Retrieving the Intrinsic Microwave Permittivity and Permeability of Ni-Zn Ferrites

Abstract

Mixing rules may be extremely useful for predicting the properties of composite materials and coatings. The paper is devoted to the study of the applicability of the mixing rules to permittivity and permeability and the possibility of retrieving the intrinsic properties of inclusions. Magnetically soft Ni-Zn ferrites are chosen as the object of the study due to their low permittivity and the negligible influence of the skin effect. Due to this, the microwave properties of bulk ferrites may be measured by standard techniques. It is suggested to perform the analysis of the microwave properties of composites filled with Ni-Zn ferrite powder in terms of the normalized inverse susceptibility defined as the volume fraction of inclusions divided by the effective dielectric or magnetic susceptibility of the composite. The measured properties of the bulk ferrite are compared with those obtained by mixing rules from composite materials. The experimental evidence for difference between the mixing rules for permittivity and permeability of a composite, which was previously predicted only theoretically, is obtained. The reason for the difference is considered to be the effect of non-ideal electrical contacts between neighboring inclusions. It is also experimentally shown that the measured permeability of the bulk material may differ from the retrieved one. The measured static permeability is 1400 and the retrieved one is 12. The reason for the discrepancy is the difference between the domain structures and demagnetizing fields of particles and bulk ferrite.

1. Introduction

Microwave technology is extensively used for telecommunications and radar engineering due to the ease of focusing into narrow beams, wide bandwidth, high data transmissions rates and small antenna sizes. Bulk ferrites and composite materials filled with ferrite powders are often used for developing novel materials and coatings for microwave applications [1,2,3]. Ferrite materials are predominantly applied for electromagnetic compatibility [4], anechoic chamber coatings [5] and radar absorbers [6] due to their high magnetic properties and low conductivity. The evaluation of their microwave permittivity ε and permeability μ is important for assessing the potential for practical use.

An important problem is the description of the effective properties of composites as a function of the properties of inclusions and the host matrix. Mixing rules [7,8] are often used to solve this task. A huge number of different mixing rules have been proposed, see, e.g., review [9]. The most commonly used of them are the Maxwell Garnet (MG) model, the effective medium theory (EMT), the asymmetric Bruggeman theory and the Landau–Lifshitz–Looyenga (LLL) formula. The comparison of the mixing rules with the experimental data can be found in [10,11,12,13,14,15,16,17].

The standard approach to the analysis of the effective properties of composites assumes that magnetic inclusions may be characterized in terms of intrinsic permeability. It is usually believed that both the permittivity ε_eff and permeability μ_eff of composites should be described by the same mixing rule [18,19,20,21] at all frequencies. Thus, the mixing rules may be extremely useful for predicting the microwave properties of materials and coatings.

In contrast to the LLL formula, EMT and MG include the form factor of inclusions, N. Strictly speaking, EMT and MG formulas are valid only for N = 1/3, because, only in this case, they are consistent with the LLL theory at μ_i → 1. In practice, the form factor N differs from 1/3 even for spherical inclusions [22].

Composite materials filled with ferromagnetic inclusions are considered in most of the works devoted to the comparison of mixing rules with the experimental data. In contrast to bulk ferromagnets, the microwave properties of bulk ferrites may be measured by standard techniques due to their low permittivity and the negligible influence of the skin effect. This makes it possible to compare the measured intrinsic permeability with that found by mixing rules. It is also important that the permeability of ferrites has a strong frequency dispersion in the microwave range. This makes it possible to evaluate the applicability of mixing rules over a wide range of frequencies. An investigation of the properties of bulk and powdered ferrites was carried out in [23,24,25]. The variation in static permeability and the frequency dispersion parameters with the concentration of inclusions was evaluated using the coherent model approximation and the MG mixing rule. However, the effective permeability was poorly described by mixing rules, and the difference between theory and the experiment was not explained.

