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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].&lt;br/&gt;&lt;br/&gt;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)&lt;br/&gt;[2] A.V. Chumak et al., J. Phys. D: Appl. Phys. 50 244001 (2017)&lt;br/&gt;[3] A.V. Chumak et al., Nat. Commun. 1, 141 (2010)&lt;br/&gt;[4] M. Vogel et al., Nat. Phys. 11, 487 (2015)&lt;br/&gt;[5] A.V. Sadovnikov et al., Phys. Rev. Lett. 120, 257203 (2018)&lt;br/&gt;[6] M. Krawczyk et al., J. Phys. Condens. Matter 26 (2014)&lt;br/&gt;[7] H.J. Qin et al., Nat. Commun. 9, 5445 (2018)&lt;br/&gt;[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 S&lt;sub&gt;12&lt;/sub&gt;) 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.



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

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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.

technical paper

It has been shown that a rapid cooling of Yttrium-Iron-Garnet (YIG)/Pt nanostructures preheated by a DC electric current pulse sent through the Pt layer leads to an overpopulation of a magnon gas and to the subsequent Bose-Einstein condensation (BEC) of magnons [1]. Hereby, the application of a sufficiently-long DC heating pulse was used to generate a high population of magnons being in equilibrium with the phononic system. The subsequent rapid decrease in the temperature resulted in the break of the equilibrium, in the over-population of magnons over the whole magnon spectrum, and, in the following, in a redistribution of magnons from higher to lower energies. Finally, if the elevated temperature was high enough, and the cooling process fast enough, the chemical potential reached the minimal energy of the system and the BEC was triggered [1]. In these experiments, a Pt or Al layer was used to heat the YIG nanostructure. For a Pt heater, in addition to the Ohmic heating effects, one expects the generation of a spin-polarized current transverse to the YIG/Pt interface due to the spin Hall effect (SHE) [2]. The spin current, consequently, is known to act on the magnetization dynamics in the YIG via spin-transfer torque (STT). Whereas in our previous studies the induced disequilibrium was found to be mainly given by the temperature increase and subsequent rapid cooling, we observe a significant contribution of the spin Hall effect (SHE) in a new set of samples. The larger effect of the SHE-STT was reached by a modification of the structures (see Fig. 1) with respect to those in the original study [1], i.e. by a decrease in the YIG thickness, and via using a sputtered Pt layer instead of an epitaxially grown Pt layer studies to enhance the spin Hall angle [3]. It was found that SHE-induced STT enhances the BEC formation at the bottom of the magnon spectrum – in the minimum frequency of the fundamental spin-wave mode, and excites a nonlinear magnon mode – the spin-wave bullet, located below the linear magnon spectrum [4]. We investigated the dynamics of these magnon states employing Brillouin light scattering (BLS) spectroscopy. A sketch of the experimental setup and SEM images of the investigated structure is depicted in Fig. 1. The application of a 50-ns-long DC-pulse to a 7-nm- thin Pt layer on top of a 34-nm-thin and 2-μm-wide YIG strip results in the SHE-STT. At sufficiently large amplitudes of the DC-pulse applied, the bullet mode is formed during the DC pulse at a frequency around 1.8 GHz below the frequency of the fundamental mode, see Fig. 2b. The pronounced magnon density peak in the left panel in Fig. 2c (solid orange line), measured at the time just before the current pulse is switched off, clearly demonstrates the bullet mode formation. The SHE-STT nature of the mode is confirmed by the variation of the magnetisation orientation with respect to the current direction. The consequent switching of the DC pulse off results in the rapid cooling-induced BEC of magnons at the frequency around 6 GHz - see Fig. 2c. The results of the control experiment with no SHE-STT contribution under periodically changing polarity of the applied current (Fig. 2d) are in agreement with the original findings presented in Ref. [1]. The interplay between the magnon condensate and the SHE-STT bullet mode, after the current pulse was gone, attracts special attention. Initially, in the rapid cooling phase of the process, thermally-induced excess magnons populate both the bullet mode and the condensate, see Fig. 2a. Further, both modes decay slower than expected from their lifteime indicating that they are still pumped from the gas. Finally, about 30 ns after the DC pulse is turned off, a qualitatively new process begins, resulting in the redistribution of magnons from the bullet mode to the condensate. This phenomenon manifests itself in an elevation of the condensate density accompanied by an increased decay of the bullet mode, and apparently occurs when the chemical potential of the magnon gas [1] decreases below the bullet mode frequency. The results suggest a path towards the application of macroscopic quantum states in spintronic devices.

