Cite as: J. Appl. Phys. 127, 093903 (2020); https://doi.org/10.1063/1.5141793
Submitted: 07 December 2019 . Accepted: 19 February 2020 . Published Online: 05 March 2020 I. A. Golovchanskiy , N. N. Abramov, V. S. Stolyarov , A. A. Golubov, V. V. Ryazanov , and A. V. Ustinov
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Cite as: J. Appl. Phys. 127, 093903 (2020);doi: 10.1063/1.5141793
View Online Export Citation CrossMark Submitted: 7 December 2019 · Accepted: 19 February 2020 ·
Published Online: 5 March 2020
I. A. Golovchanskiy,1,2,3,a) N. N. Abramov,2V. S. Stolyarov,1,3 A. A. Golubov,1,4V. V. Ryazanov,2,5,6 and A. V. Ustinov2,7,8
AFFILIATIONS
1Moscow Institute of Physics and Technology, State University, 9 Institutskiy per., Dolgoprudny, Moscow Region 141700, Russia 2National University of Science and Technology MISIS, 4 Leninsky prosp., Moscow 119049, Russia
3Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia
4Faculty of Science and Technology and MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands
5Institute of Solid State Physics (ISSP RAS), Chernogolovka 142432, Moscow Region, Russia 6Solid State Physics Department, Kazan Federal University, 420008 Kazan, Russia
7Russian Quantum Center, Skolkovo, Moscow 143025, Russia
8Physikalisches Institut, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany a)Author to whom correspondence should be addressed:golov4nskiy@gmail.com
ABSTRACT
This work is focused on the numerical investigation of spin waves that propagate in nonlinear ferromagnet/superconductor bilayered films and periodic structures. The nonlinearity in these hybrid structures emerges due to the non-monotonous dependence of magnetization of a superconducting subsystem on the magnetic field, which is characterized by the superconducting critical field. It is shown that at relatively high amplitudes of spin waves in comparison to the superconducting critical field, the wave spectrum changes drastically: the spin-wave spectral line can either bifurcate or stretch continuously depending on the type of considered superconductor. In addition, in the case of propagation of spin waves with relatively high amplitude in periodic magnonic metamaterials, additional zero-group-velocity modes appear that are known as flatbands. Overall, these findings suggest a versatile way for tunability of the spin-wave spectrum in nonlinear fer-romagnet/superconductor structures by changing the excitation signal in respect to the superconducting critical field.
Published under license by AIP Publishing.https://doi.org/10.1063/1.5141793
I. INTRODUCTION
Magnonics is a rapidly developing field of research that on the practical side studies possibilities to transfer and process signals via spin waves, while on the fundamental side deals with magnon qua-siparticles and their interaction with each other and with other species of quasiparticles. Most often, magnonics considers spin waves in a linear magnonic environment and rely on the lineariza-tion of the Landau–Lifshitz–Gilbert equation. Such a linearization holds applicable for sufficiently small angles of magnetization precession.
More sophisticated spin-wave phenomena appear when linear-ization is violated and the nonlinear magnonics emerges.1,2 In general, nonlinear magnonics is characterized by many-magnon
scattering processes. Thus, three-magnon processes1,3,4 trig para-metric phenomena,5–10 while four-magnon processes typically increase the linewidth of spectral lines. In recent years, a remark-able progress has been achieved in experimental studies that involve nonlinear magnonics in confined magnetic systems. For instance, in thin films, parametric excitation of magnons has been employed for the creation of the Bose–Einstein condensate.11 In micro-scaled ferromagnetic disks, parametric excitation of whis-pering gallery magnon states9has been achieved. In YIG-based active magnonic ring, unconventional soliton spin-wave modes12have been observed. Also, it has been shown that nonlinear damping of spin-wave activity in monodomain ferromagnetic films assists in the redistribution of the spatial profiles of dynamic magnetization.13,14
The effects of lateral confinement on spin-wave propagation at high signal powers cause the self-modulation of spin-wave beams.13 In coupled magnonic waveguides, nonlinear phenomena guide the intensity-dependent power exchange between the waveguides.15This rich variety of phenomena has been observed in conventional mag-netic systems.