The difference between mixing rules for the permittivity and permeability was theoretically predicted in [26]. The magneto-dipole interaction in single-domain particles leads to the concentration dependence of the demagnetizing factor [27], while the depolarization factor is not affected by this interaction. The experimental evidence for difference between dielectric and magnetic mixing rules for composites has not been found in the literature. This paper is devoted to the experimental study of these issues.

Thus, the aims of this study are analysis of the possibility of retrieving the intrinsic properties of inclusions, the comparison of the measured properties of bulk ferrite and properties retrieved by mixing rules from the composite materials and the study of the applicability of the same mixing rule to permittivity and permeability. The investigation of the microwave magnetic and dielectric properties of Ni-Zn ferrites [28,29,30] is carried out using the inverse susceptibility approach described in the next section.

For the first time, it is experimentally shown that the permittivity and permeability of composite materials may obey different mixing rules and the measured permeability of the bulk material may differ from the retrieved one.

2. Materials and Methods

2.1. Research Method

The formulas relating the permittivities of composite ε_eff, host matrix ε_m, and inclusions ε_i for various mixing rules are given in

Table 1. Various mixing rules.

Mixing Rule	Formula	Number
The Maxwell Garnet (MG)	${\epsilon }_{eff}={\epsilon }_m\left(1+\frac{p}{N\left(1-p\right)+\frac{1}{{\epsilon }_i/{\epsilon }_m-1}}\right)$	(1)
The effective medium theory (EMT)	$p\frac{{\epsilon }_i-{\epsilon }_{eff}}{{\epsilon }_{eff}+N\left({\epsilon }_i-{\epsilon }_{eff}\right)}+\left(1-p\right)\frac{{\epsilon }_m-{\epsilon }_{eff}}{{\epsilon }_{eff}+N\left({\epsilon }_m-{\epsilon }_{eff}\right)}=0$	(2)
The asymmetric Bruggeman theory	$\frac{{\epsilon }_i-{\epsilon }_{eff}}{{\epsilon }_i-{\epsilon }_m}=\left(1-p\right){\left(\frac{{\epsilon }_{eff}}{{\epsilon }_m}\right)}^{1/3}$	(3)
The Landau–Lifshitz–Looyenga (LLL)	${\epsilon }_{eff}={\left[\left({\epsilon }_i^{1/3}-{\epsilon }_m^{1/3}\right)p+{\epsilon }_m^{1/3}\right]}^3$	(4)

, where p is the volume fraction of inclusions, and N is the form factor of inclusions. For the permeabilities of composite µ_eff and inclusions µ_i, the formulas are written identically to the permittivities. The mixing rules differ only slightly from each other at low fractions of inclusions, and it is often difficult to choose a model that correctly describes the properties of composites.

The approach proposed in [20] can facilitate this task. It is based on the analysis of the inverse normalized susceptibility η of the composite, which is defined for the permittivity and permeability, respectively, as

$$\eta =\frac{p}{{\epsilon }_{eff}/{\epsilon }_m-1}\mathrm{or}\eta =\frac{p}{{\mu }_{eff}-1},$$

(5)

A consideration of the microwave properties of a composite in the form of normalized inverse susceptibility was shown as a useful tool for studying the effects determining the permeability of composites, such as a magnetic interaction between inclusions and a distribution of the particles in shape. It was shown in [20] that the concentration dependence of the inverse susceptibility may indicate the mixing rule that determines the properties of the composite. The calculated dependence of the susceptibility η on the volume fraction p for various mixing models is shown in Figure 1. The Figure shows the dependencies for the mixing rules given in Table 1. For example, the inverse susceptibility for the MG mixing model, see Equation (1), is written as

$$\eta =-Np+N+1/{\chi }_i,$$

(6)

where χ_i = μ_i − 1 or ε_i/ε_m − 1 is the normalized susceptibility of inclusions.

