(d)] The spin current from the YIG to the Pt is negative when the latter is hotter. In both cases, the ISHE leads to a voltage between the contacts Hi and Lo. when the tangential velocity vis along the ngers of its st. The magnetic moment of a particle with charge q is given by L=q(2m)1LLL[30]. This magnetic moment direction is also generated by two monopoles on the L- axis, the negative (south pole) just below and the pos- itive (north) pole just above the origin. The magnetic moment of the Earth points to the south, so the geo- graphic north pole is actually the magnetic south pole. Hence, the geomagnetic elds on the surface of the Earth as measured by a compass needle point to the north pole. The intrinsic angular momentum (spin) of non- relativistic electrons [31] is s=~ 2, where is the vector of Pauli spin matrices. The corresponding magnetic mo- ments=e(2m)1gsss= sss, wheregis the g-factor and is the gyromagnetic ratio. In most solids gand therefore are positive. The angular momentum and spin current tensor !Js consists of column vector elements J sthat represent the polarization of angular momentum currents in the Carte- sian-direction, while the row vectors Js;represent the ow direction of angular momentum along :The charge- (JJJc) and spin currents are both dened as particle ow (unitss1). We can then dene the spin Hall angle SH as the proportionality factor in the phenomenological re-lations Js;=SH^^^Jc (1) Jc=SHX J s^^^ (2) where the ^^^;^^^2f^ x;^ y;^ zgare Cartesian unit vectors. We now demonstrate the physical signicance of the sign ofSH. We do not intend to model the material dependence or contribute to the discussion on whether observed eects are intrinsic or extrinsic. For an external point charge Qat the origin in the weakly relativistic electron gas the bare Coulomb potential at distance ris 0(r) =1 40Q r; (3) where0is the vacuum permittivity. In metals 0(r) is screened by the mobile charge carriers to become the Yukawa potential =0er=. The screening length  serves to regularize the expectation values, but drops out of the nal results. The spin-orbit interaction of an elec- tron in a potential is equivalent to an eective magnetic eld [32] Bso=e 2m2c2 (rp); (4) wherecis the velocity of light. The force on the electron then reads Fso=r(
Bso) =e~ 4m2c2r[
(rp)]: (5) Focussing on a free electron moving in the y-direction (p=py^ y) with its spin pointing in the z-direction (=^ zz) in an ensemble of randomly distributed identi- cal point with density n, the expectation values hzi= 1 and the average force is hFyz soi=n 404eQ~2 3m2c2py^ x: (6) For our denition of a positive spin Hall angle we now choose the charge to be negative ( Q< 0, repulsive). We then arrive at the following right-hand rule: The elec- tron with its spin pointing in the z-direction (thumb) and moving in the y-direction (forenger) drifts to the negativex-direction (middle nger) [Fig. 1(a)]. A com- parison with Eq. (1) and using JJJs;z=C e2rs=C e2hFyz soi (7) and JJJc=py mneC^ y (8) wheresis the spin chemical potential, is the resistiv- ity,Cis the cross sectional area the currents are owing3 through, and neis the carrier density, leads to a spin Hall angle SH=n 404eQ~2 3m2c2m e2ne: (9) Inserting numbers for fundamental constants SH= 31010 mn=(ne) forQ=e. The time-averaged spin current injected by a steady pre- cession [26] around the equilibrium magnetization with unit vector ^ mis polarized along ^ m. This spin pumping process is associated with energy relaxation of the mag- netization dynamics that increases the magnetic moment in the direction of the eective magnetic eld. When the g-factor is positive, the spin pumping current through the interface is positive as well. In the SSE, when the temperature of the magnetization is lower than that of the electrons in the metal, the energy and, if g>0, spin current is opposite to that under FMR [27], leading to an opposite sign in the ISHE voltage compared to the FMR. Under open circuit conditions the inverse spin Hall eect [Eq. (2)] leads to a charge separation and an electrostatic eld EEEs=e ASH[JJJs;^mmm^mmm]; (10) whereAis the area of the ferromagnet jmetal interface andis the resistivity of the metal layer. This corre- sponds to an electromotive force E=EEEs
lll, wherelll is the length vector from the Loto the Hicontact in Fig. 1(c) and (d). The sign of an applied magnetic eld is related to the current direction according to Ampere's right hand law as depicted in Fig. 1(b). In practice, it is convenient to use a compass needle for comparison with the Earth's magnetic eld. Fig. 1(b) denes the positive eld di- rection from the antarctic to the arctic, i.e. along the geomagnetic eld. Typical experimental setups for spin pumping [(c)] and spin Seebeck experiments [(d)] on yt- trium iron garnet jplatinum thin lm (YIG jPt) bilayers are also sketched in Fig. 1. In the former, a ferromagnet (F)jnormal metal (N) stack is exposed to microwave ra- diation with frequency f(typically in the GHz regime), while in spin Seebeck experiments the bilayer is exposed to a thermal gradient. Sample parameters used by the dierent groups are listed in the third column of Fig. 2. For details on the sample fabrication we refer to Refs. 33 and 34 (WMI), Ref. 18 (RUG), Refs. 35 and 36 (UniKL) and Ref. 37 (IMR). Fig. 2 summarizes the results of the participating groups. Note that in each group both the spin pumping and spin Seebeck experiments were performed on the same sample, without changing the setup. At the WMI, FMR experiments (rst row) were car- ried out in a microwave cavity with xed frequency fres= 9:82 GHz as a function of an applied magnetic eldHextleading to resonance at 0Hext=270 mT. A source meter was used to drive a large ( Ih=20 mA) dccharge current through the platinum lm ( RPt= 197 ) in order to generate a temperature gradient (hot Pt, cold YIG) [23]. By summing voltages recorded for opposite Ihdirections, the magnetoresistive contributions cancel out, such that only the spin Seebeck signal remains. The ISHE voltages for both FMR and the spin Seebeck exper- iments were measured by the same, identically connected nanovoltmeter with microwave and heating current sep- arately turned on. Results from the UniKL are shown in the second row in Fig. 2. A microwave with fres= 7 GHz fed into a Cu stripline on top of the Pt lm excites the FMR at an external magnetic eld of 0Hext=175 mT. The mi- crowave current amplitude was modulated at a frequency offmod= 500 Hz to allow for lock-in detection of the induced voltages [36] that are measured by a nanovolt- meter. The spin pumping data show a small oset be- tween positive and negative magnetic elds, which stems from Joule heating in the Pt layer by the microwaves. Peltier elements on the top and bottom (separated by an AlN layer) generated a thermal gradient for the spin Seebeck experiments that were reversed for cross checks, as shown in the right graph. The third row in Fig. 2 shows the results obtained at the IMR. Here, the sample is placed on a coplanar waveguide such that at fres= 3:8 GHz FMR condition is fullled for 0Hext=70 mT. The thermal gradient for the spin Seebeck measurements was generated by an electrically isolated separate heater on top of the Pt. The fourth row in Fig. 2 shows the RUG results. A coplanar waveguide on top of the YIG was used to excite the FMR at a magnetic eld of 0Hext=6 mT. The spin Seebeck eect was detected using an ac-variant of the current heating scheme. The spin Seebeck voltage can thereby be detected as described above in the second harmonic of the ac voltage signal. In spite of dierences in samples and measurement techniques, all experiments agree on the sign for spin pumping and spin Seebeck eect. We all measure neg- ative spin pumping and positive spin Seebeck voltages for positive applied magnetic elds that all change sign when the magnetic eld is reversed, consistent with the theoretical expectations [1, 2, 26, 27]. We can now address the absolute sign of the spin Hall angle. The results in Fig. 2 were obtained with measurement congurations equivalent to the one depicted in Figs. 1(c) and (d). With external magnetic eldHextpointing in the ^ zdirection ^ m=^ zand^JJJs=^yyy for FMR spin pumping. According to Eq. (10), when SH>0^EEEs=^ x, which leads to a negative (positive) charge accumulation at the x(+x) edge of the Pt lm and a negative spin pumping voltage is expected as well as observed. In the spin Seebeck experiments with Pt hotter than YIG, the spin current ows in the opposite direction ( ^JJJs=^ y), and the voltage is inverted. Therefore, the spin Hall angle of Pt is positive if dened as above. The nature of the spin Hall eect in Pt is likely to be governed by its electronic band4 -0.20 .0+ 0.2-3-2-10+ 1+ 2+ 3- 10 + 1-10+ 1V /VsatS SEΔ T > 0l ength: 4 mmw idth: 3 mmlength: 3 mmw idth: 1 mmG GG (500 µm)Y IG (4.1 µm)P t (10 nm) G archingspin pumpings pin Seebecks ample parametersG GG (500 µm)Y IG (160 nm)P t (7 nm)V/VmaxS P fres = 9.82 GHzµ 0Hres = 268 mT - 10 + 1-10+ 1 fres = 1 GHzµ 0Hres = 5.5 mT fres = 7 GHzµ 0Hres = 175 mT V/VmaxS P- 0.30 .0+ 0.3-3-2-10+ 1+ 2+ 3 ΔT < 0 V/VsatS SEK aiserslauternΔ T > 0- 10 + 1-10+ 1H ext/Hres V/VmaxS P- 10 + 1-3-2-10+ 1+ 2+ 3 Hext/Hreslength: 600 µmw idth: 30 µmGGG (500 µm)Y IG (200 nm)P t (6 nm)ΔT > 0G roningen V/VsatS SE -10 + 1-10+ 1 fres = 3.8 GHzµ 0Hres = 70 mT V/VmaxS P- 0.50 .0+ 0.5-3-2-10+ 1+ 2+ 3Δ T > 0 V/VsatS SE G GG (500 µm)Y IG (4 µm)P t (10 nm)length: 6 mmw idth: 2 mmS endai Figure 2. Measured voltage signals for the FMR spin pumping (left column) and spin Seebeck (middle column) experiments obtained by the contributing groups. The voltage signals have been normalized to a maximum modulus of unity while the applied magnetic elds are in units of the FMR resonance eld Hresgiven in the insets. The temperature dierence
T=TPtTYIG is positive. The third column lists sample layer thicknesses and dimensions. The sign of the observed voltages is consistent between the individual groups. structure [38], but it should be a helpful to know that the sign is identical to that caused by negatively charged impurities. In summary, we present spin pumping and spin See- beck experiments for various samples and experimental conditions leading to gratifying agreement of the results obtained by dierent groups. By carefully accounting for the signs of all experimental parameters and denitions we were able to determine both the relative and the absolute signs of both eects, linking the positive spin Hall angle of Pt to a simple physical model of negative scattering centers. The relative signs of spin pumping and spin Seebeck eect are consistent with theoretical predictions [14, 27{29]. The techniques and samplesused in this letter are representative for a large number of spin pumping and spin Seebeck experiments and should serve as a reference for other materials or sample geometries. We thank M. Wagner, M. Althammer and M. Opel for sample preparation and gratefully acknowledge nancial support by the DFG via SPP 1538 \Spin Caloric Trans- port" (projects GO 944/4-1, SE 1771/4-1, BA 2954/1-1) and CH 1037/1-1, NanoLab NL and the Foundation for Fundamental Research on Matter (FOM), JSPS Grants- in-Aid for Scientic Research, EU-RTN Spinicur, EU- FET InSpin 612759, PRESTO-JST \Phase Interfaces for Highly Ecient Energy Utilization", and CREST-JST \Creation of Nanosystems with Novel Functions through Process Integration". [1] M. I. D'yakonov and V. I. Perel', JETP Lett. 13, 467 (1971).[2] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).5 [3] S. Zhang, Phys. Rev. Lett. 85, 393 (2000). [4] S. Murakami, in Advances in Solid State Physics , Ad- vances in Solid State Physics, Vol. 45, edited by B. Kramer (Springer Berlin Heidelberg, 2006) pp. 197{ 209. [5] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). [6] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). [7] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). [8] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). [9] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [10] J. Wunderlich, B.-G. Park, A. C. Irvine, L. P. Zrbo, E. Rozkotov, P. Nemec, V. Novk, J. Sinova, and T. Jung- wirth, Science 330, 1801 (2010). [11] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Homann, Phys. Rev. B 82, 214403 (2010). [12] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [13] C.-F. Pai, L. Liu, Y. Li, H. Tseng, D. Ralph, and R. Buhrman, Appl. Phys. Lett. 101, 122404 (2012). [14] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). [15] K. Ando, Y. Kajiwara, K. Sasage, K. Uchida, and E. Saitoh, IEEE Trans. Magn. 46, 3694 (2010). [16] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jung eisch, E. Saitoh, and B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011). [17] F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011). [18] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van Wees, Appl. Phys. Lett. 101, 132414 (2012). [19] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [20] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [21] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom,J. P. Heremans, and R. C. Myers, Nat Mater 9, 898 (2010). [22] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). [23] M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 103, 242404 (2013). [24] B. Jonker, A. Hanbicki, D. Pierce, and M. Stiles, J. Magn. Magn. Mater. 277, 24 (2004). [25] H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. 76, 036501 (2013). [26] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [27] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). [28] H. Adachi, J.-i. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011). [29] S. Homan, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [30] J. Jackson, Classical electrodynamics (Wiley, 1975). [31] L. I. Schi, Quantum Mechanics (McGraw-Hill, 1949). [32] H.-A. Engel, E. I. Rashba, and B. I. Halperin, \The- ory of spin hall eects in semiconductors," in Handbook of Magnetism and Advanced Magnetic Materials (John Wiley & Sons, Ltd, 2007). [33] S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wil- helm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012). [34] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). [35] M. B. Jung eisch, T. An, K. Ando, Y. Kajiwara, K. Uchida, V. I. Vasyuchka, A. V. Chumak, A. A. Serga, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 102, 062417 (2013). [36] M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, Appl. Phys. Lett. 103, 022411 (2013). [37] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Appl. Phys. Lett. 103, 092404 (2013). [38] G. Y. Guo, S. Murakami, T.-W. Chen, and N. Nagaosa, Phys. Rev. Lett. 100, 096401 (2008).

2014-04-14

We carried out a concerted effort to determine the absolute sign of the inverse spin Hall effect voltage generated by spin currents injected into a normal metal. We focus on yttrium iron garnet (YIG)|platinum bilayers at room temperature, generating spin currents by microwaves and temperature gradients. We find consistent results for different samples and measurement setups that agree with theory. We suggest a right-hand-rule to define a positive spin Hall angle corresponding to with the voltage expected for the simple case of scattering of free electrons from repulsive Coulomb charges.

Sign of inverse spin Hall voltages generated by ferromagnetic resonance and temperature gradients in yttrium iron garnet|platinum bilayers

1404.3490v2

Remote magnon entanglement between two massive ferrimagnetic spheres via cavity optomagnonics Wei-Jiang Wu,1Yi-Pu Wang,1Jin-Ze Wu,1Jie Li,1,and J. Q. You1 1Interdisciplinary Center of Quantum Information, Zhejiang Province Key Laboratory of Quantum Technology and Device, and State Key Laboratory of Modern Optical Instrumentation, Department of Physics, Zhejiang University, Hangzhou 310027, China (Dated: August 18, 2021) Recent studies show that hybrid quantum systems based on magnonics provide a new and promising platform for generating macroscopic quantum states involving a large number of spins. Here we show how to entan- gle two magnon modes in two massive yttrium-iron-garnet (YIG) spheres using cavity optomagnonics, where magnons couple to high-quality optical whispering gallery modes supported by the YIG sphere. The spheres can be as large as 1 mm in diameter and each sphere contains more than 1018spins. The proposal is based on the asymmetry of the Stokes and anti-Stokes sidebands generated by the magnon-induced Brillouin light scattering in cavity optomagnonics. This allows one to utilize the Stokes and anti-Stokes scattering process, re- spectively, for generating and verifying the entanglement. Our work indicates that cavity optomagnonics could be a promising system for preparing macroscopic quantum states. I. INTRODUCTION During the past decade, cavity magnonics [1–6] has been emerged and developed as a new and active platform for the study of strong interactions between light and matter [7]. It consists of microwave photons which reside in a resonant cav- ity and interact with magnons (i.e., collective spin excitations) in a ferrimagnetic material, e.g., yttrium iron garnet (YIG). The system exhibits its unique features and advantages, which lie in the large frequency tunability and low damping rate of the magnon mode, as well as its excellent ability to coher- ently interact with other systems, including microwave [1– 6] or optical photons [8–12], phonons [13–15], and super- conducting qubits [16–18]. These hybrid cavity magnonic systems promise potential applications in quantum informa- tion processing and quantum sensing [19]. A variety of in- teresting phenomena have been explored in cavity magnon- ics, including magnon gradient memory [20], exceptional points [21, 22], manipulation of distant spin currents [23], bi-[24] and multi-stability [25], level attraction [26–29], non- reciprocity [30], anti-PT symmetry [31, 32], among others. In addition, it has been suggested that cavity-magnon polari- tons could be used as an ultra-sensitive magnetometer [33– 35], and for searching dark matter axions [36] and detecting high-frequency gravitational waves using the gravitomagnetic eect [37]. In this article, we study an emerging field of cavity opto- magnonics [8–10, 12], where a YIG sphere simultaneously supports optical whispering gallery modes (WGMs) and a magnetostatic mode of magnons. The WGM photons are scat- tered by the GHz magnons in the form of generating sideband photons with the frequency shifted by the magnon frequency, and the high-quality WGM cavity drastically enhances this magnon-induced Brillouin light scattering (BLS). The nature of the spin-orbit coupling of the WGM photons, combined jieli6677@hotmail.comwith the geometrical birefringence of the WGM resonator, leads to a pronounced nonreciprocity and asymmetry in the Stokes and anti-Stokes sidebands generated by the magnon- induced BLS [8–10, 12]. This is the result of the selection rule [38–41] imposed by the angular momentum conservation. Because of this, this kind of BLS requires a change in optical polarization, distinctly di erent from the light scattering in cavity optomechanics [42]. The asymmetry nature of the BLS allows us to select, on demand, the Stokes or anti-Stokes scat- tering event to occur, corresponding to the process of creat- ing or annihilating magnons. Such a mechanism can be used for the manipulation of magnons, and has been adopted for preparing nonclassical states of magnons. This includes the proposals for cooling the magnons [43], preparing magnon Fock states [44], a magnon laser [45], an optomagnonic Bell test [46], and an opto-microwave entanglement mediated by magnons [47], etc. Based on this novel system and its unique properties, we provide a scheme to entangle two magnon modes in two mas- sive YIG spheres, which can be separated remotely, mani- festing the nonlocal nature of the macroscopic entanglement. Specifically, a weak laser pulse with a certain polarization is sent into an optical interferometer formed by two 50 /50 beam splitters (BSs). Each arm of the interferometer contains one cavity optomagnonic device, i.e., a YIG sphere supporting op- tical WGMs and a magnon mode [8–10, 12], and the magnon mode is cooled to its ground state using a dilution refrigerator. The pulse is detuned to be resonant with the WGM cavities in the two arms and to activate the Stokes scattering event, which yields a single magnon residing in one of the YIG spheres and a lower-frequency photon with a changed polarization in the same arm. Since the BSs are 50 /50 and the two devices are assumed identical, the probabilities of the Stokes scatter- ing event in each arm are thus equal. A single-photon detec- tion in the output of the interferometer then projects the two magnon modes onto a path-entangled state, in which the two YIG spheres share a single magnon excitation. The scheme can be regarded as the DLCZ protocol [48] ap- plied to the system of cavity optomagnonics. The DLCZ pro-arXiv:2103.10595v2 [quant-ph] 17 Aug 20212 tocol for a cavity optomechanical system has been realized using GHz mechanical resonators [49]. We would like to note that the entanglement between two magnon modes of massive ferrimagnets has been extensively studied in cavity magnon- ics [51–57]. Such entangled states involving a large number of spins are genuinely macroscopic quantum states, and are thus useful for the study of the quantum-to-classical transi- tion and the test of unconventional decoherence theories [50]. In most studies, the magnon entanglement essentially orig- inates from the nonlinearity of the system, which can be achieved from, e.g., the magnetostrictive interaction [51], the magnon Kerr e ect [52], or the coupling to a superconduct- ing qubit [57]. Alternatively, entanglement can be obtained by feeding a squeezed vacuum microwave field into the cav- ity [55, 56]. However, by now proposals for entangling two magnon modes by means of cavity optomagnonics are still missing. The magnon entanglement achieved in the present work utilizes the nonlinearity of the magnon-induced BLS, which is a unique property of the cavity optomagnonic sys- tem. II. BASIC INTERACTIONS IN OPTOMAGNONICS We start with the description of two basic interactions in cavity optomagnonics, which are key elements for realizing our protocol. They are the optomagnonic two-mode squeezing and beamsplitter interactions, which are used, respectively, to prepare and verify the magnon entanglement of two YIG spheres. The magnon-induced BLS in a cavity optomagnonic sys- tem [8–10, 12] is intrinsically a three-wave process, which can be described by the Hamiltonian H=H0+Hint; (1) where H0is the free Hamiltonian of two WGMs and a magnon mode H0=~=!1ay 1a1+!2ay 2a2+!mmym; (2) with ajandm(ay jandmy,j=1;2) being the annihilation (creation) operators of the WGMs and magnon mode, respec- tively, and!i(i=1;2;m) being their resonance frequencies, which satisfy the relation !m!jandj!1!2j=!m, imposed by the conservation of energy in the BLS. The inter- action Hamiltonian Hintof the three modes is given by Hint=~=G0ay 1a2my+a1ay 2m
; (3) where G0is the single-photon coupling rate. This coupling is weak owing to the large frequency di erence between the WGM and the magnon mode, but it can be significantly en- hanced by intensely driving one of the WGMs. To maximize the BLS scattering probability, we resonantly pump the WGM a1(a2) to activate the anti-Stokes (Stokes) scattering, which is responsible for the optomagnonic state-swap (two-mode squeezing) interaction. Note that the selection rule [38–41] causes di erent polarizations of the two WGMs. Without lossof generality, we assume a2(a1) mode to be the transverse- magnetic (TM) (transverse-electric (TE)) mode of a certain WGM orbit, and !2(TM)>! 1(TE) due to the geometrical bire- fringence of the WGM resonator [8]. We now consider that the WGM a2is resonantly pumped by a strong optical field. In this case, the strongly driven mode a2 can be treated classically as a number 2ha2i=pN2(2be- ing real for a resonant drive), with N2the intra-cavity photon number, which is determined by the pump power and the de- cay rate of the WGM. The linearized interaction Hamiltonian can then be obtained HSt: int=~G2ay 1my+a1m
; (4) where G2=G02is the e ective coupling rate. This Hamil- tonian is responsible for the two-mode squeezing interaction between the WGM a1and magnon mode m, and can be used to prepare optomagnonic entangled states. This corresponds to the Stokes scattering process, where a TM polarized photon converts into a lower-frequency TE polarized photon by creat- ing a magnon excitation. Under this Hamiltonian, the WGM a1and magnon mode mare prepared in a two-mode squeezed state (unnormalized) j ioptomag =j00ia1;m+p Pj11ia1;m+Pj22ia1;m+O(P3=2);(5) where Pis the probability for a single Stokes scattering event to occur, andO(P3=2) denotes the terms with more excitations whose probabilities are equal to or smaller than P3. The scat- tering probability Pincreases with the strength of the driv- ing field. For a su ciently weak driving field, P1 can be achieved, e.g., in analogous cavity optomechanical experi- ments, P'0:7% [49] and P'3% [58] were achieved using very weak laser pulses. In this case, the probability of creating two-magnon /photon statej2im=aand higher excitation states is negligibly small. Such a low Stokes scattering probability of generating an entangled pair of single excitations, accom- panied with very weak laser pulses, is vital for realizing the DLCZ(-like) protocols [48, 49]. This is exactly what we shall utilize and apply to optomagnonics, in Sec. III, to generate an entangled pair of a single magnon and a TE polarized photon. Similarly, when mode a1is resonantly pumped by a strong field, one obtains the following linearized interaction Hamil- tonian HA:St: int =~G1a2my+ay 2m
; (6) where G1=G01and1=pN1, with N1the intra-cavity pho- ton number of WGM a1. This Hamiltonian leads to the state- swap interaction between the WGM a2and magnon mode m, and can be used to read out the magnon state by measuring the created anti-Stokes field a2. This anti-Stokes scattering corre- sponds to the process where a TE polarized photon converts into a higher-frequency TM polarized photon by annihilating a magnon. As will be shown in Sec. IV, we shall use this anti-Stokes process to verify the magnon entanglement.3 FIG. 1: (a) Sketch of the system used for generating the magnon entanglement between two YIG spheres. It consists of an interfer- ometer formed by two 50 /50 BSs and in each arm of the interfer- ometer there is a cavity optomagnonic device, in which the magnon- induced Brillouin light scattering occurs. A weak laser pulse with a certain polarization is sent into the interferometer and a subsequent single-photon detection in the output of the interferometer projects the two magnon modes onto an entangled state. (b) Mode frequen- cies of the Stokes light scattering by magnons. A +-polarized pho- ton of frequency !+is converted into a -polarized Stokes photon of frequency !by creating a magnon of frequency !m. The in- put pulse couples to the +-polarized WGM and the generated - polarized photon goes into the detector, and meanwhile, the magnon mode gains a single excitation. III. THE PROTOCOL We now proceed to describe our protocol. The schematic diagram of the protocol is depicted in Fig. 1. Two cavity op- tomagnonic devices are placed in two arms of an optical in- terferometer formed by two 50 /50 BSs. In each device, a YIG sphere supports a magnon mode and high- Qoptical WGMs. In Refs. [8, 10, 12], a roughly 1-mm-diameter sphere was used. The YIG sphere is placed in a bias magnetic field along thezdirection, while the WGMs propagate along the perime- ter of the sphere in the x-yplane [8–10, 12]. The frequency of the magnon mode can be adjusted by varying the strength of the bias magnetic field. The two devices and two optical paths are assumed identical to erase the ‘which-device /path’ infor- mation of the scattered photons at the output of the interferom- eter (though some mismatches can be compensated via optical operations [49]), which is required by the DLCZ protocol. As a particular advantage of the system, the two magnon-mode frequencies can be tuned to be equal by altering the external bias fields. This is technically more di cult for optomechan- ical systems as one has to fabricate a large number of samples and find a pair of nearly identical mechanical resonators [49]. At the end of the interferometer, two single-photon detectors are placed in the outputs of the 50 /50 BS, and in each output a polarizer is placed in front of the detector, to select the photon with a certain polarization to be detected.We now describe the scheme first by neglecting any optical losses, including the propagation loss and the detection loss due to the nonunity detection e ciency, as well as the loss as- sociated with the magnon modes. Also, we further assume that the two magnon modes are prepared in their quantum ground state. We then analyse the e ects of various experi- mental imperfections, and finally provide a strategy for veri- fying the entanglement. A laser pulse is sent into one input port of the first BS of the interferometer. The energy of the pulse is so weak that the mean photon number is much smaller than 1 [48, 49]. It is thus in a weak coherent state ji(jj1), with the probability of all the n-photon components jni(n>1) being negligible, i.e., ji'j0i+ppj1i, where p=jj21 is the probability of the pulse being in the single-photon state. Since the BS is 50 /50, the single photon goes, with equal probability, into one of the interferometer arms (the two arms are termed as path A and B, see Fig. 1(a)), thus having the state after the BS jiopt'j00iAB+pp p 2j01iAB+j10iAB
: (7) The pulse is sent through a fiber polarization controller and coupled to the WGM resonator via a tapered silica optical nanofiber [8], or a prism coupler [10, 12]. The polarization and the frequency of the pulse is tuned to couple to a certain TM WGM, e.g., the +-polarized TM mode. Therefore, the state of the system, soon after the TM mode is excited, is j0iopt'j00i+ A+ B+pp p 2j01i+ A+ B+j10i+ A+ B
; (8) where the subscript + j(j=A, B) denotes the +-polarized TM mode of the WGM resonator in path j. The+-polarized photon is then scattered by creating a magnon, and generates a -polarized Stokes photon at frequency !=!+!minto the TE WGM, see Fig. 1(b). This is just what we introduced, in Sec. II, the low Stokes scattering probability of generating an entangled pair of a single magnon and a TE polarized photon. This magnon-induced Stokes BLS leads to the following state jioptomag'j0000imAmBAB +pp p 2j0101imAmBAB+j1010imAmBAB
;(9) wherej0101imAmBABdenotes the coexistence of a magnon re- siding in the YIG sphere and a -polarized Stokes photon in path B, and similarly for j1010imAmBAB. Note that this BLS occurs only between the TM and TE modes with the same WGM index, owing to the angular momentum conservation of photons. Because of the geometrical birefringence, which imposes a restriction on the frequencies of the TM and TE modes (of the same mode index), i.e., !TM>! TE, the Stokes scattering is preferred in the BLS, while the anti-Stokes scat- tering is prohibited, in which a -polarized photon is con- verted into an anti-Stokes photon at frequency !=!+!m by annihilating a magnon [8]. As discussed later, in the part of entanglement verification, we shall send a pulse that is cou- pled to a TE WGM to activate the anti-Stokes scattering. Such4 an asymmetry nature of the BLS by the magnons is the cor- nerstone of realizing our scheme, which o ers the possibility of separately implementing the entangling operation and the readout operation of the scheme. The generated -polarized Stokes photon, with equal prob- ability in path A or B, then couples to the nanofiber (or the prism coupler) and enters the second 50 /50 BS. The polariz- ers in the outputs of the BS select the TE Stokes photon over the TM photons that failed to e ectively activate the Stokes scattering (in practice, the experiment will be repeated many times due to the low scattering probability in the current opto- magnonic weak coupling regime [8–10, 12]). A single-photon detection in the output of the BS, which realizes the mea- surement M=j01iABj10iAB
y, then projects the two magnon modes onto the state jimag=1p 2j01imAmBj10imAmB
: (10) This is a path-entangled state of the two magnon modes in paths A and B, and ‘ ’ correspond to the detection of the TE photon in di erent outputs of the BS. In deriving the entangled state (10), we have neglected the optical losses (e.g., the propagation loss and the detection loss), and the magnon loss. Also, we have assumed that the magnon modes are initialized to their quantum ground state j00imAmBby eliminating the residual thermal excitations. We now analyse these e ects one after another. For both the optical propagation loss and the detection loss due to the nonunity detection e ciency, their e ect only re- duces the probability of obtaining the desired state (10) but does not destroy the state [59], which implies longer mea- surement time. This is only true for a su ciently weak pulse (being in a weak coherent state jiwithjj1) that produces at most a single Stokes photon in the output of the interferom- eter. Optical losses thus disable the single-photon detection. This experiment will be disregarded such that there is no ac- tual impact. However, the two-photon component in jican indeed result in unwanted additional states, such as j02imAmB, j20imAmB,j11imAmB, in the final magnon state (10). This is also true for the case without su ering any optical loss as the de- tectors are assumed to be photon-number non-resolving in our scheme. Nevertheless, the probability of the two-photon state is much smaller than that of the single-photon state for a weak coherent state with jj1. As long as this is satisfied, those additional states are negligible. For the dissipation of the magnon modes, since the timescale at which our scheme is realized (laser pulses with duration of tens of nanoseconds were used in Refs. [49, 58]) is much shorter than the magnon lifetime (typically of a mi- crosecond [8–10, 12]), during a complete run of the exper- iment the magnon modes can be assumed to have negligi- ble dissipation. However, the magnon modes cannot be per- fectly initialized to their ground state j00imAmBat typical cryo- genic temperatures. We now study the impact of the residual magnon thermal excitations. Since the frequencies of the two magnon modes are tuned to be equal, the two magnon modesare in the same thermal state under the same temperature th mag=(1S)1X n=0Snjnihnj; (11) where S=¯n=(¯n+1), with ¯ n=exp(~!m=kBT)11being the equilibrium mean thermal magnon number at the temper- ature T. In general, quantum states of macroscopic objects re- quire very low environmental temperatures. For the magnon mode with frequency of about 7 GHz [8–10, 12], the ther- mal occupation ¯ n'0:036 at T=100 mK. For ¯ n=0:036, S'0:035 and S2'0:001, therefore, high-excitation terms jniwith n>1 can be safely neglected, and we can then ap- proximate it as th mag'(1S)j0ih0j+Sj1ih1j
. The two magnon modes are thus in a mixed state of a probabilistic mix- ture of four pure states ji jimAmB(i(j)=0;1). The ratio of the probabilities is 1 : S:S:S2for the two magnon modes being initially inj00imAmB,j01imAmB,j10imAmB, andj11imAmB, respec- tively. The ground state j00imAmBis what we have assumed for obtaining the desired state jimagin (10). Combining the other initial states, we obtain the final magnon state (unnormalized), conditioned on the single-photon detection, which is final mag'j00ih00j+Sj01ih01j+j10ih10j
+S2j11ih11j; (12) wherej00ijimag, and j01i=1p 2j02imAmBj11imAmB
; j10i=1p 2j11imAmBj20imAmB
; j11i=1p 2j12imAmBj21imAmB
;(13) corresponding to the magnon modes being initially in j01imAmB,j10imAmB, andj11imAmB, respectively. The states in (13) reduce the fidelity of the desired state (10), as F= h00jfinal magj00i ' 1=(1+2S+S2). Nevertheless, these addi- tional states can be well suppressed if S1. This is well fulfilled at the temperature of 100 mK (50 mK), since S' 0:035 (0:001)1. Under this temperature, the fidelity of the state (10) isF ' 0:93 (0.998). Therefore, the impact of the residual thermal excitations under a temperature below 100 mK will be negligibly small. IV . VERIFICATION OF THE ENTANGLEMENT Lastly, we show how to verify the generated magnon en- tanglement. The magnon state can be read out by using the anti-Stokes process of the BLS. Specifically, as depicted in Fig. 2, a weak read pulse is sent into the interferometer with its polarization and frequency tuned to couple to a TE WGM to activate the anti-Stokes scattering, where a -polarized photon is converted into a +-polarized anti-Stokes photon by annihilating a magnon, satisfying the frequency relation !+=!+!m[8]. This can be regarded as the inverse pro- cess of the Stokes scattering used for preparing the entangle- ment. The generated +-polarized anti-Stokes photon then5 FIG. 2: (a) Sketch of the system for verifying the magnon entangle- ment. A weak read pulse is sent into the interferometer soon after the entangling pulse and couples to a -polarized WGM to activate the anti-Stokes scattering. The created +-polarized anti-Stokes photon, containing the magnon state information, enters one of the detec- tors after passing through a polarization rotator (PR) and a polarizer (P). An electro-optic modulator (EOM) is added in one arm to re- alize the phase o set
. (b) Mode frequencies of the anti-Stokes light scattering by magnons. A -polarized photon is converted into a+-polarized anti-Stokes photon by annihilating a magnon that was produced in the preceding entangling stage. couples to the nanofiber /prism coupler and enters the second BS. In this circumstance, the polarizers in the outputs of the BS select the TM anti-Stokes photon over the TE photons that disable the anti-Stokes scattering. In practice, the read pulse is sent after the entangling pulse with a time delay much shorter than the magnon lifetime. Within the delay, the polarizers must be quickly switched in order to select the TM photon for verifying the entanglement. Alternatively, one could keep the polarizer fixed and put a polarization rotator before the po- larizer, which quickly rotates the polarization of the photon (TM$TE). This can be realized via a high-speed waveguide electro-optic polarization modulator [60]. In view of the similarity of optomagnonics and optome- chanics [61], we adopt the following witness for the magnon entanglement [49, 64] Rm(
;j)=4g(2) A1;Sj(
)+g(2) A2;Sj(
)1  g(2) A1;Sj(
)g(2) A2;Sj(
)2; (14)where g(2) Ai;Sj=hAy iSy jAiSji=hAy iAiihSy jSji
is the second- order coherence between the TE Stokes photons (with Sjand Sy jthe annihilation and creation operators for the Stokes pho- tons going to detector j,j=1;2) and the TM anti-Stokes pho- tons (with AiandAy ithe annihilation and creation operators for the anti-Stokes photons going to detector i,i=1;2), and
is the phase o set added to the read pulse in one of the in- terferometer arms via an electro-optic modulator (EOM), see Fig. 2. The witness gives Rm(
;j)1 for all separable states of the two magnon modes for any
andj. Therefore, if there is any
andjwith which Rm(
;j)<1, the two magnon modes are then entangled. V . CONCLUSION We present a scheme for entangling two magnon modes of two massive ferrimagnetic spheres in an optical interferom- eter configuration, containing two optomagnonic devices, by using short optical pulses in a cryogenic environment. The scheme is based on the asymmetry of the Stokes and anti- Stokes sidebands in the magnon-induced Brillouin light scat- tering. The entanglement is generated on the condition of the detection of single photons with a certain polarization. We analyse the e ects of various experimental imperfections and provide a strategy for verifying the entanglement based on the second-order coherence between the Stokes and anti-Stokes photons with di erent polarizations. The magnon entangled state of two ferrimagnetic spheres, which contain more than 1018spins and can be distantly separated, is truly a macro- scopic quantum state and manifests its nonlocal nature. Our work, along with the earlier studies [43, 44, 46], show that cavity optomagnonics could become a promising platform for the study of macroscopic quantum phenomena. Acknowledgments This work has been supported by Zhejiang Province Pro- gram for Science and Technology (Grant No. 2020C01019), the National Natural Science Foundation of China (Grants Nos. U1801661, 11934010, 11774022 and 12004334), the Fundamental Research Funds for the Central Universities (No. 2021FZZX001-02), and China Postdoctoral Science Founda- tion. [1] H. Huebl et al., Phys. Rev. Lett. 111, 127003 (2013). [2] Y . Tabuchi et al., Phys. Rev. Lett. 113, 083603 (2014). [3] X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [4] M. Goryachev et al., Phys. Rev. Appl. 2, 054002 (2014). [5] L. Bai et al., Phys. Rev. Lett. 114, 227201 (2015). [6] D. Zhang et al. , npj Quantum Information 1, 15014 (2015). [7] P. Forn-Diaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Rev.Mod. Phys. 91, 025005 (2019). [8] A. Osada et al., Phys. Rev. Lett. 116, 223601 (2016). [9] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016). [10] J. A. Haigh et al., Phys. Rev. A 92, 063845 (2015); J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, Phys. Rev. Lett. 117, 133602 (2016). [11] R. Hisatomi et al., Phys. Rev. B 93, 174427 (2016).6 [12] A. Osada et al., Phys. Rev. Lett. 120, 133602 (2018). [13] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv. 2, e1501286 (2016). [14] J. Li, S.-Y . Zhu, and G. S. Agarwal, Phys. Rev. Lett. 121, 203601 (2018). [15] M. Yu, H. Shen, and J. Li, Phys. Rev. Lett. 124, 213604 (2020). [16] Y . Tabuchi et al., Science 349, 405 (2015). [17] D. Lachance-Quirion et al., Sci. Adv. 3, e1603150 (2017). [18] D. Lachance-Quirion et al., Science 367, 425 (2020). [19] D. Lachance-Quirion, Y . Tabuchi, A. Gloppe, K. Usami, and Y . Nakamura, Appl. Phys. Express 12, 070101 (2019). [20] X. Zhang et al., Nat. Commun. 6, 8914 (2015). [21] D. Zhang, X.-Q. Luo, Y .-P. Wang, T.-F. Li, and J. Q. You, Nat. Commun. 8, 1368 (2017). [22] X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Phys. Rev. Lett. 123, 237202 (2019). [23] L. Bai et al., Phys. Rev. Lett. 118, 217201 (2017). [24] Y .-P. Wang et al., Phys. Rev. Lett. 120, 057202 (2018). [25] R.-C. Shen, Y .-P. Wang, J. Li, G. S. Agarwal, and J. Q. You. To be submitted. [26] V . L. Grigoryan, K. Shen, and K. Xia, Phys. Rev. B 98, 024406 (2018). [27] M. Harder et al., Phys. Rev. Lett. 121, 137203 (2018). [28] B. Bhoi et al. , Phys. Rev. B 99, 134426 (2019). [29] W. Yu, J. Wang, H. Y . Yuan, and J. Xiao, Phys. Rev. Lett. 123, 227201 (2019). [30] Y .-P. Wang et al., Phys. Rev. Lett. 123, 127202 (2019). [31] Y . Yang et al. , Phys. Rev. Lett. 125, 147202 (2020). [32] J. Zhao et al. , Phys. Rev. Applied 13, 014053 (2020). [33] Y . Cao and P. Yan, Phys. Rev. B 99, 214415 (2019). [34] N. Crescini, C. Braggio, G. Carugno, A. Ortolan, and G. Ruoso, Appl. Phys. Lett. 117, 144001 (2020); N. Crescini, G. Carugno, and G. Ruoso, arXiv:2010.00093. [35] M. S. Ebrahimi, A. Motazedifard, and M. Bagheri Harouni, arXiv:2011.06081. [36] N. Crescini et al., Phys. Rev. Lett. 124, 171801 (2020). [37] M. L. Ruggiero and A. Ortolan, Phys. Rev. D 102, 101501(R) (2020). [38] S. Sharma, Y . M. Blanter, and G. E. W. Bauer, Phys. Rev. B 96, 094412 (2017). [39] P. A. Pantazopoulos, N. Stefanou, E. Almpanis, and N. Pa- panikolaou, Phys. Rev. B 96, 104425 (2017). [40] A. Osada, A. Gloppe, Y . Nakamura, and K. Usami, New J. Phys. 20, 103018 (2018).[41] J. A. Haigh et al., Phys. Rev. B 97, 214423 (2018). [42] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014). [43] S. Sharma, Y . M. Blanter, and G. E. W. Bauer, Phys. Rev. Lett. 121, 087205 (2018). [44] V . A. S. V . Bittencourt, V . Feulner, and S. V . Kusminskiy, Phys. Rev. A 100, 013810 (2019). [45] Z.-X. Liu and H. Xiong, Optics Letters 45, 5452 (2020). [46] H. Xie et al. , arXiv:2103.06429. [47] Q. Cai, J. Liao, and Q. Zhou, Ann. Phys. (Berlin) 532, 2000250 (2020). [48] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001). [49] R. Riedinger et al., Nature 556, 473 (2018). [50] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013). [51] J. Li and S.-Y . Zhu, New J. Phys. 21, 085001 (2019). [52] Z. Zhang, M. O. Scully, G. S. Agarwal, Phys. Rev. Research 1, 023021 (2019). [53] H. Y . Yuan, S. Zheng, Z. Ficek, Q. Y . He, and M.-H. Yung, Phys. Rev. B 101, 014419 (2020). [54] M. Elyasi, Y . M. Blanter, and G. E. W. Bauer, Phys. Rev. B 101, 054402 (2020). [55] J. M. P. Nair and G. S. Agarwal, Appl. Phys. Lett. 117, 084001 (2020). [56] M. Yu, S.-Y . Zhu, and J. Li, J. Phys. B 53065402 (2020). [57] D.-W. Luo, X.-F. Qian, and T. Yu, Optics Letters 46, 1073 (2021). [58] R. Riedinger et al., Nature 530, 313 (2016). [59] J. Li et al. , Phys. Rev. A 102, 032402 (2020). [60] R. C. Alferness and L. L. Buhl, Optics Letters 7, 500 (1982). [61] The optomagnonic interaction is formally equivalent to the op- tomechanical interaction (thus radiation pressure-like) [62, 63], under the assumption of the low-lying excitations, which allows treating the collective motion of the spins as a harmonic oscil- lator described by boson operators. [62] T. Liu, X. Zhang, H. X. Tang, and M. E. Flatt ´e, Phys. Rev. B 94, 060405(R) (2016). [63] S. V . Kusminskiy, H. X. Tang, and F. Marquardt, Phys. Rev. A 94, 033821 (2016). [64] K. Borkje, A. Nunnenkamp, and S. M. Girvin, Phys. Rev. Lett. 107, 123601 (2011).

2021-03-19

Recent studies show that hybrid quantum systems based on magnonics provide a new and promising platform for generating macroscopic quantum states involving a large number of spins. Here we show how to entangle two magnon modes in two massive yttrium-iron-garnet (YIG) spheres using cavity optomagnonics, where magnons couple to high-quality optical whispering gallery modes supported by the YIG sphere. The spheres can be as large as 1 mm in diameter and each sphere contains more than $10^{18}$ spins. The proposal is based on the asymmetry of the Stokes and anti-Stokes sidebands generated by the magnon-induced Brillouin light scattering in cavity optomagnonics. This allows one to utilize the Stokes and anti-Stokes scattering process, respectively, for generating and verifying the entanglement. Our work indicates that cavity optomagnonics could be a promising system for preparing macroscopic quantum states.

Remote magnon entanglement between two massive ferrimagnetic spheres via cavity optomagnonics

2103.10595v2

1 Current -induced in -plane magnetization switching in biaxial ferrimagnetic insulator Yongjian Zhou1,4, Chenyang Guo2,4, Caihua Wan2, Xianzhe Chen1, Xiaofeng Zhou1, Ruiqi Zhang1, Youdi Gu1, Ruyi Chen1, Huaqiang Wu3, Xiufeng Han2, Feng Pan1 and Cheng Song1* 1Key Laboratory of Advanced Materials (MOE), School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China 2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences , Chinese Academy of Sciences, Beijing 100190, China 3Institute of Microelectronics, Tsinghua U niversity, Beijing 100084, China 4These authors contributed equally: Y. Zhou, C. Guo. *songcheng@ mail. tsinghua.edu.cn 2 Ferrimagnetic insulators (FiMI) have been intensively used in microwave and magneto -optical devices as well as spin caloritronics, where their magnetization direction plays a fundamental role on the device perform ance. The magnetization is generally switched by applying external magnetic fields. Here we investigate current -induced spin -orbit torque (SOT ) switch ing of the magnetization in Y 3Fe5O12 (YIG)/Pt bilayers with in -plane magnetic anisotropy, where the switch ing is detected by spin Hall magnetoresistance. Reversible switching is found at room temperature for a threshold current density of 107 A cm–2. The YIG sublattices with antiparallel and unequal magnetic moments are aligned parallel/antiparallel to the direction of current pulses, which is consistent to the N éel order switching in antiferromagnetic system. It is proposed t hat such a switching behavior may be triggered by the antidamping -torque acting on the two antiparallel sublattices of FiMI . Our finding not only broadens the magnetizatio n switching by electrical means and promote s the understanding of magnetization switching, but also paves the way for all -electrically modulated microwave devices/ spin caloritronics with low power consumption. I. INTRODUCTION Yttrium iron garnet (Y 3Fe5O12, YIG), a ferrimagnetic insulator (FiMI) with ultralow Gilbert damping and high permeability , has long term been applied in the microwave and magneto -optical devices. It has drawn increasing interest in spintronics field, such as the investigations of the spin Seebe ck effect [1,2], spin Hall magnetoresistance (SMR) [3–5], spin pumping [4,6], non-local magnon transport [7], cavity magnon polariton [8] and magnon valves [9]. In particular, YIG is an ideal magnetic mat erial for pure spin current transport because charge current and corresponding Seebeck effect and Nernst effect can be completely eliminated [2], 3 providing a promising candidate for electronics with low energy dissipation. Note that the magnitude or even o n/off state of signals and resultant device performance are strongly modulated by the magnetization direction in YIG-based applications , such as in spin Seebeck effect [2] and magnon valve [9]. From the application of spintronics, YIG is always treated as a simple “ferromagnet” with a single net moment, though it is a typical FiMI with two sublattices, especially a magnetic primitive cell of YIG contains 20 Fe moments and a complicated spin structure [10]. In such case, the external magnetic fields are generally used to switch the magnetization direction of YIG. A remarkable miniaturization trend on electronics calls for the switching of FiMI with a more convenient and efficient method. The spin -orbit torque (SOT) in ferromagnet/heavy metal bilayers , where the angular momentum of spin current induced by charge current with spin Hall effect [11] is transferred into magnetic layer in the form of magnetic torque , provides an effective electrical means for manipulating magnetic dynamics and switching the uniform magnetization [12]. Previous studies concentrated on the SOT in metallic systems, including out -of-plane [13–16] and in -plane [17–19] magnetization switching, magnetic oscillations [20,21 ], domain wall motion [22,23 ] and skyrmions [24,25 ]. These concepts are transferred to the FiMI system, and there are several recent works reporting on the SOT switching in FiMI/heavy metal bilayers with perpendicular magnetic anisotropy [26–29], because of the lower energy dissipation and easy readout of th e switching signal, such as by anomalous Hall effect. However, almost all of the microwave and spintronics applications of YIG, as mentioned above, are based on YIG with in -plane magnetization, therefore the electrical switching of in -plane magnetized YIG is strongly pursued (TABLE I ). The experiments described here investigate the SOT switching of in -plane magnetized YIG (001) in YIG/Pt bilayer s, 4 where the two anti -parallel magnetic moments are set parallel/antiparallel to the direction of writing current. TABLE I . SOT switching in ferromagnet (FM), ferrimagnetic metal (FiM), ferrimagnetic insulator (FiMI), antiferromagnetic alloy (AFM) and antiferromagnetic insulator (AFMI) with out -of-plane and in -plane magnetic anisotropy. Out-of-plane and in -plane SOT s witching was extensively studied in FM and FiM. Out -of-plane switching of FiMI and in -plane switching of AFMI were realized recently. This work reports on the in -plane SOT switching of FiMI YIG. Magnetic anisotropy FM FiM FiMI AFM AFMI Out-of-plane Refs. [13,14 ] Refs. [15,16 ] Refs. [26–29] – – In-plane Refs. [17,18 ] Ref. [19] This work Refs. [30–34] Refs. [35–37] II. MRTHODS The YIG films with in -plane magnetic anisotropy were deposited on GGG(001) substrates using a sputtering system with a base vacuum of 1 × 10–6 Pa. After the deposition, high temperature annealing with oxygen atmosphere was carried out to further improve the crystalline quality and epitaxial relation between YIG film and GGG substrate [9]. The YIG thickness was determined using a pre -calibrated growth rate. The crystal structure was measured by x -ray diffraction (XRD). The in -plane magnetic anisotropy was recorded by vibrating sample magnetometer (VSM). The annealed YIG films were then tra nsferred into another high -vacuum magnetron sputtering chamber to ex-situ deposit 5 nm Pt top layer at room temperature. The YIG/Pt bilayers were patterned into eight -terminal devices with channel width of 5 μm through standard photolithography and Ar ion etching. The current 5 induced magnetization switching measurements were carried out at room temperature by applying current pulses of 1.4 × 107 A cm−2 with the width of 1 ms, then the transverse Hall resistance was recorded with a reading current of 1.2 × 1 06 A cm−2. And the spin Hall magnetoresistance experiments with different current densities and magnetic fields were conducted with physical properties measurement system (PPMS ). III. RESULTS AND DISSCUSION A series of YIG films ( t = 15, 20, 30, 60 nm) were grown on Gd 3Ga5O12 (GGG) (001) substrates by magnetron sputtering. In the following we focus primarily on data obtained from 20 -nm-thick YIG films at room temperature. X -ray diffraction spectra in Fig. 1(a) shows that an addition al peak from YIG (008) emerges in YIG/GGG sample, besides the diffraction peak from the GGG substrate, indicating that the YIG exhibit (001) -orientation, which serves as the basis of magnetization easy -plane (001). Figure 1 (b) presents hysteresis loops of YIG with the magnetic field ( H) applied along four in -plane directions of [100], [010], [110], and [ 1̅10], as well as out -of-plane direction of [001]. A comparison of the squared in -plane loops and slanted out-of-plane loop shows that [001] is a hard -axis. The saturation field is ~15 Oe when H is applied along [110] and [ 1̅10], in contrast to ~75 Oe along [10 0] and [010]. This observation reflects that the YIG films possess fourfold in -plane magnetic anisotropy with easy -axes along [110] and [ 1̅10] and hard -axes along [100] and [010], which results from the cubic anisotropy of bulk YIG [38]. The saturation magn etization ( MS) is around 115 emu cm–3, which is lower than the bulk value (140 emu cm–3) [39]. To perform current -induced in -plane magnetization switching measurements, the YIG were covered by 5 -nm-thick P t and then fabricated into eight -terminal devices with the channel width of 5 μm, where the writing current pulse channels are along 6 easy-axes of [ 1̅10] and [110] [Fig. 1(c)] . The in -plane switching measurements were carried out in the following way: five successive pulses (current density J = 1.4 × 107 A cm–2, 1-ms-width) were applied along [ 1̅10] (write 1), and then along its orthogonal direction [110] (write 2) at zero external magnetic field. After each writing current pulse, a small reading current ( J = 1.2 × 106 A cm–2) was applied and the transvers e Hall resistance variation (Δ Rxy) was recorded. Δ Rxy is intrinsically the spin Hall magnetoresistance of YIG/Pt system, where the spin polarization and relevant resistance in Pt can reflect the alignment of YIG moments [3]. Concomitant Δ Rxy for the magnet ization switching is displayed in Fig. 1(e), where the red and blue circles correspond to the red (write 1) and blue (write 2) arrows, respectively (consistent correspondence for the following results). The current pulse of 1.4 × 107 A cm–2 and 1-ms-width is the threshold current density for the switching [40]. The most eminent result is the two writing current pulses along [ 1̅10] and [110] lead to the variation of ΔRxy between low and high resistance states. Further inspection shows that Δ Rxy shows a sudden decrease (increase) and is almost saturated whe n the first write 1 (write 2) current pulse is applied , and the following four pulses cause a negligible variation in ΔRxy. This observation indicates step -like switching of in -plane magnetized YIG, which is most likely due to the small in -plane magnetic anisotropy of the present YIG films. Similar switching features were observed in antiferromagnetic α-Fe2O3 [42] and Mn 2Au for the switching from hard - to easy -axis [34], but different from the multidomain switching in NiO/Pt [35]. The situation differs dramatically when the current pulses are applied along in -plane hard -axes ([100] and [010]) [Fig. 1(d)] . The Δ Rxy remains constant when the current pulses of 1.4 × 107 A cm–2 are alternatively applied along [100] and [010] [Fig. 1(f)] , indicating the negligible in -plane switching between the hard -axes. A similar behavior also 7 occurs in YIG (111) films with in -plane isotropy [40]. It is then concluded that the magnetic anisotropy of YIG plays a fundamental role during in -plane switching process and the magnetic moments of in -plane biaxial YIG can be switched between the two easy -axes by a current pulse. FIG. 1. Crystal structure, magnetism and current -induced magnetization switching of 20 -nm-thick YIG. (a) X-ray diffraction spectra of GGG substrate and YIG/GGG sample. The peak from YIG (008) is marked. (b) Hysteresis loops at room temperature with magnetic field (H) applied along out -of-plane direction [001] and four in -plane directions, where [100] and [010] are in -plane hard -axes, while [110] and [ 1̅10] are in -plane easy -axes. The axe s are marked with the same colo r as their corresponding hysteresis loops. (c),(d) Measurement configurations of current -induced magnetization switching with writing current applied along in -plane easy- (c) and hard -axes (d), respectively. (e),(f) ΔRxy as a function of the number of writing current pulses applied as depicted in (c) and (d), respectively. The external magnetic field would provide an additional tool to modulate the 8 current -induced switching of YIG. Figure 2 (a) shows Δ Rxy as a function of current pulses with different additional fields ( H = 0, 5, 10, 50, and 75 Oe ). The measurement configuration is identical to the one used in Fig. 1(c), except the H is applied along hard-axis [100]. When H is fixed at a quite low value of 5 and 10 Oe, which is just below and above the coercivity of YIG [Fig. 2(b)] , there is no obv ious difference as compared with the data at H = 0. The scenario turns out to be different when H increases to 50 Oe. The amplitude of Δ Rxy variation is reduced but still evident as H is up to 50 Oe, which is close to the saturation field as shown in Fig. 2(b), suggesting that the current -induced in -plane magnetization switching is partially suppressed by H. Once H increases to the saturation field of 75 Oe, there is negligible change of Δ Rxy when the current pulses alter their directions between write 1 an d write 2, indicating that the current -induced magnetization switching is completely suppressed. This magnetic field modulated switching signals support that the present Δ Rxy variation is indeed ascribed to the SOT -induced in -plane switching of YIG ferrimagnetic sublattice magnetization , which is similar to the true Né el order switching in α-Fe2O3 with magnetic field [42]. It indicates that thermal artifacts [42–44], which are at least not sensitive to a low magnetic field , have a negligible effect in our experiments . Moreover, both the unambiguous current -induced switching signals and artifacts are found in YIG/Cu/Pt trilayer [40 ]. 9 FIG. 2. SOT-induced switching in YIG/Pt and Co/Pt. (a) Summary of SOT -induced ΔRxy as a function of writing current pulses with different additional fields H applied along [100] in YIG/Pt bilayers . Results under different H are separat ed by regions of different colo rs. (b) Hysteresis loop with H applied along [100]. The typical H employed i n SOT switching measur ements are denoted by dashed arrows. (c) ΔRxy as a function of the number of writing current pulses in Co/Pt bilayers with H = 0. The measurement configuration is identical to that of YIG/Pt sample. The polarity of Δ Rxy variation of Co/Pt is opposite to th at of YIG/Pt. (d) Schematic of current -induced magnetization switching in YIG/Pt, where M1 and M2 denote magnetic moments in two sublattices. M1 and M2 are switched toward the writing current direction. We then turn toward the current -induced switching mechanism of YIG/Pt. Note that only one ferrimagnetic insulator layer YIG is used, hence SOT rather than spin-transfer torque exists in our case. Remarkably, the switching polarity of Δ Rxy in Fig. 1(e), negative for w rite 1 and positive for write 2, reveals that the sublattice magnetizations in YIG are aligned parallel/antiparallel to the direction of writing 10 current according to SMR theory [41 ]. This feature is also supported by the longitudinal resistance variation [40]. These experimental observations unravel that the current -induced in-plane sublattice magnetizations switching of bi-axial ferrimagnet YIG/Pt is in analogy to the Né el order switching toward the writing current direction in antiferromagnet/Pt systems [35,37 ], where the antidamping -torque dominates. It is also reasonable to propose that the antidamping -torque may induce the ferrimagnetic moments switching of bi-axial YIG, where two sublattices with antiparallel magnetic moments is strongly antiferromag netic coupled , though uncompensated. Note that the polarity of current -induced switching in ferrimagnetic amorphous metal CoGd [19], which possesses negligible magnetic anisotropy, is opposite to our results in bi -axial YIG. This indicates that the magneti c anisotropy in ferrimagnet has a vital effect on current -induced in -plane switching, which is supported by the absence of SOT -induced switching in our YIG(111) samples [40]. To further study the SOT switching in biaxial ferrimagnetic YIG where the magnetization is aligned parallel/antiparallel to the current axis, we performed contro l experiment with ferromagnet and compare d the current -induced switching polarity of ΔRxy using Co (2 nm)/Pt (5 nm) bilayers where the magnetization shoul d be aligned perpendicular to the current axis [17,18,45 ]. Although Co is a conductor, which is not perfect as a comparison of YIG, Co/Pt bilayer possesses considerable SMR signal [46], from which the direction of magnetization can be readout easily. Therefore, Co/Pt bilayer us ed as a control sample is reasonable . The measurement configuration is identical to that of YIG/Pt [Fig. 1(c)] . The variation of Δ Rxy in Fig. 2(c) is opposite to the YIG/Pt case. Once an external field of 10 kOe is applied on the devices, the switching signals vanishes [40], indicating the variation of Δ Rxy is indeed due to the switching of Co moments. In the case of current -induced in -plane switching of 11 ferromagnet, the magnetic moment should be switched to the direction of spin polarization [17,18 ,45], which is perpendicular to the writing current direction. This process can be understood by the transfer of spin angular moment from the spin-polarized current to the magnetization, similar to the spin -transfer -torque scenario. This control experiment further confirms that the sublattice magnetizations of YIG are aligned parallel/antiparallel to the writing current direction, rather than being aligned along spin polarization direction. This may indicate the importance of antiferromagnetic coupled sublattices during current -induced in -plane switching in bi-axial ferrimagnet. The charge current in the Pt layer produces spin current and the resultant spin accumulation by spin Hall effect [11], exerting a t orque on the two anti-parallel magnetic sublattices ( M1 and M2) and then resulting in their switching parallel or antiparallel to the writing current direction [Fig. 2(d)] . The intriguing in-plane switching feature in our case discloses that as a typical FiMI, the anti-paralleled magnetic sublattice in YIG may play an important role , which is similar to the N éel order switching in antiferromagnets to some extent at least from the current -induced in -plane magnetization switching viewpoint. More experiments or simulation s in different FiMI are needed for further understanding of current -induced in-plane switching in FiMI system s. Based on those results and analyse s, the magnetic field manipulated true current -induced switching and co -existence of artifacts [40] make in -plane bi -axial ferrimagnet YIG a model for investigating the current -induced switching, which could promote the application of ferrimangetic spintronics. In addition to the SOT -induced magnetization switching, we have explored SOT-induced magnetic moments tilting toward the current direction by SMR measurements with different J and H. For these experiments , Hall resistance ( Rxy) was recorded when the current was applied along one of the easy -axes [110] and the 12 magnetic field was in -plane rotated from the current ( I) direction (angle αH = 0 for H // I), as depicted in Fig. 3(a). Figure 3 (b) shows the angle αH dependent Rxy with different current densities ( J = 1.2, 2, 6, 8, and 10 × 106 A cm–2) and H = 50 Oe. Remarkably, as the current density increases gradually from 1.2 to 10 × 106 A cm–2, within 0 –90° scale the high Hall resistance state (peak) shifts to a high rotation angle. On the basis of the SOT -switching results described above, magnetization is tilted toward the current direction, therefore M1 (αM) and M2 deviate from the field direction and toward the current direction ( αM = 0) [Fig. 3(c)] . Thus it is necessary to use a magnetic field with a larger rotation angle αH to compensate the tilting tendency induced by SOT, which results in the shift of SMR curves. Also visible is that in the range of 90–180° the low Hall resistance state (valley) shifts to a low rotation angle, because the SOT induces magnetization switching toward the current direction ( αM = 180° ) and then a magnetic field with a smaller rotation angle αH is employed. The variation tendencies in 180 –270° (peak) and 270–360° (valle y) scales are similar to these two scenarios, respectively. In general, the SMR curve exhibits a high and low Hall resistance states for αH = 45º /225º and 135º /315º , respectively [41]. Note that the slight deviation of the observed peak and v alley with low current density [Fig. 3(b)] from these theoretical angles is due to a low field of 50 Oe used, which is below the saturation field of the YIG film. Such a deviation vanishes when a high magnetic field is used, such as H = 5000 Oe [40]. The a ngle shift of the SMR curves as a function of J with different external fields (H = 50, 100, 1000, and 5000 Oe) is summarized in Fig. 3(d), where Δ αH is the angle difference between the valley and peak (in 0 –180° ) of Rxy. There are two striking features in the figure: (i) The angle difference Δ αH is reduced (angle shift is enhanced) with increasing J; (ii) The modulation of Δ αH is suppressed with increasing magnetic 13 field, which almost vanishes with a high H, e.g., 5000 Oe [40]. It is easy to understand that a larger current induces stronger magnetization titling toward the current direction, resulting in a larger “compensated angle” for the field, which reduces ΔαH. With increasing magnetic field, the SOT -induced magnetization tilting is negligible because the magnetization is always retained along the field. The magnetization switching phenomena deduced from these SMR results with different J and H are consistent with switching responses in Fig. 2. Thereby the current -dependent SMR measurements support SOT -induced in -plane sublattice magnetization s switching toward parallel/antiparallel to the writing current direction. In addition, we quantify the SOT -field equivalence through the summary of Δ αH resulting from SOT induced SMR tilting. An extra angle αSOT,J produced by SOT with current density J is introduced in transverse SMR equation: 𝑅xy=∆𝑅sin2(𝛼H−𝛼SOT ,J), 0°<𝛼<90° (1) 𝑅xy=∆𝑅sin2(𝛼H+𝛼SOT ,J), 90° <𝛼<180° (2) where Rxy and Δ R are transverse resistance and amplitude of transverse SMR, respectively. When αH = 45º +αSOT,J (135º –αSOT,J), Rxy is high (low) resistance state, namely peak (valley) of SMR in 0 –180°. Therefore, Δ αH in Fig. 3(d) equals 90º – 2αSOT,J. The variation of Δ αH with different J under H = 50 Oe are fitted by linear functions [dashed line in Fig. 3(d)], and the slope k is arou nd –4.1 deg/(106 A cm–2). Based on the equation of slope k: 𝑘=(90°−2𝛼SOT ,J1)−(90°−2𝛼SOT ,J2) 𝐽1−𝐽2=−2𝛼SOT ,J1+2𝛼SOT ,J2 𝐽1−𝐽2=−2∆𝛼SOT ,J ∆𝐽 (3) where αSOT,J1(J2) , ΔαSOT,J and ΔJ are angles induced by SOT with current density J1(J2), the variation of αSOT,J and the change of current density J, respectively, the value of ∆𝛼SOT ,J ∆𝐽 is calculated to be around 2.1 deg/(106 A cm–2). Based on the above analyses , 14 the sublattice magnetization is switched toward the direction of current, hence the SOT induced equivalent field HSOT is set along current axis here [40]. On the basis of magnetic field vector addition, the SOT -field equival ence in YIG is determined to be 2.6 Oe/(106 A cm–2) [40], which is comparable with or slightly larger than the value obtained in typical ferrimagneti c insulators, such as TmIG [0.6 Oe/(106 A cm–2)] [26]. FIG. 3. Current -induced magnetization tilting during SMR measurements . (a) Measurement configurations of SMR. The current I is applied along [110]. The magnetic field H rotates in the plane of device . (b) SMR curves with different current densities (marked in the inset) and H = 50 Oe. The dashed arrows are a guide to reflect the shift of peak /valley positions. (c) Schematic of the magnetization tilting of two sublattices of YIG induced by applied current. αM (αH) denotes the angle between I and M1 (H). The red ( M1) and blue ( M2) arrows denote the magnetization of two magnetic sublattices, and the thin purple arrow represents the tilting direction . (d) Summary of the angle difference Δ αH between the valley and peak (in 0 –180° ) of SMR with different J and H. The typical H used are marked. The error bars are estimated from standard deviation of three SMR measurements. The dashed line is linear fitting of Δ αH with different current densities under H = 50 Oe. 15 We then focus on the YIG -thickness (t) dependent transport measurements. Figure 4(a) presents the SOT -induced Δ Rxy variation for 15, 20, 30, and 60 nm-thick YIG. All of the YIG/Pt samples exhibit reversible Δ Rxy variation. A comparison of the Δ Rxy shows that the magnitude of the Δ Rxy is greatl y enhanced with increasing t to 30 nm, and then Δ Rxy is saturated and keeps almost unchanged even up to 60 nm. On the other side, the angle αH dependent SMR curves measured with H = 5000 Oe for different t is shown in Fig. 4(b). Remarkably, the magnitude of SMR signals are enhanced with increasing t and saturated at t = 30 nm, which coincides with the thickness -dependence of SOT -induced Δ Rxy variation. Since SOT -induced switching results in the present case are dependent on two factors, the current -based writing efficiency and the SMR -based readout capability, it is significant to exclude the influence of thickness -dependent SMR when exploring the switching efficiency. In this scenario, the ratio of Δ Rxy/ΔSMR , where ΔSMR is the Rxy difference between the peak and valley of SMR curves in Fig. 4(b), is introduced to reflect the switching efficiency in our case. The ratio of Δ Rxy/ΔSMR as a function of t is displayed in Fig. 4(c), which shows a gradual enhancement with increasing t and is almost saturated at t = 30 nm. Meanwhile, the saturation magnetization ( MS) [40] of the YIG films is also presented in Fig. 4(c) for a comparison. It is found that both of them show a similar thickness -dependence, suggesting that the switching efficiency is relevant to MS. Because of the enhancement of MS and corresponding interfacial exchange interaction, more spin current can flow into the YIG [28], which enhances switching efficiency of YIG and result ant Δ Rxy/ΔSMR. 16 FIG. 4. SOT-induced switching and SMR measurements in YIG/Pt with different YIG thicknesses ( t). (a) ΔRxy as a function of pulse numbers in YIG/Pt bilayers with t = 15, 20, 30, and 60 nm. Results of different t are separate d by regions of different color s. (b) SMR results with different t under H = 5000 Oe. The Rxy differences between peak and valley ( ΔSMR ) are denoted by (grey) arrows in the inset, and the values are 3.0, 3.2, 3.8, and 3.8 m Ω for t = 15, 20, 30, and 60 nm, respectively. (c) ΔRxy/ΔSMR and MS versus t. Both Δ Rxy/ΔSMR and MS show a similar thickness -dependence and saturate at t = 30 nm. The error bars of Δ Rxy/ΔSMR and MS are estimated from standard deviation of reversible switching and three magnetization measurements, respec tively. IV. CONCLUSION In summary, we have demonstrated the reversible in -plane magnetization switching of YIG in YIG/Pt bilayers by SOT. The switching signal is readout by spin Hall magnetoresistance. The sublattice magnetizations of YIG are found to be aligned parallel/antiparallel to the direction of writing current, and may be ascribed to the antidamping -torque for the two strongly antiferromagnetic coupled sublattices, which is similar to the N éel order switching in antiferroma gnetic system to some extent . This 17 phenomenon indicates that anti-paralleled sublattices in ferrimagnetic insulator may play an important role during the current -induced in -plane magnetization switching process, and more studies with different materials are needed to further explore switching in ferrimagnetic insulator s in the future. Our finding not only promotes the understanding of current -induced switching , but also accelerate s combination of current -induced magnetization switching and previous microwave devices/spin caloritronics to realize high energy efficient spintronic applications based on magnetic insulators modulated by all -electrical means. ACKNOWLEDGMENTS We thank D r. D. Z. Hou for helpful discussions. C.S. acknowledges the support of the Beijing Innovation Center for Future Chips, Tsinghua University and the Young Chang Jiang Scholars Programme. This work was supported by the National Key R&D Programme of China (grant nos. 2017YFB04 05704 and 2017YFA0206200 ) and the National Natural Science Foundation of China (grant nos. 51871130, 51571128, 51671110 and 5183000528). REFERENCES [1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Observation of the spin Seebeck effect, Nature (London) 455, 778 (2008). [2] K.-i. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Observation of longitudinal spin -Seebec k effect in magnetic insulators, Appl. Phys. Lett. 97, 172505 (2010). [3] H. Nakayama, M. Althammer, Y . -T. Chen, K. Uchida, Y . Kajiwara, D. Kikuchi, T. 18 Ohtani, S. Geprä gs, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh , Spin Hall magnetoresistance induced by a nonequilibrium pro ximity effec , Phys. Rev. Lett. 110, 206601 (2013). [4] C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. Ben Youssef, Comparative measurements of inverse spin Hall effects and magneto resistance in YIG/Pt and YIG/Ta, Phys . Rev. B 87, 174417 (2013). [5] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y . -T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein , Quantitative study of the spin Hall magnetoresistance in ferromagnetic insulator/normal metal hybrids , Phys. Rev. B 87, 224401 (2013). [6] Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saito h, Transmission of electrical signals by spin -wave interconversion in a mag netic insulator, Nature (London) 464, 262 (2010). [7] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Long -distance transport of magnon spin information in a magnetic insulator at room temperature , Nat. Phys. 11, 1022 (2015). [8] L. Bai, M. Harder, Y . P. Chen, X. Fan, J. Q. Xiao, and C. M. Hu, Spin pumping in electrodynamically coupled magnon -photon system s, Phys. Rev. Lett. 114, 227201 (2015). [9] H. Wu, L. Huang, C. Fang, B. S. Yang, C. H. Wan, G. Q. Yu, J. F. Feng, H. X. Wei, and X. F. Han, Magnon valve effect between two magnetic insulators , Phys. Rev. Lett. 120, 097205 (2018). 19 [10] S.Geller and M.A.Gil leo, The crystal structure and ferrimagnetism of yttrium -iron garnet, Y 3Fe2(FeO 4)3, J. Phys. Chem. Solids 3, 30 (1957). [11] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015). [12] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Current -induced spin -orbit torques in ferromagnetic and antiferromagnetic systems, Rev. Mod. Phys. 91, 035004 (2019). [13] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel, and P. Gambardella, Current -driven spin torque induced by the Rashba effect in a ferromagnetic metal layer , Nat. Mater. 9, 230 (2010). [14] L. Liu, O. J. Lee, T. J. Gudmundsen , D. C. Ralph, and R. A. Buhrman, Current -induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect , Phys. Rev. Lett. 109, 096602 (2012). [15] J. Finley and L. Liu, Spin-orbit -torque efficiency in compensated ferrimagnetic cobalt -terbium alloys , Phys. Rev. Appl. 6, 054001 (2016). [16] K. Ueda, M. Mann, C. -F. Pai, A. -J. Tan, and G. S. D. Beach, Spin-orbit torques in Ta/Tb xCo100–x ferrimagnetic alloy films with bulk perpendicular magnetic anisotropy , Appl. Phys. Lett. 109, 232403 (2016). [17] L. Liu, C. -F. Pai, Y . Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman , Spin-torque switching with the giant spin Hall effect of tantalum , Science 336, 555 (2012). [18] S. Fukami, T. Anekawa, C. Zhang, and H. Ohno, A spin -orbit torque switching scheme with collinear magnetic easy axis and current configuration , Nat. Nanotechnol. 11, 621 (2016). [19] T. Moriyama, W. Zhou, T. Seki, K. Takanashi, and T. Ono, Spin-orbit -torque 20 memory operation of synthetic antiferromagnets , Phys. Rev. Lett. 121, 167202 (2018). [20] V . E. Demidov, S. Urazhdin, H. Ulrichs, V . Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov, Magnetic nano -oscillator driven by pure spin current , Nat. Mater. 11, 1028 (2012). [21] L. Liu, C. -F. Pai, D. C. Ralph, and R. A. Buhrman, Magnetic oscillations driven by the spin Hall effect in 3 -terminal magnetic tunnel junction devices , Phys. Rev. Lett. 109, 186602 (2012). [22] S. Emor i, U. Bauer, S. M. Ahn, E. Martinez, and G. S. Beach, Current -driven dynamics of chiral ferromagnetic domain walls , Nat. Mater. 12, 611 (2013). [23] K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Chiral spin torque at magnetic domain walls , Nat. Nanotechnol. 8, 527 (2013). [24] W. Jiang, P . Upadhyaya, W. Zhang, G. Yu, M. B . Jungfleisch, F. Y . Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O . Heinonen, S . G. E. te Velthuis, and A. Hoffmann , Blowing magnetic skyrmion bubbles , Science 349, 283 (2015). [25] G. Yu, P. Upadhyaya, X. Li, W. Li, S. K. Kim, Y . Fan, K. L. Wong, Y. Tserkovnyak, P. K. Amiri , K. L. Wang , Room -Temperature Creation and Spin−Orbit Torque Manipulation of Skyrmions in Thin Films with Engineered Asymmetry , Nano . Lett. 16, 1981 (2016 ). [26] C. O. Avci, A. Quindeau, C. -F. Pai, M. Mann, L. Caretta, A. S. Tang, M. C. Onbasli, C. A. Ross, and G. S. Beach, Current -induced switching in a magnetic insulator , Nat. Mater. 16, 309 (2017). [27] P. Li, T. Liu, H . Chang, A . Kalitsov, W . Zhang, G . Csaba, W . Li, D. Richardson , A. DeMann , G. Rimal , H. Dey, J.S. Jiang, W . Porod , S. B. Field , J. Tang, M. C. Marconi, A . Hoffmann, O . Mryasov , and M. Wu, Spin-orbit torque -assisted 21 switching in magnetic insulator thin films with perpendicular magnetic anisotropy , Nat. Commun. 7, 12688 (2016). [28] Q. Shao, C. Tang , G. Yu, A. Navabi, H. Wu, C. He, J. Li, P. Upadhyaya, P . Zhang, S. A. Razav, Q. L. He, Y . Liu, P. Yang, S. K. Kim, C. Zheng, Y . Liu, L. Pan, R. K. Lake, X. Han, Y . Tserkovnyak, J. Shi, and K. L. Wang , Role of dimensional crossover on spin -orbit torque efficiency i n magnetic insulator thin films, Nat. Commun. 9, 3612 (2018). [29] C. Y . Guo, C. H. Wan, M. K. Zhao, H. Wu, C. Fang, Z. R. Yan, J. F. Feng, H. F. Liu, and X . F. Han, Spin-orbit torque switching in per pendicular Y3Fe5O12/Pt bilayer, Appl. Phys. Lett. 114, 192409 (2019). [30] P. Wadley, B. Howells, J. Železný, C. And rews, V . Hills, R. P. Campion, V . Nová k, K. Olejní k, F. Maccherozzi, S. S. Dhesi, S. Y . Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y . Mokrousov, J. Kuneš, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gal lagher, and T. Jungwirth , Electrical switching of an antiferromagnet , Science 351, 587 (2016) . [31] S.Y . Bodnar, L. Šme jkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A.A. Sapozhnik, H. -J. Elmers , M. Klä ui , and M. Jourdan , Writing and reading antiferromagnetic Mn 2Au by Né el spin -orbit torques and large anisotropic magnetoresistance , Nat. Commun. 9, 348 (2018). [32] X. F. Zhou, J. Zhang, F. Li, X. Z. Chen, G. Y . Shi, Y . Z. Tan, Y . D. Gu, M. S. Saleem, H. Q. Wu, F. Pan, and C. Song , Strong orientation -dependent spin -orbit torque in thin films of the antiferromagnet Mn 2Au, Phys. Rev. Appl. 9, 054028 (2018). [33] M. Meiner t, D. Graulich, and T. Matalla -Wagner, Electrical switching of antiferromagnetic Mn 2Au and the role of thermal activation , Phys. Rev. Appl. 9, 22 064040 (2018). [34] X. Chen, X . Zhou, R . Cheng, C . Song, J . Zhang, Y . Wu, Y. Ba, H. Li, Y . Sun, Y . You, Y . Zhao , and F . Pan, Electric field control of Né el spin -orbit torque in an antiferromagnet , Nat. Mater. 18, 931 (2019). [35] X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou, G. Y . Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, and Y . Tserkovnyak , Antidamping -torque -induced switching in biaxial antiferromagnetic insulators , Phys. Rev. Lett. 120, 207204 (2018). [36] T. Moriyama, K. Oda, T. Ohkochi, M. Kimata, and T. Ono, Spin torque control of antiferromagnetic moments in NiO , Sci. Rep. 8, 14167 (2018). [37] L. Baldrati, O. Gomonay, A. Ross, M. Filianina, R. Lebrun, R. Ramos, C. Leveille, F. Fuhrmann, T. R. Forrest, F. Maccherozzi, S. Valencia, F. Kronast, E. Saitoh, J. Sinova, and M. Klä ui, Mechanism of Né el order switching in antiferromagnetic t hin films revealed by magnetotransport and direct imaging , Phys. Rev. Lett. 123, 177201 (2019). [38] A. Kehlberger, K. Richter, M. C. Onbasli, G. Jakob, D. H. Kim, T. Goto, C. A. Ross, G. Gö tz, G. Reiss, T. Kuschel, and Mathias Klä ui , Enhanced magneto -optic Kerr effect and magnetic properties of CeY 2Fe5O12 epitaxial thin films , Phys. Rev. Appl. 4, 014008 (2015). [39] B. Lax and K. J. Button, Microwave ferrites and ferrimagnetics (McGraw -Hill, New York, 1962). [40] See Supplementary Material s at http://xxx for magentization switching with different current densities, for magnetization switching of YIG(111)/Pt, for magnetization switching of YIG (001)/Cu/Pt , for longitudinal resistance variation of YIG(001)/Pt, for magnetization switching of Co /Pt, for SMR measurements of 23 YIG(001)/Pt, for SOT -field equivalence deduced from SMR tilting and for hysteresis loops of YIG(001) with different thic knesses, which includes Ref. [41]. [41] Y . -T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goenn enwein, E. Saitoh, and G. E. W. Bauer, Theory of spin Hall magnetoresistance, Phys. Rev. B 87, 144411 (2013). [42] Y. Cheng, S . Yu, M. Zhu, J . Hwang, and F. Yang , Electrical Switching of Tristate Antiferromagnetic Né el Order in α-Fe2O3 Epitaxial Films , Phys. Rev. Lett. 124, 027202 (2020). [43] C. C. Chiang, S. Y. Huang, D. Qu, P. H. Wu, and C. L. Chien , Absence of Evidence of Electrical Switching of the Antiferromagnetic Né el Vector , Phys. Rev. Lett. 123, 227203 (2019). [44] P. Zhang, J . Finley, T. Safi, and L. Liu, Quantitative Study on Current -Induc ed Effect in an Antiferromagnet Insulator/Pt Bilayer Film , Phys. Rev. Lett. 123, 247206 (2019) . [45] S. Shi, S . Liang, Z . Zhu, K. Cai, S . D. Pollard, Y . Wang , J. Wang, Q . Wang , P. He , J. Yu, G. Eda, G . Liang , and H. Yang , All-electric magnetization switching and Dzyaloshinskii -Moriya interaction in WTe 2/ferromagnet heterostructures, Nat. Nanotechnol . 14, 945 (2019). [46] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F. Alvarado and Pietro Gambardella, Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers, Nat. Phys. 11, 570 (2015).

2020-06-18

Ferrimagnetic insulators (FiMI) have been intensively used in microwave and magneto-optical devices as well as spin caloritronics, where their magnetization direction plays a fundamental role on the device performance. The magnetization is generally switched by applying external magnetic fields. Here we investigate current-induced spin-orbit torque (SOT) switching of the magnetization in Y3Fe5O12 (YIG)/Pt bilayers with in-plane magnetic anisotropy, where the switching is detected by spin Hall magnetoresistance. Reversible switching is found at room temperature for a threshold current density of 10^7 A cm^-2. The YIG sublattices with antiparallel and unequal magnetic moments are aligned parallel or antiparallel to the direction of current pulses, which is consistent to the Neel order switching in antiferromagnetic system. It is proposed that such a switching behavior may be triggered by the antidamping-torque acting on the two antiparallel sublattices of FiMI. Our finding not only broadens the magnetization switching by electrical means and promotes the understanding of magnetization switching, but also paves the way for all-electrically modulated microwave devices and spin caloritronics with low power consumption.

Current-induced in-plane magnetization switching in biaxial ferrimagnetic insulator

2006.10313v1

Cryogenic hybrid magnonic circuits based on spalled YIG thin films Jing Xu∗,1, 2Connor Horn∗,1Yu Jiang,3Xinhao Li,2Daniel Rosenmann,2 Xu Han,2Miguel Levy,4Supratik Guha†,1and Xufeng Zhang‡3 1Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA 2Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL 60439, USA 3Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115, USA 4Department of Physics, Michigan Technological University, Houghton, MI 49931, USA (Dated: December 20, 2023) Yttrium iron garnet (YIG) magnonics has sparked extensive research interests toward harnessing magnons (quasiparticles of collective spin excitation) for signal processing. In particular, YIG magnonics-based hybrid systems exhibit great potentials for quantum information science because of their wide frequency tunability and excellent compatibility with other platforms. However, the broad application and scalability of thin-film YIG devices in the quantum regime has been severely limited due to the substantial microwave loss in the host substrate for YIG, gadolinium gallium garnet (GGG), at cryogenic temperatures. In this study, we demonstrate that substrate-free YIG thin films can be obtained by introducing the controlled spalling and layer transfer technology to YIG/GGG samples. Our approach is validated by measuring a hybrid device consisting of a superconducting resonator and a spalled YIG film, which gives a strong coupling feature indicating the good coherence of our system. This advancement paves the way for enhanced on-chip integration and the scalability of YIG-based quantum devices. The field of YIG magnonics [1] is a rapidly evolving re- search area dedicated to studying the collective spin exci- tations (magnons) in YIG (yttrium iron garnet) crystals. In recent years, it has shown extensive potential in hybrid information systems [2–5]. Thanks to its low magnetic damping, high spin density, and excellent compatibility with various physical platforms, YIG has been consid- ered as an ideal magnonic platform for hybrid quantum information processing. Researchers are actively explor- ing various YIG-based hybrid systems such as electro- magnonics [6–13], optomagnonics [14–16], and magnome- chanics [17–20] for different applications. As demand grows for scalable quantum systems, thin-film YIG de- vices are highly desired for on-chip integration over bulk YIG spheres used in earlier research. However, the development of thin film YIG devices at cryogenic temperature regimes has been severely limited. One major obstacle is the undesirable properties of the substrate used for the growth of YIG thin film. The best growth method for high quality single crystalline YIG films is epitaxial growth on gadolinium gallium gar- net (GGG) substrates which has a matched lattice con- stant with YIG, yielding a room-temperature magnon linewidth close to that of single-crystal YIG spheres. However, at cryogenic temperatures such YIG films ex- hibit very high microwave losses because the host sub- strate GGG undergoes a phase transition into a geomet- rically frustrating spin-liquid state below 5 Kelvin [21]. In this state, the short-range ordered spins in the GGG substrate shows strong absorption to external energy, a property that has found application in commercial adia- batic demagnetization cooling [22, 23]. The presence of this spin-liquid state in GGG degrades the lifetime of spin excitations in the YIG layer, as indicated by the larger FMR linewidth [24–26] compared with spheres made ofpure YIG, posing a significant impediment to its integra- tion in cryogenic quantum systems. For instance, electro- magnonic systems involving strongly coupled microwave photons and magnons have been extensively studied in the past few years [6–13, 27–29], which have enabled ad- vanced functionalities such as entanglement with super- conducting qubits [30–32], but most of previous demon- strations are based on YIG spheres while YIG thin films have been rarely used. One promising solution to this grand challenge is us- ing YIG thin films without the GGG substrate, which in- cludes two technical approaches. The first approach is to grow YIG thin films on substrates other than GGG such as silicon [25, 33–35]. This approach is straightforward and more favorable from the aspect of device integration; however, the quality of YIG thin films are usually low be- cause of the lattice mismatch between YIG and the new substrate. The second approach involves growth of YIG on GGG substrates with post-processing to detach YIG from the GGG substrate [36–39]. This method produces high quality YIG but the separation processes for YIG are usually challenging due to the similar physical and chemical properties of YIG and GGG, hindering wide application of such technologies. In this work, we show our investigation on a new method that can simplify the YIG detaching process, providing a new direction for the development of YIG thin film devices. The approach we used for detaching YIG from GGG is based on controlled mechanical spalling [40], as shown by the procedures in 1(b). The substrate we used is a commercially available single-crystal YIG film [200 nm thick, (111)-oriented] grown on a 500 µm-thick GGG substrate by liquid phase epitaxy (LPE). After cleaning the sample, a layer of 10-nm-thick chromium followed by 70-nm-thick gold are deposited on the YIG surfacearXiv:2312.10660v2 [cond-mat.mes-hall] 19 Dec 20232 FIG. 1. (a) A schematic of an integrated hybrid quantum device using the substrate-free YIG thin film. (b) Schematics of the YIG film spalling process: (I) (Optional) helium ion implantation process with implantation depth around 7 µm in the YIG/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 nm YIG/500 µm GGG substrate. (III) The separation point occurs at approximately 2 µm beneath the YIG/GGG substrate’s top surface, where stress accumulates, achieved by using a thermal release tape. (IV) Materials remaining on the tape, from bottom 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 of metal layers is accomplished using nickel, gold, and chromium etchants in sequence. (c) Optical image of the spalled sample after step II. Right side: substrate-free film adhered to the thermal release tape. Left side: the remaining GGG substrate after the spalling process. (d) Confocal microscope measurement image of the surface of the remaining GGG substrate. The outline of the spalled area is marked by the yellow dashed line. through magnetron sputtering. Using the gold layer as a seed layer, a thick layer of Ni is electroplated with a fi- nal thickness of 7 µm using the electroplating conditions in reference [41], which yields an intrinsic tensile stress of 700 MPa. The tensile stress and thickness of the Ni defines an equilibrium depth within the YIG/GGG sub- strate at which steady state crack propagation can take place [42]. Using a thermal release tape, the top layer stack (Ni/Au/Cr/YIG/GGG) is carefully spalled, result- ing in a continuous substrate-free film, as shown in Figure 1(c). Considering that this depth is larger than the thick- ness of the YIG layer, the spalled YIG is still attached to a thin layer of GGG. However, by choosing YIG wafers with larger YIG thickness (e.g., 10 µ) and optimizing the thickness and tensile stress of the YIG layers, it is pos- sible to obtain a spalled layer of pure YIG without any residual GGG. The final substrate-free device is obtained by removing the nickel, gold, and chromium layers with appropriate wet chemical etchants. We noticed that with our current conditions, the spalling process is also largely affected by ion implan- tation in the substrate. When the substrate is intrinsic YIG/GGG sample, the resulting spalling depth is typi- cally 5-10 um, whereas if the YIG/GGG sample is treated with helium ion implantation (following the conditions from Refs. [36, 43]), thinner spalling depths are obtainedwith smoother surfaces, as shown by height scan of the remaining GGG substrate using a 3D laser scanning con- focal microscope [1(d)] which reveals a spalling depth of around 2-3 µm. This may be attributed to the fact that the GGG layer above the ion implantation depth (7 µm) is damaged by the high-energy helium ions during the im- planting process and becomes easier to break under the elastic stress from the Ni layer, resulting in a shallower spalling depth. To characterize the microwave performance of the spalled YIG thin film at room temperature, it is flip- bonded to a rectangle split ring resonator (RSRR) made of copper [13] to test the ferromagnetic resonance (FMR) response, as schematically illustrated by Fig. 2(a). The reflection spectrum of the RSRR is obtained using a vec- tor network analyzer. When an out-of-plane bias mag- netic field is tuned to sweep the FMR frequency, it is expected that the magnon mode becomes visible in the spectrum when it is tuned very close to the microwave resonator frequency. However, when a 500 ×300×2µm3 flake from the spalled YIG film (which has been treated with helium ion implantation) is bonded on the RSRR, no magnon modes can be observed from the RSRR reflec- tion spectra. This is speculated as the result of the large magnon linewidth in the YIG flake and the small volume of the YIG flake (accordingly small coupling with the3 FIG. 2. (a) Schematics of a substrate-free YIG thin film bonded onto an RSRR chip. (b) Optical image of a piece of spalled YIG (approximately 500 µm×300µmaffixed to a Kapton tape. (c) Measured reflection spectra for the RSRR device with the YIG/GGG chip attached (prior to annealing). (d) Measured reflection spectra for the RSRR device with the spalled YIG (after annealing). RSRR resonator). To verify this speculation, we tested a larger piece (roughly 5mm lateral size) of unspalled YIG that was treated through the same ion implantation process, using the same RSRR reflection measurement. With the increased YIG volume, the magnon mode is successfully observed, but the measured data [Fig. 2(c)] shows a notably high dissipation rate κm/2π= 39.1 MHz at 11.6 GHz, which is one order of magnitude higher than previously reported values on single-crystal YIG thin films [13]. Such elevated disspation rates can be attributed to two possible sources of dissipation: (1) The damage to the crystalline structure caused by the high- energy ions penetrating the YIG layer, and (2) The ac- cumulated helium ions in the YIG layer. Both effects can be mitigated using an annealing process, which will restore the damaged lattice structure and repel the ac- cumulated helium ions from YIG. We carried out an an- nealing process for multiple flakes of the substrate-free YIG film in ambient air, reaching a maximum temper- ature of 850◦C. The samples is gradually heated from room temperature to 850◦C over a period of 6 hours, followed by a 3-hour hold at the maximum temperature, and then a slow cooling process over a span of 14 hours to room temperature, providing ample time for the re- pair of the YIG crystalline lattice. After the annealing process, the magnon mode shows up on a device with a small YIG flake (around 500 ×300×2µm3), as shown in Fig. 2(d). Numerical fitting shows a dissipation rage of κm/2π= 2.08 MHz at 10.5 GHz, which is comparable with those of high-quality LPE YIG thin films reported in previous articles [13, 24, 44]. To further demonstrate its potential for cryogenic FIG. 3. (a) Layout depicting the superconducting resonator coupled to a bus transmission line. (b) Zoomed-in view of the lumped element resonator design, showing the interdigital ca- pacitors on the sides and the inductor line in the middle. (c) Optical image displaying a superdoncuting resonator loaded with the spalled YIG thin film (measuring approximately 300 µm×200µm). The irregular shaded area is the GE varnish for chip bonding. (d) The measured spectrum of the chip bonded with the spalled YIG at 200 mK, revealing two mi- crowave resonances at 5.05 GHz and 9.34 GHz, respectively. (e) The measured transmission spectrum of another super- conducting resonator device at 200 mK with (red curve) and without (black curve) the unspalled YIG/GGG substrate. quantum operations, the spalled YIG thin films are fur- ther characterized at millikelvin (mk) temperature. A superconducting resonator is used to couple with the YIG flake, which is fabricated using 100-nm-thik nio- bium through photolithography and dry etching. To en- hance the magnon-photon coupling, a lumped-element resonator with interdigital capacitors is used, which is inductively couples to a bus transmission line, as shown by the layout plot in Figs. 3(a) and (b). The YIG flake is flip-bonded using GE varnish to cover the center in- ductor line where the microwave magnetic field is the strongest, which further enhances the coupling strength between the magnon and photon modes. An optical im- age of the assembled device is shown in Fig. 3(c)], where the position of the spalled YIG flake is outlined by the red dashed curve. Figure 3(d) depicts the measured microwave transmis- sion of our device at a temperature of 200 millikelvin performed within an adiabatic demagnetization refriger- ator (ADR). Two resonance modes can be observed as the two sharp dips in the transmission spectrum, simi- lar to what is observed in Ref.[28]. The inset highlights the fundamental mode at 5.05 GHz which has a lower damping rate and smaller extinction ratio. The higher-4 FIG. 4. (a) A heatmap of the measured transmission spec- trum for the superconducting resonator device shown in Fig. 3(c), showing the avoid-crossing feature. The overlaid red circles represent the calculated frequencies of magnon-photon modes. (b) A heatmap of the calculated transmission spec- trum using Eq (1) with the fitted magnon linewidth. (c) Transmission data at 1300 G where the magnon mode is far detuned. The photon linewidth is extracted as κc/2π= 3.9 MHz. (d) Transmission data at 1000 G where the magnon and the photon modes are on resonance and fully hybridized. The hybrid linewidth is fitted to be κm/2π= 15 MHz. order mode at 9.34 GHz has a larger extinction ratio and larger dissipation rate but it couples weakly with the YIG magnon mode and thus will not be discussed further. Im- portantly, the clear observation of two high-quality res- onances on the YIG-loaded superconducting resonator indicates that the effect of the GGG substrate has been significantly suppressed. As a comparison, we measured another superconducting resonator device at the same temperature (200 mK), which is loaded with a regular YIG thin film with the 500 −µm-thick GGG substrate still attached. From the measurement results in Fig. 3(e) (red curve), no clear microwave resonances can be ob- served, indicating the significantly increased loss on the superconducting resonator, while prior to the YIG/GGG bonding two resonances (at 7.4 GHz and 8.5 GHz) are clearly visible [black curve in Fig. 3(e)]. To further investigate the performance of the device shown in Fig. 3(c), a series of transmission spectra are col- lected while a magnetic field is swept to tune the magnon frequency. The magnetic field is applied parallel to the surface of the superconducting resonator chip, aligning with the direction of the inductor wire. A clear avoid- crossing feature is observed in the measured spectra, as shown in Fig. 4(a), indicating that our device has entered the strong coupling regime, where the magnon-photon coupling strength exceeds the dissipation rate of each in- dividual mode. Using numerical fitting based on rotating-wave ap- proximation (RWA) [45], the magnon-photon couplingstrength gis extracted as 62 MHz, with detailed proce- dures described in Section II of the supplemental mate- rial [46]. The two branches of the calculated magnon- photon mode frequencies, represented by the red dots in Fig. 4(a), match well with the measured spectrum. At a magnetic field of 1300 G where the magnon mode is large detuned from the microwave resonance, the dissipa- tion rate of the microwave modeis extracted as κc= 1.95 MHz, which leads to a quality factor Q= 2πfc/κc= 5.05GHz/ 1.95MHz ≈2600. Considering that the YIG flake is positions sufficiently far from the bus transmis- sion line, its direct coupling with the bus waveguide is negligible. Therefore, the magnon mode is observed only when its frequency is close to or on-resonance with the microwave resonance. We can estimate the magnon dis- sipation rate using the relation κh= (κm+κc)/2 when the two modes are maximally hybridized (when magnon and photon modes are on resonance). Here κhis the on-resonance linewidth of the hybrid mode, which is fit- ted to be 15 MHz as presented in Fig. 4(d). Accordingly, the dissipation rate of the magnon mode is determinate to be κm/2π= 13 MHz. Compared with previous de- vices which have YIG on the 500- µm-thick GGG, the linewidth of the magnon has been significantly reduced. Although it is still higher than the linewidths measured on bulk YIG samples, it may be attributed to residual GGG layer attached to the YIG thin film which can be removed through futher optimization. Such calculation result is consistent with our fitting results based on the input-output theory [47–49], as described by equation S21=ˆaout ˆain=κexi∆c+g2/(i∆m−κm) i∆c−κc+g2/(i∆m−κm),(1) where grepresent the magnon-photon coupling strength, κexrepresents the external coupling rate between the mi- crowave resonator and the bus feeding line, κc(κm) repre- sents the total dissipation rate of the resonator (magnon) mode, ∆ c=ωd−ωcand ∆ m=ωd−ωmrepresent the detuning of the driving frequency from the microwave resonance and magnon frequency, respectively. The cal- culated spectra using Eq. 1 are plotted in Fig. 4(b), which match well with the measured result in Fig. 4(a). For a comprehensive exploration of the fitting procedures and detailed results, we direct readers to the Supplemental Material [46]. In conclusion, we have demonstrated a new approach for developing substrate-free YIG thin films, and vali- dated the our method by measuring the strong magnon- photon coupling on a hybrid device. The direct compar- ison with conventional YIG/GGG devices confirmed the effectiveness of our approach. Our discovery represents the first application of the spalling technology on mag- netic garnets, which have been known as very strong and hard to process. Compared with other approaches, our spalling-based method offers distinct advantages includ- ing reduced material contamination and flexible thickness control. Although the linewidth of our measured device at cryogenic temperatures is still higher compared with5 bulk YIG, it can be further improved by completely re- moving the GGG substrate. Upon further optimization, including the thickness of the original YIG film as well as the stress and thickness of the Ni layer, our approach is promising for wafer-scale production of magnonic de- vices for quantum applications and beyond. In particu- lar, when combined with recent experimental techniques [13, 16, 50] for YIG magnonic devices, our substrate-free YIG thin films may offer unique properties for magnons to couple with a broad range of degrees of freedom, in- cluding optics, mechanics, and magnetics. ACKNOWLEDGMENTS The authors thank R. Divan, L. Stan, C. Miller, and D. Czaplewski for support in the device fabrication. X.Z. ac-knowledges support from ONR YIP (N00014-23-1-2144). Contributions by C.H. and S.G. were supported by the Vannevar Bush Fellowship received by S.G. under the program sponsored by the Office of the Undersecretary of Defense for Research and Engineering and in part by the Office of Naval Research as the Executive Manager for the grant. Work performed at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility, was supported by the U.S. DOE Office of Basic Energy Sciences, under Contract No. DE-AC02- 06CH11357. †guha@uchicago.edu ‡xu.zhang@northeastern.edu ∗J. Xu and C. Horn contributed equally to this work. [1] A. A. Serga, A. V. Chumak, and B. Hillebrands, YIG magnonics, J. Phys. D: Appl. Phys. 43, 264002 (2010). [2] D. D. Awschalom, C. R. Du, R. He, F. J. Heremans, A. Hoffmann, J. Hou, H. Kurebayashi, Y. Li, L. Liu, V. Novosad, et al. , Quantum engineering with hybrid magnonic systems and materials, IEEE Transactions on Quantum Engineering 2, 1 (2021). [3] Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Hoff- mann, and V. Novosad, Hybrid magnonics: Physics, cir- cuits, and applications for coherent information process- ing, Journal of Applied Physics 128, 130902 (2020). [4] X. Zhang, A review of common materials for hybrid quan- tum magnonics, Materials Today Electronics 5, 100044 (2023). [5] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Hybrid quantum systems based on magnonics, Applied Physics Express 12, 070101 (2019). [6] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons, Phys. Rev. Lett. 113, 156401 (2014). [7] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Spin pumping in electrodynamically coupled magnon-photon systems, Phys. Rev. Lett. 114, 227201 (2015). [8] M. Harder and C.-M. Hu, Cavity spintronics: an early review of recent progress in the study of magnon–photon level repulsion, Solid State Physics 69, 47 (2018). [9] B. Bhoi and S.-K. Kim, Roadmap for photon-magnon coupling and its applications, in Solid State Physics , Vol. 71 (Elsevier, 2020) pp. 39–71. [10] C.-M. Hu, The 2020 roadmap for spin cavitronics, in Solid State Physics , Vol. 71 (Academic Press, Cambridge, MA, USA, 2020) pp. 117–121. [11] B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Cavity magnonics, Physics Reports 979, 1 (2022). [12] J. W. Rao, S. Kaur, B. M. Yao, E. R. J. Edwards, Y. T. Zhao, X. Fan, D. Xue, T. J. Silva, Y. S. Gui, and C.-M. Hu, Analogue of dynamic Hall effect in cavity magnon polariton system and coherently controlled logic device,Nat. Commun. 10, 1 (2019). [13] J. Xu, C. Zhong, X. Han, D. Jin, L. Jiang, and X. Zhang, Coherent Gate Operations in Hybrid Magnonics, Phys. Rev. Lett. 126, 207202 (2021). [14] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Op- tomagnonic Whispering Gallery Microresonators, Phys. Rev. Lett. 117, 123605 (2016). [15] L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, On-chip optical isolation in monolithically integrated non-reciprocal optical res- onators, Nat. Photonics 5, 758 (2011). [16] N. Zhu, X. Zhang, X. Zhang, X. Han, X. Han, C.-L. Zou, C.-L. Zou, C. Zhong, C. Zhong, C.-H. Wang, C.-H. Wang, L. Jiang, L. Jiang, and H. X. Tang, Waveguide cavity op- tomagnonics for microwave-to-optics conversion, Optica 7, 1291 (2020). [17] Y.-J. Seo, K. Harii, R. Takahashi, H. Chudo, K. Oy- anagi, Z. Qiu, T. Ono, Y. Shiomi, and E. Saitoh, Fab- rication and magnetic control of Y3Fe5O12 cantilevers, Appl. Phys. Lett. 110, 10.1063/1.4979553 (2017). [18] K. An, A. N. Litvinenko, R. Kohno, A. A. Fuad, V. V. Naletov, L. Vila, U. Ebels, G. de Loubens, H. Hurd- equint, N. Beaulieu, J. Ben Youssef, N. Vukadinovic, G. E. W. Bauer, A. N. Slavin, V. S. Tiberkevich, and O. Klein, Coherent long-range transfer of angular mo- mentum between magnon Kittel modes by phonons, Phys. Rev. B 101, 060407 (2020). [19] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity magnomechanics, Sci. Adv. 2, 10.1126/sciadv.1501286 (2016). [20] J. Xu, C. Zhong, X. Zhou, X. Han, D. Jin, S. K. Gray, L. Jiang, and X. Zhang, Coherent Pulse Echo in Hybrid Magnonics with Multimode Phonons, Phys. Rev. Appl. 16, 024009 (2021). [21] O. A. Petrenko, C. Ritter, M. Yethiraj, and D. M. Paul, Spin-liquid behavior of the gadolinium gallium garnet, Physica B 241-243 , 727 (1997). [22] I. D. Hepburn, A. Smith, and I. Davenport, Adiabatic Demagnetisation Refrigerators for Space Applications on JSTOR (1994), [Online; accessed 19. Oct. 2023]. [23] D. Kwon, J. Bae, and S. Jeong, Development of the inte-6 grated sorption cooler for an adiabatic demagnetization refrigerator (ADR), Cryogenics 122, 103421 (2022). [24] S. Kosen, A. F. van Loo, D. A. Bozhko, L. Mihalceanu, and A. D. Karenowska, Microwave magnon damping in YIG films at millikelvin temperatures, APL Mater. 7, 10.1063/1.5115266 (2019). [25] S. Guo, B. McCullian, P. Chris Hammel, and F. Yang, Low damping at few-K temperatures in Y3Fe5O12 epi- taxial films isolated from Gd3Ga5O12 substrate using a diamagnetic Y3Sc2.5Al2.5O12 spacer, J. Magn. Magn. Mater. 562, 169795 (2022). [26] C. L. Jermain, S. V. Aradhya, N. D. Reynolds, R. A. Buhrman, J. T. Brangham, M. R. Page, P. C. Hammel, F. Y. Yang, and D. C. Ralph, Increased low-temperature damping in yttrium iron garnet thin films, Phys. Rev. B 95, 174411 (2017). [27] Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Di- van, J. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Hoffmann, and W. Zhang, Strong Coupling be- tween Magnons and Microwave Photons in On-Chip Ferromagnet-Superconductor Thin-Film Devices, Phys. Rev. Lett. 123, 107701 (2019). [28] J. T. Hou and L. Liu, Strong Coupling between Mi- crowave Photons and Nanomagnet Magnons, Phys. Rev. Lett.123, 107702 (2019). [29] J. Xu, C. Zhong, X. Han, D. Jin, L. Jiang, and X. Zhang, Coherent Gate Operations in Hybrid Magnonics, Phys. Rev. Lett. 126, 207202 (2021). [30] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015). [31] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura, Entanglement-based single- shot detection of a single magnon with a superconducting qubit, Science 367, 425 (2020). [32] D. Xu, X.-K. Gu, H.-K. Li, Y.-C. Weng, Y.-P. Wang, J. Li, H. Wang, S.-Y. Zhu, and J. Q. You, Quantum Con- trol of a Single Magnon in a Macroscopic Spin System, Phys. Rev. Lett. 130, 193603 (2023). [33] L. Bi, J. Hu, P. Jiang, H. S. Kim, D. H. Kim, M. C. Onbasli, G. F. Dionne, and C. A. Ross, Magneto-Optical Thin Films for On-Chip Monolithic Integration of Non- Reciprocal Photonic Devices, Materials 6, 5094 (2013). [34] M. C. Onbasli, T. Goto, X. Sun, N. Huynh, and C. A. Ross, Integration of bulk-quality thin film magneto- optical cerium-doped yttrium iron garnet on silicon ni- tride photonic substrates, Opt. Express 22, 25183 (2014). [35] S. Guo, D. Russell, J. Lanier, H. Da, P. C. Hammel, and F. Yang, Strong on-Chip Microwave Photon–Magnon Coupling Using Ultralow-Damping Epitaxial Y3Fe5O12 Films at 2 K, Nano Lett. 23, 5055 (2023). [36] M. Levy, R. M. Osgood, A. Kumar, and H. Bakhru, Crys- tal ion slicing of single-crystal magnetic garnet films, J. Appl. Phys. 83, 6759 (1998). [37] F. Heyroth, C. Hauser, P. Trempler, P. Geyer, F. Sy-rowatka, R. Dreyer, S. G. Ebbinghaus, G. Woltersdorf, and G. Schmidt, Monocrystalline Freestanding Three- Dimensional Yttrium-Iron-Garnet Magnon Nanores- onators, Phys. Rev. Appl. 12, 054031 (2019). [38] J. A. Haigh, R. A. Chakalov, and A. J. Ramsay, Sub- picoliter Magnetoptical Cavities, Phys. Rev. Appl. 14, 044005 (2020). [39] P. G. Baity, D. A. Bozhko, R. Macˆ edo, W. Smith, R. C. Holland, S. Danilin, V. Seferai, J. Barbosa, R. R. Peroor, S. Goldman, U. Nasti, J. Paul, R. H. Hadfield, S. McVi- tie, and M. Weides, Strong magnon–photon coupling with chip-integrated YIG in the zero-temperature limit, Appl. Phys. Lett. 119, 10.1063/5.0054837 (2021). [40] S. W. Bedell, D. Shahrjerdi, B. Hekmatshoar, K. Fogel, P. A. Lauro, J. A. Ott, N. Sosa, and D. Sadana, Kerf- Less Removal of Si, Ge, and III–V Layers by Controlled Spalling to Enable Low-Cost PV Technologies, IEEE J. Photovoltaics 2, 141 (2012). [41] S. W. Bedell, P. Lauro, J. A. Ott, K. Fogel, and D. K. Sadana, Layer transfer of bulk gallium nitride by con- trolled spalling, J. Appl. Phys. 122, 10.1063/1.4986646 (2017). [42] Z. Suo and J. W. Hutchinson, Steady-state cracking in brittle substrates beneath adherent films, Int. J. Solids Struct. 25, 1337 (1989). [43] M. Levy, R. M. Osgood, R. Liu, L. E. Cross, G. S. Cargill, A. Kumar, and H. Bakhru, Fabrication of single-crystal lithium niobate films by crystal ion slicing, Appl. Phys. Lett.73, 2293 (1998). [44] V. Castel, A. Manchec, and J. B. Youssef, Con- trol of Magnon-Photon Coupling Strength in a Pla- nar Resonator/Yttrium-Iron-Garnet Thin-Film Config- uration, IEEE Magn. Lett. 8, ArticleSequenceNum- ber:3703105 (2016). [45] R. W. Boyd, Nonlinear Optics (Elsevier, Academic Press, 2008). [46] Supplemental material. [47] D. I. Schuster, A. P. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. J. L. Morton, H. Wu, G. A. D. Briggs, B. B. Buckley, D. D. Awschalom, and R. J. Schoelkopf, High- Cooperativity Coupling of Electron-Spin Ensembles to Superconducting Cavities, Phys. Rev. Lett. 105, 140501 (2010). [48] D. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, Germany, 2008). [49] Q.-M. Chen, M. Pfeiffer, M. Partanen, F. Fesquet, K. E. Honasoge, F. Kronowetter, Y. Nojiri, M. Renger, K. G. Fedorov, A. Marx, F. Deppe, and R. Gross, Scattering coefficients of superconducting microwave resonators. I. Transfer matrix approach, Phys. Rev. B 106, 214505 (2022). [50] B. Heinz, T. Br¨ acher, M. Schneider, Q. Wang, B. L¨ agel, A. M. Friedel, D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, and A. V. Chumak, Prop- agation of Spin-Wave Packets in Individual Nanosized, Nano Lett. 20, 4220 (2020).Cryogenic hybrid magnonic circuits based on spalled YIG thin films Jing Xu∗,1, 2Connor Horn∗,1Yu Jiang,3Xinhao Li,2Daniel Rosenmann,2 Xu Han,2Miguel Levy,4Supratik Guha†,1and Xufeng Zhang‡3 1Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA 2Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL 60439, USA 3Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115, USA 4Department of Physics, Michigan Technological University, Houghton, MI 49931, USA (Dated: December 20, 2023) I. RESONATOR DESIGN The superconducting resonator incorporates a hanger structure to enable inductive coupling with the bus transmis- sion line, featuring two series of interdigital capacitor fingers flanking a central inductive thin wire, as illustrated in Fig. S1. Simulated transmission power (S21) results, focusing on the first two modes of the designed superconducting resonator, reveal that the low frequency mode at 4.85 GHz exhibits a lower extinction ratio and higher quality factor, indicative of well-confined RF field within the resonator. The magnetic field distribution is depicted in Fig. S1(c) using a heat map, with the red color on the central inductance wire highlighting the highly confined magnetic fields. In contrast, the high frequency mode at 8.26 GHz displays a higher extinction ratio and broader linewidth, as indicated by the field distribution in Fig. S1(d), which shows less confinement of the microwave field at that frequency. FIG. S1. (a) Experimental and (b) simulated reflection spectrum of the superconducting chip at 200 mK, respectively. (c) and (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 Based on these simulation results, precise placement of the YIG thin film on the inductance wire is imperative to ensure maximal overlap between the magnon and photon modes. This arrangement will cause the magnon mode to have stronger coupling with the low frequency resonator mode and weaker coupling with the high frequency mode. Experimental measurement, as shown in Fig. S1(a), confirms the presence of two microwave modes at 5.05 GHz and 9.34 GHz, aligning closely with the simulated values. The slight blueshift observed in the experimental data compared to the simulated values may originate from minor deviations in the designed geometrical parameters during the fabrication process. II. COUPLING STRENGTH FITTING The system Hamiltonian governing the coupled magnon and photon can be described using the rotating-wave approximation (RWA) [1]: ˆH=ℏωcˆc†ˆc+ℏωmˆm†ˆm+ℏg(ˆc†ˆm+ ˆm†ˆc), (1) where candmare associated with resonator photons and magnons, respectively. When expressed in matrix form and taken dissipations into account, the equation becomes: ˆH=/bracketleftbigg ℏωc0 0ℏωm/bracketrightbigg +/bracketleftbigg −iκcg g−iκm/bracketrightbigg (2) In this equation, ℏis the reduced Planck’s constant, while ωc(ωm) and κc(κm) are the frequencies and dissipation rates of the photon (magnon) modes, respectively. The eigenfrequencies of the system Hamiltonian are given by: ℏω±=(ωc+ωm)−i(κc+κm)±/radicalbig [(ωc−ωm)−i(κc−κm)]2+ 4g2 2(3) This equation can be utilized to fit the coupling strength of the coupled magnon-photon system. We extract the frequencies of the measurement dips from Fig. 3(e) in the main text at each scanning magnetic field point. To perform the fitting, we use QKIT [2], an open-source Python package developed by the Karlsruhe Institute of Technology. The fitting procedures and results are presented in Fig. S2. In Fig. S2(a), the measurement dips are extracted by identifying the minimum transmission power at specific magnetic fields, which are marked in red (for the upper branch) and blue (for the lower branch) colors in Fig. S2(b), respectively. The curves in Fig. S2(c) are calculated using the fitted coupling strength (62 MHz), which closely matches the extracted transmission dips from the experiment. FIG. S2. (a) Heat map of the transmission spectrum as a function of the magnetic field strengths. (b) Extracted frequencies of transmission dips at different magnetic fields overlaid with the measured spectra, with upper and lower branches denoted in red and blue, respectively. (c) Comparison between the fitted (curves) and extracted (dots) mode frequencies.3 III. MAGNON DISSIPATION RATE FITTING Equqation (1) in the main text, derived from the input-output theory [3], reveals that the transmission spectrum depends on the coupling strength gbetween the magnon and photon modes, as well as the dissipation rates κcandκm. In the preceding sections, we successfully fitted the microwave photon dissipation rate ( κc= 1.95 MHz) and coupling strength ( g= 62 MHz). Consequently, we can extract the magnon dissipation rate κmthrough single-parameter fitting based on the transmission spectrum. The simplest approach to calculate κmis to use the on-resonance relation: κh=κm+κc 2(4) Here, κhrepresents the dissipation rate of the hybridized mode, which is the average of the magnon and photon dissipation rates. In Fig. S3(b), the line cut at the on-resonance position, with a bias magnetic field strength of 1000 Gauss, reveals a fitted dissipation rate of approximately 15 MHz, indicating κmis around 13 MHz. Given the low signal-to-noise ratio of the transmission spectrum in Fig. S3(b), we performed additional calculations to validate the fitted magnon dissipation rate. In Fig. S3(c), we present a color plot of the simulated transmission spectra at various magnetic fields using Eq. (1) from the main text, with the magnon dissipation rate κmset to 28 MHz. This calculated data match well with the experimental measurements shown in Fig. S3(a). FIG. S3. (a) Heat map of the transmission spectrum as a function of magnetic field strength. (b) Transmission line cut at the coupling center where the magnon mode resonates with the photon mode, indicated by the red dashed line in (a). The linewidth of the on-resonance dip is determined to be 30 MHz. (c) Heat map displaying the calculated transmission spectrum using Eq. (1) from the main text, with the magnon dissipation rate κmset to 28 MHz. IV. SIMULATION ON TRANSMISSION SPECTRUM To assess the sensitivity of Eq. (1) in the main text with respect to the fitting parameter κm, we present a color map of the calculated spectrum with varying κmvalues in Fig. S4. As κmincreases from 1 MHz in Fig. S4(a) to 50 MHz in Fig. S4(f), the frequency regimes in which the hybridized dip is observable gradually diminish. It is evident that Fig. S4(d), where κmis set to 13 MHz, closely resembles our experimental results. [1] R. W. Boyd, Nonlinear Optics (Elsevier, Academic Press, 2008). [2] qkitgroup, qkit (2023), https://github.com/qkitgroup/qkit. [3] D. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin, Germany, 2008).4 FIG. S4. Color plot displaying the calculated transmission spectrum using Eq.(2) in the main text. The magnon dissipation rateκmin Figures (a)-(e) are set to be 1 MHz, 10 MHz, 20 MHz, 30 MHz, 40 MHz, and 50 MHz, respectively.

