Direct observation of multiband transport in magnonic Penrose quasicrystals via broadband and phase-resolved spectroscopy

Detection and imaging omnidirectional SW emission

We first present BLS microscopy performed on the aperiodic Penrose P3 lattice (sample P3-MQC) (Fig. 1) and a reference periodic lattice (Fig. 2). Physical parameters and lattice characteristics are summarized in Table 1. BLS data were taken near the CPW by which we excited SWs, and an in-plane magnetic field H was applied parallel to the CPW and along the symmetry axis (x axis) of P3-MQC. Here, a bare CPW allows for the excitation of SWs that propagate along the y direction. Before discussing MQC data in detail, it is instructive to consider the SW imaging data taken on an MC (Fig. 2A), incorporating a translationally invariant square lattice of nanotroughs (sample SQ-MC).

Figure 2 (A to C) illustrates the periodic square lattice in both real and reciprocal space. The spatial distribution of SWs in SQ-MC at f = 1.80 GHz (Fig. 2D) shows wavefronts of SWs propagating parallel to the CPW. We analyzed the excited wave vectors of SWs for SQ-MC (Fig. 2E) by fast Fourier transformation (FFT) of complex-valued SW images. Here, the highest-intensity FFT peak close to (k_x, k_y) = (0,0) is observed at k = (k_x, k_y) ∼ (0,2.4) rad/μm. This wave vector is anticipated by the SW dispersion relation for Damon-Eschbach (DE) modes at f = 1.80 GHz in the unpatterned YIG thin film (fig. S1A), and the wave vector is attributed to the CPW [k_CPW = (0, k_CPW)]. In addition to this mode, sharp peaks appear periodically with spacings of 7.0 rad/μm ∼ 2 π/(900 nm). The spacings correspond to reciprocal lattice vectors G of SQ-MC. That is, we observe SWs with wave vectors k_CPW + G, where k_CPW = (0, k_CPW) is the wave vector imposed by the CPW at a given frequency. This set of peaks is consistent with the magnonic grating coupler effect (21). It evidences coherent scattering of SWs and the backfolding of the SW dispersion via reciprocal lattice vectors. In (27), it was argued that periodically modulated demagnetization fields induced by nanoholes in an MC served as scattering potentials for Bragg scattering of SWs. At f = 1.62 GHz (Fig. 3A), the wavefronts display a longer wavelength in the y direction compared with that in Fig. 2D. At f = 1.48 GHz (Fig. 3B), the wavelength has increased further and the wavefronts appear to be distorted. Last, at f = 1.22 GHz (Fig. 3C), the wavelength in the y direction can no longer be extracted from the field of view (λ ≥ 10 μm) and the wavefronts appear to be subdivided into channels extending parallel to the y direction. Subsequently, the corresponding FFT results (Fig. 3, D to F) show that beyond the pronounced mode with (k_x, k_y) = (0, k_CPW), the SW patterns contain excitations with reciprocal lattice vectors (kx,ky)=(±p·2πa,kCPW±p·2πa), with p = 0,1. In Fig. 3 (D and E), the wavelengths of SWs correspond to discrete k values in x and y directions. At low frequency f = 1.22 GHz in Fig. 3F, FFT peaks are found near (k_x, k_y) ∼ (±7.0,0) rad/μm instead of (0,0) rad/μm. Here, SWs were elongated along the +y direction underneath nanotroughs, and standing SWs were confined in the x direction, which results in the formation of SW nanochannels (28). The SW phase within neighboring nanochannels is almost identical, while regions with large and small spin-precessional amplitudes are out of phase with each other.

The real-space SW map in the quasicrystalline structure (Fig. 1E) shows irregular SW wavefronts compared to SQ-MC at f = 1.80 GHz. The corresponding FFT image (Fig. 1F) contains a dumbbell-shaped iso-frequency contour in k-space. Unexpectedly, such iso-frequency contour agrees with the anisotropic characteristics (dispersion relations) of SWs in an unpatterned YIG film (see figs. S1B and S2A). The data indicate that the microwave current in the CPW induces an inhomogeneous radio frequency field h_rf that emits SWs into almost any direction at f. Reciprocal lattice vectors are not defined for an aperiodic Penrose P3 lattice; still, Bragg peaks are seen in diffraction pattern of a quasicrystal due to its self-similar and long-range ordered properties (Materials and Methods and Fig. 1C). The corresponding reciprocal vectors F densely fill out the reciprocal space (8). Such reciprocal vectors might provide the relevant Fourier components for h_rf to excite SWs in the nearly omnidirectional manner at fixed f as evidenced by the observed iso-frequency contour in k-space. This functionality is valid for a single frequency and thereby different from the previously reported grating couplers based on periodic lattices for which quasi-omnidirectional emission was claimed for SWs of different frequencies excited within a broad frequency band (21).

