Impact of Cr Substitution on Structural, Magnetic, Electric and Impedance Study of Mn-Ni-Zn Ferrites

Shanzida Haque¹, Sajal Chandra Mazumdar^1*, Mohammed Nazrul Islam Khan², Mithun Kumar Das¹

¹Department of Physics, Comilla University, Cumilla, Bangladesh.
²Material Science Division, Atomic Energy Centre, Dhaka, Bangladesh.

DOI: 10.4236/msa.2021.123008                

Abstract

A series of Cr³⁺-substituted Mn-Ni–Zn ferrites; Mn_0.5Ni_0.1Zn_0.4Fe_2-xCr_xO₄ (x = 0.0 - 0.4 in a step of 0.1) were prepared by traditional solid-state reaction route. The structural, magnetic, dielectric properties and impedance spectroscopy of these compositions were studied. Phase identification and lattice constant (a₀) determination were carried out by X-ray diffraction (XRD). The XRD patterns established the fabrication of a single-phase spinel structure. The FESEM micrographs exposed that the average grain size () increased slightly with chromium (Cr) substitution and then decreased for a higher concentration of chromium in the composition. The real part of initial permeability () diminished owing to the enhanced porosity of the compositions with the increase of Cr³⁺ content in the composition. The highest relative quality factor (RQF) was attained for the samples with x = 0.1. The magnetic hysteresis was investigated to know the effect of Cr³⁺ substitution in the composition of the magnetic properties. The decrease of saturation magnetization (M_s) with an enhancement in Cr³⁺ might be triggered by switching of Fe³⁺ ions from octahedral to tetrahedral site. The samples with x = 0.1 exhibited the highest anisotropy constant (K). Curie temperatures of the investigated samples were significantly modified to lower temperatures with the Cr³⁺ content. The frequency characteristics of dielectric properties and impedance spectroscopy had been investigated. The highest dielectric constant (ε') and resistivity were observed for x = 0.1 and x = 0.2 samples. The complex impedance spectra analysis reveals in-depth information about the conduction mechanism, microstructure, and orientation of the grains in the samples.

Keywords

X-Ray Diffraction, Permeability, Dielectric Property, Curie Temperature, Impedance Spectroscopy

1. Introduction

During the last few decades, ferrites are materials of great interest owing to their notable magnetoelectric properties. According to the structure, they have tetrahedral A-site and octahedral B-sites in the AB₂O₄ crystal lattice. These materials exhibit diverse magnetic and electrical properties subject to their cation distribution of the chemical compositions. Numerous cations could be located on the A-site and B-sites to modify their magnetic properties [1] [2] [3] [4]. Sivakumar et al. [1] studied the effect of milling, grain size, frequency, and temperature on the dielectric properties of nanocrystalline Ni-Zn ferrites. They found an unusual rise in dielectric loss with milling resulting from increased electrical conductivity due to oxygen vacancies introduced upon milling. On the other hand, the relaxation frequency was observed to decrease with milling due to increased interaction between charge carriers. Hossain et al. [2] observed that with the increase in Zn content in Ni-ferrite the Curie temperature (T_c), resistivity, and the activation energy dropped off whereas the magnetization, initial permeability, and the relative quality factor (Q) increased. A Hopkinson peak was obtained near T_c in the real part of the initial permeability vs. temperature curves. Chukalkin et al. [5] [6] and Mahmud et al. [6] observed the effect of Zn substitution on the magnetic properties Ni-Zn ferrites. Local angular structures of spin were evident for x = 0.45 with effective sizes of 1 - 10 nm [5] [6].Rezlescu et al. studied the effect of divalent ions [7] and rare earth ions [8] on the structural, magnetic and electric properties of Ni-Zn ferrites.

From the functional viewpoint, Mn-Zn along with Ni-Zn ferrites epitomizes the most significant types. These are utilized in various ferrite devices for instance converters, inductor cores, electromagnetic wave absorbers, recording heads, etc. Mn-Zn ferrites own high ${{\mu }^{\prime }}_i$ and magnetization but these are not appropriate for magnetic uses at higher frequencies owing to their high electrical conductivity and excessive power losses. In contrast, Ni-Zn ferrites have giant resistivity, small-dielectric loss (tanδ_E), and high Curie temperature (T_C), but they have somewhat minimal ${{\mu }^{\prime }}_i$ at higher frequencies. Several authors considered the mixture of these two ferrites to obtain encouraging magnetic characteristics with moderate losses particularly in the excessive frequency range [9] [10] [11] [12]. In earlier studies, the magnetic and electrical effects of this sort of mixture were evaluated [12] [13]. The composition Mn_0.5Ni_0.1Zn_0.4Fe₂O₄ was found to possess the optimum magnetic characteristics for auspicious functions [12] [14] [15]. Preferential occupation of Cr³⁺ ions by octahedral sites in Ni-Zn ferrites replaces Fe³⁺ ions limiting Fe²⁺-Fe³⁺ hoping conduction and increases resistivity. Accordingly, in the present work, we are interested to deal with the enhancement of the electrical properties of the optimum composition Mn_0.5Ni_0.1Zn_0.4Fe₂O₄ by substituting Cr³⁺ ions in octahedral B-site.