2023-12-17

Yttrium iron garnet (YIG) magnonics has sparked extensive research interests toward harnessing magnons (quasiparticles of collective spin excitation) for signal processing. In particular, YIG magnonics-based hybrid systems exhibit great potentials for quantum information science because of their wide frequency tunability and excellent compatibility with other platforms. However, the broad application and scalability of thin-film YIG devices in the quantum regime has been severely limited due to the substantial microwave loss in the host substrate for YIG, gadolinium gallium garnet (GGG), at cryogenic temperatures. In this study, we demonstrate that substrate-free YIG thin films can be obtained by introducing the controlled spalling and layer transfer technology to YIG/GGG samples. Our approach is validated by measuring a hybrid device consisting of a superconducting resonator and a spalled YIG film, which gives a strong coupling feature indicating the good coherence of our system. This advancement paves the way for enhanced on-chip integration and the scalability of YIG-based quantum devices.

Cryogenic hybrid magnonic circuits based on spalled YIG thin films

2312.10660v2

Direct probing of strong magnon-photon coupling in a planar geometry Mojtaba Taghipour Kaash,1Dinesh Wagle,1Anish Rai,1 Thomas Meyer,2John Q. Xiao,1and M. Benjamin Jung eisch1, 1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States 2THATec Innovation GmbH, D-67059 Ludwigshafen, Germany (Dated: February 28, 2022) We demonstrate direct probing of strong magnon-photon coupling using Brillouin light scattering spectroscopy in a planar geometry. The magnonic hybrid system comprises a split-ring resonator loaded with epitaxial yttrium iron garnet thin lms of 200 nm and 2.46 m thickness. The Brillouin light scattering measurements are combined with microwave spectroscopy measurements where both biasing magnetic eld and microwave excitation frequency are varied. The cooperativity for the 200 nm-thick YIG lms is 4.5, and larger cooperativity of 137.4 is found for the 2.46 m-thick YIG lm. We show that Brillouin light scattering is advantageous for probing the magnonic character of magnon-photon polaritons, while microwave absorption is more sensitive to the photonic character of the hybrid excitation. A miniaturized, planar device design is imperative for the potential integration of magnonic hybrid systems in future coherent information technologies, and our results are a rst stepping stone in this regard. Furthermore, successfully detecting the magnonic hybrid excitation by Brillouin light scattering is an essential step for the up-conversion of quantum signals from the optical to the microwave regime in hybrid quantum systems. The emergent properties of hybrid systems are promis- ing for a wide range of quantum information applica- tions. In particular, light-matter interaction has been at the forefront of contemporary studies on hybrid quantum systems. To this end, hybrid magnonic systems based on the coupling of magnons, the elementary excitations of magnetic media, and photons have gained increased at- tention [1{4]. Magnons display a highly tunable disper- sion, while they can be used for coherent up- and down conversion between microwave and optical photons [5{ 9]. In addition, magnons can serve in quantum memory applications owing to their collective behavior and ro- bustness [10]. A critical requirement for coherent information trans- fer based on magnons is a high cooperativity, which means that the coupling between the two disparate types of excitations, i.e., the photonic and the magnonic sub- systems, exceeds the loss rates of either subsystem. This is known as the strong coupling regime in the language of quantum information. In this strong coupling regime, information can be eciently exchanged, potentially en- abling ecient transduction applications. Another pre- requisite for large scale quantum information processing and transfer applications is the conversion between opti- cal and microwave frequencies. Previous microwave-to- optical transduction studies based on ferromagnets ei- ther employed Brillouin scattering of optical whispering gallery modes by magnetostatic modes [6{8] or coupling of the microwave eld through a cavity mode concomi- tant with the coupling of the optical eld through the Kittel mode via Faraday and inverse Faraday eects [5]. Most of these prior works relied on macroscopic samples made of bulk yttrium iron garnet (YIG) crystals. Uti- lizing YIG is advantageous as it has a large spin density mbj@udel.eduand narrow linewidth [11{14]. However, scalable on-chip solutions require device miniaturization. Therefore, pla- nar microwave resonators are advantageous for building hybrid magnonic networks and circuits [15]. They of- fer great exibility in terms of circuit design; they are compatible with lithographic fabrication processes and the prevalent complementary metal-oxide-semiconductor (CMOS) platform [16]. Furthermore, planar microwave resonators typically have a smaller eective volume than their three-dimensional counterparts and can provide an enhanced coupling with magnetic dipoles [17, 18]. In ad- dition, they potentially simplify the integration of optical components [9] enabling simplied optomagnonic device concepts. Here, we demonstrate coherent microwave-to-optical up-conversion using strong magnon-photon coupling in a split-ring resonator/YIG thin lm hybrid circuit. We directly probe the coupling in YIG lms of 200 nm and 2.46m thickness by conventional and microfocused Bril- louin light scattering (BLS) spectroscopy and compare these optical results to microwave absorption measure- ments. Clear avoided level crossings are observed evi- dencing the hybridization of the magnon and microwave photon modes in the strong coupling regime. In addition, we identify contributions of higher order magneto-static surface spin waves. The cooperativity for the 200 nm- thick YIG lms is 4.5 and 137.4 for the 2.46 m-thick YIG lm. On the one hand, we nd that BLS is advan- tageous for probing the magnonic character of magnon- photon polaritons, while microwave absorption is found to be more sensitive to the photonic character. On the other hand, detecting the magnonic hybrid excitation by Brillouin light scattering demonstrates a coherent con- version of microwave to optical photons. The coherent microwave-to-optical up-conversion pro- cess based on the strong magnon-photon coupling is il- lustrated in Fig. 1(a). The magnonic hybrid system com-arXiv:2202.12696v1 [cond-mat.mtrl-sci] 25 Feb 20222 acBLS Lightbd FIG. 1. (a) Schematic illustration of the coupling process be- tween the microwave photon (MW) mode of the split-ring res- onator (SRR) with the magnon mode of the YIG lm, where pandmare the dissipation rates of microwave photon and magnon, respectively, and geis their mutual coupling strength. Microwave-to-optical up-conversion is achieved by coupling the incident microwave photons via the the SRR to the magnon mode that interacts with the BLS laser pho- tons. (b) A typical BLS spectrum with the Rayleigh peak at 0 GHz and the Stokes signal at around -5 GHz. The ver- tical red dashed lines show the region of interest (ROI). (c) Experimental setup: The resonator consists of a square SRR patterned next to the microwave feed line. The YIG lm is placed on the top of the SRR. An external biasing magnetic eld (iny-direction) magnetizes the sample during the BLS and MW measurements. The probing BLS beam is focused onto the surface of the YIG lm. (d) Top view of the SRR with the dimensions as dened in the text. prises a split-ring resonator (SRR) loaded with epitaxial YIG thin lms. The microwave photons interact with the SRR mode that exhibits a dissipation rate of pat its resonance frequency. The SRR mode couples with the magnon mode of the YIG sample with a coupling con- stant ofge, while the YIG sample dissipates its energy at the ratem. Finally, the excited magnons interact and couple with the incident BLS probe beam. The up-conversion process is realized by two sepa- rate sets of measurements: in-plane magnetic eld de- pendent microwave (MW) absorption measurements and BLS (both microfocused and conventional) of RF driven magnetization dynamics. A typical BLS spectrum is shown in Fig. 1(b), where the region of interest (ROI) is limited to the frequency range of hybrid excitation (here: Stokes peak, I S). The elastically scattered light is centered at 0 GHz. The probing BLS laser beam is focused on the sample surface; therefore, we detect magnons modes only in the top layer [19], while both the top and bottom layers contribute in the MW absorp- tion measurements. However, since each sample is grown under the same fabrication procedure, similar properties are expected from each YIG-lm layer in each sample. Figure 1(c) depicts the experimental conguration con- sisting of the square SRR in the vicinity of an MW feed a bdc SRRFMR minℎ!"max 𝐴/𝑚 SRRFMRFIG. 2. (a) SRR resonance obtained by HFSS simulations (Qsim= 83:1) and corresponding experimentally realized res- onance (Qexp= 94:0). Data shown in blue, corresponding ts are shown by in red dashed lines. (b) RFmagnetic eld ( hrf) distribution obtained by HFSS simulations. (c) MW absorp- tion measurements of the magnon-photon hybridization (here, YIG lm thickness: 2 :46m), where the false color represents the S 12transmission parameter. (d) S 12transmission param- eter versus frequency fat selected biasing magnetic elds as shown by white dashed lines in (c). line loaded with a YIG sample placed on the top and in the presence of a biasing in-plane magnetic eld applied along they-axis. For the MW absorption measurement, a vector network analyzer (VNA) is used to record the eld-dependent transmission parameter S 12with an out- put power of +13 dBm connected to P1 and P2. We use a continuous single-mode 532-nm wavelength laser for the BLS measurements that is focused on the YIG lm's surface [see Fig. 1(c)]. A MW generator provides the a RF signal to the feed line (P1) with output pow- ers of +20 dBm for microfocused BLS and +27 dBm for conventional BLS measurements. The BLS process can be described by the inelastic scattering of laser photons with magnons [20]. Since this process is energy and mo- mentum conserving, inelastically scattered photons carry information about the probed magnons [21], which we analyze using a high-contrast tandem Fabry-P erot inter- ferometer. Two dierent objective lenses are used for the BLS measurements: for the microfocused measure- ments, a high-numerical-aperture (NA = 0.75) objective lens with a working distance of 4 mm is used, while a lens with a focal lens of 40 mm and a diameter of 1 inch is used for the conventional measurement setup. We designed and optimized the SRR via ANSYS HFSS to exhibit a resonance ( f0) at 5.1 GHz, which agrees with the experimentally observed result (4.9 GHz) as shown in Fig. 2(a). Figure 1(d) illustrates the top view of the SRR with the following dimensions: the SRR's outer and inner widths ofa= 4:5 mm andb= 1:5 mm, the gap between3 abc FIG. 3. False color-coded spectra of the magnon-photon hybridization of the 2.46 m-thick YIG lm. Results obtained by (a) microwave absorption measurements, where S12is plotted versus fand0H, (b) microfocused BLS spectroscopy, and (c) conventional BLS spectroscopy. In the BLS measurements, the Stokes peak [compare to Fig. 1(b)] is plotted in logarithmic scale versus fand0H. The black dashed lines are the ts to Eqs. (1) and (2). the SRR and the feed line g= 0:2 mm, and the feed line's width ofw= 0:4 mm. The SRR is fabricated by etching one side of Rogers RO3010 laminate with a dielectric con- stant of 10.20 0.30 and copper thickness of 35 m that is coated on both sides of the substrate. By tting the res- onance data to a Lorentzian function with full-width at half maximum (FWHM), we determine the quality factor (Q=f0=
fFWHM ) of the resonator to be Qsim=83.1 for the simulation and Qexp=94.0 for the experiment [shown with the red dashed lines in Fig. 2(a)]. The 2D prole of the modeled RF-magnetic eld hrfon resonance is shown in Fig. 2(b). hrfis the most intense and uniform at the center of the SRR. The SRR is loaded with low-loss YIG lms placed on the top of the center of the SRR [for details on broadband ferromagnetic resonance measure- ments we refer to the supplemental material (SM)]. We compare the results of two YIG-lm thicknesses: the lat- eral dimensions of the 2.46 m thick square-shaped sam- ple is 5.3 mm 5.3 mm, while the 200 nm thick-sample is parallelogram-shaped with a base and height of 10 mm and 7.5 mm, respectively. Both samples are grown on 500m-thick gadolinium gallium garnet substrates by liquid phase epitaxy on both sides of the substrates. Fig. 2(c) shows a typical false color-coded microwave absorption spectrum of the magnon-photon hybridiza- tion (here, YIG lm thickness: 2 :46m), where the color represents the transmission parameter. In the eld/frequency region where the uncoupled photon and the magnon modes would cross, we observe the behavior of an eective two-level system, where the two disparate subsystems couple electromagnetically with the coupling strengthge. The coupling is quantied by the cooper- ativityC=g2 e=mp. The mode coupling lies in the strong regime if geis larger than the loss rate of YIG, m, and the SRR, p, respectively [22]; thus, C > 1.This is shown more in detail in Fig. 2(d), where S 12is plotted versusffor dierent elds from 86 to 102 mT close tothe avoided crossing as indicated by white dashed lines in Fig. 2(c). At high elds (e.g., at 102 mT), the higher frequency mode (FMR mode) has a lower intensity than the lower frequency mode (SRR mode). By sweeping the eld from higher to lower values, the FMR mode approaches the SRR mode. In this transition regime, the modes switch the magnitude of their intensities: at 94 mT, both modes have the same intensity, and the fre- quency gap between them is almost minimum. Further decreasing the eld magnitude to 86 mT results in the modes switching their intensities and moving apart. This behavior describes an avoided level crossing indicative of the formation of magnon-photon polaritons [23, 24]. We model the photon-magnon hybridization using a coupled two harmonic oscillator model with frepre- senting the hybridized mode frequencies: f=fSRR+fFMR 2sfSRRfFMR 22 +ge 22 ; (1) wheregeis the coupling strength, fSRRis the uncou- pled SRR resonance, fFMR is the ferromagnetic resonance of YIG that increases as the eld is increased and is given by the Kittel formula fFMR = 20p H(H+Me); (2) where is the gyromagnetic ratio and Meis the eec- tive magnetization (see SM). The systematic deviations of the BLS data from the FMR tting are discussed in the SM. The microwave absorption measurement result of the 2:46m-thick YIG lm shows a clear avoided crossing which centers at 96 mT, Fig. 3(a). As is visible from the gure, the signal is particularly strong before and after the avoided crossing ( <82 mT and >110 mT). This eld-independent signal is the SRR resonance mode.4 FIG. 4. Conventional BLS spectra of the 2.46 m-thick YIG lm. Dashed lines represent ts to Eq. (3). The black bold dashed line represents the k= 0 (n= 0) mode, which is the FMR mode, while the higher-lying dashed lines are MSSW modes (n= 1,..., 12) with k=n=2l. However, a pronounced avoided crossing is observed when the eld-dependent FMR mode of YIG approaches the SRR resonance at 96 mT leading to the formation of a hybridization. The upper and lower frequency modes and the uncoupled FMR mode are tted to the experimental results according to Eq. (1) and Eq. (2), respectively. Using a similar eld/frequency sweep range as in the microwave absorption measurements, we probe the magnon-photon hybridized state by microfocused BLS [Fig. 3(b)]. Here, the Stokes BLS intensity in logarithmic scale is plotted. The eld is swept from 110 to 82 mT in 0.3 mT eld steps after saturating at 200 mT. The MW frequency excites the sample from 4.25 to 5.10 GHz in 12 MHz steps. The two hybridized modes are detectable, similar to the MW absorption measurements. Note that, in addition to the coupled resonances, we detect a con- tribution of modes directly excited by the feed line [12] in the BLS experiments. We will discuss these modes below. BLS's successful detection of the strongly cou- pled magnon-photon state demonstrates a coherent microwave-to-optical up-conversion based on the scheme shown in Fig. 1(a). Interestingly, the intensity distri- bution detected in BLS is reverse to the MW absorption technique: BLS is more sensitive in probing the magnonic character of the magnon-photon polariton compared to MW absorption measurements shown in Fig. 3(a), which is more sensitive to the photonic character of the hybrid excitation conrming previous reports [9]. By tting the experimental data to Eq. (1), we ex- tract the magnon-photon coupling strength ge. Ferro- magnetic resonance measurements (SM) yield the fol- lowing parameters: 0Me= 183:5 mT and =2=28:2 GHzT1, which we use to t the microfocused BLS results [Figs. 3(b,c)] to Eq. (1), we obtain ge=2= 114:6 MHz. By calculating the dissipation rates of the microwave photon ( p=2= 25:8 MHz) and the magnon (m=2= 3:9 MHz), we can obtain a cooperativity of C= 137:4, which fulls the conditions C > 1 and ge>p;m. We compare the microfocused BLS experiments to con- ventional BLS measurements as is shown in Fig. 3(c). We use a lens with a smaller numerical aperture than the ob- jective lens used in the microfocused setup. However, the laser beam spot size is signicantly larger, and hence, it covers a larger area of the YIG lm leading to a stronger signal intensity. Due to the stronger signal strength, we are able to detect modes inaccessible by the microfocused system as further evidenced in Fig. 4. The additional ne features revealed by the conventional BLS measure- ments lie in the anticrossing region parallel to the Kittel mode. These modes are due to the excitations of higher- order wavenumber spin-wave modes directly excited by the MW feed line. These modes occur at frequencies higher than the Kittel mode for a given magnetic eld and are identied as magnetostatic surface spin waves (MSSWs) that propagate in the lm plane in a direction perpendicular to the applied eld [12, 25{27]. We model them by: fMSSW = 20q H(H+Me) +M2 e(1e2kd)=4; (3) wheredis the thickness of the sample, k=n=2lis the spin-wave wavevector, lis the length of the square-shaped sample and nis the mode number with n= 0 being the uniform Kittel mode. =2= 28:2 GHzT1and0Me = 183.5 mT both of which are obtained from the tting of the lowest lying mode, Eq. (2). The dashed lines above the main modes in Fig. 4 shows ts of the experimental data to Eq. (3) for n= 0;1;:::;12. Here, the bold dashed line represents the k= 0 (n= 0) mode, which is the FMR mode, while the other dashed lines are the higher-order MSSW modes ( n= 1,..., 12). These higher-order MSSW modes have wavevectors of k= 3:5103rad/m for n= 12, which is within the detectable wavevector range of our conventional system ( kmax= 6:9 rad/m). While most recent works on strong-magnon photon coupling utilized micrometer-thick-YIG or YIG spheres [28{31], sample miniaturization is imperative for a scal- able on-chip solutions. In the following, we demonstrate magnon-photon coupling in a miniaturized 200 nm-thick YIG lm. Using the identical 2D planar resonator as used for studying the 2.46 m lm, we observe mode anti-crossing by the microwave absorption technique as shown in Fig. 5(a). As is shown Fig. 5(b), we are unable to detect a suciently strong signal of the hybridized excitation in the microfocused measurements. Surpris- ingly, the Kittel mode directly excited by the feed line is signicantly stronger than the magnon-photon coupled modes. However, as we switch from the microfocused to the conventional BLS setup, not only the Kittel mode5 abc FIG. 5. A typical false color-coded spectrum of the magnon-photon hybridization for the 200 nm-thick YIG lm using the (a) microwave absorption technique, where S12is plotted versus fand0H, (b) microfocused BLS technique, and (c) BLS technique with a conventional objective lens. The dashed lines are the tted plots to Eqs. (1) and (2). becomes more intense, but also the two hybridized mode can be detected in the spectra [Fig. 5(c)]. Fits to the experimental data agree reasonably well as shown by the black dashed lines. From the combined optical and mi- crowave experiments, we extract the following parame- ters:0Me= 187:3 mT, =2= 28:2 GHzT1, and ge=2= 37:3 MHz. The photon and magnon dissi- pation rates are found to be p=2= 25:8 MHz and m=2= 12:1 MHz, respectively. Therefore, the coop- erativityC= 4:5, fullling both conditions C > 1 and ge>  p;mand, hence, the mode hybridization is in the strong coupling regime. Higher modes similar to the ones observed in the 2.46 m are absent in the spectra of the 200 nm lm since the eld/frequency separation of the higher-order modes decreases as the YIG thickness decreases [32]. In summary, we showed direct probing of strong magnon-photon coupling using Brillouin light scatter- ing spectroscopy in a planar geometry. The optical measurements are combined with microwave spec- troscopy experiments where both biasing magnetic eld and microwave excitation frequency are varied. The miniaturized YIG sample of 200 nm thickness exhibits a cooperativity of 4.5, while 2.46 m-thick lm showed a larger cooperativity of 137.4. We nd that Brillouin light scattering is advantageous for probing themagnonic character of magnon-photon polaritons, while microwave absorption is more sensitive to the photonic character of the hybrid excitation. In addition, modes directly excited by the feed line signicantly contribute to the optical measurements: they are detected in the gaped region between the two coupled magnon-photon modes. The detection of the magnonic hybrid excitation by Brillouin light scattering can be understood as an up-conversion mechanism of signals from the optical to the microwave regime in the magnonic hybrid systems. The planar structure presented here enables spatially- resolved imaging of magnon-photon polaritons that can serve as a platform for studying magnonics strongly coupled to microwave photons. ACKNOWLEDGMENT We thank Prof. Matthew Doty, University of Delaware, for valuable discussions. Research supported by the U.S. Department of Energy, Oce of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0020308. The authors acknowledge the use of facilities and instrumentation supported by NSF through the University of Delaware Materials Re- search Science and Engineering Center, DMR-2011824. [1] C.-M. Hu, arXiv preprint arXiv:1508.01966 (2015). [2] M. Harder and C.-M. Hu, Solid State Physics 69, 47 (2018). [3] Y. Li, C. Zhao, W. Zhang, A. Homann, and V. Novosad, APL Materials 9, 060902 (2021). [4] B. Bhoi and S.-K. Kim, in Solid State Physics , Vol. 70 (Elsevier, 2019) pp. 1{77. [5] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. B 93, 174427 (2016).[6] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett.117, 123605 (2016). [7] J. A. Haigh, A. Nunnenkamp, A. J. Ramsay, and A. J. Ferguson, Phys. Rev. Lett. 117, 133602 (2016). [8] A. Osada, A. Gloppe, R. Hisatomi, A. Noguchi, R. Ya- mazaki, M. Nomura, Y. Nakamura, and K. Usami, Phys. Rev. Lett. 120, 133602 (2018). [9] S. Klingler, H. Maier-Flaig, R. Gross, C. M. Hu, H. Huebl, S. T. B. Goennenwein, and M. Weiler, Appl. Phys. Lett. 109, 072402 (2016).6 [10] K. Schultheiss, R. Verba, F. Wehrmann, K. Wag- ner, L. K orber, T. Hula, T. Hache, A. K akay, A. A. Awad, V. Tiberkevich, A. N. Slavin, J. Fassbender, and H. Schultheiss, Phys. Rev. Lett. 122, 097202 (2019). [11] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Physical review letters 113, 083603 (2014). [12] B. Bhoi, T. Cli, I. Maksymov, M. Kostylev, R. Aiyar, N. Venkataramani, S. Prasad, and R. Stamps, Journal of Applied Physics 116, 243906 (2014). [13] M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, Phys. Rev. Lett. 121, 137203 (2018). [14] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [15] V. Castel, A. Manchec, and J. Ben Youssef, IEEE Mag- netics Letters 8, 1 (2017). [16] A. V. Chumak, P. Kabos, M. Wu, C. Abert, C. Adel- mann, A. Adeyeye, J. Akerman, F. G. Aliev, A. Anane, A. Awad, C. H. Back, A. Barman, G. E. W. Bauer, M. Becherer, E. N. Beginin, V. A. S. V. Bittencourt, Y. M. Blanter, P. Bortolotti, I. Boventer, D. A. Bozhko, S. A. Bunyaev, J. J. Carmiggelt, R. R. Cheenikundil, F. Ciubotaru, S. Cotofana, G. Csaba, O. V. Dobro- volskiy, C. Dubs, M. Elyasi, K. G. Fripp, H. Fulara, I. A. Golovchanskiy, C. Gonzalez-Ballestero, P. Graczyk, D. Grundler, P. Gruszecki, G. Gubbiotti, K. Guslienko, A. Haldar, S. Hamdioui, R. Hertel, B. Hillebrands, T. Hioki, A. Houshang, C. M. Hu, H. Huebl, M. Huth, E. Iacocca, M. B. Jung eisch, G. N. Kakazei, A. Khi- tun, R. Khymyn, T. Kikkawa, M. Kl aui, O. Klein, J. W. K los, S. Knauer, S. Koraltan, M. Kostylev, M. Krawczyk, I. N. Krivorotov, V. V. Kruglyak, D. Lachance-Quirion, S. Ladak, R. Lebrun, Y. Li, M. Lindner, R. Mac^ edo, S. Mayr, G. A. Melkov, S. Mieszczak, Y. Nakamura, H. T. Nembach, A. A. Nikitin, S. A. Nikitov, V. Novosad, J. A. Otalora, Y. Otani, A. Papp, B. Pigeau, P. Pirro, W. Porod, F. Porrati, H. Qin, B. Rana, T. Reimann, F. Riente, O. Romero-Isart, A. Ross, A. V. Sadovnikov, A. R. San, E. Saitoh, G. Schmidt, H. Schultheiss, K. Schultheiss, A. A. Serga, S. Sharma, J. M. Shaw, D. Suess, O. Surzhenko, K. Szulc, T. Taniguchi, M. Urb anek, K. Usami, A. B. Ustinov, T. van der Sar, S. van Dijken, V. I. Vasyuchka, R. Verba, S. V. Kus-minskiy, Q. Wang, M. Weides, M. Weiler, S. Wintz, S. P. Wolski, and X. Zhang, Roadmap on spin-wave computing (2021), arXiv:2111.00365 [physics.app-ph]. [17] Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Ho- mann, and W. Zhang, Phys. Rev. Lett. 123, 107701 (2019). [18] J. T. Hou and L. Liu, Phys. Rev. Lett. 123, 107702 (2019). [19] F. Kargar and A. A. Balandin, Nat. Photon. , 1 (2021). [20] M. B. Jung eisch, Inelastic Scattering of Light by Spin Waves. Optomagnonic Structures , edited by E. Almpanis (World Scientic, 2020). [21] M. Madami, G. Gubbiotti, S. Tacchi, and G. Carlotti, in Solid state physics , Vol. 63 (Elsevier, 2012) pp. 79{150. [22] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Physical review letters 113, 156401 (2014). [23] B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, arXiv preprint arXiv:2106.09312 (2021). [24] B. Bhoi and S.-K. Kim, in Solid State Physics , Vol. 71 (Elsevier, 2020) pp. 39{71. [25] G. B. Stenning, G. J. Bowden, L. C. Maple, S. A. Gre- gory, A. Sposito, R. W. Eason, N. I. Zheludev, and P. A. de Groot, Optics Express 21, 1456 (2013). [26] D. Zhang, W. Song, and G. Chai, Journal of Physics D: Applied Physics 50, 205003 (2017). [27] N. Lambert, J. Haigh, and A. Ferguson, Journal of Ap- plied Physics 117, 053910 (2015). [28] B. Bhoi, S.-H. Jang, B. Kim, and S.-K. Kim, Journal of Applied Physics 129, 083904 (2021). [29] X. Zhang, K. Ding, X. Zhou, J. Xu, and D. Jin, Physical Review Letters 123, 237202 (2019). [30] Y. S. Ihn, S.-Y. Lee, D. Kim, S. H. Yim, and Z. Kim, Physical Review B 102, 064418 (2020). [31] I. Boventer, C. D or inger, T. Wolz, R. Mac^ edo, R. Le- brun, M. Kl aui, and M. Weides, Physical Review Re- search 2, 013154 (2020). [32] B. Kalinikos and A. Slavin, Journal of Physics C: Solid State Physics 19, 7013 (1986).

2022-02-25

We demonstrate direct probing of strong magnon-photon coupling using Brillouin light scattering spectroscopy in a planar geometry. The magnonic hybrid system comprises a split-ring resonator loaded with epitaxial yttrium iron garnet thin films of 200 nm and 2.46 $\mu$m thickness. The Brillouin light scattering measurements are combined with microwave spectroscopy measurements where both biasing magnetic field and microwave excitation frequency are varied. The cooperativity for the 200 nm-thick YIG films is 4.5, and larger cooperativity of 137.4 is found for the 2.46 $\mu$m-thick YIG film. We show that Brillouin light scattering is advantageous for probing the magnonic character of magnon-photon polaritons, while microwave absorption is more sensitive to the photonic character of the hybrid excitation. A miniaturized, planar device design is imperative for the potential integration of magnonic hybrid systems in future coherent information technologies, and our results are a first stepping stone in this regard. Furthermore, successfully detecting the magnonic hybrid excitation by Brillouin light scattering is an essential step for the up-conversion of quantum signals from the optical to the microwave regime in hybrid quantum systems.

Direct probing of strong magnon-photon coupling in a planar geometry

2202.12696v1

Unveiling the polarity of the spin-to-charge current conversion in Bi 2Se3 J. B. S. Mendes1, *, M. Gamino2,3, R. O. Cunha1,4, J. E. Abrão2, S. M. Rezende2 and A. Azevedo2 1Departamento de Física, Universidade Federal de Viçosa, 36570 -900 Viçosa, MG, Brazil 2Departamento de Física, Universidade Federal de Pernambuco, 50670 -901 Recife, PE, Brazil 3Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078 -900 Natal, RN, Brazil 4Centro Interdisciplinar de Ciências da Natureza, Universidade Federal da Integração Latino - Americana, 85867 -970 Foz do Iguaçu, PR, Brazil We report an investigation of the spin - to charge -current conversion in sputter -deposited films of topological insulator Bi2Se3 onto single crystalline layers of YIG ( Y3Fe5O12) and polycrystalline films of Permalloy ( Py = Ni81Fe19). Pure spin current was injected into the Bi 2Se3 layer by means of the spin pumping process in which the spin precession is obtained by exciting the ferromagnetic resonance of the ferromagnetic film. Th e spin -current to charge -current conversion, occurring at the Bi 2Se3/ferromagnet interface, was attribute to the inverse Rashba -Edelstein effect (IREE ). By analyzing the data as a function of the Bi 2Se3 thickness we calculated the IREE length used to characterize the efficiency of the conversion process and found that 1.2 pm ≤|𝜆𝐼𝑅𝐸𝐸|≤ 2.2 pm . These results support the fact that the surface states of Bi 2Se3 have a dominant role in the spin - charge conversion process, and the mechanism based on the spin diffusion process plays a secondary role. We also d iscovered that the spin - to charge -current mechanism in Bi 2Se3 has the same polarity as the one in Ta, which is the opposite to the one in Pt. The combination of the magnetic properties of YIG and Py , with strong spin -orbit coupling and dissipationless surface states topologically protected of Bi 2Se3 might lead to spintronic devices with fast and efficient spin-charge conversion . *Corresponding author: Joaquim B. S. Mendes, joaquim.mendes@ufv.br The investigation of new materials with strong spin -orbit coupli ng (SOC) has improve d the means for the generation and detection of spin currents in nonmagnetic materials. This study gave birth to the emergent subfield of spintronics , named spin orbitronics [1-3]. Despite being a subject of interest for many years to the investigation of magnet ocrystalline anisotropy , the SO C has been pivotal to the revolution that spintronics has undergone in the last decade . In particular, heavy metals, such as Pt and Pd, have been used as efficient materials for mutual conversion between spin and charge currents via direct and inverse spin Hall effects (SHE and ISHE , respectively ) [4-7]. In the l ast decade, there has been significant progress towards developing materials with strong SOC , which can produce current -driven torques strong enough to switch the magnetization of a ferromagnetic (FM) layer in a spin-valve structure. Such improvement in th e SHE has been observed in a wide variety of systems that include enhancement of the SOC driven by surface roughness and volume impurities [8-11], at 2D materials [12] and interfacial effects [13-15]. Indeed, many spintronics -phenomena driven by interface -induced spin-orbit interaction have been extensively investigated over the last few years . For instance, the inverse Rashba - Edelstein effect [16,17] (IREE) was considered for converting spin into charge current [13 ] in many interface systems [18-25]. Moreover , other materials with outstanding spintronics properties, the topological insulators (TIs) , stand out for the mutual conversion between charge and spin due to the large SOC in surface states that locks spin to momentum [ 26-29]. TIs are a new class of quantum materials that presen t insulating bulk, but metallic dissipationless surface states topologically protected by time reversal symmetry, opening several possibilities for practical applications in many scientific arenas including spintronics, quantum co mputation, magnetic monopoles, highly correlated electron systems , and more recently in optical tweezers experiments [30, 31 –34]. It is known that in TIs the effects of SOC are maximized because the electron ’s spin orientation is fix ed relative to its direction of propagation. Among the 3D TIs, Bi2Se3 is a unique material with large bandgap of 0.35 eV and its surface spectrum consists of single Dirac cone roughly centered within the gap [ 30]. In spite of the fact that the first investigations of spintronic s properties of TIs were performed i n samples grown by the Molecular Beam Epitaxy (MBE) [29,35], the sputtering deposition technique has been successfully used to grow high quality Bi 2Se3 [23, 28, 36]. The spin Hall angle ( 𝜃𝑆𝐻), used to quantify the mutual conversion between spin and charge current, has limited use in systems in which the cross -section of the charge -current - carrying layer is reduced. Owing to the transverse nature of the spin transport phenomena, SH E is a bulk effect occurring within a volume limited by t he spin -diffusion length (𝜆𝑠𝑑) [15]. For instance, when a 3D spin current density 𝐽𝑆 [𝐴𝑚2⁄] is injected through an interface with high SOC, 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 efficiency of conversion between spin- to charge current [2,13]. Not only the absolute value of 𝜆𝐼𝑅𝐸𝐸, but also its polarity must be of interest to understand the physics behind the interplay between spin and charge currents. Here we report an investigation of the spin - to charge current conversion in bilayers of Bi2Se3(t)/YIG (6 m), (YIG = Y 3Fe5O12, Yttrium Iron G arnet ) by means of the ferromagnetic resonance driven spin pumping (FMR -SP) technique . While the Bi 2Se3 films were grown by DC sputtering, the single -crystal YIG films were grown by Liquid Phase Epitaxy (LPE) onto (111) GGG ( =Gd 3Ga5O12) substrates . The pure spin-current density ( 𝐽𝑆), which flows across the Bi2Se3(t)/YIG interface due to the YIG magnetization precession , is converted in to a transversal charge current density (𝐽𝐶) that is detected by measuring a DC voltage between two edge contacts. The Bi 2Se3 samples were deposited on top of small pieces of YIG /GGG(111) cut from the same wafer , with thickness 6 µm, width of 1.5 mm and length of 3.0 mm. The YIG films have in -plane magnetization and thus the magnetic proximity effect is expected to shift the Dirac cone sideways along the momentum direction and does not open an exchange gap (i.e. in our heterostructures, the Dirac cone of the TI film will be preserved). The Bi 2Se3/YIG interface has the advantage over the Bi 2Se3/ferro magnetic -metal because it ensures cleaner interface and avoids current shunting as well as spurious spin rectification effects. Previously reported spin -to-charge current conversion experiments with sputtered Bi 2Se3/YIG were carried out in YIG grown by sputtering or MBE and , to the authors knowledge, there is no investigation about the polarity of 𝜆𝐼𝑅𝐸𝐸 [23, 28]. X-ray diffraction (XRD) analysis was carried out by mean s of out -of-plane scan as well as grazing incidence X -ray diffraction (GIXRD), which is more valuable for assessing ultra -thin film structures. Fig. 1(a) shows the out of plane XRD θ -2θ scan pattern of the Bi2Se3(6 nm)/ YIG(6µm)/GGG sample over a 2θ range between 20° and 70°. The pattern shown in Fig. 1(a) displays reflections associated with the (222) and (444) crystal planes of YIG, proving that the present YIG film is epitaxially grown on the GGG substrate. In the inset, we can se e the XRD spectrum at high resolution detailing the double peak corresponding to the (444) Bragg reflections of the GGG substrate and the epitaxial YIG in the (444) plane. In order to optimize the scattering contribution from the Bi2Se3 films, we used graz ing incidence X -ray diffraction (GIXRD) for investigating the Bi2Se3/YIG(6µm) /GGG samples. As shown in Fig. 1(b), the GIXRD data evidenced the diffraction peaks characteristic of the Bi2Se3 6 nm thick film, meaning that the film is polycrystalline and has a preferential texture oriented in the planes: (0 0 9), (0 0 15), (0 0 18), (0 0 21), which is in agreement with the literature [37, 38]. Figure 1(c) shows the X -ray reflectivity (XRR) data for Bi 2Se3(16.0 nm)/Si. The w ell-defined and the good periodicity of the Kiessig fringes 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 the YIG film surface with very small roughness (~0.2 nm). On the other hand, Fig. 1( e) shows the AFM image of the sputtered granular bismuth selenide thin film ( t = 4 nm) grown onto YIG/GGG substrate. The image shows that Bi 2Se3 film grown onto YIG favors the formation of a granular film, with grain sizes up to ~ 0.3 μm, and has a root -mean -square (RMS) surface roughness of about 1.0 nm. The typical energy -dispersive x -ray (EDX) spectrum of Bi2Se3 on the YIG film can be seen in Fig. 1(f). The EDX spectrum taken from an arbitrary region of the sample shows the presence only of yttrium (Y), iron (Fe), oxygen (O) of the YIG film; bismuth (Bi) and selenium (Se) of Bi 2Se3. The additional peak of the carbon (C) in the EDX spectrum is due to the presence of carbon tape used as support on which the samples are prepared for analysis. In the figure there are also the EDX -maps showing that the Bi and Se are evenly distributed over the entire surface of the film. Different regions of the sam ples were analyzed, in order to confirm the results of the EDX measurements. Fig. 1 (color online). (a) Out-of-plane XRD patterns (θ -2θ scans) of Bi2Se3 film grown on YIG/GGG substrate . The XRD spectrum at high resolution detailing the position s of the peaks of the YI G film and the GGG substrate is shown in the inset. (b) The GIXRD pattern of the Bi 2Se3(6 nm)/YIG/GGG sample. (c) XRR spectra of the Bi 2Se3 thin film (t ≈ 16 nm). The red solid line across the XRR data indicates the best fitting obtained for the thickness calibration. (d) AFM image of the YIG film surface. (e) AFM image of the surface of the Bi2Se3(4nm)/YIG(6µm)/GGG sample . (f) EDX spectrum (top) from an arbitrary region in the sample of Bi2Se3(6nm)/YIG and EDX -maps (bottom) showing that the Bi and Se are evenly distributed over the entire surface of the film. Figure 2 (a) illustrates the performed experiments of FMR -SP in which the sample w ith electrodes at the edges is mounted on the tip of a polyvinyl chloride (PVC) rod and is inserted, via a hole drilled at the bottom wall of a shorted X -band waveguide, in a position of maximum rf magnetic field and zero electric field. The loaded wavegui de is placed between the poles of an electromagnet that applies a DC magnetic field 𝐻⃗⃗ 0 perpendicular to the in -plane RF magnetic field, ℎ⃗ 𝑟𝑓. Electric contacts of silver were sputtered at the edges perpendicular to the larger sample size, so that the spin pumping voltage ( 𝑉𝑆𝑃) can be directly measured by means a nanovoltmeter. As the DC and RF magnetic field s are perpendicular to each other, the sample, attached to a goniometer, can be rotated so that we can investigate de angular dependence of both the ferromagnetic resonance (FMR) as well as 𝑉𝑆𝑃 . Field scan spectra of the derivative 𝑑𝑃𝑑𝐻⁄ , at a fixed frequency of 9.5 GHz, are obtained by modulating the field 𝐻⃗⃗ 0 with a small sinusoidal field at 1.2 kHz and using lock -in amplifier detectio n. Figure 2 (b) shows the FMR spectrum of a bare YIG sample (3.0 mm x 1.5 mm x 6.0 µm) obtained with the in -plane field applied normal to the larger length with an incident power of 54 mW . The strongest line corresponds to the uniform FMR mode ( 𝑘0≅0) in whic h the frequency is given by the Kittel’s equation 𝜔0= 𝛾√(𝐻0+𝐻𝐴)(𝐻0+𝐻𝐴+4𝜋𝑀𝑒𝑓𝑓), where 𝛾=2𝜋×2.8 𝐺𝐻𝑧𝑘𝑂𝑒⁄ and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀+ 𝐻𝑠≅1760 𝐺 for YIG. While the lines to the left of the uniform mode correspond to hybridized standing spin -wave su rface modes, the lines to the right correspond to the backward volume magnetostatic modes with quantized wave number 𝑘, subjected to the appropriated boundary conditions. All modes have similar half -width-half-maximum linewidth (HWHM ) of ∆𝐻𝑌𝐼𝐺= 1.4 Oe. As shown in Fig. 2(c), the deposition of a 4.0 nm thick film of Bi 2Se3 on the YIG layer increases the FMR linewidth to ∆𝐻𝐵𝑖2𝑆𝑖3𝑌𝐼𝐺⁄=1.7 Oe. This lin ewidth increas e is mostly due to the spin pumping process that transports spin angular moment out of the YIG layer [39,40]. As the YIG magnetization vector precesses, it injects a pure spin current density 𝐽 𝑆, that flows perpendicularly to the YIG/Bi 2Se3 interface with transverse spin polarization 𝜎̂, which is given by 𝐽 𝑆=(ℏ𝑔𝑒𝑓𝑓↑↓4𝜋𝑀𝑠2⁄)(𝑀⃗⃗ (𝑡)×𝜕𝑀⃗⃗ (𝑡)𝜕𝑡⁄), (1) where 𝑀𝑠 and 𝑀(𝑡) are the saturation and time dependent magnetization, respectively, and 𝑔𝑒𝑓𝑓↑↓ is the real part of the spin interface mixing conductance, that takes into account the forward and backward flows of the spin current [39]. It is important to mention that 𝐽𝑆 in Eq. (1) has units of (angular moment)/(time.area). As previously mentioned, 𝐽 𝑆 results in an increased magnetization damping due to the outflow of the spin an gular moment, and due to the IRE E it generates a transverse charge current in the Bi 2Se3 film. From the additional linewidth broadening, we can estimate the value of the spin mixing conductance 𝑔𝑒𝑓𝑓↑↓ of the Bi 2Se3/YIG interface. As 𝑔𝑒𝑓𝑓↑↓ is proportional to the additional linewidth broadening, i .e., 𝑔𝑒𝑓𝑓↑↓=(4𝜋𝑀𝑡𝐹𝑀ℏ𝜔⁄)(∆𝐻𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄− ∆𝐻𝑌𝐼𝐺), where 𝜔=2𝜋𝑓 and 𝑡𝐹𝑀 is the ferromagnetic (FM) layer thickness for thin FM films (or the 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 obtain 𝑔𝑒𝑓𝑓↑↓(𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄)≈5.4×1013cm−2. Figure 2 (color online) . (a) Schematic s of the FMR -SP technique in which we highlights the spin -current to charge -current conversion process at the interface . Field scan FMR absorption derivative , for a bare YIG film 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 pumping voltage measured for the bilayer Bi2Se3 (4 nm)/ YIG(6 m) at three different in -plane angles as illustrated in the inset, with an incident microwave power of 157 mW. The measurement of 𝑉𝑆𝑃 is carried out by sweeping the DC field with no AC field modulation and directly measuring 𝑉𝑆𝑃 that is generated between the two electrodes due to the spin-to-charge current conversion. Fig. 2(d) shows the spin pumping voltage, measured directly by a n anovoltmeter , in a bilayer of Bi 2Se3(4 nm)/YIG as function of the applied field for three in - plane directions given by 𝜙=0°,90°and 180° , as illustrated in the inset. As expected from the equation 𝐽 𝐶=𝜃𝑆𝐻(2𝑒ℏ⁄)(𝐽 𝑆×𝜎̂), where 𝜎̂∥𝐻⃗⃗ , the charge current flows in -plane so that the value of 𝑉𝑆𝑃 is maximum for 𝜙=0° and 𝜙=180° for blue and red curves, respectively. While it is null for 𝜙=90°, as shown by the black curve. The asymmetry between the positive and negative peaks is similar to that observed in other bilayer systems and can be attributed to a thermoelectric effect [41]. While Fig. 3(a) shows the field scans of 𝑉𝑆𝑃 for 39 𝑚𝑊 ≤𝑃𝑟𝑓≤157 𝑚𝑊, Fig. 3(b) shows the RF -power dependence of the peak voltage measured at 𝜙=180° . The linear dependence of the 𝑉𝑆𝑃 as a function of 𝑃𝑟𝑓 confirms that we are exciting the FMR in the linear regime. On the other hand, the dependence of the peak voltage as a function of the Bi 2Se3 layer thickness ( 𝑡𝐵𝑖2𝑆𝑒3) exhibits a more challe nging behavior. It decreases as 𝑡𝐵𝑖2𝑆𝑒3 increases in a clear opposition with results shown by materials in which the spin - to charge current conversion occurs in the bulk, as in Pt, for example. This decrease in the peak voltage was also observed in crystalline Bi 2Se3 grown by MBE [35]. We could try to explain the origin of the voltage in Bi2Se3/YIG as due to the spin pumping ISHE mechanism, by means spin diffusion model where the spin pumping voltage is given by [42 -44], 𝑉𝑆𝑃(𝐻)=𝑅𝑁𝑒𝜃𝑆𝐻𝜆𝑁𝑤𝑝𝑥𝑧𝜔𝑔𝑒𝑓𝑓↑↓ 8𝜋tanh(𝑡𝑁 2𝜆𝑁)(ℎ𝑟𝑓 ∆𝐻)2 𝐿(𝐻−𝐻𝑅)cos𝜙. (2) Here, 𝑅𝑁, 𝑡𝑁, 𝜆𝑁 and w, are respectively the resistance, thickness, spin diffusion length and width of the Bi 2Se3 layer, considering the microwave frequency = 2f, and 𝑝𝑥𝑧 is a factor that expresses the ellipticity and the spatial variation of the rf magnetization of the FMR mode. Also, ℎ𝑟𝑓 and ∆𝐻 are the applied microwave field and FMR linewidth, and 𝐿(𝐻−𝐻𝑅) represents the Lorentzian function. By assuming that 2𝜆𝑁≫𝑡𝑁, thus tanh(𝑡𝑁2𝜆𝑁⁄)≈𝑡𝑁2𝜆𝑁⁄ . Therefore, Eq. (2) can be written as 𝑉𝑆𝑃=(𝑅𝑁𝑓𝑒𝜃𝑆𝐻𝑤𝑝𝑥𝑧𝑔𝑒𝑓𝑓↑↓𝑡𝑁8⁄)(ℎ𝑟𝑓∆𝐻⁄)2. This expression does not depend on 𝜆𝑁, as expected for TIs, so that 𝑡𝑁 can be interpreted as an effective thickness attributed to the Bi 2Se3. From the measured quantities for the bilayer 𝐵𝑖2𝑆𝑒3(4 𝑛𝑚)𝑌𝐼𝐺⁄ , 𝑅𝑁=173 𝑘Ω, 𝑔𝑒𝑓𝑓↑↓≈5.4×1013𝑐𝑚−2, ℎ𝑟𝑓=0.055 𝑂𝑒, ∆𝐻=1.7 𝑂𝑒, 𝑉𝑆𝑃=44.7 𝜇𝑉 and 𝜃𝑆𝐻≅0.11 [as reported in Ref. [27] for average value of 𝜃𝑆𝐻], the effective thickness of the Bi 2Se3 layer is 𝑡𝑁= 0.46 Å. This small value is certainly unphysical for an effective layer that converts a 3D spin current density in a 3D charge current, as happens in the SHE effect. However, it provides an evidence that the spin-to-charge current conversion is dominated by surface states of the sputtered Bi2Se3 layer. To further verify that the spin - to charge -current conversion in 𝐵𝑖2𝑆𝑒3𝑌𝐼𝐺⁄ is dominated by the surface states, we can calculate the effective length 𝜆𝐼𝑅𝐸𝐸=(ℏ2𝑒⁄)𝐽𝐶𝐽𝑆⁄, where 𝑉𝐼𝑅𝐸𝐸= 𝑅𝐵𝑖2𝑆𝑒3𝑤𝐽𝐶 and 𝐽𝑆=(𝑒𝜔𝑝𝑥𝑧𝑔𝑒𝑓𝑓↑↓16𝜋⁄)(ℎ𝑟𝑓∆𝐻⁄)2𝐿(𝐻−𝐻𝑅), with 𝑝11=0.31, see Ref . [45]. Therefore, 𝜆𝐼𝑅𝐸𝐸 is given by, 𝜆𝐼𝑅𝐸𝐸=4𝑉𝐼𝑅𝐸𝐸 𝑅𝐵𝑖2𝑆𝑒3𝑒𝑤𝑓𝑔𝑒𝑓𝑓↑↓𝑝𝑥𝑧(ℎ𝑟𝑓∆𝐻⁄)2. (3) Figure 3 (color online). (a) Field scans of 𝑉𝑆𝑃 for several values of the incident microwave power. (b) Peak voltage value as a function of the incident microwave p ower measured for the bilayer of Bi 2Se3(4 nm)/YIG . (c) Peak voltage value measured as a function of the Bi 2Se3 thickness for an incident power of 157 mW. The inset shows the dependence of the spin pumping current ( 𝐼𝑆𝑃=𝑉𝑆𝑃𝑝𝑒𝑎𝑘𝑅⁄). (d) Field scans of VSP for the bilayer of Ta(2nm) /YIG obtained at same experimental configuration used to measure VSP in Bi2Se3/YIG . By comparing Fig. 3(d) with Fig. 2(d) we concluded that the VSP polarization of Bi 2Se3 is the same as in Ta. Using the physical quantities for the bilayer 𝐵𝑖2𝑆𝑒3(4 nm)𝑌𝐼𝐺⁄ , given above, we obtained |𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=4 nm)=(2.2±0.4)×10−12 m. For the other two bilayers we obtained, |𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=6 nm)=(2.0±0.5)×10−12 m, and |𝜆𝐼𝑅𝐸𝐸|(𝑡𝐵𝑖2𝑆𝑒3=8 nm)=(1.2± 0.1)×10−12 m. Where o nly th ree parameters varied from sample to sample, which are: resistance (R), average voltage <𝑉𝑆𝑃>, and the FMR linewidth ∆𝐻𝐵𝑖2𝑆𝑒3/YIG. The error bars were incorporated in 𝜆𝐼𝑅𝐸𝐸 by taking into account the variation of 𝑉𝑆𝑃 measured at 𝜙=0° and 180°. Therefore, we found values that varies in the range of 0.012 𝑛𝑚≤|𝜆𝐼𝑅𝐸𝐸|≤0.022 𝑛𝑚, and in the literature there are values reported in the range of 0.01 𝑛𝑚<𝜆𝐼𝑅𝐸𝐸≤0.11 𝑛𝑚 [23,35]. Although we cannot rule out the spin diffusion mechanism, the values of 𝜆𝐼𝑅𝐸𝐸 strongly support the role played by the surface states in the spin - to charge -current conversion process occurring in sputtered Bi 2Se3 layers. Indeed, granular Bi 2Se3 films grown by sputtering keep the topological insulator properties even in the nanometer size regime. The basic mechanisms explaining the existence of topological surface states in granular films of Bi 2Se3 is based on the electron tunneling between grain surfaces. Also, the electron quantum confinement in nanometer sized grains, has been considered as the reason of the high charge -to-spin conversion effect in granular TIs [36]. Figure 4 (color online). (a) Sketch of the bilayer sample Py/Bi2Se3. In order to minimize shunting effect s, the Py film partially covers the Bi 2Se3 film surface. (b) Derivative FMR field scan for the bilayer of Py (12 nm)/ Bi2Se3 (4 nm) obtained by inserting the sample in a microwave rectangular cavity operating at 9.4 GHz. The inset show s the derivative FMR absorption field scan for the bare Py (12 nm) film. The increase of the linewidth (HMHM ) for the bilayer Py (12 nm)/ Bi2Se3 (4 nm) is mostly due to the spin pumping process. (c) Field scan of VSP for two in -plane angles, measured at the same experimental configuration. The result confirms that the sign of the VSP in the bilayer Py (12 nm)/ Bi2Se3 (4 nm), is the same as the one measured in Bi 2Se3/YIG. (d) Decomposition of the symmetric and antisymme tric components of VSP obtained by fitting the data of (c). The inset shows the dependence of the peak value of the symmetric component as a function of the microwave power, measured at 𝜙=180° . To further study the polarization of the spin -to-charge current conversion process in Bi2Se3, we investigated the spin -pumping voltage in the bilayer of Bi 2Se3(4 nm)/Py(12 nm), where Py is Permalloy (Ni 81Fe19). The investigated sample is illustrated in Fig. 4(a), where the layer of Py partially covers the Bi 2Se3 surface, so that the electrodes are attached out of the Py layer. The sample was sputter grown onto SiO 2(300nm)/Si(001) where a passivation layer of MgO(2nm) was grown underneath the Bi 2Se3 layer. Fig. 4(b) shows the FMR spectrum of the bilayer Py(12 nm)/ Bi2Se3(4 nm) in which the sample is placed in a microwave cavity resonating at 9.4 GHz, with 𝑄≈2000 and an incident microwave power of 17 mW. Due to the spin pumping effect, the FMR linewidth (HWH M) increased to 32 Oe in comparison with the linewidth of 27 Oe of a bare Py(12 nm) layer, shown in the inset of Fig. 4(b). Figure 4(c) shows the spin pumping voltage measured between the electrodes for 𝜙=0° (blue curve) and 𝜙=180° (red curve), with an incident power of 170 mW. The 𝑉𝑆𝑃 lineshape is descri bed by the sum of symmetric and antisymmetric components, 𝑉𝑆𝑃(𝐻)=𝑉𝑠(𝐻−𝐻𝑅)+𝑉𝐴𝑆(𝐻−𝐻𝑅), where 𝑉𝑠(𝐻−𝐻𝑅) is the (symmetric) Lorentzian function and 𝑉𝐴𝑆(𝐻−𝐻𝑅) is the (antisymmetric) Lorentzian derivative centered at the FMR resonance field ( 𝐻𝑅). Figure 4(d) s hows the corresponding symmetric (red) and antisymmetric (green) components of the 𝑉𝑆𝑃 line shape for 𝜙=180° , obtained by fitting the data (black symbols) with a sum of a Lorentzian function and Lorentzian derivative (given by the cyan curve). The inset of Fig. 4(d) shows the linear dependence of the peak value of the symmetric component as a function of the i ncident power. The symmetric component of 𝑉𝑆𝑃 in Py/Bi2Se3, whic h is attributed to the spin -to-charge current conversi on process, has the same polarity as the one observed for 𝑉𝑆𝑃 measured in Bi 2Se3/YIG and Ta/YIG bilayers. In conclusion, we report an investigation of the spin - to charge -current conversion process in bilayers of YIG/Bi 2Se3 and Py/Bi 2Se3, where the Bi 2Se3 layer was grown by sputtering. The results obtained by means of the ferromagnetic resonance driven spin pumping technique has shed light in some aspects not investigated by previous papers. We discovered that the spin - to charge - current mechanism in topological insulator Bi 2Se3 has the same polarity as the one of Ta, and opposite to the one in Pt. By interpreting the spin pumping voltage as due to the inverse Rashba - Edelstein effect , we calculated the value of 𝜆𝐼𝑅𝐸𝐸 as a function of B i2Se3 thickn ess and the values found demonstr ate that the surface states have a dominant role in the spin -charge conversion process . Thus, the spin -charge conversion mechanism based on the spin diffusion process plays a secondary role. We expect that our results will be useful for app lications in spintronic devices and understanding the spin - to charge -current mechanism in sputter -deposited films of topological insulator Bi 2Se3. This research was supported by Conselho Nacional de Desenvolvimento Cient ífico e Tecnol ógico (CNPq), Coordena ção de Aperfei çoamento de Pessoal de N ível Superior (CAPES), Financiadora de Estudos e Projetos (FINEP), Fundaç ão de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE), Fundaç ão Arthur Bernardes (Funarbe), Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) - Rede de Pesquisa em Materiais 2D and Rede de Nanomagnetismo. References : 1. A. Manchon, A. Belabbes, Solid State Physics, Volume 68, Chap. 1, (2017). 2. A. Soumyanarayanan, N. Reyren, A. Fert, C. Panagopoulos, Nature. 539 (7630) 509 – 1517 (2016) . 3. A. Hoffmann and S. D. Bader, Phys. Rev. Appl., 4, 047001 (2015). 4. M. I. Dyakonov, and V. I. Perel, Phys. Lett. A 35, 459 –460 (1971). 5. J. E. Hirsch, Phys. Rev. Lett. 83, 1834 –1837 (1999). 6. A. Azevedo, L. H. Vilela Leão, R. L. Rodriguez -Suarez, A. B. Oliveira, and S. M. Rezende, J. Appl. Phys. 97, 10C715 (2005). 7. E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 8. L. Zhou, V. L. Grigoryan, S. Maekawa, X. Wang, and J. Xiao, Phys. Rev. B 91, 045407 (2015) 9. Alves -Santos, E. F. Silva, M. Gamino, R. O. Cunha, J. B. S. Mendes, R. L. Rodríguez - Suárez, S. M. Rezende, and A. Azevedo, Phys. Rev. B 96, 060408(R) (2017). 10. Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, Y. Otani, Phys. Rev. Lett. 2012, 109, 156602. 11. Y. Niimi and Y. Otani, Rep. Prog. Phys. 78, 124501 (2015). 12. J. Sławińska, F. T. Cerasoli, H. Wang, S. Postorino, A. Supka, S. Curtarolo, M. Fornari and M. B. Nardelli, 2D Mater. 6, 025012, (2019). 13. O. J. C . Rojas Sánchez, L. Vila, G. Desfonds, S. Gambarelli, J. P. Attané, J. M. De Teresa, C. Magén, and A. Fert, Nat. Commun. 4, 2944 (2013). 14. Kyoung -Whan Kim, Kyung -Jin Lee, Jairo Sinova, Hyun -Woo Lee, and M. D. Stiles, Phys. Rev. B, 96, 104438 (2017). 15. J. Sklen ar, W. Zhang, M. B. Jungfleisch, W. Jiang, H. Saglam, J. E. Pearson, J. B. Ketterson, and A. Hoffmann, J. Appl. Phys., 120, 180901 (2016). 16. Y. A. Bychkov and E. I. Rashba, Sov. Phys. JETP Lett. 39 (2), 78 (1984). 17. V. M. Edelstein, Solid State Commun. 73, 233 (1990). 18. J. B. S. Mendes, O. Alves Santos, L. M. Meireles, R. G. Lacerda, L. H. Vilela -Leão, F. L. A. Machado, R. L. Rodríguez -Suárez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 115, 226601 ( 2015). 19. A. Nomura, T. Tashiro, H. Nakayama and K. Ando, Appl. Phys. Lett. 106, 212403 (2015). 20. E. Lesne, Yu Fu, S. Oyarzun, J. -C. Rojas -Sánchez, D. C. Vaz, H. Naganuma, G. Sicoli, J.-P. Attan´e, M. Jamet, E. Jacquet, J. -M. George, A. Barth´el´emy, H. Jaffr`es, A. Fert, M. Bibes, and L. Vila, Nat. Mater. 15, 1261 (201 6). 21. S.Karube, K.Kondou, and Y.Otani, Appl. Phys. Express 9, 033001 (2016). 22. M. Matsushima, Y . Ando, S . Dushenko, R . Ohshima, R . Kumamoto, T . Shinjo, and M . Shiraishi, Appl. Phys. Lett. 110, 072404 (2017) 23. Mahendra DC, T. Liu, J .-Y. Chen, T. Peterson, P. Sahu , H. Li, Z. Zhao, M. Wu, and J . - P. Wang , Appl. Phys. Lett. 114, 102401 (2019). 24. J. B. S. Mendes, A. Aparecido -Ferreira, J. Holanda, A. Azevedo, and S. M. Rezende. Appl. Phys. Lett. 112, 242407 (2018). 25. J. B. S. Mendes et al, Phys. Rev. B 99, 214446 (2019). 26. A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E. -A. Kim, N. Samarth and D. C. Ralph, Nature , 511 , 449−451 (2014 ). 27. M. Jamali, J.S. Lee, J.S. Jeong, F. Mahfouzi, Y. Lv, Z. Zhao, B. K. Nikolic , K. A. Mkhoyan, N. Samarth and J. P. Wang, Nano Lett., 15 (10), 7126−7132 (2015 ). 28. Mahendra DC, Jun -Yang Chen, Thomas Peterson, Protuysh Sahu, Bin Ma, Naser Mousavi, Ramesh Harjani, and Jian -Ping Wang, Nano Lett., 19, 4836−4844 (2019). 29. J. B. S. Mendes, O. Alves Santos, J. Holanda, R. P. Loreto, C. I. L. de Araujo, C. -Z. Chang, J. S. Moodera, A. Azevedo, and S. M. Rezende. Phys. Rev. B 96, 180415(R) (2017). 30. M. Hasan and C. Kane, Rev. Mod Phys. 82, 3045 –3067 (2010). 31. C. Nayak et al., Rev. Mod. Phys. 80, 1083 (2008). 32. X.-L. Qi, R. Li, J. Zang, and S. -C. Zhang, Science 323, 1184 (2009). 33. L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). 34. W. H. Campos, J. M. Fonseca, V. E. de Carvalho, J. B. S. Mendes, M. S. Rocha, and W. A. Moura -Melo, ACS Photonics. 5, 74 1 (2018). 35. H. Wang, J. Kally, J. S. Lee, T. Liu, H. Chang, D. R. Hickey, K. A. Mkhoyan, M. Wu, A. Richardella, and N. Samarth, Phys. Rev. Letts., 117, 076601 (2016). 36. Mahendra DC et al, Nat. Mater. 17(9), 800 -807 (2018). 37. Y. F. Lee, R. K umar, F. Hunte, J. Narayan, J. Schwartz , J. Appl. Phys. 118, 125309 (2015) . 38. A. Banerjee, O. Deb, K. Majhi, R. Ganesan, D. Sen, P. S. Anil Kumar , Nanoscale, 9, 6755 (2017 ). 39. Y. Tserkovnyak, A. Brataas, G . E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 40. S. M. Rezende, R. L. Rodríguez -Suárez, and A. Azevedo, Phys. Rev. B, 88, 014404 (2013). 41. H. Emoto, Y. Ando, G. Eguchi, R. Ohshima, E. Shikoh, Y. Fuseya, T. Shinjo, M. Shiraishi. Phys. Rev. B 93, 174428 (2016). 42. A. Hoffmann, IEEE Trans. Mag. 49, 5172 (2013). 43. O. Mosendz et al., Phys. Rev. B 82, 214403 (2010). 44. A. Azevedo, L. H. Vilela -Leão, R. L. Rodríguez -Suárez, A. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011). 45. M. Arana, M. Gamino, E. F. Silva, V. M. T. S. Barthem, D. Givo rd, A. Azevedo, and S. M. Rezende , Phys. Rev. B 98, 144431 (2018).

2020-08-29

We report an investigation of the spin- to charge-current conversion in sputter-deposited films of topological insulator $Bi_2Se_{3}$ onto single crystalline layers of YIG $(Y_{3}Fe_{5}O_{12})$ and polycrystalline films of Permalloy $(Py = Ni_{81}Fe_{19})$. Pure spin current was injected into the $Bi_{2}Se_{3}$ layer by means of the spin pumping process in which the spin precession is obtained by exciting the ferromagnetic resonance of the ferromagnetic film. The spin-current to charge-current conversion, occurring at the $Bi_{2}Se_{3}/$ferromagnet interface, was attribute to the inverse Rashba-Edelstein effect (IREE). By analyzing the data as a function of the $Bi_{2}Se_{3}$ thickness we calculated the IREE length used to characterize the efficiency of the conversion process and found that 1.2 pm $\leq|{\lambda}_{IREE}|\leq$ 2.2 pm. These results support the fact that the surface states of $Bi_{2}Se_{3}$ have a dominant role in the spin-charge conversion process, and the mechanism based on the spin diffusion process plays a secondary role. We also discovered that the spin- to charge-current mechanism in $Bi_{2}Se_{3}$ has the same polarity as the one in Ta, which is the opposite to the one in Pt. The combination of the magnetic properties of YIG and Py, with strong spin-orbit coupling and dissipationless surface states topologically protected of $Bi_{2}Se_{3}$ might lead to spintronic devices with fast and efficient spin-charge conversion.

Unveiling the polarity of the spin-to-charge current conversion in $Bi_2Se_3$

2008.12900v1

1Whopumpsspincurrentintononmagnetic ‐metal(NM)layerin YIG/NMmultilayers atferromagnetic resonance? Yun Kang1, Hai Zhong1, Runrun Hao1, Shujun Hu1, Shishou Kang1★, Guolei Liu1, Y. Zhang2, X. R. Wang2★, Shishen Yan1, Yong Wu3, Shuyun Yu1, Guangbing Han1, Yong Jiang3 and Liangmo Mei1 1School of physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong, 250100, China. 2Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. 3State Key Laboratory for Advanced Metals and Materials, School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China. ★email: skang@sdu.edu.cn; phxwan@ust.hk 2Spin pumping in Yttrium-iron-garnet (YIG)/nonmagnetic-metal (NM) layer systems under ferromagnetic resonance (FMR ) conditions is a popular method of generating spin current in the NM l ayer. A good understanding of the spin current source is essential in extracti ng spin Hall angle of the NM and in potential spintronics applications. It is widely believed that spin current is pumped 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 spin current in NM came from the magnet ized NM surface (in contact with YIG) due to the magnetic proximity effect (MPE), rather than the precessing YIG magnetization. This conclusion is reached through our unique detecting method where the FMR microwave absorpti on of the magnetized NM surface, hardly observed in the conventional FMR experiments, was greatly amplified when the electrical detection circuit was switched on. Spin current generation, de tection, and manipulation i nvolve fundamental science as well as the key-technologies 1,2 in spintronics. Spin pumping from precessing magnetization under ferromagnetic resonanc e (FMR) conditions is an attractive method for generating cohere nt pure spin current3-8. Pure spin current is the key resource in spintronics as well as a base for studying the inverse spin Hall effect (ISHE) characterized by the spin Hall angle αSH. Ferromagnetic-insulator (FI)/nonmagnetic-metal (NM) bilayers are believed to be the ideal settings for measuring αSH of the non-magnetic metals. One of the widely studied such systems is Yttrium-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 strong spin-orbit interacti on. The conventional understanding of the system under FMR conditions is that magnetization of YI G precesses coherently, and the precessing magnetization pumps pure spin current into Pt layer9-12 across the YIG/Pt interface. The spin current in the Pt layer is then c onverted into a transver se (normal to spin flow direction and spin polarization) char ge current that can be detected as an electrical voltage signal. YIG/ Pt bilayer is believed to be a clean system for studying spin pumping and ISHE since YIG cannot conduct the electric current so that whatever the DC-voltage measured in Pt must be from the ISHE. The signature of spin pumping is the broadening of the FMR peak width of YIG as well as a detected DC-voltage in the Pt layer. Controversially, the extracted values of αSH from different groups differ by two orders of magnitude , inconsistent with each other10,16,17 in this “clean” system. It arises the question of who is responsible for the spin pumping, YIG or other magnetic sources? The answer to the question requires a good understanding of the interfacial phenomena between the YIG and a NM. Here, we use YIG(16nm)/Pt(10nm) bila yer and YIG(16nm)/Cu(5nm)/Pt(10nm), YIG(16nm)/Cu(5nm)/Ta(10nm), YIG(16 nm)/Al( 5nm)/Pt(10nm) trilayer systems in a rectangular cavity to investigate these issues by a combined measurements of microwave absorption and DC-voltage. Cont rary to the popular belief, the spin current in Pt or Ta was not pumped fro m the precessing YIG magnetization, but from the magnetized NM surface (in contact with YIG) originated from the magnetic proximity effect (MPE) 18-24. Interestingly and surprisingl y , the FMR signal from the magnetized NM surface was greatly amplif ied when the electrical measurement 4circuit was connected (otherwise, the signa l could hardly be observed). When the MPE was absent, such as in YIG(16nm)/A l(5nm)/Pt(10nm) samples, no DC-voltage signal 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 effective, in comparison with that from a magnetized metallic surface into Pt or Ta. Results YIG(16nm)/Pt(10nm) systems. The black circles in Fi g. 1a are the derivative microwave absorption spectrum ( dI/dH ) of a pure YIG sample as a function of external magnetic field H. The lineshape of the FMR derivative absorption spectrum of the pure YIG sample follows a standard di fferential Lorentzian line with FMR peak at H=2.497 kOe, and peak width of Γ=10 Oe which shows high quality of our YIG samples16. The green/blue circles in Fig. 1a are th e FMR derivative absorption spectra of a YIG(16nm)/Pt(10nm) bilayer strip when the electrical detection circuit is switched on/off (see the left inset of Fig. 1a and the methods below). In contrast, the FMR derivative microwave absorption spectrum of YIG/Pt bilayer sample appears to shift to a lower field with a seem ingly broaden peak width that was observed in previous studies5,16,25 and was used as an essential evidence of spin pumping from YIG. More strikingly, the microwave absorption signals are substantially different when the electrical detection circuit was switched on (green circles) and off (blue circles). Obviously, the absorption signal was greatly amplified when the electrical detection circuit was switched on. A more careful examination showed that the absorption curves 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 has the same peak position and peak width of H1=2.497 kOe and Γ1=10 Oe as the free YIG. The other with peak position H2=2.488 kOe , peak width Γ2=12 Oe, and relative amplitude A2=39.5% was naturally attr ibuted to the YIG covered by Pt. Because Pt modifies magnetic pr operties of YIG25, the peak position and peak width of the second FMR signal differ slightly from those of the free YIG . It is worthy to note that the FMR absorpti on curves of switch-on circuit were best fitted by three independent FMR signals as shown in Fig. 1b. Among the three signals in Fig. 1b, two signals (with A 1=54.5% and A2=36.4% ) are exactly those of free YIG and Pt-covered YIG, and the third signal of H3=2.477 kOe, Γ3=14 Oe, and A 3=9.1% came from the amplification of a very weak signal (hidden in the blue circles) originated from the MPE-induced magnetized Pt surface that was in contact with YIG. Furthermore, the corresponding DC-voltage detected in Pt was from the magnetized Pt surface since their peak positions and peak widths match exactly with each other as shown in Fig 1b. To substantiate this interp retation, we fabricated also YIG/Pt bilayer samples in which YIG was fully covered by Pt layer. As shown in Fig. 1c, the blue and green circles are the FMR absorption signals when the electrical meas urement circuit was switched on (green) and off (blue). As e xpected, the signal from the free YIG was absent. In Fig. 1d, the FMR absorption curves of switch-on circuit now consist of two signals respectively from the Pt-covered YI G and the magnetized Pt surface. Again, the DC-voltage relates to the signal of the magnetized Pt surface since their peak positions and widths match exactly with each other, and cannot be from the spin 6pumping of YIG. The above conclusion could also be reach ed from the change of the shape of voltage-H curves as angle θ between ac-magnetic field and sample long edge varies. The DC-voltage originated from the spin pumping of YIG should follow a Lorentzian lineshape since the FMR absorption is described by the Lorentzian function26. Thus DC-voltage lineshape would be symmetric about its peak for any angle θ if the spin pumping was from YIG. However, as plotted in Fig. 2a, it is clear that most spectra consist of a superposition of a Lorentz- and a dispersive-type resonance lineshape26,27 For a given θ, the DC-voltage curve was fitted to Eq. (1) so that both symmetric and asymmetric components of the DC-voltages Usym and Uasy were obtained. Their angle-dependences were plotted in Fig. 2b that fit well with theoretical prediction of Eq. (2) (see the Methods below). V oltages s SRU (for symmetric component) and a SRU (for asymmetric component) due to the spin rectification are 0.065 V=s SRU μand 0.568 V=a SRU μ. V oltage U sp from the ISHE due to spin pumping is 1.02 V=spU μ. Thus it shows that substantial amount of the DC-voltage came from the AMR and AHE of a ferromagnetic metal that resulted in an asymmetric lineshape, and the only possibility is that the Pt surface in contact with YIG wa s magnetized and generated a spin rectification voltage28. Obviously, the spin pumping effect generated a larger DC-voltage than the spin rectification. Th ese results further confirmed the MPE and spin precession of magnetized Pt surface in YIG/Pt system. Figure 3 is the H -dependence of the derivative microwave absorption with the switch-on circuit and DC voltage of a t ypical YIG/Pt strip sample for various frequencies 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 sources. This implies that only one FMR source pumped spin current into Pt and generated the DC-voltage. Clearly, the p eak position and width of the DC-voltage match well with those of the tiny FMR signa l. Furthermore, Fig. 3c shows that the peak field of the DC-voltage increased with the frequency that fits well with the Kittel formula29,30, f=(γ/2π)(H res*(H res+4πMs))1/2, with the gyromagnetic ratio γ= gμB/h=1.738×1011 T−1 s−1 and the saturation magnetization Ms =0.248 T . I t g a v e a Lander factor g=2.08 for magnetized Pt surface. The fitted Ms of magnetized Pt is very closed to the observed value in Ni/Pt system by XMCD measurement19. Thus, this result further supports our assertion that the precessing magnetization of YIG did not pump spin current to Pt laye r, contrary to the popular belief5,14-16,31. YIG(16nm)/Cu(5nm)/Pt(10nm) and YIG(16nm)/Cu(5nm)/Ta(10nm) systems. To further substantiate our claim that the DC -voltage in YIG/Pt bilayer is due to the spin 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 YIG and no MPE is possible for Pt (Ta). The upper panel of Fig. 4 is the typical FMR derivative microwave absorption spectrum of one of our YIG/Cu/Pt samples. The blue circles are the results when the electrical detection circuit was switched off while the green circles are those when the circ uit was switched on. The signal from the magnetized Pt surface was obviously absent , and was replaced by a new FMR signal at an even lower field of H=2.46 kOe, far below YIG resonance peak, and with a peak width of Γ=12 Oe. Although this new signal was seen in a 50 times enlarged figure as shown 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 can be clearly observed when the electric al detection circuit was switched on. As expected this time, the DC voltage of YI G/Cu/Pt and/or YIG/Cu/Ta (middle and lower panels of Fig. 4) was observed at H=2.46 kOe, exactly corresponding to the new FMR signal. The peak widths of the FMR a nd DC-voltage signals matched again with each other as shown in the middle (for YIG/ Cu/Pt) and lower (for YIG/Cu/Ta) panels. In contrast, there were no DC-voltage signa ls at the YIG resonant fields, confirming that the DC-voltage was not due to the spin pumping of YI G, but due to that of the magnetized Cu surface (in contact with YIG) 32,33,34,35. The signs of the DC-voltages of YIG/Cu/Pt and YIG/Cu/Ta samples are opposite due to the opposite sign of spin Hall angle for Pt (positive) and Ta (negative)16. Again, above experiments do not support th e general belief that precessing YIG magnetization at FMR pumps spin current into Pt or Ta in YIG/Pt, YIG/Cu/Pt, and YIG/Cu/Ta systems. Our results are consiste nt with the assertion that it was the magnetized NM surface pumping spin current into the Pt or Ta layer. The clear evidences include 1) DC-voltage peaks were far from the FMR peaks of the YIG, but were exactly overlapped with the FMR peak of MPE-induced magnetized NM surface; 2) the angle-dependence of DC-voltage lin eshape that shows big contribution from AMR and AHE of a magnetized metal. Furthe rmore, the MPE of both Pt and Cu were confirmed by our first-principl e calculations (see the Me thods). It was found that a few Cu atomic layers adjacent to Ni wa s magnetized with average moment about -0.02 μB/atom, which was only about 1/5 of the average moment of Pt (0.11 μB/atom) in 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 current 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 that in YIG/Pt system as shown in Fi gs. 1 and 2. The small negative Cu magnetic moment is also consistent with lower reso nance field for the magnetized Cu due to the negative exchange field at FMR 29. YIG(16nm)/Al(5nm)/Pt(10nm) systems. To further verify the assertion that spin current in NM layer(s) was not from YI G in YIG/NM1 bilayer or YIG/NM1/NM2 multilayer samples, but from magnetized NM 1 surface (in contact with YIG), we did a controlled experiment with YIG(16nm)/Al( 5nm)/Pt(10nm) samples. Al has no MPE, in consistent with our first-principle calcu lations on Ni/Al system. As shown in Fig. 5, the derivative microwave absorption spect rum of a typical YIG/Al/Pt samples does not change whether the electri cal detection circuit was switch ed on or off, in contrast to that for YIG/Pt or YIG/Cu/Pt(Ta) system s. Consistent with our assertion, there was no spin current in Pt layer since YIG could not pump detectable spin current and no DC-voltage was observed as shown in the bottom panel of Fig. 5. Discussion A magnetic metallic film at its FMR can gene rate not only a DC signal but also a radio-frequency ac field28,36 by the AHE and the AMR. As illustrated in Fig. 6 (see the Methods) when the sample is connected to Cu connection-pads through two Al wires, the whole structure becomes a patch ante nna and a high fre quency pass filter. According to the patch antenna theory37,38, the ac signals from the AMR and AHE of the 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 amplification of the FMR signal of the metallic film. Here we termed this amplification as “antenna effect” when the Al wires were connected to Cu-Pads (see experimental setup, switch-on ci rcuit). Fig. 6d show s clearly this antenna effect since the microwave absorption of a 3nm-thick Pe rmalloy (Py) film sample was greatly enhanced when Al wires were connected to Cu-Pads. One possible reason, that the precessing YIG could not inject a detectable spin current 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 broadening and additional damping mechan ism observed previously may be due to the overlap of resonant peak s of both YIG and magnetized Pt 12,14. Thus one should extract the spin Hall angle αSH by taking into the account of the new findings reported here. Conclusion In summary, our experiments on YIG/Pt bilayer and YIG/Cu/Pt (YIG/Cu/Ta) trilayer 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 switched 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 pumping 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 layer. 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 9-11,16. Methods Sample preparation and experimental procedure. YIG [Y3Fe2(FeO 4)3] films (16 nm) were fabricated on Gd 3Ga5O12 (GGG) wafers by pulsed laser deposition (PLD). X-ray diffraction (XRD) and atomic force microscopy (AFM) showed that our YIG are high quality (See supplementary Figure 1). Py, Pt , Cu, Ta or Al with high purity (4N) was then deposited on YIG by magnetron sputtering to create a NM strip with a mask of 0.2 mm × 2.3 mm. All samples were cut into 1 mm × 3 mm for DC-voltage and microwave absorption measurements in a homemade X-band microwave absorption spectrometer. The experimental setup is shown in Fig. 6. FMR microwave absorption and DC-voltage were measured at frequency f=9.7 GHz of TE10 mode in the X-band cavity. The sample with size of 1 mm x 3 mm was mounted in th e middle of a shorted copper plate at one end of the cavity that can rotate in the XY-plane as illustrated in Fig. 6. The angle between the Y-axis and the long edge of the sample is denoted as θ. Two thin rectangular copper sheets of 1.5 mm × 8 mm were symmetrically placed on the both sides ( 1 mm away from sample) of sample in the cavity as illustrated in Fig. 6. These two small copper sheets were isol ated from the shorted copper plate and acted as electrical connecti on pads that connected to a SR-530 lock-in amplifier of Stanford Research Systems or Keithley- 2182 nanovoltmeter. It should be pointed out that 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). Two Al wires of diameter about 30 micromet ers were attached to two long-edge ends of 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 external field and ac microwave field were always orthogonal with each other in order to have a maximal precessing magnetization. Angular dependence: The DC-voltage from ISHE, AMR and/or AHE of a ferromagnetic metal near the FMR has a symmetric Lorentz-lineshape and an asymmetric dispersive-lineshape 26,27,28 00 (H, H , ) U (H, H , )sym asy UUL D=Γ + Γ 2 0 22 0(H, H , )(H H )LΓΓ=−+Γ (1) 0 0 22 0(H H )D(H, H , )(H H )Γ−Γ=−+Γ Here, H 0 and Γ are respectively the resonance field and resonant peak width. Usym and Uasy are the voltages of the symmetry a nd asymmetry components of DC-voltage that depend on angle θ as39 sin(2 )sin( ) U cos( )s sym SR SPUU θθθ =+ (2) sin(2 )sin( )a asy SRUU θθ = Here,s SRU, a SRUare the voltages due to th e spin rectification. U sp is the voltage from the ISHE due to spin pumping. First-Principle calculations: Because our computation resources do not allow us to perform reliable calculations on YIG/Pt a nd/or YIG/Cu systems due to the huge unit cells, 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. References 1. Wolf, S. A. et al. Spintronics: A spin -based electronics vi sion for the future. Science 294, 1488 (2001). 2. Źutić, I., Fabian, J. & Sarma, S. D. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004). 3. Urban, R., Woltersdorf, G. & Heinrich, B. Gilbert damping in single and multilayer ultrathin films: Role of interfaces in nonlocal spin dynamics. Phys. Rev. Lett. 87, 217204 (2001). 4. Mizukami, S., Ando,Y . & Miyazaki, T. Effect of spin diffusion on Gilbert damping for a very thin permalloy layer in Cu/permalloy/Cu/Pt films. Phys. Rev. B 66, 104413 (2002). 5. Tserkovnyak, Y ., Brataas, A. & Bauer, G . E. W. Enhanced Gilbert damping in thin ferromagnetic films. Phys. Rev. Lett. 88, 117601 (2002). 6. Brataas, A., Tserkovnyak, Y ., Bauer, G. E. W. & Halperin, B. I. Spin battery operated by ferromagnetic resonance. Phys. Rev. B 66, 060404(R) (2002). 7. Heinrich, B. et al. Dynamic exchange coupling in magnetic bilayers. Phys. Rev. Lett. 90, 187601 (2003). 8. Zhang, Y ., Zhang, H. W. & Wang, X. R. New galvanomagnetic effects of polycrystalline magnetic films. Europhysics Lett. 113, 47003 (2016) . 9. Kajiwara, Y . et al. Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. Nature ( London) 464, 262 (2010). 10. Ando, K. et al. Inverse spin-Hall effect induced by spin pumping in metallic system. J. Appl. Phys. 109, 103913 (2011). 11. Takahashi, R. et al. Electrical determination of spin mixing conductance at metal/insulator interface using inverse spin Hall effect. J. Appl. Phys. 111, 07C307 (2012). 12. Du, C. H. et al. Probing the spin pumping mechanism: Exchange coupling with exponential decay in Y3Fe5O12/Barri er/Pt heterostructures. Phys. Rev. Lett. 111, 247202 (2013). 13. Sandweg, C. W. et al. Spin pumping by parametrically excited exchange magnons. Phys. Rev. Lett. 106, 216601 (2011). 14. Castel, V ., Vlietstra, N., Ben Youssef, J. & van Wees, B. J. Platinum thickness dependence of the inverse spin-Hall voltage from spin pumping in a hybrid yttrium iron garnet/platinum system. Appl. Phys. Lett. 101, 132414 (2012). 15. Hahn, C. et al. Comparative measurements of inverse spin Hall effect s and magnetoresistance in YIG/Pt and YIG/Ta. Phys. Rev. B 87, 174417 (2013). 16. Wang, H. L. et al. Scaling of spin Hall angle in 3d, 4d, and 5d metals from Y 3Fe5O12/Metal spin pumping. Phys. Rev. Lett. 112, 197201 (2014). 17. Sinova, J. et al. Spin Hall effect. arXiv:1411.3249v1 (2014). 18. Antel, W. J. et al. Induced ferromagnetism and anisotropy of Pt layers in Fe/Pt(001) multilayers. Phys. Rev. B 60, 12933 (1999). 19. Wilhelm, F. et al. Layer-resolved magnetic moments in Ni/Pt multilayers. Phys. Rev. Lett. 85, 413 (2000). 20. Huang, S. Y . et al. Transport magnetic proximity effects in platinum. Phys. Rev. Lett. 109, 107204 (2012). 21. Lu, Y . M. et al. Pt magnetic polarization on Y 3Fe5O12 and magnetotransport characteristics. Phys. Rev. Lett. 110, 147207 (2013). 22. Guo, G. Y ., Niu, Q. & Nagaosa, N. Anomalous Nernst and Hall effects in magnetized platinum 14and palladium. Phys. Rev. B 89, 214406 (2014). 23. Ryu, K.-S., Yang, S.-H., Thomas, L. & Parkin, S. S. P. Chiral spin torque arising from proximity-induced magentization. Nat. Commun. 5, 3910 (2014). 24. Zhou, X. et al. Tuning magnetotransport in PdPt/Y 3Fe5O12: Effect of magnetic proximity and spin-orbit coupling. Appl. Phys. Lett. 105, 012408 (2014). 25. Sun, Y . et al. Damping in yttrium iron garent nanoscale films capped by platinum. Phys. Rev. Lett. 111, 106601 (2013). 26. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect. Appl. Phys. Lett. 88, 182509 (2006). 27. Zhang, Y . et al. Dynamic magnetic susceptibility and electrical detection of ferromagnetic resonance. arXiv:1512.01913 (2015) . 28. Mecking, N., Gui, Y . S. & Hu, C. M. Microwave photovoltage and photoresistance effects in ferromagnetic microstrips. Phys. Rev. B 76, 224430 (2007). 29. Kittel, C. On the theory of ferromagnetic resonance absorption. Phys. Rev. 73, 155 (1948). 30. Jiang, S. W. et al. Exchange-dominated pure spin current transport in Alq3 molecules. Phys. Rev. Lett. 115, 086601 (2015). 31. Rao, J. et al. Observation of spin rectification in Pt/yttrium iron garnet bilayer. J. Appl. Phys. 117, 17C725 (2015). 32. Carbone, C. et al. Exchange split quantum well states of a noble metal film on a magnetic substrate. Phys. Rev. Lett. 71, 2805 (1993). 33. Garrison, K., Chang, Y . & Johnson, P. D. Spin polarization of quantum well states in copper thin films deposited on a Co(001) substrate. Phys. Rev. Lett. 71, 2801 (1993). 34. Pizzini, S., Fontaine, A., Giorgetti, C. & Dartyge, E. Evidence for the spin polarization of copper in Co/Cu and Fe/Cu multilayers. Phys. Rev. Lett. 74, 1470 (1995). 35. Hirai, K. X-ray magnetic circular dichroism at the K-edge and proximity effects in Fe/Cu multilyers. Physica B 345, 209 (2004). 36. Jiao. H. & Bauer, G. E. W. Spin backflow and ac voltage generation by spin pumping and the inverse spin Hall effect. Phys. Rev. Lett. 110, 217602 (2013). 37. Carver, K. R. & Mink, J. W. Microstrip antenna technology. IEEE Trans. Antennas Propag. 39, 2-24 (1981). 38. Stutzman,W. L. & Thiele, G. A. Antenna Theory and Design (Wiley, 1981). 39. Azevedo, A. et al. Spin pumping and anisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment. Phys. Rev. B 83, 144402 (2011). Acknowledgements This work was supported by the National Basic Research Program of China under Grant No. 2015CB921502 and 2013CB922303, the National Natural Science Foundation 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 Kong RGC Grants No. 163011151 and No. 605413. YW and YJ were supported by the National Natural Science Foundation of China under Grant No. 51501007. Author contributions X.R.W. and S.K. contributed to project de sign and manuscript writing; Y .K., H.Z. and R.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. 16 Figure legends Fig. 1 (a) The FMR derivative absorption spectra of free YIG and YIG/Pt bilayer strip with θ=0o(θ is the angle between microwave field and the sample long edges). The lower panel is the corresponding 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 FMR signals respectively for free YIG, YIG covere d with Pt, and magnetized Pt surface in contact with YIG. The DC signal agrees with the a ssertion of spin pumping from the magnetized Pt surface. (c) The FMR derivative microwave absorpti on (upper) and DC-voltage (lower) spectra of fully covered YIG/Pt bilayer (illustrated in the lower left) with θ=170o. The inset shows the YIG sample fully covered by Pt. (d) The FMR spectr um of fully covered YIG/Pt bilayer with the antenna effect is best fitted by two FMR sign als from YIG covered with Pt and magnetized Pt surface. The corresponding DC-voltage si gnal was from the magnetized Pt surface. Fig. 2 (a) The DC-voltage spectra of a YIG(16nm)/Pt (10nm) strip at various angle θ. The symmetrical (b) and asymmetrical (c) components of DC-voltages defined in Eq. (1) were extracted. The solid lines are the best fits of Eq. (2) with 0.065 V,=s SRU μ 0.568 V=a SRU μand 1.02 V=spU μ. The inset illustrates experimental setup and angle θ. Fig. 3 The H-dependence of DC-voltage (a) and FMR spectra (b) of YIG/Pt stripe line at θ=0o and for various frequencies with antenna effect. (c) The frequency dependence of peak position of DC-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 YIG/Cu/Pt (middle panel) and YI G/Cu/Ta (lower panel) with θ=0o. In the upper panel, the green (blue) circles are the FMR derivative microwave ab sorption spectra when the electrical detection circuit is switched on (off). A weak signal, as shown by the black circles that is the zoom-in (50 times 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). 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. 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 about 1 mm. (c) Th e in-plane rotation geometry of sample. The microwave of frequency f=9.7 GHz propagated along the Z-axis, and the external field H was along the X-axis. e and h are the electric and magnetic components of the microwave, respectively along the X-axis and the Y-axis. The angle between the Y-axis and the long edge of sample was denoted as θ that varies as the shorted copper plate rotates in the XY-plane. (d) FMR signals of 3 nm Py thin film strip when the Al wires were connected/disconnected to the Cu-pads (with/without antenna effect). The inset is the equivalent circuit with antenna effect. 17 Fig. 1 Kang et al . 18 FIG. 2 Kang et al . 19 FIG. 3 Kang et al . 20 FIG. 4 Kang et al . 21 FIG. 5 Kang et al . 22 FIG 6 Kang et al .

2016-04-24

Spin pumping in Yttrium-iron-garnet (YIG)/nonmagnetic-metal (NM) layer systems under ferromagnetic resonance (FMR) conditions is a popular method of generating spin current in the NM layer. A good understanding of the spin current source is essential in extracting spin Hall angle of the NM and in potential spintronics applications. It is widely believed that spin current is pumped 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 spin current in NM came from the magnetized NM surface (in contact with YIG) due to the magnetic proximity effect (MPE), rather than the precessing YIG magnetization. This conclusion is reached through our unique detecting method where the FMR microwave absorption of the magnetized NM surface, hardly observed in the conventional FMR experiments, was greatly amplified when the electrical detection circuit was switched on.

Who pumps spin current into nonmagnetic-metal (NM) layer in YIG/NM multilayers at ferromagnetic resonance?

1604.07025v1

Investigation of Magnetic Proximity Effect inTa/YIG Bilayer Hall Bar Structure Yumeng Yang1,2, Baolei Wu1, Kui Yao2,Santiranjan Shannigrahi2, Baoyu Zong3, Yihong Wu1, a) 1Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583 2Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 Research Link, Singapore 117602 3Temasek Laboratories, National University of Singapore, T -Lab B uilding, 5A, Engineering Drive 1, #09-02, Singapore 117411 In this work, the investigation of magnetic proximity effect was extended to Ta which has been reported to have a negative spin Hall angle. Magnetoresistance and Hall measurements for in- plane a nd out -of- plane applied magnetic field sweeps were carried out at room temperature. The size of the MR ratio observed (~10 -5) and its magnetization direction dependence are similar to that reported in Pt/YIG, both of which can be explained by the spin Hall magnetoresistance theory . Additionally, a flip of magnetoresistance polarity is observed at 4 K in the temperature dependent measurements, which can be explained by the magnetic proximity effect induced anisotropic magnetoresistance at low temperature. Our findings suggest that both magnetic proximity effect and spin Hall magnetoresistance have contribution to the recently observed unconventional magnetoresistance effect. a) Author to whom correspondence should be addressed: elewuyh@nus.edu.sg 1 I. INTRODUCTIO N Platinum (Pt) has been investigated as a promising spin current detector in recent reports on spin pumping effect1-5 and spin Seebeck effect6-8 involving ferromagnet (FM)/non- ferromagnet (NM) structures. In these investigations, the generated spin current is converted into an electromotive force via inverse spin Hall effect (ISHE), which can then be detected electrically. On the other hand, recently anisotropic -magnetoresistance (AMR) -like effect in Pt on the insulating ferromagnet yttrium iron garnet (YIG) was reporte d by Weiler et al .9 and Huang et al .10, which is attributed to the induced magnetic dipole moment in Pt due to magnetic proximity effect (MPE). This “pseudo- magnetization” in Pt was later supported by X- ray magnetic circular dichroism (XMCD) measurement11. Further investigation by Nakayama et al.12 found that the magnetization direction dependence of this resistance change is distinct from any of the known MR effect. They proposed a spin Hall magnetoresistance (SMR) theory13 to explain the experimental observations by taking into account the influence of spin current to the charge resistance through spin -orbit coupling (SOC). Some of the follow- up works14-19 are in supportive of this model, in the way that the governing parameters such as spin mixing conductance of the interface, spin diffusion length and spin Hall angle of Pt could be extracted and appeared to be comparable to those of previous studies. Although the presence of unconventional MR effect is confirmed, its underly ing mechanism remains to be unveiled. Contradictory XMCD results showing the presence of only negligible Pt magnetic polarization in Pt/YIG structure 20,21 cast some doubts o n the strength of MPE. On the other hand, the incompleteness of SMR theory was also indicated by the absence of similar effect in Au/YIG22 and its preservation in Pt/surface modified YIG23 structures. Therefore, in order to reveal the true mechanism of spintronic effects at FM/NM interf aces, it is necessary to extend the study to different FM/NM material systems. So far, with most investigations focusing on Pt/YIG, only a few have extended the bilayer structure to Au/YIG 22, Ta/YIG17, Pt/Py24, Py/YIG23, Fe 3O4/Pt, NiFe 2O4/Pt14, and CoFe 2O4/Pt19. In this work, we focus on the Ta/YIG interface and discuss for the first time the Hall measurement results. The MR effect observed at room temperature is similar to that observed in Pt/YIG. The out -of- plane Hall resistance was found to be influenced by YIG substrate’s magnetization at magnetic field below its saturation value of around 2000 Oe. Both observations could be explained by the SMR theory 2 However, we observed an additional dip centered at zero- field between around - 200 Oe and 200 Oe, which could not be attributed to YIG’s in -plane magnetization rotation15. The shape of the in- plane Hall resistance is different from that predicted by the SMR theory nor the MPE -induced planar Hall effect (PHE), but instead similar to that of the MR curve. The additional dip and shape difference might be due to the possible influence from MR. In the MR measurement with out -of-plane magnetic field, a flip of the MR polarity was observed at 4 K, implying the enhancement of MPE at low temperature. The results of the 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 observed MR effect. II. SAMPLE FABRICATION AND CHARACTERIZATION Samples were fabricated on (111) oriented YIG (8 μm)/ GGG (500 μm) single crystal wafer with a size of 5mm × 5mm. Fig. 1(a) shows the X- ray diffraction (XRD) pattern of an unpatterned reference Ta thin film (10 nm) deposited on YIG/GGG substrate. Comparison with the XRD pattern of bare substrate (inset of Fig. 1(a)) confirms that the Ta layer is in β -phase. To f abricate the Hall bars, the substrates were properly cleaned with acetone and isopropyl alcohol before coated with a Microposit S1805/PMGI SF6 bilayer resist. The resist -coated substrate was subsequently exposed using a Microtech laser writer to form a 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 transverse electrodes is 0.1 mm × 1 mm. A Ta (5 nm, 7 nm, 10 nm) layer was subsequently deposited using a sputter with a base pressure below 5× 10 -5 Pa, and in an Ar working pressure of 5× 10-1 Pa, followed by liftoff to form the Hall bar. Before electrical measurements, the as -fabricated samples were bonded to chip carriers using a wire bonder. The resistivity of the samples (5 nm, 7 nm, 10 nm) falls into the range of 6.6× 102 to 1.1× 103 µΩ cm, which is in good agreement with the reported values for β -phase Ta17. Figs. 2(a) -(b) and Figs. 3(a)- (b) are the schematic configurations for MR and Hall measurements. For room temperature electrical measurement s, the samples were placed in the ambient environment with in-plane ( x, y directions) and out -of-plane ( z direction) magnetic field applied, respectively. Low temperature 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 3 Hall measurement was performed using a DC technique with a current of 100 µ A. The hysteresis loops for YIG/GGG were measured using vibrating sample magnetom eter (VSM) as a reference to assist the interpretation of electrical transport data. III. RESULTS AND DISCUSSION A. Magnetic properties of YIG Fig. 1(b) shows the in- plane and out -of-plane M -H loops for YIG/GGG substrates. The observation of an in- plane saturati on field below 100 Oe and out -of- plane saturation field of around 2000 Oe suggests an in -plane anisotropy for (111) -oriented YIG/GGG. The inset shows a superimposition of the in- plane M-H loop and the M -H loop obtained with a 90° in -plane rotation of the sample with respect to its original position. The close overlap of two loops suggests no preferred easy axis in the (111) plane. Notably, for both cases, the coercivity (H c) for YIG is below 10 Oe, which is reasonable for a ferrimagnetic material. B. Room te mperature MR and Hall measurements Figs. 2(c), (e) and (g) are the MR (hereafter referred to as the MR observed in the measurements of this work regardless of its origin) ratio of Ta (5 nm)/YIG for three applied magnetic field directions (H x, Hy, H z), respectively. Conventionally, for NM, its resistance can increase slightly as the applied field increases due to the Lorentz force felt by the conduction electrons25. This effect is named as ordinary magnetoresistance (OMR) with a positive polarity. In the present case, since the field applied is small (maximum 500 Oe), OMR is negligible in the field sweeps. However, a dip or peak is observed during all sweeps in the field region between -50 Oe and +50 Oe, indicating the presence of interfacial coupling in Ta/YIG. The MR ratio of 2 ×10-5 is comparable to that reported in Pt/YIG10. If we only focus on the MR in x and y directions, it seems that it can be explained by the proximity effect, i.e., the anisotropic magnetoresistance (AMR) arises from the magnetized Ta layer (hereafter named as MPE -induced AMR). However, if this is the case, the MR in z direction should be negative as well instead of being positive as observed experimentally. Instead of the MPE -induced AMR, the experimental data can be understood as follows using the SMR theory. According to this th eory, the MR is determined by the angle between spin polarization 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 sense, the resistance response to applied field in x and z direction should be similar except that the MR 4 curve in z direction is broader than that in x direction due to shape anisotropy of the YIG layer. Figs. 2(d), (f) are the Hall resistance of Ta (5 nm)/YIG for in -plane applied magnetic field directions (H x, H y), respectively. Conventionally, for in -plane field, the Hall effect in NM should vanish with only th e presence of a background offset resistance coming from electrodes asymmetry. Similar to the longitudinal resistance, we observed a dip or peak in the in- plane Hall resistance located at around -50 Oe and +50. The magnitude of around 10- 20 mΩ is on the sa me order of those reported by N. Vlietstra et al.15. However, its shape is different from that predicted by SMR or MPE -induced PHE, which should be an odd functi on of the applied magnetic field. One possible reason is that due to the relatively large size of the pattern and current distribution in the electrodes, the contribution from the longitudinal MR to the Hall resistance cannot be excluded. As the longitudinal MR is one order of magnitude larger than the Hall resistance, it is possible that the MR signal covers the true Hall signal. This can also be inferred from the similarity between the shape of the MR and Hall curves. The out -of-plane Hall resi stance of Ta (5 nm)/YIG , as shown in Fig. 2(h), is different from the linear ordinary Hall effect for NM. It has a nonlinear region between -2000 Oe and 2000 Oe, which coincides with YIG’s saturation field. This is presumably caused by the fact that the total field experienced by electrons inside Ta is the sum of externally applied field and the stray field from the YIG. The latter is large when the in -plane magnetization of YIG is oriented into the vertical direction by an external field, while it is relat ively small when it is saturated in -plane. The additional stray field dominates in the field range below saturation, causing the nonlinear region. While, when the field is above the saturation, the contribution from the applied field dominates, resulting i n the linear region. Noticeably, an additional dip is observed in the center between - 200 Oe and 200 Oe. N. Vlietstra et al .15 related it to the in-plane magnetization rotation at H c, which is unlikely the origin of the dip observed here as the H c of YIG i s below 10 Oe, as shown in the M -H loop. Considering its similar shape and field range as that of the out -of-plane MR curve in Fig. 2(g), we believe that it is also related to the influence of MR signal as discussed above. C. Temperature -dependent out -of-plane MR and Hall measurements So far, it seems that most of our results at RT can be explained by the SMR theory. However, we still cannot completely exclude the MPE contribution in Ta/YIG bilayers, as the proximity effect may be 5 enhanced at low temperature. To this end, temperature -dependent MR and Hall measurements were performed in temperatures from 4 K to 300 K. As discussed in Part B in the room temperature measurements, the polarity of MPE -induced AMR and SMR differs from each other only when the magneti c field is applied in z direction. This means that if the magnetic field is applied in x or y direction, even if both types are present at low temperature, it may still be difficult to separate them because of the same polarity. Therefore, we chose to apply the field in the out -of-plane direction ( z direction) in the temperature dependent measurem ents. As our measurement system does not allow in -situ rotation of sample direction, for the temperature -dependent measurements in z -direction, we have changed the 5- nm- thick sample to samples with a thickness of 7 nm and 10 nm, respectively. As all the sa mples show similar MR and Hall behavior at room temperature, the change of samples will not compromise the consistency of discussion. Fig. 3(c) is the out -of-plane MR of Ta (10 nm)/YIG at temperatures from 4 K to 300 K (similar results are obtained for Ta (7 nm)/YIG which are not shown here). The curves are clearly a superposition of two types of effects. For the curves at 10 - 300 K, a small positive MR is observed at large field region. This background MR comes from OMR as discussed in Part B, which i s a result of the Lorentz force felt by the conduction electrons. Additionally, sharp central dips are observed in the range between - 2000 Oe and 2000 Oe, which are consistent with the polarity and field range predicted by the SMR theory. This large positi ve MR is the result of the SMR caused by the change in YIG’s magnetization direction below its saturation field. Therefore, the MR curves obtained at 10 - 300 K is dominantly a superposition of the OMR and SMR effect. For the curve at 4 K, the central dip polarity remains the same, indicating again the SMR contribution. However, the background MR polarity is flipped to be negative. This negative MR follows the polarity predicted by MPE, i.e., the AMR arises from the magnetized Ta layer, suggesting the large enhancement of MPE at low temperature. In this sense, the MR curve at 4 K is dominantly a superposition of the MPE -induced AMR and SMR effect. A schematic illustration of the above scenario is shown in Fig. 3(e). Same MR measurement was performed on Pt (5 nm) /Ta (5 nm) multilayer structure on SiO 2/Si to exclude the contribution from magnetic impurities in the as -sputtered Ta. The polarity remains the same for this sample at 4 K, indicating the flip in Ta/YIG is due to the presence of the ferrimagnetic YIG and MPE from YIG. Based on these different origins, the background MR is subtracted, and 6 temperature -dependent SMR ratio is shown in Fig. 3(f) for both the 7 nm and 10 nm Ta samples. The size of the SMR ratio (~10-5), is comparable to room temperature value for the 5 nm Ta sample. Despite some fluctuations, the overall trend for the SMR ratio is that it increases when the temperature and Ta thickness decrease. The former might be due to the enhancement of spin diffusion length at low temperature, while the latter confirms the nature of SMR as an interface effect. Fig. 3(d) is the out -of-plane Hall resistance of Ta (10 nm)/YIG at temperatures from 4 K to 300 K (similar results are obtained for Ta (7 nm)/YIG which are not shown here). The shapes of the curve s are similar to that of the 5 nm Ta sample at room temperature. After subtracting the linear ordinary Hall effect contribution, the Hall resistance ratio is shown in Fig. 3(g). The size (~10-6) is one order of magnitude smaller than the SMR ratio, and has a similar temperature and thickness dependence as that of the SMR ratio, indicating the same origin for both MR and Hall resistance. IV. SUMMARY In conclusion, unconventional MR effect was observed in Ta/YIG bilayer with a comparable size to Pt/YIG. Our MR and Hall results show that the electron transport property in the Ta overlayer is greatly influenced by YIG’s magnetization, which could be explained by the SMR theory. Additionally, the observation of a flip of polarity in the temperature dependent MR measurements suggests that MPE - induced AMR is enhanced at low temperature. Our findings suggest that both MPE and SMR may contribute to the unconventional MR effect, in particular at low temperatures. A theory combining the features of MPE and SMR may be h elpful in fully understanding the experimental results observed in NM/YIG bilayers. ACKNOWLEDGM ENTS This work was supported by the National Research Foundation of Singapore (Grants No. NRF -G-CRP 2007 -05 and R -263-000-501-281). The authors wish to acknowledge Wei Zhang from National University of Singapore for her assistance during sample preparation, and Wei Ji, Meysam Sharifzadeh Mirshekarloo from Institute of Materials Research and Engineering (IMRE) for their assistances during structural characterization. K. Y. and S. S. wish to acknowledge the support from IMRE under project number IMRE/10 -1C0107. 7 REFER ENCES 1 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 2 K. Uchida, T. An, Y. Kajiw ara, M. Toda, and E. Saitoh, Appl. Phys. Lett. 99, 212501 (2011). 3 M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 176601 (2012). 4 Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizu guchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 5 K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109 , 103913 (2011). 6 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 7 K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010). 8 K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, Nat. Mater. 10, 737 (2011). 9 M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I. M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). 10 S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J . Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012). 11 Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). 12 H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 13 Y. T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013). 14 M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprägs, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J. -M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y. T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 15 N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Physical Review B 87, 184421 (2013). 16 N. Vlietstra, J. Shan, V. Castel, J. Ben Youss ef, G. E. W. Bauer, and B. J. van Wees, Appl. Phys. Lett. 103, 032401 (2013). 17 C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). 18 M. Weiler, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y. T.Chen, H. J. Jiao, G. E. W. Bauer, S. T. B. Goennenwein, arXiv:1306.5012v1 (2013). 19 M. Isasa, F. Golmar, F. Sánchez, L. E. Hueso, J. Fontcuberta, F. Casanova, arXiv:1307.1267v2 (2013). 20 S. Geprägs, M. Schneid er, F. Wilhelm, K. Ollefs, A. Rogalev, M. Opel, R. Gross, arXiv:1307.4869v1 (2013). 21 S. Geprägs, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012). 22 D. Qu, S. Y. Huang , J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). 23 B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 111, 066602 (2013). 24 Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien, Phys. Rev. B 87, 220409( R) (2013). 25 T. R. Mcguire and R. I. Potter, IEEE Trans. on Mag. 11 , 1048 (1975). 8 FIGURE CAPTIONS: FIG. 1. (a) Grazing incidence angle XRD pattern for the unpatterned reference Ta (10 nm)/YIG/GGG sample; (b) in - plane and out -of-plane M -H loop for YIG/GGG substrate. Inset in (a): XRD pattern for YIG/GGG wafer with (111) orientation; inset in (b): Superimposition of in-plane M -H loop and M -H loop obtained with a 90o in-plane rotation of the sample with respect to its original position. FIG. 2. (a) - (b) Schematic configuration for MR and Hall measurements; MR and Hall resistance measured with field applied in different directions for Ta (5 nm)/YIG: MR in x-direction (c), Hall effect in x -direction (d), MR in y - direction (e), Hall effect in y -directi on (f), MR in z -direction (g), Hall effect in z -direction (h). An offset resistance of 3-4 mΩ is subtracted in Hall resistance. All measurements were performed at room temperature. FIG. 3. (a)- (b) Schematic configuration for temperature -dependent MR an d Hall measurements; (c) Temperature- dependence of MR with field applied in z direction for Ta (10 nm)/YIG; (d) Temperature -dependence of Hall resistance with field applied in z direction for Ta (10 nm)/YIG; (e) Schematic illustration for the change of MR origin at 4 K; (f) Temperature -dependence of SMR ratio for Ta (7 nm)/YIG and Ta (10 nm)/YIG; (g) Temperature - dependence of Hall resistance ratio for Ta (7 nm)/YIG and Ta (10 nm)/YIG. An offset resistance of 3 -4 m Ω i s subtracted in Hall resistance and the MR and Hall curves are vertically shifted for clarity. 9 FIG. 1 Journal of Applied Physics Yumeng Yang Ta YIG GGG YIG GGG -3.0 -1.5 0.0 1.5 3.0-1.0-0.50.00.51.0 Hy Hx Hz HxM/Ms Applied Field (kOe)(b) -0.2 0.0 0.2-101 35 40 45 50 550.000.030.060.09 YIG (444) GGG (444)Ta (312)Intensity (a. u.) 2θ (degree )Ta (510)(a) 50.5 51.0 51.5101102103104105 10 FIG. 2 Journal of Applied Physics Yumeng Yang (a) (b) (h) (g)(f) (e)(d) (c)I VHz HxHy -300 -150 0150 300-12-8-404 0102030Hx Applied Field (Oe)Ta(5nm)/YIG Ta(5nm)/YIG HyRxy (mΩ) -200 -100 0100 200-3-2-101 -10123Hx Applied Field (Oe)Ta(5nm)/YIGTa(5nm)/YIG Hy∆Rxx/R (×10−5) -500 -250 0250 500-3-2-101 Ta(5nm)/YIG Applied Field (Oe)∆Rxx/R (×10−5)HzHz I VHxHy -4 -2 0 2 4-120-60060120 Applied Field (kOe)Rxy (mΩ)HzTa(5nm)/YIG 11 FIG. 3 Journal of Applied Physics Yumeng Yang (g)(f)(b) I VHz(e) (c)(a) 0 100 200 300234567 Ta (10nm)/YIGTa (7nm)/YIGSMR Ratio (×10-5) Temperature (K) -10 -5 0 5102548254925502551 150 K120 K100 K50 K 80 K 200 K10 K 300 K20 KRxx (Ω) Applied Field (kOe)4 KHz I V -10 -5 0 510-1000-800-600-400-2000 20 K10 K 150 K 200 K 300 K100 K 120 K50 K4 KRxy (mΩ) Applied Field (kOe)80 K (d) 0 100 200 30012345 Ta (10nm)/YIGTa (7nm)/YIGRxy Ratio (×10-6) Temperature (K)Polarized Ta YIGTaOMR +SMR AMR +SMR Ta T: 300K 4 K YIG 12

2013-11-06

In this work, the investigation of magnetic proximity effect was extended to Ta which has been reported to have a negative spin Hall angle. Magnetoresistance and Hall measurements for in-plane and out-of-plane applied magnetic field sweeps were carried out at room temperature. The size of the MR ratio observed (~10-5) and its magnetization direction dependence are similar to that reported in Pt/YIG, both of which can be explained by the spin Hall magnetoresistance theory. Additionally, a flip of magnetoresistance polarity is observed at 4 K in the temperature dependent measurements, which can be explained by the magnetic proximity effect induced anisotropic magnetoresistance at low temperature. Our findings suggest that both magnetic proximity effect and spin Hall magnetoresistance have contribution to the recently observed unconventional magnetoresistance effect.

