Introduction

A recursive sequence is one of the most fundamental growth mechanisms in nature, interest in it having grown significantly for its potential quantum applications1,2,3. 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 nature4. Those fractals are classified as statistical and random fractals5. 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 structure6,7,8 and quantization9,10,11,12 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 bottleneck13. 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 confinements15. A variety of types of antidot-lattices have been employed that include bi-component (of different materials16 or different sizes of holes17), different Bravais type18, and defective lattices19. On the other hand, non-trivial magnonic dynamic behaviors were observed in aperiodic structures of antidots such as magnonic quasicrystals of Fibonacci structure20,21,22, Penrose and Ammann tilings23, and Sierpiński carpet24,25,26. In fact, Sierpiński fractals have led to unique phenomena in electronics27,28 and photonics29,30.

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 objects31. 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 Dn and Ln of the nth antidot-lattice (An) are exactly half of those of An−1. For example, A2 has \({\text{L}}_{2} = {\text{L}}_{1} /2\) and \({\text{D}}_{2} \,{ = }\,{\text{D}}_{1} /2\). Next, the nth fractal Sn is constructed by the superposition of A1 + A2 + ⋯ + An as follows: S1 = A1, S2 = A1 + A2, S3 = A1 + A2 + A3, and S4 = A1 + A2 + A3 + A4, as depicted in the series up to S4 (see Fig. 1b). In detail, A1 with D = D1 and L = L1 corresponds to the initiator (mother). A1 and A2 make up S2. Since the D2 and L2 of A2 are half those of A1, the number of antidots for A2 is increased by 4 times, and thus the Hausdorff dimension is log24 = 2. Following An are scaled copies of previous An−1 in the same manner. Due to the self-similarity of the fractals, a recursive sequence arises inside the geometry of the motifs: let Fn denote the geometrical sequence. Fn (\(0 \le x \le L/2\)) of each Sn is a summation of Fn−1 (\(L/4 \le x \le L/2\)) and Gn−1 (\(0 \le x < L/4\)). The appearance of Fn−1 in Sn is a scaled recursion of Fn−1 (\(0 \le x \le L/2\)) in Sn−1. In Sn, Gn−1 is a variation of Fn−1 and can be viewed as the A1 antidot superimposed onto Fn−1.

Figure 1
figure 1

(a) Sequence of antidot-lattices with self-similar geometry. The capital letters of D and L correspond to the diameter of a hole and the size of a square Bravais lattice, respectively. (b) Evolution of antidot-lattice fractals. Each motif of Sn denotes the superposition of the individual lattices of A1, A2, … and An. (c) Fractal magnonic crystal of 100 μm × 100 μm dimensions where periodic boundary condition was applied using the area marked by blue-dashed box (S2 as an example). For excitations of all possible spin-wave modes, dc-bias and sinc-function magnetic fields were applied in the + x direction and along the z-axis, respectively.

Full size image

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 S1 (= A1) exhibited three normal standing spin-wave modes as indexed by E1, C1, and CV112. The very weak mode (E1) at 1.77 GHz corresponds to the edge mode, and the strongest mode (C1) at 5.02 GHz to the center mode. The two modes are periodically excited along the bias field direction. The last minor mode (CV1) 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.

Figure 2
figure 2

Modes’ spectra in magnonic crystals of antidot-lattice fractals, S1, S2, S3, and S4, with L1 = 1400 nm and D1 = 300 nm. En and Cn denote the edge and center mode of Sn. A bias magnetic field of 30 mT was applied in the + x direction.

Full size image

For S2, while keeping the E1 mode at a similar frequency, an additional doublet (E2) of the edge mode appeared, which originated from A2. The doublet (C2) of the center mode appeared as its substitution for the singlet C1 in S1 (see Fig. 2b). The higher mode (5.61 GHz) of the doublet C2 was hybridized with the CV1 mode (5.80 GHz) in S1. In a similar manner, for S3, an additional triplet (E3) came from A3, while the doublet C2 in S2 then became the triplet C3. The center-vertical mode (CV2) of A2 was hybridized with the highest mode (6.38 GHz) of C3. To sum up, by adding A2 and A3 to S2 and S3, respectively, each En mode was newly updated while the Cn mode substituted for Cn−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 An and An−1 remained intact with each other, while the channels of An were impacted by the channels of An−1.

In the case of A4, 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 A4. Therefore, for S4, the hybrid mode (E4 + C4 \(\to\) EC4) appeared instead of the individual E4 and C4 modes. A total of five EC4 modes (pink-colored peaks) substituted for the C3 triplet.