2. Materials and Methods

The Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ (where x = 0.0 - 0.4) samples were fabricated by standard solid-state reaction technique. High purity (99.9%) crushes of MnCO₃, NiO, ZnO, Cr₂O_3, and Fe₂O₃ were utilized as untreated substances for the synthesis of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ compositions. The constituents in requisite stoichiometric amounts were mixed mechanically by hand milling in acetone media using a ceramic mortar for 6 hours and calcined in ambient air at 700˚C for 3 hours. Afterward, the calcined powders were shaped into pellets and rings by a hydraulic press. After that, the samples were sintered at 1100˚C, 1150˚C and 1200˚C for 3 hours. The temperature variation for sintering was 5˚C/min during heating and 10˚C/min during cooling. The compositions sintered at 1150˚C were reserved for characterization based on optimum density. Some of these samples were polished for complex initial permeability and electrical characterization.

The XRD of the samples was performed using PHILIPS PW 3040 X’pert PRO X-ray diffractometer. The Nelson-Riley process was utilized for determining the precise a₀of the samples. The Nelson-Riley (N-R) function F(θ) is specified by

$F\left(\theta \right)=\frac{1}{2}\left(\frac{{\cos }^2\theta }{\sin \theta }+\frac{{\cos }^2\theta }{\theta }\right)$ (1)

where θ demonstrates the Bragg angle. The values of a₀for all the peaks of a sample were plotted as a function of F(θ). After that utilizing the least square fitting method precise lattice constants were found out. The point where the least square fitting straight line meets the y-axis (i.e. at F(θ) = 0) is the real a₀of the sample.

Bulk density (ρ_B) of all the samples was measured by the formulae [16]:

${\rho }_B=\frac{m}{\pi r^{2}t}$ (2)

where m, r, and t represent the mass, radius, and thickness of the pellet or ring, individually. The X-ray density (ρ_x) was computed applying the following expression:

${\rho }_x=\frac{nM}{N_{A}V}$ (3)

where n,M,N_A, and V reflect the number of atoms in a unit cell, the molar mass, Avogadro’s number, and unit cell volume, separately. The porosity (P) was determined from the expression

$P\left(\%\right)=\frac{{\rho }_x-{\rho }_B}{{\rho }_x}\times 100\%$ (4)

The electromagnetic properties were characterized at ambient temperature as a function of frequency using WAYNE KERR 6500B Impedance Analyzer. The ${{\mu }^{\prime }}_i$ was calculated with the following relations [17]:

${{\mu }^{\prime }}_i=\frac{L_s}{L_0}$ (5)

and $L_0=\frac{{\mu }_0{N}^{2}S}{\pi \bar{d}}$ (6)

where, L_s is the self-inductance of the sample core and L₀ is the winding coil inductance in absence of sample core, N is the number of windings of the coil (N = 5), S is the cross-sectional area of the ring-shaped sample, $S=d\times h$, where, $d=\frac{d_2-d_1}{2}$ and $\bar{d}=\frac{d_1+d_2}{2}$ is the mean diameter of the ring-shaped sample. The RQF was settled from the ratio, $\frac{{{\mu }^{\prime }}_i}{\tan {\delta }_M}$. The T_C of the compositions was computed from the temperature-controlled ${{\mu }^{\prime }}_i$.

For dielectric measurements, disk-shaped samples have been painted with silver paste on both sides for better electrical contact. The ɛ' was estimated using the given formula [3]:

${\epsilon }^{\prime }=\frac{C}{C_0}$ (7)

and ${\epsilon }^{″}={\epsilon }^{\prime }\tan {\delta }_E$ (8)

where C is the capacitance of the dielectric materials and $C_0=\frac{{\epsilon }_{0}A}{t}$ is resulting geometrically. Here C₀ is the capacitance of the capacitor in the absence of dielectric specimens, ε₀ is the permittivity of free space (ε₀ = 8.85 × 10⁻¹² Fm⁻¹),t is the thickness and $A\left(=\pi r^2\right)$ is the cross-sectional area of the sample painted with silver paste.

