Spintronic devices operating on spin waves (SW) can be more efficient than oxide semiconductors due to their high frequencies and low energy cost. Therefore, SW based technology may prove to be more powerful than CMOS devices due to its intrinsic characteristics [1]. Magnetic field is the primary stimuli controlling magnetism and SW dynamics. Thus, the study of the influence of the magnetic field on ferromagnetic nanostructures with different geometries is critical for modern technologies, especially for the development of 3D-magnonic circuits, where the geometric constraints generate a strong relationship between the fields of demagnetization, exchange and bias.
Recently, it has been experimentally demonstrated that the 3D network formed from the crescent-shape nanorods (CSN) is promising for development of 3D ferromagnetic systems for various applications [2]. As yet, CSN-s have not been investigated for their effect on SW spectra and manipulations in relation to magnitude and magnetic field orientation. Untypical results make our analysis relevant for future investigations and applications. In particular, we show that there are edge localized and bulk SW modes. The edge modes decouple when the external magnetic field rotates, Fig. 1. Thus, the two edges of the CSN behave as two distinct channels for SW guiding, whose frequency can be controlled by the magnetic field magnitude and orientation. Interestingly, both the polar and azimuthal rotations of the field significantly affect the ferromagnetic resonance spectra. It alters the number of most intense lines in the spectra and their frequency range. When additional magnetic layer is added to the CSN, the magnetostatic coupling between the elements leads to additional modes splitting, Fig. 2. Thus, the CSN-s can support 3D magnonic circuits at nanoscale, which operate at wide range of frequencies, offer multimode operation, and promise nonreciprocal effects. 
The research has received founding from the NCN Poland, project no 2020/39/I/ST3/02413.
**References** [1] A. V. Chumak et al., Advances in Magnetics Roadmap on Spin-Wave Computing, IEEE Trans. Magn. 58, 1 (2022).
[2] A. May, et al., Realisation of a frustrated 3D magnetic nanowire lattice. Commun. Phys. 2, 13 (2019). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6188d565ba7c5e1f93f8bd16ea6986c0.png)

Control of the spin wave spectra by magnetic field orientation and magnitude in crescent

Wood's anomaly is a phenomenon discovered in 1902 [1], manifesting as a decrease of the reflected waves amplitude due to the excitation of a surface mode that is still investigated in photonics [2]. Another curious effect observed in the reflection of obliquely incident waves is the Goos-Hanchen effect (GHE) [3,4], which causes a spatial shift of the reflected waves. One of the configurations suitable for observing both phenomena for spin waves (SWs) is the Gires-Tournois interferometer (GTI) proposed in Ref. [5,6], made of a magnetic stripe placed above the edge of a magnetic layer. We use micromagnetic simulations to study the reflection of an oblique incident SW beam at such a GTI. We identified the resonance conditions required to efficiently excite SW modes in the GTI by the incident SW beam, which give rise to magnonic Wood's anomaly in our system. The amplitude of these modes is confined in the stripe and emits SWs back to the layer; therefore, these modes can be classified as 'leaky-modes'. The consequence of the leaky-mode is the creation of multiple spatially shifted SW beams in the layer parallel to one another. Furthermore, the excitation of the leaky-modes is accompanied by a significant GHE for the primary reflected beam with values spanning between -175 nm and 225 nm with respect to the non-resonant case. In Fig. 1a and 1b we show the distributions of SW intensity for the case with the resonant excitation of a leaky-mode and without the excitation, respectively. In Fig. 1b multiple reflected SW beams are evident. Our findings contribute to understanding and utilizing the interaction of propagating SWs in thin films with the localized leaky-modes. Additionally, an incident beam's excitation of modes in GTI provides a platform for investigations of SWs inelastic scattering on interferometer's modes. We acknowledge the funding from the Polish National Science Centre project No. UMO-2019/33/B/ST5/02013.

Magnonic interferometer’s leaky mode influence on spin

Magnetic multilayers of Ir/Co/Pt with interfacial Dzyaloshinskii-Moriya interaction (IDMI) [1] were deposited by magnetron sputtering with Co thickness d=1.8nm and repetitions N=6. Ferromagnetic resonance spectroscopy was applied to determine the values of effective magnetic uniaxial anisotropy, Hu = -0.45 T. Exploiting magneto-optical Kerr effect in longitudinal mode microscopy, magnetic force microscopy and vibrating sample magnetometery the magnetic field driven evolution of domain structures and hysteresis loops have been studied. Existence of weak stripe domains structure was deduced – large micrometer size domains with in-plane "core" magnetization modulated by small nano-domains with out-of-plane magnetization. Quantitative evaluation of IDMI was carried out using Brillouin light scattering (BLS) spectroscopy [2,3] from the difference between Stokes and anti-Stokes peak frequencies Δf. Due to the additive nature of IDMI, the asymmetric combination of Ir and Pt covers led to large values of surface DMI energy density Ds. Fig. 1 shows Stokes and anti-Stokes peaks for magnon scattering. It was found that Stokes and anti-Stokes frequencies as well as Δf, measured as a function of in-plane applied magnetic field, show hysteresis. These results are explained under the consideration of the influence of IDMI on the dynamics of in-plane magnetized “core” with weak stripe domains. In order to understand the spatial magnetization distribution and magnetization process, the micro-magnetic simulations were performed. Our findings leverage the impact of IDMI on domain structure. Large values of Ds can be useful for the stabilization of skyrmions [1] and studying the dynamical behavior of magnetic skyrmions in the in-plane magnetized sample [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/92e6a5942842efc48ac7a6e5bad30c22.jpg)

