## Abstract

We explored spin-wave multiplets excited in a different type of magnonic crystal composed of ferromagnetic antidot-lattice fractals, by means of micromagnetic simulations with a periodic boundary condition. The modeling of antidot-lattice fractals was designed with a series of self-similar antidot-lattices in an integer Hausdorff dimension. As the iteration level increased, multiple splits of the edge and center modes of quantized spin-waves in the antidot-lattices were excited due to the fractals’ inhomogeneous and asymmetric internal magnetic fields. It was found that a recursive development (F_{n} = F_{n−1} + G_{n−1}) of geometrical fractals gives rise to the same recursive evolution of spin-wave multiplets.

## Introduction

A recursive sequence is one of the most fundamental growth mechanisms in nature, interest in it having grown significantly for its potential quantum applications^{1,2,3}. That is, a successive descendant of the nth generation is an aggregation of one or more preceding ascendants and their variations. More intriguing sequences are involved in fractal geometries in nature^{4}. Those fractals are classified as statistical and random fractals^{5}. Although the random fractals are relatively sporadic and weakly self-similar, the deterministic (exact) fractals have regularity and strong self-similarity. The recursive ordering has been discovered in the context of those two different fractal growths. Despite their close relationship, the recursive sequences in fractal growths have barely been studied in ordered spin systems.

Meanwhile, spatial periodicities of spin ordering in lattice crystals lead to the modifications of magnonic properties such as band structure^{6,7,8} and quantization^{9,10,11,12} of spin-waves. The antidot-lattice, a periodic array of many holes in a continuous film, has been a basic and promising two-dimensional (2D) magnonic crystal due to its scalability and good hysteric effect without superparamagnetic bottleneck^{13}. The alteration of internal magnetic fields in the antidot-lattices results in several non-propagating eigenmodes even in forbidden bands. For example, edge modes, which resemble a “butterfly state”^{10}, are localized around the boundary of each antidot, while center modes are extended along the channel in between the neighboring holes (antidots)^{12,14}. Furthermore, standing spin-wave modes can be excited by different field conditions as well as in geometric confinements^{15}. A variety of types of antidot-lattices have been employed that include bi-component (of different materials^{16} or different sizes of holes^{17}), different Bravais type^{18}, and defective lattices^{19}. On the other hand, non-trivial magnonic dynamic behaviors were observed in aperiodic structures of antidots such as magnonic quasicrystals of Fibonacci structure^{20,21,22}, Penrose and Ammann tilings^{23}, and Sierpiński carpet^{24,25,26}. In fact, Sierpiński fractals have led to unique phenomena in electronics^{27,28} and photonics^{29,30}.

### Model system of antidot-lattice fractals

Here, we propose magnonic crystals composed of ferromagnetic antidot-lattice fractals, arranged similarly to one of Sierpiński aperiodic motifs, as studied by micromagnetic simulations along with a delicate analysis of multiplet spin-wave modes. The overlap of scaled antidot-lattices in a regular routine yields deterministic fractals, e.g., periodic structures with a local aperiodicity. This controllable non-statistical geometry provides a self-similarity in a part of the structure at every magnification, and can be designated according to the Hausdorff dimension of \({\mathrm{log}}_{\mathrm{S}}\mathrm{N}\), where S is the scale factor and N is the number of scaled objects^{31}. That is, we used a series of scaled antidot-lattices (Fig. 1a) to construct antidot-lattice fractals deterministically with iterations, as illustrated in Fig. 1b. The antidot-lattices are self-similar in their geometric parameters of diameter D and lattice constant L. The D_{n} and L_{n} of the nth antidot-lattice (A_{n}) are exactly half of those of A_{n−1}. For example, A_{2} has \({\text{L}}_{2} = {\text{L}}_{1} /2\) and \({\text{D}}_{2} \,{ = }\,{\text{D}}_{1} /2\). Next, the nth fractal S_{n} is constructed by the superposition of A_{1} + A_{2} + ⋯ + A_{n} as follows: S_{1} = A_{1}, S_{2} = A_{1} + A_{2}, S_{3} = A_{1} + A_{2} + A_{3}, and S_{4} = A_{1} + A_{2} + A_{3} + A_{4}, as depicted in the series up to S_{4} (see Fig. 1b). In detail, A_{1} with D = D_{1} and L = L_{1} corresponds to the initiator (mother). A_{1} and A_{2} make up S_{2}. Since the D_{2} and L_{2} of A_{2} are half those of A_{1}, the number of antidots for A_{2} is increased by 4 times, and thus the Hausdorff dimension is log_{2}4 = 2. Following A_{n} are scaled copies of previous A_{n−1} in the same manner. Due to the self-similarity of the fractals, a recursive sequence arises inside the geometry of the motifs: let F_{n} denote the geometrical sequence. F_{n} (\(0 \le x \le L/2\)) of each S_{n} is a summation of F_{n−1} (\(L/4 \le x \le L/2\)) and G_{n−1} (\(0 \le x < L/4\)). The appearance of F_{n−1} in S_{n} is a scaled recursion of F_{n−1} (\(0 \le x \le L/2\)) in S_{n−1}. In S_{n}, G_{n−1} is a variation of F_{n−1} and can be viewed as the A_{1} antidot superimposed onto F_{n−1}.

