Magnetization dynamics of a Fibonacci-distorted kagome artificial spin ice

We present results of ferromagnetic resonance (FMR) experiments and micromagnetic simulations for a distorted, two-dimensional (2D) kagome artificial spin ice. The distorted structure is created by continuously modulating the 2D primitive lattice translation vectors of a periodic honeycomb lattice, according to an aperiodic Fibonacci sequence used to generate 1D quasicrystals. Experimental data and micromagnetic simulations show that the Fibonacci distortion causes broadening and splitting of FMR modes into multiple branches, which accompany the increasing number of segment lengths and orientations that develop with increasing distortion. When the applied field is increased in the opposite direction to the net magnetization of a segment, spin wave modes appear, disappear, or suddenly shift, to signal segment magnetization reversal events. These results show that the complex behavior of reversal events, as well as well-defined frequencies and frequency-field slopes of FMR modes, can be precisely tuned by varying the severity of the aperiodic lattice distortion. This type of distorted structure could therefore provide a tool for the design of complicated magnonic systems.

Figure 1

Fibonacci spacing in third generation Penrose P2 tiling. L is “long” and S is “short.” The ratio of L/S is 1.62. The Fibonacci word in the figure is $S4=01001010=\mathrm{LSLLSLSL}$. Letters in the right margin show the expansion of the Fibonacci word from the center of symmetry in either the up or down directions.

Figure 2

Geometry of the KASI distortion. (a) Undistorted KASI of third generation ($r=1$); $a$ and $b$ are the undistorted primitive vectors. Green dots (type-$A$ lattice point) are primitive lattice points, and violet dots (type-$B$ lattice point) define the basis positions. (b)–(d) In the distorted lattice, the length of the primitive vectors is either long (L with blue color) or short (S with red color), according to the Fibonacci sequence. Distorted ASIs are shown for $r=L/S=1.15, r=1.3$, and $r=1.45$. (e) Distorted ASI for $r=1.62$. Different segment types are shown in different colors and labeled by roman numerals. For example, the blue segments aligned along the $x$ axis are assigned the numeral I. We can see a sixfold rotational symmetry of the undistorted KASI is reduced to one mirror plane that is indicated by the orange vector along the $x$ axis.

Figure 3

SEM image of the FKASI for $r=1.62$ patterned on a quartz substrate. The orange line denotes the $+x$ direction and the location of a mirror plane. The width of the segments is $w\sim 140\mathrm{nm}$ (note the 1-micron scale bar).

Figure 4

FMR results for applied field $H$ = 1000 Oe for different distortion ratios $r$. (a) ${\mathrm{S}}_{12}$ (transmission from VNA port 1 to port 2) as a function of frequency for each value of $r$. (b) Numerical simulations of the absorption spectrum as a function of frequency for each value of $r$. Note that modes are labeled for $r=1.0$ and 1.62. The mode labeling is consistent with corresponding segments labeled I–V. (c)–(g) Mode profiles for $r=1.62$ for five peaks of the absorption spectrum. Segments in (c)–(g) are labeled corresponding to the mode labels. The color bar represents absorption intensity, where red is maximum absorption intensity and blue is zero absorption intensity.

Figure 5

Experimental (top panels) and simulation (bottom panels) results for field vs frequency for samples with the following distortions: (a) $r=1.0$, (b) $r=1.3$, and (c) $r=1.62$. High frequency modes are labelled I–V, and assigned colors. Modes labeled I-$R$ and II-$R$ are created after reversal. The color bar defines 1 (dark) for maximum absorption intensity and 0 (light) for no absorption.

Figure 6

Simulated magnetization textures for three values of applied field (a) $H=-390\mathrm{Oe}H$ = −390 Oe, (b) $H$ = −440 Oe, and (c) $H$ = −460 Oe, during a sweep from +1000 to −600 Oe. Magnetization reversal of several segments begins at −390 Oe followed by additional segment reversals at −440 Oe. All segments are reversed at −460 Oe. Blue and red indicate reversed and nonreversed regions, respectively. Small arrows indicate magnetization direction; curved arrows indicate chiral cells.

Figure 7

Mode profiles at $H$ = −440 Oe for $r=1.3$. The mode profile for frequency 9.7 GHz (right) exhibits a resonance of nonreversed segments, and the mode profile at 12.6 GHz (left) exhibits a resonance of reversed segments. The color bar represents absorption intensity, where red is maximum intensity and blue is zero intensity.

Figure 8

Mode profiles for LDW modes. Panels (a) and (b) show the mode profiles for an applied field $H$ = 1000 Oe for $r=1.0$ at 8.2 GHz (VCM) and 5.2 GHz (LDW mode), respectively. Panels (c)–(e) show mode profiles for $r=1.62$ at 8.4 GHz (VCM), 4.6 GHz (LDW), and 4.2 GHz (LDW), respectively. Panels (d) and (e) show that two different vertex types resonate at different frequencies.

Figure 9

(a) Magnetization texture of a representative vertex in KASI for $r=1$. (b) The corresponding texture for a distortion ratio $r=1.62$. DW are shown by green lines. The applied field is $H$ = 1000 Oe, oriented horizontally to the right.

Figure 10

Setup for BB FMR measurements. The top figure shows a cross-sectional view. Green denotes the sample. and placed as a flip-chip on the red microstrip line deposited on the Kapton substrate, which is attached on the solid copper plane using silicone adhesive. The bottom figure shows the top view of the setup. The red on the lower-left image is the microstrip line, and green dots represent arrays of ASIs. The bottom right shows an optical image of an ASI sample patterned on a quartz substrate, which is placed as a flip-chip on the microstrip line and aligned with the narrow, 20-micron strips.

Figure 11

${\mathrm{S}}_2$ as a function of frequency for applied fields $-500\mathrm{Oe}\le H\le 400\mathrm{Oe}$ for (a) $r=1.0$, (b) $r=1.3$, and (c) $r=1.62$.

Figure 12

${\mathrm{S}}_2$ as a function of frequency for applied fields $1000\mathrm{Oe}\le H\le 500\mathrm{Oe}$ for (a) and (b) $r=1.0$, (c) $r=1.3$, and (d) $r=1.62$.

Figure 13

Different types of vertices created by the Fibonacci distortion. Angles are shown for $r=1.62$.