Fig. 1. BLS spectrum for d=1.8 nm where Stokes and anti-Stokes peaks are shown. Measurement was done for wavevector k=11.81 μm-1 and applied in-plane magnetic field µ0H=72.8 mT. Experimental data (black line) and its mirror curve (red line, for easy comparison) determine the frequency difference Δf ≈1.5 GHz.

Magnetization statics and dynamics in Ir/Co/Pt multilayers with Dzyaloshinskii-Moriya interaction

Spin waves (SWs) are precessional magnetization disturbances propagating in magnetic media in the form of waves at microwave frequencies [1] that are the most commonly used frequencies range for wireless communication. Wavelengths of SWs are several orders of magnitude shorter than the lengths of electromagnetic waves of corresponding frequencies. The dynamics of SWs even in uniformly magnetized films is extraordinarily rich. It is easy to obtain effects difficult to be achived for other type of waves, such as nonreciprocity, excitation caustic-beam that are characteristic for hyperbolic media, easy external control over propagation (e.g., by inhomogeneous magnetic field) [2]. Non-linear phenomena are also easily available[1,3]. All these make SWs to be a promising candidate for information carriers, especially in the context of beyond-CMOS applications [4]. SWs may be confined to certain areas of the thin film, e.g., in a potential well created by an inhomogeneity of the static demagnetizing field in the vicinity of film’s edge. Such localized modes are called edge SWs (E-SWs). Usually, the frequencies of E-SWs are downshifted with respect to bulk-type SWs (B-SWs), i.e., SWs propagating far from the edge where the static effective field is practically homogenous. Here, we theoretically study by means of micromagnetic simulations [5] and analytical modelling the dynamics of the E-SWs localized at the edge of thin permalloy film and their interaction with B-SWs. Firstly, we focus on the utilization of the E-SWs to excite short B-SWs propagating in the form of plane waves. Secondly, we study the inelastic non-linear scattering of B-SW beam by the E-SWs. We consider propagation of the E-SW along the edge of 10 nm thick permalloy film that is uniformly magnetized by the external field of value 300 mT directed perpendicularly to the film’s edge (see Fig. 1a, where the system is presented along with exemplary result of a simulation). The E-SW is excited by the point-source-like microwave field of frequency 12 GHz located at the film’s edge. There are visible two prominent peaks in the simulated spectrum (Fig. 1b). The first one corresponds to the frequency of the microwave field (12 GHz) and propagation of the E-SWs, whereas the second peak has doubled frequency (24 GHz) and corresponds to the B-SWs propagating obliquely outwards the edge in the form of plane waves. It indicates excitation of the B-SWs by the propagating E-SWs in the process of second harmonic generation. Noteworthy, the wavelength of the B-SWs is over twice shorter than the wavelength of the E-SWs shown in Fig. 1a. The B-SW of wavelength equal to 78 nm is excited by the 285 nm long E-SW. For the same sample, we analytically and numerically analyze the interaction of an obliquely incident B-SW beam (at frequency f) with an E-SWs at frequency n. We find that due to the inelastic scattering the secondary B-SW beams with up- and down-shifted frequency (f-n, f+n) can be excited. In Fig. 2 an exemplary result of simulation for a B-SW beam incident at the films edge where E-SW propagates and the resulting secondary scattered beam with frequency (f+ν)=42 GHz are shown. Moreover, we observe angular shifts between the primary incident and the scattered beams with shifted frequencies and analyze the efficiency of the scattering process. Finally, we present an application of this effect for SW demultiplexing. Acknowledgments. The research was supported by the National Science Centre of Poland, project no. 2019/35/D/ST3/03729 (P.G. and M.K.); COST action under project CA17123 MAGNETOFON (I.L.); the Spanish MCIU grant PID2019-108075RB-C3-3 (K.G.). K.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science). The simulations were performed at the Poznan Supercomputing and Networking Center (Grant No. 398).