Then, the 2D periodic lattice of magnonic crystals has a square Bravais symmetry, as shown in Fig. 1c.

## Results

### Recursive evolution of spin-wave multiplets

Figure 2 reveals that the spin-wave eigenmodes in the antidot-lattice fractals split into multiplets according to the recursive sequence (for better spectra, see also Supplementary Fig. S1). The first ordinary crystal denoted as S_{1} (= A_{1}) exhibited three normal standing spin-wave modes as indexed by E_{1}, C_{1}, and C_{V1}^{12}. The very weak mode (E_{1}) at 1.77 GHz corresponds to the edge mode, and the strongest mode (C_{1}) at 5.02 GHz to the center mode. The two modes are periodically excited along the bias field direction. The last minor mode (C_{V1}) at 5.80 GHz is a center-vertical mode (or a fundamental-localized mode) at the center between the neighboring antidots along the axis perpendicular to the bias field direction.

For S_{2}, while keeping the E_{1} mode at a similar frequency, an additional doublet (E_{2}) of the edge mode appeared, which originated from A_{2.} The doublet (C_{2}) of the center mode appeared as its substitution for the singlet C_{1} in S_{1} (see Fig. 2b). The higher mode (5.61 GHz) of the doublet C_{2} was hybridized with the C_{V1} mode (5.80 GHz) in S_{1}. In a similar manner, for S_{3}, an additional triplet (E_{3}) came from A_{3}, while the doublet C_{2} in S_{2} then became the triplet C_{3}. The center-vertical mode (C_{V2}) of A_{2} was hybridized with the highest mode (6.38 GHz) of C_{3}. To sum up, by adding A_{2} and A_{3} to S_{2} and S_{3}, respectively, each E_{n} mode was newly updated while the C_{n} mode substituted for C_{n−1}. This happened because the edge mode is strongly localized at the boundary of each antidot while the center mode is extended through the channels between the neighboring holes in the antidot-lattices. Whenever the next scaled antidot-lattices were overlapped to the previous one, each boundary of antidot entity of A_{n} and A_{n−1} remained intact with each other, while the channels of A_{n} were impacted by the channels of A_{n−1}.

In the case of A_{4}, the edge mode and the center mode were hybridized into one mode, because the hole-to-hole distance was smaller than the previous antidot-lattices: the dipolar and exchange interactions were equally dominant at the narrow channel of A_{4}. Therefore, for S_{4}, the hybrid mode (E_{4} + C_{4} \(\to\) EC_{4}) appeared instead of the individual E_{4} and C_{4} modes. A total of five EC_{4} modes (pink-colored peaks) substituted for the C_{3} triplet.

