Magnonic Crystal with Strips of Magnetic Nanoparticles: Modeling and Experimental Realization via a Dip-Coating Technique

: In this article, we describe a magnonic crystal formed by magnetite nanoparticles. The periodic strip-like structure of the nanoparticles was fabricated on the surface of thin yttrium iron garnet single-crystal ﬁlm grown on a gallium–gadolinium garnet substrate via dip-coating techniques. It was shown that such periodic structure induces the formation of the bandgaps in the transmission spectra of magnetostatic surface spin-waves (MSSW). The structure was simulated by the transfer matrix method. Spin-wave detection has been carried out by using a pair of microwave antennas and a vector network analyzer.


Introduction
Magnonic crystals (MCs) and the behavior of spin-waves propagating inside them have attracted significant attention in recent years [1][2][3][4][5]. The different techniques for producing such magnonic devices have been based on chemical etching, metal deposition, ion implantation, and many other methods that can introduce a periodic variation of any magnetic parameter [6][7][8][9][10][11][12][13][14]. Similarly to their phononic and photonic counterparts, the presence of a periodic structure in MCs results in the formation of artificially tailored bandgaps, wherein spin-wave propagation is not permitted. The formation of MCs by a periodic variation in the local temperature at the surface of a ferromagnetic film has also been demonstrated [15]. Up to now, the most studied MC is built from thin yttrium iron garnet (YIG) films with a chemically etched array of parallel grooves on the surface that has been the most popular media to implement MCs due to their particular characteristics and performance [9][10][11]. The theoretical investigation and experimental observation of band gaps have been reported, including current-controlled magnonic crystals [8,16], optically controlled structures [15], one-dimensional magnonic crystals of different magnetic heterostructures [17], MC created by the ion implantation [18], and different kinds of overlayers [19][20][21]. In this study, we show that MC also can be created by the deposition of periodic micro-structured strips of magnetite nanoparticles (NPs).
It has been demonstrated that spin-waves are very sensitive to the boundary conditions of the interfaces of magnonic waveguides [19,22]. Recently, alternative devices such as magnonic gas analyzers based on the combination of thin layers of magnetic nanoparticles deposited on YIG films have also been demonstrated [23,24]. These magnonic gas sensors take advantage of the high sensitivity of magnetostatic surface waves (MSSW) with respect to the changes in magnetic permeability of magnetic nanopowders interacting with gases. Such devices are capable of detecting low concentrations of different volatile organic compounds at room temperature with high sensitivity, short response time, and good reproducibility thanks to a very large surface-to-volume ratio of the sensing layer. We suppose that such a magnetic bi-layer opens the possibility of creating tunable and functional spin-wave band-gap MCs similar to the photonic crystals [25], which can be used in spin-wave electronics for biomedical applications. The formation of the bandgaps in the transmission spectra of magnetostatic surface spin-waves and the possibility of tuning frequency, width, and depth of these bandgaps enable a set of potential applications of periodic structures with sensitive magnetic nanoparticles. The technological developments of MC devices with magnetic nanoparticles could be an essential tool for spin-waves research, such as band-pass or notch filters, single-mode waveguides that can increase transmission efficiency, or bending waveguides with suppressed energy leakage similar to what has been obtained in photonic waveguides [25].
In this paper, we report a transfer matrix approach and experimental results on the bandgaps in the transmission spectra of MSSW propagated in a magnonic delay line containing MC that was fabricated by the deposition of periodic micro-structured magnetite nanoparticles (NPs) via the dip-coating technique. The structure was analyzed by the transfer matrix method developed in references [9,11,15]. This approach has been proved to provide good coincidence with experimental data both for MC with etched grooves on the YIG surface [9][10][11] and for MC with smooth thermomagnetic grating [15]. It should be noted that the method is quite universal and can be applied to any kind of physical wave systems. Here, in particular, it describes the periodic magnonic structure as a longitudinal sequence of microwave transmission lines with different parameters, involving complex reflection and transmission coefficients at the interfaces between them. In our case, NP strips induce a significant local decrease in the effective bias magnetic field (H) in YIG in the regions where they are deposited and, consequently, modify the propagation constant of MSSW (i.e., we consider only the static effects of NP on MSSW). In turn, this results in the appearance of the bandgaps in MSSW transmission spectra. The size of the deposited particles should be in the range of nanometers in order to obtain the NPstrips' border roughness below the minimum excited MSSW wavelength. We suppose that the described magnonic crystal possesses several advantages since the "magnetic contrast" between segments in MC can be controlled by the bias magnetic field, considering that the magnetization (M) of the magnetite NP layer shows practically a linear dependence on H, up to values of M of ≈ 100 kA/m [26].

