Long-range spin-wave propagation in transversely magnetized nano-scaled conduits
Björn Heinz, Qi Wang, Michael Schneider, Elisabeth Wei?, Akira Lentfert, Bert Lägel, Thomas Brächer, Carsten Dubs, Oleksandr V. Dobrovolskiy, Philipp Pirro, Andrii V. Chumak
LLong-range spin-wave propagation in transversely magnetized nano-scaledconduits
Björn Heinz, a) Qi Wang, Michael Schneider, Elisabeth Weiß, Akira Lentfert, Bert Lägel, Thomas Brächer, Carsten Dubs, Oleksandr V. Dobrovolskiy, Philipp Pirro, b) and Andrii V. Chumak c) Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, D-67663 Kaiserslautern,Germany Faculty of Physics, University of Vienna, A-1090 Wien, Austria Nano Structuring Center, Technische Universität Kaiserslautern, D-67663 Kaiserslautern,Germany INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany (Dated: 26 January 2021)
Magnonics attracts increasing attention in the view of novel low-energy computation technologies based on spin waves.Recently, spin-wave propagation in longitudinally magnetized nano-scaled spin-wave conduits was demonstrated, prov-ing the fundamental feasibility of magnonics at the sub-100 nm scale. Transversely magnetized nano-conduits, whichare of great interest in this regard as they offer a large group velocity and a potentially chirality-based protected trans-port of energy, have not yet been investigated due to their complex internal magnetic field distribution. Here, wepresent a study of propagating spin waves in a transversely magnetized nanoscopic yttrium iron garnet conduit of50 nm width. Space and time-resolved micro-focused Brillouin-light-scattering spectroscopy is employed to measurethe spin-wave group velocity and decay length. A long-range spin-wave propagation is observed with a decay lengthof up to ( . ± . ) µm and a large spin-wave lifetime of up to ( . ± . ) ns. The results are supported with micro-magnetic simulations, revealing a single-mode dispersion relation in contrast to the common formation of localizededge modes for microscopic systems. Furthermore, a frequency non-reciprocity for counter-propagating spin wavesis observed in the simulations and the experiment, caused by the trapezoidal cross-section of the structure. The re-vealed long-distance spin-wave propagation on the nanoscale is particularly interesting for an application in spin-wavedevices, allowing for long-distance transport of information in magnonic circuits, as well as novel low-energy devicearchitectures.Magnonics, the research field of spin-wave based datatransport and information processing, aims to complementCMOS-based computation technology by replacing thecharge-based binary logic with a wave-based logic .Utilizing spin waves as information carriers provides avariety of advantages such as a high energy efficiency dueto the absence of ohmic losses and additional degreesof freedom since frequency and phase of a spin wave arereadily accessible . Moreover, spin-wave systems inherita multitude of nonlinear mechanisms while beingscaleable to the nanoscale , allowing for a novel devicearchitecture and simultaneously reducing the devicefeature size to sizes comparable or even smaller than theirCMOS-based equivalent . A large number of spin-wavebased devices and logic elements have been realized in therecent years such as transistors , majority gates ,directional coupler , half-adder or de-/multipexer , withmany more theoretical concepts proposed . While beingwell investigated on the macro- and microscale, the study ofnano-sized magnonic elements has just scratched the surfacedue to the limited spin-wave propagation distance on thisscale and the difficulties accompanying the fabrication ofnano-sized magnetic elements. Only recent progress pushedthe go-to material of magnonics, yttrium iron garnet (YIG), a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] to the nanoscale revealing a reasonably large exponentialdecay length of 1 . . Although a further perfection of mate-rial growth or the usage of new materials with improvedcharacteristics might allow for an increase of the spin-wavepropagation length, it is already apparent that the dominatingloss channels in such systems are not intrinsic processes ,but extrinsic scattering processes mediated by lattice defects,surface roughness or magnetic inhomogeneities . Thus,novel approaches to avoid and eliminate these scatteringprocesses are of great interest. Among them, the transport ofinformation using topological protected (e.g. based on theDzyaloshinskii-Moriya interaction ) or backscatteringimmune (caused by the intrinsic spin-wave mode chirality)spin-wave states is investigated. As it has been