Precessional dynamics of geometrically scaled magnetostatic spin waves in two-dimensional magnonic fractals

The control of spin waves in periodic magnetic structures has facilitated the realization of many functional magnonic devices, such as band stop filters and magnonic transistors, where the geometry of the crystal structure plays an important role. Here, we report on the magnetostatic mode formation in an artificial magnetic structure, going beyond the crystal geometry to a fractal structure, where the mode formation is related to the geometric scaling of the fractal structure. Specifically, the precessional dynamics was measured in samples with structures going from simple geometric structures toward a Sierpinski carpet and a Sierpinski triangle. The experimentally observed evolution of the precessional motion could be linked to the progression in the geometric structures that results in a modification of the demagnetizing field. Furthermore, we have found sets of modes at the ferromagnetic resonance frequency that form a scaled spatial distribution following the geometric scaling. Based on this, we have determined the two conditions for such mode formation to occur. One condition is that the associated magnetic boundaries must scale accordingly, and the other condition is that the region where the mode occurs must not coincide with the regions for the edge modes. This established relationship between the fractal geometry and the mode formation in magnetic fractals provides guiding principles for their use in magnonics applications.

Figure 1

(a) SEM image of the square (SQ) and triangular (TRI) structures. (b) Schematic of the TRSKM setup and the sample. (c), (d) TRSKM measurements of the sample TRI-0 at $B=15$ mT and $f_{\mathrm{RF}}=3.7$ GHz. (c) Normalized time-resolved Kerr rotation. (d) Five normalized scanning Kerr rotation images, in the range $[-1,1]$, measured from $t_1$ to $t_5$ at 360, 465, 570, 675, and 780 ps, respectively, which are indicated by the solid lines in (c).

Figure 2

TRSKM measurements of triangular structures. (a) Frequency spectra of the triangular structures TRI-0 to TRI-2. (b) SEM images of structures TRI-0 to TRI-2, with the magnetic field direction indicated next to TRI-2. (c) Normalized scanning Kerr images measured for the three samples. The mode is indicated by red markers in (a).

Figure 3

TRSKM measurements of square structures. (a) Frequency spectra of square structures SQ-0 to SQ-2. (b) SEM images of structures SQ-0 to SQ-2. For SQ-2, L1–L4 are the four edges with width of $9\mu \mathrm{m}$, s1 is the central $3\mu \mathrm{m}\times 3\mu \mathrm{m}$ square hole, and s2 corresponds to the eight $1\mu \mathrm{m}\times 1\mu \mathrm{m}$ square holes. (c) Normalized scanning Kerr images of the different modes for the three structures. The modes corresponding to different frequencies are indicated by different colored markers in (a).

Figure 4

Comparison between the simulated and experimental results for SQ-2. (a) Simulated time-resolved precession using two single-frequency excitations at 3.66 and 4.56 GHz. (b) Simulated $m_z$ distributions that are taken at the time indicated by the red dashed lines in (a). The color maps are set to optimize the image contrast for the spatial distribution. (c) Experimental scanning Kerr images at the phase value of $\pi /2$ for the corresponding sinusoidal oscillation at frequencies of 3.7 and 4.4 GHz. The color scale for (c) is similar to that used in Fig. 3.

Figure 5

Analysis of simulated modes for SQ-2. (a) Frequency spectrum of SQ-2 at 15 mT, with the edge modes (E), the first-order localized modes (M), and the higher-order modes (HO) indicated with different color shaded areas. (b) Amplitude distributions of the three modes M1 (3.66 GHz), M2 (4.18 GHz) and M3 (5.08 GHz). (c) Total field distribution $B_{\mathrm{tot}}$ within the magnetic structure of SQ-2. The external magnetic field is applied along the horizontal direction, as indicated by the arrow. (d) Five line profiles of the total field distribution along the horizontal direction, with positions indicated by dashed lines in (c).

Figure 6

Simulated mode evolution from SQ-1 to SQ-3. (a) Amplitude distributions of the three modes for SQ-3 at 4.30 GHz ($\mathrm{M}{1}^{\prime }$), 7.55 GHz ($\mathrm{M}{3}^{\prime }$), and 6.04 GHz. The white boxes indicate the regions where the edge modes occur. (b) Amplitude distributions of the three modes M1 to M3 for SQ-2. (c) Amplitude distributions of the two modes at 3.10 and 4.00 GHz for SQ-1. (d) Schematics of structures SQ-2 and SQ-3, with different color shaded areas indicating the eight scaled SQ-2 structures $\mathrm{SQ}\text{−}2\left(j\right),j=1–8$.