Propagation Matrix
Following the schematics in Figure 1, spin-wave transmission through MC is described in terms of a propagation matrix in each section of the MC and a reflection (o transmission) matrix at the boundaries between the sections. Each period of the MC consists of two sections indicated in Figure 1b: (1) a region with magnetite NPs deposited on the surface of YIG film and (2) a section of plane YIG film (Plane Film-PF) with only air on the surface. Within these sections, MSSW possesses different propagation constants (k). As we mentioned above, in this approach, we consider that the presence of NPs on the surface of the YIG film modifies the effective H inside of YIG; hence, this changes the MSSW wavenumber in region (1) shown in Figure 1b.

Reflection Matrix
Since the presence of NPs modifies MSSW propagation constant, the waves will be partially reflected when they pass through the interface between the different sections. This reflection at the interface can be described by the following matrices: where is the reflectivity, and is the direction index. = +1 is used for the edge where the spin-wave is coming from the PF section of the film towards the NP section, and = −1 is for the edge where the wave is coming from the NP strip towards the PF section.
Meanwhile, the structure has been treated as a periodic sequence of microwave transmission strips surrounded by different environments. It is reasonable to define Γ in terms of a difference in the "characteristic impedance" of the YIG film in the different regions. Assuming that the influence of the material deposited on the surface of the YIG film can be considered as a variation of the film's effective inductance, can be written as follows: as refs. [9][10][11]15]. Finally, by using the propagation and transmission matrix properties, the propagation through a complete period could be obtained by multiplying matrices = [ • • • ], and, for the complete MC with periods, the total matrix is given by the product of transfer matrices . The matrix contains the information about the two steps necessary for the computation analysis of NPs regions (as is shown by the green dashed line in Figure 1b). Then, where , is the matrix associated with the first step of length = , , and , is the matrix associated with the second step of length = , . Finally, the power transmission coefficient of the magnonic crystal can be determined as = 1 | | ⁄ = 1 | | ⁄ [11,15]. The matrices that describe the MSSW phase (the propagation matrix) either in the PF regions or in the NP regions are given by the following [11,15].
Here, j could be either NP or PF, L j is the length of the corresponding region, and k j and k j are the spin-wave wavenumber and spatial damping rate, respectively. The spin-wave wavenumber for MSSW is calculated by using the following [27]: where k NP and k PF will have different values in each section of the MC and depend on ω 0 j . For PF regions, ω 0PF = 2π|γ|H 0 , where H 0 is the applied magnetic bias field and γ the gyromagnetic ratio. For NP regions (bi-layer), ω 0NP,i = 2π|γ|H 0 ξ i , where the parameter ξ i is introduced to describe the changes coming from the perturbation of H inside the YIG waveguide caused by the magnetite NPs. Sub-index i is used to distinguish the two sections of the dashed strip in Figure 1b, i = 1 to the first section, and i = 2 to the second one. The thickness of the plain YIG film is given by d and ω M = 2π|γ|M s , where M s is the saturation magnetization of YIG. For the evaluation of parameter ξ i , we have numerically simulated the structure shown in Figure 1 by using COMSOL ® Multiphisics software (5.6, COMSOL Inc., Burlington, MA, USA). The result for the effective bias magnetic field (H 0eff ) in YIG film for the section containing NP is the following: for the NP layer magnetization for M = 300 kA/m, H 0 − H 0eff = 4.3 Oe, the saturation magnetization of NP is M s ≈ 410 kA/m (the experimental M(H) can be found in [26]). This value of the magnetic field was used as the reference for calculating fitting parameter ξ i .
In order to calculate the spatial damping rate in each region, we consider the spinwave loss as a result of intrinsic magnetic damping only, which provides us with the following: k j = 2π|γ|∆Hζ j / v g j , where ∆H is the ferromagnetic resonance half-power linewidth. The spin-wave group velocity, v g j = ∂ω/∂k j , for MSSW can be calculated as in [27], and the regions are given by the following.
Parameter ζ j is introduced to consider the losses due to the presence of magnetite NPs on the surface. For PF regions ζ PF = 1 and for NP regions, ζ NP > 1. This means that nanoparticles result in more significant damping. This accounts for the larger contribution of two-magnon scattering processes in the regions with NP [9].

