Quasiperiodic Magnonic Crystals Overview

• Quasiperiodic Magnonic Crystals are artificial magnetic media with deterministic, aperiodic order that produce fractal spectra, multiple minibands, and critically localized modes.
• They employ Fibonacci and Penrose tilings to generate a hierarchy of Bragg-like scattering conditions and complex, multifractal spectral gaps.
• These systems facilitate advanced magnonic filtering and reprogrammable mode control, paving the way for innovative device applications in spin-wave technology.

Searching arXiv for relevant papers on quasiperiodic magnonic crystals and closely related quasicrystalline magnonics. Searching arXiv for "quasiperiodic magnonic crystals Fibonacci Penrose phasonic defects spin waves". Quasiperiodic magnonic crystals are artificial magnetic media whose structural order is long-range but lacks translational periodicity. In contrast to periodic magnonic crystals, their reciprocal-space spectra are discrete yet nonperiodic, so spin waves experience a hierarchy of Bragg-like scattering conditions rather than a single lattice-defined zone-boundary condition. The resulting spectra typically contain dense sets of minibands, numerous partial and full gaps, fractal-like or multifractal structure, and mode profiles that localize on aperiodic motifs without requiring random disorder (Mehta et al., 2 Sep 2025). Within this class, the most extensively studied realizations are one-dimensional Fibonacci systems and two-dimensional quasicrystalline tilings such as Penrose and Ammann lattices, implemented in exchange-coupled strips, dipolarly coupled nanowires, continuous films with antidots, or nanobar networks (Mieszczak et al., 2022).

1. Conceptual framework and defining characteristics

Periodic magnonic crystals are built from a unit cell repeated periodically in space, and their band structures show band gaps at Brillouin-zone boundaries set by a single lattice constant. A magnonic quasicrystal is instead aperiodic but ordered: its structure factor exhibits a countable set of Bragg peaks with incommensurate ratios, and multiple scattering across many wavevectors produces a hierarchy of gaps and a fractal-like spectrum of spin-wave eigenfrequencies (Mieszczak et al., 2022). In the broader QPMC literature, this is often expressed by stating that reciprocal space is spanned by incommensurate vectors rather than a single lattice, that spectra show multifractal features and dense sets of minibands, and that modes localize at specific aperiodic motifs while exhibiting unconventional rotational symmetries such as eightfold and tenfold patterns without Bloch periodicity (Mehta et al., 2 Sep 2025).

In one-dimensional Fibonacci media, quasiperiodicity is commonly generated by the substitution rule $A \to AB$, $B \to A$, or equivalently $L \to LS$, $S \to L$. The golden ratio $\tau=(1+\sqrt{5})/2$ organizes both the real-space inflation rules and the reciprocal-space hierarchy. For a finite Fibonacci approximant, one convenient generator is

$$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$

with $\chi_n=\pm1$ selecting the two magnetic constituents at position $n$ (Mieszczak et al., 2022). In the review literature, the associated reciprocal vectors are described as $G_{m,n}=(2\pi/a_0)(m+n\tau)$, and the resulting spectral organization is discussed in terms of pseudo-Brillouin zones and staircase-like IDOS plateaus that label gaps (Mehta et al., 2 Sep 2025).

A central distinction from random disorder is that quasiperiodic order preserves deterministic long-range correlations. This deterministic order produces intrinsic localization and rich gap hierarchies, whereas strong randomization suppresses Bragg peaks and removes the quasiperiodic spectral structure. In the Fibonacci phason-disorder study, the limiting case $\Delta\phi=\pi$ yields sequences statistically equivalent to random order, for which Bragg peaks vanish and gaps disappear (Mieszczak et al., 2022).

2. One-dimensional Fibonacci magnonic architectures

The canonical one-dimensional experimental and theoretical platforms are Fibonacci arrays of two magnetic components. One realization uses Co and permalloy strips, each $B \to A$0 wide, $B \to A$1 thick, and infinitely long, placed side-by-side in direct contact to form a continuous ferromagnetic layer under $B \to A$2 along the strip axis. The material parameters are $B \to A$3, $B \to A$4, $B \to A$5, and $B \to A$6 (Mieszczak et al., 2022). A closely related platform used to study damping and lifetime consists of exchange-coupled Py/Co wire arrays with thickness $B \to A$7, widths $B \to A$8 or $B \to A$9, and a Fibonacci sequence $L \to LS$0 containing $L \to LS$1 wires; the periodic reference is an alternating Py/Co crystal with the same materials and field geometry (Rychły et al., 2019).