shownrecently, magnetostatic surface waves (MSSW ) featuresuch a backscattering immunity to surface defects due to thepresence of energy gaps in the volume mode spectrum, ren-dering them insensitive to significant structural defects .In addition, in nanometer thick films, MSSWs are knownto provide a much larger group velocity in comparison toother spin-wave modes , which renders these waves aninteresting subject of investigation regarding a long-distancedata transport in magnonic circuits. However, MSSWrequire a transverse magnetization state leading to a stronglynon-uniform internal magnetic field distribution in laterallyconfined nanostructures, which causes the formation oflocalized edge-mode states and separated volume states .For nano-scaled conduits this field non-uniformity is even a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n more pronounced and, in addition, a strong quantization ispresent, resulting in a complex interplay. Therefore, the modestructure in transversely magnetized nano-scaled systems isstill an open question.Here, we report on the investigation of propagating spinwaves in a transversely-magnetized nano-sized YIG conduitof 50 nm width. We show that spin waves can propagatein such a waveguide in spite of the strong non-uniformityof the internal magnetic field, observing a large spin-wavedecay length up to 8 µm. In addition, a non-reciprocity forcounter-propagating spin waves is observed, which opensup the path for new device architectures. The experimentalfindings are supported with micro-magnetic simulations,revealing a single-mode dispersion relationship in contrastto the common formation of edge modes for microscopicsystems .In this study, a thin lanthanum-doped ( ) YIG filmof 44 nm thickness is used which is grown by liquidphase epitaxy on top of a 500 µm thick ( ) gadolin-ium gallium garnet substrate. A characterization of theplain film by means of vector network analyzer ferro-magnetic resonance spectroscopy and micro-focusedBrillouin-light-scattering (BLS) spectroscopy revealed thefollowing material parameters: saturation magnetization M s =( . ± . ) kA m − , Gilbert damping parameter α =( . ± . ) × − , inhomogeneous linewidth broad-ening µ ∆ H =( . ± . ) mT and exchange constant A ex =( . ± . ) pJ m − . These parameters are within thetypical range for high-quality thin YIG films . Nanoscopicwaveguides were fabricated using a hard mask ion beammilling procedure resulting in conduits with a trapezoidalcross-section. The bottom of the structure is 85 nm widewhile the top width is as narrow as 50 nm, which wasdetermined by scanning electron microscopy, see Fig. 1.Afterwards, a gold coplanar waveguide (CPW) antennawas added on top of the waveguide with a center-to-centerdistance of ground and signal line of 1 . µ H ext =
270 mT is applied in-planealong the short axis of the structure to ensure a transverselymagnetized state. Radio-frequency (RF) continuous-wave(cw) currents or pulses with 50 ns length and 350 ns repetitiontime are generated and fed into the CPW antenna using anRF generator in combination with a fast switch and filterelements. Subsequently, propagating spin-wave packetsare excited in the YIG waveguide and are detected usingmicro-focused BLS spectroscopy. A single-frequency laseroperating at 457 nm is used which is focused through thesubstrate of the sample on the structure using a compensat-ing microscope objective (magnification 100 × , numericalaperture NA = . k =
24 rad µm − can be detected. The laser spot diameter isapproximately 300 nm and the effective laser power on thesample is 5 mW.To support the findings, micro-magnetic simulations areperformed using the MuMax3 open source framework .The trapezoidal waveguide is modeled with the following
100 nm50 nm spin wave
GGG Y I G RFeneratorg PulseeneratorgTriggeringTriggering ~ SpectrumTime trace µ H C op l ana r w a v egu i de an t enna PhotodetectorFabry-PérotinterferometerLaserSwitchFilter ×2xy
FIG. 1. Experimental setup and scanning electron microscopy mi-crograph of the structure under investigation. A large bias magneticfield of µ H ext =
270 mT is used to magnetize the waveguide alongthe short axis ( y -direction). Spin-wave packets are generated by feed-ing RF current pulses into a CPW antenna on top of the YIG waveg-uide. Frequency, spatial and time-resolved scans are performed us-ing micro-focused BLS. The structure exhibits a trapezoidal cross-section with a top and bottom width of 50 nm and 85 nm respectively,caused by the fabrication process. Spin-wave wavelength and laserspot size not up to scale. dimensions: 20 µm length, 85 nm and 50 nm bottom andtop width respectively and 44 nm thickness. The cell size is10 × . × . , thus introducing 8 thickness layers withdiscrete varying width. The material parameters of the plainfilm are used in the simulations, since it has been shownthat the structuring process only has a moderate influenceon the structures properties . An external magnetic fieldof µ H ext =
