In this work, the propagation of spin waves in translational symmetry broken irregular magnonic waveguide is investigated. The mechanism of the transfer of the magnetic moment and the conversion of spin waves from backward volume to surface waves is revealed. Mechanisms for controlling spin-wave transport by changing the direction of an external magnetic field are investigated by the method of micromagnetic modeling. The results of studies of structures with broken translational symmetry open up new possibilities for the formation of multilayer magnonic networks of various topologies and miniaturization of computing devices based on the principles of magnonics.

## INTRODUCTION

Nowadays the interest in condensed matter physics is arisen to the field of dielectric magnonics - the study of the mechanisms of excitation and control of magnetostatic spin waves (MSW) in magnetic materials.^{1–3} One of the key problems in the development of spin-wave devices is associated with the damping of MSWs.^{4} The solution to this problem can be achieved by using films of yttrium iron garnet (YIG).^{4} In YIG it is possible to excite and observe to propagation of MSWs and exchange dominaed spin waves with wavelengths from tens of nanometers to several millimeters.^{2} Improvement of the technological process of forming thin magnetic YIG films made it possible to fabricate multilevel micro- and nanostructures for the implementation of magnon networks (MN),^{5–8} on the basis of which it is possible to fabricate an element base for information signal processing devices.^{9,10} One of the promising areas of application of magnon networks is the processing of information signals in a wide frequency range based on spin-wave interference effects and amplitude-phase coding.^{11,12} The amplitudes and phases of MSWs in magnonic devices can be controlled in various ways, for example, by creating periodic structures — magnonic crystals or using nonlinear spin-wave effects.^{13,14} One of the approaches for controlling spin-wave transport is based on the transformation of an effective static magnetic field inside a magnetic microstructure. Such a transformation can be realized, for example, by a local change in the material parameters of magnetic media, geometric dimensions of magnon waveguides or spatial variation of external magnetic fields.^{14} Investigation of the mechanisms of spin-wave transport in three-dimensional magnonic structures makes it possible to increase the number of functional elements in multilevel MN.^{6,15} We proposes and investigates a variant of implementing interlevel spin-wave transport between parallel layers of MN based on orthogonally articulated magnetic microwave guides. With the help of numerical studies, the features of the transmission of the spin-wave signal at orthogonal coupling of the waveguide sections are revealed. It is shown that, based on the effect of the transformation of SW types in the irregular region of the waveguide structure, it is possible to effectively control the spin-wave transport by deviating the magnetization angle of the structure. Changing the geometric parameters of the sections allows to effectively control the wavelength of the MSW. The proposed element for interconnecting the functional blocks of multilayer MS also makes it possible to implement functional processing of information signals (frequency filtering and phase correction) when transmitting a signal between parallel layers of 3D magnonic network.

## RESULTS AND DISCUSSION

To create in micromagnetic modeling (MM) a structure with the possibility of interlevel transmission of a spin-wave signal, thin YIG films were used as magnetic microwaveguides $[Y3Fe2(FeO4)3(111)]$ with thickness from 10 *μ*m and saturation magnetization 4*πM*_{s} = 1750 G on a 500 *μ*m thick gallium-gadolinium garnet substrate [(*GGG*)*Gd*_{3}*Ga*_{5}*O*_{12}(111)]. The dimensionless dissipation parameter was set equal to *α* = 10^{−5}, and the exchange stiffness *A*_{ex} = 3^{−7} erg/cm. Each of the three orthogonally connected microwave guides (*A*, *B*, *C*) had the width *w* = 500 *μ*m. The length of each section was *l*_{1} = *l*_{2} = *l*_{3} = 2500 *μ*m. The initial structure (Fig. 1(a)) can be represented as a sequential connection of two partial systems *AB* in the *xy* plane and *BC* in the *xz* plane placed in an external uniform magnetic field, *H*_{0} = 1200 Oe, directed along axis *y*, providing effective excitation of backward bulk magnetostatic waves.^{4} Such a configuration of microwave guides and external magnetic field **H**_{0} provides conditions for propagation of a backward volume magnetosatic spin wave (BVMSW) in the *A* section and a magnetostatic surface spin wave (MSSW)^{4} in the *B* and *C* sections.

To study spin-wave transport, micromagnetic modeling (MM) was carried out in the *MuMax*3^{16} program based on the numerical solution of the Landau-Lifshitz-Hilbert equation by the Dorman-Prince method:

which describes the precession of the magnetic moment **M** in the effective magnetic field **H**_{eff} = **H**_{0} + **H**_{demag} + **H**_{ex} + **H**_{a}, where **H**_{0} – external magnetic field, **H**_{demag} – demagnetization field, **H**_{ex} – exchange field, **H**_{a} – anisotropy field. Solution (1) was carried out at **H**_{a} = 0, i.e. for the case of isotropic magnetic media. Micromagnetic modeling based on solution (1) includes the solution of two problems: static (distribution of internal static magnetic fields in the structure) and dynamic (propagation of spin-wave excitations).