**References:**

[1] M. Schneider, T. Brächer, D. Breitbach, V. Lauer, P. Pirro, D. A. Bozhko, H. Y. Musiienko- Shmarova, B. Heinz, Q. Wang, T. Meyer, F. Heussner, S. Keller, E. T. Papaioannou, B. Lägel, T. Löber, C. Dubs, A. N. Slavin, V. S. Tiberkevich, A. A. Serga, B. Hillebrands, and A. V. Chumak, Bose-Einstein condensation of quasiparticles by rapid cooling, Nature Nanotechnology 15, 457 (2020). [2] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016). [3] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Tuning the spin hall effect of pt from the moderately dirty to the superclean regime, Phys. Rev. B 94, 060412 (2016). [4] A. Slavin and V. Tiberkevich, Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet, Phys. Rev. Lett. 95, 237201 (2005).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/60e5026f5c65e8d34354a4da266411ef.jpg)
Colored SEM images of the investigated structures and sketch of the experimental setup. The structures consists of a 2-μm-wide and 34-nm-thick YIG strip. A 3-μm-long Platinum-heater (7 nm) on top is contacted by Ti/Au leads. The magnon intensity below the Pt-covered region is measured using time-resolved BLS spectroscopy. The shape of the Ti/Au contact pads attached to the Pt injector results in a DC current parallel to the short axis of the strip. Qualitatively the same results were obtained for another set of samples with a DC current flowing parallel to the short axis of the strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/534ab70ea9e9adb9ae113bed737dcfc7.jpg)
(a) Integrated BLS intensity from (b) as a function of the time within the two indicated frequency ranges. (b) BLS intensity color-coded as a function of time and the BLS frequency. During the pulse (0 ns < t < 50 ns and U = 2.0 V) the formation of a bullet mode is visible. The inset shows a sketch of the structure. (c) BLS intensity as a function of the BLS frequency at three different time steps, indicated by the dashed lines in (b). The black dashed lines depict the thermal BLS spectrum without a DC pulse present. (d) Integrated BLS intensity color coded as a function of time and the BLS frequency. In contrast to (b), the current polarity is alternating during the pulse, effectively switching off the contribution of the SHE-STT while kepping the heating constant.

 The interplay between the rapid cooling-induced BEC and SHE-STT-driven bullet mode

Quantum magnonics is a rapidly growing field offering myriad of potential applications[1,2] particularly in the realm of coherent information processing. Magnons which are the collective excitations of spins offer exotic functionalities such as the ability to couple to various dynamic systems. Realisation of strong coupling , owing to large spin densities, which is tunable into the ultrastrong coupling regime makes them suitable for coherent information exchange at much faster speeds. Apart from this the easily accessible gigahertz frequency range, simple fabrication and miniaturization provides an ideal platform for incorporation in various microwave electronic circuits. Recently[3,4,5], Synthetic Antiferromagnets(SyAF) have attracted prodigious interest due to the controllable exchange coupling in the GHz regime making them a test bed for various magnonic applications[6]. Moreover, the substantially large mode volume overlapping gives them an advantage over light-matter interactions in attaining higher a coupling strength. A SyAF consists of two ferromagnetic thin films (FM) with an antiferromagnetic exchange coupling separated by a non-magnetic spacer (NM). When the two FMs are symmetric, a mode crossing occurs between symmetry protected in-phase and out- of -phase resonance precessions via in-plane external magnetic field tuning. This is reminiscent of degeneracy between even and odd parity magnons. This degeneracy and subsequent mode crossing can be lifted by breaking the symmetry either extrinsically or intrinsically. Extrinsic symmetry breaking (SB) can be exerted by tilting the external magnetic field out-of-plane which shifts the rotational symmetry axis from the SyAF plane, and the system is no longer symmetric under two-fold rotation. This leads to hybridization of the two modes and an anticrossing gap is generated. Additionally, a coupling gap can be induced by dynamic dipolar interaction[3]. The strength of the coupling between two magnon modes is determined by the size of the gap. Maximizing the coupling efficiency is thus easily attainable by the controllability of the coupling gap by external magnetic field orientation. Apart from an extrinsic control of coupling strength the symmetry of the system can be broken intrinsically by using either two different FMs or the same FMs with different thickness. This would lead to a coupling gap even for the external applied fields applied in-plane, owing to a change of magnetization under two-fold rotation. In this work we demonstrate the control of magnon-magnon coupling in SyAFs via intrinsic and extrinsic SB. The schematic of the two modes in SyAF system is shown in Fig 1(a). We first consider a symmetrical SyAF structure CoFeB/Ru/CoFeB as in Fig 2(a). The apparent mode crossing results from a degeneracy of two non-interacting modes which is protected by a two-fold rotational symmetry. By tilting the external field (see Fig 2(b)) we break this symmetry and an anticrossing gap appears. This demonstrates the extrinsic SB. When two FMs with different thickness are considered an inherent SB exists in the system which opens a gap even for an in-plane field as in Fig 2(c). Further, breaking the symmetry as in 2(d) by tilting the field leads to an increase of the gap and thus the coupling strength. This effect can be further enhanced when using two FMs of different materials, exhibiting a large coupling gap for an in-plane field as in Fig 2(e). Here, the coupling strength far exceeds the losses, described by the frequency half-width at half maximum of the individual modes. The absolute coupling strength reaches 30% of the mode frequencies, indicating the onset of the ultrastrong coupling regime. By calculating the coupling gap as a function of external field tilt angle we can putatively claim that both intrinsic and extrinsic SB can realise strong magnon-magnon coupling in SyAFs. Further, the strong coupling leads to an entanglement between even and odd parity modes, making it attractive for quantum information applications. We generalize a theoretical model in the macro spin limit by solving the two coupled LLG(Landau Lifshitz Gilbert) equations to describe the microwave absorption spectroscopy for the SB induced coupling in SyAFs. This study should shed light on the potential of SyAFs for studying and controlling magnon-magnon coherent coupling at room temperature. Moreover large tunable coupling strengths in this system will provide a novel platform for increased control of quantum systems.