Investigation of Magnetic Proximity Effect inTa/YIG Bilayer Hall Bar Structure

1311.1262v1

Chiral charge pumping in graphene deposited on a magnetic insulator Michael Evelt,1Hector Ochoa,2Oleksandr Dzyapko,1Vladislav E. Demidov,1Avgust Yurgens,3Jie Sun,3 Yaroslav Tserkovnyak,2Vladimir Bessonov,4Anatoliy B. Rinkevich,4and Sergej O. Demokritov1;4 1Institute for Applied Physics and Center for nanotechnology, University of Muenster, 48149 Muenster, Germany 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 3Department of Microtechnology and Nanoscience-MC2, Chalmers University of Technology, SE-41296, Gothenburg, Sweden 4Institute of Metal Physics, Ural Division of RAS, Yekaterinburg 620041, Russia We demonstrate that a sizable chiral charge pumping can be achieved at room temperature in graphene/Yttrium Iron Garnet (YIG) bilayer systems. The eect, which cannot be attributed to the ordinary spin pumping, reveals itself in the creation of a dc electric eld/voltage in graphene as a response to the dynamic magnetic excitations (spin waves) in an adjacent out-of-plane magnetized YIG lm. We show that the induced voltage changes its sign when the orientation of the static magnetization is reversed, clearly indicating the broken spatial inversion symmetry in the studied system. The strength of eect shows a non-monotonous dependence on the spin-wave frequency, in agreement with the proposed theoretical model. I. INTRODUCTION The generation of spin-polarized currents is of techno- logical interest in order to transmit information encoded in the spin degrees of freedom. A spin current can be in- jected into a nonmagnetic conductor by spin pumping.1,2 In this process, the reservoir of the angular momentum is a ferromagnet where a ferromagnetic resonance (FMR) is excited by a microwave eld. The spin pumping with- out net charge transport stems from the exchange of angular momentum between the itinerant spins in the metal and the collective magnetization dynamics in the adjacent ferromagnet. The dynamical generation of spin currents is of special relevance since alternative meth- ods based on driving an electrical current through the interface are limited by the conductance mismatch.3The spin pumping-induced currents have been detected as the long-ranged dynamic interaction between two ferromag- nets separated by a normal metal4,5or as a dc-voltage signal in bilayer systems.6In the case of nonmagnetic metals with a strong spin-orbit coupling, the spin cur- rent through the interface may engender a charge current along the metallic layer as result of the inverse spin-Hall eect.7,8As the thickness of the metallic layer is reduced, additional interfacial eects must be considered. The dif- ferent contributions may be identied by analyzing the symmetries of the voltage signal with respect to dier- ent adjustable parameters, such as the orientation of the saturated magnetization controlled by the applied static eld.9Additionally, we can use propagating spin waves instead of FMR and vary the direction of propagation. Among other materials, the unique candidate for inves- tigations of interface-induced phenomena is single layer graphene (SLG). Due to its unique mechanical, optical, and electronic properties, graphene has attracted enor- mous attention since its discovery in 2004.10,11Nowa- days, one can produce large-area high-quality SLG by using, e.g., chemical vapor deposition on metalcatalysts.12{14For the observation of spin pumping ef- fects, SLG should be brought in contact with a mag- netic material. Yttrium Iron Garnet (YIG) holds a spe- cial place among all magnetic materials. Specically, it shows an unprecedentedly small magnetic damping re- sulting in the narrowest known line of the FMR and en- abling propagation of spin waves over long distances.15,16 Due to these unique characteristics, YIG lms have been recently considered as a promising material for spintronic and magnonic applications.17,18By combining YIG and SLG in a bilayer, one obtains a unique model system for investigations of the spin-wave induced interfacial eects. Here, we report the experimental observation of a chi- ral pumping eect in out-of-plane magnetized YIG/SLG bilayer and show that spin waves propagating in the YIG lm induce a sizable electric voltage in the adjacent SLG. We study the symmetry of the observed phenomenon by reversing the static magnetic eld and the direction of spin-wave propagation and nd that the corresponding changes in the sign of the electric voltage clearly indicate a broken re ection symmetry about the plane normal to the interface. We associate this symmetry breaking with the presence of screw dislocations in the crystalline lat- tice of YIG, which are expected to have a strong eect on the two-dimensional electron gas in SLG. We present a theoretical model, which shows that such crystalline defects can result in a chiral pumping eect with the observed symmetry. It is worth mentioning that the conventional spin pumping-induced voltages obtained for in-plane magnetized lms19{21are rooted in the re ec- tion symmetry breaking about the plane of the interface. They present dierent symmetries and are therefore unre- lated to the signal reported here. Moreover, in agreement with the experiment, our model predicts a non-monotonic dependence of the induced voltage on the spin-wave fre- quency. The structure of the manuscript is the following: We present the microwave and voltage measurements in Sec. II. Special attention is paid to the inversion sym-arXiv:1609.01613v2 [cond-mat.mes-hall] 7 Sep 20162 metry breaking, conrmed by further FMR experiments in YIG, revealing the chiral nature of the eect. The the- oretical model is discussed in Sec. III. Technical details of the model are set-aside in the Appendix. We summarize our ndings in Sec. IV. II. FERROMAGNETIC RESONANCE EXPERIMENTS A. Sample preparation and characterization The experimental layout is shown in Fig. 1(a). Graphene was placed on top of a monocrystalline YIG lm of 5.1m thickness grown by means of liquid phase epitaxy on 0.5 mm thick gallium gadolinium garnet (GGG) substrate. The saturation magnetization of the YIG lm dening the saturation eld in the out-of-plane geometry was 4 MS= 1:75 kG. The SLG sample was grown on a 50 m thick 99.99% pure copper foils in a cold-wall low-pressure CVD reactor (Black Magic, AIX- TRON). After the Cu foil had been cleaned in acetone and then shortly etched in acetic acid to remove surface oxide, it was placed on a graphitic heater inside the reac- tor and annealed at 1000C during 5 min in a ow of 20 sccm H 2and 1000 sccm Ar. Then, a ow of pre-diluted CH4(5% in Ar, 30 sccm) was introduced during 5 min to initiate the growth of graphene. After the growth, the gas ow was shut down, the system evacuated to <0:1 mbar and cooled down to room temperature.22SLG was trans- ferred to YIG/GGG by using the bubbling delamina- tion technique23,24and lithographically patterned into a multi-terminal Hall-bar structure with Au(100 nm)/Cr(5 nm) pads at the edges. The lateral dimensions of the sample were 1.5 by 25 mm. The Hall-eect mobility of the resulting devices was found to be in the range from 600 to 1300 cm2(V
s)1at T=10 K. The local resistance maximum at zero magnetic eld H00, usually ascribed to the weak localization eect, allowed for an estimation of the phase coherence length L100 nm at the same temperature. B. Microwave and voltage measurements The YIG/SLG sample was built onto a massive metal holder whose temperature was controlled and kept con- stant at 2950:02 K. The holder with the sample was placed between the poles of an electromagnet, which cre- ates a static magnetic eld H0with the inhomogeneity below 2105over 1 mm3. In all described experi- ments we kept H0>2 kOe ensuring the collinear ori- entation of H0andMS. To excite spin waves in the YIG layer, broadband microstrip lines ending with 50 m wide antennas were used, which were attached to the sample. Microwave (MW) eld with a xed frequency in the rangef= 38 GHz was applied to the antennas. A directional coupler was used to monitor the re ected b) c)a)FIG. 1: Chiral charge pumping in YIG/SLG. a) Schematic of the experiment. Single layer graphene (SLG) is transferred to yttrium-iron-garnet (YIG) lm grown on gallium gadolin- ium garnet (GGG) substrate. A, B, C, D are gold contact patches for electric measurements. The system is magnet- ically saturated normally to the sample plane by means of an applied magnetic eld H0. Spin waves are excited by mi- crowave (MW) current owing in one of the two narrow an- tennae, producing a local MW eld. b) Field dependence of the MW absorption for the up(H+) and down (H) orien- tation of the static eld. Dierent peaks on the dependences indicate excitation of dierent non-uniform spin-wave modes. Note the similarity of the two curves. c) Field dependence of the dc-voltage detected between the pads A and B for the up (H+) and down (H) orientation of the applied eld. Note dierent signs of the voltages for the two orientation of the eld. The red curves in (b) and (c) are shifted vertically for the sake of clarity. MW power, allowing for the determination of the reso- nance absorption in the sample due to the excitation of the spin waves. By keeping fconstant and varying H0, we recorded Pabs(H0) { curves characterizing the eld- dependent excitation of spin waves in YIG, see Fig. 1 b). The curves exhibit several maxima corresponding to the FMR, higher-order standing, as well as propagating spin3 FIG. 2: Symmetry properties of the observed eect. Both the induced electric eld Eand the wavevector of the spin wave ksware in the plane of the sample, while the applied eld H0and the YIG-magnetization MSare oriented normally to the sample plane. The + and signs indicate the polarity of the measured dc-voltage. Rotation of the system by 180around thez-axis reverses the directions of both Eandkswin agreement with the experiment. Mirroring with respect to the vertical plane (y!y) does not change Eandksw, whileH0andMSare reversed, in contrast to the experimental ndings. waves in YIG. Simultaneously with the microwave measurements, we recorded the dc-voltage between the electrodes. A lock- in technique was applied in order to detect the dc-voltage caused by the spin waves propagating in the YIG lm. Microwaves were modulated by a square wave with the repetition frequency of 10 kHz and the voltage dier- ence between dierent leads was measured by the lock- in amplier. During the measurements, the static mag- netic eld was swept and the dc-voltage along with the re ected MW power signal as a function of H0were recorded. Figure 1(c) shows a typical dependence of VAB(H0) { the voltage between the electrodes A and B. Comparing Figs. 1(b) and 1(c), one sees that the dc-voltage in SLG is induced at the same values of H0, where the ecient excitation of spin waves takes place. This clearly indicates its direct connection to the high- frequency magnetization precession in YIG. C. Symmetry breaking The most striking feature of the observed phenomenon is its unusual symmetry with respect to the inversion of H0. As seen from Fig. 1(c), the detected voltage changes its sign if the orientation of H0(andMS) is reversed. We emphasize that this inversion is not accompanied by visible changes in the microwave absorption curve (Fig.1(b)). Further measurements show that the induced volt- age is the same for both sides of the sample, VAB=VCD. It is proportional to the intensity of the spin wave (the absorbed microwave power) and changes sign if the di- rection of the spin-wave propagation is reversed. To obtain better insight into the underlying physics, we consider the symmetry of the phenomenon in more detail. As shown in Fig. 2, the induced electric eld E is oriented in the sample plane parallel to the direction of the spin-wave propagation indicated in Fig. 2 by a vector ksw, whereas the magnetization/magnetic eld is perpendicular to the sample plane. The + and signs in- dicate the polarity of the measured dc-voltage at a given orientation of the magnetization. As seen in Fig. 2, rota- tion by 180around the normal to the lm plane reverses both the direction of the spin-wave propagation and that of the electric eld, whereas the direction of the magneti- zation stays unchanged, all in agreement with the exper- imental ndings. However, mirroring of the system with respect to the vertical plane parallel to the direction of the spin-wave propagation ( y!y) does not change nei- ther the direction of the spin-wave propagation nor that of the electric eld, since they are real vectors, whereas the magnetization, which is an axial vector, is reversed. We emphasize that this contradicts to the experiment (see Fig. 1 c)). In other words, the phenomenon respon- sible for the appearance of the voltage in the YIG/SLG bilayer requires that the inversion symmetry is broken.4 More specically, the observed eect would be forbidden if the clockwise and counter-clockwise directions in the sample plane were equivalent, which reveals its chiral na- ture. To check whether the observed non-equivalence is a characteristic feature of the YIG lm itself, we performed high-precision FMR measurements on YIG lms of small lateral dimensions (5.1 m thick circle with the 0.5 mm radius and 6 m thick 0:50:5 mm2square) without SLG. In these experiments, the FMR in the YIG sample was excited by a quasi-uniform dynamic magnetic eld. The entire test device was mounted on a non-magnetic ro- tatable sample holder, which was placed in a static mag- netic eld with a homogeneity better than 0.1 Oe, with thex-axis being the rotation axis of the holder. Note here that the actual rotation axis (the x-axis) in the described experiments diers from the rotation axis of the thought experiment (the z-axis) shown in Fig. 2. The sample holder was designed to keep the position of the sample constant with an accuracy of 0.2 mm over the entire ro- tation. By sweeping the microwave frequency at a xed magnetic eld, the transmission coecient T=Pout=Pin was measured as a function of the frequency. Then, the holder with the sample was rotated by 180, and the measurements were repeated. The results of the mea- surements for the two orientations of the sample were averaged over several measurement cycles. The obtained FMR curves for the directions of the static magnetic eld parallel (H+) and antiparallel ( H) to the normal to the lm surface demonstrating a clear frequency dierence of
f= 5 MHz are shown in Fig. 3 (a). The value of
f is signicantly larger compared to the estimated uncer- tainty of 0.5 MHz originating from the inhomogeneity of the static eld and the stray elds of the sample holder. The measurements were repeated for dierent values of the applied magnetic eld resulting in the eld depen- dence of
fshown in Fig. 3 (b). One clearly sees that
f systematically increases as H0approaches 4 MS= 1:75 kG. Qualitatively similar results were obtained for dier- ent samples except that the maximum values
fwere found to vary from 6 to 13 MHz. The results presented in Fig. 3 show that the frequency of the FMR in YIG lms depends on whether the static magnetic eld is parallel or antiparallel to the normal to the lm surface ^z, i.e. it depends on the direction (clock- wise or counter-clockwise with respect to ^z) of the mag- netization precession. This indicates that the inversion symmetry is broken, since the clockwise and the counter- clockwise directions in the plane of the lm are not equiv- alent. Although the microscopic origin of this breaking is not yet fully clear, it should be connected with the de- fects in the crystallographic structure of YIG, since the ideal high-symmetry cubic structure of YIG is denitely incompatible with this symmetry breaking. It is known that the dominating defects in high-quality epitaxial YIG lms are growth dislocations. Their typical lateral den- sity of about 108cm2is connected with the typical mis- t between YIG and GGG lattices of 103(Refs. 15,25). FIG. 3: Ferromagnetic resonance (FMR) in a YIG lm. a) FMR curves measured at a constant H0for the up(H+) and down (H) orientation of the eld. Note a dierence in the FMR frequencies
f. b) Field dependence of
f. The dash line is a guide for the eye. It is also known that such dislocations strongly in u- ence the magnetic dynamics in YIG.26Possible mech- anisms of the defect-mediated symmetry breaking can include a growth-induced misbalance between clockwise and counter-clockwise screw dislocations and eects sim- ilar to the antisymmetric surface Dzyaloshinskii-Moriya- like interactions27in combination with a dislocation. We emphasize that, although the symmetry breaking is present in YIG lms without SLG on top, its in uence on the magnetization dynamics is very small and can only be detected in high-precision measurements. This is not surprising, since the symmetry breaking appears to be a surface phenomenon, which has vanishing in u- ence on the magnetization in the bulk of the lm. On the contrary, this symmetry breaking is expected to have sig- nicant in uence on conduction electrons in SLG placed on the surface of the YIG lm. III. PHENOMENOLOGICAL MODEL Next, we provide a phenomenological model for the ob- served eect based on the existence of a nite density of screw dislocations in YIG. The voltage signal is related with the electromotive forces induced by the magnetiza- tion dynamics. A screw dislocation creates a distortion of the graphene lattice as shown in Fig. 4 (a), which cou- ples to the electron spin through the spin-orbit interac- tion. As a result, the exchange eld seen by the itinerant electrons is tilted with respect to the magnetization in YIG, Fig. 4 (b), modifying the longitudinal response in the non-adiabatic regime. We emphasize that this mech- anism must be taken as a suggestive explanation for the observed phenomenon since there is no direct observation of the proposed mechanical deformations. A. Hamiltonian We assume that the exchange interaction couples the spins of itinerant electrons of SLG to the localized mag-5 )b )a FIG. 4: Distorted SLG near a dislocation in a YIG-lm. a) Top view of the distortion induced by a screw dislocation (the dislocation line is represented by the black dot). The red arrows illustrated the in-plane component of the eective exchange eld. b) Isometric illustration of the tilt of the ef- fective exchange eld (red arrows) along a path encircling the dislocation line (black arrow). The original eld points along the normal to the sample plane, and the tilting is generated by the spin-lattice coupling. netic moments in YIG. The Hamiltonian reads as H=~vF
p+
exm(t;r)
s+Hs-l: (1) The rst term is the Dirac Hamiltonian describing elec- tronic states with momentum paround the two inequiv- alent valleys Kin graphene, where = (x;y) is a vector of Pauli matrices associated to the sub-lattice de- gree of freedom of the wave function and vFis the Fermi velocity. The second term corresponds to the exchange coupling,
ex, where s= (sx;sy;sz) are the itinerant spin operators and m(t;r) is a unit vector along the lo- cal magnetization in YIG. The last term is a spin-lattice interaction that couples the mechanical degrees of free- dom with the spins of electrons. As it is well known,11graphene can support large me- chanical distortions that change dramatically the dynam- ics of Dirac electrons. When an SLG is placed on top of disturbed YIG, the carbon atoms near a screw disloca- tion can be expected to follow its distorted prole, which introduces a torsion in the honeycomb lattice, as repre- sented in Fig. 4 (a). This deformation couples to the electron spin as Hs-l=
so(@xuy@yux)(xsx+ysy); (2) where@xuy@yuxis the anti-symmetrized strain tensor {u= (ux;uy) represents the displacements of the carbon atoms { that parameterizes the torsion of the graphene lattice and can on average be related to the density of screw dislocations. The form of this coupling is dictated by the C 6vsymmetry of the honeycomb lattice, and its microscopic origin resides in the hybridization of and orbitals of carbon atoms, which enhances the spin-orbit eects within the low-energy bands.28Note here that the proposed deformation just models the observed symme- try breaking to provide a proof of principle for the chiral pumping eect. We expect our model to be qualitatively correct regardless of the actual deformations of the lattice close to dislocations.The eective exchange eld seen by itinerant electrons, n(t;r), is tilted away from the local magnetization fol- lowing the torsion of the lattice, as illustrated in Fig. 4b. This is implemented by the spin-lattice coupling in Eq. (2), which can be understood on geometry grounds as the non-abelian connection that transports the spin quantization axis along the distorted graphene lattice. The details are provided in the Appendix. To the lowest order in
so, we obtain n(t;r)m(t;r)2a
so ~vFb(r)m(t;r);(3) where b(r) is dened as b(r) =a1Zr r0dr0(@xuy@yux): (4) Here r0is the origin of the dislocation and we have in- troduced a microscopic length scale ain order to make b(r) dimensionless. B. Electromotive forces The magnetization dynamics induces electromotive forces in SLG along the direction of propagation of spin waves. At low frequencies, majority electrons along the quantization axis dened by n(t;r) experience an eec- tive electric eld of the form29{31 Ei=~ 2e_n
(n@in+@in): (5) The rst term is purely geometrical, strictly valid in the limit
ex~j_nj,~vFj@inj, when the electron spins fol- low adiabatically the direction of eective exchange eld. Thecorrection is related to a slight misalignment due to spin relaxation caused by the spin-orbit interaction. The electromotive force in the adiabatic limit, which is rooted in the associated geometrical Berry phase, re- spects the structural symmetries of the device. Indeed, the correction due to the magnetization tilting expressed in Eq. (3) averages to 0 when integrated upon the period of the spin wave excitation. The introduction of screw dislocations in YIG breaks the structural symmetries of the transport signals in SLG owing only to non-adiabatic corrections to the electromotive force. These appear as higher order expansions in~
ex_nand spatial gradients ~vF
ex@inthat capture spin-orbital eects. The new terms compatible with C 6vsymmetry read Ei=~3vF 2e
2ex(1@in+2n@in)
^zr(_n)2;(6) where1;2are dimensionless phenomenological parame- ters. The tilting of the exchange eld translates the eect of the symmetry breaking in YIG into the electronic re- sponse of graphene, driving a voltage along the direction6 02 4 6 8 10-0.20.00.20.40.60.81.06.2GHz5 .8GHz5 .0GHz8 .0GHzV(µV)P abs (mW)a)4 .0GHz4 5 6 7 8 0.00.10.20.3V / Pabs ( µV / mW )F requency (GHz)b) FIG. 5: Frequency dependence of the eect. a) The detected dc-voltage as a function of the absorbed MW-power for dif- ferent frequencies as indicated. The dash lines are linear ts of the experimental data used for calculation of the eciency coecientV=P abs. b) The eciency coecient V=P absas a function of the spin-wave frequency. Note that a sizable ef- fect is observed between 5 and 7 GHz only. The dash line is a guide for the eye. of propagation of the spin waves of the form32 V=1~2
so
2exZ dx(@xuy@yux)^z
m@x(_m)2:(7) As seen from this equation, the proposed theoretical model reproduces the experimentally observed inversion of the sign of the induced voltage accompanying the reversal of the magnetization ( ^z
m! ^z
mwhen H0!H0) or the direction of the spin-wave propaga- tion (@x(_m)2!@x(_m)2whenksw!ksw). The contribution in Eq. (7) should saturate when the theory crosses over to the strongly non-adiabatic regime, at frequencies of the order of
ex=~. We expect for the electromotive force to be suppressed at larger frequencies due to a dynamic averaging eect, similarly to the mo- tional narrowing in the spin diusion problem: the mag- netization precession is so fast that its time-dependent component eectively averages out from the view of elec- tron spins. C. Frequency dependence The model predicts that the eect should exist only within a certain range of spin-wave frequencies around
ex=~. Although
excannot be directly measured in the experiment, the theoretical prediction can be veried by analyzing the frequency dependence of the observed eect. Figure 5(a) shows the dependences of the voltage onPabsproportional to the intensity of the excited spin waves, recorded for dierent spin-wave frequencies. As seen from Fig. 5(a), the voltage is linearly proportionalto the spin-wave intensity and the proportionality coef- cientV=Pabsis indeed strongly frequency dependent. Moreover, in agreement with the theoretical results, the frequency dependence of V=Pabs(Fig. 5(b)) clearly ex- hibits a resonance-like behavior with the maximum at about 6 GHz. The relatively small exchange constant
ex=~30 eV obtained from the observed resonant fre- quency nicely agrees with the recently determined ex- change eld of 0.2 T.33 IV. CONCLUSIONS In conclusion, we experimentally observe a chiral charge pumping in YIG/SLG bilayers caused by the sym- metry breaking at the YIG surface. The eect provides a novel, ecient mechanism for detection of magnetization dynamics for spintronic and magnonic applications. The developed theoretical model, taking into account the ex- change interaction between localized magnetic moments in YIG and itinerant spins in graphene, predicts that the strength of the eect can be increased by making use of two-dimensional conductors with a strong spin-orbital coupling, such as MoS 2.34Additionally, the found sym- metry breaking can result in other, not yet observed, eects. In particular, it can enable the reciprocal ef- fect consisting in the excitation of spin-waves of unusual symmetry via an electric current in graphene. These ef- fects provide essentially new opportunities for the direct electrical detection and manipulation of magnetization in insulating magnetic materials and open new horizons for the emerging technologies. Acknowledgements This work was supported in part by the Deutsche Forschungsgemeinschaft, the Swedish Research Coun- cil, the Swedish Foundation for Strategic Research, the Chalmers Area of Advance Nano, the Knut and Alice Wallenberg Foundation, the U.S. Department of Energy, Oce of Basic Energy Sciences under Award No. DE- SC0012190 (H.O. and Y.T.), and the program Megagrant No. 14.Z50.31.0025 of the Russian Ministry of Education and Science. J. S. thanks the support of STINT, SOEB, and CTS. S.O.D is thankful to L.A. Melnikovsky for fruit- ful discussions and to R. Sch afer for domain mapping in YIG. Appendix: Tilting of the exchange eld The spin rotational symmetry of graphene Hamilto- nian is expressly broken by the spin-lattice coupling in Eq. (2), but it can be approximately restored in certain limit by means of a local unitary rotation of the spinor7 wave function. Such a gauge transformation reads as U= Exp [ib(r)
s]Iib(r)
s; (A.1) where b(r) is an axial vector, dened in Eq. (4), and is a small, dimensionless parameter that we identify with =a
so ~vF1: (A.2) Notice that there is a gauge ambiguity in the denition of U. In Eq. (4), the gauge is xed by choosing the origin of the dislocation as the lower limit of integration, but this is arbitrary. As a result, physical eects induced by the tilting of the eective exchange eld appear as derivatives ofb(r), making this approach fully consistent. Next, we see that the unitary transformation in Eq. (A.1) gauges away the last term in the Hamiltonian of Eq. (1), generating new terms that are second order inand we can safely neglect. First, we have that UyHs-lUH s-li[Hs-l;b
s] +O 2
; (A.3) but note that the strength of Hslis already rst order inand therefore the second term in this last equation is actually a second order term; then, we can write UyHs-lUH s-l+O 2
: (A.4) On the other hand, if we apply the unitary transforma- tion to the kinetic term in Eq. (1) we obtain i~vFUy
@U~vF a(@xuy@yux) (xsx+ ysy) +O 2
=Hs-l+O 2
: (A.5) Then, the subsequent gauge eld cancels out the spin- lattice coupling. To the leading in 1 (therefore, inthe strength of the interaction), the spin-lattice coupling in Eq. (2) can be understood as the non-abelian connec- tion that transports the electron spin through the dis- torted crystal, following the torsion created by the screw dislocations in YIG. We have to apply the transformation to the exchange coupling. Formally, we have
exUym
sU=
exn
s; (A.6) where n=Rm, andRis a SO(3) rotation whose matrix elements read Rij=1 2Tr siUysjU : (A.7) More precisely,Rcan be written as R= Exp [ib
`]; (A.8) where the SO(3) generators `= (`x;`y;`z) in this basis read `x=0 @0 0 0 0 02i 0 2i01 A; `y=0 @0 0 2i 0 0 0 2i0 01 A; `z=0 @02i0 2i0 0 0 0 01 A: To the leading order in we have35 nm2bm+O 2
; (A.9) which corresponds to Eq. (3). 1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 2A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404(R) (2002). 3G. Schmidt, D. Ferrand, L. W. Mollenkamp, A. T. Filip, B. J. van Wees, Phys. Rev. B 62, R4790(R) (2000). 4Bret Heinrich, Yaroslav Tserkovnyak, Georg Woltersdorf, Arne Brataas, Radovan Urban, and Gerrit E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). 5K. Lenz, T. Toli nski, J. Lindner, E. Kosubek, and K. Baberschke, Phys. Rev. B 69, 144422 (2004). 6M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Wal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). 7E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 8O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Homann, Phys. Rev. Lett. 104 046601 (2010).9A. Azevedo, L. H. Vilela-Le~ ao, R. L. Rodr guez-Su arez, A. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011). 10K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 11A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). 12S. K. Keun et al., Nature 457, 706 (2009). 13X. Li et al., Science 324, 1312 (2009). 14S. Bae et al., Nature Nanotechnology 5, 574 (2010). 15V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep. 229, 81 (1993). 16A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D43, 264002 (2010). 17Y. Kajiwara et al., Nature 464, 262 (2010). 18A. Hamadeh et al, Phys. Rev. Lett. 113, 197203 (2014). 19Zhenyao Tang, Eiji Shikoh, Hiroki Ago, Kenji Kawahara,8 Yuichiro Ando, Teruya Shinjo, and Masashi Shiraishi, Phys. Rev. B 87, 140401(R) (2013). 20J. B. S. Mendes, O. Alves Santos, L. M. Meireles, R. G. Lacerda, L. H. Vilela-Le~ ao, F. L. A. Machado, R. L. Rodr guez-Su arez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 115, 226601 (2015). 21S. Singh, A. Ahmadi, C. T. Cherian, E. R. Mucciolo, E. del Barco, and B. Ozyilmaz, Appl. Phys. Lett. 106, 032411 (2015). 22J. Sun, et al., IEEE Trans. Nanotechnol. 11, 255 (2012). 23L. Gao, et al., Nat Commun. 3, 699 (2012). 24C. J. Lockhart de la Rosa, et al., Appl. Phys. Lett. 102, 022101 (2013). 25S. A. Nikitov, Relaxation Phenomena of Magnetic Exci- tations in Ferromagnetic Media , in Advances in Chemi- cal Physics: Relaxation Phenomena in Condensed Matter , Vol. 87, ed. by W. Coey (John Wiley, New Jersey, 1994). 26P. Nowik-Boltyk, O. Dzyapko, V. E. Demidov, N. G. Berlo, and S. O. Demokritov, Sci. Rep. 2, 482 (2012). 27Ioan Mihai Miron, Gilles Gaudin, St ephane Auret, Bernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vo-gel, and Pietro Gambardella, Nat. Mater. 9, 230 (2010). 28H. Ochoa, A. H. Castro Neto, V. I. Fal'ko, and F. Guinea, Phys. Rev. B 86, 245411 (2012). 29G. E. Volovik, J. Phys. C: Solid State Phys. 20, L83 (1987). 30R. A. Duine, Phys. Rev. B 77, 014409 (2008). 31Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407 (2008). 32By plugging Eq. (3) into Eq. (6), the 1term generates the longitudinal voltage in Eq. (7), whereas the 2term leads to a transverse voltage. The latter is even in the applied magnetic eld, in agreement with symmetry arguments. This prediction is in fact supported by preliminary exper- iments. A careful study of the induced transverse voltage will be a topic our further investigations. 33C. Leutenantsmeyer, A. A. Kaverzin, M. Wojtaszek, and B. J. van Wees, arXiv:1601.00995 [cond-mat.mes-hall]. 34K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 36805 (2010). 35The action of the SO(3) generators over a vector vsatisfy the following property: ( u
`)v= 2iuv.

2016-09-06

We demonstrate that a sizable chiral charge pumping can be achieved at room temperature in graphene/Yttrium Iron Garnet (YIG) bilayer systems. The effect, which cannot be attributed to the ordinary spin pumping, reveals itself in the creation of a dc electric field/voltage in graphene as a response to the dynamic magnetic excitations (spin waves) in an adjacent out-of-plane magnetized YIG film. We show that the induced voltage changes its sign when the orientation of the static magnetization is reversed, clearly indicating the broken spatial inversion symmetry in the studied system. The strength of effect shows a non-monotonous dependence on the spin-wave frequency, in agreement with the proposed theoretical model.

Chiral charge pumping in graphene deposited on a magnetic insulator

1609.01613v2

Ultra-High Cooperativity Interactions between Magnons and Resonant Photons in a YIG sphere J. Bourhill,1,N. Kostylev,1M. Goryachev,1D. L. Creedon,1and M.E. Tobar1 1ARC Centre of Excellence for Engineered Quantum Systems, University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia (Dated: September 6, 2018) Resonant photon modes of a 5mm diameter YIG sphere loaded in a cylindrical cavity in the 10- 30GHz frequency range are characterised as a function of applied DC magnetic eld at millikelvin temperatures. The photon modes are conned mainly to the sphere, and exhibited large mode lling factors in comparison to previous experiments, allowing ultrastrong coupling with the magnon spin wave resonances. The largest observed coupling between photons and magnons is 2g=2= 7:11GHz for a 15.5 GHz mode, corresponding to a cooperativity of C= 1:510:47107. Complex modications beyond a simple multi-oscillator model, of the photon mode frequencies were observed between 0 and 0.1 Tesla. Between 0.4 to 1 Tesla, degenerate resonant photon modes were observed to interact with magnon spin wave resonances with dierent couplings strengths, indicating time reversal symmetry breaking due to the gyrotropic permeability of YIG. Bare dielectric resonator mode frequencies were determined by detuning magnon modes to signicantly higher frequencies with strong magnetic elds. By comparing measured mode frequencies at 7 Tesla with Finite Element modelling, a bare dielectric permittivity of 15:960:02of the YIG crystal has been determined at about 20mK. INTRODUCTION Hybrid photon-magnon systems in ferromagnetic spheres have recently emerged as a promising approach towards coherent information processing [1{8]. Due to the large exchange interaction between spins in ferromag- nets, they will lock together to form a macrospin that can be utilised for coherent information processing protocols [8, 9]. The quantised excitation of the collective spin is referred to as a magnon. Yttrium Iron Garnet (YIG) based magnon systems are attractive due to very high spin density, resulting in signicant cooperativity as well as relatively narrow linewidths [4, 5, 10, 11]. Further- more, due to the possibility of coupling magnon modes to photons at optical frequencies [10{13], magnon systems may be considered as a candidate for coherent conver- sion of microwave and optical photons [10, 11]. In addi- tion, magnons interact with elastic waves [14, 15] open- ing a window for combining mechanical and magnetic systems. These systems therefore possesses great poten- tial as an information transducer that mediates inter- conversion between information carriers of dierent phys- ical nature thus establishing a novel approach to hybrid quantum systems [9, 16{18]. Among all magnon systems the central role is de- voted to YIG, a material that possesses exceptional mag- netic and microwave properties and has been used in microwave systems such as tuneable oscillators and l- ters for many decades [19, 20]. Although, only recently Soykal and Flatt e proposed and modelled the photon- magnon interaction based on YIG nano-spheres with ap- plication to quantum systems [21, 22]. As predicted by the authors, extremely large coupling rates, g, could be achieved in YIG spheres, which is favourable for coher- ent information exchange and has been demonstrated ex-perimentally later [4, 5, 23]. For these experiments, the interaction is observed between photon and magnon reso- nances created correspondingly by photon cavity bound- ary conditions and spin precession under external DC magnetic eld. A commonly used method is to place a relatively small YIG sphere in a local maxima of the mag- netic eld inside a much larger microwave cavity. This is done to achieve quasi uniform distribution of the cavity eld over the sphere volume to avoid spurious magnon modes. Cavities can take oval [3, 4] or spherical shapes [21, 22, 24], and even re-entrant cavities with multiple posts have been used in an attempt to focus the mi- crowave energy over the sphere [5, 23]. In this work we investigate a completely dierent regime in which the magnon and photon wavelengths are comparable, lead- ing to considerably larger coupling strengths, but addi- tional couplings to higher order modes. In general, for this case the strength of the photon-magnon interaction will be determined by an overlap integral of the two re- spective mode shape functions. Given the magnon mode shape is limited to the sphere's volume, this integral will be maximised when the photon mode is conned to the same volume. To achieve the latter, we utilise an ex- ceptionally large YIG sphere with diameter d= 5 mm, matching magnon and microwave photon mode volumes, unlike previous microwave cavity experiments. In order to investigate this regime we use common mi- crowave spectroscopy techniques [5, 23, 25{28] to directly observe the mode splitting caused by the magnon inter- action to determine the coupling values. Similar systems have been extensively utilised not only in the eld of spin- tronics to investigate the interaction between microwave photons and paramagnetic spin ensembles [25{29], but also to realise optical comb generation [30], ultra-low threshold lasing [31], cavity-assisted cooling, control andarXiv:1512.07773v5 [quant-ph] 26 Apr 20162 573061.38YIG sphereSapphiresubstrateMicrowavecoaxial cableSMA connectorMagnetic loopprobeCopper cavityzx Applied Magnetic Field FIG. 1: (Color online) Cross section of the copper cavity that houses the 5 mm YIG sphere. The sphere sits on a sapphire disk, and microwaves are coupled in and out of it via loop probes which produce and detect an Hcomponent. A variable DC magnetic eld is applied along the zaxis. measurement of optomechanical systems [32, 33], and ex- tremely stable cryogenic sapphire oscillator clock technol- ogy [34, 35]. To date, exciting internal, highly-conned photonic modes in a YIG sphere has only recently been demonstrated in the optical regime using Whispering Gallery Modes (WGM) [10, 11] but has never before been achieved in the microwave domain. This is due to the typical sub-millimetre diameter of the spheres. As such, interactions with magnons must be observed via Bril- louin scattering [36], which has yielded high quality fac- tors and also demonstrated a pronounced nonreciprocity and asymmetry in the sideband signals generated by the magnon-induced scattering. Extremely large mode splittings ( g=! > 0:1) cause si- multaneous coupling to a higher density of modes, with an overlap of avoided level crossings. Therefore, the model proposed by Soykal and Flatt e [21] becomes no longer applicable, as it assumes the interaction occurs between a single photonic and magnon mode. More re- cently, a paper by Rameshti et al. [24] simulated a sim- ilar scenario of the presented experiment, in which the ferromagnetic sphere is itself the microwave cavity. Our observed results may appear to be in good agreement with this work's predictions, however, what is apparent is that in this specialised case, one must consider more than just the magnetostatic, uniform Kittel magnon mode, a limitation of [24]. Indeed, due to the nonuniformity of both the microwave mode magnetic eld energy density across the sphere, which is unique to this experiment, and the nonuniformity of the sphere parameters arising due to cryogenic cooling, the assumption that only the uni- form Kittel magnon resonance participates is no longer valid. Despite this, in this paper we use a two mode model to obtain estimations of coupling strengths, and demonstrate how this results in inconsistent susceptibil- ity values. Frequency (GHz) Magnetic Field (T)00.20.40.60.811.21015202530 101520 5Normalised S21 (dB) 02530FIG. 2: (Color online) Transmission data as magnetic eld is swept. PHYSICAL REALIZATION Thed= 5 mm YIG sphere was manufactured by Ferri- sphere, Inc. with a quoted room temperature saturation magnetisation of 0M= 0:178 T. It is placed on a small sapphire disk, with a concavity etched out using a dia- mond tipped ball grinder, to keep the sphere from rolling out of position, and reduce dielectric losses that would arise if the YIG were in direct contact with the conduc- tive copper housing. Sapphire was chosen over te on as an intermediary between the YIG and copper to improve the thermal conductivity to the sphere. Together, the sapphire and YIG are housed inside a copper cavity with dimensions specied in gure 1. A loop probe constructed from exible subminiature ver- sion A cable launchers is used to input microwaves and a second is used to make measurements, allowing the de- termination of Sparameters. The entire cavity is cooled to about 20 mK by means of a dilution refrigerator (DR) with a cooling power of about 500 W at 100 mK. The cavity is attached to a copper rod bolted to the mixing chamber stage of the DR that places it at the eld center of a 7 T superconducting magnet, whose applied eld is oriented in the zdirection of the cavity. The magnet is attached to the 4 K stage of the DR, with the copper cav- ity mounted within a radiation shield of approximately 100 mK that sits within the bore of the magnet. EXPERIMENTAL OBSERVATIONS The transmission spectrum of the YIG was recorded for DC magnetic elds swept from 0 { 7 T using a vec- tor network analyser (VNA), with partial results shown in gure 2. A host of magnon resonances/higher or-3 der magnon-polaritons can be observed originating from (0 T, 0 GHz) with an approximate gradient of 28 GHz/T. The more-or-less horizontal lines approaching the magnon resonances from either side correspond to resonant photon modes of the sphere. Importantly, we can observe that in the dispersive regime, far removed from any microwave resonant mode, there still exist mul- tiple magnon modes. We observe that the anticrossing gaps are populated by unperturbed modes, which are remnant \tails" of both \higher" and \lower" mode in- teractions, as predicted by Rameshti et al. in the ultra- strong coupling regime [24]. For the remainder of this paper, we will focus on the six lowest frequency photon modes, whose resonant frequen- cies may only be accurately determined at large magnetic elds, when the entire spin ensemble has been detuned, as shown by gure 3. The modes have been categorised into three distinct classes: mode \ x" is the lowest frequency and lowest Q factor mode, the two highest Qfactor modes; \ i" and \ii", and the three remaining highest frequency modes; \1", \2" and \3". Their asymptotic frequencies as B!7 T are summarised in table I. The behaviour of these modes as the magnon reso- nances are tuned via the applied magnetic eld is shown in gure 4. It has been shown previously [5] that a standard model of two interacting harmonic oscillators can accurately determine the coupling values from such avoided crossings. However, we observe strong distortion around 0 T, and also an asymmetry of the mode split- tings about the central magnon resonances due to the ultrastrong coupling of the photon modes to the magnon modes, as was observed previously in Ruby [29]. There- Frequency (GHz)1013161514121112Magnetic Field (T)34567 xi123ii 1618Normalised S21 (dB)2021412108640Normalised S21 (dB) B = 7 Txi1ii23(a)(b) FIG. 3: (Color online) (a) Asymptotic frequency values of the six lowest order photon modes as B!7 T. (b) Transmission spectra at B= 7 T, from which mode linewidths may be measured.Mode!jjB!7 T=2j=gj=Cjgj=!j (GHz) (MHz) (GHz) (105) (%) x 12.779 11.84 4.79 5.971.85 18.7 i 15.506 1.029 7.11 15147.0 22.9 ii 15.563 1.197 4.19 45.214.0 13.5 1 15.732 5.355 6.15 21.86.76 19.5 2 15.893 2.965 3.04 9.602.98 9.56 3 15.950 2.965 0.78 0.6320.196 2.45 TABLE I: Measured and calculated results for each mode showing couplings gjand cooperativities, Cj. Frequency (GHz)Magnetic Field (T)00.10.20.30.40.50.60.70.80.91020 131619181715141211xi1ii23i’1’2’3’ii’x’ FIG. 4: (Color online) Two harmonic oscillator model tting to modes i,ii, 1, 2 and 3. From the curved lineshapes, one can determine the coupling value g. fore we t only the curves to the right of the magnon res- onance. These ts are shown as the dashed lines in gure 4. From these ts we can approximate the values of gfor each mode, as summarised in table I. The linewidths, j and frequencies, !j=2of the photon modes are deter- mined from the transmission spectra taken at high eld values (gure 3 (b)), whilst the magnon linewidth, mag can be determined by analysing the transmission spec- tra in the dispersive regime. We take a frequency sweep atB= 0:2475 T from 5.75{9 GHz in order to view the magnon resonance peaks far away from any interaction with the dielectric microwave modes, as shown in g- ure 5. There is a level of variation amongst the magnon linewidths as calculated by tting the peaks with Fano resonance ts, as shown in gure 6. This variation and the presence of multiple peaks demonstrates the presence of higher order magnon modes. Taking the average and standard deviation of these linewidths gives a nal es- timate of magnon linewidth as mag== 3:2470:493 MHz. Cooperativity is calculated as Cj=g2 j=magj. The cooperativity values in table I demonstrate that all modes are strongly coupled to the magnons, and all with the exception of modes 2 and 3 are in the ultrastrong coupling regime (i.e. gj=!j0:1 [24]). The largest co-4 FIG. 5: S21 transmission spectra showing a host of magnon resonance peaks at B= 0:2475 T. ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=2.95 MHz6.1006.1026.1046.1066.1086.11025303540 Frequency(GHz)Normalised S21(dB) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=4.12 MHz6.3606.3656.3706.3756.3806.385152025303540 Frequency(GHz)Normalised S21(dB) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=2.81 MHz6.5206.5256.5306.5356.54051015202530 Frequency(GHz)Normalised S21(dB) ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●● ●● ●●●●● ●● ●●●●●●●●●● ●●●●●●2Γmag/2π=2.89 MHz6.6406.6456.6506.6556.6606.6656.670-5051015 Frequency(GHz)Normalised S21(dB) ● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●2Γmag/2π=3.46 MHz6.836.846.856.866.87-10-50510 Frequency(GHz)Normalised S21(dB) FIG. 6: Fitting the magnon resonances with Fano ts. operativity value obtained is that of mode i, which is, to the authors' knowledge, the largest value ever reported to date in any previously studied spin system. A transmission spectrum taken at B= 0:6425 T is shown in gure 7, demonstrating the mode splitting of mode 1, symmetric about the magnon resonance. Over- laid in red is the bare photon resonance at 7 T, i.e. the microwave mode unperturbed by the magnon modes. From this red curve, 2 1=2is determined to be 5.355 MHz, as shown in table I. When one takes the average of 2 mag=2= 3:247 MHz and 2 1=2, one obtains the line width of the resulting hybrid state when the magnon resonance is tuned coincident in frequency with the pho- ton mode, as depicted by the dashed blue curve in gure 7, i.e.,4.4 MHz. This excellent agreement indicates that at this particular Beld, mode 1 exists as a hybrid magnon-polariton. AroundB= 0 T, we observe a severe distortion of the cavity mode frequency dependence on magnetic eld as demonstrated by Fig. 8. Around 16 GHz, we see there exist ve modes, corresponding to modes i,iiand 1{ 3, on the \left" side of the magnon resonances. These modes have been given a primed nomenclature to indi- g/!≈ 6.2 GHz Transmission spectrum for H = 0.6425 TBare dielectricresonance for H = 7 T (overlay)Magnon-polariton fit●●●●●●●●●●●●●●●●●●●●●●●●●12.70512.71512.725354045505560●●●●●●●●●●●●●●●●●18.8418.8518.863638404244464850 15.7415.72 2Γ=2Γmag+2ΓcavB = 0.6425 T4.4 MHz 2Γ=2Γmag+2Γcav2Γ=2Γmag+2Γcav4.4 MHz2Γcav5.4 MHz22Γ=2Γmag+2ΓcavFIG. 7: (Color online) Transmission spectrum at B= 0:6425 T. At this applied magnetic eld the magnon resonance is tuned coincident in frequency with mode 1, and the strong coupling between the two results in a mode splitting of 6.2 GHz. The high density of resonant peaks in the centre of the gure suggest a large number of higher order magnon modes are present in this system. cate their existence at Belds lower than that required to tune the magnons to their frequencies. This phenomenon has been previously observed in single crystal YAG [37] highly doped with rare-earth Erbium ions, and is ex- plained by the in uence on the ferromagnetic phase of the impurity ions on degenerate modes. The eect can be explained by the in uence of the ensemble of strongly coupled spins on the centre-propagating waves of the near degenerate mode doublet. For large spin-photon inter- actions, tails of Avoided Level Crossings (ALCs) from the positive half plane ( B > 0) should still exist on the Magnetic Field (T)00.10.20.150.05-0.1-0.2-0.15-0.05Frequency (GHz)2021 1619181715142223 5Normalised S21 (dB) 01015202530 -5i’1’2’3’ii’ FIG. 8: (Color online) Behaviour of the photon modes aroundB= 0 T, a result of the internal magnetisation of YIG.5 negative half plane ( B < 0) and vice a versa. Although, instead of gradual change of direction, the system demon- strates an abrupt transition to a \no coupling" state. It is worth mentioning that such eect has not been observed in photonic systems interacting with paramagnetic spin ensembles [25, 26, 38]. In the present case, the eect is much more pronounced, with fractional frequency de- viations and the magnetic eld range of the eect both orders of magnitude larger than observed previously [37], a result of the magnetic spin density. DISCUSSION COMSOL 3.5's electromagnetic package was used to model the system. A 3D model was used so as to analyse the degeneracies in the axis of the dielectric modes. The internal copper wall of the cavity is modelled as a perfect electrical conductor, which for the purposes of the desired eigenfrequency study, is an appropriate simpli- cation. The results of the FEM using a value of YIG=0= 15:965 andr0= 3:71 mm, where r0is the radius of curva- ture of the sapphire support's concavity, are summarised in gure 10 and in table II. The measured frequency of the doublet modes has been taken as the average of the two constituent's frequencies at B= 7 T. From the FEM and the analytical mode shapes of spherical dielectric resonances described by [39], we can identify mode xas ann= 0 mode with no degener- acy. Therefore it is present as a singular resonance. The other ve modes appear as n= 1 modes. There should only exist a 2 n+ 1 fold degeneracy for resonant spheri- cal photon modes, which can be broken by internal im- purities or by asymmetric boundary conditions set by a cylindrical enclosure, microwave loop probes, and the sapphire substrate, collectively termed \backscatterers". This degeneracy arises from a Legendre polynomial in the mode's HandEeld analytical expressions of the formPm n(cos)n cos(m) sin(m)o , wherem= 0;:::;n . The in- tegersmandnrepresent the number of maxima of the mode's energy density in the direction over 180, and the number in the direction over 180, respectively. This would imply that for n= 1 we should observe three distinct modes corresponding to a single ( n;m) = (1;0) and two (1;1) modes, rather than ve modes. However, Modefmeas (GHz)fsim(GHz) (n;m) x 12.779 12.785 (0,0) i&ii 15.534 15.286 (1,1) 1 15.732 15.736 (1,0) 2 & 3 15.922 15.921 (1,1) TABLE II: Comparison of FEM and measured frequencies and hence mode identication.the FEM demonstrates that the use of the sapphire sup- port base introduces a further degeneracy to the (1 ;1) modes depending on the amount of eld that permeates the sapphire. The modelling predicts four (1 ;1) modes, existing as two sets of two, which are separated by ap- proximately 500 MHz. This is in fair agreement with the separation of modes i,iiwith modes 1{3. Therefore it is apparent that modes iandiiare a doublet pair with (n;m) = (1;1). Given that modes 2 and 3 approach relatively similar frequencies at high magnetic elds, it is reasonable to as- sume that these modes correspond to the second (1 ;1) doublet pair, which FEM predicts will have a larger pro- portion of microwave eld inside the sapphire support. This means that mode 1 must be the (1 ;0) single mode. From gure 4, we can see that both the doublet pairs demonstrate a gyrotropic response when interacting with the magnon resonances, i.e., one mode interacts more than its doublet pair. This is a common occurrence in spin ensemble systems and has been observed in para- magnetic systems such as Fe3+in sapphire [26, 27, 40]. This asymmetric interaction strength for doublet pairs has also been observed in ferromagnets by Krupka et al. [41, 42] and predicted by Rameshti et al. [24], with the latter stating that gn;m=n> gn;m=n, where a dier- ent notation to that used here is employed, in which m=n;:::; 0;:::;n . The notations are equivalent as anm=ndoublet in [24] corresponds to an cos(m) sin(m)o doublet pair here. The gyrotropic response is a result of the anisotropy of a ferromagnet's permeability tensor; the same reason why these materials are used in circulators. The permeability tensor, containing o diagonal terms appears as ~ =00 @1 +i0 i 1 +0 0 0 11 A; (1) where0is the permeability of free space, and is the magnetic susceptibility of the ferromagnet, which is related to the magnetic permeability tensor by ~ = 0 ~1 +~  . When any resonant photonic mode exists as a doublet, it is because then cos(m) sin(m)o degeneracy has been broken by some backscatterer, and the two resulting modes ex- ist as counter propagating travelling waves [26, 40]. The overall eect is that one travelling wave will see an eec- tive permeability of +=0(1 ++), whilst the other will see=0(1 +), which can be rewritten as =0(1+), and we can state that ( ++)=2 =, whereis the \unperturbed" magnetic susceptibility that a standing wave would observe. The eective susceptibility that a mode experiences will determine the interaction strength of that mode with6 a magnon resonance according to [5]: g2 i=e!2; (2) whereis the total magnetic lling factor of the mode; i.e., the proportion of magnetic eld within the ferro- magnetic material compared to the entire system. This parameter is used in an attempt to quantify the overlap of the magnon and photon modes and is calculated as =RRR VYIG0~H~HdV YIGRRR V0~H~HdV: (3) It should be noted that typically it is only the magnetic eld energy density perpendicular to the external mag- netic eld that is considered to interact with the spin system [5, 21]. However, the interaction of mode 1 is far larger than its perpendicular lling factor of 0.075 would suggest. So, in an attempt to account for the interac- tion with nonuniform magnon modes, the total magnetic lling factor has been used. These have been calculated from the FEM and the resulting values of eare dis- played in table III. Given our assumption that mode 1 represents the (1,0) dielectric mode, which will exist as a standing wave given no possible degeneracy, the calculated evalue for this mode should represent the unperturbed magnetic suscep- tibility of the YIG. Taking the average of the evalues for the doublet modes i(+) andii() yields a value of= 0:0595; in reasonable agreement with the value obtained from mode 1. The FEM predicts that modes x, 2 and 3 will each contain a signicant proportion of magnetic eld energy within the sapphire support, so one would expect these modes to observe a lower eective magnetic susceptibil- ity, which would appear true for the latter two modes (their average susceptibility yields an unperturbed sus- ceptibility of0:01). However, mode xdemonstrates a much larger coupling strength than what should be af- forded a mode with its lling factor, hence a evalue approximately three times larger than the unperturbed value obtained from modes i,iiand 1. This suggests that our approximation of using the total magnetic lling fac- tor to quantify the overlap of the magnon and photon Mode!jjB!7 T=2gj=je (GHz) (GHz) x 12.779 4.79 0.221 0.159 i 15.506 7.11 0.594 0.0885 ii 15.563 4.19 0.594 0.0305 1 15.732 6.15 0.728 0.0525 2 15.893 3.04 0.493 0.0185 3 15.950 0.78 0.493 0.00121 TABLE III: Calculated magnetic lling factors, jand eective magnetic susceptibilities, efor each of the photon modes. δf1 δf 2&δf 3 δf xϵ/ϵ0=15.965 15.4 15.6 15.8 16.0 16.2 16.4-0.2-0.10.00.10.20.3 ϵ/ϵ0δf=f sim-f meas (GHz)FIG. 9: Frequency dierence between simulated and measured results as the relative permittivity of YIG is varied in the FEM software. The radius of curvature of the sapphire support used here was r0= 3:7 mm, which for mode 1 is largely irrelevant, but for modes x, 2 and 3 gives good agreement. modes is not entirely accurate. To accurately explain the origins of the diering interaction strengths of each mode, knowledge of higher order, nonuniform magnon mode shapes are required, in order to replace the ll- ing factor approximation with an overlap value. Un- like Zhang et al. 's [4] ultrastrong coupling results with ad= 2:5 mm YIG sphere, in which higher order magnon modes mostly couple weakly with the microwave cavity, here we excite internal, nonuniform electromagnetic res- onances, so it is more likely than not that these modes will couple more strongly to nonuniform magnon modes if their mode shapes match up well spatially. The de- rived values of susceptibility in table III agree within an order of magnitude to previously measured results [41] but have been underestimated due to the use of lling factor as opposed to a mode overlap integral. Finally, we can use the predicted mode frequencies of the FEM to determine the permittivity of the YIG sample, by varying YIG=0until the frequencies match the asymptotic values measured at high magnetic elds. At these magnetic eld values, the matrix in equation (1) becomes the identity matrix [41]. By measuring the depth of the sapphire concavity and its width at the surface, the radius of curvature was determined to be r0= 3:710:2 mm. With this information, an iterative simulation was conducted mapping mode frequencies versus relative permittivity of YIG. It was found that mode 1 is relatively insensitive to the radius of curvature of the sapphire support. This is due to the absence of electric eld density outside the YIG for this particular mode. Given that r0contains a signicant amount of uncertainty, this mode is used to match fsimwith fmeas. A plot of f=fsimfmeas versus permittivity is shown in gure 9. From this result, we can state thatYIG=0= 15:960:02. This value agrees well7 with previous measurements taken using the so-called \Courtney" technique with YIG samples [42]. In conclusion, we observe ultrastrong coupling between internal dielectric microwave resonances and magnons in- side ad= 5 mm YIG sphere. The large diameter of the sphere results in not only an increased number of spins, but also the accessibility of the internal electromagnetic resonances due to their existence below K-band frequen- cies. The use of internal microwave modes instead of an external cavity resonance results in far larger magnetic lling factors than ever before achieved in such an exper- iment, hence the coupling values and cooperitivity values observed are, to the authour's knowledge, the largest ever reported, with a maximum g== 7:11 GHz, or7000 mode linewidths, and C= 1:5107. This implies an ex- tremely high level of coherence in this system. Most im- portantly however, the numerous resonant magnon peaks in the dispersive regime and the discrepancies in calcu- lated susceptibilities suggest that higher order magnon modes participate in this system. This implies that the previously theoretically analysed models of such systems are incomplete. jeremy.bourhill@uwa.edu.au [1] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, \Magnon spintronics," Nat Phys 11, 453{ 461 (2015). [2] Yutaka Tabuchi, Seiichiro Ishino, Atsushi Noguchi, Toyofumi Ishikawa, Rekishu Yamazaki, Koji Us- ami, and Yasunobu Nakamura, \Coherent cou- pling between a ferromagnetic magnon and a su- perconducting qubit," Science 349, 405{408 (2015), http://www.sciencemag.org/content/349/6246/405.full.pdf. [3] Yutaka Tabuchi, Seiichiro Ishino, Toyofumi Ishikawa, Rekishu Yamazaki, Koji Usami, and Yasunobu Naka- mura, \Hybridizing ferromagnetic magnons and mi- crowave photons in the quantum limit," Phys. Rev. Lett. 113, 083603 (2014). [4] Xufeng Zhang, Chang-Ling Zou, Liang Jiang, and Hong X. Tang, \Strongly coupled magnons and cav- ity microwave photons," Phys. Rev. Lett. 113, 156401 (2014). [5] Maxim Goryachev, Warrick G. Farr, Daniel L. Cree- don, Yaohui Fan, Mikhail Kostylev, and Michael E. Tobar, \High-cooperativity cavity qed with magnons at microwave frequencies," Phys. Rev. Applied 2, 054002 (2014). [6] Lihui Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, \Spin pumping in electrodynamically cou- pled magnon-photon systems," Phys. Rev. Lett. 114, 227201 (2015). [7] Hans Huebl, Christoph W. Zollitsch, Johannes Lotze, Fredrik Hocke, Moritz Greifenstein, Achim Marx, Rudolf Gross, and Sebastian T. B. Goennenwein, \High coop- erativity in coupled microwave resonator ferrimagnetic insulator hybrids," Phys. Rev. Lett. 111, 127003 (2013).[8] Xufeng Zhang, Chang-Ling Zou, Na Zhu, Florian Mar- quardt, Liang Jiang, and Hong X. Tang, \Magnon dark modes and gradient memory," Nat Commun 6(2015). [9] Atac Imamo glu, \Cavity qed based on collective mag- netic dipole coupling: Spin ensembles as hybrid two-level systems," Phys. Rev. Lett. 102, 083602 (2009). [10] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, \Optomagnonic whispering gallery microresonators," arXiv:1510.03545 (2015). [11] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya- mazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, \Cavity optomagnonics with spin- orbit coupled photons," arXiv:1510.03545 (2015). [12] Y. R. Shen and N. Bloembergen, \Interaction between light waves and spin waves," Phys. Rev. 143, 372{384 (1966). [13] S.O. Demokritov, B. Hillebrands, and A.N. Slavin, \Bril- louin light scattering studies of conned spin waves: lin- ear and nonlinear connement," Physics Reports 348, 441 { 489 (2001). [14] C. Kittel, \Excitation of spin waves in a ferromagnet by a uniform rf eld," Phys. Rev. 110, 1295{1297 (1958). [15] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, \Cavity magnomechanics," arXiv:1511.02680v2 (2015). [16] L. Tian, P. Rabl, R. Blatt, and P. Zoller, \Interfacing quantum-optical and solid-state qubits," Phys. Rev. Lett. 92, 247902 (2004). [17] J. Verd u, H. Zoubi, Ch. Koller, J. Majer, H. Ritsch, and J. Schmiedmayer, \Strong magnetic coupling of an ultra- cold gas to a superconducting waveguide cavity," Phys. Rev. Lett. 103, 043603 (2009). [18] Ze-Liang Xiang, Sahel Ashhab, J. Q. You, and Franco Nori, \Hybrid quantum circuits: Superconducting cir- cuits interacting with other quantum systems," Reviews of Modern Physics 85, 623{653 (2013). [19] V. Cherepanov, I. Kolokolov, and V. Lvov, \The saga of yig: Spectra, thermodynamics, interaction and relax- ation of magnons in a complex magnons," Physics Re- ports229, 81{144 (1993). [20] A.G. Gurevich, Ferrites at Microwave Frequencies (Con- sultants Bureau, New York, 1963). [21] O. O. Soykal and M. E. Flatt e, \Strong eld interactions between a nanomagnet and a photonic cavity," Phys. Rev. Lett. 104, 077202 (2010). [22] O. O. Soykal and M. E. Flatt e, \Size dependence of strong coupling between nanomagnets and photonic cav- ities," Phys. Rev. B 82, 104413 (2010). [23] N. Kostylev, M. Goryachev, and M. E. Tobar, \Super- strong coupling of a microwave cavity to yig magnons," arXiv:1508.04967 (2015). [24] Babak Zare Rameshti, Yunshan Cao, and Gerrit E. W. Bauer, \Magnetic spheres in microwave cavities," Phys. Rev. B 91, 214430 (2015). [25] Warrick G. Farr, Daniel L. Creedon, Maxim Goryachev, Karim Benmessai, and Michael E. Tobar, \Ultrasensi- tive microwave spectroscopy of paramagnetic impurities in sapphire crystals at millikelvin temperatures," Phys. Rev. B 88, 224426 (2013). [26] Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon, and Michael E. Tobar, \Spin-photon interaction in a cav- ity with time-reversal symmetry breaking," Phys. Rev. B 89, 224407 (2014). [27] J. Bourhill, K. Benmessai, M. Goryachev, D. L. Cree-8 FIG. 10: Spherical coordinate eld components of the six lowest dielectric modes in the YIG/sapphire/air system. Each mode is viewed parallel to the z-axis (top row) and in the x;yplane (bottom row), except where no eld is present. We can readily identify modes xand 1 as (0,0) and (1,0) spherical dielectric modes, respectively. These two modes appear as \pure" dielectric modes containing only three eld components. The two doublet modes ( i,ii, 2 and 3) appear to contain energy density in all six eld components, and are greatly aected by the sapphire support, which is what appears to lead to the additional degeneracy; splitting two modes in to four modes. From the radial components we can identify these modes as being modied (1,1) spherical dielectric modes.9 don, W. Farr, and M. E. Tobar, \Spin bath maser in a cryogenically cooled sapphire whispering gallery mode resonator," Phys. Rev. B 88, 235104 (2013). [28] Jeremy Bourhill, Maxim Goryachev, Warrick G. Farr, and Michael E. Tobar, \Collective behavior of cr3+ions in ruby revealed by whispering gallery modes," Phys. Rev. A 92, 023805 (2015). [29] Warrick G. Farr, Maxim Goryachev, Daniel L. Creedon, and Michael E. Tobar, \Strong coupling between whis- pering gallery modes and chromium ions in ruby," Phys. Rev. B 90, 054409 (2014). [30] P. Del/'Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, \Optical frequency comb generation from a monolithic microresonator," Na- ture450, 1214{1217 (2007). [31] T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, \Demonstration of an erbium-doped microdisk laser on a silicon chip," Phys. Rev. A 74, 051802 (2006). [32] Markus Aspelmeyer, Tobias J. Kippenberg, and Florian Marquardt, \Cavity optomechanics," Rev. Mod. Phys. 86, 1391{1452 (2014). [33] J. Bourhill, E. Ivanov, and M. E. Tobar, \Precision measurement of a low-loss cylindrical dumbbell-shaped sapphire mechanical oscillator using radiation pressure," Phys. Rev. A 92, 023817 (2015). [34] Eugene N Ivanov and Michael E Tobar, \Microwave phase detection at the level of 10(-11) rad." Rev Sci In- strum 80, 044701 (2009). [35] E.N. Ivanov and M.E. Tobar, \Low phase-noise sapphire crystal microwave oscillators: current status," Ultrason- ics, Ferroelectrics, and Frequency Control, IEEE Trans- actions on 56, 263{269 (2009). [36] A. A. Serga, C. W. Sandweg, V. I. Vasyuchka, M. B. Jung eisch, B. Hillebrands, A. Kreisel, P. Kopietz, and M. P. Kostylev, \Brillouin light scattering spectroscopyof parametrically excited dipole-exchange magnons," Phys. Rev. B 86, 134403 (2012). [37] Warrick G. Farr, Maxim Goryachev, Jean-Michel le Floch, Pavel Bushev, and Michael E. Tobar, \Evi- dence of dilute ferromagnetism in rare-earth doped yt- trium aluminium garnet," Applied Physics Letters 107, 122401 (2015), http://dx.doi.org/10.1063/1.4931432. [38] Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon, and Michael E. Tobar, \Controlling a whispering-gallery- doublet-mode avoided frequency crossing: Strong cou- pling between photon bosonic and spin degrees of free- dom," Phys. Rev. A 89, 013810 (2014). [39] Jean-Michel le Floch, James David Anstie, Michael Ed- mund Tobar, John Gideon Hartnett, Pierre-Yves Bour- geois, and Dominique Cros, \Whispering modes in anisotropic and isotropic dielectric spherical resonators," Physics Letters A 359, 1 { 7 (2006). [40] Karim Benmessai, Michael Edmund Tobar, Nicholas Bazin, Pierre-Yves Bourgeois, Yann Kersal e, and Vin- cent Giordano, \Creating traveling waves from standing waves from the gyrotropic paramagnetic properties of fe3+ions in a high- qwhispering gallery mode sapphire resonator," Phys. Rev. B 79, 174432 (2009). [41] J. Krupka, \Measurements of all complex permeability tensor components and the eective line widths of mi- crowave ferrites using dielectric ring resonators," Mi- crowave Theory and Techniques, IEEE Transactions on 39, 1148{1157 (1991). [42] Jerzy Krupka, Stephen A Gabelich, Krzysztof Derza- kowski, and Brian M Pierce, \Comparison of split post dielectric resonator and ferrite disc resonator techniques for microwave permittivity measurements of polycrys- talline yttrium iron garnet," Measurement Science and Technology 10, 1004 (1999).

2015-12-24

Resonant photon modes of a 5mm diameter YIG sphere loaded in a cylindrical cavity in the 10-30GHz frequency range are characterised as a function of applied DC magnetic field at millikelvin temperatures. The photon modes are confined mainly to the sphere, and exhibited large mode filling factors in comparison to previous experiments, allowing ultrastrong coupling with the magnon spin wave resonances. The largest observed coupling between photons and magnons is $2g/2\pi=7.11$ GHz for a 15.5 GHz mode, corresponding to a cooperativity of $C=1.51\pm0.47\times10^7$. Complex modifications beyond a simple multi-oscillator model, of the photon mode frequencies were observed between 0 and 0.1 Tesla. Between 0.4 to 1 Tesla, degenerate resonant photon modes were observed to interact with magnon spin wave resonances with different couplings strengths, indicating time reversal symmetry breaking due to the gyrotropic permeability of YIG. Bare dielectric resonator mode frequencies were determined by detuning magnon modes to significantly higher frequencies with strong magnetic fields. By comparing measured mode frequencies at 7 Tesla with Finite Element modelling, a bare dielectric permittivity of $15.96\pm0.02$ of the YIG crystal has been determined at about $20$ mK.

Ultra-High Cooperativity Interactions between Magnons and Resonant Photons in a YIG sphere

1512.07773v5

1 Interplay between nonlinear spectral shift and nonlinear damping of spin waves in ultrathin YIG waveguides S. R. Lake1 , B. Divinskiy2*, G. Schmidt1,3, S. O. Demokritov2, and V. E. Demidov2 1Institut für Physik, Martin -Luther -Universität Halle -Wittenberg, 06120 Halle, Germany 2Institute of Applied Physics, University of Muenster, 48149 Muenster, Germany 3Interdisziplinäres Zentrum für Material wissenschaften , Martin -Luther -Universität Halle - Wittenberg, 06120 Halle, Germany We use the phase -resolved imaging to directly study the nonlinear modification of the wavelength of spin waves propagating in 100-nm thick , in-plane magnetized YIG waveguides. We show that, by using moderate microwave powers, one can realize spin waves with large amplitudes corresponding to precession angles in excess of 10 degrees and nonlinear wavelength variation of up to 18% in this system . We also find that , at large precession angles , the propagation of spin waves is strongly affected by the onset of nonlinear damping, which results in a strong spati al dependence of the wavelength. This effect leads to a spatially - dependent controllability of the wavelength by the microwave power . Furthermore, it leads to the saturation of nonlinear spectral shift’ s effects several micrometers away from the excitation point. These findings are important for the development of nonlinear , integrated spin-wave signal -processing devices and can be used to optimize their characteristics. *Corresponding author, e -mail: b_divi01@uni -muenster.de 2 I. INTRODUCTION Nonlinear phenomena accompanying propagation of spin waves in magnetic films have been known for many decades to enable implementing a large variety of advanced signa l- processing devices [ 1,2]. One fundamental nonlinear phenomen on that is frequently applied is the nonlinear transformation of the spin wave’s dispersion spectrum that occurs with increasing spin wave intensity . This phenomenon can be regarded as a frequency shift of the dispersion spectrum of the spin waves . This shift results from the decrease in the magnitude of the static magnetization caused by the increase in the magnetization precession angle [1,2]. Because of this frequency shift, the wavelen gth of a spin wave at a given frequency becomes dependen t on its amplitude, which allows one to implement , for example, nonlinear magnonic phase shifters [ 3,4], interferometers [ 5], couplers [ 6,7], and switches [ 8]. Additionally, these effects hold promise for implementation of spin -wave logic devices [ 9-11] and neuromorphic computing with spin waves [12,13 ]. The vast majority of previous work on nonlinear spin waves utilized millimeter -scale magnetic structures fabricated from low -loss magnetic insulator – yttrium iron garnet (YIG) [14]. In recent years, the miniaturization of spin -wave devices down to micrometer and sub - micrometer dimensions has become the main trend and challenge in the field of magnonics. Such m iniatur ization has been greatly advanced by the advent of high -quality ultrathin YIG films [ 15-17] that can be structured on the nanometer scale [18-20]. This breakthrough not only provide s the possibility to implement low -loss propagation of spin waves in the linear regime , but also brings novel opportunities to investigate dynamic nonlinear phenomena in ultrathin films and their utilization in signal -processing applications . In particular, the strong quantization of the spectrum of magnetic excitations in microscopic YIG structures can substantially suppress the spin-wave scattering effects and help achieve very large amplitudes of magnetic dynamics [21] that have not been observed on the macroscopic scale. 3 Although nonlinear spin -wave dynamics in microscopic YIG structures remain largely unexplored, it has already been demonstrated experimentally that the nonlinear shift of the dispersion spectrum of propagating spin waves can be used, for example, to control switching in coupled spin -wave waveguides via intensity [7] and to indirectly excite propagati ng spin waves [22]. It has also been theorized to be the underlying physical phenomenon for the realization of nonlinear nano -ring resonators [ 23] and nanoscale neural networks [ 13]. However, until now it has remained unclear how large nonlinear shift is practically achievable and which factors can limit this effect in real devices . In this work, we study the propagation of intense spin waves in microscopic ultrathin - YIG waveguide s using a wide range of power for the excitation signal . By using high - resolution , phase -sensitive , magneto -optical detection, we directly measure the spin wave’s spatial dependencies of intensity and wavelength. Our experimental findings indicate that the nonlinear spectral shif t is strongly affected by the onset of the nonlinear magnetic damping: at large amplitudes, spin waves start to exhibit strongly enhanced spatial attenuation leading to a spatially -dependent magnitude of the nonlinear spectral shift. As a result, the wavel ength (wavevector) of intense spin waves varies strongly along the propagation path . While the wavelength exhibits good controllability by the applied microwave power at the beginning of the propagation path , this controllability is almost completely lost several micrometers away from the excitation point. O ur findings provide insight into nonlinear propagation of spin waves in microscopic waveguides and are of decisive importan ce for the development of efficient magnonic devices utilizing the effect of the nonlinear spectral shift. II. EXPERIMENT Figure 1(a) shows the schematics of our experiment. We study propagation of spin waves in a 2-m wide spin-wave waveguide patterned from a 100 -nm thick YIG film. The YIG 4 film is characterized by a saturation magne tization of 4π MS = 1.75 kG and a Gilbert damping constant α = 4×10-4, as determined from ferrom agnetic resonance measurements. The spin waves are excited by using a 500 -nm wide and 150 -nm thick inductive Au antenna carrying microwave current with a frequen cy f and a power P. The YIG waveguide is magnetized to saturation by a static magnetic field, H0=1000 Oe , which is applied in-plane along the Au antenna. For patterning , a double -layer PMMA resist is deposited on a <111> oriented GGG substrate. The resist is exposed by e -beam lithography using a RAITH Pioneer system and developed in isopropanol. Subsequently 110 nm of YIG are deposited at room temperature by pulsed laser deposition using a recipe by Hauser et al. [17]. After lift -off in acetone , the sample is annealed in a pure oxygen atmosphere. Wet etching in Phosphoric acid is used to remove 10 nm of YIG to smoothen the edges of the remaining structures. The process is completed by patterning a microstrip antenna (10 nm Ti, 150 nm Au) on top of the structures using electron beam lithography, e -beam evaporation , and lift -off. We study the propagation of spin waves in the YIG waveguide with spatial and phase resolution by using micro -focus Brillouin light scattering (BLS) spectroscopy [ 24]. We focus the probing laser light with a wavelength of 473 nm and a power of 0.25 mW into a diffraction - limited spot on the sample surface [Fig. 1(a)] and analyze the modulation of the probing light due to its interaction with the magnetization dynamics . The intensity of this modulation , or BLS intensity, is proportional to the intensity of spin waves at t he position of the probing spot. This allows us to record two -dimensional maps of the spin -wave intensity by rastering the spot over the sample surface . Additionally, we use the interference of the modulated light with the reference light to measure the spatial maps of cos( ), where is the phase diffe rence between the spin wave and the microwave signal applied to the antenna. Analysis of the phase maps 5 provides information about the wavelength of spin waves at a given frequency f and allows us to directly address the effect of the nonlinear spectral sh ift. First, we characterize the expected effect by using micromagnetic simulations (Figs. 1(b) and 1 (c)). We calculate amplitude -dependent dispersion curves using the simulation software MuMax3 [ 25] and the approach developed in Ref. [26]. We consider a 2-m wide and 100 -nm thick waveguide with the length L=20 m discretized into 10 nm 10 nm 10 nm cells with periodic boundary conditions at the ends. The standard for YIG exchange constant of 3.66×10-7 erg/cm is used. The Gilbert damping parameter is se t to an artificially small value 10-12 to fix the amplitude of the magnetization dynamics at the chosen level. We excite magnetization dynamics by initially deflecting magnetic moments from their equilibrium orientation by an angle . The deflection is spa tially periodic with the period L/n (where n is an integer) , which defines the wavelengt h of the excited wave. By analyz ing the free dynamics of magnetization, we determine the frequency corresponding to the given spatial period and obtain the frequency vs wavenumber relations (solid curves in Fig. 1( b)) for a given spin-wave amplitude characterized by the precession angle . As seen from the data of Fig. 1(b), the increase in the angle from 0.1 to 15° leads to a noticeable shift of the dispersion cu rve down toward smaller frequencies and a slight overall decrease in its slope. Both effects are consistent with the changes expected fo r a decrease in the static component of magnetization , MST, with the increase of the amplitude of the magnetization precession: 𝑀ST=√(𝑀S2−𝑚2)≈𝑀S−1 2𝑀Ssin2(𝜃), where MS is the saturation magnetization, m is the amplitude of the dynamic magnetization, and is the mean precession angle. Because MST enters the expression for the frequency of ferromagnetic resonance: 𝑓FMR=𝛾√𝐻0(𝐻0+4𝜋𝑀ST), its decrease causes the overall decrease in the frequency of spin waves. Additionally, the decrease of MST is known to lead to a decrease in the group velocity of spin waves [2], which is consistent with the overall decrease of the slope 6 of the dispersion curve in Fig. 1(b). The nonlinea r transformation of the spectrum can be quantitatively characterized by the variation of the wavenumber d k due to the increase in from 0.1 to 15° (dashed curve in Fig. 1(b)). These data indicate that the dispersion’s shift becomes significantly stronge r as wavelength decreases . As can be seen from the data of Fig. 1(c), dk varies approximately linearly with sin2(𝜃), which in turn is proportional to the spin- wave intensity. This linear dependence makes it straight forward to calibrate the precession angle achieved in the experiment. We also emphasize that the dispersion curve calculated for = 0.1° coincides well with that measured by BLS at low excitation power P = 0.1 mW. This provides clear proof of the validity of the results obtained from the micromagnetic simulations. III. RESULTS AND DISCUSSION To experimentally address the effect of the nonlinear spectral shift, we perform phase - resolved BLS measurements at a fixed frequency of the excitation signal f and analyze the variation of the wavele ngth of spin waves with the increase in the power P. Figure 2(a) shows representative spin-wave phase maps recorded at f = 4.8 GHz and P = 0.1 and 3 mW , whereas Fig. 2(b) shows the spatial dependence of the wavelength of spin waves , obtained from the Fourier analysis of these maps. As seen from these data, at P = 0.1 mW , the spin waves exhibit a well -defined constant wavelength = 1.38 m, which is in good agreement with the dispersion curve calculated for the small precession angle = 0.1° (Fig. 1(b)). At P = 3 mW, the situation changes drastically. First, the wavelength is reduced , explained by the effect of the spectral shift. Second, the wavelength strongly changes in space exhibiting the maximum reduction of about 18% close to the antenn a, which becomes as small as 3% at the propagation distance of 18 m. This spatial variation could be associated with the decrease of the intensity of the spin wave along the propagation path due to the damping. However, taking into account the linear dependence in Fig. 1(c), this would require that the intensity decrease s by more than 7 a factor of five over the propagation interval x = 2 – 18 m, which is inconsistent with the small damping in the YIG film. We quantify t he spatial dec ay of spin waves in the waveguide by analyzing the spin - wave intensity maps (Fig. 3(a)). The direct comparison of the maps recorded at P = 0.1 mW and 3 mW clearly shows that the spatial decay becomes significantly stronger, as the power is increased. In Fi g. 3(b) , we show in the log -linear coordinates the spatial dependence of the BLS intensity integrated over the waveguide width. At small power, P = 0.1 mW, this dependence is exponential and is characterized by the decay length = 29 m, where is the distance, at which the amplitude decreases by a factor of e . This value agrees reasonably well with = 34 m obtained from micromagnetic simulations using experimentally determined α = 4×10-4. In contrast to these simple behaviors, a t large powers, the s patial decay is not described by a single exponential and strongly exceeds that observed in the linear propagation regime. This fact explains the strong spatial variation of the wavelength in Fig. 2(b). To characterize the modification of the decay with the increase of P in detail, we plot in Fig. 4(a) the power dependence of the BLS intensity recorded at x = 0 and 20 m. We note that, in the linear propagation regime, the se dependence s are expected to grow linearly with power . As seen from the data of Fig. 4(a), at x = 0, the BLS intensity remains proportional to P at P < 1 mW and then starts to saturate. At approximately the same power, the intensity recorded at x = 20 m exhibits a drop reflecting an increase in the spatial decay. This indicates an onset at the threshold power P = 1 mW of the so -called nonlinear damping [ 26-32], which is associated with the energy transfer from the coherent propagating spin wave into other incoherent sh ort-wavelength spin -wave modes . These sh ort-wavelength spin -wave modes do not contribute to the BLS intensity and cannot be directly accessed in the experiment. The contribution of this phenomenon relative to the normal linear damping can be estimated based on the data presented in Fig. 4(b) . The figure shows the decrease of the spin 8 wave intensity I over a propagation length of 1 µm expressed by an attenuation factor I(x)/I(x+1 µm). This factor is plotted over the used range of microwave power for positions x = 0 and 20 um. At large distance s from the antenna ( x = 20 m), the attenuation factor is about 1.08 and remains approximately constant within the entire studied interval of P (point -down triangles in Fig. 4(b)) . This value is consistent with that expected for the effects of the linear da mping for α = 4×10-4. We note that it does not increase at large P, because even for P = 3 mW, the strong nonlinear decay at the initial propagation stage has already severely diminished the intensity of spin waves at x = 20 m. This initial strong decay is well characterized by the attenuation factor measured at x = 0 (point -up triangles in Fig. 4(b)) . In the linear propagation regime ( P < 1 mW), the attenuation factor remains constant and coincides with the observed value at x= 20 m. However, at P > 1 mW, it increases significantly and reaches the value of 1.93 at P = 3 mW. Comparing this value with 1.08, one can conclude that the nonlinear energy transfer into short -wavelength spin -wave modes causes the increase of the effective damping of the coherent wave by nearly a factor of two. The results presented above suggest that, due to the effects of the nonlinear damping, the efficient controllability of the wavelength (wavenumber) by the microwave power can only be achieved near the excitation point, while at large propagation distances this controllability becomes relatively poor . This is evidenced by the data in Fig. 5, which shows how the change in wavenumber, d k, depends on the applied power for various distances from the antenna. Clos e to the antenna, d k changes li nearly with the microwave power reaching the value of 0. 9 m-1 at P = 3 mW (point -up triangles in Fig. 5) . In contrast, a t a distance x = 8 m (circles in Fig. 5), dk saturates to the value of about 0. 4 m-1 and remains nearly constant for P > 1.5 mW . Finally, at x = 18 m (point -down triangles) , the maximum achieved shift never exceeds 0.1 m-1 within the entire range of P. 9 It is instructive to discuss approaches to suppress the detrimental effects of the nonlinear damping. The dominating mechanism of the nonlinear damping is the energy transfer from a coherent spin wave into incoherent modes possessing the same frequency. This process can be controlled by varying the geometrical parameters of the waveguide , which allows one to modify the spin -wave dispersion spectrum using the effects of spin -wave quantization and avoid the detrimental spectral degeneracy (see, e.g., Refs. 32, 33). In particular, this can be achieved by reducing the width of the waveguide b elow a certain critical value. For the 100 - nm thick YIG film used in our study, we estimate the critical width of about 200 nm. Reduction of the width below this value can allow a significant suppression of the nonlinear damping within the addressed range of the wavelength of spin waves. However, this reduction is also expected to result in an increase of the linear damping due to the increasing contribution of the spin-wave scatting caused by the roughness of the waveguide edges. We note that, due to the competing effects of the dipolar and the exchange interaction, the critical width weakly depends on the thickness of the YIG film. It remains nearly unchanged within the practically important range of thicknesses 50 -200 nm. Additionally, since the nonlinear damping is caused by the parametric interactions of spin -wave modes, its threshold can be increased by increasing the linear damping. However, this approach is impractical, since it leads to a faster decay of spin waves. Finally , the threshold power of the nonlinear damping can be increased by reducing the ellipticity of the magnetization precession, which can be achieved by using magnetic films with perpendicular magnetic anisotropy [26,34]. We now turn to the estimation of characteristic precession angl es in our experiment s. We base our analysis on the comparison of the experimental dependence d k(P) measured close to the antenna (point -up triangles in Fig. 5) with the dependence d k(sin2) obtained from micromagnetic simulations ( red curve in Fig. 1(c)). Because both of these relationships are linear with their respective variable , one can conclude that sin2 is proportional to the excitation 10 power P. Additionally, the intensity of the spin wave excited by the antenna is also proportio nal to P at P < 1 mW, where the nonlinear damping is not active (point -up triangles in Fig. 4(a)). These facts allow us to estimate the critical precession angle corresponding to the onset of the nonlinear damping at the threshold power P=1 mW: 9°. At this angle, the nonlinear shift of the wavevector d k = 0.27 m-1, which corresponds to the modification of the wavelength by about 6%. At powers P > 1 mW , the estimation of precession angles is less straightforward. Due to the influence of the nonlinear d amping, the energy becomes transferred from the directly excited coherent spin wave into incoherent short -wavelength spin -wave modes each of them reducing the saturation magnetization accordingly . Under these conditions, the reduction of the static magneti zation causing the spectral shift is determined not only by the intensity of the coherent wave but also by those of the incoherent modes. Therefore, we introduce an effective total precession angle , tot, which is generally larger than the coherent spin-wave precession angle, SW. The former can be directly found from the analysis of the shift of the dispersion spectrum (point -up triangles in Fig. 5) : at the maximum power P=3 mW, the effective total angle tot17°. The angle SW can be estimated based on the analysis of the power dependence of the intensity of the coherent spin wave (point -up triangles in Fig. 4(a)). W e extrapolate the linear fit of the experimental data (line in Fig. 4(a)) to P=3 mW and find the ratio of this value to the experimentally observed intensity at this power , 1.7. This ratio is approximately equal to sin2(tot)/sin2(SW), where sin2(SW) is proportional to the measured BLS intensity while sin2(tot) is proportional to the microwave power. This allows us to estimate SW13°. As seen from the data of Figs. 4(a) and Fig. 5, further increase in the excitation power above 3 mW is expected to result in the further increase of tot, while SW shows a clear tendency to saturation . We emphasize that, although the spectral shift is det ermined by tot, the linear increase of its 11 value with the increase in P is only observed near the antenna, while at distance s of several micrometers , this angle also exhibits saturation ( circles and point -down triangles in Fig. 5). We finally discuss possible effects of the heating of the sample by the intense microwave radiation. According to the theory of spin -wave interactions (Ref. 2), the increase of the temperature is not expected to noticeably affect the nonlinear damping. Thi s was also shown experimentally in, e.g., Ref. 35. However, the heating can potentially contribute to the observed shift of the spin -wave spectrum. Since the heating results in an increase of the intensities of incoherent spin -wave modes, it contributes to the reduction of the saturation magnetization and causes an additional frequency shift. Thermally induced shift can be distinguished based on its slow temporal dynamics. Therefore, we performed additional time - resolved measurements, where the excitation w as applied in the form of pulses with the duration of 5 -50 s and the spectral shift was analyzed in the time domain with the resolution down to 2 ns. The measurements at the largest microwave power of 3 mW did not reveal any slowly varying contribution su ggesting that the heating effects are negligible in the studied system. IV. CONCLUSIONS In conclusion, we have shown that the effect of the nonlinear spectral shift enabling the controllability of the wavelength of spin waves by their intensity is a compl ex phenomenon, which is strongly affected by the nonlinear spin -wave damping. This effect becomes pronounced at precession angles exceeding 9° and results in a spatially -dependent controllability of the wavelength. T he efficient controllability can only be achieved at small distances from the excitation point, wh ereas the controllability becomes relatively poor several micrometers away . Additionally, the fast spatial decay caused by the nonlinear damping results in the strong spatial variation of the wavele ngth near the excitation point. These findings are 12 critically important for the development of efficient nonlinear magnonic devices utilizing the effect of the nonlinear spectral shift. ACKNOWLEDGMENTS This work was supported in part by the Deutsche Fors chungsgemeinschaft (DFG, German Research Foundation) – Project -ID 433682494 – SFB 1459 and TRR227 TP B02. [1] P. E. Wigen, Nonlinear Phenomena and Chaos in Magnetic Materials (World Scientific , Singapore, 1994). [2] A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC, New York, 1996). [3] A. B. Ustinov and B. A. Kalinikos, A microwave nonlinear phase shifter , Appl. Phys. Lett. 93, 102504 (2008). [4] U.-H. Hansen, V. E. Demidov, and S. O. Demokritov , Dual -function phase shifter for spin-wave logic applications, Appl. Phys. Lett. 94, 252502 (2009). [5] A. B. Ustinov and B. A. Kalinikos, Ferrite -film nonlinear spin wave interferometer and its application for power -selective suppression of pulsed microw ave signals , Appl. Phys. Lett. 90, 252510 (2007). [6] A. V. Sadovnikov, E. N. Beginin, M. A. Morozova, Yu. P. Sharaevskii, S. V. Grishin, S. E. Sheshukova, and S. A. Nikitov, Nonlinear spin wave coupling in adjacent magnonic crystals, Appl. Phys. Lett. 109, 042407 (2016). [7] Q. Wang, M. Kewenig, M. Schneider, R. Verba, F. Kohl, B. Heinz, M. Geilen, M. Mohseni, B. Lägel, F. Ciubotaru, C. Adelmann, C. Dubs, S. D. Cotofana, O. V. 13 Dobrovolskiy, T. Brächer, P. Pirro, and A. V. Chumak, A magnonic directional cou pler for integrated magnonic half -adders, Nat . Elect r. 3, 765 –774 (2020). [8] A. V. Sadovnikov, S. A. Odintsov, E. N. Beginin, S. E. Sheshukova, Yu. P. Sharaevskii, and S. A. Nikitov, Toward nonlinear magnonics: Intensity -dependent spin -wave switching in insulating side -coupled magnetic stripes , Phys. Rev. B 96, 144428 (2017). [9] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Spin -wave logical gates , Appl. Phys. Lett. 87, 153501 (2005). [10] A. Khitun, M. Bao, and K. L. Wang, Mag nonic logic circuits, J. Phys. D 43, 264005 (2010). [11] A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V. Chumak, S. Hamdioui, C. Adelmann, and S. Cotofana, Introduction to spin wave computing, J. Appl. Phys. 128, 161101 (2020). [12] M. Zahedinejad, A. A. A wad, S. Muralidhar, R. Khymyn, H. Fulara, H. Mazraati, M. Dvornik, and J. Åkerman, Two -dimensional mutually synchronized spin Hall nano - oscillator arrays for neuromorphic computing, Nat. Nanotech. 15, 47-52 (2020). [13] A. Papp, G. Csaba, and W. Porod, Cha racterization of nonlinear spin -wave interference by reservoir -computing metrics , Appl. Phys. Lett. 119, 112403 (2021). [14] A. A. Serga, A. V. Chumak, and B. Hillebrands, YIG magnonics , J. Phys. D: Appl. Phys. 43, 264002 (2010). [15] Y. Sun, Y. Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, and A. Hoffmann, Growth and ferromagnetic resonance properties of nanometer -thick yttrium iron garnet films, Appl. Phys. Lett. 101, 152405 (2012). [16] O. d’Allivy K elly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J. -C. Mage, G. de 14 Loubens, O. Klein, V. Cros, and A. Fert, Inverse spin Hall effect in nanometer -thick yttrium iron garnet/Pt system, Appl. Phys. Lett. 103, 082408 (2013). [17] C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. G. Ebbinghaus, and G. Schmidt, Yttrium iron garnet thin films with very low damping obtained by r ecrystallization of amorphous material, Sci. Rep. 6, 20827 (2016). [18] S. Li, W. Zhang, J. Ding, J. E. Pearson, V. Novosad, and A. Hoffmann, Epitaxial patterning of nanometer -thick Y3Fe5O12 films with low magnetic damping, Nanoscale 8, 388 (2016). [19] B. Heinz, T. Br ächer, M. Schneider, Q. Wang, B. L ägel, A. M. Friedel, D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, and A. V. Chumak, Propagation of spin-wave packets in individual nanosized yttrium iron garnet magnonic conduits , Nano Letters 20, 4220−4227 (2020). [20] G. Schmidt, C. Hauser, P. Trempler, M. Paleschke, E. T. Papaioannou, Ultra thin films of yttrium iron garnet with very low damping: a review , Phys. Stat. Sol. B 257, 1900644 (2020). [21] Y. Li, V. V. Naletov, O. Klein, J . L. Prieto, M. Muñoz, V. Cros, P. Bortolotti, A. Anane, C. Serpico, and G. de Loubens, Nutation spectroscopy of a nanomagnet driven into deeply nonlinear ferromagnetic resonance, Phys. Rev. X 9, 041036 (2019). [22] A. Papp, M. Kiechle, S. Mendisch , V. Ahrens, L. Sahin, L. Seitner, W. Porod, G. Csaba, and M. Becherer, Experimental demonstration of a concave grating for spin waves in the Rowland arrangement, Sci. Rep. 11, 14239 (2021). [23] Q. Wang, A. Hamadeh, R. Verba, V. Lomakin, M. Mohseni, B. Hi llebrands, A. V. Chumak, and P. Pirro, A nonlinear magnonic nano -ring resonator , npj Computational Materials 6, 192 (2020). 15 [24] V. E. Demidov and S. O. Demokritov, Magnonic waveguides studied by micro -focus Brillouin light scattering, IEEE Trans. Mag. 51, 0800215 (2015). [25] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. Van Waeyenberge, The design and verification of MuMax3, AIP Adv. 4, 107133 (2014). [26] B. Divinskiy, S. Urazhdin, S. O. Demokritov, and V. E. Demidov, Co ntrolled nonlinear magnetic damping in spin -Hall nano -devices , Nat. Com m. 10, 5211 (2019). [27] M. M. Scott, C. E. Patton, M. P. Kostylev, and B. A. Kalinikos, Nonlinear damping of high-power magnetostatic waves in yttrium –iron–garnet films, J. Appl. Phys. 95, 6294 (2004). [28] V. E. Demidov, J. Jersch, K. Rott, P. Krzysteczko, G. Reiss, and S. O. Demokritov, Nonlinear propagation of spin waves in microscopic magnetic stripes, Phys. Rev. Lett. 102, 177207 (2009). [29] H. G. Bauer, P. Majchrak, T. Kachel , C. H. Back, and G. Woltersdorf, Nonlinear spin- wave excitations at low magnetic bias fields, Nat. Comm. 6, 8274 (2015). [30] I. Barsukov, H. K. Lee, A. A. Jara, Y. -J. Chen, A. M. Gonçalves, C. Sha, J. A. Katine, R. E. Arias, B.A. Ivanov, and I.N. Krivorotov, Giant resonant nonlinear damping in nanoscale ferromagnets. Science Advances 5, eaav6943 (2019). [31] T. Hula, K. Schultheiss, A. Buzdakov, L. Körber, M. Bejarano, L. Flacke, L. Liensberger, M. Weiler, J. M. Shaw, H. T. Nembach, J. Fassbender, and H. Schultheiss, Nonlinear losses in magnon transport due to four -magnon scattering, Appl. Phys. Lett. 117, 042404 (2020). [32] M. Mohseni, Q. Wang, B. Heinz, M. Kewenig, M. Schneider, F. Kohl, B. Lägel, C. Dubs, A. V. Chumak, and P. Pirro, Controlling the nonlinear relaxation of quantized propagating magnons in nanodevices, Phys. Rev. Lett. 126, 097202 (2021). 16 [33] A. Smith, K. Sobotkiewich, A. Khan, E. A. Montoya, L. Yang, Z. Duan, T. Schneider, K. Lenz, J. Lindner, K. An, X. Li, and I. N. Krivorotov, Dimensional crossover in spin Hall oscillators, Phys. Rev. B 102, 054422 (2020) [34] J. Gückelhorn, T. Wimmer, M. Müller, S. Geprägs, H. Huebl, R. Gross, and M. Althammer, Magnon transport in Y 3Fe5O12/Pt nanostructures with reduced effective magnetization, Phys. Rev. B 104, L180410 (2021) [35] I. Lee, C. Zhang, S. Singh, B. McCullian, P. C. Hammel, Origin of nonlinear damping due to mode coupling in auto -oscillatory modes strongly driven by spin -orbit torque, arXiv:2107.00150 (2021) 17 FIG. 1. (a) Schematics of the experiment. (b) Solid curves – calculated dispersion curves of spin waves in the YIG waveguide corresponding to different angles of magnetization precession, as labeled. Dashed curve – variation of the wavenumber caused by the increas e in the precession angle from 0.1 to 15°. Symbols – dispersion curve measured by BLS at low excitation power P = 0.1 mW. (c) Dependence of the nonlinear variation of the wavenumber on the precession angle calculated at different frequencies, as label ed. T he data are obtained at H0=1000 Oe. 18 FIG. 2. (a) Representative spin -wave phase maps recorded by BLS at f = 4.8 GHz and P = 0.1 and 3 mW , as label ed. (b) Spatial dependence of the wavelength of spin waves obtained from the Fourier analysis of the pha se maps. Symbols – experimental data. Curves – guide for the eye. The data are obtained at H0=1000 Oe. 19 FIG. 3. (a) Spin -wave intensity maps recorded by BLS at f = 4.8 GHz and P = 0.1 and 3 mW , as label ed. (b) Normalized s patial dependence of the BLS intensity integrated over the waveguide width. Note the logarithmic scale of the vertical axis. Symbols – experimental data. Curves – exponential fit of the data obtained at P = 0.1 mW and double -exponential fit of the data obt ained at P = 3 mW. The data are obtained at H0=1000 Oe. 20 FIG. 4. (a) Power dependence of the BLS intensity recorded at x = 0 and 20 m, as labe led. Line – linear fit of the experimental data at P < 1 mW . (b) Power dependence of the factor, by which the spin -wave intensity is attenuated over a propagation distance of 1 m, at x = 0 and 20 m, as label ed. Vertical dashed line in (a) and (b) marks the threshold power, at which the nonlinear damping emerges . The data are obtained at H0=1000 Oe. 21 FIG. 5. Power dependences of the nonlinear variation of the wavenumber of spin waves at different dista nces from the antenna, as label ed. Symbols – experimental data. Solid line – linear fit of the data ob tained at x = 2 m. Horizontal dashed lines mark the saturation values at x = 8 and 18 m. The data are obtained at H0=1000 Oe.

2022-03-08

We use the phase-resolved imaging to directly study the nonlinear modification of the wavelength of spin waves propagating in 100-nm thick, in-plane magnetized YIG waveguides. We show that, by using moderate microwave powers, one can realize spin waves with large amplitudes corresponding to precession angles in excess of 10 degrees and nonlinear wavelength variation of up to 18 percent in this system. We also find that, at large precession angles, the propagation of spin waves is strongly affected by the onset of nonlinear damping, which results in a strong spatial dependence of the wavelength. This effect leads to a spatially dependent controllability of the wavelength by the microwave power. Furthermore, it leads to the saturation of nonlinear spectral shift's effects several micrometers away from the excitation point. These findings are important for the development of nonlinear, integrated spin-wave signal processing devices and can be used to optimize their characteristics.

Interplay between nonlinear spectral shift and nonlinear damping of spin waves in ultrathin YIG waveguides

2203.04018v1

Mechanical Bistability in Kerr-modified Cavity Magnomechanics Rui-Chang Shen,1Jie Li,1,Zhi-Yuan Fan,1Yi-Pu Wang,1,yand J. Q. You1,z 1Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device, School of Physics, Zhejiang University, Hangzhou 310027, China Bistable mechanical vibration is observed in a cavity magnomechanical system, which consists of a mi- crowave cavity mode, a magnon mode, and a mechanical vibration mode of a ferrimagnetic yttrium-iron-garnet (YIG) sphere. The bistability manifests itself in both the mechanical frequency and linewidth under a strong microwave drive field, which simultaneously activates three di erent kinds of nonlinearities, namely, magne- tostriction, magnon self-Kerr, and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr nonlinearity is first predicted and measured in magnomechanics. The system enters a regime where Kerr- type nonlinearities strongly modify the conventional cavity magnomechanics that possesses only a radiation- pressure-like magnomechanical coupling. Three di erent kinds of nonlinearities are identified and distinguished in the experiment. Our work demonstrates a new mechanism for achieving mechanical bistability by combining magnetostriction and Kerr-type nonlinearities, and indicates that such Kerr-modified cavity magnomechanics provides a unique platform for studying many distinct nonlinearities in a single experiment. Introduction.— Bistability, or multistability, discontinuous jumps, and hysteresis are characteristic features of nonlin- ear systems. Bistability is a widespread phenomenon that exists in a variety of physical systems, e.g., optics [1–3], electronic tunneling structures [4], magnetic nanorings [5], thermal radiation [6], a driven-dissipative superfluid [7], and cavity magnonics [8]. Its presence requires nonlinearity in the system. To date, bistability has been studied in vari- ous mechanical systems, including nano- or micromechani- cal resonators [9–11], piezoelectric beams [12], mechanical morphing structures [13], and levitated nanoparticles [14]. Bistable mechanical motion finds many important applica- tions: It is the basis for mechanical switches [15, 16], mem- ory elements [17, 18], logic gates [19], vibration energy har- vesters [12, 20], and signal amplifiers [11, 14], etc. Dierent mechanisms can bring about nonlinearity in the system leading to bistable mechanical motion. Most com- monly, a strong drive can induce bistability of a mechanical oscillator, of which the dynamics is described by the Du - ing equation [11, 21–23]. Mechanical bistability can also be caused by the Casimir force [9], nanomechanical e ects on Coulomb blockade [10], magnetic repulsion [24], and intrin- sic nonlinearity in the optomechanical coupling [25], etc. Here we introduce a mechanism to induce mechanical bistability, distinguished from all the above mechanisms, by exploiting rich nonlinearities in the ferrimagnetic yttrium- iron-garnet (YIG) in cavity magnomechanics (CMM). In the CMM [26–29], magnons are the quanta of collective spin excitations in magnetically ordered materials, such as YIG. They can strongly couple to microwave cavity pho- tons by the magnetic-dipole interaction, leading to cavity po- laritons [30–35]. They can also couple to deformation vi- bration phonons of the ferrimagnet via the magnetostrictive force [26, 28, 36]. Such a radiation-pressure-like magnome- chanical coupling provides necessary nonlinearity, enabling a number of theoretical proposals, including the preparation of entangled states [37–41], squeezed states [42–44], mechanical quantum-ground states [45–47], slow light [48, 49], thermom-etry [50], quantum memory [51, 52], exceptional points [53], and parity-time-related phenomena [54–57], etc. In contrast, the experimental studies on this system are, by now, very lim- ited: Magnomechanically induced transparency and absorp- tion [26] and mechanical cooling and lasing [28] have been demonstrated. In this Letter, we report an experimental observation of bistable mechanical vibration of a YIG sphere in the CMM. We show that both the frequency and linewidth of the me- chanical mode exhibit a bistable feature as a result of the combined e ects of the radiation-pressure-like magnetostric- tive interaction [26, 28], the magnon self-Kerr [58, 59], and the magnon-phonon cross-Kerr nonlinearities. Three di erent kinds of nonlinearities are simultaneously activated by apply- ing a strong drive field on the YIG sphere. Their respective contributions to the mechanical frequency and linewidth are discussed. Kerr-modified CMM.— The CMM system consists of a mi- crowave cavity mode, a magnon mode, and a mechanical vi- bration mode, see Fig. 1. In the experiment, we use the oxygen-free copper cavity with dimensions of 42 228 mm3. The cavity TE 101mode has a frequency !a=2=7:653 GHz, and a total decay rate a=2=2:78 MHz. The cavity decay rates associated with the two ports are 1;2=2=0:22 MHz and 1:05 MHz, respectively. The magnon and mechanical modes are supported by a 0.28 mm-diameter YIG sphere. The frequency of the magnon mode can be tuned by ad- justing the bias magnetic field B0via!m= B0, with being the gyromagnetic ratio. The magnon dissipation rate ism=2=2:2 MHz. The magnon mode couples to the cavity magnetic field by the magnetic-dipole inter- action with the coupling strength gma=2=7:37 MHz, and to a vibration mode by the magnetostrictive (radiation- pressure-like) interaction with the bare magnomechanical coupling strength gmb=2=1:22 mHz. Here we consider the lower-frequency mechanical mode (with a natural frequency !b=2=11:0308 MHz and linewidth b=2=550 Hz) in our observed two adjacent mechanical modes, which has aarXiv:2206.14588v2 [quant-ph] 3 Sep 20222 x yzPort 1 gmbgma B0MW Port 2 VNA x y minmaxB0 FIG. 1. Device schematic. Left panel: Schematic of the CMM system. A 0.28 mm-diameter YIG sphere is placed (free to move) in a horizontal 0.9 mm-inner-diameter glass capillary and at the antinode of the magnetic field of the cavity mode TE 101. The cavity has two ports: Port 1 is connected to a microwave source (MW) to load the drive field, and Port 2 is connected to a vector network analyzer (VNA) to measure the reflection of the probe field with the power of 5 dBm. We set the direction of the bias magnetic field B0 as the zdirection, and the vertical direction as the xdirection. Right panel: Schematic of the coupled three modes. stronger coupling gmb. The magnomechanical coupling can be significantly enhanced by applying a pump field on the magnon mode [37]. In our experiment, this is realized by strongly driving the cavity, which linearly couples to the magnon mode. In Ref. [60], we provide a list of parameters and the details of how they are extracted by fitting the experi- mental data. Under a strong pump, the Hamiltonian of the CMM system is given by [60] H=~=!aaya+!mmym+!bbyb+gma(aym+amy) +gmbmym b+by +HKerr=~+p1"d ayei!dt+H:c: ; (1) where a,m, and b(ay,my, and by) are the annihilation (cre- ation) operators of the cavity mode, the magnon mode, and the mechanical mode, respectively. The last term is the driving Hamiltonian, where 1is the cavity decay rate associated with the driving port (Port 1), and "d=pPd=(~!d), with Pd(!d) being the power (frequency) of the microwave drive field. The novel part of the Hamiltonian, with respect to the conventional CMM, is the Kerr nonlinear term HKerractivated by the strong pump field [60] HKerr=~=Kmmymmym+Kcrossmymbyb; (2) where Kmis the magnon self-Kerr coe cient, and Kcrossis the magnon-phonon cross-Kerr coe cient. The magnon self-Kerr eect is caused by the magnetocrystalline anisotropy [58, 59], and the cross-Kerr nonlinearity originates from the magne- toelastic coupling by including the second-order terms in the strain tensor [61, 62], i j=1 20BBBBB@@ui @lj+@uj @li+X k@uk @li@uk @lj1CCCCCA; (3)where uiare the components of the displacement vector, and li=i(i=x;y;z). The first-order terms lead to the conven- tional radiation-pressure-like interaction Hamiltonian [63], ~gmbmym b+by . Under a moderate drive field, the second- order terms are negligible [26, 28], but can no longer be ne- glected when the drive becomes su ciently strong, as in our experiment, yielding an appreciable magnon-phonon cross- Kerr nonlinearity. As will be seen later, the cross-Kerr nonlin- earity is indispensable in the model for fitting the mechanical frequency shift. Both the magnon self-Kerr and magnon-phonon cross- Kerr terms, as well as the radiation-pressure-like term, cause a magnon frequency shift !m=2KmjMj2+KcrossjBj2+ 2gmbRe[B], where O=hoi(o=m;b;a) denote the average of the modes. In our experiment, the dominant contribution is from the self-Kerr nonlinearity [60], which gives a bistable magnon frequency shift [8]. Also, the cross-Kerr nonlinearity causes a mechanical frequency shift !b=KcrossjMj2. Using the Heisenberg-Langevin approach, we obtain the equation for the steady-state average M[60] a1g2 ma"2 d=jMj2 "
mag2 ma
a+2KmjMj22+m 2+ag2 maa 22# ;(4) where
a (m)=!a (m)!d,a=1
2a+(a=2)2. In deriving Eq. (4), we neglect contributions from the mechanical mode to the magnon frequency shift, because of a much smaller mechan- ical excitation number compared to the magnon excitation number for the drive powers used in this work. It is a cubic equation of the magnon excitation number jMj2. In a suitable range of the drive power, there are two stable solutions, lead- ing to the bistable magnon and phonon frequency shifts by varying the drive power. The radiation-pressure-like coupling gives rise to an e ec- tive susceptibility of the mechanical mode [60] b;e(!)= 1 b(!)2ijGmbj2ma(!) ma(!)
1;(5) whereb(!) is the natural susceptibility of the mechanical mode, but depends on the modified mechanical frequency ˜!b=!b+KcrossjMj2, which includes the cross-Kerr induced frequency shift. The e ective coupling Gmb=gmbM, and ma(!)=h 1 m(!)+g2 maa(!)i1, wherem(!) anda(!) are the natural susceptibilities of the magnon and cavity modes, with the magnon detuning in m(!) modified as ˜
m=
m+ 2KmjMj2, which includes the dominant magnon self-Kerr in- duced frequency shift. See [60] for the explicit expressions of the susceptibilities. The e ective mechanical susceptibility yields a frequency shift of the phonon mode (the so-called “magnonic spring” eect [28], in analogy to the “optical spring” in optomechan- ics [69]) !b=Reh 2ijGmbj2ma(!) ma(!)
i +KcrossjMj2;(6) where we write together the frequency shift induced by the cross-Kerr nonlinearity. Moreover, it leads to a mechanical3 -6-4-20 7.64 7.66 Frequency (GHz)-6-4-20 -11.04 -11.03 -11.02 pd (MHz)-0.2-0.10-6.2-6-5.8-4-20 -0.24-0.23 7.64 7.66 Frequency (GHz)-8-6-4-20 Reflection (dB) 11.02 11.03 11.04-8-6-4 pd (MHz)(a) (b) Reflection (dB)4.7 dBm 29.7 dBm ωm ωa ωaωbωb Zoom-inZoom-in 4.7 dBm 23.7 dBmωmωb ωb ωbωb ωb ωb FIG. 2. (a) Left panel: Measured reflection spectra under a red-detuned drive. The frequency of the drive field !d=2=7:645 GHz (black ar- row). By increasing the drive power, the magnon frequency shift is neg- ative:!L m=2=7:658 GHz (light green dashed line) at a lower power Pd=4:7 dBm (upper panel) and !H m=2=7:640 GHz (dark green dashed line) at Pd=29:7 dBm (lower panel). The green arrow indicates the direc- tion in which the magnon frequency shifts by increasing the power. Right panel: Zoom-in on the red shaded areas in the left panel shows detailed spec- tra of the magnomechanically induced resonances, where
pd=!p!d. The black lines are the fitting curves. (b) Left panel: Measured reflection spec- tra under a blue-detuned drive. The drive frequency !d=2=7:660 GHz. By adjusting the bias magnetic field, the magnon frequency is tuned close to the drive frequency !L m=2'!dat the power Pd=4:7 dBm. Increas- ing the power to 23 :7 dBm, the magnon frequency !H m=2=7:645 GHz. Right panel: Zoom-in on the blue shaded areas in the left panel shows de- tailed spectra of the magnomechanically induced resonances. We observe two adjacent mechanical modes with the frequencies !b=2=11:0308 MHz and!0 b=2=11:0377 MHz. Due to their similar behaviors, we focus on the lower-frequency mode in the text. linewidth change b=Imh 2ijGmbj2ma(!) ma(!)
i : (7) Clearly, this linewidth change is only caused by the radiation- pressure-like coupling, distinguished from the frequency shift caused by the self-Kerr or cross-Kerr nonlinearity. By apply- ing a red- or blue-detuned drive field, we can choose to operate the system in two di erent regimes, where either the mag- nomechanical anti-Stokes or Stokes scattering is dominant. This yields an increased ( b>0) or a reduced ( b<0) mechanical linewidth, corresponding to the cooling or ampli- fication of the mechanical motion [28, 69]. In our system, due to the strong coupling gma>  m;a, the magnon and cavity modes form two cavity polariton (hy- bridized) modes (Fig. 2, left panels) [30–35]. Here, a red (blue)-detuned drive means that the drive frequency is lower (higher) than the frequency of the cavity-like polariton mode, i.e., the “deeper” polariton in the spectra close to the cavity resonance. For the red (blue)-detuned drive, we show the anti-Stokes (Stokes) sidebands associated with two mechanical modes for two drive powers in the zoom-in plots of Fig. 2(a) (Fig. 2(b)). When the detuning between the drive field and the deeper polariton matches the mechanical frequencies, the anti- Stokes (Stokes) sidebands are manifested as the magnome- chanically induced transparency (absorption) [26]. Red-detuned drive.— To implement a red-detuned drive, we drive the cavity with a microwave field at frequency !d=2=7:645 GHz. By adjusting the bias magnetic field, we tune the magnon frequency to be !L m=2=7:658 GHz at the drive power Pd=4:7 dBm (see Fig. 2(a)). We have the [110] axis of the YIG sphere aligned parallel to the static magnetic field, which yields a negative self-Kerr coe cient Km=2=6:5 nHz. An increase in power thus results in a negative magnon frequency shift !m=2KmjMj2, and the magnon frequency reduces to !H m=2=7:640 GHz when the power increases to Pd=29:7 dBm (Fig. 2(a)), which yields an eective coupling Gmb=2=45:8 kHz. Under these con- ditions, the magnon excitation number jMj2shows a bistable behavior through variation of the power. Equation (6) indicates that the radiation-pressure-like cou- pling results in a mechanical frequency shift, and so does the cross-Kerr nonlinearity. This is confirmed by the exper- imental data in Fig. 3(a). It shows that the cross-Kerr plays a dominant role because of a large magnon excitation num- ber, and both the frequency shifts caused, respectively, by the cross-Kerr and the radiation-pressure-like coupling show a bistable feature with the forward and backward sweeps of 0 5 10 15 20 Drive power (dBm)-6-4-20b (kHz) 0 5 10 15 20 Drive power (dBm)04080b (Hz)Scan forward Scan backward Radiation-pressure-like Cross-Kerr Sum Scan forward Scan backward Theory(a) (b) FIG. 3. Bistable mechanical frequency and linewidth under a red-detuned drive. (a) The mechanical frequency shift versus the drive power. The red (green) curve is the fitting of the frequency shift induced by the radiation- pressure-like coupling (cross-Kerr e ect) using Eq. (6), and the black curve is the sum of the two contributions. (b) The mechanical linewidth variation versus the drive power. The black curve is the fitting of the linewidth change using Eq. (7). In both figures, the blue (orange) triangles are the experimental data obtained via forward (backward) sweep of the drive power.4 the drive power. This is because both of them originate from the bistable magnon excitation number jMj2, c.f. Eq. (6). Note that for the spring e ect, the bistability of jMj2is mapped to the magnon frequency shift ˜
m, then to the polariton suscep- tibilityma(!), and finally to the mechanical frequency. By increasing the drive power from 4 :7 dBm to 19 :7 dBm, the cross-Kerr causes a maximum frequency shift of 4:6 kHz (Fig. 3(a), the green line). The fitting cross-Kerr coe - cient is Kcross=2=5:4 pHz. Under the red-detuned drive, the magnonic spring e ect yields a negative frequency shift !b=Reh 1 b;e(!)1 b(!)i <0 (Fig. 3(a), the red line), and the maximum frequency shift is 200 Hz. Adding up these two frequency shifts gives the total mechanical frequency shift (Fig. 3(a), the black line), which fits well with the experimen- tal data (Fig. 3(a), triangles) when the power is not too strong. Another interesting finding is the bistable feature of the me- chanical linewidth (Fig. 3(b)). The magnomechanical backac- tion leads to the variation of the mechanical linewidth b= Imh 1 b;e(!)1 b(!)i . For a red-detuned drive, the anti- Stokes process is dominant, resulting in an increased mechan- ical linewidth b>0 and the cooling of the motion. The bistable mechanical linewidth is also induced by the bistable jMj2(see Eq. (7)), similar to the mechanical frequency. The theory fits well with the experimental results, and the discrep- ancy appears only in the high-power regime. This is because the considerable heating e ect at strong pump powers can broaden the mechanical linewidth [70, 71], which is not in- cluded in our model. Blue-detuned drive.— When a blue-detuned drive is applied, the system enters a regime where the Stokes scattering is dom- inant. The magnomechanical parametric down-conversion amplifies the mechanical motion with the characteristic of a reduced linewidth. Furthermore, the mechanical frequency shift induced by the spring e ect will move in the opposite direction compared with the red-detuned drive. To implement a blue-detuned drive, we drive the cavity with a microwave field at frequency !d=2=7:66 GHz, and tune the magnon frequency close to the drive frequency (see Fig. 2(b)). We attempted to make the Stokes sideband of the drive field resonate with the “deeper” polariton at a high pump power , such that the Stokes scattering rate is maximized and the magnomechanical coupling strength Gmbbecomes strong. However, to meet the drive conditions for a bistable magnon excitation number jMj2, the drive frequency is restricted to a certain range, which hinders us to make the Stokes side- band and the “deeper” polariton resonate. Therefore, we only achieve this at lower drive powers, giving a faint magnome- chanically induced absorption (Fig. 2(b), upper panels). From the red to the blue detuning, we only adjust the magnon and the drive frequencies. Because the direction of the crystal axis is unchanged, the magnon self-Kerr coe - cient Kmis still negative, so again a negative frequency shift by increasing the power (green arrow in Fig. 2(b)). For the power up to 23.7 dBm, which yields Gmb=2=42:7 kHz, the frequency of the cavity-like polariton is always lower than the drive frequency, so the system is operated under a blue- 0 5 10 15 20 25 Drive power (dBm)-8-6-4-20b (kHz) Scan forward Scan backward Radiation-pressure-like Cross-Kerr Sum 0 5 10 15 20 25 Drive power (dBm)-120-80-400b (Hz) Scan forward Scan backward Theory(a) (b)FIG. 4. Bistable mechanical frequency and linewidth under a blue-detuned drive. (a) The mechanical frequency shift and (b) the mechanical linewidth change versus the drive power. The curves and the triangles are shown in the same manner as in Fig. 3. detuned drive. Figure 4 displays the bistable mechanical frequency shift !band linewidth change b. For the frequency shift, both the contributions from the cross-Kerr and the radiation- pressure-like coupling should be considered, but the former plays a dominant role (Fig. 4(a), the green line), as in the case of the red-detuned drive, yielding a frequency shift of 6:5 kHz at the power of 23.7 dBm. Di erently, the spring ef- fect induced frequency shift (370 Hz at 23 :7 dBm) is positive (Fig. 4(a), the red line). The opposite frequency shifts by the spring e ect in the blue and red-detuned drives agree with the finding of Ref. [28], but no bistability was observed in their work. The reduced mechanical linewidth b<0 under a blue- detuned drive is confirmed by Fig. 4(b). However, unlike the bistable curve in the red-detuned drive case (Fig. 3(b)), b manifests the bistability in an alpha -shaped curve by sweep- ing the drive power. This is the result of the trade-o be- tween the growing coupling strength Gmb(which enhances the Stokes scattering rate, yielding an increasing jbj) and the larger detuning between the Stokes sideband and the “deeper” polariton (Fig. 2(b)) (which reduces the Stokes scattering rate, resulting in a decreasing jbj) by raising the drive power. These two e ects are balanced when the power is in the range of 12 dBm to 15 dBm. Conclusions.— We have observed bistable mechanical fre- quency and linewidth and the magnon-phonon cross-Kerr nonlinearity in the CMM system. The mechanical bistability results from the magnomechanical backaction on the mechan- ical mode and the strong modifications on the backaction due to the magnon self-Kerr and magnon-phonon cross-Kerr non-5 linearities. The e ects of the magnon self-Kerr, the magnon- phonon cross-Kerr, and the radiation-pressure-like interac- tions can be identified by measuring primarily the magnon frequency shift, the mechanical frequency shift, and the me- chanical linewidth, respectively. The new mechanism for achieving bistable mechanical motion revealed by this work promises a wide range of applications, such as in mechanical switches, memories, logic gates, and signal amplifiers. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grants Nos. 11934010, U1801661, 12174329, 11874249), Zhejiang Province Program for Science and Technology (Grant No. 2020C01019), and the Fundamental Research Funds for the Central Universities (No. 2021FZZX001-02). jieli007@zju.edu.cn yyipuwang@zju.edu.cn zjqyou@zju.edu.cn [1] P. D. Drummond, and D. F. Walls, Quantum theory of optical bistability. I. Nonlinear polarisability model. J. Phys. A 13, 725 (1980). [2] P. D. Drummond, and D. F. Walls, Quantum theory of opti- cal bistability. II. Atomic fluorescence in a high-Q cavity. Phys. Rev. A 23, 2563 (1981). [3] L. A. Lugiato, Theory of Optical Bistability. Prog. Opt, 2169 (1984) and references therein. [4] V . J. Goldman, D. C. Tsui, and J. E. Cunningham, Observation of intrinsic bistability in resonant tunneling structures. Phys. Rev. Lett. 58, 1256 (1987). [5] F. Q. Zhu, G. W. Chern, O. Tchernyshyov, X. C. Zhu, J. G. Zhu, and C. L. Chien, Magnetic bistability and controllable re- versal of asymmetric ferromagnetic nanorings. Phys. Rev. Lett. 96, 027205 (2006). [6] V . Kubytskyi, S.-A. Biehs, and P. Ben-Abdallah, Radiative bistability and thermal memory. Phys. Rev. Lett. 113, 074301 (2014). [7] R. Labouvie, B. Santra, S. Heun, and H. Ott, Bistability in a driven-dissipative superfluid. Phys. Rev. Lett. 116, 235302 (2016). [8] Y .-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, Bistability of cavity magnon polaritons. Phys. Rev. Lett. 120, 057202 (2018). [9] H. B. Chan, V . A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, Nonlinear micromechanical Casimir oscillator. Phys. Rev. Lett. 87, 211801 (2001). [10] S. Sapmaz, Ya. M. Blanter, L. Gurevich, and H. S. J. van der Zant, Carbon nanotubes as nanoelectromechanical systems. Phys. Rev. B 67, 235414 (2003). [11] R. L. Badzey, and P. Mohanty, Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance. Nature 437, 995 (2005). [12] F. Cottone, H. V occa, and L. Gammaitoni, Nonlinear energy harvesting. Phys. Rev. Lett. 102, 080601 (2009). [13] Z. Chen, Q. Guo, C. Majidi, W. Chen, D. J. Srolovitz, and M. P. Haataja, Nonlinear geometric e ects in mechanical bistable morphing structures. Phys. Rev. Lett. 109, 114302 (2012). [14] F. Ricci, R. A. Rica, M. Spasenovi ´c, J. Gieseler, L. Rondin, L. Novotny, and R. Quidant, Optically levitated nanoparticle as amodel system for stochastic bistable dynamics. Nat. Commun. 8, 15141 (2017). [15] Q. P. Unterreithmeier, T. Faust, and J. P. Kotthaus, Nonlinear switching dynamics in a nanomechanical resonator. Phys. Rev. B81, 241405(R) (2010). [16] R. J. Dolleman, P. Belardinelli, S. Houri, H. S.J. van der Zant, F. Alijani, and P. G. Steeneken, High-frequency stochastic switch- ing of graphene resonators near room temperature. Nano Lett. 19, 1282 (2019). [17] R. L. Badzey, G. Zolfagharkhani, A. Gaidarzhy, and P. Mo- hanty, A controllable nanomechanical memory element. Appl. Phys. Lett. 85, 3587 (2004). [18] D. Roodenburg, J. W. Spronck, H. S. J. van der Zant, and W. J. Venstra, Buckling beam micromechanical memory with on- chip readout. Appl. Phys. Lett. 94, 183501 (2009). [19] D. N. Guerra, A. Bulsara, W. Ditto, S. Sinha, K. Murali, and P. Mohanty, A noise-assisted reprogrammable nanomechanical logic gate. Nano Lett. 10, 1168 (2010). [20] R. L. Harne, and K. W. Wang, A review of the recent re- search on vibration energy harvesting via bistable systems. Smart Mater. Struct. 22, 023001 (2013). [21] J. S. Aldridge, and A. N. Cleland, Noise-enabled precision mea- surements of a du ng nanomechanical resonator. Phys. Rev. Lett. 94, 156403 (2005). [22] R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, Noise squeezing in a nanomechanical du ng resonator. Phys. Rev. Lett. 98, 078103 (2007). [23] I. Katz, A. Retzker, R. Straub, and R. Lifshitz, Signatures for a classical to quantum transition of a driven nonlinear nanome- chanical resonator. Phys. Rev. Lett. 99, 040404 (2007). [24] M. L ´opez-Su ´arez, and I. Neri, Micro-electromechanical mem- ory bit based on magnetic repulsion. Appl. Phys. Lett. 109, 133505 (2016). [25] H. Seok, L. F. Buchmann, E. M. Wright, and P. Meystre, Mul- timode strong-coupling quantum optomechanics. Phys. Rev. A 88, 063850 (2013). [26] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity mag- nomechanics. Sci. Adv. 2, e1501286 (2016). [27] D. Lachance-Quirion, Y . Tabuchi, A. Gloppe, K. Usami, and Y . Nakamura, Hybrid quantum systems based on magnonics. Appl. Phys. Express 12, 070101 (2019). [28] C. A. Potts, E. Varga, V . Bittencourt, S. V . Kusminskiy, and J. P. Davis, Dynamical backaction magnomechanics. Phys. Rev. X11, 031053 (2021). [29] H. Y . Yuan, Y . Cao, A. Kamra, R. A. Duine, and P. Yan, Quan- tum magnonics: When magnon spintronics meets quantum in- formation science. Phys. Rep. 965, 1 (2022). [30] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, High Coopera- tivity in Coupled Microwave Resonator Ferrimagnetic Insulator Hybrids. Phys. Rev. Lett. 111, 127003 (2013). [31] Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, Hybridizing ferromagnetic magnons and mi- crowave photons in the quantum limit. Phys. Rev. Lett. 113, 083603 (2014). [32] X. Zhang, C. L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons. Phys. Rev. Lett. 113, 156401 (2014). [33] M. Goryachev, W. G. Farr, D. L. Creedon, Y . Fan, M. Kostylev, and M. E. Tobar, High-cooperativity cavity QED with magnons at microwave frequencies. Phys. Rev. Applied 2, 054002 (2014). [34] L. Bai, M. Harder, Y . P. Chen, X. Fan, J. Q. Xiao, and C. M. Hu, Spin pumping in electrodynamically coupled magnon-photon6 systems. Phys. Rev. Lett. 114, 227201 (2015). [35] D. Zhang, X.-M.Wang, T.-F. Li, X.-Q. Luo, W.Wu, F. Nori, and J. Q. You, Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium-iron-garnet sphere. npj Quantum Inf.1, 15014 (2015). [36] C. Kittel, Interaction of spin waves and ultrasonic waves in fer- romagnetic crystals. Phys. Rev. 110, 836 (1958). [37] J. Li, S.-Y . Zhu, and G. S. Agarwal, Magnon-photon-phonon entanglement in cavity magnomechanics. Phys. Rev. Lett. 121, 203601 (2018). [38] J. Li, and S.-Y . Zhu, Entangling two magnon modes via mag- netostrictive interaction. New J. Phys. 21, 085001 (2019). [39] H. Tan, Genuine photon-magnon-phonon Einstein-Podolsky- Rosen steerable nonlocality in a continuously-monitored cavity magnomechanical system. Phys. Rev. Res. 1, 033161 (2019). [40] M. Yu, H. Shen, and J. Li, Magnetostrictively induced station- ary entanglement between two microwave fields. Phys. Rev. Lett. 124, 213604 (2020). [41] J. Li, and S. Gr ¨oblacher, Entangling the vibrational modes of two massive ferromagnetic spheres using cavity magnomechan- ics. Quantum Sci. Technol. 6, 024005 (2021). [42] J. Li, S.-Y . Zhu, and G. S. Agarwal, Squeezed states of magnons and phonons in cavity magnomechanics. Phys. Rev. A 99, 021801(R) (2019). [43] W. Zhang, D.-Y . Wang, C.-H. Bai, T. Wang, S. Zhang, and H.- F. Wang, Generation and transfer of squeezed states in a cavity magnomechanical system by two-tone microwave fields. Opt. Express 29, 11773 (2021). [44] J. Li, Y .-P. Wang, J. Q. You, and S.-Y . Zhu, Squeezing Mi- crowaves by Magnetostriction. arXiv:2101.02796. Nat. Sci. Rev. (to be published). [45] M.-S. Ding, L. Zheng, and C. Li, Ground-state cooling of a magnomechanical resonator induced by magnetic damping. J. Opt. Soc. Am. B 37, 627 (2020). [46] Z.-X. Yang, L. Wang, Y .-M. Liu, D.-Y . Wang, C.-H. Bai, S. Zhang, and H.-F. Wang, Ground state cooling of magnome- chanical resonator in PT-symmetric cavity magnomechanical system at room temperature. Front. Phys. 15, 52504 (2020). [47] M. Asjad, J. Li, S.-Y . Zhu, and J. Q. You, Magnon squeez- ing enhanced ground-state cooling in cavity magnomechanics. arXiv:2203.10767. [48] C. Kong, B. Wang, Z.-X. Liu, H. Xiong, and Y . Wu, Magnet- ically controllable slow light based on magnetostrictive forces. Opt. Exp. 27, 5544 (2019). [49] K. Ullah, M. Tahir Naseem, and ¨O. E. M ¨ustecaplioglu, Tunable multiwindow magnomechanically induced transparency, Fano resonances, and slow-to-fast light conversion. Phys. Rev. A 102, 033721 (2020). [50] C. A. Potts, V . A. S. V . Bittencourt, S. V . Kusminskiy, and J. P. Davis, Magnon-phonon quantum correlation thermometry. Phys. Rev. Applied 13, 064001 (2020). [51] S.-F. Qi, and J. Jing, Magnon-assisted photon-phonon conver- sion in the presence of structured environments. Phys. Rev. A 103, 043704 (2021). [52] B. Sarma, T. Busch, and J. Twamley, Cavity magnomechani- cal storage and retrieval of quantum states. New J. Phys. 23, 043041 (2021). [53] T.-X. Lu, H. Zhang, Q. Zhang, and H. Jing, Exceptional-point- engineered cavity magnomechanics. Phys. Rev. A 103, 063708 (2021). [54] S.-N. Huai, Y .-L. Liu, J. Zhang, L. Yang, and Y .-X. Liu, Enhanced sideband responses in a PT-symmetric-like cavity magnomechanical system. Phys. Rev. A 99, 043803 (2019). [55] M. Wang, D. Zhang, X.-H. Li, Y .-Y . Wu, and Z.-Y . Sun,Magnon Chaos in PT-Symmetric Cavity Magnomechanics. IEEE Photonics J. 11, 5300108 (2019). [56] Y .-T. Chen, L. Du, Y . Zhang, and J.-H. Wu, Perfect transfer of enhanced entanglement and asymmetric steering in a cavity- magnomechanical system. Phys. Rev. A 103, 053712 (2021). [57] M.-S. Ding, X.-X. Xin, S.-Y . Qin, and C. Li, Enhanced entan- glement and steering in PT-symmetric cavity magnomechan- ics. Opt. Commun. 490, 126903 (2021). [58] Y .-P. Wang, G. Q. Zhang, D. Zhang, X. Q. Luo, W. Xiong, S. P. Wang, T. F. Li, C. M. Hu, and J. Q. You, Magnon Kerr e ect in a strongly coupled cavity-magnon system. Phys. Rev. B 94, 224410 (2016). [59] R.-C. Shen, Y .-P. Wang, J. Li, S.-Y . Zhu, G. S. Agarwal, and J. Q. You, Long-Time Memory and Ternary Logic Gate Using a Multistable Cavity Magnonic System. Phys. Rev. Lett. 127, 183202 (2021). [60] See Supplemental Materials for additional proofs, which in- clude Refs. [26, 28, 61–68]. [61] L. Landau, E. Lifshitz, A. Kosevich, J. Sykes, L. Pitaevskii, and W. Reid, Theory of Elasticity: V olume 7, Course of Theoretical Physics (Elsevier, Amsterdam, 1986) [62] K. S. U. Kansanen, C. Tassi, H. Mishra, M. A. Sillanp ¨a¨a, and T. T. Heikkil ¨a, Magnomechanics in suspended magnetic beams. Phys. Rev. B 104, 214416 (2021). [63] Z.-Y . Fan, R.-C. Shen, Y .-P. Wang, J. Li, and J. Q. You. Optical sensing of magnons via the magnetoelastic displacement. Phys. Rev. A 105, 033507 (2022). [64] A. G. Gurevich, and G. A. Melkov, Magnetization Oscillations and Waves (CRC, Boca Raton, FL, 1996). [65] O. O. Soykal, and M. E. Flatte, Strong field interactions be- tween a nanomagnet and a photonic cavity. Phys. Rev. Lett. 104, 077202 (2010). [66] T. Holstein, and H. Primako , Field dependence of the intrinsic domain magnetization of a ferromagnet. Phys. Rev. 58, 1098 (1940). [67] C. Kittel, Physical theory of ferromagnetic domains. Rev. Mod. Phys. 21, 541 (1949). [68] D. Vitali, S. Gigan, A. Ferreira, H. R. Bohm, P. Tombesi, A. Guerreiro, V . Vedral, A. Zeilinger, and M. Aspelmeyer, Op- tomechanical entanglement between a movable mirror and a cavity field. Phys. Rev. Lett. 98, 030405 (2007). [69] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity op- tomechanics. Rev. Mod. Phys. 86, 1391 (2014). [70] Y . Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, Ultraco- herent nanomechanical resonators via soft clamping and dissi- pation dilution. Nat. Nanotech. 12, 776 (2017). [71] G. S. MacCabe, H. J. Ren, J. Luo, J. D. Cohen, H. Y . Zhou, A. Sipahigil, M. Mirhosseini, and O. Painnter, Nano-acoustic res- onator with ultralong phonon lifetime. Science 370, 840 (2020).7 SUPPLEMENTAL MATERIALS I. SYSTEM PARAMETERS Quantity Symbol Value Gyromagnetic ratio 22.8 MHz /Oe Frequency of the cavity TE 101mode !a 27.653 GHz Frequency of the magnon mode !m B0 Frequency of the lower-frequency phonon mode !b 211.0308 MHz Frequency of the higher-frequency phonon mode !0 b211.0377 MHz Total cavity decay rate a 22.78 MHz Cavity decay rate via Port 1 1 20.22 MHz Cavity decay rate via Port 2 2 21.05 MHz Linewidth of the magnon mode m 22.2 MHz Linewidth of the lower-frequency phonon mode b 2550 Hz Linewidth of the higher-frequency phonon mode 0 b2180 Hz Cavity-magnon coupling strength gma 27.37 MHz Bare magnomechanical coupling strength (for the lower-frequency mode) gmb 21.22 mHz Bare magnomechanical coupling strength (for the higher-frequency mode) g0 mb20.62 mHz Magnon self-Kerr coe cient Km -26.5 nHz Magnon-phonon cross-Kerr coe cient Kcross -25.4 pHz TABLE S1. List of system parameters. Table S1 provides a list of the system parameters. In the experiment, the frequencies and the dissipation rates (linewidths) of the cavity and magnon modes and their coupling strength gmaare extracted by fitting the cavity-magnon polariton in the reflection spectra using Eq. (S30). The mechanical frequency and linewidth and the magnomechanical coupling strength gmb are obtained by fitting the spectra of the magnomechanically induced resonances using also Eq. (S30). The cavity decay rates 1 and2associated with the two ports need to be fitted by measuring the reflection spectrum through the two ports, respectively. The magnon self-Kerr coe cient Kmof the crystal axis [110] is determined by measuring the magnon frequency shift, which is consistent with the value calculated from Eq. (S4). The magnon-phonon cross-Kerr coe cient Kcrossis obtained by fitting the mechanical frequency shift !busing Eq. (S26). Specifically, the first term of Eq. (S26) is the frequency shift caused by the magnon-phonon radiation-pressure-like coupling, which can be calculated (e.g., -200 Hz under a red-detuned drive with the power 19.7 dBm). The magnon excitation number jMj2can also be calculated by using Eq. (S20). With all these at hand, the cross-Kerr coe cient Kcrosscan then be determined by fitting the mechanical frequency shift. II. DERIV ATION OF THE HAMILTONIAN FOR KERR NONLINEARITIES In this section, we provide a detailed derivation of the Hamiltonian HKerrassociated with the two Kerr nonlinear terms, namely, the magnon self-Kerr nonlinearity and the magnon-phonon cross-Kerr nonlinearity. These two terms become appreciable in the system when the pump field is su ciently strong, such that the system enters a regime where the Kerr-type nonlinearities strongly modify the conventional cavity magnomechanics (CMM), which we term as the Kerr-modified CMM. Here the conventional CMM means that there is only a radiation-pressure-like coupling between magnons and vibration phonons [26, 28]. A. Magnon self-Kerr nonlinearity The magnon self-Kerr nonlinearity originates from the anisotropic field. When the bias magnetic field is aligned along the [110] axis of the YIG sphere, the anisotropic field is given by [64] Han=3KanMx 0M2ex+9KanMy 40M2ey+KanMz 0M2ez; (S1) where Kanis the first-order magnetocrystalline anisotropy constant, and for the YIG at room temperature Kan=610 J=m3. M=(Mx;My;Mz) denotes the magnetization of the YIG sphere, Mis the saturation magnetization, and 0is the permeability8 of vacuum. The anisotropy Hamiltonian reads Han=0 2Z VmM
Hand; =KanVm 8M2 12M2 x+9M2 y+4M2 z ;(S2) where Vmis the volume of the YIG sphere. By using the relation S=MVm=~ (Sx;Sy;Sz) [65], with Sbeing the macrospin operator and being the gyromagnetic ratio, the anisotropy Hamiltonian Hancan be written as Han=~=3~Kan 2 2M2VmS2 x9~Kan 2 8M2VmS2 y~Kan 2 2M2VmS2 z: (S3) Using the Holstein-Primako transformation [66]: S+=p 2Smymm,S=myp 2Smym, and Sz=Smym, where Sis the total spin number of the macrospin and my(m) is the creation (annihilation) operator of the magnon mode, we obtain Han=~13~S K an 2 8M2Vmmym+13~Kan 2 16M2Vmmymmym: (S4) The first term of the anisotropy Hamiltonian will modify the magnon frequency !0 m=!m13~S K an 2 8M2Vm, and the second term accounts for the magnon self-Kerr nonlinearity, which can be written in the form of Hself–Kerr=~=Kmmymmym, with the self- Kerr coe cient Km=13~Kan 2 16M2Vm. Note that in deriving the Hamiltonian (S4), we have omitted the constant term and the higher order terms. B. Magnon-phonon cross-Kerr nonlinearity The magnetoelastic coupling describes the interaction between the magnetization and the elastic strain of the magnetic ma- terial. Depending on the distance between magnetic atoms (or ions), there are di erent kinds of interactions: the spin-orbital interaction, the exchange interaction between magnetic atoms (or ions), and the magnetic dipole-dipole interaction [64]. In a cubic crystal, the magnetoelastic energy density is given by [67] fme=b1 M2 M2 xxx+M2 yyy+M2 zzz +2b2 M2 MxMyxy+MxMzxz+MyMzyz ; (S5) where b1andb2are the magnetoelastic coupling constants, and the strain tensor i jis given in the nonlinear Euler-Bernoulli theory by [61, 62] i j=1 20BBBBB@@ui @lj+@uj @li+X k@uk @li@uk @lj1CCCCCA; (S6) with uibeing the components of the displacement vector. The two first-order terms in i jlead to the magnon-phonon radiation- pressure-like coupling (see [63] for a strict derivation). In typical CMM experiments using a moderate drive field [26, 28], the second-order terms are neglected. However, for an intense drive field as used in our experiment, those terms will produce a noticeable e ect. As we will derive below, those second-order terms in the strain tensor are responsible for the magnon-phonon cross-Kerr nonlinearity. In the magnetoelastic energy density (S5), the second term accounts for the parametric magnon generation when the phonon frequency is twice the magnon frequency, or the linear magnon-phonon coupling when they are nearly resonant [26, 63]. Thus, this term is negligible for our system with a low mechanical frequency !b!m, and we can only consider the first term in (S5). Integrating over the whole volume of the YIG sphere, the interaction Hamiltonian can be written as H1=b1 M2Z dl3 M2 xxx+M2 yyy+M2 zzz ; 'b1 M~ VmmymZ dl3 xx+yy2zz :(S7) To quantize the above Hamiltonian, we express the displacement vector uias a superposition ~u=X n;m;ld(n;m;l)~(n;m;l)(x;y;z); (S8)9 where~(n;m;l)(x;y;z)is the displacement eigenmode with the corresponding amplitude d(n;m;l), and the superscript n;m;ldenote the mode indices. The mechanical displacement can be quantized as d(n;m;l)=d(n;m;l) zpm (bn;m;l+by n;m;l), where d(n;m;l) zpm is the amplitude of the zero-point motion, and bn;m;l(by n;m;l) is the bosonic annihilation (creation) operator of the mechanical mode. Substituting (S8) into the Hamiltonian (S7), we obtain H1'X n;m;l~g(n;m;l) mbmym bn;m;l+by n;m;l +X n;m;l~K(n;m;l) cross mymby n;m;lbn;m;l;(S9) where we use the rotating-wave approximation by neglecting the fast-oscillating terms in deriving the second term. This is valid when!bKcrossjMj2, which is well satisfied in our experiment. The first term describes the magnon-phonon radiation-pressure- like interaction, and the second term accounts for the cross-Kerr interaction between the magnon and mechanical modes. The radiation-pressure-like coupling strength g(n;m;l) mband the cross-Kerr coe cient K(n;m;l) cross are given by g(n;m;l) mb=b1 M VmZ dl3d(n;m;l) zpm0BBBBB@@(n;m;l) x @x+@(n;m;l) y @y2@(n;m;l) z @z1CCCCCA; K(n;m;l) cross =b1 M VmZ dl3d(n;m;l)2 zpmX k2666666640BBBBB@@(n;m;l) k @x1CCCCCA2 +0BBBBB@@(n;m;l) k @y1CCCCCA2 20BBBBB@@(n;m;l) k @z1CCCCCA2377777775:(S10) When considering a specific mechanical mode as in our experiment, the interaction Hamiltonian takes a simple form of H1=~=gmbmym b+by +Kcrossmymbyb: (S11) III. HAMILTONIAN OF THE KERR-MODIFIED CA VITY MAGNOMECHANICAL SYSTEM The CMM system under study consists of a microwave cavity mode, a magnon (Kittel) mode, and a mechanical vibration mode. The magnon mode couples to the cavity mode by the magnetic-dipole interaction, and to the mechanical mode by the magnetostrictive interaction. There is no direct coupling between the cavity and the mechanics. The drive field is applied to the cavity 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 self-Kerr and magnon-phonon cross-Kerr nonlinearities are activated in the system. The total Hamiltonian of the CMM system is given by H=~=!aaya+!mmym+!bbyb+gma(aym+amy)+gmbmym b+by +Kmmymmym +Kcrossmymbyb+p1"d aei!dt+ayei!dt +p2"p aei!pt+ayei!pt ;(S12) where1(2) is the cavity decay rate associated with the driving (probe) port, "d=q Pd ~!dand"p=q Pp ~!p, with Pd(Pp) and!d (!p) being the power and frequency of the drive (probe) microwave field. Since in the experiment the probe field is of a much smaller power than that of the drive field and thus can be treated as a perturbation to the system, we therefore omit the probe term in the Hamiltonian, giving rise to the Hamiltonian (1) that is provided in the main text H=~=!aaya+!mmym+!bbyb+gma(aym+amy)+gmbmym b+by +Kmmymmym+Kcrossmymbyb+p1"d aei!dt+ayei!dt :(S13) IV . DETERMINATION OF THE MAGNON EXCITATION NUMBER From the Hamiltonian (S13), we can obtain the Heisenberg-Langevin equations by including the dissipation and input noise of each mode. In the frame rotating at the drive frequency, they are given by da dt= i
a+a 2 aigmamip1"d+paain; dm dt= i
m+2Kmmym+Km+Kcrossbyb +igmb(b+by)+m 2 migmaa+pmmin; db dt= i !b+Kcrossmym +b 2 bigmbmym+pbbin;(S14)10 where
a=!a!d, and
m=!m!d, whilea,mandb(ain,min, and bin) are the dissipation rates (input noises) of the three modes. Since the cavity mode is strongly driven, this leads to a large amplitude jhaij 1 in the steady state, and further due to the cavity-magnon coupling, the magnon mode also has a large amplitude jhmij 1. This allows us to linearize the system dynamics around the classical average values by writing the mode operators as aA+a,mM+m, and bB+b, and neglecting small second-order fluctuation terms [68]. Substituting these mode operators into Eq. (S14), the equations are then separated into two sets of equations, respectively, for classical averages ( A,M,B) and for quantum fluctuations ( a,m,b). The equations for the classical averages in the steady state are as follows: 
aia 2 A+gmaM+p1"d=0; 
m+2KmjMj2+Km+KcrossjBj2+gmb(B+B)im 2 M+gmaA=0;  !b+KcrossjMj2ib 2 B+gmbjMj2=0:(S15) From the first equation of Eq. (S15), we get A=agma
a+ia 2 Map1
a+ia 2 "d; (S16) wherea=1
2a+(a 2)2. Substituting Ainto the second equation of Eq. (S15), we obtain 
mag2 ma
a+2KmjMj2+Km+KcrossjBj2+2gmbRe[B]im 2+ag2 maa 2 M+ap1gma(
a+ia 2)"d=0: (S17) Multiplying Eq. (S17) with its complex conjugate, we obtain the equation for the magnon excitation number "
mag2 ma
a+2KmjMj2+Km+KcrossjBj2+2gmbRe[B]2+m 2+ag2 maa 22# jMj2=a1g2 ma"2 d: (S18) Similarly, we get the equation for the phonon excitation number " !b+KcrossjMj22+b 22# jBj2=g2 mbjMj4: (S19) In our experiment, the drive power is scanned from 4.7 dBm to 23.7 dBm, which gives the magnon excitation number jMj22 [1012;1015], and the phonon excitation number jBj22[106;1010]. Thus, the phonon excitation number is much smaller than that of the magnon. Using our parameters gmb=2=1:22 mHz, Km=2=6:5 nHz, and Kcross=2=5:4 pHz, the magnon frequency shift !m=2KmjMj2+Km+KcrossjBj2+2gmbRe[B]2KmjMj2. Therefore, Eq. (S18) is reduced to "
mag2 ma
a+2KmjMj22+m 2+ag2 maa 22# jMj2=a1g2 ma"2 d: (S20) This is a cubic equation of the magnon excitation number jMj2, and given as Eq. (4) in the main text. Under certain conditions, all the three solutions of jMj2are real, among which there are two stable solutions. The stable solutions can be measured in the experiment, and it shows a hysteresis loop by varying the drive power. V . EFFECTIVE SUSCEPTIBILITY OF THE MECHANICAL MODE The magnon-phonon radiation-pressure-like coupling gives rise to the magnomechanical backaction on the mechanical mode, which is manifested as the mechanical frequency shift (i.e., the magnonic spring e ect) and the increased (reduced) linewidth associated with the cooling (amplification) of the mechanical mode. The frequency shift and the linewidth variation can be eval- uated from the e ective susceptibility of the mechanical mode. In what follows, we show in detail how the e ective mechanical susceptibility is derived. The linearization of the Langevin equations (S14) yields a set of linearized quantum Langevin equations for the quantum fluctuations ( m;a;x;p), wherex=(b+by)=p 2 andp=i(byb)=p 2 denote the fluctuations of two mechanical11 quadratures (position and momentum). By taking the Fourier transform, we obtain the following equations in the frequency domain: i!m= i˜
m+m 2 migmaaip 2Gmbx+pmmin; i!my= i˜
m+m 2 my+igmaay+ip 2G mbx+pmmy in; i!a= i
a+a 2 aigmam+paain; i!ay= i
a+a 2 ay+igmamy+paay in; i!x=˜!bp; i!p=˜!bxbpp 2 G mbm+Gmbmy +;(S21) where ˜
m'
m+2KmjMj2includes the magnon frequency shift dominantly caused by the magnon self-Kerr e ect, and ˜!b= !b+KcrossjMj2includes the mechanical frequency shift due to the magnon-phonon cross-Kerr e ect.Gmb=gmbMis the e ective radiation-pressure-like coupling strength. Note that we adopt an equivalent model of dealing with the mechanical damping and input noise, where the damping rate band the Hermitian Brownian noise operator are added only in the momentum equation [68]. Solving separately the two equations for each mode, we obtain the following equations: m=m(!) igmaaip 2Gmbx+pmmin ; my= m(!) igmaay+ip 2G mbx+pmmy in ; a=a(!) igmam+paain ; ay= a(!) igmamy+paay in ; x=b(!) p 2G mbmp 2Gmbmy+ ; p=i! ˜!bx;(S22) where we define the natural susceptibilities of the magnon, cavity, and mechanical modes as m(!)=1 i˜
m! +m 2;  m(!)=1 i˜
m+! +m 2; a(!)=1 i(
a!)+a 2;  a(!)=1 i(
a+!)+a 2; b(!)=˜!b ˜!2 b!2ib!:(S23) Solving the first four equations in Eq. (S22) for mandmy, and inserting their solutions into the equation of x, we obtain x=b;e(!)0BBBBBB@+ip 2G mbm(!) ipmmin+pagmaa(!)ain 1+g2maa(!)m(!)ip 2Gmb m(!) ipmmy in+pagma a(!)ay in 1+g2maa(!)m(!)1CCCCCCA; (S24) whereb;eis the e ective mechanical susceptibility, defined as b;e(!)= 1 b(!)2ijGmbj2ma(!) ma(!)
1; (S25) withma(!)=1 1m(!)+g2maa(!). The change of the mechanical frequency and linewidth can be extracted from the e ective suscep- tibility: the real part of 1 b;e(!)1 b(!) corresponds to the mechanical frequency shift !b=Reh 2ijGmbj2ma(!) ma(!)
i +KcrossjMj2; (S26) where we write together the frequency shift due to the cross-Kerr e ect, and the imaginary part of 1 b(!)1 b;e(!) yields the variation of the mechanical linewidth b=Imh 2ijGmbj2ma(!) ma(!)
i : (S27)12 VI. REFLECTION SPECTRUM OF THE PROBE FIELD Here we show how to derive the reflection spectrum of the probe field under the strong drive field. The Hamiltonian including the probe field is given in Eq. (S12). The reflection spectrum can be conveniently solved by including the strong pump e ects into the linearized Langevin equations. Following the linearization approach used in Sec. IV and Sec. V , the Hamiltonian (S12) leads to the following Langevin equations for the classical averages in the frequency domain: i!M= i˜
m+m 2 MigmaAip 2GmbX; i!A= i
a+a 2 AigmaMip2"p(!p!d!); i!X=˜!bP; i!P=˜!bXbPp 2G mbM;(S28) where X=(B+B)=p 2 and P=i(BB)=p 2 denote the classical averages of the mechanical position and momentum. Note that, same as Eqs. (S14) and (S21), the above equations are provided in the frame rotating at the drive frequency !d. Solving the above equations, we obtain A(!)=ip2 12ijGmbj2b(!)m(!) g2 mb(!)m(!)1a(!)12ijGmbj2b(!)m(!)
"p: (S29) Using the input-output theory, Aout="p+ip2A, we therefore achieve the reflection spectrum of the probe field r(!)Aout "p=12 12ijGmbj2b(!)m(!) g2 mb(!)m(!)1a(!)12ijGmbj2b(!)m(!)
: (S30)