It is seen that the dependence of the real part of η on the volume fraction is linear, and the imaginary part does not depend on the fraction. The inclination of the line is determined by the effective form factor of the inclusions. The concentration dependence of the permittivity or permeability of the composite can be presented in a form that is more convenient for the analysis. Due to this, the distinctive features of the dependence become more pronounced and useful for understanding the phenomena that determine the properties of the composite.

The inverse susceptibility approach is used to describe the microwave properties of composites filled with ferrite powders for the first time. This approach makes it possible to clearly determine which mixing rule is applicable to the measured permittivity and permeability. A schematic illustration of the research approach is shown in Figure 2.

2.2. Materials under Study

The object of the study is magnetically soft Ni-Zn ferrite of 2000 NN brand. The chemical composition is Ni_0.32Zn_0.68Fe₂O₄. This ferrite was chosen for the research because of the high static permeability and the strong difference in microwave properties between the powder and bulk materials. In general, any material with low permittivity and conductivity can be suitable for this study. However, a small difference in the properties of the powder and bulk material could be attributed to measurement inaccuracy.

The ferrite is made at the National University of Science and Technology MISIS (Moscow, Russia) according to standard ceramic technology [31]. The mixture of initial oxides Fe₂O₃, ZnO and NiO is used as the starting material. The initial oxides are mixed and crushed in a rotary mill, the mixture is calcined, and the synthesized powders are comminuted in the vibration mill. Then, the obtained powder is sintered and pressed into the coaxial sample with an inner diameter of 7 mm and an outer one of 16 mm. The thickness of the sample is 8.9 mm. The density of the ferrite bulk is found from the volume and mass of the sample and is 5.0 g/cm³. The found density coincides with the typical values for Ni-Zn ferrites.

Composites filled with both spherical and stone-like powders of ferrite are investigated. The spherical powder for the composite is obtained during the manufacturing process of the bulk ferrite before pressing and annealing. However, the color of the powder differs from that of the bulk ferrite, which may be due to a change in the material during annealing. The stone-like powder is obtained by grinding bulk ferrite with a diamond disk and sieving with a mesh size of 20 μm. The stone-like powder is obtained directly from the measured sample of the bulk ferrite, so the similarity between the powder and bulk materials is beyond doubt.

The photos of the studied powders are obtained using a scanning electron microscope. The SEM images of spherical and stone-like ferrite powders are shown in Figure 3a and Figure 3b, respectively. It is seen that the size of the spherical particles is less than 2 µm, and the stone-like particles are strongly distributed in size.

The samples of composites are manufactured by mixing calculated amounts of ferrite powder and paraffin wax. The components are heated until the paraffin melts; then, the powder is blended with the melted paraffin until it cools and solidifies. The resulting mixture is pressed into a sample to fit a 7/3 mm measuring coaxial cell. The volume fraction of ferrite powders is calculated from the thickness of the sample and the masses of the paraffin, filler, and the obtained sample. The concentration of inclusions in the samples is approximately 15, 27, 35, 48 and 61 vol.%. The permittivity of the paraffin host matrix is 2.25.

The elemental analysis of the materials under study is carried out by energy-dispersive X-ray spectroscopy (EDX) using a JEOL JCM-7000 scanning electron microscope. The penetration depth of the beam during analysis is less than 5 µm. The ferrite powders are dispersed on a conductive carbon adhesive tape, placed on an aluminum substrate and mounted in a microscope chamber. The powders covered the adhesive tape in a thick layer to neglect the errors during the EDX analysis. A sample of the bulk ferrite is split, and the analysis is carried out at the split site. The elemental analysis data for bulk ferrite are given in

Table 2. The elemental analysis of the studied Ni-Zn ferrite.

Materials	O, atom%	Fe, atom%	Ni, atom%	Zn, atom%
bulk ferrite	54	32	4.9	8.4

. The chemical compositions are the same for powder and bulk ferrite and coincide with the chemical formula.