**References:**

[1] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proc. Natl. Acad. Sci. U.S.A. 112, 3866 (2015). [2] M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Appel, and A. Rubio, Nature Reviews Chemistry, 2(3), pp.1-16 (2018). [3] Y. Shiota, T. Taniguchi, M. Ishibashi, T. Moriyama, and Teruo Ono, Phys. Rev. Lett. 125, 017203 (2020). [4] A. Sud, C. W. Zollitsch, A. Kamimaki, T. Dion, S. Khan, S. Iihama, S. Mizukami, and H. Kurebayashi, Phys. Rev. B 102, 100403(R) (2020). [5] M. Ishibashi, Y. Shiota, T. Li, S. Funada, T. Moriyama, and T. Ono, Sci. Adv. 6, eaaz6931 (2020). [6] A. Guedes, M. J. Mendes, P. P. Freitas, and J. L. Martins, Journal of applied physics 99,08B703 (2006).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fda936d0116f14aa194f61f7d6f95187.jpg)
Fig 1. (a) Schematic of resonance modes in SyAFs. Two moments (m1 and m2) are coupled antiferromagnetically and canted at equilibrium. Under microwave irradiation, they precess in-phase (acoustic mode) and out-of-phase (optical mode) at different angular frequencies ωac and ωop respectively .We define θB as in the figure. (b) Schematics of the magnon-magnon coupling phenomena. When the exchange field (Bex) is small or two moments are within the film plane, the coupling strength (g) is zero and we can valve g by tuning Bex and θB.

![](https://assets.underline.io/uploads/markdown_image/1/image/7eaef5b95c81b3bc4450cc2312714f60.jpg)
Fig 2. Microwave transmission as a function of frequency.(a) For CoFeB(5)/Ru(0.5)/CoFeB(5) at θB = 90°, the two modes show mode crossing (b) At θB = 25° a gap appears. (c) In NiFe(5)/Ru(0.4)/NiFe(3) intrinsic SB occurs and the gap is seen for θB = 90° which increases as in (d) for θB = 30°. (e) CoFeB(3)/Ru(0.5)/NiFe(3) showing large gap at (e) θB = 90° and (f) θB = 20°. (g-h) Coupling strength g/2π as a function of θB which increases as the tilt angle is increased.* The numbers in brackets are the thickness of the FM layer in nm.