A second important realization employs dipolarly coupled Py nanowires of two widths in a Fibonacci arrangement. In that system, narrow wires have $L \to LS$2, wide wires have $L \to LS$3, the air gap is $L \to LS$4, the thickness is $L \to LS$5, and the length is $L \to LS$6. An $L \to LS$7 Fibonacci array contains $L \to LS$8 wide and $L \to LS$9 narrow nanowires, and the average lattice parameter is

$S \to L$0

with $S \to L$1 the golden ratio (Lisiecki et al., 2018).

These architectures produce spectra with both broad gaps and finer mini-gaps. In the dipolarly coupled Py nanowire Fibonacci array at $S \to L$2, the first main band spans approximately $S \to L$3–$S \to L$4, the main band gap spans $S \to L$5–$S \to L$6, and the second main band lies near $S \to L$7–$S \to L$8. Within the first band, sub-bands contain $S \to L$9, $\tau=(1+\sqrt{5})/2$0, and $\tau=(1+\sqrt{5})/2$1 modes, following Fibonacci numbers (Lisiecki et al., 2018). In the 377-strip Co/Py Fibonacci approximant, the largest gap spans $\tau=(1+\sqrt{5})/2$2–$\tau=(1+\sqrt{5})/2$3, with additional narrower gaps around $\tau=(1+\sqrt{5})/2$4 and $\tau=(1+\sqrt{5})/2$5 (Mieszczak et al., 2022).

Across these 1D platforms, the essential mechanism is the same: quasiperiodic long-range order generates a dense hierarchy of scattering vectors, so the spectrum fragments into multiple gaps and supports critically localized or motif-localized states. A plausible implication is that material contrast, geometric contrast, and the specific coupling mechanism—exchange, dipolar, or mixed exchange-dipolar—determine the absolute frequency scale, while quasiperiodicity determines the multiscale spectral organization.

3. Two-dimensional quasicrystalline and artificial-quasicrystal platforms

Two-dimensional quasiperiodic magnonic crystals have been studied in both exchange-dominated continuous films and artificial nanobar networks. In the Penrose-tiling film platform, circular magnetic disks are placed at the centers of rhombi of a P3 Penrose tiling. The rhombus side length is $\tau=(1+\sqrt{5})/2$6, the disk radius is $\tau=(1+\sqrt{5})/2$7, the thickness is $\tau=(1+\sqrt{5})/2$8, and the filling fraction in the largest supercell is $\tau=(1+\sqrt{5})/2$9. Two material combinations were analyzed: Ni disks in an Fe matrix and Py disks in a Co matrix. The high-contrast Ni/Fe system exhibits more distinctive magnonic gaps than Py/Co, and reducing the filling fraction below $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$0 diminishes gap widths (Rychły et al., 2017).

A different 2D class consists of artificial magnetic quasicrystals made from interconnected Py nanobars arranged on Penrose P2, Penrose P3, and Ammann tilings. In one implementation, the arrays are $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$1, the nanobar length is $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$2, the thickness is $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$3, and the widths are $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$4 for P2, P3, and square lattices and $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$5 for Ammann lattices. Broadband spin-wave spectroscopy at $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$6 shows tenfold magnetic symmetry for P2 and P3, eightfold symmetry for Ammann, and fourfold symmetry for a periodic square lattice (Bhat et al., 2018). Micromagnetic simulations identify mirror-symmetric mode profiles and allocate resonance branches to bar-orientation families such as $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$7, $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$8, and $$\chi_n(\phi)=\mathrm{sign}\!\left[\cos\!\left(\frac{2\pi n}{\tau}+\phi\right)-\cos\!\left(\frac{\pi}{\tau}\right)\right],$$9 in Penrose or $\chi_n=\pm1$0, $\chi_n=\pm1$1, and $\chi_n=\pm1$2 in Ammann (Bhat et al., 2018).

The role of connectivity has been isolated by comparing exchange-coupled and dipolar-only quasicrystal nanobar lattices. In that study, Ni$\chi_n=\pm1$3Fe$\chi_n=\pm1$4 bars of thickness $\chi_n=\pm1$5, width $\chi_n=\pm1$6, and lengths $\chi_n=\pm1$7, $\chi_n=\pm1$8, or $\chi_n=\pm1$9 were fabricated on Penrose and Ammann tilings in fully interconnected, partially connected, and disconnected versions. Only exchange-coupled lattices show aperiodicity-specific collective phenomena, non-stochastic switching, cooperative reversal, and extended domains linked to flux-closure loop formation and low vertex charges; dipolar-only lattices retain narrow-linewidth reprogrammable modes in minor loops but lack cooperative avalanches and extended domains (Bhat et al., 2022).