270 mT is applied in-plane along the short axisof the structure and a ground state is prepared by relaxinga random magnetization distribution. Afterwards, a drivingmagnetic field is applied with a spacial field distributioncorresponding to the CPW antennas magnetic field to excitespin-wave packets. Time- and spacial Fourier transformationsallow to extract the dispersion relationship and connectedparameters.First, the spin-wave spectra are investigated for the case ofa cw microwave excitation. The results are presented in Fig.2 for different distances to the CPW antenna, 1 . N o r m a li z ed B L S i n t en s i t y ( a r b . u . ) AB x = 1.5 µm x = 7.5 µm FIG. 2. Spin-wave spectra in dependency of the applied microwavepower. Continuous-excitation spin-wave spectra are measured at adistance of ( A ) 1 . B ) 7 µm from the CPW antenna. Lowfrequency states close to the ferromagnetic resonance exhibit arestrongly attenuated. The spectra are normalized with respect to thethermal noise level. nels and influence a measurement of the decay length of thesystem. As shown in Fig. 2A, the shape of the spectrum isconserved for different applied powers, thus indicating thatno strong nonlinear effects arise in the selected microwavepower range. The observed spectrum has a spectral widthof 300 MHz and shows a complex behaviour, which mightbe attributed to microwave transmission characteristics of theused CPW antenna. Comparing the spectral distribution closeto the antenna to the distribution after several micrometer ofpropagation (Fig. 2B) reveals that the low-frequency part ofthe spectrum is strongly attenuated. These states are likelyclose to the ferromagnetic resonance of the structure and thus,as it is shown in the following, exhibit only a small group ve-locity.To support the findings, micro-magnetic simulations are con-ducted using a sinc-function pulse to realize a broadband ex-citation of the whole dispersion relation. In Fig. 3A the result-ing excited spectrum is shown for the experimentally accessi-ble wavevector range. A clear single mode state is observedfollowing a monotonous function. However, the absolute fre-quency is shifted by approximately 300 MHz with respect tothe experimental results of Fig. 2, which indicates that themagnetic parameters are slightly alternated by the structur-ing process . In addition, a small variation of the magneticwidth of the structure can also significantly impact the spin-wave frequency on the present scale. The various small gapsin the dispersion relationship are caused by the wavevectorselective excitation efficiency of the CPW antenna, as shownin Fig. 3B. Here, the efficiency is approximated by the spa-cial Fourier transformation of the in-plane field distribution,for a detailed discussion see . The simulated internal fielddistribution of the ground state is shown in Fig. 3C. As ex-pected, large demagnetizing fields arise leading to a stronglynon-uniform internal field distribution along the external mag-netic field direction. Similar to micron-sized conduits distinctregions of a significantly reduced effective magnetic field are F r equen cy ( G H z ) Wavevector (rad µm ) -1 -25 -20 -15 -10 -5 0 5 10 15 20 25 00.20.40.60.81 E xc i t a t i one ffi c i en cy ( a r b . u . ) AB Group velocity k - Group velocity k + Dispersion relationship k - Dispersion relationship k + F r equen cy ( G H z ) G r oup v e l o c i t y ( µ m n s ) - -1 F I n t en s i t y ( a r b . u . ) k - k + Frequency (GHz) E -40 0 40Width (nm)18Layer-40 0 40Width (nm)150170190210230 N o r m . | m | z ( a r b . u . ) C D I n t en s i t y ( a r b . u . ) E ff e c t i v e fi e l d ( m T ) d y n FIG. 3. Micro-magnetic simulations of the investigated structure.( A ) The dispersion relationship shows a distinct single-mode be-haviour in the experimentally accessible wavevector range. ( B ) Ex-citation efficiency of the CPW antenna, approximated by the spa-cial Fourier transformation of the in-plane field distribution. Thewavevector selectivity causes several gaps, visible in ( A ). ( C ) Y -component of the internal magnetic field of the ground state. Thedistribution of the internal magnetic field along the y -direction isstrongly inhomogeneous. ( D ) Normalized absolute value of the dy-namic out-of-plane magnetization component | m dynz | for a frequencyof 7 .