2022-06-29

Bistable mechanical vibration is observed in a cavity magnomechanical system, which consists of a microwave cavity mode, a magnon mode, and a mechanical vibration mode of a ferrimagnetic yttrium-iron-garnet (YIG) sphere. The bistability manifests itself in both the mechanical frequency and linewidth under a strong microwave drive field, which simultaneously activates three different kinds of nonlinearities, namely, magnetostriction, magnon self-Kerr, and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr nonlinearity is first predicted and measured in magnomechanics. The system enters a regime where Kerr-type nonlinearities strongly modify the conventional cavity magnomechanics that possesses only a radiation-pressure-like magnomechanical coupling. Three different kinds of nonlinearities are identified and distinguished in the experiment. Our work demonstrates a new mechanism for achieving mechanical bistability by combining magnetostriction and Kerr-type nonlinearities, and indicates that such Kerr-modified cavity magnomechanics provides a unique platform for studying many distinct nonlinearities in a single experiment.

Mechanical Bistability in Kerr-modified Cavity Magnomechanics

2206.14588v2

Superconductivity Induced by Interfacial Coupling to Magnons Niklas Rohling, Eirik Lhaugen Fjrbu, and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim We consider a thin normal metal sandwiched between two ferromagnetic insulators. At the in- terfaces, the exchange coupling causes electrons within the metal to interact with magnons in the insulators. This electron-magnon interaction induces electron-electron interactions, which, in turn, can result in p-wave superconductivity. In the weak-coupling limit, we solve the gap equation nu- merically and estimate the critical temperature. In YIG-Au-YIG trilayers, superconductivity sets in at temperatures somewhere in the interval between 1 and 10 K. EuO-Au-EuO trilayers require a lower temperature, in the range from 0.01 to 1 K. The interactions between electrons in a conductor and ordered spins across interfaces are of central importance in spintronics [1, 2]. Here, we focus on the case in which the magnetically ordered system is a ferromagnetic insu- lator (FI). The interaction at an FI-normal metal (NM) interface can be described in terms of an exchange cou- pling [3{6]. In the static regime, this coupling induces eective Zeeman elds near the boundary [7{10]. The magnetization dynamics caused by the coupling can be described in terms of the spin-mixing conductance [4{ 6]. Such dynamics can include spin pumping from the FI into the NM [11, 12] and its reciprocal eect, spin- transfer torques [5, 13]. These spin-transfer torques en- able electrical control of the magnetization in FIs [14]. One important characteristic of FIs is that the Gilbert damping is typically small. This leads to low-dissipation magnetization dynamics [15], which, in turn, facilitates coherent magnon dynamics and the long-range transport of spin signals [5, 13]. These phenomena should also en- able other uses of the quantum nature of the magnons. Here, we study a previously unexplored eect that is also governed by the electron-magnon interactions at FI- NM interfaces but is qualitatively dierent from spin pumping and spin-transfer torques. We explore how the magnons in FIs can mediate superconductivity in a metal. The exchange coupling at the interfaces between the FIs and the NM induces Cooper pairing. In this scenario, the electrons and the magnons mediating the pairing reside in two dierent materials. This opens up a wide range of possibilities for tuning the superconduct- ing properties of the system by combining layers with the desired characteristics. The electron and magnon disper- sions within the layers as well as the electron-magnon coupling between the layers in uence the pairing mecha- nism. Consequently, the superconducting gap can also be tuned by modifying the layer thickness, interface quality, and external elds. Since the interactions occur at the interfaces, the con- sequences of the coupling are most profound when the NM layer is thin. We therefore consider atomically thin FI and NM layers. This also reduces the complexity of the calculations. For thicker layers, multiple modes exist along the direction transverse to the interface ( x), withdierent eective coupling strengths. We expect a qual- itatively similar, but somewhat weaker, eect for thicker layers. Paramagnonic [16] or magnonic [17] coupling may ex- plain experimental observations of superconductivity co- existing with ferromagnetism in bulk materials [18{20]. Paramagnons [16, 21] and magnons [17, 22] are predicted to mediate triplet p-wave pairing with equal and antipar- allel spins, respectively. High-quality thin lms oer new possibilities for super- conductivity [23]. Consequently, the emergence of super- conductivity at interfaces has recently received consid- erable attention [23{31]. Theoretical studies have been conducted on interface-induced superconductivity medi- ated by phonons [27{29], excitons [32], and polarizable localized excitations [31, 33]. A model of interface-induced magnon-mediated d-wave pairing has been proposed to explain the observed super- conductivity in Bi/Ni bilayers [34]. A p-wave pairing of electrons with equal momentum|so-called Amperean pairing|has been predicted to occur in a similar sys- tem [35]. Importantly, the electrons that form pairs in these models reside in a spin-momentum-locked surface conduction band. By contrast, we consider a spin-degenerate conduction band in an FI-NM-FI trilayer system. We nd interfa- cially mediated p-wave superconductivity with antipar- allel spins and momenta. These pairing symmetries are distinct from those of the 2D systems mentioned above. We assume that the equilibrium magnetization of the left (right) FI is along the ^ z(^z) direction; see Fig. 1. We consider matching square lattices, with lattice constant a, in all three monolayers. The interfacial plane com- prisesNsites with periodic boundary conditions. The Hamiltonian is H=HA FI+HB FI+HNM+Hint; (1) where we use A(B) to denote the left (right) FI. The Heisenberg Hamiltonian HA FI=J ~2X iX j2NN(i)SA i
SA j (2)arXiv:1707.03754v1 [cond-mat.mes-hall] 12 Jul 20172 FI NM FI FIG. 1. A trilayer formed of a normal metal between two fer- romagnetic insulators. The magnetizations are antiparallel. At the interfaces, conduction electrons couple to magnons. This results in eective electron-electron interactions in the metal. describes the left FI. Here, iis an in-plane site, NN( i) is the set of its nearest neighbors, Jis the exchange interac- tion, and SA iis the localized spin at site i. The expression forHB FIis similar. For the time being, we assume that the conduction electron eigenstates in the NM are plane waves of the formcq;=P jexp(irj
q)cj=p N. Here,c(y) jannihi- lates (creates) a conduction electron with spin at site jin the NM, and qis the wavevector. For now, the NM Hamiltonian is HNM=P qP Eqcy qcq, and the dispersion is quadratic, Eq=~2q2=(2m): (3) Here,mis the eective electron mass. Below, when esti- mating the coupling JIat YIG-Au interfaces, we consider another Hamiltonian with dierent eigenstates and a dif- ferent dispersion. We model the coupling between the conduction elec- trons and the localized spins as an exchange interaction of strength JI: Hint=2JI ~X 0X jX L=A;Bcy j0cj0
SL j;(4) where = (x;y;z) is a vector of Pauli matrices. After a Holstein-Primako transformation, we expand the Heisenberg Hamiltonian given in Eq. (2) up to second order in the bosonic operators and diagonalize it. We rep- resent SA jbySA jx+iSA jy=~p 2saj,SA jxiSA jy=~p 2say j, andSA jz=~(say jaj), wheresis the spin quantum num- ber of the localized spins and a(y) jis a bosonic annihila- tion (creation) operator at site j. The magnons in layer A, with the form ak=P j2Aexp(irj
k)aj=p N, are the eigenstates of the resulting Hamiltonian. Analogously, the magnons in layer Bare denoted by bk. The magnon dispersion is "k= 4sJ[2cos(kya)cos(kza)]: (5) We disregard second-order terms in the bosonic operatorsfrom the interfacial coupling and obtain H=X k"k(ay kak+by kbk) +X qEqcy qcq +X kqV(akcy q+k;#cq"+bkcy q+k;"cq#) + h.c.;(6) whereV=2JIps=p 2Nis the coupling strength be- tween the electrons in the NM and the magnons in the FI layers. There is no induced Zeeman eld in the NM since the magnetizations in the FIs are antiparallel. Analogously to phonon-mediated coupling in conventional supercon- ductors, the magnons mediate eective interactions be- tween the electrons. For electron pairs with opposite mo- menta, we obtain Hpair=X kk0Vkk0cy k#cy k"ck0"ck0#; (7) with the interaction strength Vkk0= 2jVj2"k+k0 "2 k+k0(EkEk0)2: (8) We dene the gap function in the usual way:
k=P k0Vkk0hck0"ck0#i. The gap equation becomes
k=X k0Vkk0
k0 2~Ek0tanh ~Ek0 2kBT! ; (9) where ~Ek=p (EkEF)2+j
kj2andEFis the Fermi energy. In the continuum limit, we replace the discrete sum over momenta kwith integrals over E=Ekand the angle', where k=k[sin(');cos(')]. We assume that only the conduction electrons close to the Fermi surface form pairs. The magnon energy that appears in Eq. (8) is then given by "k+k0"('0;'), where "('0;') =4sJf2cos(kFa[sin'+ sin'0]) cos(kFa[cos'+ cos'0])g:(10) Here,kF=p2mEF=~is the Fermi wavenumber. We assume that the NM is half lled, kF=p 2=a. We introduce the energy scale E= 4sJk2 Fa2= 8sJ, which is associated with the FI exchange interaction. Then, we scale all other energies with respect to E:=
=E, =kBT=E,x= (EEF)=E, ~x=~E=E, and= "=E. In this way, the gap equation presented in Eq. (9) simplies to (x;') =p 2 xBZ xBdx02Z 0d'0('0;')(x0;'0) tanhh ~x0 2i ~x0[2('0;')(xx0)2]; (11) with the dimensionless coupling constant = J2 I=(16p 2EFJ) =J2 Ima2=(16p 22~2J). In Eq. (11),3 we have restricted the energy integral to the range [EFxBE;EF+xBE]. We choose xB|based on the value of |in the following way. xBmust be suf- ciently large that all contributions to the gap from re- gions outside this range are vanishingly small. In the weak-coupling limit ( 1), the gap function has a nar- row peak near x= 0, and therefore, xBcan be much smaller than 1. To gain a better understanding, we rst assume a quadratic dispersion for the magnons, which matches that of Eq. (5) in the long-wavelength limit. Conse- quently, the dimensionless magnon energy ('0;') be- comesq('0;') = 1 + cos( '0'). Below, we numerically check the correspondence between the solutions result- ing from the full dispersion versus the solutions obtained with the quadratic approximation assumed here. For the quadratic magnon dispersion, the gap equation has a so- lution with p-wave symmetry, (x;') =f(x) exp(i'). Applying this ansatz to Eq. (11), we calculate the in- tegral over the angle '0in the weak-coupling limit [36]. The gap equation becomes f(x) =xBZ xBdx0V(xx0)f(x0) tanhp x02+f(x0)2 2 p x02+f(x0)2; (12) whereV(xx0)1=p jxx0j2p 2. Using a Gaussian centered at x= 0 as an initial guess, we solve Eq. (12) numerically through iteration [37]. Fig. 2 shows the results. For a xed coupling , the maximum value occurs when x= 0 and= 0. The dimensionless critical temperature cis the temperature at which the gap vanishes. As in the BCS theory, the gap equation can also be solved analytically by approximat- ingV(x) as a constant with a cuto centered at x= 0. In this constant-potential approximation, the ratio fmax=c is approximately 1 :76, which is slightly lower than what we nd numerically; see Fig. 2 (c). Let us check that the numerical solutions to Eq. (12), for the quadratic magnon energy, resemble the solutions to Eq. (11) for the full magnon energy of Eq. (10). To this end, we numerically iterate Eq. (11), starting from the solution to Eq. (12) as the initial guess [38]. We consider the case of zero temperature, = 0. The symmetries (x;') =(x;') =i(x;'+=2) =(x;'), where is the complex conjugate of , imply that we need to consider only x > 0 and 0< ' < = 4. We show the results of these iterative calculations in Fig. 3. The third iteration of is shown in Fig. 3 (a,b). After only three iterations, the dierences between consecutive functions are already nearly imperceptible; see Fig. 3 (c,d). The gap as a function of energy still exhibits a peak at the Fermi energy. Compared with the results obtained for a quadratic magnon dispersion, this peak is of a similar shape but is slightly lower and narrower; see the inset of Fig. 3 (c). There are also additional features of (x;') 0510 x×1030246f×104 (a) 0123 τ×1040246f(x=0)×104 (b) -3-2-1 log10(α)1.922.1fmax/τc (c) -3-2-1 log10(α)-4-3-2-1log10(fmax)(d)FIG. 2. Numerical solutions to the gap equation (12) deter- mined through iteration. (a) Gaussian-shaped initial guess (dashed line) and the results of the rst eight iterative cal- culations of the gap f(x) (from light blue to red) when the dimensionless temperature is = 0 and the coupling constant is= 0:005. Note that f(x) =f(x) and that the energy cutoxB0:03 lies outside the range of the plot. (b) Gap fat energyx= 0 as a function of for= 0:005. (c) Ratio between the maximum gap value, fmax, and the dimensionless critical temperature cas a function of . (d)dependence of fmax. The gray line corresponds to a quadratic dependence, fmax2. at positions ( x;') = (('0;');') in the parameter space where the derivative of ('0;') with respect to '0van- ishes. Next, we estimate the critical temperatures Tcfor two possible experimental realizations, one in which the FI is yttrium-iron-garnet (YIG) and one in which the FI is europium oxide (EuO). The NM layer is gold in both cases. We consider the YIG-Au-YIG trilayer rst. For the FIs, we assume|encouraged by the results pre- sented in Fig. 3|that the low-energy magnons dominate the gap. The relevant magnons can therefore be well de- scribed by a quadratic dispersion. Our model assumes that the FI and NM layers have the same lattice struc- ture. However, in reality, the unit cell of YIG is much larger than that of Au. To capture the properties of YIG in our model, we t the parameters such that the FIs have the same exchange stiness ( D=kB= 71 K nm2[39]) and saturation magnetization ( Ms= 1:6
105A/m [39]) as those of bulk YIG. We assume that each YIG layer has a thickness equal to the bulk lattice constant of YIG (aYIG12A [39]). We use the thickness, the saturation magnetization and the electron gyromagnetic ratio eto4 0246 00.10.20.3 x0π 8π 4ϕ (a)|δ3|×104 0π2π 00.10.20.3 x0π 8π 4 ϕ (b)arg/braceleftbig δ3/bracerightbig 00.050.1 x0246|δi(x,ϕ=0)|×104 (c) 0.002.0050246 0π/8π/4 ϕ0246 |δi(x=0,ϕ)|×104 (d) 0π 8π 4 arg/braceleftbig δi(x=0,ϕ)/bracerightbig FIG. 3. Numerical iteration of the gap equation (11), starting from the solution to Eq. (12) as an initial guess, for = 0 and= 0:005. (a,b) Absolute value and phase of 3(x;'), where the index 3 indicates the number of iterations. (c) ji(x;' = 0)jfori= 0 (orange line), i= 2 (black dashed line), andi= 3 (gray circles). (d) ji(x= 0;')j(left axis) fori= 0;2;3, with the same colors as in (c), and the phase ofi(x= 0;') (right axis) for i= 0 (purple), i= 2 (blue, dashed),i= 3 (cyan, wide). Note that the dierence from the second to the third iteration is nearly indiscernible. estimate the spin quantum number s=MsaYIGa2=(~ e). Using the quadratic dispersion approximation, we deter- mine the exchange interaction to be J=D=(2a2s). The lattice spacing aremains undetermined. In the bulk, gold has an fcc lattice and a half-lled con- duction band. We use experimental values of the Fermi energy (EB F= 5:5 eV [40]) and the Sharvin conductance (gSh= 12 nm2[6]) to determine the eective mass, m= 2gSh~2=EB F. We assume that the monolayer is half lled and has the same eective electron mass as that of bulk gold. We consider the case in which the monolayer lattice constant ais equal to the lattice constant atof a simple cubic tight-binding model for gold. atis ap- proximately 20% smaller than the bulk nearest-neighbor distance of actual gold. We calculate the interfacial exchange coupling JIfor a YIG-Au bilayer in terms of the spin-mixing conduc- tance, which has been experimentally measured. In doing this calculation, we use the same model for the YIG as in the trilayer case; however, for the gold, we employ a tight-binding model of the form Ht= ttP P iP j2NN(i)cy icj, with a simple cubic lat- tice. The Hamiltonian of the bilayer is HB=Ht+ HA FI+Hint. We assume that JIstt, which al-lows us to disregard the proximity-induced Zeeman eld. The energy eigenstates ct qand the dispersion Et q= 4tt(3cos(qxat)cos(qyat)cos(qzat)) ofHtare well known. Under the assumption of half lling, we nd that tt=EB F=12 andat=p 0:63=gSh. We use the same ex- perimental values for EB FandgSh(from Ref. 6 and 40) as before. We set the lattice constant of the trilayer, a, equal to the lattice constant of the bilayer, at. This ensures that both models have the same lattice structure at the in- terface and, consequently, that the interfacial exchange interaction Hamiltonian Hinthas the same form in both cases. To rst order in the bosonic operators, Hint=P kqVtakcty q+k;#ct q". The coupling strength Vtis propor- tional to the amplitudes of the tight-binding-model eigen- states at the interface: Vt= 2Vsin(qxat) sin([kx+qx]at). The spin-mixing conductance can now be calculated for the ferromagnetic resonance (FMR) mode, resulting [41] ing"#= 4a2 tV0sN=(2)2, where V0=ZZ jVj2sin(qxat)2sin(q0 xat)2 qyq0 y
(qzq0 z) Et qEF
 Et q0EF
d3qd3q0:(13) We numerically evaluate V0and estimate the bilayer in- terfacial exchange coupling JI=p (2)2g"#t2 ta2 t=(9:16s2) using measured values of the spin-mixing conductance g"#. We assume that JIhas the same value in the tri- layer case. Using E= 8sJ, we nd that Eis approx- imately 1:5 eV. We nd the coupling constant from the relation =J2 Ima2=(16p 22~2J). The reported ex- perimental values for the spin-mixing conductance range from 1:2 nm2to 6 nm2[42{44]. In turn, this implies thatlies in the range of [0 :0014{0:007]. The corre- sponding critical temperatures range from 0 :5 K to 10 K. Next, we consider a EuO-Au-EuO trilayer. Europium oxide has an fcc lattice structure with a lattice constant of 5:1A, a spin quantum number of s= 7=2 and a nearest- neighbor exchange coupling of J=kB= 0:6 K [45]. The nodes on a (100) surface of an fcc lattice form a square lattice in which the lattice constant is equal to the dis- tance between nearest neighbors in the bulk. We assume that the monolayer has the same structure and therefore setaequal to the distance between nearest neighbors in bulk EuO. We use the same eective mass as for the YIG- Au-YIG trilayer. Then, the Fermi energy is EF= 1:8 eV, and the energy scale E=kBis approximately 53 K. Val- ues on the order of 10 meV have been reported for the interfacial exchange coupling strengths JI[46] in EuO/Al [7], EuO/V [8], and EuS/Al [9, 10]. These estimates were based on measurements of a proximity-induced eective Zeeman eld. Under the assumption that JIis in the range of [5{15] meV, we nd a wide range of values of [0:004{0:03] for. We estimate the corresponding critical temperatures numerically using the quadratic dispersion approximation. Finally, we nd a range of [0 :01{0:4] K as possible values for Tc.5 In conclusion, interfacial coupling to magnons induces p-wave superconductivity in metals. The critical temper- atures are experimentally accessible in the weak-coupling limit. The gap size strongly depends on the magnitude of the interfacial exchange coupling. The thickness depen- dence, the robustness against disorder, and the physics beyond the weak-coupling limit should be explored in the future. This work was partially supported by the European Research Council via Advanced Grant No. 669442 \In- sulatronics" and the Research Council of Norway via the Centre of Excellence \QuSpin". [1] P. Bruno, Phys. Rev. B 52, 411 (1995). [2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [3] T. Tokuyasu, J. A. Sauls, and D. Rainer, Phys. Rev. B 38, 8823 (1988). [4] S. M. Rezende, R. L. Rodr guez-Su arez, and A. Azevedo, Phys. Rev. B 88, 014404 (2013). [5] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanash, S. Maekawa, and E. Saitoh, Nature 464, 7286 (2010). [6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [7] J. E. Tkaczyk, Ph.D. thesis, MIT (1988). [8] G. M. Roesler, M. E. Filipkowski, P. R. Broussard, Y. U. Idzerda, M. S. Osofsky, and R. J. Soulen, Proc. SPIE 2157 , 285 (1994). [9] X. Hao, J. S. Moodera, and R. Meservey, Phys. Rev. Lett. 67, 1342 (1991). [10] G.-X. Miao, J. Chang, B. A. Assaf, D. Heiman, and J. S. Moodera, Nature Communications 5, 3682 (2014). [11] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, J. App. Phys. 109, 103913 (2010). [12] M. B. Jung eisch, A. V. Chumak, A. Kehlberger, V. Lauer, D. H. Kim, M. Onbasli, C. A. Ross, M. Kl aui, and B. Hillebrands, Phys. Rev. B 91, 134407 (2015). [13] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 12 (2010). [14] C. O. Avci, A. Quindeau, C. Pai, M. Mann, L. Caretta, A. S. Tang, M. C. Onbasli, C. A. Ross, and G. S. D. Beach, Nat. Mater. 16, 4812 (2016). [15] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D.43, 264002 (2010). [16] T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 67, 024515 (2003). [17] N. Karchev, Phys. Rev. B 67, 054416 (2003). [18] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. Julian, P. Monthoux, G. G. Lonzarich, A. H. I. Sheikin, D. Braithwaite, and J. Flouquet, Nature (Lon- don) 406, 6796 (2000). [19] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J. Brison, E. Lhotel, and C. Paulsen, Nature (London) 413, 6856 (2001).[20] C. P eiderer, M. Uhlarz, S. M. Hayden, R. Vollmer, H. v. L ohneysen, N. R. Bernhoeft, and G. G. Lonzarich, Nature (London) 412, 6842 (2001). [21] D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980). [22] N. Karchev, EPL (Europhysics Letters) 110, 27004 (2015). [23] Y. Saito, T. Nojima, and Y. Iwasa, Nat. Rev. Mater. 2, 16094 (2016). [24] S. Gariglio, M. Gabay, J. Mannhart, and J.-M. Triscone, Physica C 514, 189 (2015). [25] N. Reyren, S. Thiel, A. D. Caviglia, L. F. Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.- S. R uetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J. Mannhart, Science 317, 1196 (2007). [26] Q. Y. Wang, Z. Li, W. H. Zhang, Z. C. Zhang, J. S. Zhang, W. Li, H. Ding, Y. B. Ou, P. Deng, K. Chang, J. Wen, C. L. Song, K. He, J. F. Jia, S. H. Ji, Y. Y. Wang, L. L. Wang, X. Chen, X. C. Ma, and Q. K. Xue, Chin. Phys. Lett. 29, 037402 (2012). [27] H. Boschker, C. Richter, E. Fillis-Tsirakis, C. W. Schnei- der, and J. Mannhart, Sci. Rep. 5, 12309 (2015). [28] S. N. Klimin, J. Tempere, J. T. Devreese, and D. van der Marel, Phys. Rev. B 89, 184514 (2014). [29] B. Li, Z. W. Xing, G. Q. Huang, and D. Y. Xing, J. Appl. Phys. 115, 193907 (2014). [30] X.-X. Gong, H.-X. Zhou, P.-C. Xu, D. Yue, K. Zhu, X.- F. Jin, H. Tian, G.-J. Zhao, and T.-Y. Chen, Chinese Physics Letters 32, 067402 (2015). [31] C. Stephanos, T. Kopp, J. Mannhart, and P. J. Hirschfeld, Phys. Rev. B 84, 100510(R) (2011). [32] D. Allender, J. Bray, and J. Bardeen, Phys. Rev. B 7, 1020 (1973). [33] V. Koerting, Q. Yuan, P. J. Hirschfeld, T. Kopp, and J. Mannhart, Phys. Rev. B 71, 104510 (2005). [34] X. Gong, M. Kargarian, A. Stern, D. Yue, H. Zhou, X. Jin, V. M. Galitski, V. M. Yakovenko, and J. Xia, Science Advances 3(2017), 10.1126/sciadv.1602579. [35] M. Kargarian, D. K. Emkin, and V. Galitski, Phys. Rev. Lett. 117, 076806 (2016). [36] For the integration, we use the Cauchy principle value and the fact that
x=jxx0j1, V(
x) =p 2 Z2 0d~'(1 + cos ~') cos ~' (1 + cos ~')2
x2 =1 +j
xjp
x2=2 +
x2p 21p
x2p 2: [37] To eliminate the singularity in V(xx0) atx0=xfor the numerical integration, we replace V(xx0) in Eq. (12) withRx 0d~xV(~xx0) andf(x) on the left-hand side of Eq. (12) with F(x) =Rx 0d~xf(~x). In each iteration, we numerically evaluate the integral over x0in the resulting equation and obtain f(x) by numerically dierentiating F. [38] In the same way as for the iteration of Eq. (12), we eliminate singularities from the integral over x0in Eq. (11). Hence, we replace the factor U('0;';x0;x) = 1=[2('0;')(xx0)2] in the inte- grand withRx 0d~XR~X 0d~xU('0;';x0;~x) and(x;') with D(x;') =Rx 0d~XR~X 0d~x(~x;'). In each iteration, we nd Dby numerically integrating over x0and then nd by numerically dierentiating Dtwice. [39] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi,6 C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, and A. Conca, J. Phys. D. 48, 015001 (2015). [40] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson, 1976). [41] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Phys. Rev. Lett. 108, 246601 (2012). [42] B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y. Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). [43] C. Burrowes, B. Heinrich, B. Kardasz, E. . Montoya,E. Girt, Y. Sun, Y. Song, and M. Wu, App. Phys. Lett. 100, 092403 (2012). [44] M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr ags, H. Huebl, S. T. B. Goennenwein, and G. Woltersdorf, Phys. Rev. B 92, 054437 (2015). [45] A. Mauger and C. Godart, Physics Reports 141, 51 (1986). [46] Note that the strength of the exchange coupling is dened dierently in Refs. [7, 8]; the values reported there must be divided by 2 to obtain the value of JIas it is dened in this Letter.

2017-07-12

We consider a thin normal metal sandwiched between two ferromagnetic insulators. At the interfaces, the exchange coupling causes electrons within the metal to interact with magnons in the insulators. This electron-magnon interaction induces electron-electron interactions, which, in turn, can result in p-wave superconductivity. In the weak-coupling limit, we solve the gap equation numerically and estimate the critical temperature. In YIG-Au-YIG trilayers, superconductivity sets in at temperatures somewhere in the interval between 1 and 10 K. EuO-Au-EuO trilayers require a lower temperature, in the range from 0.01 to 1 K.

Superconductivity Induced by Interfacial Coupling to Magnons

1707.03754v1

1. Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom. 2. ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Campus , Didcot OX11 0QX, United Kingdom 3. Laboratory for Neutron Scattering and Imaging, Paul Scherrer Institut, CH -5232 Villigen, Switzerland. 4 INNOVENT e.V., Technologieentwicklung , Pruessingstrasse. 27B , D- 07745 Jena , GERMANY The Final Chapter I n The Saga O f YIG A. J. Princep1*, R. A. Ewings2, S. Ward3, S. T óth3, C. Dubs4, D. Prabhakaran1, A. T. Boothroyd1 The magnetic insulator Y ttrium Iron Garnet can be grown with exceptional quality , has a ferrimagnetic transition temperature of nearly 600 K, and is used in microwave and spintronic devices that can operate at room temperature1. The most accurate prior measurements of the magnon spectrum date back nearly 40 years, but cover only 3 of the lowest energy m odes out of 20 distinct magnon branches2. Here we have used time -of-flight inelastic neutron s cattering to measure the full magnon spectrum throughout the Brillouin zone . We find that the existing model of the excitation spectrum , well known from an earlier work titled “The Saga of YIG” 3, fails to describe the optical magnon modes . Using a very general spin Hamiltonian , we show that the magnetic interactions are both longer -ranged and more complex than was previously understood . The result s provide the basis for accurate microscopic models of the finite temperature magnetic properties of Yttrium Iron Garnet, necessary for next - generation electronic devices. Yttrium Ir on G arnet (YIG) is the ‘miracle material’ of microwave magnetics. Since its synthesi s by Geller and Gilleo in 19574, it is widely acknowledged to have contributed more to the understanding of electronic spin -wave and magnon dynamics than any other substance3. YIG (chemical formula Y3Fe5O12, crystal structure depicted in Fig. 1 ) is a ferrimagnetic insula ting oxide with the lowest magnon damping of any known material. Its exceptionally narrow magnetic resonance linewidth — orders of magnitude lower than the best polycrystalline metals — allows magnon propagation to be obse rved over centimetre distances. This makes YIG both a superior model system for the experimental study of fundamental aspects of microwave magnetic dynami cs5 (and indeed, general wave and quasi - particle dynamics6,7), and an ideal platform for the development of microwave magnetic technologies , which have already resulted in the creation of the magnon transistor and the first magnon logic gate5,8. The unique p roperties of YIG have underpinned the recent emergence of new fields of research, including magnonics : the study of magnon dynamics in magnetic thin-films and nanostructures5, and magnon spintronics , concerning structures and devices that involve the interconversion between electronic spin currents and magnon currents1. Such systems exploit the established toolbox of electron -based spintronics as well as the ability of magnons to be decoupled from th eir environment and efficiently manipulated both magnetically and electrically5,9,10. Significant interest has also developed in quantum aspects of magnon dynamics, using YIG as the basis for new solid -state quantum measurement and info rmation processing technologies including cavity-based QED, optomagnonics, and optomechanics11. It has also recently been realised that one can stimulate strong coupling between the magnon modes of YIG and a superconducting qubit, potentially as a tool for q uantum informatio n technologies12. Spin Figure 1. Crystal structure and magnetic exchange paths in YIG. Left: First octant of the unit cell of YIG, indicating the two different Fe3+ sites , with the tetrahedral sites in green and the octahedral sites in blue . Exchange pathways used in the Heisenberg effective Hamiltonian are labelled. Right: Unit cell of YIG, with the majority tetrahedral sites in green and the minority octahedral sites in blue . Black spheres are yttrium, red spheres are oxygen. Figure 2. Neutron scattering intensity maps of the magnetic excitation spectrum of YIG. a-c) Measured magnon spectrum along (H,H,4), (H,H,3), and (H,H,H) directions in reciprocal space , recorded in absolute units of mb sr-1 meV-1 f.u-1. d-f) resolution convoluted best fit to the model presented in the text , currently the basis for theoretical models of YIG. No scaling factors were used in the model. caloritronics has also recently emerged as a potential application of YIG, utilising the spin Seebeck effect (SSE) and the spin Peltier effect (SPE) to interconvert between magnon and thermal currents, either for efficient large -scale energy harvesting, or the generation of spin cu rrents using thermal gradients13. If the research into classical and quantum aspects of spin wave propagation in YIG is to achieve its potential, it is absolutely clear that the community requires the deep understanding of its mode structure , which only neutron scattering measurements can offer. In many theories and experiments , YIG is treated as a ferromagnet with a single, parabolic spin wave mode14,15, simply because the influence of YIG ’s complex electronic and magnetic structure on spin transport is not known in sufficient detail . Such approaches must break down at high temperature when the optical modes are appreciably populated and a detailed knowl edge of the structure of the optic al modes is a necessary first step in any realistic model of the magnetic properties of YIG in this operational regime. Despite this, surprisingly little data exists relating to the detail of its magnon mode structure. The key previous work in this area is due to Plant et. al. 2, and dates back to the 70 s. Using a triple -axis spectrometer , these early measurements were able to record 3 of the spin wave modes up to approximately 55 meV, but crucially there are 20 such modes and they are predicted to extend up to approximately 90meV (22 THz) 3. Data were collected (see methods section) as a large, 4 -dimensional hypervolume in frequency and momentum space, covering the complete magnon dispersion over a large number of Brillouin zones. Fig. 2 shows two -dimensional energy -momentum slices from this hyper volume with the wave -vector along three high -symmetry directions , normalised to a measurement on vanadium (see methods section) . A large number of modes can be seen up to an energy of 80meV , whilst data in other slices show modes extending up to nearly 100 meV. The spectrum is dominated by a strongly dispersing and well-isolated acoust ic mode at low energies (the so -called ‘ferromagnetic ’ mode), and a strongly dispersing optical mode separated from this by a gap of approximately 30meV at the zone centre. Intersecting this upper mode is a large number of more weakly dispersing optic al modes in the region of 30 -50meV . We model the data using a Heisenberg effective spin Hamiltonian, appropriate to YIG as it is both a good insulator and the Fe3+ ions (S = 5/2, g = 2) possess a negligible magnetic anisotropy due to the quenched orbital moment. 𝐻=∑𝐒𝑖𝑇𝐽𝑖𝑗𝐒𝑗 𝑖,𝑗+∑𝐒𝑖𝑇𝐴𝑖𝐒𝑖 𝑖 We nevertheless include a magnetic anisotropy Ai in our analysis to take into account crystal field effects, but find this term to be vanishingly small, consistent with previous results. The exchange matrix Jij is a general 3x3 matrix, whose elements are restricted by the symmetry of the bond connecting the spins Si and Sj. Following common practice, Jij is restricted to having only identical diagonal components (i.e. isotropic exchange) since anisotropic and off -diagonal contributions are likely to be small due to the lack of significant orbital angular momentum . The spin Hamiltonian was diagonalized using the SpinW software package16 and th e calculated magnon dispersion was fit ted by a constrained nonlinear least squares method to 1D cuts taken through the 2D intensity slices . We do not include any scaling factors for the magnon intensity, so the agreement between the model and the data in terms of absolute units is indicative of the quality of the model. Our final /best -fit model includes isotropic exchange 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 for 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 octahedral (a) sites . Due to the extremely large number of magnetic atoms (20) within the primitive cell, and the consideration of so many exchange pathways, this analysis would be impossible without the use of sophisticated software such as SpinW as the construction of an analytic model would be prohibitively time consuming. During the fitting process, f eatures in the spectrum were weighted so that weak but meaningful features in the data were considered as significant as strong features. The Spin W model output is then convoluted with the calculated experimental resolution of the MAPS spectrometer, including all features of the neutron flight path and associated focussing/defocussing effects, as well as the detector coverage and effects from symme trisation (see Supplementary Information for details). The final fitted values of the exchanges are listed in Table 1. An important difference between our results and those of previous authors2,3 is that there are two symmetry -distinct 3rd-nearest -neighb our bonds (the so -called J aa in the literature , which we label J 3a and J 3b) which have identical length but differ in symmetry . The J 3b exchange lies precisely along the body -diagonal of the crystal , and thus has severely limited symmetry -allowed components owing to the high symmetry of the bond (point group D3). The J 3a exchange connects the same atoms with the same radial separation, but represents a different Fe -O-Fe exchange pathway as a result of the different point group symmetry (point group C2), so it is distinguished from J 3b by the environment around the Fe atoms. Models including anisotropic exchange or Dzyaloshinskii -Moriya interactions on the 1st -4th neighbour bonds were tested, but such interactions were found to destabilise the magnetic structure for arbitrarily small perturbations . We also find that J2 (Jdd) is much smaller than previously supposed — the main effect of this exchange is to increase the bandwidth and split the optic modes clustered around 40 meV in a way contradicted by the data. It has been pointed out17 that the magnetic structure of YIG is incompatible with the cubic crystal symmetry, although to date no measurements have found any evidence for departures from the ideal cubic structure. Nevertheless, it is necessary to refine the magnetic structure in a trigonal space group ( a symmetry that is experimentally observed in terbium rare earth garnets where the magnetoelastic coupling is much stronger18), in order to obtain a satisfactory goodness of fit, and a magnetic moment which agrees with bulk magnetome try19. Treating the unit cell of YIG in this fashion for the purposes of the SpinW simulation would be feasible, but would introduce a large number of free parameters which (given the very small size of the departure from cubic symmetry) would nevertheless be expected to change very little from a cubic model. This expectation is borne out by the excellent agreement between the data and the cubic spinwave model (see Fig. 2 and the supplementa ry materials for more details) . Features absent from the data which are relevant to technological applications (such as the conversion of microwave photons into magnons) include any strong indications of magnon -phonon or magnon -magnon coupling. The data are well described by a linear spinwave model, although the size o f the 5th-neighbour exchange is perhaps indicative of some small deviations not easily captured without such couplings . A strong magnon -phonon coupling would be expected to cause both broadening and anomalies in the dispersion of the magnon modes20. We do not observe any such effects, although our measurements would not be sensitive to any magnon -phonon coupling that shifts or broadens th e spin wave signal by less than 3meV (the instrumental resolution) . Our results require a substantial revision of the impact of the optical modes on the room -temperature magnetic properties . The differences compared with the existing model are illustrated in Fig. 3, in which we plot the antisymmetric combination of transverse scattering function s Sxy(Q,ω) − Syx(Q,ω), which is proportional to the sign and magnitude of the measured spin -Seebeck effect arising from the associated magnon mode21. Most strikingly the absence of spectral weight in the flat mode at ~35meV, as well as a compression and shift of the ‘positively’ polarised (red) optical modes i.e. those modes which would precess counterclockwise with respect to an applied field . As has recen tly been shown, the thermal population, broadening , and softening of these modes at elevated temperatures substantially modifies the magnitude of the measured spin-Seebeck effect, which places limitations on device performance and determines the optimum operating temperature . Our results show that the distribution of optic al modes is very different from what had previously been assumed , which has consequences for the temper ature -Table 1 . Fitted exchange parameters for YIG and their statistical uncertainties. The exchange constants are defined in Fig. 1 Exchange This work (meV) Ref. 3 (meV) J1 6.8(2) 6.87 J2 0.52(4) 2.3 J3a 0.0(1) 0.65 J3b 1.1(3) 0.65 J4 -0.07(2) - J5 0.47(8) - J6 -0.09(5) - dependent broad ening21. Our new measurements can therefore be used as the basis for a precise microscopic model of the temperature dependent dynamical magnetic properties in YIG. We have also estimated the parabolicity of the lowest lying ‘ferromagnetic’ magnon mode i.e. the point where the error of a quadratic fit becomes greater than 5 %. We find this region to extend 14.8% of the way towards the Brillouin zone boundary along the H direction ( the (0,0,1) direction in t he centred unit cell) , represe nting about 0.3% of the entire B rillouin zone . The departure from a purely parabolic acoustic magnon dispersion, as well as the population of optic magnon modes directly generates the temperature dependence of the spin Seebec k effect and our model can be used to fully understand such effects even at elevated temperatures through extension to a multi - magnon picture following the procedure in Refs. 21, and 22. We have presented the most detailed and comp lete measurement of the magnon dispersion of YIG in a pristine, high quality crystal. Using linear spin -wave theory analysis we are able to reproduce the entire magnon spectrum across a large number of brillouin zones including a reproduction of the absolute intensities of the mo des. We confirm the importance of the near est-neighbour exchange, but are forced to radically reinterpret the nature and hierarchy of longer -ranged interactions. Our work has uncovered substantial discrepancies between previous models and the measured disp ersion of the optical magnon modes in the 30 -50 meV region, as well as the total magnon bandwi dth, and the detailed nature of the magnetic exchanges . Through a detailed consideration of the symm etries of the exchange pathways and long -ranged interactions , we are able to fully reproduce the entire measured spectrum . Technological applications of YIG, particularly those utilising the spin Seebeck effect, are very sensitive to the optical magnons in the region of 30 -40meV. This work overturns 40 years of estab lished work on magnons in YIG and will be an essential tool for accurate modelling of the optical magnon modes in the room temperature regime. Acknowledgements This work was supported by the Engineering and Physical Sciences Research Council of the Unite d Kingdom (grant nos. EP/J017124 /1 and EP/M020517/1 ). We wish to acknowledge useful discussions with A. Karenowska , B. Hillebrandts , and T Hesje dal. We are grateful to S. Capelli (ISIS Facility) for use of the SXD instrument to characterise the crystal s used in the experiments . Furthermore , C.D. would like to acknowledge the technical assistance of R. Meyer in the crystal growth of YIG . Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beamtime allocation from the United Kingdom Science and Technology Facilities Council. Author Contributions DP produced the preliminary optical floating zone crystal, and CD grew the flux crystal used in the final experiment. AJP and RAE performed the neutron experiments, with AJP responsible for the data reduction and RAE performed the instrumental resolution convolution . SW and AJP performed the data analysis and spinwave simulation, with advice and input from ST. AJP prepared the manuscript with input from all the co -authors. ATB supervised the project and assisted in the planning and editing of the manuscript. References 1 A.V. Chumak, V. I. Vasyuchka, A. A. Serga & B. Hillebrands . Magnon Spintronics. Nature Physics 11, 453–461 ( 2015 ); [2] J.S. Plant., Spinwave dispersion curves for yttrium iron garnet . J. Phys. C . 10, 4805 -4814 (1977) [3] V. Cherepanov , I. Kolokolov & V. L’Vov . The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet . Phys. Reps. 229, 81 - 144 (1993) [4] S. Geller & M. A. Gilleo . Structure and ferrimagnetism of yttrium and rare -earth -iron garnets . Acta Cryst. 10 (1957) 239 Figure 3: Simulations of the magnon dispersion in YIG. (Right) the previous model3,21 and ( left) our new model of the magnon dispersion, w here the colour and intensity correspond to the sign and magnitude of the correlator Sxy(Q,ω) − Syx(Q,ω), which is responsible for the spin Seebeck effect21. The horizontal line indicates k BT at room temperature. [5] A. A. Serga, A.V. Chumak, & B Hillebrands . YIG Magnonics. J. Phys. D: Appl. Phys . 43 264002 (2010) [6] A.V. Chumak , , A. A. Serga & B.Hillebrands . Magnon transistor for all -magnon data processing . Nat. Commun . 5 4700 (2014) [7] A. A. Serga et. al. Bose –Einstein condensation in an ultra -hot gas of pumped magnons. Nat. Commun . 5 3452 (2014) [8] Y. Kajiwara et. al. Transmission of electrical signals by spin - wave interconversion in a magnetic insulator . Nature, 464 262-266 (2010) [9] M. B. Jungfleisch et. al . Temporal evolution of inverse spin Hall effect voltage in a magnetic insulator -nonmagnetic metal structure . Appl. Phys. Lett ., 99 182512 (2011) [10] J. Flipse et. al. Observation of the Spin Peltier Effect for Magnetic Insulators . Phys. Rev. Lett. 113 027601 (2014) [11] D. Zhang et. al. Cavity quantum electrodynamics with ferromagnetic magnons in a small yttrium -iron-garnet sphere . NPJ Quant. Inf . 1, 15014 (2015) [12] Y. Tabuchi, et. al. Coherent coupling between a ferromagnetic magnon and a superconducting qubit . Science 349 405-408 (2015) [13] A. Hoffmann & S. D. Bader . Opportunities at the Frontiers of Spintronics . Phys. Rev, Appl . 5 047001 (2015) [14] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, & S. Maekawa , Theory of magnon -driven spin Seebeck effect . Phys. Rev. B 81, 214418 (2010) [15] U. Ritzmann, D. Hinzke, & U. Nowak . Propagation of thermally induced magnonic spin currents . Phys. Rev. B 89, 024409 (2014) [16] S. Toth & B. Lake . Linear spin wave theory for single -Q incommensurate magnetic structures . J. Phys.: Condens. Matt . 27, 166002 (2015). [17] D. Rodic, M. Mitric, R. Tellgren, H. Rundlof, A. Kremenovic . True magnetic structure of the ferrimagnetic garnet Y 3Fe5O12 and magnetic moments of iron ions . J. Magn. Magn. Mat. 191 137-145 (1999) [18] R. Hock, H. Fuess, T. Vogt, M. Bonnet. Crystallographic distortion and magnetic structure of terbium iron garnet at low temperatures . J. Solid State Chem. 84 39-51 (1990) . 19 G. Winkler. Magnetic Garnets , 5th ed. (Vieweg, Braunschweig/ Wiesbaden, 1981). [20] P. Dai et al. Magnon damping by magnon -phonon coupling in manganese perovskites . Phys Rev B 61 9553 (2000) [21] J. Barker & G. E. W. Bauer Thermal Spin Dynamics of Yttrium Iron Garnet . Phys. Rev. Lett . 117 217201 (2016) [22] H. Jin, S. R. Boona, Z. Yang, R. C. Myers, & J.P. Heremans . Effect of the magnon dispersion on the longitudinal spin Seebeck effect in yttrium iron garnets . Phys. Rev. B 92, 054436 (2015) Methods Crystal Growth. YIG crystal growth was carried out in high -temperature solutions a pplying the slow cooling method23. Starting compounds of yttrium oxide (99.999%) and iron oxide (99.8%) as solute and a boron oxide - lead oxide solvent were placed in a platinum crucible and melted in a tubular furnace to obtain a high -temperature solution24. Using an appropriate temperature gradient only a few single crystals nucleate spontaneously at the cooler crucible bottom and forced convect ion, obtained by accelerated crucible rotation technique (ACRT), allows a stable growth which results in nearly defect -free large YIG crystals25. The YIG crystal used in this study exhibits a size of 25 mm x 20 mm x 11 mm and a weight of 12 g. It was confirmed by neutron and X -ray diffraction that the YIG crystal was a single grain with a crystalline mosaic of approximately 0.07 degree s FWHM. Preliminary measurements were made on a crystal that was grown by the optical floating -zone method, starting fr om a pure powder of YIG. Neutron Scattering Data Collection and Reduction. Data were collected on the MAPS time -of-flight neutron spectrometer at the ISIS spallation neutron source at the STFC Rutherford Appleton Laboratory, UK. On direct geome try spectro meters such as MAPS , monochromatic pulses of neutrons are selected using a Fermi chopper with a suitably chosen phase. In our experiment neutrons with an incident energy (E i) of 120 meV were used with the chopper spun at 350 Hz, giving energy resolution of 5.4 meV at the elastic line, 3.8 meV at an energy transfer of 50 meV, and 3.1 meV at an energy transfer of 90 meV. The spectra were normalized to the incoherent scattering from a stand ard vanadium sample measured with the same incident energy, enabling us to present the data in absolute units of mb sr-1 meV-1 f.u. -1 (where f.u. refers to one formula unit of Y 3Fe5O12). Neutrons are scattered by the sample on to a large area detector on which their time of flight, and hence final energy , and position are recorded. The two spherical polar angles of each detector element, time of flight, and sample orientation allow the scattering function S( Q, ω) to be mapped in a four dimensional space (Q x,Qy,Qz,E). In our experimen t the sample was oriented with the (HHL) -plane horizontal, while the angle of the (00L) direction with respect to the incident beam direction was varied over a 120 degree range in 0.25 degree steps . This resulted in coverage of a large number of Brillouin zones , which was essential in order to disentangle the 20 different magnon modes. D ue to the complex structure factor resulting from the number of Fe atoms in the unit cell, the mode intensity varies considerably throughout recip rocal space . The large datasets recording the 4 D space of S(Q,ω), ~100 GB in this case, were redu ced using the Mantid framework26, and both visualized and analyzed using the Horace software package27. Taking advantage of the cubic symmetry of YIG, t he data were folded into a single octant of reciprocal space (H>0, K>0, L>0) , adding together data points that are equivalent in order to produce a better signal -to- noise. 2D slices were taken from the 6 reciprocal space directions depicted in Fi gure 2 and in Sup plementary Information . 23 P. Görnert & F. Voigt, in: Current Topics in Materials Science, Vol. 11, Ed. E. Kaldis (North -Holland, Amsterdam, 1984) ch. 1 24] S. Bornmann, E. Glauche , P. Görnert , R. Hergt , & C. Becker , Preparation and Properties of YIG Single Crystals . Krist. Tech . 9, 895 (1974) [25 C. Wende & P. Görnert . Study of ACRT Influence on Crystal Growth in High -Temperature BY Solutions by the “High -Reso lution Induced Striation Method “. Phys. Stat. S ol. (a) 41, 263 (1977) [26] J. Taylor et al. Mantid, A high performance framework for reduction and analysis of neutron scattering data . Bulletin of the American Physical Society 57 (2012) [27] R. A. Ewings et. al. HORACE: software for the analysis of data from single crystal spectroscopy experiments at time -of-flight neutron instruments . Nucl. Inst rum. Meth . A 834 132 (2016) Supplementary Information For: The Final Chapter In The Saga Of YIG A. J. Princep1*, R. A. Ewings2, S. Ward3, S. Tóth3, C. Dubs4, D. Prabhakaran1, A. T. Boothroyd1 1. Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, United Kingdom. 2. ISIS Facility, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot OX11 0QX, United Kingdom 3. Laboratory for Neutron Scattering and Imaging, P aul Scherrer Institut, CH -5232 Villigen, Switzerland. 4 INNOVENT e.V., Technologieentwicklung, Pruessingstrasse. 27B, D -07745 Jena, GERMANY Contents: 1) Additional information on the fitting procedure 2) Symmetry of the 3rd neighbour Exchange 3) Additional data slices simulated, and detailed comparison with previous model. 4) Instrumental broadening and simulation 5) Supplementary References 1. Additional information on the fitting procedure To extract the exchange parameters fitting was performed by a bounded non -linear least squares fit to the following cuts of the experimental data set: Q-BASIS Q RANGE (R.L.U) ENERGY RANGE (MEV) H H 4 0.0 0.05 3.5 40 1.5 80 2.5 0.05 5.0 08 1.5 90 H H H 0.0 0.05 3.5 20 1.5 60 3.0 0.05 5.0 08 1.5 90 H H 3 0.0 0.05 5.0 09 1.5 90 2 2 L 0.0 0.05 1.0 40 1.5 50 3.0 0.05 5.0 20 1.5 90 3 3 L 0.0 0.05 5.0 09 1.5 90 Initial fitting was performed on exchanges J 2, J3a, J3b, J4, J5 and J 6 where the results were subsequently used as starting points for when the exchanges J 1, and the anisotropy parameter D were allowed to vary. Due to the exceptional quality of the data, the background signal was approximated as a constant value, which was un ique to each cut. As well as this, a common intensity factor and convolution width was used for all cuts. The convolution width was not fitted 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 energy binned between 9.0 and 90 meV in steps of 1.5 meV. The ferromagnetic features around (4, 4, 4) were found to be dominant over the intricate higher energy features in the basic fitting approach. Masking this feature led to unsatisfactory param eter convergence, so to overcome this a weighting factor was introduced, which allowed a parameter convergence describing both low and high energy features. During the fitting procedure parameters were allowed to vary between their boundary conditions: Exchange Value Boundary condition J1 6.7600 ( 0.2025) [3 10] J2 0.5207 (0.0370) [0 1] J3a 0.0001 (0.1362) [-2 2] J3b 1.0539 ( 0.3216) [-2 2] J4 -0.0686 (0.0152 ) [-1 1] J5 0.4736 (0.0840) [0 1] J6 -0.0930 (0.0499) [-1 1] D 0.01000 (0) [-1 1] Starting parameters were randomly selected within the limits and were free to vary within the boundary conditions. It was found that only those around the presumed starting parameters gave a meaningful convergence. 2. Symmetry of the 3rd neighbour Exchange The difference between J 3a and J 3b can be understood easily from a projection of the crystal structure along the axis of the bond, depicted in figure S1. Figure S1. The left image depicts the projection of the crystal stru cture along the J3a bond, from which the 2 -fold symmetry can be seen. The right image depicts the projection of the crystal structure along the J3b bond, from which the symmetry can be seen to obey the higher symmetry D3 point group. These images were gene rated using VESTAS1. 3. Additional data slices simulated, and detailed comparison with previous model. Further 2D slices to the data that were used for fitting are found below in figure S2. Figure S3 depicts a detailed comparison of the optical magnon modes in the region 30 -50meV, contrasting the model presented in this work with that of previous authors3. Figure S2 . Supplementary neutron scattering intensity maps along (H,H,4), (H,H,3), and (H,H,H) directions in recipr ocal space. Left: Experimental data. Right: Model. Figure S3. Zoomed figures emphasising comparison between the model in this paper (middle column) and the model owing to previous work3 (right column). The left column s hows the corresponding data in the same region of reciprocal space. 4. Instrumental broadening and simulation The broadening of the signal due to instrumental resolution arises due to several effects. The main contributions are: Deviations in neutron arriva l time (at both the chopper and detectors) due to the finite depth of the source moderator and the angle of the moderator face with respect to the beamline collimation, and hence deviations in the departure time of the neutron burst compared to the notiona l zero time. Deviations in neutron arrival time due to the finite sweep time, length and slit width of the monochromating Fermi chopper. The divergence of the scattered beam due to the finite size of the sample subtended at each detector element. The divergence of the incident neutron beam due to the finite size of the beam and the geometry of the beamline collimation These effects are computed in the reference frame of the spectrometer and then converted into the reference frame of the sample and convoluted. The convolution is performed by Monte -Carlo sampling over the resolution width of each of the terms described above. In general each Q-energy bin in the images shown in Fig. 2 of the main article (as well as supplementary figures S2 and S3) con tains data from many detector elements taken with many different sample orientations. The resolution for each element was accounted for in the corresponding simulations shown in the same figures. 4. Supplementary references [S1] K. Momma & F. Izumi. VESTA: a three -dimensional visualization system for electronic and structural analysis . J. Appl. Crystallogr. , 41, 653 -658 (2008)

2017-05-18

The magnetic insulator Yttrium Iron Garnet can be grown with exceptional quality, has a ferrimagnetic transition temperature of nearly 600 K, and is used in microwave and spintronic devices that can operate at room temperature. The most accurate prior measurements of the magnon spectrum date back nearly 40 years, but cover only 3 of the lowest energy modes out of 20 distinct magnon branches. Here we have used time-of-flight inelastic neutron scattering to measure the full magnon spectrum throughout the Brillouin zone. We find that the existing model of the excitation spectrum, well known from an earlier work titled "The Saga of YIG", fails to describe the optical magnon modes. Using a very general spin Hamiltonian, we show that the magnetic interactions are both longer-ranged and more complex than was previously understood. The results provide the basis for accurate microscopic models of the finite temperature magnetic properties of Yttrium Iron Garnet, necessary for next-generation electronic devices.

The Final Chapter In The Saga Of YIG

1705.06594v1

1 Scaling of the spin Seebeck effect in bulk and thin film By K. Morrison,1* A.J Caruana,1,2 C. Cox1 & T.A. Rose1 [*] Dr K. Morrison 1Department of Physics, Loughborough University, Loughborough LE11 3TU (United Kingdom) 2ISIS Neutron and Muon Source, Didcot, Oxfordshire, OX11 0QX E-mail: k.morrison@lboro.ac.uk Keywords: Magnetic materials, thermoelectrics, thin films PACS: 72.25.Mk, 72.25.-b, 75.76.+j Whilst there have been several reports of the spin Seebeck effect to date, comparison of the absolute voltage(s) measured, in particular for thin films, is li mited. In this letter we de monstrate normalization of the spin Seebeck effect for Fe 3O4:Pt thin film and YIG:Pt bulk samp les with respect to the heat flux Jq, and temperature difference ΔT. We demonstrate that the standard normalization procedures for these measurements do not account for an une xpected scaling of the measured volta ge with area that is observed in both bulk and thin film. Finally, we present an alterna tive spin Seebeck coefficien t for substrate and sample geometry independent characterization of the spin Seebeck effect. I. INTRODUCTION The spin Seebeck effect (SSE) is defined as the production of a spin polarized current in a magnetic material subject to a thermal gradient. It was first highlighted by Uchida et al. [1], and led to the development 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 typically achieved by placing a heavy metal such as Pt in contact 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 current for spintronics applications, paving the way for a host 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 conventional thermoelectrics for harvesting of waste heat, due to decoupling of the electric and thermal conductivities 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 chosen to have low thermal conductivity independent of the 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 consider in this geometry is the impact of other magneto-thermal contributions to the measured voltage, V ISHE, when assessing the LSSE. This includes the anomalous Nernst effect (ANE) and proximity induced ANE (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 PANE is the only potential artefact of the measurement. Whilst there have been extensive studies of the spin Seebeck effect over the last 5-10 years, this has predominantly been in the YI G:Pt system, where for thin films, the substrate is often GGG [14] [15]. Reports of the spin Seebeck coefficient fo r this material system can, however, 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]. Despite indications of the importance of heat flux in measurements [16], [17] [19], typically, LSSE measurements 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 problematic, especially for thin films, where the temperature 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 in metrology so that meaningful comparison of the material 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 temperature difference across the active (magnetic) layer, however, this is limited by the unknown thermal conductivity of the thin films (likely to differ from the 2 bulk), and thermal interface resistances. By measuring the voltage generated with respect to the heat flux, J Q, we can obtain a robust measurement of the temperature gradient across the active material ( ܶ ൌ ܬ ொ/ߢ .)In this case we are only limited by our knowledge of the thermal conductivity, κ. In this paper we present measurement of the spin Seebeck voltage with respect to the temperature difference, ΔT, and heat flux, JQ. We demonstrate that the JQ method is indeed more reliable than the ΔT method and can be used to compare thin films with varying substrate thicknesses and thermal conductivity. In addition we find the surprising result that the measured spin Seebeck voltage, V ISHE, is proportional not only to the contact separation, Ly, and temperature gradient, ܶሺൌ ܬ ொ/ߢሻ, but also the sample area, AT. This manifests as an increase in VISHE/Lyܶ in both thin film and 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 coefficient suitable for normalizing the measured voltage for generic sample geometries. II. EXPERIMENTAL METHOD The thin film samples tested here were part of a series of 80:5 nm thick Fe 3O4:Pt bilayers deposited using pulsed laser deposition (PLD) in ultra-high vacuum (base pressure 5 x 10 -9 mbar) onto 0.15 mm, 0.3 mm and 0.5 mm thick borosilicate glass and 0.6 mm or 0.9 mm fused silica substrates. The laser fluence and substrate temperature for Fe 3O4 and Pt deposition were 1.9 ± 0.1 J cm-2 and 3.7 ± 0.2 J cm-2, and 400 °C and 25 °C, respectively. Fe 3O4 was chosen due to its large spin polarization (~80% [20]), low thermal conductivity (at 300 K: κ thin film ~2-3.5 W m-1 K-1, [21] κbulk ~ 2-7 W m-1 K- 1 [22]), and relative abundance of the constituent elements. Of particular note are the following: X-ray reflectivity data indicated a typical Pt thickness across the series of 4.5 0.5 nm and a roughness σ = 1.5 ± 0.2 nm. Fe 3O4 thickness was varied from 20 to 320 nm. In addition, these films have been shown to demonstrate highly textured, columnar growth, with grain sizes of the order of 100 μm. Further details are given in [19] and the supplementary information. Bulk YIG was prepared by the solid state method [23]. Stoichiometric amounts of Y 2O3 and Fe2O3 starting powders (Sigma Aldrich 99.999% and 99.995% trace metals basis, respectively) were ground and mixed together before calcining in air at 1050 ᵒC for 24 hours. Approximately 0.5g of the calcined powder was then dry pressed into a 13 mm diameter, 1.8±0.2 mm thick cylindrical pellet. The pellet was then sintered at 1400 ᵒC for 12 hours, after which, it was checked by XRD 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 IsoMet low speed precision cutter. Average grain size for this polycrystalline sample (determined using scanning electron microscopy) was 14.5 μm, with an open 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 characterization was obtained using a Quantum Design Magnetic Property Measurement System (SQuID) as a function of temperature and field. Magnetometry of the Fe 3O4 films at room temperature typically exhibited a coercive field Hc = 197±5 Oe, saturation magnetization Ms = 90 emu g-1, and a remanent moment of Mr = 68 emu g-1. Magnetometry of the YIG pellet exhibited a coercive field Hc = 7 Oe, saturation magnetization Ms = 25 emu g-1, and a remanent moment of Mr = 3 emu g-1 Spin Seebeck measurements were obtained from a set-up similar to that of Sola et al. [16] and optimized for 12x12, and 40x40 mm samples. The thin film was sandwiched between 2 Peltier ce lls, where the top Peltier cell (1) acted as a heat source, and the bottom Peltier cell (2) monitored the heat Q, passing through the sample. Additional measurements we re obtained in set-ups optimized for 10x10, 20x20 and 40x40 mm samples, where 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 voltage V P, generated as a current was passed through a resistor fixed to the top side of the Peltier cell (where the sensitivity S p of the set-ups varied from 0.11 to 0.264 V W-1 at 300K). Two type E thermocouples mounted on the surface of the Peltier cells (such that they are in contact with the sample during the measurement) monitored the temperature difference across the sample, ΔT; a schematic of this set-up is given in Figure 1(c). Further information on calibration is given in the supplementary information. The voltage generated by the inverse spin Hall effect, V ISHE, was determined by taking the saturation values 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 ISHE is expected to increase linearly with ΔT and Q. Finally, ΔT and Q were plotted as a function of one another (see the example in Figure 1(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 the Fe 3O4:Pt thin films we demonstrated that these contributions were negligible in a previous work by measuring thin films with a Au spacer layer between the Fe 3O4 and the Pt. In this case, the V ISHE was similar to corresponding PM layer thicknesses without the spacer 3 layer[19]. In addition, a separate work by Ramos et al. , has shown that the ANE contribution was ~3% of the total signal in their epitaxially grown Fe 3O4 thin films and obtained an upper limit of contribution due to PANE of 7.5 nV/K.[12] This was an order of magnitude smaller than their observed VISHE, which is comparable to the measurements 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 the Pt layer is a potential contribution to V ISHE. In this case, 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 alternate geometries to separate the two contributions [13] or by introducing spacer layers between the YIG and the Pt. [24] III. THEORY It is useful at this point to note some key characteristics of 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 Q) [16]. Secondly, it is known to increase linearly with contact separation ( L y) [10]. Lastly, for thin films, there have been indications of a length scale of the order of the magnon free path length above which the voltage generated saturates [25] [26]. In response to these observations, some of the first attempts to quantify the SSE were to normalize the voltage measured, V ISHE, to the temperature difference ΔT, 1ISHEVST (1) where the units are ( μV K-1). More accurately, this could also be normalized to contact separation Ly, 2ISHE yVSLT (2) with units of ( μV K-1 m-1). Given that the spin Seebeck effect is often defined in terms of the thermal gradient across the magnetic material ܶ ,the spin Seebeck coefficient is more often defined as, ܵ ଷൌିாೄಹಶ ்ൌೄಹಶ ∆் (3) Figure 1. (a) Longitudinal spin Seebeck measurement geometries. Thermally generated spin current Js, is produced in the magnetic layer (dark grey) and converted to a measureable voltage ( EISHE) in the spin convertor layer (light grey) by the inverse spin Hall effect. (b) Typical longitudinal sp in Seebeck thin film device; the length scales Lx, Ly, and Lz denote device dimensions with respect to the thermal gradient (alo ng z axis). Individual thicknesses of each layer { d1, d2, d3} are also indicated. (c) Schematic of the experimental set-up used in this work. (d) Top view of a typical thin film device where active material may not cover the entire substrate. In this case, the thermal contact area is quantified by lengthscales LT x and LT y, active material width by Lx, and contact separation by Ly±σs, where σs denotes contact size. (e) Example of the linear relationship between heat flux, JQ, and temperature difference, ΔT, in these measurements. (f) Example spin Seebeck data, VISHE (symbols) from 80 nm Fe 3O4:Pt thin film plotted alongside corre sponding SQuID magnetometry (line). 4 where the units are ( μV K-1), and the thermal gradient ܶ can be described by th e temperature difference ΔT, divided by the thickness of the sample Lz. It was shown recently by Sola et al. , that there is the added complication of thermal resistance between the sample and the hot and cold baths (i.e. the interface across which ΔT is measured) [17]. Here they showed in measurements of the same sample in an experimental set-up at both INRIM and Bielefield that the measurement as a function of ΔT was unreliable – differing by a factor of 4.6 (gave S 3 = 0.231 µV K-1 and 0.0496 µV K-1, respectively), whereas by normalizing to heat flux, both set-ups obtained the same value to within 4% (S 3 = 0.685 and 0.662 µV K-1, respectively). In this work they measured the heat flux, JQ, using a calibrated Peltier cell and initially determined the normalized voltage generated per unit of heat flux, 4ISHE y TVS QLA (4) where Q is the heat passing through a sample with cross- sectional area surface AT, and the units are ( μV m W-1). Given that the thermal conductivity can be defined by, / TzQT A L (5) where JQ = Q/A T, they argued that S3 (equation 3) could be estimated by multiplying S4 (equation 4) by κ. This was based on using a simple linear model that assumed that the thermal conductivity of the substrate and magnetic 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 – the magnetic layer – in a thin film device [19]. This highlights a significant disadvantage of using the spin Seebeck coefficients outline d in equations (1)-(3) for thin film devices, where the measurement of ΔT is a poor indicator of the temperature gradient across the active material. To circumvent this problem, we argued that in the equilibrium condition a simple thermal model can be used to determine the heat flux through the entire sample, 3 12 123TATQd dd (6) where { d1,d2,d3} and {1,2,3} are the thicknesses and thermal conductivities of the top layer (1), FM layer (2) and substrate (3), respectively; and ΔT is the temperature difference across the entire device. It can be seen from this, that the temperature gradient is a function of the thermal conductivities of each layer and that the temperature 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 3 and glass Fe 3O4:Pt in [19]). Note that this treatment of the thermal profile is, an oversimplification as it does not take into account any temperature drops at interfaces. Given that J Q will be constant across the sample once thermal equilibrium has been reached, and that it will be proportional to the temperature gradient across the active magnetic layer, of thickness d 2, we can write the thermal gradient as, 2 22 2 2QQJJd TTdd (7) where for the magnetic layer Lz=d2 and we now have a direct relationship between the thermal gradient, ܶ ,and the heat flux JQ in the active material . Finally, whilst there has been limited discussion of the scaling of the spin Seeb eck effect with area, it was argued by Kirihara et al. , that the spin Seebeck effect may also scale with area [27] based on the change in internal resistance of the paramagnetic layer, ܴ ൌ ఘ ೣ௧ು (8) where ρ is the resistivity, tPt is the thickness, and Lx is the width of the Pt layer. This suggests that the maximum power extracted is, ܲ ௫∝మ ோబ∝ܮ௫ܮ௬ (9) Note that this observation would only indicate an increase in power output of the bilayer (not the voltage). IV. RESULTS AND DISCUSSION To test normalization of the spin Seebeck measurements for both heat flux and sample area, we prepared several Fe 3O4:Pt samples on glass substrates of varying thickness, where the Pt thickness was kept constant at 5 nm (0.5 nm) in order to minimize any variation in the observed voltage, VISHE, due to the inverse spin Hall effect [19]. VISHE was then measured for various thermal gradients as a function of applied magnetic field. We first took one of the Fe 3O4:Pt samples from our study (22x22x0.5 mm glass substrate, 80 nm Fe 3O4, 5 nm Pt) and measured it in various orientations, as summarized in Figure 2(a)&(e). 5 To increase the area of the measured sample, a buffer layer between the two Peltier cells was introduced (substrate of same thickness as the sample – on the assumption that control of J Q by the magnetic layer is largely negligible). This effectively increased the thermal contact area AT, by increasing { LT x, LT y} whilst {Lx, Ly, Lz} were fixed (see Figure 1(d)). Secondly, the 22x22 mm sample was cleaved in two and measured along the long and short sides, thus changing AT, the aspect ratio and Ly. Errors were determined from the combination of uncertainty in contact separation Ly, thermal contact area AT, and the noise floor of the VISHE voltage. Figure 2(a) shows the calculated spin Seebeck coefficients S 2 (left) and S4 (right axis) as the area, AT, was increased. Initially this data suggests that there appears to be an increase in the coefficient measured by the heat flux method ( S 4), but not by the temperature difference method ( S2). In other words, for the same sample, the aspect ratio and total area seem to have an impact on the determination of S 4. In addition, by plotting S4 against various combinations of LT x, LT y, and Figure 2 Summary of normalization measurements for thin film Fe 3O4:Pt and bulk polycrystalline YIG:Pt. (a)&(b) S2 (open symbols) 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. (c) S2 (open symbols) and S4 (closed symbols) as a function of A T0.5 for the YIG sample. (d) Simulated change in the measurement 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 the thin film and bulk sample, respectively. Inset sketches indicate aspect ratio for the individual data points. For the thin film measurements a buffer layer (same thickness glas s substrate) was inserted in order to increase AT; as indicated by the inset sketch with dotted outline. Where x-axis errors are not visible they are smaller than the symbol used. 6 AT (as shown in Appendix A, Figure A1), we found that the most likely way to re duce the measurement to a constant value was to divide through by AT0.5. This suggests that the data requires the following normalization relation in order to produce a geometry independent coefficient, S 5, ܵହൌೄಹಶ ൬ೂ ඥಲ൰ ( 1 0 ) This is further demonstrated by the linear dependence observed for the thin film data in Figure 2(e). Note that for this series of measurements, as A T exceeded 484 mm2 the area of active material was no longer increasing (as AT was further increased by introducing a buffer layer with matched thermal conductivity). So whilst the heat flux across the active material was constant (by definition of S 4 in equation (4)) there was an apparent increase in the voltage per unit heat flux. It could be argued that this is simply due to heat losses, however, as will be shown later, this trend was still observed for samples where AT was varied and measured in a setup with matching Peltier surface area. Given that in the steady state, S 4κ=S2 (equations 2-7), the mismatch between the values of S2 and S4 determined for the thin film samples indicates that there is a measurement artefact that needs to be resolved. To test for this, we also measured bulk YIG samples as a function of A T, where the temperature difference across the active material would now be an order of magnitude larger (than our thin films) and thus, less prone to errors such as interfacial thermal resi stance. In addition, due to the insulating nature of YIG, any contribution to V ISHE due to the anomalous Nernst effect (ANE) will no longer be present. Figure 2(b), (c) & (f), shows the same normalization measurements for the bulk YIG:Pt sample as a function of A T. Here, the sample was measured in both the 12x12 mm and 40x40 mm sample holders, and cut from the original 13 mm pellet to various sizes. Notice that the data from the 12x12 mm and 40x40 mm measurement set-ups are the same within error, and that it still indicates possible scaling with A T0.5. For this dataset, plotting S4 as a function of LT x also indicated a linear trend (as seen in Appendix A, Figure A2), but this is for the case where A T=LxLy (as LxT=Lx and LyT = Ly), i.e. it does not account for non-standard geometries, where the contacts are not necessarily at the edges of the sample. In either case, th ere is still a pronounced increase in the measured voltage as the sample size increases that cannot be resolved by normalizing simply by contact separation and/or resistance. For the bulk samples, however, there is now an indication of scaling of the temperature difference method ( S 2) with AT. This can be seen in Figure 2(b) and (c) where S 2 is plotted alongside S4 as a function of AT and AT0.5. The difference between these measurements and that of the thin films is firstly that the measurement of ΔT across the (active) magnetic sample is now direct, and secondly, that the increased thickness of the sample (1.8 mm rather than 0.15 – 0.9 mm) limits the impact of temperature drops at the Peltier:sample interface due to thermal resistance. We argue that these are the reasons why the increase of S 2 with AT is now obvious for the bulk samples. In order to demonstrate this, in Figure 2(d), we present a simple model of the expected trend for S 2, as the sample area is increased. We first rewrite equation (2) to include the systematic error in measurement of ΔT, ܵ ଶ′ൌೄಹಶ ሺ∆்ೞା∆்ሻ ( 1 1 ) where ΔTs is the actual temperature drop across the sample and ΔTi is the temperature drop at the sample:thermocouple interface(s). (For the true measurement of S 2, the temperature offset, ΔTi, should be subtracted.) We then make the assumption that ΔTi will be approximately constant (for the same Q). We argue that if the individual measurements are well controlled (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 properties of the interface to which the thermocouples are 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 2, both here and in the literature. In this case, according to equation (5), we can define the heat, Q, passing through the sample and the interface as follows, ܳൌ ∆் ௗܣ ( 1 2 ) ܳൌ∆்ೞೞ ௗೞܣ௦ ( 1 3 ) where κi & κs are the thermal conductivities, Ai & As are the cross sectional areas, and di & ds are the thicknesses of the interface and the sample, respectively. If we combine equations (12) & (13), we can write ΔT i in terms of ΔTs as, ∆ܶൌ∆ ܶ௦ቀௗೞೞ ௗೞቁ ( 1 4 ) and finally the measured temperature difference, ΔT, as, ∆ܶ ൌ ∆ܶ ௦ ∆ܶൌ∆ ܶ௦ቀ1ௗೞೞ ௗೞቁ (15) From 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 iAi/di remains approximately constant) increases. Similarly, 7 for the thin film samples, as ds is of the order of 80 nm, the impact of ΔTi will be more pronounced. This was shown previously, where ΔTs was found to be 0.01% of the total measured ΔT in our thin film Fe 3O4. [19] Equation (15) was used to simulate S2’, i.e. how the temperature dependent spin Seebeck coefficient ( S2) is modified as a result of ΔTi. In this case we used the measured values of S4, and the known thermal conductivity and thickness of the sample (3 W/K/m and 1.8 mm) to estimate S 2 where ΔTi was negligible . We then multiplied this by ΔTs/ (ΔTs + ΔTi) to obtain S2’ for various values of κeff (=κiAi/di). Note that as AT is increased, S2’ appears to saturate, and that this occurs sooner for higher values of κeff (as seen in Figure 2(d)). To summarize, there is competition between an increase in S 2 as AT increases (which is observed with the heat flux method), and a decrease in S2 due to ΔTi. As stated earlier, the advantage of defining a spin Seebeck coefficient dependent on JQ rather than ΔT is that 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 matched to the sample area, we repeated the thin film measurement 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 resistor (R) mounted onto the top surface of the top Peltier 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 thickness variation on the measurement, we took 3 10x10 mm pieces from corners of the sample, to measure the scatter in the spin Seebeck coefficient. This can be seen in Figure 3, where for one of the 10x10 mm samples S 4 = 51.63 nV.m/K compared to 30.1 and 37.43 nV.m/K, and gives an upper limit of the expected variation in the film (see supplementary information for more information). The result of these measurements is given alongside a tabulated example of the key figures for a subset of the measurements (i.e. one heat flux for each area) in Figure 3 . Notice here, that the difference in S4 is greater than a factor of 3 between the smallest and largest sample(s). Even when scaling by the power supplied to the resistive heater (where we expect heat losses), this increase in S 4 is observed. In addition, we also performed measurements on a sample with fixed area, A T, but modifying Lx and Ly by scribing away areas of the sample (so that it can be considered as thermally connected but electrically isolated). In this case, V ISHE always scaled with Ly, as typically expected fo r a measurement where AT does not change. To summarize, we observed, for fixed thickness of the magnetic and Pt layers: 1. At constant A T and JQ: Linear scaling of VISHE with Ly, which is independent of sample geometry (as expected). 2. At constant AT and Ly: Linear scaling of VISHE with ΔT and JQ (as expected). 3. At constant AT and JQ: No observable change, when areas 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, I c). Figure 3 - Spin Seebeck measurement(s) of Fe 3O4:Pt 50x50 mm thin film which was cleaved to match measurement areas of 10x10, 20x20, and 40x40mm Peltier cells. The line is a guide for the eye and x-error was smaller than the symbols used. The table shows a set of values used to obtain this plot where Sp is the Peltier calibration coefficient, Iresistor , the current supplied to a resistor, R, mounted onto the top surf ace of the top Peltier cell. Qresistor is the resulting heat source due to the resistor, Vp is the measured voltage for the top Peltier cell, and Qtop the measured heat from the top Peltier (=Vp/Sp). 8 4. At constant JQ: An increase in the spin Seebeck coefficient, S4 (=VISHEAT/LyQ) when AT was increased. 5. At constant JQ and constant Ly: An increase in the spin Seebeck coefficient, S4 (=VISHEAT/LyQ) when AT was increased. The above observations indicate that there is an additional parameter that is being overlooked when evaluating the magnitude of voltage generation due to the inverse spin Hall effect, or thermal pumping of spin current, I s, into the Pt layer. At this stage it is not clear what this mechanism is, but we speculate that it could be due to the assumption that the spin current density, J S, injected in the Pt layer at the Ferromagnet:Paramagnet interface is directly proportional to JQ. Finally, this suggests that S5, as defined in Equation (10), is the most appropriate coefficient to use when comparing different bilayer systems, with arbitrary area, A T. Overall, for the thin film Fe 3O4 samples we can compare our results to that of Ramos et al. ,[12]. In this work they reported S2 = 150 μV/K/m for 50 nm Fe 3O4 deposited epitaxially onto a 0.5 mm thick, 8x4 mm SrTiO 3 substrate, with a 5 nm Pt layer. If we assume that the temperature gradient is controlled by the thermal conductivity of the substrate so that S4 = S2κ/d (kSrTiO3 =11.9 W/m/K for SrTiO 3) this would give S4 = 6.3 nV.m/W. For the 0.5 mm thick SiO 2 substrate sample in Figure 2 of this work (80 nm Fe 3O4, 5 nm Pt, 10x10 mm) applying a similar approach ( kSiO2~1 W/m/K, d = 0.5 mm) would give S4 ~ 12 nV.m/W, where we measure approximately 30 nV.m/W by the heat flux method. This data shows that for these Fe 3O4:Pt bilayers S5 = 3±0.2 µV W-1. The difference between the values obtained by the heat flux and temperature difference methods are not unexpected as Sola et al .[17] showed that the ΔT method can routinely underestimate the spin Seebeck coefficient. However, it does demonstrate the comparable quality of our thin films. For bulk YIG measurements we obtain S 5 = S4/AT0.5 = 4.8±0.34 μV/W and S2/AT0.5 = 8±0.24 mV/K/m2 (using κ = 3 W/m/K and d = 1.8 mm). These values are reasonable given the porosity of this YIG pellet. For comparison, measurements of bulk polycrystalline YIG by Saiga et al. , [28] found S 1 of up to 5 μV/K, when annealing to improve the interface equality. Given the dimensions L x = 5 mm, Ly = 2 mm, Lz = 1 mm, this amounts to S2 = 1000 μV/K/m and S2/AT0.5 = 316 mV/K/m2, or S5 = 45 μV/W (assuming κ = 7 W/m/K). If we compared to direct heat flux measurements by Sola et al. , with thin film YIG on GGG [17], they measured S 4 = 0.11 μV.m/W for a 5x2 mm sample. Hence, S5 = 34 μV/W. Whilst these values are an order of magnitude larger than our measurement, for these examples care was taken to maximize the in terface quality. In addition, the Pt thickness was smaller ( t Pt < 3 nm), which would be expected to increase VISHE further. To further demonstrate the impact of thermal resistance on the measurement of S2, we show in Figure 4, results of spin Seebeck measurements for our Fe3O4:Pt bilayers as a function of substrate and Fe 3O4 thickness ( tsubstrate and tFe3O4 respectively). Note that there was a scatter in our individual data points for S2 that can be attributed to varying thermal resistance between repeated measurements for the same sample (see supplementary information for individual measurements); this led to larger error bars. In Figure 4(a) we show a summary of all measurements as a function of substrate thickness. As can be seen, S 2 decreased linearly with increasing thickness, as a larger proportion of ΔT was shunted across the substrate. This is not unexpected. As soon as the data was normalized according 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 Pt and the resistivity). With regards to the thickness dependence of the spin Seebeck effect (Figure 4(b)) we also see collapse of Figure 4 Summary of measurements as a function of substrate and Fe 3O4 thickness, tsubstrate and tFe3O4 , respectively. (a) Average value for S2 (open symbols) and S5 (=VISHEAT0.5/LyQ) (closed symbols) as a function of the substrate thickness, tsubstrate . (d) S5 as a function of tFe3O4. Dashed line shows a fit according to linear response theory[25], with a magnon accumulation length of 17 nm. 9 the data onto the expected saturation at around 80 nm. Again, this is not unusual: saturation of the signal would be expected as the thickness of the Fe 3O4 layer increases, given that the volume magnetization also saturates at around 80 nm (see supplementary information, Fig. S3). The dashed line in this figure shows the corresponding fit to this data using the approach of Anadón et al. ,[25] who found a magnon accumulation length of 17 (± 3) nm at 300K from this fit. The slight decrease in the spin Seebeck coefficient for the th ickest sample could be due to a change in the morphology of the thicker film and is the focus of current work. V. CONCLUSION To summarize, we have demonstrated that the heat flux method is suitable for obtaining substrate independent measurements of the spin Seebeck effect, where the thermal conductivity of the thin film need not be known, and have proposed an alternative spin Seebeck coefficient ( S 5) for comparison of different material systems. By measuring the spin Seebeck effect as the thickness of the substrate, Fe 3O4 layer and sample dimensions { Lx, Ly, AT} were varied, we demonstrated that for the same material system (assuming tPt is constant at 5 nm) this method reliably returned a value of 3.0±0.2 µV W-1. Since this result holds for different values of tsubstrate , the heat flux normalization method could, therefore, be used to compare similar samples where the substrate thickness or type were varied. This would thus be a useful metric for comparison of prospective 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 proportional to the available energy per unit length ( Q), which we speculate could be a result of thermal spin injection into the paramagne tic layer before detection. The exact nature of this scaling will be the topic of future studies. ACKNOWLEDGEMENTS This 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 sample fabrication and M Greenaway and J Betouras for useful discussions. Supporting data will be made available via the Loughborough data repository under doi 10.17028/rd.lboro.5117578. APPENDIX A: ADDITIONAL PLOTS OF S 4 FOR THE THIN FILM AND BULK SAMPLES Figures A1 and A2 show additional plots of the data presented in Figure 2 (e) & (f), as AT was varied. This includes parameters such as the horizontal and vertical lengths, LxT, LyT, their ratio, and the area AT. The dashed lines are guides for the eye. APPENDIX B: HEAT LOSS CONSIDERATIONS In an ideal case, the sample area would be chosen such that it matches the area of the hot and cold bath(s) in the measurement. Given that this paper studies the impact of sample size (where A T is varied), it could be argued that for samples where AT is less than the area of the top and bottom Peltier cells, there are significant thermal losses (radiation and convection). This was monitored during measurement by plotting the change in the measured J Q as a function of ΔT (see Figure 1(e)). It was also confirmed by obtaining comparable measurements in air and under vacuum, as detailed in the supplementary information. We consider here the potential impact of radiative losses, as defined by: Q = εσA(T h4- Tc4) ( 1 6 ) where ε is the emissivity of the radiating surface ( ε = 1 for a perfect radiator), σ is the Stefan-Boltzman constant (σ = 5.67x10-8 W/m2/K4), A is the radiating surface area, and Th and Tc are the temperatures of the hot and cold (ambient) surfaces respectively. For the sample environment used in these measurements there will be 3 sources of radiative heat loss: 1) Top surface of the Peltier cell where the sample is not connected (forming a conductive path). 2) Bottom surface of the Peltier cell where sample is not connected. 3) The sides of the sample. Sources 1&2 can be considered simultaneously given that A will be the same, and it is reasonable to assume that the majority of the heat lost from the top surface will be measured by the bottom surface (i.e. overestimating J Q). Thus for 1&2: Q rad = εσ(AP-AT)(T h4-Tc4) (17) where Ap is the Peltier area. For the worst case scenar io for radiative losses 1&2 the smallest samples measured in the 40x40mm2 Peltier setup had AT=100 mm2. In this case, AP-AT=1500 mm2. Inserting this into equation (17), the radiative heat loss would be Q = 0.045 W for a measured heat flow of 2.5 W (1.8%). 10 The worst case scenario for radiative loss via the sides of the sample was for the thickest samples, with largest 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 2. For a typical temperature difference of 5 K this would amount to radiative loss of approximately 0.0045 W (0.2% of Q). For the YIG samples, 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 temperature differences, radiative loss is negligible with respect to the changes seen as a function of A T0.5. As a general rule of thumb, radiative heat losses in this system can be quickly assessed by monitoring J Q vs. ΔT. Given equations (5) and (16), for the ideal case the heat flux through the sample should be linear with respect to ΔT. As soon as the radiative heat losses become significant with respect to the heat flow through the sample, this linearity breaks down as a larger J Q will be observed per unit ΔT. Conversely, for much thicker samples, where radiative loss from the sides of the sample starts to become significant, such as is the case with the bulk YIG samples, heat loss would result in a lower measured J Q per unit ΔT as well as non-linearity of JQ(ΔT). We only present data for values of ΔT where this linearity was preserved. Figure A1 (a) – (f) Spin Seebeck measurement(s) for the Fe 3O4:Pt thin film as AT (=LT x.LT y), Lx and Ly were varied, plotted against LT x, LT y, LT y/LT x, AT/LT y, AT and AT0.5. Where x-axis errors are not visible th ey are smaller than the symbol used. 11 REFERENCES [1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). [3] T. Jungwirth, J. Wunderlich, and K. Olejník, Nat. Mater. 11, 382 (2012). [4] A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013). [5] H.-H. Fu, D.-D. Wu, L. Gu, M. Wu, and R. Wu, Phys. Rev. B 92, 045418 (2015). [6] J. Ren, J. Fransson, and J. X. Zhu, Phys. Rev. B - Condens. Matter Mater. Phys. 89, 1 (2014). [7] L. Gu, H. H. Fu, and R. Wu, Phys. Rev. B - Condens. Matter Mater. Phys. 94, 1 (2016). [8] S. Borlenghi, W. Wang, H. Fangohr, L. Bergqvist, and A. Delin, Phys. Rev. Lett. 112, 047203 (2014). [9] K. I. Uchida, H. Adachi, T. Kikkawa, A. Kirihara, M. Ishida, S. Yorozu, S. Maekawa, and E. Saitoh, Proc. IEEE 104, 1946 (2016). [10] K. Uchida, M. Ishida, T. Kikkawa, a Kirihara, T. Murakami, and E. Saitoh, J. Phys. Condens. Matter 26, 343202 (2014). [11] S. R. Boona, S. J. Watzman, and J. P. Heremans, APL Mater. 4, (2016). [12] R. Ramos, T. Kikkawa, K. Uchida, H. Adachi, I. Lucas, M. H. Aguirre, P. Algarabel, L. Morellón, S. Maekawa, E. Saitoh, and M. R. Ibarra, Appl. Phys. Lett. 102, 072413 (2013). [13] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Figure A2 (a) – (f) Spin Seebeck measurement(s) of YIG:Pt as AT (=LT xLT y except where sample was 13 mm pellet) , LT x and LT y were varied, plotted against LT x, LT y, LT y/LT x, AT/LT y, AT and AT0.5. 12 Hou, D. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev. Lett. 110, 067207 (2013). [14] Z. Qiu, D. Hou, K. Uchida, and E. Saitoh, J. Phys. D. Appl. Phys. 48, 164013 (2015). [15] T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. I. Uchida, Z. Qiu, G. E. W. Bauer, and E. Saitoh, Phys. Rev. Lett. 117, 207203 (2016). [16] A. Sola, M. Kuepferling, V. Basso, M. Pasquale, T. Kikkawa, K. Uchida, and E. Saitoh, J. Appl. Phys. 117, 17C510 (2015). [17] A. Sola, P. Bougiatioti, M. Kuepferling, D. Meier, G. Reiss, M. Pasquale, T. Kuschel, and V. Basso, Sci. Rep. 7, 1 (2017). [18] R. Iguchi, K. Uchida, S. Daimon, and E. Saitoh, Phys. Rev. B 95, 174401 (2017). [19] A. J. Caruana, M. D. Cropper, J. Zipfel, Z. Zhou, G. D. West, and K. Morrison, Phys. Status Solidi - Rapid Res. Lett. 10, 613 (2016). [20] Y. S. Dedkov, U. Rüdiger, and G. Güntherodt, Phys. Rev. B 65, 064417 (2002). [21] N.-W. Park, W.-Y. Lee, J.-A. Kim, K. Song, H. Lim, W.-D. Kim, S.-G. Yoon, and S.-K. Lee, Nanoscale Res. Lett. 9, 96(8pp) (2014). [22] G. A. Slack, Phys. Rev. 126, 427 (1962). [23] P. Grosseau, a. Bachiorrini, and B. Guilhot, Powder Technol. 93, 247 (1997). [24] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, AIP Adv. 6, 015018 (2016). [25] A. Kehlberger, U. Ritzmann, D. Hinzke, E.-J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jungfleisch, B. Hillebrands, U. Nowak, and M. Kläui, Phys. Rev. Lett. 115, 096602 (2015). [26] A. Anadon, R. Ramos, I. Lucas, P. A. Algarabel, L. Morellon, M. R. Ibarra, and M. H. Aguirre, Appl. Phys. Lett. 109, 012404 (2016). [27] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, Nat. Mater. 11, 686 (2012). [28] Y. Saiga, K. Mizunuma, Y. Kono, J. C. Ryu, H. Ono, M. Kohda, and E. Okuno, Appl. Phys. Express 7, 093001 (2014). 1 Supplementary information for “Scaling of the spin Seebeck effect in bulk and thin film” K. Morrison1, A J Caruana1,2, C. Cox1, T.A Rose1 1 Department of Physics, Loughborough Univ ersity, Loughborough, LE11 3TU, United Kingdom 2 ISIS Neutron and Muon Source, Didcot, Oxfordshire, OX11 0QX Keywords Magnetic materials, thermoelectrics, thin films 1 Introduction Extensive experimental details and additiona l characterization of the films presented in “Scaling of the spin Seebeck effect in bulk and thin film” is given in this supplementary information. Additional X-ray diffraction (XRD), resistivity and X-ra y reflectivity (XRR) data is used to demonstrate the quality of Fe 3O4 thin films – exhibiting the characteristic Verw ey transition for this phase[1], as well as the expected magnetic characteristics as a function of te mperature. This is followed by additional XRD, and XRR data for the Fe 3O4:Pt bilayers to further demonstrate the magne tic and structural quality of the films, as well as XRD and SEM data of the YIG pellet chosen for this study. Finally, further details of the spin Seebeck measurements are given, alongsid e the additional ‘scribing’ study data. 2 Experimental Details 2.1 Sample preparation The series of films prepared for this study are listed in Table S1 . They were prepared in vacuum (base pressure of 5x10-9 mbar) by pulsed laser deposition (PLD) using a frequency doubled Nd:YAG laser (Quanta Ray GCR-5) with a wavelength of 5 32 nm and 10 Hz repetition rate. The films were deposited onto 10 x 10 mm and 22 x 22 mm glass slides (Agar Scientific bor osilicate coverglasses), or 24 x 32 and 50 x 50 mm fused silica, which were baked out at 400 °C prior to the growth of the Fe 3O4 layer at the same substrate temperature. The samples were then left to cool in vacuum until they reached room temperature, at which point the Pt layer was deposited. The Fe 3O4 and Pt layers were deposited from stoichiometric Fe 2O3 (Pi-Kem purity 99.9%) and elemental Pt (Testbourne purity 99.99%) targets using ablation fluences of 1.9±0.1 and 3.7±0.2 J cm - 2 respectively. The target to substrate distance was 110 mm. The Pt layer thickness was kept constant at 5±0.5 nm for all films, whilst the substrate and Fe 3O4 thicknesses were varied. Bulk YIG was prepared by the solid state method. Stoichiometric amounts of Y 2O3 and Fe 2O3 starting powders (Sigma Aldrich 99.999% and 99.995% trace metals basis, respectively) were ground and mixed together before calcining in air at 1050 ᵒC for 24 hours. Approximately 0.5g of the calcined powder was then dry pressed into a 13 mm diameter, 1.8±0.2 mm thick cylindrical pellet. The pellet was then sintered at 1400 ᵒC for 12 hours, after 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 coater. Samples were cut to size thereafter, using an IsoMet low speed precision cutter. 2.2 Structural characterization XRD was obtained using a Bruker D2 Phaser in the FIG. S1 Schematic of the spin Seebeck measurement system. Top and bottom Peltier cells act as heat source and heat flux monitor, respec tively. Copper plates on the sample facing side of the Peltier cells provide high thermal conductivity at th e interface, thus reducing lateral temperature gradients. GE varnish and mica are used to electrically isol ate the copper surface from voltage contacts. Thermocouples monitor the temperature (with respect to the cold sink) at the top and bottom of the sample. Finally, the voltage, V ISHE is monitored as Q and B are varied. Note that the thermal contact area, A T, is defined as the area of sample connecting the top and bottom Peltier cells. 2 standard Bragg-Brentano geometry using Cu K radiation; a Ni filter was used to remove the Cu K line. The samples were rotated at a constant 15 rpm throughout the scan to increase the number of crystallites being sampled. XRR measurements were taken using a modified Siemens D5000 diffractometer comprising a graphite monochromator reflecting only Cu K into the detector. The incident beam was collimated with a 0.05 mm divergence slit, while the reflected beam w as passed through a 0.2 mm anti-scatter slit followed by a 0.1 mm resolution slit. XRR data was fitte d using the GenX software package [2]. 2.3 Magnetic and electric characterization Magnetization as a function of applied magnetic fiel d was measured using a Quantum Design Magnetic Properties Measurement System (SQUID). Several bilayer samples were characterized as a function of field at room temperature, whilst the Fe 3O4 film (without Pt contact) was measu red at 5 K, 100-130 K, and 295 K in order to observe the characteristic magnetic Verwey transition at 115 K. Room temperature sheet resistance was obtained using the Van der Pauw method with a Keithley 6221/2182 nanovoltmeter and current source in conjunction with a Keithley 705 scanner. 2.4 Thermoelectric characterization Spin Seebeck measurements were obtained from a set-up similar to that of Sola et al. [3], and outlined in Figure S1. There were 4 separate set-ups, optimi zed for 10x10, 12x12, 20x20 and 40x40 mm samples. The majority of data presented in the main manuscr ipt were obtained using the 12x12 and 40x40mm setups Table S1 Summary of the thin film samples measured. The contact separation and corresponding thermal contact area, AT, are also indicated. There were 3 distinct series of sample: (a) where the Fe 3O4 and Pt thicknesses, tFe3O4 and tPt respectively, were kept constant (at a nominal 80 nm:5 nm) and the substrate thickness tsubstrate , was varied, (b) where the Fe 3O4 thickness was varied from 20 – 320 nm, whilst the Pt thickness was kept constant, and (c) a large area sample (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 Fe3O4 exceeded 80 nm. Sam ple A T (mm2) L y (mm) Substrate t substrate (mm) t Pt (nm) t Fe3O4 (nm) FP1 768 28.8 Glass 0.17 ~4.8 80 FP3 (G6) (FP3 frag) 136.8 203 8.47 12.4 Glass 0.3 4.9 80 FP5 (G34) 484, 684, 880, 1530 214 270 19.3 19.3 7.62 19.44 Glass 0.5 5.5 80 FP6 772 28.56 Fused silica 0.67 4.8 80 0.67+0.67 FP9 983 36.66 Fused silica 0.9 5.4 80 F320P (G27) 66.6 6.58 Glass 0.3 4.4 320 F40P (G30) 100 6.26 Glass 0.3 5.2 40 F160P (G31) 100 5.5 Glass 0.3 4.5 160 F20P (G32) 100 7.25 Glass 0.3 5 20 G48 2500 1400 400 100 - 37 20 8.5 Glass 0.5 5 80 3 unless otherwise stated. The thin film was sandwiched between 2 Peltier cells, where the top Peltier cell (1) acted as a heat source, and the bottom Peltier cell (2 ) monitored the heat Q, through the sample. A copper sheet was affixed to the surface of the Peltie r cells to promote uniform heat transfer. Two type E thermocouples were mounted on the su rface of these copper sheets such that they were in contact with the sample during the measurement. These thermocouples were connected in differential mode to monitor the temperatur e difference across the sample, ΔT. The Peltier cells were calibrated by comparing the voltage generation, V p, in two scenarios: (1) where the Peltiers were clamped together, with a resist or attached to the top of one, and the heat flow passing through both was assumed to be constant, such that: Q top = Q bottom ( S . 1 ) (2) where the resistor was clamped between the two Pe ltier cells and the total power measured by the Peltier cells was assumed to be equal to the heat output of the resistor, such that:. Qtop + Q bottom = P resistor ( S . 2 ) Solving these simultaneous equations produced the sensitivity, Sp, which was found to be 0.11 - 0.26 V/W at 300K. The heat flux, JQ, is then determined by: JQ = V p/(SpAT) ( S . 3 ) where A T is the contact area between the top and bottom Pe ltier cells. This can be quickly related to the measured temperature difference ΔT, assuming that there are no major thermal losses or temperature drops at the sample:Peltier interface(s): JQ = k effΔT/L z ( S . 4 ) where k eff is the effective thermal conductivity of the sample. Given that this paper studies the impact of sample size (thus A T is varied), it could be argued that for samples where A T is less than the area of the top and bottom Pe ltier cells, there are signi ficant thermal losses (radiation and convection). This is monitored by plotting the change in the measured J Q as a function of ΔT, as was seen in Figure 1. In addition, Figure 6 of the main manuscript (and Fig. S7 here) shows data where the size of the sample was matched to the size of the Peltier cell for measurement of J Q. Finally, Figure S9 here shows similar spin Seebeck measurements in ai r and vacuum, where both Peltier cells are used to monitor heat flow. 3 Results 3.1 Characterization of the Fe 3O4 layer Additional characterization data for the Fe 3O4 thin films (deposited under id entical conditions as the devices measured, but without the Pt top layer) is given in Fig. S2. The Verwey transition is a characteristic feature of the Fe 3O4 phase of iron oxide and manifests as a sharp increase in resistivity and hysteresis below the Verw ey transition temperature (typically between 115 and 120 K)[1]. This feature is clearly seen in Fig. S2 (a), where magnetometry shows a sudden increase in hysteresis below 120 K. In addition, XRD indicates a preference for <111> text ure that is moderately sensitive to the direction of the plume during PLD (see repeated film depositions of Fig. S2 (c)). Due to the instability of the Fe 3O4 phase, some Fe is also present. Finally, an example of the X-ray Reflectivity (XRR) fits used to obtain film thickness is given in Fig. S2 (d). 3.2 Characterization of the Fe 3O4:Pt bilayers Additional characterization data for some of the Fe 3O4:Pt bilayers is given in Fig. S3. Previous work showed that for t Pt>2 nm a thin continuous paramagnetic Pt layer deposited on top of the Fe 3O4 layer follows the wavy surface of the Fe 3O4 layer. The Pt stacks on top of the Fe 3O4 (111) planes and the two grains have an orientation relationship of [011]Pt//[011]Fe 3O4, (111)Pt//(111)Fe 3O4, which minimizes the interface energy due to minimal lattice mismatch of d(111) Pt (0.226 nm; JCPDS card 4-802) and d(222) Fe3O4 (0.242 nm; JCPDS card 19-629). XRD data for the sample series where the Fe 3O4 and substrate thickness was va ried is given in Fig. S3. For the substrate study, XRR (S3b) indicated similar Pt thickness (low frequency ‘bumps’ in the data) as 4 well as some variation in the interface quality (whether high frequency fringes can be observed indicates the relative roughness of the substrate, Fe 3O4 and Pt layers). Fig. S3 also shows the XRD and magnetization measurements of the sample series where the Fe 3O4 thickness was varied. Of particular note is the saturation of the magnetic moment of Fe 3O4 measured at 1 T for thicknesses > 80 nm. 3.3 Characterization of YIG XRD of the YIG pellet presented here is given in Figure S4. No evidence of unreacted starter powders (Fe 2O3, Y 2O3) or potential secondary phase YIP (YFeO 3) was seen, as highlighted in Figure S4(b). Scanning electron microscopy images show grain structure with a porosity expected from 30% estimated open porosity as measured by the Archimedes method. 3.4 Additional spin Seebeck data Fig. S5 shows some of the raw data for FP5 (analysed datapoints shown in Figure 2&3 of the main article). The 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. S7 shows more detailed data for the large area thin film measurement given in Figure 6 of the main text. Fig. S8 shows the results of the scribing study, where A T was fixed, but L x, L y (not L xT, L yT) were varied by scribing away sections of the thin film, as illustrated in Fig. S8(a). In this case, the voltage always scaled with the contact separation, L y, indicating that the behavior we see is not due to a change in the film properties (such as resistance), but scaling of the h eat flux. Figure S9 shows additional spin Seebeck measurements where the rig was altered to monitor heat flux above and below the sample, with a resistor as the heat source. In this example, the Peltier area was 20x20 mm and the resistor was 57 Ohms. Measurements taken in air and under vacuum indicate a difference of approximately 3.9% due to convective losses (from the sides of the sa mple). Differences between Q top and Q bottom indicate radiative losses of the order of 1.3%, which is comparable to the estimate given in the main text. Finally, Figure S10 shows expanded data from Figure 6 of the main manuscript, where scatter in individual datapoints due to poor thermal contact, or variation in substrate thickness or area, which was not accounted for. References [1] F. Walz, J. Phys. Condens. Mat . 14, R285-R340 (2002). [2] M. Björck, and G. Andersson , J. Appl. Cryst. 40, 1174-1178 (2007). [3] A. Sola, M. Kuepferling, V. Basso, M. P asquale, T. Kikkawa, K. Uchida, and E. Saitoh, J. Appl. Phys. 117, 17C510 (2015) 5 FIG. S2 Characterization of the Fe 3O4 film. a) SQUID magnetometry above and below the Verwey transition, TV. b) Resistivity as a function of temperature. c) XRD of a set of 4 separately prepared Fe 3O4 films. The inset shows a close-up of the (311), (222) peaks. d) Example XRR data (symbols) and fit (solid line), indicating thickness = 79 nm, roughness = 1.5 nm. FIG. S3 Characterization of the Fe 3O4:Pt films, where Fe 3O4 and substrate thickness was varied. a) XRD for Fe 3O4 thickness study, b) XRR for substrate study, and c) SQUID magnetometry as a function of Fe 3O4 thickness at 300K. d) Saturation magnetisation, M, as a function of Fe 3O4 thickness. 6 FIG. S4 (a)&(b) X-ray diffraction data for the YIG pelle t. (a) Normalized data alongside reference for Y 3Fe5O12 (YIG). (b) Zoomed in data selec tion plotted alongside possible impurity phases Fe 3O4 and Y 2O3 (starting powders) and YFeO 3 (YIP). (c) Scanning Electron Microscope image of the pellet, wh ere grain size was found to be an average of 14.5 μm, with open porosity of 30%. 7 FIG. S5 Raw voltage measurements for FP5 at 3 different areas, A T. The contact separation, area, temperature difference, heat flux, JQ, and resultant spin Seebeck coefficient, S 4, are shown for each sample. Data has been offset for clarity. 8 FIG. S6 Raw voltage measurements for the bulk YIG pellet as it was cut down to different sizes, A T. The contact separation, area, temperature difference, heat flux, J Q, and resultant spin Seebeck coefficient, S 4, are shown for each sample. Inset shows the linear relationship between heat flux, J Q, and temperature difference ΔT, which starts to break down for ΔT>10K due to radiation losses. 9 FIG. S7 Summary of the voltage measured as a function of heat flux for the (a) – (c) 10x10, (d) 20x20, (e) 40x40 mm Peltier cell measuremen ts. (f) S4 plotted as a function of A T0.5 for these samples. Note the scatter for the 100mm2 samples is indicative of thickness variation over the 50x50 mm film, which was sample d by selecting pieces from 3 corners of the sample. Due to plume direction during the pulsed laser deposit ion process, we usually expe ct thickness variation of th e Pt layer to differ by 10- 15% 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 nm thinner. To avoid this difference in standard measur ements we constrain the sample deposition area to 40x40mm. 10 FIG. 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 such that it is still thermally connected, but electrically isolat ed), the voltage always scaled the same. Large noise is due t o reducing contact separation and additional electrical noise during some of the measuremen ts (due to a loose wire). FIG. S9 Additional spin Seebeck meas urements for the 20x20 mm Peltier arrangement. (a) & (b) Final calibration data for the Peltier cells in air and under vacuum (P~1x10-4 mBar), respectively. Q top is the heat measured by the top Peltier (W), and Q bottom 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 voltage measurements (data offset for clarity) at various heating powers (driven by a resistive heater) in air and vacuum, respectively. (e) schematic of the altered measurement. (f) V ISHE as a function of heat flux (extracted from data in (c) and (d)) for the air and vacuum measurements. The fit to this data indicates a deviation of approximately 2%. 11 FIG. S10 Data from the substrate and Fe 3O4 thickness study plotted to show scatter in datapoints. (a) Impact of poor thermal contact 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 thickness measurements. (d) S 4 (blue symbols) versus S 5 (black symbols) for the substrate measurements.

