Ferromagnetic Resonance and Antiresonance in Composite Medium with Flakes of Finemet-Like Alloy

nanomaterials

Article Ferromagnetic Resonance and Antiresonance in Composite Medium with Flakes of Finemet-Like Alloy

Dmitry V. Perov and Anatoly B. Rinkevich *

M.N. Miheev Institute of Metal Physics UB RAS, Soﬁa Kovalevskaya St., 18, 620108 Ekaterinburg, Russia; [email protected] * Correspondence: [email protected]

Abstract: Propagation of microwaves is studied in a composite material containing flakes of Fe- Si-Nb-Cu-B alloy placed into an epoxyamine matrix. The theory is worked out, which permits to calculate the coefficients of the dynamic magnetic permeability tensor and the effective magnetic permeability of the transversely magnetized composite. The measurements of magnetic field dependences of the transmission and reflection coefficients were carried out at frequencies from 12 to 38 GHz. Comparison between calculated and measured coefficients were conducted, which show that the calculation reproduces all main features of the resonance variations caused by ferromagnetic resonance and antiresonance. The dissipation of microwave power was calculated and measured. It is shown that the penetration depth of the electromagnetic field increases under antiresonance condition and decreases under resonance.

Keywords: magnetic composites; microwaves; absorption; transmission and reﬂection coefﬁcients

Citation: Perov, D.V.; Rinkevich, A.B. Ferromagnetic Resonance and 1. Introduction Antiresonance in Composite Medium Composite media can have properties which are absent for their composing parts. with Flakes of Finemet-Like Alloy. Composite materials containing metallic particles in the dielectric matrix present important Nanomaterials 2021, 11, 1748. areas of application [1]. The microwave magnetic properties of composites look promising, https://doi.org/10.3390/nano11071748 but, so far, they have not been adequately explored. The laws governing the frequency dispersion in the magnetic composites are discussed in detail [2]. The mixing rules for Academic Editor: Julian Maria the constitutive parameters are presented in the monograph [3]. The complex magnetic Gonzalez Estevez permeability is measured in the frequency range of 0.01–10 GHz for magnetically structured granular systems using the transmission/reﬂection waveguide method [4]. In order to Received: 28 May 2021 secure a high level of electromagnetic screening, the particles in the form of ﬂakes placed in Accepted: 1 July 2021 Published: 2 July 2021 a polymer matrix are used [5]. The composite with Fe16Ni82Mo2 alloy particles shows fairly good shielding results in the frequency range from 2 to 18 GHz [6]. Microwave magnetic

Publisher’s Note: MDPI stays neutral permeability for the composites with carbonaceous particles is studied [7,8]. with regard to jurisdictional claims in Resonance-type phenomena, such as ferromagnetic resonance and antiresonance, published maps and institutional affil- can be observed on microwaves for a ferromagnetic medium placed in a DC magnetic iations. field. Ferromagnetic resonance (FMR), as known, leads to absorption of electromagnetic waves [9]. Phenomenon of ferromagnetic antiresonance (FMAR) has been observed earlier if electromagnetic wave transmits through a thin ferromagnetic metallic film. In this case, the FMAR condition leads to a sharp increase in the transmission of microwaves through a film [10]. This phenomenon occurs providing that the real part of the dynamic permeability Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. equals to zero in a magnetic field lower than the FMR field [9,11]. In paper [12], the This article is an open access article FMAR phenomenon is observed experimentally in a nanocomposite medium. The FMAR distributed under the terms and phenomenon in a composite medium demonstrates a number of specific features. For conditions of the Creative Commons nanocomposites not having the DC conductivity, the skin effect is absent. Instead of an Attribution (CC BY) license (https:// increase in microwave transparency of the sample, in non-conductive nanocomposites, creativecommons.org/licenses/by/ the FMAR usually causes an increase in the reflection coefficient. Nonetheless, some 4.0/). features of antiresonance preserve in non-conductive nanocomposites. In particular, the

Nanomaterials 2021, 11, 1748. https://doi.org/10.3390/nano11071748 https://www.mdpi.com/journal/nanomaterials Nanomaterials 2021, 11, 1748 2 of 21

FMAR realizes only at frequencies above a certain value, which depends on the saturation magnetization of a material. Due to high magnetic permeability, the finemet-type alloys are regarded as a suit- able material for radio- and microwave devices. Improved electromagnetic shielding and absorbing properties are obtained for the flake-shaped and spherical-like form of Fe16Ni82Mo2 alloy particles [6]. The microwave complex refraction coefficient of a composite consisting of flakes of Fe-Si-Nb-Cu-B alloy placed into epoxy resin matrix has been studied [13]. The material under consideration is a dielectric at DC and a lossy dielectric at microwave frequencies in zero magnetic fields. Near the field of ferromagnetic resonance, however, the real and imaginary parts of the complex refraction coefficient are of the same order of magnitude, such as in a conductive medium. The magnetic properties of powdered nanostructured alloy Fe73.5Cu1Nb3Si13.5B9 have been studied in the frequency range 0.2–10.2 GHz [14]. The investigation of magnetic resonances is carried out in this paper for the composites containing flakes of finemet-type alloy at frequencies of the centimeter and millimeter wavebands. These frequency ranges were chosen in order to observe the FMAR phenomenon. The calculation was carried out for magnetic field dependences of the microwave transmission and reflection coefficients. These coefficients were also measured in the frequency range from 12 to 38 GHz. The theory of dynamic magnetic permeability of the composite media was worked out. The complexity of this problem consists, in particular, in the necessity to take into consideration the shape, dimensions and space orientation of ferromagnetic particles. The calculation of the dynamic magnetic permeability is discussed in papers [15,16]. The dependence of the effective permeability on the fraction of magnetic component was calculated. For the periodical magnetic inclusions, averaging was realized for both magnetic field and magnetic induction within the assigned cells [17–19]. A pro- posed cluster model was used in order to determine analytically the effective magnetic permeability of a composite system when this permeability is due to generation of eddy currents in an alternating magnetic field near the percolation threshold [20]. Magnetic field dependence of the relative magnetic permeability was calculated using a nonlinear homogenization approach [21]. A model based on the Landau–Lifshitz–Gilbert equation was developed to calculate the complex permeability of magnetic particles/insulator matrix composite [22]. In this paper, a new method for the calculation of the effective magnetic permeability is offered for composite magnetic materials which contain ferromagnetic particles in the form of ellipsoid with arbitrarily directed axes. Comparison between the results of experiments and theoretical calculations is outlined for transmission and reflection of microwaves. The dissipation of microwave power was calculated and the penetration depth of the wave into the sample was also estimated.

2. Materials and Methods Samples of the composites were prepared of Fe-Si-Nb-Cu-B alloy particles and the epoxyamine matrix. This matrix was chosen due to its moderate dielectric permittivity and ease of preparation of the composite. Metallic particles are in the form of flakes of the finemet-type alloy with the following composition: Fe—80.1%, Si—8.5%, Nb—8.4%, Cu—1.1%, B—1.2%, Cr—0.2%, Mn—0.1%, Ni—0.1% and Co—0.3%. As known, the nanocrystalline finemet-type alloy has high magnetic permeability [14]. The composite material was prepared by mechanical mixing of particles in the epoxyamine matrix. After mechanical mixing, the additional mixing in the ultrasonic bath was undertaken and then hardening of the mixture occurred during several hours. Two series of composite samples were prepared with 15 wt.% and 30 wt.% particles. The X-ray phase analysis carried out with a “Pananalytical” spectrometer shows that the main phases are two BCC-type ones, with the slightly different lattice parameters of 2.871 Å and 2.841 Å. In Figure1, structure of the composite with 15% particles is shown. The samples for microscopy were prepared both from the inner part of the sample (Figure1a) and near NanomaterialsNanomaterials 2021 2021, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 33 ofof 2222

Nanomaterials 2021, 11, 1748 3 of 21

InIn Figure Figure 1, 1, stru structurecture of of the the composite composite with with 15% 15% particles particles is is shown. shown. The The samples samples for for mi- mi- croscopycroscopy were were prepared prepared both both from from the the inner inner part part of of the the sample sample (Figure (Figure 1a) 1a) and and near near the the topthetop surface topsurface surface (Figure (Figure (Figure 1b). 1b).1 Itb).It should should It should bebe noted benoted noted that that that particlesparticles particles werewere were arbitrararbitrar arbitrarilyililyy orientedoriented oriented in in the the in innertheinner inner ppartart part ofof samplesample of sample,, , butbut but somesome some preferablepreferable preferable orientationorientation orientation inin in parallelparallel parallel toto to thethe the sample sample surface surfacesurface presentpresentedpresenteded near nearnear the thethe top toptop surface. surface.surface. The TheThe preferable preferablepreferable orientation orientationorientation of ofof particles particlesparticles in inin the thethe upper upperupper part partpart of ofof thethethe sample samplesample appear appearedappeareded duringduring during thethe the hardeninghardening hardening process process process and and and approximately approximately approximately 20% 20% 20% of of theof the the particles parti- parti- cleshadcles had thehad preferable thethe preferablepreferable orientation orientationorientation in parallel inin parallelparallel to the toto surface, thethe surfacesurface while, , whilewhile 80% of80%80% the ofof particles thethe particlesparticles were wereorientedwere oriented oriented arbitrarily. arbitrarily. arbitrarily. The DC The The conductivity DC DC conductivity conductivity of the of of samples the the samples samples was verywas was low.very very low. low.