Dielectric relaxation and conduction processes were accomplished in the complex modulus (M^*) spectra. The real (M') and imaginary (M'') segments of electric modulus were picked up from the dielectric data from the relations:

$M^{\prime }=\frac{{\epsilon }^{\prime }}{{{\epsilon }^{\prime }}^2+{{\epsilon }^{″}}^2}$ (9)

and $M^{″}=\frac{{\epsilon }^{″}}{{{\epsilon }^{\prime }}^2+{{\epsilon }^{″}}^2}$ (10)

3. Results and Discussion

3.1. X-Ray Diffraction, Lattice Constant, Density, and Porosity

XRD patterns for the samples Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ (x = 0.0, 0.1, 0.2, 0.3, 0.4) sintered at 1150˚C for 3 hours are represented in Figure 1. The peaks in the XRD patterns disclose the single-phase spinel structure. The position of the XRD peaks is in good concurrence with other reports [17]. The a₀, density, and P of the samples are presented in Table 1. To find the precise value of a₀, all the values of a₀ are plotted against the N-R function. The exact value of a₀of the sample is that point where the least square fitting straight line cuts the Y-axis. By plotting

Figure 1. XRD patterns of different Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

Table 1. The a₀, ρ_B, ρ_x,P, and $\bar{D}$ of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

Figure 2. Lattice constant of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ as a function of Cr content.

the a₀against Cr content in Figure 2, it appears that the value of the a₀drops up to x = 0.3 with increasing the value of Cr content showing inconsistency for x = 0.2 and then rises with Cr content. This drop of the a₀with increasing Cr content establishes the fact that the a₀is proportional to the rise of the ionic radius as the ionic radius of Cr³⁺ (0.64 Å) is less than that of Fe³⁺ (0.67 Å) [18]. After x = 0.3, the a₀rises with increasing Cr content. The rise in the a₀ for a higher concentration of Cr was also observed by Li et al. [19] The rise in the a₀ for a higher concentration of Cr may be due to the fact that instead of trivalent the divalent Cr ion substitutes Fe²⁺ in the A-site which results in the increase of lattice constant as the lattice constant of Cr²⁺ is larger than that of Fe²⁺ [19]. Another possible reason is: the oxides of Cr reside at the grain boundary (as seen from the micrographs in section 3.2) thereby obstructing the grain growth and exerts a stress on the unit cell. To compensate for the stress exerted by the grain boundary, the a₀rises for higher Cr³⁺ concentration. The inconsistency from the declining trend for x = 0.2 may be caused by the inhomogeneous diffusion of Cr³⁺ in the lattice.

The alteration of ρ_B and X-ray density (ρ_x) of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ is understood from Figure 3. It is perceived that the ρ_xrises with Cr content linearly up to x = 0.2 and this is because the reduction in the volume of the unit cell increases the density of the sample. Unit cell volume is calculated using the formula $V=a_0^3$. Unit cell volume is reduced with rising Cr content until x = 0.2 as it depends on the a₀which is also reduced with reducing Cr content. From the graph, it is perceived that the ρ_xis higher than the ρ_Bone which means that when the samples are sintered, they might contain cracks and pores on the microscopic scale and vacancies in the lattice on the atomic scale [20]. It is observed from Table 1 that the variation of P of the sample behaves exactly in the opposite manner to that of the ρ_B. The ρ_Bis diminished with increasing Cr³⁺ because the atomic mass of Cr³⁺ is less than that of Fe³⁺. Moreover, the influence of the residual stress decreases the $\bar{D}$ and increases P.

3.2. Microstructure

For the morphological study, a microstructural image of Mn-Ni-Zn-Cr ferrite samples sintered at 1150˚C was taken using FESEM (Model no. JEOL JSM 7600F) so that an insight of grain structure can be understood. The linear intercept technique was employed to determine the mean grain diameter of the samples.

It is evident from Figure 4 that the $\bar{D}$ is increased with Cr content until x = 0.2 because the doped Cr₂O₃ becomes liquid phase in optimized oxygen potential and that liquid phase stimulates the grain growth. The $\bar{D}$ decreases for x = 0.3 and x = 0.4 which may be due to the fact that higher compounds can act as obstacles for visco-plastic deformation and Cr³⁺ resides at the grain boundary and hinders the grain growth. Grain growth and grain boundary mobility are very closely related to each other. Once the homogeneous driving force of grain

Figure 3. Variation of ρ_B, ρ_x, and P of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of Cr content.

