Periodic and Aperiodic NiFe Nanomagnet/Ferrimagnet Hybrid Structures for 2D Magnon Steering and Interferometry with High Extinction Ratio

Magnons, quanta of spin waves, are known to enable information processing with low power consumption at the nanoscale. So far, however, experimentally realized half‐adders, wave‐logic, and binary output operations are based on few µm‐long spin waves and restricted to one spatial direction. Here, magnons with wavelengths λ down to 50 nm in ferrimagnetic Y3Fe5O12 below 2D lattices of periodic and aperiodic ferromagnetic nanopillars are explored. Due to their high rotational symmetries and engineered magnetic resonances, the lattices allow short‐wave magnons to propagate in arbitrarily chosen on‐chip directions when excited by conventional coplanar waveguides. Performing interferometry with magnons over macroscopic distances of 350 × λ without loss of coherency, unprecedentedly high extinction ratios of up to 26 (±8) dB [31 (±2) dB] for a binary 1/0 output operation at λ = 69 nm (λ = 154 nm) are achieved in this work. The reported findings and design criteria for 2D magnon interferometry are particularly important in view of the realization of complex neuronal networks recently proposed for interfering spin waves underneath nanomagnets.


Introduction
Magnons (spin waves, SWs) are collective spin excitations in magnetically ordered materials, enabling information processing without the flow of electrons. [1,2] In comparison to the complementary metal-oxidesemiconductor circuits based on the electron's charge, wave-logic devices based on magnons avoid energy losses by Joule heating and offer high speed. The wavelength of SWs is reduced by a factor of about 10 5 with respect to the wavelength of an electromagnetic wave in free space at the same frequency. Hence, on-chip wavelengths of sub-100 nm could be realized at 25 GHz used for 5G communication devices. However, wave-logic experiments have so far been conducted on SWs with macroscopic wavelengths of a few μm. [3][4][5][6][7][8] They propagated over distances less than 6 × , and binary 1/0 output operations exhibited an extinction ratio (ER) of 19 dB in ref. [8]. At the same time the operations were restricted to a single direction on the chip, that is, perpendicular to a microwave antenna. This restriction to unidirectional transport was still true for both a half-adder exploiting SWs of ≈ 2 μm at a frequency f of 3.5 GHz in the yttrium iron garnet Y 3 Fe 5 O 12 (YIG) [2] and a spin-wave interferometer which operated over 2 μm with spin waves of f up to 22 GHz ( down to about 50 nm). [9] Toward technologically relevant devices, excitation and detection of short-wavelength magnons into arbitrarily chosen onchip directions are key. Magnetic vortices and domain walls have been reported to act as versatile emitters in the low GHz frequency regime. Indeed, spin waves emitted from two angled domain walls in metallic ferromagnets created 2D interference patterns. They spatially extended over about 7 μm, that is, several times their wavelength of 450 nm. [11] For both emission and detection along multiple directions, the magnonic grating coupler (GC) effect has proven to be particularly advantageous. [12] Corresponding microwave-to-magnon transducers have used 2D lattices of ferromagnetic nanoelements integrated to a SW guiding medium such as the single-crystalline ferrimagnetic insulator YIG. [13] However, excitation of short-wave magnons near 25 GHz and zero field was not yet demonstrated by 2D grating couplers. www.advancedsciencenews.com www.advmat.de Only a 1D lattice of 100 μm-long ferromagnetic stripes on YIG excited and detected corresponding spin waves. [9,14] Their form factor contradicted, however, the creation of nanoscale devices with magnons propagating in arbitrarily chosen directions.
Recently, it was reported that nanopillars prepared from the polycrystalline ferromagnet Ni 81 Fe 19 (Permalloy, Py) exhibited a characteristic series of resonant modes over a broad frequency regime. [15][16][17] Integrated to YIG, their low-frequency fundamental mode in the vortex state ( Figure S1A,B, Supporting Information) was used to create a metasurface for spin waves near 1 GHz. [18] However, the higher-order modes have not yet been considered for microwave-to-magnon transducers on YIG, neither in 2D periodic nor aperiodic (quasicrystalline) lattices of nanopillars. The latter ones promise excitation and steering of magnons into a vast number of on-chip directions. [19][20][21] However, SW interference has not yet been evidenced for SWs with deep-submicron wavelengths over macroscopic distances and arbitrary directions.