FigureFigureFigure 1. 1. Electron ElectronElectron microscopic microscopicmicroscopic images imagesimages of ofof str structurestructureucture of ofof the thethe composite compositecomposite with withwith 15% 15%15% f ﬂakes:lakesflakes: :inside insideinside the thethe sample samplesample ( (a(aa););); near nearnear the thethe top toptop surfacesurfacesurface ( (b(bb).).). Structure StructureStructure of ofof the thethe composite compositecomposite was waswas studied studiedstudied with withwith aa Vega3Vega3 Tescan Tescan microscopemicroscope microscope atat at anan an acceleratingaccelerating accelerating voltagevoltage voltage ofof of 3030 30 kV.kV. kV.

TheThe distribution distributiondistribution of of the thethe diameters diametersdiameters of ofof particles particlesparticles is isis shown shown in in FigureFigure 2 2aa.2a. . The TheThe maximal maximalmaximal MartinMartin diameter diameter,diameter, ,that thatthat is is,is, ,the the maximal maximal length length of of a a midline midline progressing progressing through through the the particle particle particle,, , waswaswas 51 51 µm µmµm in in in average. average. average. The The The distribution distribution distribution of of the ofthe themaximal maximal maximal Martin Martin Martin diameters diameters diameters is is shown shown is shown in in Fig- Fig- in ureFigureure 2b. 2b. 2The b.The The minimal minimal minimal Martin Martin Martin diameter diameter diameter was was was 27.4 27.4 27.4 µm µm µin inm average. average. in average.

FigureFigureFigure 2. 2.2. Distribution Distribution of of the thethe mean meanmean diameters diametersdiameters of ofof particles particlesparticles ( (a(aa)) )and andand distribution distributiondistribution of ofof the thethe maximal maximalmaximal Martin MartinMartin diameters diametersdiameters ( (b(b).).). The microwave measurements were carried out at frequencies from 12 to 38 GHz, TheThe microwavemicrowave measurementsmeasurements werewere carriedcarried outout atat frequenciesfrequencies fromfrom 1212 toto 3838 GHzGHz, , using the methods described in [12]. All microwave measurements were performed at room usingusing thethe methodsmethods describeddescribed inin [12].[12]. AllAll microwavemicrowave measurementsmeasurements werewere performedperformed atat temperature. The scheme of the experiment is presented in Figure3. The sample was placed roomroom temperature.temperature. TheThe schemescheme ofof thethe experimentexperiment isis presentedpresented inin FigureFigure 3.3. TheThe samplesample in the cross-section of a rectangular waveguide 1, in order to completely overlap the cross- waswas placedplaced in in the the cross cross--sectionsection of of a a rectangular rectangular waveguide waveguide 11, , inin order order to to completely completely section. The thickness of the sample was from 1.5 to 2 mm. The waveguide operates with a TE10 mode and its dimensions were deﬁned by the frequency ranges: 16 mm × 8 mm,

Nanomaterials 2021, 11, x FOR PEER REVIEW 4 of 22

overlap the cross-section. The thickness of the sample was from 1.5 to 2 mm. The waveguide operates with a TE10 mode and its dimensions were defined by the frequency ranges: 16 mm × 8 mm, for 12–17 GHz (WR-62); 11.5 mm × 5 mm, for 17–26 GHz (WR-42); 7.2 mm × 3.4 mm, for 27–38 GHz (WR-28). The wave fell normally to the surface of sample. The external DC magnetic field vector H lay in the plane of the plate and was directed per- pendicularly to the microwave magnetic field H~ , H ⊥ H~. This configuration of vectors is called transversely magnetized media. The measurements were performed with the scalar network analyzer. The ampli- tudes of transmitted and reflected waves were determined with the directional couplers 2. The modules of the transmission T and reflection R coefficients and their frequency and magnetic field dependences were measured. The measurements of the coefficients in zero Nanomaterials 2021, 11, 1748 magnetic field were used to determine the complex dielectric permittivity  = − 4i of 21. T(H) − T(0) The relative variations of the transmission coefficient are defined as d = , m T(0) wherefor 12–17 T(H GHz) is (WR-62); the module 11.5 of mm transmission× 5 mm, for coefficient 17–26 GHz in (WR-42);the magnetic7.2 mm field× 3.4H. mmAnalo-, for 27–38 GHz (WR-28). The wave fell normally to the surface of sample. The externalR(H) − R(0 DC) gously, the relative variations of the reflection coefficient are defined as r = , magnetic field vector H lay in the plane of the plate and was directed perpendicularlym R(0) to the microwave magnetic field H , H⊥H . This configuration of vectors is called transversely where R(H) is the module of∼ the reflection∼ coefficient in the magnetic field H. magnetized media.

FigureFigure 3. 3. SchemeScheme of of microwave microwave measurements: measurements: 1 1—waveguide,—waveguide, 2 2—electromagnet,—electromagnet, 3 3—directional—directional coupler,coupler, 4 4—sample—sample and and 5 5—absorber.—absorber.

3. ResultsThe measurements were performed with the scalar network analyzer. The ampli- 3.1.tudes Resonance of transmitted Phenomena and in reflected Magnetized waves Composite were determined Media with the directional couplers 2. TheThe modules theory of of the microwave transmission propagationT and reflection is here describedR coefficients in a transversely and their frequency magnetized and magnetic field dependences were measured. The measurements of the coefficients in zero composite medium. On the assumption of symmetry, the tensor components. of the mag- ε = ε0 − iε00 neticmagnetic susceptibility field were of useda composite to determine medium the are complex defined. dielectric The effective permittivity magnetic parameter. Thes = |T(H)|−|T(0)| relative variations of the transmission coefficient are defined as dm |T(0)| , where |T(H)| is the module of transmission coefficient in the magnetic field H. Analogously, |R(H)|−|R(0)| the relative variations of the reflection coefficient are defined as rm = , where |R(0)| |R(H)| is the module of the reflection coefficient in the magnetic field H.

3. Results 3.1. Resonance Phenomena in Magnetized Composite Media The theory of microwave propagation is here described in a transversely magnetized composite medium. On the assumption of symmetry, the tensor components of the magnetic susceptibility of a composite medium are defined. The effective magnetic parameters of the composite medium are further introduced, in which magnetic particles with form of ellipsoid are placed into a nonmagnetic dielectric material. The resonance phenomena are studied in the case of the electromagnetic wave propagation in a magnetized composite medium. The numerical calculations of the transmission and reflection coefficients, as well Nanomaterials 2021, 11, 1748 5 of 21

as their magnetic ﬁeld dependences, are carried out. The magnetic ﬁeld dependences of the wavenumber are drawn as well.