26 GHz and k = . − . ( E ) The extracted spectral dis-tribution shows a large intensity difference for counter-propagatingspin waves. Comparison of the spectral width to Fig. 2 indicates thatthe experiment is limited to the first two excitation efficiency max-ima of the CPW antenna. ( F ) A small frequency non-reciprocity forcounter-propagating waves is found in the dispersion relation whichis caused by the trapezoidal cross-section of the structure. The groupvelocity is much larger than for waves of the corresponding longitu-dinal magnetized state. formed at the edges of the structure. However, since the struc-ture is too small to allow for a homogeneous field region inthe center, a single mode behaviour is observed in contrastto the typically observed appearance of localized edge modestates. This is validated by the mode profile (normalized ab-solute value of the dynamic out-of-plane magnetization com-ponent | m dynz | ) shown in Fig. 3D for a frequency of 7 .
26 GHzand k = . − , which extents fully into the edge regimeof the structure.Extracting the respective spectral distribution, see Fig. 3E,shows an intensity difference for counter-propagating waves( k − and k + ), which is caused by a non-reciprocal excitationefficiency of MSSW when using a microwave antenna .Moreover, comparing the spectra to Fig. 2 indicates thatthe experiment is limited to the first two excitation efficiencymaxima of the CPW antenna. Excitation maxima of higher or-der are likely not observed due to the limited sensitivity of theused BLS setup, especially due to the small amount of probedmaterial and the background of thermal spin waves leading toan increased noise level in the experiment.In Fig. 3F the dispersion relationship, derived from a 6 th -orderpolynomial approximation of Fig. 3A, and the a group veloc-ity, extracted as the derivative of the dispersion relationship,are shown. Similar to microscopic systems, the transverselymagnetized state offers a high and much larger group velocitythan the waves of the corresponding longitudinal magnetizedstate possess .In the following, the group velocity and the decay length aredetermined experimentally by a direct measurement of prop-agating spin-wave packets to further characterize the systemand investigate whether a long range spin-wave propagationcan be realized in such nano-scaled systems. Thus, a pulsedexcitation with 50 ns pulse length and 350 ns repetition timeis used, choosing an applied microwave power of −
10 dBm toensure an operation within the linear regime. In Fig. 4A time-resolved BLS measurements of the excited spin-wave packetsare shown for different positions along the conduit. The re-spective center-of-mass arrival time of the pulse is determinedby subtracting the thermal noise and calculating the weightedaverage of the packet. A linear approximation of the packetarrival time, as shown in Fig. 4B, yields the group velocity v g . Additionally, the decay length can be extracted from theintegrated pulse intensity of each position by approximatingthe decay as follows: I = I exp ( − x / λ D ) + I . (1)Here, I denotes the initial intensity, I the offset intensity, x the position, and λ D the decay length. Extracted accord-ingly to this principle, the resulting group velocities and de-cay lengths are presented in Fig. 4C. The velocity lies withinthe expected range predicted by the simulations when com-pared for wavevectors up to 10 rad µm − , see Fig. 3F, and fol-lows the rising trend to higher frequencies. However, an unex-pected large velocity is observed for the smallest investigatedfrequency not covered by the prediction of the simulations.In contrast, the decay length follows a steep increase up to amaximum of ( . ± . ) µm, slightly dropping off for higherfrequencies. For comparison, the corresponding cw excita- G r oup v e l o c i t y ( µ m n s ) - D e c a y l eng t h ( µ m ) Group velocityDecay lengthSpectrum01020304050 Time (ns) B L S i n t en s i t y ( a r b . u . )
25 50 75 100 125 15000510 0510 P o s i t i on ( µ m ) Arrival timeLinear approximation
A C
Frequency (GHz)7.3 7.35 7.4 7.45 7.5 7.55 7.6 L i f e t i m e ( n s ) BD v g = 0.18 rad/µm -1 plain film FIG. 4. ( A ) Spin-wave pulses detected at different distances fromthe CPW antenna for a chosen excitation frequency of 7 .
