Magnons in a Quasicrystal: Propagation, Localization and Extinction of Spin Waves in Fibonacci Structures

Magnonic quasicrystals exceed the possibilities of spin wave (SW) manipulation offered by regular magnonic crystals, because of their more complex SW spectra with fractal characteristics. Here, we report the direct x-ray microscopic observation of propagating SWs in a magnonic quasicrystal, consisting of dipolarly coupled permalloy nanowires arranged in a one-dimensional Fibonacci sequence. SWs from the first and second band as well as evanescent waves from the band gap between them are imaged. Moreover, additional mini-band gaps in the spectrum are demonstrated, directly indicating an influence of the quasiperiodicity of the system. The experimental results are interpreted using numerical calculations and we deduce a simple model to estimate the frequency position of the magnonic gaps in quasiperiodic structures. The demonstrated features of SW spectra in one-dimensional magnonic quasicrystals allows utilizing this class of metamaterials for magnonics and makes them an ideal basis for future applications.


Introduction
Magnonic crystals are periodically modulated magnetic structures, which enable tailoring of the magnonic band structure and formation of allowed and forbidden bands in the spin wave (SW) spectrum. 1,2 Additionally, the SW spectrum can easily be modified by an external magnetic field or a change of the magnetization configuration in the structure. 3 This offers fine tuning and re-programmability, 4 which are desirable properties for potential applications. 5 Apart from periodic modulations, defects in the regular structure can also alter SW propagation. Defects can cause the appearance of localized SW modes with their amplitude confined to the disturbed area.
Equally, they can introduce additional magnonic branches at frequencies inside the band gaps of the SW spectrum. [6][7][8][9] In a periodically arranged array of magnetic stripes, a stripe with differing dimensions can be considered and designed as such a defect. Thereby, providing an opportunity for the design of cavity resonators and ultra-narrow band filters. 10,11 Quasicrystals, unlike crystals, do not have periodicity albeit possessing long-range order, which results in a discrete diffraction pattern. 12 They are characterized by more complex dispersion relations than those of periodic systems with an increased number of forbidden band gaps and narrow allowed bands. Spectra of quasicrystals also include localized excitations concentrated in various parts of the self-similar structure, which is a characteristic feature of such aperiodic systems. 13 Previously, these properties have been widely studied in the context of photonics and phononics for one-dimensional (1D) quasicrystals designed by using the Fibonacci sequence. [14][15][16] Quasiperiodicity has also been applied to magnonic systems. [17][18][19][20][21][22][23][24][25][26][27][28] Fundamental theoretical studies used Fibonacci structures comprised of multilayers [17][18][19][20][21] or bi-component stripes 22,23 to investigate the spectrum of exchange or dipolar SWs respectively. However, these hypothetical structures were far from experimental realization. First experimental realizations of quasiperiodicity in magnonics were based on groves in a micrometer thick YIG film. 24 All electrical spin wave spectroscopy measurements indicated an influence of the quasiperiodicity on the SW spectrum. 24 A key experimental realization of flexible magnonic quasicrystals were 2D arrays of Py nanobars in Penrose, Ammann, or Fibonacci arrangement. [25][26][27][28] However, these studies were focused on the magnetization reversal processes and only demonstrated a rich ferromagnetic resonance (FMR) spectrum of these systems, hinting at a more complex SW band structure. [24][25][26][27][28] So far, there has been no study of the microscopic behavior, propagation, nor localization of SWs in magnonic quasicrystals.
Here, we present the direct microscopic observation of propagating SWs in a 1D quasiperiodic magnonic structure formed by thin Py stripes arranged in a Fibonacci sequence. This is achieved by using scanning transmission X-ray microscopy (STXM) that provides ultimate spatial (<20 nm) and temporal (<50 ps) resolution. Propagating modes from the first and second bands were detected with a band gap between them. Furthermore, the existence of a mini-band gap within the first band was demonstrated, showing the influence of the quasiperiodicity on the dispersion relation. We complement the experimental results with calculations that show good agreement with the resonant frequencies and profiles of the investigated modes and allow their interpretation. Moreover, we propose a simple model based on diffraction structure factor calculations, which allows to predict the frequency positions of most magnonic bandgaps in the SW spectra of 1D magnonic quasicrystals.

Sample characteristics
Nanowires (NWs, length L = 10 m) were fabricated by e-beam lithography and subsequent liftoff in a thin Py film (Ni 80 Fe 20 , thickness d = 30 nm) on a Si 3 N 4 (100 nm)/Si substrate with membrane windows for X-ray transmission measurements. 100 µm wide arrays of narrow (W N = 700 nm) and wide (W W = 2W N = 1400 nm) NWs were arranged quasiperiodically using Fibonacci's inflation rule. According to this rule, a sequence of higher order n is determined by the sum of the two previous structures (S n = S n-1 + S n-2 ) as shown in Fig. 1a. To ensure magnetostatic coupling between the NWs an air gap (W G = 80 nm) was introduced between adjacent NWs. 29 For SW excitation a coplanar waveguide (CPW) made from Cu(150 nm)/Al(10 nm) with a 2 m wide signal line was fabricated on top of the structure using direct laser lithography 30  were illuminated under perpendicular incidence by circularly polarized light in an applied inplane field of up to 240 mT that was generated by a set of four rotatable permanent magnets. 31 The photon energy was set to the absorption maximum of the Fe L3 edge to get optimal XMCD contrast for imaging. A lock-in like detection scheme allows sample excitation at arbitrary frequencies at a time resolution of 50 ps using all photons emitted by the synchrotron.