Figure 4. FESEM images of various Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

boundary is attained, the sintered body achieves a unified grain size distribution. The compositional variation of the samples affects the grain size distribution. The grain size reflects the presence of a lesser grain boundary area. The pores are left behind the fast-moving grain boundaries and are confined within the grains only when the grain expansion rate is very high.

3.3. Magnetic Characterization

3.3.1. Complex Initial Permeability

The ${{\mu }^{\prime }}_i$ provides a precise description of the stored energy by expressing magnetic induction with the magnetic field. The ${{\mu }^{\prime }}_i$ has been measured against frequency in the range 10 kHz - 120 MHz at ambient temperature for all the samples of the series Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄. From Figure 5, it is noted that with the rise in frequency, the ${{\mu }^{\prime }}_i$ remains stable up to a particular frequency, and after that ${{\mu }^{\prime }}_i$ falls rapidly. Generally, the ferri-magnetic order of ferrite develops from anti-parallel and unequal spin moments in the neighboring magnetic sub-lattices which is determined by the distribution of the transition metal ions [21]. A substantial amount of Fe³⁺ ions reside in the octahedral sub-lattice in Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄. So, the Cr³⁺ ion leads the Fe³⁺ ions to enter into the tetrahedral sub-lattice. Thus the doped sample will show the antiferromagnetic character and the remarkable reduction of ${{\mu }^{\prime }}_i$ with the rise of Cr³⁺ results due to this antiferromagnetic behavior of the samples as in Figure 5.

The frequency dependence of the magnetic loss tangent (tanδ_M) of the composition Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ sintered at 1150˚C at ambient temperature over the range 10 kHz - 120 MHz is illustrated in Figure 6. The value of tanδ_M decreases with frequency arriving at the lowest value and after that becomes nearly constant. The grounds behind reducing tanδ_M with increasing frequency is that, beyond the definite frequency of the electric field, the domain wall motion cannot go along with the external electric field. It is perceived that the tanδ_M reduces significantly with the Cr content at a very high frequency. The low value of the tanδ_Mis the main requirement for the high-frequency magnetic application [16].

The frequency dependence of the RQF of the compositions of Mn_0.5Ni_0.1Zn_0.4 Fe_2−xCr_xO₄ sintered at 1150˚C at ambient temperature over the range 10 kHz - 120 MHz is illustrated in Figure 7. It is perceived that RQF enhances with frequency and after a certain frequency RQF slightly reduces with the rise of frequency.

Figure 5. Variation of ${{\mu }^{\prime }}_i$ of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

Figure 6. Variation of tanδ_M of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

RQF has the highest magnitude at a certain frequency where the tanδ_M has the lowest magnitude [22]. RFQ’s improvement can be attributed to the 10% of Cr addition and this addition is expected to decrease the M_s. The RQF is often used as a measure of performance for inductors used in filtering action.

3.3.2. Temperature-Dependent Complex Initial Permeability

Curie temperature (T_c) measurement is one of the most significant measurements for magnetic materials. The T_cgives substantial information on the magnetic status of a substance for the strength of exchange interaction. The temperature-dependent ${{\mu }^{\prime }}_i$ for Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ is depicted in Figure 8. The ${{\mu }^{\prime }}_i$ is measured at a fixed frequency of 100 kHz. The ${{\mu }^{\prime }}_i$ drops rapidly at the temperature when the magnetism of the samples shifts from ferromagnetic to paramagnetic (at Curie point). The sharp fall of ${{\mu }^{\prime }}_i$ with temperature represents the degree of homogeneity of the samples [23] [24]. In the present case instead of sharp fall ${{\mu }^{\prime }}_i$ drops rapidly owing to the presence of crack and pores within the samples. It is revealed from Figure 8 that when Cr³⁺ is doped in Mn_0.5Ni_0.1Zn_0.4Fe₂O₄, T_c decreases. The decline in T_C is mainly attributable to the weakening of the A-B interaction, which is confirmed from that the ionic radius of Cr³⁺ (0.64 Å) is less than Fe³⁺ (0.67 Å). Due to the replacement of Fe³⁺ by Cr³⁺, the longest distance between moments causes a decrease in A-B interaction in

Figure 7. Variation of RQF of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

Figure 8. Variation of ${{\mu }^{\prime }}_i$ with temperature of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

the Mn_0.5Ni_0.1Zn_0.4Fe₂O₄ relative to all other samples, and consequently, T_cdecreases [4].