47 GHz. ( B )Center-of-mass arrival time of the spin-wave pulses of ( A ). A linearapproximation yields the group velocity of the wave packets. ( C )Group velocity for various excitation frequencies extracted accord-ingly to the principle presented in ( A ) and ( B ). The decay length iscalculated from the integrated intensity of the spin-wave pulses. Thelight gray curve is the associated cw excitation spectrum, slightlyshifted to lower frequencies due to a magnetic field difference of0 . D ) Experimental lifetime derived from the decay length andgroup velocity. All data measured for an applied power of −
10 dBm. tion spectrum is displayed in light gray, which is, however,slightly shifted to smaller frequencies due to a small mag-netic field difference of 0 . . . . This potentially enables complexnano-sized integrated spin-wave circuits consisting of multi-ple elements without any means of intermediate amplification,severely lowering the energy consumption of such circuits. Frequency (GHz) N o r m a li z ed B L S i n t en s i t y ( a r b . u . ) µ H = +270 mT-270 mT µ H =7.2 7.3 7.4 7.5 7.6 7.7024681012 75 MHz FIG. 5. Frequency non-reciprocity of counter-propagating spinwaves. Inverting the field polarity equals the switching of the propa-gation direction for the spin waves. A frequency shift of 75 MHz isobserved. The spectra are recorded for an applied power of 0 dBmat a distance of 7 µm from the CPW antenna. Normalization withrespect to the thermal noise level.
In the following, the lifetime τ of the propagating waves,shown in Fig. 4D, is derived from the experimental resultsusing the expression τ = λ D / v g . (2)Here, a rather large lifetime of up to ( . ± . ) ns is found.Calculating the lifetime of the ferromagnetic resonance ofthe plain film as a comparison , considering the decreasedinternal magnetic field of the transverse magnetization state,results in 112 . . and potentially reduces the lifetime compared to the plainfilm. On the other hand, the non-local inhomogeneouslinewidth broadening extracted from the plain film overesti-mates the effective inhomogeneity on the length scale of thenanostructure . Thus, a comparison to the full linewidth ofthe plain film can yield a lifetime smaller than the lifetimeof the nanostructure. Nonetheless the measured lifetimeexceeds other commonly used materials such as permalloy by far, even when comparing to microscopic or macroscopicsystems.Finally, we would like to discuss a peculiarity of the investi-gated system, observed in Fig. 3 F. The simulated dispersionrelationship exhibits a small frequency non-reciprocity, whichis caused by the trapezoidal cross-section of the structureand the associated internal field distribution (Fig. 3C). Thisintroduces an additional spacial symmetry break, similar tothe case of magnetic bilayers . In Fig. 5 two measuredspectra for normal and inverted field polarity are shown,which equals a switch of the dispersion branch from k − / + to k + / − . Indeed, a distinct non-reciprocity with a frequencyshift of 75 MHz is found, which is substantially larger thanthe predicted shift of 20 MHz–30 MHz for spin waves inthe range of 7 rad µm − –10 rad µm − , see Fig. 3F. It should be noted that, the observed frequency shift is likely influ-enced by the non-reciprocal excitation efficiency for bothconfigurations, leading to different spin-wave densities, andthus to a different nonlinear frequency downshift potentiallyincreasing the observed frequency gap. Nonetheless, sucha pronounced non-reciprocity is of particular interest for anapplication in spin-wave devices since it allows for a noveldevice architecture, e.g. allows for the construction of anano-sized spin-wave diode.To conclude, we presented a study of propagating spin-wave