Here we report about the strain-tuned dipolar spin-wave coupling in the adjacent system of yttrium iron garnet stripes, which were strain-coupled with the patterned piezoelectric layer. Spatially-resolved laser ablation technique was used for structuring the surface of the piezoelectric layer and electrodes on top of it. Using a phenomenological model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes. The features of the tunable spin-wave coupling by changing the geometric parameters and the type of magnetization is demonstrated.

## I. INTRODUCTION

The using elementary quanta of magnetic excitations (magnons or spin waves (SW)) as carriers of information signals has attracted more interest in recent years due to the possibility of transferring the magnetic moment (spin) of an electron without transferring an electric charge and without generating Joule heat inherent in semiconductor technologies.^{1–5} The properties of SWs are determined by the dipolar and exchange interactions in magnetic medias^{6–8} and can change significantly during structuring of magnetic films. SWs are used for generation,^{9} transmission^{10} and the processing^{11,12} of the information signals in micro- and nanoscale structures. A thin ferrites films is used for these purposes, for example, yttrium iron garnet (YIG) which demonstrates record low values of the spin-wave damping parameter.^{6}

The voltage-controlled tunability of the spin-wave spectra in the thin magnonic films carried out due to the transformation of the effective internal magnetic field. The latter changes due to inverse magnetostriction (Villari effect) as a result of local deformation of the magnetic film. It has been experimentally demonstrated that electrical field tunability of spin-wave coupling can be effectively used to control the magnon transport,^{13,14} which led to the creation of a class of spin-wave devices, such as two-channel directional couplers,^{15} spin-wave splitters.^{16,17} This demonstration of spin-wave coupling phenomena opens the possibility of study the nonlinear dynamics of SW^{18} and the mechanisms for the spin-wave coupling tunability. The using of YIG films opens a possibility to create functional elements of magnonic networks based on the study the properties of spin waves propagating along irregular magnetic stripes with broken translational symmetry.^{19}

Here, we use a numerical and experimental techniques to demonstrate the effects of voltage-controlled the spin-wave coupling in a system of three lateral magnetic stripes with a patterned piezoelectric layer. Spatially-resolved laser ablation technique was used for structuring the surface of the piezoelectric layer and electrodes on top of it. We show an effective tuning of the spin-wave characteristics using an electric field due to local deformation of the piezoelectric layer and the inverse magnetostriction effect in YIG stripes. Using a phenomenological model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes. The features of the tunable spin-wave coupling by changing the geometric parameters and the type of magnetization is demonstrated.

A lateral structure consisted of three parallel-oriented magnonic stripes *S*_{1}, *S*_{2} and *S*_{3} (Fig. 1(a)). Using the laser ablation technique based on fiber YAG:Nd laser with the high precision 2D galvanometric scanning module (Cambridge Technology 6240H) magnonic structure was fabricated from *t* = 10 *μm*-thick a yttrium iron garnet (YIG) film [(YIG) *Y*_{3}*Fe*_{5}*O*_{12} (111)], grown on a 500 *μm*-thick gallium-gadolinium garnet [(GGG) *Gd*_{3}*Ga*_{5}*O*_{12} (111)] substrate. A system of *w* = 500 *μm*-width with a distance of *d* = 40 *μm* between each other forms three spin-wave channels. The length of the magnonic stripes was 6 mm for *S*_{1,3} and 8 mm for *S*_{2}. The SW was excited using a microstrip antenna with 1 *μm*-thick and 30 *μm*-width. The structure is placed in external static magnetic field, *H*_{0} = 1100 Oe. Field is oriented along the *x* axis. This configuration is allows to excite the magnetostatic surface wave (MSSW) in stripe *S*_{2}. A 200 *μm*-thick lead zirconate titanate (PZT) is used as a piezoelectric material. A 1 *μm*-thick copper electrode is placed (“GND” in Fig. 1(a)) on a top side of PZT, which does not have a significant effect on the propagation of SW in magnetic stripes. On the bottom side of PZT a 100 nm-thick titanium electrodes *G*_{1} and *G*_{2} were deposited above *S*_{1} and *S*_{3}, respectively. For more efficient magneto-electric coupling, we use a spatial resolution laser ablation technique to patterned a piezoelectric layer on 25 *μm*-thick. In the upper inset in Fig. 1(a) shows a SEM image of the edge of a piezoelectric layer in direct contact with a YIG stripes. A voltage $Vg1,2$ was separately applied to each of the electrodes in the experiment. We use a two-component epoxy strain gauge adhesive (labeled “EA” - epoxy adhesive in bottom inset in Fig. 1(a)) to connect the magnonic stripes and the PZT layer.