The wavelengths of SWs are found to increase in P3-MQC (Fig. 3, G and H, for f = 1.48 and f = 1.22 GHz, respectively) when the excitation frequency is reduced; this trend is similar to the SWs excited in SQ-MC. The extension of the dumbbell-shaped iso-frequency contour at f = 1.48 GHz for P3-MQC (Fig. 3I and fig. S2B) is smaller, because the anisotropy of SWs at f = 1.48 GHz involves smaller wave vectors compared to f = 1.80 GHz. The dumbbell-shaped contour is now closed (Fig. 3I), as the maximum wave vector k resides in the sensitivity range of the BLS.

The FFT maps at f = 1.22 GHz for SQ-MC (Fig. 3F) and P3-MQC (Fig. 3J) show wide intensity distributions along the k_x direction. These indicate that in real space, the SW excitations are confined along the x direction, i.e., in both samples, nanochannels are formed and extend in the y direction (orthogonal to H). The nanochannel formation is due to the inhomogeneous internal field, which takes locally small values because of the demagnetization effect of the nanotroughs, and resonant spin precession occurs at correspondingly small excitation frequency. The nanochannels in P3-MQC appear to be irregular, consistent with (12). In the following, we present broadband SW spectroscopy based on a vector network analyzer (VNA) to explore the frequency dependence of SW absorption and transmission in P3-MQC in detail and compare the results with SQ-MC and a plane film.

Manifold SW emission

We conducted angular-dependent SW spectroscopy in a rotating in-plane field H for the broadband characterization of SW properties in P3-MQC (Fig. 4). The field value of 90 mT was applied at various in-plane angles θ between the field direction and the CPW long axis. The SW resonances were determined using scattering parameters (S parameters) in absorption and transmission configuration using a VNA (see Materials and Methods). In the transmission configuration, two CPWs were used: one for excitation and one for detection of propagating SWs. The propagating SWs induced microwave voltages (S₂₁), which, in real and imaginary part, oscillated as a function of frequency due to the phase accumulated by SWs between the CPWs. The period ∆f of the oscillating S₂₁ signal scales linearly with the group velocity v_g according tovg=Δf·s(1)where s represents the distance between the two signal lines of the CPWs (29). The most prominent excitation for the CPWs used in this work is centered around k₁ = 1.92 rad/μm (Fig. 4A). The second and third most prominent excitations of a bare CPW occur near k₂ and k₃, respectively. In the following, we present SW spectroscopy data taken by the VNA on three different samples: (i) an unpatterned YIG thin film (Fig. 4, B and C), (ii) the YIG film containing a periodic lattice (SQ-MC) of nanotroughs (Fig. 4, D to F), and (iii) quasi-crystalline lattice (P3-MQC) of nanotroughs (Fig. 4, G to I). All the three samples were obtained from the same wafer.

The S₂₁ signals obtained on the plane YIG film (PF) are displayed in Fig. 4B. The oscillating S₂₁ signals of propagating SWs appear in three pronounced frequency bands reflecting the wave vector distribution of the CPW around k₁, k₂, and k₃. The signals are most pronounced near θ = 0°, and the bands reside at the lowest frequency near θ = 90°. The observed SW propagation modes follow the angular dispersion expected for dipole dominated SWs in a plane film. In Fig. 4C, we show analytically calculated SW spectra considering the excitation spectrum of CPWs (Fig. 4A) and the dispersion relation of the YIG thin film (30). Resonance frequencies of the prominent modes supported by the CPW follow a sinusoidal-like angular dependence due to the anisotropic dispersion relations of SWs. Oscillations in S₂₁ vanish near θ ∼ 55° and 125°, indicating a group velocity close to zero. Correspondingly, the decay length l_d = v_gτ is small (τ is the lifetime of SWs), and therefore, the amplitude of S₂₁ measured at the detector CPW is low. SWs with wave vectors k₁, k₂, and k₃ are weak at θ = 180° due to the nonreciprocity of the DE-type SWs in a YIG film (25). The nonreciprocity parameter κ is defined in (31) as the ratio κ = A₀(+H)/A₀(−H) of S₂₁ oscillation amplitudes A₀(H) for positive and negative fields H. The maximum κ amounts to κ = A₀(+90 mT)/A₀(−90 mT) = 25 at f = 4.8 GHz in the unpatterned YIG plane film.