3.1.1. Propagation of Electromagnetic Waves in Transversely Magnetized Medium Let the electromagnetic wave propagate in an inﬁnite macroscopically uniform medium along the 0y axis. The magnetic ﬁeld H is directed along the 0z axis. Following to [9], let us consider the solution of Maxwell’s equations:

∂ b rot e = − ∂ t ∂ d (1) rot h = ∂ t jointly with the constitutive relations:

d = ε0ε · e ↔ (2) b = µ0 µ · h

where ε0 and µ0 are the electric and magnetic constants, respectively. Note that the second of Equation (1) takes into account both the conduction and displacement currents when we are using the complex dielectric permittivity ε. In Equations (1) and (2), b and d are the magnetic and electric induction and e and h are the electric and magnetic field, respectively. It is assumed that the dielectric permittivity of the medium ε is a scalar value and it does not depend on magnetic field. The magnetic ↔ permeability is defined by the tensor µ. At chosen directions of the fields, this tensor has a view [9]:  µ µ 0  ↔ xx xy µ =  µyx µyy 0  (3) 0 0 1 We will assume here that all the fields do not change in the direction of the x axis and the electromagnetic wave propagates along the y axis. Thus, the vectors h, b, e ^ ^ and d can be expressed in the form h = h(z) exp[i(ωt − ky)], b = b(z) exp[i(ωt − ky)], ^ ^ ^ ^ ^ ^ e = e(z) exp[i(ωt − ky)] and d = d(z) exp[i(ωt − ky)], where h(z), b(z), e(z) and d(z) are the vector complex amplitude factors for the corresponding alternating fields, ω = 2π f is the cyclic frequency and k is the propagation constant. The solution of the system of Equations (1) and (2) in a coordinate view is reduced to the following equations:

2 ˆ ∂ hx h 2 2 2 i − k − εµxxk hˆ x − εµxyk hˆ y = 0 (4) ∂ z2 0 0 2 ˆ ∂ hy h 2 2 2 2 i − µyx k − ε k hˆ x + µyy k − ε k hˆ y = 0 (5) ∂ z2 0 0 where k is the wavenumber in the magnetized medium and k = ω , c = √ 1 is the speed 0 c ε0µ0 of electromagnetic wave in vacuum. The condition of consistency of the algebraic equations systems (4) and (5) relatively to hˆ x and hˆ y is the equality to zero of the determinant of the following form: 2 2 2 k − εµxxk0 −εµxyk0 det 2 2 2 2 = 0 (6) µyx k − ε k0 µyy k − ε k0 From here, two solutions for the wavenumbers are the following: The relation √ k = k0 ε

corresponds to the first solution. This solution is related to the wave in which the polariza- tion of the alternating magnetic field h coincides with the direction of the DC magnetic field H. The wavenumber of this wave does not depend on the DC magnetic field. The second Nanomaterials 2021, 11, 1748 6 of 21

solution corresponds to the wave in which the alternating magnetic ﬁeld is perpendicular to the magnetizing ﬁeld. From (6), it follows that

s µxyµyx p k = k0 ε µxx − = k0 ε µe f f (7) µyy

where the effective magnetic permeability is introduced as:

µxyµyx µe f f = µxx − (8) µyy

Equation (8) can be used for both the uniform magnetic medium and the composite medium. The following relations are valid for the classical Polder tensor: µxx = µyy = µ, µxy = iµa and µyx = −iµa. Equation (8) takes the known view in this case [9]:

µ2 µ = µ − a (9) e f f µ

3.1.2. Propagation Tensor of Magnetic Permeability of the Media with a Single Particle Before turning to the composite medium, let us consider an alone magnetic particle in the non-magnetic medium. Assume that the particle has the form of an ellipsoid with arbitrarily directed axes in relation to the DC and microwave fields. Denote Hi the DC magnetic field inside the ellipsoid and Hi differs from the external magnetic field H. The connection between these fields is given by:

↔ Hi = H − NM (10) ↔ where M is the magnetization and N is the tensor of the demagnetizing factors. This tensor, written for any particle, defines the difference of the magnetic field Hi inside it from the outer magnetic field H in the surrounding non-magnetic matrix. Its off-diagonal elements obey to the following conditions: Nij = Nji at i 6= j, i, j ↔ x, y, z; the diagonal ones to the ↔ other condition: Nxx + Nyy + Nzz = 4π, i.e., trN = 4π. Let us write the Landau–Lifshitz equation for the lossy magnetic medium [9]:

↔ ↔ iαω iω m + γµ m × H − NM + γµ Nm × M + m × M = −γµ M × h (11) 0 0 M 0

where M is the length of the DC magnetization vector M, h and m are the alternating magnetic ﬁeld and magnetization, α is the dissipation parameter and γ is the gyromagnetic ratio. Equation (11) can be rewritten as:

 + + = − +  a11mx a12my a13mz γµ0 Myhz γµ0 Mzhy, a21mx + a22my + a23mz = −γµ0 Mzhx + γµ0 Mxhz, (12)  a31mx + a32my + a33mz = −γµ0 Mxhy + γµ0 Myhx,

with the following elements of the matrix aij: ω a11 = i + Nxy Mz − Nxz My γµ0

iαω a12 = Hz + Mz − Nxz Mx − 2Nyz My − Nzz − Nyy Mz γµ0 M iαω a13 = − Hy + My − Nxy Mx − Nyy − Nzz My − 2Nyz Mz γµ0 M Nanomaterials 2021, 11, 1748 7 of 21

iαω a21 = − Hz + Mz − 2Nxz Mx − Nyz My − (Nzz − Nxx)Mz γµ0 M ω a22 = i + Nyz Mx − Nxy Mz γµ0 iαω a23 = Hx + Mx − (Nxx − Nzz)Mx − Nxy My − 2Nxz Mz γµ0 M iαω a31 = Hy + My − 2Nxy Mx − Nyy − Nxx My − Nyz Mz γµ0 M iαω a32 = − Hx + Mx − Nxx − Nyy Mx − 2Nxy My − Nxz Mz γµ0 M ω a33 = i + Nxz My − Nyz Mx γµ0 The solution of Equation (12) lets one disclose the connection between the elements of ↔ the vectors m = mx my mz and h = hx hy hz as m = χ · h, therefore, deﬁne the elements of the magnetic susceptibility tensor:

 χ χ χ  ↔ xx xy xz χ =  χyx χyy χyz  (13) χzx χzy χzz

its elements have the view: Mz A21 + My A31 χ = xx deta M A − M A χ = z 11 x 31 xy deta

My A11 + Mx A21 χ = − xz deta

Mz A22 + My A32 χ = − yx deta M A − M A χ = − z 12 x 32 yy deta

My A12 + Mx A22 χ = yz deta

Mz A23 + My A33 χ = zx deta M A − M A χ = z 13 x 33 zy deta

My A13 + Mx A23 χ = − zz deta

Here, the definitions Aij are used. They are the minors of the matrix a, which are deter- h i = = mined by the relation Aij det apq p6=i,q6=j . It is assumed, here, that m mx my 0 and h = hx hy 0 . If H = 0 0 Hz and H > Hs, where Hs is the field of magnetic saturation, the inequalities Mx << Mz and My << Mz are valid for the projections of the vector M. Therefore, in this case, one can suppose that M||H, that is, M = 0 0 Mz , at least at first approximation. Then the tensor (13) becomes:

 χ χ 0  ↔ xx xy χ =  χyx χyy 0  (14) 0 0 0 Nanomaterials 2021, 11, 1748 8 of 21

ωM ωH + iωα − Nzz − Nyy ωM χ = xx D ωM iω − NxyωM χ = xy D ωM iω + NxyωM χ = − yx D ω [ω + iωα − (N − N ) ω ] χ = M H zz xx M yy D 2 2 D = [ωH + iωα − (Nzz − Nxx) ωM] ωH + iωα − Nzz − Nyy ωM − NxyωM − ω

where ωH = γµ0 Hz, ωM = γµ0 Mz. ↔ Using the magnetic susceptibility tensor χ, it is possible to ﬁnd the magnetic permeability tensor: ↔ ↔ ↔ µ = I + χ (15) Inserting (14) into (15), one gets, according to (3):

 µ µ 0  ↔ xx xy µ =  µyx µyy 0  (16) 0 0 1 h i ωM ωH + iωα − Nezz − Neyy ωM µxx = 1 + De h i ωM iω − NexyωM µxy = De h i ωM iω + NexyωM µyx = − De h i ωM ωH + iωα − Nezz − Nexx ωM µyy = 1 + De 2 h i h i 2 De = ωH + iωα − Nezz − Nexx ωM ωH + iωα − Nezz − Neyy ωM − Nexy ωM − ω ↔ ↔ ~ ↔ ~ 1 The designation N = 4π · N is used here, at the deﬁnition trN = 1.