The number of multiplets in the serial spectra followed the recursive sequence, F_{n} = F_{n−1} + G_{n−1}. The two eigenmodes (E_{n} and C_{n}) appeared as a singlet, doublet, triplet, and quintet for S_{1}, S_{2}, S_{3} and S_{4}, respectively. Since there is only one zeroth (n = 0) standing spin-wave mode in unpatterned (continuous) thin film (i.e., S_{0}), the number of the split modes corresponded to 1, 1, 2, 3, and 5 for n = 0, 1, 2, 3, and 4, respectively, for both E_{n} and C_{n}. The difference sequence (G_{n}) is 1, 1, and 2 for n = 1, 2, and 3, respectively.

In order to identify all of the excited modes represented by the FFT power-vs.-frequency spectra shown in Fig. 2, we performed FFTs on every single unit cell (or mesh) at the indicated resonance frequencies of the modes. Figure 3 shows the spatial distributions of FFT power in the bottom-right quarter of each motif (\(0 \le x \le L/2\), \(- L/2 \le y \le 0\)) for each resonance frequency of the excited modes (for the corresponding phase profiles, see Supplementary Fig. S2). For S_{1}, the major modes of E_{1} and C_{1} were visualized at 1.77 and 5.02 GHz, respectively. The edge mode was excited at the edge (or end) of the antidot (Fig. 3a), while the center mode was excited at the center (or channel) of the neighboring antidots (Fig. 3b). The higher mode (5.61 GHz) of the C_{2} doublet, was vertically localized between the neighboring holes of A_{1} in which C_{V1} was also localized in S_{1} (Fig. 3c). This explains why the higher mode of C_{2} was hybridized with C_{V1}, as mentioned earlier. On the other hand, the lower (4.94 GHz) one of C_{2} remained extended along the channel between S_{2}. The doublets of both E_{2} and C_{2} are antiphase with each other in temporal oscillation; S_{2} can be considered to be a two-dimensional nano-oscillator. For S_{3}, the highest C_{3} at 6.38 GHz was fully localized in between the A_{2} antidots along the y-axis: it was hybridized with C_{V2} due to their shared localization area. The lowest C_{3} (5.35 GHz) remained extended along the channel between S_{3}. Similarly, the lowest EC_{4} (4.04 GHz) for S_{4} was the only extended mode, while the others were localized in different local regions as noted by the red color. Since the distance between the antidots of A_{3} and A_{4} are close, some localized modes (4.30 GHz and 4.98 GHz) were excited at antidots of A_{3} together with certain antidots of A_{4}. On the other hand, some of the E_{3} modes (4.04 GHz and 4.70 GHz) of S_{4} were excited at the same frequencies as those of the EC_{4} modes. Those E_{3} and EC_{4} modes become separated when the intensity of the bias field increased (see Supplementary Fig. S3).

The gap between the split modes narrowed down and finally merged into a singlet as the inhomogeneity of the magnetic energy decreased. In the other direction, the gap became wide and the corresponding different modes crossed over each other (showed conversion) as the inhomogeneity of the magnetic energy increased.