Taken together, these 2D studies show that quasiperiodicity can manifest either as exchange- and contrast-driven gap formation in continuous media or as rotationally organized mode families, self-biasing, and field-programmable microstates in nanobar networks. This suggests that “quasicrystalline magnonics” is not tied to a single microscopic mechanism, but to deterministic aperiodic order acting through whatever coupling channels dominate a given geometry.

4. Modeling strategies and spectral diagnostics

The standard dynamical framework is the linearized Landau–Lifshitz or Landau–Lifshitz–Gilbert equation. In the Co/Py Fibonacci-strip study, damping is neglected and the dynamics are governed by

$n$0

with $n$1 and $n$2 (Mieszczak et al., 2022). For damping studies, the linearized LLG with spatially varying $n$3 is used, and finite-element eigenproblems return complex frequencies $n$4, with $n$5 defining the lifetime $n$6 (Rychły et al., 2019).

Three numerical approaches dominate. First, the plane-wave method is applied to large periodic supercells that approximate quasiperiodic order; this was used both for 1D Fibonacci strips and for 2D Penrose tilings (Mieszczak et al., 2022). Second, finite-element calculations in COMSOL were used for periodic and quasiperiodic Py/Co wire arrays, including exchange and magnetostatic fields and interface boundary conditions requiring continuity of the dynamical magnetization and of $n$7 (Rychły et al., 2019). Third, OOMMF-based micromagnetics were used to interpret angle-dependent spectra and field-dependent mode families in artificial quasicrystals (Bhat et al., 2018).

Spectral analysis relies heavily on the integrated density of states,

$n$8

or equivalently on staircase-like cumulative mode counts. In large 1D approximants, $n$9 behaves like $G_{m,n}=(2\pi/a_0)(m+n\tau)$0, plateaus identify gaps, and near-plateau clustering indicates van Hove singularity-like behavior at gap edges (Mieszczak et al., 2022). In Penrose films, recurring IDOS plateaus across $G_{m,n}=(2\pi/a_0)(m+n\tau)$1 to $G_{m,n}=(2\pi/a_0)(m+n\tau)$2 supercells demonstrate that the gaps are rooted in quasiperiodic long-range order rather than supercell artifacts (Rychły et al., 2017).

Localization diagnostics vary across studies. The phasonic-disorder work uses

$G_{m,n}=(2\pi/a_0)(m+n\tau)$3

with uniform profiles giving $G_{m,n}=(2\pi/a_0)(m+n\tau)$4 and strongly localized profiles giving large negative values (Mieszczak et al., 2022). The damping study introduces the concentration factor $G_{m,n}=(2\pi/a_0)(m+n\tau)$5, which tracks how strongly a mode resides in low-loss Py and therefore how effectively the composite suppresses damping (Rychły et al., 2019). Standard alternatives such as the inverse participation ratio are also explicitly discussed in the literature (Mieszczak et al., 2022).

5. Localization, phasons, damping, and transport

Intrinsic localization is one of the defining physical consequences of quasiperiodicity. In the defect-free 377-strip Fibonacci Co/Py approximant, multiple gaps and near-edge clusters appear in $G_{m,n}=(2\pi/a_0)(m+n\tau)$6, and critically localized bulk modes occur naturally near gap edges. Their profiles are not exponentially localized to a point but exhibit quasi self-similar envelopes and enhanced amplitude on recurring local motifs (Mieszczak et al., 2022). In the Penrose-film study, modes localize on selected Ni disks forming pentagons, on doublets of Ni disks, or in Fe-matrix voids enclosed by pentagonal disk clusters, depending on frequency relative to major gaps (Rychły et al., 2017).

Phasons are structural degrees of freedom unique to quasicrystals. In the 1D Fibonacci setting, phasonic rearrangements correspond to local swaps $G_{m,n}=(2\pi/a_0)(m+n\tau)$7, and static phasonic defects were modeled as uncorrelated site-dependent perturbations of the phase parameter, $G_{m,n}=(2\pi/a_0)(m+n\tau)$8 with $G_{m,n}=(2\pi/a_0)(m+n\tau)$9. Disorder levels $\Delta\phi=\pi$0, $\Delta\phi=\pi$1, and $\Delta\phi=\pi$2 were explored over ensembles of $\Delta\phi=\pi$3 random configurations. Even at $\Delta\phi=\pi$4, new in-gap modes appear, especially in narrower gaps; at $\Delta\phi=\pi$5 and $\Delta\phi=\pi$6, gaps progressively fill and many small gaps vanish, while the largest gap remains more robust (Mieszczak et al., 2022). The widest gaps host defect-induced localized modes near $\Delta\phi=\pi$7 and $\Delta\phi=\pi$8, many of them concentrated on “double Py” motifs created by phasonic flips (Mieszczak et al., 2022).