2017-05-06

Whilst there have been several reports of the spin Seebeck effect to date, comparison of the absolute voltage(s) measured, in particular for thin films, is limited. In this letter we demonstrate normalization of the spin Seebeck effect for Fe$_3$O$_4$:Pt thin film and YIG:Pt bulk samples with respect to the heat flux J$_q$, and temperature difference $\Delta$T. We demonstrate that the standard normalization procedures for these measurements do not account for an unexpected scaling of the measured voltage with area that is observed in both bulk and thin film. Finally, we present an alternative spin Seebeck coefficient for substrate and sample geometry independent characterization of the spin Seebeck effect.

Scaling of the spin Seebeck effect in bulk and thin film

1705.02491v3

Magnon confinement in an all-on-chip YIG cavity resonator using hybrid YIG/Py magnon barriers† Obed Alves Santos∗,‡,¶and Bart J. van Wees‡ ‡Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, Groningen, AG 9747, The Netherlands ¶Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom E-mail: oa330@cam.ac.uk Abstract Confining magnons in cavities can introduce new functionalities to magnonic devices, en- abling future magnonic structures to emulate established photonic and electronic components. As a proof-of-concept, we report magnon confinement in a lithographically defined all-on-chip YIG cavity created between two YIG/Permalloy bilayers. We take advantage of the modified magnetic properties of covered/uncovered YIG film to define on-chip distinct regions with boundaries capable of confining magnons. We confirm this by measuring multiple spin pump- ing voltage peaks in a 400 nm wide platinum strip placed along the center of the cavity. These peaks coincide with multiple spin-wave resonance modes calculated for a YIG slab with the corresponding geometry. The fabrication of micrometer-sized YIG cavities following this tech- nique represents a new approach to control coherent magnons, while the spin pumping voltage in a nanometer-sized Pt strip demonstrates to be a non-invasive local detector of the magnon resonance intensity. †Keywords: Magnonics, Spin pumping, Spin waves, YIG resonators, On-chip cavity magnonics. 1arXiv:2306.14029v2 [cond-mat.mes-hall] 6 Oct 2023Magnonics aims to transmit, store, and process information in micro- and nano-scale by means of magnons, the quanta of spin waves.1,2Typical studies in this field use the ferrimagnet Yttrium Iron Garnet (Y 3Fe5O12- YIG) as a key material, due to its desirable properties, such as very low magnetic losses,2large applicability in mainstream electronics,3long magnon spin relaxation time,4and high magnon spin conductivity.5,6The magnetic proprieties of YIG can be modified by the presence of strong exchange/dipolar coupling caused by the deposition of a ferromagnetic layer onto the YIG film.7,8These hybrid magnonic systems, such as YIG/Py,8–13YIG/CoFeB,14,15and YIG/Co,16,17draw attention from fundamental and applied physics towards the control of coherent magnon excitations for information processing.18Alternatively, systems in which the magnetic material strongly interacts with electrodynamic cavities also provide a good platform for next- generation quantum information technologies, using the dual magnon-photon nature to enable new quantum functionalities.19,20The field called cavity magnonics, among many applications, can offer good compatibility with CMOS, room temperature operation, and GHz-to-THz spin-wave transducers.21–24 The possibility of confining magnons in cavities may enable future magnonic devices that emulate established electronic and photonic components, such as magnonic quantum point con- tacts, magnonic crystals, magnonic quantum bits, magnonic frequency combs, among others.25–33 Building upon similar ideas, recent theoretical results suggest all-on-chip structures to produce magnonic cavities by magneto-dipole interaction with a chiral magnonic element,34using an array of antiferromagnets on a ferromagnet,35or by proximity with superconductors.36 One approach to confining magnons involves the fabrication of rectangular or circular nano- and micro-structures using YIG.37–40Many of these structures are made by etching the YIG film or sputtering YIG from a target on defined micro-structures, adding an extra step in the fabrication process and limiting the options of available structures. Recently, Qin et al. , demonstrated the (par- tial) confinement of magnons in a region of the YIG film covered by a ferromagnetic metal.29This confinement arises from the difference in magnetic properties between the covered and uncovered regions of the YIG film, resulting in magnon reflections at the boundaries. Such phenomena en- 2abled the fabrication of an on-chip nanoscale magnonic Fabry-Pérot cavity,28,29observed by the transmission of magnons through the YIG film.30 (c) 𝑽𝒓𝑺𝑷+ − 𝑽𝒄𝑺𝑷+ − 𝒉𝒓𝒇z xy(b) lw𝑽𝒑𝑺𝑷 + −𝑩 𝑩(a) 10𝜇mw Pt PyPtPy Py Figure 1: (a) Schematic illustration of the lateral view of the cavity formed by confining the YIG film between two YIG/Py bilayers. The Pt strip in the middle of the cavity can be used as a local magnon detector. (b) Schematic illustration of the waveguide stripline and sample arrangement of the microwave excitation setup for the FMR absorption and spin pumping measurements. (c) Optical image of the fabricated devices, and the electrical connections used in the experiments. The cavity is formed in the YIG region constrained between the Py squares, defined by w×l. Multiple peaks are observed with the platinum strip placed along the center of the cavity, VSP c, and absent in both Pt strips placed outside of the cavity, VSP randVSP p. In this letter, we complement these studies by achieving an order of magnitude higher magnon confinement in a YIG film region which is not covered by the ferromagnetic metal, preserving the optimal magnetic properties of YIG within the cavity. We report, as a proof-of-concept, the fabrication of an all-on-chip micrometer-sized YIG cavity, by partially covering 100 nm thick LPE- grown YIG film with two square-shaped 30 nm thick permalloy (Py) layers. The exchange/dipolar 3interactions in the YIG/Py bilayer define on-chip, magnetically distinct regions, effectively forming reflecting boundaries for magnons, resulting in a magnonic cavity. We confirm this by measuring multiple spin pumping voltage (VSP)peaks with a 400 nm wide Pt strip placed along the center of the cavity. This voltage is proportional to the intensity of the FMR-excited magnon resonances in theuncovered YIG film, indicating the formation of standing-wave resonance modes. We assign these peaks to multiple standing backward volume spin wave modes (BVSWs) and magnetostatic surface spin wave modes (MSSWs), calculated from the spin-wave theory for a YIG slab with similar dimensions. Multiple spin pumping voltage peaks were not observed for Pt strips placed outside of the cavity. All the measurements were performed at room temperature. The presence of BVSWs and MSSWs modes in micrometer YIG cavities has already been observed.37,38,41–44However, the YIG cavities were produced by sputtering or wet-etching tech- nique, and the modes were measured by a local FMR antenna or time-resolved magneto-optical Kerr effect. To the best of our knowledge, this work represents the first observation of cavity res- onant modes measured by spin pumping voltages using all-on-chip hybrid magnonic structures, illustrated in Figure 1 a). The device was fabricated on a high-quality 100 nm thick YIG film grown by liquid phase epitaxy on a GGG substrate. Electron beam lithography was used to pattern the device, consisting of multiple strips of Pt and two Py squares with edge-to-edge distance of w=2µm, Figure 1 b) and c). The square shape was chosen to avoid effects of shape anisotropy in the Py film.45,46The sample was placed on top of a stripline waveguide and connected to a vector network analyzer. The stripline waveguide was then placed between two poles of an electromagnet in such a way that the DC external magnetic field, B=µ0H, where µ0is the vacuum permeability, and the microwave field, hr f, were perpendicular to each other and both were in the plane of the YIG film in all the measurements, see Figure 1 a) and b). See supporting information section I for more details on sample fabrication and experimental setup.47 TheB-field scan of the microwave absorption, S21, which measures the overall magnetic re- sponse of 4 ×3 mm sized YIG film, is shown in the top panel of Figure 2 a), for 1 to 9 GHz. The FMR absorption peak of the 100 nm thick YIG film has a typical linewidth of ≈0.2 mT, demon- 40.00.51.01.5 1 GHz 5 GHz 8 GHz 9 GHz 2 GHz 6 GHz 3 GHz 7 GHz 4 GHzSP voltage ( mV) 0 50 100 150 200 25002468 Magnetic field (mT) Kittel equation Remote strip Frequency (GHz)(a) (b) -0.2-0.10.0 1 GHz 2 GHz 5 GHz 3 GHz 6 GHz 4 GHz 7 GHz 8 GHz 9 GhzS21(a.u) 0 50 100 150 200 25002468 Magnetic field (mT) Kittel equation m0HFMR of bulk YIGFrequency (GHz) 2 4 6 8 100.00.10.20.3Linewidth (mT) Frequency (GHz)Figure 2: Comparison between FMR absorption of the bulk YIG and the spin pumping voltage in theremote strip. (a) Top panel shows B-field scan of the microwave transmission absorption, S21. The bottom panel shows the field position of FMR peaks for different microwave frequencies. The inset shows the linewidth vs.resonance frequency. Note that we do not see the magnon spectra because the magnetic field is uniform on a relatively long scale. Therefore, only the uniform mode is excited and measured. (b) The top panel shows the B-field scans of the VSPof the remote Pt strip for different microwave frequencies. The bottom panel shows the field position of the maximum VSP ras a function of the microwave frequency. 5strating the high-quality of the YIG film. The FMR spectra are fit using an asymmetric Lorentizan function, obtaining the B-field value of the FMR peak and the linewidth (FWHM), see details in Supporting Information section I. The bottom panel of Figure 2 a) shows the microwave frequency versus the B-field value of the FMR peak. The solid blue curve corresponds to the best fit to the Kittel equation, f=γµ0/2π/radicalbig H(H+M),3where γis the gyromagnetic ratio. The best fit was obtained for γ/2π=27.2±0.1 GHz/T and M=142.4±0.8 kA/m. The inset in Figure 2 a) shows the linewidth as a function of the microwave frequency. The solid blue curve corresponds to the best linear fit, obtaining a Gilbert damping of α≈5.0×10−4and the inhomogeneous linewidth of µ0∆H0=0.06 mT. In this letter, we address the uniform FMR resonance of the YIG film as "bulk" YIG resonance, or µ0HFMR, to distinguish the FMR absorption measurement of the mm-range size YIG film from the local spin pumping voltage measurements for different platinum strips. The upper panel of Figure 2 b) shows the B-field scan of the spin-pumping voltage for the remote Pt strip, VSP r, for different microwave frequencies. The bottom panel presents the magnetic field of theVSP rpeak for different rffrequencies. Again, we fit the results using the Kittel equation with γ/2π=27.2±0.1 GHz/T and M=140.2±0.9 kA/m. It is important to emphasize that albeit Figures 2 a) and b) look effectively identical, they cor- respond to two completely different experiments. Both measure the intensity of the ferromagnetic resonance, but in one case we measure the FMR absorption of the bulk YIG film, and in the other we measure the local spin pumping voltage, as a result of the injection of spin current by means of the spin pumping effect,48and the conversion of the spin current into charge current in the Pt strip by the inverse spin Hall effect.49,50We did not observe a significant peak broadening caused by the spin absorption due to the presence of the Pt layer. This is because the width of the strip is 400 nm, such that it covers only a fraction of the YIG film resulting in a spin pumping response proportional to the FMR absorption of the bulk YIG.51These results show that the platinum strip is a local and non-invasive intensity detector of the magnon excitation. Figure 3 compares the B-field scan of the spin pumping voltage measured on the remote strip, VSP r, solid black lines, and on the cavity strip, VSP c, solid blue lines, for different rffrequencies, 640 45 50 55 60 653 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV 100 105 110 115 120 125 1305 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV 175 180 185 190 195 200 2057 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV 215 220 225 230 235 2408 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV 250 255 260 265 270 275 2809 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV 15 20 25 30 352 GHzSpin Pumping voltage Magnetic field (mT) Cavity strip Remote strip BVSWs MSSWs1 mV(a) (b) (c) (d) (e) (f)Figure 3: The B-field scan of the spin pumping voltage measured with the remote strip VSP rand cavity strip VSP cfor different frequencies is shown from (a) to (f). The resonance modes of the cavity strip occur before the FMR of the bulk YIG resonance field for low frequency, 2 GHz, and are present after the FMR bulk resonance for 9 GHz. The resonance frequencies of the BVSWs and MSSWs modes obtained using equations 1 and 2 is shown as blue and pink star-symbol, respectively, for each frequency. One can notice a secondary broad peak in the remote strip at 7 GHz. This peak is less evident or absent in other remote strip. We discuss more on that in the supporting information section III.47 7from 2 GHz to 9 GHz. The remote strip shows a single resonance peak, while the cavity strip shows multiple resonances. The average linewidth of the spin pumping voltage peaks on the cavity strip is slightly broader than the remote strip, suggesting an additional damping contribution. Note that there is no pronounced peak on the cavity strip B-field scan corresponding to the bulk YIG resonance at 2 GHz, and the corresponding peak is small for 8 and 9 GHz, Figure 3 a), e) and f), respectively. This indicates that the spin pumping voltage on the cavity strip is dominated by magnons excited inside the cavity itself, not by magnons generated outside of the cavity, corre- sponding to bulk FMR values, which could be transmitted into the cavity. As mentioned above, VSP cis proportional to the resonance intensity of the YIG film in the region between the Py squares. This means that the series of resonance modes present underneath the Pt strip, indicate the exis- tence of a cavity supporting standing magnonic waves. We can explain these modes by calculating the spin-wave dispersion relation for a YIG slab with dimensions w×lwith magnetization in the plane of the film, given by52–54 f=γµ0 2π/parenleftig/parenleftbig H+Ha+λexk2 totM/parenrightbig/parenleftbig H+Ha+λexk2 totM+MF/parenrightbig/parenrightig1/2 , (1) where λex=2A/µ0M2is the exchange constant with A=3.5 pJ/m, ktot2=kn2+km2is the total quantized wavenumber defined by kn=nπ/wandkm=mπ/l, where nandmare the mode numbers along the width and length of the cavity, respectively. The function Fcan be written as F=P+/parenleftbigg 1−P/parenleftbig 1+cos2(φk−φM)/parenrightbig +MP(1−P)sin2(φk−φM) H+Mλexktot2/parenrightbigg , (2) where P=1−1−edktot dktot, for a YIG film with thickness d. In Eq. 2, φk=arctan (km/kn), and φM is the angle between the magnetization and the spin-wave propagation or direction of ⃗k. Two magnetostatic modes can be accessed considering the symmetry of the device, the magnetostatic surface spin wave modes (MSSWs), where ⃗k⊥⃗M,i.e.,φM=π/2 and the backward volume spin waves modes (BVSWs) where ⃗k∥⃗M,i.e.,φM=0. The best fit to Eq. 1 and 2 reproducing the majority of the spin pumping peaks for different 8frequencies was obtained using M=130 kA/m, d=100 nm, w=2.5µm,l=30µm,µ0Ha=10 mT, and γ/2π=26.5 GHz/T. The calculated values for (m=1)of the n= (1,2,3,...6)mode of the BVSWs and first and second mode of MSSWs are shown in Figure 3 a) to f), blue, and pink star symbols, respectively. We obtained a good agreement between the peaks present in VSP c and the calculated modes. Since l≫w, modes with m>1 are hard to distinguish since they are superimposed on the m=1 mode due to their proximity in frequency. One important feature obtained from the fit is an anisotropy field of µ0Ha=10 mT. The anisotropy may originate from the stray fields produced by the magnetization of Py, leading to a local increase of the effective DC-magnetic field applied on the cavity.55Py stray fields may also be responsible for inducing even modes in the cavity, observed in our results.55These even modes are not expected for a YIG slab when a homogeneous DC magnetic field and rffield are applied.37,38 One can see that the modes appear in Figure 3 at a lower field than the bulk YIG resonance for 2 GHz and a higher field than the bulk resonance at 9 GHz. We can analyze the frequencies of the cavity resonance modes by simultaneously plotting the SP-FMR resonance field of the remote strip (black circles) and the resonance field of each magnon mode in the cavity strip (vertical blue stripes), Figure 4 a). Overall, the resonance field distribution of cavity modes evolves linearly in frequency, with a slope of ≈28 GHz/T, solid red curve in Figure 4 a). The linear behavior was also reported in previous YIG cavity results.38,43,56 To emphasize the localized magnon detection characteristic of the Pt strip, we normalize the spin pumping voltage of the cavity strip, VSP c, between 0 and 1, based on the lowest and highest spin pumping voltage value in each B-field scan. We centered the B-field scans with respect to the resonance field of the bulk YIG obtained from the FMR measurements, µ0HFMR, for each frequency. Figure 4 b) shows the dispersion of the resonant modes of the cavity strip compared with FMR of the bulk YIG. The BVSWs modes, up to (n=6), and the first and second modes of the MSSWs calculated from Eq. 1 and 2 are shown in Figure 4 b) as solid black and dashed pink lines, respectively. The model of Eq. 1 and 2 is very useful for confirming the mode position as a function of field and frequencies, and the peak spacing between each peak. Presenting the voltages 9-20 -15 -10 -5 0 5 10 15 2023456789 BVSWs MSSWsFrequency (GHz) m0H - m0HFMR (mT)01 0 50 100 150 200 2500123456789 Magnon modes in the cavity strip Remote strip Kittel equation 28 GHz/TFrequency (GHz) Magnetic field (mT)(a) (b)Figure 4: (a) Distribution of the identified magnon modes in the cavity as a function of the mag- netic field for each frequency, plotted as vertical blue stripes. The solid red curve corresponds to 28 GHz/T. The resonance of the remote strip and the best fit of the Kittel equation are addressed as a black symbol and a dashed black line, respectively. (b) Spin pumping intensity spectra of the Pt strip within the cavity. The solid black and dashed pink lines correspond to the resonant modes for a YIG slab with similar dimensions as the build cavity. 10as intensity spectra demonstrates that the 400 nm wide Pt strip can be used as a localized FMR detector in future magnonic cavity studies. The spectrum at 1 GHz is not shown in Fig. 4 b) due to the absence of a prominent spin-pumping voltage peak. Additional intensity spectra with other cavity widths are shown in the supporting information section II,47confirming the reproducibility of the fabrication technique. We do not observe multiple voltage peaks with the proximity strip, VSP p, placed close to the Py square but still outside of the cavity. This confirms that the Pt layer alone does not induce sufficient magnetic changes in the YIG to create a cavity. Additionally, we do not observe multiple peaks in a device where the Py film was replaced with gold, ruling out microwave artifacts and confirming the requirement of confinement by YIG/Py bilayers on both sides to create the cavity, see supporting information section III.47We hypothesize that the difference between the magnetization dynamics of covered and uncovered YIG regions by the Py film creates a magnon barrier, as ilustrated in Figure 1 a), and discussed in previous reports.28,29 In analogy to a (lossy) cavity resonator, we can estimate a finesse given by Φm=∆Bspacing /∆B, where ∆Bspacing is the B-field peak spacing and ∆Bis the linewidth of the resonance peak.57As a figure of merit, the frequency dependence of the average ∆Bspacing and calculated finesse is shown in Figure 5 a). Although the peak spacing increases as a function of frequency, the finesse fluctuates around a value of Φm≈10, about one order of magnitude higher than the previous report.29This value of finesse corresponds to a reflectance of R≈0.73, approximately constant through the entire frequency range. Figure 5 b) shows the average peak spacing and the finesse as a function of the cavity width, calculated from the B-field scans at 5 GHz, see supporting information section II.47 Although only three sets of data are presented, a clear correlation between the average peak spacing and the finesse is observed. This correlation arises because the cavity is formed in a uncovered YIG region, preserving the optimal magnetic properties of YIG. In fact, the peak linewidth decrease with decreasing cavity width, w, indicating that the additional broadening may stem from the superposition of higher modes ( m>1) along the cavity length, see supporting information section II.47The frequency spacing between these modes increases as a function of the aspect ratio (l/w). 112 3 4 5 6 7 8 91234 Frequency (GHz)Avarage peak spacing (mT) 0510152025 Finesse, Fm(a) 1.5 2.0 2.5 3.0 3.5 4.01234 Cavity width ( mm)Avarage peak spacing (mT) 0510152025 Finesse, Fm (b) 𝑤=2𝜇m5GHzFigure 5: (a) Frequency dependence of the average ∆Bspacing and the finesse Φm. The error bar is calculated as the standard deviation of ∆Bspacing . (b) Average peak spacing and finesse as a function of cavity width calculated from the B-field scan at 5 GHz. Ultimately, a maximum finesse of Φm≈21, or R≈0.86, is achieved for a cavity with w=1.6µm, which demonstrates a high potential for magnon confinement. We consider Eq. 1 and 2 as an approximate model since it describes a YIG slab with di- mensions w×l, where the magnetization amplitude is minimum at the boundary, i.e., an infinite potential well. In our case, the cavity is a consequence of the exchange and dipolar interaction in the YIG/Py bilayer.7,8,29,58This means that a finite height potential well would better describe the system. This discrepancy can be the origin of a minor deviation between the measured VSP cpeaks and the calculated cavity modes. Accurate modeling of the modes should be performed using micromagnetic simulations, taking into account the exchange and dipolar interaction with Py in further investigation. The potential height of the cavity barriers should be dependent on the thick- ness of the YIG film and the exchange/dipolar interaction with the top ferromagnetic layer. Further investigation using the present technique should be performed for different adjacent ferromagnetic layers in which strong exchange interaction has already been reported, such as YIG/CoFeB15and YIG/Co.16YIG films thinner than 100 nm with the damping below than 5 .0×10−4are good can- didates to produce magnonic cavities with higher reflectance factors keeping the magnetic losses close to those reported in this letter.59 In summary, we take advantage of the difference in the magnetic dynamics between the YIG 12film and the YIG/Py bilayers, to fabricate an all-on-chip magnonic cavity supporting standing magnon modes in a uncovered YIG film between two YIG/Py bilayers. This approach enables the confinement of magnons while preserving the optimal magnetic properties of the YIG cavity. The spin pumping voltage of a 400 nm wide Pt strip proved to be a reliable technique to detect the magnon resonance modes of the cavity. Following this idea, 1D and 2D magnonic crystals could be obtained by having a regular array of magnetic strips onto YIG, with the possibility of measuring the magnon modes locally by means of spin pumping. Moreover, further investigations should involve designing coupled cavities by placing two cavities side by side, where the coupling strength could be controlled by the width of the central YIG/Py bilayer. This cavity fabrication process opens new possibilities for investigating and characterizing micron-sized YIG cavities with a wide range of arbitrary shapes. It also allows for the implementation of on-chip magnonic computation structures, serving as a printed circuit board for magnons. These results demonstrate a promising combination of hybrid magnonics and cavity magnonics, which has the potential to drive the integration of future all-on-chip magnonic devices into mainstream microwave electronics. Acknowledgement We acknowledge the technical support from J. G. Holstein, T. J. Schouten and H. de Vries, F. A. van Zwol, A. Joshua. We are grateful to A. Azevedo, C. Ciccarelli and C. M. Gilardoni for the valuable discussion. We acknowledge the financial support of the Zernike Institute for Advanced Materi- als and the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open Grant No. 618083(CN- TQC). This project is also financed by the NWO Spinoza prize awarded to Prof. B. J. van Wees by the NWO, and ERC Advanced Grant 2DMAGSPIN (Grant agreement No. 101053054). 13Supporting Information Available Device fabrication and experimental setup details (section I). Additional cavities measurements (Section II) and control samples measurements (section III). References (1) Kruglyak, V .; Demokritov, S.; Grundler, D. Magnonics. Journal of Physics D: Applied Physics 2010 ,43, 264001. (2) Rezende, S. M. Fundamentals of Magnonics ; Springer, 2020; V ol. 969. (3) Pozar, D. M. Microwave engineering ; John wiley & sons, 2011. (4) Cornelissen, L. J.; Liu, J.; Duine, R. A.; Youssef, J. B.; van Wees, B. J. Long-distance trans- port of magnon spin information in a magnetic insulator at room temperature. Nature Physics 2015 ,11, 1022–1026. (5) Wei, X.-Y .; Alves Santos, O.; Lusero, C. H. S.; Bauer, G. E. W.; Ben Youssef, J.; van Wees, B. J. Giant magnon spin conductivity in ultrathin yttrium iron garnet films. Nature Materials 2022 ,21, 1352–1356. (6) Jungfleisch, M. B. Two-dimensional magnon spin transport. Nature Materials 2022 ,21, 1348–1349. (7) Grünberg, P. Magnetostatic spin-wave modes of a heterogeneous ferromagnetic double layer. Journal of Applied Physics 1981 ,52, 6824–6829. (8) Li, Y .; Cao, W.; Amin, V . P.; Zhang, Z.; Gibbons, J.; Sklenar, J.; Pearson, J.; Haney, P. M.; Stiles, M. D.; Bailey, W. E.; others Coherent spin pumping in a strongly coupled magnon- magnon hybrid system. Physical Review Letters 2020 ,124, 117202. 14(9) Das, K.; Feringa, F.; Middelkamp, M.; Van Wees, B.; Vera-Marun, I. J. Modulation of magnon spin transport in a magnetic gate transistor. Physical Review B 2020 ,101, 054436. (10) Fan, Y .; Quarterman, P.; Finley, J.; Han, J.; Zhang, P.; Hou, J. T.; Stiles, M. D.; Grutter, A. J.; Liu, L. Manipulation of coupling and magnon transport in magnetic metal-insulator hybrid structures. Physical Review Applied 2020 ,13, 061002. (11) Xiong, Y .; Li, Y .; Hammami, M.; Bidthanapally, R.; Sklenar, J.; Zhang, X.; Qu, H.; Srini- vasan, G.; Pearson, J.; Hoffmann, A.; others Probing magnon–magnon coupling in exchange coupled Y 3Fe5O12/Permalloy bilayers with magneto-optical effects. Scientific Reports 2020 , 10, 1–8. (12) Alves Santos, O.; Feringa, F.; Das, K.; Youssef, J. B.; van Wees, B. Efficient modulation of magnon conductivity in Y 3Fe5O12using anomalous spin Hall effect of a permalloy gate electrode. Physical Review Applied 2021 ,15, 014038. (13) Li, Y .; Zhao, C.; Amin, V . P.; Zhang, Z.; V ogel, M.; Xiong, Y .; Sklenar, J.; Divan, R.; Pearson, J.; Stiles, M. D.; others Phase-resolved electrical detection of coherently coupled magnonic devices. Applied Physics Letters 2021 ,118, 202403. (14) Cramer, J.; Ross, A.; Jaiswal, S.; Baldrati, L.; Lebrun, R.; Kläui, M. Orientation-dependent direct and inverse spin Hall effects in Co 60Fe20B20.Physical Review B 2019 ,99, 104414. (15) Qin, H.; Hämäläinen, S. J.; Van Dijken, S. Exchange-torque-induced excitation of perpendic- ular standing spin waves in nanometer-thick YIG films. Scientific Reports 2018 ,8, 1–9. (16) Klingler, S.; Amin, V .; Geprägs, S.; Ganzhorn, K.; Maier-Flaig, H.; Althammer, M.; Huebl, H.; Gross, R.; McMichael, R. D.; Stiles, M. D.; others Spin-torque excitation of per- pendicular standing spin waves in coupled YIG/CO heterostructures. Physical Review Letters 2018 ,120, 127201. 15(17) Wang, H.; Chen, J.; Yu, T.; Liu, C.; Guo, C.; Liu, S.; Shen, K.; Jia, H.; Liu, T.; Zhang, J.; others Nonreciprocal coherent coupling of nanomagnets by exchange spin waves. Nano Re- search 2021 ,14, 2133–2138. (18) Li, Y .; Zhang, W.; Tyberkevych, V .; Kwok, W.-K.; Hoffmann, A.; Novosad, V . Hybrid magnonics: Physics, circuits, and applications for coherent information processing. Journal of Applied Physics 2020 ,128, 130902. (19) Elyasi, M.; Blanter, Y . M.; Bauer, G. E. W. Resources of nonlinear cavity magnonics for quantum information. Physical Review B 2020 ,101, 054402. (20) Harder, M.; Yao, B. M.; Gui, Y . S.; Hu, C.-M. Coherent and dissipative cavity magnonics. Journal of Applied Physics 2021 ,129, 201101. (21) Barman, A. et al. The 2021 Magnonics Roadmap. Journal of Physics: Condensed Matter 2021 ,33, 413001. (22) Pirro, P.; Vasyuchka, V . I.; Serga, A. A.; Hillebrands, B. Advances in coherent magnonics. Nature Reviews Materials 2021 ,6, 1114–1135. (23) Sheng, L.; Chen, J.; Wang, H.; Yu, H. Magnonics Based on Thin-Film Iron Garnets. Journal of the Physical Society of Japan 2021 ,90, 081005. (24) Chumak, A.; others Advances in Magnetics Roadmap on Spin-Wave Computing. IEEE Transactions on Magnetics 2022 ,58, 1–72. (25) Zakeri, K. Magnonic crystals: towards terahertz frequencies. Journal of Physics: Condensed Matter 2020 ,32, 363001. (26) Yu, T.; Wang, H.; Sentef, M. A.; Yu, H.; Bauer, G. E. W. Magnon trap by chiral spin pumping. Physical Review B 2020 ,102, 054429. 16(27) Zare Rameshti, B.; Viola Kusminskiy, S.; Haigh, J. A.; Usami, K.; Lachance-Quirion, D.; Nakamura, Y .; Hu, C.-M.; Tang, H. X.; Bauer, G. E.; Blanter, Y . M. Cavity magnonics. Physics Reports 2022 ,979, 1–61. (28) Hong, I.-S.; Kim, S. K.; Lee, K.-J.; Go, G. Tunable magnonic cavity analogous to Fabry–Pérot interferometer. Applied Physics Letters 2021 ,119, 202401. (29) Qin, H.; Holländer, R. B.; Flajšman, L.; Hermann, F.; Dreyer, R.; Woltersdorf, G.; van Di- jken, S. Nanoscale magnonic Fabry-Pérot resonator for low-loss spin-wave manipulation. Nature Communications 2021 ,12, 1–10. (30) V ogel, M.; Chumak, A. V .; Waller, E. H.; Langner, T.; Vasyuchka, V . I.; Hillebrands, B.; von Freymann, G. Optically reconfigurable magnetic materials. Nature Physics 2015 ,11, 487– 491. (31) Wang, Z.; Yuan, H. Y .; Cao, Y .; Li, Z.-X.; Duine, R. A.; Yan, P. Magnonic Frequency Comb through Nonlinear Magnon-Skyrmion Scattering. Physcal Review Letters 2021 ,127, 037202. (32) Hula, T.; Schultheiss, K.; Gonçalves, F. J. T.; Körber, L.; Bejarano, M.; Copus, M.; Flacke, L.; Liensberger, L.; Buzdakov, A.; Kákay, A.; Weiler, M.; Camley, R.; Fassbender, J.; Schultheiss, H. Spin-wave frequency combs. Applied Physics Letters 2022 ,121, 112404. (33) Yuan, H.; Cao, Y .; Kamra, A.; Duine, R. A.; Yan, P. Quantum magnonics: When magnon spintronics meets quantum information science. Physics Reports 2022 ,965, 1–74. (34) Kruglyak, V . V . Chiral magnonic resonators: Rediscovering the basic magnetic chirality in magnonics. Applied Physics Letters 2021 ,119, 200502. (35) Xing, Y . W.; Yan, Z. R.; Han, X. F. Magnon valve effect and resonant transmission in a one-dimensional magnonic crystal. Physical Review B 2021 ,103, 054425. (36) Yu, T.; Bauer, G. E. W. Efficient Gating of Magnons by Proximity Superconductors. Physcal Review Letters 2022 ,129, 117201. 17(37) Trempler, P.; Dreyer, R.; Geyer, P.; Hauser, C.; Woltersdorf, G.; Schmidt, G. Integration and characterization of micron-sized YIG structures with very low Gilbert damping on arbitrary substrates. Applied Physics Letters 2020 ,117, 232401. (38) Costa, J. D.; Figeys, B.; Sun, X.; Van Hoovels, N.; Tilmans, H. A.; Ciubotaru, F.; Adel- mann, C. Compact tunable YIG-based RF resonators. Applied Physics Letters 2021 ,118, 162406. (39) Lee-Wong, E.; Xue, R.; Ye, F.; Kreisel, A.; van der Sar, T.; Yacoby, A.; Du, C. R. Nanoscale Detection of Magnon Excitations with Variable Wavevectors Through a Quantum Spin Sen- sor.Nano Letters 2020 ,20, 3284–3290. (40) Srivastava, T. et al. Identification of a Large Number of Spin-Wave Eigenmodes Excited by Parametric Pumping in Yttrium Iron Garnet Microdisks. Physical Review Applied 2023 ,19, 064078. (41) Heyroth, F.; Hauser, C.; Trempler, P.; Geyer, P.; Syrowatka, F.; Dreyer, R.; Ebbinghaus, S.; Woltersdorf, G.; Schmidt, G. Monocrystalline freestanding three-dimensional yttrium-iron- garnet magnon nanoresonators. Physical Review Applied 2019 ,12, 054031. (42) Zhu, N.; Zhang, X.; Han, X.; Zou, C.-L.; Zhong, C.; Wang, C.-H.; Jiang, L.; Tang, H. X. Waveguide cavity optomagnonics for microwave-to-optics conversion. Optica 2020 ,7, 1291– 1297. (43) Chang, K.-W.; Ishak, W. Magnetostatic Forward V olume Wave Straight Edge Resonators. 1986 IEEE MTT-S International Microwave Symposium Digest. 1986; pp 473–475. (44) Ishak, W.; Kok-Wai, C.; Kunz, W.; Miccoli, G. Tunable microwave resonators and oscillators using magnetostatic waves. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 1988 ,35, 396–405. 18(45) Mruczkiewicz, M.; Graczyk, P.; Lupo, P.; Adeyeye, A.; Gubbiotti, G.; Krawczyk, M. Spin- wave nonreciprocity and magnonic band structure in a thin permalloy film induced by dy- namical coupling with an array of Ni stripes. Physical Review B 2017 ,96, 104411. (46) Talapatra, A.; Adeyeye, A. Linear chains of nanomagnets: engineering the effective magnetic anisotropy. Nanoscale 2020 ,12, 20933–20944. (47) Device fabrication and experimental setup details, (section I). Additional cavities measure- ments, (Section II) and control samples measurements (section III). (48) Tserkovnyak, Y .; Brataas, A.; Bauer, G. E. Enhanced Gilbert damping in thin ferromagnetic films. Physical Review Letters 2002 ,88, 117601. (49) Azevedo, A.; Vilela-Leão, L.; Rodríguez-Suárez, R.; Santos, A. L.; Rezende, S. Spin pump- ing and anisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment. Physical Review B 2011 ,83, 144402. (50) Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C.; Jungwirth, T. Spin Hall effects. Re- views of Modern Physics 2015 ,87, 1213. (51) Cheng, Y .; Lee, A. J.; Wu, G.; Pelekhov, D. V .; Hammel, P. C.; Yang, F. Nonlocal Uniform- Mode Ferromagnetic Resonance Spin Pumping. Nano Letters 2020 ,20, 7257–7262. (52) Guslienko, K. Y .; Slavin, A. Boundary conditions for magnetization in magnetic nanoele- ments. Physical Review B 2005 ,72, 014463. (53) Mahmoud, A.; Ciubotaru, F.; Vanderveken, F.; Chumak, A. V .; Hamdioui, S.; Adelmann, C.; Cotofana, S. Introduction to spin wave computing. Journal of Applied Physics 2020 ,128, 161101. (54) Zheng, L.; Jin, L.; Wen, T.; Liao, Y .; Tang, X.; Zhang, H.; Zhong, Z. Spin wave propagation in uniform waveguide: effects, modulation and its application. Journal of Physics D: Applied Physics 2022 ,55, 263002. 19(55) Zhang, Z.; V ogel, M.; Holanda, J.; Ding, J.; Jungfleisch, M. B.; Li, Y .; Pearson, J. E.; Di- van, R.; Zhang, W.; Hoffmann, A.; Nie, Y .; Novosad, V . Controlled interconversion of quan- tized spin wave modes via local magnetic fields. Physical Review B 2019 ,100, 014429. (56) Dai, S.; Bhave, S. A.; Wang, R. Octave-tunable magnetostatic wave YIG resonators on a chip. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 2020 ,67, 2454–2460. (57) Ismail, N.; Kores, C. C.; Geskus, D.; Pollnau, M. Fabry-Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity. Opt. Express 2016 ,24, 16366–16389. (58) Kalinikos, B.; Slavin, A. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. Journal of Physics C: Solid State Physics 1986 ,19, 7013. (59) Schmidt, G.; Hauser, C.; Trempler, P.; Paleschke, M.; Papaioannou, E. T. Ultra Thin Films of Yttrium Iron Garnet with Very Low Damping: A Review. physica status solidi (b) 2020 ,257, 1900644. 20TOC Graphic w 215 220 225 230 235 2400.00.20.40.6 8 GHzSpin Pumping voltage ( mV) Magnetic field (mT) magnon cavity modes 10𝜇m𝑉𝑆𝑃 𝐵𝑒𝑥 21SUPPORTING INFORMATION Magnon confinement in an all-on-chip YIG cavity resonator using hybrid YIG/Py magnon barriers Obed Alves Santos∗ Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, Groningen, AG 9747, The Netherlands and Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom Bart J. van Wees Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, Groningen, AG 9747, The Netherlands 1arXiv:2306.14029v2 [cond-mat.mes-hall] 6 Oct 2023I. SAMPLE FABRICATION AND EXPERIMENTAL SETUP The samples consists of a high-quality 100 nm thick YIG film (Y3Fe5O12)grown by liquid phase epitaxy on a GGG substrate, obtained commercially from Matesy GmbH, measuring approximately (4×3)mm. Electron beam lithography (EBL) was used to pattern the device, which consists of multiple strips of Pt with 35 µm length and 400 nm wide. The Permalloy (Py) squares have dimensions (30×30µm2), the square shape was chosen to avoid shape anisotropies of the Py film.[1, 2] The Pt and Py layers were deposited by DC sputtering in an Ar+ plasma with thicknesses of 8 nm and 30 nm, respectively. The deposition of Ti(5 nm)/Au(75 nm) leads is made by e-beam evaporation. The last step consists of mounting the sample onto a non-resonant stripline waveguide, which can be used from 0.1 to 9 GHz with a characteristic impedance of 50 Ω, 0.030" RO4350, GCPWG, with a signal line 45 mil (1.14 mm) wide. Figure S1 a) show an optical image of the final YIG film sample containing multiple devices, while b) shows a zoom-in image of an individual device. The wire bonding is made using AlSi (Al 99%, Si 1%) wires on the sample holder and connected to a lock-in amplifier. (a) (b) 1 mm FIG. S1. a)Show the optical image of the final YIG sample with multiple devices. b)Show a zoom-in optical image of an individual device. ∗Correspondence should be addressed: oa330@cam.ac.uk, or obed.alves.santos@gmail.com 2The microwave field excitation is sent connecting the stripline waveguide to a vector network analyzer (VNA). The FMR absorption of the YIG film is obtained by scanning the absorption (S21) or reflection (S11) in the VNA for a fixed microwave frequency as a function of the magnetic field. To obtain the µ0HFMR value and the FMR linewidth (∆H), each S21 B-field scan is fitted using an asymmetric Lorentzian function, L(H−HFMR) =S∆H2/[(H−HFMR) +∆H2] +A[∆H(H− HFMR]/[(H−HFMR) +∆H2]. Where SandAare the symmetric and antisymmetric amplitudes. The spin pumping voltage is measured at the end of the Pt strips by a lock-in amplifier, triggered with the VNA. The microwave is then switched from low power Plow r f=−25 dBm to high power Phigh r f=16 dBm, with a modulation frequency of 27 .71 Hz. The (waveguide stripline + sample) is positioned between two poles of an electromagnet such that the external magnetic field (H)and the microwave field (hr f)are perpendicular to each other, and both are in applied in the plane of the YIG film. All the measurements were performed at room temperature. 5 6 7 8 9 10-0.04-0.03-0.02-0.010.000.01S21 (a.u) Magnetic field (mT) FMR at 1 GHz Asymmetric Lorentzian 0 20 40 60 80 100012345 Bulk YIG FMR Kittel equationFrequency (GHz) Magnetic field (mT)(a) (b) 113 114 115 116 117 118-0.18-0.16-0.14-0.12-0.10-0.08-0.06-0.04-0.020.000.020.04S21 (a.u) Magnetic field (mT) FMR at 5 GHz Asymmetric Lorentzian (c) FIG. S2. a)Field resonances of the bulk YIG FMR for low frequency. b)andc)are S21 measurements of the bulk YIG FMR at 1 GHz and 5 GHz, respectively. Our external magnetic field step corresponds to 0.1 mT. The error bar corresponds to the standard deviation from the multiple VNA readings. In spintronics, the FMR drives the spin pumping effect (SPE),[3] where a flow of spin current, occurs from the ferromagnetic/ferrimagnetic layer towards an adjacent layer at the peak of the microwave absorption during the FMR process.[4] Therefore, the YIG film injects a spin current by means of the SPE into the Pt strip.[3] That spin current can be expressed as Js=g↑↓ e f f¯hω 4π/parenleftbigghr f ∆H/parenrightbigg2 L(H−HFMR), (S1) where g↑↓ e f fis the effective spin mixing conductance, ω=2πfis the r ffrequency, ¯his the reduced Planck constant, and L(H−HFMR)is the FMR absorption, usually a Lorentizian-like line shape. The spin current along ˆ zwith spin polarization ⃗σalong ˆ xis converted into charge current along ˆ y 3direction by the inverse spin Hall effect (ISHE), following ⃗Jc∝θPt(⃗Js×⃗σ).[5] Where θPtis the spin Hall angle, which quantifies the conversion efficiency between spin and charge currents. The total voltage build-up at the edge of the Pt strip can be expressed by[5, 6] VSP=θPtRPtλPtwPt tPt/parenleftbigg2e ¯h/parenrightbigg tanh/parenleftbiggtPt 2λPt/parenrightbigg Js, (S2) where RPt,tPt,wPtandλPtare the resistance, thickness, width, and the spin diffusion length of the Pt strip. The spin pumping process typically leads to an increase in the FMR linewidth as it introduces an additional component to the magnetic losses of the ferromagnetic layer. However, in our measurements, we did not observe any significant broadening of the linewidth in the spin pumping voltages measured on the Pt strip. This absence of pronounced linewidth broadening in VSP rcan be attributed to the fact that the Pt strip only covers a fraction of the YIG film. Cheng et al.demonstrated that for Pt strips with widths w<5µm, the spin pumping voltage is dominated by the spin current injected in the proximity of the Pt strip, within a YIG region unaffected by the presence of the Pt layer. In this region, the damping and linewidth correspond to the bulk values. Hence, the spin pumping voltage measured with the Pt strip effectively acts as a localized FMR absorption detector or an rf antenna.[7] One can realize that an additional spin pumping voltage could be detected by the Pt strip within the field range corresponding to the resonance of the Py layer, resulting from spin pumping from Py towards the YIG film. However, in our measurements, we observed that the voltage remained below the noise level throughout the resonance field of Py. As a result, our main focus in this study is on the resonance of the YIG film. II. ADDITIONAL CA VITIES MEASUREMENTS Additional spin pumping measurements were conducted on several other cavities to demonstrate the reproducibility of the method. Optical microscope images of the cavities with different widths are shown in Figure S3 a) and b). The main results presented in the text are based on the cavity denoted as "Cavity A," which has a width of w=2.0µm. Subsequently, we present results for two separate cavities: "Cavity B" and "Cavity C," both with a width of w=1.6 µm, as well as "Cavity D", which has a width of w=3.7µm. Figure S3 c) shows the average peak spacing, average peak linewidth, and finesse as a function of the inverse square of the cavity 4105 110 115 120 125 130 1350.00.51.01.52.0SP voltage ( mV) Magnetic field (mT) Cavity strip 105 110 115 120 125 130 1350.00.51.01.5SP voltage ( mV) Magnetic field (mT) Cavity strip 105 110 115 120 125 130 1350.00.51.01.5SP voltage ( mV) Magnetic field (mT) Cavity strip𝑤=1.6𝜇m 𝑤=3.7𝜇m (a) (b)10 𝜇m 10 𝜇m (d) (e) (f) 𝑤=2.0𝜇m𝑤=2.0𝜇m 𝑤=3.7𝜇m 𝑤=1.6𝜇m(c) 0.0 0.1 0.2 0.3 0.41234Average peak spacing (mT) 1/w2(mm-2)0.10.20.30.4 Average peak linewidth (mT) 0510152025 Finesse, Fm 3.7 µm 2.0 µm 1.6 µmFIG. S3. a)andb)show optical images from different devices fabricated with distinct distance cavity widths, indicated in the figure. c)Average peeks spacing in (mT), average peak linewidth, and calculated finesse as a function of the inverse square of the cavity width (w).d),e)andf)shows B-field scan of the spin pumping voltage in the central platinum strip at 5 GHz, for a cavity width of w=1.6µm,w=2.0µm, andw=3.7µm, respectively. width. It can be observed that the peak spacing shows a linear dependence as function of 1 /w2 within the error bar, following the expected behavior for a magnonic cavity.[8, 9] The linear behavior is also evident in the average peak linewidth, suggesting that increasing the aspect ratio of the cavity ( l/w) leads to a more homogeneous distribution of cavity modes. However, it should be noted that our results are limited to only three different cavity widths, making it challenging to make a definitive statement about the observed behavior. Figure S3 d) to f) show the B-field scan of the spin pumping voltage for three cavities widths for 5 GHz. As the width decreases, a few observations can be made. Firstly, the peak linewidth decreases, indicating a narrower spectral distribution of the spin pumping voltage. Secondly, the peak spacing increases, suggesting a larger separation in corresponding excitation frequencies. Finally, the spin pumping voltage height becomes more equal for the cavity with w=1.6µm. Figure S4 a) to c), shows the B-field spin pumping voltage for the remote cavity and cavities B and C. All three measurements were obtained simultaneously, with each cavity strip connected to an independent lock-in amplifier. Both cavities, B and C, have been designed with the same 555 60 65 70 75 80SP Voltage ( mV) Magnetic field (mT) Remote strip Cavity strip B Cavity strip C1 mV3.5 GHz 140 145 150 155 160 165SP Voltage ( mV) Magnetic field (mT) Remote strip Cavity strip B Cavity strip C6.0 GHz 1 mV 180 185 190 195 200 205SP Voltage ( mV) Magnetic field (mT) Remote strip Cavity strip B Cavity strip C1 mV7.0 GHz(a) (b) (c) (d) (e) Cavity strip B Cavity strip C -20 -15 -10 -5 0 5 10 15 2023456789Frequency (GHz) m0H - m0HFMR (mT)01 -20 -15 -10 -5 0 5 10 15 2023456789Frequency (GHz) m0H - m0HFMR (mT)01FIG. S4. a)toc)shows the B-field scan of the spin pumping voltage obtained for the remote strip and the Pt strips placed along the center of the cavity B and C, respectively. d)ande)shows the spin pumping intensity spectra for two different cavities with the same width distance of w=1.6µm. width of w=1.6µm. This explains why the field position of almost every peak aligns with each other. Once again, it is evident that the corresponding peak for the bulk YIG resonance is absent or reduced to the noise level at 6 GHz and 7 GHz. This observation highlights that the voltage peaks are primarily determined by the magnon modes confined within the cavity. One can also observe a small alternation of the intensity between consecutive peaks in Figure S4 a) to c). A similar trend seams to be present in Figure 3 of the main text. This behavior might occur due to the difference in microwave excitation corresponding to odd and even modes. However, we cannot explain them in detail without more detailed micromagnetic simulations, taking into account the exchange and dipolar interactions in the YIG/Py bilayer, and then correctly addressing each peak to the corresponding magnon mode and intensity. 6The spin pumping intensity spectra for cavities B and C are present in Figure S4 d) and e), with a frequency spacing of 0.5 GHz. Although the peak between 5.5 GHz and 6.5 GHz in cavity C is difficult to identify due to the noise, both cavities exhibit a similar peak dispersion, with almost every peak matching at each frequency. The solid black lines in Figure S4 d) and e) are the modes calculated by equation (1) and (2) from the main text, up to n=6 using the following parameters: w=1.6µm,l=30µm,M=130 kA/m, µ0H=14 mT, and γ/2π=26.5 GHz/T. We believe the small diminishment in the gyromagnetic ratio could be due to other factors and effects not included in the model, such as the boundary interface with the YIG/Py bilayer. These results confirm the reproducibility of the technique. III. CONTROL SAMPLES In Figure 3 d) on the main text, one can identify a secondary peak at 7 GHz. This secondary “peak” at the left side of the spin-pumping-FMR peak might be caused by the excitation of the finite kbulk-magnons, corresponding to the spin wave modes in the whole YIG sample. The corresponding peak is also not observed in the FMR absorption. This secondary structure is less evident or absent in another remote Pt strip using a different YIG sample, shown in Figure S5, as well as in Figure S6. We believe this structure could be due to the magnon interferences in the bulk-YIG region. This does not affects the analysis of the resonance modes of the cavity. 110 112 114 116 118 1200.00.30.60.91.2SP voltage ( mV) Magnetic field (mT) 5 GHz 178 180 182 184 186 1880.00.30.60.91.2SP voltage ( mV) Magnetic field (mT) 7 GHz 214 216 218 220 222 2240.00.30.60.91.2SP voltage ( mV) Magnetic field (mT) 8 GHz(a) (b) (c) FIG. S5. Less intense or not present "secondary" peak in a Remote Pt strip on a different 100 nm thick YIG sample at a)5 GHz, b)7 GHz, and a)8 GHz. Further experiments using rectangular microwave waveguides may avoid surface mode excitations in the bulk YIG film. Figure S6 a) shows the absence of magnetostatic modes in a control device where an Au film replaces the Py film. This result rules out the possibility of (multiple) peaks originating from 70 50 100 150 200 2500246810 Bulk FMR Spin pumping of remote strip Spin pumping of gold squares Kittel curveFrquency (GHz) Magnetic field (mT) 140 160 180 200 220 2400.00.20.40.60.8Spin Pumping voltage ( mV) Magnetic field (mT) 6 GHz 7 GHz 140 160 180 200 220 2400.00.20.40.60.8Spin Pumping voltage ( mV) Magnetic field (mT) 7 GHz 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 of Kittel fitting between FMR of YIG film, spin pumping voltage in remote strip, and when Py square is replaced with Au. c)Spin pumping voltage with a proximity strip, no evidence of cavity resonant modes was observed. microwave artifacts caused by the proximity of a metallic film to the Pt strip.. Figure S6 b) shows the direct comparison of the Kittel equation for FMR of the YIG bulk, the spin pumping voltage of theremote Pt strip, and the spin pumping voltage of the Pt strip when Py is replaced with gold. The adjusted Kittel curves show only a slight deviation. These results also suggest that the magnetic nature of the exchange/dipolar interaction between the YIG and Py films is crucial for constructing the cavity through engineered-design lithography. The spin pumping voltage was also measured in a Pt strip located near the left side of the Py square, VSP pin Figure 1 b) of the main text. However, despite the higher noise level, the spin pumping voltage as a function of the B-field did not exhibit any evidence of (multiple) peaks, as shown in Figure S6 c). It becomes evident that the presence of YIG|Py interfaces on both sides is necessary to create the magnon resonant modes within the cavity. In Figure 1 c) of the main text, one can identify a Pt strip placed underneath the middle of the Py square. The electrical resistance of that strip was in the M Ωrange, much higher than the usual 3 .5kΩfor 400 nm wide, 35 µm long, and 8 nm thick Pt strip. This suggests that the Pt strip may have been damaged during the sputtering and lift-off process involved in the fabrication of the Py square. As a result, we do not have reliable data for the electrical contact of that particular strip. [1] M. Mruczkiewicz, P. Graczyk, P. Lupo, A. Adeyeye, G. Gubbiotti, and M. Krawczyk, Spin-wave nonreciprocity and magnonic band structure in a thin permalloy film induced by dynamical coupling 8with an array of Ni stripes, Physical Review B 96, 104411 (2017). [2] A. Talapatra and A. Adeyeye, Linear chains of nanomagnets: engineering the effective magnetic anisotropy, Nanoscale 12, 20933 (2020). [3] Y . Tserkovnyak, A. Brataas, and G. E. Bauer, Enhanced gilbert damping in thin ferromagnetic films, Physical Review Letters 88, 117601 (2002). [4] S. Mizukami, Y . Ando, and T. Miyazaki, Ferromagnetic resonance linewidth for NM/80NiFe/NM films (NM= Cu, Ta, Pd and Pt), Journal of Magnetism and Magnetic Materials 226, 1640 (2001). [5] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Spin Hall effects, Reviews of Modern Physics 87, 1213 (2015). [6] A. Azevedo, L. Vilela-Leão, R. Rodríguez-Suárez, A. L. Santos, and S. Rezende, Spin pumping and anisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment, Physical Review B83, 144402 (2011). [7] Y . Cheng, A. J. Lee, G. Wu, D. V . Pelekhov, P. C. Hammel, and F. Yang, Nonlocal uniform-mode ferromagnetic resonance spin pumping, Nano Letters 20, 7257 (2020). [8] T. Yu and G. E. W. Bauer, Efficient gating of magnons by proximity superconductors, Physcal Review Letters 129, 117201 (2022). [9] Y . W. Xing, Z. R. Yan, and X. F. Han, Magnon valve effect and resonant transmission in a one- dimensional magnonic crystal, Physical Review B 103, 054425 (2021). 9

2023-06-24

Confining magnons in cavities can introduce new functionalities to magnonic devices, enabling future magnonic structures to emulate established photonic and electronic components. As a proof-of-concept, we report magnon confinement in a lithographically defined all-on-chip YIG cavity created between two YIG/Permalloy bilayers. We take advantage of the modified magnetic properties of covered/uncovered YIG film to define on-chip distinct regions with boundaries capable of confining magnons. We confirm this by measuring multiple spin pumping voltage peaks in a 400 nm wide platinum strip placed along the center of the cavity. These peaks coincide with multiple spin-wave resonance modes calculated for a YIG slab with the corresponding geometry. The fabrication of micrometer-sized YIG cavities following this technique represents a new approach to control coherent magnons, while the spin pumping voltage in a nanometer-sized Pt strip demonstrates to be a non-invasive local detector of the magnon resonance intensity.

Magnon confinement in an all-on-chip YIG cavity resonator using hybrid YIG/Py magnon barriers

2306.14029v2

1 Leveraging symmetry for an accurate spin-orbit torques characterization in ferrimagnetic insulators Martín Testa-Anta1,*, Charles- Henri Lambert2, Can Onur Avci1,* 1Institut de Ciència de Materials de Barcel ona (ICMAB-CSIC), Campus de la UAB, 08193 Bellaterra, Spain 2Department of Materials, ETH Züri ch, Hönggerbergring 64, CH-8093 Zürich, Switzerland Abstract Spin-orbit torques (SOTs) have emerged as an e fficient means to electrically control the magnetization in ferromagnetic het erostructures. Lately, an in creasing attention has been devoted to SOTs in heavy metal (HM)/magnetic insulator (MI) bilayers owing to their tunable magnetic properties and insulating natu re. Quantitative characterization of SOTs in HM/MI heterostructures are, thus, vita l for fundamental understandi ng of charge-spin interrelations and designing novel devices. Howe ver, the accurate det ermination of SOTs in MIs have been limited so far due to sma ll electrical signal outputs and dominant spurious thermoelectric effects caused by Joule heating. Here, we report a simple methodology based on harmonic Hall voltage det ection and macrospin simulations to accurately quantify the damping-like and field-lik e SOTs, and thermoelectric contributions separately in MI-based systems. Experiment s on the archetypical Bi-doped YIG/Pt heterostructure using the developed method yi eld precise values for the field-like and damping-like SOTs, reaching -0.14 and -0.15 mT per 1.7 ൈ1011 A/m2, respectively. We further reveal that current-induced Joule heat ing changes the spin transparency at the interface, reducing the spin Hall m agnetoresistance and damping-like SOT, simultaneously. These results and the devis ed method can be beneficial for fundamental understanding of SOTs in MI- based heterostructures and designing new devices where accurate knowledge of SOTs is necessary. 2 I. INTRODUCTION Spin-orbit torques (SOTs) are the generic nam e given to the current -induced torques in ferro-/ferrimagnetic heterostructures with la rge spin-orbit couplin g and broken inversion symmetry mainly driven by, but not limited to, bulk spin Hall (SHE) and interfacial Rashba- Edelstein effects.1 During the past decade, SOTs have become the state-of-the-art magnetic manipulation met hod by electrical currents and enabled magnetization switching,2,3,4 domain wall and skyrmion motion,5,6,7 steady-state magnetic oscillations,8,9 and magnon generation/suppression10,11 in convenient device geometries. These experimental achievements have given rise to a multitude of spintronics device concepts with memory, logic, signal transmission and co mputing functionalities suitable for the complementary metal-oxide-se miconductor (CMOS) industry and the post-CMOS computing era.12,13,14,15 SOTs have been originally discove red and extensively studied in all-conducting heavy metal (HM)/ferromagnetic (F M) bilayers such as Pt/Co, Ta/CoFeB and W/CoFeB.16,17,18,19,20 However, the research has rapidly expanded into other materials such as antiferromagnets,21,22,23 ferrimagnets,24,25,26 topological insulators27,28,29 and magnetic insulators (MIs)4,30,31, among others. Recently, the interest in MIs in the SOTs context has rapidly grown.5,32,33,34 MIs offer many advantages over their conducting counterparts thanks to thei r tunable magnetization and anisotropy, low Gilbert damping35, and long spin diffusion lengths36. These attributes, together with their insulating nature leading to reduced Ohmi c losses when integrated into microelectronic circuits, make MIs a highly a ttractive material pl atform for low power magnonic and spintronic applications.37,38,39 Within the family of MIs, yttrium iron garnet (Y3Fe5O12, YIG) is particularly appealing due to its record low Gilbert damping (~10-4 - 10- 5), making it ideal for nano-oscillators,40,41 and magnonic devices,42,43,44,45 with SOT- enabled operation. Despite the vi tal importance of SOTs in the integration of MIs in potential spintronic concepts, their accurate and straightforward characterization by simple electrical methods is intrinsi cally difficult. Harmonic Hall voltage (HHV)4,16,17,46 and spin-torque ferromagnetic resonance47 measurements are the tw o established methods in this context but rely on the Hall resi stance and magnetoresistance output signals, respectively, which are orders of magnitude sm aller in HM/MI systems with respect to all- metallic systems. The few existing studies have considered only the damping-like (DL)- SOT characterization,4,46,48,49,50 but the separate quantification of the field-like (FL)-SOT, equally important for the SOT- driven magnetization dynamics,51,52,53,54,55,56 and a proper account of current-induced Joule heating in t hese measurements and interfacial spin transport properties remained elusive thus far. In this article, we show a simple method fo r an accurate SOTs quanti fication in MIs relying on the combination of HHV m easurements and macrospin simu lations. We develop, and test with simulations, a HHV measurement scheme using a non-standard geometry, which allows us to exploit simple symmetry arguments to disentangle the effective fields due to the damping-like ( 𝐵) and field-like ( 𝐵ி) SOTs, and thermoelectric contributions precisely. The quantification of both SOTs components and ther moelectric effects yield 3 an intrinsic theoretical error as low as ~0.2% on the simulated data. As a proof-of- concept, we apply the proposed a pproach on a Bi-doped YIG/Pt bilayer with in-plane (IP) magnetic anisotropy obtaining 𝐵 = -0.15 mT and 𝐵ி = -0.14 mT and a thermoelectric field of 𝐸 = 0.27 V/m per 𝑗 = 1.7ൈ1011 A/m2 injected current, all values in the range expected of such systems. Finally, current -dependent measurements show that Joule heating reduces the spin mixing conductance ( 𝐺↑↓) at the HM/MI and results in a systematic reduction of the spin Hall magnetoresistance (SMR) and 𝐵 up to ~15%, simultaneously. II. EXPERIMENTAL DETAILS Sample preparation . An ~18-nm-thick Bi-doped YIG (Bi:YIG from hereon) layer was rf- sputtered from a stoichiometric target ont o a single-crystal Sc-substituted gadolinium gallium garnet substrate (Gd 3Sc2Ga3O12, GSGG) with a base pressure <5 ൈ10-8 Torr. The growth temperature and the Ar partial pressure during th e Bi:YIG deposition were 800 ℃ and 1.5 mTorr, respectively. After the deposit ion, the samples were annealed for 30 min at the deposition temperatur e and subsequently cooled down to room temperature in vacuum. In order to have a clean interface, a 4 nm-thick Pt was dc-sputtered in-situ at room temperature without break ing the vacuum. The Ar pre ssure during the Pt deposition was 3 mTorr. After the Pt deposition, the c ontinuous Bi:YIG/Pt layers were patterned into Hall bar structures by means of standar d photolithography followed by top-down ion milling. The lateral dimensi ons of the Hall bars were 10 μm (current line width) ൈ 30 μm (distance between two Hall crosses). Hall effect measurements . HHV measurements were performed by applying an AC voltage along the current line modulated at 𝜔/2𝜋 = 1092 Hz and simultaneously measuring the first and second harmonic Hall vo ltage responses with a lock-in amplifier. The current amplitude through t he device under test was determined by connecting a 10 Ω resistor in series and reading the voltage drop across with a digita l multimeter before or during the measurements. The acquired HHVs were converted into Hall resistances using the relation 𝑅 ఠு = 𝑉ఠு/𝐼 for comparability with the liter ature. For the field scans, an out-of-plane (OOP) or in-plane (IP) DC magnetic field was swept in the േ600 mT or േ200 mT range, respectively, while keeping the azimuthal angle ( 𝜑) fixed. For the polar ( 𝜃) angle scans, the sample was rotated in the pres ence of a constant DC field in the 0º-360º range using a motorized rotation stage at an angular speed of 3 degrees/second. All measurements were performed at room temperature and a ll the data presented were averaged over five scans to impr ove the signal-to-noise ratio, unless specified otherwise. 4 III. RESULTS AND DISCUSSION A. Structural and ma gnetic characterization A structural characterization of the as-dep osited Bi:YIG/Pt heterostructure was carried out by means of high-resolution X-ray di ffraction (XRD). Figure 1a shows a symmetric 𝜃- 2𝜃 scan around the (444) diffraction peak of the GSGG substrate, occurring at 50.44º. The data do not show a clear emergence of t he Bragg reflection for the Bi:YIG phase, but only a small shoulder at the right side of the substrate peak. Such observation may be attributed to a reduced crystallinit y of the magnetic film or, more likely in our case, to the overlap between the substrate and Bi:YIG (444) reflections. Indeed, in the absence of interfacial strain, Bi-substitution leads to an increase in the cubic lattice parameter, resulting in a lower 2 𝜃 position compared to the bul k YIG (expected at 51.09º)57. Additionally, the fact th at the Bi:YIG peak is not shifted to lower 2 𝜃 values with respect to the substrate indicates that: i) its OOP lattice parameter is smaller than that of GSGG, and ii) the tetragonal distortion induced by the latti ce mismatch with th e substrate is not significant.58 Owing to the negative (111) magnetostricti on constant for this particular iron garnet,59 this absence of moderate interfacial strain will favor the occurrence of IP magnetic anisotropy. X-ray reflectivity (XRR) analysis was performed in order to check the thickness of Bi:YIG confirming the antici pated value of 18 nm. The sample topography was addressed through atomic force micr oscopy (AFM) before and after the Pt deposition. Figure 1b is a repr esentative AFM image acquired from the Bi:YIG/Pt sample. The sample surface is rather flat with an RMS roughness of 0.66 േ0.13 nm, similar to the roughness before Pt deposition (<1 nm, not s hown). This number was estimated by averaging measurements of five different regions on the sa mple, which highlights the overall good quality of the films. The IP hysteresis loop recorded at 300 K us ing the superconducting quantum interference device (SQUID) magnetometry is displa yed in Figure 1c. The large remanence accompanied by a negligible coercivity (<1 mT) are characteristic of IP magnetized garnets due to their small magnet ocrystalline anisotropy. Prefer ential IP magnetization of Bi:YIG was also corroborated by magneto-optical Kerr effect (MOKE) measurements as shown in Figure 1d. Data recorded in polar (signal proportional to the OOP component of the magnetization) and longit udinal (signal proportional to the IP component of the magnetization) MOKE c onfigurations with corresponding fi eld sweeps clearly show that the IP and OOP are the easy and hard axes, res pectively. Additional longitudinal MOKE measurements at different az imuthal angles showed a negligible difference (data not shown), hence, the sample is hereafter assu med to exhibit easy-plane anisotropy. Notably, the experimental value of the saturation magnetization ( 𝑀௦~85 emu/cm3) lies well below that of the bulk YIG ( ~140 emu/cm3 at 300 K).60,61 Such reduction can be explained considering the Bi3+ ↔ Y3+ substitution at the dodecahedral sites, as this will expand the crystal lattice and weaken the superexchange interact ions between the Fe3+ cations located at the tetrahedral and octahedr al sites, primarily responsible for the ferrimagnetic behavior in iron garnets.62 Another possibility re lates to an oxygen off-5 stoichiometry, occurring due to the annealing at very low O 2 pressures. Under these conditions, the formation of oxygen va cancies has been reported to proceed via Fe3+ to Fe2+ reduction, giving rise to a decrease in magnetization.63 Nevertheless, our results are consistent with literature values for Bi :YIG thin films grown on GGG or GSGG substrates,62,64 and exploring the underlying cause of the reduced magnetization is beyond the scope of the present study. Figure 1. Structural and magnetic characterization of the Bi:YIG/Pt sample: (a) 𝜃-2𝜃 scan of the Bi:YIG (18 nm)/Pt (4 nm) bilayer deposited onto a (111)-oriented GSGG substrate; (b) Representative AFM image of the prev ious film, revealing a rms roughness ( 𝑆) value of 0.66േ0.13 nm (averaged over different regi ons in the sample); (c) In-plane magnetization as a function of an external magnetic field (a fter subtraction of the linear paramagnetic background), obtained by SQUID magnetometry; (d) MOKE response as a function of an in-plane (red, top axis) or out-of-plane (bl ue, bottom axis) magnetic field in their respective measurement ge ometries (i.e., longitudinal and polar), which confirms the in-plane anisotropy of the prepared sample. 6 B. First harmonic Hall characterization The magnetotransport pr operties of Bi:YIG/Pt were in ferred through SM R measurements and reported in Figure 2. In Figure 2a, a curr ent injection through t he Pt layer along the x-axis produces a pure spin current due to the SHE and other po ssible charge-spin conversion mechanisms with polarization along the y-axis and flowing along the z-axis reaching Bi:YIG. The absorption and reflection of this spin current at the Bi:YIG/Pt interface modulates the longitudinal and transverse resistance of Pt depending on the relative orientation between the magnetization and spin accu mulation, which lies at the origin of SMR in HM/MI bilayers.65 The SMR manifests itself on the Hall voltage (which we mainly use in this study) wit h signals proportional to both IP ( ∝𝑚௫𝑚௬) and OOP ( ∝ 𝑚௭) components of t he magnetization.65,66 Therefore, these two SM R contributions exhibit the very same symmetry as the planar and anomalous Hall resistances in conducting magnetic materials, respectively, but with ty pical values several orders of magnitude lower. According to the angle definitions given in Figure 2a, the Hall resistance at the first harmonic of the current frequency, 𝑅ఠு, can be expressed as follows: 𝑅ఠுൌ𝑅ௌெோsinଶ𝜃ெsinሺ2𝜑ெሻ𝑅ௌெோିுா cos𝜃ெ𝑅ைுா𝐵௫௧cos𝜃 ሺ1ሻ where 𝑅ௌெோ, 𝑅ௌெோ ,ுா and 𝑅ைுா represent the resistance m odulation due to SMR, SMR- induced anomalous Hall effect (SMR-AHE ) and the ordinary Ha ll effect (OHE), respectively. 𝐵௫௧ and 𝜃 are the magnitude and polar angle of the external magnetic field whereas 𝜃ெ and 𝜑ெ are the polar and azimuthal angles of the magnetizat ion vector. Note that the OHE is t he property of the Pt itse lf and has, in principle, no relationship with the magnetic layer underneath. Figure 2b shows the first harmo nic Hall resistance in a swept OOP field. Besides the linear OHE contribution, the signal difference at positive and negative field at high amplitudes reflect the magnetiz ation vector switching betw een the up and down states. By symmetry operations with respect to 𝐵௭ = 0, we find that the magnetization saturates along the z-axis at around 𝐵௭ = 45 mT, which we denote as 𝐵௦௧. Since the Bi:YIG layer possesses in-plane anisotropy, this value is equivalent to the su m of the demagnetizing field and the effective perpendicular anisotropy field possibly due to the reminiscent OOP anisotropy driven by a small lattice mism atch undetected in the XRD measurements. From the experimental data we find 𝑅ௌெோିுா = -1.36 m Ω for an rms cu rrent density of 1.7ൈ1011 A/m2, which yields 𝜌ௌெோିுா = -5.4ൈ10-12 Ωꞏm. A symmetric component is likewise visible in the OOP fi eld-scan, responsible for t he inverse U-shape signal and spikes observed at low fields. This additional contri bution is ascribed to the reorientation of the magnetization vector within the film plane for 𝐵௭<𝐵௦௧ and possibly domain formation at very small fields le ading to extra contributions in 𝑅ఠு due to the non-zero 𝑚௫𝑚௬ components. 7 IP field sweep measurements at different azimuthal angles ( 𝜑) are depicted in Figure 2c. The modulation of 𝑅ఠு due to SMR shows a maximum (minimum) at 𝜑 = 45º (135º), consistent with the positive 𝑅ௌெோ coefficient in YIG/Pt.4 The 𝜑-dependence of the data is displayed in Figure 2d. The fitting following eq. 1 reveals a 𝑅ௌெோ value of 11.23 m Ω, corresponding to 𝜌ௌெோ = 4.5ൈ10-11 Ωꞏm. Note that we assume 𝜑ൎ𝜑ெ due to easy-plane anisotropy, which is further corroborated by t he similar saturation fields observed for all IP field-scans. Our results show therefore that 𝑅ௌெோ is much larger than 𝑅ௌெோିுா (a factor 8.25), as customary in MI/Pt bilayers. Figure 2. Harmonic Hall characterization of the Bi:YIG/Pt sample: (a) Schematic representation of the device geometry and corresponding coordinate system; Transverse Hall resistance measured upon sweeping an exte rnal (b) OOP or (c ) IP (at different 𝜑 angles) magnetic field; (d) 𝜑-dependence of the transverse resistance measurements summarized in c, fitted to a sin2 𝜑 function in accordance with eq. 1. All measurements were conducted at a current density (rms) of 𝑗 = 1.7ൈ1011 A/m2. Note that a constant offset has been subtracted fr om the raw data shown in b and c. 8 C. Description of the SOTs and macrospin model Macrospin simulations are a useful tool not onl y to verify that the experimental data can be reproduced with existing theoretic al models, but also to a scertain the symmetry of the different contributions to the current-induced fields. It is esta blished that SOTs consist of two distinct components acting on the magnetization such as a field-like torque with symmetry 𝐓𝐅𝐋∝𝐦ൈ𝐲 , and a damping-like torque defined as 𝐓𝐃𝐋∝𝐦ൈሺ𝐲ൈ𝐦ሻ. Here, 𝐦 stands for the unit m agnetization vector and 𝐲 the in-plane axis transverse to the current flow (see Figure 2a). The effect of the 𝐓𝐅𝐋 is therefore equivalent to an in-plane field 𝐁𝐅𝐋 acting along 𝐲, whereas that of 𝐓𝐃𝐋 is an effective field 𝐁𝐃𝐋 perpendicular to the magnetization, with rotati onal symmetry within the 𝑧𝑥-plane. Upon applying an AC current, these current-induced SOTs will induce periodic oscillations to the magnetization about its equilibrium position at the same AC frequency, dict ated by the balance between the demagnetizing ( 𝐁𝐝𝐞𝐦), anisotropy ( 𝐁𝐚𝐧𝐢) and external ( 𝐁𝐞𝐱𝐭) fields. In a macrospin approximation, we can then write the total torque ( 𝐓𝐭𝐨𝐭) as: 𝐓𝐭𝐨𝐭ൌ𝐓𝐝𝐞𝐦𝐓𝐚𝐧𝐢𝐓𝐞𝐱𝐭𝐓𝐅𝐋𝐓𝐃𝐋 ൌ 𝑀௦ሺ𝐦ൈ𝐁 𝐝𝐞𝐦𝐦ൈ𝐁 𝐚𝐧𝐢𝐦ൈ𝐁 𝐞𝐱𝐭െ𝐵ி𝐦ൈ𝐲െ𝐵 𝐦ൈ𝐲ൈ𝐦 ሻ ሺ2ሻ In the present work, we simulated the equiva lent first and second harmonic signals using the Scilab open source software for numerical computations.67 More specifically, we performed a polar angle scan of the external field ( 𝜃) in the 0º-360º range beyond the OOP saturation field of the magnetizat ion, applied at a fi xed azimuthal angle 𝜑. The equilibrium configuration ( 𝜃ெ, 𝜑ெ) is calculated for each set of ( 𝜃, 𝜑, 𝐁𝐞𝐱𝐭) by minimizing eq. 2. Subsequently, the Hall resistance is com puted according to the following relations: 𝑅ூାுൌ𝑅ௌெோsinଶ𝜃ெsin2𝜑ெ𝑅ௌெோିுா cos𝜃ெ𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ3ሻ 𝑅ூିுൌെ 𝑅ௌெோsinଶ𝜃ெsin2𝜑ெെ𝑅ௌெோିுா cos𝜃ெ𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ4ሻ where 𝑅ூାு and 𝑅ூିு constitute the Hall resistances corresponding to a positive and negative DC current, respectively. Here we note that reversing the current direction changes the sign of the Hall effect coefficients but not the last term related to the current- induced thermoelectric contribution, which re mains constant irrespective of the current direction. The unavoidable occurrence of J oule heating primarily creates a vertical temperature gradient, owing to the large t hermal conductivity differences between the substrate and the air surrounding the device (see Figure 3a, right panel). This will create spin Seebeck effect (SSE) in the MI68 and subsequently an inve rse SHE voltage in Pt, whose contribution to the Hall resist ance is quantified through the parameter 𝑟ௌௌா and added to eq. 3 and 4. We then find the equival ent first and second harmonic resistances by applying the followi ng operations to the a bove computed signals: 𝑅ఠுൌ1 2ሺ𝑅ூାுെ𝑅ூିுሻ ሺ5ሻ 𝑅ଶఠுൌ1 2ሺ𝑅ூାு𝑅ூିுሻ ሺ6ሻ 9 D. Simulations of the second harmonic response Based on the model detailed above, we first discuss the harmonic signals occurring during 𝜃-scans at 𝜑 = 90º, which is the standard geomet ry for quantifying SOTs in MI- based heterostructures.4,17 The simulations were carried out using the experimental values of 𝐵௦௧, 𝑅ௌெோ and 𝑅ௌெோିுா (previously indicated in Section III-B), and, for symmetry comparison, the individual DL -SOT, FL-SOT and SSE contributions are separately depicted in Figures 3b- d (black traces). In this geometry, the SSE contribution vanishes since it is proportional to cos𝜑ெ, making thus 𝜑 = 90º an ideal geometry for the SOTs quantification. In this configuration 𝐵ி (𝐵) drives the OOP (IP) magnetization oscillations, such that they scale with 𝑅ௌெோିுா (𝑅ௌெோ), respectively (see Figure 3a). This justifies the much smaller contribution of 𝐵ி to the second harmonic response, which is about one order of magnitude lower than that of 𝐵 even though the same field amplitudes are inputted. In terms of symmetry, they both display a similar 𝜃-dependence as explained below. In the limit of small oscill ations of the magnetization and assuming a linear relationship between the field and the current, the sec ond harmonic resistance (for an angle-scan) can be expressed as follows:17 𝑅ଶఠுൌሾ𝑅ௌெோିுா െ2𝑅ௌெோcos𝜃ெsinሺ2𝜑ெሻሿ𝑑cos𝜃ெ 𝑑𝜃𝐵ூఏ cosሺ𝜃െ𝜃ெሻ𝐵 𝑅ௌெோsinଶ𝜃ெ𝑑sinሺ2𝜑ெሻ 𝑑𝜑𝐵ூఝ sin𝜃cosሺ𝜑െ𝜑ெሻ𝐵௫௧𝑟ௌௌாsin𝜃ெcos𝜑ெ ሺ7ሻ where 𝐵ூఏ (𝐵ூఝ) represents the component of the current-induced field ( 𝐵ூ) that induces a change in 𝜃ெ (𝜑ெ). The first term in eq. 7 accounts for the OOP oscill ations and will be counteracted by the demagnetiz ing field, hence the effect ive field is defined as 𝐵ൌ 𝐵௫௧𝐵ௗ. Note that, due to the negligible ani sotropy field of the Bi:YIG sample, 𝐵ௗ is herein approximated as ൎ𝐵௦௧. During the simulation of the 𝜃-scans, we set 𝐵௫௧ൌ120 mT, which is beyond the OOP saturation field of 45 mT. Thus, the magnetizat ion can be assumed to be saturated along 𝐵௫௧ throughout the entire angle-scan ( 𝜃ெൎ𝜃). Recalling as well that the sample displays easy-plane anisotropy ( 𝜑ெൎ𝜑), in the vicinity of 𝜑 = 90º eq. 7 reads: 𝑅ଶఠுൌ𝑅ௌெோିுா𝑑cos𝜃ெ 𝑑𝜃ெ𝐵ூఏ 𝐵௫௧𝐵ௗ𝑅ௌெோsin𝜃ெ𝑑sinሺ2𝜑ெሻ 𝑑𝜑ெ𝐵ூఝ 𝐵௫௧ ሺ8ሻ Since 𝐁𝐅𝐋 = 𝐵ி𝐲 and 𝐁𝐃𝐋 = 𝐵𝐲ൈ𝐦 , in the framework of a spherical coordinate system (defined by unit vectors 𝐞𝐫, 𝐞𝛉 and 𝐞𝛗) 𝐵ூఏ and 𝐵ூఝ are given by: 𝐵ூఏൌcos𝜃ெsin𝜑ெ𝐵ிcos𝜑ெ𝐵 ሺ9ሻ 𝐵ூఝൌcos𝜑ெ𝐵ிെcos𝜃ெsin𝜑ெ𝐵 ሺ10ሻ 10 Again, for an azimuthal angle 𝜑 = 90º one obtains 𝐵ூఏ = cos𝜃ெ𝐵ி and 𝐵ூఝ = െcos𝜃ெ𝐵. Taking these relations into account and computi ng the derivative terms in eq. 8, it follows that: 𝑅ଶఠுൌെ1 2𝑅ௌெோିுா sinሺ2𝜃ெሻ𝐵ிାை 𝐵௫௧𝐵ௗ𝑅ௌெோsinሺ2𝜃ெሻ𝐵 𝐵௫௧ ሺ11ሻ Equation 11 shows that, for the standar d geometry discussed above, both the 𝐵ி (including the Oersted field) and 𝐵 components acquire a sinሺ2𝜃ெሻ dependence, which is well reproduced by the macrospin simulati ons. An accurate discrimination between the DL and FL-SOT signals at the 𝜑 = 90º geometry is thus impossible by symmetry considerations. We note that the small di fference in the field dependences are very difficult to discriminate due to low signal output leve ls typical of this system. Hence, the reported approaches typi cally assume that 𝐵ிൎ𝐵.4,46 On that basis, and considering that for most MIs 𝑅ௌெோିுா ≪𝑅ௌெோ, the second harmonic resist ance is approximated as arising solely due to the DL-S OT, leaving FL-SOT unquantified. The estimation of the FL- SOT can be normally achieved by using the other standard geometry 𝜑 = 0º, where 𝐵ி drives the IP oscillatio ns and its contribution to 𝑅ଶఠு is maximum. However, since the SSE contribution is proportional to cos𝜑ெ, its contribution dominates the second harmonic signal making the 𝐵ி quantification highly inaccurate, if not impossible. In the above context, we suit ably find the Hall signals at 𝜑 = 85º and 95º as an alternative to circumvent the aforementione d issues. Indeed, deviations from 𝜑 = 90º induce a significant asymmetry in the 𝐵ி contribution, thereby provid ing a convenient tool for its quantification. We note that this will occur at the expense of nonzero thermoelectric effects, so that a reasonabl e compromise is found at 𝜑 = 85º and 95º fo r the parameter set considered in this study. 11 Figure 3. Symmetry-based toolbox herein proposed to disentangle the interplay between the DL, FL and SSE components. (a) Schemati c representation depicting the symmetry of the current-induced fields at 𝜑 = 90º geometry. Owing to t heir different symmetry with respect to 𝜑 = 90º, the second harmonic oscillati ons originating from the previous individual contributions were simulated as a function of the azimuthal angle at 𝜑 = 85º and 95º (top panel, b-d), and their average (m iddle panel, e-g) and difference (bottom panel, h-j) are herein proposed as reference si gnals for spin-orbit torque quantification. 12 E. Separation of the DL-SOT, FL-SOT and thermoelectric signals Figures 3b-d show simulated second harmonic 𝜃-scan curves at 𝜑 = 85º, 90º and 95º (red, black, and blue lines). We discussed the standard case of 𝜑 = 90º in the previous section and the problems associ ated with its consideration fo r the SOT quantification. For the 𝜑 = 85º and 95º case, the first important obse rvation is the nearly insensitivity of the DL-SOT component (Fig.3b) to small 𝜑 variations around 𝜑 = 90º. These changes account for 0.6% and will fall well below the detection limit in any given harmonic measurement. The contribution from the FL-SOT is, however, significantly distorted with respect to the 𝜑 = 90º data and is amplified by a factor of 2.7. Moreover , and importantly, it has an opposite sign considering the 𝜃 ~90º and 270º data points as reference. Finally, the SSE contribution becomes now finite and has an opposite sign between 𝜑 = 85º and 95º. In an actual measurement, the signal will be a convolution of these three contributions. To separate the individual components, we proceed with the addition/ subtraction of the transverse signals at 𝜑 = 85º and 95º as the key step to disentangle the signals with different origins. The average (difference) of the second ha rmonic response at these two 𝜑 angles, hereafter denoted as 𝑅ଶఠ଼ହାଽହ (𝑅ଶఠ଼ହିଽହ) for simplicity, is depicted in Figures 3e- g (Figures 3h-j). As expected, 𝑅ଶఠ଼ହାଽହ exhibits the same sy mmetry and magnitude as in the 𝜑 = 90º case for all three components, with a calculated variation as small as 0.6% and 0.3% for the DL and FL-SOT contributions. On the other hand, 𝑅ଶఠ଼ହିଽହ effectively suppresses the contribution from the DL-SOT, resulting in a combined response of the FL-SOT and SSE effects. It can be observed that 𝑅ଶఠ଼ହିଽହ displays a maximum (minimum) at 𝜃 = 90º (270º) for the latter two components. Note that in the SSE case it is a direct consequence of the sin𝜃ெ dependence as 𝜃ெൎ𝜃 holds throughout the entire scan. Further separation between the FL-SOT and SSE can be achieved analyzing the 𝑅ଶఠ଼ହିଽହ variation with the external field amplitude. For increasing 𝐵௫௧, the amplitude of the 𝐵ி- induced oscillations are reduc ed, as the magnetization beco mes more strongly coupled to the field direction. The SSE constitutes instead a static effect, which depends on the magnetization orient ation but not on t he field amplitude.17 The exact external field dependence of 𝑅ଶఠ଼ହିଽହ at 𝜃 = 90º can be ascertained from the second harmonic analysis. In the vicinity of 𝜃 = 90º, eq. 7 reduces to: 𝑅ଶఠுൌ𝑅ௌெோିுா𝑑cos𝜃ெ 𝑑𝜃ெ𝐵ூఏ 𝐵௫௧𝐵ௗ𝑅ௌெோ𝑑sinሺ2𝜑ெሻ 𝑑𝜑ெ𝐵ூఝ 𝐵௫௧𝑟ௌௌாcos𝜑ெ ሺ12ሻ At this 𝜃 angle, the angular components of the curr ent-induced fields in eq. 12 are given by 𝐵ூఏ = cos𝜑ெ𝐵 and 𝐵ூఝ = cos𝜑ெ𝐵ி. Upon calculating the derivative terms, we obtain: 𝑅ଶఠுൌെ 𝑅ௌெோିுா cos𝜑ெ𝐵 𝐵௫௧𝐵ௗ2𝑅ௌெோcosሺ2𝜑ெሻcos𝜑ெ𝐵ிାை 𝐵௫௧𝑟ௌௌாcos𝜑ெ ൌ𝑅ଶఠ𝑅ଶఠி𝑅ଶఠௌௌா ሺ13ሻ 13 Note that the contributions of the DL-S OT, FL-SOT and SSE to the transverse second harmonic resistance ( 𝑅ଶఠ, 𝑅ଶఠி and 𝑅ଶఠௌௌா, respectively) appear as separate terms. Summarizing these relations, and taking into account that cosሺ90°െ𝑥ሻ = െcosሺ90°𝑥ሻ, the following expression is derived for the 𝑅ଶఠ଼ହିଽହ parameter: 𝑅ଶఠ଼ହିଽହൌെ 𝑅ௌெோିுா𝐵 𝐵௫௧𝐵ௗcos85°2𝑅ௌெோ𝐵ிାை 𝐵௫௧ሺ2cosଷ85°െcos85°ሻ 𝑟ௌௌாcos85° ሺ14ሻ The discrimination between the DL and FL-SOT can be then accomp lished through their different dependencies on 𝐵௫௧. In fact, since 𝑅ௌெோିுா ≪2𝑅ௌெோ and the effective field acting against 𝑅ଶఠ is larger than that of 𝑅ଶఠி because of 𝐵ௗ, the second term in eq. 14 significantly dominates over the first one. This leads to a linear relationship between 𝑅ଶఠ଼ହିଽହ and 1/𝐵௫௧, according to which the slope is modulated by the product of 𝑅ௌெோ and 𝐵ி, and 𝑟ௌௌா provides a constant offset independent of the external field. We verified the above calculations by m eans of macrospin simulations. Figure 4a summarizes the magnetic field dependence of 𝑅ଶఠ଼ହିଽହ, simulated with parameters of 𝐵 = 𝐵ி = 1 mT and 𝑟ௌௌா = 1 mΩ. For a better appreciation, a clos e-up view of the same plot about 𝜃 = 90º is displayed in Figure 4b. T he amplitude modulation of the peak at 𝜃 = 90º with 𝐵௫௧ is predominantly due to the FL-SOT. The SSE contribution to the data is constant at 𝜃 = 90º or 270º and only the width of the peak changes due to stronger coupling of the magnetizat ion vector with the exte rnal field at higher 𝐵௫௧ (see supplementary Figure S1c). Exploiting this particular behavior, the peak amplitude is plotted as a function of the inverse of the external field in Figur e 4c. In agreement with eq. 14 the data can be fitt ed to a straight line and 𝐵ி can be extracted by dividing the slope by 2𝑅ௌெோሺ2cosଷ85°െcos85°ሻ. The y-axis intercept of the linear fitting yields 𝑟ௌௌா upon normalizing over cos85° . The quantification of 𝐵 can be finally accomplished through the 𝑅ଶఠ଼ହାଽହ data or the second harmoni c resistance measured at 𝜑 = 90º (𝑅ଶఠଽ). In both data the 𝑟ௌௌா contribution vanishes, such that 𝐵 can then be deduced by quantitatively comparing the macrospi n simulations (per formed with fixed 𝐵ி and variable 𝐵 parameters) with the experimental data. The as-described methodology will hold for moderate 𝐵/𝐵ி ratios, as demonstrated by t he error estimation in these two parameters (included in Figur e S1e). Nevertheless, when 𝐵>>𝐵ி a strong non-linearity will be observed in the 𝑅ଶఠ଼ହିଽହ vs. 1/𝐵௫௧ data, leading to an increas ing systematic error in the determination of 𝐵ி. For such scenario an iterative approach should be followed, so that the calculated 𝐵 is recursively used as an input parameter for 𝐵ி quantification via fitting of the 𝑅ଶఠ଼ହିଽହ data according to eq. 14. This pr ocedure allows for an intrinsic error in 𝐵ி of only 0.6% even when consideri ng the unfavorable sc enario in which 𝐵/𝐵ி=10 (refer to Figure S1f for estima tion errors upon conducting the pr evious iterative analysis). 14 Figure 4. (a) External field dependence of 𝑅ଶఠ଼ହିଽହ and (b) a close-up view of the same plot around 𝜃= 90°; (c) 𝑅ଶఠ଼ହିଽହ (measured at 𝜃= 90°) as a function of the inverse external field. A linear fit to this data allows for the quantification of 𝐵ி and 𝑟ௌௌா by dividing the slope and intercept over 2𝑅ௌெோሺ2cosଷ85°െcos85°ሻ and cos85° , respectively. F. Experimental results and discussion a. Second harmonic measurement s of SOTs in Bi:YIG/Pt As a proof-of-concept, the described met hodology has been applied to the Bi:YIG/Pt device described in Section III-A. The seco nd harmonic Hall resistance was measured while sweeping the polar field angle 𝜃 for a fixed magnitude of 120 mT and a current density of 𝑗 = 1.7ൈ1011 A/m2. The measurements were performed at a 𝜑 angle of 85º or 95º, as shown in Figure 5a (in red and blue, respectively). The lineshape of the as- measured signals greatly differs from that simulated in Fi gure 3. In fact, from the simulations it can be seen that all current-i nduced effects are antisymmetric with respect to 𝜃 = 180º. This distinctive property allows us to identify and separate relevant SOT and SSE signals from other spurious signals based on symmetry operations. Figures 5b,c show the antisymmetric and symmetric component s of the raw data where only the former is considered for the HHV analysis. The li neshape of the additional (spurious) symmetric component resembles that of the firs t harmonic signal originating from 𝑅ௌெோିுா and 𝑅ைுா. Therefore, we believe that the signal in the second ha rmonic is related to an in- plane temperature gradient in the device along the current injection line, producing the thermal counterparts of 𝑅ௌெோିுா and 𝑅ைுா. The external field dependence of 𝑅ଶఠ଼ହିଽହ, determined from the ex perimental data in Figure 5b, is displayed in Figure 5d. Notice that the sign of 𝑅ଶఠ଼ହିଽହ is opposite to the simulations and that the 𝜃 = 90º peak amplitude becomes less negative as increasing 𝐵௫௧. This is consistent with negative 𝑟ௌௌா and positive 𝐵ிାை parameters. The dependence of this peak amplitude as a function of 1/ 𝐵௫௧ is plotted in Figure 5e, whose linear fitting reveals 𝑟ௌௌா and 𝐵ிାை values of -0.39( േ0.01) mΩ and 0.29( േ0.04) mT respectively. Upon 15 subtraction of the Oersted field, which is as sumed to be linearly pro portional to the current (𝐵ை = 0.43 mT), we obtain 𝐵ி = -0.14( േ0.04) mT. The quantification of 𝐵 was then addressed by measuring the se cond harmonic resistance at 𝜑 = 90º (i.e. 𝑅ଶఠଽ, see Figure 5f). Rescaling the simulated 𝜃-scans for the DL and FL components (shown in Figures 3b,c respectively) with the as-calculated 𝐵ி and variable 𝐵 parameters, a quantitative comparison of the simulations wi th the experimental data reveals a 𝐵 value of -0.15 mT. This result was also verified with the 𝑅ଶఠ଼ହାଽହ data (not shown). Ignoring 𝐵ி instead results in a non-negligible overestimation of 𝐵 of about 6.7%, proving the necessity and relevance of the hereby described method. Figure 5. Quantification of the SO Ts via second harmonic Ha ll measurements: (a) Raw second harmonic transverse resistance as a function of the azimuthal angle (at 𝜑 = 85°, 90° and 95°), applying an external field of 120 mT; (b) Antisymmetric and (c) symmetric components of the data shown in a, being the former hereafter employed for further quantitative analysis; (d) Variation of the 𝑅ଶఠ଼ହିଽହ signal and (e) its amplitude (at 𝜃= 90°) with the inverse of the exter nal field. (f) Fitting of the 𝑅ଶఠு response at 𝜑 = 90° taking into account the damping-like and field-like cont ributions simulated in Figures 3b,c. All measurements were conducted at a current density (rms) of 𝑗 = 1.7ൈ1011 A/m2. The values of 𝐵 and 𝐵ி reported in Section III-F,a are significantly lower than those reported in metallic films16,17,69 and in the specific case of 𝐵, it is somewhat lower than those found in other MI/Pt systems4,46. For quantitative comparison, we convert 𝐵 into an effective spin Hall angle (SHA) assuming t he SHE as the sole spin current generating mechanism:70 16 𝜃ௌுൌ2𝑒 ℏ𝑀௦𝑡:ூீ𝐵 𝑗 ሺ15ሻ where 𝑒 stands for the electron charge, ℏ the reduced Planck constant, and 𝑀௦ and 𝑡:ூீ the saturation magnetization and thickness of the Bi:YIG layer. We note that eq.15 does not take into account the effect of spin diffusion length and assume full absorption of the spin current without cancelation effect due to the opposite spin accumulation at the counter interface. We find a SHA value of ~0.4%, which is lower than the common values reported for YIG/Pt bilayers.35 Assuming that Pt deposited in our chamber has comparable bulk properties to the systems r eported in literature, a relatively small effective SHA in our system can have multiple interfacial origins. One common difference is due to the inefficiency of the conversion of the spin current into damping-like spin- torque. This can be due to a low spin mixi ng conductance or large spin memory loss71,72 at the Bi:YIG/Pt interface. Another potential orig in is related to the density of magnetic ions at the interface. It has been specul ated that a reduced magnetization would also reduce the capability of the magnetic layer to absorb the spin current and convert it into spin-torque.48,73,74 Due to Bi3+ doping and consequently low 𝑀௦ value, it is plausible that we obtain a lower SHA with respect to other garnet systems studied thus far. Another remarkable observation is that the magnitude of 𝐵ி is comparable to that of 𝐵. Typically, the damping-like and field-like torques are associ ated with the real ( 𝐺↑↓) and imaginary ( 𝐺↑↓) parts of the spin mixing conductances, re spectively. In this picture, judging from the 𝑅ௌெோ (proportional to 𝐺↑↓) and 𝑅ௌெோିுா (proportional to 𝐺↑↓), 𝐵 should be about a factor 8 larger than 𝐵ி. The large 𝐵ி in our system suggests that, either this commonly accepted picture should be revised (if SHE is assumed to be the sole SOT source) or some additional SOT-generating mec hanisms exist at the Bi:YIG/Pt interface. Plausible mechanisms include the occurrence of the Rashba-Edelst ein effect generating additional SOTs with a stronger field-like contribution or magnet ic proximity effect in Pt acting as an additional source of spin scatte ring near the interface favoring field-like SOT generation.75 Nevertheless, investigating the SOT anomalies and deviations with respect to the literature are beyond t he scope of the present work. b. Current dependence of SOTs in Bi:YIG/Pt In typical metallic SOTs systems (e.g., Pt/C o), current-induced Joule heating plays a minor role in the first and second harm onic responses, hence generally neglected in measurements with moderat e current densities ( 𝑗 < 2ൈ1011 A/m2).16 It is typically assumed that the parameters giving rise to the firs t harmonic response are independent of the current, meanwhile the SOTs and thermoelectric effects (expressed in resistance) depends linearly in current. Howeve r, in MI/Pt, comparable cu rrent densities can create a larger Joule heating due to lowe r thermal conductivity of the MI and the substrate, which can cause a significant modulation to t he SMR parameters thr ough the temperature-17 dependence of interfacial sp in mixing conductance,76 as well as a decrease of the magnetization, magnetocr ystalline or magnetoelastic anisotropies.77 In Figure 6a, we report the current-dependence of the 𝑅ௌெோ, 𝑅ௌெோିுா and 𝐵 in Bi:YIG/Pt in the current range of 𝑗 = 0.5-2ൈ1011 A/m2. Assuming that the first data point at 𝑗 =0.5ൈ1011 A/m2 is representative of room tem perature values, we find that the magnitude of 𝑅ௌெோ progressively decreases by about 16% when increasing the current density to 2 ൈ1011 A/m2, whereas 𝑅ௌெோିுா increases by 43% in the same current range. Surprisingly, 𝐵 departs from its expec ted linear trend at about 𝑗 = 1.25ൈ1011 A/m2 and follows a decreasing tendency at elevated cu rrent densities. At first approximation, 𝑅ௌெோ and 𝐵 are both directly related to the real part of the spin-mixing conductance of the Bi:YIG/Pt interface. In order to compare them on equal grounds, we subtract a linear background from the 𝐵 values using the data points corresponding to 𝑗 ≤ 1.0ൈ1011 A/m2 and then normalize both 𝐵 and 𝑅ௌெோ dataset by dividing them by the value obtained at the lowest current density, i.e., 𝑗 = 0.5ൈ1011 A/m2. Figure 6c shows the comparison of the normalized data displaying a strong correla tion between these two sets of data and pinpointing their expec ted common origin. These results collectively show that the sp in-dependent paramet ers of Bi:YIG/Pt critically depend on the measurement te mperature even for modest current densities. This behavior is tentatively attri buted to the strong sensitiv ity of magnetic and spin-dependent interfacial properties of Bi:YIG to temper ature variations created by Joule heating. Especially, in light of the literature,78,79 we believe that the decreasing spin-mixing conductance upon increasing temper ature is the likely cause of the reduced damping-like SOT efficiency and 𝑅ௌெோ collectively. Figure 6. (a) Modulation of the 𝑅ௌெோିுா and 𝑅ௌெோ parameters as a func tion of the current density (in rms) due to Joule heating and (b) as-derived current dependence of 𝐵. A small deviation from linearit y is observed at large 𝑗, even after considering the current- corrected values of 𝑅ௌெோିுா and 𝑅ௌெோ, which highlights a non- negligible temperature- dependence of 𝐵. Note that the linear fitting in b has been performed considering only the experimental data for 𝑗 ≤ 1.0ൈ1011 A/m2. (c) Correlation between the current 18 modulation of the 𝑅ௌெோ coefficient and the linear deviation in 𝐵. For that, the 𝑅ௌெோ and 𝐵 parameters were normalized over the experimental val ues obtained at 𝑗 = 0.5ൈ1011 A/m2 (𝑅ௌெோ , and 𝐵,, respectively), where no moderat e Joule heating is expected. IV. CONCLUSIONS In summary, the study herein presented offe rs an alternative scheme for SOT vector characterization in MIs by shi fting to a non-standard geometry at 𝜑 = 85º and 95º. Supported by macrospin simulations and sy mmetry arguments, we have developed a consistent methodology that allows to accu rately quantify the damping-like and field-like SOTs and thermoelectric contributions separat ely, with an inherent error on the simulated data well below 1% for all three components. As a proof-of-concept, this strategy was tested on an IP-magnetized Bi:YIG/P t heterostructure, obtaining 𝐵 and 𝐵ி values of - 0.15 and -0.14 mT per 𝑗 = 1.7ൈ1011 A/m2 injected current, respectively. Despite the appropriateness of the devised method for this sample, we further show how a recursive procedure can be equally applied without compromi sing the accuracy of the resultant data to systems displaying higher 𝐵/𝐵ி ratios, where the 𝐵ி quantification becomes remarkedly 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 to be stable against moderate azimuthal angle misalignments. Finally, we demonstrate that Joule heating can modulate the spin transparency at th e interface, even for modest current densities. This leads to a systematic reduction of the SMR and the DL-SOT, which is often neglected in most studies up to date. Ov erall, this study reinforces the need of an adequate SOT characterization, which is of paramount importance fo r the design of MI- based spin Hall nano-osc illators and novel magnoni cs devices, among others. ACKNOWLEDGEMENTS The authors acknowledge funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project MAGNEPIC, grant agreement No . 949052) and from the Spanish Ministry of Science and Innovation through grant reference No. PI D2021-125973OA-I00. M. T.-A. acknowledges financial support from the Spanish Ministry of Science and I nnovation under grant FJC2021-046680-I. CONFLICT OF INTEREST The authors declare no conflict of interest. 19 REFERENCES (1) Manchon, A.; Železný, J. ; Miron, I. M.; Jungwirth, T. ; Sinova, J.; Thiaville, A.; Garello, K.; Gambardella, P. Curr ent-Induced Spin-Orbit Torques in Ferromagnetic and Antiferromagnetic Systems. Rev. Mod. Phys. 2019 , 91 (3), 035004. (2) Miron, I. M.; Garello, K.; Gaudin, G.; Zermatten, P.-J .; Costache, M. V.; Auffret, S.; Bandiera, S.; Rodmac q, B.; Schuhl, A.; Gambardella, P. Perpendicular Switching of a Single Ferromagnetic Layer Induced by In-Plane Current Injection. Nature 2011 , 476 (7359), 189–193. (3) Liu, L.; Pai, C.-F.; Li, Y.; Tseng, H. W.; Ralph, D. C.; Buh rman, R. A. Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum. Science 2012 , 336 (6081), 555–558. (4) Avci, C. O.; Quindeau, A .; Pai, C.-F.; Mann, M.; Caretta , L.; Tang, A. S.; Onbasli, M. C.; Ross, C. A.; Beach, G. S. D. Current-Induced Switch ing in a Magnetic Insulator. Nat. Mater. 2017 , 16 (3), 309–314. (5) Avci, C. O.; Rosenberg, E.; Caretta, L.; Büttner, F.; Mann, M.; Marcus, C.; Bono, D.; Ross, C. A.; Beach, G. S. D. Inte rface-Driven Chiral Magnetism and Current- Driven Domain Walls in Insulating Magnetic Garnets. Nat. Nanotechnol. 2019 , 14 (6), 561–566. (6) Woo, S.; Song, K. M.; H an, H.-S.; Jung, M.-S.; Im, M.-Y .; Lee, K.-S.; Song, K. S.; Fischer, P.; Hong, J.-I.; Choi , J. W.; Min, B.-C.; Koo, H. C.; Chang, J. Spin-Orbit Torque-Driven Skyrmion Dynamics Revealed by Time-Resolved X-Ray Microscopy. Nat. Commun. 2017 , 8 (1), 15573. (7) Ryu, K.-S.; Thomas, L. ; Yang, S.-H.; Parkin, S. Chir al Spin Torque at Magnetic Domain Walls. Nat. Nanotechnol. 2013 , 8 (7), 527–533. (8) Demidov, V. E.; Urazhdin, S.; Ulrichs, H.; Tiberkevich, V.; Slavin, A.; Baither, D.; Schmitz, G.; Demokritov, S. O. Magnet ic Nano-Oscillator Driven by Pure Spin Current. Nat. Mater. 2012 , 11 (12), 1028–1031. (9) Liu, L.; Pai, C.-F.; Ralph, D. C.; B uhrman, R. A. Magnetic Oscillations Driven by the Spin Hall Effect in 3-Terminal Magnetic Tunnel Junction Devices. Phys. Rev. Lett. 2012 , 109 (18), 186602. (10) Divinskiy, B.; Demidov, V. E.; Uraz hdin, S.; Freeman, R.; Rinkevich, A. B.; Demokritov, S. O. Excita tion and Amplification of Spin Waves by Spin-Orbit Torque. Adv. Mater. 2018 , 30 (33), 1802837. (11) Fulara, H.; Zahedinejad, M.; Khymyn, R.; Awad, A. A.; Muralidhar, S.; Dvornik, M.; Åkerman, J. Spin-Orbit Torque- Driven Propagating Spin Waves. Sci. Adv. 2019 , 5 (9), eaax8467. (12) Baek, S. C.; Park, K.-W .; Kil, D.-S.; Jang, Y.; Park , J.; Lee, K.-J.; Park, B.-G. Complementary Logic Operati on Based on Electric-Field Controlled Spin-Orbit 20 Torques. Nat. Electron. 2018 , 1 (7), 398–403. (13) Luo, S.; Song, M.; Li, X .; Zhang, Y.; Hong, J.; Yang, X .; Zou, X.; Xu, N.; You, L. Reconfigurable Skyrmion Logic Gates. Nano Lett. 2018 , 18 (2), 1180–1184. (14) Luo, Z.; Hrabec, A.; Dao, T. P.; Sala, G.; Finizio, S.; Feng, J.; Mayr, S.; Raabe, J.; Gambardella, P.; Heyderman, L. J. Curr ent-Driven Magnetic Domain-Wall Logic. Nature 2020 , 579 (7798), 214–218. (15) Finocchio, G.; Di Ventra, M.; Camsari, K. Y.; Everschor-S itte, K.; Khalili Amiri, P.; Zeng, Z. The Promise of Spintroni cs for Unconventional Computing. J. Magn. Magn. Mater. 2021 , 521, 167506. (16) Garello, K.; Miron, I. M.; Avci, C. O.; Freimuth, F.; Mokrousov, Y.; Blügel, S.; Auffret, S.; Boulle, O.; G audin, G.; Gambardella, P. Symmetry and Magnitude of Spin-Orbit Torques in Ferromagnetic Heterostructures. Nat. Nanotechnol. 2013 , 8 (8), 587–593. (17) Avci, C. O.; Garello, K.; Gabureac, M. ; Ghosh, A.; Fuhrer, A.; Alvarado, S. F.; Gambardella, P. Interplay of Spin-Orbit Torque and Thermoelectric Effects in Ferromagnet/Normal-Metal Bilayers. Phys. Rev. B 2014 , 90 (22), 224427. (18) Hao, Q.; Xiao, G. Giant Spin Hall Effect and Switching Induced by Spin-Transfer Torque in a W/Co 40Fe40B20/MgO Structure with Pe rpendicular Magnetic Anisotropy. Phys. Rev. Appl. 2015 , 3 (3), 034009. (19) Pai, C.-F.; Liu, L.; Li , Y.; Tseng, H. W.; Ralph, D. C.; Buhrman, R. A. Spin Transfer Torque Devices Utilizing the Giant Spin Hall Effect of Tungsten. Appl. Phys. Lett. 2012 , 101 (12), 122404. (20) Qiu, X.; Deorani, P.; Narayanapillai, K.; Lee, K.-S.; Lee, K.-J .; Lee, H.-W.; Yang, H. Angular and Temperature Dependence of Current Induced Spin -Orbit Effective Fields in Ta/CoFeB/MgO Nanowires. Sci. Rep. 2014 , 4 (1), 4491. (21) Zhang, P.; Chou, C.-T.; Y un, H.; McGoldrick, B. C.; Ho u, J. T.; Mkhoyan, K. A.; Liu, L. Control of Néel Vector with Sp in-Orbit Torques in an Antiferromagnetic Insulator with Tilted Easy Plane. Phys. Rev. Lett. 2022 , 129 (1), 017203. (22) Kim, T. H.; Hwang, S.; Hamh, S. Y.; Yoon, S.; Han, S. H.; Cho, B. K. Spin Orbit Torque Switching in an Antiferromagnet through Néel Reorientation in a Rare- Earth Ferrite. Phys. Rev. B 2021 , 104 (5), 054406. (23) DuttaGupta, S.; Kur enkov, A.; Tretiakov, O. A .; Krishnaswamy, G.; Sala, G.; Krizakova, V.; Maccherozzi, F.; Dhesi, S. S.; Gambardella, P.; Fukami, S.; Ohno, H. Spin-Orbit Torque Switching of an An tiferromagnetic Metallic Heterostructure. Nat. Commun. 2020 , 11 (1), 5715. (24) Cai, K.; Zhu, Z. ; Lee, J. M.; Mishra, R. ; Ren, L.; Pollard, S. D.; He, P.; Liang, G.; Teo, K. L.; Yang, H. Ultraf ast and Energy-Efficient Spin-Orbit Torque Switching in Compensated Ferrimagnets. Nat. Electron. 2020 , 3 (1), 37–42. (25) Finley, J.; Liu, L. Spin-Orbit-Tor que Efficiency in Compensated Ferrimagnetic 21 Cobalt-Terbium Alloys. Phys. Rev. Appl. 2016 , 6 (5), 054001. (26) Roschewsky, N.; Matsum ura, T.; Cheema, S.; Hellman, F.; Kato, T.; Iwata, S.; Salahuddin, S. Spin-Orbit Torques in Ferrimagnetic GdFeCo Alloys. Appl. Phys. Lett. 2016 , 109 (11), 112403. (27) Binda, F.; Avci , C. O.; Alvarado, S. F.; Noël, P .; Lambert, C.-H.; Gambardella, P. Spin-Orbit Torques and Magnet otransport Properties of α − Sn and β − Sn Heterostructures. Phys. Rev. B 2021 , 103 (22), 224428. (28) Bonell, F.; Goto, M.; S authier, G.; Sierra, J. F.; Figuer oa, A. I.; Costache, M. V.; Miwa, S.; Suzuki, Y.; Valenzuela, S. O. C ontrol of Spin-Orbit Torques by Interface Engineering in Topological In sulator Heterostructures. Nano Lett. 2020 , 20 (8), 5893–5899. (29) Wang, Y.; Zhu, D.; Wu, Y.; Yang, Y.; Yu, J.; Ramaswam y, R.; Mishra, R.; Shi, S.; Elyasi, M.; Teo, K.-L.; Wu, Y.; Yang, H. Room Temper ature Magnetization Switching in Topological In sulator-Ferromagnet Heterost ructures by Spin-Orbit Torques. Nat. Commun. 2017 , 8 (1), 1364. (30) Li, P.; Riddiford, L. J.; Bi, C.; Wisser, J. J.; Sun, X.-Q .; Vailionis, A.; Veit, M. J.; Altman, A.; Li, X.; DC, M. ; Wang, S. X.; Suzuki, Y .; Emori, S. Charge-Spin Interconversion in Epitaxial Pt Probed by Spin-Orbit Torques in a Magnetic Insulator. Phys. Rev. Mater. 2021 , 5 (6), 064404. (31) Li, P.; Liu, T.; Chang, H.; Kalitsov, A.; Zhang, W.; Csaba, G.; Li, W.; Richardson, D.; DeMann, A.; Rimal, G.; Dey, H.; Jiang, J. S.; Porod, W.; Fiel d, S. B.; Tang, J.; Marconi, M. C.; Hoffmann, A .; Mryasov, O.; Wu, M. Sp in-Orbit Torque-Assisted Switching in Magnetic Insulator Thin Films with Perpendicular Magnetic Anisotropy. Nat. Commun. 2016 , 7 (1), 12688. (32) Vélez, S.; Schaab, J.; Wörnle, M. S .; Müller, M.; Gradauska ite, E.; Welter, P.; Gutgsell, C.; Nistor, C.; Degen, C. L.; Tr assin, M.; Fiebig, M.; Gambardella, P. High-Speed Domain Wall Racetra cks in a Magnetic Insulator. Nat. Commun. 2019 , 10 (1), 4750. (33) Ding, S.; Ross, A.; Lebrun, R.; Becke r, S.; Lee, K.; Boventer, I.; Das, S.; Kurokawa, Y.; Gupta, S.; Y ang, J.; Jakob, G.; Kläui, M. Interfacial Dzyaloshinskii- Moriya Interaction and Chiral Magnetic Textures in a Ferrimagnetic Insulator. Phys. Rev. B 2019 , 100 (10), 100406. (34) Vélez, S.; Ruiz-Gómez , S.; Schaab, J.; Gradauskaite, E.; Wörnle, M. S.; Welter, P.; Jacot, B. J.; Degen, C. L.; Trassin, M.; Fiebig, M.; Gambardella, P. Current- Driven Dynamics and Ratchet Effect of Skyrmion Bubbles in a Ferrimagnetic Insulator. Nat. Nanotechnol. 2022 , 17 (8), 834–841. (35) Hahn, C.; de Loubens, G.; Klein, O.; Vi ret, M.; Naletov, V. V.; Ben Youssef, J. Comparative Measurements of Inverse Spin Hall E ffects and Magnetoresistance in YIG/Pt and YIG/Ta. Phys. Rev. B 2013 , 87 (17), 174417. (36) Cornelissen, L. J.; Liu, J.; Duine, R. A.; Ben Youssef, J.; van Wees, B. J. Long-22 Distance Transport of Magnon Spin Info rmation in a Magnetic Insulator at Room Temperature. Nat. Phys. 2015 , 11 (12), 1022–1026. (37) Chumak, A. V.; Ser ga, A. A.; Hillebrands, B. M agnon Transistor for All-Magnon Data Processing. Nat. Commun. 2014 , 5 (1), 4700. (38) Ganzhorn, K.; Klingl er, S.; Wimmer, T.; Geprägs, S.; Gross, R.; Huebl, H.; Goennenwein, S. T. B. Magnon-Based Logic in a Multi-Terminal YIG/Pt Nanostructure. Appl. Phys. Lett. 2016 , 109 (2), 022405. (39) Guo, C. Y.; Wan, C. H.; Wang, X.; Fang, C.; Tang, P .; Kong, W. J.; Zhao, M. K.; Jiang, L. N.; Tao, B. S. ; Yu, G. Q.; Han, X. F. Magnon Valves Based on YIG/NiO/YIG All-Insulating Magnon Junctions. Phys. Rev. B 2018 , 98 (13), 134426. (40) Zhang, H.; Ku, M. J. H .; Casola, F.; Du, C. H. R.; v an der Sar, T.; Onbasli, M. C.; Ross, C. A.; Tserkovnyak, Y.; Yacoby , A.; Walsworth, R. L. Spin-Torque Oscillation in a Magnetic Insulator Probed by a Single-Spin Sensor. Phys. Rev. B 2020 , 102 (2), 024404. (41) Evelt, M.; Safranski, C.; Aldosary, M.; Demidov, V. E.; Barsukov, I.; Nosov, A. P.; Rinkevich, A. B.; Sobotkiewich, K.; Li, X.; Sh i, J.; Krivorotov, I. N.; Demokritov, S. O. Spin Hall-Induced Auto-Oscillations in Ultrathin YIG Grown on Pt. Sci. Rep. 2018 , 8 (1), 1269. (42) Evelt, M.; Demidov, V. E.; Bessonov, V.; Demokritov, S. O.; Prieto, J. L.; Muñoz, M.; Ben Youssef, J.; Naletov, V. V.; de Loubens, G.; Klein, O.; Collet, M.; Garcia- Hernandez, K.; Bortolotti, P.; Cros, V.; A nane, A. High-Efficiency Control of Spin- Wave Propagation in Ultra-Thin Yttrium Iron Garnet by the Spin-Orbit Torque. Appl. Phys. Lett. 2016 , 108 (17), 172406. (43) Navabi, A.; Liu, Y.; Up adhyaya, P.; Murata, K.; Ebrahi mi, F.; Yu, G.; Ma, B.; Rao, Y.; Yazdani, M.; Montazeri, M. ; Pan, L.; Krivorotov, I. N.; Barsukov, I.; Yang, Q.; Khalili Amiri, P.; Tserkovnya k, Y.; Wang, K. L. Contro l of Spin-Wave Damping in YIG Using Spin Currents from Topological Insulators. Phys. Rev. Appl. 2019 , 11 (3), 034046. (44) Wang, Z.; Sun, Y.; Wu, M.; Tiberkevich, V.; Sl avin, A. Control of Spin Waves in a Thin Film Ferromagnetic Insulator through Interfacial Spin Scattering. Phys. Rev. Lett. 2011 , 107 (14), 146602. (45) Shan, J.; Cornelissen, L. J.; Vlietstr a, N.; Ben Youssef, J.; Kuschel, T.; Duine, R. A.; van Wees, B. J. Influence of Yttr ium Iron Garnet Thi ckness and Heater Opacity on the Nonlocal Transport of Elec trically and Thermally Excited Magnons. Phys. Rev. B 2016 , 94 (17), 174437. (46) Avci, C. O.; Quindeau, A .; Mann, M.; Pai, C.-F.; Ross, C. A.; Beach, G. S. D. Spin Transport in As-Grown and Annealed Thuliu m Iron Garnet/Platinum Bilayers with Perpendicular Magnetic Anisotropy. Phys. Rev. B 2017 , 95 (11), 115428. (47) Pai, C.-F.; Ou, Y.; Vile la-Leão, L. H.; Ralph, D. C.; Buhrman, R. A. Dependence of 23 the Efficiency of Spin Hall Torque on the Transparency of Pt/Ferromagnetic Layer Interfaces. Phys. Rev. B 2015 , 92 (6), 064426. (48) Shao, Q.; Tang, C.; Yu, G.; Navabi, A.; Wu, H.; He, C.; Li , J.; Upadhyaya, P.; Zhang, P.; Razavi, S. A.; He, Q. L.; Liu, Y.; Yang, P.; Kim, S. K.; Zheng, C.; Liu, Y.; Pan, L.; Lake, R. K.; Han, X.; Tserkovn yak, Y.; Shi, J.; W ang, K. L. Role of Dimensional Crossover on Spin-Orbit Torque Efficiency in Magnetic Insulator Thin Films. Nat. Commun. 2018 , 9 (1), 3612. (49) Ren, Z.; Qian, K.; Aldos ary, M.; Liu, Y.; Cheung, S. K.; Ng, I.; Shi, J.; Shao, Q. Strongly Heat-Assisted Spin-Orb it Torque Switching of a Ferrimagnetic Insulator. APL Mater. 2021 , 9 (5), 051117. (50) Li, Y.; Zheng, D.; Liu, C.; Zhang, C.; Fang, B.; C hen, A.; Ma, Y.; Manchon, A.; Zhang, X. Current-Induced Magnetization S witching Across a Nearly Room- Temperature Compensation Point in an Insulating Compensated Ferrimagnet. ACS Nano 2022 , 16 (5), 8181–8189. (51) Yoon, J.; Lee, S.-W.; Kwon, J. H.; Lee, J. M.; Son, J.; Qiu, X.; Lee, K.-J.; Yang, H. Anomalous Spin-Orbit Torque Switchi ng Due to Field-like Torque-Assisted Domain Wall Reflection. Sci. Adv. 2017 , 3 (4), e1603099. (52) Kim, C.; Chun, B. S.; Y oon, J.; Kim, D.; Kim, Y. J.; C ha, I. H.; Kim, G. W.; Kim, D. H.; Moon, K.-W.; Kim, Y. K.; Hwang, C. Spin-Orbit Torque Driven Magnetization Switching and Precession by Mani pulating Thickness of CoFeB/W Heterostructures. Adv. Electron. Mater. 2020 , 6 (2), 1901004. (53) Krizakova, V.; Hoffmann, M.; Kateel, V.; Rao, S.; Couet, S. ; Kar, G. S. ; Garello, K.; Gambardella, P. Tailoring the S witching Efficiency of Magnetic Tunnel Junctions by the Fieldlike Spin-Orbit Torque. Phys. Rev. Appl. 2022 , 18 (4), 044070. (54) Legrand, W.; Ramaswam y, R.; Mishra, R.; Yang, H. Coherent Subnanosecond Switching of Perpendicular Magnetization by the Fieldlike Spin-Orbit Torque without an External Magnetic Field. Phys. Rev. Appl. 2015 , 3 (6), 064012. (55) Chiba, T.; Schreier, M.; Bauer, G. E. W.; Takahashi , S. Current-Induced Spin Torque Resonance of Magnetic Insulators Affected by Field-like Spin-Orbit Torques and Out-of-Plane Magnetizations. J. Appl. Phys. 2015 , 117 (17), 17C715. (56) Lee, J. M.; Kwon, J. H.; Ramaswamy, R.; Yoon, J. ; Son, J.; Qiu, X.; Mishra, R.; Srivastava, S.; Cai, K.; Yang, H. Oscilla tory Spin-Orbit Torque Switching Induced by Field-like Torques. Commun. Phys. 2018 , 1 (1), 2. (57) Nakatsuka, A.; Yoshiasa, A.; Takeno, S. Site Preference of Cations and Structural Variation in Y 3Fe5− xGaxO12 (0 ≤x≤ 5) Solid Solutions with Garnet Structure. Acta Crystallogr. 1995 , 51 (5), 737–745. (58) Vu, N. M.; Meisenheimer , P. B.; Heron, J. T. T unable Magnetoelastic Anisotropy in Epitaxial (111) Tm 3Fe5O12 Thin Films. J. Appl. Phys. 2020 , 127 (15), 153905. 24 (59) Li, G.; Bai, H.; Su, J.; Zhu, Z. Z.; Zhang, Y.; Ca i, J. W. Tunabl e Perpendicular Magnetic Anisotropy in Epitaxial Y 3Fe5O12 Films. APL Mater. 2019 , 7 (4), 041104. (60) Lax, B.; Button, K. J. Microwave Ferrites and Ferrimagnetics ; McGraw-Hill: New York, 1962. (61) Su, T.; Ning, S.; Cho, E.; Ross, C. A. Magnetism and Site Occupancy in Epitaxial Y-Rich Yttrium Iron Garnet Films. Phys. Rev. Mater. 2021 , 5 (9), 094403. (62) Lin, Y.; Jin, L.; Zhang, H.; Zhong, Z.; Yang, Q.; Rao, Y.; Li, M. Bi-YIG Ferrimagnetic Insulator Nanometer F ilms with Large Perpendicular Magnetic Anisotropy and Narrow Ferromagnetic Resonance Linewidth. J. Magn. Magn. Mater. 2020 , 496, 165886. (63) Dumont, Y.; Keller, N.; Popova, E.; Sc hmool, D. S.; Tessier, M.; Bhattacharya, S.; Stahl, B.; Da Silva, R. M. C.; Guyo t, M. Tuning Magnetic Properties with Off- Stoichiometry in Oxide Thin Films: An Experiment with Yttrium Iron Garnet as a Model System. Phys. Rev. B 2007 , 76 (10), 104413. (64) Fakhrul, T.; Tazlaru, S.; Beran, L.; Zhang, Y.; Ve is, M.; Ross, C. A. Magneto- Optical Bi:YIG Films with High Figure of Merit for Nonreciprocal Photonics. Adv. Opt. Mater. 2019 , 7 (13), 1900056. (65) Nakayama, H.; Althammer, M.; Chen, Y.-T.; Uchida, K. ; Kajiwara, Y.; Kikuchi, D.; Ohtani, T.; Geprägs, S.; Opel, M.; Tak ahashi, S.; Gross, R.; Bauer, G. E. W.; Goennenwein, S. T. B.; Saitoh, E. Spin Hall Magnetoresistance Induced by a Nonequilibrium Proximity Effect. Phys. Rev. Lett. 2013 , 110 (20), 206601. (66) Chen, Y.-T.; Takahashi, S.; Nakayama , H.; Althammer, M. ; Goennenwein, S. T. B.; Saitoh, E.; Bauer, G. E. W. Theory of Spin Hall Magnetoresistance. Phys. Rev. B 2013 , 87 (14), 144411. (67) INRIA-ENPC; Scilab. Copyright © 1989-2005. Scilab Is a Trademark of INRIA. www.scilab.org. (68) Uchida, K.; Adachi, H.; Ota, T.; Nakayama, H.; Maekawa, S.; Saitoh, E. Observation of Longitudinal Spin-Seebe ck Effect in Magnetic Insulators. Appl. Phys. Lett. 2010 , 97 (17), 172505. (69) Martini, M.; Avci, C. O.; Tacchi, S.; Lambert, C.-H .; Gambardella, P. Engineering the Spin-Orbit-Torque Efficiency and Magnet ic Properties of Tb/Co Ferrimagnetic Multilayers by Stacking Order. Phys. Rev. Appl. 2022 , 17 (4), 044056. (70) Liu, L.; Lee, O. J.; Gudmundsen, T. J.; Ralph, D. C.; Buhrman, R. A. Current- Induced Switching of Per pendicularly Magnetized M agnetic Layers Using Spin Torque from the Spin Hall Effect. Phys. Rev. Lett. 2012 , 109 (9), 096602. (71) Rojas-Sánchez, J.-C.; Reyren, N.; La czkowski, P.; Savero, W.; Attané, J.-P.; Deranlot, C.; Jamet, M.; George, J.-M.; Vila, L.; Jaffrès, H. Spin Pumping and Inverse Spin Hall Effect in Platinum: T he Essential Role of Spin-Memory Loss at Metallic Interfaces. Phys. Rev. Lett. 2014 , 112 (10), 106602. 25 (72) Gupta, K.; Wesselink, R. J. H.; Liu, R.; Yuan, Z.; Kelly, P. J. Disorder Dependence of Interface Spin Memory Loss. Phys. Rev. Lett. 2020 , 124 (8), 087702. (73) Wahada, M. A.; Şaşıoğlu, E.; Hoppe, W.; Zhou, X.; Deniz, H.; Rouzegar, R.; Kampfrath, T.; Mertig, I.; Pa rkin, S. S. P.; Woltersdorf, G. Atomic Scale Control of Spin Current Transmission at Interfaces. Nano Lett. 2022 , 22 (9), 3539–3544. (74) Catalano, S.; Gomez-Pe rez, J. M.; Aguilar-Pujol, M. X.; Chuvilin, A.; Gobbi, M.; Hueso, L. E.; Casanova, F. Spin Hall Magnetoresistance Effe ct from a Disordered Interface. ACS Appl. Mater. Interfaces 2022 , 14 (6), 8598–8604. (75) Li, J.; Yu, G.; Tang, C.; Liu, Y.; Shi, Z.; Liu, Y.; Nav abi, A.; Aldosary, M.; Shao, Q.; Wang, K. L.; Lake, R.; Shi, J. Deficiency of the Bulk Spin Hall Effect Model for Spin-Orbit Torques in Magnetic-Insul ator/Heavy-Metal Heterostructures. Phys. Rev. B 2017 , 95 (24), 241305. (76) Wang, S.; Zou, L.; Zhang, X.; Cai, J.; Wang, S.; Shen, B.; Sun, J. Spin Seebeck Effect and Spin Hall Magnetoresistance at High Temperatures for a Pt/Yttrium Iron Garnet Hybrid Structure. Nanoscale 2015 , 7 (42), 17812–17819. (77) Quindeau, A.; Avci, C. O. ; Liu, W.; Sun, C.; Mann, M.; Tang, A. S.; Onbasli, M. C.; Bono, D.; Voyles, P. M.; Xu, Y.; Robinson , J.; Beach, G. S. D.; Ross, C. A. Tm 3Fe5O12/Pt Heterostructures with Perpendicular Magnetic Anisotropy for Spintronic Applications. Adv. Electron. Mater. 2017 , 3 (1), 1600376. (78) Atsarkin, V. A.; Boris enko, I. V.; Demidov, V. V.; S haikhulov, T. A. Temperature Dependence of Pure Spin Current and Spin-Mixing Conductance in the Ferromagnetic-Normal Metal Structure. J. Phys. D. Appl. Phys. 2018 , 51 (24), 245002. (79) Marmion, S. R.; Ali, M.; McLaren, M.; Williams, D. A. ; Hickey, B. J. Temperature Dependence of Spin Hall Magnetoresis tance in Thin YIG/Pt Films. Phys. Rev. B 2014 , 89 (22), 220404. 26 Supporting Information Leveraging symmetry for an accurate spin-orbit torques characterization in ferrimagnetic insulators Martín Testa-Anta1,*, Charles-Henri Lambert2, Can Onur Avci1,* 1Institut de Ciència de Materials de Barcel ona (ICMAB-CSIC), Campus de la UAB, 08193 Bellaterra, Spain 2Department of Materials, ETH Züri ch, Hönggerbergring 64, CH-8093 Zürich, Switzerland Figure S1 . External field dependence of 𝑅ଶఠ଼ହିଽହ considering the individual (a) DL, (b) FL and (c) SSE contributions; (d) Zoom- out of the same plot shown in a, displaying the residual contribution to 𝑅ଶఠ଼ହିଽହ due to the OOP oscillations induced by 𝐵. This contribution was neglected in the SOT analysis, as being ~20 and ~120 times smaller than those of the FL-SOT and SSE, respectively. (e) Estimated errors in the field-like and 27 damping-like fields for different 𝐵/𝐵ி ratios owing to t he contribution shown in d. Considering a system where 𝐵 = 𝐵ி = 1 mT and 𝑟ௌௌா = 1 mΩ, the estimated error is calculated to be 0.26, 3.54 and 0.13% for 𝐵, 𝐵ி and 𝑟ௌௌா respectively, which is expected to be hidden within the experimental noise. Note that the error in 𝐵 (𝐵ி) will slightly decrease (significantly increase) as increasing the 𝐵/𝐵ி ratio, as depicted in e. For such scenario an iterative process is proposed, in which the calculated 𝐵 is used as an input parameter for the estimation of 𝐵ி via fitting of the 𝑅ଶఠ଼ହିଽହ vs. 𝐵௫௧ experimental data (according to eq. 14 in the manuscript) . The remarked reduction of the estimated error in the 𝐵ி parameter after conducting this iterative process is shown in f. Figure S2. (a) 𝑅ଶఠ଼ହିଽହ, (b) 𝑅ଶఠ଼ହାଽହ and (c) 𝑅ଶఠଽ as a function of the azimuthal angle for different 𝜑-angle off-centerings. A critical fact or when registering the second harmonic response is the Hall bar alignment with respect to the external field. Owing to the large SSE intrinsic to the Bi:YIG laye r, small off-centerings from 𝜑 = 90º (even <<1º) will lead to a significant SSE contribution that will mani fest through changes in the relative intensity of the peaks at 𝜃 ~ 28º and 208º, as shown in c. An identical parasitic sin𝜃ெ contribution is observed in the 𝑅ଶఠ଼ହାଽହ response for different off-centerings (see graph b), provided that the measurements at the two 𝜑 angles span a constant 10º range. In any case, in light of the antisymmetric behavior of the SOTs with respect to 𝜃 = 180º, the experimental occurrence of a 𝜑 off-centering can be corrected in both 𝑅ଶఠଽ and 𝑅ଶఠ଼ହାଽହ by taking the average of the peaks at 𝜃 ~ 28º and 208º as the actual am plitude of the second harmonic resistance. This average is indica ted by horizontal dotted lines in b and c. Alternatively, 𝑅ଶఠ଼ହିଽହ sees a negligible influence on a potential off-centering (see graph a), meaning that neither 𝐵ி nor 𝑟ௌௌா will be affected by the misalignment. 28 Figure S3. Current-dependence of the (a) firs t and (b) second harmonic resistance measured at 𝜑 = 90º (under a 120 mT exte rnal field). At this c onfiguration, the second harmonic resistance encompas ses the DL and FL-SOT contributions, without the influence of the thermoelectric SSE effect. A remarked deviation from linearity can be observed at high 𝑗, which stems from the J oule heating modulation of 𝑅ௌெோ and, subsequently, of the in-plane oscillations. 29 SciLab script for macrospin simulations clear; function T_tot=T_min(angle) theta_M=angle(1); phi_M=angle(2); zeta_M=angle(3); //---Current-induced fields scaling with the injected current--------------- H_FL=I_DC*HFL; H_DL=I_DC*HDL; //---Torques acting on the magnetization--------------- 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))); H*M_s*(-sin(theta_M)*cos(phi_M)*co s(theta_H(ii))+cos(theta_M)*cos(phi_H)*sin(theta_H(ii))); 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))]; T_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); H_dem*M_s*0]; T_FL=[H_FL*M_s*(cos(theta_M)); H_FL*M_s*0; -H_FL*M_s*sin(theta_M)*cos(phi_M)] T_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); -H_DL*cos(theta_M)*sin(theta_M)*sin(phi_M)]; T_tot(1)=T_ext(1)+T _dem(1)+T_FL(1)+T_DL(1); T_tot(2)=T_ext(2)+T _dem(2)+T_FL(2)+T_DL(2); T_tot(3)=T_ext(3)+T _dem(3)+T_FL(3)+T_DL(3); endfunction //---External parameters--------------- I_app=1; //Scaling parameter related to the applied current theta_Hdeg=[0:0.1:360]; //Out-of-plane field angle with respect to z-axis phi_H=95*%pi/180; //In-plane field angle with respect to current inje ction direction Hext=[120]; //External field amplitude (in mT) //---Magnetic, electrical and thermal parameters--------------- H_dem=45; //Demagnetizing field (in mT) M_s=1; //Scaling parameter related to the saturation magnetization SMR=11.23; //Spin Hall magnetoresistance (in mOhm) SMR_AHE=-1.36; //Anomalous Hall-like spin Hall magnetoresistance (in mOhm) HFL=1; //Field-like SOT effective field (in mT) HDL=1; //Damping-like SOT effective field (in mT) r_SSE=1; //SSE coefficient (in mOhm) for a temperature gradient along z-axis 30 //---Data initialization--------------- theta_H=[]; R_xy1=[]; R_xy2=[]; //Transverse resistance for I+ and I- Rth_xy1=[]; Rth_xy2=[]; //Thermal contribution to transverse resistance for I+ and I- Ph1=[]; Ph2=[]; //Equilibrium phi_M Th1=[]; Th2=[]; //Equilibrium theta_M R_xy=[]; //First harmonic transverse resistance Delta_R_xy=[]; //Second harmonic transverse resistance init1=theta_Hdeg(1)*%pi/180; //Initial value of theta_M init2=phi_H; //initial value of phi_M //---Computation--------------- for hh=1:length(Hext) //Loop for different external field amplitude values H=Hext(hh); for ii=1:length(theta_Hdeg); //Loop for different external field theta angle values theta_H(ii)=theta_Hdeg (ii)*%pi/180; //--- Computation for positive current I+ --------------- I_DC=+I_app; for jj=0:10; [t]=fsolve([init1;init2;0],T_min,1e-15); init1=t(1); init2=t(2); end Ph1(ii)=2*t(2); Th1(ii)=t(1); Rth_xy1(ii)= r_SSE*sin(t(1))*cos(t(2)); R_xy1(ii)=I_app*(SMR*(sin(t(1))^2)*sin(2*t(2)))+ SMR_AHE *cos(t(1))+Rth_xy1(ii); //--- Computation for negative current I- --------------- I_DC=-I_app; for kk=0:10; [t]=fsolve([init1;init2;0],T_min,1e-15); init1=t(1); init2=t(2); end Ph2(ii)=2*t(2); Th2(ii)=t(1); Rth_xy2(ii)=-(r_SSE*si n(t(1))*cos(t(2))); R_xy2(ii)=I_app*(SMR*(sin(t(1))^2)*sin(2*t(2)))+SMR_AHE*cos(t(1))+Rth_xy2(ii); 31 //--- Computing the average and the difference between I+ and I- --------------- R_xy(ii)=(R_xy1(ii)+R_xy2(ii))/2; //Results for the first harmonic transverse resistance Delta_R_xy(ii)=(R_xy1(ii)-R_xy2(ii))/2; //Results for the second harmonic transverse resistance end //---Data saving - Output--------------- out=[theta_Hdeg' R_xy Delta_R_xy]; writefile='Data saving directory\filename.dat' fprintfMat(writefile,out,'%.10g'); end