Reflection Matrix
Since the presence of NPs modifies MSSW propagation constant, the waves will be partially reflected when they pass through the interface between the different sections. This reflection at the interface can be described by the following matrices: where Γ is the reflectivity, and ν is the direction index. ν = +1 is used for the edge where the spin-wave is coming from the PF section of the film towards the NP section, and ν = −1 is for the edge where the wave is coming from the NP strip towards the PF section. Meanwhile, the structure has been treated as a periodic sequence of microwave transmission strips surrounded by different environments. It is reasonable to define Γ in terms of a difference in the "characteristic impedance" of the YIG film in the different regions. Assuming that the influence of the material deposited on the surface of the YIG film can be considered as a variation of the film's effective inductance, Γ can be written as follows: Γ = |k NP − k PF |/(k NP + k PF ) as refs. [9][10][11]15].
Finally, by using the propagation and transmission matrix properties, the propagation through a complete period could be obtained by multiplying matrices T = [T PF ·T + ·T NP ·T − ], and, for the complete MC with N periods, the total matrix is given by the product of N transfer matrices T. The matrix T NP contains the information about the two steps necessary for the computation analysis of NPs regions (as is shown by the green dashed line in Figure 1b). Then, T NP = T NP,1 ·T + ·T NP,2 ·T + ·T NP,1 , where T NP,1 is the matrix associated with the first step of length L j = L NP,1 , and T NP,2 is the matrix associated with the second step of length L j = L NP,2 . Finally, the power transmission coefficient of the magnonic crystal can be determined as P tr = 1/ T MC  Figure 1 shows a schematic view of the delay structure used to measure the amplitudefrequency characteristic of MSSW in the magnonic crystal. The static external field H = 770 Oe was applied parallel to the surface of the YIG film (y-axis) and perpendicular to the spin-wave propagation direction (x-axis). The field was set up at a nominal value of 770 Oe. In order to excite (input) and detect (output) the spin-waves, a pair of 20 µm diameter gold wire microwave antennas were placed directly on the surface of the YIG film with a separation of 6 mm so that the patterned array of NPs was entirely localized between them. A vector network analyzer connected to these antennas was used to measure the amplitude-frequency characteristic of the magnonic crystal.

Experiment
The amplitude-frequency characteristic was measured by using a vector network analyzer connected to the input and output antennas. Figure 2a shows experimental transmission spectra of MSSW, measured for NP-free YIG film (blue thin line) and for the MC sample (black thick line). Figure 2b shows the simulated transmission curve for the fabricated MC. The values used for the simulation are as follows: N = 12, d = 6.3 µm, L PF = 340 µm, L NP,1 = 5 µm, and L NP,2 = 20 µm in such a manner that L NP = L NP,1 + L NP,2 + L NP,1 = 30 µm, as shown in Figure 1b. In order to obtain the best coincidence between the experimental and calculated curves, we used the following fitting parameters: ξ 1 = 0.985, ξ 2 = 0.96, and ζ NP = 3.5.
to measure the amplitude-frequency characteristic of the magnonic crystal.
The amplitude-frequency characteristic was measured by using a vector network analyzer connected to the input and output antennas. Figure 2a shows experimental transmission spectra of MSSW, measured for NP-free YIG film (blue thin line) and for the MC sample (black thick line). Figure 2b shows the simulated transmission curve for the fabricated MC. The values used for the simulation are as follows: = 12, = 6.3 μm, = 340 μm , , = 5 μm, and , = 20 μm in such a manner that = , + , + , = 30 μm, as shown in Figure 1b. In order to obtain the best coincidence between the experimental and calculated curves, we used the following fitting parameters: = 0.985, = 0.96, and = 3.5.