In this paper, we report on ultrashort magnons in YIG undergoing spin-wave interference underneath Py nanopillar lattices in arbitrarily chosen on-chip directions. We demonstrate unprecedentedly large extinction ratios of about31 and 26 dB for a binary 1/0 operation at small wavelengths of 155 and 69 nm, respectively. These ratios are promising in view of nanomagnonics and considerably larger than 19 dB achieved by the unidirectional long-wavelength spin waves exploited before. [8] Our magnonic circuits are based on periodically and aperiodically arranged Py nanopillars which steer short-wave magnons into different directions in YIG films. We use SW spectroscopy [22] and micromagnetic simulations [23] to evidence the emission, steering and detection of spin waves down to a wavelength of about 50 nm. The ferromagnetic nanopillars are arranged at the vertices of a square lattice and a Penrose P3 tiling (Figure 1a) which is a 2D analogue of a natural quasicrystal. [24,25] Coplanar waveguides (CPWs) are integrated on top of the nanopillars which then operate as GCs for magnon emission and detection when microwaves are applied. [12] In case of the Penrose P3 lattice, wave vectors k = 2 / of SWs in YIG are related to reciprocal vectors F describing the different diffraction spots in reciprocal space ( Figure 1c). [26] The magnetic field is applied in-plane with an angle . In small in-plane magnetic fields, the relatively thick nanopillars incorporate the vortex state (Figure 1g (inset)). Simulated spectra of eigenfrequencies in an individual nanopillar (Experimental Section) show resonant spin-precessional motion which correspond to higher-order confined modes similar to refs. [15][16][17] (peaks in Figure 1g (blue curve)). Spectra S 22 measured on the hybrid samples contain consistent resonances (marked by arrows in the black and gray curves in Figure 1g).
We exploit the resonant and nonresonant GC effect [13] to perform on-chip SW interference experiments at unprecedentedly small wavelength down to = 63.1 nm and in arbitrary directions over macroscopic propagation lengths corresponding to 354 × . Our findings are key for magnonic networks enabling microwave technologies which operate at high frequencies on the nanoscale and in arbitrary on-chip directions like electronic circuits. Omnidirectional propagation, punchcard-like patterns of nanomagnets on YIG and interference over macroscopic distances are of key importance for nanomagnonic holographic memory and neural networks for neuromorphic computing. [27,28] Figure 1. Ferromagnetic nanopillars on a ferrimagnetic insulator for steering ultrashort magnons between coplanar waveguides. a) Penrose P3 tiling consisting of acute (red) and obtuse rhombi (green). b) Sketched experimental setup representing two CPWs (dark yellow) for emission and detection of propagating spin waves. Aperiodically arranged ferromagnetic Py nanopillars (brown) of diameter D reside on top of the ferrimagnetic YIG film (yellow) and in an applied magnetic field H. is the angle with the x-direction. c) Penrose P3 lattice in reciprocal space. Via the grating coupler effect, SWs corresponding to reciprocal vectors F are first emitted and then steered via the further nanopillars toward the detector CPWs. d) Resist mask after electron beam exposure and development of openings. e) Scanning electron microscopy and f) atomic force microscopy images of Py nanopillars on GGG after lift-off processing. g) Power spectral density (intensity) of simulated spin dynamics of a 100 nm-thick Py nanopillar (blue) with a diameter of 155 nm in the vortex state at μ 0 H = 5 mT (inset, illustrated via Mayavi [10] ). Reflection measurement (real part (Re) of S 22 ) of the SQ (black) and P3 lattice (gray) showing resonances (arrows) attributed to the Py nanopillars at 5 mT (Experimental Section).

Multifold Symmetry for Omnidirectional Spin Wave Emission
For the present study, we integrated multiple CPWs on both periodic square (SQ) and aperiodic Penrose (P3) lattices of  ferromagnetic nanopillars on top of 100 nm-thick YIG (Figure 1a-f). The ferromagnetic nanopillars were made from 100-nm-thick Py with different diameters D for different samples (see Table S1, Supporting Information). The images shown in Figure 1d-f were taken on a nominally identical electron beam resist mask and Py nanopillars which we had fabricated on an insulating GGG substrate prior to transferring the optimized nanofabrication process to YIG on GGG. The spacing between nanopillars did not vary and amounted to a = 480 nm. The CPWs were separated by s = 12 μm. A vector network analyzer (VNA) was connected to the CPWs in order to measure the SW properties by applying/detecting microwave signals using scattering parameters S (S-parameters, see Experimental Section).
