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Geometrically confined standing spin wave resonance (SSWR) has attracted intense research interests as a fundamental physics and practical applications [1][2]. The magnetostatic surface wave (MSSW) mode has been reported owning larger signal intensity than magnetostatic backward volume wave (MSBVW) mode[3]. These two modes are controlled by bias field continuously applied to magnetic strip, where the bias field is applied in the vertical or parallel directions according to MSSW or MSBVW modes, respectively. In this study, we successfully deposited Co&lt;sub&gt;2&lt;/sub&gt;FeAl films with 4-fold in-plane anisotropy and studied SSWR properties of the fabricated magnetic strips. Due to the 4-fold anisotropy, we succeeded in generating MSSW and MSBVW modes under zero bias field and confirmed absorption intensity difference between these two modes. Considering MSSW mode as ‘ON’ state and MSBVW mode as ‘OFF’ state , the absorption intensity difference demonstrated a spin wave switch or sensor device.&lt;br/&gt;Co&lt;sub&gt;2&lt;/sub&gt;FeAl films were deposited on MgO (001) substrate at 400°C by sputtering method. The angular dependence of the squareness are plotted in Fig. 1(a), which showed strong 4-fold in-plane anisotropy (an easy axis corresponds to MgO[001]) when film thickness was 60 nm. Co&lt;sub&gt;2&lt;/sub&gt;FeAl (400) peak at 65 degree of the corresponding XRD translated the 60-nm-thick film was well crystallized. The magnetic strip was fabricated by lithography techniques, where the lateral coordinates were defined to coincide with the anisotropy directions. 100-nm-thick SiO&lt;sub&gt;2&lt;/sub&gt; insulating layer was deposited on the strips and Grand-Signal-Grand coplanar waveguide (CPW) was fabricated on the SiO&lt;sub&gt;2&lt;/sub&gt; layer (Fig. 1(b)). The transmission characteristics of S&lt;sub&gt;21&lt;/sub&gt; was measured by VNA under the bias field, &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt;, ranged from 600 Oe to 0.&lt;br/&gt;TThe S&lt;sub&gt;21&lt;/sub&gt; of MSSW and MSBVW at different &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt; are shown in Fig. 2(a). The broad absorption dip includes multiple MSSW resonance modes, which are shown by the Lorenzian fitting. The absorption intensity of MSSW mode at various &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt; were greater than 0.5 dB, while the absorption intensity of MSBVW mode were less than 0.2 dB. The absorption intensity ratio of MSSW to MSBVW exceed 20 dB when &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt; = 600 Oe, which is large enough to distinguish the ON/OFF states. The resonance frequency estimated from the Lorenzian fitting were shown in Fig. 2(b). Micromagnetic simulation explains that the discontinuous &lt;i&gt;H&lt;/i&gt;&lt;sub&gt;b&lt;/sub&gt; dependence of SSWR frequency is reasoned by the absorption intensity balance of multiple modes. The absorption intensity of each order was different and the dominant order changes with the bias field. The dominant order changed from 7th order to 5th order at strong bias field, and from 5th order to 3rd order at weak bias field.

**References:**

[1] T. Schneider, A. A. Serga, B. Leven, and B. Hillebrands, Appl. Phys. Lett., vol. 92, p. 022505-1-3 (2008)&lt;br/&gt;[2] X. Ya, H. Chen, S. Oyabu, B. Peng, H. Otsuki, T. Tanaka, and K. Matsuyama, J. Appl. Phys., vol. 117, p. 17A719-1-3 (2015).&lt;br/&gt;[3] Kazuto Yamanoi, Satoshi Yakata, Takashi Kimura, and Takashi Manago, JJAP, vol 52, p. 083001 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9f8588c516a813a13f4ad7b26d48d415.jpg)
Fig. 1 (a) Squareness for the CoFeAl film deposited at various film thickness. (b) schematic of a CPW on magnetic strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/c1f07eb8d3cdf0324e9fb175770a2c61.jpg)
Fig. 2 (a)S&lt;sub&gt;21&lt;/sub&gt; parameter at different bias field, (b) dependence of resonance frequency on the bias field.

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 Standing spin wave resonance properties of multiple spin wave modes on Co2FeAl magnetic strip under zero bias field

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Geometrically confined standing spin wave resonance (SSWR) has attracted intense research interests as a fundamental physics and practical applications [1][2]. The magnetostatic surface wave (MSSW) mode has been reported owning larger signal intensity than magnetostatic backward volume wave (MSBVW) mode[3]. These two modes are controlled by bias field continuously applied to magnetic strip, where the bias field is applied in the vertical or parallel directions according to MSSW or MSBVW modes, respectively. In this study, we successfully deposited Co2FeAl films with 4-fold in-plane anisotropy and studied SSWR properties of the fabricated magnetic strips. Due to the 4-fold anisotropy, we succeeded in generating MSSW and MSBVW modes under zero bias field and confirmed absorption intensity difference between these two modes. Considering MSSW mode as ‘ON’ state and MSBVW mode as ‘OFF’ state , the absorption intensity difference demonstrated a spin wave switch or sensor device. Co2FeAl films were deposited on MgO (001) substrate at 400°C by sputtering method. The angular dependence of the squareness are plotted in Fig. 1(a), which showed strong 4-fold in-plane anisotropy (an easy axis corresponds to MgO[001]) when film thickness was 60 nm. Co2FeAl (400) peak at 65 degree of the corresponding XRD translated the 60-nm-thick film was well crystallized. The magnetic strip was fabricated by lithography techniques, where the lateral coordinates were defined to coincide with the anisotropy directions. 100-nm-thick SiO2 insulating layer was deposited on the strips and Grand-Signal-Grand coplanar waveguide (CPW) was fabricated on the SiO2 layer (Fig. 1(b)). The transmission characteristics of S21 was measured by VNA under the bias field, Hb, ranged from 600 Oe to 0. TThe S21 of MSSW and MSBVW at different Hb are shown in Fig. 2(a). The broad absorption dip includes multiple MSSW resonance modes, which are shown by the Lorenzian fitting. The absorption intensity of MSSW mode at various Hb were greater than 0.5 dB, while the absorption intensity of MSBVW mode were less than 0.2 dB. The absorption intensity ratio of MSSW to MSBVW exceed 20 dB when Hb = 600 Oe, which is large enough to distinguish the ON/OFF states. The resonance frequency estimated from the Lorenzian fitting were shown in Fig. 2(b). Micromagnetic simulation explains that the discontinuous Hb dependence of SSWR frequency is reasoned by the absorption intensity balance of multiple modes. The absorption intensity of each order was different and the dominant order changes with the bias field. The dominant order changed from 7th order to 5th order at strong bias field, and from 5th order to 3rd order at weak bias field.