3.1.3. Tensor of Magnetic Permeability of Composite Medium Let us now discuss the problem of the effective magnetic parameters of the composite medium, which consists of identical magnetic particles in the form of ellipsoid with the same spatial orientation placed into the non-magnetic matrix. This composite medium is regarded as a macroscopically uniform medium, since the sizes of any elementary volume are significantly less than the characteristic scales, such as the wavelength of the propagating wave and the dimensions of the sample, but larger than the sizes of any magnetic particle. Magnetization of the composite medium is characterized by a volume fraction θv of the ferromagnetic phase, which is assumed constant for any elementary volume of the medium. ↔ The tensor µ, in accordance with (16), takes into account the demagnetizing fields both for the DC and alternating fields. Following to [3], the expression for permeability ↔m tensor of composite µ can be represented as the Silberstein formula:

↔m ↔ ↔ µ = (1 − θv) · I + θv · µ (17) Nanomaterials 2021, 11, 1748 9 of 21

As is shown in [23,24], variations of the field in the composite medium at θv < 1 ↔ ~ can be taken into account by introducing the effective demagnetizing tensor L, which ↔ ↔ ~ ~ has the following limiting cases: at θv → 0, the approximate equality L ≈ N is valid, ↔ ~ where the tensor N is defined for the single particle of a given form. At θv → 1, the ↔ ~ ↔ effective demagnetizing tensor is close to zero, L ≈ 0 , that corresponds to the unbounded magnetic medium. The paper [25] describes a mathematical model enabling the calculation of the effective permeability tensor of heterogeneous magnetic materials. The model uses a definition similar to Equation (17) and gives all the complex components of the permeability tensor. According to [23,24], the effective demagnetizing tensor can be determined as:

↔ − ↔ ~ ↔ ↔m ↔m ↔ ↔ 1 ~ L = µ − µ · µ · µ − I · N (18)

↔ ~ Taking (17) into account, one gets the equation for L:

↔ − ↔ ~ ↔ ↔ ↔ 1 ~ L = (1 − θv) · I + θv · µ − I · N (19)

↔ ~ ↔ Equation (19) shows that L is, generally speaking, dependent on µ. In the ﬁrst approximation for θv << 1 and H > Hs, it is reasonable to consider this dependence as negligible and Equation (19) takes the simpler view:

↔ ↔ ~ ~ L ≈ (1 − θv) · N (20)

↔m Therefore, the magnetic permeability tensor µ of the composite medium with iden- tically directed particles is determined by Expression (20), where the components of the ↔ ↔ ~ tensor µ are the same as in (19), but with replacement of the components of N by the ↔ ~ components of L. Finally, it can be written:

 µm µm 0  m ↔ xx xy ↔ ↔ m m µ = (1 − θv) · I + θv · µ =  µyx µyy 0  (21) 0 0 1

 h i  ωM ωH + iωα − Nezz − Neyy (1 − θv) ωM m µ = 1 − θv + θv1 +  xx Dˆ h i θvωM iω − Nexy(1 − θv) ωM µm = xy Dˆ h i θvωM iω + Nexy(1 − θv) ωM µm = − yx Dˆ  h i  ωM ωH + iωα − Nezz − Nexx (1 − θv) ωM m µ = 1 − θv + θv1 +  yy Dˆ

2 h ih i 2 Dˆ = ωH + iωα − Nezz − Nexx (1 − θv) ωM ωH + iωα − Nezz − Neyy (1 − θv) ωM − Nexy(1 − θv) ωM − ω Nanomaterials 2021, 11, 1748 10 of 21

The foregoing formulas for the components of the magnetic permeability tensor are a generalization of formulas for a composite medium specified in [9]. They describe the resonance variations of permeability near FMR. Of course, these resonance features of frequency or magnetic field dependences of the component of the tensor (21) have to manifest themselves in the dependences of transmission and reflection coefficients and absorption of the wave. Besides, variations of non-resonant type are possible, which are caused by magnetization of the composite. These variations allow, for example, the so- called low field absorption [26]. In order to take into account non-resonant variations, the factor X is introduced, which corrects the values of the elements of the magnetic permeability tensor. One of the simplest ways to introduce X is the expression:

χ − 1 = + 11 X 1 2 (22) 1 + κHz

where χ11 is the diagonal component of the susceptibility tensor at Hz = 0. The contribution to dynamic magnetic permeability coming from the domain walls motion is calculated [27]. The magnetic ﬁeld dependence of this contribution is very similar to our Equation (22). ↔m The components of the tensor µ can be rewritten as:

θvω [ω +iωα−(Nezz−Neyy) (1−θv) ω ] µm = 1 + X M H M , xx Dˆ θvω [iω−Nexy(1−θv) ω ] µm = X M M , xy Dˆ (23) θvω [iω+Nexy(1−θv) ω ] µm = −X M M , yx Dˆ θvω [ω +iωα−(Nezz−Nexx) (1−θv) ω ] µm = 1 + X M H M . yy Dˆ

Notice that, according to (22), under condition χ11 =1, the equality X = 1 is valid. The 2 value of X is frequency dependent. Under magnetic saturation, at 1 + κHz >> χ11 − 1, X almost equals to 1.

3.1.4. Magnetic Permeability of Ensemble of Arbitrarily Directed Particles Let us set the vector of rotation angles of a ferromagnetic particle relative to the axes x, y, z as Θ = α β γ . It conditions, ﬁrst, variation of the components of ↔ ~ demagnetizing tensor N(Θ) of the particle if its orientation varies and, second, variation of D↔mE Mz, therefore, ωM(Θ). In order to get the effective magnetic permeability tensor µ of a composite medium in the case of arbitrarily oriented particles, it is necessary to perform the statistical averaging of its components:

* µm (Θ) µm (Θ) 0 + m ↔ xx xy D↔ E D↔E m m µ = (1 − θv) · I + θv · µ =  µyx(Θ) µyy(Θ) 0  (24) 0 0 1

The effective magnetic permeability of the transversely magnetized medium µe f f (Θ), corresponding to a given value of the vector Θ, is expressed by a relation similar to (8), m which includes the components of the tensor µij (Θ). The averaged value of the effective magnetic permeability for the transversely magnetized medium is determined by the formula: * + D E D E µm (Θ) · µm (Θ) = ( ) = m ( ) − xy yx µe f f µe f f Θ µxx Θ m (25) µyy(Θ) If particles have the form of flakes, their space orientation can be characterized by the direction of normal to its plane. If the normal is directed along to the y axis, then it can be defined by the vector n = 0 1 0 . Let us believe that this the initial or base orientation of a ferromagnetic particle, for which the vector of rotation angles can be defined as Θ0 = 0 0 0 . All other possible space orientations of particles have Nanomaterials 2021, 11, 1748 11 of 21

to be determined by the vectors Θ = α β γ . In the case of arbitrary orientation of a great number of particles, every element from the multitude of vectors Θ belongs to the independent multitudes of uniformly distributed random numbers which get into the intervals: α ∈ [−π ; π], β ∈ [−π ; π] and γ ∈ [−π ; π]. The discrete number of the vector Nanomaterials 2021, 11, x FOR PEER REVIEW 12 of 22 of rotation angles Θp corresponds to the discrete multitudes of numbers αp, βp and γp. The calculation of the effective magnetic permeability of a magnetized composite has been carried for 10,000 particles made of material with the saturation magnetization M900s = kA/m 900 kA/m and magneticand magnetic damping damping constant constant α = 0.05.α Every= 0.05. particle Every has particle the form has of the ellipsoid form ofwith ellipsoid the axes with a = theb = 25 axes µma and= b =c = 25 1 µµm.m andThe ccalculation= 1 µm. The is carried calculation out for is frequency carried out f = for 32 frequencyGHz. The fresults= 32 GHz. for several The results values for of severalv are valuesshown ofinθ Figurev are shown 4. Thein variations Figure4. of The the variationseffective permeability of the effective caused permeability by FMR are caused present by in FMR the graphs. are present It is evidently in the graphs. seen, from It is evidentlyFigure 4, seen,that the from resonance Figure4, thatoccupied the resonance a wide region occupied of magnetic a wide region fields of despite magnetic the ﬁelds small despitevalue of the α. smallIncrease value in the of αvolume. Increase fraction in the of volume ferromagnetic fraction particles of ferromagnetic leads to more particles pro- leadsnounced to more resonance pronounced variations. resonance variations.