The inhomogeneous radiofrequency magnetic field around the emitter CPW exerts torques on the spins and emits spin waves with wave vectors k into YIG. For the relevant microwaveto-magnon transduction, [13] the two kinds of nanopillar lattices underneath the CPWs offer a different performance. On the one hand, there are advantages of P3 over the SQ lattice: i) The high rotational symmetry of the P3 tiling enhances the number of directions of excited/detected spin-wave modes; [21] ii) The number of lattice points in a given area is higher than that of a square lattice, that is, the number of excited modes is enlarged. On the other hand, the peaks in reciprocal space (Figure 2a) are partly weaker for the P3 lattice compared to the SQ lattice (Figure 2d), meaning that GC modes are less efficiently excited. It is instructive to first benchmark the directionality and rotational symmetries of the different Py nanopillar lattices when operated as microwave-to-magnon transducers. Then we explore interference effects of short-wave magnons. In our case the nanopillars filled the region between the CPWs in order to steer the magnons. Thereby, we avoided the recently discovered wavelength conversion effect between a grating coupler underneath a CPW and neighboring bare YIG. [29] Following the original work on grating couplers, [12] we apply an in-plane field μ 0 H of 90 mT to samples SQ-ND95 and P3-ND95 (i.e., samples with a nanopillar diameter (ND) of 95 nm). For both samples, we vary the angle in steps of Δ from 135 to 0 deg (indicated by the blue arrows labeled sweep direction in Figure 2b,c). At each we measure S 21 . We display angle-derivative spectra in that we evaluate and color-code ΔS 21 = S 21 ( ) − S 21 ( −Δ ). This analysis enhances the signal-to-noise ratio (SNR), and similar derivatives are applied to further displayed datasets (Experimental Section). Propagating spin waves appear as black-white-black contrast oscillations. In Figure 2b,c we observe pronounced angular-dependent frequency bands of propagating spin waves appearing between about 4 GHz at = 90 deg and 5.5 GHz at = 0 deg. These signals are attributed to spin waves excited directly by the CPWs at wave vectors k CPW due to their inhomogeneous magnetic radiofrequency fields. These bands are similar for SQ and P3 samples and also seen for a thin YIG film without nanopillars ( Figure S2, Supporting Information). Intentionally, we present the consistently measured spectra in Figure 2b,c in mirror symmetry to highlight the quantitative agreement of these spin wave bands near = 0 deg. The angle dependency of the broad bottom-most band follows the anisotropic SW dispersion relation of a thin YIG film (compare Figure 3a and Figure S2, Supporting Information) when the angle between the wave vector k CPW defined by the CPW and the magnetization vector M YIG in YIG is varied. [30] The vector k CPW is in-plane and perpendicular to the CPW.
Both, aperiodic ( Figure 2b) and periodic ( Figure 2c) nanopillars allow us to excite high-frequency branches beyond about 5 GHz due to the grating coupler effect. For < 30 deg, these additional branches are superimposed by the directly excited perpendicular standing spin waves (PSSWs) existing in the YIG film of thickness t ( Figure S2, Supporting Information). Their discrete wave vectors in z-direction amount to k PSSW(n) = n ∕t with n = 0, 1, 2, .. (n = 0 indicates uniform precession). For ⩾ 40 deg, we highlight several GC modes and their characteristic angular dependencies by dashed lines. For the periodic GC sample, these excited SWs are known to possess a wave vector k CPW provided by the CPW plus a reciprocal lattice vector G (Figure 2d). The oscillating black-white-black signals in Figure 2c are consistent with ref. [12] where the authors investigated a square-lattice GC in an all-metal thin-film sample and with ref. [31] where GCs on 200 nm-thick YIG were studied. Their specific frequencies and angle-dependencies reflected the anisotropy of spin-wave dispersion relations and the fourfold rotational symmetry, respectively, of the periodic Py nanopillar lattice. [12] Note that signal strengths of individual branches, but not their wave vectors, are expected to depend on the exact diameters of grating elements.
In Figure 2c, a minimum in a branch (dashed lines) corresponds to the backward volume (BWV) spin-wave configuration with k∥M YIG . [12] When G is parallel to k CPW (and the unit vectorŷ) the minimum is at = 90 deg. If G is misaligned withŷ, the GC mode propagates diagonally through the YIG and the minimum resides at a different for which k∥M YIG ∥H.