Considering previous studies on periodic antidot lattices and the magnonic grating coupler effect (21), we show in Fig. 4D relevant wave vectors (colored arrows) that are excited due to backfolding according to k₁ + G_ij [k₁ is the wave vector provided by the CPW (Fig. 4A) and i, j = 0, ±1]. We detect more complex SW spectra on SQ-MC (Fig. 4E) compared to PF (Fig. 4B). Oscillating SW propagation signals in sample SQ-MC appear even at θ ∼ 125° (purple arrow), where the vanishing group velocity in PF prohibited a propagation signal. These additional propagation signals highlight the modified SW wave vectors due to the grating coupler effect. The colored arrows in Fig. 4E indicate modifications in SW spectra due to wave vectors displayed in Fig. 4D in the corresponding colors. Figure 4F displays the calculated angular dependencies of backfolded SW modes using a color code consistent with Fig. 4D, and we compare these with the k_i modes in the plane film (gray colored lines). The width of a line related to the grating coupler effect (colored line) represents the expected SW amplitude considering the nonreciprocity of SWs. At the intersections of k₁ and k₂ modes (gray) with the backfolded modes, angular-dependent SW propagation signals are expected to be modified strongly as observed in the experiments. Note that some backfolded modes such as k₁ + G₀₍₋₁₎, k₁ + G₁₍₋₁₎, and k₁ + G_(−1)(−1) do not reach the detector CPW due to the propagation direction opposite to k₁. However, the CPW also offers −k₁, and modes like −k₁ − G_ij can be detected. Resonance frequencies of k₁ + G_ij and −k₁ − G_ij are identical.

In Fig. 4G, we display the Bragg peaks (black circles) and color-coded vectors, which represent k₁ + F expected for P3-MQC. The Bragg peaks exhibit a manifold rotational symmetry, and vectors F are hence numerous. We focus on basic reciprocal vectors F⁽¹⁾ and the second prominent reciprocal vectors F⁽²⁾ illustrated using differently colored arrows (compare Fig. 1C, and see Materials and Methods). Similarly colored arrows are used in Fig. 4H to highlight where SW modes k₁ + F intersect with the k₁ branch. Analytic calculations of a large set of SW modes corresponding to k₁ + F, where F are different reciprocal vectors F⁽¹⁾ and F⁽²⁾, are shown in Fig. 4I. Note that there are many additional reciprocal vectors F, of which the magnitudes are smaller than the basic reciprocal vectors F⁽¹⁾ (Materials and Methods) (8), and therefore, more backfolded modes are possible than we indicate in Fig. 4I. The detailed inspection of the data shown in Fig. 4H underlines that the aperiodic quasi-crystalline lattice allows one to excite SWs propagating in many more directions than the periodic SQ-MC, which exhibits a lower rotational symmetry compared to P3-MQC.

Formation of magnonic band structure in MQC

In the following, we present absorption spectra. The absorption strength is known to scale with the frequency-dependent magnon DOS. Figure 5 (A to C) displays measured SW absorption spectra of PF, SQ-MC, and P3-MQC, respectively (dark indicates large absorption). The overall sinusoidal characteristics of the dark bands of excited SWs are consistent with SWs expected for the PF. We find specific narrow regions for SQ-MC and P3-MQC (colored arrows in Fig. 5, B and C), where the signal strength is reduced or vanishes. We attribute these features to forbidden frequency gaps in SQ-MC and P3-MQC, which are not present in PF. Note that the modifications in the SW absorption spectra of SQ-MC and P3-MQC are different.

To understand the origin of the forbidden frequency gaps, the DOS of SWs is calculated from the SW dispersion relation for each sample. Here, the oscillator strength related to the projection of magnon eigenfunction and the excitation spectrum of the magnetic field h_rf is considered (see Materials and Methods). The angular-dependent weighted projected DOS (wPDOS) for PF is shown in Fig. 5D. At θ ∼ 55°, the wPDOS is high, as the slope of the SW dispersion relation in the dipolar regime is almost flat. The large wPDOS is consistent with the fact that the group velocity v_g at θ ∼ 55° is almost zero, consistent with the results discussed for Fig. 4B.