Currently, a hybrid magnonics is developing, where spin dynamics interplays with superconducting phenomena in ferromagnet/superconducting (FM/SC) systems resulting in mod-ulation of a spin-wave spectrum. For instance, the interaction of spin waves with the hexagonally ordered magnetic perturbations that are induced by the superconducting vortex phase results in the formation of forbidden bands for spin-wave propagation.16In contrast, the interplay of spin dynamics with disordered pinned vortex phase17 leads to a magnetic hysteresis in the resonance
spectrum. In the case of ferromagnet/superconductor bi-layered structures in the absence of the vortex phase, the magnetostatic interaction of spin waves with superconducting Meissner screen-ing currents takes place, which enhances phase velocity of spin waves in continuous bilayers18and leads to the formation of for-bidden bands for spin-wave propagation in periodic bi-layered hybrid metamaterials.19
So far, studies of the interplay of spin waves with Meissner currents in FM/SC considered the linear response of superconduc-tors on the magnetic field, i.e., the ideal diamagnetic Meissner effect. However, the Meissner state is terminated upon reaching the superconducting critical field, which is relatively low as compared to field scales for conventional ferromagnets. This makes the dependence of diamagnetic moment of SC on the magnetic field a nonlinear non-monotonous characteristic. This work aims for the development of a general understanding and expectations for non-linear spin-wave phenomenon in FM/SC bilayered structures, which occurs when the amplitude of stray fields exceeds this critical field. This work is organized as follows. Simulation details are pro-vided in Sec.II. In Sec.III, first, we compare dispersion relations of spin waves in FM/SC bilayers that are obtained using two different numerical methods. Next, we discuss the nonlinear behavior of spin waves in FM/SC bilayers. Finally, we discuss the nonlinear behavior of spin waves in periodic FM/SC magnonic structures. Concluding remarks are given in Sec.IV.
II. SIMULATION DETAILS
The dispersion relation for spin waves in ferromagnetic media can be studied numerically using micromagnetic simulations20,21as demonstrated in Refs.22–24. For the excitation of spin waves in a thin ferromagnetic film in a wide frequency range, a local magnetic field pulse of a sinc temporal profile is applied to the film. After the excitation, the time evolution of local magnetic moments in the film ~M(~r, t) is recorded. The maximum of the amplitude of the space-time Fourier transform spectrum of ~M(~r, t) provides the dispersion relation f (~k). In the case of a specified spin-wave mode in thin film geometry, it is sufficient to perform micromagnetic simulations employing a 1D mesh in order to capture correctly the spin-wave activity.25 For the purpose of conciseness, only the
magnetostatic surface spin-wave (MSSW) mode25–27is in focus of this study.
In general, the magnetostatic problem of the FM/SC hybrid can be treated with micromagnetic simulations as magnetostatic interaction of two ferromagnets in two ways. The magnetostatic problem of a finite-sized ferromagnet placed near infinite super-conducting surface can be treated with the method of images.18,28 The method of images implies the magnetostatic interaction of magnetic moments of the ferromagnet ~M(x, y, z)¼ (Mx, My, Mz)
located at a distance z above the superconducting surface x y with their mirror image moments ~Mim(x, y,z) ¼ (Mx, My,Mz). The
magnetostatic interaction of a ferromagnet with a finite-sized super-conducting structure can be treated as magnetostatic interaction of magnetic moments of the ferromagnet ~M(x, y, z) with local magnetic moments of a superconductor ~Msc(x, y, z), which are induced as a
diamagnetic response on the external field and on stray magnetic fields induced by the ferromagnet,19i.e., ~M
sc(x, y, z)¼ ~H(x, y, z).
In both cases, FM and SC subsystems are considered as electrically isolated, implying purely magnetostatic interaction and absence of superconducting proximity. The numerical representation of a super-conductor in micromagnetic simulations is realized (i) by the expan-sion of the micromagnetic mesh on superconducting subsystem for inclusion of“superconducting” macro-spins (~Mscor ~Mim); (ii) by the
addition of an intermediate integration step for calculation of stray fields and corresponding superconducting magnetic moments in “superconducting” cells; and (iii) by the calculation of additional dipole component of the effective field that is caused by the super-conducting subsystem at each time step of integration of the Landau–Lifshitz–Gilbert equation.