**References:**

[1] T. Schneider, A. A. Serga, B. Leven, and B. Hillebrands, Appl. Phys. Lett., vol. 92, p. 022505-1-3 (2008) [2] X. Ya, H. Chen, S. Oyabu, B. Peng, H. Otsuki, T. Tanaka, and K. Matsuyama, J. Appl. Phys., vol. 117, p. 17A719-1-3 (2015). [3] Kazuto Yamanoi, Satoshi Yakata, Takashi Kimura, and Takashi Manago, JJAP, vol 52, p. 083001 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9f8588c516a813a13f4ad7b26d48d415.jpg)
Fig. 1 (a) Squareness for the CoFeAl film deposited at various film thickness. (b) schematic of a CPW on magnetic strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/c1f07eb8d3cdf0324e9fb175770a2c61.jpg)
Fig. 2 (a)S21 parameter at different bias field, (b) dependence of resonance frequency on the bias field.

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The early development of photonic devices [1, 2] is leading the way for their magnonic analogs [3] such as spin-wave (SW) generators [4], couplers [5, 6], and phase shifters [7]. Such basic functional components are necessary for the development of miniaturized and energy-effcient magnetic circuits for processing high-frequency microwave signals. Our study goes along further development of SW devices, to demonstrate circulation [8] and other functionalities using ferromagnetic conduits coupled by resonators in analogy to photonic channel add-drop filters [1]. We investigate the propagation behavior of SW in a cobalt (Co) bi-layer structure separated by a non-magnetic material and coupled with a permalloy (Py) stripe placed between two layers. The layers have a thickness of 5 nm, the separation between the stripe and the layers is of 10 nm and the stripe has a thickness of 50 nm. All elements are saturated in the same in-plane direction, perpendicular to the propagation of SWs. We investigate numerically the SW dynamics governed by the Landau-Lifshitz-Gilbert equation, solved numerically using the software COMSOL Multiphysics. We study the SWs dynamics with two different approaches. In the first approach, we investigate the SW dispersion relation of the system in the frequency domain. The lattice constant of the system is set at 500 nm. To analyze the origin of different type of interactions and possible functionalization we calculate the dispersion relations for two layers, the stripe only, three layers, and finally for the two layers coupled with the Py stripe. The dispersion relations allow to determine propagation properties, the coupling between the modes and their hybridization. We found that the coupling between the elements is strong at some frequencies at which suitable conditions for control of propagating SWs are formed. As a second approach, we used the time-domain simulations to demonstrate functionality of the proposed system. We excite the system with an antenna generating SWs with a single frequency. We found interesting propagation properties for a 19-GHz spin wave. The excitation of a wave in the top layer generates an edge mode in the Py resonator (Fig. 1a) and the SW is fully transmitted to the bottom layer propagating in the left. The excitation in the bottom layer (Fig. 1b) is not transferring to the top layer even if the Py resonator is weakly excited. Due to inversion symmetry with respect to the horizontal mid-plane, the analogous properties are for SW excited on the right side of the resonator. Thus, we demonstrated the functionality of a circulator in the system which is suitable for further miniaturization through decreasing the size of a resonant element. The proposed system composed of two thin-film SW conduits coupled by a ferromagnetic resonator can operate with different functionalities in dependence on the frequency. The next investigations, at other frequencies, positioned at proximity of other Py resonances, offer other functionalities, like magnetostatic screening from the direct coupling between the waveguides. The operation can be further tuned by changing the magnetization characteristics, magnetic field, and by exploring the third dimension. The research leading to these results has received funding from the Polish National Science Centre, project No. UMO-2018/30/Q/ST3/00416.

**References:**

[1] S. Fan et al. In: Opt. Exp. 3 (Aug. 1998), pp. 4-11. [2] S. Fan et al. In: Phys. Rev. Lett. 80 (Feb. 1998), pp. 960-963. [3] A. Barman and A. Haldar. In: Solid State Phys. 65 (May 2014), pp. 1-108. [4] P. Gruszecki et al. In: Solid State Phys. 6 (Mar. 2016), p. 22367. [5] A. Sadovnikov et al. In: Appl. Phys. Lett. 107 (Nov. 2015), p. 202405. [6] Q. Wang et al. In: Nature Electronics 3 (Dec. 2020), pp. 1-10. [7] Z. Zhang et al. In: J. Phys. D 53.10 (Dec. 2019), p. 105002. [8] K. Szulc et al. In: Phys. Rev. Appl. 14 (Sept. 2020), p. 12.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8ea59d302777763a68e7427139755587.jpg)
Fig.1