Gallardo et al. considered the plane-wave method (32) for calculating the dynamic response and bandgaps of periodically modulated magnetic materials (33, 34). We applied this method to our surface-modulated SQ-MC (see Materials and Methods). Figure 5E shows the wPDOS of SWs in SQ-MC when we consider the reciprocal lattice vectors G_ij, where i, j = 0, ±1, ±2, ±3 for the basis set in magnon band calculations. Modifications of the DOS are indicated by arrows. These changes are traced back to the partial bandgap openings in the SW dispersion due to the formation of BZs (see Materials and Methods). The calculated modifications are similar to the experimental results (displayed on the left), which indicates that the forbidden gaps observed in the experiments are derived from the backfolding effect and miniband formation in the periodically modulated MC.

Aperiodically arranged nanostructures do not have Brillouin zones because reciprocal lattice vectors G do not exist in quasicrystals. However, one can still choose the reciprocal vectors F of quasicrystals and assume corresponding backfolding of the SW dispersion (33, 34). For our calculation, we consider reciprocal vectors F⁽¹⁾, F⁽²⁾, and F⁽³⁾ for the basis set. In the backward volume wave (BVW) regime near θ ∼ 90°, the modification of the wPDOS is pronounced (Fig. 5F). This is because the original SW dispersion relation of BVW is relatively flat, and thus, backfolded SW dispersion relations interact with the original SW dispersion strongly (34). Consistent modifications are observed in the experiments on P3-MQC. The qualitative agreement indicates the formation of a p-BZ in a real MQC, which has not been reported before. The resolved p-BZ is defined by the basic reciprocal vectors F⁽¹⁾ shown in Fig. 1C (8). The purple arrows indicate the angle-dependent modifications of SW absorption in the BVW regime.

Figure 5 (G to I) shows the line spectra of SW absorption and propagation signals at θ = 35° for PF, SQ-MC, and P3-MQC, respectively. The strong oscillation of Re(S₂₁) due to SW propagation between emitter and detector CPW is detected at the frequency for which the SW absorption peaks in sample PF (Fig. 5G) are large. According to Eq. 1, the group velocity is 0.30 km/s. In sample SQ-MC (Fig. 5H), mode k₁ and a backfolded mode k₁ + G coexist in the same frequency regime. Here, a dip is observed in the SW absorption spectra at f = 4.6 GHz, with a width of about 20 MHz. The oscillation signal of SW propagation at this frequency is modified as well. We attribute this modification to the partial gap opening of the magnonic band structure of SQ-MC. The group velocity of SWs for sample SQ-MC below and above f = 4.6 GHz is 0.26 and 0.36 km/s, respectively, which indicates that different modes dominate the SW propagation above and below the frequency. A similar type of modification is also seen in sample P3-MQC (Fig. 5I).

At θ = 45°, observed modifications are of a different kind. In Fig. 5J, we report a strong absorption peak for sample PF, but the amplitude of SW propagation is very small due to the low group velocity of 0.17 km/s. For SQ-MC (Fig. 5K), a clear dip occurs in the absorption spectrum. The dip is more than half of the SW absorption strength. The SW propagation appears to be completely suppressed, most likely due to a pronounced gap opening in the SW band. For sample P3-MQC (Fig. 5L), the measured absorption peak is featureless, and contrary to PF and SQ-MC, a relatively strong SW propagation signal with a group velocity of 0.32 km/s is detected. We attribute this observation to backfolded modes forming an allowed miniband with an improved group velocity. At θ = 86°, BVW-SWs with a group velocity of 0.28 km/s were emitted in PF. Similar to the angle 45°, SW absorption is suppressed at f = 4.26 GHz in sample SQ-MC. At the same frequency, a slight modification of SW absorption is detected in sample P3-MQC. Here, the SW group velocity amounts to 0.3 km/s. Note that sample SQ-MC shows a very similar group velocity of 0.31 km/s above the frequency of f = 4.26 GHz, at which the absorption is suppressed.