Discussion
As observed in Figure 2a, there are two frequency regions with high transmission losses centered at the frequencies 4.15 GHz and 420 GHz, which can be interpreted as the magnonic bandgaps. The presence of small oscillations in the transmission curve denotes the interference between the spin waves and the electromagnetic waves induced in the output antenna due to a direct parasitic coupling between the antennas. The experimental and calculated decrease in the deepness of the band notches takes place due to the increase in MSSW attenuation as the frequency increases.

Discussion
As observed in Figure 2a, there are two frequency regions with high transmission losses centered at the frequencies 4.15 GHz and 420 GHz, which can be interpreted as the magnonic bandgaps. The presence of small oscillations in the transmission curve denotes the interference between the spin-waves and the electromagnetic waves induced in the output antenna due to a direct parasitic coupling between the antennas. The experimental and calculated decrease in the deepness of the band notches takes place due to the increase in MSSW attenuation as the frequency increases.
Although the calculated curve in Figure 2b shows qualitative agreement with the experiment, some quantitative differences between them are observed. It can be caused by the above-mentioned parasitic coupling between the antennas and some imperfections in the periodicity of the deposited NP strips. Moreover, in the theoretical model, the interfaces between the MC sections were approximated as straight borders, which, in reality are smooth. In other words, NP strips create a potential well in YIG with a smooth spatial profile that still must be investigated. Moreover, in the model, we did not evaluate a possible resonant interaction of MSSW with the ferromagnetic resonance (FMR) response of NP. In the present study, the interaction between MSSW field and NP spin excitations was only described by loss parameters. Nevertheless, the obtained results show that the structure can provide a magnonic bandgap with acceptable characteristics. We suppose that the proposed NP based magnonic crystal can be a good candidate for sensing applications, taking into account that the dip-coating technique allows one to create periodical patterns from a different kind of magnetic nano powder, which can be chemically functionalized to interact and detect some biomaterials or gases, such as the phononic crystal of photonic crystal counterparts [28][29][30][31][32].
From the technological point of view, the dip-coating technique can be considered as an alternative tool in the fabrication of MCs mainly because it has the micrometric reso-lution needed for this kind of structure [33]. Dip-coated deposition depends on interface and parameters used in the fabrication process, such as substrate temperature, solvent or composition of the nanofluid, and finally the shape of the nanostructures [34]. This fabrication method of magnonic structures has multiple advantages, which are led by the possibility of inducing significant changes in the amplitude-frequency characteristic of the magnonic waveguides. The design of magnonic periodic structures with a wide range of different sizes, shapes, and properties of magnetite nanoparticles could be used to develop high-quality resonant sensors with the extraordinary capability to reuse the YIG strip by removing the nanoparticles from its surface without producing any permanent changes. One possible application of MCs is as components for processing analog and digital information [35]. Furthermore, since MCs are sensitive to the external magnetic field, they have been demonstrated as extremely sensitive magnetic field sensors at room temperature [36]. The MCs presented here are of great interest for research from a wide range of fields, including magnetic fluids, catalysis, biotechnology, biomedicine, biosensing applications, magnetic resonance imaging, data storage, and environmental remediation [37]. The applications in biomedicine stand out and include tumor tissue targeting, local hyperthermia effect, drug delivery, and magnetic resonance imaging diagnostics [38]. The great potential for use in medicine is due to their biocompatibility, biodegradability, easy synthesis, and ease with which they may be tuned and functionalized for specific applications [39].