 Spin-wave Circulation in a Ferromagnetic Resonator with Two Adjacent Layers

Spin-wave based data transport and information processing aims to complement computation technology by exploiting wave logic, with the benefits of potentially reducing the feature size, as well as allowing for data multiplexing and more energy efficient operations [1]. Recent studies provide advanced theoretical models for nano-scaled spin-wave conduits and revealed a reasonably long propagation distance for longitudinally magnetized conduits, thus proving the fundamental feasibility of magnonics at the sub-100nm scale [2]. In contrast, little focus has been put on transversely magnetized nano-conduits. However, these systems are of great interest offering a large group velocity and a potentially protected transport of energy and information caused by the intrinsic chirality of the spin-wave modes [3]. Thus, they promise a significantly enhanced spin-wave decay length. Here, we present a study of propagating spin-wave packets in a transversely magnetized nanoscopic yttrium iron garnet conduit of 50 nm width and 44 nm thickness. Pulsed spin-wave packets are exicted using a coplanar waveguide antenna. Space and time-resolved micro-focused Brillouin-light-scattering spectroscopy is employed to measure the spin-wave group velocity and decay length. As shown in Fig. 1A, a long-range spin-wave propagation is observed with a decay length of up to 8 µm, which is multiple times larger than reported values for the corresponding longitudinal magnetized state [2]. In addition, a large spin-wave lifetime of up to 44.7 ns is found, see Fig. 1B. The results are supported with micro-magnetic simulations, revealing a frequency non-reciprocity for counter-propagating spin waves, see Fig. 1 C, caused by the trapezoidal cross-section of the studied structure and the associated curved internal field distribution which introduces a spacial symmetry break. The presence of this non-reciprocity is experimentally verified and is particularly interesting for an application in spin-wave devices since it allows for a novel device architecture and low-loss unidirectional data transport.

**References:**

[1] A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V. Chumak, S. Hamdioui, C. Adelmann and S. Cotofana J. of Appl. Phys. 128, 161101 (2020). [2] 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, and A. V. Chumak, “Propagation of spin-wave packets in individual nano-sized yttrium iron garnet magnonic conduits,” Nano Lett. 20, 4220–4227 (2020). [3] M. Mohseni, R. Verba, T. Brächer, Q. Wang, D. A. Bozhko, B. Hillebrands, and P. Pirro, “Backscattering immunity of dipole-exchange magnetostatic surface spin waves,” Phys. Rev. Lett.122, 197201 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4526d049eeed65a5eb4fdfcbe73371b6.jpg)
Fig. 1 (A) Group velocity and decay length measured via microfocused BLS spectroscopy. The light gray curve is the associated cw excitation spectrum. Frequency shifted due to a magnetic field difference of 0.2 mT. (B) Experimental lifetime derived from the decay length and group velocity. (C) Results of the micro-magnetic simulations. A frequency non-reciprocity is observed for counter-propagating spin waves.

 Long-range propagation of spin waves in transversely magnetized nano-scaled yttrium iron garnet conduits

A slot line waveguide that consists of two equal-width lines induces local magnetization dynamics around each line. Spin waves emitted from the local magnetization dynamics propagate and interact with the other side of the local magnetization dynamics. We have studied the high frequency waveguide’s size dependence of magnetization dynamics to understand the interaction. In our previous study, we reported the interaction for the case of which the magnetic field applied to the film plane [1,2]. We have revealed that the interaction induces the resonantly excited magnetization dynamics, and the excitation condition is determined by the design of lines. In this paper, we study the magnetization dynamics induced by a slot line waveguide with the applied magnetic field perpendicular to the plane. The spin wave mode for our previous study was magnetostatic surface spin waves (MSSW) [3]. When the magnetic field is applied normally to the film plane, forward volume magnetostatic spin waves (FVMSWs) are induced [3]. YIG (111) single crystal thin films with the thickness of 10 µm were epitaxially grown on gadolinium gallium garnet (GGG) (111) single crystal substrates. All the slot line waveguides were fabricated directly on the YIG (111) single crystal thin film using a combination of photolithography, Ar ion milling and sputtering techniques (as) shown in Fig. 1 (a). The line width and the gap between the lines in the waveguides were systematically varied. The high frequency power response was evaluated with a vector network analyzer. The input signal strength was fixed at 0 dBm for all the measurements and the frequency was set at a frequency between 3.0 and 13.0 GHz for each measurement. Figs 2 (a) and (b) show the maginetization dynamics at 7.0 GHz dependent on the line width and the gap, respectively. Multiple peaks were observed for all the measurement conditions. The magnetic field at the peak of magnetization lowered with the increase of the line width. The number of peaks also increased with the line width as well as the distance between lines. It should be noted that the number of peaks is much larger compared to the measurement for in plane [2]. We assume this is caused by the shorter wavelength of spin waves for FVMSW compared to MSSW, which enhances the periodicity of the condition for the same phase between local magnetization dynamics and spin waves. Those experimental results indicate that artificially controlled magnetization dynamics can be effectively induced with the use of FVMSW. We thank GRANOPT Co,Ltd. for their support of YIG single crystal thin films. This work was supported in part by JSPS KAKENHI Grant Number JP18K14114. This work was supported in part by CSRN, Tohoku University. A part of this work was carried out at Nano-Fabrication Support Laboratory, Yamaguchi University supported by "Nanotechnology Platform Project" of the Ministry of Education，Culture，Sports，Science and Technology (MEXT)，Japan. A part of this work was carried out at Nano-Fabrication Support Laboratory, Yamaguchi University supported by "Nanotechnology Platform Project" of the Ministry of Education，Culture，Sports，Science and Technology (MEXT)，Japan.