**References:**

 [1] A. Gurevich and G. Melkov, Magnetization oscillations and waves (CRC Press, Boca Raton, 1996). [2] A. Barman et al., J. Appl. Phys. 128, 170901 (2020). [3] A. V. Chumak et al., Nat. Phys. 11, 453 (2015). [4] B. Murmann, B. Hoefflinger, "IRDS—International Roadmap for Devices and Systems, Rebooting Computing, S3S" in NANO-CHIPS 2030 (Springer, Cham, 2020). [5] A. Vansteenkiste et al., AIP Advances 4, 107133 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6f369e1f64f9a7c41706068472bcb0f0.jpg)
Fig. 1. (a) System under consideration with the snapshot of the out-of-plane magnetization component (mz) in the steady state with E-SW excitation at 12 GHz. The source of E-SWs is located in the area indicated by the white triangle. (b) Simulated SW frequency spectrum.

![](https://assets.underline.io/uploads/markdown_image/1/image/3d4019bb873d5286156ad035fb24cd24.jpg)
Fig. 2. Scheme of the system under investigation with the result of micromagnetic simulation performed for B-SW beam at frequency f = 30 GHz incident under the angle 60° and scattered on an E-SW at ν =12 GHz. Red (at the edge), yellow and blue (beams) corresponds to presence of the SWs of frequencies ν, f, and f+ν, respectively. The white color denotes areas, where at least two frequencies are mixed.

 Inelastic scattering of spin wave beam at the edge localized spin waves and second harmonic generation of spin waves

Very recently magnetic stripe domains, characterized by alternating up and down out-of-plane orientation of the magnetization, have received great interest due to the possibility to use stripe patterns to manipulate spin-wave (SW) propagation as in artificial magnonic crystals [1]. In this work, we demonstrate the control of the SW propagation by using reconfigurable regular stripe-pattern domain structure in the hybrid system. The investigated system consists of 64-nm-thick NdCo layer and 10-nm-thick NiFe layer, coupled through an Al layer of different thicknesses. Magnetic force microscopy (MFM) measurements show that, due to the perpendicular magnetic anisotropy of the NdCo film, the system develops stripe domains aligned with the last in-plane saturation direction [2]. The domain pattern is found to have a period of about 140 nm, which is almost independent on the thickness of the Al layer. The magnetization reversal of the trilayer system was investigated by vibrating sample magnetometer, showing that the hysteresis loop is characterized by a two-step process, due to the different coercivity of the NiFe and NdCo films. Detailed analysis of the hysteresis loops along with micromagnetic simulations indicates that the stray magnetic field coming from the NdCo layer induces a regular domain structure also in the NiFe layer, which is tuned by the thickness of Al spacer. In addition, upon reversing the applied magnetic field, an antiparallel state, characterized by an antiparallel alignment of the magnetization component parallel to the domain axis in the NdCo and NiFe stripes, is formed. Then, Brillouin light scattering spectroscopy has been used to measure the spectra of the SWs propagating in the direction perpendicular to stripe domains for the parallel and the antiparallel state. For both configurations, the dispersion relation shows a strongly nonreciprocal mode (Figs. 1-2). However, in the parallel state SWs propagating with positive and negative wavevector are both characterized by a positive dispersion, while in the reversed state SWs propagating with negative wavevector show a negative dispersion. The above experimental results have been satisfactorily reproduced by numerical simulations. The latter show that the detected SW mode is mainly localized in the NiFe layer and its frequency nonreciprocity can be ascribed to the static magnetization configuration as well as to the interaction with the NdCo induced by the SWs via the dynamic stray field.

**References:**

[1] C. Banerjee et al., Phys. Rev. B 96, 024421 (2017) [2] D. Markó et al., Appl. Phys. Lett. 115, 082401 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1f952ef7a0b2bb8cf6cafa5252632c82.jpg)
Fig. 1. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the parallel state. Empty and full dots mark the results of Brillouin light scattering measurements and simulations, respectively. Size and color intensity of the numerical results indicate the predicted intensity in the measurements.

![](https://assets.underline.io/uploads/markdown_image/1/image/dbd287cdf366b5549a79bdda3afc82b8.jpg)
Fig. 2. Dispersion relation in the NdCo(64 nm)/Al(5)/NiFe(10) in the antiparallel state.

 Reconfigurable spin-wave propagation in magnetic stripe domains in hybrid system

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

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.

 Subwavelength resonant control of the spin-wave phase in thin ferromagnetic films

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Presentations

Control of the spin wave spectra by magnetic field orientation and magnitude in crescent

Magnonic interferometer’s leaky mode influence on spin

Magnetization statics and dynamics in Ir/Co/Pt multilayers with Dzyaloshinskii-Moriya interaction

Inelastic scattering of spin wave beam at the edge localized spin waves and second harmonic generation of spin waves

Reconfigurable spin-wave propagation in magnetic stripe domains in hybrid system

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

Subwavelength resonant control of the spin-wave phase in thin ferromagnetic films

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