3.3.3 Magnetic Hysteresis Loop

Measurement of magnetic hysteresis loop and magnetization for all the samples of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ were carried out by using Vibrating Sample Magnetometer. Observation of Figure 9 showed that M_sreduces with the rise of Cr³⁺ concentration. This happens since the substituted Cr³⁺ ion has a site preference to occupy B-site which makes the iron enter into the A-site from B-site. As a consequence, the exchange interaction between A and B sites turns weaker resulting in intensifying the B-B interaction and weakening the A-B interaction, which in turn reduces the M_sof the samples. Besides, Cr has less magnetic moment than Fe, which may be another reason for the reduction of magnetization with Cr content. Similar results were reported in other reports [25] [26]. The samples having x = 0.1 confirmed improved ferromagnetic property as shown in Table 2.

3.4. Electrical Characterization

3.4.1. Dielectric Property

The variation of frequency-dependent ε' of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ at ambient temperature is illustrated in Figure 10. It can be realized that the ε' drops rapidly with a rise in frequency thereby showing dispersion in the low-frequency range

Figure 9. M-H loops of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites at room temperature.

Table 2. The M_s, coercivity (H_c), K, remanent magnetization (M_r) of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

Figure 10. Variation of ε' for various Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

Figure 11. Variation of ε' with Cr content of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

and become constant at the high-frequency range. The observed dispersion at lower frequencies can be interpreted applying Maxwell-Wagner interfacial polarization which plays a crucial role in such a heterogeneous system [27] [28] [29]. The high values of ε' observed at low-frequency range are because of the presence of the heterogeneity in the composition. Heterogeneities in the samples initiate space charge polarization and contribute towards the high values of ε'. As Cr content rises, the amount of space charge also rises. The reduction of Fe³⁺ concentration causes the transport of some Mn²⁺ in the octahedral site which causes an enhancement of hopping charge carriers and consequently provides a high value of ε'. Figure 11 displays the variation of ε' with Cr content at a frequency of 100 kHz. The highest ε' is obtained for the sample with x = 0.4. The grain boundary impedance decreases due to the presence of the second phase in between grains for high Cr content which in turn increases ε'.

The variation of tanδ_E as a function of the frequency of the composition Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ is shown in Figure 12 sintered at 1150˚C. The tanδ_Eunveils the similar nature of ɛ'. This means that at low frequency the tanδ_Eis high and drops with frequency. The tanδ_Eis proportional to the “loss of energy” from the applied field in the ferrite (in fact this energy is wasted as heat) and hence designated as a tanδ_E. The tanδ_E arises in materials due to the phase difference between polarization and applied field in the high-frequency region.

3.4.2. AC Resistivity

The significant electrical property resistivity is an intrinsic property of magnetic material. The resistivity of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ sintered at 1150˚C was calculated and plotted against frequency as shown in Figure 13. It is observed that AC resistivity also shows significant dispersion as ɛ' at lower frequencies due to the dominant space charge polarization. At low-frequency AC resistivity rises with Cr content up to x = 0.2 and then reduces with a further rise of Cr content because the ɛ' is high for samples with higher Cr concentration i.e., for x = 0.3 and x = 0.4. It is found that ac resistivity reduces with frequency showing significant dispersion and this dispersion is more at low-frequency range which might be due to the variation in the concentration of Fe³⁺/Fe²⁺ and other ions in various samples [30].

3.5. Complex Impedance Spectra Analysis

Transport properties are strongly dependent on microstructure in polycrystalline solids, and impedance spectra generally include functionalities that may be

Figure 12. Variation of tanδ_Eof Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

Figure 13. Resistivity of various Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

specifically linked to microstructure. Figure 14(a) demonstrates the alteration of resistance (Z') with the frequency of the samples at ambient temperature. The higher value of Z' at lower frequency is due to the dominant space charge polarization [31]. Observation confirms that with the rise in frequency up to a specific limit (~1 kHz), the value of Z' rapidly decreases which means conductivity of the ferrite samples increases [32]. As all types of polarization are present at the lower frequency it gives higher magnitudes of Z'. But at the high-frequency region, the values of Z' decreases as the contribution from different parts of polarization ceases, this is because only electronic polarization can carry over the fast variation of field frequency [33] [34]. The alteration of reactance (Z") with the frequency is illustrated in Figure 14(b). The frequency dependence of Z" [Figure 14(b)] behaves similarly as Z' in addition to the appearance of some relaxation peaks due to immobile charges.