**References:**

[1] Tetsunori Koda, Sho Muroga, and Yasushi Endo. “Artificial control of magnetization dynamics via spin waves by slot line waveguide.” in Digests of Intermag 2020, session GE-5, Montreal, Canada, 4-8 May, 2020. [2] Tetsunori Koda, Sho Muroga, and Yasushi Endo, “Synchronized excitation of magnetization dynamics via spin waves in Bi-YIG thin film by slot line waveguide” Appl. Phys. Lett. 116, 102403 (2020). [3] A A Serga, A V Chumak, and B Hillebrands. “YIG magnonics” J. Phys. D: Appl. Phys. 43 264002 (2010).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/110cf3dc099d22f66347a0a16090ef60.jpg)
Schematic View of measurement set up for magnetization dynamics using a slot line waveguide. Magnetic field is applied to perpendicular to the plane.

![](https://assets.underline.io/uploads/markdown_image/1/image/5f11489dab4811544ec888fa8595d88e.jpg)
Magnetization dynamics measured at 7.0GHz for (a) Line’s width dependence and (b) Line’s gap dependence.

 Enhancement of magnetization dynamics via spin waves by slot line waveguide for perpendicularly magnetized yttrium iron garnet

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.

 Anomalous Refraction of Spin Waves as a Way to Guide Signals in Curved Magnonic Multimode Waveguides

It is now well established that the propagation of spin waves can be controlled by using approaches similar to those used in optics. In recent years particular attention was given to the possibility to controllably focus propagating spin waves. Such focusing can be achieved relatively easily in confined geometries. In the case of extended magnetic films, the implementation of spin-wave focusing appears to be less straightforward. Several approaches have been suggested. However, all of them are rather complex in terms of practical implementation, particularly on the nanoscale. Here, we experimentally demonstrate tight focusing of a spin-wave beam excited in extended nanometer-thick films of YIG by a simple microscopic antenna functioning as a single-slit near-field lens. We show that the focal distance and the minimum transverse width of the focal spot can be controlled in a broad range by varying the frequency/wavelength of spin waves and the antenna geometry. The experimental data are in good agreement with the results of numerical simulations. Our findings provide a simple solution for the implementation of magnonic nanodevices requiring a local concentration of the spin-wave energy.

**References:**

B. Divinskiy, N. Thiery, L. Vila, O. Klein, N. Beaulieu, J. Ben Youssef, S. O. Demokritov, and V. E. Demidov, Appl. Phys. Lett. 116, 062401 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/678744363a23cd07e63934a39dba2cab.jpg)
Fig.1 Schematic of the experiment.

![](https://assets.underline.io/uploads/markdown_image/1/image/224225623364fe1dd737aeb316342533.jpg)
Fig. 2 Measured intensity map of spin waves radiated by the antenna.

 Sub-Micrometer Near-Field Focusing of Spin Waves in Ultrathin YIG Films

Magnonics utilizing spin waves for information transport, storage, and processing offers an alternative for CMOS-based technology free of detrimental Joule heating at high operation frequency [1]. The realization of magnonic devices requires active control of spin waves and low transmission losses. Magnonic crystals, i.e., metamaterials with periodically modulated magnetic properties, made of yttrium iron garnet (YIG) are promising candidates for performing such tasks because of ultralow magnetic damping [2]. Thus far, several types of micrometer-thick YIG magnonic crystals have been explored including crystals made of periodic shallow airgrooves, metallic stripe arrays on top of a YIG film, and width-modulated YIG structures [2]. On the other hand, programmable control of spin-wave transmission in micrometer-thick YIG films has been demonstrated using current-carrying meander structures [3], optical absorbers [4], and strain coupling to a piezoelectric layer [5]. However, micrometer-thick YIG magnonic crystals with a large number of crystal units are not compatible with miniaturized on-chip device integration. In addition, magnonic crystals based on thick YIG films exhibit small bandgaps in the range 5 MHz – 50 MHz, which limits the manipulation of propagating spin waves. Low-loss nanomagnonics requires smaller YIG magnonic crystals with larger tunable bandgaps [6]. Here, we present the experimental realization of magnonic crystals made of only 2 - 4 discrete 260-nm-thick YIG stripes with robust bandgaps up to 200 MHz for Damon-Eshbach spin waves [7]. The YIG stripes of the magnonic crystal are separated by airgrooves or grooves filled by CoFeB. Compared to micrometer-thick YIG films, the opening of larger bandgaps with less crystal units in arrays of thin YIG stripes is explained by strong Bragg reflection on the individual scatterers. Moreover, efficient dipolar coupling between the YIG stripes via in-plane dynamic fields facilitates low-loss transmission in the allowed bands of the magnonic crystal. The bandgaps of the nanometer-thick YIG magnonic crystals are tunable through a variation of the groove depth, lattice constant, and film thickness [8]. For instance, the bandgaps of a 260-nm-thick YIG crystal widen upon an increase of the groove depth from half to full film thickness. Importantly, low-loss spin-wave transmission in the allowed bands of the magnonic crystal is hardly affected by the patterning of fully discrete YIG stripes. Downscaling of the YIG film thickness to 35 nm decreases the bandgap size through a flattening of the spin-wave dispersion relation. We show that a reduction in the lattice constant effectively compensates for this trend. For 65-nm-thick YIG crystals, we realized 100–200 MHz bandgaps by decreasing the lattice constant from 50 µm to 10 µm. Finally, we show that robust bandgaps up to 220 MHz in combination with low-loss spin-wave transmission at allowed frequencies are attained in a 45-nm-thick YIG crystal with a lattice constant of 2 µm (see Fig. 1). We anticipate that further downscaling of YIG magnonic crystals to the nanoscale is possible through an optimization of the fabrication process and efficient spin-wave excitation over a broad range of wave vectors.