In this work, a L W ¼ 900 900 μm2 rectangular
permal-loy film is considered of thickness d¼ 100 nm, placed in the x y plane. The film is 1D meshed with L 100 d nm3 cell size
along the y axis. For excitation of MSSWs, the magnetic field is applied along the x-axis and the sinc field pulse is applied along the z axis. The following micromagnetic parameters, typical for permalloy films, have been used for simulations: saturation magnetization Ms¼ 9:3 105A/m, zero anisotropy field,
external magnetic field H¼ 5:6 103A/m, the exchange stiffness
constant A¼ 1:3 1011J/m, and the gyromagnetic ratio μ0γ ¼ 2:21 105m/A/s. The excitation field pulse has the
maximum frequency fmax¼ 30 GHz, the Gaussian spatial profile
with the width at half-maximum of 200 nm, and the amplitude of 0:001 Ms. The typical amplitude of spin-wave precession that is
excited by such field pulse is about 104Ms.
III. RESULTS AND DISCUSSIONS A. Diamagnetic representation of SC
Figure 1 compares dispersion relations for spin waves that propagate in the plain FM film and FM/SC bilayers. The latter are obtained using both the method of images and the diamagnetic representation of SC. In general, dispersion curves for FM/SC bilay-ers are nonreciprocal: at positive wavenumbbilay-ers, the phase velocity /f λ of spin waves in FM/SC bilayers is enhanced substantially as compared to the plain FM films and remains practically unchanged at negative wavenumbers. The nonreciprocity emerges for the MSSW mode due to the localization of the wave energy at a particular surface of the film depending on the orientation of the wavevector in respect to the applied field28,29and, subsequently, the efficient
interaction of the spin wave with a superconductor for a specific direction of spin-wave propagation. Basically, any symmetry break across ferromagnetic film that can be induced by covering of the film with a non-ferromagnetic conducting metal30 or with a
superconductor,28 by the formation of a bilayer film structure with different ferromagnetic materials and antiferromagnetic alignment,31 or by engineering the interfacial Dzyaloshinskii– Moriya interaction32 leads to the formation of nonreciprocity in
the dispersion of spin waves.
Figure 1 indicates that the phase velocity of spin waves in FM/SC bilayers is higher when it is obtained using the method of images as compared to the diamagnetic representation of SC. This discrepancy between dispersions that are obtained with the method of images and with the diamagnetic representation of SC has rather straightforward origin. It emerges due to a difference in volume of SC that is considered in these two models. Thus, the method of images represents a complete diamagnetic screening of all stray fields that are distributed beyond the infinite SC surface, while the diamagnetic representation of SC implies diamagnetic screening of stray fields that are distributed in the specified volume of the finite-sized SC structure. This dependence of the dispersion relation on dimensions of SC is well illustrated in Fig. 1, where for positive wavenumbers, the phase velocity of the FM/SC bilayer with a 400 nm thick SC layer (blue curve inFig. 1) is enhanced as com-pared to the FM/SC bilayer with a 100 nm thick SC layer (red curve inFig. 1).
It should be noted that the diamagnetic representation of superconductors using the specified local screening rule
~
Msc(x, y, z)¼ ~H(x, y, z) is not quite accurate since the
supercon-ducting Meissner state is a non-local state and is represented by the zero magnetic flux inside a SC rather than by its ideal local diamag-netism. For instance, in a uniform external field, the Meissner state of a superconductor is induced by the nonuniform distribution of screening currents and corresponding local magnetizations that “squeeze out” both the external field and stray self-fields to edges of a finite-sized superconductor.33This state accounts demagnetizing effects. Such a state cannot be achieved using micromagnetic simu-lations and without implementation of additional physical princi-ples, for instance, London equations. Nevertheless, the diamagnetic representation of SC is easily integrated into micromagnetic simu-lations and, according to Fig. 1, it provides adequate solution. Below, we use the diamagnetic representation of superconductors to study spin waves in FM/SC hybrids in presence of nonlinear response of SC.