Numerical calculations
To calculate SW spectra we solve the Landau-Lifshitz (LL) equation: where t is time and r is the position vector. Damping has been neglected in the calculations.
( ) is an effective magnetic field, which is assumed to be the sum of three terms: problem, which is solved by using a FEM approach with COMSOL 5.1 to obtain the dispersion relation and profiles of the SWs. For more details concerning this computation, we refer to Ref. 33 . In numerical calculation we used A/m, the exchange constant J/m, and gyromagnetic ratio for Py.  From the measurements, we are able to estimate the decay length () of SWs propagating through the structure, it is from an exponential fit of the spatial amplitude distribution according where A is the SW amplitude and x is the distance from the excitation source. We calculated the decay length to be  = 14±2 m, which is in good agreement with the literature value for Py waveguides. 34 To aid interpretation of the experimental results, we solved the linearized Landau-Lifshitz (LL) equation complemented by the magnetostatic Maxwell equations in the frequency domain, using a finite element method (FEM) approach as described earlier. Thereby, we obtained the resonance frequencies and amplitude distributions of the standing SW oscillations in the system.
In the calculations we assumed an array of infinitely long Py NWs, keeping the thickness, widths and the Fibonacci arrangement (cf. Fig. 1a) from the experiment. The total size of the structure,

Estimation of the band gaps and mini-gaps positions.
In the SW spectrum presented in For a wave with wavenumber k, the Bragg condition may be written as: where h 1 and h 2 are integer numbers. The 1D reciprocal lattice vectors ( ) are defined for the Fibonacci lattice as: 36,37 ( where: is an averaged period of the Fibonacci structure, and √ is the so called golden ratio. The where and are characteristic frequencies corresponding to the applied field and the saturation magnetization (expressed in frequency units) respectively. and are the magnetic permeability of vacuum and the gyromagnetic ratio respectively. For the same parameters as in the numerical simulations, we were able to shift the FMR frequency Apart from the frequencies for which gaps open, we can also estimate the relative widths of these gaps. This can be done with the help of the diffraction spectra of the structure. The diffraction properties of any kind of structure are given by the structure factor ( ) which can be estimated by Fourier transform of the spatial dependence of the material parameter, e.g. the saturation magnetization , in the composite systems. We calculated the discrete Fourier transform of : where is the location of ( ) in the sample in real space and is the discretized reciprocal lattice number. The modulus | ( )| obtained for the structure considered here is presented in Fig. 3c. The height of the bars representing | ( )| is related to the intensity of the Bragg peaks of the scattered waves. The location of Bragg peaks with large intensity corresponds to the frequency gaps opened at given by ( ) , for which the relation ( ) can be found from the dispersion relation (6). The position of the highest peaks from the numerically calculated Fourier spectrum of the quasiperiodic structure strictly coincides with the wavenumbers fulfilling the analytical formula (3).
The procedure described above allowed us to connect the structure of the magnonic quasicrystal with the frequency gaps in the SW spectrum, which are in good agreement with the results of the numerical calculations and experimental data obtained from STXM measurements.

Conclusion
We have experimentally observed propagating SWs in real space and time domain in 1D Fibonacci quasicrystals of dipolarly coupled Py NWs of two different sizes and fully recovered and explained this system using numerical calculations. Thereby, we have demonstrated the existence of propagating SW modes in such quasicrystals, crucial for future magnonic data processing applications. We have shown that SW propagation is not restricted to the longwavelength limit, for which the structure can be considered as an effective medium, but also occurs at higher frequencies, for which the structure's long-range quasiperiodic order is critical.
Additionally, we have experimentally proven the existence of mini-band gaps that are a direct consequence of the collective SW effects in magnonic quasicrystals. Furthermore, a simple analytical method has been derived for the estimation of the magnonic gaps and mini-gaps in the SW spectra of 1D quasicrystals, providing a powerful tool for designing quasiperiodic systems.
The mini-gaps are wide enough to prohibit propagation of SWs despite the finite SW damping, thus, offering usefulness for potential applications in the filtering of microwave signals.
Moreover, these propagating SWs in quasicrystal structures featuring mini-gaps originating from a dense spectrum of diffraction peaks in reciprocal space, offering unprecedented flexibility in the design of effective non-linear processes, 39 which is one of the main challenges in the application of magnonics to transfer and process information. 40