The number of multiplets in the serial spectra followed the recursive sequence, Fn = Fn−1 + Gn−1. The two eigenmodes (En and Cn) appeared as a singlet, doublet, triplet, and quintet for S1, S2, S3 and S4, respectively. Since there is only one zeroth (n = 0) standing spin-wave mode in unpatterned (continuous) thin film (i.e., S0), 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 En and Cn. The difference sequence (Gn) 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 S1, the major modes of E1 and C1 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 C2 doublet, was vertically localized between the neighboring holes of A1 in which CV1 was also localized in S1 (Fig. 3c). This explains why the higher mode of C2 was hybridized with CV1, as mentioned earlier. On the other hand, the lower (4.94 GHz) one of C2 remained extended along the channel between S2. The doublets of both E2 and C2 are antiphase with each other in temporal oscillation; S2 can be considered to be a two-dimensional nano-oscillator. For S3, the highest C3 at 6.38 GHz was fully localized in between the A2 antidots along the y-axis: it was hybridized with CV2 due to their shared localization area. The lowest C3 (5.35 GHz) remained extended along the channel between S3. Similarly, the lowest EC4 (4.04 GHz) for S4 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 A3 and A4 are close, some localized modes (4.30 GHz and 4.98 GHz) were excited at antidots of A3 together with certain antidots of A4. On the other hand, some of the E3 modes (4.04 GHz and 4.70 GHz) of S4 were excited at the same frequencies as those of the EC4 modes. Those E3 and EC4 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.

Figure 3
figure 3

Spatial distribution of power of FFTs in bottom-right quarter area of motifs of S1, S2, S3 and S4 at indicated frequencies of specific modes.

Full size image

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 En modes, let E1 in S1 be F1. The two E2 modes in S2 are F2. The right part (L/4 \(\leq\) x \(\le\) L/2) of the E2 (2.93 GHz) profile is the same as the E1 profile in S1. E2 (2.93 GHz) is F1 in S2. The left part (0 \(\le\) x \(<\) L/4) of the E2 (3.65 GHz) profile is a variation of the E2 (2.93 GHz) profile. E2 (3.65 GHz) is G1 in S2. The three E3 modes in S3 are F3. The right parts of the E3 (4.24 GHz and 5.09 GHz) profiles are similar to the E2 profiles in S2. The two E3 modes are F2 in S3. The left part of the E3 (4.72 GHz) profile is a variation of the E3 (4.24 GHz) profile. E3 (4.72 GHz) is G2 in S3. The five EC4 modes in S4 are F4. The right parts of the EC4 (4.04 GHz, 4.98 GHz, and 5.49 GHz) profiles are similar to the E3 profiles in S3. The three EC4 modes are F3 in S4. The left parts of the EC4 (4.30 GHz and 4.70 GHz) profiles are variations of the EC4 (4.98 GHz and 4.04 GHz, respectively) profiles. The two EC4 (4.30 GHz and 4.70 GHz) modes are G3 in S4. In the same way, let C1 in S1 be F1. The two C2 modes in S2 are F2. The right part of the C2 (4.94 GHz) profile is the same as the C1 profile in S1. C2 (4.94 GHz) is F1 in S2. The left part of the C2 (5.61 GHz) profile is a variation of the C2 (4.94 GHz) profile. C2 (5.61 GHz) is G1 in S2. The three C3 modes in S3 are F3. The right parts of the C3 (5.35 GHz and 6.38 GHz) profiles are the same as the C2 profiles in S2. C3 (5.35 GHz and 6.38 GHz) are F2 in S3. The left part of the C3 (5.72 GHz) profile is a variation of the C3 (5.35 GHz) profile. C3 (5.72 GHz) is G2 in S3. The EC4 modes are considered in the same way as mentioned above.

Figure 4
figure 4

(Upper row) Spatial distributions of total magnetic energy density for thin film (S0) and magnonic crystals of An and Sn. The energy densities were plotted by cross-section along the y-axis where corresponding antidots were located. Each color of the plot matches with the index of the y-slice (black box) at the bottom of the figure. The gray-colored regions inside the plots denote the locations of the antidots. (Bottom row) Contour plots of FFT power on frequency and x position.

Full size image

Origin of recursive evolution

Next, in order to identify the splits of the spin-waves excited in Sn with respect to An, 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

$$\varepsilon_{tot} = - \frac{{\mu_{0} }}{{2V_{mesh} }}\int {\textbf{\textit{M}} \cdot ({\textbf{\textit {H}}_{\textbf{\textit {Zeem}}}} + {\textbf{\textit {H}}_{\textbf{\textit {demag}}}})dV_{mesh} + A_{exch} \int {\left| {{\varvec \nabla} \textbf{\textit{M}}} \right|} }^{2} dV_{mesh} ,$$
(1)

where HZeem and Hdemag are the Zeeman and demagnetization fields, respectively, M is the magnetization, Aex is the exchange constant, and Vmesh 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 S0) at 5.10 GHz, as indicated by the homogeneous total energy distribution. The recursive sequence (Fn) 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 Sn+1 is Fn. The appearance of Fn in Sn+1 is similar to Fn in Sn. The left region (\(0 \le x < L/4\)) of Sn+1 is Gn, which is a variation of Fn in Sn+1.

