A magnetoactive metamaterial based on a structured ferrite

Subject and Purpose. The use of spatially structured ferromagnets is promising for designing materials with unique predetermined electromagnetic properties welcome to the development of magnetically controlled microwave and optical devices. The paper addresses the electromagnetic properties of structured ferrite samples of a di ﬀ erent shape (spatial geometry) and is devo-ted to their research by the method of electron spin resonance (ESR). Methods and Methodology . The research into magnetic properties of structured ferrite samples was performed by the ESR method. The measurements of transmission coe ﬃ cient spectra were carried out inside a rectangular waveguide with an external magnetic ﬁ eld applied. Results. We have experimentally shown that over a range of external magnetic ﬁ eld strengths, the frequency of the ferromagnetic resonance (FMR) of grooved ferrite samples (groove type spatial geometry) increases with the groove depth. The FMR frequency depends also on the groove orientation relative to the long side of the sample. We have shown that as the external static magnetic ﬁ eld approaches the saturation ﬁ eld of the ferrite, the FMR frequency dependence on the external static magnetic ﬁ eld demonstrates “jump-like” behavior. And as the magnetic ﬁ eld exceeds the ferrite saturation ﬁ eld, the FMR frequency dependence on the groove depth gets a monotonic character and rises with the further growth of the ﬁ eld strength. Conclusion. We have shown that the use of structured ferrites as microwave electronics components becomes reasonable at magnetic ﬁ eld strengths exceeding the saturation ﬁ eld of the ferrite. At these ﬁ elds, such a ferrite o ﬀ ers a monotonically increasing dependence of the resonant frequency on the external magnetic ﬁ eld and on the depth of grooves on the ferrite surface. Structured ferrites are promising in the microwave range as components of controlled ﬁ lters, polarizers, anisotropic ferrite res-onators since they can provide predetermined e ﬀ ective permeability and anisotropy. 12

It is well known that magnetoactive metamaterials are prospective base for frequency fi lters. In addition, magnetoactive metamaterials can fi nd application in computer technology, microwave and optical engineering. The main advantage of the abovementioned fi lters is wideband-frequency tunability. The fi lters based on magnetically con-trolled photonic crystals with their magnetoactive elements constructed of ferromagnetic materials are eff ective in the microwave and optical ranges [1][2][3][4][5][6][7][8].
However, modern electronics of high and extremely high frequencies requires brand-new fi lters where application of classical ferrodielectrics A magnetoactive metamaterial based on a structured ferrite is problematic. It is well known that electrical and magnetic properties of ferrodielectrics are determined by the crystal structure. Any modifi cation in the crystal structure leads to a change in its properties. The development of such materials (magnets) requires large costs.
An alternative solution to the problem is the use of spatially structured ferromagnets, such as magnetoactive metamaterials. The electromagnetic properties of a magnetoactive metamaterial depend not only on the properties of the metamaterial constituents but on their spatial arrangement, too. On this basis, the development of materials with predetermined electromagnetic properties (constitutive parameters -permittivity and permeability) is possible. In this case, it is natural to turn attention to magnetoactive metamaterials under electron spin resonance (ESR) conditions. For this, a magnetic fi eld of the kOe order should be applied to the magnetoactive metamaterial [9,10] for the microwave range. The ESR area is the most attractive as the frequency dispersion of magnetic permeability is at its maximum there [9,10]. This structuring contributes to the formation of spatial electrodynamic resonances appearing in the sample and thus greatly controls formation of the effective magnetic permeability of the entire structure. Besides, this structuring modifi es demagnetizing factors [9] and substantially transforms the electromagnetic properties (eff ective constitutive parameters) of the sample. So, the magnetoactive metamaterial research using the ESR-method notable for its great sensitivity to magnetic structural changes is really necessary in the frequency range of the metamaterial prospective application.
On this basis, the aim of the work is the ESR-method research into the electromagnetic properties of structured ferrite samples of a diff erent shape (spatial geometry).