2.3. Measurement Techniques

The frequency dependencies of microwave permittivity ε(f) and permeability μ(f) are measured with a vector network analyzer. The parameters of the bulk ferrites are measured by the coaxial technique [32,33] in a 16/7 mm coaxial line in the frequency range of 10 kHz to 1 GHz. The parameters of the composite materials filled with ferrite powders are measured by the transmission-reflection (Nicolson–Ross–Weir [34,35]) technique in a 7/3 mm coaxial line in the frequency range of 0.1 to 20 GHz. Additionally, the permeability of the bulk ferrite is determined from the quasi-static measurements of the inductance [32] of the coil with a ferrite ring in the frequency range of 1 kHz to 10 MHz.

3. Results

The measured frequency dependencies of permittivity and permeability of the composite materials filled with stone-like ferrite particles are shown in Figure 4. The color indicates the volume fraction of the ferrite powder in the sample. The frequency dependence of the imaginary permittivity is smoother than that characteristic of a conductive material.

The frequency dependencies of permittivity and permeability of the bulk ferrite are shown in Figure 5. The permeability measured by the inductance technique is shown as gray lines. It is seen that measurements by two different techniques give the same result. It is worth noting that the measurement accuracy of microwave permittivity of the bulk ferrite is low due to gaps between the sample and the walls of the coaxial cell. The obtained frequency dependencies of permeability for the composite and for the bulk ferrite are quite different, even for the volume fraction of 60%. Both the magnitude and position of the magnetic loss differ greatly.

4. Discussion

The inverse susceptibility η approach mentioned above is used to analyze the measured frequency dependencies of permittivity and permeability. The real and imaginary parts of inverse normalized susceptibility are derived from the measured data. The dependencies of inverse electric susceptibility on the volume fraction are shown in Figure 6a, and the data for the magnetic susceptibility are shown in Figure 6b. The dependencies are shown at frequencies of 0.5 to 5 GHz. The dependence of the real inverse magnetic susceptibility at a frequency of 0.5 GHz is close to that at a frequency of 1 GHz.

The dependence of the real inverse magnetic susceptibility on the concentration is linear. The imaginary part is constant. This suggests that the permeability obeys the MG mixing rule, see Equation (6). The effective form factor of inclusions is found from the coefficient of linear dependence. The form factor is the same for all studied frequencies and is approximately N = 0.15 ± 0.01.

The electric susceptibility dependence behaves differently from the magnetic one. The real part of the inverse dielectric susceptibility has a linear dependence on concentration, which is typical for the Maxwell–Garnett mixing rule, see Figure 1. However, the imaginary part is not constant and has a systematic concentration dependence that is the same for all observed frequencies. Therefore, the MG mixing rule is not applicable to the permittivity. Moreover, the form factor found from the electric inverse susceptibility changes with the frequency, see Figure 7, and is different from the form factor obtained from the permeability data. Thus, it is experimentally shown that the permittivity and permeability obey different mixing rules.

It is not clear which mixing rule to use to find the intrinsic permittivity of the ferrite. We retrieve it using the MG formula (1), but the obtained permittivity depends on the concentration and has an implausible frequency dependence. However, the measured and retrieved permittivities of the ferrite are close in order of magnitude.

The permittivity of composites depends not only on the geometric distribution of particle or cluster sizes but also on the effect of non-ideal electrical contacts. The resistance of the cluster should be determined mainly by the resistance of the contacts between the contiguous inclusions that make up the cluster. This resistance, as a rule, is significantly higher than the intrinsic resistance of the conductive particles. This effect is usually not taken into account within the percolation theory and mixing rules and may lead to their inapplicability.