To demonstrate a processes of piezo-magnetic coupling, we are developed a numerical program based on a finite element method (FEM). First, we calculate elastic deformations caused by an external electric field in the piezoelectric layer. Next, we obtain the profiles of the internal magnetic field in lateral magnetic stripes. Then, the obtained profiles of the internal magnetic field were used in the micromagnetic simulation.^{20}

The relative transformation in the size of the PZT layer is shown in Fig. 1(b), where the colour gradient is shows the distribution of the mechanical stress tensor component *S*_{yy} in the case of *V*_{g1} = 250 V. It means that the deformation of the piezoelectric layer occurs in the local region of the PZT layer under the electrode *G*_{1}, which leads to a change in the value of the internal magnetic field *H*_{int} in the stripe *S*_{1} due to the inverse magnetostrictive effect. In addition, we estimate the effective deformations in case of patterned piezoelectric layer (see Fig. 1(c)) and of unpatterned piezoelectric layer (see Fig. 1(b)). It should be noted that in the case of a patterned piezoelectric layer we obtain the amplification of local deformations in the region of contact of the piezoelectric layer with the YIG stripe.

We use the phenomenological model based on the idea of the coupling of the co-propagating spin wave. In the case of multimode spin-wave transport the intermodal coupling coefficients between the magnetic stripes can be obtained from experimentally observed beating of spin waves propagating in adjacent channels:^{21}

where coupling coefficients *κ*_{11}, *κ*_{12}, *κ*_{21}, *κ*_{22} represents the intermodal interaction between 1st and 2nd transverse modes,^{6,22}$Aij$ - is the dimensionless SW amplitude along *x* coordinate, *β*_{i} is the wave number of SW propagating in the single magnonic waveguide of the same width as the widht of one of the magnonic channel in the lateral structure, *κ* is the coupling coefficient between the SW in the adjacent stripes. Figure 2(a) shows mode profiles of each separated magnetic stripe *S*_{2,3} at *E*_{1} = *E*_{3} = 0 kV/cm for 1st transverse mode in *S*_{2} (blue solid curve), 1st transverse mode in *S*_{3} (red solid curve) and 2nd transverse mode in *S*_{3} (red dashed curve). The shaded area in this case is called the overlap integral *C*(*f*, *E*). In terms of the phenomenological theory of coupled waves, we can introduce the value *C*(*f*, *E*), the numerical value of which is equal to the integral of the overlap of the eigenmodes of two separate YIG stripes:

where Φ_{i}(*x*, *f*, *E*_{i}) is the distribution of the field of the eigenmodes of the SW propagating in the *i*th stripe. An exact calculation of the coupling coefficient *C*, including for a system of three stripes, can be performed using the FEM^{23} and is beyond the scope of this work.

Figures 2(b), (c) shows the solution of the system (1), it should be noted that the SW intensity in the stripes is redistributed periodically along the stripes *S*_{1,2,3}. In case of *E*_{1} = 10 kV/cm and *E*_{3} = 0 kV/cm (see Fig. 2(C)) spin-wave energy ceases to be transmitted in the stripe *S*_{1} due to decrease internal magnetic field and in terms of this model due to changing *β*_{1}.

To demonstrate the voltage-control distribution of the spin-wave signal a numerical simulation was performed based on the solution of the Landau-Lifshitz-Hilbert equation (LLG):^{24–26}

where **M** is the magnetization vector, *M*_{s} = 139 G is the saturation magnetization of the YIG film, *α* = 10^{−5} is the damping parameter of SW, **H**_{eff} = **H**_{0} + **H**_{demag} + **H**_{ex} + **H**_{a}(*E*) is the effective magnetic field, **H**_{0} is the external magnetic field, **H**_{demag} - demagnetization field, **H**_{ex} - exchange field, **H**_{a}(*E*) is the anisotropy field, which includes taking into account the external electric field, *γ* = 2.8 MHz/Oe is the gyromagnetic ratio in the YIG film.

In order to reduce signal reflections from the boundaries of the computational domain, regions (0 < *y* < 0.3 mm and 3.7 < *y* < 4.0 mm) were introduced in numerical simulation with the damping parameter *α* with exponential decreasing. Figure 3 shows the intensity distribution $I(x,y)=my2+mz2$ in the case of the MSSW propagation (see Figs. 3(a, b, e, f)) and in the case of propagation backward volume magnetostatic waves (BVMSW) (see Figs. 3(c, d, g, h)).