 Modulation of magnon-magnon coupling by inherent symmetry breaking in synthetic antiferromagnets

Kagome artificial spin ice (KASI) is a network of Ising type nanobars on a kagome lattice [1,2]. Great progress has been made in understanding defects including disordered states in KASI via quasistatic imaging techniques [1,3]. However, the magnetodynamic study of disordered states has been confined to global magnetodynamic (e.g. broadband spin wave spectroscopy) and micromagnetic simulations [4]. Furthermore, the simulation study on KASI has shown the presence of novel magnetodynamic microstates that offer excellent insights into the KASI’s disordered regime [4]. From the fundamental physics side, dynamically controlled microstates may offer a way to create Dirac strings via microwave-assisted switching interior to the KASI lattice in a controlled manner and study the disordered regime systematically. From the magnonic application perspective, experimental studies of microstates in KASIs are key towards their usage as a new type of microwave filter [5] and reprogrammable magnonic crystal [6]. We investigate spin dynamics of a KASI consisting of Ni81Fe19 nanomagnets arranged on an interconnected kagome lattice using broadband spin wave spectroscopy (Fig. a,b), magnetic force microscopy (MFM) (Fig. c), and micro-focus Brillouin light scattering (BLS) microscopy (Fig. e-f). Micro-focus BLS performed on magnetically disordered states exhibit a series of magnon resonances that depend on topological defect configurations that we image by magnetic force microscopy. Nanomagnets on a Dirac string and between a monopole-antimonopole pair show pronounced modifications in magnon frequencies. Our work is key for the creation and annihilation of Dirac strings via microwave-assisted switching and reprogrammable magnonics based on ASIs.

**References:**

1) S. Ladak, D. Read, G. Perkins, L. Cohen, and W. Branford, Nature Physics 6, 359 (2010). 2) R. Wang, C. Nisoli, R. Freitas, J. Li, W. McConville, B. Cooley, M. Lund, N. Samarth, C. Leighton, V. Crespi, et al., Nature 439, 303 (2006). 3) E. Mengotti, L. J. Heyderman, A. F. Rodriguez, F. Nolting, R. V. Hugli, and H.-B. Braun, Nat. Phys. 7, 68 (2011). 4) V. Bhat, F. Heimbach, I. Stasinopoulos, and D. Grundler, Physical Review B 93, 140401 (2016). 5) X. Zhou, G.-L. Chua, N. Singh, and A. O. Adeyeye, Advanced Functional Materials 26, 1437 (2016). 6) M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter 26, 123202 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a84d0d93b7a280959772b4805fbd9680.jpg)
Grayscale AESWS spectra map of KASI for a field protocol of (a) -90 mT → +90 mT and (b) -90 mT → 45 mT → 0 mT. (c) Magnetic force microscopy images of KASI after a field protocol of −100 mT → +44 mT → 0 mT. Stray fields detected by MFM are colored in blue and red consistent with charges Q = +q and Q = −q, respectively. (d) Sketches of charge configurations (blue and red spheres) and magnetization vectors M (arrows) for configurations C1 to C5. (e) BLS intensities measured at the central position on horizontal nanobars belonging to the configurations displayed above the spectra at µ0H = 25 mT. (f) Spatially resolved BLS intensity maps at 25 mT for fixed frequencies in configurations C1 to C5 displayed above.

 Direct Observation of Magnon Modes In Kagome Artificial Spin Ice with Topological Defects