In S1, at a similar frequency to that of the FMR mode, the C1 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 C1 mode started to be excited at the end of the E1 mode (1.77 GHz) in terms of the x position. This profile well matches the total energy distribution of S1 (= A1). The magnetic energy variation inside the magnonic crystal corresponds to the localization of the quantized spin-wave modes. The mode arrangement of A2 was similar to that of A1. The C2 mode of A2 was excited at 5.03 GHz, whereas the E2 mode was excited at 3.00 GHz. With regard to S2, the E2 and C2 modes were split into doublets. Due to the existence of the A1 antidot on the left side of the A2 antidot, the energy profile on the left side is different from the right side of the A2 antidot. The CV1 mode and the shifted C2 mode were hybridized together, as shown in Fig. 3. In S3, the magnetic energy profile inside the motif was divided into three distinct regions. Therefore, the E3 mode at 4.42 GHz in A3 became split into three modes at 4.24, 4.72, and 5.09 GHz in S3. Similarly, the C3 mode at 5.50 GHz in A3 split into triplets (5.35, 5.72, and 6.38 GHz) in S3. In A4, the EC4 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 S4, the EC4 mode was split into 5 modes (quintets). Since the E3 mode of A3 and the EC4 mode of A4 were excited at almost an equal frequency, a total of 8 split modes (the E3 triplets plus the EC4 quintets) in S4 were mixed up. The total energy distribution of S4 is complicated compared with those of the previous motifs, because of the complex arrangement of antidots inside S4.

The self-similarity of the fractal motifs introduces aperiodic arrangements of antidots in recursive order. In Sn, the total magnetic energy (most dominantly demagnetization energy) of the nth antidot array (An) became aperiodic since the previous antidots modulated the magnetization configurations around An 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 array32,33 or bi-component antidot-lattice16,17. To avoid confusion, the term ‘2D NaCl type’ is employed to describe the antidot-lattice with alternating different diameters. The two antidot sublattices of A1 and A2 compose S2 as well as 2D NaCl type. In S2, 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 A1 antidot is asymmetric to that of the A2 antidot in the S2 motif, while it is symmetric in the 2D NaCl type motif. In both structures, antidots of L1 (L) = 1400 nm and D1 (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 E1, E2 and C2 modes appeared noticeably in both S2 and 2D NaCl type. Both of the E1 modes were independent singlets derived from the A1 antidots in both patterns. The only difference was that the E1 mode in S2 was excited at a slightly higher frequency, since the total magnetic field near the ends of the A1 antidots in S2 was higher than that of 2D NaCl type. The E2 and C2 modes appeared as doublets only in S2, 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 S2, the non-fractal 2D NaCl type has the same energy distribution at both sides of the A2 hole; i.e., it shows a mirror symmetry about \(x = L/4\). For S2, the asymmetry of the total magnetic energy inside the fractal magnonic crystal is the origin of the spin-wave multiplets.

Figure 5
figure 5

Comparison between S2 fractal and 2D NaCl type (non-fractal) magnonic crystals: (Upper row) spatial distribution of total magnetic energy density in bottom-right quarter area of given motifs. (Bottom row) Spatial distribution of frequency spectra of FFT power in x-direction. Doublets of E2 and C2 appeared only at the S2 fractal.

Full size image

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, MuMax334, which incorporates the Landau-Lifshitz-Gilbert equation35,36 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 S2 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: Ni80Fe20) were as follows: gyromagnetic ratio \(\upgamma\) = 2.211 × 105 [m/A s], saturation magnetization Ms = 8.6 × 105 [A/m], exchange stiffness Aex = 1.3 × 10–11 [J/m], damping constant \(\mathrm{\alpha }\) = 0.01, and zero magnetic anisotropy constant, K1 = K2 = 0 [J/m3].

In order to excite spin-wave modes in the given magnonic crystals, we used a sinc (sine-cardinal) field as expressed by h(t) = h0sin[2πf0(t − t0)]/[2πf0(t − t0)] with \(\mu_{0} {\text{h}}_{0}\) = 1 mT, \({\text{f}}_{0}\) = 20 GHz, t0 = 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).