Spin-wave localization on phasonic defects in a one-dimensional magnonic quasicrystal

We report on the evolution of the spin-wave spectrum under structural disorder introduced intentionally into a one-dimensional magnonic quasicrystal. We study theoretically a system composed of ferromagnetic strips arranged in a Fibonacci sequence. We considered several stages of disorder in the form of phasonic defects, where different rearrangements of strips are introduced. By transition from the quasiperiodic order towards disorder, we show a gradual degradation of spin-wave fractal spectra and closing of the frequency gaps. In particular, the phasonic defects lead to the disappearance of the van Hove singularities at the frequency gap edges by moving modes into the frequency gaps and creating new modes inside the frequency gaps. These modes disperse and eventually can close the gap, with increasing disorder levels. The work reveals how the presence of disorder modifies the intrinsic spin-wave localization existing in undefected magnonic quasicrystals. The paper contributes to the knowledge of magnonic Fibonacci quasicrystals and opens the way to study of the phasonic defects in two-dimensional magnonic quasicrystals.

Figure 1

(a) All possible approximates of a Fibonacci crystal composed of 21 elements. As the phase $\varphi$ changes [see Eq. (1)], we obtain 21 possible sequences of Co (light yellow) and Py (dark blue)—note that Co strips can appear in doublets. The solid red line at $\varphi =\pi /\tau$ corresponds to the approximate generated by standard substitution rules: $\mathrm{Co}\to \mathrm{Co}|\mathrm{Py}, \mathrm{Py}\to \mathrm{Co}$, presented in (b)—see also Fig. 2. The red dashed lines show the range in which the parameter $\varphi$ is randomly changed at each position $n$. The changes of $\varphi$ which induces the phasonic defects are marked by green bars. They are responsible for the substitution $\mathrm{Py}\to \mathrm{Co}$ at position 7 and swap between positions 12 and 13 ($\mathrm{Co}|\mathrm{Py}\to \mathrm{Py}|\mathrm{Co}$). A sequence with defects is presented in (c); note that position of swaps are marked by arrows.

Figure 2

The approximate of a Fibonacci quasicrystal corresponding to the $\varphi =\pi /\tau$ (see Fig. 1), i.e., resulting from the standard substitution rules: $\mathrm{Co}\to \mathrm{Co}|\mathrm{Py}, \mathrm{Py}\to \mathrm{Co}$. This exemplary structure is composed of Py and Co flat strips (30 nm thick and 300 nm wide), aligned side-by-side and being in direct contact. The field ${\mu }_0{H}_0=0.1$ T is applied along the strips. The sequence of tilted arrows and line in front of them visualizes the spin-wave mode profile.

Figure 3

Top row: (a), (b), and (c) Integrated density of states as a function of frequency plotted in the inverse form $f\left(\mathrm{IDOS}\right)$ (blue/green color), and the dispersion relation for SWs in a homogeneous film with weight averaged material parameters (black color). Please note that IDOS and dispersion relation has their own abscissa, and shares a common ordinate. The abscissa of each plot has the same scale, indicated only in the leftmost plot. (a) Results obtained for a perfect Fibonacci sequence composed of 377 strips. (b) Results obtained for a defected sequence with amplitude $\mathrm{\Delta }\varphi /\left(2\pi \right)=$ 5% and (c) $\mathrm{\Delta }\varphi /\left(2\pi \right)=$ 10%. Bottom row: (d), (e), and (f) Bar plot of reciprocal lattice vector intensities, corresponding to the Bragg peaks, for the structures from (a), (b), and (c), respectively. Phasonic defects destroy the fine structure of Bragg peaks that in consequence lead to modification of the density of states at the edges of frequency gaps and creates new modes inside the gaps.

Figure 4

Top row: (a), (b), and (c) Integrated density of states as a function of frequency plotted in the inverse form $f\left(\mathrm{IDOS}\right)$ calculated for the Fibonacci sequence with introduced defects. Gray areas represents the frequency gaps in an ideal Fibonacci sequence. The amplitude of phasonic defects $\mathrm{\Delta }\varphi /\left(2\pi \right)$ are (a) $5\%$, (b) $10\%$, and (c) $25\%$. A histogram of integrated density of states (IDOS) is obtained from 100 configurations of differently introduced defects. Intensity of the green color reflects how often a given position is occupied by an SW mode. Bottom row: (d), (e), and (f) Localization measure ${\lambda }_i$ as a function of frequency for an SW in a 1D Fibonacci sequence with phasonic defects. The values of ${\lambda }_i$ are calculated for structures with (d) $5\%$, (e) $10\%$, and (f) $25\%$ of defects. Every plot aggregates 100 different system configurations. ${\lambda }_i$ increases significantly even if a small amount of defects is introduced (d), and consequently increases with amount of defects (e) and (f).

Figure 5

The evolution of the bulk mode under the presence of the defects. (a) In the absence of defects the mode is not localized, its amplitude is more concentrated in Py than in Co. (b) For $\mathrm{\Delta }\varphi /\left(2\pi \right)=10\%$ the defects (marked by arrows below the plot) lead to the formation of double Py strips and can concentrate the SW dynamics.

Figure 6

The transition of the mode No. 136 from critically localized at the edge of the frequency gap to strongly localized in the frequency gap induced by phasonic defects. The critically localized mode (a) enters into the gap and (b) becomes strongly localized, due to the presence of defects $\mathrm{\Delta }\varphi /\left(2\pi \right)=10\%$ which is accompanied by a significant change in frequency from 12.57 to 13.29 GHz.

Figure 7

(a) The critically localized mode (No. 359 at 16.96 GHz) which increases its localization and slightly changes its frequency to $f=16.92$ GHz due to partial confinement between defects (b) at $\mathrm{\Delta }\varphi /\left(2\pi \right)=10\%$.

Figure 8

The defect modes from the two largest frequency gaps shown in Figs. 3 and 4. (a) and (b) Modes No. 135 and No. 138 with frequency 13.28 and 13.29 GHz. (c) and (d) Modes No. 291 and No. 292 with frequency 16.03 GHz. The modes are strongly localized at one of few locations but have the same profile, differing only in phase (flipped upside down) [see (a) and (b)] or reversed along with the structure (flipped left-right) [see (c) and (d)]. Due to strong localization, the modes are practically degenerated. The results are shown for the structure with $\mathrm{\Delta }\varphi /\left(2\pi \right)=10\%$.

Figure 9

The illustration of the cut-and-projection (C&P) method and the induction of phasonic defect. The array of dots represent the square lattice in a 2D hyperspace. The Fibonacci lattice (black and blue crosses) is generated by the projection of a square lattice from the belt between solid and dashed lines onto the line $y={\tau }^{-1}x+c$ of irrational slope, being the inverse of the golden ratio $\tau$. The visible (21-element) section of the Fibonacci lattice corresponds to the selection of $\varphi =2\pi c/a=0.8$, see Fig. 1. For a defect-free Fibonacci lattice the belt (between solid blue and dashed blue line) is straight. By bending the belt (limited here by solid green and dashed green lines), we can induce the phasonic defects in the Fibonacci lattice (black and green crosses).

Figure 10

(a) Integrated density of states for SWs in the randomly generated sequence of Co and Py, where the ratio between types of strips is kept as for a Fibonacci quasicrystal, i.e., it corresponds to the golden ratio. The dark-green color represents a histogram of aggregated results obtained from 100 different random sequences (the color scale is the same as in Fig. 4). Light-green points stand for one specific structure for which the bar of the Fourier transform (b) are plotted.