Similar to the recursive sequence in the geometrical fractals, we also found the recursive sequence in the evolution of the eigenmodes’ spatial profiles, as shown in Fig. 4. In detail, for the E_{n} modes, let E_{1} in S_{1} be F_{1}. The two E_{2} modes in S_{2} are F_{2}. The right part (L/4 \(\leq\) x \(\le\) L/2) of the E_{2} (2.93 GHz) profile is the same as the E_{1} profile in S_{1}. E_{2} (2.93 GHz) is F_{1} in S_{2}. The left part (0 \(\le\) x \(<\) L/4) of the E_{2} (3.65 GHz) profile is a variation of the E_{2} (2.93 GHz) profile. E_{2} (3.65 GHz) is G_{1} in S_{2}. The three E_{3} modes in S_{3} are F_{3}. The right parts of the E_{3} (4.24 GHz and 5.09 GHz) profiles are similar to the E_{2} profiles in S_{2}. The two E_{3} modes are F_{2} in S_{3}. The left part of the E_{3} (4.72 GHz) profile is a variation of the E_{3} (4.24 GHz) profile. E_{3} (4.72 GHz) is G_{2} in S_{3}. The five EC_{4} modes in S_{4} are F_{4}. The right parts of the EC_{4} (4.04 GHz, 4.98 GHz, and 5.49 GHz) profiles are similar to the E_{3} profiles in S_{3}. The three EC_{4} modes are F_{3} in S_{4}. The left parts of the EC_{4} (4.30 GHz and 4.70 GHz) profiles are variations of the EC_{4} (4.98 GHz and 4.04 GHz, respectively) profiles. The two EC_{4} (4.30 GHz and 4.70 GHz) modes are G_{3} in S_{4}. In the same way, let C_{1} in S_{1} be F_{1}. The two C_{2} modes in S_{2} are F_{2}. The right part of the C_{2} (4.94 GHz) profile is the same as the C_{1} profile in S_{1}. C_{2} (4.94 GHz) is F_{1} in S_{2}. The left part of the C_{2} (5.61 GHz) profile is a variation of the C_{2} (4.94 GHz) profile. C_{2} (5.61 GHz) is G_{1} in S_{2}. The three C_{3} modes in S_{3} are F_{3}. The right parts of the C_{3} (5.35 GHz and 6.38 GHz) profiles are the same as the C_{2} profiles in S_{2}. C_{3} (5.35 GHz and 6.38 GHz) are F_{2} in S_{3}. The left part of the C_{3} (5.72 GHz) profile is a variation of the C_{3} (5.35 GHz) profile. C_{3} (5.72 GHz) is G_{2} in S_{3}. The EC_{4} modes are considered in the same way as mentioned above.

### Origin of recursive evolution

Next, in order to identify the splits of the spin-waves excited in S_{n} with respect to A_{n}, we plotted the contours of FFT power for the frequency and the longitudinal x-direction, as shown in the bottom row of Fig. 4. In the upper row of Fig. 4, we also plotted the spatial distributions of the total magnetic energy density (\({\mathcal{E}}_{\mathrm{tot}}\)), as expressed by

where **H**_{Zeem} and **H**_{demag} are the Zeeman and demagnetization fields, respectively, **M** is the magnetization, A_{ex} is the exchange constant, and V_{mesh} is the volume of the mesh. The gray-colored regions depict the locations of the antidots inside each motif. The color of each plot matches with the y-slice index (the black box) at the bottom of Fig. 4. The FFT powers along the x distance (\(0 \le x \le L/2\)) agree well with the spatial distributions of the total energy density in terms of the x position. For example, the FMR mode was excited in the thin film (denoted as S_{0}) at 5.10 GHz, as indicated by the homogeneous total energy distribution. The recursive sequence (F_{n}) marked at the top of Fig. 4 denotes the evolution of both the total magnetic energy and the frequency of the eigenmodes. The right region (\(L/4 \le x \le L/2\)) of S_{n+1} is F_{n}. The appearance of F_{n} in S_{n+1} is similar to F_{n} in S_{n}. The left region (\(0 \le x < L/4\)) of S_{n+1} is G_{n}, which is a variation of F_{n} in S_{n+1}.

In S_{1}, at a similar frequency to that of the FMR mode, the C_{1} mode was excited at 5.02 GHz in the region of \(D/2 < x \le L/2\). To be specific, the FFT power spatially informed that the C_{1} mode started to be excited at the end of the E_{1} mode (1.77 GHz) in terms of the x position. This profile well matches the total energy distribution of S_{1} (= A_{1}). The magnetic energy variation inside the magnonic crystal corresponds to the localization of the quantized spin-wave modes. The mode arrangement of A_{2} was similar to that of A_{1}. The C_{2} mode of A_{2} was excited at 5.03 GHz, whereas the E_{2} mode was excited at 3.00 GHz. With regard to S_{2}, the E_{2} and C_{2} modes were split into doublets. Due to the existence of the A_{1} antidot on the left side of the A_{2} antidot, the energy profile on the left side is different from the right side of the A_{2} antidot. The C_{V1} mode and the shifted C_{2} mode were hybridized together, as shown in Fig. 3. In S_{3}, the magnetic energy profile inside the motif was divided into three distinct regions. Therefore, the E_{3} mode at 4.42 GHz in A_{3} became split into three modes at 4.24, 4.72, and 5.09 GHz in S_{3}. Similarly, the C_{3} mode at 5.50 GHz in A_{3} split into triplets (5.35, 5.72, and 6.38 GHz) in S_{3}. In A_{4}, the EC_{4} mode was excited at 4.45 GHz, and the excitation profile showed that the edge and center modes had been hybridized into a single mode. Then, for S_{4}, the EC_{4} mode was split into 5 modes (quintets). Since the E_{3} mode of A_{3} and the EC_{4} mode of A_{4} were excited at almost an equal frequency, a total of 8 split modes (the E_{3} triplets plus the EC_{4} quintets) in S_{4} were mixed up. The total energy distribution of S_{4} is complicated compared with those of the previous motifs, because of the complex arrangement of antidots inside S_{4}.