**References:**

[1] A.V. Chumak et al., Nat. Phys. 11, 453 (2015) [2] A.V. Chumak et al., J. Phys. D: Appl. Phys. 50 244001 (2017) [3] A.V. Chumak et al., Nat. Commun. 1, 141 (2010) [4] M. Vogel et al., Nat. Phys. 11, 487 (2015) [5] A.V. Sadovnikov et al., Phys. Rev. Lett. 120, 257203 (2018) [6] M. Krawczyk et al., J. Phys. Condens. Matter 26 (2014) [7] H.J. Qin et al., Nat. Commun. 9, 5445 (2018) [8] H.J. Qin et al., Appl. Phys. Lett. 116, 202403 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8e63382b69aa64034d71dd49d355d495.jpg)
Fig. 1: (a) Contour plot of spin-wave transmission spectra (amplitude of S12) for a YIG magnonic crystal as a function of external magnetic field. The crystal consists of four 0.5-µm-wide airgrooves with a separation of 2 µm. (b) Line profile of spin-wave transmission (blue curve) for an external magnetic field of 5 mT and a reference spectrum (orange curve) measured on an unpatterned part of the same YIG film. (c) Dispersion relation of Damon-Eshbach spin waves in YIG for a magnetic field of 5 mT. The horizontal and vertical dashed lines indicate the bandgap center frequencies and corresponding Bragg scattering wave vectors. (d) Measured gap size of the first bandgap as a function of external magnetic field.

 Low-loss magnonic crystals based on nanometer-thick YIG films

Brillouin Light Scattering (BLS) [1, 2] experiment has been used to determine the spin waves in Co-rich cobalt phosphorous (CoP) exchange spring [3, 4] thin films prepared by the electrodeposition techniques [5, 6] at room temperature. The static magnetic measurement using hysteresis loop tracer (ShB Instruments, USA) shown in FIG. 1 (a)-(b) reveals the hysteresis loop structures for the electrodeposited thin films – the staircase like exchange spring nature with the presence of two coercivities in the hysteresis loops – composed by the soft and hard magnetic phases in a single material. Contrasted with the conventional single-phase magnetic material having a comparatively low coercivity – such kind of exchange spring nanomagnetic frameworks are promising for cutting edge, forefront magnetic applications due to their relatively higher energy product – the blend of magnetization and permanent magnetic field [7]. The differential of the magnetization shown in FIG. 1 (c)-(d) confirms the presence of two coercivities in the 2.8 μm, 5.7 μm thick thin films, respectively. Utilizing a (3+3)-pass tandem Fabry-Perot interferometer, and considering the backscattering geometry, the BLS spectra of the thermally excited SWs were measured. The BLS Spectra of the 2.8 µm thick CoP thin film shows that the peaks are very broad. The field dependent BLS spectra of 5.7 µm thick CoP thin film electrodeposited for 60 min is shown in FIG. 2(a). The BLS study yields two main modes present in the BLS spectrum: the lowest frequency one is a band of bulk modes (B) while the one at the higher frequency (sharp peak) is the so-called Damon-Eshbach surface spin wave (S) [8]. On the Stokes-side of the spectra, two well resolved peaks (S and B, as mentioned in the figure) are observed while the anti-Stokes side of the spectra shows only one peak (B). The S mode is observed only on one side of the spectrum, depending on the direction of the external magnetic field. This is the consequence of the nonreciprocal character of the S wave and of the micrometer thickness of the investigated films. The external magnetic field dependent BLS spectra shows an almost linear dependence of the mode frequencies vs the magnetic field intensity. The BLS spectra shown in FIG. 2(b) reveals the effect of magnetization reversal at constant polarization which unveils the predicted side change of Damon Eshbach (DE) surface mode. The change in Stokes/anti-Stokes ratio has also been affirmed because of the associated transfer of the S and B mode interaction with this magnetization reversal [8].

**References:**

[1] G. Gubbiotti, X. Zhou, A. O. Adeyeye, Phys. Rev. B 101, 224431 (2020). [2] S. Tacchi, G. Gubbiotti, M. Madami, J. Phys. Condens. Matter 29, 73001 (2016). [3] E. E. Fullerton, J. S. Jiang, M. Grimsditch, Phys. Rev. B 58, 12193 (1998). [4] F. Magnus, M. E. Brooks-Bartlett, R. Moubah, Nat. Commun. 7, ncomms11931 (2016). [5] P. McCloskey, B. Jamieson, T. O’Donnell, J. Magn. Magn. Mater. 320, 2509 (2008). [6] S. Roy, A. Connell, M. Ludwig, J. Magn. Magn. Mater. 290–291, 1524 (2005). [7] H. Zeng, J. Li, J. P. Liu, Nature 420, 395 (2002). [8] J. R. Sandercock and W. Wettling, J. Appl. Phys. 50, 7784 (1979).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/824d97b4d904e77f940a3d8f4d27e808.jpg)
FIG. 1: Hysteresis loop of the electrodeposited CoP thin films measured ShB Hysteresis loop tracer in XY plane, and here, dir-1+90°=dir-2. The thickness of the films is (a) 2.8 μm, (b) 5.7 μm, for the electrodeposited time (a) 30 min, (b) 60 min. (c)-(d) The differential of the magnetization of the (a) 2.8 μm, (b) 5.7 μm thick thin films, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/5584003a2099c29290540314fe3a0987.jpg)
FIG. 2 (a) BLS spectra of the 5.7 thick CoP thin film (electrodeposited for 60 min) at different in-plane bias magnetic fields measured from 1.50 kOe to 0 Oe with the backscattering geometry. (b) BLS Spectra of 5.7 in a bias magnetic field of +1.0 and -1.0 kOe revealing the effect of the reversal of the magnetization on the frequency position of the Damon-Eshbach wave.