Structures under study and theoretical approach.
For the magnetoactive metamaterial research, six parallelepipedic samples of ferrite (1SCh4 brand, 4  M s  4 750 Gs, g  2.14) are used. Each sample is of the size a  b  c   7.2  3.4  1.0 mm. By the term "spatial geometry" we mean parallel grooves cut on one side of the sample so that the sample represents a comb structure ( Fig. 1, a, b).
All the samples are separated into two series (see the Table). The fi rst series samples have grooves parallel to the long side (a) of the sample (see Fig. 2, a). The second series samples have grooves perpendicular to the long side (a) of the sample (see Fig. 2 To have samples with a structured surface and avoid thermally-induced demagnetization, we resorted to laser ablation using a femtosecond laser system Spitfi re Pro 35F-XP, Spectra-Physics. The laser system generates 50 fs pulses at a 1 kHz repetition rate, the center wavelength is 800 nm. The resulting laser beam with an average power of 1.4 W is focused on the sample surface using a 50 mm lens. The scanning velocity along the surface is 5 mm/s. The groove period is 0.4 mm, the ridge width is 0.2 mm. The three groove depths are obtained using three dedicated positioning programs. In the case of collinear magnetic materials, the ESR phenomenon observed for ferromagnetic materials is commonly called [9,11] the Ferromagnetic Resonance (FMR) phenomenon described by the motion equation where  is the gyromagnetic ratio, M  is the magnetization of the sample unit volume, and eff H  is the total magnetic fi eld with the components [9] , where H x , H y , H z are the total fi eld components in the x, y, z directions, respectively, with both external and internal fi elds taken into account, 0 H  is the external static magnetic fi eld directed along the is the high-frequency alternating magnetic fi eld directed along the x-axis, and M x , M y , M z are the components of the alternating magnetization .
M  In (2), N x , N y , N z are the components of the eff ective demagnetizing factor N [9, 10] of the sample. N governs the FMR frequency f FMR and directly depends on the shape of the sample. Since the samples diff er in their groove depths, the demagnetizing fi elds of the structured and unstructured ferrite samples of the same size should differ, too.
A numerical modelling was made in order to determine the FMR frequency of the studied samples, with demagnetizing factors and groove depth d taken into account.
2. Experiment and data analysis. Each sample is placed inside a metal rectangular hollow waveguide. Its cross-section is A  B 23  10 mm, the main waveguide mode is TE 10 . The external static magnetic fi eld 0 H  is directed along the OY axis ( Fig. 2) [6,7,11,12]. The waveguide is positioned so that the sample is located between the electromagnet poles to satisfy the FMR-conditions . A Vector Network Analyzer N5230A is connected to the waveguide by coaxial cables, the spectrum registration is performed in the frequency domain mode at several values of the static magnetic fi eld strength H 0 . Fig. 3, a, b shows the experimental f FMR frequency dependences on the external magnetic fi eld H 0 for the samples located as in Fig. 2 (the insets in Fig. 3, a, b). As seen from Fig. 3, a, b, the resonant frequency f FMR increases with H 0 for all the   Fig. 4. Let us analyze these dependences in more detail. According to Fig. 4, the dependences can be conventionally separated into two -nonmonotonic (grey colored) and monotonic (white colored) -areas. The nonmonotonic behavior can be explained by the fact that the ferrite (in the "grey" area in Fig. 4) is unsaturated, H 0  H sat , where H sat is the saturation fi eld of the ferrite sample [9,10]. When H 0  H sat , the ferrite passes to the saturated state. As a result, the dependence f FMR  f FMR (d ) becomes monotonic (the "white" area in Fig. 4). In other words, when H 0  H sat and the magnetic saturation occurs in the structured ferrite, the magnet turns out to be collinear. In this case, an internal fi eld is formed inside it, and the magnetic fi eld value substantially exceeds the surface anisotropy fi eld caused by the structuring. In this case, a groove depth change acts as a small magnetic-fi eld perturbation and provides a monotonic f FMR  f FMR (d ) dependence. When the magnet is unsaturated, H 0  H sat (Fig. 4) the internal fi eld is basically determined by the surface anisotropy fi eld caused by the magnet structuring. So, the groove depth d largely perturbs the f FMR , leading to a nonmonotonic nonlinear dependence f FMR  f FMR (d ), which is typical for any large perturbation.
It should be noted that the FMR frequency is affected by the groove arrangement relative to the direction of the external static magnetic fi eld, 0 H  . The fi eld 0 H  is directed along the OY axis, i.e. parallel to the long side a of the sample. Thus, the value of the FMR frequency for the samples with grooves parallel to the long side a of the sample (series 1, Fig. 2, a) is greater than for the samples with grooves perpendicular to the side a (series 2, Fig. 2, b), compare the FMR frequency values for the curves with d  0.2 mm and d  0.4 mm in Figs. 3, a and b. The observed phenomenon is caused by the action of the demagnetizing fi elds N x M x ; N y M y ; N z M z in the given structured ferrite sample [9].
Note that in Figs. 3, a, b, there are "jumps" on the curve f FMR  f FMR (H 0 ) (experiment) near H 0  2 500 Oe. We suggest that the observed "jumps" are caused by the passage from one mode arising in the given electrodynamic structure to ano-ther one. This eff ect occurs when the wavelength is comparable with the sample size. A numerical calculation using the model described by formula (1) has been performed. Fig. 3, c presents the : a -sample orientation inside a waveguide according to Fig. 2, a (series 1, experiment), b -according to Fig. 2, b  (series 2, experiment), and c -according to Fig. 2, a (series 1 magnetic fi eld (marked by the solid-line square in Fig. 3, a, b). Also, Fig. 3, c indicates the passages from "Mode 1" to "Mode 2". To confi rm the assumption above, the spatial distribution of the h x -component of the electromagnetic fi eld in sample No. 3 is presented in Fig. 5. A numerical modeling was carried out for the mode arising at frequency f FMR  13.824 GHz and magnetic fi eld intensity H 0  1 900 Oe, the sample location is as in Fig. 2, a. From Fig. 5, it can be seen that the wavelength  in the sample approximately equals the length of its short side b at the given frequency (the distance between the two dashed lines).
That is, indeed, the "jumps" on the dependences f FMR  f FMR (H 0 ) in Fig. 3 are observed when the geometric dimensions of the sample are comparable with the wavelength as was assumed earlier.
In other words, at the given magnetic fi elds, the transitions from one spatial mode (Mode 1) excited in the studied sample to another (Mode 2) (Fig. 3, c) occur in the form of "jumps" on the dependences f FMR  f FMR (H 0 ) as observed in the graphs in Fig. 3. That the electromagnetic fi eld spatial distribution shown in Fig. 5 is similar to the electromagnetic fi eld distributions for the other samples at the other resonant frequencies (near the "jumps") and other groove depth d values confi rms this fact.
Conclusions. We have established that: 1. The FMR frequency of the structured ferrite sample depends on the depth of grooves on the ferrite surface. Namely, the FMR frequency increases with the groove depth over some range of the external magnetic fi eld.
2. The FMR frequency of the investigated structured ferrite depends on the groove orientation relative to the long side of the sample and to the direction of the external magnetic fi eld 0 .
H  When the grooves are perpendicular to the long side of the sample, the FMR frequency is lower than when the grooves are parallel to it due to the infl uence of the demagnetizing fi elds of the structured ferrite.