The self-similarity of the fractal motifs introduces aperiodic arrangements of antidots in recursive order. In S_{n}, the total magnetic energy (most dominantly demagnetization energy) of the nth antidot array (A_{n}) became aperiodic since the previous antidots modulated the magnetization configurations around A_{n} antidots. The aperiodic energy variation inside the antidot-lattice fractals gives rise to the multiplets of the spin-wave eigenmodes under the recursive evolution. The energy aperiodicity can be reduced according to the geometric parameters or the externally applied magnetic fields in order to make the magnons’ multiplets degenerate. As the dot-to-dot distance increased, the extent of aperiodicity decreased, and then the magnons’ modes became degenerated (see Supplementary Fig. S5). In the same way, as the strength of the external magnetic field increased, the split modes were reunited (see Supplementary Fig. S3).

### Origin of spin-wave multiplets

Finally, in order to examine the difference of the magnonic excitations between the fractal and non-fractal structures, we conducted the same simulation for the non-fractal, 2D type of NaCl lattice where two different radius holes are arranged alternately. This type of antidot-lattice has been studied under different terminologies, either composite-antidot array^{32,33} or bi-component antidot-lattice^{16,17}. To avoid confusion, the term ‘2D NaCl type’ is employed to describe the antidot-lattice with alternating different diameters. The two antidot sublattices of A_{1} and A_{2} compose S_{2} as well as 2D NaCl type. In S_{2}, they satisfy the initiator-generator relationship with \({\text{L}}_{2} = {\text{L}}_{1} /2\) and \({\text{D}}_{2} \,{ = }\,{\text{D}}_{1} /2\). In 2D NaCl type geometry, both antidots exist in the 1:1 ratio, since \({\text{L}}_{2} = {\text{L}}_{1}\) but \({\text{D}}_{2} \,{ = }\,{\text{D}}_{1} /2\). The location of the A_{1} antidot is asymmetric to that of the A_{2} antidot in the S_{2} motif, while it is symmetric in the 2D NaCl type motif. In both structures, antidots of L_{1} (L) = 1400 nm and D_{1} (D) = 300 nm were used, while a 30 mT strength of magnetic field was applied in the + x-direction.

Figure 5 shows the FFT power versus frequency and the x position (\(0 \le x \le L/2\)) along with the total energy (\({\upvarepsilon }_{\mathrm{tot}}\)) density distribution. In the range of f = 1 ~ 6 GHz, the E_{1}, E_{2} and C_{2} modes appeared noticeably in both S_{2} and 2D NaCl type. Both of the E_{1} modes were independent singlets derived from the A_{1} antidots in both patterns. The only difference was that the E_{1} mode in S_{2} was excited at a slightly higher frequency, since the total magnetic field near the ends of the A_{1} antidots in S_{2} was higher than that of 2D NaCl type. The E_{2} and C_{2} modes appeared as doublets only in S_{2}, whereas those modes were typical singlets in 2D NaCl type. In a comparison of the energy distributions between the two structures, the above difference resulted from the asymmetry of the internal energy about \(x = L/4\). Unlike S_{2}, the non-fractal 2D NaCl type has the same energy distribution at both sides of the A_{2} hole; i.e., it shows a mirror symmetry about \(x = L/4\). For S_{2}, the asymmetry of the total magnetic energy inside the fractal magnonic crystal is the origin of the spin-wave multiplets.