2023-02-02

Spin-orbit torques (SOTs) have emerged as an efficient means to electrically control the magnetization in ferromagnetic heterostructures. Lately, an increasing attention has been devoted to SOTs in heavy metal (HM)/magnetic insulator (MI) bilayers owing to their tunable magnetic properties and insulating nature. Quantitative characterization of SOTs in HM/MI heterostructures are, thus, vital for fundamental understanding of charge-spin interrelations and designing novel devices. However, the accurate determination of SOTs in MIs have been limited so far due to small electrical signal outputs and dominant spurious thermoelectric effects caused by Joule heating. Here, we report a simple methodology based on harmonic Hall voltage detection and macrospin simulations to accurately quantify the damping-like and field-like SOTs, and thermoelectric contributions separately in MI-based systems. Experiments on the archetypical Bi-doped YIG/Pt heterostructure using the developed method yield precise values for the field-like and damping-like SOTs, reaching -0.14 and -0.15 mT per 1.7x$10^{ 11}$ A/$m^2$, respectively. We further reveal that current-induced Joule heating changes the spin transparency at the interface, reducing the spin Hall magnetoresistance and damping-like SOT, simultaneously. These results and the devised method can be beneficial for fundamental understanding of SOTs in MI-based heterostructures and designing new devices where accurate knowledge of SOTs is necessary.

Leveraging symmetry for an accurate spin-orbit torques characterization in ferrimagnetic insulators

2302.01141v2

arXiv:1103.3764v2 [cond-mat.mtrl-sci] 15 Sep 2011epl draft Spin transfer torque on magnetic insulators Xingtao Jia1, Kai Liu1andKe Xia1 (a)and Gerrit E. W. Bauer2,3 1Department of Physics, Beijing Normal University, Beijing 100875, China 2Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 3Delft University of Technology, Kavli Institute of NanoSci ence, 2628 CJ Delft, The Netherlands PACS72.25.Mk – Spin transport through interfaces PACS72.10.-d – Theory of electronic transport; scattering mechanisms PACS85.75.-d – Magnetoelectronics; spintronics: devices exploiting sp in polarized transport or integrated magnetic fields Abstract –Recent experimental and theoretical studies focus on spin -mediated heat currents at interfaces between normal metals and magnetic insulator s. We resolve conflicting estimates for the order of magnitude of the spin transfer torque by first -principles calculations. The spin mixing conductance G↑↓of the interface between silver and the insulating ferrimag net Yttrium Iron Garnet (YIG) is dominated by its real part and of the orde r of 1014Ω−1m−2,i.e.close to the value for intermetallic interface, which can be explain ed by a local spin model. Introduction. – It has recently been reported that the magnetism of insulators can be actuated electrically and thermally by normal metal contacts [1,2]. The mate- rial of choice is the ferrimagnet Y 3Fe5O12(YIG), because of its extremely small magnetic damping [3–5]. The low- lying excitations of magnetic insulators are spin waves, which carry heat and angular momentum [6]. Existing experiments use Pt contacts, which by means of the in- verse spin Hall effect are effective spin current detectors [7]. Slonczewski [8] reports that the thermal spin transfer torquein magneticnanopillars[9,10] canbemuch moreef- ficientthanthe electricallygeneratedspintorquein metal- lic structures. The electrical and thermal injection of spin and heat currents into insulating magnets is governed by the spin transfer torque at the metal |insulator interface [11,12], which is parameterized by the spin-mixing conductance G↑↓=e2Tr/parenleftBig I−r† ↑r↓/parenrightBig /h,whereIandrσare the unit matrix and the matrix of interface reflection coefficients for spinσspanned by the scattering channels at the Fermi energy of the metal [13]. Crude approximations such as a Stoner model with spin-split conduction bands [11] and parameterized exchange between the itinerant metal elec- trons and local moments of the ferromagnet [1,8,12] have been used to estimate G↑↓for YIG interfaces1. Experi- (a)E-mail: kexia@bnu.edu.cn 1The spin mixing conductance is governed by the reflection coe ffi- cients only and remains finite when the transmission coefficie nts van-ments and initial theoretical estimates found very small spin torques that are at odds with Slonczewski’s predic- tions [8]. Here we report calculations of the spin mixing conduc- tanceforthe Ag |YIGinterfacebasedonrealisticelectronic structures. Silver is a promising material[14] fornon-local spin current detection [15], which should be more efficient than the inverse spin Hall effect in nanostructures. We demonstrate that the calculated G↑↓for the Ag |YIG inter- face is much larger than expected from the Stoner model and better described by local-moment exchange fields. Free-electron model. – We start with a reference structureconsistingofanAg |FI|Ag(001)junction inwhich the ferromagnetic insulator (FI) is modeled by a spin- split vacuum barrier, i.e.,the free-electron Stoner model. The vacuum potential is chosen to be spin-split by 0 .3 and 3.0 eV, whereas the barrier height is adjusted to 0.3, 1.4, 2.6 and 2 .85 eV, respectively. The barrier thick- ness (1.2 nm) is chosen here such that electron transmis- sion is negligible. Table 1 lists the corresponding G↑↓of Ag|FI|Ag. Both Re G↑↓and ImG↑↓decrease with increas- ing barrier height, as expected [11]. Band structure. – We calculate the electronic struc- ture of YIG using the tight-binding linear-muffin-tin- ish. This is not a breach of the scattering theory of transpor t,since the incoming and outgoing scattering states are well defined as prop- agating states in the metallic contacts. p-1X. Jiaet al. Table 1: Spin-dependent and spin mixing conductances of a Ag|FI|Ag(001) junction with different barrier heights and spin splitting ∆ = 0 .3 and 3.0 eV. The mixing conductances for the (111) orientation differs by less than 20%. The Sharvin conductance (GSh) of Ag(001) is 4 .5×1014Ω−1m−2. Barrier G↑/GshG↓/GshReG↑↓/GshImG↑↓/Gsh ∆ = 0.3eV 0.3 6.3E-5 5.1E-6 0.009 -1.1E-1 1.4 3.3E-8 7.1E-9 0.003 -7.4E-2 2.6 3.5E-9 1.3E-9 0.001 -4.0E-2 2.85 0* 0 0.001 -5.1E-2 ∆ = 3.0 eV 0.3 7.0E-6 0 0.15 -0.45 1.4 7.4E-10 0 0.08 -0.35 2.6 0 0 0.05 -0.28 2.85 0 0 0.04 -0.27 * 0 means a transmission probability of less than 10−10 orbital code in the augmented spherical wave approxi- mation as implemented in the Stuttgart code [16–18] us- ing the generalized gradient correction (GGA) to the lo- cal density approximation (LDA). The cubic lattice con- stanta= 12.2˚A is chosen 1.6% smaller than the exper- imental one [19]. We use 136 additional empty spheres (ES) for better space filling and reduced overlap between neighboring atomic spheres. YIG is a ferrimagnetic insu- lator with band gap of 2 .85 eV [20,21]. Magnetism is carried by majority and minority spin Fe atoms (tetrago- nal Fe(T) and octahedral Fe(O) sites in Fig.1(a), respec- tive.) with a net magnetic moment of 5 µBper formula unit [19,22–24]. The magnetic moments are 3.95 and −4.06µBfor majorityand minority spin Fe atoms, respec- tively. Both Y and O atoms show small positive magnetic moments of 0.03 and 0.09 µB, respectively, while those on the empty spheres do not exceed 0.007 µB. The common problem of density-functional theory to predict the energy gap of insulators can be handled by an on-site Coulomb correction (LDA/GGA+U) [25,26] or a scissor operator (LDA/GGA+C) [27]. Figure 1(b) is a plot of the band structureofGGA with a fundamental band gapof0 .33 eV between the valence band edge of the majority-spin chan- nel and conductance band edge of minority-spin channel. The GGA+C method can be used to increase the band gap depending on the scissor parameters C. A GGA+C band structure with a band gap of ∽1.25 eV is shown in Fig. 1(c). The GGA+U method applied to the YIG band structure using the parameters from Ref. [25,26] leads to the band structure plotted in 1(d) with the same energy gap∽1.25 eV. While a visual comparison of the band structures in Figs. 1(c) and(d) assurestheequivalenceofthe twometh- ods, we can assess the differences quantitatively by com- paring the effective masses at the band edges as shown inTable 2: Band gap ( Eg) and effective masses (in unit of me) of the band structure of YIG at the Γ point as calculated by the GGA, GGA+U, and GGA+C methods. CB and VB denote conductance and valence bands, respectively. Eg(eV) Majority-spin Minority-spin VB CB VB CB GGA 0.33 0.10 0.52 0.40 0.17 GGA+Ua1.25 0.13 0.60 0.37 0.19 GGA+Cb1.25 0.17 1.00 0.31 0.27 GGA+Cc1.4 0.17 1.00 0.28 0.31 GGA+Cd1.4 0.18 1.46 0.25 0.25 aU= 3.5 eV,J= 0.8 eV bC(Fe,Y) = 6.1 eV,C(ES) = 3.05 eV cC(Fe,Y) = 7.2 eV dC(Fe,Y) = 7.5 eV,C(ES) = 3.75 eV Table 2. For band gaps of ∽1.25 eV the effective mass at the conductance band edge of majority-spin as obtained by the GGA+U and GGA+C methods differ by up to 67%. This seems significant, but the effects on the mixing conductance, which is the quantity of our main interest here, is small, as discussed in the next section. Ag|YIG interface. – We study the spin mixing con- ductance in Ag |YIG|Ag with a 3 ×3 and 6×6 lateral su- percell of fcc Ag to match a cubic YIG unit cell along the (001) and (111) directions with lattice mismatch ∼1%. Interfacescanbeclassifiedaccordingtotheirmagneticsur- face properties into three types of terminations. For the (001) texture, one cut is terminated by Y as well as ma- jority and minority spin Fe atoms with compensated mag- netic moment (“YFe-termination”). Another cut yields onlymajorityFeatomsattheinterface(“Fe-termination”) with total magnetic moment of 7 .90µBper lateral unit cell. The third interface covered by O atoms is obtained by removing Fe and Y atoms from the YFe-termination. The oxygen layer is separated from adjacent Fe atoms by only∼0.3˚A. Including the latter, the “O-termination” also corresponds to a net interface magnetic moment of 7.90µB. The interfaces for the (111) direction can be clas- sified analogously. The “YFe-termination” cut has now a net interface magnetic moment of 23 .70µB. The Fe- termination contains now minority-spin Fe atoms with net magnetic moment of −16.24µB, while the O-terminated surface has the same magnetic moment when including the shallowly buried Fe layer. We chose a YIG film of 4 unit cell layers, because its electric conductance does not exceed 10−10e2/hper unit cell.G↑↓is therefore governedsolely by the single Ag |YIG interface. First, we inspect G↑↓of Ag|YIG interfaces computed with and without scissor corrections. We find that the difference of Re G↑↓is as small as 21% when increasing the band gap of YIG from its GGA value of 0 .33 eV to p-2Spin transfer torque on magnetic insulators Fig. 1: (a): 1/8 of the cubic YIG cell; the full structure can b e obtained by symmetry operations. Here, Fe(T) and Fe(O) are Featoms at tetragonal and octahedronal sites, respectivel y. (b- d): Band structures of YIG with GGA, GGA+C, and GGA+U method with band gap of 0.33, 1.25, and 1 .25 eV, respectively. a GGA+C band gap of 2 .1 eV as shown in Table 3. We conclude that the precise band gap is a parameter that hardly affects the G↑↓of the Ag |YIG interface. Table 3: Spin mixing conductance of Ag |YIG(001) with differ- ent YIG band gaps modulated by the GGA+C methods. We pin the Fermi level of Ag at mod-gap of YIG. 0 .33 eV is the band gap of GGA (without a scissor operator). Eg(eV) 0.33 0.65 0.95 1.4 1.8 2.1 ReG↑↓ (1014Ω−1m−2) 3.46 3.94 3.43 3.01 2.82 2.74 By scanning the Fermi energy of Ag (or the YIG work function), we can obtain information similar to that when varyingthe band gap. Scanning the Ag Fermi energyfrom the valence to conductance band edges for a band gap of 1.4 eV, we obtain results equivalent to a mid-gap Fermi energy and band gaps varying from zero to 2 .8 eV, but without changing the details of the band dispersion. In Figure 2 we plot the mixing conductance of Ag |YIG(001) with YFe termination as a function of YIG’s work func- tion. Here we consider two kinds of band dispersions with the same band gap of 1.4eV obtained by different scissor operator implementations as shown in Table.2. We find that the mixing conductance does not depend sensitively on (i) the YIG work function or interface potential bar- rier as well as (ii) the band dispersion when fixing the Fermi energy of Ag in the middle of the band gap of YIG; the difference in effective mass of 46% causes changes in Fig. 2: Effect of band dispersion and band alignment on the spin mixing conductance of Ag |YIG|Ag(001) with YFe- termination. We use YIG with same band gap of 1.4eV with different implementations of scissor operator (a) C(Fe,Y) = 7.2 eV; (b) C(Fe,Y) = 7 .5 eV, and C(ES) = 3 .75 eV to see the effect of band dispersion. We fix the Fermi energy of Ag while scanning the YIG work function. ReG↑↓of only 13%. These deviations are within the error bars due to other approximations (see below). We there- fore conclude that the transport properties in the present system are sufficiently well represented by the scissor op- erator or on-site Coulomb correction methods for the gap problem. Besides the band alignment discussed in the previous paragraph, two more properties are difficult to compute self-consistently for large unit cells, viz. the atomic inter- face configuration and the ferromagnetic proximity effect: (i): We determine the distance between Ag |YIG by mini- mizing ASA overlap while keeping the space filled. We es- timate that the differences in G↑↓for configurations with maximum and minimum ASA overlap is less than 30% (ii): We assess the ferromagnetic proximity effect by us- ing the self-consistent electronic structure of Ag atom at the Ag|Fe interface. We find that the Ag atoms closest to Fe acquire a magnetic moment of 0.025 µBand the ef- p-3X. Jiaet al. Fig. 3: Spin mixing conductance of Ag |YIG(001) (left) and Ag|YIG(111) (right) with a YIG band gap of 1 .4 eV. We fix the Fermi energy of Ag while scanning the YIG work function. fect is observable up to the 4th Ag layer. The spin mixing conductance is found to be enhanced by about 10% in this system. In the following wedisregardsuch an effect. From various checks of these and other issues, the magnitude of a possible systematic error in the mixing conductance is estimated to be <40%. Results. – Fig. 3 summarizes our results for G↑↓ of Ag|YIG(111) and Ag |YIG(001) junction with different YIG interface-terminations. We find G↑↓≃1014Ω−1m−2 for both Ag |YIG(111) and Ag |YIG(001), with the real part dominating over the imaginary one, in stark con- trast to the Stoner model (cf. Table 1). G↑↓depends only weakly on exposing different YIG(111) surface cuts toAg, whichweattributetothehomogeneousdistribution of magnetic atoms. The YFe termination of YIG(001) is a nearly compensated magnetic interface, but we still calcu- late a large spin mixing conductance. Finally, our results are two orders of magnitude larger than the experimental value found for Pt |YIG(111) [1]! The difference between Ag and Pt cannot account for this discrepancy: one would ratherexpect a larger G↑↓for Pt because of its higher con- duction electron density. The difference between the Stoner model and the first-principles calculations indicate that the spin-transfer torquephysicsat normalmetal interfaceswith YIG is very different from those with transition metals. Spin-transfer is equivalent to the absorption of a spin current at an in- terface that is polarized transversely to the magnetization direction. Magnetism in insulators is usually described in a local moment model. The physical picture of spin trans- ferappropriateformetals, viz.thedestructiveinterference ofprecessingspinsintheferromagnet,thenobviouslyfails. When the spin transfer acts locally on the magnetic ions, we expect no difference for the spin absorbed by a fully ordered interface with a large net magnetic moment or a Fig. 4: Re G↑↓as a function of interface magnetic moment den- sity in Ag |YIG compared with that of Fe atoms at the Ag |Vac interface. Insert: Crystal-planeresolvedspindensity /angbracketleftσy/angbracketrightinar- bitraryunitsfor Ag |YIG(001) withYFe-terminationwhenfully polarized electrons are injected from theAgside at mid-gap en- ergy. The maximum magnetic moment density is 39 µB/nm2as estimated from a full monolayer of Fe at the interface, in whi ch the Fe atoms adopt the Ag structure with magnetic moment of 2.81µB. compensated one, in which the local moments point in op- posite directions, as is indeed born out of our calculations. In orderto test the localmoment paradigm, we consider non-conducting Ag |Vac(4L)|Ag(111) junctions. We now sprinkle one vacuum interface randomly with Fe atoms. At low densities the Fe atoms are weakly coupled and form local moments. The electronic structure is gener- ated using the Coherent Potential Approximation (CPA) for interface disorder. A 10 ×10 lateral supercell with 100 atoms in one principle layer is used to model a mag- netic impurity range from 1% to 80%. The high den- sity limit is a monolayer of Fe atoms in the fcc struc- ture: Ag|Fe(1L)|Vac(3L)|Ag(111)with totalmagneticmo- ment of 2 .81µBper Fe atom. So, the maximum mag- netic moment density here is 39 µB/nm2. The results for the mixing conductances is summarized in Fig. 4. We find that the ratio of G↑↓to the (Ag) Sharvin conduc- tances monotonically increases with the Fe density at the Ag|Vac interface. The increase is linear at small densi- ties and saturates around 30 µB/nm2due to interactions between neighboring moments. We find that Re G↑↓of Ag|YIG and Ag |Fe|vacuum agrees well for corresponding Fe densities at the interface, in strong support of the local moment model. Since the mixing conductance is dominated by the local moments at the interface, we understand that the results are relatively stable against the difficulties density func- tional theory has for insulators. The variation of the band gap of the insulator as well as the band alignment with respect to normal metal changes the penetration of the p-4Spin transfer torque on magnetic insulators spin accumulation, but since only the uppermost layers contribute this is of little consequence. Table 4: G↑↓of a disordered Ag |YIG(001) interface with YFe- termination. Directional disorder is introduced by flippin g three majority Fe spins in the 2 ×2 super cell. ReG↑↓(1014Ω−1m−2) ImG↑↓(1014Ω−1m−2) clean 3.010 0.302 disorder 3.145 0.382 Table 4 shows the effect of directional disorder of magnetic moments on the mixing conductance for Ag|YIG(001) with YFe-termination, for which the in- tegrated surface magnetic moment density is close to zero. Here, we use a 2 ×2 lateral YIG supercell in which three magnetic moments are flipped to a negative value, amounting to a total surface magnetic moment of −23.7µBper lateral unit cell. The directional disorder of magnetic moments at the interface slightly enhances ReG↑↓(around 5%), as indeed expected from the local moment picture. Fig. 5: Spin mixing conductance of Ag |Fen|YIG|Fen|Ag(001) with YFe termination for (a) ballistic and (b) diffusive tran s- port (i.e. in the presence of Schep correction), where nis the number for Fe monolayers inserted between Ag and YIG. Inserting a thin ferromagnetic metallic layer betweenthe normalmetalandYIG shouldenhancethe spinmixing conductance. In Fig. 5 (a), we show that inserting Fe atomic layers indeed increases Re G↑↓by 40-65% up to the intermetallic Ag |Fe value, which is close to the Ag Sharvin conductance. In these calculation, the Ag reservoirshas been assumed to be ballistic. When the spin mixing conductanceis small relative to the Sharvin conductance, this is a valid approx- imation, but otherwise the diffusive nature of transport may not be be neglected. Since Re G↑↓turns out to be of the same order as GSh Agwe have to introduce the diffu- sive transport correction as introduced by Schep et al.as [28,29] 1 ˜G↑↓=1 G↑↓−1 2GSh Ag. (1) The results are shown in Fig. 5(b). We observe that the ”Schep” correction enhances the spin mixing conductance by 20% for for the and about 90% for 4 nonolayers Fe insertions between Ag and YIG. The spin transfer can be maximized by a high den- sity of magnetic ions at the interfaces. In YIG we could not identify interface directions or cuts that are espe- cially promising, but this could be different for other magnetic insulator, such as ferrites [8]. Slonczewski [8] uses a local moment model with a somewhat smaller ex- change splitting (0.5eV) than found here; when defined as△/parenleftBig /vectorR/parenrightBig =/integraltext ΩWS/parenleftBig V↓ xc/parenleftBig /vectorR,/vector r/parenrightBig −V↑ xc/parenleftBig /vectorR,/vector r/parenrightBig/parenrightBig ρ/parenleftBig /vectorR,/vector r/parenrightBig d/vector r, whereρ/parenleftBig /vectorR,/vector r/parenrightBig is the density of the evanescent wave func- tion in YIG at mid-gap energy disregarding its spin split- ting [30], Ω WSthe Wigner-Seitz sphere at the lattice site /vectorR, andV↑(↓) xcdenotes the exchange-correlation potentials for spin-up (down) electronsthe exchange splitting felt by the Ag conduction electrons at the YIG interface is up to ∼3.0 eV. Since Slonzcewski focusses on the magnetiza- tion dynamics of the magnetic insulator we cannot carry out a quantitative comparison with his model here. Conclusion. – In conclusion, we computed the spin mixing conductance G↑↓of the interface between sil- ver and the insulating ferrimagnet Yttrium Iron Garnet (YIG). Re G↑↓is found to be ofthe orderof1014Ω−1m−2, which is much larger than expected for a Stoner model, which indicates the importance of the local magnetic ex- change field at the interface. On the other hand, G↑↓is not very sensitive to crystal orientation and interface cut. ReG↑↓can be enhanced to around 40-65% of the fully metallic limit by inserting a monolayers of iron between Ag and YIG. The discrepancy between the measured and calculated mixing conductance might indicate previously unidentified interfacecontaminationsthat, when removed, would greatly improve the usefulness of magnetic insula- tors in spintronics. ∗∗∗ We would like to thank Burkard Hillebrands, Eiji p-5X. Jiaet al. Saitoh, and Ken-ichi Uchida for stimulating discussions. This work was supported by National Basic Research Program of China (973 Program) under the grant No. 2011CB921803 and NSF-China grant No. 60825404, the EC Contract ICT-257159 “MACALO” and the Dutch FOM foundation. This research was supported in part by the Project of Knowledge Innovation Program (PKIP) of Chinese Academy of Sciences, Grant No. KJCX2.YW.W10. Additional remark: After first submission of our manuscript arXiv:1103.3764, a manuscript was submitted and accepted by Physical Review Letters (Heinrich B. et al., Phys. Rev. Lett., 107 (2011) 066604) that reports a mixing conductance that is clearly enhanced compared to ref. [1], but still an order of magnitude smaller than our predictions. REFERENCES [1]Kajiwara Y., Harii K., Takahashi S., Ohe J., Uchida K., Mizuguchi M., Umezawa H., Kawai H., Ando K., Takanashi K., Maekawa S. andSaitoh E. , Nature,464(2010) 262. [2]Uchida K., Xiao J., Adachi H., Ohe J., Takahashi S., Ieda J., Ota T., Kajiwara Y., Umezawa H., Kawai H., Bauer G. E. W., MaekawaS. andSaitoh E.,Nat. Mater. ,9(2010) 894. [3]Geller S. andGilleo M. A. ,Acta Crystallogr. ,10 (1957) 239. [4]Cherepanov V., Kolokolov I. andLvov V. ,Phys. Rep.-Rev. Sec. Phys. Lett. ,229(1993) 81. [5]Serga A. A., Chumak A. V. andHillebrands B. ,J. Phys. D: Appl. Phys. ,43(2010) 264002. [6]Schneider T., Serga A. A., Leven B., Hillebrands B., Stamps R. L. andKostylev M. P. ,Appl. Phys. Lett.,92(2008) 022505. [7]Saitoh E., Ueda M., Miyajima H. andTatara G. , Appl.Phys. Lett. ,88(2006) 182509. [8]Slonczewski J. C. ,Phys. Rev. B ,82(2010) 054403. [9]Hatami M., Bauer G. E. W., Zhang Q. andKelly P. J.,Phys. Rev. Lett. ,99(2007) 066603. [10]Yu H., Granville S., Yu D. P. andAnsermet J.-Ph. , Phys. Rev. Lett. ,104(2010) 146601. [11]Xiao J., Bauer G. E. W., Uchida K.-C., Saitoh E. andMaekawa S. ,Phys. Rev. B ,81(2010) 214418. [12]Adachi H., Ohe J., Takahashi S., Maekawa S. ,Phys. Rev. B,83(2011) 094410. [13]Brataas A.,Bauer G. E. W. andKelly P. J. ,Phys. Rep.,427(2006) 157. [14]Kimura T. andOtani Y. ,Phys. Rev. Lett. ,99(2007) 196604. [15]Jedema F. J., Heersche H. B., Filip A. T., Basel- mans J. J. A. andvan Wees B. J. ,Nature,416(2002) 713. [16]Andersen O. K., Jepsen O. andGl¨otzel D. ,High- lights of Condensed Matter Theory , edited by Bassani F., Fumi F. andTosi M. P. (North-Holland, Amster- dam) 1985, p. 59 [17]Andersen O. K. andJepsen O. ,Phys. Rev. Lett. ,53 (1984) 2571[18]Gunarson O., Jepsen O. andAndersen O. K. ,Phys. Rev. B,27(1983) 7144. [19]Baettig P. andOguchi T. ,Chem. Mater. ,20(2008) 7545. [20]Metselaar R. andLarsen P. K. ,Solid State Commun. , 15(1974) 291. [21]Wittekoek S., Popma T. J. A., Robertson J. M. and Bongers P. F. ,Phys. Rev. B ,12(1975) 2777. [22]Rodic D., Mitric M., Tellgren R., Rundlof H. and Kremonovic A. ,J. Magnet. Magnet. Mater. ,191(1999) 137. [23]Pascard H. ,Phys. Rev. B ,30(1984) 2299. [24]Gilleo M. A. ,Ferromagnetic Materials , edited by Wohlfarth E.P. , Vol.2(North-Holland, Amsterdam) 1980, p. 1 [25]Ching W. Y., Gu Z.-Q. andXu Y.-N. ,J. Appl. Phys. , 89(2001) 6883. [26]Rogalev A., Goulon J., Wilhelm F., Brouder Ch., Yaresko A., Youssef J. Ben andIndenbom M. V. ,J. Magn. Magn. Mater. ,321(2009) 3945. [27]Fiorentini V. andBaldereschi A. ,Phys. Rev. B ,1995 (51) 17196. [28]Schep K. M., van Hoof J. B. A. N., Kelly P. J., Bauer G. E. W., andInglesfield J. E. ,Phys. Rev. B , 56(1997) 10805. [29]Bauer G. E. W., Schep K. M., Kelly P. J., andXia K.,J. Phys. D: Appl. Phys. ,35(2002) 2410. [30]Gunnarsson O. ,J. Phys. F: Metal Phys. ,6(1976) 587. p-6

2011-03-19

Recent experimental and theoretical studies focus on spin-mediated heat currents at interfaces between normal metals and magnetic insulators. We resolve conflicting estimates for the order of magnitude of the spin transfer torque by first-principles calculations. The spin mixing conductance G^\uparrow\downarrow of the interface between silver and the insulating ferrimagnet Yttrium Iron Garnet (YIG) is dominated by its real part and of the order of 10^14 \Omega^-1m^-2, i.e. close to the value for intermetallic interface, which can be explained by a local spin model.

Spin transfer torque on magnetic insulators

1103.3764v2

Finite-frequency spin conductance of a ferro-/ferrimagnetic-insulator jnormal-metal interface David A. Reiss1and Piet W. Brouwer1 1Dahlem Center for Complex Quantum Systems and Physics Department, Freie Universit at Berlin, Arnimallee 14, 14195 Berlin, Germany The interface between a ferro-/ferrimagnetic insulator and a normal metal can support spin cur- rents polarized collinear with and perpendicular to the magnetization direction. The ow of angular momentum perpendicular to the magnetization direction (\transverse" spin current) takes place via spin torque and spin pumping. The ow of angular momentum collinear with the magnetization (\longitudinal" spin current) requires the excitation of magnons. In this article we extend the ex- isting theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal spin conductance non-perturbatively (but in the low-frequency limit) and at nite frequency (but in the limit of low interface transparency). For the paradigmatic spintronic material system YIG jPt, we nd that non-perturbative eects lead to a longitudinal spin conductance that is ca. 40% smaller than the perturbative limit, whereas nite-frequency corrections are relevant at low temperatures .100 K only, when only few magnon modes are thermally occupied. I. INTRODUCTION In magnetic insulators, transport of angular momen- tum is possible via spin waves, collective wave-like excursions of the magnetization from its equilibrium direction.1{3A spin wave | or its quantized counter- part, a \magnon" | carries both an oscillating angu- lar momentum current with polarization perpendicular (transverse) to and a non-oscillating angular momentum current with polarization parallel (longitudinal) to the magnetization direction. The magnitude of the trans- verse spin current is proportional to the amplitude of the spin wave; the magnitude of the longitudinal spin current is quadratic in the spin wave amplitude, i.e., it scales pro- portional to the number of excited magnons.4{6 Both components of the spin current couple to con- duction electrons at the interface between a ferro- /ferrimagnetic insulator (F) and a normal metal (N). Microscopically, the coupling of the transverse compo- nent can be understood in terms of the interfacial spin torque and spin pumping,7{11which both give an an- gular momentum current perpendicular to the magneti- zation direction, see Fig. 1 (left). A longitudinal spin current across the interface is obtained from the spin torque acting on or spin pumped by the small thermally- induced transverse magnetization component.12Alterna- tively and equivalently, a longitudinal interfacial spin cur- rent results from magnon-emitting or -absorbing scat- tering at the interface, as shown schematically in Fig. 1 (right). The transverse component of the interfacial spin current is relevant for coherent eects, such as the spin-torque diode eect13,14or the spin-torque induced ferromagnetic resonance.15{17The longitudinal compo- nent governs incoherent eects, such as the interfacial contribution to the spin-Seebeck eect,18{21the spin- Peltier eect,22or non-local magnonic spin-transport eects.23{25The spin-Hall magnetoresistance26{31de- pends on a competition between both components of the FIG. 1. Illustration of the microscopic mechanisms underly- ing the transverse (left) and longitudinal (right) components of the spin current through the interface between a ferro- /ferrimagnetic insulator F and a normal metal N. The trans- verse spin current is mediated by the spin torque and spin pumping involving electrons (red) with spins perpendicular to the magnetization and spin waves (blue) with frequency !equal to the frequency at which the spin accumulation s in N is driven. The longitudinal component arises from spin- ip scattering of conduction electrons (red), combined with the creation or absorption of thermal magnons of frequency (blue). (The thermal magnon frequency is not related to the driving frequency !.) Alternatively, the longitudinal component can be seen as arising from the spin torque exerted on/spin pumped by the transverse magnetization component induced by thermal uctuations in F (not shown schemati- cally). spin current.32,33 In the linear-response regime, the transverse spin cur- rent density jx s?through the FjN interface (directed from N to F) is proportional to the dierence of the trans- verse spin accumulation s?in N and the time deriva- tive of the transverse magnetization amplitude m?at thearXiv:2208.01420v1 [cond-mat.mes-hall] 2 Aug 20222 interface,11 jx s?(!) =g"# 4[s?(!) +~!m?(!)]: (1) The coecient of proportionality g"#is complex and known as the \spin-mixing conductance" per unit area.34 Omitting Seebeck-type contributions that depend on the temperature dierence across the F jN interface, the lon- gitudinal spin current density jx skis proportional to the dierence of the longitudinal spin accumulation skin N and the \magnon chemical potential" m,35 jx sk(!) =gsk 4[sk(!)m(!)]: (2) In the limit of weak coupling across the F jN interface the longitudinal interfacial spin conductance is proportional to the real part of the spin-mixing conductance,12,35,36 gsk=4 Reg"# sZ d m( )  dfT( ) d  : (3) Heresis the spin per volume in F, m( ) the density of states (DOS) of magnon modes at frequency , and fT( ) the Planck distribution at temperature Tof the magnons. The availability of high-quality THz sources, combined with spin-orbit-mediated conversion of electric into mag- netic driving, as well as of femtosecond laser pulses for pump-probe spectroscopy has made it possible to exper- imentally access spin transport across F jN interfaces on ultrafast time scales.37{43Whereas Eq. (1) is valid for fre- quencies small in comparison to the frequencies of acous- tic magnons at the zone boundary,10which reach well into the THz regime, Eq. (3) requires driving frequencies much smaller than the frequencies of thermal magnons, i.e.,!=2.kBT=h6:3 THz for 300 K.12At room temperature, the two conditions roughly coincide for the magnetic insulator YIG, which is the material of choice for many experiments, or for ferrites, such as CoFe 2O4 and NiFe 2O4, see Refs. 44 and 45. But at low temper- atures, the condition for the applicability of Eq. (3) is stricter and may be violated for suciently fast driving for these materials.46An example of a magnetic mate- rial for which the two conditions do not coincide already atroom temperature is Fe 3O4(magnetite), for which the frequency of acoustic magnons at the zone boundary is well above the frequency of thermal magnons at room temperature.47 In this article, we present two calculations of the lon- gitudinal interfacial spin conductance gsk(!) per area that go beyond the low-frequency weak-coupling regime of validity of Eq. (3): (i) We calculate gskin the low- frequency limit, but without the assumption of weak cou- pling across the F jN interface, and (ii) we calculate the nite-frequency longitudinal spin conductance gsk(!) per area in the weak-coupling limit. Our nite-frequency re- sult is applicable in the same frequency range as Eq. (1), i.e., within the entire frequency range of acousticmagnons. Additionally, the temperature Tmust be low enough such that only acoustic magnons are thermally excited. For YIG this condition amounts to the require- ment thatT.300 K.48Comparing our non-perturbative low-frequency calculation to the weak-coupling result in Eq. (3), we nd that the latter is a good order- of-magnitude estimate for most material combinations, whereas quantitative deviations are possible. This article is organized as follows: In Sec. II we report our non-perturbative calculation of the longitudinal spin conductance gskat zero frequency, using scattering the- ory for the re ection of spin waves from the F jN interface. In Sec. III we present our perturbative calculation of the nite-frequency longitudinal spin conductance gsk(!), us- ing the method of non-equilibrium Green functions. We give numerical estimates for material combinations in- volving the magnetic insulator YIG in Sec. IV and we conclude in Sec. V. Appendices A and B contain further details of the calculations. II. NON-PERTURBATIVE CALCULATION AT ZERO FREQUENCY Central to our non-perturbative calculation is the am- plitude( ) that a magnon with frequency incident on the FjN interface is re ected back into F. The \transmis- sion coecient"j( )j2= 1j( )j2is the probability that the magnon is not re ected and, instead, transfers its angular momentum ~to the conduction electrons in N. As we show below, knowledge of ( ) is sucient for the calculation of the longitudinal interfacial spin con- ductancegsk(!) per area in the low-frequency limit. Magnon re ection amplitude .|To keep the notation simple, we describe our calculation for a one-dimensional geometry and switch to three dimensions in the presen- tation of the nal results. We consider an F jN interface with coordinate xnormal to the interface and a magnetic insulator F for x>0, see Fig. 1. Magnetization dynamics in F is described by the Landau-Lifshitz equation _m=!0ekm+1 ~s@ @xjx s; (4) where mis a unit vector pointing along the direction of the magnetization, !0is the ferromagnetic resonance fre- quency, ekthe equilibrium magnetization direction, and jx s=~sDexm@m @x(5) the spin current density, with Dexthe spin stiness of di- mension length2
time1. (We recall that the gyromag- netic ratio is negative, so that the angular momentum density corresponding to the magnetization direction m is~sm.) The spin current density through the F jN3 interface is10,11,34,49,50 jx s=1 4(Reg"#m+ Img"#) [(ms) +~_m] +~r Reg"# 2mh0; (6) whereg"#is the complex spin-mixing conductance51and h0is proportional to a stochastic magnetic eld repre- senting the spin torque due to current uctuations in N. If the normal metal is in equilibrium at temperature TN, the correlation function of the stochastic term h0is given by the uctuation-dissipation theorem,49 hh0 ( 0)h0 ( )i= fTN( )( 0); (7) wherefT( ) = 1=(e~ =kBT1) is the Planck function and the Fourier transform is dened as h0(t) =1p 21Z 1d h0( )ei t: (8) We parameterize the magnetization direction mas m(x;t) =p 12jm?(x;t)j2ek +m?(x;t)e?+m?(x;t)e ?; (9) where the complex unit vectors e?and e ?span the di- rections orthogonal to the equilibrium magnetization di- rection ekand satisfy the condition e?ek=ie?. The solution of the Landau-Lifshitz equation (4), up to linear order in the magnetization amplitude m?, then reads m?(x;t) =1Z 1d ei t p4sD exkx  ain( )eikxx+aout( )eikxx ;(10) where kx( ) =r !0 Dex(11) andain( ) andaout( ) are ux-normalized amplitudes for spin waves moving towards the F jN interface at x= 0 and away from it, respectively. (The amplitudes ain( ) andaout( ) may be interpreted as magnon annihilation operators in a quantized formulation.) The spin current density jx scan be decomposed into transverse and longi- tudinal contributions analogous to Eq. (9), jx s(x;t) =jx sk(x;t)ek+jx s?(x;t)e?+jx s?(x;t)e ?:(12) In the same way, the spin accumulation sand the stochastic term h0can be decomposed into transverse and longitudinal contributions. We rst consider the transverse spin current density jx s?to linear order in the magnetization amplitude m?. From Eqs. (5) and (10), one nds that the magnonictransverse spin current density jx s?(0;t) at the FjN inter- facex= 0 is jx s?(0;t) =i~sDex@m?(x;t) @x(13) =~ 41Z 1d ei tp 4sD exkx( ) [ain( )aout( )]: Equation (6) implies that the transverse spin current den- sity through the interface is given by jx s?(0;t) =g"# 4[s?(t) +i~_m?(0;t)sk(t)m?(0;t)] i~r Reg"# 2h0 ?(t): (14) Imposing continuity of the transverse spin current at the FjN interface allows us to express the amplitude aoutof magnons moving away from the interface in terms of the amplitude ainof incident magnons and the stochastic eldh0 ?. Inserting Eqs. (10) and (13) into the bound- ary condition (14), we get aout( ) =( )ain( ) +0( )h0 ?( ); (15) with ( ) =4sD exkx( )( sk=~)g"# 4sD exkx( ) + ( sk=~)g"#; 0( ) =2p 4sD exkx( ) Reg"# 4sD exkx( ) + ( sk=~)g"#: (16) The coecient ( ) is the amplitude that a magnon with frequency incident on the F jN interface is re ected. One therefore may interpret j( )j2= 1j( )j2 = ( sk=~)j0( )j2(17) as the probability that a magnon is annihilated at the FjN interface while exciting a spinful excitation in N. Longitudinal interfacial spin conductance.| The lon- gitudinal spin current is quadratic in the magnetization amplitude. From Eqs. (5) and (12) one nds jx sk(0;t) =m?(0;t)jx s?(0;t) +jx s?(0;t)m?(0;t);(18) so that continuity of jx s?at the FjN interface to linear order inm?also ensures continuity of jx sk. In terms of the magnon amplitudes, we nd from Eqs. (5) and (10) that jx sk(0;t) =~1Z 1d! 2ei!t1Z 1d (19) [aout( )aout( +)ain( )ain( +)];4 where we abbreviated = !=2 and omitted terms that drop out in the limit !!0. The correlation func- tion of the magnon amplitudes is given by the (quantum- mechanical) uctuation-dissipation theorem,52 hain( )ain( +)i=fTF( m=~)(!): (20) HereTFis the (magnon) temperature of the magnetic insulator and fTF( ) = 1=(e~ =kBTF1)( !0) the Planck function, with  the Heaviside step function. To obtain the correlation function of the stochastic eld h0 in the presence of a spin accumulation s=skek, we use the equilibrium result in Eq. (7) and make use of the fact that a spin accumulation scan be shifted away by transforming to a spin reference frame that rotates at angular frequency !=s=~, see App. A. Denoting the stochastic eld in the rotating frame by ~h0, we then have ~h0 ?( ) =h0 ?( +sk): (21) In the rotating frame there is no spin accumulation in N, so that the correlation function of ~h0 ?is given by Eq. (7). It follows that hh0 ?( )h0 ?( +)i= ( sk=~) fTN( sk=~)(!):(22) Inserting this result as well as Eqs. (15), (17), and (20) into Eq. (19), we nd for the longitudinal spin current jx sk=~ 21Z !0d j( )j2 [fTN( sk=~)fTF( m=~)]: (23) Equation (23), together with Eq. (17) for j( )j2, illus- trates the equivalence of the two pictures of longitudinal spin transport mentioned in the introduction: as aris- ing from magnon-emitting/absorbing scattering at the FjN interface (see rst line in Eq. (17)) as well as from stochastic spin torques due to thermal uctuations (see second line in Eq. (17)). In three dimensions the calculation of the longitudinal spin current density involves an integration over modes with transverse wavenumbers ( ky;kz). For each trans- verse mode the previous calculation applies, but with kx( ) replaced by kx( ;k?) =r !0 Dexk2 ?; (24) withk2 ?=k2 y+k2 z. In particular, the mode-dependent re ection amplitude ( ;k?) and transmission coecient j( ;k?)j2are found by substituting kx( ;k?) forkx( ) in Eq. (16). For the steady-state longitudinal spin current density we then nd jx sk=~ 2(2)21Z !0d kx( )2Tm( ) [fTN( sk=~)fTF( m=~)]; (25)wherekx( ) is given by Eq. (11) and Tm( ) is the mode- averaged magnon transmission coecient, Tm( ) =2 kx( )2kx( )Z 0dk?k?j( ;k?)j2: (26) The validity of Eqs. (23) and (25) is not restricted to linear response or to weak coupling across the F jN in- terface. For comparison with the literature and with the perturbative calculation of the next section, it is never- theless instructive to expand Eqs. (23) and (25) to linear order in the interfacial spin-mixing conductance, which gives jx sk=1 sReg"#1Z !0d m( )(~ sk) [fTN( sk=~)fTF( m=~)]; (27) wherem( ) is the magnon density of states, which equals1D m( ) = 1=2Dexkx( ) in the one-dimensional case and 3D m( ) =kx( )=42Dexin the three- dimensional case. One veries that this expression is consistent with Eq. (3) to linear order in skm. III. PERTURBATIVE CALCULATION AT FINITE FREQUENCIES In this section we again consider the longitudinal spin current density jx skthrough the interface between a ferro- /ferrimagnetic insulator F and a normal metal N, but now with a time-dependent spin accumulation sk(t) in N. We calculate jx skto leading order in the spin-mixing conductance per unit area, g"#. To keep the notation simple, we present the calculation for a one-dimensional FjN junction. To generalize to the three-dimensional case it is sucient to replace the magnon density of states m( ) by3D m( ). Starting point of our calculation is the Hamiltonian coupling conduction electrons in N and magnons in F, ^H=J^ y "^ #^a+J^ y #^ "^ay: (28) Here ^ is the annihilation operator for a conduction electron with spin at the FjN interface, Jis the (suit- ably normalized) interfacial exchange (s-d) interaction strength, and the raising and lowering operators ^ ayand ^adescribe the transverse magnetization amplitude at the FjN interface at x= 0. (They are the Fourier transforms of the second-quantization counterparts of the amplitude ain( ) +aout( ) of the previous section.) The spin cur- rent through the F jN interface is ^jx sk=i[J^ y "^ #^aJ^ y #^ "^ay]: (29)5 Calculating the expectation value jx skto leading order inJusing Fermi's Golden rule, one nds jx sk= 2jJj221Z 1d"1Z !0d m( ) (30) fn"(")[1n#("~ )][1 +fTF( m=~)] [1n"(")]n#("~ )fTF( m=~)g; wherenis the distribution function of electrons with spinin N,the electron density of states at the Fermi energy, and m( ) the magnon density of states at the in- terface. (We assume that the electronic density of states is constant within the energy window of interest.) Tak- ing a Fermi-Dirac distribution with chemical potential  and temperature TNfor the electron distribution function nand performing the integration over the electron en- ergy", one obtains jx sk= 2jJj221Z !0d m( )(~ sk)  fTN( sk=~)fTF( m=~) ;(31) wheresk="#andfTis the Planck distribution as before. This result is identical to Eq. (27) if we identify12 jJj22=Reg"# 22s: (32) To obtain the spin current density for an oscillating spin accumulation, we set (t) = +1Z 1d!(!)ei!t;  m(t) = m;(33) with(!) =(!). Hence, we impose oscillat- ing chemical potentials on top of a time-independent background  in N and a time-independent background min F. We use the method of non-equilibrium Green functions to calculate the expectation value jx skin the presence of the chemical potentials of Eq. (33). To linear order insk(!) ="(!)#(!), we nd (see App. B for details) jx sk(t) =jx sk+1Z 1d!jx sk(!)ei!t; (34) with jx skequal to the steady-state spin current density of Eq. (31) with m= m;sk= skand jx sk(!) =gsk(!) 4sk(!): (35)Heregsk(!) is the nite-frequency longitudinal spin con- ductance per unit area, gsk(!) =i2 Reg"# s1Z 1d (36) fD( ) [fTF( m=~)F N( ;!)] D( )[fTF( m=~)F N( ;!)]g; where we dened FN( ;!) =1 ~! (~ sk)fTN( sk=~) (37) (~ ~!sk)fTN( !sk=~) ; and where D( ) =1Z 1d 0m( 0) +i 0(38) is the (retarded) magnon Green function, with a posi- tive innitesimal. One veries that Eq. (36) reproduces the perturbative result in Eq. (27) for the limit !!0 and that it satises the Kramers-Kronig relation gsk(!) =1 iZ d!0Regsk(!0) !0!i: (39) IV. DISCUSSION Zero-frequency limit.| We evaluate the results of our calculations in Secs. II and III for the paradigmatic spintronic material combination YIG jPt. Longitudinal spin transport through the F jN interface is expected to play an important role for the ferrimagnetic insulator YIG, since at room temperature the longitudinal spin conductance gskis comparable to the (transverse) spin- mixing conductance g"#for this material. (This leads, e.g., to a prediction of a remarkable frequency depen- dence of the spin-Hall magnetoresistance for this mate- rial combination.33) To facilitate a comparison with the literature, we use the same material parameters as Cor- nelissen et al. in Ref. 35 (if applicable). We summarize the material parameters in Tab. I. Our non-perturbative calculation of the longitudinal spin conductance uses the magnon dispersion of the Landau-Lifshitz equation (4). This is a good approxima- tion at long wavelengths, for which the magnon disper- sion is quadratic as in Eq. (11). The use of the quadratic approximation to the magnon dispersion is justied if kBT~ max, where maxis the frequency of acoustic magnons at the zone boundary, max!0+12Dex a2m; (40) withamthe size the of the magnetic unit cell. For YIG, one has max=21013Hz,48,55so that the condition6 material experimental parameters ref. YIG !0=2= 7:96
109Hz29 am= 1:2
109m35 Dex= 8:0
106m2s1 35 s= 5:3
1027m3 35 YIGjPt (e2=h)Reg"#= 1:6
1014 1m2 29,53 (e2=h)Img"#= 0:08
1014 1m2 TABLE I. Typical values for the relevant material parameters of YIG and YIGjPt interfaces considered in this article. The last column states the references used for our estimates. The spin density s=S=a3 m, whereS= 10 is the magnitude of the spin in each magnetic unit cell with lattice constant am. The frequency of acoustic magnons at the zone boundary is max=2(12Dex=a2 m)=21:01013Hz. The imaginary part of the spin-mixing conductance is found from the esti- mate Img"#=Reg"#0:05, see Refs. 31 and 54. kBT~ maxis only weakly obeyed at room tempera- ture. The result (26) for the mode-averaged transmission co- ecientTm( ), which is the probability that a magnon is annihilated at the F jN interface and excites a spin- ful excitation of the conduction electrons in N, is shown in Fig. 2 for sk= 0. At the lowest magnon fre- quency!0, the magnon wave vector k= 0 ( i.e.,kx= k?= 0) and thus the re ection coecient (!0;k?) = 1, so that Tm(!0) = 0. However, upon increas- ing above !0,j( ;k?)jrst very quickly drops to approximately 0 and then reaches a maximum; corre- spondingly, Tm( ) rst features a maximum and then reaches a minimum upon increasing the magnon fre- quency above !0. The maximum is at a frequency ( !0)=!0!0jg"#j2=(4s)2Dex1; the minimum is at 2!0. Upon further increasing the frequency, Tm( ) increases monotonously with . In this frequency range, a good approximation for Tm( ) is obtained by expandingj( ;k?)j2to rst order in g"#, which gives T(p) m( ) =8Reg"# s( sk=~)3D m( ) kx( )2(41) as shown by the blue dashed curve in Fig. 2. The perturbative approximation for Tm( ) remains valid for (4s)2Dex=jg"#j2, a condition that is obeyed as long as  max. (The condition (4s)2Dex=jg"#j2 becomes equal to the condition  max if one uses the Sharvin approximation for the spin-mixing conductance34,56and takes the Fermi wavelength of elec- trons in N to be of the same order of magnitude as the sizeamof the magnetic unit cell, so that jg"#j1=a2 m.) Now we are ready to discuss the dierential longitudi- nal spin conductance per unit area gsk= 4@jx sk @sk: (42) From Eq. (25) we nd for T=TN=TFand=sk= 10101011101210130.00.20.40.60.81.0 7.95 8.00 8.05 Ω/2π (GHz)0.00.51.0 0.0 0.2 0.4 0.6 0.8 1.0 Ω/2π (Hz)0.00.20.40.60.81.0TmFIG. 2. Mode-averaged magnon transmission coecient Tm( ) at a YIGjPt-interface. The solid red curve shows the non-perturbative result of Eq. (26) and the blue dashed curve the weak-coupling approximation T(p) m( ) of Eq. (41). Both curves are based on the quadratic approximation to the magnon dispersion, which breaks down for magnon frequen- cies  max, which is the frequency of acoustic magnons at the zone boundary. Parameter values are taken from Tab. I. m, that gsk=1 21Z !0d kx( )2Tm( ) @fT( =~) @  :(43) In the perturbative limit of small g"#this result simplies to g(p) sk=4Reg"# s1Z !0d 3D m( )( =~)  @fT( =~) @  : (44) The perturbative result for the ratio gsk=Reg"#depends on the magnetic properties of bulk YIG only and not on the choice of the normal metal N or the transparency of the interface, whereas the non-perturbative result shows a (quantitative, but not qualitative) dependence on the interface properties. The results of Eqs. (43) and (44) are shown in Fig. 3 as functions of temperature Tfor the material parameters of a YIG jPt interface, see Tab. I. (We assume no temperature dependence of the spin densitysand the spin stiness Dex.) The green dashed straight line in Fig. 3 is the perturbative result with the additional approximation ~!0kBT, which gives35 g(p0) sk=cReg"# s"kBTF ~Dex3=2 +1 2kBTN ~Dex3=2# (45) withc= (1=2)(3=2)1:31. The dierence between the perturbative and non-perturbative results increases7 with temperature and reaches a factor 1:7 at room temperature, whereby the non-perturbative result for gsk is always below the small- g"#approximation, see Fig. 3 (upper left inset). Since the perturbative nite-frequency expression for the longitudinal spin conductance, discussed below, can not be evaluated using a magnon density of states m( ) of a continuum magnon model, we compare the zero- frequency longitudinal spin conductance for a quadratic magnon dispersion (as is used in the main panel of Fig. 3) with that for a magnon dispersion of a Heisenberg model on a simple cubic lattice (see Eq. (46) below). This comparison is shown in the lower right inset of Fig. 3. Whereas the dierence between the two cases is small for low temperatures and near room temperature, the Heisenberg model leads to a longitudinal spin conduc- tance that is up to a factor 1:45 larger than that of the quadratic approximation at intermediate temperatures. This is consistent with the absence of van Hove peaks in the magnon density of states in the quadratic approxi- mation. In principle, the dierential longitudinal spin conduc- tance per unit area, gsk, also depends on the chemical potentialsskandm. Such dependence governs the in- terfacial spin current beyond linear order in skm. Because the driving potentials skandmmust remain below ~!0| otherwise the magnon system is unstable |, the range of admissible values for skandmremains well below kBTat most temperatures, so that apprecia- ble nonlinear eects can be found only for extremely low temperatures T.1 K. At those low temperatures ther- mal magnons are as good as absent, so that the longitudi- nal spin conductance is negligibly small in comparison to the transverse spin conductance. For a further discussion we refer to the discussion of nonlinear eects in the con- text of the nite-frequency longitudinal spin conductance below. Finite-frequency longitudinal spin transport.| For a discussion of the nite-frequency longitudinal spin con- ductance per unit area, gsk(!), the quadratic approxi- mation of the magnon dispersion is not sucient even at temperatures kBT~ max. The reason is that at nite frequencies, gsk(!) acquires a nite imaginary part, which depends on the full magnon spectrum. (The real part of gsk(!), which describes the dissipative re- sponse, can still be calculated within the quadratic ap- proximation.) For temperatures of the order of room temperature and below and for frequencies !. max it is sucient to consider the lowest-lying magnon band and neglect higher magnon bands in YIG.48The lowest magnon band can be described eectively by a Heisen- berg model of spins on a simple cubic lattice with nearest- neighbor interactions.55,57The resulting dispersion rela- tion is given by (k) =!0+2Dex a2mX =x;y;z(1cos(kam)); (46) with maximal magnon frequency max, given in Eq. 10010110210-510-410-310-210-1100 1001011020.51.0gs g(p0) s 1001011021.001.25g(pH) s g(p) s 0.0 0.2 0.4 0.6 0.8 1.0 T (K)0.00.20.40.60.81.0gs/Reg FIG. 3. Zero-frequency longitudinal spin conductance per unit area,gsk, at a YIGjPt-interface as function of the temper- atureT=TN=TFfor=sk=m= 0. The red solid curve shows the non-perturbative result gsk=Reg"#of Eq. (43), the blue dashed curve the perturbative result g(p) sk=Reg"#of Eq. (44), and the thin green dot-dashed curve the approximation g(p0) sk=Reg"#of Eq. (45). The upper left inset shows the ratios gsk=g(p0) sk(red solid curve) and g(p) sk=g(p0) sk(blue dashed curve). The lower right inset shows the ratio g(pH) sk=g(p) sk, whereg(pH) sk is the result of Eq. (44) for the magnon density of states ob- tained from a Heisenberg model, see Eq. (46), and g(p) skthat of Eq. (44) for the quadratic approximation of the magnon dispersion. Parameter values are taken from Tab. I. (40), and agrees with the quadratic approximation for  max. The nite magnon bandwidth regular- izes the integrations for the imaginary part of gsk(!). In the numerical evaluations of the real and imaginary parts ofgsk(!) that are discussed below we therefore use the magnon density of states corresponding to the dispersion in Eq. (46). We veried that as long as !, kBT=~ maxthe results for real and imaginary parts ofgsk(!) depend only weakly on the precise form of the magnon density of states at frequencies !kBT=~. Figures 4 and 5 show the real and imaginary parts of the nite-frequency spin conductance gskat an FjN in- terface with F=YIG as function of the driving frequency !and for dierent temperatures T=TN=TFand sk= m= 0. In the perturbative regime, the ratio gsk=Reg"#is independent of the choice of the normal metal N or the quality of the F jN interface. For driving frequencies !kBT=~, the real part Regskapproaches the zero-frequency limit discussed above. (Note that there may be small deviations between the zero-frequency limit obtained from the quadratic approximation of the magnon dispersion and from the magnon dispersion of Eq. (46), see Fig. 3, lower right in- set.) ForT= 300 K, the real part Re gskdoes not show an appreciable frequency dependence. At this temperature, the Planck distribution fTNmay be well approximated8 10101011101210131014101510-610-510-410-310-210-1100 T=0KT=3KT=10KT=30KT=100KT=300K 101210140.00.20.40.60.81.0 0.0 0.2 0.4 0.6 0.8 1.0 ω/2π (Hz)0.00.20.40.60.81.0Regs/Reg FIG. 4. Real part Re gskof the nite-frequency longitudi- nal spin conductance of an F jN interface in the perturbative regime of weak coupling as a function of the driving frequency !=2for various temperatures T=TN=TF(solid colored lines). Material parameters are taken for an F jN interface with F=YIG and N an arbitrary normal metal, see Tab. I. The black dashed curve shows the result of the Rayleigh-Jeans ap- proximation, see Eq. (47). The inset displays the same curves. The time-independent background magnon chemical potential and spin accumulation have been set to zero,  m= sk= 0. by the Rayleigh-Jeans distribution fTN( ) =kBTN ~ : (47) In this limit one nds that FN( ;!) = 0 in Eq. (36), so thatgsk(!) is independent of frequency !, temperature TN, and background spin accumulation  skin N. At lower temperatures, Re gskshows an increase with frequency for!&kBTN=~, followed by a saturation at ! max. One may obtain an analytical expression for Re gsk(!) in the limit!kBTN=~(setting sk= m= 0): Regsk(!)2Reg"# s2 421Z !0d m( )fTF( m) +!+sk=~Z !0d m( )! sk !3 75:(48) (To keep the notation simple, we drop the superscript \(p)" because all nite-frequency spin conductances are obtained in the perturbative limit of small g"#.) The rst line in Eq. (48) is a frequency-independent oset which depends on the temperature TFand magnon chem- ical potential  mof the ferro-/ferrimagnetic insulator only. Using the quadratic approximation for the magnon dispersion and assuming !0kBTF=~, this term is found to be equal to the rst term in Eq. (45). For !0, kBTN=~! maxwe may also use the quadratic approximation for the magnon dispersion in the secondterm and nd Regsk(!)Reg"# s" ckBTF ~Dex3=2 +8 15! Dex3=2# ; (49) wherec= (1=2)(3=2)1:31 as below Eq. (45). In the limit! max(but stillkBT~ max) we nd similarly Regsk(!)Reg"# s" ckBTF ~Dex3=2 +2 a3m# : (50) whereamis the lattice constant of the magnetic unit cell. (Note that, up to a numerical factor of order unity in the second term, Eq. (50) is what one obtains when kBTN=~ in Eq. (45) is replaced by max.) With respect to the high-frequency limit !& max and/or the high-temperature limit kBT&~ max, it should be kept in mind that our calculation only consid- ers the contribution from the lowest-lying magnon band. For such high frequencies, other magnon bands are likely to contribute to gsk(!) as well and such a contribution is not included in our theory. Hence, Eq. (50) and anal- ogously Eq. (53) for Im gsk(!) discussed below should be interpreted as the contribution of the lowest-lying magnon band to the longitudinal spin conductance only. The imaginary part Im gsk(!) increases linearly with!for small frequencies, reaches a maximum at max( max;kBT=~), and decreases with !in the high- frequency limit, see Fig. 5. The linear increase with ! for frequencies !. maxis given by the expression Imgsk(!)!2Reg"# sZ d m( + sk=~) hTN( ); (51) with hT( ) =1Z 1d 0 0@2 @ 02[ 0fT( 0)]: (52) This function behaves as hT( )!1 for kBT=~ andhT( )!0 for kBT=~. Hence, eectively only frequencies &kBT=~contribute to the integra- tion in Eq. (51), which explains the decreasing slope ofImgsk(!) vs.!|i.e., the intercept with the ver- tical axis in Fig. 5 | with increasing temperature T. The decay of Im gsk(!) in the limit of large frequencies ! maxis described by Imgsk(!)1 !4Reg"# sZ d m( + sk~) h 1 + ln! +h0 TN( )i ; (53) with h0 T( ) =1Z 1d 0 0 ( 0 )[fT( 0) + ( 0)];(54)9 10101011101210131014101510-510-410-310-210-1 T=0K, 3K, 10K, 30K T=100K T=300K 0.0 0.2 0.4 0.6 0.8 1.0 ω/2π (Hz)0.00.20.40.60.81.0− Imgs/Reg FIG. 5. Same as Fig. 4, but for the imaginary part Im gskof the nite-frequency spin conductance. The black dashed lines show the limiting behavior for small and large !according to Eqs. (51) and (53) for TN!0. where ( 0) is the Heaviside step function. The temperature-dependent term proportional to h0 TNis sub- leading for ! max, so that Im gsk(!) becomes eec- tively temperature-independent for suciently high fre- quency!, as seen in Fig. 5. The role of the time-independent background magnon chemical potential  mand spin accumulation  skis ad- dressed in Fig. 6. The gure shows Re gsk(!) as function of!, as in Fig. 4, but for dierent values of  mand sk, while satisfying the bound  m, sk<~!0. As the magnon chemical potential and spin accumulation ap- pear in Eq. (36) only in the combinations  m=kBTFand sk=kBTNand since ~!0is much smaller than kBTfor most temperatures considered, we only show results for T=TN=TF= 0:03 K andT=TN=TF= 3 K. As can be seen in Fig. 6, the dependence of Re gsk(!) on m and skdisappears, when ~!0kBT(as forT= 3K in Fig. 6) or when ~!becomes large in comparison to mand sk. The imaginary part of gsk(!) does not show any appreciable dependence on  skin the full parameter range considered (not shown) and is independent of  m. Spin-Seebeck coecient.| Our non-perturbative cal- culation of the longitudinal spin current through the F jN interface also describes the longitudinal spin current in response to a temperature dierence Tacross the in- terface. We set TF=T+T,TN=T,m=sk, and expandjx skin Eq. (25) to linear order in T, resulting in jx sk=LSSE TT (55) 10810101012101410-1410-1110-810-510-2 T=3K T=0.03K 0.0 0.2 0.4 0.6 0.8 1.0 ω/2π (Hz)0.00.20.40.60.81.0Regs/Reg FIG. 6. Real part Re gskof the nite-frequency spin con- ductance of an FjN interface in the weak-coupling regime as function of the driving frequency for two values of the tem- peratureT=TN=TF. Material parameters are taken for an FjN interface with F=YIG and N is an arbitrary normal metal, see Tab. I. Curves are shown for three combinations of the time-independent background potentials  mand sk: The solid colored curves correspond to  m= sk= 0; the dashed curves correspond to  m= sk= 0:5~!0, and the dot-dashed ones to m= sk=0:5~!0. with the spin-Seebeck coecient LSSE58 LSSE=~ 2(2)21Z !0d kx( )2Tm( )( sk=~)  @fT( sk=~) @  : (56) In the weak-coupling limit of Eq. (27), one recovers the spin-Seebeck coecient obtained by Cornelissen et al. ,35 L(p) SSE=~Reg"# sZ d 3D m( )( sk=~)2  @fT( sk=~) @  : (57) In the limit !0kBT=~the frequency integration may be performed and one nds35 L(p0) SSE=c0Reg"#kBT skBT ~Dex3=2 ; (58) withc0= 15(5=2)=320:63. All three expressions are evaluated in Fig. 7 as a function of Tfor mate- rial parameters of a YIG jPt interface. Like in the case of the longitudinal spin conductance, we observe that there are small quantitative dierences between the non- perturbative and perturbative results. These dierences are small at low temperatures, but the perturbative weak-coupling result deviates from the non-perturbative one at higher temperatures, the dierence reaching a fac- tor2:3 at room temperature, see the upper left inset of Fig. 7.10 10010110210-2910-2710-2510-2310-21 1001011020.51.0 LSSE L(p0) SSE 1001011020.51.01.5L(pH) SSE L(p) SSE 0.0 0.2 0.4 0.6 0.8 1.0 T (K)0.00.20.40.60.81.0LSSE/Reg (J) FIG. 7. Spin-Seebeck coecient LSSEat a YIGjPt-interface in the linear-response regime as function of the temperature T=TN=TF. All three curves in the main panel are ob- tained using the parabolic approximation of the magnon dis- persion. The red solid curve shows LSSEaccording to the non- perturbative theory, see Eq. (56), the blue dashed curve the perturbative result L(p) SSEof Eq. (57), and the thin green dot- dashed curve includes the approximation L(p0) SSEof Eq. (58). The upper left inset shows the ratios LSSE=L(p0) SSE(red solid curve) and L(p) SSE=L(p0) SSE(blue dashed curve). The lower right inset shows the ratio L(pH) SSE=L(p) SSE, whereL(pH) SSE is the result of Eq. (57) for the magnon density of states obtained from a Heisenberg model, see Eq. (46), and L(p) SSEthat of Eq. (57) for the quadratic approximation of the magnon dispersion. Parameter values are taken from Tab. I. V. CONCLUSIONS AND OUTLOOK The spin angular momentum current from a normal metal N into a ferro-/ferrimagnetic insulator F in general has a component collinear with the magnetization, which is carried by thermal magnons in F. In this article, we pre- sented two calculations of the longitudinal interfacial spin conductance: At zero frequency, but for arbitrary trans- parency of the interface, and at nite frequencies, but to leading order in the interface transparency. In general, one expects the longitudinal interfacial spin conductance to acquire a dependence on the driving frequency !, when !exceedskBT=~. In the case of typical parameters for the material combination YIG jPt and at room tempera- ture, we nd that the resulting frequency dependence of the interfacial spin conductance is rather weak, not more than a factor1:1 between the low- and high-frequency limits. Also, we nd that (at zero frequency) the dier- ence between the spin conductance in a non-perturbative treatment of the coupling across the F jN interface and the perturbative result to leading order in the spin-mixing conductance g"#is not more than a factor 1:7 at room temperature, despite the fact that g"#of a good YIGjPt interface (see Tab. I) is only slightly below the Sharvin limit (e2=h)g"#=e2=h2 e6:8
1014 1m2,56whereeis the Fermi wavelength of Pt.59{61In that sense, for FjN interfaces involving the ferrimagnetic insulator YIG, our two calculations may seen as a conrmation of the ex- isting low-frequency weak-coupling theory.12,23,35A sim- ilar conclusion applies to the interfacial spin-Seebeck coecient, for which we compared the existing weak- coupling zero-frequency theory35,36,62with a calculation non-perturbative in the interface transparency. Of course, one may turn the question around and ask, under which experimental conditions or for which mate- rial combinations a frequency dependence of the inter- facial longitudinal spin conductance or a deviation from the perturbative weak-coupling approximation will be- come signicant. To see an appreciable frequency de- pendence of gsk, it is necessary that the temperature is signicantly below the maximum energy ~ maxof acous- tic magnons. For YIG, this means that the temperature must be well below room temperature. Our numerical estimates based on material parameters for YIG indicate thatgsk(!) may increase by a factor &10 between low- and high-frequency regimes if T.30 K and that the eect can be larger at lower temperatures, whereas the frequency dependence of gsk(!) is small for T&100 K. An experimental technique to measure these eects is the spin-Hall magnetoresistance, which depends on the competition of longitudinal and transversal spin trans- port across the F jN interface. Measurements of the spin-Hall magnetoresistance up to the lower GHz range63 have already been performed. Since the longitudinal and transversal interfacial spin conductances are of compa- rable magnitude in the high-frequency limit, one may thus expect a visible frequency dependence of the spin- Hall magnetoresistance eect for frequencies in the THz range, if the temperature is low enough that not all magnon modes are thermally excited. (This eect is ad- ditional to a frequency dependence of the spin-Hall mag- netoresistance in the GHz range predicted in Ref. 33.) However, since the spin-Hall magnetoresistance eect in- volves the dierence of two contributions of comparable magnitude, a more precise material-specic modeling is required to reach a rm prediction. Another experimental platform in which the lon- gitudinal interfacial spin conductance plays a role is that of non-local magnonic spin transport.23{25In this case, the interfacial spin conductance directly determines the coupling between the magnon system in a ferro- /ferrimagnetic insulator and the electrical currents in ad- jacent normal-metal contacts used to excite and detect the magnon currents. Our predictions directly trans- late to a frequency dependence of the electron-to-magnon and magnon-to-electron conversion in such experiments. Furthermore, the dierence between the weak-coupling and strong-coupling predictions may quantitatively af- fect estimates of the spin-mixing conductance based on a measurement of the longitudinal spin conductance or the spin-Seebeck coecient.28{31,53,64,65. We predict that the longitudinal spin conductance depends on the temperatures TFandTNof the ferro-11 /ferrimagnetic insulator and the normal metal in dier- ent ways, see, e.g., Eqs. (45) and (50). Whereas the lon- gitudinal spin current in F is carried by thermal magnons if F and N are close to equilibrium, the longitudinal spin conductance does not vanish if TF= 0, as long as TN is non-zero. In this case, the spin current is carried by magnons in F excited by spin- ip scattering of thermally excited electrons at the F jN interface. Apart from the diculty that a large temperature dierence between F and N is dicult to realize experimentally, a large tem- perature dierence across an F jN interface also leads to a large steady-state spin current via the interfacial spin- Seebeck eect. However, this DC spin current can be easily distinguished experimentally from the AC signal, which is caused by time-dependent driving of the spin accumulation in N. At the interface between a normal metal and a ferro- /ferrimagnetic metal, there are two contributions to the longitudinal spin current: A contribution from conduc- tion electrons in the ferro-/ferrimagnetic metal and a magnonic contribution. The results derived in this article also apply to the magnonic contribution at such an inter- face. However, at metal-metal interfaces, the magnonic contribution to the spin current is typically much smaller than the electronic contribution so that the frequency and temperature dependence of the magnonic contribu- tion is a sub-leading eect at such interfaces. We close with two remarks on possible further ex- tensions of our work. An important limitation of our theory is the restriction to the lowest magnon band. On the one hand, this limitation enters into our non- perturbative calculation for low frequencies, because the calculation relies on the continuum limit of the Landau- Lifshitz-Gilbert equation. On the other hand, this lim- itation enters both calculations, because the boundary condition at the F jN interface implicitly assumes that the coupling between electronic degrees of freedom in N and the magnonic degrees of freedom in F at the inter- face is local. For acoustic magnons at the zone boundary and for higher magnon bands, electrons in N re ecting o the ferro-/ferrimagnetic insulator F penetrate F suf- ciently deep such that they are in uenced by the non- uniformity of m, violating the assumption of a local cou- pling between magnonic and electronic degrees of free- dom. The rst problem can be partially addressed by replacing the quadratic magnon dispersion by the dis- persion of a Heisenberg model on a simple cubic lattice, as we have done in Sec. IV, but this replacement does not account for the non-uniformity of the magnetization near the interface. A rough estimate for the frequency at which the non-uniformity becomes relevant is the max- imum frequency maxof acoustic magnons, where for YIG max=21013Hz. It is an open task for the future to extend our theory to appropriate couplings be- tween electron spins and short-wavelength magnons, op- tical magnons, and antiferromagnons in antiferromagnets and ferrimagnets. Our nite-frequency calculations assume that it is onlythe electronic distribution in the normal metal N that is driven out of equilibrium. Experiments exciting directly the phonons of an insulating magnet F such as YIG, e.g., by an ultrashort THz laser pulse, might also create a time-dependent magnon chemical potential in F on ul- trafast time scales.35Time-dependent magnon chemical potentials may also appear in ultrafast versions of non- local magnon transport experiments or in the ultrafast spin-Hall magnetoresistance eect with an ultrathin mag- netic insulator F.33Investigating the ultrafast response to a change of magnon chemical potential is another in- teresting avenue for future research. Acknowledgements.| We thank T. Kampfrath, T. S. Seifert, U. Atxitia, F. Jakobs, and S. M. Rouzegar for stimulating discussions. This work was funded by the German Research Foundation (DFG) via the collabora- tive research center SFB-TRR 227 \Ultrafast Spin Dy- namics" (project B03). Appendix A: Transformation to a rotating frame Here we consider the transformation to a reference sys- tem for the spin degree of freedom that rotates with an- gular frequency !(t) = !(t)ek. We discuss how the lon- gitudinal spin accumulation skin N, the magnetization amplitudem?, and the stochastic transverse spin current js?transform upon passing to the rotating frame. We re- strict the discussion to linear response in skand!. We use a prime to denote creation and annihilation operators and observables in the rotating reference system. We rst consider the transformation to a frame rotat- ing at constant angular velocity != !ek. The transfor- mation relation for the electron annihilation operators in N is ^ 0(t) =ei!
t=2^ (t); (A1) where (t) is a two-component column spinor for the wavefunction of the conduction electrons. Solving for the annihilation operator c(") in energy representation, we have ^ 0(") =^ ("+~!
=2) (A2) and, similarly, ^ 0(")y=^ ("+~!
=2)y: (A3) It follows that the distribution function f0(") in the ro- tating frame is f0 TN(") =fTN("+~!
=2); (A4) wherefTNis the distribution function in the original ref- erence frame. We thus conclude that, in linear response, upon transforming to a rotating frame the spin accumu- lation changes as 0 s=s~!: (A5)12 Appendix B: Weak-coupling spin current at nite frequency We rst discuss the expression (29) for the longitudi- nal spin current through the F jN interface. From the Heisenberg equation of motion, the spin current into the magnetic insulator is ^jx sk(t) =i 2[^H;^N"^N#]; (B1) where ^His the Hamiltonian and ^Nis the number of conduction electrons with spin ,=",#. The only contribution to ^Hthat does not commute with ^Nis the term (28) describing the coupling via the F jN interface. Inserting Eq (28) into the above expression gives Eq. (29) of the main text. We next turn to the calculation of the expectation valuejx sk(t) of the interfacial longitudinal spin current. Calculating jx sk(t) to leading order in perturbation the-ory inJgives jx sk(t) =ijJj2 ~Z cdt0[G"(t0;t)G#(t;t0)D(t;t0) G"(t;t0)G#(t0;t)D(t0;t)]; (B2) wheret0is integrated along the Keldysh contour ( i.e., forward and backward integrations along the real time axis), G(t0;t) =ihTc^ (t0)^ y (t)i (B3) is the contour-ordered Green function for the conduction electrons, evaluated at the interface, and D(t0;t) =ihTc^a(t0)^ay(t)i (B4) is the contour-ordered magnon Green function, again evaluated at the interface. Equation (B2) may be written as jx sk(t) =ijJj2 ~1Z 1dt0h (GR "(t0;t) +G< "(t0;t))(GR #(t;t0) +G< #(t;t0))(DR(t;t0) +D<(t;t0)) (GR "(t;t0) +G< "(t;t0))(GR #(t0;t) +G< #(t0;t))(DR(t0;t) +D<(t0;t)) G> "(t0;t)G< #(t;t0)D<(t;t0) +G< "(t;t0)G> #(t0;t)D>(t0;t)i : (B5) In this expression, the integration variable t0is a time, not a contour time. We rst evaluate Eq. (B5) for the case that the three subsystems | conduction electrons with spin up, con- duction electrons with spin down, and magnons | are separately in equilibrium at chemical potentials and mand temperatures TandTm, respectively. In this case, all Green functions depend on the time dierence tt0only. Changing to the integration variable t0tfor the rst term and third term and tt0for the second and fourth term in Eq. (B5), one nds that the rst and third terms in Eq. (B5) cancel, whereas the second and fourth terms give, after Fourier transform, jx sk=ijJj2 ~Zd" 2Zd 2h G> "(")G< #(" )D<( ) G< "(")G> #(" )D>( )i : (B6) According to the uctuation-dissipation theorem, one has G> (") =2i~(1n(")); G< (") = 2i~n("): (B7)Similarly, for the magnon Green function, one has D>( ) =2i(fTF( m=~) + 1)m( ); D<( ) =2ifTF( m=~)m( ): (B8) Hence, we nd that the spin current is jx sk= 2jJj2"#Z d"Z d m( ) (B9) [n"(")(1n#("~ ))(1 +fTF( )m=~) (1n"("))n#("~ )fTF( m=~)]: SettingT"=T#=TNand performing the integration over", one reproduces Eqs. (29) and (30) of the main text, which was derived from Fermi's Golden Rule. We now consider an additional oscillating component of the chemical potential as in Eq. (33) of the main text. In the presence of the oscillating chemical potential the electron Green function G(t;t0) reads13 G(t;t0) =G0(t;t0)eiRt t0d(0)=~ =G0(t;t0) 1 +Z d!(!) ~!ei!t[1ei!(tt0)] +:::; (B10) for the greater and lesser Green functions, where, in the second line, the subscript \0" indicates the equilibrium Green function and the dots indicate terms of higher order in (!). Similarly, one has G(t0;t) =G0(t0;t) 1Z d!(!) ~!ei!t[1ei!(tt0)] +:::: (B11) To nd the spin current, we nd it advantageous to cast the rst two terms of Eq. (B5) into a dierent form, making repeated use of the identities GR+G<=GA+G>andDR+D<=DA+D>, jx sk(t) =ijJj2 ~Z dt0h (GA "(t0;t) +G> "(t0;t))(GR #(t;t0) +G< #(t;t0))(DR(t;t0) +D<(t;t0)) (GR "(t;t0) +G< "(t;t0))(GA #(t0;t) +G> #(t0;t))(DA(t0;t) +D>(t0;t)) G> "(t0;t)G< #(t;t0)D<(t;t0) +G< "(t;t0)G> #(t0;t)D>(t0;t)i : (B12) For the linear-response correction to the spin current, we then obtain jx sk(!) =ijJj2 ~2!Z dt0[ei!(tt0)1] n [G< "0(t0t)GR #(tt0)"(!)GA "(t0t)G< #0(tt0)#(!)][DR(tt0) +D<(tt0)] + [GA #(t0t)G< "0(tt0)"(!)G< #0(t0t)GR "(tt0)#(!)][DA(t0t) +D>(t0t)] +G> "0(t0t)G< #0(tt0)("(!)#(!))DR(tt0) +G< "0(tt0)G> #0(t0t)("(!)#(!))DA(t0t)o : (B13) Writing the Green functions in terms of their Fourier representations, we write this as jx sk(!) =ijJj2 ~!Zd" 2Zd 2 nh [G< "0("!)G< "0(")]GR #(" )"(!)GA("+ )[G< #0("+!)G< #0(")]#(!)i [DR( ) +D<( )] +h G< "0("+!)G< "0(")]GA #(" )"(!)GR("+ )[G< #0("!)G< #0(")]#(!)i [DA( ) +D>( )] + [G> "0("!)G> "0(")]G< #0(" )("(!)#(!))DR( ) + [G< "0("+!)G< "0(")]G> #0(" )("(!)#(!))DA( )o : (B14) Again we use the uctuation-dissipation theorem, see Eqs. (B7) and (B8). For the electrons we assume that the spectral density is independent of energy and we set GR (") =GA (") =i~. For the magnons we use that DR( ) +D<( ) =DA( ) +D>( ) =DR( )(fTF( m=~) + 1)DA( )fTF( m=~): (B15) We then nd jx sk(!) =ijJj2 2~!Z d"Z d "#f[[n"("~!)n"("+~!)]"(!)[n#("~!)n#("+~!)]#(!)] [DR( )(fTF( m=~) + 1)DA( )fTF( m=~)]2("(!)#(!))  [n"("~!)n"(")]n#("~ )DR( )[n"("+~!)n"(")][1n#("~ )]DA( ) =ijJj2 ~!sk(!)Z d  DR( ) ![fTF( m=~) + 1]Z d"[n"("~!)n"(")]n#("~ ) +DA( ) (!)[fTF( m=~) + 1]Z d"[n"("+~!)n"(")]n#("~ ) : (B16)14 IfT"=T#=Tthis may be further simplied as jx sk(!) =ijJj2 ~!"#sk(!)Z d  DR( ) (~ sk!)fTN( !sk=~)(~ sk)fTN( sk=~) +~!fTF( m=~) +DA( ) (~ sk+~!)fTN( +!sk=~)(~ sk)fTN( sk=~)~!fTF( m=~) : (B17) The retarded and advanced magnon Green functions can be obtained from the Krppmers-Kronig relations, DR( ) =DA( ) =Z d 0m( 0) +i 0; (B18) whereis a positive innitesimal. In the main text the superscript \R" for the retarded magnon Green function is omitted. In the limit !!0, Eq. (B17) simplies to jx sk(0) = 2jJj2"#sk(0)Z d m( )( fTF( m=~)fTN( sk=~)( sk=~)dfTN d  sk=~) ;(B19) which is consistent with Eq. (31). 1N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt College Publishers, Inc., 1976). 2C. Kittel, Introduction to Solid State Physics , 8th ed. (John Wiley & Sons, Inc., 2005). 3L. D. Landau, E. M. Lifschitz, and L. P. Pitaevski, Sta- tistical Physics, part 2 (Pergamon, Oxford, 1980). 4A. G. Gurevich and G. A. Melkov, Magnetization Oscilla- tions and Waves (CRC Press, Inc., 1996). 5N. Majlis, The Quantum Theory of Magnetism (World Sci- entic Publishing, 2007). 6S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon Press, 1966). 7L. Berger, Phys. Rev. B 54, 9353 (1996). 8J. C. Slonczewski, J. Magn. Magn. Mater. 159, 1 (1996). 9J. C. Slonczewski, J. Magn. Magn. Mater. 195, 261 (1999). 10Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 11Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 12S. A. Bender and Y. Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015). 13A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub- ota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005). 14J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivo- rotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 (2006). 15L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 16K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and S. Kasai, Appl. Phys. Express 5, 073002 (2012). 17A. Ganguly, K. Kondou, H. Sukegawa, S. Mitani, S. Kasai, Y. Niimi, Y. Otani, and A. Barman, Appl. Phys. Lett.104, 072405 (2014). 18K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 19C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Materials 9, 898 (2010). 20K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. Bauer, S. Maekawa, and E. Saitoh, Nature Materials 9, 894 (2010). 21G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Materials 11, 391 (2012). 22J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). 23S. S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). 24S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, , and H. Huebl, App. Phys. Lett. 107, 172405 (2015). 25R. Schlitz, S. V elez, A. Kamra, C.-H. Lambert, M. Lam- mel, S. T. B. Goennenwein, and P. Gambardella, Phys. Rev. Lett. 126, 257201 (2021). 26M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). 27S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012). 28H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel,15 S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goen- nenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 29C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). 30N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). 31M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.- M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 32X.-P. Zhang, F. S. Bergeret, and V. N. Golovach, Nano Lett. 19, 6330 (2019). 33D. Reiss, T. Kampfrath, and P. Brouwer, Phys. Rev. B 104, 024415 (2021). 34A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000). 35L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). 36R. Schmidt, F. Wilken, T. S. Nunner, and P. W. Brouwer, Phys. Rev. B 98, 134421 (2018). 37A. J. Schellekens, K. C. Kuiper, R. R. J. C. de Wit, and B. Koopmans, Nature Comm. 5, 4333 (2014). 38I. Razdolski, A. Alekhin, N. Ilin, J. P. Meyburg, V. Rod- datis, D. Diesing, U. Bovensiepen, and A. Melnikov, Na- ture Comm. 8, 15007 (2017). 39T. S. Seifert, S. Jaiswal, J. Barker, S. T. Weber, I. Razdol- ski, J. Cramer, O. Gueckstock, S. F. Maehrlein, L. Nad- vornik, S. Watanabe, C. Ciccarelli, A. Melnikov, G. Jakob, M. M unzenberg, S. T. B. Goennenwein, G. Woltersdorf, B. Rethfeld, P. W. Brouwer, M. Wolf, M. Kl aui, and T. Kampfrath, Nature Comm. 9, 2899 (2018). 40L. Brandt, U. Ritzmann, N. L. M. Ribow, I. Razdolski, P. W. Brouwer, A. Melnikov, and G. Woltersdorf, Phys. Rev. B. 104(2021). 41P. Jim enez-Cavero, O. Gueckstock, L. N advorn k, I. Lu- cas, T. S. Seifert, M. Wolf, R. Rouzegar, P. W. Brouwer, S. Becker, G. Jakob, M. Kl aui, C. Guo, C. Wan, X. Han, Z. Jin, H. Zhao, D. Wu, L. Morell on, and T. Kampfrath, arXiv:2110.05462 (2021). 42J. Kimling, G.-M. Choi, J. T. Brangham, T. Matalla- Wagner, T. Huebner, T. Kuschel, F. Yang, and D. G. Cahill, Phys. Rev. Lett. 118, 057201 (2017). 43F. N. Kholid, D. Hamara, M. Terschanski, F. Mertens, D. Bossini, M. Cinchetti, L. McKenzie-Sell, J. Patchett, D. Petit, R. Cowburn, J. Robinson, J. Barker, and C. Ci- ccarelli, arXiv:2103.07307 (2021). 44H. C. Teh, M. F. Collins, and H. A. Mook, Can. J. Phys. 52, 396 (1973). 45J. Shan, A. V. Singh, L. Liang, L. J. Cornelissen, Z. Galazka, A. Gupta, B. J. van Wees, and T. Kuschel, Appl. Phys. Lett. 113, 162403 (2018).46The eects discussed here in principle arise in magnetic metals , too. However, in that case these eects are dicult to be distinguished experimentally from the much larger spin conductance carried by electrons in the magnet.33 This is the reason why we restrict the present discussion to magnetic insulators. 47S. Krupi cka and P. Nov ak, Handbook of Magnetic Mate- rials, edited by E. P. Wohlfarth, Vol. 3, Ch. 4 (North- Holland, Amsterdam, 1982). 48J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117, 217201 (2016). 49J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 95, 016601 (2005). 50G. Tatara and S. Mizukami, Phys. Rev. B 96, 064423 (2017). 51Equation (6) contains two terms proportional to the imaginary part of the spin mixing conductance, (1=4)Img"#(ms) and(1=4)Img"#(~_m). In a mi- croscopic theory of the F jN interface, the coecient mul- tiplying ~_mmay be dierent from the coecient multi- plying ms, see Ref. 50. Since the numerical values for the imaginary part of the spin mixing conductance used to estimate the longitudinal spin conductance in Sec. IV are much smaller than those for its real part and the longitu- dinal spin conductance is mainly determined by Re g"#, we ignore this dierence here. 52L. D. Landau and E. M. Lifschitz, Statistical Physics, part 1(Pergamon, Oxford, 1980). 53Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Appl. Phys. Lett. 103, 092404 (2013). 54X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys. Lett. 96, 17005 (2011). 55V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep. 229, 81 (1993). 56M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005). 57J. S. Plant, J. Phys. C: Solid State Phys. 16, 7037 (1983). 58R. Schmidt and P. W. Brouwer, Phys. Rev. B 103, 014412 (2021). 59J. B. Ketterson, F. M. Mueller, and L. R. Windmiller, Phys. Rev. 186(1969). 60F. M. Mueller, J. W. Garland, M. H. Cohen, and K. H. Bennemann, Ann. Phys. 67, 19 (1971). 61F. Y. Fradin, D. D. Koelling, A. J. Freeman, and T. J. Watson-Yang, Phys. Rev. B 12, 5570 (1975). 62J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). 63J. Lotze, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B 90, 174419 (2014). 64C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013). 65H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014).

2022-08-02

The interface between a ferro-/ferrimagnetic insulator and a normal metal can support spin currents polarized collinear with and perpendicular to the magnetization direction. The flow of angular momentum perpendicular to the magnetization direction ("transverse" spin current) takes place via spin torque and spin pumping. The flow of angular momentum collinear with the magnetization ("longitudinal" spin current) requires the excitation of magnons. In this article we extend the existing theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal spin conductance non-perturbatively (but in the low-frequency limit) and at finite frequency (but in the limit of low interface transparency). For the paradigmatic spintronic material system YIG|Pt, we find that non-perturbative effects lead to a longitudinal spin conductance that is ca. 40% smaller than the perturbative limit, whereas finite-frequency corrections are relevant at low temperatures < 100 K only, when only few magnon modes are thermally occupied.