FigureFigure 4.4. CalculatedCalculated magnetic magnetic field field dependence dependence of ofthe the effective effective magnetic magnetic permeability permeability for the for com- the compositeposite containing containing 10,000 10,000 particles particles of material of material with with the the saturation saturation magnetization magnetization MsM = s900= 900 kA/m kA/m and andthe themagnetic magnetic damping damping constant constant α = 0.05.α = 0.05. Every Every particle particle has hasthe form the form of ellipsoid of ellipsoid with with the axes the axes a = b = 25 µm and c = 1 µm. The calculation is carried out for frequency f = 32 GHz. The continuous lines a = b = 25 µm and c = 1 µm. The calculation is carried out for frequency f = 32 GHz. The continuous correspond to the real part of permeability and the dashed lines correspond to the imaginary part lines correspond to the real part of permeability and the dashed lines correspond to the imaginary of permeability. part of permeability. Let us consider the case when a group of particles with definite orientation is present Let us consider the case when a group of particles with definite orientation is present inin the the composite composite besides besides the the group group of of arbitrarily arbitrarily oriented oriented particles. particles. Denote Denote the the whole whole number of ferromagnetic particles as L ; the number of arbitrarily oriented particles as number of ferromagnetic particles as L; the number of arbitrarily oriented particles as L1; theL1 number; the number of particles of particles with the with definite the definite orientation orientation (in the plane (in the of aplane sample, of a for sample, example) for asexample)L2, L = asL1 +L2L,2 .L In= theL1 + limitingL2. In the case limiting when Lcase2 = when0, all particlesL2 = 0 , areall particles oriented are arbitrarily, oriented whenarbitrarL2il=y, Lwhen, all particles L = L are, all oriented particles in are the oriented plane of in a sample.the plane The of formulaa sample. for The calculation formula 2 D↔mE of the components of the tensor µ can be writtenm for the presence of several groups of for calculation of the components of the tensor μ can be written for the presence of differently oriented particles. Their mean values are determined by the equation: several groups of differently oriented particles. Their mean values are determined by the m m L1 m m equation: m µxy(Θ0)·µyx(Θ0) m µxy(Θp)·µyx(Θp) L2 µxx(Θ0) − m + ∑ µxx Θp − m µyy(Θ0) L µyy(Θp) D E  m (Θ ) m (Θ )p=1 1  m Θ  m Θ  µe f f =  m xy 0 yx 0   m xy ( p ) yx ( p ) (26) L2 xx (Θ0 )− L + +L xx (Θ p )−  m (Θ ) 1  2 m (Θ )  (26)  yy 0  p=1  yy p  eff = L1 + L2 The results of the calculation of the effective magnetic permeability following to Equation (27) are shown in Figure 5 for the composite with 5000 particles of ellipsoidal form with the same sizes as in Figure 4. The saturation magnetization of particles is Ms = 900 kA/m, the magnetic damping constant α = 0.19 and the volume fraction of

Nanomaterials 2021, 11, 1748 12 of 21

Nanomaterials 2021, 11, x FOR PEER REVIEWThe results of the calculation of the effective magnetic permeability following13 of to22 Equation (27) are shown in Figure5 for the composite with 5000 particles of ellipsoidal form with the same sizes as in Figure4. The saturation magnetization of particles is Ms = 900 kA/m, the magnetic damping constant α = 0.19 and the volume fraction of ferromagnetic particles  = 0.15 . In Figure 5, curve 1 corresponds to the case  = 1, ferromagnetic particles θv v= 0.15. In Figure5, curve 1 corresponds to the case χ11 = 1, thatthat is,is, the non non-resonant-resonant absorption absorption is is not not taken taken into into account. account. It is It assumed is assumed,, in this in case this , case,that L that1 = 4000,L1 = L 4000,2 = 1000L2 =and, 1000 namely,and, namely, 20% particles 20% particles are oriented are orientedby the manner by the when manner the whenDC magnetic the DC magneticfield is in ﬁeldparallel is into paralleltheir plane. to their The plane.remainder The 80% remainder particles 80% are particles oriented arearbitrar orientedily. C arbitrarily.urve 2 corresponds Curve 2 correspondsto the case when to the all caseparticles when are all oriented particles to are the oriented plane of tothe the sample plane and of the the sampleDC magn andetic the field DC lies magnetic in this plane. ﬁeld lies The in resonant this plane.-type The variations resonant- are −9 2 typeexpressed variations much are more expressed strongly much in this more case. strongly The values in this 11 case. =1.4 The and values κ = 1.184χ11·10= 1.4 (m/A) and −9 2 κare= 1.184chosen× 10for curve(m/A) 3, soare the chosen non-resonant for curve contribution 3, so the non-resonant is taken into contribution account. The is takencalcu- intolation account. is carried The out calculation for frequency is carried f = 32 out GHz. for frequency f = 32 GHz.

FigureFigure 5.5. MagneticMagnetic field ﬁeld dependence dependence of ofthe the effective effective magnetic magnetic permeability permeability calculated calculated for the for com- the compositeposite with with 5000 5000 particles particles with withthe saturation the saturation magnetization magnetization of particles of particles Ms = 900M kA/m,s = 900 the kA/m, magnetic the damping constant α = 0.19 and the volume fraction of ferromagnetic particles  = 0.15 . Curve 1 magnetic damping constant α = 0.19 and the volume fraction of ferromagnetic particlesv θv = 0.15. Curvecorresponds 1 corresponds to the case to the where case where11 =χ 1,11 L=1 1,= L4000,1 = 4000, L2 = L10002 =. 1000. For curve For curve 2,  2,11χ 11=1= as 1 aswell well,, but but all allparticles particles are are oriented oriented in in the the plane plane of of the the sample sample and and the the DC DC magnetic magnetic field ﬁeld lies in the samesame plane.plane. −9−9 22 CurveCurve 3 3 is is calculated calculated at atχ 1111= 1.4 = 1.4 and andκ = κ 1.184 = 1.184× 10× 10 (m/A) (m/A) ,, thatthat is,is, thethe non-resonantnon-resonant contributioncontribution inin permeability permeability isis taken taken into into account. account. The The calculation calculation is is carried carried out out for for frequency frequencyf f= = 32 32 GHz. GHz. The The continuouscontinuous lineslines correspond correspond to to the the real real part part of of permeability permeability and and the the dashed dashed lines lines correspond correspond to to the the imaginaryimaginary part part of of permeability. permeability.

TheThe resultsresults ininFigures Figures4 4and and5 show5 show that that well-known well-known procedures procedures of of homogenization homogenization basedbased onon thethe constitutiveconstitutive parametersparameters andand concentrationsconcentrations of of itsits componentscomponents areare insufﬁcientinsufficient inin orderorder toto describedescribe thethe resonanceresonance phenomenaphenomena inin thethe heterogeneousheterogeneous magneticmagnetic medium;medium; seesee [[3]3] for example. example. The The conception conception of ofthe the mean mean demagnetizing demagnetizing factor factor is sometimes is sometimes used usedinstead instead of the of procedure the procedure of averaging of averaging of contributions of contributions in the components in the components of the magnetic of the magneticpermeability permeability tensor from tensor particles from particleswith different with different demagnetizing demagnetizing factor [16]. factor This [16 concep-]. This conceptiontion is capable, is capable, for instance, for instance, to describe to describe the FMR the FMRspectrum spectrum of the of composite the composite material material con- containingtaining spherical spherical particles particles [24]. [ 24For]. media For media in which in which the value the value of the of demagnetizing the demagnetizing factor factorof particles of particles varies varies essentially essentially,, it is it prescribed is prescribed to to apply apply the the afore afore-mentioned-mentioned calculationcalculation procedure. Notice that division of particles into separate groups with identical value of the demagnetizing factor used in our calculation in Figure 5 is similar to the method applied in [16].

Nanomaterials 2021, 11, 1748 13 of 21

procedure. Notice that division of particles into separate groups with identical value of the demagnetizing factor used in our calculation in Figure5 is similar to the method applied in [16].