 Brillouin Light Scattering Study of Spin Waves in CoP Exchange Spring Thin Films

Spin waves (SWs) are emerging as a potential alternative to electrical signals for computing and signal processing. Because SWs have relatively short wavelengths at microwave frequencies, they have the potential for applications in high-speed chip-scale electronics [1]. This work seeks to design and fabricate a spin-wave-based real-time spectrum analyzer that exploits SW interference in analogy to an optical spectrometer [2]. Here, microwave electrical signals are converted into SWs. All processing is then done by the diffraction and interference of SWs as they propagate in a magnetic thin film. After the interference pattern forms, SWs are then converted back into electrical signals. This type of device requires SWs having an isotropic dispersion relation, which, in turn, requires an out-of-plane DC bias field. These spin waves are forward-volume spin waves (FVSW). The focus of the work presented here is to explore the design space of a microwave-to-spin-wave transducer for FVSWs, where the microwave signal is introduced by a coplanar waveguide (CPW). The SW excitation spectrum of an external RF magnetic field acting on a magnetic thin film is the result of the interference between SWs launched from all points of the film that are excited by the magnetic field. For a magnetic field from a CPW, this interference results in a comb-like spatial-frequency response, which is equivalent to the Fourier transform (FT) of the spatial distribution of the magnetic field that acts on the magnetic film [3]. However, because the device described above requires a relatively constant transfer function over the full bandwidth, i.e., without nulls, the comb-like excitation spectrum from a CPW would not be suitable for our applications. Launching spin waves from the edge of the film can be used to improve the spin-wave excitation spectrum for a couple of reasons. First, it has been shown that the edge of a film can be an efficient broadband source of spin-waves [4]. And second, because only part of the CPW’s field acts on the magnetic film, the FT of only that part describes the excitation spectrum; the FT of the spatial distribution is more delta-function-like and therefore is more broadband than the previous case. To design the edge-launched SW transducer, micromagnetic simulations were done in Mumax3 [5] using the magnetic field from CPWs simulated in Ansys HFSS. Our simulations (Fig. 1) show the frequency response of this simulated structure with the edge 2 μm away from the CPW. These simulations verify that the transducer’s excitation spectrum is improved, i.e., made more uniform, by placing the CPW beside the edge of the film rather than on top of the film. This improved bandwidth comes at the cost of transducer efficiency, since only the field to the side of the CPW acts on the magnetic film. Because the magnetic field decreases rapidly with distance from the CPW, the edge of the film should be as close to the CPW as possible to increase coupling between the CPW and the film. Representative waveforms given in Fig. 2 show that because the small out-of-plane component of the RF magnetic field is parallel to the large DC-bias magnetic field, it contributes very little to the excitation of a SW compared to the in-plane components of the field. The results of the simulations show that for the same magnetic field, raising the bottom of the CPW increases the in-plane component of the field and increases the amplitude of the SW. Yet another configuration would be to place the CPW on the magnetic film and create an edge-launcher by etching a gap in the film to the side of the CPW. Excessively large fields can produce unstable spin-waves with additional unwanted undesirable frequency components [6]. With a gap in the magnetic material, the internal magnetic field crosses the gap at a reduced field strength, removing the unwanted structure and resulting in SWs at the desired wavelength, as well as a smoother frequency response (Fig. 1.). The structure of the previous paragraph was simulated with the properties of an YIG thin film having different thicknesses (30 nm and 243 nm). It is found that increasing the distance between the gap and the CPW increases the amplitude of the resulting SW with diminishing returns, as shown in Fig. 2. With a gap that is 2 μm wide and 6 μm away from the CPW, we find that for the thinner film this structure produces a SW with a smaller amplitude than when the material underneath the CPW is nonmagnetic and the edge is flush with CPW. However, with the thicker film (Fig. 2), the structure with the magnetic film under the CPW produces a larger SW than the structure without the magnetic material under the CPW.

**References:**

[1] G. Csaba, Á. Papp and W. Porod, Physics Letters A, Vol. 381, no. 17, pp. 1471-1476 (2017) [2] Á. Papp, et al., Scientific Reports, vol. 7, no. 1, pp.9245-9245 (2017) [3] S. Maendl, I. Stasinopoulos and D. Grundler, Applied Physics Letters, vol. 111, no. 1, p. 012403 (2017) [4] F. Mushenok, et al., Applied Physics Letters, Vol. 111, p. 042404 (2017) [5] A. Vansteenkiste, et al., AIP Advances, vol. 4, no. 10, p. 107133 (2014) [6] H. Aquino, et al., poster presented at the 78th Device Research Conference

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/09cf474cf41a0ae00817c2d3cf2617d2.jpg)
Fig. 1. The spatial frequency response of simulated microwave to SW transducers with the CPW placed to the side of the film with and without magnetic material underneath the CPW. The same magnetic field was used for both transducers. The FT of the field is shown to illustrate the comb-like response of a CPW on the film.

![](https://assets.underline.io/uploads/markdown_image/1/image/67403ddf0bb41726426c09e479eb943c.jpg)
Fig. 2. Simulation results with SWs (in-plane magnetization, Mx) launched from the edge of a 243 nm thick YIG film. All plots begin at the edge of the magnetic film. 1) The edge is flush to the CPW and the bottom of the 300 nm thick CPW is in the same plane as the bottom of the magnetic film; 2) the CPW is on the magnetic film with a 2 μm gap etched to the side of the CPW; 3) the CPW is raised on nonmagnetic material such that the bottom of the CPW is in the same plane as the top of the magnetic film; and 4) the CPW is on the magnetic film with the gap 6 μm to the side.