Finite-frequency spin conductance of a ferro-/ferrimagnetic-insulator|normal-metal interface

2208.01420v1

arXiv:2308.15826v1 [quant-ph] 30 Aug 2023Chiral cavity-magnonic system for the unidirectional emis sion of a tunable squeezed microwave field Ji-kun Xie, Sheng-li Ma,∗Ya-long Ren, Shao-yan Gao, and Fu-li Li MOE Key Laboratory for Non-equilibrium Synthesis and Modul ation of Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an 710049 , China Unidirectional photon emission is crucial for constructin gquantumnetworks and realizing scalable quantum information processing. In the present work an effici ent scheme is developed for the unidirectional emission of a tunable squeezed microwave fie ld. Our scheme is based on a chiral cavity magnonic system, where a magnon mode in a single-crys talline yttrium iron garnet (YIG) sphereisselectively coupledtooneofthetwodegeneratero tatingmicrowave modesinatorus-shaped cavity with the same chirality. With the YIG sphere driven by a two-color Floquet field to induce sidebands in the magnon-photon coupling, we show that the un idirectional emission of a tunable squeezed microwave field can be generated via the assistance of the dissipative magnon mode and a waveguide. Moreover, the direction of the proposed one-way emitter can be controlled on demand by reversing the biased magnetic field. Our work opens up an aven ue to create and manipulate one-way nonclassical microwave radiation field and could find potent ial quantum technological applications. I. INTRODUCTION Chiral quantum optics has become a burgeoning field due to its potential applications in quantum informa- tion processing and quantum networks [ 1–5]. It seeks to exploit new approaches and systems exhibiting chi- ral light-matter interactions [ 6–16]. On this subject, nanophotonic systems, such as nanoscale waveguides and whispering-gallery-mode(WGM) resonators[ 17–19], have emerged as attractive candidates, where the light field is strongly transversely confined in a subwavelength spaceandexhibitstheopticalspin-orbitcoupling[ 20,21]. The chiral interaction can be achieved by coupling the spin-momentum-locked light to quantum emitters with polarization-dependent dipole transitions [ 22–26]. And a large number of chiral devices have been proposed the- oretically and demonstrated experimentally, like single- photon diodes [ 27,28], single-photon routing [ 29–31], cir- culators [ 28,32], isolators [ 32–34] and etc. These ele- ments are key components for building large-scale quan- tum networks. In parallel, the chiral light-matter interactionscan also be utilized to control the direction of photon emission [35–38]. By designing the suitable chiral coupling be- tween quantum emitters and evanescent fields, the de- terministic and the highly directional photon emission can be achieved along only one of the selected directions [17,22,38]. Recently, thechiral-waveguide-basedandthe chiral-cavity-based systems have been explored to gener- ate the unidirectional emission of single photons [ 39–43]. In addition, the unidirectional laser emission has also been demonstrated through the non-Hermitian scatter- ing induced chiralityin the WGM resonator[ 44,45]. The unidirectional photon emission can facilitate the free- spacecoupling ofenergy, improvethe collection efficiency ∗msl1987@xjtu.edu.cnof weak optical signals and on-demand distribute quan- tum information [ 46–50]. In recent years, the hybrid quantum system of yttrium iron garnet (YIG) and superconducting circuit has been considered as a powerful platform for quantum informa- tion applications [ 51–54]. Due to the high spin density and lowdamping rate ofthe ferrimagneticinsulator YIG, the strong and even ultrastrong coupling between a mag- netostatic mode in a YIG sphere and a microwave cavity mode have been experimentally observed [ 55–60]. Many intriguing phenomena have been studied in this hybrid system, such as the generation of various quantum states [61–72], nonreciprocity [ 73–75], non-Hermitian physics [76–78], and Floquet engineering [ 79–82]. Remarkably, magnonsalsocarryangularmomentumor“spin”. Forin- stance, the Kittel mode magnetization precessescounter- clockwise around the applied magnetic field [ 83]. Anal- ogous with the chiral coupling of spin-polarized atoms or quantum dots with spin-momentum-locked light [ 17– 19], the Kittel mode can only couple to photons with the same polarization in ring-shaped cavity or waveguide [84–88]. Therefore, this hybrid magnonic system can act as a promising architecture for studying chiral quantum optics [89–93]. In this paper, we develop a method for the unidirec- tional emission of a tunable squeezed microwave field basedonachiralcavitymagnonicsystem. In ourscheme, a torus-shaped microwave cavity that supports a pair of degenerate counter-rotating microwave modes is ex- ploited to chirally coupled to a YIG sphere. The chiral interaction can be achieved by locating the YIG sphere on a special chiral line in the cavity, where the circular polarization of the cavity mode is locked to its propa- gation direction [ 86–90]. As a result, the Kittel mode will only couple to one of the microwave modes with the same polarization. By applying a two-color Floquet driving field to the magnetic sphere to induce sideband transitions, we can engineer a squeezing-type interaction between the magnon mode and the selected microwave2 YIG waveguide FIG. 1. (Color online) Schematic diagram of the hybridquan- tum setup. A torus-shaped cavity supports both CCW ( a) and CW ( b) rotating microwave modes, which are evanes- cently coupled to a nearby microwave waveguide. A small fer- romagnetic YIG sphere is placed inside the cavity and biased perpendicularly by a static magnetic field B0to establish the magnon-photon coupling. Additionally, the magnon mode is driven by a two-color Floquet driving field (not shown) along the bias field direction. mode, which can be steered into a squeezed state with the aid of large decay rate of the magnon mode. By further considering the cavity evanescently coupled to a microwave waveguide, we can generate a unidirectionally squeezed microwave source. Notably, the emission direc- tion can be conveniently adjusted by reversing the bias magneticfield, and the amountofsqueezingis tunable by changing the external parameters. Moreover, the numer- ical simulations indicate that the unidirectional emission is robust against the imperfect chiral coupling. Our work could find a wide range of practical applications in quan- tum networks and quantum sensing. II. MODEL As schematically shown in Fig. 1, the hybrid cavity magnonic system under consideration consists of a torus- shaped microwave cavity with a small highly polished YIG sphere placed inside [ 87,88]. Due to the geomet- rical rotational symmetry, the cavity supports a pair of degeneratecounterclockwise(CCW) andclockwise(CW) microwave modes, which are evanescently coupled to a nearbymicrowavewaveguide. Underastaticout-of-plane bias magnetic field at a strength B0, many magnetostatic modes will be excited in the YIG sphere [ 55–57]. In our scheme, we focus only on the Kittel mode, which has the uniform spin precessions in the whole volume of the magnetic sphere. The two microwave modes are cou- pled to the Kittel mode through the collective magnetic- dipoleinteraction,respectively. Now,wecangivethefree HamiltonianoftheKittelmode andthe cavitymodes(we set/planckover2pi1= 1 hereafter) H0=ωmm†m+ω0/summationdisplay α=a,bα†α. (1)Here,m(m†) is the boson operator of the magnon mode with the resonance frequency ωm=γB0, where γ= 2π×28 GHz/T is the gyromagnetic ratio. a(a†) and b(b†) represent the annihilation (creation) operators for the CCW and CW microwave modes with the resonance frequency ω0. We then consider the chiral photon-magnon interac- tion, which takes the form [ 86–90] Hint=/summationdisplay α=a,bgα(α†+α)(m†+m) (2) withgα(α=a,b) describing the coupling strength be- tweenthe cavitymode αand the magnonmode m. To be specific, the Kittel mode magnetization precesses around the effective magnetic field in an anticlockwise manner and couples preferentially to photons with the same po- larization. Here, the magnetic field of each TE mode is transversallyconfinedinthecavity,sothatithasastrong longitudinal polarization component along the propaga- tion direction [ 1,29,94]. Moreover, the longitudinal and transverse polarization components oscillate ±90◦out of phase with each other, where + ( −) sign depends on the propagationdirection of the cavity modes. At the special radial locations, the magnetic field can be circularly po- larized when the longitudinal and transversepolarization components have the same amplitudes. Also, since the polarization direction is locked to the sign of their lin- ear momentum, the counterpropagated CW and CCW modes can thereby possess mutually orthogonal polar- izations and opposite chiralities [ 86]. As a result, the magnon mode can only couple to one of them with the same chirality, i.e., ga/ne}ationslash= 0 and gb= 0, or gb/ne}ationslash= 0 and ga= 0. In addition, the chiral coupling can be well controlled byreversingthedirectionofthebiasedmagneticfield, be- cause it is accompanied with the reversion of the preces- sionoftheKittelmode. Alternatively,eachofthetwode- generate counter-propagatingmicrowavemodes can have opposite chirality at different radial locations [ 87,88], so the chiral coupling can also be controlled by shifting the magnet position. Furthermore, a two-color Floquet driving field is ap- plied to the YIG sphere along the bias magnetic field direction, under which the oscillating frequency of the magnon mode is modulated accordingly. So the coupling between the Floquet driving field and the Kittel mode is described by [ 79–82] Hd(t) = [Ω 1cos(ωd1t)+Ω2cos(ωd2t+θ)]m†m,(3) where Ω j(j= 1,2) denotes the driving amplitudes with thedrivingfrequencies ωdj,θistherelativephase. Exper- imentally,thismanipulationcanbeimplementedthrough a small coil being looped tightly around the magnetic sphere to modulate the bias magnetic field [ 79]. Under this kind of periodical modulation, the desired paramet- ric magnon-photon interaction can be sculpted to realize the one-way squeezed microwave source.3 FIG. 2. (Color online) Schematic diagram of sideband tran- sitions of the coupled magnon-photon modes, where ωd1= ωm−ω0andωd2=ωm+ω0correspond to the red and blue sidebands, respectively. III. UNIDIRECTIONAL EMISSION OF A TUNABLE SQUEEZED MICROWAVE FIELD In this section, we will detail the procedure for unidi- rectionally creating a tunable squeezed microwave field based on above introduced chiral cavity magnonic sys- tem. The chirality stems from the selective coupling of the magnon mode to one of the two degenerate rotating microwavemodeswiththesamepolarization. Combining with the Floquet driving to engineer a desired squeezing- type interaction, we can generate a chiral squeezed mi- crowave radiation under the assistance of a dissipative magnon mode and a waveguide. A. The effective Hamiltonian for the generation of a chiral squeezed state We start to derive the effective Hamiltonian to selec- tively generate a chiral squeezed state; that is, only one of the two cavity modes will be squeezed. To this end, the Floquet driving to the magnonmode plays a keyrole. We define the following rotating transformation: U(t) =Texp{−i/integraldisplayt 0[H0+Hd(τ)]dτ} = exp{−i(ωmm†m+ω0/summationdisplay α=a,bα†α)t −i[ξ1sin(ωd1t)+ξ2sin(ωd2t+θ)]m†m},(4) whereTis the time order operator and ξj= Ωj/ωdj(j= 1,2). Inthe rotatingframewith respectto U(t), the total Hamiltonian Ht=H0+Hint+Hd(t) of the system will become HI=U†(t)HtU(t)+idU†(t) dtU(t) =/summationdisplay α=a,bgαm†ei[ωmt+ξ1sin(ωd1t)+ξ2sin(ωd2t+θ)] ×(αe−iω0t+α†eiω0t)+H.c. (5)By using the Jacobi-Anger identity eixsinφ=n=∞/summationdisplay n=−∞Jn(x)einφ(6) with the nth Bessel functions of the first kind Jn(x), the Hamiltonian can be rewritten as HI=/summationdisplay α=a,bgαm†eiωmt∞/summationdisplay n=−∞Jn(ξ1)einωd1t∞/summationdisplay k=−∞Jk(ξ2) ×eik(ωd2t+θ)(αe−iω0t+α†eiω0t)+H.c. (7) To produce the desired magnon-photon coupling, as displayed in Fig. 2, we choose the Floquet driving fre- quencies satisfying ωd1=ωm−ω0andωd2=ωm+ω0, which correspond to the red and blue sideband transi- tions, respectively. Furthermore, under the condition gα≪ {ωdj,ωm,ω0}, we can only keep the resonant terms in Eq. (7), but safely discard those fast oscillating terms by means of the rotating-wave approximation. Then, we can obtain the effective Hamiltonian of the system Heff=/summationdisplay α=a,bGαm†(α+εe−iθα†)+H.c., (8) where we have Gα=gαJ−1(ξ1)J0(ξ2), and εe−iθ=J0(ξ1)J−1(ξ2)e−iθ/J−1(ξ1)J0(ξ2) is the ra- tio of the parametric-amplifier interaction to the Jaynes-Cummings-type coupling. In what follows, we exclusively consider the situation of |ε|<1, which ensures that Eq. ( 8) is stable. It is now clear that Heff describes a squeezing-type interaction, and can be used to produce the squeezed state. However, due to the chiral coupling, i.e., Ga/ne}ationslash= 0 and Gb= 0, orGb/ne}ationslash= 0 and Ga= 0, we can only generate a squeezed state in one of the two cavity modes. By introducing a hybridized mode A= (Gaa+ Gbb)//radicalbig G2a+G2 b, the Hamiltonian in Eq. ( 8) can be rewritten as Heff=/radicalBig G2a+G2 b[m†(A+εe−iθA†)+H.c.].(9) To clearly reveal the mechanism for generating the chiral squeezed state, we perform the unitary squeezing trans- formation HS=S† A(ζ)HeffSA(ζ) [95], where SA(ζ) = exp[(ζA2−ζ∗A†2)/2] represents the single-mode squeez- ingoperatorwith ζ=reiθ. Hererissqueezingparameter defined by tanh r=εandθdenotes the squeezing angle, both of which can be controlled on-demand by adjust- ing the two-color Floquet driving field. In the squeezed frame, the effectiveHamiltonian ( 9) is transformedto the form HS=/radicalBig (G2a+G2 b)(1−ε2)(m†A+mA†).(10) Obviously, HSdescribes a beam-splitter interaction. If we setGa/ne}ationslash= 0 and Gb= 0 (Gb/ne}ationslash= 0 and Ga= 0) via the4 selective coupling rule, the mode A=a(A=b) can be cooled down to the vacuum state in the squeezed picture via the dissipation of the magnon mode. Reversing the squeezing transformation, the cavity mode a(b) is actu- ally steered into a single-mode squeezed state. In this way, a chiral squeezed state is created. By further con- sidering the microwave cavity coupled to a waveguide, the squeezed output field can be emitted along the right (left) direction of the waveguide. This is the basic idea for the unidirectional emission of a squeezed source. B. Unidirectional squeezing emission We now study the unidirectional squeezing emission by considering the coupling of the microwave cavity cou- pled to a transmission line that acts the role of a waveg- uide. According to the Hamiltonian ( 8) and the stan- dardinput–outputtheory, thegeneralquantumLangevin equations of the system is given by ˙α=−κ 2α−iGα(m+εe−iθm†)+√κexαin,ex +√κ0αin,0, (11a) ˙m=−γm 2m−i/summationdisplay α=a,bGα(α+εe−iθα†)+√γmmin.(11b) Without loss of generality, the cavity modes aandbare assumed to have the same damping rate κ=κ0+κex, whereκ0is the intrinsic decay rate, and κexdenotes the external coupling between the cavity mode and the waveguide. In addition, αin,0andαin,ex(α=a,b) repre- sent the noise operators. minis the noise operator of the magnon mode, and γmis the associated damping rate. These noise operators obey the following correlation re- lations /angbracketleftBig αin,ex(t)α† in,ex(t′)/angbracketrightBig =δ(t−t′),(12a) /angbracketleftBig αin,0(t)α† in,0(t′)/angbracketrightBig =δ(t−t′),(12b)/angbracketleftBig min(t)m† in(t′)/angbracketrightBig =δ(t−t′), (12c) where we have neglected the thermal effects. Because, at the experimental working temperature T= 20 mK, the average thermal excitation number of a boson mode with resonance frequency 6 .5 GHz is about 10−7, which is thus negligible. By introducing the Fourier transformation o(t) =/integraltext+∞ −∞o(ω)e−iωtdω/2πfor an arbitrary operator o, we can rewritethe Eqs. ( 11a)and(11b) in thefrequency domain as α(ω) =1 κ 2−iω{−iGα[m(ω)+εe−iθm†(−ω)] +√κexαin,ex(ω)+√κ0αin,0(ω)},(13a) m(ω) =1 γm 2−iω{−i/summationdisplay α=a,bGα[α(ω) +εe−iθα†(−ω)]+√γmmin(ω)}.(13b) Correspondingly, the correlation functions of Eqs. ( 12a)- (12c) in the frequency domain yield /angbracketleftBig αin,ex(ω)α† in,ex(−ω′)/angbracketrightBig = 2πδ(ω+ω′),(14a) /angbracketleftBig αin,0(ω)α† in,0(−ω′)/angbracketrightBig = 2πδ(ω+ω′),(14b) /angbracketleftBig min(ω)m† in(−ω′)/angbracketrightBig = 2πδ(ω+ω′).(14c) Our interest is the output fields of the cavity. Hence, in terms of the standard input-output relation αout,ex=√κexα−αin,ex, (15) wecanderiveoutthefrequencycomponentsoftheoutput field operators aout,ex(ω) = [Nb(ω)κex D(ω)−1]ain,ex(ω)+1 D(ω){Nb(ω)√κexκ0ain,0(ω)+8GaGb(1−ε2) ×[κexbin,ex(ω)+√κexκ0bin,0(ω)]+4iGaκ−√κexγm[min(ω)+εe−iθm† in(−ω)]},(16a) bout,ex(ω) = [Na(ω)κex D(ω)−1]bin,ex(ω)+1 D(ω){Na(ω)√κexκ0bin,0(ω)+8GaGb(1−ε2) ×[κexain,ex(ω)+√κexκ0ain,0(ω)]+4iGbκ−√κexγm[min(ω)+εe−iθm† in(−ω)]}(16b) withκ−=κ−2iω,γ−=γm−2iω,Nα(ω) = 8G2 α(ε2−1)−2κ−γ−andD(ω) = 4κ−(ε2−1)(G2 a+G2 b)−κ2 −γ−. Now, the spectrum of the output microwave field for the αmode is given by Sα,out(ω) =1 4π/integraldisplay+∞ −∞dω′e−i(ω+ω′)t[/angbracketleftbig Xθ α,out(ω)Xθ α,out(ω′)/angbracketrightbig +/angbracketleftbig Xθ α,out(ω′)Xθ α,out(ω)/angbracketrightbig ], (17)5 whereXθ α,out(ω) =αout,ex(ω)eiθ 2+α† out,ex(−ω)e−iθ 2is the associated quadrature operator of the output fields. Sub- stituting Eqs. ( 16a) and (16b) into Eq. ( 17), we can obtain the squeezing spectra of the output fields Sa,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleNb(ω)κex D(ω)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingleNb(ω)√κexκ0 D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)κex D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)√κexκ0 D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gaκ−(1−ε)√κexγm D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , (18a) Sb,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleNa(ω)κex D(ω)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingleNa(ω)√κexκ0 D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)κex D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle8GaGb(1−ε2)√κexκ0 D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gbκ−(1−ε)√κexγm D(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (18b) To quantify the amount of squeezing of the output fields, we now define the quantum noise reduction Fα,out(ω) =−10log10[Sα,out(ω)/Svac α,out(ω)] in dB units withSvac α,out(ω) = 1 being the squeezing spectrum of the vacuum state. Note that if the output field is squeezed, the quantum noise reduction Fα,out(ω) is larger than zero. Clearly, to achieve a perfect one-way squeezing emitter, the condition Fa,out(ω)>0 andFb,out(ω) = 0 (Fb,out(ω)>0 andFa,out(ω) = 0) should be satisfied, which indicates that the squeezed field is only emitted along the right (left) direction of the waveguide. To well grasp the main result of this work, we start our analysis with the ideal chiral coupling, i.e., ga/ne}ationslash= 0 andgb= 0. Then, the output squeezing spectra in Eqs. (18a) and (18b) yield Sa,out(ω) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleκ−γ−(2κex−κ−)−4κ−(1−ε2)G2 a κ2 −γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle2κ−γ−√κexκ0 κ2 −γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle4Gaκ−(ε−1)√κexγm κ2 −γ−+4κ−(1−ε2)G2a/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ,(19a) Sb,out(ω) = 1, (19b) It is now clear that the spectra Sa,out(ω)<1 and Sb,out(ω) = 1 can be satisfied under the situation of per- fect chirality, which is equivalent to Fa,out(ω)>0 and Fb,out(ω) = 0. So, a desired one-way squeezed source can be created; that is, the squeezed microwave field can only be emitted in one direction. To exhibit the unidi- rectional squeezing emission, we choose ε= 0.95, and showFa,out(ω) andFb,out(ω) versus the dimensionless frequency ω/γmin Fig. 3. It is observed that Fa,out(ω) is larger than zero within a large frequency range, where the largest value of Fa,out(ω) is about 13 .45 dB at the central resonance frequency. On the contrary, Fb,out(ω) is zero over the whole frequency range. This means that we can obtain a squeezed output field in the right side of the waveguide, but there is no squeezing of the output(a) (b) FIG. 3. (Color online) The squeezing spectra Fa,out(ω) [(a)] andFb,out(ω) [(b)] versus the dimensionless frequency ω/γm under the different coupling ratios gb/ga= 0, 0.05, 0.1. The related parameters are chosen as Ga/γm= 2.5,ε= 0.95, κ0/γm= 0.05,κex/γm= 2.5 andγm/2π= 2 MHz. field in the left side of the waveguide. Thus, the unidi- rectional emission of a squeezed source can be achieved. Notably, we can readily switch the emission direction of the squeezed field by reversing the direction of the exter- nal biased magnetic field. In practice, our system will inevitably suffer from the non-ideal chiral coupling, i.e., ga/ne}ationslash= 0 and gb/ne}ationslash= 0, such that that the completely one-way squeezing emission will be spoiled. In the presence of non-ideal chiral coupling, we know from Eq. ( 10) that the hybridized mode Awill be steered into a squeezed state via the dissipation of the magnon mode m, i.e., both the modes aandbare thereby squeezed. As a result, the squeezed fields will be emitted in both directions of the waveguide. With the increase of gb, the associated output squeezing Fa,out(ω) will be reduced, while the output squeezing Fb,out(ω) is gradually increased. Nonetheless, for gb/ga= 0.1, the value of Fa,out(ω) only has a negligible reduction, and the maximal value of Fb,out(ω) is about 0 .04 dB [See Fig. 3]. So, our scheme is robust against the imperfect chiral coupling, indicating that we can still implement a high- performance unidirectional squeezing emitter. Finally, we emphasize here that the proposed one-way emitter with a tunable amount of squeezing can also be6 (a) (b) FIG. 4. (Color online) The squeezing degree Fa,out(0) [(a)] andFb,out(0) [(b)] versus the parameter εunder the different coupling ratios gb/ga= 0, 0.05, 0.1. The other parameters are the same as in Fig. 3. generated in our scheme. Fig. 4 displays the Fa,out(0) andFb,out(0) versus the parameter ε. We can see that Fa,out(0)behavesasanonmonotonicfunctionof ε,imply- ing that a tunable squeezed output field can be created by tuning the parameter ε. This can be implemented by adjusting the Floquet driving parameters Ω j(j= 1,2). As presented in Fig. 4(a), there exists an optimum value of ε, where a maximal amount of output squeez- ing is produced. To illustrate this point, we recall that the intra-cavity squeezing depends on the parameter ε. Meanwhile, according to the standard input-output re- lation, the intra-cavity squeezing determines the output one. So, the output squeezing is also determined by the parameter ε. Based on Eqs. ( 9) and (10), we know that the dissipative magnon mode can be used to cool the cavity mode down to a squeezed state. So, it seems like that the intra-cavity squeezing will be continuously in- creased with the increase of ε. However, as the param- eterεincreasing, the effective magnon-photon coupling, which plays the role of cooling, is gradually decreased in the squeezed representation [See Eq. ( 10)]. Besides, thecavity vacuum noise is transformed to the effective ther- mal noise, which is also be amplified. As a result, the cavity can not approach a squeezed vacuum state, which is actually a thermal squeezed state. Therefore, the com- petition ofthese two processesleads to an optimum value ofεthat can generate a peak of output squeezing. IV. CONCLUSION In summary, we have presented an approach for the unidirectional emission of a tunable squeezed microwave field in a chiral cavity magnonic system of a YIG sphere and a torus-shaped cavity. The chirality comes from the selective coupling between the Kittel mode to one of the two counter-propagating microwave modes with the same polarization. Under a bichromatic Floquet driv- ing field to induce sidebands, the unidirectional emission of a tunable squeezed microwave source can be gener- ated with the help of a dissipative magnon mode and a waveguide. Moreover, the emission direction can be con- venientlychangedbyvaryingthedirectionoftheexternal applied magnetic field. With the rapid development of hybrid cavity magnonic system, our scheme is expected to be implemented in a realistic experiment, which can stimulate a wide range of quantum technological appli- cations. ACKNOWLEDGEMENT This work was supported by the National Natural Sci- ence Foundation of China (Grant Nos. 11704306 and 12074307). [1] Peter Lodahl, Sahand Mahmoodian, Søren Stobbe, Arno Rauschenbeutel, Philipp Schneeweiss, J¨ urgen Volz, Hannes Pichler, and Peter Zoller, “Chiral quantum op- tics,”Nature541, 473–480 (2017) . [2] Jan Petersen, J¨ urgen Volz, and Arno Rauschenbeutel, “Chiral nanophotonic waveguide interface based on spin- orbit interaction of light,” Science346, 67–71 (2014) . [3] Immo S¨ ollner, Sahand Mahmoodian, Sofie Lind- skov Hansen, Leonardo Midolo, Alisa Javadi, Gabija Kirˇ sansk˙ e, Tommaso Pregnolato, Haitham El-Ella, Eun Hye Lee, Jin Dong Song, Søren Stobbe, and Peter Lodahl, “Deterministic photon–emitter coupling in chiral photonic circuits,” Nature Nanotechnology 10, 775–778 (2015). [4] Sahand Mahmoodian, Peter Lodahl, and Anders S. Sørensen, “Quantum networks with chiral-light–matter interaction in waveguides,” Physical Review Letters 117, 240501 (2016) . [5] Lei Tang, Jiang shan Tang, and Keyu Xia, “Chi- ral quantum optics and optical nonreciprocity based onsusceptibility-momentum locking,” Advanced Quantum Technologies 5, 2200014 (2022) . [6] Tom´ as Ramos, Hannes Pichler, Andrew J. Daley, and Peter Zoller, “Quantum spin dimers from chiral dissipa- tion in cold-atom chains,” Physical Review Letters 113, 237203 (2014) . [7] Su-Hyun Gong, Filippo Alpeggiani, Beniamino Sciacca, Erik C. Garnett, and L. Kuipers, “Nanoscale chiral valley-photon interface through optical spin-orbit cou- pling,”Science359, 443–447 (2018) . [8] Eduardo S´ anchez-Burillo, Chao Wan, David Zueco, and Alejandro Gonz´ alez-Tudela, “Chiral quantum optics in photonic sawtooth lattices,” Physical Review Research 2, 023003 (2020) . [9] Fan Zhang, Juanjuan Ren, Lingxiao Shan, Xueke Duan, Yan Li, Tiancai Zhang, Qihuang Gong, and Ying Gu, “Chiral cavity quantum electrodynamics with cou- pled nanophotonic structures,” Physical Review A 100, 053841 (2019) .7 [10] Sabyasachi Barik, Aziz Karasahin, Sunil Mittal, Edo Waks, and Mohammad Hafezi, “Chiral quantum optics using a topological resonator,” Physical Review B 101, 205303 (2020) . [11] Benoˆ ıt Vermersch, Tom´ as Ramos, Philipp Hauke, and Peter Zoller, “Implementation of chiral quantum optics with rydberg and trapped-ion setups,” Physical Review A93, 063830 (2016) . [12] Mahmoud Jalali Mehrabad, Andrew P. Foster, Ren´ e Dost, Edmund Clarke, Pallavi K. Patil, A. Mark Fox, Maurice S. Skolnick, and Luke R. Wilson, “Chiral topo- logical photonics with an embedded quantum emitter,” Optica7, 1690 (2020) . [13] John Clai Owens, Margaret G. Panetta, Brendan Saxberg, Gabrielle Roberts, SrivatsanChakram, Ruichao Ma, Andrei Vrajitoarea, Jonathan Simon, and David I. Schuster, “Chiral cavity quantum electrodynamics,” Na- ture Physics 18, 1048–1052 (2022) . [14] JunHwan Kim, Mark C. Kuzyk, Kewen Han, Hailin Wang, and Gaurav Bahl, “Non-reciprocal brillouin scattering induced transparency,” Nature Physics 11, 275–280 (2015) . [15] Chun-Hua Dong, Zhen Shen, Chang-Ling Zou, Yan- Lei Zhang, Wei Fu, and Guang-Can Guo, “Brillouin- scattering-induced transparency and non-reciprocal light storage,” Nature Communications 6(2015), 10.1038/ncomms7193 . [16] Sheng-Li Ma, Ya-Long Ren, Ming-Tao Cao, Shou-Gang Zhang, and Fu-Li Li, “Optical isolator based on back- ward brillouin scattering,” Applied Physics Letters 120, 051109 (2022) . [17] R. J. Coles, D. M. Price, J. E. Dixon, B. Royall, E. Clarke, P. Kok, M. S. Skolnick, A. M. Fox, and M. N. Makhonin, “Chirality of nanophotonic waveg- uide with embedded quantum emitter for unidirec- tional spin transfer,” Nature Communications 7(2016), 10.1038/ncomms11183 . [18] Christian Junge, Danny O’Shea, J¨ urgen Volz, and Arno Rauschenbeutel, “Strong coupling between single atoms and nontransversal photons,” Physical Review Letters 110, 213604 (2013) . [19] Elisa Will, Luke Masters, Arno Rauschenbeutel, Michae l Scheucher, and J¨ urgen Volz, “Coupling a single trapped atom to a whispering-gallery-mode microresonator,” Physical Review Letters 126, 233602 (2021) . [20] K. Y. Bliokh, F. J. Rodr´ ıguez-Fortu˜ no, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nature Photonics 9, 796–808 (2015) . [21] Todd Van Mechelen and Zubin Jacob, “Universal spin- momentum locking of evanescent waves,” Optica3, 118 (2016). [22] D. L. Hurst, D. M. Price, C. Bentham, M. N. Makhonin, B. Royall, E. Clarke, P. Kok, L. R. Wilson, M. S. Skol- nick, and A. M. Fox, “Nonreciprocal transmission and reflection of a chirally coupled quantum dot,” Nano Let- ters18, 5475–5481 (2018) . [23] R. Mitsch, C. Sayrin, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Quantum state-controlled direc- tional spontaneous emission of photons into a nanopho- tonic waveguide,” Nature Communications 5(2014), 10.1038/ncomms6713 . [24] Sebastian Pucher, Christian Liedl, Shuwei Jin, Arno Rauschenbeutel, andPhilipp Schneeweiss, “Atomic spin- controlled non-reciprocal raman amplification of fibre-guided light,” Nature Photonics 16, 380–383 (2022) . [25] Fam Le Kien and A. Rauschenbeutel, “Anisotropy in scattering of light from an atom into the guided modes of a nanofiber,” Physical Review A 90, 023805 (2014) . [26] Dominic Hallett, Andrew P. Foster, David Whittaker, Maurice S. Skolnick, and Luke R. Wilson, “Engineering chiral light–matter interactions in a waveguide-coupled nanocavity,” ACS Photonics 9, 706–713 (2022) . [27] Wei-Bin Yan, Wei-Yuan Ni, Jing Zhang, Feng-Yang Zhang, and Heng Fan, “Tunable single-photon diode by chiral quantum physics,” Physical Review A 98, 043852 (2018). [28] Michael Scheucher, Ad` ele Hilico, Elisa Will, J¨ urgen Volz, and Arno Rauschenbeutel, “Quantum optical circulator controlled by a single chirally coupled atom,” Science 354, 1577–1580 (2016) . [29] Itay Shomroni, Serge Rosenblum, Yulia Lovsky, Orel Bechler, Gabriel Guendelman, and Barak Dayan, “All- optical routing of single photons by a one-atom switch controlled by a single photon,” Science 345, 903–906 (2014). [30] Cong-Hua Yan, Yong Li, Haidong Yuan, and L. F. Wei, “Targeted photonic routers with chiral photon-atom in- teractions,” Physical Review A 97, 023821 (2018) . [31] Carlos Gonzalez-Ballestero, Esteban Moreno, Fran- cisco J. Garcia-Vidal, and Alejandro Gonzalez-Tudela, “Nonreciprocal few-photon routingschemes based on chi- ral waveguide-emitter couplings,” Physical Review A 94, 063817 (2016) . [32] Keyu Xia, Franco Nori, and Min Xiao, “Cavity-free opti- cal isolators and circulators using a chiral cross-kerr non - linearity,” Physical Review Letters 121, 203602 (2018) . [33] Cl´ ement Sayrin, Christian Junge, Rudolf Mitsch, Bern - hard Albrecht, Danny O’Shea, Philipp Schneeweiss, J¨ urgen Volz, and Arno Rauschenbeutel, “Nanophotonic optical isolator controlled by the internal state of cold atoms,” Physical Review X 5, 041036 (2015) . [34] Keyu Xia, Guowei Lu, Gongwei Lin, Yuqing Cheng, Yueping Niu, Shangqing Gong, and Jason Twamley, “Reversible nonmagnetic single-photon isolation using unbalanced quantum coupling,” Physical Review A 90, 043802 (2014) . [35] Ryan Jones, Giuseppe Buonaiuto, Ben Lang, Igor Lesanovsky, and Beatriz Olmos, “Collectively enhanced chiral photon emission from an atomic array near a nanofiber,” Physical Review Letters 124, 093601 (2020) . [36] Diego Martin-Cano, Harald R. Haakh, and Nir Roten- berg, “Chiral emission into nanophotonic resonators,” ACS Photonics 6, 961–966 (2019) . [37] C.A. Downing, J.C.L´ opez Carre˜ no, F.P. Laussy, E. del Valle, and A.I. Fern´ andez-Dom´ ınguez, “Quasichiral in- teractions between quantum emitters at the nanoscale,” Physical Review Letters 122, 057401 (2019) . [38] B. le Feber, N. Rotenberg, and L. Kuipers, “Nanopho- tonic control of circular dipole emission,” Nature Com- munications 6(2015), 10.1038/ncomms7695 . [39] Lei Tang, Jiangshan Tang, Weidong Zhang, Guowei Lu, Han Zhang, Yong Zhang, Keyu Xia, and Min Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Physical Review A 99, 043833 (2019). [40] Lucas Ostrowski, Scott Parkins, Morito Shirane, and Mark Sadgrove, “Interference-induced directional emis- sion from an unpolarized two-level emitter into a circu-8 lating cavity,” Physical Review A 105, 063719 (2022) . [41] Wen ju Gu, Lei Wang, Zhen Yi, and Li hui Sun, “Generation of nonreciprocal single photons in the chi- ral waveguide-cavity-emittersystem,” Physical Review A 106, 043722 (2022) . [42] R. J. Coles, D. M. Price, B. Royall, E. Clarke, M. S. Skol- nick, A. M. Fox, and M. N. Makhonin, “Path-dependent initialization of a single quantum dot exciton spin in a nanophotonic waveguide,” Physical Review B 95, 121401 (2017). [43] Liuyang Sun, Chun-Yuan Wang, Alex Krasnok, Junho Choi, Jinwei Shi, Juan Sebastian Gomez-Diaz, Andr´ e Zepeda, Shangjr Gwo, Chih-Kang Shih, Andrea Al` u, and Xiaoqin Li, “Separation of valley excitons in a MoS2 monolayer using a subwavelength asymmetric groove ar- ray,”Nature Photonics 13, 180–184 (2019) . [44] Bo Peng, S ¸ahin Kaya ¨Ozdemir, Matthias Liertzer, Wei- jian Chen, Johannes Kramer, Huzeyfe Yılmaz, Jan Wier- sig, Stefan Rotter, and Lan Yang, “Chiral modes and di- rectional lasing atexceptional points,” Proceedings ofthe National Academy of Sciences 113, 6845–6850 (2016) . [45] Stefano Longhi and Liang Feng, “Unidirectional lasing in semiconductor microring lasers at an exceptional point [invited],” Photonics Research 5, B1 (2017) . [46] Babak Bahari, Abdoulaye Ndao, Felipe Vallini, Ab- delkrim El Amili, Yeshaiahu Fainman, and Boubacar Kant´ e, “Nonreciprocal lasing in topological cavities of a r- bitrary geometries,” Science358, 636–640 (2017) . [47] Xue-Feng Jiang, Chang-Ling Zou, Li Wang, Qihuang Gong, and Yun-Feng Xiao, “Whispering-gallery micro- cavities with unidirectional laser emission,” Laser & Pho- tonics Reviews 10, 40–61 (2015) . [48] Hannes Pichler, Tom´ as Ramos, Andrew J. Daley, and Peter Zoller, “Quantum optics of chiral spin networks,” Physical Review A 91, 042116 (2015) . [49] H. J. Kimble, “The quantum internet,” Nature453, 1023–1030 (2008) . [50] Jan Wiersig and Martina Hentschel, “Combining direc- tional light output and ultralow loss in deformed mi- crodisks,” Physical Review Letters 100, 033901 (2008) . [51] Dany Lachance-Quirion, Yutaka Tabuchi, Arnaud Gloppe, Koji Usami, and Yasunobu Nakamura, “Hybrid quantum systems based on magnonics,” Applied Physics Express 12, 070101 (2019) . [52] Babak Zare Rameshti, Silvia ViolaKusminskiy, James A. Haigh, Koji Usami, Dany Lachance-Quirion, Yasunobu Nakamura, Can-Ming Hu, Hong X. Tang, Gerrit E.W. Bauer, and Yaroslav M. Blanter, “Cavity magnonics,” Physics Reports 979, 1–61 (2022) . [53] A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, andY.Nakamura,“Hybridquantumsystemswith circuit quantum electrodynamics,” Nature Physics 16, 257–267 (2020). [54] H.Y. Yuan, Yunshan Cao, Akashdeep Kamra, Rem- bert A. Duine, and Peng Yan, “Quantum magnonics: When magnon spintronics meets quantum information science,” Physics Reports 965, 1–74 (2022) . [55] Yutaka Tabuchi, Seiichiro Ishino, Toyofumi Ishikawa, Rekishu Yamazaki, Koji Usami, and Yasunobu Naka- mura, “Hybridizing ferromagnetic magnons and mi- crowave photons in the quantum limit,” Physical Review Letters113, 083603 (2014) .[56] Xufeng Zhang, Chang-Ling Zou, Liang Jiang, and Hong X. Tang, “Strongly coupled magnons and cav- ity microwave photons,” Physical Review Letters 113, 156401 (2014) . [57] Yi Li, Volodymyr G. Yefremenko, Marharyta Lisovenko, Cody Trevillian, Tomas Polakovic, Thomas W. Cecil, Peter S. Barry, John Pearson, Ralu Divan, Vasyl Ty- berkevych, Clarence L. Chang, Ulrich Welp, Wai-Kwong Kwok, and Valentine Novosad, “Coherent coupling of two remote magnonic resonators mediated by supercon- ducting circuits,” Physical Review Letters 128, 047701 (2022). [58] Maxim Goryachev, Warrick G. Farr, Daniel L. Creedon, Yaohui Fan, Mikhail Kostylev, and Michael E. Tobar, “High-cooperativity cavity QED with magnons at mi- crowave frequencies,” Physical Review Applied 2, 054002 (2014). [59] Yi-Pu Wang, Guo-Qiang Zhang, Dengke Zhang, Tie- Fu Li, C.-M. Hu, and J.Q. You, “Bistability of cav- ity magnon polaritons,” Physical Review Letters 120, 057202 (2018) . [60] H. Maier-Flaig, M. Harder, R. Gross, H. Huebl, and S. T. B. Goennenwein, “Spin pumping in strongly cou- pled magnon-photon systems,” Physical Review B 94, 054433 (2016) . [61] Jie Li, Shi-Yao Zhu, and G.S. Agarwal, “Magnon- photon-phonon entanglement in cavity magnomechan- ics,”Physical Review Letters 121, 203601 (2018) . [62] Zhedong Zhang, Marlan O. Scully, and Girish S. Agarwal, “Quantum entanglement between two magnon modes via kerr nonlinearity driven far from equilibrium,” Physical Review Research 1, 023021 (2019) . [63] Jie Li, Shi-YaoZhu, andG. S. Agarwal, “Squeezedstates of magnons and phonons in cavity magnomechanics,” Physical Review A 99, 021801 (2019) . [64] Jayakrishnan M. P. Nair and G. S. Agarwal, “Determin- istic quantum entanglement between macroscopic ferrite samples,” Applied Physics Letters 117, 084001 (2020) . [65] H.Y. Yuan, Peng Yan, Shasha Zheng, Q.Y. He, Ke Xia, and Man-Hong Yung, “Steady bell state generation via magnon-photon coupling,” Physical Review Letters 124, 053602 (2020) . [66] Deyi Kong, Xiangming Hu, Liang Hu, and Jun Xu, “Magnon-atom interaction via dispersive cavities: Magnon entanglement,” Physical Review B 103, 224416 (2021). [67] Ji kun Xie, Sheng li Ma, and Fu li Li, “Quantum- interference-enhanced magnon blockade in an yttrium- iron-garnet sphere coupled to superconducting circuits,” Physical Review A 101, 042331 (2020) . [68] Zeng-Xing Liu, Hao Xiong, and Ying Wu, “Magnon blockade in a hybrid ferromagnet-superconductor quan- tum system,” Physical Review B 100, 134421 (2019) . [69] Chengsong Zhao, Xun Li, Shilei Chao, Rui Peng, Chong Li, and Ling Zhou, “Simultaneous blockade of a pho- ton, phonon, and magnon induced by a two-level atom,” Physical Review A 101, 063838 (2020) . [70] Kun Wu, Wen xue Zhong, Guang ling Cheng, and AixiChen,“Phase-controlled multimagnon blockadeand magnon-induced tunneling in a hybrid superconducting system,” Physical Review A 103, 052411 (2021) . [71] Ya long Ren, Ji kun Xie, Xin ke Li, Sheng li Ma, and Fu li Li, “Long-range generation of a magnon-magnon entangled state,” Physical Review B 105, 094422 (2022) .9 [72] HuatangTan and Jie Li, “Einstein-podolsky-rosen enta n- glement and asymmetric steering between distant macro- scopic mechanical and magnonic systems,” Physical Re- view Research 3, 013192 (2021) . [73] Yi-Pu Wang, J.W. Rao, Y. Yang, Peng-Chao Xu, Y.S. Gui, B.M. Yao, J.Q. You, and C.-M. Hu, “Nonreciproc- ity and unidirectional invisibility in cavity magnonics,” Physical Review Letters 123, 127202 (2019) . [74] Cui Kong, Hao Xiong, and Ying Wu, “Magnon-induced nonreciprocitybasedonthemagnonkerreffect,” Physical Review Applied 12, 034001 (2019) . [75] Ya long Ren, Sheng li Ma, and Fu li Li, “Chiral coupling between a ferromagnetic magnon and a superconducting qubit,”Physical Review A 106, 053714 (2022) . [76] Dengke Zhang, Xiao-Qing Luo, Yi-Pu Wang, Tie-Fu Li, and J. Q. You, “Observation of the exceptional point in cavity magnon-polaritons,” Nature Communications 8(2017), 10.1038/s41467-017-01634-w . [77] Tian-XiangLu, HuilaiZhang, QianZhang, andHuiJing, “Exceptional-point-engineered cavity magnomechanics,” Physical Review A 103, 063708 (2021) . [78] M. Harder, Y. Yang, B.M. Yao, C.H. Yu, J.W. Rao, Y.S. Gui, R.L. Stamps, and C.-M. Hu, “Level attrac- tion due to dissipative magnon-photon coupling,” Phys- ical Review Letters 121, 137203 (2018) . [79] Jing Xu, Changchun Zhong, Xu Han, Dafei Jin, Liang Jiang, and Xufeng Zhang, “Floquet cavity electro- magnonics,” PhysicalReviewLetters 125,237201 (2020) . [80] Feng-Yang Zhang, Qi-Cheng Wu, and Chui-Ping Yang, “Non-hermitian shortcut to adiabaticity in floquet cav- ity electromagnonics,” Physical Review A 106, 012609 (2022). [81] Shi fan Qi and Jun Jing, “Chiral current in floquet cavity magnonics,” Physical Review A 106, 033711 (2022) . [82] Jikun Xie, Huaiyang Yuan, Shengli Ma, Shaoyan Gao, Fuli Li, and Rembert A Duine, “Stationary quantum entanglement and steering between two distant macro- magnets,” Quantum Science and Technology 8, 035022 (2023). [83] Tao Yu, Zhaochu Luo, and Gerrit E.W. Bauer, “Chiral- ity as generalized spin–orbit interaction in spintronics, ” Physics Reports 1009, 1–115 (2023) . [84] TaoYu,Yu-XiangZhang, SancharSharma, XiangZhang, Yaroslav M. Blanter, and Gerrit E.W. Bauer, “Magnon accumulation in chirally coupled magnets,” Physical Re-view Letters 124, 107202 (2020) . [85] Tao Yu, Xiang Zhang, Sanchar Sharma, Yaroslav M. Blanter, and Gerrit E. W. Bauer, “Chiral coupling of magnons in waveguides,” Physical Review B 101, 094414 (2020). [86] Xufeng Zhang, Alexey Galda, Xu Han, Dafei Jin, and V. M. Vinokur, “Broadband nonreciprocity enabled by strong coupling of magnons and microwave photons,” Physical Review Applied 13, 044039 (2020) . [87] Jeremy Bourhill, Weichao Yu, Vincent Vlaminck, Ger- rit E. W. Bauer, Giuseppe Ruoso, and Vincent Castel, “Generation of circulating cavity magnon polaritons,” Physical Review Applied 19, 014030 (2023) . [88] Weichao Yu, Tao Yu, andGerrit E. W. Bauer, “Circulat- ing cavity magnon polaritons,” Physical Review B 102, 064416 (2020) . [89] Na Zhu, Xu Han, Chang-Ling Zou, Mingrui Xu, and Hong X. Tang, “Magnon-photon strong coupling for tun- able microwave circulators,” Physical Review A 101, 043842 (2020) . [90] Ying-Ying Wang, Sean van Geldern, Thomas Connolly, Yu-Xin Wang, Alexander Shilcusky, Alexander McDon- ald, Aashish A. Clerk, and Chen Wang, “Low-loss ferrite circulator as a tunable chiral quantum system,” Physical Review Applied 16, 064066 (2021) . [91] YalongRen, Shengli Ma, Ji kunXie, XinkeLi, Mingtao Cao, andFuliLi,“Nonreciprocal single-photonquantum router,” Physical Review A 105, 013711 (2022) . [92] Yi Xu, Jin-Yu Liu, Wenjing Liu, and Yun-Feng Xiao, “Nonreciprocal phonon laser in a spinning microwave magnomechanical system,” Physical Review A 103, 053501 (2021) . [93] Huiping Zhan, Lihui Sun, and Huatang Tan, “Chirality-induced one-way quantum steering between two waveguide-mediated ferrimagnetic microspheres,” Physical Review B 106, 104432 (2022) . [94] Andrea Aiello, Peter Banzer, Martin Neugebauer, and Gerd Leuchs, “From transverse angular momentum to photonic wheels,” Nature Photonics 9, 789–795 (2015) . [95] Ji kun Xie, Sheng li Ma, Ya long Ren, Xin ke Li, and Fuli Li, “Dissipative generation ofsteady-state squeezin g of superconducting resonators via parametric driving,” Physical Review A 101, 012348 (2020) .

2023-08-30

Unidirectional photon emission is crucial for constructing quantum networks and realizing scalable quantum information processing. In the present work an efficient scheme is developed for the unidirectional emission of a tunable squeezed microwave field. Our scheme is based on a chiral cavity magnonic system, where a magnon mode in a single-crystalline yttrium iron garnet (YIG) sphere is selectively coupled to one of the two degenerate rotating microwave modes in a torus-shaped cavity with the same chirality. With the YIG sphere driven by a two-color Floquet field to induce sidebands in the magnon-photon coupling, we show that the unidirectional emission of a tunable squeezed microwave field can be generated via the assistance of the dissipative magnon mode and a waveguide. Moreover, the direction of the proposed one-way emitter can be controlled on demand by reversing the biased magnetic field. Our work opens up an avenue to create and manipulate one-way nonclassical microwave radiation field and could find potential quantum technological applications.

Chiral cavity-magnonic system for the unidirectional emission of a tunable squeezed microwave field

2308.15826v1

Temporal evolution of auto-oscillations in a YIG/Pt microdisc driven by pulsed spin Hall effect-induced spin-transfer torque Viktor Lauer1, Michael Schneider1, Thomas Meyer1, Thomas Brächer1*, Philipp Pirro1, Björn Heinz1, Frank Heussner1, Bert Lägel1, Mehmet C. 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. 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. 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. The 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. This might be attributed to the fact that the system does not reach a quasi-equilibrium state during the first few tens of nanoseconds of an applied current pulse. ACKNOWLEDGMENT This research has been supported by the EU-FET Grant InSpin 612759, and by the ERC Starting Grant 678309 MagnonCircuits. REFERENCES Ando K. et al. (2008), “Electric manipulation of spin relaxation using the spin Hall effect,” Phys. Rev. Lett., vol. 101, p. 036601. Berger L (1996), “Emission of spin waves by a magnetic multilayer traversed, by a current,” Phys. Rev. B, vol 54, pp. 9353-9358. Chumak A V, Vasyuchka V I, Serga A A, Hillebrands B (2015), “Magnon spintronics,” Nature Phys., vol. 11, p. 453. Collet M, et al. (2016), “Generation of coherent spin-wave modes in yttrium iron garnet microdiscs by spin-orbit torque,” Nat. Commun., vol. 7, p. 10377. Demidov V E, Urazhdin S, Edwards E R J, Demokritov S O (2011), “Wide-range control of ferromagnetic resonance by spin Hall effect,” Appl. Phys. Lett., vol. 99, p. 172501. Demidov V E, et al. (2012), “Magnetic nano-oscillator driven by pure spin current,” Nature Mater., vol. 11, pp. 1028-1031. Demidov V E, et al. (2016), “Direct observation of dynamic modes excited in a magnetic insulator by pure spin current,” Sci. Rep., vol. 6, 32781. Dyakonov M I, Perel V. I. (1971), “Current-induced spin orientation of electrons in semiconductors,” Phys. Lett. A, vol. 35, pp. 459-460. Hamadeh A, et al. (2014), “Full control of the spin-wave damping in a magnetic insulator using spin-orbit torque,” Phys. Rev. Lett., vol. 113, p. 197203. Hirsch J E (1999), “Spin Hall effect,” Phys. Rev. Lett., vol. 83, pp. 1834-1837. Jungfleisch M B, Lauer V, Neb R, Chumak A V, Hillebrands B (2013) “Improvement of the yttrium iron garnet/platinum interface for spin pumping-based applications,” Appl. Phys. Lett., vol. 103, p. 022411. Jungfleisch M B, et al. (2016) “Large spin-wave bullet in a ferrimagnetic insulator driven by the spin Hall effect,” Phys. Rev. B, vol. 116, p. 057601. Kajiwara Y., et al. (2010), “Transmission of electrical signals by spin-wave interconversion in a magnetic insulator,” Nature, vol. 464, pp. 262-266. Lauer V, et al. (2016A), “Spin-transfer torque based damping control of parametrically excited spin waves in a magnetic insulator,” Appl. Phys. Lett., vol. 108, p. 012402. Lauer V, et al. (2016B), “Auto-oscillations in YIG/Pt nanostructures due to thermal gradients,” AG Magnetism, Annual Report 2016. Onbasli M C, et al. (2014), “Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert damping and bulk-like magnetization,” APL Mater., vol. 2, p. 106102. Pirro P, et al. (2014), “Spin-wave excitation and propagation in microstructured waveguides of yttrium iron garnet/Pt bilayers,” Appl. Phys. Lett., vol. 104, p. 012402. Safranski C, et al. (2016), “Spin caloritronic nano-oscillator,” https://arxiv.org/abs/1611.00887 Sebastian T, Schultheiss K, Obry B, Hillebrands B, Schultheiss H (2015) “Micro-focused Brillouin light scattering: imaging spin waves at the nanoscale,” Cond. Matter. Phys., vol. 3, p. 35. Serga A A, Chumak A V, Hillebrands B (2010) “YIG magnonics,” J. Phys. D: Appl. Phys., vol. 43, p. 264002. Slonczewski J C (1995), “Current-driven excitation of magnetic multilayers,” J. Magn. Magn. Mater., vol. 159, pp. L1-L7. Schreier M, et al. (2015) “Sign of inverse spin Hall voltages generated by ferromagnetic resonance and temperature gradients in yttrium iron garnet|platinum bilayers,” J. Phys. D: Appl. Phys., vol. 48, p. 025001. Uchida K, Kikkawa T, Miura A, Shiomi J, Saitoh E (2014) “Quantitative temperature dependence of longitudinal spin Seebeck effect at high temperatures,” Phys. Rev. X, vol. 4, p. 041023.

2016-11-18

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.

Temporal evolution of auto-oscillations in a YIG/Pt microdisc driven by pulsed spin Hall effect-induced spin-transfer torque

1611.06054v1

TEMPERATURE DEPENDENCE OF SPIN PINNING AND SPIN-WAVE DISPERSION IN NANOSCOPIC FERROMAGNETIC WAVEGUIDES 1B. HEINZ*,1, 2Q. WANG*,1R. VERBA,3V. I. VASYUCHKA,1M. KEWENIG,1P. PIRRO,1M. SCHNEIDER,1T. MEYER,1, 4B. L¨AGEL,5C. DUBS,6T. BR¨ACHER,1 O. V. DOBROVOLSKIY,7A. V. CHUMAK**1, 8 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern (Erwin-Schr¨ odinger-Stra ße 56, 67663 Kaiserslautern, Germany) 2Graduate School Materials Science in Mainz (Staudingerwerg 9, 55128 Mainz, Germany) 3Institute of Magnetism (Vernadskogo blvd. 36-b, 03142 Kyiv, Ukraine) 4THATec Innovation GmbH (Augustaanlage 23, 68165 Mannheim, Germany) 5Nano Structuring Center, Technische Universit¨ at Kaiserslautern (Erwin-Schr¨ odinger-Stra ße 13, 67663 Kaiserslautern, Germany) 6INNOVENT e.V., Technologieentwicklung (Pr¨ ussingstra ße 27B, 07745 Jena, Germany) 7Physikalisches Institut, Goethe-Universit¨ at Frankfurt (Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany) 8Nanomagnetism and Magnonics, Faculty of Physics, University of Vienna (Boltzmanngasse 5, A-1090 Wien, Austria) Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides The field of magnonics attracts significant attention due to the possibility of utilizing informa- tion coded into the spin-wave phase or amplitude to perform computation operations on the nanoscale. Recently, spin waves were investigated in Yttrium Iron Garnet (YIG) waveguides with widths ranging down to 50 nm and aspect ratios thickness over width approaching unity. A critical width was found, below which the exchange interaction suppresses the dipolar pin- ning phenomenon and the system becomes unpinned. Here we continue these investigations and analyse the pinning phenomenon and spin-wave dispersions as a function of temperature, thickness and material of choice. Higher order modes, the influence of a finite wavevector along the waveguide and the impact of the pinning phenomenon on the spin-wave lifetime are discussed as well as the influence of a trapezoidal cross section and edge roughness of the waveguides. The presented results are of particular interest for potential applications in magnonic devices and the incipient field of quantum magnonics at cryogenic temperatures. Keywords : spin waves, yttrium iron garnet, Brillouin light scattering spectroscopy, low temperatures c○B. HEINZ*, Q. WANG*, R. VERBA, V. I. VASYUCHKA, M. KEWENIG, P. PIRRO, M. SCHNEIDER, T. MEYER, B. L ¨AGEL, C. DUBS, T. BR¨ACHER, O. V. DOBROVOLSKIY, A. V. CHUMAK**arXiv:2002.00003v1 [physics.app-ph] 31 Jan 2020B. Heinz et al. 1. Introduction Thefieldofmagnonicsproposesapromisingapproach for a novel type of computing systems, in which magnons, the quanta of spin waves, carry the infor- mation instead of electrons [1–12]. Since the phase of a spin wave provides an additional degree of free- dom efficient computing concepts can be used result- ing in a valuable decrease in the footprint of logic units. Moreover, the scalability of magnonic struc- tures down to the nanometer scale and the possibility to operate with spin waves of nanometer wavelength are additional advantages of the magnonics approach. The further miniaturization will, consequently, result in an increase in the frequency of spin waves used in the devices from the currently employed GHz range up to the THz range. In classical magnonics, spin- wave modes in thin films or rather planar waveguides with thickness-to-width aspect ratios 𝑎r=ℎ/𝑤≪1 have been utilized. In the case of a waveguide, edge magnetostatic charges arise, which can be accounted for by the introduction of boundary conditions [13]. Therefore thin waveguides demonstrate the effect of "dipolar pinning" at the lateral edges, and for its the- oretical description the thin strip approximation was developed, in which only pinning of the much-larger- in-amplitudedynamicin-planemagnetizationcompo- nent is taken into account [14–19]. The recent progress in fabrication technology leads to the development of nanoscopic magnetic devices in which the width 𝑤and the thickness ℎbecome com- parable [20–27]. The description of such waveguides is beyond the thin strip model of effective pinning, because the scale of nonuniformity of the dynamic dipolar fields, which is described as "effective dipo- lar boundary conditions", becomes comparable to the waveguide width. Additionally, both, in-plane and out-of-plane dynamic magnetization components, be- come involved in the effective dipolar pinning, as they become of comparable amplitude. Thus, a more gen- eral model should be developed and verified experi- mentally. In addition, such nanoscopic feature sizes imply that the spin-wave modes bear a strong ex- change character, since the widths of the structures are now comparable to the exchange length [28]. A proper description of the spin-wave eigenmodes in nanoscopic strips which considers the influence of the exchangeinteraction, aswellastheshapeofthestruc-ture, was recently performed in [29] and is fundamen- tal for the field of magnonics. Very recently, the fields of quantum magnonics and magnonics at cryogenic temperatures were es- tablished. Among the highlights, one could mention the first realization of a coherent coupling between a ferromagnetic magnon and a superconducting qubit [30], the first observation of the interaction between magnons and Abrikosov fluxes in supercondutor- ferromagnet hybrid structures [31] and the investi- gation of the interplay of magnetization dynamics with a microwave waveguide at cryogenic tempera- tures [32]. Thus, the understanding of the influence of temperature on spin pinning conditions and on spin-wavedispersionsinnano-structuresisofhighde- mand. In this Article, we continue the investigation car- riedoutinPhys. Rev. Lett. 122, 247202(2019). The evolution of the frequencies and profiles of the spin- wave modes in Yttrium Iron Garnet (YIG) waveg- uides with a thickness of 39nm and widths rang- ing down to 50nm are discussed in detail. The phenomenon of unpinning and the underlying the- ory as well as the experimental proof are outlined. A throughout discussion of the effective width and the critical width, at which the system becomes un- pinned, in dependency of the thickness and the ma- terial of choice is presented. Moreover, the temper- ature dependency is analysed theoretically. Higher order modes up to 𝑛= 2, the influence of a finite wavevector along the waveguide and the impact of the pinning phenomenon on the spin wave lifetime are discussed. To account for the imperfections of a real system, the influence of a trapezoidal cross sec- tion and edge roughness on the effective width and the critical width are investigated. 2. Methodology 2.1.Sample fabrication A39nm thick Yttrium Iron Garnet (YIG) film has been grown on a 1 inch (111) 500𝜇m thick Gadolin- iumGalliumGarnet(GGG)substratebyliquidphase epitaxy from PbO-B 2O3based high-temperature so- lutionsat 860∘Cusingtheisothermaldippingmethod (see e.g. Ref. [33]). A pure Y 3Fe5O12film with a smooth surface was obtained by rotating the sub- strate horizontally with a rotation rate of 100rpm. The saturation magnetization of the YIG film is 2Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides Fig. 1. (a) Schematically depicted main steps in the nanos- tructuring process. (b) Sketch of the sample and the exper- imental configuration: a set of YIG waveguides is placed on a microstrip line to excite the quasi-FMR in the waveguides. BLS spectroscopy is used to measure the local spin-wave dy- namics. (c) and (d) SEM micrograph of a 1𝜇m and a 50-nm wide YIG waveguide of 39-nm thickness. The color code shows the simulated amplitudes of the fundamental mode at quasi- ferromagnetic resonance, i.e., 𝑘𝑥= 0, in the waveguides. The mode in the 50𝑛m waveguide is almost uniform across the width of the waveguide evidencing the unpinning directly. [29] 1.37×105A/m and its Gilbert damping is 𝛼= 6.41×10−4, as it was extracted by ferromagnetic res- onance spectroscopy [34]. The nanostructures were fabricated by utilising a hard mask ion milling procedure. The key steps in the fabrication process are shown in Fig. 1(a). First a double layer of polymethyl methacrylate (PMMA) was spin coated on the YIG film and a Chromium/Titanium hard mask was fabricated using electron beam lithography and electron beam evapo- ration. This hard mask acts as a protective layer in a successive Ar+ion milling step. In a final step, any residualChromiumisremovedusinganacidthatYIG is inert to. 2.2.Microfocused Brillouin Light Scattering (BLS) spectroscopy measurements BLS spectroscopy is a unique technique for measur- ing the spin-wave intensities in frequency, space, andtimedomains. Itisbasedontheinelasticscatteringof an incident laser beam from a magnetic material. In ourmeasurements,alaserbeamof 457nmwavelength and a power of 1.8mW is focused through the trans- parent GGG substrate on the center of the respective individual waveguide using a 100×microscope objec- tive with a large numerical aperture ( NA = 0 .85). The effective spot-size is 350nm. The scattered light was collected and guided into a six-pass Fabry-P´ erot interferometer to analyse the frequency shift. 2.3.Numerical simulations Themicromagneticsimulationsofthespace-andtime dependent magnetization dynamics were performed bytheGPU-acceleratedsimulationprogramMumax3 using a finite-difference discretisation [35]. The struc- ture is schematically shown in Fig. 1(b). The follow- ing material parameters were used in the simulations: the saturation magnetization 𝑀s= 1.37×105A/m and the Gilbert damping 𝛼= 6.41×10−4were extracted by ferromagnetic resonance spectroscopy measurements of the plain film before patterning [36]. A gyromagnetic ratio of 𝛾= 175 .86rad/(ns·T) and an exchange constant of 𝐴= 3.5pJ/m for a stan- dard YIG film were assumed. An external field 𝐵= 108 .9mT is applied along the waveguide long axis. Three steps were performed to calculate the spin-wave dispersion curve: (i) The external field was applied along the waveguide, and the magnetization was allowed to relax into a stationary state (ground state). (ii) A sinc field pulse 𝑏𝑦=𝑏0sinc(2 𝜋𝑓𝑐𝑡), with oscillation field 𝑏0= 1mT and cut-off frequency 𝑓𝑐= 10GHz, was used to excite a wide range of spin waves. (iii) The spin-wave dispersion relations were obtained by performing the two-dimensional Fast Fourier Transformation of the time- and space- dependent data. Furthermore, the spin-wave width profileswereextractedfromthe 𝑚𝑧componentacross the width of the waveguides using a single frequency excitation. 2.4.Quasi-analytical theoretical model In order to accurately describe the spin-wave char- acteristics in nanoscopic longitudinally magnetized waveguides, a more general semi-analytical theory is provided which goes beyond the thin strip ap- proximation [29]. Here, we assume a uniform spin wave mode profile across the waveguide thickness (no 3B. Heinz et al. 𝑧-dependency), which is valid for the fundamental thickness mode in thin waveguides. In a general case thez-dependencyofanyhigherthicknessmodecanbe included in a similar way as shown here. Also, please note that the theory is not applicable in transversely magnetized waveguides due to their more involved internal field landscape [20]. The lateral spin-wave mode profile m𝑘𝑥(𝑦)and frequency can be found as solutions of the linearized Landau-Lifshitz equation [37,38] −𝑖𝜔𝑘𝑥m𝑘𝑥(𝑦) =×(︁ ^Ω𝑘𝑥·m𝑘𝑥(𝑦))︁ (2.1) with appropriate exchange boundary conditions, which take into account the surface anisotropy at the edges. Here, is the unit vector in the static mag- netization direction and ^Ω𝑘𝑥is a tensorial Hamilton operator, which is given by ^Ω𝑘𝑥·m𝑘𝑥(𝑦) =(︂ 𝜔H+𝜔M𝜆2(︂ 𝑘2 𝑥−𝑑2 𝑑𝑦2)︂)︂ m𝑘𝑥(𝑦) +𝜔M∫︀^G𝑘𝑥(𝑦−𝑦′)·m𝑘𝑥(𝑦′)𝑑𝑦′. (2.2) Here, 𝜔H=𝛾𝐵,𝐵is the static internal magnetic field that is considered to be equal to the external field due to the negligible demagnetization along the 𝑥-direction, 𝜔M=𝛾𝜇0𝑀s,𝛾is the gyromagnetic ra- tio and ^G𝑘𝑥is the Green’s function (see next sub- section). A numerical solution of Eq. (2.1) gives both, the spin-wave profile m𝑘𝑥and frequency 𝜔𝑘𝑥. In the fol- lowing, we will regard the ouf-of-plane component 𝑚𝑧(𝑦)to show the mode profiles representatively. In the past, it was demonstrated that in microscopic waveguides, the fundamental mode is well fitted by the function 𝑚𝑧(𝑦) =𝐴0cos(𝜋𝑦/𝑤 eff)with the am- plitude 𝐴0and the effective width 𝑤eff[16,17]. This mode, as well as the higher modes, are referred to as “partially pinned”. Pinning hereby refers to the fact that the amplitude of the mode at the edges of the waveguideisreduced. Inthatcase, theeffectivewidth 𝑤effdetermines where the amplitude of the modes would vanish outside the waveguide [9,16,27]. With this effective width, the spin-wave dispersion relation can also be calculated by the analytical formula [9]: 𝜔0(𝑘𝑥) = √︁ (𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑦𝑦 𝑘𝑥))(𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑧𝑧 𝑘𝑥)),(2.3) where 𝐾=√︀ 𝑘2𝑥+𝜅2and𝜅=𝜋/𝑤 eff. The ten- sor^𝐹𝑘𝑥=1 2𝜋∫︁∞ −∞|𝜎𝑘|2 ˜𝑤^N𝑘𝑑𝑘𝑦accounts for the dy- namic magnetization, 𝜎𝑘=∫︁𝑤/2 −𝑤/2𝑚(𝑦)𝑒−𝑖𝑘𝑦𝑦𝑑𝑦is the Fourier-transform of the spin-wave profile across the width of the waveguide and ˜𝑤=∫︁𝑤/2 −𝑤/2𝑚(𝑦)2𝑑𝑦 is the normalization of the mode profile 𝑚(𝑦). 2.5.Numerical solution of the eigenproblem In this sub-section we discuss the details of the nu- merical solution of the eigenproblem. The eigenprob- lem Eq. (2.1) should be solved with proper boundary conditions at the lateral edges of the waveguide. We use a complete description of the dipolar interaction via Green’s functions: ^G𝑘𝑥(𝑦) =1 2𝜋∫︁∞ −∞^N𝑘𝑒𝑖𝑘𝑦𝑦𝑑𝑘𝑦. (2.4) Here, ^N𝑘=⎛ ⎜⎝𝑘2 𝑥 𝑘2𝑓(𝑘ℎ)𝑘𝑥𝑘𝑦 𝑘2𝑓(𝑘ℎ) 0 𝑘𝑥𝑘𝑦 𝑘2𝑓(𝑘ℎ)𝑘2 𝑦 𝑘2𝑓(𝑘ℎ) 0 0 0 1 −𝑓(𝑘ℎ)⎞ ⎟⎠,(2.5) where 𝑓(𝑘ℎ) = 1−(1−exp(−𝑘ℎ))/𝑘ℎ,𝑘=√︁ 𝑘2𝑥+𝑘2𝑦 and it is assumed that the waveguides are infinitely long. The boundary condition (2.6) only accounts for the exchange interaction and surface anisotropy (if any) and reads [39] m×(𝜇0𝑀s𝜆2𝜕m 𝜕n−∇M𝐸a) = 0 , (2.6) where nis the unit vector defining the inward normal direction to the waveguide edge, and 𝐸a(m)is the en- ergy density of the surface anisotropy. In the studied case of a waveguide magnetized along its long axis, the conditions (2.6) for the dynamic magnetization components can be simplified to ±𝜕𝑚𝑦 𝜕𝑦+𝑑𝑚𝑦|𝑦=±𝑤/2= 0,𝜕𝑚𝑧 𝜕𝑦|𝑦=±𝑤/2= 0, (2.7) 4Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides where 𝑑=−2𝐾s/(𝑚0𝑀2 s𝜆2)isthepinningparameter [18] and 𝐾sis the surface anisotropy constant at the waveguide lateral edges. More complex cases like, e.g., diffusive interfaces can be considered in the same manner [40]. For the numerical solution of Eq. (2.1) it is con- venient to use finite element methods and to dis- cretize the waveguide into 𝑛elements of the width Δ𝑤=𝑤/𝑛, where 𝑤is the width of the wave- guide. The discretization step should be at least sev- eral times smaller than the waveguide thickness and the spin-wave wavelength 2𝜋/𝑘𝑥for a proper descrip- tion of the magneto-dipolar fields. The discretization transforms Eq. (2.1) into a system of linear equations for magnetizations m𝑗,𝑗= 1,2,3,···𝑛 𝑖×((𝜔M+𝜔M𝜆2𝑘2 𝑥)m𝑗 −𝜔M𝜆2m𝑗−1−2m𝑗+m𝑗+1 Δ𝑤2 +𝜔M∑︀𝑛 𝑗′=1^G𝑗−𝑗′·m𝑗′) =𝜔m𝑗(2.8) where dipolar interaction between the discretized el- ements is described by ^G𝑘𝑥,𝑗(𝑦) = 1 Δ𝑤∫︀Δ𝑤/2 −Δ𝑤/2𝑑𝑦∫︀Δ𝑤/2 −Δ𝑤/2𝑑𝑦′^G𝑘𝑥(𝑦−𝑦′−𝑗Δ𝑤).(2.9) The direct use of Eq. (2.9) is complicated since the Green’s function ^G𝑘𝑥(𝑦)is an integral itself. Using the Fourier transform it can be derived as ^G𝑘𝑥,𝑗(𝑦) =Δ𝑤 2𝜋∫︁ sinc(𝑘𝑦Δ𝑤/2)^N𝑘𝑒𝑖𝑘𝑦𝑗Δ𝑤𝑑𝑘𝑦, (2.10) with sinc(𝑥) =sin(𝑥) 𝑥. This can be easily calcu- lated, especially using fast Fourier transform. Equa- tion (2.8) is, in fact, a 2𝑛-dimensional linear alge- braic eigenproblem (since m𝑗is a 2-component vec- tor),whichissolvedbystandardmethods. Thevalues m0andm𝑛+1in Eq. (2.8) are determined from the boundary conditions (2.7). In particular, for negligi- ble anisotropy at the waveguide edges one should set m0=m1andm𝑛+1=m𝑛. 3. Results and discussions 3.1.Original experimental findings In these studies, we consider rectangular magnetic waveguides as shown schematically in Fig. 1(b). Inthe experiment, a spin-wave mode is excited by a stripline that provides a homogeneous excitation field over the sample containing various waveguides etched from a ℎ= 39nm thick YIG film. The widths of the waveguides range from 𝑤= 50nm to 𝑤= 1𝜇m and the length is 60𝜇m. The waveguides are uniformly magnetized along their long axis by an external field 𝐵(𝑥-direction). Figure 1(c) and 1(d) show scanning electronmicroscopy(SEM)micrographsofthelargest and the narrowest waveguide studied in the experi- ment. The intensity of the magnetization precession is measured by microfocused BLS spectroscopy [41] (see Methodology section) as shown in Fig. 1(b). Black and red lines in Fig. 3(a) show the frequency spectra for a 1𝜇m and a 50-nm wide waveguide, re- spectively. No standing modes across the thickness were observed in our experiment, as their frequen- cies lie higher than 20GHz due to the small thick- ness. The quasi-FMR frequency is 5.007GHz for the 1𝜇m wide waveguide. This frequency is compara- ble to 5.029GHz, the value predicted by the classi- cal theoretical model using the thin strip approxima- tion [16–18]. In contrast, the quasi-FMR frequency is 5.35GHz for a 50nm wide waveguide which is much smaller than the value of 7.687GHz predicted by the same model. The reason is that the thin strip approx- imation overestimates the effect of dipolar pinning in waveguides with aspect ratio either 𝑎r= 1or close to one, forwhichthenonuniformityofthedynamicdipo- lar fields is not well-localized at the waveguide edges. Additionally, in such nanoscale waveguides, the dy- namicmagnetizationcomponentsbecomeofthesame order of magnitude and both affect the effective mode pinning, in contrast to thin waveguides, in which the in-plane magnetization component is dominant. 3.2.Spin pinning in nano-structures In the following, the experiment is compared to the theory and to micromagnetic simulations. The bottom panels of Fig. 2(a) and 2(b) show the spin-wave mode profile of the fundamental mode for 𝑘𝑥= 0, which corresponds to the quasi-FMR, in a 1𝜇m (a) and 50nm (b) wide waveguide which have been obtained by micromagnetic simulations (blue dots) and by solving Eq. (2.1) numerically (black lines) (higher width modes are discussed in the next sections). The top panels illustrate the mode pro- file and the local precession amplitude in the wave- 5B. Heinz et al. Fig. 2. Schematic of the precessing spins and simulated pre- cession trajectories (ellipses in the second panel) and spin-wave profile 𝑚𝑧(𝑦)of the quasi-FMR. The profiles have been ob- tained by micromagnetic simulations (blue dots) and by the quasi-analytical approach (black lines) for a (a) 1𝜇m and a (b) 50nm wide waveguide. (c), (d): Corresponding normalized square of the spin-wave eigenfrequency 𝜔′2/𝜔2 Mas a function of𝑤/𝑤 effand the relative dipolar and exchange contributions. [29] guide. As it can be seen, the two waveguides feature quite different profiles of their fundamental modes: in the 1𝜇m wide waveguide, the spins are partially pinned and the amplitude at the edges of the wave- guide is reduced compared to the maximal value of 𝑚𝑧= 1. This still resembles the cosine-like profile of the lowest width mode 𝑛= 0that has been well es- tablished in investigations of spin-wave dynamics in waveguides on the micron scale [23,27,42] and that can be well-described by the simple introduction of a finite effective width 𝑤eff> 𝑤(𝑤eff=𝑤for the case of full pinning). In contrast, the spins at the edges of the narrow waveguide are completely unpinned and the amplitude of the dynamic magnetization 𝑚𝑧of the lowest mode 𝑛= 0is almost constant across the width of the waveguide, resulting in 𝑤eff→∞. To understand the nature of this depinning, it is in- structive to consider the spin-wave energy as a func- tion of the geometric width of the waveguide normal- ized by the effective width 𝑤/𝑤 eff. This ratio cor- responds to some kind of pinning parameter taking values in between 1 for the fully pinned case and 0for the fully unpinned case. According to the Ritz’s variational principle, the profiles of the spin wave modes correspond to the respective minima of the spin wave frequency (energy) functional. Since only one minimization parameter – 𝑤eff– is used, the min- imization as a function of 𝑤/𝑤 effyields only the ap- proximate spin wave profiles. Nevertheless, this is sufficient for the qualitative understanding. To illus- trate this, Figs. 2(c) and 2(d) show the normalized square of the spin-wave eigenfrequencies 𝜔′2/𝜔2 Mfor thetwodifferentwidthsasafunctionof 𝑤/𝑤 eff. Here, 𝜔′2refers to a frequency square, not taking into ac- count the Zeeman contribution (𝜔2 H+𝜔H𝜔M), which only leads to an offset in frequency. The minimum of𝜔′2is equivalent to the solution with the lowest energy corresponding to the effective width 𝑤eff. In addition to the total 𝜔′2(black), also the individ- ual contributions from the dipolar term (red) and the exchange term (blue) are shown, which can only be separated conveniently from each other if the square of Eq. (2.3) is considered for 𝑘𝑥= 0. The dipolar contribution is non-monotonous and features a mini- mum at a finite effective width 𝑤eff, which can clearly be observed for 𝑤= 1𝜇m. The appearance of this minimum, which leads to the effect known as “effec- tive dipolar pinning” [17,18], is a result of the inter- play of two tendencies: (i) an increase of the volume contribution with increasing 𝑤/𝑤 eff, as for common Damon-Eshbach spin waves, and (ii) a decrease of the edge contribution when the spin-wave amplitude at the edges vanishes ( 𝑤/𝑤 eff>1). This minimum is also present in the case of a 50nm wide waveguide (red line), even though this is hardly perceivable in Fig. 2(d) due to the scale. In contrast, the exchange leads to a monotonous increase of the frequency as a function of 𝑤/𝑤 eff, which is minimal for the unpinned case, i.e., 𝑤/𝑤 eff= 0implying 𝑤eff→∞, when all spins are parallel. In the case of the 50nm wave- guide, the smaller width and the corresponding much larger quantized wavenumber in the case of pinned spins would lead to a much larger exchange contribu- tion than this is the case for the 1𝜇m wide waveguide (please note the vertical scales). Consequently, the system avoids pinning and the solution with lowest energy is situated at 𝑤/𝑤 eff= 0. In contrast, in the 1𝜇m wide waveguide, the interplay of dipolar and ex- change energy implies that the energy is minimized at a finite 𝑤/𝑤 eff. 6Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides Fig. 3. (a) Frequency spectra for 1𝜇m and 50nm wide waveguides measured for a respective microwave power of 6dBm and 15dBm. (b) Experimentally determined resonance frequencies (black squares) together with theoretical predictions and micromagnetic simulations. (c) Inverse effective width 𝑤/𝑤 effas a function of the waveguide width. (d) The critical width (𝑤crit) as a function of thickness ℎ. (e) Inverse effective width 𝑤/𝑤 effas a function of waveguide width for different materials at fixed thickness of 39 nm. (f) The critical width 𝑤critas a function of exchange length 𝜆for different thicknesses. (a) - (c) [29] 3.3.The dependence of the spin-wave frequency on the spin pinning and the critical width of the exchange unpinning As it is evident from Figs. 2(c) and 2(d), the pin- ning and the corresponding effective width have a large influence on the spin-wave frequency. This al- lows for an experimental verification of the presented theory, since the frequency of partially pinned spin- wave modes would be significantly higher than in the unpinned case. Black squares in Fig. 3(b) summa- rize the dependence of the frequency of the quasi- FMR on the width of the YIG waveguide. The ma- genta line shows the expected frequencies assuming pinned spins, the blue (dashed) line gives the reso- nance frequencies extrapolating the formula conven- tionally used for micron-sized waveguides [43] to the nanoscopic scenario, and the red line gives the result of the theory presented here, together with simula- tion results (green dashed line). As it can be seen, the experimentally observed frequencies can be well reproduced if the real pinning conditions are taken into account.As has been discussed alongside with Fig. 2, the unpinning occurs when the exchange interaction con- tribution becomes so large that it compensates the minimum in the dipolar contribution of the spin-wave energy. Since the energy contributions and the de- magnetization tensor change with the thickness of the investigated waveguide, the critical width below which the spins become unpinned is different for dif- ferent waveguide thicknesses. This is shown in Fig. 3(c),wheretheinverseeffectivewidth 𝑤/𝑤 effisshown for different waveguide thicknesses. Symbols are the results of micromagnetic simulations, lines are calcu- lated semi-analytically. As can be seen from the fig- ure, the critical width linearly increases with increas- ing thickness. This is summarized in Fig. 3(d), which shows the critical width (i.e. the maximum width for which 𝑤/𝑤 eff= 0) as a function of thickness. The critical widths for YIG, Permalloy, CoFeB and the Heusler compound Co 2Mn0.6Fe0.4Si with differ- ent thicknesses are investigated. Figure 3(e) shows the inverse effective width 𝑤/𝑤 effas a function of the waveguide width for these materials which can be considered as typical materials used in magnonics. 7B. Heinz et al. Width coordinate (nm) y Width coordinate (nm) y Width coordinate (nm) yNormalized mzNormalized mzNormalized mz Fig. 4. The spin-wave profile representatively depicted using the 𝑚𝑧component of the dynamic magnetization for the three lowest width modes obtained by micromagnetic simulation (black solid lines) and numerical calculation (red dots) for (a) 5𝜇m, (b)1𝜇m and (c) 50nm wide waveguides. Figure 3(f) shows the critical width ( 𝑤crit) as a func- tion of the exchange length 𝜆for different thicknesses. A simple empirical linear formula is found by fitting the critical widths for different materials in a wide range of thicknesses to estimate the critical width: 𝑤crit= 2.2ℎ+ 6.7𝜆 (3.1) where ℎis the thickness of the waveguide and 𝜆is the exchange length given by 𝜆=√︀ 2𝐴/(𝜇0𝑀2s)with the exchange constant 𝐴, the vacuum permeability 𝜇0, and the saturation magnetization 𝑀s. 3.4.Profiles of higher-order width modes In [29], only the profile of the fundamental mode (𝑛= 0) has been discussed, therefore the mode pro- files of higher width modes are shown in Fig. 4 for the widthsofthewaveguidesof 5𝜇mcorrespondingtothe practically fully pinned case (Fig. 4(a)), 50nm rep- resenting fully unpinned case (Fig. 4(c)), and 1𝜇m which can be considered as an intermediate case (Fig. 4(b)). For a 5𝜇m wide waveguide all higher width modes are clearly partially pinned due to the large widthandaninsufficientcontributionoftheexchangeenergy. In contrast to this the higher modes of a 1𝜇m widewaveguideareclearlyunpinnedformodes 𝑛 >2. Since the fundamental mode is already unpinned for a50nm wide waveguide also all higher width modes are fully unpinned. 3.5.Temperature dependence of spin pinning and frequencies of the spin-wave modes In the following, the quasi-analytic theory is used to study the influence of the temperature on the dis- cussed phenomena. There are two main parameters that introduce the temperature dependence of the spin-wave dispersion, the pinning condition and the pinning parameter: the saturation magnetization 𝑀s and the exchange constant 𝐴. Furthermore, the tem- perature dependence of the surface anisotropy con- stant 𝐾sat the lateral edges of the waveguide, can lead to an additional temperature dependence of the pinning parameter 𝑑(see Eq. (2.7)). However, typi- cally this dependency is rather weak and is therefore neglected in the following. The calculated saturation magnetization 𝑀sfor YIG is shown in Fig. 5(a) as a function of the temperature and was obtained using 8Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides Fig. 5. Temperature dependence of the saturation magnetization (a) and exchange constant (b) of YIG. (c) The temperature dependencies of the frequencies of the first three modes for a YIG waveguide with ℎ= 20nm,𝑤= 200nm and 𝐵0= 108 .9mT. (d) The temperature dependence of the inverse effective width (left axis) and critical width of the exchange unpinning (right axis). the theoretical model developed in [44]. The exper- imentally measured temperature dependence of the exchange constant 𝐴taken from [45] is shown in Fig. 5(b). Figure 5(c) shows the resulting temperature de- pendency of the frequencies of the first three modes for a YIG waveguide of thickness ℎ= 20nm, width 𝑤= 200nm, and for an external magnetic field 𝐵0= 108 .9mT applied along the stripe. One can clearly see that the frequencies of all modes decrease with the increase in temperature due to the decrease in the saturation magnetization. The critical width, at which the unpinning takes place, depends on both the saturation magnetization and the exchange con- stant as it can be seen e.g. from the empirical Eq. (3.1). The interplay between the both dependen- cies results in the increase of the critical width with the increase in temperature from the value of around 140nm for zero temperature up to around 200nm for 500K - see Fig. 5(d). At the same time, the spin pinning, which is shown in the same figure in terms of inverse effective width of the waveguide 𝑤/𝑤 eff, de- creases with the increase in temperature. This hap- pens due to the dominant contribution of the tem-perature dependence of the saturation magnetization which, consequently, defines the strength of the dipo- lar pinning phenomenon. To conclude, if one con- ducts low temperature experiments which rely or re- quire a fully unpinned state of the system a careful design of the structure dimensions is necessary. 3.6.Spin-wave dispersion in nano-strucutres and the dependence of spin pinning of spin-wave wavenumber Up to now, the discussion was limited to the spe- cial case of 𝑘𝑥= 0. In the following, the influence of finite wave vector will be addressed. The spin- wave dispersion relation of the fundamental ( 𝑛= 0) mode obtained from micromagnetic simulations (color-code) together with the semi-analytical solu- tion (white dashed line) are shown in Fig. 6(a) for the YIG waveguide of 𝑤= 50nm width. The figure also shows the low-wavenumber part of the disper- sion of the first width mode ( 𝑛= 1), which is pushed up significantly in frequency due to its large exchange contribution. Bothmodesaredescribedaccuratelyby the quasi-analytical theory. As it is described above, the spins are fully unpinned in this particular case. 9B. Heinz et al. n n Fig. 6.(a) Spin-wave dispersion relation of the first two width modesfrommicromagneticsimulations(color-code)andtheory (dashed lines). (b) Inverse effective width 𝑤/𝑤 effas a function of the spin-wave wavenumber 𝑘𝑥for different thicknesses and waveguide widths, respectively. [29] In order to demonstrate the influence of the pinning conditions on the spin-wave dispersion, a hypothetic dispersion relation for the case of partial pinning is shown in the figure with a dash-dotted white line (the case of 𝑤/𝑤 eff= 0.63is considered that would result from the usage of the thin strip approximation [16]). One can clearly see that the spin-wave frequencies in this case are considerably higher. Figure 6(b) shows the inverse effective width 𝑤/𝑤 effas a function of the wavenumber 𝑘𝑥forthreeexemplarywaveguidewidths of𝑤= 50nm,300nm and 500nm. As it can be seen, theeffectivewidthand,consequently,theratio 𝑤/𝑤 eff shows only a weak nonmonotonic dependence on the spin-wave wavenumber in the propagation direction. This dependence is a result of an increase of the in- homogeneity of the dipolar fields near the edges forlarger 𝑘𝑥, which increases pinning [18], and of the simultaneous decrease of the overall strength of dy- namic dipolar fields for shorter spin waves. Please note that the mode profiles are not only important for the spin-wave dispersion. The unpinned mode profiles also greatly improve the coupling efficiency between two adjacent waveguides [9,46–48]. 3.7.Spin-wave lifetime in magnetic nanostructures The spin-wave lifetime depends on the ellipticity of the magnetization precession, and, thus, on the spin pinning conditions. The top panel of Fig. 2(b) shows an additional feature of the narrow waveguide: as the aspect ratio of the waveguides approaches unity, the ellipticity of the precession, a well-known feature of micron-sized waveguides which still resemble a thin film [27, 39], vanishes and the precession becomes nearly circular. In addition, in nanoscale waveguides, the ellipticity is constant across the width, while it can be different at the waveguide center and near its edges for a 1𝜇m wide waveguide. In general, the def- inition of the ellipticity 𝜖of the precession is given by the ratio of the precession components as follows: 𝜖= 1−𝑚min 𝑚max, (3.2) where 𝑚minand𝑚maxdenote the respective ampli- tudes of the smaller and larger component of the pre- cession. Calculating the average relation between the magnetization components 𝑚𝑦and𝑚𝑧it follows ⃒⃒⃒⃒𝑚𝑦 𝑚𝑧⃒⃒⃒⃒=⎯⎸⎸⎷(︃ (𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑧𝑧 𝑘𝑥) (𝜔H+𝜔M(𝜆2𝐾2+𝐹𝑦𝑦 𝑘𝑥)))︃ , (3.3) from which the ellipticity can be calculated for any widthindependencyofthespin-wavewavenumber 𝑘𝑥 as it is shown in Fig. 7(a). The relaxation lifetime 𝜏of the uniform precession mode in an infinite medium (without inhomogeneous linewidth Δ𝐵0) is simply defined as 𝜏= 1/(𝛼𝜔), where 𝜔is the angular frequency of the spin wave and𝛼is the damping. However, the dynamic demag- netizing field has to be taken into account in finite spin-wave waveguide. The lifetime can be found by the phenomenological model [49–51] 𝜏=(︂ 𝛼𝜔𝜕𝜔 𝜕𝜔H)︂−1 . (3.4) 10Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides The dispersion relation has been shown in the manuscript (Eq. (2.3)). The demagnetization ten- sors are independent of 𝜔H. Differentiating Eq. (2.3) yields the lifetime as 𝜏=(︂1 2𝛼(2𝜔H+ 2𝜔M𝜆2𝐾2+𝜔M(𝐹𝑧𝑧 𝑘𝑥+𝐹𝑦𝑦 𝑘𝑥)))︂−1 . (3.5) This formula clearly shows that the lifetime of the uniform precession ( 𝑘𝑥= 0) depends only on the sum of the dynamic 𝑦𝑦and𝑧𝑧components of demagneti- zation tensors. Figure 7(b) shows the cross-section, spin precession trajectory (red line) and the dynamic components of the demagnetization tensors of different sample geometries. The spin precession trajectory changes from elliptic for the thin film ( 𝑎r≪1) to circular for the nanoscopic waveguide ( 𝑎r= 1). The spin preces- sion trajectory in the bulk material is also circular (in the geometry when spin waves propagate parallel to the static magnetic field, the same geometry as stud- ied for nanoscale waveguides). The dependence of the lifetime on the wavenumber is shown in Fig. 7(c) for YIG with a damping constant 𝛼= 2×10−4. The inhomogeneous linewidth is not taken into account. Thelifetimeoftheuniformprecession( 𝑘𝑥= 0)forthe bulk material is much large than that in the thin film and nanoscopic waveguide, another consequence of the absence of dynamic demagnetization in the bulk (𝐹𝑧𝑧 0=𝐹𝑦𝑦 0). Moreover, the lifetimes of the uniform precession ( 𝑘𝑥= 0) for a thin film (red line) and for a nanoscopic waveguide (black line) have the same value, because the lifetime depends only on the sum of the two components, which is the same for both cases. Moreover, the 𝑦𝑦and𝑧𝑧components of the de- magnetization tensor decrease with an increase of the spin-wave wavenumber (instead, the 𝑥𝑥component, which does not affect the spin wave dynamic in our geometry, increases). Thelifetimeisinverselypropor- tional to the square of the wavenumber and the sum of the dynamic demagnetization components. In the exchange region, the lifetime is, thus, dominated by the wavenumber. Therefore, the lifetimes for short- wave spin-waves are nearly the same for the three different geometries. Fig. 7. (a) Ellipticity as a function of the waveguide width for different spin-wave wavenumbers 𝑘𝑥for a thickness of ℎ= 39nmandanexternalmagneticfieldof 𝐵0= 108 .9mT(b)The spin precession trajectories (red lines) and the components of the demagnetization tensor 𝐹𝑦𝑦 0and𝐹𝑧𝑧 0for different sample geometries. (c) The spin-wave lifetime as a function of the spin-wave wavenumber. The lines and dots are obtained from Eq. (3.5) and micromagnetic simulation, respectively. 3.8.Dependence of the spin pinning on a trapezoidal form of the waveguides A perfect rectangular form is not achievable in the experiment due to the involved patterning technique. As a result of the etching, the cross-section of the waveguides is always slightly trapezoidal. In this sec- tion, the influence of such a trapezoidal form on the spin pinningconditions isstudied. Inour experiment, 11B. Heinz et al. Fig. 8. (a) Trapezoidal cross section of the simulated wave- guide with the normalized spin-wave profile for the different layers. (b) The inverse effective width 𝑤/𝑤 effas a function of the width of the waveguide for trapezoidal and rectangular form. the trapezoidal edges extent for approximately 20nm on both sides for all the patterned waveguides as it can be seen from Fig. 1(d). We performed addi- tionalsimulationonwaveguideswithsuchtrapezoidal edges. The simulated cross-section is shown in the top of Fig. 8. The thickness of the waveguide is di- vided into 5 layers with different widths ranging from 90nm to 50nm. The steps at the edges are hard to be avoided due to the finite difference method used in MuMax3. The spin-wave profiles in the different 𝑧-layers are shown at the bottom of Fig. 8(a). The results clearly show that the spin-wave profiles are fully unpinned along the entire thickness. This is due to the fact that the largest width ( 90nm) isstill far below the critical width. Hence, the influence of the trapezoidal form of the waveguide on the spin pinning condition is negligible for very narrow waveg- uides. For large waveguides, it also does not have a large impact as the ratio of the edge to the wave- guide area becomes close to zero. Quantitatively, the quasi-ferromagnetic resonance frequency in a 50nm wide waveguide decreases from 5.45GHz for the rect- angular shape to 5.38GHz for the trapezoidal form due to the increase of the averaged width which, in fact, even closer to the experiment results ( 5.35GHz). The inverse effective width 𝑤/𝑤 effas a function of the width of the waveguides is simulated for a trape- zoidal and a rectangular form and the result is shown in Fig. 8(b). Here, the width is defined by the min- imal width for the trapezoidal form, i.e., the width of the top layer. In the case of trapezoidal form, the inverse effective width is averaged over all 5layers. The critical width slightly decreases from 200nm for the rectangular cross-section to 180nm for the trape- zoidal form due to the increase of the averaged width. The difference between the inverse effective widths decreases with increasing width of the waveguide and vanishes when the width is larger than 300nm. Furthermore, it should be noted that the results of the multilayer simulations demonstrate that the as- sumption of a uniform dynamic magnetization distri- bution across the thickness that is used in our analyt- ical theory and micromagnetic simulations featuring only one cell in the z dimension is valid. 3.9.Influence of edge roughness on the spin pinning Perfectly smooth edges are also hard to obtain in the experiment. Therefore, we have considered the in- fluence of edge roughness on the spin pinning. We performed additional simulations on waveguides with rough boundaries for a fixed thickness of 39nm. 5nm (for50nm to 100nm wide waveguides) or 10nm (for 100nm to 1000nm wide waveguides) wide rectangu- lar nonmagnetic regions with a random length are introduced randomly on both sides of the waveguides to act as defects. The introduction of roughness re- sults in a slight increase of the critical width from 200nm to 240nm, as is shown in Fig. 9(a). These re- sults demonstrate that edge roughness does not have a large influence on spin pinning condition. 12Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides Fig. 9.(a) Top: Schematic of the rough waveguide and close- up image of the introduced edge roughness. A single random- ized defect pattern is generated for each structure width. Bot- tom: Inverse effective width 𝑤/𝑤 effas a function of the wave- guide width for rough and smooth edges. (b) The normalized spin-wave intensity as a function of the propagation length for 50nm wide waveguides with smooth and rough edges. Additional simulations are performed to study the influence of a rough edge on the propagation length of spin waves with frequency 6.16GHz ( 𝑘𝑥= 0.03rad/nm). Figure9(b)showsthenormalizedspin- wave intensity as a function of propagation length for smooth and rough edged waveguide of 450nm width. The decay length slightly decreases from 15.96𝜇m for smooth edges to 15.76𝜇m for rough edges. Since the spins in nanoscopic waveguides are already unpinned, the effect of such an edge roughness is not too impor- tant anymore and the propagation length is essen- tially unaffected.4. Conclusions To conclude, an in-detail investigation of the pinning phenomenon based on the theoretical description of [29] is presented and the quasi-analytical model is outlined. The dependency of the effective width on the thickness and the material of choice is analysed and a simple empirical formula is found to predict the critical width for a given system. In addition to [29], higher order width modes up to 𝑛= 2are anal- ysed. An investigation of the effective width for finite wavevectors along the waveguide yields only a weak nonmonotonic dependence. It is shown, that assum- ing a more realistic trapezoidal cross section of the structures rather than the ideal rectangular shape re- sults in a small decrease of the quasi-FMR frequency and a slight reduction of the critical width. More- over, the influence of edge roughness is studied which shows a small increase of the critical width compared to the case of smooth edges. Here, also the impact on the decay length of propagating waves is investigated and only a small reduction is found. The tempera- ture dependence of the pinning phenomenon shows that the dependencies of the saturation magnetiza- tion and exchange constant of YIG result in the de- crease of spin pinning with the increase in tempera- ture and in the increase in the critical width of the exchange unpinning. This assumes that low temper- atures are favourable for the dipolar pinning and the sizes of the structures have to be decreased further in order to operate with fully-unpinned uniform spin- wave modes. The presented results provide valuable guidelines for applications in nano-magnonics where spin waves propagate in nanoscopic waveguides with aspect ra- tios close to one and lateral sizes comparable to the sizes of modern CMOS technology. Acknowledgement. The authors thank Burkard Hillebrands and Andrei Slavin for valuable discus- sions. This research has been supported by ERC Starting Grant 678309 MagnonCircuits and by the DFG through the Collaborative Research Center SFB/TRR-173“Spin+X” (ProjectsB01)andthrough theProjectDU1427/2-1. B.H.acknowledgessupport by the Graduate School Material Science in Mainz (MAINZ). R. V. acknowledges support by the Na- tional Academy of Sciences of Ukraine Grant No. 23- 04/01-2019. 13B. Heinz et al. *These authors have contributed equally to this work. **chumak@physik.uni-kl.de 1. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille- brands, Magnon spintronics, Nat. Phys. 11, 453 (2015), DOI:http://dx.doi.org/10.1038/nphys3347. 2. V. V. Kruglyak, S. O. Demokritov, and D. Grundler, Magnonics, J. Phys. D 43, 264001 (2010), DOI:http://dx. doi.org/10.1088/0022-3727/43/26/264001. 3. C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Cher- topalov, M. T. Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, Towards graded-index magnonics: Steering spin waves in magnonic networks, Phys. Rev. B 92, 020408 (2015), DOI: http://dx.doi.org/10.1103/PhysRevB.92.020408. 4. A. Khitun, M. Bao, and K. L. Wang, Magnonic logic cir- cuits, J. Phys. D: Appl. Phys. 43, 264005 (2010), DOI: http://dx.doi.org/10.1088/0022-3727/43/26/264005. 5. M. Schneider, T. Br¨ acher, V. Lauer, P. Pirro, D. A. Bozhko, A. A. Serga, H. Y. Musiienko-Shmarova, B. Heinz, Q. Wang, T. Meyer, F. Heussner, S. Keller, E. T. Pa- paioannou, B. L¨ agel, T. L¨ ober, V. S. Tiberkevich, A. N. Slavin, C. Dubs, B. Hillebrands, and A. V. Chumak, Bose- Einstein condensation of quasi-particles by rapid cooling, arXiv 1612.07305 (2016). 6. M. Krawczyk and D. Grundler, Review and prospects of magnonic crystals and devices with reprogrammable band structure, J. Phys.: Condens. Matt. 26, 123202 (2014), DOI:http://dx.doi.org/10.1088/0953-8984/26/12/123202. 7. S. Wintz, V. Tiberkevich, M. Weigand, J. Raabe, J. Lind- ner, A. Erbe, A. Slavin, and J. Fassbender, Magnetic vor- tex cores as tunable spin-wave emitters, Nat. Nanotechnol. 11, 948 EP (2016), DOI:http://dx.doi.org/10.1038/nnano. 2016.117. 8. T. Br¨ acher and P. Pirro, An analog magnon adder for all-magnonic neurons, J. Appl. Phys. 124, 152119 (2018), DOI:http://dx.doi.org/10.1063/1.5042417. 9. Q. Wang, P. Pirro, R. Verba, A. Slavin, B. Hillebrands, and A. V. Chumak, Reconfigurable nanoscale spin-wave directional coupler, Science Advances 4(2018), DOI:http: //dx.doi.org/10.1126/sciadv.1701517. 10. O. Zografos, B. Soree, A. Vaysset, S. Cosemans, L. Amaru, P. Gaillardon, G. De Micheli, R. Lauwereins, S. Sayan, P. Raghavan, I. P. Radu, and A. Thean, Design and benchmarking of hybrid cmos-spin wave device circuits compared to 10nm cmos pp. 686–689 (2015), DOI:http: //dx.doi.org/10.1109/NANO.2015.7388699. 11. S. Manipatruni, D. E. Nikonov, and I. A. Young, Be- yond CMOS computing with spin and polarization, Nature Physics 14, 338 (2018), DOI:http://dx.doi.org/10.1038/ s41567-018-0101-4. 12. A. Chumak, Fundamentals of magnon-based computing, arXiv 1901.08934 (2019).13. B. A. Ivanov and C. E. Zaspel, Magnon modes for thin circular vortex-state magnetic dots, Appl. Phys. Lett. 81, 1261 (2002), DOI:http://dx.doi.org/10.1063/1.1499515. 14. G. T. Rado and J. R. Weertman, Spin-wave resonance in a ferromagnetic metal, J. Phys. Chem. Sol. 11, 315 (1959), DOI:https://doi.org/10.1016/0022-3697(59)90233-1. 15. R. W. Damon and J. R. Eshbach, Magnetostatic modes of a ferromagnet slab, J. Phys. Chem. Sol. 19, 308 (1961), DOI:https://doi.org/10.1016/0022-3697(61)90041-5. 16. K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Effective dipolar boundary conditions for dynamic magnetization in thin magnetic stripes, Phys. Rev.B 66, 132402(2002), DOI:http://dx.doi.org/10.1103/ PhysRevB.66.132402. 17. K. Y. Guslienko and A. N. Slavin, Boundary conditions for magnetizationinmagneticnanoelements, Phys.Rev.B 72, 014463 (2005), DOI:http://dx.doi.org/10.1103/PhysRevB. 72.014463. 18. K. Y. Guslienko and A. N. Slavin, Magnetostatic green’s functions for the description of spin waves in finite rectan- gular magnetic dots and stripes, J. Magn. Magnet. Mater. 323, 2418 (2011), DOI:https://doi.org/10.1016/j.jmmm. 2011.05.020. 19. R. E. Arias, Spin-wave modes of ferromagnetic films, Phys. Rev.B 94, 134408(2016), DOI:http://dx.doi.org/10.1103/ PhysRevB.94.134408. 20. T. Br¨ acher, O. Boulle, G. Gaudin, and P. Pirro, Creation of unidirectional spin-wave emitters by utilizing interfa- cial Dzyaloshinskii-Moriya interaction, Phys. Rev. B 95, 064429 (2017), DOI:http://dx.doi.org/10.1103/PhysRevB. 95.064429. 21. V. E. Demidov and S. O. Demokritov, Magnonic waveg- uides studied by microfocus Brillouin light scattering, IEEETrans.Magnet. 51, 1(2015), DOI:http://dx.doi.org/ 10.1109/TMAG.2014.2388196. 22. F. Ciubotaru, T. Devolder, M. Manfrini, C. Adelmann, and I. P. Radu, All electrical propagating spin wave spec- troscopy with broadband wavevector capability, Applied Physics Letters 109, 012403 (2016), DOI:http://dx.doi. org/10.1063/1.4955030. 23. P. Pirro, T. Br¨ acher, A. V. Chumak, B. L¨ agel, C. Dubs, O. Surzhenko, P. G¨ arnert, B. Leven, and B. Hillebrands, Spin-wave excitation and propagation in microstructured waveguides of yttrium iron garnet/Pt bilayers, Appl. Phys. Lett. 104, 012402 (2014), DOI:http://dx.doi.org/10.1063/ 1.4861343. 24. M. Mruczkiewicz, P. Graczyk, P. Lupo, A. Adeyeye, G. Gubbiotti, and M. Krawczyk, Spin-wave nonreciproc- ity and magnonic band structure in a thin permalloy film induced by dynamical coupling with an array of ni stripes, Phys. Rev. B 96, 104411 (2017), DOI:http://dx.doi.org/ 10.1103/PhysRevB.96.104411. 25. A. Haldar and A. O. Adeyeye, Deterministic control of magnetization dynamics in reconfigurable nanomagnetic networks for logic applications, ACS Nano 10, 1690 14Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides (2016), DOI:http://dx.doi.org/10.1021/acsnano.5b07849, pMID: 26738567. 26. R. Verba, V. Tiberkevich, E. Bankowski, T. Meitzler, G. Melkov, and A. Slavin, Conditions for the spin wave nonreciprocity in an array of dipolarly coupled magnetic nanopillars, Appl. Phys. Lett. 103, 082407 (2013), DOI: http://dx.doi.org/10.1063/1.4819435. 27. T. Br¨ acher, P. Pirro, and B. Hillebrands, Parallel pumping formagnonspintronics: Amplificationandmanipulationof magnon spin currents on the micron-scale, Phys. Rep. 699, 1 (2017), DOI:https://doi.org/10.1016/j.physrep.2017.07. 003. 28. G. S. Abo, Y. Hong, J. Park, J. Lee, W. Lee, and B. Choi, Definition of magnetic exchange length, IEEE Trans. Magnet. 49, 4937 (2013), DOI:http://dx.doi.org/ 10.1109/TMAG.2013.2258028. 29. Q. Wang, B. Heinz, R. Verba, M. Kewenig, P. Pirro, M. Schneider, M. Thomas, B. L¨ agel, C. Dubs, T. Br¨ acher, and A. V. Chumak, Phys. Rev. Lett. (2019), DOI:https: //doi.org/10.1103/PhysRevLett.122.247202. 30. Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015), DOI:http://dx.doi.org/10. 1126/science.aaa3693. 31. O. V. Dobrovolskiy, R. Sachser, T. Br¨ acher, T. B¨ ottcher, V. V. Kruglyak, R. V. Vovk, V. A. Shklovskij, M. Huth, B. Hillebrands, and A. V. Chumak, Magnon-fluxon in- teraction in a ferromagnet/superconductor heterostruc- ture, Nat. Phys. (2019), DOI:http://dx.doi.org/10.1038/ s41567-019-0428-5. 32. I. Golovchanskiy, N. Abramov, M. Pfirrmann, T. Piskor, J. Voss, D. Baranov, R. Hovhannisyan, V. Stolyarov, C. Dubs, A. Golubov, V. Ryazanov, A. Ustinov, and M.Weides, Interplayofmagnetizationdynamicswithami- crowave waveguide at cryogenic temperatures, Phys. Rev. Applied 11, 044076(2019), DOI:10.1103/PhysRevApplied. 11.044076. 33. C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Br¨ uckner, and J. Dellith, Sub-micrometer yttrium iron garnet LPE films with low ferromagnetic resonance losses, J. Phys. D: Appl. Phys. 50, 204005 (2017), DOI:http: //dx.doi.org/10.1088/1361-6463/aa6b1c. 34. I. S. Maksymov and M. Kostylev, Broadband stripline ferromagnetic resonance spectroscopy of ferromagnetic films, multilayers and nanostructures, Physica E: Low- dimensional Systems and Nanostructures 69, 253 (2015), DOI:https://dx.doi.org/10.1016/j.physe.2014.12.027. 35. A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, The de- sign and verification of mumax3, AIP Advances 4, 107133 (2014), DOI:http://dx.doi.org/10.1063/1.4899186. 36. I. S. Maksymov and M. Kostylev, Broadband stripline fer- romagnetic resonance spectroscopy of ferromagnetic films, multilayers and nanostructures, Physica E 69, 253 (2015),DOI:https://doi.org/10.1016/j.physe.2014.12.027. 37. R. Verba, G. Melkov, V. Tiberkevich, and A. Slavin, Collective spin-wave excitations in a two-dimensional ar- ray of coupled magnetic nanodots, Phys. Rev. B 85, 014427 (2012), DOI:http://dx.doi.org/10.1103/PhysRevB. 85.014427. 38. R. Verba, Spin waves in arrays of magnetic nanodots with magnetodipolar coupling, Ukr. J. Phys. 58, 758 (2013), DOI:http://dx.doi.org/10.15407/ujpe58.08.0758. 39. A. Gurevich and G. Melkov, Magnetization Oscillations and Waves (New York: CRC Press) (1996). 40. V. V. Kruglyak, O. Y. Gorobets, Y. I. Gorobets, and A. N. Kuchko, Magnetization boundary conditions at a ferromagnetic interface of finite thickness, J. Phys.: Cond. Matt. 26, 406001 (2014), DOI:http://dx.doi.org/10.1088/ 0953-8984/26/40/406001. 41. T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, Micro-focused Brillouin light scattering: imaging spin waves at the nanoscale, Front. Phys. 3, 35 (2015), DOI:http://dx.doi.org/10.3389/fphy.2015.00035. 42. M. B. Jungfleisch, W. Zhang, W. Jiang, H. Chang, J. Sl- lenar, S. M. Wu, J. E. Pearson, A. Bhattacharya, J. B. Ketterson, M. Wu, and A. Hoffmann, Spin waves in micro- structured yttrium iron garnet nanometer-thick films, J. Appl. Phys. 117, 17D128 (2015), DOI:http://dx.doi.org/ 10.1063/1.4916027. 43. B. A. Kalinikos and A. N. Slavin, Theory of dipole- exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions, J. Phys. C 19, 7013 (1986). 44. P. Hansen, P. R¨ oschmann, and W. Tolksdorf, Saturation magnetization of gallium-substituted yttrium iron garnet, J. Appl. Phys. 45, 2728 (1974), DOI:http://dx.doi.org/10. 1063/1.1663657. 45. R. C. LeCraw and L. R. Walker, Temperature dependence of the spin-wave spectrum of yttrium iron garnet, J. Appl. Phys. 32, S167 (1961), DOI:http://dx.doi.org/10.1063/1. 2000390. 46. A. V. Sadovnikov, E. N. Beginin, S. E. Sheshukova, D. V. Romanenko, Y. P. Sharaevskii, and S. A. Nikitov, Di- rectional multimode coupler for planar magnonics: Side- coupled magnetic stripes, Appl. Phys. Lett. 107, 202405 (2015), DOI:http://dx.doi.org/10.1063/1.4936207. 47. A. V. Sadovnikov, A. A. Grachev, S. E. Sheshukova, Y. P. Sharaevskii, A. A. Serdobintsev, D. M. Mitin, and S. A. Nikitov, Magnon straintronics: Reconfigurable spin- waveroutinginstrain-controlledbilateralmagneticstripes, Phys. Rev. Lett. 120, 257203 (2018), DOI:http://dx.doi. org/10.1103/PhysRevLett.120.257203. 48. A. V. Sadovnikov, S. A. Odintsov, E. N. Beginin, S. E. Sheshukova, Y. P. Sharaevskii, and S. A. Nikitov, To- ward nonlinear magnonics: Intensity-dependent spin-wave switching in insulating side-coupled magnetic stripes, Phys. Rev. B 96, 144428 (2017), DOI:http://dx.doi.org/ 10.1103/PhysRevB.96.144428. 15B. Heinz et al. 49. D. D. Stancil, Phenomenological propagation loss theory for magnetostatic waves in thin ferrite films, J. Appl. Phys. 59, 218 (1986), DOI:http://dx.doi.org/10.1063/1.336867. 50. D. D. Stancil and A. Prabhakar, Spin Waves. Theory and Applications (Berlin: Springer) (2009). 51. R. Verba, V. Tiberkevich, and A. Slavin, Damping oflinear spin-wave modes in magnetic nanostructures: Lo- cal, nonlocal, and coordinate-dependent damping, Phys. Rev.B 98, 104408(2018), DOI:http://dx.doi.org/10.1103/ PhysRevB.98.104408. 16

2020-01-31

The field of magnonics attracts significant attention due to the possibility of utilizing information coded into the spin-wave phase or amplitude to perform computation operations on the nanoscale. Recently, spin waves were investigated in Yttrium Iron Garnet (YIG) waveguides with widths ranging down to 50 nm and aspect ratios thickness over width approaching unity. A critical width was found, below which the exchange interaction suppresses the dipolar pinning phenomenon and the system becomes unpinned. Here we continue these investigations and analyse the pinning phenomenon and spin-wave dispersions as a function of temperature, thickness and material of choice. Higher order modes, the influence of a finite wavevector along the waveguide and the impact of the pinning phenomenon on the spin-wave lifetime are discussed as well as the influence of a trapezoidal cross section and edge roughness of the waveguides. The presented results are of particular interest for potential applications in magnonic devices and the incipient field of quantum magnonics at cryogenic temperatures.