Damping and lifetime in quasiperiodic bi-material structures are governed not only by the material fractions but by the mode profiles. In Py/Co arrays, $\Delta\phi=\pi$9 and $B \to A$00, so an amplitude-weighted effective damping $B \to A$01 explains why modes concentrated in Py live longer. Lifetimes rise from the low-frequency metamaterial limit to a maximum near $B \to A$02, then fall at higher frequencies; within each magnonic band, $B \to A$03 increases with frequency, and quasiperiodic localization yields more modes near the lifetime maximum than in periodic arrays (Rychły et al., 2019).

A distinct but related transport picture emerges in the Fibonacci XY-chain analysis of critical magnons. There, the middle-energy wavefunction obeys $B \to A$04, the transmittance is

$B \to A$05

and the set of distances with perfect transmission $B \to A$06 is self-similar with Hausdorff dimension $B \to A$07. For system size $B \to A$08, the two-magnon decay rate $B \to A$09 is maximal in the periodic limit and monotonically suppressed as $B \to A$10 increases, despite an increase in the two-particle DOS from spectral flattening and fractal minibands (Jeon et al., 2022). This suggests that critical, intermediate localization can simultaneously reduce interaction-induced decay and preserve nontrivial transmission pathways.

6. Comparison with periodic systems, reconfigurability, and outlook

Compared with periodic magnonic crystals, quasiperiodic systems offer more and narrower band gaps, denser spectral textures, richer localization patterns, and unconventional rotational symmetries. Periodic crystals support a few gaps at conventional zone boundaries and extended Bloch waves except near deliberate defects, whereas Fibonacci quasicrystals support many gaps and critically localized bulk states even without added defects (Mieszczak et al., 2022). At the opposite extreme, when quasiperiodic order is randomized strongly enough, Bragg peaks disappear and the spectrum approaches that of an averaged homogeneous film (Mieszczak et al., 2022).

Reconfigurability is especially prominent in artificial quasicrystal nanobar lattices. Angle-dependent spectroscopy shows that mode branches emerge from distinct bar-orientation families and can be selected by field azimuth; in the reversal regime, resonances disappear and reappear systematically as magnetic subgroups switch, yielding reprogrammable mode sets (Bhat et al., 2018). In connected Penrose and Ammann lattices, exchange coupling stabilizes low-charge, loop-rich microstates and enables non-stochastic switching, cooperative reversal, and mode splitting near $B \to A$11; in disconnected lattices, sharp reprogrammable modes remain, but the reversal is fragmented and lacks extended domains (Bhat et al., 2022).

These properties translate directly into device concepts already articulated in the literature. Fibonacci quasicrystals support fine-grained filtering, frequency-selective extinction in both main gaps and mini-gaps, and mode-dependent localization that can switch spin-wave amplitude between spatially separated outputs (Lisiecki et al., 2018). Phason engineering can add defect passbands or selectively close smaller gaps while preserving robust wide stop bands (Mieszczak et al., 2022). Penrose channels in low-damping YIG, fractal lattices, Fibonacci-distorted artificial spin ice, and quasiperiodically gated structures have all been identified as routes toward waveguides, demultiplexers, resonators, splitters, and reprogrammable magnonic components (Mehta et al., 2 Sep 2025).

Several open directions recur across the field. Extending phasonic-defect studies to two-dimensional quasicrystals is expected to introduce directional localization, anisotropic defect bands, and tunable angular filtering because dipolar interactions are long range and anisotropic in 2D (Mieszczak et al., 2022). The interplay between quasiperiodicity and topology is repeatedly identified as a natural extension, particularly through gap labeling, generalized Harper-model constructions, and the possibility of topological interface states, although explicit topological invariants are not yet quantified in the cited 1D phason study (Mieszczak et al., 2022). The review literature also notes that no 3D quasiperiodic magnonic crystals have yet been reported, and highlights DMI-enabled nonreciprocity, voltage-controlled quasiperiodic gating, thermal-magnon and magnon-phonon physics, and unified modeling of localization and transport as major unresolved problems (Mehta et al., 2 Sep 2025).