In Figures S3a and S3b, Supporting Information we demonstrate how we use the formalism by Kalinikos and Slavin (KS) [30] to find k of spin wave branches that exist near 19 and 16.5 GHz, respectively, in a small field of 5 mT. In Figure S3a, Supporting Information a maximum near = 60 deg is seen and the wave vector k = k 1 + G (− 6)3 + k PSSW(2) is adequate to model the resolved branch (k 1 = 1.92 radμm −1 is provided by the CPW). Its magnitude amounts to 108.7 rad μm −1 , providing a value of = 2 /k = 57.8 nm. In the following, values of arise from the described analysis of angular-dependent branches based on the KS formalism (if not otherwise stated). In Figure S3b, Supporting Information we show two curves that indicate short-wave magnons with of 62.0 and 62.1 nm which vary around 16.5 GHz as a function of . From this comparison, we estimate an error of 0.2% in values of when analyzing angle-dependent datasets at large frequencies.
We observe GC modes also for the Penrose P3 lattice (Figure 2b). Its reciprocal vectors F densely fill the reciprocal space [21] as can be seen in the calculated diffraction pattern in Figure 2a. Local minima in the angular-dependent spectra are denoted by the corresponding reciprocal vectors F and indicated by blue dashed lines. For F (4) 2 we have identified only parts of the relevant branch. By the yellow transparent belt we highlight vectors F (G) in Figure 2a (Figure 2d) which give rise to SW wave vectors of around 30 rad μm −1 . Due to the high rotational symmetry of the Penrose P3 lattice, spin wave modes can be emitted more isotropically for a specific microwave frequency [21] than in ref. [12]. But their detection is not complete in the given experiment due to scattering and dephasing as discussed below. Before exploring the interference of short-wave magnons propagating in diagonal directions between CPWs, we explain signal enhancement for magnon emission by intentionally multi-resonant Py nanopillars. For this, we discuss the grating coupler effect and magnon steering in samples SQ-ND155 and P3-ND155.

Emission, Steering, and Detection of Ultrashort Spin Waves by Multi-Resonant Nanopillars Near-Zero Magnetic Field
It is instructive to first study SW propagation between two CPWs in a small magnetic field applied parallel to the CPW long axis (x-direction, in-plane). Under this condition, the investigated nanopillars host vortex states ( Figure S1A, Supporting Information). In Figure 1g we show absorption spectra based on S 22 that demonstrate that such nanopillars exhibit higher-order eigenresonances (standing spin waves, Figure S1B, Supporting Information). The thick nanopillars are hence adequate to make use of the resonant grating coupler effect and thereby enhance amplitudes of short wavelength magnons at high frequency. [31] In Figure 3a we depict Damon-Eshbach (DE) (defined by k⊥M YIG ) and BVW-type dispersion relations calculated by the KS formalism for an in-plane field of 5 mT applied at = 0 deg. [30] Figure 3b to d displays the field-dependent SW propagation spectra of SQ-ND155 below 40 mT (Data of P3-ND155 can be found in Figure S4, Supporting Information). Propagating SWs www.advancedsciencenews.com www.advmat.de (oscillating signals) are detected in many frequency bands (branches) from below 1 to up to 25.5 GHz. The linescan of S 21 (red) shown in Figure 3e contains several frequency bands of large amplitudes (seen near 6, 7.5, 9, 10, 11, and 11.5 GHz). We attribute the large oscillating signals between about 6 and 9.5 GHz to GC modes which are pronounced due to the nanopillar eigenresonances of high order (Figure 1g). The RF magnetic field of the CPW excites spin-precessional motion in the nanopillars which then enlarges the magnetic field components exerting torques on spins in the underlying YIG. The green line in Figure 3e reflects the calculated signal strength without considering the resonance effect in the nanopillars. Considering this line, the resonant GC effect is seen also between about 10.5 and 12 GHz in the red spectrum. Overall, we detect propagating SWs up to 25.3 GHz at 5 mT in Figure 3d. The measured group velocity at 25.3 GHz amounts to 2.0 km s −1 . According to the dispersion relations depicted in Figure 3a, the wave vector is 126 rad μm −1 at 25.3 GHz assuming the DE-type configuration. The corresponding wavelength amounts to 49.9 nm. would be smaller in case of a diagonal propagation direction between emitter and detector CPWs. In the extreme case of BWV-type propagation, would amount to 47.5 nm at 25.3 GHz. At this high frequency, we were not able to follow the angular dependency of the spin-wave branch and fit a specific wave vector via the KS formalism as was done in Figure S3, Supporting Information. Considering the remaining uncertainty about the wave vector direction, we estimate the value = 48.7(± 1.2) nm at 25.3 GHz as the shortest wavelength studied in this work.