The MG model (1) and the found form factor are applied to retrieve the intrinsic permeability of the ferrite. The retrieved permeability is the same with high accuracy for all studied volume fractions of inclusions, see Figure 8. Frequency dependencies at low frequencies have significant noise due to the peculiarities of measurement and calibration techniques, especially for samples with a low concentration of inclusions. Retrieving the intrinsic permeability from the measured data leads to an additional increase in the noise. Therefore, the curves are smoothed to better represent the results obtained. However, it does not coincide at all with the measured permeability of the bulk ferrite. The values of static permeability differ by two orders of magnitude, although the permeability of the composite is perfectly described by the mixing rule. Thus, it is experimentally shown that the retrieved permeability of the material and the measured one may not match.

The reasons for such a strong discrepancy may be the influence of the skin effect, the errors in concentration calculations, the difference between the materials of the powdered and bulk ferrite, the poor retrieving accuracy and the different magnetic structures of particles and bulk ferrites.

We assume that the powder has the same density as the bulk ferrite. The densities may vary, for example, due to the presence of air pores in the bulk ferrite sample. This leads to an error in the determination of the volume concentration and, consequently, in the calculation of intrinsic properties. However, these errors cannot lead to a discrepancy in the permeability by two orders of magnitude. The found density coincides with the typical values for Ni-Zn ferrites.

The powder is obtained directly from the bulk ferrite sample to avoid material change. The appearance of an oxide shell layer can lead to the difference between powder and bulk materials, since the shell can occupy a significant part of a small particle. It is known that an oxide film appears on the surface of metals. The microstructure of Ni-Zn ferrites was studied in [36], and the presence of an oxide film was not found.

The composites filled with the spherical ferrite inclusions are additionally investigated. The SEM image of the particles is shown in Figure 3a. They are studied in the same way as the composites with stone-like particles. The frequency dependencies of the material parameters are measured, the inverse susceptibilities are calculated, and the intrinsic permeability of the inclusions is found. Its frequency dependence is shown in Figure 9. The intrinsic permeability is also retrieved by the MG model (1) with high accuracy but also does not coincide with that of the bulk ferrite, see Figure 10. Moreover, it does not match the permeability of the stone-like inclusions.

It is known that the skin effect leads to a distortion in the frequency dependence of the microwave permeability [37]. The difference between the intrinsic permeability of the ferromagnetic inclusions of various shapes and sizes is usually attributed to the influence of the skin effect. The resistivity $\rho$ of the studied ferrite is quite high, approximately 1000 Ohm × cm, and the skin effect should have a rather weak influence. The skin depth δ may be estimated using the formula:

$$\delta =\sqrt{2\rho /\left(2\pi f\mu \right)}.$$

(7)

It is calculated to be approximately 5 cm at a frequency of 100 MHz, which is larger than the thickness of the sample and much larger than the particle size. Moreover, the permeability measured by the coaxial and the inductance techniques is the same. Therefore, the skin effect may be neglected.

Another reason for the discrepancy may be the poor retrieving accuracy if the intrinsic permeability is large. It is difficult to make a composite with a volume fraction above 70% using simple mixing. Figure 11 shows the theoretical dependencies of the static permeability of a composite on the volume fraction of inclusions at the permeability of inclusions of 1000, 50, and 10. The dependencies are calculated using the MG model. The greatest increase in permeability is observed at high concentrations. If the measurement error of the composite effective properties is approximately 10%, it is simply impossible to retrieve intrinsic permeability above 50. However, the permeability of the studied bulk ferrite reaches high values at low frequencies. In the region of 100 MHz, it is still rather small. Therefore, this cannot be the reason for the discrepancy.

The main reason for the discrepancy between permeabilities may be the different magnetic structure of the bulk ferrite and particles. The studied ferrite particles are rather small, due to which their domain structure may differ from that of bulk ferrite, which leads to a difference in magnetic properties. The inapplicability of the same mixing rule for permittivity and permeability may also be explained by the influence of the domain structure. Permittivity is determined by particles and permeability by domains.

