The frequencies and wave vectors of dipolar dominated spin waves (SW) and surface acoustic waves (SAW) cover the same ranges. Therefore, the propagating waves of both kinds can potentially scatter each other - in other words, the necessary condition for the interaction in a linear regime can be satisfied. The simplest structure in which the magnetoelastic coupling between SW and SAW can be observed is the magnetostrictive layer deposited on the substrate. The confinement of SWs in the magnetic layer is necessary to induce the dipolar interaction and surface localization of SAW ensures the co-existence of both kinds of waves in the magnetostrictive layer deposited on the surface. The strength of the magnetoelastic interaction depends on the orientation of the wave vector concerning the direction of the static magnetic field[1]. Moreover, this interaction is different for different types of SAWs, specifically, Rayleigh-SAWs (R-SAW) and Love-SAWs (L-SAW)[1,4] – see Fig.1. Thus, the coupling is strongly anisotropic and cannot be observed for arbitrary selected SAWs and SWs, even if their frequencies and wave vectors match. This effect is well-known and broadly discussed in the literature[1]. Our study shows an additional factor limiting the interaction between SAWs and SWs. The SAW/SW coupling proves to require an appropriate profile of the elastic wave near the surface of the magnetostrictive structure, at distances much smaller than the wavelength. For R-SAWs the tangential component of displacement ux can have nodes within the magnetic layer, resulting in a reduction of the net strength of magnetoelastic interaction even if the strain εxx is locally significant. In an L-SAW the displacement uy does not have any nodes (uy changes monotonously in the normal direction). We have shown that this additional factor plays a role for some types of surface acoustic waves (R-SAWs), while other types (L-SAWs) are insensitive to it. We think that the studies on magnon-phonon interaction in confined geometries (surfaces, cavities) are very promising and can reveal unusual interaction mechanisms. Sample We studied [Co20Fe60B20/Au]20 multilayer of the thickness 60 nm as a magnetostrictive medium. In Co20Fe60B20 layers (2.1 nm) the magnetization is oriented in-plane and the presence of Au layers (0.9 nm) reduces the SWs’ frequencies due to out-of-plane anisotropy. The multilayer was deposited on top of a 4 nm titanium (Ti) and a 15 nm gold (Au), which buffers the naturally oxidized (001) silicon substrate. Method We measured the dispersion relations of thermally excited SAWs and magnetostatic SWs using a six-pass tandem Brillouin spectrometer (Scientific Instrumentsc©TFP2-HC), which ensures a contrast of 1015. A frequency-stabilized diode-pumped solid-state laser (Coherentc©VERDI V5) operating at λ0= 532 nm was used as a source of incident light. The measurements were performed in the 180o backscattering geometry with crossed (p-s)polarization of the incident and scattered light for SWs and non-crossed (p-p) polarization for SAWs. Using the finite element method in COMSOL Multiphysics©, we solve numerically the coupled equations of motion for magnetization and mechanic displacement. Acknowledgement The authors hereby acknowledge the financial support from the National Science Centre, Poland, project No. UMO-2016/21/B/ST3/00452. S. Mies. would like to additionally acknowledge the support from the National Science Centre, Poland, project No. UMO-2020/36/T/ST3/00542.

![](https://assets.underline.io/uploads/markdown_image/1/image/0d5d93d1592e64189ced2b9a3d060b55.jpg)

Fig.1 (left) - the illustration of the Rayleigh and Love surface acoustic waves (R SAW and L SAW). (right) -the numerically calculated profiles of the strain tensor playing the most significant role for magnetoelastic interaction of spin waves with R SAW (magnetic field applied the 45 deg) and L SAW (magnetic field applied at 0 deg). Please note that the interaction with R SAW can be canceled due to the nodal line.