packets in a transversely magnetized nano-scaled YIGconduit of 50 nm width. Micro-magnetic simulations areperformed to support the experimental findings, revealing asingle-mode dispersion relationship in contrast to the commonformation of localized edge modes for microscopic systems.It is shown that the observed mode is not localized withinthe central part of the structure. A large spin-wave groupvelocity is measured and a long-range spin-wave propagationis observed with a decay length of up to ( . ± . ) µm, whichis multiple times larger than reported values for the corre-sponding longitudinal magnetized state . In addition, a largespin-wave lifetime of up to ( . ± . ) ns is found. Further-more, a frequency non-reciprocity for counter-propagatingspin waves is observed and experimentally verified, which iscaused by the trapezoidal cross-section of the structure andthe associated internal field distribution. This non-reciprocityand the revealed long-distance spin-wave propagation on thenanoscale are particularly interesting for an application inspin-wave devices, allowing for long-distance transport ofinformation in magnonic circuits, as well as novel low-energydevice architectures potentially opening up the path tomulti-element circuits without intermediate amplification.This research has been funded by the European ResearchCouncil project ERC Starting Grant 678309 MagnonCircuits,by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) - 271741898, by the Collaborative Re-search Center SFB/TRR 173-268565370 (Project B01), andby the Austrian Science Fund (FWF) through the projectI 4696-N. B.H. acknowledges support from the GraduateSchool Material Science in Mainz (MAINZ). The authorsthank Burkard Hillebrands for support and valuable discus-sions. A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys ,264002 (2010). A. Khitun, M. Bao, and K. L. Wang, J. Phys. D: Appl. Phys , 264005(2010). V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys , 260301 (2010). A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. , 453 (2015). A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V. Chumak, S. Hamdioui,C. Adelmann, and S. Cotofana, J. Appl. Phys. , 161101 (2020). Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi,H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, andE. Saitoh, Nature , 262 (2010). H. Yu, O. d. Kelly, V. Cros, R. Bernard, P. Bortolotti, A. Anane, F. Brandl,R. Huber, I. Stasinopoulos, and D. Grundler, Sci. Rep. , 6848 (2014). C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz,J. Grenzer, R. Hübner, and E. Wendler, Phys. Rev. Mater. , 024416 (2020). T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, andM. P. Kostylev, Appl. Phys. Lett. , 022505 (2008). P. Krivosik and C. E. Patton, Phys. Rev. B , 184428 (2010). V. E. Demidov, J. Jersch, K. Rott, P. Krzysteczko, G. Reiss, and S. O.Demokritov, Phys. Rev. Lett. , 177207 (2009). A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat. Commun. , 4700(2014). A. V. Sadovnikov, E. N. Beginin, M. A. Morozova, Y. P. Sharaevskii, S. V.Grishin, S. E. Sheshukova, and S. A. Nikitov, Appl. Phys. Lett. ,042407 (2016). Q. Wang, B. Heinz, R. Verba, M. Kewenig, P. Pirro, M. Schneider,T. Meyer, B. Lägel, C. Dubs, T. Brächer, and A. V. Chumak, Phys. Rev.Lett. , 247202 (2019). B. Heinz, T. Brächer, M. Schneider, Q. Wang, B. Lägel, A. M. Friedel,D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, andA. V. Chumak, Nano Lett. , 4220 (2020). T. Brächer and P. Pirro, J. Appl. Phys. , 152119 (2018). Á. Papp, W. Porod, Á. I. Csurgay, and G. Csaba, Sci. Rep. , 9245 (2017). Q. Wang, M. Kewenig, M. Schneider, R. Verba, F. Kohl, B. Heinz,M. Geilen, M. Mohseni, B. Lägel, F. Ciubotaru, and et al., Nat. Electron. , 765–774 (2020). 