 Design of a Coplanar-Waveguide-Based Microwave-to-Spin-Wave Transducer

Signal processing, conditioning, filtering and are becoming critical parts of RF communications. We have designed a magnonic Bragg filter with a tailored frequency response. A systematic procedure for arriving at the period of Bragg grating, excitation, probe region etc., has been formulated to simulate the expected spectral response. We simulate and analyze a 1D magnonic Bragg narrow band rejection filter at 28 GHz, the frequency allotted to India, in the 5G spectrum. The magnonic filter with 10 grooves has a pass band from 12 GHz to 50 GHz and a tuneable band gap with attenuation as high as 36 dB. Modulating the depth and the duty cycle of the magnonic crystal, we demonstrate that we could tune parameters like, the bandgap, rejection ratio and group velocity of the spin waves. The rejection efficiency of the filter at the frequency of 28 GHz is 90% higher compared to a signal of 40 GHz, in the pass band. INTRODUCTION 5G mobile telecommunications operate above 24 GHz band and these signals suffer much higher path loss [1]. Magnons, or quanta of spin waves (SW) have been explored as information carriers and for computing [2] with the help of magnonic crystals (MCs). Experimental demonstrations show tuning of bandgaps in MCs, by varying parameters such as the applied magnetic field (HDC), width, thickness [3, 4] etc. With extensive experimental investigations being pursued, we arrive at an optimized set of design procedures to estimate different simulation parameters. A magnonic Bragg filter (MBF) is designed to operate at 28 GHz and simulated using the micromagnetic simulation tool, MuMax3. The tunability of bandgaps, rejection and group velocity with modulations of duty cycle and depth of the gratings are studied extensively to finally realize an optimized and highly efficient 1D MBF. DESIGN PROCEDURE The nonlinear dispersion relation between frequency(ω) and wavenumber(kSW) introduces a constraint in the filter design. We must invert ω(kSW), find kSW for a given excitation geometry, and then pick the Bragg wavelength(Λ). We must also use a broadband excitation to capture the full spectrum. Micromagnetics thus become computationally intensive, and a simple MBF simulation occupies terabytes of storage. Hence, we developed custom Go routines to extract the relevant spatially averaged and temporally sampled data from MuMax3 simulations [5]. For our purposes, we assume a strip of permalloy, with saturation magnetization 800 kA/m, exchange constant 1.3×10−11J/m and damping constant α= 0.006 [6]. The backward volume spin waves configuration excites SWs along x, parallel to HDC, as in Fig.1(a). At ω=28 GHz, and with HDC=54.9 kA/m (a convenient field value chosen for our future experiments), we obtained kSW = 0.145 rad/nm from the dispersion relation [7, Eqn(18)]. This corresponds to a wavelength λSW = 43.33 nm, and we choose Λ=21.6 nm to match this at normal incidence. A broadband small signal excitation of hy(x,t) with cut off values of kc = ±0.3 rad/nm and fc = ±50 GHz is applied using sinc pulses in space and time. This yields sinc pulses with a peak centered at 100 ps and spread over 108 nm. In the probe region, the magnetization component my is collected for post processing. We chose a probe region 6×λSW long, and locate it sufficiently far away from gratings to minimize transient interferences. Absorbing boundary regions are defined at either boundaries with a parabolic variation of α from 1 to 0.006 to absorb the reflections from edges[8]. With all these estimated parameters, the dimensions of the MC with 10 grooves on an 8 nm thick permalloy film is 1843.2×8×100 nm3. DUTY CYCLE & DEPTH MODULATIONS While the grating period decides the reflected kSW, the duty cycle (DC) of the grating affects the bandgap. We simulated gratings with depths of 2 nm and DCs of 50%, 33.3% and 16.7%. 1D spectral plots in Fig.1(b) show sharp dips of the order of 20 dB consistently, at kSW= 0.145 rad/nm for all MCs except the plain film, indicating that this wavevector is rejected irrespective of the DC variation. We also study the effect of depth modulation for gratings of depths 1 nm, 2 nm and 4 nm, all with a DC of 50%. The 1D-FFT plot in Fig.1(c), shows the occurrence of bandgaps in the designed frequency range. The effective thickness of the film allows propagation of both exchange and dipolar coupled SWs. From Fig.2(a), we observe a narrower bandgap, a reduced group velocity of the SWs, and an increase in frequency rejection ratio with increasing groove depth thickness. Based on these studies, the MBF is optimized to have 2 nm depth and a DC of 50%. Finally, to confirm the performance, a single frequency sinusoidal excitation at 28 GHz (in the bandgap), is compared with that of 40 GHz (not in bandgap) and observed that the rejection efficiency is about 90% higher at 28 GHz, as in Fig.2(b). CONCLUSION For a given frequency response of a magnonic device, we developed a systematic set of procedures to plan out various simulation parameters. The magnonic Bragg filter is designed and realized to reject frequency of 28 GHz efficiently. This study highlights the effect of variations, in duty cycle and depth, of MCs with sub-micron features, and assists in the fabrication process. We also expect that the outlined design procedure, and the computational routines, will be helpful to the magnonic community.