## Discussion

The proposition of novel magnonic crystals composed of antidot-lattice fractals enlarges a basic understanding of quantized spin-wave modes. The fractals of 2-Hausdorff dimensions were constructed by the superposition of self-similar antidot-lattices. Local asymmetries inside the aperiodic magnonic motifs result in the split of the spin-wave eigenmodes: the edge mode, the center mode and the center-vertical mode (see Supplementary Fig. S5b,c). Due to the recursive sequence from the geometrical fractal growth, the local asymmetries inside the antidot-lattice fractals split the spin-waves into multiplets in the frequency spectra, showing the same recursive development. The split modes were finely localized into their own characteristic regions enabling selective excitation of the local area inside the magnonic crystals. Some of those split modes were reunited (or even duplicated) by the variations of the strength and direction of applied bias magnetic fields (see Supplementary Figs. S3 and S4, respectively). The reunion and the crossover among those finely divided modes would make the most of an active control with the bias magnetic field and the crystal geometry design (see Supplementary Fig. S5).

The proliferous standing spin-wave modes with fine localizations would be good candidates for magnonic devices that require diminutive excitation in a certain area of 2D nano-oscillators, memory devices, and sensors.

## Methods

### Micromagnetic simulation procedure

In the present simulations, we used an open-source software, MuMax3^{34}, which incorporates the Landau-Lifshitz-Gilbert equation^{35,36} along with GPU acceleration to solve the dynamic motions of individual magnetizations in given magnonic crystals, for example, as shown in Fig. 1c. There, the motif, the S_{2} fractal marked by a dashed square box, was extended to sufficiently large dimensions (100 μm × 100 μm × 10 nm) with a periodic boundary condition in order to avoid the distortions of the static and dynamic magnetizations at the discontinuous boundaries of its finite dimensions. The sizes of unit cells in the simulations were set up to 5 nm × 5 nm × 10 nm. The material parameters used for Permalloy (Py: Ni_{80}Fe_{20}) were as follows: gyromagnetic ratio \(\upgamma\) = 2.211 × 10^{5} [m/A s], saturation magnetization M_{s} = 8.6 × 10^{5} [A/m], exchange stiffness A_{ex} = 1.3 × 10^{–11} [J/m], damping constant \(\mathrm{\alpha }\) = 0.01, and zero magnetic anisotropy constant, K_{1} = K_{2} = 0 [J/m^{3}].

In order to excite spin-wave modes in the given magnonic crystals, we used a sinc (sine-cardinal) field as expressed by h(t) = h_{0}sin[2πf_{0}(t − t_{0})]/[2πf_{0}(t − t_{0})] with \(\mu_{0} {\text{h}}_{0}\) = 1 mT, \({\text{f}}_{0}\) = 20 GHz, t_{0} = 1 ns, and t = 100 ns. This pumping field was applied along the film normal under a dc bias field of \(\mu_{0} {\text{H}}_{{{\text{bias}}}}\) = 30 mT applied in the + x direction on the film plane (The eigenmodes of antidot-lattice fractals were stabilized at magnetic fields of greater strength than 20 mT; see Supplementary Fig. S3). The temporal oscillations of local magnetizations at each cell were transformed into the frequency domain via Fast Fourier Transforms (FFTs).

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## Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (Grant No. I-AM35-21(0417-20210049)). The Institute of Engineering Research at Seoul National University provided additional research facilities for this work.

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S.K.K. and G.P. conceived the main idea and the conceptual design of the experiments. G.P. performed the micromagnetic simulations. G.P., J.Y., and S.K.K. analyzed the data. S.K.K. and G.P. led the work and wrote the manuscript.

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Park, G., Yang, J. & Kim, SK. Recursive evolution of spin-wave multiplets in magnonic crystals of antidot-lattice fractals.
*Sci Rep* **11, **22604 (2021). https://doi.org/10.1038/s41598-021-00417-0

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DOI: https://doi.org/10.1038/s41598-021-00417-0

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