Figure 6 displays SW band structures and wPDOS of SQ-MC and P3-MQC (blue curves) compared to the SW dispersion relation of the YIG film (red) at the field of the experiment applied with an angle of θ = 45°. The diameter of dots forming the blue curves in Fig. 6 (A and C) represents the projection amplitude of the SW eigenfunction ∣Ψ_k⟩ onto a basis plane-wave mode ∣G⟩ (∣F⟩) corresponding to the reciprocal lattice vector G (reciprocal vector F) of the MC (MQC), relevant for backfolding wave vectors k into the first BZ (p-BZ) according to ∣P_G(k)∣ = ∣〈G∣Ψ_k〉∣(∣P_F(k)∣ = ∣〈F∣Ψ_k〉∣) (see Materials and Methods; fig. S3). The original SW dispersion relations (red dashed lines) fall onto a region with many backfolded modes in each of the two samples in Fig. 6 (A and C). At the angle of 45°, the group velocity of plane-film SWs is low. However, the numerous backfolded modes interact with the original dispersion relation and can have higher group velocities than the original mode. Such backfolded modes then carry the pronounced SW propagation signals observed experimentally in SQ-MC and P3-MQC (Fig. 4, E and H). At the same time, hybridization of modes occurs and novel allowed minibands form separated by small frequency gaps (highlighted by vertical arrows). Accordingly, the wPDOSs of SQ-MC and P3-MQC in Fig. 6 (B and D; light blue colored area) show fine structures of local maxima and dips (horizontal arrow), while the unstructured film (dashed red line) exhibits a smooth wPDOS. In fig. S4, we depict for direct comparison the band structures and DOS calculated at the four angles 0°, 35°, 45°, and 86° for SQ-MC and P3-MQC side by side. In the experiments, the absorption spectra do not feature all the predicted dips. We attribute this to SW damping in the real samples, which was not considered for the calculations.

Sample fabrication

The single-crystalline 100-nm-thick YIG films were grown on (111) GGG substrate by liquid phase epitaxy treated by Matesy GmbH (Jena, Germany). Al/Ti layers were evaporated on top of the YIG films with a thickness of 10/150 nm. The MCs and MQC masks were patterned on hydrogen silsesquioxane negative electron beam resists using electron beam lithography (EBL). Inductively coupled plasma etching and ion beam etching were performed to form a decagonal mesa and to introduce nanotroughs in the YIG film for the fabrication of the MCs and the MQC as summarized in Table 1. Samples were immersed into hydrofluoric acid bath to remove Al/Ti completely. Consecutively, CPWs were patterned on the MCs and MQC via EBL, and Ti/Au (4/200 nm) were sputtered before lift-off processes. Symmetry axes of SQ-MC and P3-MQC are parallel to the long axis of the CPWs. The width of a signal line and ground lines of the CPWs are 800 nm, and the gap between the signal and ground lines is 640 nm. A center-to-center separation between two CPWs is 12 μm.

Note that samples SQ-MC used in SW spectroscopy and BLS experiments are different. As indicated in Table 1, the diameters and depths of nanotroughs are different for each sample, while the characteristic length a stays the same. SQ-MC used for BLS measurement misses a column of nanotroughs indicated in fig. S6. As the missing column is far from the region of interest (ROI) for BLS experiment, we expect that the backfolding effect of the square lattice for SWs is still observed.

For the investigation of SW properties in quasicrystals, an identical P3-MQC was used. First, SW transport measurement was conducted by broadband SW spectroscopy technique. Afterward, one of the CPWs that used to be connected to port 2 of VNA was removed by injecting a large current. Then, spatial distribution of SWs was obtained by the BLS measurement. Note that a part of ROI is overlapped with the place where the CPW was mounted as shown in fig. S7.

Broadband SW spectroscopy

SW excitations/propagations were studied via all-electrical SW spectroscopy. The two ends of the CPWs (patterned on top of the devices; see fig. S8) were connected to a VNA to apply microwaves. The in-plane angle θ of the applied field H was varied. A two-port VNA allowed us to generate a microwave magnetic field with frequencies ranging from 10 MHz to 26.5 GHz. The microwave with a power of −25 dBm was applied at the port 1(2) of the CPW to excite magnetization precession. The precession-induced voltage was detected at port 2 by reading the scattering parameter S₂₁(S₂₂), where the numbers 2 and 1(2) in the subscript denote the detection and excitation port. An external magnetic field μ₀H of up to 90 mT was applied under an angle θ between the external field H and the CPW’s long axis. To increase signal-to-noise ratio, ΔS₂₁ = S₂₁(H) − S₂₁(Ref) was evaluated, where S₂₁(H) and S₂₁(Ref) represent scattering parameters measured at a given field H and at 90 mT along θ = 90°, respectively.