**References:**

 [1] Jack Browne, https://www.mwrf.com/technologies/systems/article/21120263/surveying-the-status-of-5g-technology, (2020) [2] A. Mahmoud, F. Ciubotaru and F. Vanderveken, J. Appl. Phys., Vol. 128,p. 161101 (2020) [3] P. Frey, A. A. Nikitin and D. A. Bozhko, Commn. Phys., Vol. 3, p.17(2020) [4] A. V. Chumak A. A. Serga and S. Wolff, Appl. Phys. Lett, Vol. 94,p.172511 (2009) [5] https://github.com/LekhaRam/Mumax310AveragedSampling (2020) [6] Y. Zhao, Q. Song and S. H. Yang, Sci. Rep., Vol. 6, p.22890 (2016) [7] B. A. Kalinikos, IEE Proc.H.,Vol. 127, p.4 (1980) [8] G. Venkat and A. Prabhakar, J. Magn. Magn. Mater., Vol. 450, p.34 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5fbbbf88907c50bb875d36f1018d0bb3.jpg)
(a) Structure of MBF with dimensions in nm (b) Variation of my in k space with DC (c) Spectrum of depth modulated SW

![](https://assets.underline.io/uploads/markdown_image/1/image/67db7f859951c9d5171facd2890a6a12.jpg)
(a) Results of depth modulated MC (b) Sinusoidal excitation at 28 GHz & 40 GHz for the optimized 1D MBF

 Magnonic Bragg Filter Design for India's 5G Spectrum

In contemporary developing world demand for small and less energy-consuming devices rapidly increases. To meet this requirement a new generation of technology has to be developed which overcome problems known from the usage of electronics such as frequency limit and heat dissipation. One possible answer promises spin waves (SW) which have a short wavelength and do not suffer from Joule heating as electrons [1]. To successfully use SWs in widespread devices there is a need to find methods that allow handling this type of low energy consuming information carrier. The usage of metasurfaces consisting of arrays of nano-resonators was an innovative concept in modern optics, as made it possible to design a new class of optical systems with properties unheard in classical optics as for example flat lenses [2]. Such devices become available because metasurfaces enable complete control of electromagnetic wave’s phase and amplitude at subwavelength distances. However, such a concept has not been yet widespread in the realm of magnonics, although it would be extremely useful in the design of SW devices [3]. Especially because SW propagation is significantly limited by damping, so requires the usage of technology that works well in the constrain of ultra-short distances. Here, we present extensive theoretical research on the influence of the position and geometry of a ferromagnetic stripe on the reflection and transmission of the SWs from the edge and middle, respectively, of a permalloy film (Fig. 1) [4]. We have focused on how properties of SW change with the resonator width and separation between both elements. Interactions between the stripe and the layer, which depend on the value of separation between the elements and dimensions of the stripe, affect the propagation and reflection of SW. To check the scope of the interactions we use three numerical methods. Namely, finite element method in frequency domain, micromagnetic simulations [5], and finite element modal method. Employing three different methods allowed us to cross-check their results to confirm their correctness. We show that the phase shift as a function of the stripe’s width is characterized by intervals of small and steady growth which are separated by narrow areas of resonances in which phase increases linearly (Fig. 1 inset a). In the areas of resonances there is a visible increase of amplitude and resonances appear periodically in function of a stripe width with a period of about 160 nm. Simulations with increasing separation show (Fig.1 inset b) that resonances also occur with changes of separation but then the function is not periodic. Additionally above certain separation resonances do not appear anymore. The behavior of phase changes is explained with analytical model in which the reflection coefficient consists of two terms of different origins. The first one is responsible for a slow increase of phase with increasing width of the resonator and is a result of phase accumulated in the reflection of the long-wavelength mode. The second term has origin in constructive interference of the short-wavelength mode in the bilayer part of the system, playing a role of the resonator. This term gives significant input only at certain widths of the resonator, which can be predicted from Fabry-Pérot resonance conditions [6]. Intervals of rapid phase changes (Fig. 1 inset a) are caused by the presence of the second term. Because of the properties of the second term, the system used in the simulations can be considered as a realization of the Gires-Tournois type magnonic interferometer [7]. Furthermore, we extend the idea of resonance control of SW phase to the transmission mode. Acquired knowledge allows designing a new class of SW devices based on the Gires-Tournois interferometer concept and manipulations of SW with the usage of magnetic elements smaller than the wavelength of SW. Control of the SW phase can be used for the creation of SW-optical elements such as flat lenses and phase shifters. Especially interesting idea is utilization of proposed interferometer in the system to detect magnetic particles in the vicinity of the bilayer part. The research leading to these results has received funding from the Polish National Science Centre projects No. UMO-2015/17/B/ST3/00118, UMO-2019/33/B/ST5/02013, and UMO-2019/35/D/ST3/03729. The simulations were partially performed at the Poznan Supercomputing and Networking Center (Grant No. 398).

**References:**

 [1] M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter 26, 123202 (2014) [2] N. Yu, et al., Nat. Mater. 13, 139–150 (2014) [3] M. Zelent, et al., Nanoscale 11, 9743-9748 (2019) [4] K. Sobucki, et al., arXiv preprint arXiv:2007.15226 (2020) [5] A. Vansteenkiste et al., AIP Advances 4, 107133 (2014) [6] F. Fabry, A. Perot, Ann. Chim. Phys. 16, 115 (1899) [7] F. Gires, P. Tournois, R. Acad. Sci. Paris. 258, 6112–6115 (1964)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c03f5d01e1f59b26517f8a0224a9f829.jpg)
Fig. 1 Geometry of the system used in the simulations. Parameters w and s were manipulated in the simulations, where d1 = 5 0nm, d2 = 40 nm and μH0 = 0.1 T were kept constant. In simulations frequency of 11 GHz was assumed. Between coordinates x = 0 and x = w the bilayered part is regarded as resonator. Inset (a) Phase shift as a function of stripe’s width. Resonances appear periodically with increasing stripe’s width with period around 160 nm. Inset (b) Phase shift as a function of stripe-film separation.

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