Temperature dependence of spin pinning and spin-wave dispersion in nanoscopic ferromagnetic waveguides

2002.00003v1

arXiv:1904.04167v1 [quant-ph] 8 Apr 2019Quantum entanglement between two magnon modes via Kerr nonl inearity Zhedong Zhang,1,∗Marlan O. Scully,1, 2, 3and Girish S. Agarwal1, 4,† 1Institute for Quantum Science and Engineering, Texas A &M University, College Station, TX 77843, USA 2Quantum Optics Laboratory, Baylor Research and Innovation Collaborative, Waco, TX 76704, USA 3Department of Mechanical and Aerospace Engineering, Princ eton University, Princeton, NJ 08544, USA 4Department of Biological and Agricultural Engineering, Department of Physics and Astronomy, Texas A &M University, College Station, TX 77843, USA (Dated: April 9, 2019) We propose a scheme to entangle two magnon modes via Kerr nonl inear effect when driving the systems far-from-equilibrium. We consider two macroscopic yttriu m iron garnets (YIGs) interacting with a single-mode microcavity through the magnetic dipole coupling. The Kitt el mode describing the collective excitations of large number of spins are excited through driving cavity with a str ong microwave field. We demonstrate how the Kerr nonlineraity creates the entangled quantum states between the two macroscopic ferromagnetic samples, when the microcavity is strongly driven by a blue-detuned microw ave field. Such quantum entanglement survives at the steady state. Our work o ffers new insights and guidance to designate the experiments f or observing the entanglement in massive ferromagnetic materials. It can al so find broad applications in macroscopic quantum effects and magnetic spintronics. Introduction.– Recent advance in ferromagnetic materials draw considerable attention in the studies of quantum natur e in magnetic systems, as the limitations of electrical circu itry are reached. Thanks to the low loss of the collective exci- tations of spins known as magnons in magnetic samples, the magnons offer a new paradigm for developing future gener- ation of spintronic devices and quantum engineering [1–6]. The yttrium iron garnet (YIG) with the size of ∼100µm as fabricated in recent experiments provides new insights f or studying the macroscopic quantum e ffects, such as entangle- ment and squeezing that have raised widespread interest in different branches of physics during decade [7–12]. Quan- tum entanglement between massive mirror and optical cav- ity photons has been explored, in both theoretical and ex- perimental aspects [13–18]. Several ideas follow-on sugge st the extension of such entangled quantum state towards the magnons in microwave regime, due to their great potential fo r macroscopic spintronic devices. Much experimental e fforts have been devoted to the quantum nature of magnon states, through hybridizing the spin waves with other degrees of fre e- doms, e.g., superconducting qubits and phonon modes [19– 22]. Compared to atoms and photonics, magnonics holds the potential for implementing quantum states in more massive objects. This can be seen from the 320 µm-diam YIG spheres implemented in recent experiments [23]. As a powerful platform for investigating the light-matter i n- teraction [23–30], ferromagnetic materials are taking the ad- vantage of reaching strong and ultrastrong coupling regime s, along with the fact of their high spin density as well as low dissipation rate. The strong coupling results in the cavity magnon-polariton, serving as a potential candidate for im- plementing quantum information transducers and memories [30, 31]. To achieve the quantum regime in magnon polari- tons, the macroscopic quantum e ffects are essentially wor- thy of being explored. The most recent work using driven- dissipation theory suggest the magnon-photon-phonon enta n- glement and also the squeezing of magnon modes in whichboth the entanglement and squeezing are essentially trans- ferred into the mechanical mode [32–34]. From a theoreti- cal view-point, this macroscopic quantum nature of magnon modes stems from the nonlinearity that can be enhanced by driving the systems far-from-equilibrium. Two prominent schemes are responsible for introducing such nonlinearity : the magnetostrictive interaction and the Kerr e ffect, where the latter results from the magnetocrystalline anisotropy. Ap art from the magnon-phonon interaction, Kerr nonlinearity pla ys a significant role in magnon spintronics [5]. Recent exper- iments in YIG spheres demonstrated the multistability and photon-mediated control of spin current, due to the Kerr ef- fect [35, 36]. In this Letter, we propose a scheme of entangling magnon modes in two massive YIG spheres via the Kerr nonlinearity. The two magnon modes interact with a microcavity through the beam-splitter-like coupling, which cannot produce any en- tanglement. Nevertheless, activating the Kerr nonlineari ty via strong driving results in squeezing-like coupling which ma y let magnon get entangled with cavity photons. The subse- quent entanglement transfer between photons and the other magnon mode will lead to the entanglement between magnon modes. The condition for optimizing the magnon-magnon en- tanglement is found and is confirmed by our numerical cal- culations. By taking into account the experimentally feasi ble parameters, we show the considerable magnon-magnon entan- glement can be created. Such entanglement is also shown to be robust against cavity leakage. Our work o ffers new insight and perspective for studying the quantum e ffects in complex molecules. These have been manifested by the excited-state dynamics in dye molecules and even bacterias implying the entangled quantum states when interacting with microcavit ies [37–41]. Model and equation of motion.– We consider a hybrid magnon-cavity system consisting of two bulk ferromagnetic materials and one microwave cavity mode. The ferromagnetic sample contains dispersive spin waves, in which only the spa -2 FIG. 1: Schematic of cavity magnons. Two YIG spheres are interacting with the basic mode of microcavity in which the right mirror is made of high-reflection material so that pho- tons leak from the left side. The static magnetic field for pro - ducing Kittel mode is along z-axis whereas the microwave driving and magnetic field inside cavity are along x-axis. tially uniform mode (Kittel mode [42]) is assumed to strongl y interact with cavity photons. The full Hamiltonian of this c av- ity magnonics system reads [43] H=−/integraldisplay MzB0dr−µ0 2/integraldisplay MzHandr +1 2/integraldisplay/parenleftigg ε0E2+B2 µ0/parenrightigg dr−/integraldisplay M·Bdr(1) where B0=B0ezis the applied static magnetic field and M=γS/Vmwithγ=e/medenoting the gyromagnetic ra- tio.Sstands for the collective spin operator and Vmis volume of ferromagnetic material. Hanis the anisotropic field due to the magnetocrystalline anisotropy and has zcomponent only owing to the crystallographic axis being aligned along the a p- plied static magnetic field. Thereby the anisotropic field is given by Han=−2KanMz/M2where KanandMdenote the dominant 1st anisotropy constant and the saturation magnet i- zation, respectively. One can recast the Hamiltonian in Eq. (1) into H=−γ2/summationdisplay j=1Bj,0Sj,z+γ22/summationdisplay j=1µ0K(j) an M2 jVj,mS2 j,z +/planckover2pi1ωca†a−γ2/summationdisplay j=1Sj,xBj,x(2) by assuming the magnetic field inside cavity is along x- axis. The Holstein-Primako fftransform yields to Si,z=Si− m† imi,Si,+=(2Si−m† imi)1/2mi,Si,−=m† i(2Si−m† imi)1/2 where Si,±≡Si,x±iSi,yandmirepresents the bosonic anni- hilation operator [44]. For the yttrium iron garnets (YIGs) with diameter d=40µm, the density of ferrum ion Fe3+ isρ=4.22×1027m−3, which leads to the total spin S= 5 2ρVm=7.07×1014. This is often much larger than the num- ber of magnons, so that we can safely approximate Sj,+≃/radicalbig2Sjmj,Sj,−≃/radicalbig2Sjm† j. In the presence of external mi- crowave driving field, the e ffective Hamiltonian of hybrid magnon-cavity system is of the form Heff=/planckover2pi1ωca†a+/planckover2pi12/summationdisplay j=1/bracketleftig ωjm† jmj+gj(m† ja+mja†) +∆ jm† jmjm† jmj/bracketrightig +i/planckover2pi1Ω(a†e−iωdt−aeiωdt)(3) where the rotating-wave approximation was employed and cavity frequency is denoted by ωc. The frequency of Kit- tel mode isωj=γBj,0withγ/2π=28GHz/T.gjgives the magnon-cavity coupling and ∆j=µ0K(j) anγ2/M2 jVj,mgives the Kerr nonlinearity, resulting from the on-site magnon-magn on scattering. The Rabi frequency Ω=/radicalbig 2Pdγc//planckover2pi1ωdin last term quantifies the strength of the field inside microcavity drive n by the microwave magnetic field, where Pdandωdrepresent the power and frequency of the microwave field, respectively . γcis the cavity leaking rate. In the rotating frame of mi- crowave field, the dynamics of hyrid cavity-magnon system is governed by the quantum Langevin equations (QLEs) ˙ms=−(iδs+γs)ms−2i∆sm† smsms−igsa+/radicalbig 2γsmin s(t) ˙a=−(iδc+γc)a−i2/summationdisplay j=1gjmj+Ω+/radicalbig 2γcain(t)(4) whereγsquantifies the magnon dissipation. δs=ωs− ωd, δc=ωc−ωd.min s(t) and ain(t) are the input noise op- erators having zero mean and white noise: /angbracketleftmin,† s(t)min s(t′)/angbracketright= ¯nsδ(t−t′),/angbracketleftmin s(t)min,† s(t′)/angbracketright=(¯ns+1)δ(t−t′);/angbracketleftain,†(t)ain(t′)/angbracketright= 0,/angbracketleftain(t)ain,†(t′)/angbracketright=δ(t−t′) where ¯ ns=[exp(/planckover2pi1ωs/kBT)−1]−1 denotes the Planck factor of the s-th magnon mode. Since the microcavity is under strong driving by the microwave field, the beam-splitter-like coupling between magnons and cavity leads to the large amplitudes of both magnon and cavity modes, namely, |/angbracketleftms/angbracketright|,|/angbracketlefta/angbracketright| ≫ 1. In this case, one can safely introduce the expansion ms= /angbracketleftms/angbracketright+δms,a=/angbracketlefta/angbracketright+δain the vicinity of steady state, by neglecting the higher-order fluctuations of the operator s. We thereby obtain the linearized QLEs for the quadratures δXs,δYs,δX,δYdefined asδX1=(δm1+δm† 1)/√ 2, δY1= (δm1−δm† 1)/i√ 2, δX2=(δm2+δm† 2)/√ 2, δY2=(δm2− δm† 2)/i√ 2,δX=(δa+δa†)/√ 2,δY=(δa−δa†)/i√ 2 ˙σ(t)=Aσ(t)+f(t) (5) whereσ(t)=[δX1(t),δY1(t),δX2(t),δY2(t),δX(t),δY(t)]Tand f(t)=[/radicalbig 2γ1Xin 1(t),/radicalbig 2γ1Yin 1(t),/radicalbig 2γ2Xin 2(t),/radicalbig 2γ2Yin 2(t),/radicalbig 2γcXin(t),/radicalbig 2γ1Yin(t)]Tare the vectors for quantum fluctuations and3 (a) (b)(c) (d)(e) (f) FIG. 2: 2D plots for (top) magnon-magnon entanglement Em1m2and (bottom) magnon-cavity entanglement Em1awhen turning offthe coupling between cavity and the 2nd sphere ( g2=0). (a) g1,2/2π=41MHz,δc/2π=−0.03GHz; (b)g1/2π=41MHz, g2=0,δc/2π=−0.03GHz; (c) F1,2=−0.048GHz, g1,2/2π=41MHz; (d) F1,2= −0.048GHz, g1/2π=41MHz, g2=0 and (e,f) F1,2=−0.048GHz,δc/2π=−0.03GHz. Other param- eters areω1,2/2π=10GHz,δ1,2/2π=−1MHz,γ1,2/2π=8.8MHz,γc/2π=1.9MHz and T=10mK. noise, respectively. The drift matrix reads A=F1−γ1˜δ1−G1 0 0 0 g1 −˜δ1−G1−F1−γ1 0 0−g10 0 0 F2−γ2˜δ2−G20 g2 0 0−˜δ2−G2−F2−γ2−g20 0 g1 0 g2−γcδc −g1 0−g2 0−δc−γc (6) with magnetocrystalline anisotropy quantified by Gs= 2∆sRe/angbracketleftms/angbracketright2,Fs=2∆sIm/angbracketleftms/angbracketright2and the effective detuning of magnons ˜δs=δs+2/radicalbig G2s+F2s=δs+4∆s|/angbracketleftms/angbracketright|2, which in- cludes the frequency shift caused by Kerr nonlinearity. The mean/angbracketleftm1,2/angbracketrightare given by /angbracketleftm1/angbracketright=ig1Ω (˜δ1−iγ1)(δc−iγc)−g2 1−g2 2(˜δ1−iγ1) ˜δ2−iγ2, and (1↔2)(7) Before the study of entanglement, it is essential to elucida te the mechanism for optimizing the entanglement via Kerr non- linearity. To this end, we proceed via the e ffective Hamilto-nian for quantum fluctuations Hqf=/planckover2pi12/summationdisplay s=1/bracketleftig˜δsδm† sδms+˜∆sδm† sδm† s+˜∆∗ sδmsδms +gs/parenleftig δm† sδa+δmsδa†/parenrightig/bracketrightig +/planckover2pi1δcδa†δa(8) where ˜∆s= (Gs+iFs)/2. The quadratic terms δm† sδm† s, δmsδmsimply the effective magnon-magnon inter- action induced by the magnetocrystalline anisotropy, whic h may be significantly enhanced by strong driving. This, in fact, is responsible for the entanglement. To make it elabo- rate, let us introduce the Bogoliubov transformation [45, 4 6] δβs=usδms−v∗ sδm† s, δβ† s=−vsδms+u∗ sδm† swhere us=/radicalig 1 2/parenleftig˜δs εs+1/parenrightig ,vseiα=−/radicalig 1 2/parenleftig˜δs εs−1/parenrightig ,α=arctan( Fs/Gs) and εs=/parenleftig˜δ2 s−4|˜∆s|2/parenrightig1/2. Inserting these into Eq.(8) we find Hqf=/planckover2pi12/summationdisplay s=1/bracketleftig εsδβ† sδβs+gs/parenleftig (vsδβs+usδβ† s)δa +(u∗ sδβs+v∗ sδβ† s)δa†/parenrightig/bracketrightig +/planckover2pi1δcδa†δa(9) which showsεs≃−δcis optimal for the entanglement, due to the magnon-photon squeezing term gs(vsδβsδa+v∗ sδβ† sδa†).4 This will be confirmed by the latter numerical results when taking into account of experimental parameters. Entanglement between magnon modes.– Since we are us- ing the linearized quantum Langevin equations, the Gaussia n nature of the input states will be preserved during the time evolution of systems. The quantum fluctuations are thus the continuous three-mode Gaussian state, which is completely characterized by an 6 ×6 covariance matrix (CM) defined asCi j(t,t′)=1 2/angbracketleftσi(t)σj(t′)+σj(t′)σi(t)/angbracketright; (i,j=1,2,···,6) where the average is taken over the system and bath degrees of freedoms. Suppose the drift matrix Ais negatively defined, the solution to Eq.(5) is σ(t)=M(t)σ(0)+/integraltextt 0M(s)f(t−s)ds where M(t)=exp(At). This enables us to find the equation which CM obeys ˙C(t+τ,t)=AC(t+τ,t)+C(t+τ,t)AT+eAτD (10) forτ≥0. Thus the stationary CM can be straightforwardly obtained by letting τ=0,t→∞ in Eq.(10) that yields to the Lyapunov equation AC∞+C∞AT=−D (11) where the diffusion matrix is D=diag[γ1(2¯n1+1),γ1(2¯n1+ 1),γ2(2¯n2+1),γ2(2¯n2+1),γc,γc] defined through/angbracketleftfi(t)fj(t′)+ fj(t′)fi(t)/angbracketright=2Di jδ(t−t′). We adopt the logarithmic negativity ENto quantify the magnon-magnon and magnon-photon en- tanglements by comupting the 4 ×4 CM related to the two magnon modes. This can be achieved by defining EN= max[0,−ln2v−] where v−=min|eig⊕2 j=1(−σy)P12C∞P12|and σyis the Pauli matrix [47, 48]. The matrix P12=σz⊕1 realizes the partial transposition at the level of CM. In wha t follows, we will work in the monostable scheme of magnons. Furthermore, we will focus on the case of two identical magnons having G1,2=G,F1,2=F,˜δ1,2=˜δ,∆1,2= ∆,g1,2=g. Fig.2 shows the magnon-magnon entanglement versus some key parameters of the system. Here we have taken into account the experimentally feasible parameters [35]: ω1,2/2π=10GHz,δ1,2/2π=−1MHz,γ1,2/2π=8.8MHz andγc/2π=1.9MHz for the YIG bulk at low temperature T=10mK. First of all we observe from Fig.2(a,b) that the Kerr nonlinearity is responsible for creating the stead y- state entanglement between two magnon modes, evident by the fact that the entanglement dies out when G=F=0. This results from the dominated beam-splitter-interactio n be- tween magnon mode and cavity photons, once G=F=0. Thereby no magnon-cavity entanglement can be created, as seen in Fig.2(b). We take the condition εs≃ −δcfor op- timizing the magnon-photon entanglement, as illustrated i n Fig.2(d) whereεs≃/radicalbig 3(G2s+F2s). The two-mode squeez- ing term gs(vsδβsδa+v∗ sδβ† sδa†) squeezes the joint state be- tween one magnon mode and cavity photons, which results in the partial entanglement in between. Because the same type of interaction occurs when coupling the other magnon mode with cavity, the two distanced magnon modes are expected(a) (b) FIG. 3: Entanglement between two magnon modes varies with (a) cavity detuning and (b) driving power. (a,b) Solid blue, dotdashed purple and dashing red lines are for the cavity leakageγc/2π=1.9MHz,20MHz and 70MHz, re- spectively; g1,2/2π=41MHz. (a) Solid blue, dotdashed purple and dashing red lines also correspond to driving power Pd=393mW,38mW and 11mW, respectively; (b) δc/2π=−30MHz. Other parameters are the same as Fig.2. to be entangled. This is confirmed in Fig.2(c) manifesting the optimal magnon-magnon entanglement in the vicinity of εs≃ −δc. The elaborate transfer from magnon-photon en- tanglement to magnon-magnon entanglement is subsequently evident by Fig.1S in supplementary material (SM) that the considerable reduction of magnon-photon entanglement as the coupling of cavity to another sphere is turned on. Since the biparticle entanglement is originated from the Kerr non - linearity quantified by G1,2and F1,2, there must be the in- terplay between the couplings Gs,Fsandgswhich is de- picted in Fig.2(e,f). In Fig.2(c) we take g1,2/2π=41MHz and it impliesδc/2π≃−0.03GHz for the optimal entangle- ment Em1m2. We then adopt the magnitude of δcfor plot- ting Fig.2(a,b). Using√ G2+F2=2∆|/angbracketleftm/angbracketright|2and Eq.(7) for the 40µm-diam YIG spheres, the optimal entanglement with |G|=0.038GHz,|F|=0.028GHz (see Fig.2(a)) yields to the Rabi frequencyΩ= 1.06×1015Hz, corresponding to the drive power Pd=314mW. Indeed, the stronger nonlinearity will create more entanglement between the magnon modes. But we have to ensure the negatively defined matrix Agiven in Eq.(6). Also, the experimental feasibility of ultrastrong drive using microwave field needs the consideration. Since we are working with the strong driving, it is worthy of checking the validity of the results obtained above. The magnon description for magnetic materials is e ffective only when/angbracketleftm† jmj/angbracketright ≪ 2Njs=5Njwhere Nj=ρjVj,mdenotes the total number of spins in the bulk material. For the 40 µm- diam YIG sphere, Nj≃1.41×1014and the drive power Pd= 393mW results in|/angbracketleftmj/angbracketright|≃2.3×106, giving/angbracketleftm† jmj/angbracketright≃5.28× 1012≪5N=7.07×1014. Hence the condition /angbracketleftm† jmj/angbracketright≪ 2Njsis fulfilled. Fig.3 illustrates the entanglement between two magnon modes versus some controllable parameters by considering the 40µm-diam YIG-sphere experiment, where ω1,2/2π= 10GHz,δ1,2/2π=−1MHz,∆1,2/2π=1µHz, g1,2/2π= 41MHz,γ1,2/2π=8.8MHz andγc/2π=1.9MHz have been5 taken according to Ref.[35]. We observe in Fig.3(a) that for fixed driving power, the magnon-magnon entanglement is quite sensitive to cavity detuning δc≡ωc−ωd, reaching its maximum atδc/2π≃−0.03GHz. This is consistent with the conditionεj≃−δcas clarified for optimizing the entan- glement. Fig.3(b) shows the considerable entanglement whe n the system is driven far-from-equilibrium. This is reasona ble because the strong external driving significantly enhances the Kerr nonlinearity that is responsible for both magnon-cavi ty squeezing and entanglement, as elucidated in Eq.(7) and (8) . Furthermore, Fig.3 shows that the weaker magnon-magnon entanglement is observed when increasing the cavity leakag e. By noting the magnitude, we can still obtain some entangle- ment, even with a low-quality cavity showing weak magnon- cavity coupling where γc=8γ1,2>g1,2denoted by red dashed lines. This regime is crucial for detecting the entanglemen t used in Refs[17, 18]. in which an additional cavity has a beam-splitter-like interaction with the magnon mode for re ad- ing out the magnon states associated with the CM. The trans- ferred entanglement can then be measured through the homo- dyne detection by sending a weak microwave probe. This ap- proach requires much larger cavity leakage than the magnon dissipation, namely, γc≫γ1,2, so that the magnon states can remain almost unchanged when switching o ffthe laser driv- ing. The time-resolved detection of the photons emitting o ffthe cavity axis may offer an alternative scheme for entanglement measurement. The quadrature information of magnon modes can be transferred to the time-gated emitted photons, which can be homodynely detected by interfering with an extra mi- crowave field. This quantum-light-probe scheme may take the advantage of being noninvasive detection for the entanglem ent measurement. Conclusion and remarks.– In conclusion, we have proposed a protocol for entangling the magnon modes in two mas- sive YIG spheres, through the Kerr nonlinearity that origi- nates from the magnetocrystalline anisotropy. We shew that such nonlinearity has to be essentially included, for produ c- ing the entanglement. Our work demonstrated the stationary entanglement between two macroscopic YIG spheres driven far-from-equilibrium, within the experimentally feasibl e pa- rameter regime. The amount of entanglement is quantified by the logarithmic negativity and surprisingly robust agai nst the cavity leakage: entangled quantum state may persist wit h low-quality cavity giving weak magnon-cavity coupling. Th is may be helpful to the experimental design for the entangle- ment measurement. We should note that our idea for entangling magnon modes may be potentially extensive to other complex systems, such as molecular aggregates and clusters, along with the fact of similar forms of nonlinear couplings b†bqand∆b†bb†b. With the scaled-up parameters, the long-range entanglemen t in molecular aggregates would be anticipated, in that the exciton-exciton interaction is of several orders of magnit ude higher than the Kerr nonlinearity resulting from the magne- tocrystalline anisotropy. For instance, the two-exciton c ou-pling in J-aggregate and light-harvesting antenna take the value of∼50cm−1which is∼0.3% of the exciton frequency. This is much stronger nonlinearity than that in YIGs with Ker r coefficient K∼0.1nHz that is∼10−11of its Kittel frequency. Recent development in both ultrafast spectroscopy and syn- thesis have revealed the important role of quantum coher- ence which may significantly modify the functions of complex molecules and may help the design of polaritonic molecular devices as well as polariton chemistry. Hence entangling th e molecular aggregates may help the studies of quantum phe- nomena in complex molecules. We gratefully acknowledge the support of AFOSR Award FA-9550-18-1-0141, ONR Award N00014-16-1-3054 and Robert A. Welch Foundation (Award A-1261 & A-1943- 20180324). We also thank Jie Li and Tao Peng for the useful discussions. ∗zhedong.zhang@tamu.edu †girish.agarwal@tamu.edu [1] Y . Kajiwara, et al. , Nature 464, 262-266 (2010) [2] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef and B. J. van Wees, Nat. Phys. 11, 1022-1026 (2015) [3] N. Zhu, et al. , Appl. Phys. Lett. 109, 082402-082406 (2016) [4] T. An, et al. , Nat. Mater. 12, 549-553 (2013) [5] A. V . Chumak, V . I. Vasyuchka, A. A. Serga and B. Hillebran ds, Nat. Phys. 11, 453-461 (2015) [6] A. V . Chumak, A. A. Serga and B. Hillebrands, Nat. Commun. 5, 4700-4707 (2014) [7] M. Collet, et al. , Nat. Commun. 7, 10377-10384 (2016) [8] M. Ho, E. Oudot, J.-D. Bancal and N. Sangouard, Phys. Rev. Lett. 121, 023602-023607 (2018) [9] R. Riedinger, et al. , Nature 556, 473-477 (2018) [10] H. Y . Yuan and M.-H. Yung, Phys. Rev. B 97, 060405-060410 (2018) [11] T. Morimae, A. Sugita and A. Shimizu, Phys. Rev. A 71, 032317–32328 (2005) [12] C. F. Ockeloen-Korppi, et al. , Nature 556, 478-482 (2018) [13] S. Gr¨ oblacher, K. Hammerer, M. Vanner and M. Aspelmeye r, Nature 460, 724-727 (2009) [14] M. Aspelmeyer, T. Kippenberg and F. Marquardt, Rev. Mod . Phys. 86, 1391-1452 (2014) [15] C. Genes, D. Vitali, P. Tombesi, S. Gigan and M. Aspelmey er, Phys. Rev. A 77, 033804-033812 (2008) [16] E. Verhagen, S. Deleglise, S. Weis, A. Schliesser and T. J. Kip- penberg, Nature 482, 63-67 (2012) [17] T. A. Palomaki, J. D. Teufel, R. W. Simmonds and K. W. Lehn - ert, Science 342, 710-713 (2013) [18] D. Vitali, et al. , Phys. Rev. Lett. 98, 030405-030408 (2007) [19] B. Julsgaad, A. Kozhekin and E. S. Polzik, Nature 413, 400-403 (2001) [20] A. J. Berkley, et al. , Science 300, 1548-1550 (2003) [21] D. Lachance-Quirion et al. , Sci. Adv. 3, e1603150 (2017) [22] X. Zhang, C.-L. Zou, L. Jiang and H. X. Tang, Sci. Adv. 2, e1501286 (2016) [23] D. Zhang, et al. , NPJ Quantum Inf. 1, 15014-15019 (2015) [24] Y . Tabuchi, et al. , Phys. Rev. Lett. 113, 083603-083607 (2014) [25] ¨O. O. Soykal and M. E. Flatte, Phys. Rev. Lett. 104, 077202- 077205 (2010)6 [26] X. Zhang, C.-L. Zou, L. Jiang and H. X. Tang, Phys. Rev. Le tt. 113, 156401-156405 (2014) [27] J. Bourhill, N. Kostylev, M. Goryachev, D. L. Creedon an d M. E. Tobar, Phys. Rev. B 93, 144420-144427 (2016) [28] Y . Tabuchi, et al. , Science 349, 405-408 (2015) [29] M. Harder, et al. , Phys. Rev. Lett. 121, 137203-137207 (2018) [30] B. Yao, et al. , Nat. Commun. 8, 1437-1442 (2017) [31] X. Zhang, et al. , Nat. Commun. 6, 8914-8920 (2015) [32] J. Li, S.-Y . Zhu and G. S. Agarwal, Phys. Rev. Lett. 121, 203601-203606 (2018) [33] J. Li, S.-Y . Zhu and G. S. Agarwal, Phys. Rev. A 99, 021801- 021806 (2019) [34] J. Li and S.-Y . Zhu, arXiv:1903.00221v1 [quant-ph] [35] Y .-P. Wang, et al. , Phys. Rev. Lett. 120, 057202-057207 (2018) [36] L. Bai, et al. ,118, 217201-217205 (2017) [37] M. Sarovar, A. Ishizaki, G. R. Fleming and K. Birgitta Wh aley, Nat. Phys. 6, 462-467 (2010) [38] Z. D. Zhang and J. Wang, Sci. Rep. 6, 37629-37637 (2016) [39] D. M. Coles, et al. , Nat. Mater. 13, 712-719 (2014)[40] Z. D. Zhang, P. Saurabh, K. E. Dorfman, A. Debnath and S. Mukamel, J. Chem. Phys. 148, 074302-074314 (2018) [41] M. Kowalewski, K. Bennett and S. Mumakel, J. Phys. Chem. Lett. 7, 2050-2054 (2016) [42] C. Kittel, Phys. Rev. 73, 155-161 (1948) [43] S. Blundell, Magnetism in Condensed Matter (Oxford Univer- sity Press, Oxford, 2001) [44] O. Madelung and B. C. Taylor, Introduction to Solid-State The- ory(Springer, Berlin, 1978) [45] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics: The- ory of the Condensate State (part 2) , revised ed. (Butterworth- Heinemann, Oxford, 1980) [46] A. L. Fetter and J. D. Walecka, Quantum Theory of Many- Particle Systems (Dover Publications, Mineola, 2003) [47] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314-032325 (2002) [48] R. Simon, Phys. Rev. Lett. 84, 2726-2729 (2000)

2019-04-08

We propose a scheme to entangle two magnon modes via Kerr nonlinear effect when driving the systems far-from-equilibrium. We consider two macroscopic yttrium iron garnets (YIGs) interacting with a single-mode microcavity through the magnetic dipole coupling. The Kittel mode describing the collective excitations of large number of spins are excited through driving cavity with a strong microwave field. We demonstrate how the Kerr nonlineraity creates the entangled quantum states between the two macroscopic ferromagnetic samples, when the microcavity is strongly driven by a blue-detuned microwave field. Such quantum entanglement survives at the steady state. Our work offers new insights and guidance to designate the experiments for observing the entanglement in massive ferromagnetic materials. It can also find broad applications in macroscopic quantum effects and magnetic spintronics.

Quantum entanglement between two magnon modes via Kerr nonlinearity

1904.04167v1

Bistability in dissipatively coupled cavity magnonics H. Pan,1, 2,Y. Yang,2Z.H. An,1, 3, 4,yand C.-M. Hu2,z 1State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, People's Republic of China 2Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 3Institute of Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai 200433, People's Republic of China 4Shanghai Qi Zhi Institute, 41th Floor, AI Tower, No. 701 Yunjin Road, Xuhui District, Shanghai, 200232, People's Republic of China (Dated: June 6, 2022) Dissipative coupling of resonators arising from their cooperative dampings to a common reservoir induces intriguingly new physics such as energy level attraction. In this study, we report the nonlin- ear properties in a dissipatively coupled cavity magnonic system. A magnetic material YIG (yttrium iron garnet) is placed at the magnetic eld node of a Fabry-Perot-like microwave cavity such that the magnons and cavity photons are dissipatively coupled. Under high power excitation, a nonlinear eect is observed in the transmission spectra, showing bistable behaviors. The observed bistabili- ties are manifested as clockwise, counterclockwise, and butter y-like hysteresis loops with dierent frequency detuning. The experimental results are well explained as a Dung oscillator dissipatively coupled with a harmonic one and the required trigger condition for bistability could be determined quantitatively by the coupled oscillator model. Our results demonstrate that the magnon damping has been suppressed by the dissipative interaction, which thereby reduces the threshold for conven- tional magnon Kerr bistability. This work sheds light upon potential applications in developing low power nonlinearity devices, enhanced anharmonicity sensors and for exploring the non-Hermitian physics of cavity magnonics in the nonlinear regime. I. INTRODUCTION Nonlinearities are ubiquitous phenomena in various physics elds. For instance, Kerr nonlinearity and res- onant two-level nonlinearity [1], leading to anharmonici- ties, have been widely investigated in the context of op- tics [2]. One signature of nonlinear dynamics is bista- bility for a given input exceeding a threshold power and manifests itself as a foldover eect [3{7]. The eects of nonlinearity have technological implications in sophisti- cated optical devices for controlling light with light [8], novel magnetic data storage devices [9], switches with bistable metamaterial [10] as well as developed mechani- cal devices for emergent applications like energy harvest- ing [11, 12]. Hybrid quantum systems, which have aroused tremen- dous interest for application in quantum information pro- cessing [13, 14], have the potential to push the develop- ment of the realm of nonlinearity. Among various hybrid subsystems, cavity magnonics appears to be an excep- tional candidate (see, e.g., [15{30]), which utilizes fer- rimagnetic materials like YIG (yttrium iron garnet) to create collective spin excitations. This cavity magnon- ics system has resulted in a variety of semiclassical and quantum phenomena including, cavity-magnon polari- tons [31{33], magnon bistability [29, 34, 35], bidirectional microwave-optical conversion mediated by ferromagnetic hpan19@fudan.edu.cn yanzhenghua@fudan.edu.cn zhu@physics.umanitoba.camagnons [36], magnon dark mode [19], synchronization via spin-photon coupling [37], and non-Hermitian exotic properties [25, 38, 39]. Recent work has attempted to utilize the photon-magnon coupling freedom to tune the nonlinearity. The magnon-polariton bistability has been successfully observed in a coherent hybrid system [29] and the bistable behavior could also be directly measured with the coupled cavity being pumped [35]. However, in these reported works the photon-magnon are typically co- herently coupled, and their coherent coupling enhances the damping of magnon states [40]. Due to the cubic dependence of the threshold for nonlinearity on the ef- fective damping of magnon [35], the coherent coupling of cavity-magnon hybrid system therefore increases the nonlinearity threshold which may impede the device ap- plication of nonlinearity. Coherent coupling originates from the direct spatial overlap between photon and magnon modes and forms hybrid states with repulsed energy levels and attracted damping rates. In contrast, a dissipative form of magnon- photon interaction [41] requires no direct mode over- lap and has been demonstrated to cause level attraction and a damping repulsion eect[42{44]. This dissipative coupling is indirect as it is mediated through a shared reservoir, resulting in an imaginary spin-photon coupling strength. Dissipatively coupled systems have important applications such as nonreciprocal transport [45], en- hanced sensing [46], and non-Hermitian singularities [47, 48]. So far, however, reported work with dissipative cou- pling have been focused on the linear regime and the nonlinearity with dissipative coupling have not been ex- perimentally examined, except a theoretical prediction ofarXiv:2206.01231v1 [cond-mat.mes-hall] 2 Jun 20222 a lower threshold if the (imaginary) coupling strength is suciently large in an anti-PT regime [49]. Inspired by this, we study a dissipatively coupled cavity-magnon system where a YIG sphere is imbed- ded in a Fabry-Perot-like cavity. The coupled system is directly pumped through the cavity and the pumping power is suciently high to create the nonlinear eect. We experimentally demonstrate that, in the dissipatively coupled nonlinear system, the threshold power for bista- bility is lower than the corresponding bound in coherent scenario. This is due to the suppression of magnon damp- ing on-resonance induced by dissipative coupling. In ad- dition, such a dissipatively coupled hybridized system results in dierent magnetic eld and power dependent bistable behaviors as the magnon frequency is detuned for on and o cavity resonant frequency. By introducing the model based on a Dung oscillator dissipatively cou- pled with a harmonic oscillator, the experimental results can be well explained. This method allows us to de- termine the required experimental condition for trigger- ing bistable behavior in a dissipatively coupled system. Our work shows that dissipative coupling in a hybrid sys- tem with a nonlinear eect may be utilized to engineer a lower power threshold for nonlinearity and enhanced anharmonicity sensors. II. THEORETICAL MODEL We begin with a model considering the nonlinear Kerr eect of the magnon excitation in a YIG sphere which dissipatively interacts with the cavity photons. Here the cavity is directly pumped. This dissipatively cou- pled nonlinear system is characterized by a Hamiltonian (the details shown in Appendix A and with reduced units ~1): Htot=!ca+a+!mb+b+Kb+bb+b+i a+b+ab+
+ d a+ei!t+aei!t
; (1) wherea+(a) andb+(b) correspond to the creation (an- nihilation) operators of the cavity photons at frequency !cand of the Kittel-mode magnons at !m, respectively. Here, the Kerr eect of magnons term Kb+bb+borig- inates from magnetocrystalline anisotropy in the YIG material [29], in which Kis the Kerr coecient and is positive in our experiment specically (see the Appendix A).is the dissipative coupling strength between the cavity photon and the magnon via their common reser- voir. The last term in the above equation describes that the cavity is pumped by the oscillating microwave eld, with an amplitude d[50]. By adopting the Heisenberg-Langevin approach [51], the dynamics of the coupled cavity photon-magnon hy-brid nonlinear system can be described by the equations: da dt=i(!ci)a+ bi d; (2) db dt=i(!mi )bi(Kbb+b+Kb+bb) + a: (3) Under the mean-eld approximation [46], the higher- order expectations can be decoupled, so that the non- linear termhb+bbiandhbb+bican be simplied as jbj2b, then the dynamics of the dissipatively coupled magnon- photon system follows as, da dt=i(!ci)a+ bi d; (4) db dt=i !m+ 2Kjbj2i
b+ a; (5) where=i+eis the total damping of the cavity mode, withi(e) being the intrinsic (external) damping of the cavity mode. is the damping rate of the Kittel-mode. Suppose the photon and magnon mode have time depen- dence ofei!t, then Eqs. (4) and (5) can be simplied as (!!c+i)aib= d; (6) !!m2Kjbj2+i  b=ia: (7) Eq. (6) can be expressed as a= (ib+ d)=(!!c+i). By substituting such an expression into Eq. (7), we get jbj m0i !m02Kjbj2
 = d !c+i: (8) Here,!m0=!m+!cwith!m=!!m,!c= !!cdenotes eective frequency shift, and m0= ; (9) denotes the eective damping of the magnon for the dis- sipative coupling system, where the negative sign shows the suppression eect of the magnon damping. We note that for the coherent coupling, the sign is positive rep- resenting the enhancement eect of the magnon damp- ing [29, 35]. The coecient = 2=(!2 c+2) stands for the transfer eciency of the excitation power Pfrom the input port into the magnon system, and is dependent on the dissipative coupling strength , the frequency de- tuning!c, and the total damping of the cavity. Taking the squared modulus of Eq. (8) and dening
m2Kjbj2, which is the shift of the magnon fre- quency [34], we have
mh 0 m2+ (!0 m
m)2i = 2K 2 d: (10) Equation (10) has a similar form to the uncoupled Du- ing oscillator [3], except that 0 mand!0 mare the eective damping and frequency shift of the magnon, respectively, which result from dissipative coupling with the cavity photon. The right term of Eq. (10) denotes the eective drive eld of the magnon via dissipative interaction with the cavity photon as the eld directly pumps the cavity rather than the YIG sphere.3 A. Bistability and the transition point Equation (10) describes an oscillator with cubic nonlin- earity, called the Dung equation. As for this equation, there are three real roots for a nite range of frequen- cies when dexceeds a specic value called the critical eld t. Among the three roots, one is unstable and the other two are stable, called bistability, which is a signa- ture of the anharmonic oscillator. We will focus on the two boundaries (hereafter termed as up border and down border) inbetween which bistability occurs. As there are abrupt transitions between two stable states at up and down borders, the transition points of bistability are de- termined by the condition d d=d
m= 0, i.e., 3
2 m4!0 m
m+!0 m2+ 0 m2= 0: (11) According to the root discriminant of the quadratic equa- tion, when 4 !0 m212 0 m2= 0 , i.e., !0 m=p 3 0 m; (12) there is only one root of Eq. (11), which corresponds to the critical condition of bistable behavior. When 4!0 m212 0 m2>0, there are two real roots which correspond to the up border and down border of the bistability. By adopting the approximationp 3 0 m !0 mto solve Eq. (11), then the upper border satises !m= 33r KS2P 2!c;(up) (13) and the down border satises !m=2KS2P ( )2!c;(down) (14) where d=Sp P, withSdescribing the conversion ef- ciency from input power Pto the eld ddriving the cavity resonance mode. The magnitude of Sdepends on the frequency, phenomenologically introduced exter- nal loss of the cavity eld, and the loss in the cable which connects the device. Based on Eqs. (13) and (14), we can come to conclusions that both the upper border and down border have input power dependence, with the up- per border satisfying !m/P1=3compared with down border satisfying !m/P. These dependencies are dierent from those of coherently coupled anharmonic oscillators [35], where the upper border has !m/P dependence and the down border !m/P1=3depen- dence. The dierence arises from that the magnon shifts to higher frequencies with negative Kerr coecient K while that of our work shifts to lower frequencies with positive Kerr coecient. In fact, the two cases can be unied where the border with a large frequency shift has aPdependence and the border with a small frequency shift has aP1=3dependence. The above discussion is based on tuning the magnetic eld to obtain bistability, called eld bistability, and itstransition point has power dependence. On the other hand, we can adjust input power to obtain bistability, called power bistability, and its transition point has mag- netic eld dependence. According to Eqs. (13) and (14), we can get borders of power bistability by regarding mag- netic eldHas an independent variable P=2(!m+!c)3 27KS2;(up) (15) P=(!m+!c)( 2) 2KS2:(down) (16) As the magnon resonance detuning !m= 0Hwith denitionH=HHr(Hris the resonant magnetic eld and 0is gyromagnetic ratio), the up and down bor- der of power bistability have respective jHj3andjHj dependence when up-sweeping and down-sweeping power as shown in Eqs. (15) and (16), respectively. B. Critical condition of bistability When Eq. (11) has only one real solution, the bista- bility vanishes because the two transition points collapse to one point. The critical driving eld of the cavity cor- responds to the threshold beyond which the bistability appears. Thus, by substituting the critical condition of bistability shown in Eq. (12) and the only solution
m= 2=3!minto Eq. (10), the critical eld t(critical powerPt) of eld bistability is obtained: 2 t=4p 3 9K 03 m ; (17a) Pt=4p 3 9KS2 03 m : (17b) These equations imply that the threshold has cubic de- pendence on the eective damping of magnon 0 m, which is similar to that of coherently coupled nonlinear sys- tem [35]. The threshold of nonlinearity for the hybrid sys- tem have similar dependence with that of uncoupled ferri- magnetic resonance which has cubic dependence upon the damping of magnon (see Table I). Next, we compare the threshold value for the dissipatively coupled system with that of the coherently coupled system. For this purpose, the threshold for a generic two-mode system involving both coherent and dissipative coupling is generated by substituting 0 m= +with= (g+i)2=(!2 c+2) into Eqs. (17a) and (17b) where gis the coherent cou- pling strength. Supposing scenarios = 0 and g= 0, we will get the threshold for coherently coupled and dissipa- tively coupled systems, respectively. By comparing the thresholds in relation to the nature of coupling (setting !c= 0), we have ( td)2 ( tc)2=Ptd Ptc=g2 212= 1 +g2= 3 ; (18)4 where we have assumed that the two kinds of coupling systems have the same damping coecient and td(Pd t), tc(Pc t) denote the critical eld (power) of bistability for dissipative and coherent coupling respectively. TABLE I. Thresholds of the ferrimagnetic materials in the coupled and uncoupled scenarios. sample threshold eective dampinga YIG sphere [52] 2 0h2 t=8p 3 9 3/ 3b = 0
Hc Py lm [6] 2 0h2 t=16p 3 9 0Ms 3/ 3 = 0
H YIG (Coh.)d[35] 2 t=4p 3 9K 03 m/ 03 m 0 m= + YIG (Dis.)e 2 t=4p 3 9K 03 m/ 03 m 0 m=  aEective damping of magnon of each system. bHere,htis the critical drive magnetic eld, and is related to crystalline anisotropy [52]. c
His the linewidth of the ferromagnetic resonance. dYIG sphere is placed in a microwave cavity, and they interact coherently. eYIG sphere is placed in a microwave cavity, and they interact dissipatively. Notably, the expression of Eq. (18) is always less than 1 for nonzero and g, revealing a consistently lower threshold in the dissipatively coupled system than that in the coherently coupled system. In order to make a fair comparison, we assume identical magnitudes of coupling strength, i.e., =2=g=2= 21 MHz. Then, by substi- tuting the value of =2= 5:1 MHz and =2= 126 MHz into Eq. (18), the dissipative-coherent threshold ratio be- comes a nite value of 0 :0064, implying that the dissipa- tive coupling may lead to very low power threshold of the bistability. Such a low threshold arises intrinsically from the suppression of the magnon damping on-resonance [see Eq. (9)] and the cubic dependence of the threshold upon the eective magnon damping [indicated by Eqs. (17a) and (17b)]. On the other hand, we can obtain the requirement to observe the power bistability. The only one root of Eq. (11) described by Eq. (12) corresponds to the criti- cal magnetic eld to generate power bistability. Combin- ing the relation !m= 0Hand Eq. (12), the critical magnetic eld is given by H=1 0[!cp 3( )]: (19) Compared with the critical magnetic eld condition H Hr=p 3 = 0of uncoupled magnetic systems [6, 53], the extra term of the magnetic eld shift p 3= 0in Eq. (19) results from the eective damping of the magnon resonance near !=!c[18]. The factor !c= 0rep- resents the additional resonance shift of the magnon which arises from the interaction between magnon and cavity.C. Transmission spectra with bistability The bistability can be detected experimentally via mi- crowave transmission spectra of the cavity. In this sec- tion, we show the magnon frequency shift
m(due to the Kerr nonlinearity) is observed in the transmission spec- tra of the cavity. By considering that the cavity mode couples with the input energy from the port, the dynamic equation Eq. (4) can be rewritten as da dt=i(!ci)a+ b+pecin; (20) wherecinis the input eld. From Eq. (7), the amplitude of the cavity eld can be derived b=ia !!m
m+i : (21) According to the input-output theory [51], the relation of input-output eld can be described as coutcin=pea; (22) wherecoutis the output eld. Combining Eqs. (20), (21), (22), with the denition of transmission coecient S21= cout=cin, we can obtain the transmission coecient S21= 1 +e i(!!c)2=[i(!!m
m) ]: (23) The transmission of nonlinear dissipatively coupled sys- tem in Eq. (23) can be reduced to that of two cou- pled linear oscillators if we perform the transformation !m+
m!!m. Here,
mis the nonlinear magnon res- onance frequency shift which is the solution of the Du- ing equation as shown in Eq. (10). The nonlinear eect can be observed through cavity transmission due to the interaction between the cavity photon and magnon. III. EXPERIMENTAL RESULTS AND DISCUSSION A. The hybridized cavity-magnon mode in the linear range The experimental setup is sketched in Fig. 1(a). A pol- ished YIG sphere with 1 mm diameter is placed at the magnetic eld node of a Fabry-Perot-like cavity, which is an assembled apparatus with a circular waveguide con- necting to coaxial-rectangular adapters [32, 35], such that the excited magnon and cavity photon are dissipatively coupled. The cavity is pumped by a microwave gener- ator, and the transmission is detected by a signal ana- lyzer. The embedded YIG sphere is placed in a static magnetic eld Hproduced by tunable electromagnets at room temperature, which is not depicted in Fig. 1(a). We rst conduct the experiment under linear condi- tions, using a vector network analyzer to measure the5 transmission of the cavity with input power below the threshold to create the nonlinear eect, so that the Kerr nonlinear eect is negligible. Our cavity resonance is at!c=2= 12:8888 GHz. The resonance frequency of Kittel-mode in our experiments follows the dispersion !m= 0(Hr+HA), where 0=2= 29:860GHz/T is the gyromagnetic ratio, 0HA=6:1 mT is the anisotropy eld, andHris the biased static magnetic eld at reso- nance. When the frequency of the Kittel-mode is tuned in resonance with the cavity microwave photons, the stan- dard level attraction of the hybridized modes, which is the signature of dissipative coupling, was measured and is shown in Fig. 1(b). On the left side of this level attrac- tion, an additional mode split caused by the high order spin wave is not of immediate interest for the discussion of dissipatively coupled nonlinear bistable eect. The dis- persion of the hybridized cavity mode and magnon mode is shown in Fig. 1(c), where points A and E correspond to far o-resonance condition with j!!cj2, and point C indicates on-resonance condition with j!!cj= 0, and points B and D indicate an intermediate frequency condi- tion withj!!cj2. The dissipative coupling strength can be determined by the separated gap at !m=!c in Fig. 1(d), i.e., =2= 21 MHz. The intrinsic and extrinsic linewidth of cavity-mode i=2= 2:58 MHz, e=2= 126 MHz are obtained by tting the transmis- sion coecient spectra when j!c!mj2 where the coupling eect is negligible. As seen from Fig. 1(d), the tting agrees well with the experimental results. The intrinsic and extrinsic dampings of the magnon are 1 :6 MHz and 3 :5 MHz respectively. Hence, the total damp- ing of magnon is =2= 5:1 MHz. B. Field foldover hysteresis loop In this section, nonlinear eects in our coupled cavity- magnon system for on- and far o-resonance frequencies are measured by sweeping the magnetic eld. Here high microwave powers provided by a microwave generator are used to drive the large angle precession of the magnon, while the transmission signal is measured by a signal an- alyzer, as depicted in Fig. 1(a). We start our measurements in the linear range by setting the output power of the microwave generator to be 0:1 mW. The transmission was measured at an on-resonance frequency !=2= 12:8888 GHz indicated by point C in Fig. 2(c), and o-resonance frequen- cies!=2= 12:6000 GHz, 13 :5000 GHz indicated by points A and E in Fig. 2(c), when we perform up- sweeping and down-sweeping magnetic eld, as shown in Figs. 2(a), 2(c) and 2(e), respectively. At conditions !=2= 12:6000 GHz and !=2= 13:5000 GHz, where the magnon mode is dominant, the spectra show a min- imum transmission at the resonance condition H=Hr because of strong absorption due to the magnon exci- tation as shown in Figs. 2(a) and 2(e). In contrast, the spectrum shows a maximum transmission at condi- FIG. 1. (a) Illustration of the experimental setup, where a YIG sphere is imbedded in an assembly cavity. Here, the YIG is at the node of microwave magnetic eld. The cavity is pumped by the microwave generator and the transmission is measured by signal analyzer. (b) Transmission coecient mapping of the hybridized cavity-magnon system, with level attraction implying dissipative coupling. White dashed line indicates the cut at the coupling point with !m=!c. (c) Dispersion relation of hybridized cavity and magnon mode, where points A and E indicate o-resonance with j!!cj 2, and point C indicates on-resonance with j!!cj= 0, and points B and D indicate intermediate frequencies with j! !cj2. (d) The xed eld cut of transmission coecient mapping at the coupling condition !m=!cindicated by white dashed line in (b), with symbols for the experimental data and solid line for calculation based on Eq. (23). tion!=!cas shown in Fig. 2(c). This peak origi- nates from two factors: (1) the cavity strongly absorbs microwaves at resonance near !=!cwith the small transmission producing the at background; (2) when the cavity mode dissipatively couples to a YIG magnon mode at!m=!c, the hybrid system will thus result in a maximum transmission signal at H=Hr. The peak of transmission spectra indicates half-photon and half- magnon mode. In order to study the nonlinear eect, we increase power above the threshold. By up- and down-sweeping the magnetic eld, we observe that two abrupt jumps of transmission spectra occur at dierent static magnetic eld biases H, corresponding to abrupt transitions be- tween these two stable states. In the range of the tran- sition, a hysteresis loop is clearly seen in the up- and down-sweeping traces of transmission spectra shown in Figs. 2(b), 2(d) and 2(f). This behavior can be explained by Eq. (10), which predicts the behavior of bistability and transitions between the two stable states when the power is above the threshold. The hysteresis loop becomes more evident with the increasing power, because the transition points will depart from each other as microwave power6 increases, as shown in Fig. 4. The bistability of o- and on-resonance have distinctly dierent behaviors in our magnon-cavity system. When the system is o-resonance, i.e., j!!cj2, the eld hysteresis loops (Figs. 2(b) and 2(f)) are clockwise when considering the up- and down-sweeping direction of the static magnetic eld. In contrast, when the system is on-resonance, i.e., at !=!c, the eld hysteresis loop in Fig. 2(d) is counterclockwise. This behavior is quite dif- ferent from the bistability of coherently coupled magnon- photon system as measured in Ref. [35]. The work in Ref. [35] demonstrated when K < 0 there is clockwise hysteresis for on-resonance (with peak background) and anti-clockwise hysteresis for o-resonance (with dip back- ground). The direction of hysteresis depends on the sign of K and the background shape of the resonance. For any nonlinear mode with a Lorentzian dip or peak resonance that is excited with high power, the trace of the trans- mission spectrum will jump at the last transition point FIG. 2.jS21j2versus H with (a) P= 0:1 mW and (b) P= 200 mW at o-resonant frequency !=2= 12:6000 GHz. jS21j2versusHwith (c)P= 0:1 mW and (d) P= 200 mW at on-resonant frequency !=2= 12:8888 GHz.jS21j2versus Hwith (e)P= 0:1 mW and (f) P= 200 mW at o-resonant frequency!=2= 13:5000 GHz. Blue (orange) circle symbols are experimental data by up-sweeping (down-sweeping) static magnetic eld H. The solid curves are calculated. Dashed green lines with arrows indicate the transition process of bista- bility and dashed green lines without arrows indicate the un- stable state. FIG. 3.jS21j2versus H with (a) P= 0:1 mW and (b) P= 200 mW at intermediate frequency !=2= 12:7688 GHz. jS21j2versusHwith (c)P= 0:1 mW and (d) P= 200 mW at intermediate frequency !=2= 12:9288 GHz. Blue (orange) circle symbols are experimental data by up-sweeping (down- sweeping) the static magnetic eld H. The solid curves are calculated. Dashed green lines with arrow indicate the tran- sition process of bistability and dashed green lines without arrow indicate the unstable state. along the sweeping direction because of the hysteresis phenomena. For instance, suppose K > 0, the trace of the transmission spectrum with peak background will jump at the right transition point from a low to high am- plitude by up sweeping the magnetic eld, while it will jump at the left transition point from a high to low am- plitude by down sweeping the magnetic eld. Thus, the hysteresis for K > 0 and peak background lineshape is counterclockwise. By the same approach, the direction of hysteresis can be summarized in Table II, through which the dierence in the direction of hysteresis among our measurement and the work of [29, 35] is well explained. TABLE II. Direction of hysteresis. peak dip K > 0 counterclockwise clockwise K < 0 clockwise counterclockwise Whenj!!cj2 indicated by points B and D shown in Fig. 2(c), a dierent foldover hysteresis loop is seen at intermediate frequencies above and below !c. Gener- ally the lineshape of transmission spectrum is symmetric when the power is below the nonlinear threshold, for in- stance, a typical Lorentzian peak characteristic at !=!c and a Lorentzian dip j!!cj>>2. However, the line- shape of transmission spectrum is asymmetric when the frequency is tuned to the region among j!!cj2, shown in Figs. 3(a) and 3(c) with input power 0 :1 mW. This asymmetric lineshape, similar to that in the coher-7 ent scenario [35], is due to Fano-like resonance [54]. How- ever, their polarities are opposite because of dierent coupling mechanism. As microwave power is increased to 200 mW, in contrast to the general hysteresis loop on- and far o-resonance, a butter y-like hysteresis loop appears and the polarities of the butter y-like hysteresis loop are opposite when the microwave frequency is set at intermediate frequencies above and below !cas shown in Figs. 3(b) and 3(d). This dierence of shape results from the transition direction of bistability. For on- and o-resonance frequency, the direction of two transitions are opposite, where one is from the low to high transmis- sion, and another is from the high to low transmission. While, for the intermediate frequencies, the direction of two transitions are the identical, i.e., both from high to low transmission or from low to high transmission. The eective dampings of the magnon are 0 m=2= 4:50;0:20;0:10;0:15;5:10 MHz by tting the transmission spectra when P= 0:1 mW for dier- ent frequencies A-E, respectively. The tted eec- tive damping of the magnon reveals that the damp- ing of the magnon is suppressed on-resonance due to the dissipative interaction between the cavity and the magnon, which is consistent with the result of Ref. [40]. Then, with the damping parameters of the cavity and magnon extracted from the linear process and the so- lution of Eq. (10), we obtain the tted parameter KS2= 4:5109;6:2108;8:7108;9:9 108;7:0109GHz3=mW at cavity frequency !=2= 12:6000;12:7688;12:8888;12:9688;13:5000 GHz, re- spectively. (Here, KandScan not be determined in- dividually.) In addition, the tted transmission spec- trum as a function of magnetic eld can reproduce the eld-sweeping bistability as shown with green line in Figs. 2 and 3. This agreement veries the validity of our generalized model which describes dissipatively coupled Dung oscillator and linear oscillator in this quasi-one- dimensional cavity [32]. C. Power dependence of transition and threshold for eld foldover hysteresis loop We have observed that the resonance gradually shifts toward lower Hwhen increasing the microwave power be- cause the Kerr coecient Kis positive (see Appendix A). While this is dierent from the negative Kerr coecient system where the resonance gradually shifts toward high H[35]. When the power is above the threshold, the bista- bility appears, and its two transitions will shift depending on the power. Figures. 4(a)-4(e) show the jump positions as a function of the microwave power at !=2= 12:6000, 12:7688, 12:8888, 12:9288, and 13 :5000 GHz, respectively. As predicted by Eqs. (13) and (14), the up-sweeping tran- sition (purple symbols) follows a P1=3dependence (solid line) and the down-sweeping transition (blue symbols) follows a linear Pdependence (solid line) with tted pa- rametersKS2= 4:3109, 5:8108, 8:9108, 8:7 FIG. 4. The jump position of eld foldover hysteresis versus P at cavity frequencies (a) !=2= 12:6000 GHz, (b) !=2= 12:7688 GHz, (c) !=2= 12:8888 GHz, (d) !=2= 12:9288 GHz, (e) !=2= 13:5 GHz. Purple (blue) circle sym- bols are experimental results of forward (backward) H eld sweeping. The yellow and cyan solid curves are tted using Eqs. (13) and (14) for forward and backward sweeping, respec- tively. (f) The threshold for coherent and dissipative coupling system when o- and on-resonance. The threshold for co- herent coupling is from Ref. [35]. For simplicity, we consider each threshold relative to that of o-resonance in each system. Here we have utilized the relation 2 t= 2 t(off)=Pt=Pt(off), where t(off)andPt(off)are the critical driving eld and power o-resonance, respectively. 108, 6:5109GHz3=mW, respectively. This KS2 magnitudes determined by tting the transition points with the power dependence in Eq. (14) is comparable to that tted by transmission spectrum with Eq. (23). The agreement of KS2magnitudes tted by two dier- ent methods is gratifying in view of the error. The power dependencies are dierent from those of a coherently coupled nonlinear system, where the down- sweeping jump follows a P1=3dependence and the up- sweeping jump follows a linear Pdependence [35]. Because the positive Kerr coecient corresponds with !m<0 and negative Kerr coecient corresponds with !m>0, these opposing frequency shifts will lead to the reversed power dependence of two transition points. Figures 4(a)-4(e) record the up-sweeping and down- sweeping transition point when increasing power for on- and o-resonance frequencies A-E. We can extract the8 FIG. 5. The power foldover hysteresis with xed static magnetic eld at cavity frequencies (a) !=2= 12:6000 GHz, (b) !=2= 12:7688 GHz, (c) !=2= 12:8888 GHz, (d) !=2= 12:9288 GHz, (e) !=2= 13:5000 GHz. The recorded power foldover hysteresis transition points versus magnetic eld detuning HHrat cavity frequencies (f) !=2= 12:6000 GHz, (g) !=2= 12:7688 GHz, (h) !=2= 12:8888 GHz, (i) !=2= 12:9288 GHz, (j) !=2= 13:5 GHz. threshold for each frequency and plot them in Fig. 4(f) for the dissipatively [marked by orange arrows in Figs. 4(a)- 4(e)] and coherently (data from Ref. [35]) coupled sys- tems with coupling strength =2= 21 MHz, g=2= 18 MHz, respectively. The results in Fig. 4(f) reveal that the threshold of bistability on-resonance is about 2/3 of those o-resonance in a dissipatively coupled system. This bears stark disparities with a coherently coupled system, where the threshold of the cavity on-resonance is 2.5-fold of that of o-resonance in Ref. [35]. The results imply that the dissipative coupling indeed re- duces the threshold despite the fact that our coupled system deviates from anti-PT conditions. In fact, the observed improvement of lower threshold originates from the threshold's cubic dependence on eective magnon damping which remains valid in the dissipative coupling case (see Table I) and the suppressed magnon damping in our dissipative coupling condition [(see Eq. (9)]. D. Power foldover hysteresis loop and critical magnetic eld for power bistability The hysteresis loops can also be observed by up- sweeping and down-sweeping power at xed !and Has shown in the upper panels (a)-(e) of Fig. 5, where we set the biasing magnetic eld Hto be 0:80,1:04,0:54,0:64,0:76 mT at cavity fre- quency!=2= 12:6000, 2:7688, 12:8888, 12:9288, 13:5000 GHz, respectively. In contrast to the case of K < 0 with opposite direction for power and eld hys- teresis [35], here for K > 0, they have identical direction at each frequency A-E as shown in Figs. 2(b), 2(d), 2(f), 3(b), 3(d), and 5(a)-5(e). The bottom panels (f)-(j) of Fig. 5 show that the tran- sition points of power bistability have jHj3andjHj dependence for up-sweeping and down-sweeping power respectively, which agree with the theoretical predictionof Eqs. (15) and (16). As seen in Figs. 5(f)-5(j), the two transition points depart from each other, and the area of power hysteresis loops becomes larger when the bias magnetic eld is tuned to be away from Hr, and vice versa. At a critical magnetic eld Hc, the power hystere- sis loops disappear. This phenomenon can be explained by Eq. (19), which implies the requirement of produc- ing power hysteresis loops is that the biasing magnetic eld should be below the critical magnetic eld at each frequency A-E. IV. CONCLUSIONS To summarize, we have observed both the eld and power bistability of the dissipatively coupled cavity- magnon system. A theoretical model is studied in which a Dung and linear oscillator are dissipatively coupled to explain the bistable behaviors. Such a dissipatively coupled hybridized system results in distinctly dier- ent bistable behaviors, like butter y-like, clockwise, and counterclockwise hysteresis which are visualized through transmission spectra. For the eld bistability, the tran- sition points show P1=3andPrespective dependence when up-sweeping and down-sweeping the magnetic eld. Correspondingly, for the power bistability, the transi- tion points show jHj3andjHjdependence when up- sweeping and down-sweeping power, respectively. Mean- while, the critical condition required for observing eld and power bistability is obtained. With the suppressed magnon damping and therefore lowered threshold for bistability, our system may lay the foundation for wide applications of very low power nonlinearity devices. Be- sides, beneting from exible tunability with, e.g., the magnon frequency, the interaction strength between cav- ity and magnon, the drive power, the bistability of the cavity magnonics system may be potentially applied in emergent applications like memories and switches.9 ACKNOWLEDGMENTS This work has been funded by NSERC Discov- ery Grants and NSERC Discovery Accelerator Supple- ments (C.-M. H.). Z.H. A. acknowledges the nancial support from the National Natural Science Foundation of China under Grant Nos. 12027805/11991060, and the Shanghai Science and Technology Committee un- der Grant Nos. 20JC1414700, 20DZ1100604. H. Pan was supported in part by the China Scholarship Council (CSC). The authors thank Garrett Kozyniak and Bentley Turner for discussions and suggestions. Appendix A: Hamiltonian of the coupled hybrid system The cavity-magnon hybrid system shown in Fig. 1(a) includes a small YIG sphere with Kerr nonlinearity which is dissipatively coupled to a Fabry-Perot-like cavity, and the cavity is driven by a microwave eld. The Hamil- tonian of such a system consists of four parts (setting ~1): Htot=Hc+Hm+HI+Hd; (A1) hereHc=!c+corresponds to the bare Hamiltonian of the cavity mode, with the creation (annihilation) op- erator+() at frequency !c. In our experiment, we apply a uniform static magnetic eldH=Hezorientating along the z-axis, and the YIG sphere has volume Vm. The static magnetic eld is used to align the magnetization and tune the frequency of the magnon mode. When Zeeman energy and magnetocrys- talline anisotropy energy are included, the Hamiltonian of the YIG sphere reads: Hm=0Z VmM
Hd0 2Z VmM
Hand; (A2) where0is the vacuum magnetic permeability, M= (Mx;My;Mz) is the macrospin magnetization of the YIG sphere, and Hanis the magnetocrystalline anisotropy eld in the YIG crystal. We have neglected the con- tribution of demagnetization energy of the YIG sphere in Eq. (A2) as it is a constant term [34, 55]. For a uniformly magnetized YIG sphere, which is mag- netized along z-axis with its anisotropy eld along z-axis in our experiment, the anisotropy eld can be written as Han=mM zez, wheremis dependent upon the domi- nant rst-order anisotropy constant and the saturation magnetization [56]. Since HA<0 in our experiment, we can obtain that m < 0. Thus, the Hamiltonian of Eq. (A2) turns out to be Hm=0HM zVm+1 20mM2 zVm: (A3) Since the relation between macrospin magnetization Mand macrospin operator S[29, 34, 57] is M= 0S Vm= 0 Vm(Sx;Sy;Sz); (A4) where 0is the gyromagnetic ratio. By inserting such relation indicated by Eq. (A4) into Eq. (A3), we obtain Hm=0 0HSz0m 2 0S2 z 2Vm: (A5) The rst term of the above equation corresponds to Zee- man energy. The macrospin operators and the magnon operators are related via the Holstein-Primako transfor- mation [58] S+= (p (2Sb+b))b; (A6) S=b+(p (2Sb+b)); (A7) Sz=Sb+b; (A8) whereSis the total spin number of the YIG sphere, b+(b) the creation (annihilation) operator of the magnon at frequency!m, andSSxiSyare the raising and lowering operators of the macrospin. Through inserting Eq. (A8) into Eq. (A5), the Hamiltonian Hmcan be writ- ten as Hm=!mb+b+Kb+bb+b; (A9) here!m=0 0H+0 2 0mS=V mdenotes the frequency of the magnon mode and K=0 2 0m=(2Vm) the Kerr co- ecient. Since m< 0 for our experiment, the Kerr coef- cient K is positive. Notably, the Kerr eect of magnons termKb+bb+barises from magnetocrystalline anisotropy. The Hamiltonian representing the interaction between the magnon and the cavity mode is HI=i(b++b)(a++a); (A10) where denotes the dissipative coupling strength be- tween the magnon and the cavity mode. With the rotating-wave approximation, we can neglect the fast os- cillating terms [51], and the cavity-magnon interaction Hamiltonian can be reduced as HI=i(b+a+ba+): (A11) The interaction between the cavity photon and the drive eld can be expressed as [50] Hd= d(a+ei!t+aei!t); (A12) where dis the amplitude of the driving eld. Finally, we have the total Hamiltonian of the nonlinear cavity magnonics system where the cavity and magnon are dis- sipatively coupled, and the cavity is directly pumped Htot=!ca+a+!mb+b+Kb+bb+b+i a+b+ab+
+ d a+ei!t+aei!t
: (A13)10 [1] A. Camara, R. Kaiser, G. Labeyrie, W. Firth, G.-L. Oppo, G. Robb, A. Arnold, and T. Ackemann, Opti- cal pattern formation with a two-level nonlinearity, Phys. Rev. A 92, 013820 (2015). [2] L. Lugiato, F. Prati, and M. Brambilla, Nonlinear optical systems (Cambridge University Press, Cambridge, 2015). [3] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon Press, Oxford, 1969). [4] K. D. McKinstry, C. E. Patton, and M. Kogekar, Low power nonlinear eects in the ferromagnetic resonance of yttrium iron garnet, J. Appl Phys. 58, 925 (1985). [5] Y. S. Gui, A. Wirthmann, N. Mecking, and C.-M. Hu, Direct measurement of nonlinear ferromagnetic damping via the intrinsic foldover eect, Phys. Rev. B 80, 060402 (2009). [6] Y. S. Gui, A. Wirthmann, and C.-M. Hu, Foldover fer- romagnetic resonance and damping in permalloy mi- crostrips, Phys. Rev. B 80, 184422 (2009). [7] P. A. P. Janantha, B. Kalinikos, and M. Wu, Foldover of nonlinear eigenmodes in magnetic thin lm based feed- back rings, Phys. Rev. B 95, 064422 (2017). [8] H. Gibbs, Optical bistability: controlling light with light (Elsevier, New York, 2012). [9] C. Thirion, W. Wernsdorfer, and D. Mailly, Switching of magnetization by nonlinear resonance studied in single nanoparticles, Nat. Mater. 2, 524 (2003). [10] O. R. Bilal, A. Foehr, and C. Daraio, Bistable metamate- rial for switching and cascading elastic vibrations, Proc. Natl. Acad. Sci. USA 114, 4603 (2017). [11] F. Cottone, H. Vocca, and L. Gammaitoni, Nonlinear energy harvesting, Phys. Rev. Lett. 102, 080601 (2009). [12] R. L. Harne and K. Wang, A review of the recent re- search on vibration energy harvesting via bistable sys- tems, Smart Mater. Struct. 22, 023001 (2013). [13] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85, 623 (2013). [14] G. Kurizki, P. Bertet, Y. Kubo, K. Mlmer, D. Pet- rosyan, P. Rabl, and J. Schmiedmayer, Quantum tech- nologies with hybrid systems, Proc. Natl. Acad. Sci. USA 112, 3866 (2015). [15] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. B. Goennenwein, High cooperativity in coupled microwave resonator ferri- magnetic insulator hybrids, Phys. Rev. Lett. 111, 127003 (2013). [16] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Hybridizing ferromagnetic magnons and microwave photons in the quantum limit, Phys. Rev. Lett. 113, 083603 (2014). [17] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Strongly coupled magnons and cavity microwave photons, Phys. Rev. Lett. 113, 156401 (2014). [18] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Spin pumping in electrodynamically coupled magnon-photon systems, Phys. Rev. Lett. 114, 227201 (2015). [19] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Magnon dark modes and gradient memory, Nat. Commun. 6, 1 (2015).[20] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Coherent coupling between a ferromagnetic magnon and a superconducting qubit, Science 349, 405 (2015). [21] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015). [22] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity magnomechanics, Sci. Adv. 2, e1501286 (2016). [23] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya- mazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Cavity optomagnonics with spin-orbit coupled photons, Phys. Rev. Lett. 116, 223601 (2016). [24] L. Bai, M. Harder, P. Hyde, Z. Zhang, C.-M. Hu, Y. P. Chen, and J. Q. Xiao, Cavity mediated manipulation of distant spin currents using a cavity-magnon-polariton, Phys. Rev. Lett. 118, 217201 (2017). [25] D. Zhang, X.-Q. Luo, Y.-P. Wang, T.-F. Li, and J. You, Observation of the exceptional point in cavity magnon- polaritons, Nat. Commun. 8, 1 (2017). [26] J. Li, S.-Y. Zhu, and G. S. Agarwal, Magnon-photon- phonon entanglement in cavity magnomechanics, Phys. Rev. Lett. 121, 203601 (2018). [27] Z. Zhang, M. O. Scully, and G. S. Agarwal, Quantum en- tanglement between two magnon modes via kerr nonlin- earity driven far from equilibrium, Phys. Rev. Research 1, 023021 (2019). [28] J. M. P. Nair and G. S. Agarwal, Deterministic quantum entanglement between macroscopic ferrite samples, Appl. Phys. Lett. 117, 084001 (2020). [29] Y.-P. Wang, G.-Q. Zhang, D. Zhang, T.-F. Li, C.-M. Hu, and J. Q. You, Bistability of cavity magnon polaritons, Phys. Rev. Lett. 120, 057202 (2018). [30] J. M. Nair, Z. Zhang, M. O. Scully, and G. S. Agar- wal, Nonlinear spin currents, Phys. Rev. B 102, 104415 (2020). [31] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, and G. E. W. Bauer, Exchange magnon-polaritons in mi- crowave cavities, Phys. Rev. B 91, 094423 (2015). [32] B. M. Yao, Y. S. Gui, Y. Xiao, H. Guo, X. S. Chen, W. Lu, C. L. Chien, and C.-M. Hu, Theory and experi- ment on cavity magnon-polariton in the one-dimensional conguration, Phys. Rev. B 92, 184407 (2015). [33] P. Hyde, L. Bai, M. Harder, C. Dyck, and C.-M. Hu, Linking magnon-cavity strong coupling to magnon- polaritons through eective permeability, Phys. Rev. B 95, 094416 (2017). [34] Y.-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. Q. You, Magnon kerr eect in a strongly coupled cavity- magnon system, Phys. Rev. B 94, 224410 (2016). [35] P. Hyde, B. M. Yao, Y. S. Gui, G.-Q. Zhang, J. Q. You, and C.-M. Hu, Direct measurement of foldover in cav- ity magnon-polariton systems, Phys. Rev. B 98, 174423 (2018). [36] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Bidirectional conversion between microwave and light via ferromagnetic magnons, Phys. Rev. B 93, 174427 (2016). [37] V. L. Grigoryan, K. Shen, and K. Xia, Synchronized spin- photon coupling in a microwave cavity, Phys. Rev. B 98,11 024406 (2018). [38] M. Harder, L. Bai, P. Hyde, and C.-M. Hu, Topological properties of a coupled spin-photon system induced by damping, Phys. Rev. B 95, 214411 (2017). [39] B. Wang, Z.-X. Liu, C. Kong, H. Xiong, and Y. Wu, Magnon-induced transparency and amplication in pt- symmetric cavity-magnon system, Opt. Express 26, 20248 (2018). [40] B. Yao, T. Yu, X. Zhang, W. Lu, Y. Gui, C.-M. Hu, and Y. M. Blanter, The microscopic origin of magnon- photon level attraction by traveling waves: Theory and experiment, Physical Review B 100, 214426 (2019). [41] M. Harder, Y. Yang, B. M. Yao, C. H. Yu, J. W. Rao, Y. S. Gui, R. L. Stamps, and C.-M. Hu, Level attraction due to dissipative magnon-photon coupling, Phys. Rev. Lett. 121, 137203 (2018). [42] B. Bhoi, B. Kim, S.-H. Jang, J. Kim, J. Yang, Y.-J. Cho, and S.-K. Kim, Abnormal anticrossing eect in photon- magnon coupling, Phys. Rev. B 99, 134426 (2019). [43] Y. Yang, J. Rao, Y. Gui, B. Yao, W. Lu, and C.-M. Hu, Control of the magnon-photon level attraction in a planar cavity, Phys. Rev. Applied 11, 054023 (2019). [44] J. W. Rao, C. H. Yu, Y. T. Zhao, Y. S. Gui, X. L. Fan, D. S. Xue, and C.-M. Hu, Level attraction and level re- pulsion of magnon coupled with a cavity anti-resonance, New J. of Phys. 21, 065001 (2019). [45] Y.-P. Wang, J. Rao, Y. Yang, P.-C. Xu, Y. Gui, B. Yao, J. You, and C.-M. Hu, Nonreciprocity and unidirectional invisibility in cavity magnonics, Phys. Rev. Lett. 123, 127202 (2019). [46] J. M. P. Nair, D. Mukhopadhyay, and G. S. Agarwal, En- hanced sensing of weak anharmonicities through coher- ences in dissipatively coupled anti-pt symmetric systems, Phys. Rev. Lett. 126, 180401 (2021).[47] Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Adv. Phys. 69, 249 (2020). [48] Y. Yang, Y.-P. Wang, J. W. Rao, Y. S. Gui, B. M. Yao, W. Lu, and C.-M. Hu, Unconventional singularity in anti-parity-time symmetric cavity magnonics, Phys. Rev. Lett. 125, 147202 (2020). [49] J. M. P. Nair, D. Mukhopadhyay, and G. S. Agarwal, Ul- tralow threshold bistability and generation of long-lived mode in a dissipatively coupled nonlinear system: Appli- cation to magnonics, Phys. Rev. B 103, 224401 (2021). [50] G. S. Agarwal and S. Huang, Electromagnetically in- duced transparency in mechanical eects of light, Phys. Rev. A 81, 041803 (2010). [51] D. F. Walls and G. J. Milburn, Quantum optics (Springer, Berlin, 2008). [52] P. Gottlieb, Nonlinear eects of crystalline anisotropy on ferrimagnetic resonance, J. Appl. Phys 31, 2059 (1960). [53] B. Heinrich, J. Cochran, and R. Hasegawa, Fmr line- broadening in metals due to two-magnon scattering, J. Appl. Phys. 57, 3690 (1985). [54] U. Fano, Eects of conguration interaction on intensities and phase shifts, Phys. Rev. 124, 1866 (1961). [55] S. Blundell, Magnetism in Condensed Matter , Oxford Master Series in Condensed Matter Physics (Oxford Uni- versity Press, Oxford, 2001). [56] A. Prabhakar and D. D. Stancil, Spin waves: Theory and applications , Vol. 5 (Springer, Berlin, 2009). [57] O. O. Soykal and M. Flatt e, Strong eld interactions be- tween a nanomagnet and a photonic cavity, Phys. Rev. Lett. 104, 077202 (2010). [58] T. Holstein and H. Primako, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev.58, 1098 (1940).

2022-06-02

Dissipative coupling of resonators arising from their cooperative dampings to a common reservoir induces intriguingly new physics such as energy level attraction. In this study, we report the nonlinear properties in a dissipatively coupled cavity magnonic system. A magnetic material YIG (yttrium iron garnet) is placed at the magnetic field node of a Fabry-Perot-like microwave cavity such that the magnons and cavity photons are dissipatively coupled. Under high power excitation, a nonlinear effect is observed in the transmission spectra, showing bistable behaviors. The observed bistabilities are manifested as clockwise, counterclockwise, and butterfly-like hysteresis loops with different frequency detuning. The experimental results are well explained as a Duffing oscillator dissipatively coupled with a harmonic one and the required trigger condition for bistability could be determined quantitatively by the coupled oscillator model. Our results demonstrate that the magnon damping has been suppressed by the dissipative interaction, which thereby reduces the threshold for conventional magnon Kerr bistability. This work sheds light upon potential applications in developing low power nonlinearity devices, enhanced anharmonicity sensors and for exploring the non-Hermitian physics of cavity magnonics in the nonlinear regime.

Bistability in dissipatively coupled cavity magnonics

2206.01231v1

1 Secondary Excitation of Spin-Waves: How Electromagnetic Cross-Talk Impacts on Magnonic Devices Johannes Greil1, Matthias Golibrzuch1, Martina Kiechle1,´Ad´am Papp2, Valentin Ahrens1, Gy¨orgy Csaba2, and Markus Becherer1 1School of Computation, Information and Technology, Technical University of Munich, Germany 2Faculty of Information Technology and Bionics, P ´azm´any P ´eter Catholic University, Budapest, Hungary This work examines the impact of electromagnetic cross-talk in magnonic devices when using inductive spin-wave (SW) transducers. We present detailed electrical SW spectroscopy measurements showing the signal contributions to be considered in magnonic device design. We further provide a rule of thumb estimation for the cross-talk that is responsible for the secondary SW excitation at the output transducer. Simulations and calibrated electrical characterizations underpin this method. Additionally, we visualize the secondary SW excitation via time-resolved MOKE imaging in the forward-volume conguration in a 100 nm Yttrium-Iron-Garnet (YIG) system. Our work is a step towards fast yet robust joint electromagentic-micromagnetic magnonic device design. Index Terms—Magnonic Devices, Electromagnetic Cross-Talk, Spin-Wave Spectroscopy, Secondary Spin-Wave Excitation I. I NTRODUCTION MAGNONICS has become aeld of science in which the focus is no longer only on fundamental research but also on devices and their commercial usability. The promising properties of magnonic devices—tunability, ultra-low losses, GHz-regime applications—make them attractive for being miniaturized, integrated, and manufactured on a large scale. Despite the convenient inductive excitation and pick-up of spin-waves (SWs), the electromagnetic cross-talk between the SW transducers is a major hurdle when it comes to device design and benchmarking [1]. Additionally, the secondary excitation of SWs at the output transducer due to inductive coupling deteriorates the overall device performance. An essential step between the functional design of a magnonic device and the layout of its electrical input/output (I/O) and matching network is the examination of the elec- tromagnetic cross-talk that inevitably arises using inductive SW transducers. The parasitic cross-talk results in a signi- cant modication of the transmitted SW signals in electrical measurements, most prominent in AESWS. While in magnon- transport evaluation the cross-talk is calibrated out via refer- ence measurements at no or off-resonance DC magnetic bias eld, the absolute magnitude of the direct coupling is crucially important for the practical use in real-time signal processing. This work focuses on the origin of signal contributions of typical all-electrical SW spectroscopy (AESWS) [2]–[4] measurements in the forward-volume (FV) conguration [5]. These contributions can be explained by a straightforward cross-talk model based on the Hertzian dipole [6] for mi- crostrip line (MSL) transducers and easy-to-access 2D sim- ulations for coplanar waveguides (CPWs) using the mag- netostatics solver FEMM [7]. Further, we provide time- resolved magneto-optical Kerr effect (trMOKE) images con- rming the secondary SW excitation resulting from the electromagnetic cross-talk. Corresponding author: J.Greil (email: johannes.greil@tum.de).3.1003.1253.1503.1753.2003.2253.2503.2753.3003.3253.3503.3753.400−2−10 m0m1 ∆fp Frequency in GHz ∆S-Parameters in dB∆S11 ∆S22 4·∆S 21 4·∆S 12 Fig. 1.∆S-Parameter for a CPW transducer pair with60µmcenter-to-center distance on a100 nmYIGlm in the forward-volume conguration (see top view sketch in the inset). The cross-talk of about30 dBis subtracted. Both reections show a dip at3.285 GHzresulting from the power absorption in the YIGlm. In transmission, the main signal contribution is still the power absorption indicated by the large dip in∆S 21and∆S 12. The ripple on top of those signals is due to the varying spin-wave group velocity during a frequency sweep. Based on the calibrated VNA measurements it is possible to assess that the propagating SW signal has an amplitude of about0.2 dB. II. E LECTRICAL SPIN-WAVESPECTROSCOPY SIGNAL CONTRIBUTIONS Typical AESWS measurements in the FV conguration show three main signal contributions (in descending order of their amplitude): First, the electromagnetic cross-talk between the SW transducers due to the inductive coupling. Second, the absorption of RF power from the input transducer into the magnetic layer underneath. This absorption comes from the excitation of SWs that lowers the electromagnetic cross- talk measured via the transmission signals in the form ofS 21 orS 12. Third, a signal contribution corresponding to SWs propagating from the input to the output transducer. In Fig. 1 the latter two contributions are shown for the characterization of a100 nmthin plain YIGlm. The almost constant cross-talk of40 dBin the evalu- ated frequency span is subtracted for better visualization (∆S xx=S xx,H ext̸=0−S xx,H ext=0). The inset shows the2 measurement conguration using a pair of shorted CPWs with equally broad signal (S) and ground (G) lines and gaps wS=w G=w ap= 4µm, and a center-to-center distance of D= 60µm. The transducers have a length of1 mmand are tapered towards contact pads for bonding them to a chip carrier. The RF excitation frequencyf RFis swept from 31 GHzto34 GHzwith a step size of625 kHzfor an out-of- plane (OOP) DC magnetic biaseldµH xt= 260 mT. The reection signals∆S 11and∆S 22are used to determine the frequency of the most efcient SW excitation which is indi- cated by the dip at3285 GHzin both signals. Their different heights are caused by slightly varying bond connections to the carrier. A comparison to the transmission signal shows that the reection signal peaks are located at about2 MHzhigher frequencies. The peak shift arises because in reection mea- surements mainly thelm characteristics directly underneath the CPW are probed, while in transmission the spin precession is enforced by additional but weakereld components up to several micron distances [6], [8]. Moreover, therst higher- order excitation modem 1of the CPW transducer—therst spatial harmonic of the cross-sectionaleld prole of the CPW geometry—generates another small dip in the reection signals at334 GHz[1], [8], [9]. The characterization of propagating SWs is obtained via the transmission parameters∆S 21and∆S 12. Besides the subtracted electromagnetic cross-talk, the power absorption into the YIGlm for exciting SWs is the strongest contribution and shows up as the large dip in∆S 21and∆S 12in Fig. 1. This implies that one can estimate the overall absorbed power along the transducer via calibrated VNA measurements. Especially in the FV geometry not more than half of the accepted power for SW excitation is transported towards the output transducer due to the isotropic wave propagation. For the example, from Fig. 1 we can determine that about07 dB, relative to the input powerP in=−10 dBm, are absorbed into thelm. The maximum amplitude of the ripple on top of the large dip is in turn02 dB. This ripple in∆S 21and ∆S12denes the third signal contribution which represents the propagating SW signal. The ripple stems from the changing SW wavelength while sweeping the excitation frequency for axedH xtand constant transducer distanceDsuch that more SWst between the transducers. From the frequency spacingf pbetween two adjacent peaks we can determine the group velocity via [3], [5], [9] v=dωSW dk≈∆f pD(1) tov ≈660 ms. Therst higher-order excitation of propagat- ing spin-wavesm 1is also clearly visible in the transmission signal at3325 GHz. Summing up, calibrated AESWS measurements allow for a qualitative and quantitative distinction between the discussed signal contributions. Those device-specic characteristics set the basis for highly optimized magnonic devices, whereas the cross-talk reduction is the core challenge for a practically useful on-chip post processing of SW signals.III. E LECTROMAGNETIC CROSS -TALK BETWEEN SPIN-WAVETRANSDUCERS The electromagnetic cross-talk is a severe signal contribu- tion in the electrical evaluation of magnonic devices since it buries almost all SW-related signals. Thus, it is of great practical use to tackle the electromagnetic cross-talk in a straightforward and easy-to-use rule of thumb manner pre- sented in the following. The SW excitation using inductive transducers is described via the in-plane RF magneticelds that force the spins to precess around the effective magneticeldH [5]. This excitation is only applicable in very close proximity to the transducer lines because, as with conventional antennas, these eld components correspond to the so-called reactive neareld and decay fast compared to the far-eld components [6]. For SW transducers with lengths in the range of several ten to hun- dred microns, the near-eld character is preserved also for the electromagnetic cross-talk since for conventional, electrically small antennas the far-eld distanced is dened as [10] d≈2λ0(2) In the neareld, reactive power oscillates between the antenna structure and the surrounding space such that the Poynting vector is zero and no radiation takes place. In contrast, in the fareld, a locally plane wave is formed because electric and magneticeld components are in phase [6]. From (2) we see, thatd is not dependent on the physical size of the transducer because it acts as a point-like source for the electromagnetic waves with comparably long wavelengthsλ 0in free-space. For applications up to20 GHzthe fareld is thus reached rst atd ≈30 mmwhich is much larger than a reasonable magnonic device dimension. The radiation characteristics of electrically small rod antennas can be modeled via the Hertzian Dipole (HD) if their cross-section is small compared to the transducer length [6]. The azimuth magneticeld component forϑ= 90◦, i.e. in thexy-plane, of a HD aligned to thez-axis is written in polar coordinates as [6] Hφ= ikIl 4πr 1 +1 ikr e−ikr,(3) whereIis the impressed current,lis the length of the dipole,ris the radial distance to the origin, andkis the (electromagnetic) wave number in azimuth direction. When ipping the HD over to be aligned with they-axis, the azimuth eld component describes the OOPeld of an MSL in the yx-plane such thatH φ=Hz. From (3) we see that for small values ofrtheeld amplitude decays in good approximation with1r3. In the fareld, it decreases with1ras the second summand in (3) vanishes. Simulations in FEMM underpin that the HD is a good approximation for electrically short MSLs used as SW transducers. The widely used shorted CPW SW transducer [2], [8], [9], [11] can be constructed by a superposition of three sucheld components, writ- ten asH z,CPW ≈ −1 2Hφ(x+w) +H φ(x)−1 2Hφ(x−w), wherewis the center distance between the signal line and the ground lines. The factors−12stem from the fact that half the signal-line current returns in each of the ground lines. Thus, from the superposition, the near-eld amplitude of a3 Fig. 2. a) Cross-sectional view of CPW transducers withw S=w gap=w G= 4µmand a center-to-center distance ofD= 60µmfabricated on100 nm thin 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 the 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 precession starts aroundP in=−15 dBm, while for the secondarily excited SWsP in≈5 dBmis required. The non-linear excitation is visible by the power-dependent increase of the SW wavelength. The minimum input power to observe SWs is−35 dBm(not shown) and−5 dBmfor the primary and secondary excitation, respectively. CPW resembles a decay proportional to1r4. While this rough approximation does not perfectly go in line with the simulatedeld proles due to the neglected lateral dimension of the CPW, it offers a straightforward order-of-magnitude estimation of those. As a consequence, the inductive coupling entails not only a strong cross-talk but also the secondary SW excitation at the output transducer. The shorted CPW forms two loops that pick up not merely the SW-induced signal but also the perme- ating magnetic out-of-planeelds from the input transducer. The resulting magneticux through the loop surface can be estimated via Faraday’s law of induction Φ= AlBz·dA l,(4) whereA l= 2w lis the area of the loop formed by the shorted CPW with lengthland gapsw . The induced voltage computes toU in=−dΦdt. Further, the ohmic resistance of the electrically small CPW lines can be estimated via the general DC resistance model R=ρl A,(5) with the specic electrical resistivityρand the cross-sectional area of a single lineA . Hence, the induced current in the signal line isI S,in=U inR, where by denition IG,in=IS,in2for a shorted CPW. Finally, approximating each of the CPW lines again as a thin conductor rod, the magnitude of the induced secondary IP magneticeld inthe magneticlm underneath the output transducer can be estimated via Amp `ere’s law |Hs,IP|≈Iin 2πr,(6) whereris the radial distance to the surface of the trans- ducer line. Let us discuss a specic example: for a pair of CPW transducers as shown in Fig. 3 with lengthl= 1 mm, wS=w =w G= 4µm, a center-to-center distanceD= 60µm, a metallization thickness of300 nm, and an input powerP in=−10 dBmthe following values can be estimated: For a remaining OOP magneticeld of about6µT(FEMM simulation) at the output transducer forming two loops with an overall area ofA l= 2w l= 0008 mm2, the induced voltage Uin≈30 mV. From (5) the induced current in300 nmthick aluminum transducers is then6µAwhich in turn induces a secondary IP magneticeld ofµ 0Hs,IP ≈20µTat half the thickness of the100 nmYIGlm. This secondarily induced IP eld is sufcient to linearly excite SWs in the same manner as the supplied transducer does but at much lower power levels. In a perfect50ΩRF system this corresponds to an input power level ofP in≈ −34 dBmwhich is close to the minimum required input power in our trMOKE setup to resolve SW signals. In summary, the secondary excitation of SWs can be estimated in a rule-of-thumb manner via (4)-(6) showing reasonable results, also validated by the measurements in the following section. Thereby, the loop areaA lof the CPW has a major inuence on the strength of the secondary excitation.4 Fig. 3. Top view sketches of the sample settings discussed in Sec. IV. The distancesD= 60µmandd= 35µmand the transducer lengthl= 1 mmare the same for both congurations. The trMOKE picture in Fig.2 corresponds to conguration a) for an excitation at the left transducer. The measurements shown in Fig. 4 and Fig. 5 correspond to conguration b) for an excitation at the right and left transducer, respectively. IV. S PATIAL VISUALIZATION OF SECONDARILY EXCITED SPIN-WAVES USING TIME-RESOLVED MOKE A. Both Transducers on YIG To substantiate the presented method, we have consid- ered sample settings that can be used to quantify the sec- ondary SW excitation resulting from the electromagnetic cross-talk via trMOKE images. Arst test conguration consists of two parallel shorted CPWs with lengthl= 1 mm,w S=w ap=w G= 4µmand a center-to-center dis- tance of60µmfabricated on two areas of YIG separated by a35µmwide gap as shown in Fig. 3 a. Forf RF= 205 GHz andµH xt≈214 mT, sweeping the input powerP infrom −27 dBmto15 dBmin steps of2 dBmresults in the trMOKE image shown in Fig. 2. The left transducer is supplied by an RF source while the right one is not supplied and open at the tapered end. To avoid possible deviations arising from possiblelm inhomogeneities the same sample area is re- peatedly scanned for the different input powers. Due to the large range of input powers, it is necessary to underscale the high-amplitude Kerr signals to resolve weaker signals. For the primary excitation, the minimum input power generating a resolvable Kerr signal is−35 dBm, while non-linear excitation begins at aroundP in=−15 dBm. The transition to non- linear excitation is determined by the increase in wavelength compared to the scan atP in=−13 dBm. Looking at the secondarily excited SWs on the right, it is possible to detect wave fronts starting fromP in≈ −5 dBmat the supplied transducer. The non-linear secondary excitation starts at about Pin= 5 dBm. In this conguration, the difference between the accepted power for primarily and secondarily excited SWs is about10 dBwhich corresponds to the overall cross-talk loss between the transducers. This loss contains the radiated but uncoupledeld components of the supplied transducer that do not permeate the unsupplied CPW and the—in this sense unused—power that is absorbed at the left CPW for the primary excitation of SWs. 0 20 40 60 80 100 120 140 x-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 -36-16-4 0 4 16 36 Kerr signal in a.u.100 nm YIG 500m GGGPin̸= 0 Fig. 4. TrMOKE picture of primarily excited SWs for an input power sweep from−21 dBmto15 dBmin steps of2 dBmat the right transducer in Fig. 3b. The lowest possibleP inis−35 dBm(not shown), while the non-linear excitation starts atP in≈ −19 dBmdenoted by the increase in SW wavelengthλ SW. The intermediate bend ofλ SWis attributed to a connement effect between the sharp YIG edge and the CPW. B. One Transducer on GGG, One Transdcuer on YIG A second conguration is shown in Fig. 3 b, where the left transducer is fabricated directly on the GGG substrate leaving only a100 nmYIG layer underneath the right trans- ducer. The CPW dimensions are the same as for the previous measurement, the excitation frequency and externaleld are set tof RF= 205 GHzandµH xt≈223 mT, respectively. The trMOKE picture of the primarily excited SWs is shown in Fig. 4 for input powers of−27 dBmto15 dBmin steps of2 dBm. Similar to the former conguration a minimum input power of−35 dBmgenerates a resolvable Kerr sig- nal, while the non-linear excitation now starts at around Pin=−19 dBm. The intermediate bend of wavelengths for power levels between−7 dBmand−1 dBmis attributed to a connement effect that occurs from the cavity-like structure formed by the sharp YIG edge and the CPW but is not yet investigated in detail. The trMOKE picture of the secondarily excited SWs is shown in Fig. 5 for the same input power levels but this time applied to the transducer on the bare GGG substrate. Again, the same area is repeatedly scanned for all power levels to avoid inuences oflm inhomogeneities. The minimum required input power to detect SW signals is−27 dBm, while the non-linear excitation begins atP in≈1 dBm. In this way we see that non-linearity is reached at4 dBless input power compared to the conguration in Sec. IV-A where both transducers are fabricated on YIG. This difference shows in turn how much power is absorbed into the YIGlm for the primary SW excitation at the left transducer in Fig. 2. It reects the fact that in the FV geometry a maximum of50 %of the absorbed power can be transported by the spin waves and at least50 %propagate in other directions. Thus, comparing the5 0 20 40 60 80 100 120 140 x-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 -36-16-4 0 4 16 36 Kerr signal in a.u.100 nm YIG 500m GGGPin= 0 Fig. 5. TrMOKE picturce of secondarily excited SWs at the right CPW in Fig. 3b for an input power sweep at the left CPW from−27 dBmto15 dBm in steps of2 dBm. The minimum required input power to detect a SW signal is around−23 dBm, while the non-linear excitation starts atP in≈1 dBm. congurations in Fig. 3 a & b, we can measure the fraction of power that is absorbed for primary SW excitation and propagates away from the (removed) SW channel between the transducers. C. Distant Excitation of Spin-Waves In a third conguration we omit the right transducer in Fig.3b such that only the left transducer is fabricated on the GGG substrate with distances of20µmto60µmto the YIG. In this experiment, it was not possible to excite SWs or near-FMR modes. It shows that there is not a distant FMR-like excitation [8] but the non-driven output transducer is necessary for the secondary SW excitation. At those relatively long distances between the transducer and the YIG layer the IPelds are too much conned underneath the supplied CPW to be strong enough to force the spins in the YIG to precession. Following the discussion in [12] e.g. an MSL in close proximity to the YIG edge could be a feasible transducer geometry to excite a relatively broad spectrum of SWs without the need to fabricate the transducer directly on YIG. However, to our knowledge the excitation efciency of this method was not yet investigated experimentally. Even more, the fact that the secondary SWs are excited by the output transducer and not by a distanteld component of the input CPW is indicated by the equally long SW wavelengths in the linear regime: From Fig. 2 wend thatλSW≈7µmfor a linear excitation on both transducer sides. And from the comparison of Fig. 4 and Fig. 5 wend λSW≈10µm. The comparison shows that the wavelength for linear excitation depends on the cross-sectional geometry of the CPW, whereas the difference between the measured wavelengths in the two experiments results from the changed Hxtat axedf RF[11].V. C ONCLUSION Our measurements show that using inductive SW transduc- ers leads to two major challenges when it comes to high- speed signal transmission requiring lowest distortions: First, the electromagnetic cross-talk is the strongest contribution to the transmission signal and often buries the SW-related signals almost completely. The commonly used cross-talk subtraction in AESWS measurements is somewhat sufcient for investi- gating magnon-transport phenomena but is not applicable for real-time measurements in an I/O circuitry. And second, the electromagnetic cross-talk is such strong that we observe a secondary excitation of SWs at the output transducer due to the inductive coupling. These secondarily excited SWs interfere with the intentionally excited SWs and lower the magnonic signal-to-noise ratio as well as the overall detection sensitivity of the output transducer due to its own reaction on varying frequency or magneticeld. Thus, one of the main goals for paving the way for magnonic devices towards integration is to minimize the electromagnetic cross-talk between inductive SW transducers. This can be achieved either by shielding them from each other while maintaining the SW excitation efciency as well as the electrical RF properties. Or by introducing asymmetric transducer structures, e.g. with a CPW as input and several loop antennas as outputs which are optimized for the magnonic device characteristics. A fully coupled electromagnetic-micromagentic design approach is therefore essential for practically useful magnonic devices. ACKNOWLEDGMENT The authors acknowledge funding from the European Union within HORIZON-CL4-2021-DIGITAL-EMERGING-01 (No. 101070536, MandMEMS), German Research Foundation (DFG No. 429656450), and the German Academic Exchange Service (DAAD, No. 57562081). REFERENCES [1] D. A. Connelly, G. Csabaet al., “Efcient electromagnetic transducers for spin-wave devices,”Scientic Reports, vol. 11, no. 1, Sep. 2021. [2] V. Vlaminck and M. Bailleul, “Current-induced spin-wave doppler shift,” Science, vol. 322, no. 5900, pp. 410–413, Oct. 2008. [3] S. Neusser, G. Duerret al., “Anisotropic propagation and damping of spin waves in a nanopatterned antidot lattice,”Physical Review Letters, vol. 105, no. 6, Aug. 2010. [4] J. Chen, T. Yuet al., “Excitation of unidirectional exchange spin waves by a nanoscale magnetic grating,”Physical Review B, vol. 100, no. 10, Sep. 2019. [5] A. Prabhakar and D. D. Stancil,Spin Waves: Theory and Applications. Springer US, 2009. [6] D. M. Pozar,Microwave Engineering 4th Edition. John Wiley & Sons, 2011. [7] D. Meeker, “Finite element method magnetics.” [Online]. Available: https://www.femm.info [8] M. Sushruth, M. Grassiet al., “Electrical spectroscopy of forward volume spin waves in perpendicularly magnetized materials,”Physical Review Research, vol. 2, no. 4, Nov. 2020. [9] V. Vlaminck and M. Bailleul, “Spin-wave transduction at the submi- crometer scale: Experiment and modeling,”Physical Review B, vol. 81, no. 1, Jan. 2010. [10] H. A. Wheeler, “The Radiansphere Around a Small Antenna,”Proceed- ings of the IRE, vol. 47, Aug. 1959. [11] J. Lucassen, C. F. Schipperset al., “Optimizing propagating spin wave spectroscopy,”Applied Physics Letters, vol. 115, no. 1, p. 012403, Jul. 2019. [12] ´A. Papp, W. Porodet al., “Nanoscale spectrum analyzer based on spin- wave interference,”Scientic Reports, vol. 7, no. 1, Aug. 2017.

2023-03-20

This work examines the impact of electromagnetic cross-talk in magnonic devices when using inductive spin-wave (SW) transducers. We present detailed electrical SW spectroscopy measurements showing the signal contributions to be considered in magnonic device design. We further provide a rule of thumb estimation for the cross-talk that is responsible for the secondary SW excitation at the output transducer. Simulations and calibrated electrical characterizations underpin this method. Additionally, we visualize the secondary SW excitation via time-resolved MOKE imaging in the forward-volume configuration in a 100nm Yttrium-Iron-Garnet (YIG) system. Our work is a step towards fast yet robust joint electromagentic-micromagnetic magnonic device design.

Secondary Excitation of Spin-Waves: How Electromagnetic Cross-Talk Impacts on Magnonic Devices

2303.11303v2

arXiv:0903.0373v1 [nlin.PS] 2 Mar 2009Magneto-optical control of light collapse in bulk Kerr medi a Y. Linzon∗,1K. A. Rutkowska,1,2B. A. Malomed,3and R. Morandotti1 1Universit´ e du Quebec, Institute National de la Recherche S cientifique, Varennes, Quebec J3X 1S2, Canada 2Faculty of Physics, Warsaw University of Technology, Warsa w PL-00662, Poland 3Department of Physical Electronics, School of Electrical E ngineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978 , Israel (Dated: November 3, 2018) Magneto-optical crystals allow an efficient control of the bi refringence of light via the Cotton- Mouton and Faraday effects. These effects enable a unique comb ination of adjustable linear and circular birefringence, which, in turn, can affect the propa gation of light in nonlinear Kerr media. We show numerically that the combined birefringences can ac celerate, delay, or arrest the nonlinear collapse of (2+1)D beams, and report an experimental observ ation of the acceleration of the onset of collapse in a bulk Yttrium Iron Garnet (YIG) crystal in an e xternal magnetic field. PACS numbers: 42.65.Sf, 42.65.Jx, 42.81.Gs, 78.20.Ls Waves propagating in multidimensional self-focusing media are subject to instabilities that lead to the catas- trophic collapse after a finite propagation distance, fol- lowed by beam filamentation or material damage [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In the (2+1)D [(2+1)- dimensional)] setting, above a certain threshold value of the input power, the critical collapse driven by fo- cusing nonlinearities is a universal scenario, observed in high-power excitations of plasmas [1, 2], hydrodynami- cal systems [3], Bose-Einstein condensates (BECs) [4, 5], and optical media [2, 6, 7]. In particular, in optical pulse propagation through amorphous media and crys- tals without special symmetries, the dominant nonlin- earity is the Kerr (cubic) effect, which is modeled by the nonlinear Schr¨ odinger equation (NLSE) [12]. Collapsing beams in Kerr media were studied in detail, especially in the course of the past decade [6, 7, 8, 9, 10, 11]. The challenge to controlthe wave collapse (and in par- ticular to mitigate its detrimental effects) has recently drawn much attention [5, 9, 10, 11]. As recently demon- strated, the collapse distance of ultra-intense laser pulses in air can be controlled by passive optical elements [9], and in BECs the collapse time is strongly affected by the so-calledFeshbach-resonancetechnique [5]. In condensed optical media, where the collapse occurs with pulse en- ergies far below the creation of plasma [6, 7], a recently proposed scheme for collapse management relies on the use of a layered structure with the nonlinearity strength alternating in the longitudinal direction [10] (a similar mechanism was proposed for the stabilization of BECs in the 2D case [11]). However, this scheme is difficult to implement in bulk media, and such structures may give rise to linear losses induced by the reflection of light from interfaces between the alternating layers. An alternative approach to control the transition to the collapse may be provided by optical birefringence, which promotes energy and phase transfer between the polarization components of the beam. The interplay be- tween birefringence and nonlinearity is known to inducecoupling of the polarization rotation and a characteristic temporal evolution of solitons in optical fibers [13]. In this paper, we explore the birefringence as a tool for the “management” of the collapse of (2+1)D beams, which may be relatively easily implemented in affordable ex- perimental conditions by using magneto-optical(MO) ef- fects [14, 15, 16, 17], while avoiding reflection losses. To this end, we present numerical and experimental studies of the combined effects of linear and circular birefrin- gences on the dynamics of collapsing beams in a bulk self-focusing Kerr medium. We find that the onset of collapse can be accelerated, delayed, or even suppressed, at certain values of the combined birefringence strengths. We also show that the required linear and circular bire- fringence can be induced in a transparent MO Yttrium Iron Garnet (YIG, Y 3Al5O12) crystal by the application ofan external dc magnetic field, thus opening up the per- spective of using MO effects to generate and control var- ious nonlinear phenomena in optics. In our experiments, anadjustablebalanceoflinearandcircularbirefringences wasrealizedviaacombination[14]oftheCotton-Mouton (orVoigt) [15] and Faraday[16] MO effects in abulk YIG crystal. Following the propagation of femtosecond pulses in the crystal, we observed a controllable decrease of the threshold input power necessary for the onset of collapse at the output facet as a function of the magnetic field. This also constitutes a first experimental study in non- linear optics where birefringence can be switched on and varied continuously in an adjustable fashion. The evolution of the complex electric-field amplitudes, urandul, representing the right- and left-circular polar- izations (RCP and LCP), in the presence of a Kerr non- linearity and combined linear and circular birefringences, obeys the coupled NLSEs, in the scaled form [13]: i∂ur ∂z+1 2∇2 ⊥ur+bur+cul+(|ur|2+2|ul|2)ur= 0, i∂ul ∂z+1 2∇2 ⊥ul−bul+cur+(|ul|2+2|ur|2)ul= 0 (1) wherezis the propagation axis, ∇2 ⊥is the transverse2 Laplacian, while bandcare the strengths of the circular and linear birefringences, respectively. As evident from Eqs. (1) the linear and circular birefringences account for, respectively, the ratesof linearamplitude mixing and phase shift between the RCP and LCP fields. In terms of the RCP and LCP, the ratio between the cross- and self-phase modulation coefficients (XPM/SPM) is 2 for cubically-symmetric crystals, including YIG [12, 13]. Assuming vorticity-freesolutions with circular symme- try, (2+1)D Eqs. (1) reduce to a (1+1)D form, in terms ofzand the radial coordinate R. We consider input Gaussian beams, ur(R,z= 0) =Aexp/parenleftBig −R2 2ρ2/parenrightBig cosθand ul(R,z= 0) =Aexp/parenleftBig −R2 2ρ2/parenrightBig sinθ, with normalized in- put width ρ=1, amplitude A, and symmetricpolarization contentθ=π/4. This choice corresponds to an unchirped Gaussian profile launched with an horizontal linear po- larization. Since the input does not carry vorticity, the circular-symmetric structure of the solutions is not sub- ject to an azimuthal modulational instability [7]. Figure 1 shows propagation maps obtained by direct simulations of Eqs. (1). In the absence of birefringence [Figs. 1(a)-(d)], the RCP and LCP components are cou- pled only via the XPM term, which becomes significant only close to the collapse point. While a low input power excitation [Figs. 1(a,b)] results in beam diffraction, the collapse occurs after a finite propagation distance [for instance, z= 7.3 in Fig. 1(c)] when the input power ex- ceeds the critical level [8]. A substantial converging por- tion of the phase fronts emerges near the collapse point, see Fig. 1(d). The collapse can be accelerated by the in- 0 2 4 6 8-10-50510 0246810 0 2 4 6 8-10-50510 0246810 0 2 4 6 8-10-50510 0246810 0 2 4 6 8-10 -5 0 5 10 -3-2-10123 0 2 4 6 8-10 -5 0 5 10 -3-2-10123 0 2 4 6 8-10-50510 00.020.040.06 (c)(a) zR (b) (d) $=1.13,b=0,c=0.2 (e)$=1.13,b=0.08,c=0.2 (f)S S >10 >10 >10$=0.1,b=0,c=0 $=1.13,b=0,c=0R z zR R z FIG.1: (Color online)Propagation mapsof ur(R,z)obtained from direct solutions of Eq. (1). (a)-(d): Zero birefringen ces, with (a),(b) low and (c),(d) high input powers. The pairs of panels (a),(c) and (b),(d) display the evolution of the inte n- sity,|ur(R,z)|2, and phase of ur(R,z), respectively. (e), (f): Intensity maps in the presence of the birefringences. Numer - ical parameters are indicated above each respective panel. 00.10.20.30.40 0.1 0.2 0.3 0.4 0.5 5101520 00.10.20.30.40 0.1 0.2 0.3 0.4 0.5 345678910(b) (a) YIG crystal c bc>10 CollapseacceleratedCollapsedel ayed/supp ressed b0.5 0.5 FIG. 2: (Color online) Evolution maps in the parameter space (b,c) (see the text), as obtained from direct simulations of Eqs. (1), for A=1.13 and 0 ≤z≤50, up to z= 50 or the valuezcoll<50 in which the collapse occurs. The solutions that do not collapse up to z= 50 are represented in yellow areas. The dashed (green) curves show the calculated depen- dence between bandccorresponding to crystalline YIG in an external magnetic field. (a) zcoll, and (b) beam width at zcoll. troduction of linear birefringence, as shown in Fig. 1(e). Circular birefringence, if acting alone, does not affect the collapse, since the respective terms in Eqs. (1) can be eliminated by a straightforward transformation. When both of the birefringences are present, the propagation distance necessary for the onset of collapse can be ex- tended, i.e., the onset of the collapse is delayed [Fig. 1(f)] due to the interplay between amplitude and phase mix- ing. For low bandcvalues, such as those used in Fig. 1(e,f) and typically achievable in experiment, the differ- ences in the evolution of urandulare marginal, i.e., the RCP and LCP beam components feature the same collapse dynamics. With larger birefringence parameters the differences between the components become substan- tial; however, such large birefringence values were not accessible in the current experiment, see below. The results of systematic simulations are summarized in Fig. 2 by means of maps in the plane of ( b,c). Panels (a) and (b) show, respectively, the values of zcollwhere the beam collapses, if zcoll<50, and the final values of the beam width, ρ(z= 50); in cases where the collapse occurs at zcoll<50, the final width is set as ρ= 0 [black areas in (b)]. While Figs. 1 and 2 display the results for the RCP component, the LCP maps are similar in the entire range considered. As seen in Fig. 2, for a given inputpowerthedominationofamplitudemixingbetween RCP and LCP ( c > b) usually results in an acceleration of the collapse, while dominant phase mixing ( b > c) can lead to the delay or effective suppression of the collapse. In the experiment, a bulk YIG single-crystal was placed with the easy crystallographic axis [100] paral- lel toz, cf. Ref. [17]. These cubic dielectric crystals are highly transparent for optical signals in the near-infrared and exhibit large MO transmission coefficients, owing to their ferrimagnetic phase [15, 16]. We chose to work at the wavelength of 1 .2µm, which offers an optimal trade- off between the magnitudes of the MO coefficients and the absorption losses [16]. The temporal dispersion in YIG is normal and weak in the near-infrared [18], and hence a high power beam is free from the development of3 V VC POPA 800C Laser O=1.2 Pm ND (a) TL2 QWP(b)Vidicon Camera 10PmL1 Energy Meter xyInput pulseOutput pulse FIG. 3: (Color online) Experimental setup schematics. OPA, optical parametric amplifier laser; T, cylindrical telesco pe; QWP, quarter wave plate; P, polarizer; ND, neutral density attenuators; L1,L1, aspheric coupling and imaging lenses; C, crystal sample. Insets: (a) Photograph of the magnet’s inte - rior with lenses L1, L2 and crystal C; (b) The beam’s waist profile at the input facet of the crystal. temporal modulational instabilitiess [7]. The application of an external magnetic field Hper- pendicular to the propagation axis renders the crystal optically uniaxial, with the optical axis parallel to H and the respective linear birefringence proportional to H [14, 15]. The largest phase retardation associated with the linear birefringence, for components of the wavevec- torkperpendicular to the magnetic field, is 0 .45µm/cm in YIG [15] at a saturation field of H= 800 G. This cor- responds to a saturation value of c= 0.1 in terms of Eq. (1) for our crystal. For wavevector components parallel to the magnetic field, the Faraday effect induces different refractiveindices for the RCP and LCP waves(i.e., circu- larbirefringence),witharespectivepolarizationphasere- tardation of 2 µm/cm at the saturation level [16]. While the Faraday effect is predominant in the k/ba∇dblHgeome- try [17], it is weaker in the present setting, as only near the collapse point the wavevectors feature significant lat- eral components. An estimation corresponding to the case shown in Figs. 1(c),(d) yields an effective saturation value of b= 0.05. Below saturation, the Faraday coeffi- cient is proportional to the magnetic field, b∼H, while the Cotton-Mouton coefficient depends on Hquadrati- cally, i.e. c∼H2[14, 15, 16], implying a parabolic de- pendence betweenthe birefringencecoefficientsinagiven external magnetic field, b∼c2. Calculated relations be- tweenbandcfor the YIG crystal at different magnetic fields are shown by the dashed curves in Fig. 2. The experimental setup is sketched in Fig. 3. We used a Spectra Physics model 800C Optical Parametric Am- plifier laser system, delivering pulses of 200 fs duration at a repetition rate of 1 KHz with peak powers ≤50 MW. The input beam was shaped by means of a cylin- drical telescope (T), followed by the combination of a quarter wave plate (QWP) and a polarizer (P), which were used to fix the input linear polarization parallel to H(i.e. horizontal). Variable neutral density filters (ND) were used to control the input power. The YIG sample 2.1 MW (a)2.3 MW 2.9 MW 3.6 MW (c) (d) (e)10Pm Position ( Pm) Normalized intensity (a.u.) 01 xy FIG. 4: (Color online) (a) Output beam profiles as a func- tion of the input peak power, in the absence of an external magnetic field. (b) Line-outs along x(full circles) and y(hol- low rectangles) around the central spot of the collapsed (2 .9 MW) output beam. (c)-(e) Output beam images, as function of the input power (horizontal) and the external magnetic field (vertical), for the fields: H= 200 G (c), 400 G (d), 800 G (e). was placed inside an electromagnet (GMW model 3470). The application of a driving voltage between the poles (±V) induced a uniform out-of-plane dc magnetic field in the crystal. A digital Tesla-meter (Group3 DTM-133) was used to calibrate the free-space magnetic field and verify its homogeneity in the sample. A combination of aspheric lenses, L1 and L2, was used for coupling and imaging, respectively. Importantly, the lenses and crys- tal were mounted on nonmagnetic pyrex holders, see Fig. 3(a). Standard metallic mounts, which usually hold mi- croscopeobjectivelenses, werenotused, aswefoundthat they strongly affect the uniformity of the magnetic field within the sample. The sample input facet was placed at the focal plane of L1. The beam’s profile at this plane is shown in Fig. 3(b). The FWHMs of this elliptical input beam were 30 and 15 µm along the major ( x) and minor (y) axes. The beam profile at the sample output facet was imaged by L2 onto a Vidicon infrared camera. Experimental results are summarized in Fig. 4. Im- ages of the output beam as a function of the input peak power, in the absence of a magnetic field, are shown in Fig. 4(a). At the critical power of 2 .9 MW, a central high-power spot appears in the diffracting background. This spot is circularly symmetric [see Fig. 4(b)], al- though the input beam profile was elliptical, with the above-mentioned major-to-minor axes ratio of 2 : 1. The spot fits a Townes profile [6], which indicates that it is generated by the onset of the collapse. The correspond- ing critical power complies with the theoretical predic- tion for collapse [8], using λ0= 1.2µm, the linear re- fractive index of YIG n0= 1.83 and its Kerr coefficient n2= 7.2×10−16cm2/W [18]. The corresponding col- lapse distance prediction with A= 1.13,zcoll= 7.3 [Figs. 1(c),(d)], when rescaled back into physical units using the well-known transformations [6, 7, 8] and assuming a circular input beam of 10 µm diameter, matches the4 0 200 400 600 8002.32.52.72.9 (a) Magnetic field (G)(b)Critical input power (MW)b 2 crA 0.00 0.01 0.02 0.03 0.04 0.05-0.10-0.08-0.06-0.04-0.020.00 c(c) Field (G)Critical power (MW)800 G0 G b FIG. 5: (Color online) (a) Measured critical power required for the onset of the collapse in the output facet, as a func- tion of the applied magnetic field. (b) Numerically calculat ed squared critical amplitude of the input beam ( A2 cr) necessary for the onset of the collapse at z= 7.3, as a function of the birefringences corresponding to crystalline YIG. (c) The c al- culated dependence between bandcin YIG, used in (b). sample length, 3 mm. At higher powers, filaments are observed in the output facet images, indicating that the collapse occurred earlier in the crystal. Following the ap- plication of a magnetic field, the onset of collapse at the output facet is observed at lowerpowers [Figs. 4(c-e)], with the largest collapse-acceleration effect recorded at a magnetic field of H= 400 G [see Fig. 4(d)], corre- sponding to half the saturation field of YIG [17]. This observation agrees with the fact that, when YIG is ex- cited by light traveling perpendicular to the magnetic field, the linear birefringence is stronger than its circu- lar counterpart[14, 15]. A qualitative characterizationof the output beam polarization state has also shown that without an external magnetic field the beam remained linearly polarized, while with an application of the field the beam became elliptically polarized, with the ellip- ticity growing with the field. This indicates the certain presence of magnetically-induced polarization dynamics in the crystal. Figure 5(a) shows the measured critical power, corre- sponding to the data of Fig. 4, that was required for the onset of collapse at the output facet, as a function of the magnetic field. In Fig. 5(b), this is compared to results obtained by the numerical solutions of Eqs. (1), where for each set of birefringence values (corresponding to a given magnetic field) we find the critical amplitude Acr of the input beam for which the collapse occurs at the propagation distance corresponding to the output facet (z= 7.3). Figure5(c)againdisplaystherelationbetween the normalizedbirefringence parametersofthe YIG crys- tal used, cf. the dashed curves in Fig. 2. The behavior of the collapse detuning is similar in Figs. 5(a) and 5(b), even though the theoretical model did not take into con- sideration magnetically-induced losses. Specifically, the Cotton-Mouton effect is always accompanied by linear magnetic dichroism [15], and the Faraday effect entails circular magnetic dichroism [16, 17], both of which are weak but present at λ= 1.2µm. In conclusion, we have investigated the combined ef- fects of circular and linear birefringences on the propa-gation of collapsing (2+1)D beams in self-focusing bulk Kerr media, and have shown that the onset of collapse can be accelerated, delayed, or suppressed, depending on the relative birefringence strengths. Experimentally, we have demonstrated a controlled acceleration of the col- lapse at the output facet of a ferrimagnetic YIG crys- tal, following the application of an external magnetic field which induces the birefringences. The direct ob- servation of magnetization-induced effects in collapsing beams provides a unique demonstration of an all-optical magnetically-controlled lensing mechanism, pioneering the use of MO crystals in nonlinear optics experiments. Finally, sinceEqs. (1)arealsoGross-Pitaevskiiequations describing a binary BEC with linear interconversion [19], accountedforbythecoefficient c, similarphenomenamay also be observed in nonlinear matter-wave dynamics. This research was supported by NSERC and TeraXion (Canada). YL and KR respectively acknowledgesupport from FQRNT-MELS and IOF Marie Curie fellowships. ∗Corresponding author: yoli@emt.inrs.ca. [1] P. A. Robinson, Rev. Mod. Phys. 69, 507 (1997). [2] L. Berg´ e, Phys. Rep. 303, 260 (1998). [3] B. W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, Nature (London) 403, 401 (2000). [4] E. A. Donley et al., Nature 412, 295 (2001); M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, Nature 419, 51 (2002); T. Lahaye et al., Phys. Rev. Lett. 101, 080401 (2008). [5] J. L. Roberts et al., Phys. Rev. Lett. 86, 4211 (2001); T. Kochet al., Nature Phys. 4, 218 (2008). [6] K. D. Moll, A. L. Gaeta, and G. Fibich, Phys. Rev. Lett. 90, 203902 (2003). [7] L. T. Vuong et al., Phys.Rev. Lett. 96, 133901 (2006); T. D. Grow, A. A. Ishaaya, L. T. Vuong, and A. L. Gaeta, ibid.99, 133902 (2007); L. Berg´ e et al., Rep.Progr. Phys. 70, 1633 (2007). [8] G. Fibich and A. L. Gaeta, Opt. Lett. 25, 335 (2000). [9] G. Fibich et al., Opt. Express 14, 4946 (2006); S. Eisen- mannet al., Opt. Express 15, 2779 (2007). [10] I. Towers and B. A. Malomed, J. Opt. Soc. Am. B 19, 537 (2002). [11] H. Saito and M. Ueda, Phys. Rev. Lett. 90, 040403 (2003); F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A 67, 013605 (2003). [12] R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 2008), 3rd ed., Chaps. 1-4. [13] S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Commun. 70, 166 (1989); Y. Bar-Ad and Y. Silber- berg, Phys. Rev. Lett. 78, 3290 (1997). [14] R. Kurzynowski and W. A. Wozniak, Optik 115, 473 (2004). [15] J. F. Dillon, J. P. Remeika, and C. R. Staton, J. Appl. Phys.41, 4613 (1970). [16] G. B. Scott, D. E. Lacklison, H. I. Ralph, and J. L. Page, Phys. Rev. B 12, 2562 (1975). [17] Y. Linzon et al., Opt. Lett. 33, 2871 (2008). [18] M. J. Weber, Handbook of optical materials (CRC Press,5 2002). [19] R. J. Ballagh, K. Burnett, and T. F. Scott, Phys. Rev.Lett.78, 1607 (1997).

2009-03-02

Magneto-optical crystals allow an efficient control of the birefringence of light via the Cotton-Mouton and Faraday effects. These effects enable a unique combination of adjustable linear and circular birefringence, which, in turn, can affect the propagation of light in nonlinear Kerr media. We show numerically that the combined birefringences can accelerate, delay, or arrest the nonlinear collapse of (2+1)D beams, and report an experimental observation of the acceleration of the onset of collapse in a bulk Yttrium Iron Garnet (YIG) crystal in an external magnetic field.

Magnetooptical control of light collapse in bulk Kerr media

0903.0373v1