3.2. Reflection, Transmission Coefficients and Dissipation of Microwaves Let us discuss how the variations of magnetic permeability can disturb the quantities, which can be measured in an experiment. Let the electromagnetic wave of centimeter or millimeter wavebands fall from vacuum (that is medium 1) onto the plate of composite with thickness d2 (that is medium 2). After the wave passes the plate, it enters the vacuum again. The complex transmission T and reflection R coefficients can be calculated via the equations [12,28]:

2Z Z = 1 2 T 2 2 (27) 2Z1Z2 cos (k2d2) + i Z1 + Z2 sin(k2d2)

i Z2 − Z2 sin(k d ) = 2 1 2 2 R 2 2 (28) 2Z1Z2 cos (k2d2) + i Z1 + Z2 sin(k2d2) r q µ hµ i Here, Z = µ0 is the impedance of the medium 1, Z = 0 e f f is the impedance 1 ε0 2 ε0ε of the composite with the dielectric permittivity ε = ε0 − iε00 and the mean value of D E D E0 D E00 the effective magnetic permeability is µe f f = µe f f − i µe f f . Assume that the dielectric permittivity is a scalar value and it does not depend on the magnetic ﬁeld. 0 00 In Equations (27) and (28), k2 = k 2 − ik 2 is the complex wavenumber which real and imaginary parts can be calculated, following to the formulas [9]: v u D E D E0 D E00 u | | + 0 − 00 ω t ε µe f f ε µe f f ε µe f f k0 = 2 c 2 v u D E D E0 D E00 u | | − 0 + 00 ω t ε µe f f ε µe f f ε µe f f k00 = 2 c 2

The power transmission TP and reflection RP coefficients can also be introduced. They show the portions of microwave power in the wave that passed the plate and was reflected from one: ∗ TP = T · T , (29) ∗ RP = R · R . (30) Dissipation of microwaves D can be calculated from these coefficients as follows:

D = 1 − TP − RP. (31)

Dissipation shows which fraction of the incident wave power is lost inside the plate. The losses come from absorption, scattering on the inner inhomogeneities and from transfor- mation into the other waves besides TE10 at the sample boundaries. Several contributions, such as absorption, are magnetic field dependent but the others are not. It was seen, from Equations (27) and (28), that the peculiarities in the effective magnetic permeability coming from FMR, for example, also lead to variations of the transmission and reflection coefficients and wavenumber.

4. Microwave Results The results of the measurements show that the microwave transmission and reflection coefficients experience large variations under the action of a magnetic field. The results of measurements of the magnetic field dependences of the transmission and reflection Nanomaterials 2021, 11, x FOR PEER REVIEW 15 of 22

results of measurements of the magnetic field dependences of the transmission and reflection coefficients for the composite with 30% flakes are shown in Figure 6. The measurements were performed at several frequencies of millimeter waveband. The characteristic feature of all the dependences in Figure 6a is the presence of a minimum. If frequency increases, then this minimum shifts to higher magnetic fields. The absolute value of the minimum increases if frequency increases. The minimum is caused by absorption of microwaves under FMR condition [13]. At the highest frequency of 38 GHz, a maximum also presents in the field of 0H = 0.35 T, i.e., less than the field of the minimum. As shown in [13], this maximum is caused by ferromagnetic antiresonance (FMAR). The results of

Nanomaterials 2021, 11, 1748 the measurements of magnetic field dependence of the reflection coefficient in Figure14 of 6b 21 confirm the abovementioned statements. The minimum caused by FMR is presented in the dependences. The most distinctive feature of the reflection coefficient variations seems to be the maximum caused by FMAR. These variations reach 300% in magnitude; there- forecoefficients, they could forthe be regarded composite as with giant 30% ones. flakes are shown in Figure6. The measurements wereAnalogous performed measurements at several frequencies were performed of millimeter for the waveband. composite The with characteristic 15% flakes and feature the resultsof all the are dependences shown in Figure in Figure 7. The6a isminimum the presence caused of aby minimum. FMR andIf the frequency maximum increases, caused bythen FMAR this minimumwere also observed, shifts to higher but their magnetic magnitudes fields. are The less absolute than in valuethe previous of the minimum case. The resonanceincreases ifvariations frequency were increases. also observed The minimum at lower frequencies is caused by of absorption centimeter ofwaveband; microwaves see Figureunder FMR8. In conditionthis case, the [13]. fields At the of highestthe minimum frequency are ofless 38 than GHz, in a maximumFigure 7, in also accordance presents within the the field typical of µ0 spectrumH = 0.35 T, of i.e., the lessuniform than theFMR field mode. of the minimum. As shown in [13], this maximumThe weak is caused maximum by ferromagnetic caused by FMAR, antiresonance was observed (FMAR). only The for results frequency of the 26.4 measure- GHz. Really,ments ofthe magnetic FMAR phenomenon field dependence cannot of present the reflection in frequencies coefficient less in than Figure some6b confirmfrequenc theies definedabovementioned by the saturation statements. magnetization The minimum [9]. Therefore, caused by in the FMR composites is presented containing in the depen- flakes ofdences. finemet The-type most alloy, distinctive the resonance feature ofvariations the reflection are present coefficient, which variations were caused seems toby be both the FMRmaximum and FMAR. caused For by the FMAR. composite These with variations 30% flakes reach, 300%the variations in magnitude; of the reflection therefore, coef- they ficientcould beunder regarded FMAR as one giant can ones. regard as giant ones.

Figure 6. Magnetic field field dependences of the transmission ( a)) and and the the reflection reflection ( (b)) coeffic coefficientsients for the the composite with 30% 30% flakesflakes measured at several frequencies of millimeter waveband.waveband.

Analogous measurements were performed for the composite with 15% flakes and the results are shown in Figure7. The minimum caused by FMR and the maximum caused by FMAR were also observed, but their magnitudes are less than in the previous case. The resonance variations were also observed at lower frequencies of centimeter waveband; see Figure8. In this case, the fields of the minimum are less than in Figure7, in accordance with the typical spectrum of the uniform FMR mode. The weak maximum caused by FMAR, was observed only for frequency 26.4 GHz. Really, the FMAR phenomenon cannot present in frequencies less than some frequencies defined by the saturation magnetization [9]. Therefore, in the composites containing flakes of finemet-type alloy, the resonance variations are present, which were caused by both FMR and FMAR. For the composite with 30% flakes, the variations of the reflection coefficient under FMAR one can regard as giant ones. NanomaterialsNanomaterials 20212021,,, 1111,,, 1748xx FORFOR PEERPEER REVIEWREVIEW 1615 of of 2122

FigureFigure 7.7. Magnetic fieldfield dependences of the transmission (a) and the reflection reflection (b) coeffic coefficientsients forfor thethe composite with 15% flakesflakes measuredmeasured atat severalseveral frequenciesfrequencies ofof millimetermillimeter waveband.waveband.

FigureFigure 8.8. Magnetic fieldfield dependences of the transmission (a) and the reflection reflection (b) coeffic coefficientsients forfor thethe composite with 15% flakesflakes measuredmeasured atat severalseveral frequenciesfrequencies ofof centimetercentimeter waveband.waveband.

5. Discussion In thisthis section,section, let let us us compare compare the the experimental experimental results results with with the the theoretical theoretical calculations calcula- tionsfulfilled fulfilled before, before in Section, in Section3. We consider 3. We consider several several versions versions of calculations, of calculation whichs, differ which from dif- fereach from other each by theother portion by the of portion flakes, the of presenceflakes, the or, presence in contrast, or, absence,in contrast of, the absence group, flakesof the withgroup identical flakes with space identical orientation space and orientation with taking and into with account, taking orinto lacking, account the, or non-resonant lacking, the variationnon-resonant of the variation dynamic of magnetic the dynamic permeability. magnetic In permeability. order to calculate In order the transmission to calculate andthe transmissionreflection coefficients, and reflection according coefficients to the Formulas, according (27) to andthe Formulas (28), it is necessary(27) and (28), to know it is nec- the essarydielectric to permittivityknow the dielectric of the samples. permittivity The samples of the samples. used in thisThepaper samples are used the same in this as inpaper [29] areand the we same take theas in values [29] and of permittivity we take the from values that of work.permittivity The mean from values that work. of the The dielectric mean permittivityvalues of the for dielectric the three permittivity frequency rangesfor the andthree the frequency values of ranges the microwave and the values conductivity of the microwavfor the compositee conductivity with 15% for flakesthe composite are presented with 15% in Table flakes1. Forare presented the composite in Table with 1. 30% For theflakes composite in the frequency with 30% range flakes from in the 26 frequency to 38 GHz, range the mean from values 26 to 38 are GHz, the following: the mean εvalues0 = 38, areε” = the 13 andfollowing:σ = 22 ε S/m.′′ = 38, ε″ = 13 and σ = 22 S/m.

Nanomaterials 2021, 11, x FOR PEER REVIEW 17 of 22

Table 1. Dielectric permittivity and microwave conductivity of samples.