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. , 097205 (2018). J. Cramer, F. Fuhrmann, U. Ritzmann, V. Gall, T. Niizeki, R. Ramos,Z. Qiu, D. Hou, T. Kikkawa, J. Sinova, U. Nowak, E. Saitoh, and M. Kläui,Nat. Commun. , 1089 (2018). S. Klingler, P. Pirro, T. Brächer, B. Leven, B. Hillebrands, and A. V. Chu-mak, Appl. Phys. Lett. , 152410 (2014). T. Fischer, M. Kewenig, D. A. Bozhko, A. A. Serga, I. I. Syvorotka, F. Ciub-otaru, C. Adelmann, B. Hillebrands, and A. V. Chumak, Appl. Phys. Lett. , 152401 (2017). A. Mahmoud, F. Vanderveken, C. Adelmann, F. Ciubotaru, S. Hamdioui,and S. Cotofana, AIP Adv. , 035119 (2020). Q. Wang, P. Pirro, R. Verba, A. Slavin, B. Hillebrands, and A. V. Chumak,Sci. Adv. (2018). F. Heussner, G. Talmelli, M. Geilen, B. Heinz, T. Brächer, T. Meyer,F. Ciubotaru, C. Adelmann, K. Yamamoto, A. A. Serga, B. Hillebrands,and P. Pirro, Phys. Status Solidi Rapid Res. Lett. , 1900695 (2020). Q. Wang, A. Hamadeh, R. Verba, V. Lomakin, M. Mohseni, B. Hillebrands,A. V. Chumak, and P. Pirro, npj Comput. Mater. , 192 (2020). Q. Wang, A. V. Chumak, and P. Pirro, “Inverse-design magnonic devices,”(2020), arXiv:2012.04544. A. G. Gurevich and G. A. Melkov,
Magnetization oscillations and waves (CRC press, 1996). L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. B , 144101 (2013). R. Shindou, J.-i. Ohe, R. Matsumoto, S. Murakami, and E. Saitoh, Phys. Rev. B , 174402 (2013). E. Iacocca and O. Heinonen, Phys. Rev. Appl. , 034015 (2017). X. S. Wang, H. W. Zhang, and X. R. Wang, Phys. Rev. Appl. , 024029(2018). K. Yamamoto, G. C. Thiang, P. Pirro, K.-W. Kim, K. Everschor-Sitte, andE. Saitoh, Phys. Rev. Lett. , 217201 (2019). A. Prabhakar and D. D. Stancil,
Spin waves: Theory and applications ,Vol. 5 (Springer, 2009). M. Mohseni, R. Verba, T. Brächer, Q. Wang, D. A. Bozhko, B. Hillebrands,and P. Pirro, Phys. Rev. Lett. , 197201 (2019). M. Mohseni, B. Hillebrands, P. Pirro, and M. Kostylev, Phys. Rev. B ,014445 (2020). U. K. Bhaskar, G. Talmelli, F. Ciubotaru, C. Adelmann, and T. Devolder,J. Appl. Phys. , 033902 (2020). C. Bayer, J. P. Park, H. Wang, M. Yan, C. E. Campbell, and P. A. Crowell,Phys. Rev. B , 134401 (2004). G. Gubbiotti, M. Conti, G. Carlotti, P. Candeloro, E. D. Fabrizio, K. Y.Guslienko, A. Andre, C. Bayer, and A. N. Slavin, J. Phys. Condens. Matter , 7709 (2004). P. Pirro, T. Brächer, A. V. Chumak, B. Lägel, C. Dubs, O. Surzhenko,P. Gärnert, B. Leven, and B. Hillebrands, Appl. Phys. Lett. , 012402(2014). C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Brückner, and J. Del-lith, J. Phys. D: Appl. Phys , 204005 (2017). S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Ka-bos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. , 093909 (2006). I. S. Maksymov and M. Kostylev, Physica E Low Dimens. Syst. Nanostruct. , 253 (2015). T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss,Front. Phys. , 35 (2015). A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,and B. Van Waeyenberge, AIP Adv. , 107133 (2014). V. E. Demidov, M. P. Kostylev, K. Rott, P. Krzysteczko, G. Reiss, and S. O.Demokritov, Appl. Phys. Lett. , 112509 (2009). C. Hahn, V. V. Naletov, G. de Loubens, O. Klein, O. d’Allivy Kelly,A. Anane, R. Bernard, E. Jacquet, P. Bortolotti, V. Cros, J. L. Prieto, andM. Mu ˜ n oz, Appl. Phys. Lett. , 152410 (2014). K. Yamanoi, S. Yakata, T. Kimura, and T. Manago, Jpn. J. Appl. Phys. ,083001 (2013). M. Grassi, M. Geilen, D. Louis, M. Mohseni, T. Brächer, M. Hehn, D. Sto-effler, M. Bailleul, P. Pirro, and Y. Henry, Phys. Rev. Appl. , 024047(2020). K. Vogt, F. Y. Fradin, J. E. Pearson, T. Sebastian, S. D. Bader, B. Hille-brands, A. Hoffmann, and H. Schultheiss, Nat. Commun. , 3727 (2014). T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, and M. P. Kostylev,Phys. Rev. B77