To study the characteristics of SW propagation in the waveguide system, regions for excitation and detection of spin waves are defined. To reduce SW reflections from waveguide boundaries, ABL regions (absorbing boundary layers, Fig. 1(a)) with an exponentially increasing dissipation parameter *α* = 10^{−5} ÷ 1 were introduced in the numerical simulation.^{17} In the input section of the microwave guide *A*, a 50 = *μ*m wide excitation antenna with a length equal to the waveguide width was located. The external magnetic field of linear polarization in the antenna region was set in the form: b_{z}(*t*) = b_{0}*sinc*(2*π*f_{c}*t*), where f_{c} = 6 GHz, b_{0} = 10 mOe. Color gradations in Fig. 1(b) encoded the two-dimensional distribution of the quantity $D(k,f)=1N\u2211i=1N\theta 2[m(x,y,z,t)]2$, where *k* is the wave number, *θ*_{2} is the two-dimensional Fourier transform operator, *i* is the cell number, and *N* = 256 is the number of cells along the *ξ*-coordinate (length of the curve drawn along the longitudinal axes of the sections from the inlet to the outlet of the structure) into the middle of *A*, *B*, *C* waveguides. The *D*(*k*, *f*) map is the squared modulus of the magnetization amplitude and makes it possible to restore the effective dispersion characteristics for spin-wave modes in sections *A* and *B*, while on the plane (*f*, *k*) it turns out to be possible to select a set of points corresponding to local maximum of **D( k, f)** corresponding to nth order width modes with transverse wave numbers

*κ*

_{eff}=

*nπ*/

*w*. On the resulting map of dispersion characteristics, one can see the frequency region Δ(

*f*), in which two types of waves can simultaneously coexist - BVMSW and MSSW. Curves

*F*

_{n}and

*G*

_{n}(

*n*= 1, 2, 3) mark the width modes for waves propagating in sections

*A*and

*B*along the y and x axes, respectively. The SW propagation is anisotropic in nature and has a dispersion that substantially depends on the mutual orientation of the magnetization and the equilibrium magnetization vector.

^{4}In an unbounded film, BVMSW and MSSW have different frequency ranges of existence, separated by the frequency

*f*

_{0}. A change in the geometric parameters of transversely bounded waveguide structures leads to a transformation of the internal magnetic field profile, which makes it possible to observe a shift in the frequency range of BVMSW and MSSW. Fig. 2 (c) shows the dependence of the regions of coexistence of both types of waves on the width and thickness of the waveguide sections. It is seen that the decrease in the spatial dimensions of the waveguide leads to the increase of the region Δ

*f*. Next, the discrete time series were obtained in the regions

*S*

_{A},

*S*

_{B},

*S*

_{C}acting as receiving antennas, for the components of high-frequency magnetization oriented perpendicular to the planes of the each magnetic stripe $m(i)=\u222bVm(x,y,z,t)dV$, where

*V*- the volume of the receiving antenna area, Δ = 75 ps is the sampling step,

*T*= 300 ns is the time series length,

*i*= (1…

*T*/Δ). From time realizations using the discrete Fourier transform, the spectral power densities of magnetization oscillations were obtained

^{5}for different orientations of a uniform external magnetic field

**H**

_{0}relative to the structure under study:

*P*

_{out}(

*f*) = 20

*log*

_{10}|

*θ*(

*m*)|, where

*θ*is the discrete Fourier transform operator.

Figures 2(a, d, g) show the frequency dependence of the transmission coefficient of the output signal in the sections *S*_{A}, *S*_{B}, *S*_{C}, respectively, in the case, when the external magnetizing field is rotated by an angle *φ* in the *xy* plane and *β* in the *xz* plane. It can be seen that, depending on the change in the bias field, the value of frequency of the ferromagnetic resonance of the BVMSW is shifted (Fig. 2(a)). When the external magnetizing field is deflected by an angle *β*, the decrease in the power of the signal received in the *S*_{B} antenna area is about 10-30 dB (Fig. 2(c)). There is also a general decrease in power of the order of 10-15 dB. It can be seen from Fig. 2(g) that a small value of deviation of the external magnetizing field around (∼0-15°) does not affect the SW transmission through the bend between sections *A* − *B*, but a large value of deviation (∼15-30°) significantly affects the power value of the detected signal. At *φ* = 30° at frequencies from 5.15 GHz and above, almost complete signal attenuation is observed. A deviation of *β* = 30° almost completely prevents signal propagation in the *S*_{C} section. Figures 2(b, e, h) and 2(c, f, i) show the result of calculating the effective wavenumbers in the frequency range of the coexistence of two types of waves (BVMSW and MSSW): *k*_{eff}(*f*) = (*ψ*(*f*) − *ψ*_{s}(*f*))/*L*, where $\psi (f)=\u222b0Lk(o)do$ is the SW phase shift, which occurs on the length *L* between the input and output sections and *ψ*_{s}(*f*) is the initial phase of the SW source at the frequency *f*. It can be seen that, in section *A*, the excited wave type (BVMSW) leads to the dependence *k*_{eff}(*f*), which qualitatively coincides with the dispersion dependence for the BVMSW *F*_{n} modes calculated for the input section (Fig. 2(b, c)). However, for sections *B* and *C*, it can be seen that the phase incursion and effective wavenumber begin to increase with the increase of the frequency, which corresponds to the case of MSSW, while an increase in the bias angle leads to an increase in the phase incursion.