B. Nonlinear dependence of magnetization of SC on the magnetic field
In previous works,18,19,28as well as inFig. 1, the diamagnetic response of a superconductor on the magnetostatic field was con-sidered as ideal, i.e., as a linear diamagnetic response. However, the dependence of magnetization of superconductors on the magnetic field is essentially nonlinear and can be characterized by the critical field Hc(seeFig. 2). For type-I superconductors, the ideal
diamag-netic moment is induced at magdiamag-netic fields below the critical field Hcand drops to zero above this critical field. For type-II
supercon-ductors, the ideal diamagnetic moment is induced likewise at mag-netic fields below the so-called lower (or the first) critical field, which is also indicated as Hc inFig. 2. Upon increasing the
mag-netic field above the first critical field, the diamagmag-netic moment of type-II superconductors reduces progressively due to the penetra-tion of Abrikosov vortices and approaches zero at the so-called upper (or the second) critical field Hc2. For calculations in this
work, magnetization of a type-II SC at H. Hcis considered
cons-tant. This consideration holds for a case of Hc Hc2 and
Hc, H Hc2; it has been verified in Ref.19. A finite Hc2 can be
FIG. 1. Comparison of dispersion relations for spin-wave propagation in the FM film and FM/SC bilayers in MSSW geometry. Solid black curves show the dis-persion for plain FM film. Dashed black curves show the disdis-persion for the FM film placed on infinite SC surface. The dispersion is obtained using the method of images. Solid red curves and blue curves show the dispersion for the FM/SC bilayered film with a 100 nm thick SC layer and 400 nm thick SC layers, respectively.
FIG. 2. Schematic illustration of dependencies of the magnetization of type-I and type-II superconductors on the magnetic field. AtH , Hc, magnetization shows an ideal diamagnetic response withM ¼ H for both types of SC. At H . Hc, magnetization of the type-I SC isM ¼ 0 and of the type-II SC, it is M ¼ Hc. Arrows indicate reversibility of M(H). These dependencies are employed in simulations of nonlinear spin-wave spectra.
taken into account by the substitution of constant M(H) at H. Hc with a linear paramagnetic dependence that reduces M(H)
upon increasing H. Hcand reaches zero at H¼ Hc2.
Typically, the critical field and the first critical field of differ-ent superconductors at zero temperature range from 1 mT up to 100 mT. Also, these critical fields decay with temperature T follow-ing a known parabolic law Hc/ (1 (T=Tc)2), where Tc is the
superconducting critical temperature. In addition, macroscopic Hc
is reduced for superconductors with a finite demagnetizing factor33 and for disordered superconducting materials. On the contrary, the stray fields of ferromagnets are scaled with the saturation magneti-zationμ0Ms, which ranges conventionally from about 200 mT to
1 T and above. Therefore, in a FM/SC hybrid above a certain threshold of an excitation field Hac, stray fields that are induced by
precessing FM moments in SC can exceed the critical field Hc. In
this case, the nonlinear response of SC on alternating stray fields should modify the spin-wave spectrum. In this work, the depen-dence of the spin-wave spectrum on the ratio of the excitation pulse to the critical field Hac=Hc is studied in a wide range of the
ratio from 0 up to 500. The most representative cases are discussed below for both FM/SC bilayers and for FM/SC periodic structures.
It should be noted that M(H) dependencies that are given in
Fig. 2are a greatly simplified model of magnetization of a super-conductor in the AC magnetic field. For instance, dissipation pro-cesses that are induced by microwave excitation of quasiparticles in a superconductor34–36 are not considered in this work. Typically, such excitations would either contribute to the linewidth of a spin-wave signal due to additional resistive losses, which does not alter the position of a spectral line, or result in heating of a SC subsys-tem, which would reduce the critical field Hc. An opposite process
that is mediated by microwave excitation of quasiparticles and result in enhancement of superconducting characteristics, namely, the microwave-stimulated superconductivity34,37is not considered as well. The effect of the superconducting domain/vortex state in SC subsystem on magnetization dynamics is also beyond the scope
of this work. Pinned DC magnetic flux in a SC subsystem induces magnetic inhomogeneities in an FM subsystem resulting in the reduction and hysteresis of spin-wave signal,17 while the moving
magnetic flux in SC subsystem realizes an additional channel for resistive dissipation.36Importantly, M(H) in Fig. 2are considered as reversible, implying the absence of hysteresis losses. Therefore, the model of SC inFig. 2should be considered as a toy-model for general understanding of possible nonlinear phenomena in FM/SC hybrids rather than for accurate calculations.