The typical size of the domains in Ni-Zn ferrites is 1–4 µm [1,38,39]. Stone-like particles are quite strongly distributed in size, and among them should be present both single-domain and multi-domain particles. The interaction between inclusions in a composite with single-domain magnetic inclusions may have specific features. This problem was theoretically investigated in [26], where it was assumed that there are significant demagnetizing fields in single-domain particles. The interaction of these fields with the surrounding particles is not described by mixing rules. Note that demagnetizing fields may also be observed on the surface of multi-domain particles, which can cause differences in the effective magnetic properties of composites from the results of mixing rules. On the other hand, single-domain particles may form magnetic agglomerations during the preparation of composite samples, forming multi-domain clusters.

In any case, no experimental confirmation of the difference in the properties of magnetic composites containing single-domain and multi-domain inclusions was found in the literature. The differences in the microwave magnetic properties of composites containing micro- and nano-sized Fe₂O₃ powders are experimentally illustrated in [17]. However, it may be simply explained by the fact that these two types of powders contained different Fe₂O₃ phases, as can be concluded based on the difference in the color of the powders described in the paper.

According to [26], an increase in the concentration of magnetic inclusions in a composite significantly increases the interaction between these inclusions, and each particle should be considered as placed in some magnetic effective medium. The presence of this medium changes the demagnetization of the particle and, hence, the gyromagnetic spin resonance conditions. However, no change in the demagnetization factor up to volume concentrations of 60% is observed in our study. These effects likely appear at higher concentrations, which leads to a difference between the retrieved permeability and the measured one. The discrepancy should disappear for larger powders with a significant number of domains per particle.

The main methods for revealing the domain structure of the magnetic samples are magneto-optical techniques. In addition, to assess the influence of the domain structure on microwave magnetic properties of ferrite particles, the dependence of the effective and intrinsic permeability on the size of inclusions may be studied. In this case, the influence of the shape of the inclusions should be excluded. This can be performed by studying composites with spherical inclusions of various sizes. Solving these problems may be a topic for future research.

The experimental evidence for difference between mixing rules for permeability and permittivity for composites is obtained in this paper. This difference was also theoretically predicted in [26]. The magneto-dipole interaction in single-domain particles leads to the concentration dependence of the demagnetizing factor [27], while the depolarization factor is not affected by this interaction. However, the difference between the material parameters is observed at the studied concentrations, while the distortion of the demagnetization factor is not. Therefore, the reason for the difference is most likely the effect of non-ideal electrical contacts between neighboring inclusions, which is not taken into account by the mixing rules. This is supported by the fact that the permittivity increases more slowly with increasing concentration than the permeability.

The results obtained in the paper may be used in practice for the fabrication of composite materials for microwave absorbers [40], structured metasurfaces [41], infrared thin-film absorbers [42] and advanced optical composite materials [43].

5. Conclusions

The normalized inverse susceptibility approach is applied to analyze the microwave properties of composite materials filled with Ni-Zn ferrite powder. The applicability of the mixing rules to permittivity and permeability and the possibility of retrieving the intrinsic properties of inclusions are studied. The measured properties of the bulk ferrite are compared with those obtained by mixing rules from composite materials. The experimental evidence for difference between dielectric and magnetic mixing rules for composites, which was previously predicted only theoretically, is obtained. The reason for the difference in the mixing rules for the permittivity and permeability is considered to be the effect of non-ideal electrical contacts between neighboring inclusions.

It is also experimentally shown that the measured permeability of the bulk material may differ from the retrieved one. The influence of the skin effect, the errors in concentration calculations, the difference between the materials of powdered and bulk ferrite, the poor retrieving accuracy and the different magnetic structures of particles and bulk ferrites are considered as reasons for the discrepancy. The reason for the discrepancy is the difference between the domain structures and demagnetizing fields of the particles and bulk ferrite. There are significant demagnetizing fields in single-domain particles. The interaction of these fields with the surrounding particles is not described by the mixing rules.