If voltage is applied to the electrode *G*_{1}, the spin-wave intensity distribution is transformed. So for *E*_{1} = 10 kV/cm and *E*_{3} = 0 kV/cm, a spin-wave power is transfer between *S*_{2} and *S*_{3} (see Fig. 3(e-h)). In this case, *L* numerically coincides with the coupling length in two identical laterally magnetic stripes.^{21,27} It should be noted that changing of geometric parameters affects to the internal magnetic field distribution. Herewith the effective tuning of the coupling length via local strains changes in such a way that when the distance between the stripes changes from 20 *μm* (see Fig. 3(a, c)) to 60 *μm* (see Fig. 3(b, d)) the *L* increases by 1.25 times.

To fully describe the dipolar SWs in the lateral structure, the dipolar coupling efficiency was calculated for forward volume magnetostatic waves (FVMSW) propagating in a laterally stripes in the case of equilibrium magnetization direction normal to the surface of the structure (along the *z* axis). In this case, the internal magnetic field in the YIG stripes *H*_{int} ≈ *H*_{0} − 4*πM*_{s}.^{6} In case of propagating of the FVMSW in two lateral magnetic stripes,^{28} it was found that the propagation regime of the FVMSW in the lateral geometry is ineffective due to the pronounced increase in the value of the coupling length *L* in the long-wavelength part of the spectrum. In a system of three lateral stripes, this leads to the fact that in the case of excitation of the FVMSW in *S*_{2}, there is no directional coupling of SW into the lateral stripes *S*_{1} and *S*_{3}.

To understand the influence of local elastic strains on the stationary distribution of the MSSW, using a Brillouin light spectroscopy (BLS) technique of magnetic materials.^{22,29} A probe laser beam with a wavelength of 532 nm was focused on the transparent side of the GGG composite structure, as shown by the arrow in the inset in Fig. 1(a). We obtain the frequency dependence of *I*_{BLS} in the section along the axis *x* at *y* = 5.0 mm, in the case of *E*_{1} = 10 kV/cm and *E*_{3} = 0 kV/cm (see Fig. 4(a)), which demonstrate the transformation of spin-wave intensity, when an electric field is applied. In the frequencies from *f*_{1} = 4.925 GHz to *f*_{2} = 4.985 GHz, we observe a damping of SW in *S*_{2}, which corresponds the regimes when the spin-wave energy is localised in *S*_{3}. It should be noted, in this case, the we observe the edge mode^{30} propagation in the stripe *S*_{2}. We see a coupling between the edge modes propagating along *S*_{1} and *S*_{2}. To prove a strain-tuned SW switching we use the micromagnetic simulations and obtain a frequency dependence of the SW intensity (see Fig. 4(b)). We see a good agreement with the experimental data.

Let us consider the effects of gradual variation of the electric field polarity on the dynamic magnetization component *m*_{z}, which can determine the SW phase in a section along the *x* axis at *y* = 3.0 mm. With a gradual variation of *E*_{1} (see Fig. 4(c)), a change in the magnitude and sign is observed (phase change exceeds the value *π*) *m*_{z} in the stripes *S*_{1} and *S*_{2}. When the field *E*_{3} = −5 kV/cm is applied to the electrode *G*_{2}, the internal fields in the *S*_{2} and *S*_{3} become equal and the *E*_{1} changes in the range − 10…+ 10 kV/cm leads to a change in the sign of *m*_{z} in all three stripes of the YIG.

Thus, we observe the strain-tuned dipolar spin-wave coupling in the adjacent system of ferromagnetic stripes. As an experimental demonstration of the investigated physical processes, a configuration of the magnonic structure with a piezoelectric layer and structured electrodes on its surface is proposed. We use the spatial resolution laser ablation technique for structuring the piezoelectric layer. The latter is thus created by structuring the surface of the magnetic film and creating irregular waveguiding channels on it. As a demonstration of this physical effect, using numerical and experimental methods we show a voltage-controlled spin-wave transport along a three-channel lateral structure. We demonstrate that the variation in the geometric parameters of adjacent magnonic stripes leads to a change in the internal field and the effectiveness of the influence of elastic strains on the properties of propagating coupled SWs and characteristic features are revealed that manifest themselves in a change in the modes of spin-wave transport. Using a phenomenalogical model based on coupled modes equation, we demonstrate a voltage-controlled intermodal coupling in lateral magnonic stripes.

## ACKNOWLEDGMENTS

This work is supported by grant of the RFBR (19-37-90145).

## DATA AVAILABILITY

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