From the solution of the static problem, we obtained the spatial distribution of the internal static magnetic field **H**_{int} = **H**_{0} + **H**_{demag} in the waveguide system at different angles of deflection *φ* and *β* (*φ* specifies the deflection of the field in the *xy* plane, *β* in the *xz* plane) of the external magnetic field **H**_{0} relative to the *x* axis. Figure 3(a, b, c) shows the distributions of the y-component of the field **H**_{int} in sections *A* − *B* and *C*, respectively, when the field **H**_{0} is oriented along the *y*-axis (*φ* = *β* = 0°). In section *A*, the internal field **H**_{int} practically coincides in magnitude with **H**_{0}, since in this case magnetization occurs along the long side of the section (*l* > *w*) and the y-component of the demagnetization field **H**_{demag} is small. In sections *B* and *C*, magnetization occurs along the short sides of the sections, the influence of demagnetizing fields is significant, and the internal fields are less than the external one by an amount of the order of *δH* = 50 Oe. Note that a decrease in the value of the internal magnetic field in sections *B* and *C* of microwaveguides leads to a shift of the characteristic frequencies of the beginning of the SW spectrum $f0=\gamma Hint(Hint+4\pi Ms)$ downward relative to similar frequencies of the spectrum in section *A*.

Let us consider the transformation of the profile of the internal magnetic field in the bends of the sections of the structure with a change in the orientation angles *φ* and *β* of the external magnetic field. Figure 2(d–g) shows the dependences of the internal field **H**_{int}(*ξ*) at various angles of deflection of the external magnetic field **H**_{0}. The deviation of **H**_{0} in the *xy* plane leads to a decrease in the internal fields in all sections of the system while maintaining the value of *δH*. The deviation of **H**_{0} in the *xz* plane practically does not affect the value of the internal magnetic field **H**_{int} in the sections of the structure. A change in the magnitude of the internal field in the bends is a potential barrier for the spin-wave signal. When passing through the first bend between the *S*_{A} and *S*_{B} sections, the SW signal passes unhindered through the bend and the attenuation due to the change in the internal field is small. An increase in the internal magnetic field is observed in the bend between the *S*_{B} and *S*_{C} sections, and at large deflection angles in the *xz* plane, the spin-wave signal cannot propagate into the *S*_{C} section. Thus, for the efficient propagation of the spin-wave signal and control of the internal magnetic fields of the structure under consideration, it is necessary to change the direction of the external magnetic field in the *xy* plane.

## CONCLUSIONS

Using the methods of micromagnetic modeling to calculate the equilibrium distribution of magnetization, the possibility of transforming the profile and the magnitude of the internal magnetic field at orthogonal coupling of magnon microwave guides was shown. By calculating the characteristics of spin-wave transport, it was demonstrated that in a three-dimensional structure formed by two L-shaped partial sections, SW transmission is possible, accompanied by a simultaneous transformation of the wave type from the backward volume to the surface magnetostatic wave due to the violation of translational symmetry in the bended magnonic structure. In this case, a variation in the bias angle leads to a transformation of the internal magnetic field profile due to the shape anisotropy and, as a consequence, to a change in the amplitude and phase of the signal in the output section of the structure. The latter is due to the violation of translational symmetry and as a consequence of the transformation of one type of magnetostatic waves into another. The proposed structure allows the facility of the frequency selection of spin-wave signals, which makes it promising as a functional interconnection element in multilayer information processing magnonic architecture. At the same time, on the basis of the sequential interconnection of the magnonic waveguides, both in lateral and in the vertical planes, it is possible to increase the density of the arrangement of the functional elements of the magnonic networks.

## ACKNOWLEDGMENTS

The work is supported by Russian Science Foundation (Project No. 20-79-10191).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.