For quasicrystalline structures, reciprocal vectors F are different in position and strength in reciprocal space (Figure 2a) compared to G for square lattices (Figure 2d). Consequently, SW propagation amplitudes are different. Figure 3f shows the experimental line spectrum (red) extracted at 5 mT for the quasicrystalline P3-ND155 sample. Up to about 15 GHz, the grating coupler effect in P3-ND155 gives rise to broader ranges of excited SW frequencies, both, in experiment (red) and calculated (green) spectra compared to the SQ lattice. Overall, we observe SW propagation up to 19 GHz in the P3 lattice ( Figure S4, Supporting Information).
The measured propagation amplitudes (red lines in Figure 3e,f) at the very high frequencies are typically lower than predicted. We attribute this observation to the dephasing between SWs. At a fixed frequency f, SWs with several GC wave vectors according to ±k CPW + G uv + k PSSW(n) or can be emitted simultaneously in different directions (with u, v = .. − 2, −1, 0, 1, 2, ..). Since their 1) wave vectors, 2) group velocities and 3) effective propagation distances s eff are different, the accumulated phase differences of the SWs at the detector CPW are different. The different phases lead to a (partial) cancellation effect (dephasing) of voltages in CPW2. Hence, measured SW propagation amplitudes are low. We expect the partial cancellation effect to increase with f due to more and more combinations of reciprocal (lattice) vectors giving rise to numerous diagonally propagating spin waves and dephasing.
The calculated curves (Experimental Section) assume the dispersion relation of a thin YIG film and the same relaxation time for the different spin waves. The calculations do not consider two-magnon scattering by the inhomogeneous stray fields of nanopillars and their unintentional roughness. We speculate that the aperiodic P3 lattice induces both more two-magnon scattering and dephasing effects compared to the SQ lattice due to the irregular (disordered) placement of nanopillars. The relaxation time would be shortened, making voltage signals at the detector CPW smaller for P3 than SQ as observed beyond 8 GHz in Figure 3e,f. These considerations might explain also why the number of branches in Figure 2b is not clearly larger than in Figure 2c.

Spin Wave Interferometry for Logic Operations by Ultrashort Magnons Steered into Arbitrary Directions
In the following we study interferometry with short-wave magnons steered inside YIG. The YIG is decorated with the Py nanopillars underneath and between three CPWs. We thereby address the basic principle of a wave-logic operation along an arbitrarily chosen on-chip direction. For wave-based logic, the phasecoherent superposition of SWs and a large extinction ratio [8] ER = 20log 10 ( are key, where V s is the signal at the detector port of the VNA. Three parallel CPWs on top of the GC samples (Figure 4a) are connected to three ports of a VNA: Two CPWs (CPW1 and CPW3) for SW emission and one central CPW (CPW2) for the detection of interfering SWs. We applied the same microwave voltage amplitude V 0 to CPW1 and CPW3. The two microwave signals were phase-coherent with an adjustable phase difference Δϕ. Assuming phase-coherent SWs propagating to CPW2 the detector signal V s was expected to depend on Δϕ according to [6] where V i 0 represented the voltage amplitude induced by an individual SW in CPW2. Assuming identical V i 0 for the two SWs, destructive interference with V s = 0 could be observed. Before conducting the interference experiments reported below, we recorded individual signals S 21 and S 23 and optimized the parameters frequency, field magnitude and angle in such a way that, within the noise level, the separately detected SW transmission signals were similar, suggesting (nearly) identical values V i 0 at CPW2 (Figure S5a-d, Supporting Information). Thereby we aimed at intentionally avoiding that non-reciprocity of SWs [22] played a role in Equation (2).