Conditions for effective coupling between surface acoustic waves and spin waves

The interaction between different types of wave excitation in hybrid systems is usually anisotropic. Magnetoelastic coupling between surface acoustic waves and spin waves strongly depends on the direction of the external magnetic field. However, in the present study we observe that even if the orientation of the field is supportive for the coupling, the magnetoelastic interaction can be significantly reduced for surface acoustic waves with a particular profile in the dirR/S-SAWection normal to the surface at distances much smaller than the wavelength. We use Brillouin light scattering to investigate thermally excited phonons and magnons in a magnetostrictive CoFeB/Au multilayer deposited on a Si substrate – see Fig.1. The experimental data interpretation is based on a linearized model of interaction and finite element method computations[1]. For propagating waves, the coupling requires matching both the frequencies and the wavelengths[2]. This condition can be fulfilled for spin waves (SWs) and surface acoustic waves (SAWs) existing in the same range of frequencies and wave vectors. However, the strength of the magnetoelastic interaction depends on the orientation of the wave vector with respect to the direction of the static magnetic field[3]. Moreover, this interaction is different for different types of SAWs, specifically, Rayleigh-SAWs (R-SAW) and Love-SAWs (L-SAW)[1,4] – see Fig.2. Thus, the coupling is strongly anisotropic and cannot be observed for arbitrary SAWs and SWs, even if their frequencies and wave vectors match. Our study reveals an additional factor limiting the interaction between SAWs and SWs. The SAW/SW coupling proves to require an appropriate profile of the elastic wave near the surface of the magnetostrictve structure, at distances much smaller than the wavelength. For R-SAWs the tangential component of displacement ux can have nodes within the magnetic layer, resulting in a reduction of the net strength of magnetoelastic interaction even if the strain εxx is locally significant. In an L-SAW the displacement uy does not have any nodes (uy changes monotonously in the normal direction). The location of R-SAW nodes depends on elastic properties of the system, specifically, the elastic material parameters of the multilayer and the substrate, and the thickness of the multilayer. We have shown that this additional factor plays a role for some types of surface acoustic waves (R-SAW), while other types (L-SAW) are insensitive to it. We believe that the studies on magnon-phonon interaction in confined geometries (surfaces, cavities) are very promising and can reveal unusual interaction mechanisms. Sample We used naturally oxidized (001) silicon as a substrate supporting the studied [Co20Fe60B20/Au]20 multilayers deposited on top of a 4 nm titanium (Ti) and a 15 nm gold (Au) buffer layers. The multilayers were deposited by magnetron sputtering in argon atmosphere. Method We studied the dispersion relations of thermally excited SAWs and magnetostatic SWs using a six-pass tandem Brillouin spectrometer (Scientific Instrumentsc©TFP2-HC),which ensures a contrast of 1015. A frequency-stabilized diode-pumped solid-state laser (Coherentc©VERDI V5) operating at λ0= 532 nm was used as a source of incident light. The measurements were performed in the 180o backscattering geometry with crossed (p-s)polarization of incident and scattered light for SWs and non-crossed (p-p) polarization for SAWs. Using the finite element method in COMSOL Multiphysics©, we solve numerically the coupled equations of motion for mechanic displacement and magnetization. The CoFeB/Au multilayer is treated as a 60-nm-thick effective magnetic layer, and the CoFeB/Au multilayer together with the Ti/Au buffer as an 80-nm-thick effective acoustic layer on a Si substrate. Acknowledgement The research leading to these results has received funding, hereby gratefully acknowledged, from the National Science Centre, Poland, project No. UMO-2016/21/B/ST3/00452. S. Mies. would like to additionally acknowledge the financial support from the National Science Centre, Poland, project No. UMO-2020/36/T/ST3/00542.

**References:**

 [1] N. K. P. Babu, A. Trzaskowska, P. Graczyk, et al., Nano Lett. (2020) https://doi.org/10.1021/acs.nanolett.0c03692 [2] Y. Hashimoto, D. Bossini, T. H. Johansen, et al., Phys. Rev., Vol. 97, 14040 (2016) [3] L. Dreher, M. Weiler, M. Pernpeintner, et al., Phys. Rev. B 86, 134415 (2012) [4] M. Geilen, F. Kohl, A. Nicoloiu, et al., Appl. Phys. Lett. Vol. 117, 213501 (2020) [5] N.E. Khokhlov, P.I. Gerevenkov, L.A. Shelukhin, et al., Phys. Rev. Applied Vol. 12, 044044 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/052c823eb22ae084daf2c19fe75deaf0.jpg)
Fig.1 In the studied system acoustic waves propagate along the surface of the silicon substrate and are mostly concentrated in the CoFeB/Au multilayer cover. The multilayer can be regarded as an effective magnetostrictive medium where surface acoustic waves can interact with spin waves. The anisotropy of magnetoelastic interaction (i.e. the dependence on the orientation f of magnetic field) is different for Love and Rayleigh/Sezawa acoustic waves (L-SAW, R/S-SAW); presented on the left are the respective polar plots of the in-plane component of magneto elastic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/bc1badb1dc20cc3a670ea3ed7a09b463.jpg)
Fig.2 The BLS experimental magnetoelastic dispersion relation measured for magnons (redlines) and phonons (blue lines) with an in-plane applied magnetic field of (a) 30 mT and (b) 50 mT for Φ=0 deg . The dashed shows the regions where the frequencies of fundamental spin wave fundamental mode (F-SW) and L-SAW or R-SAW are agreed. The interactions is seen for L-SAW only. (c) For Φ=45 deg observe the lack of R-SAW/F-SW interaction, which cannot be explained based on the interaction anisotropy (see hme,‖(Φ) for R-SAW in Fig.1). The R-SAW/F-SW interaction is suppressed to zero due to the presence of a nodal line for the εxx component of the strain tensor, which averages to zero the magnetoelastic coupling.