In this work, we report our micromagnetic simulation results on the correlation between spin wave modes and emergent microstates in square artificial spin ice (S-ASI), consisting of elliptical and stadium shape magnetic nanoislands. S-ASI1 is a lithographically patterned engineered dipolar coupled nanostructure array, consisting of orthogonal square sublattices of highly shape anisotropic nanoislands. These islands behave as a macrospin due to high shape anisotropy where each macrospin, according to the dumbbell model, can be considered as a magnetic dipole of charge separated by the length of the nanoisland. Recently, there is a growing interest in the low-energy collective spin excitation of magnetic nanostructures of vastly different shapes resulting in the development of the field of magnonics. These excitations or spin waves are the magnetic analogs of electromagnetic waves, phonons, and plasmons which can be treated as a perturbation in terms of phase-coherent precession of magnetization vector that propagates in coupled magnetic media. The field of magnonics offers new functionalities in modern technological devices e.g., magnonics logic, phase shifters, microwave antenna, directional coupler, SW-based multiplexer, grating, and neuromorphic computing, etc. S-ASI exhibit geometrical frustration and therefore, it shows ground state degeneracy. The microstates in S-ASI enable the possibility of tunability of associated spin-wave modes and alter the spin-wave band gap without structural deformation. This key feature of S-ASI makes it a potential candidate for reconfigurable magnonics2. The interesting dynamic behavior of ASI suggests that it can be perceived as magnonic crystals with potential applications in logic devices3, data storage4, microwave filter5, spin logic gate3, etc. Hence, detailed investigation6, 7 of spin wave dynamics and evolution of modes in ASI systems involving highly shape anisotropic nanoislands of different shapes is very important. The question of the correlation between underlying magnetic states of dipolar coupled nanoislands of shapes such as ellipse and stadium, etc., and the spin wave modes is far from properly understood. In this work, we have performed a detailed study of the dynamical behavior of emergent microstates in S-ASI, using micromagnetic simulations. For these studies, we have considered S-ASI to be consisting of four square ring-type structures with elliptical and stadium shaped nanoislands of permalloy with dimensions 300 nm × 100 nm × 25 nm and lattice constant 150 nm (edge-to-edge distance of nanoislands). The magnetization reversal behavior of stadium-shaped magnetic nanoislands in S-ASI geometry involves the formation of vortices, antivortices, etc. whereas for the case of elliptical nanoislands the reversal involves sharp switching of spins. In order to investigate the dependence of the behavior of spin wave dynamics on local magnetic behavior in the S-ASI of two different shapes, we have investigated the spin wave modes and emergent microstates in both the shapes as a function of a number of square-type rings (i.e., single ring, double ring, triple ring, and quadruple ring e.g., S-ASI) of nanoislands. For S-ASI involving elliptical nanoislands, our studies reveal that emergent microstates at remanence with net local magnetic charges at the junction of the nanoislands for head-to-head or tail-to-tail configuration enhances spin wave generation in the proximity islands up-to ∼95% as compared to islands in the proximity of zero local charges ( in head-to-tail or tail-to-head configuration). Thus, we observe that the power profiles in remanence map the local charges present in the S-ASI or ring-type structures. We observe the presence of ubiquitous switching identifier mode at ~ 18 GHz (saturation) in nanoislands with an easy axis along the external bias field. This mode shows marked power variation in the switched nanoislands with respect to un-switched nanoislands. Our simulation results show that observed spin wave spectra and calculated mode profiles remain consistent as a function of rings. Our studies of S-ASI involving stadium shaped nanoislands further show that the spin wave spectra are significantly different than for the elliptical case (see Fig. 1). In the case of stadium-shaped nanoislands, the demagnetization field is inhomogeneous which creates variation in the local field within the nanoislands whereas, in the case of elliptical nanoisland, demagnetization field variation is observed near the edges. However, for a similar magnetization configuration, we have observed several new spin-wave modes with complex mode profiles in the case of stadium shape nanoislands (Fig. 2). Thus, our study gives new insight in understanding the origin of spin wave mode behavior in terms of the evolved microstates and demonstrate spin wave modes as a tool to investigate magnetization reversal and local magnetic charges in artificial spin ice system.

**References:**

(1) S. T. Bramwell et al., Science 294, 1495–1501 (2001). (2) E. Iacocca et al., Phys. Rev. B 93, 134420 (2016). (3) H. Arava et al., Nanotech. 29, 265205 (2018). (4) Y.-L. Wang et al., Science 352, 962–966 (2016). (5) I. Huynen et al., IEEE Microw. Guided W. 9, 401–403 (1999). (6) S. Gliga et al., Phys. Rev. Lett. 110, 117205 (2013). (7) E. Iacocca et al., Phys. Rev. Appl. 13, 044047 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8fa292a767f18a77cc35d89a92aba095.jpg)
The power profile of the spin wave mode excited at ∼ 6.5 GHz in S-ASI with (left) elliptical and (right) stadium-shape nanoislands at 200mT external field.

![](https://assets.underline.io/uploads/markdown_image/1/image/16c49e9a6920a28ab2b3c0ed74f242a4.jpg)
(left) Spin wave spectra of S-ASI consisting of stadium shape and elliptical shape nanoislands saturated along the x-axis at 200mT. (right) magnetization configuration of S-ASI with the stadium and elliptical shape nanoislands.