Samples P3-ND95 ( Figure 4a) and SQ-ND155 (Figure 5a) were used for SW interference experiments. Following the results of Figure 2, the field H was applied at different and gave rise to the excitation of a specific GC mode at a given VNA frequency. Thereby we explored different propagation directions and wave vectors k. Note that the phase difference Δϕ vna between the two voltage signals applied by the VNA contained a frequencydependent internal phase offset ϕ 0 due to the specific calibration of the VNA. Therefore the phase difference of SWs at CPW2 was     (Figure 4d). The oscillations arise because, at fixed f the SWs vary synchronously their field-dependent wave vector k [22] and thereby accumulate different phase differences along their path s eff . We fit a sinusoidal function ( Figure S5e, Supporting Information) to the symbols in each row between 84.7 and 87.7 mT (vertical dashed lines), and extract the oscillation amplitude V s for the different phase differences Δϕ vna . These signal strengths V s (symbols) are summarized in Figure 4e. They follow the dependence described by Equation 2 (red dashed line). The data provide ER = 31.1(± 2.2) dB. This value clearly outperforms the logic operation reported earlier for μm-long SWs. [8] The experiments and data summarized in Figures 4f-j and 4k-o refer to magnons of almost the same frequency and wavelength but different propagation directions as indicated in Figure 4f (k along = 90 deg) and Figure 4k (k along = 127.5 deg). They were realized by applying the magnetic field at = 90 and 126 deg, respectively, and led to differently long effective path lengths s eff along which the SWs accumulated phase differences depending on their wave vector k(H). We evaluated values ER of 22.5(±0.5) dB and 7.5(±0.4) dB by analyzing the data between the vertical dashed lines in Figures 4i and 4n, respectively, as discussed for Figure S5e, Supporting Information.
Interference of magnons with sub-100-nm wavelengths was explored in sample SQ-ND155 (Figure 5a,f). For the following experiments we lowered the applied field by an order of magnitude and increased the applied frequencies to about 16.7 and 19.2 GHz. The summarizing datasets are shown in panels (b-e) and (g-j) in Figure 5, respectively. The magnons of wavelengths 68.7 and 63.1 nm propagated along = 38.2 and 32.2 deg, respectively. At, for example, 19.2 GHz, the magnons emitted from CPW1 (CPW3) possessed a wave vector k CPW + G + k PSSW (2) with G = G (−5)3 (G 53 ) ( Figure S6, Supporting Information). Due to the diagonal propagation direction the propagation distance (Equation (5)) for each of the interfering SWs amounted to s eff = 22.5 μm, that is, 354 times the wavelength. The values ER corresponded to 25.6(± 8.3) dB (panel e) and 13.3(± 3.2) dB (panel j). In Figure S7, Supporting Information we show further binary 1/0 output experiments performed with magnons steered in different directions in the periodic nanopillar device with D = 95 nm at smaller frequencies (longer wavelengths).
The results of Figures 4 and 5; Figure S7, Supporting Information evidence pronounced interference effects for short-wave magnons which propagate over macroscopic distances in arbitrary directions. The extracted values ER amount to 31 dB at maximum, but vary with the applied field direction and applied VNA frequency. Considering the complex angular dependencies of GC modes (Figure 2) it is likely that more than the two assumed modes propagated through the hybrid samples and that dephasing of different modes and/or remaining nonreciprocity of spin waves led to smaller values ER. Future magnonic circuits are expected to build upon nanostructured SW channels which possess a modified band structure and, consequently, less SW minibands which could dephase and reduce the value ER. Hence, the observed maximum value is very encouraging. It even outperforms earlier results obtained by long-wavelength spin waves in an unpatterned SW guiding medium. The high extinction ratio reported here is a prerequisite toward nanoscale holographic memories, neural networks and all-magnon in-memory computation based on SW-induced magnetic-bit reversal. [27,28,32] Vanderveken et al. presented a lumped circuit model that allowed them to quantify signals in a complete microwave circuit including the magnetic system with its magnon band structure and the VNA. [33] Their model considers magnetic relaxation and describes arbitrary static magnetization orientations. Thereby, the wave impedance of a CPW inductively coupled to a specific magnonic circuit as considered here can be optimized. The magnon band structure for YIG subjected to scatterers arranged on a P3 tiling has been studied in ref. [21].
In our sample, we achieve magnons with large group velocity of 2 km s −1 and small wavelength = 48.7(±1.2) nm by using a 2D lattice of Py nanopillars near-zero field. In an earlier work, 100 μm-long CoFe nanostripes were used to emit similar magnons. In their case, the magnons were restricted to unidirectional (1D) propagation only. [14] We note that the observation of multidirectional (2D) magnon signals up to high frequencies (short wavelengths) in Figsures 3 and 4 is encouraging in view of the neural network proposed in ref. [28]. It is based on coherently excited and detected magnons in YIG underneath a large array of nanomagnets. The aperiodic lattices investigated here introduce irregular wave fronts [21] and are expected to mimic the spatially varying scattering processes assumed in ref. [28]. Neural networks exploiting sub-50-nm magnons should hence be within experimental reach by further increasing the eigenfrequencies of resonant grating coupler elements via optimized shape and magnetocrystalline anisotropy in case of, for example, hexaferrite nanomagnets.