BLS microscopy

SW eigenmodes were imaged via BLS microscopy with or without phase resolution at room temperature using a setup similar to (38, 39). Figure S9 shows a sketch of the experimental configuration. The end of the CPW was electrically bonded to a printed circuit board, and the circuit board was connected to a signal generator for a microwave application. The microwave excited spin precession in the MC/MQC near the CPW at a fixed frequency. A magnetic field of 10 mT was applied parallel to the long axis of the CPW via a permanent magnet for BLS measurement, and after that, a field of 90 mT was applied to first saturate the samples. A lens with a numerical aperture of 0.85 was used to focus a 473-nm wavelength laser with a spot size of 300 nm onto the samples. Laser power was set to 1 mW. Energy shifts of reflected laser light due to the inelastic magnon-photon scattering were detected by a triple-tandem Fabry-Pérot interferometer. The device was positioned under the laser spot using a x, y piezo-positioning system. The step size to acquire the SW images was 100 nm. We exploited phase-resolved inelastic light scattering while exciting SWs phase-coherently at the straight CPW. The nanotrough positions were reconstructed in that we overlaid (i) atomic force microscopy images taken on the studied MC/MQC, (ii) the BLS data, and (iii) the exposure masks for CPWs and EBL of nanotroughs.

Bragg peaks of crystal/quasicrystal structure

Figure S10 (A and B) shows the square and Penrose P3 lattice used for the fabrication of the MC and MQC, respectively. The Penrose P3 tiling is composed of acute and obtuse rhombi structures with the same side length a, and the real space is filled out with these rhombi in a self-similar manner. The FFT of the crystal structure shows Bragg peaks as shown in fig. S10A. Bragg peaks of the square lattice correspond to its reciprocal lattice vectors G_ij, which are expressed as followsGij=iA+jB(2)where A = (2π/a,0) and B = (0,2π/a) and i and j are integer numbers. The lines bisecting the reciprocal lattice vectors G₁₀, G₍₋₁₎₀, G₀₁, and G₀₍₋₁₎ form the square-shaped BZ boundary in square-lattice based crystals as shown in fig. S10A.

In contrast to the periodic crystals, quasicrystals do not have reciprocal lattice vectors in k-space. A Bragg peak pattern is expressed with basic reciprocal vectors C_p = [2π/(a cos (π/10))] × ( cos (pπ/5), sin (pπ/5)), where p = 0,1,2,3,4. The reciprocal vectors are expressed by a linear combination of the basic reciprocal vectorsF=m0C0+m1C1+m2C2+m3C3+m4C4(3)which are chosen such that they (i) are related to the strongest Fourier coefficients describing the distribution of the material parameters in the quasicrystal, (ii) cover the reciprocal space with the appropriate density to describe the possible distribution of the magnon eigenmodes, and (iii) have the rotational symmetry of the diffraction pattern (8). m_p are integer numbers. Here, Fq=p(1)=Cp and Fq=p+5(1)=−Cp, where q = 0,1, …,9. Note that Fq+5(1)=−Fq(1), and therefore, Fq+10(1)=Fq(1). The other intense peaks are expressed as F⁽⁰⁾ = (0,0), Fq(2)=Fq−1(1)+Fq+2(1), and Fq(3)=Fq(1)+Fq+1(1), with the magnitude of ∣F⁽⁰⁾∣ = 0, ∣F⁽²⁾∣ = 2π/(a cos (π/5)), and ∣F⁽³⁾∣ = π/a, respectively. The intensity of F⁽⁰⁾, F⁽²⁾, and F⁽³⁾ in diffraction patterns of the Penrose P3 lattice is higher than the intensity of the basic reciprocal vectors C_p. Note that Bragg peaks of the Penrose P3–based quasicrystals exhibit 10-fold rotational symmetry. The lines bisecting the reciprocal vectors Fq(1) form the decagon-shaped p-BZ boundary in Penrose P3–based quasicrystals as shown in fig. S10B.