Frequency Range (GHz) Sample ε′ ε″ σ (S/m) 12–18 Composite 15% 7.5 3.13 2.45 18–26 Composite 15% 8.2 1.5 1.9 26–38 Composite 15% 5.4 1.1 2.0

Nanomaterials 2021, 11, 1748 In Figure 9, the experimental dependences of the transmission and reflection coeffi-16 of 21 cients for the composite with 15% flakes are shown by black line with symbols. The calculations are carried out using Formulas (27) and (28) and the magnetic permeability is Tablecalculated 1. Dielectric via permittivityFormula (26). and The microwave dependence conductivity, measured of samples. at frequency f = 12 GHz, corresponds fairly well to the dependence which calculated taking into account the non-reso- 0 Frequency Range (GHz)nant contribution Sample in magnetic permeabilityε ( 11 1), theε” presence of the groupσ (S/m) of flakes 12–18oriented Composite in parallel 15% to the top surface 7.5 ( L2  0) and the group 3.13 of flakes oriented 2.45 arbitrarily. 18–26If frequency Composite increases, 15% the calculated 8.2 dependences, however, 1.5 differ from the experimental 1.9 26–38 Composite 15% 5.4 1.1 2.0 ones. For example, the results of the calculation of the reflection coefficient are shown in Figure 9b at the frequency of 38 GHz. The magnitude of variations in all variants of calcu- lationIn satisfactor Figure9,il they corresponds experimental to dependencesthe experimental of the data transmission, but the field and of reflectionmaximum coef- dif- fersficients substantially. for the composite The FMAR with field 15% is le flakesss than are the shown FMR field, by black as expected, line with and symbols. it is located The outsidecalculations of the are FMR carried line. outNote using that the Formulas magnetic (27) field and dependence (28) and the of magnetic the coefficients permeabil- at a fixedity is frequency calculated is via mainly Formula defined (26). by The dependence dependence, of measuredmagnetic permeability. at frequency fComparing= 12 GHz, thecorresponds foregoing fairlyresults, well it ca ton the be dependenceconcluded that which the method calculated of calculation taking into of account magnetic the non-per- meabilityresonant contribution developed above in magnetic describes permeability fairly well( χthe11 6= area1), of the FMR presence resonance of the but group gives of worseflakes fit oriented for weaker in parallel fields. toThe the probabl top surfacee reason (L2 for6= this0) and discrepancy the group consists of flakes in orientedthe very rougharbitrarily. approximation If frequency (22) increases, of the non the-resonant calculated contribution dependences, in the however, magnetic differ field fromdepend- the enceexperimental of magnetic ones. permeability. For example, the results of the calculation of the reflection coefficient are shownThe results in Figure presented9b at thein Figure frequency 9 show of 38 the GHz. relative The variations magnitude of of the variations coefficients in all in thevariants magnetic of calculation field. The satisfactorily absolute values corresponds of the coefficients to the experimental are shown data, in Figure but the 10. field The of maximum differs substantially. The FMAR field is less than the FMR field, as expected, experimental data are compared to the calculations for the composite with v =0.15 , for and it is located outside of the FMR line. Note that the magnetic field dependence of the several frequencies, and for the composite with  =0.3 , at the frequency of 32 GHz, in coefficients at a fixed frequency is mainly defined byv dependence of magnetic permeability. FigureComparing 10a. In the Figure foregoing 10b, the results, type itof can magnetic be concluded field dependence that the method of the reflection of calculation coeffi- of cientmagnetic and the permeability presence of developed FMAR are above reproduced describes in fairlythe calculation. well the area From of FMRthis fact resonance, it can bebut concluded gives worse that fit qualitative for weaker correspondence fields. The probable is achieved reason for between this discrepancy the experiment consists and in calculation.the very rough The approximation presence of the (22) resonant of the non-resonantvariations is reproduced contribution in in calculation the magnetic and field the magnitudesdependence of of the magnetic resonant permeability. variations are reproduced for the composite with 15% flakes.

Figure 9. Relative variations of the transmission (a) and reflection (b) coefficients for the composite with 15% flakes.

Designations of the lines correspond to the following parameters: 1—χ11 = 1.3, L1 = 4000, L2 = 1000; 2—χ11 = 1, L1 = 4000, −9 2 L2 = 1000, 3—χ11 = 1.3, L1 = 5000, L2 = 0; 4—χ11 = 1, L1 = 0, L2 = 5000 and κ = 1.184·10 (m/A) .

The results presented in Figure9 show the relative variations of the coefficients in the magnetic field. The absolute values of the coefficients are shown in Figure 10. The experimental data are compared to the calculations for the composite with θv = 0.15, for several frequencies, and for the composite with θv = 0.3, at the frequency of 32 GHz, in Figure 10a. In Figure 10b, the type of magnetic field dependence of the reflection coefficient and the presence of FMAR are reproduced in the calculation. From this fact, it can be concluded that qualitative correspondence is achieved between the experiment and Nanomaterials 2021, 11, x FOR PEER REVIEW 18 of 22

Nanomaterials 2021, 11, 1748 17 of 21

Figure 9. Relative variations of the transmission (a) and reflection (b) coefficients for the composite with 15% flakes. Des- ignations of the lines correspond to the following parameters: 1—  =1.3, L1 = 4000, L2 = 1000; 2— =1, L1 = 4000, L2 calculation. The presence of the resonant11 variations is reproduced in calculation and the = 1000, 3— =1.3, L1 = 5000,magnitudes L2 = 0; 4— of the =1, resonant L1 = 0, L2 variations= 5000 and areκ = 1.184 reproduced·10−9 (m/A) for2. the composite with 15% ﬂakes.

Figure 10. Magnetic fieldfield dependences of the transmission ( a) and reflection reflection ( b) coefficientscoefficients for the composites with 15% and 30% flakes. flakes. Designations Designations of of the the lines: lines: theory theory—continuous—continuous lines; lines; experiment experiment—lines—lines with with symbols. symbols. Designations Designations of the of

thelines lines in F inigure Figure (b) correspond (b) correspond to the to thefollowing following parameters: parameters: 1— 1—χ =1.3,11 = 1.3,L1 =L 4000,1 = 4000, L2 = L1000;2 = 1000; 2— 2— χ=1,11 =L1 1, = L4000,1 = 4000, L2 = −9 2 L10002 =, 1000, 3— 3— χ=11 1.3,= 1.3, L1 =L 5000,1 = 5000, L2 =L 0;2 =4— 0; 4—χ =11 1,= L 1,1 =L 10,= L 0,2 =L 50002 = 5000 andand κ = 1.184κ = 1.184·10−9· 10(m/A)(m/A)2. .

In Figure Figure 11 11 the the magnetic magnetic field field dependences dependences of the of real the and real imaginary and imaginary parts of parts wave- of wavenumber for the same cases as presented in Figure 10a are shown. It is not surprising number for the same cases as presented in Figure 10a are shown. It is not surprising that that the resonance variations are seen most brightly for the case θ = 0.3. In Figure 12, the resonance variations are seen most brightly for the case  = 0v.3. In Figure 12, the the magnetic field dependences of microwave dissipation determinedv via Formula (32) magnetic field dependences of microwave dissipation determined via Formula (32) for for the composite with θv = 0.15 are shown. Coincidence of the type of dependence and themaximums composite caused withby v absorption= 0.15 are under shown FMR. Coincidence condition ofoccur the both type in of calculation dependence and and in maximumsexperiment. caused The absolute by absorption value ofunder the dissipatedFMR condition power occur in calculation both in calculation corresponds and toin experiment.experiment onlyThe inabsolute order of value magnitude. of the dissipated Certainly, neitherpower in in calculation experimental corresponds nor in calculated to ex- perimentdependences, only therein order are of no magnitude. peculiarities Certainly, caused by neither FMAR. in experimental nor in calculated dependencesThe magnetic, there field are no dependence peculiarities of thecaused penetration by FMAR. depth of the electromagnetic field insideThe the magnetic composite field sample dependence is shown of the in penetration Figure 13. depth The penetration of the electromagnetic depth, which field is 1 insidedetermined the composite as δ = 00 sample, is introduced is shown as in theFigure distance 13. The from penetration the surface depth, at which which the is electro- deter- k 2 magnetic field amplitude1 decreases by e times in comparison to that at the surface. In the mined as  = k , is introduced as the distance from the surface at which the electromag- case of metallic2 conductivity of a sample, δ is the skin depth. For the composite sample, neticwhich field is a amplitude lossy dielectric, decreasesδ is theby e depth times wherein comparison electromagnetic to that at field the surface. decreases In becausethe case of metallic absorption conductivity and scattering. of a sample, This clarification δ is the skin is depth. essential, For becausethe composite the composites sample, which with isferromagnetic a lossy dielectric, flakes δ demonstrate is the depth propertieswhere elect ofromagnetic lossy dielectric field indecreases zero magnetic because field of ab- as sorptionwell as in and fields scattering. that are This significantly clarification less is than essential the field, because of FMR. the composites Near the area with of ferro- FMR, magneticthe real and flakes imaginary demonstrate parts properties of the wavenumber of lossy dielectric become in the zero same magnetic order of field magnitude as well asthat in isfields characteristic that are significantly for metals less [13]. than The the dependences field of FMR shown. Near the in Figurearea of 13FMR, once the more real andconfirm imaginary the fact parts that of the the resonance wavenumber variations become are the caused same order by FMAR of magnitude and FMR. that Actually, is char- acteristicat frequencies for metals of 26 [13]. GHz The and dependences higher, if the shown magnetic in Figure field 13 increases, once more a maximum confirm the of fact the thatpenetration the resonance depth occursvariations that isare typical caused for by FMAR. FMAR A and further FMR. increase Actually, of the at frequencies magnetic field of 26leads GH toz and a decrease higher, and if the minimum magnetic of field the penetration increases, a depth maximum due to of absorption the penetration of the wavesdepth occursunder FMR.that is Attypical frequencies for FMAR. less A than further 26 GHz, increase there of isthe no magnetic maximum field of leadsδ, as to expected a decrease for andmagnetic minimum antiresonance. of the penetration As expected, depth the due penetration to absorption depth of for the the waves composite under withFMR. 30% At frequenciesflakes is less less than than the same26 GHz for, thethere composite is no maximum with 15% of flakes.δ, as expected for magnetic antiresonance. As expected, the penetration depth for the composite with 30% flakes is less than the same for the composite with 15% flakes.