Conclusions
Combining ferromagnetic Py nanopillars and thin ferrimagnetic YIG we demonstrated emission and detection of interfering SWs underneath periodic and aperiodic nanomagnet lattices. The Py nanopillars exhibited magnetic resonances which enhanced SW transmission up to about 12 GHz in low fields H. Magnons with a wavelength = 48.7(± 1.2) nm at 25.3 GHz propagated between two CPWs with a separation of 12 μm. The superposition of SWs with wavelengths down to 63.1 nm was demonstrated along different spatial directions. Their interference was detected after effective distances of more than 20 μm. The large ER value of up to 31 dB substantiated the coherency of the short-wave magnons underneath both types of nanomagnet lattices. This ER was 12 dB larger (that is, more than an order of magnitude larger) compared to logic operations reported for SWs of μm-long wavelength in unpatterned YIG. The underlying interference occurred over 350 × compared to 6 × , respectively. The multidirectional long-distance coherence of short-wave magnons is a key asset for functional wave-based logic, neural networks, and the emerging in-memory computation. Shape and materials optimization paves the way to sub-50 nm coherent magnons in 2D multi-branched circuits.

Experimental Section
Sample Fabrication: The single-crystalline 100 nm-thick YIG film was grown on a (111) GGG substrate by liquid phase epitaxy and received by Adv. Mater. 2023, 35, 2301087 the company Matesy GmbH in Jena, Germany. The film was characterized in ref. [31]. Using electron beam lithography (EBL) and lift-off processing, ferromagnetic nanopillars were prepared from evaporated Py after performing oxygen plasma cleaning to ensure the solid contact between ferromagnetic nanopillars and the YIG films. Physical parameters and the arrangement of ferromagnetic nanopillars forming the grating couplers (GCs) are summarized in Table S1, Supporting Information. The design of the quasicrystalline samples followed the approach outlined in ref. [20] in that the resist masks with openings at the vertices of Penrose P3 tilings were used for lift-off processing of evaporated Py (instead of etching nanoholes like in ref. [20]). Subsequently, CPWs for each GC sample were patterned via EBL, and Ti/Au (4/120 nm) was evaporated before lift-off processing. Symmetry axes of the periodic and aperiodic GC samples were parallel to the long axis of the CPWs. The width of a signal line, ground lines of the CPWs and the gap between the signal and ground lines were 800, 800, and 640 nm, respectively. The CPW excited pronouncedly spin waves at wave vectors k CPW such as k 1 = 1.92 radμm −1 and k 2 = 5.94 radμm −1 . The center-to-center separation s between two CPWs wass 12 μm.
Broadband Spin-Wave Spectroscopy: SW properties were studied via all electrical SW spectroscopy using a VNA (Figure 1a), allowing to generate a microwave with frequencies ranging from 10 MHz to 26.5 GHz. The microwave with a power of 0 dBm was applied at the port 1(2) of the CPW1(2) in order to excite spin precession. Spin precessional motion induced voltages on CPWs, and therefore SW properties were characterized electrically using scattering parameters (S-parameters). S ij represents the signal detected from the CPWi with respect to the signal applied to the CPWj (i, j = 1, 2, 3). For example, S 11 (S 22 ) indicates absorption spectra of spin precessional motion detected by CPW1(2), and S 21 (S 12 ) the SW propagation from CPW1(2) to CPW2(1). When analyzing the real (Re) part of S ii (like in Figure 1) resonances were identified on the rising edge of a local maximum, and not at the local maximum as it was done for the power spectral density. The most prominent excitation (detection) strength of the bare CPWs was related to the wave vector k CPW = k 1 which is collinear witĥ y. [21] For the spin wave interference experiment, CPW3 was connected to the VNA, and microwave signals from port 1 and port 3 were applied simultaneously with a phase difference Δϕ vna . The field μ 0 H of up to 90 mT was applied under an angle between the field H and the long axis of the CPWs. For field-dependent (angular-dependent) SW spectroscopy, the angle (the field H) was fixed.
S 21 represents the SW propagation signal from CPW 1 and 2. ΔS 21 reflects the difference in scattering parameters taken at successive angles or fields H. Such a difference enhances the SNR. In order to increase the SNR of, for example, SW transmission spectra, we evaluate either ΔS ij = S ij (H) − S ij (Ref). S ij (H) is the scattering parameter measured at a given field H, and S ij (Ref) at 90 mT along = 90 deg. Or, the field-derivative (ΔS ij = S ij (H) − S ij (H − ΔH)) or angle-derivative (ΔS ij = S ij ( ) − S ij ( −Δ )) of SW propagation spectra was evaluated. Oscillations of ΔS 21 (black and white) originated from the phase accumulation between CPWs. ΔH is 1 mT and Δ is 1 deg. In order to enhance the SNR of superposed SWs, the frequency-derivative (ΔS ij = S ij (f) − S ij (f − Δf)) was considered (Δf is 30 MHz for P3-ND95 and 40 MHz for SQ-ND155 in Figure 5).