C. Propagation of spin waves in nonlinear FM/SC bilayers
Figure 3(a)shows the spectrum of MSSW that propagates in a linear FM/SC bilayer (i.e., Hac=Hc¼ 0) with a 100 nm thick FM
layer and a 300 nm thick SC layer. As discussed in Fig. 1, the SC layer enhances substantially the phase velocity of spin waves for positive wavenumbers. The spectrum is unchanged for all values of Hac=Hc, 3 for both types of SC. On the contrary, at
Hac=Hc! 1, the response of the SC layer is absent and the
disper-sion recovers to one shown inFig. 1for the plain FM film. Spectra that reflect the nonlinear response of SC layer in the most explicit way are obtained with Hac=Hc¼ 100 and are shown
inFig. 3(b)for type-I SC and inFig. 3(c)for type-II SC. For type-I SC [Fig. 3(b)] at positive wavenumbers, the spectral line bifurcates. The upper spectral line matches the dispersion curve for spin waves in the presence of a linear SC response [Fig. 3(a)], while the lower spectral line matches the dispersion curve in the absence of super-conductor. For type-II SC [Fig. 3(c)], the spectral line at positive wavenumbers is stretched continuously between dispersion curves for spin waves in linear FM/SC bilayer and in the plain FM film.
In general, a broadening/split of a spin-wave spectral line might have negative consequences for applications. Indeed, a majority of spin-wave applications are designed employing micro-wave antennas (see, for instance, Refs. 38–40) and, consequently,
FIG. 3. Spin-wave spectra for the FM/SC bilayer with a 100 nm thick FM layer and a 300 nm thick SC layer. The colorbar is scaled linearly and is given in arbitrary units, and the maximum of the amplitude in a spectrum is shown in red. (a) The spectrum for the FM/SC bilayer with a linear SC response. (b) The spectrum for the FM/SC bilayer with type-I SC andHac=Hc¼ 100. (c) The spectrum for the FM/SC bilayer with type-II SC and Hac=Hc¼ 100.
are engineered for a specified wavelength of operation. When the spectral line is broadened/split, it reduces the quality factor of the transmitted spin-wave signal, which is one of the most important practical characteristics. However, a broadening or a split of a spin-wave spectral line shown inFig. 3may appear to be useful in non-linear magnonics1,2 that study multi-magnon scattering processes and rely typically on momentum and energy conservations.3,4 Importantly, to enter the nonlinear regime in FM/SC bilayers, it is required to reduce sufficiently the superconducting critical field rather than to achieve high spin precession angles, as in conven-tional nonlinear magnonics.
D. Propagation of spin waves in nonlinear FM/SC periodic structures
Next, we study the effect of a nonlinear diamagnetic response of SC on spin waves in periodic magnonic metamaterials.19
Figure 4(a)shows a schematic illustration of the FM/SC magnonic crystal in the MSSW geometry. The magnonic crystal consists of a 100 nm thick FM layer and a 300 nm thick one-dimensional SC grid; the grid has the lattice parameter a¼ 1 μm and the width of SC stripes 700 nm.Figure 4(b)shows the spectrum of MSSW that propagates in a linear FM/SC magnonic crystal. The spectrum is nonreciprocal and it consists of indirect bandgaps that are opened at positive wavenumbers at f . 16 GHz. At f , 16 GHz, the spec-trum is continuous in the frequency domain. In addition to bandg-aps, periodicity of the FM/SC magnonic crystal is manifested in the repetition of spectral lines at wavenumbers that are multiple to the Bragg wavenumber 1=a, which is indicated by an additional branch at 1=λ . 1 μm and f 10 GHz.
Figure 5 shows spin-wave spectra of the same FM/SC magnonic crystal in the presence of type-I and type-II nonlinearity. At relatively small amplitude of excitation pulse, Hac=Hc¼ 5
[Figs. 5(a)and5(b)], the band structure at f . 16 GHz is distorted FIG. 4. (a) Schematic illustration of the FM/SC magnonic crystal that operates in the MSSW geometry. (b) Spin-wave spectrum for the FM/SC magnonic crystal in (a) a 100 nm thick FM layer and a 300 nm thick SC layer, with the lattice parameter along the y axis a ¼ 1 μm, and the width of SC stripes 700 nm. The spectrum is calculated considering the linear SC response. The colorbar is scaled linearly and is given in arbitrary units. The maximum of the amplitude in a spectrum is shown in red and corresponds to the spin-wave dispersion.