 The interaction between surface acoustic waves and spin waves: the role of anisotropy and spatial profiles of the modes

Phase and amplitude are the fundamental characteristics of waves. The processing of any wave relies on the interference effects, which depend on these characteristics. Thus, the control of spin waves’ (SWs’) phase and amplitude is essential in magnonics to perform both analog and digital SW-based computing [1]. Equally important is the transmission of the information (i.e. guiding and routing of the signal). We present a method for a coherent spin-wave guiding within the narrow magnonic nanostripe. In the magnonic multimode waveguide, scattering between the modes is one of the most important factors for the SW decoherence at the bends. Therefore, the question arises: can we modify the properties of the bending region to block the redistribution of incoming mode into the perpendicularly quantized modes, keeping the transmission as high as possible? Our proposed solution is to fill the bending region with the material of spatially tailored properties, which refract the SW and redirect its propagation strictly along the outgoing section of the waveguide (fig. 1). In other words, we are looking for the particular kind of graded-index (GRIN) element for SWs [2]. To design the GRIN element, we proposed an algorithm which achieving desired properties in a few step. In the first step, we developed the analytical theory for the scattering of exchange SWs on the homogeneous ferromagnetic slab of finite width embedded in a ferromagnetic layer. Minimizing the total energy, we derived the boundary conditions on the interfaces between the slab and its surroundings. As a results, we obtained the complete relations between the phases and amplitudes of the incident and scattered SWs. Later, we used our findings to demonstrate both analytically and numerically anomalous refraction for the purely exchange SWs incident from a waveguide to a semi-infinite film through a flat magnonic GRIN slab (fig. 2). We treat the GRIN element as an inhomogeneous slab linking the waveguide's input and output branches at the bend. For anomalous refraction, the wavefronts of refracted waves are tilted at a desirable angle with respect to wavefronts of the incident waves. The effect can be observed even at normal incidence. This phenomenon requires a linear change of the transmitted waves phase alongside the interface, where the refraction takes place. To describe it, we used the generalization of Snell’s law [3]. From previous step, we knew the phase which is acquired at specific positions so we can conduct analytical calculation of generalized Snell’s law and get the refraction angle. In the ultimate step, we could design the magnonic waveguide with the GRIN element at the bend. Knowing the SW bending, we design the properly shaped curved magnonic waveguide with GRIN element and run micromagnetic simulations. Fig. 1 presents a snapshot from simulations, where results are compared to the structure with the neglected GRIN element. We demonstrate that our findings can be used to guide the spin waves smoothly in curved waveguides, even through sharp bends, without reflection and scattering between different waveguide’s modes, preserving the phase, the quantity essential for wave computing. The systems In all the simulations we consider SW propagation in 5-nm-thick Co-Fe-B film (MS = 1200 kA/m, Aex = 27 pJ/m, neglected dumping) in the presence of the out-of-plane magnetic field (applied along the z-axis) of value μ0H =0.5 T. The SW frequency was 25 GHz. The width of the slab had 150 nm. In the considerations, we neglected dipolar interaction, however we checked and discussed the impact of this kind of interaction on the proposed system. In the first step, we considered reduced magnetization saturation in the slab in the range of 300–800 kA/m or additional anisotropy with anisotropy constant in the range of 0–490 kJ/m3. Anomalous refraction was demonstrated for two-dimensional system (fig. 2) composed of three sections. 100 nm wide waveguide, GRIN element with induced The gradient of the magnetic parameters was induced along the y-axis. Saturation magnetization changes in the range of 300–800 kA/m on the distance 100 nm, with direct contact with waveguide. Finally, using the previous step's knowledge, we design a curved waveguide, which supports coherent SW propagation alongside the waveguide [4].

**References:**

1. G. Csaba, Adam Papp, and W. Porod, Physics Letters A 381, 1471 (2017). 2. C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Chertopalov, M. T. Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, Physical Review B 92, 020408 (2015). 3. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, Science 334, 333 (2011). 4. S. Mieszczak, O. Busel, P. Gruszecki, A. N. Kuchko, J. W. Klos and M Krawczyk, Physical Rewiew Applied, 13, 054038 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c5b2b10d64e4866454fdc601539b7621.jpg)
Propagation of SWs in the Co-Fe-B waveguide, where the GRIN slab is placed at the bend (a). The color map represents a dynamical component of the magnetization in the y-direction. A snapshot is taken at the moment when a steady state is reached. At the beginning of the horizontal branch of the waveguide, the microwave antenna excites the SWs and at the end of the tilted branch, the SWs are damped to avoid reflections. After taking a turn, the SWs propagate smoothly. (b) SWs’ propagation in the waveguide without a gradient of magnetic parameters. After taking a turn, the SWs show a complex behaviour due to interference between different modes of the waveguide.

![](https://assets.underline.io/uploads/markdown_image/1/image/3fa7beebd86a61c5827762d94196f890.jpg)
The geometry of the simulation system with the GRIN element used to demonstrate anomalous refraction. The system consists of the ferromagnetic slab (B) characterized by the gradient of magnetic parameters (e.g., saturation magnetization MS or anisotropy field Ha) which links the straight section of the waveguide and the semi-infinite plane made of the homogeneous material A. The magnetic parameters in the slab B are changing in the y-direction. At each x-z cross-section, the phase shift of transmitted wave is different, which allows refracting the plane wave propagating initially in the x-direction.

Jaroslaw Klos

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