Calculation of Grating Coupler Modes: Some wave vectors of SWs k = k CPW + G in 2D GC samples were not parallel to k CPW imposed by the CPW, and thus several GC modes would be detected in SW propagation at a fixed frequency f. The phase of SWs for each GC mode was different since the amplitude of wave vectors k and effective SW propagation distance s eff were different. Furthermore, the decay length l d related to the group velocity v g and damping parameter of YIG played a role for the observed intensity of propagated GC modes. SW propagation spectra were modeled using the relevant lattice in reciprocal space as well as parameters mentioned above. The intensity of a fast Fourier transformation (FFT) peak was calculated from dots following a Gaussian profile with a diameter of D arranged in the lattices. SW intensity was calculated as follows: Iso-frequency contour at fixed frequency f derived from dispersion relation for SWs at 5 mT relevant to Figure 3e,f is plotted on top of the FFT image shifted by −k CPW . Along the contour one acquired an FFT intensity profile I fft (f, ,k CPW ), group velocity v g (f, ), and effective distance s eff ( ). Here, the SW relaxation time (f) = 1/(2 f) [34] contributed to the decay length l d (f, ) = v g (f, ) (f), which allowed one to calculate the decay amplitude I d (f,ϕ) = exp(−s eff ( )/l d (f, )). With concerning accumulated phase ϕ a (f, ) = exp (ik(f, )s eff ( )), the SW propagation spectrum with a fixed k CPW is calculated as follows With considering the weight factor I CPW (k CPW ) (Excitation spectrum of k, see Figure S2b, Supporting Information), the SW propagation spectrum is as follows The real part of G sw is plotted in Figure 3e,f to compare with the experimental results. An amplitude of the wave vector k CPW up to 4.0 rad μm −1 was considered corresponding to the wave vector distribution around the k 1 mode of the CPW. SWs with k CPW + F (or k CPW + G) might not be parallel to k CPW and travel a longer distance s eff from CPW1 to CPW2 than SWs with k CPW . The different SW propagation distances s eff , defined by the angle between k CPW and k CPW + F (or k + G), were evaluated according to [31] s eff = s∕| cos | where ( + 90 deg) is the angle between the SW wave vector and the CPW. Attenuation was large for SWs propagating for a long distance s eff . Hence, slow GC modes with | | larger than 45 deg might be below the noise level. This was also true for modes with −k CPW − G and −k CPW − F. It was noted that resonance frequencies of SWs with k CPW + G (k CPW + F) and −k CPW − G (−k CPW − F) were the same. Simulations: Micromagnetic simulations using MuMax3 [23] were performed to obtain a microscopic insight into SW excitations in ferromagnetic nanopillars. A grid of 2.5 nm × 2.5 nm × 5 nm (x × y × z) was used. A global external field was applied in-plane. It amounted to 5 mT along the +x-direction after that a field of 90 mT was applied, and the equilibrium magnetization configuration M eq was determined. Note that a misalignment angle of = 2 deg was set. Subsequently, a spatially homogeneous sinc pulse of 0.15 mT with the frequency 30 GHz was applied along outof-plane. Simulated M was recorded 512 times every 5 ps. The input parameters of Py used in the simulations were as follows: saturation magnetization M s = 800 kA m −1 , exchange constant A = 13 × 10 −12 J m −1 , and damping constant 0.01. Note that the magnetic moments were randomly oriented before applying the field.
Dynamic magnetization m = (m x , m y , m z ) = M − M eq was recorded as a function of x, y, z in time domain. A FFT was performed on the magnetization for each grid along the time axis to obtain the resonance spectrum in frequency domain by summing up a magnitude m z for in-plane bias fields over all grid cells. For the illustration of spatial characteristics of quantized SW modes, an amplitude and phase of SWs were defined by the size of dots and color in hue, respectively, using Mayavi. [10] In this simulation, the ferromagnetic nanopillar was in a vortex state ( Figure S1A, Supporting Information) for the in-plane field of 5 mT, and thus, the m z component was utilized to investigate the spatial profile of quantized magnon.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.