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FigureFigure 11.11. RealReal andand imaginaryimaginary partsparts ofof thethe wavenumberwavenumber forfor thethe compositescomposites with with 15% 15% and and 30% 30% ﬂakes flakes atatFigure thethe waveswaves 11. Real of of centimeter centimeterand imaginary and and millimeter partsmillimeter of the wavebands. wavebands.wavenumber TheThe for continuous continuousthe composites lineslines with correspondcorrespond 15% and to to30% the the flakes real real at the waves of centimeter and millimeter wavebands. The continuous lines correspond to the real partpart of of thethewavenumber wavenumber and and the the dasheddashed lines lines correspondcorrespond to to the the imaginary imaginary part part of ofthe thewavenumber. wavenumber. part of the wavenumber and the dashed lines correspond to the imaginary part of the wavenumber.

Figure 12. Magnetic field dependence of microwave dissipation for the composite with 15% flakes; Figure 12. Magnetic field dependence of microwave dissipation for the composite with 15% flakes; Figurecalculation 12. Magnetic—the continuous ﬁeld dependence lines; experiment of microwave—the lines dissipation with symbols. for thecomposite with 15% ﬂakes; calculation—the continuous lines; experiment—the lines with symbols. calculation—the continuous lines; experiment—the lines with symbols.

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FigureFigure 13. 13.Magnetic Magnetic ﬁeld field dependence dependence of of microwave microwave penetration penetration depth depth for for the the composites composites with with 15% 15% andand 30% 30% ﬂakes. flakes.

6.6. Conclusions Conclusions TheThe transmission transmission of of microwaves microwaves through through the the composite composite medium medium containing containing metallic metallic ferromagneticferromagnetic particles particles is is discussed discussed and and so so is is the the reflection reflection from from this this medium. medium. The The theory theory isis worked worked out out which which permits permits to to calculate calculate the the components components of theof the tensor tensor of magnetic of magnetic perme- per- abilitymeability for thefor compositethe composite containing containing the aggregatethe aggregate of particles of particles in the in form the form of ellipsoid of ellipsoid the axesthe axes of which of which are directed are directed in an in arbitrary an arbitrary manner manner in relation in relation to the to coordinate the coordinate axes. Theaxes. expressionThe expression is obtained is obtained for the for calculation the calculation of the effectiveof the effective magnetic magnetic permeability permeability of trans- of verselytransversely magnetized magnetized composite composite medium, medium, which which contains contains both randomly both randomly oriented oriented particles par- andticles a groupand a group of particles of particles with a w preferentialith a preferential direction. direction. The calculation The calculation is carried is carried outfor out magneticfor magnetic field field dependency dependency of the of effectivethe effective permeability, permeability, which which shows shows that, that in the, in casethe case of randomof random orientation, orientation a range, a range of magnetic of magnetic fields fields occurs, occurs where, where resonant-type resonant-type variations variations are realized,are realized, instead instead of a singleof a single FMR FMR line. line. TheThe experiments experiments are are carried carried out out with with the the composites composites containing containing 15% 15% and and 30% 30% flakes flakes ofof Fe-Si-Nb-Cu-B Fe-Si-Nb-Cu-B alloy. alloy. The The microwave microwave measurements measurements are are performed performed at at frequencies frequencies from from 1212 to to 38 38 GHz. GHz. The The magnetic magnetic field field dependences dependences are are measured measured for for the the transmission transmission and and reflection coefficients moduli. The peculiarities of resonance type are observed connected reflection coefficients moduli. The peculiarities of resonance type are observed connected with absorption of microwaves under FMR condition. For frequencies above 26 GHz, with absorption of microwaves under FMR condition. For frequencies above 26 GHz, the the peculiarities caused by FMAR are also observed. For the composite with 30% flakes peculiarities caused by FMAR are also observed. For the composite with 30% flakes at the at the frequency of 38 GHz, in the region of FMAR, the reflection coefficient maximum frequency of 38 GHz, in the region of FMAR, the reflection coefficient maximum is ob- is observed reaching 300%. Note that ferromagnetic antiresonance is observed in the served reaching 300%. Note that ferromagnetic antiresonance is observed in the composite composite medium which has no DC conductivity. medium which has no DC conductivity. The calculations have been performed to allow a comparison between the calculated The calculations have been performed to allow a comparison between the calculated magnetic field dependences of the transmission and reflection coefficients and the measured magnetic field dependences of the transmission and reflection coefficients and the meas- ones. The calculations are carried out for several frequencies of microwaves and for several ured ones. The calculations are carried out for several frequencies of microwaves and for cases of orientation of flakes—for a group of particles with a preferential direction and several cases of orientation of flakes—for a group of particles with a preferential direction for random orientation of particles. The non-resonant contribution into the magnetic permeabilityand for random is taken orientation into consideration. of particles. It The has non been-resonant shown that contribution the calculation into the reproduces magnetic thepermeability presence of is FMR taken and into FMAR considerat resonancesion. It has and been the typeshown of magneticthat the calculation field dependence reproduces of coefficientsthe presence is presentedof FMR and correctly. FMAR Anresonances approximate and the quantitative type of magnetic correspondence field dependence is obtained of forcoefficients the magnitude is presented of the resonance correctly.variations An approximate of the transmission quantitative coefficient correspondence under FMR is ob- condition.tained for Forthe magneticmagnitude fields of the that resonance are less than variations the field of ofthe FMR, transmission where FMAR coefficient is realized, under qualitativeFMR condition. correspondence For magnetic is reached fields that between are less experiment than the field and theory.of FMR, The where microwave FMAR is realized, qualitative correspondence is reached between experiment and theory. The

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power is calculated and it is shown that a maximum of dissipation occurs near FMR. In the region of FMAR, there is no strong variation of the dissipation, neither in calculation, nor in experiment. The magnetic ﬁeld dependence of the penetration depth of electromagnetic waves is calculated and it is shown that an increase in depth is realized near FMAR and a decrease near FMR.

Author Contributions: Conceptualization, A.B.R.; methodology, D.V.P.; software, D.V.P.; validation, A.B.R. and D.V.P.; formal analysis, D.V.P.; investigation, A.B.R.; writing—original draft preparation, A.B.R.; writing—review and editing, A.B.R. and D.V.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Russian Ministry of Science and Education, under the theme “Function” No AAAA-A19-119012990095-0. The calculation of the transmission and reflection coefficients and Discussion were carried out with support of the Russian Science Foundation, grant No 17-12-01002. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: The authors are grateful to Yu.V. Korkh, for the analysis of distribution the dimensions of flakes, and E.A. Kuznetsov, for help with the microwave measurements. Conflicts of Interest: The authors declare no conflict of interest.

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