 Dependence of spin wave modes on the geometry of nanomagnets in square artificial spin ice vertices

Spin waves, and their quanta magnons, are of great interest as potential data carriers in future low-energy computing devices [1]. The phase of a spin wave provides an additional degree of freedom, while the scalability of structures and wavelengths down to the nanometer regime [2] are further advantages. Recently, a set of magnonic data processing units was demonstrated. However, the development of each of them requires specialized investigations and, usually, one device design is suitable for one function only. Moreover, an integrated all-magnonic circuit, which is suitable for the cascading of multiple magnonic units, has not yet been demonstrated. In the first half of my talk, I will present briefly the experimental realization of a nanoscale magnonic directional coupler, which consists of two separated single-mode waveguides with a width of 350 nm as shown in Fig. 1a [3]. A U-shaped antenna is used to excite spin waves and space-resolved Brillouin Light Scattering (BLS) spectroscopy is exploited for detection. It is shown that the data is coded into the spin-wave amplitude is guided towards one of its two outputs depending on the signal frequency, magnitude, and on the magnetic field. Using micromagnetic simulations, we also propose an integrated magnonic half-adder that consists of two directional couplers and we investigate its functionality for information processing within the magnon domain. In the second part of the talk, I will present the method of inverse-design magnonics, in which any functionality can be specified first, and a feedback-based computational algorithm is used to obtain the device design [4]. Our proof-of-concept prototype is based on a rectangular ferromagnetic area which can be patterned using square shaped voids as shown in Fig. 1b. To demonstrate the universality of this approach, we explore linear, nonlinear and nonreciprocal magnonic functionalities and use the same algorithm to create a magnonic (de-)multiplexer, a nonlinear switch and a circulator. Thus, inverse-design magnonics can be used to develop highly efficient rf applications as well as Boolean and neuromorphic computing building blocks.

**References:**

A. V. Chumak, V. I. Vasyuchka, A. A. Serga, & B. Hillebrands, Nat. Phys. 11, 453 (2015). Q. Wang, B. Heinz, R. Verba, et al. Phys. Rev. Lett. 122, 247202 (2019). Q. Wang, M. Kewenig, M. Schneider, et al. Nat. Electron. 3, 765-774 (2020). Q. Wang, P. Pirro, A. V. Chumak, arXiv:2012.04544.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/12c3053bde7dbdc6eb5b8bbe6696770d.jpg)
Fig. 1 The evolution from the conventional design (a) to inverse-design (b) magnonic demultiplexers.

 Nanoscale magnonic devices: From conventional to inverse-design magnonics INVITED

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 waves (SWs) are promising quanta for transmission and processing of information for future communication devices. Excitation and detection of beyond 20 GHz magnons with sub-100 nm wavelength are of key importance [1]. We report on the magnonic grating coupler (MGC) effect [2] for which prepared specific nanopillars. Depending on the type of MGC lattice and nanopillar diameters, magnons are emitted resonantly on the chip in a multitude of directions with different efficiency. We demonstrate the interference effect of short-wave magnons independent of propagation directions. We engineered grating couplers for broadband magnon emission by preparing 100-nm-thick permalloy nanopillars in a periodic square lattice and an aperiodic (Penrose P3) quasicrystalline arrangement [3] on insulating ferrimagnetic yttrium iron garnet (YIG) films. Characteristic lengths amounted 480 nm. Several sub-micron-wide coplanar waveguides (CPWs) with separations of 12 micrometers were integrated on top of the MGC samples to excite/detect SWs by broadband spectroscopy. The MGC sample based on the quasicrystalline arrangement allowed for SW emission omnidirectionally with a broad range of wave vectors. SWs of up to 25.3 GHz were detected in the square lattice-based MGC sample. These parameters were achieved in very small magnetic fields of only a few mT in contrast to previous studies on grating couplers [2]. SWs simultaneously excited by two emitters interfered constructively/destructively at a separate detector antenna depending on the phase difference of the two phase-coherent SWs. Thereby a binary 1/0 output operation could be realized with 70 nm-wavelength magnons over distances of more than 10 micrometers. The applied field direction and specific microwave frequency defined the SW emission direction for the interference experiment. Our study and findings on sub-100 nm magnons are important for future magnonic devices targeting at information processing via wave-logic on the nanoscale. The work was supported by SNSF grant 163016.

**References:**

[1] A.V. Chumak, V. I. Vasyuchka, A. A. Serga et al., Nature Physics, Vol.11, p. 453 (2015) [2] H. Yu, G. Duerr, R. Huber et al., Nat. Commun., Vol. 4, p. 2702 (2013) [3] S. Watanabe, V.S. Bhat, K. Baumgaertl et al., Adv. Funct. Mater., Vol. 30, p. 2001388 (2020)

**Images:**

 Towards Wave-based Logic Operation using Multi-directional sub-100 nm Magnons

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.

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