Calculation of DOS for MC/MQC

To calculate the DOS of SW, the band structure of SWs in the MCs and MQCs needs to be calculated. Considering (33, 34), the magnonic band structures for the MCs and MQCs are calculated using the plane wave method (32). For calculating the magnon bands, 49 smallest reciprocal lattice vectors, G_ij, with i, j = 0, ±1, ±2, ±3, are considered in the case of SQ-MC. In the case of P3-MQC, we considered 31 high-intensity Bragg peaks, F⁽⁰⁾ and Fq(i), with i = 1,2,3 and q = 0,1, …,9. The material and physical parameters used for the calculation are as follows: a saturation magnetization μ₀M_s = 180 mT, an exchange constant A = 3.65 × 10⁻¹² J/m, a gyromagnetic ratio γ/2π = 28.024 GHz/T, characteristic length a = 900 nm, a diameter of nanotroughs D = 295 nm, a depth of nanotroughs t = 23 nm, and the thickness of the YIG film is 100 nm. The wave number of the first BZ (p-BZ) boundary of sample SQ-MC (P3-MQC) is 3.49 rad/μm (3.96 rad/μm) along the k_y direction. An angular-dependent external field μ₀H(θ) = (93 + 3 cos (2θ)) mT is considered for the analytic calculation to match the results with experimental data. The Fourier coefficients for circle defined in (33) are used for the calculation of magnonic band structures. The calculation result of the band structure for SQ-MC inside the first BZ at μ₀H = 93 mT and θ = 45° is shown in fig. S3A. Eigenfrequencies f_η(k) (η denotes the backfolded mode number) from this calculation display the band structure of the SQ-MC. Gap openings in an SW band structure of 1D MCs in DE configuration only appear at the BZ boundary. Here, 2D MCs even have partial bandgaps inside the BZ due to the hybridization of the backfolded branches of the BVW mode. Using the projection of the magnon eigenfunction, ∣Ψ_η,k⟩, onto the plane wave mode ∣G > (∣F⟩), we calculated the PDOS of SWs for the MC (MQC) as followsρG(F)(f)=−1π∑η∫dk Im[∣Pη,G(F)(k)∣2f−fη(k)+i0+](4)where 0⁺ is a positive infinitesimal value and P_{η, G(F)}(k) = 〈G(F)∣Ψ_η,k〉 is the projection amplitude. Depending on the MC or MQC, we either used G or F, respectively, in Eq. 4. η runs over the number of magnon bands (1 to 49 for MC and 1 to 31 for MQC). The limits of integration over k are 0 to 8.8 rad/μm. The size of blue dots in band structure graphs (see fig. S3) indicate the projection amplitude ∣P_{η, G(F)}(k)∣, which corresponds to the spin precession amplitude. For the comparison to the SW spectra in reflection geometry using VNA, we consider the Fourier transformation of the excitation spectrum of the CPW-induced microwave magnetic field, I(k) (green transparent in Fig. 6A), as a weight factor and the wPDOS is expressed asρG(F)w(f)=−1π∑η∫dk Im[∣Pη,G(F)(k)∣2∣I(k)∣2f−fη(k)+i0+](5)

The range of the wave vectors covering the k₁ and k₂ modes of I(k) is up to 8.8 rad/μm, which is beyond the wave vector of the boundary of the first BZ of the periodic SQ-MC. To extend the dispersion relation beyond its first BZ (fig. S3), a magnon band structure with respect to the reciprocal lattice vectors G is considered, which are responsible for the backfolding of wave vectors k into the first BZ (vector G₀₁ for our study; see fig. S3C). Then, the results are concatenated to the original band structure (Fig. 6A and fig. S3D). Figure 6B shows the calculated wPDOS of the SQ-MC at μ₀H = 93 mT and θ = 45°.

For the aperiodic P3-MQC, we calculated the dispersion relations in that we considered wave vectors 0 ≤ k ≤ 8.8 rad/μm along the k_y direction. The responsible reciprocal vector for the projection of magnon eigenfunctions was F⁽⁰⁾. Note that one cannot simply extend the dispersion relation of P3-MQC by further concatenating the band structure due to the aperiodicity. Figure 6C shows the dispersion relation of the P3-MQC at μ₀H = 93 mT and θ = 45°. The most pronounced excitation around the k₁ mode of the CPW (Fig. 4A) covers the p-BZ (vertical dashed line in Fig. 6C). The number of SW branches and characteristics of the gaps are different from the ones of the MC (Fig. 6A). The corresponding wPDOS is displayed in Fig. 6D. The calculations have been performed for different field orientation angles θ for SQ-MC and P3-MQC. The angular-dependent DOSs considering our experimental setting are shown in Fig. 5 (E and F).