FIG. 5. Spin-wave spectra for the same FM/SC magnonic crystal as inFig. 4in the presence of a nonlinear response of SC. The colorbar is scaled linearly and is given in arbitrary units, and the maximum of the amplitude in a spectrum is shown in red. (a) The spectrum for the FM/SC magnonic crystal with type-I SC andHac=Hc¼ 5. (b) The spectrum for the FM/SC magnonic crystal with type-II SC andHac=Hc¼ 5. (c) The spectrum for the FM/SC magnonic crystal with type-I SC and Hac=Hc¼ 100. (d) The spectrum for the FM/SC magnonic crystal with type-II SC andHac=Hc¼ 100.
while the continuous part of the spectrum at f , 16 GHz remains unchanged. Interestingly, in the case of type-II nonlinearity [Fig. 5(b)], an additional set of zero-group-velocity modes with df=dk ¼ 0 appears, also known as dispersionless modes or isolated flatbands. The flatbands appear at positive wavenumbers in the fre-quency range from about 16 GHz up to 22 GHz at frequencies inside bandgaps of the linear FM/SC system. The set of flatbands appears for a range of excitation pulses of 5, Hac=Hc, 20 (not
shown). These modes are reproducible in respect to the variation of the time step or the cell size. In general, artificial systems with flatband represent a new class of metamaterials41 that “mimic”
electronic condensed matter systems and include a variety of dif-ferent artificial lattices: periodic cold atom lattices,41 electronic lattices,41,42 and photonic lattices.43 In the case of a magnonic crystal inFig. 5(b), these flatbands indicate a formation of compact localized magnon states that do not propagate or propagate slowly. This property can be used for development of systems with slow-propagating spin waves.
At a moderate amplitude of excitation pulse, Hac=Hc¼ 100
[Figs. 5(c)and5(d)], eigenstates at f . 16 GHz are practically van-ished for both types of nonlinearity. In the case of type-I nonlinear-ity [Fig. 5(c)], the spectral line at positive wavenumbers at f, 16 GHz bifurcates following the same trend as for the continu-ous FM/SC bilayer [Fig. 3(b)]: the upper spectral line matches the dispersion curve for spin waves in the presence of a linear SC response, while the lower line corresponds to the dispersion curve of the plain FM film. In the case of type-II nonlinearity [Fig. 5(d)], the spectral line at positive wavenumbers at f, 16 GHz also follows the trend for the FM/SC bilayer [Fig. 3(d)]: the line stretches continuously between dispersion curves for spin waves in the linear FM/SC magnonic crystal and for the plain FM film. At Hac=Hc. 200, spin-wave spectra for FM/SC magnonic crystals
with both types of nonlinearity restore to the spectrum of the plain FM film.
IV. CONCLUSION
Summarizing, this work reports the numerical investigation of magnetostatic surface spin waves that propagate in ferromagnet/superconductor bilayers and periodic magnonic struc-tures. The impact of the superconducting subsystem is implemented into numerical simulations employing the local diamagnetic repre-sentation of a superconductor. The focus is made on the modifica-tion of the spin-wave spectrum due to the presence of a nonlinear response of superconducting subsystem on stray magnetic fields. Both type-I and type-II superconductors are considered; the nonline-arity is characterized by the superconducting critical field.
It is shown that in the case of a spin-wave propagation in FM/SC bilayers when the amplitude of the excitation pulse exceeds substantially the critical field, the spin-wave spectrum changes drastically. In the case of type-I nonlinearity, the spectral line bifur-cates; the upper-frequency line corresponds to the dispersion of spin waves in the linear FM/SC bilayer while the lower-frequency line corresponds to the dispersion of the plain FM film. In the case of type-II nonlinearity, the spectral line spreads continuously between the two limits mentioned above.
In the case of spin-wave propagation in the FM/SC periodic structure when stray fields exceed weakly the critical field, the band structure is distorted. In addition, in the case of a type-II supercon-ductivity, a set of additional flatbands appear at frequencies within the bandgaps. When the amplitude of the excitation pulse exceeds substantially the critical field, the part of the spectrum with the band structure vanishes and the part of the spectrum with continu-ous dispersion undergoes the same modification as in the case of FM/SC bilayers. Overall, these findings suggest a versatile way for the tunability of the spin-wave dispersion in FM/SC structures.
ACKNOWLEDGMENTS
The authors benefited from fruitful discussions with Professor G. Tsironis. The authors acknowledge the Russian Science Foundation (RSF) (Project No. 18-72-00224) and the Ministry of Education and Science of the Russian Federation (Project No. K2-2018-015 in the framework of the Increase Competitiveness Program of NUST MISiS) for support of this research study.
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