Materials and Methods
The magnonic structure was fabricated by using a uniform long and narrow YIG film strip epitaxially grown in the (111) crystallographic plane on a gallium gadolinium garnet (GGG) substrate of thickness 500 µm. The YIG strip was 35 mm long, 1.7 mm wide, and 6.3 µm thick. The high quality of the YIG film is highlighted by the ferromagnetic resonance half-power line width (∆H), which is about 0.5 Oe at 5 GHz. The patterned structure was carried out by controlled deposition on magnetite nanoparticles on the sample's surface quality via dip coating. Commercial magnetite nanoparticles with a diameter of 30 ± 2 nm were used (777408, Sigma Aldrich, St. Louis, MO, USA). Figure 3a shows the experimental and schematic view of the setup mounted to deposit NPs strips on YIG film. A magnetite solution of 8 × 10 −4 % weight per volume diluted in 1-propanol was sonicated for one hour to minimize particle aggregation [40]. For homogeneity and reproducibility, a motorized stage was used for controlling regular immersion and retrieval, while the choice of solvent enhanced wetting and evaporation in the meniscus region [41]. In order to obtain gratings, the translation stage in the immersion was scanned to deposit a series of parallel stripes that comes from the meniscus of the magnetite solution of approximately 30 microns wide and less than 1.7 mm (the width of the YIG film), leaving a non-excess stripe of 340 widths in between them. The period was 370 µm (30/340) and 30 µm per region, with magnetite nanoparticles (NP) deposited on the surface of YIG film and 340 µm for a section of plane YIG film. The motorized stage was moved with different displacements steps corresponding to the 30/340 period. The 30 µm region with NPs was created with six displacement steps of 5 µm, run at a stage speed of 0.01 mm·s −1 , and each step had a dwell time of one minute. For the 340 µm section, the motorized stage was moved using one step, run at a stage speed of 1 mm·s −1 . Figure 3b presents an image of the top view of a sample fabricated in this manner. One side of the substrate was carefully cleaned, leaving the strips of the nanoparticles only on one side. From Figure 3b, the NPs strips are approximately straight in the middle of the YIG film, and there is a discontinuous of such strip close to the lateral borders. Such discontinuous phenomena are due to the surface tension of the dip-coating solvent in a combination of the inhomogeneities of the border at the YIG waveguide surface. The patterned array characterized here corresponds to twelve NPs strips, meaning that MC had a lattice constant of 370 µm. this manner. One side of the substrate was carefully cleaned, leaving the strips of the nanoparticles only on one side. From Figure 3b, the NPs strips are approximately straight in the middle of the YIG film, and there is a discontinuous of such strip close to the lateral borders. Such discontinuous phenomena are due to the surface tension of the dip-coating solvent in a combination of the inhomogeneities of the border at the YIG waveguide surface. The patterned array characterized here corresponds to twelve NPs strips, meaning that MC had a lattice constant of 370 μm.
(a) (b) Figure 3. (a) Experimental and schematic view of the process for depositing a periodic nanoparticle pattern on the YIG waveguide surface via the deep coating technique. The first strip of NPs is deposited when the holder is at its initial position, and the subsequent strips are deposited as it proceeds upward. (b) Image of a sample fabricated by using the experimental process in (a).

Conclusions
In this study, we have shown the possibility of creating a magnonic crystal by using magnetite nanoparticles that are periodically patterned on the YIG surface. The periodic structure of the nanoparticles was created by using the dip-coating deposition method. The structure is efficient in providing the bandgaps in the transmission spectrum of MSSW. The experimental spectrum qualitatively agrees with the theoretical curve calculated with the help of the transfer matrix method. It is expected that the reported structure can be a good candidate for biosensing applications.

Patents
This section is not mandatory but may be added if there are patents resulting from the work reported in this manuscript.

Conclusions
In this study, we have shown the possibility of creating a magnonic crystal by using magnetite nanoparticles that are periodically patterned on the YIG surface. The periodic structure of the nanoparticles was created by using the dip-coating deposition method. The structure is efficient in providing the bandgaps in the transmission spectrum of MSSW. The experimental spectrum qualitatively agrees with the theoretical curve calculated with the help of the transfer matrix method. It is expected that the reported structure can be a good candidate for biosensing applications.