According to the brick-layer model, an equivalent circuit comprising of three parallel RC circuits can explain polycrystalline ceramics [35]. Each RC component of the equivalent circuit produces a semicircle in the Z' vs Z'' plot (Cole-Cole diagram). A single semicircle at a higher frequency in Z' vs Z'' plot specifies grain effect, the second semicircle at a lower frequency specifies the presence of grain boundary effect, and a third semicircular arc or spike is often observed which is due to the electrode effect. Figure 15 represents the Cole-Cole plot of various Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites. It is noticed that the first two compositions (x = 0.0 and 0.1) show two semicircular arcs one at high frequency for grain interior and another at a lower frequency for grain boundary effect. The rest of the compositions show a single semicircular arc that specifies the materials have only grain contribution to the conduction process as the semicircle appearing at higher frequency correlates to the resistance of grain only but the semicircle at low frequency is because of the resistance of grain boundaries [36]. The grain boundary impedance arises due to the random alignment of grains existing two-dimensional conductivity. The non-existence of grain boundary impedance for x = 0.2, 0.3, and 0.4 may be due to the presence of the second phase dispersed along the grain boundary area which efficiently short

Figure 14. Variation of (a) Z΄ and (b) Z˝ of Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites as a function of frequency.

Figure 15. Nyquist plots of different Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites: (a) x = 0.0, (b) x = 0.1, (c) x = 0.2, (d) x = 0.3 and (e) x = 0.4 samples, and (f) equivalent circuit [39].

circuits the grain boundary impedance. Depressed semicircular arcs are perceived for every composition. Experimentally it is unusual to find a complete semicircle with its center lying on the axis of the Z'. The agitations which result in depressed semicircular arcs in the Z' axis are: 1) the arcs do not go through the origin because of the presence of other arcs at high frequency and/or the bulk resistance is larger than zero, 2) the centers of the investigational arcs are often relocated below the real axis due to the occurrence of dispersed elements in the material–electrode arrangement [37]. Therefore the relaxation time (τ) is not single-valued and is spread constantly or separately across a mean ${\tau }_m\left(={\omega }_m^{-1}\right)$ and 3) arcs might be considerably twisted by additional relaxations. The arcs at higher frequency side constitute bulk resistance and those at the lower frequency side constitute grain boundary resistance [38] [39]. These responses can be explained by an equivalent circuit comprising of two sub-circuits attached in series as depicted in Figure 15(f) [39]. In general, the impedance is expressed as:

Table 3. The ɛ', grain, and grain boundary resistances of various Mn_0.5Ni_0.1Zn_0.4Fe_2−xCr_xO₄ ferrites.

$Z^{*}=R_g-1/jw{C}_g+R_{gb}-1/jw{C}_{gb}$ (11)

where R_g is the grain resistance, C_g is the grain capacitance, R_gb is the grain boundary resistance and C_gb is the grain boundary capacitance. The arcs in low and high-frequency areas conform to R_gbC_gb and R_gC_g responses, respectively. The grain resistance decreases rapidly with the rise in Cr³⁺ in the composition (Table 3) due to the increased space charge polarization.

4. Conclusion

The XRD patterns identified a single-phase spinel ferrite structure. The FESEM images revealed that the $\bar{D}$ of the samples increased up to x = 0.2 owing to the development of Cr₂O₃ liquid phase which promoted the grain growth and after x = 0.2, $\bar{D}$ decreased due to the shrinkage of the lattice. Among the studied samples, it is clear that 10% of Cr³⁺ doped sample has the highest ɛ' with low loss, highest AC resistivity, good magnetizing capability, and highest RQF. The well saturated magnetic hysteresis loop indicated the ferromagnetic character of the present samples. The T_C is found to be decreased with Cr content in the composition resulting from the smaller ionic radius of Cr than Fe. The impedance spectra analysis established a close correlation of the conduction mechanism with the microstructure of the samples. Both grain and grain boundaries contributed to the total conduction process for the first two compositions. The rest of the compositions had only grain contribution to the conduction process because the second phase dispersed along the grain boundary area efficiently short-circuited the grain boundary impedance. As in the field of electronics and communication devices have to deal with electric as well as magnetic properties, this ferrite composition may be used for the fabrication of different electronic and communication devices.

Acknowledgements

The authors are thankful to the Dept. of Physics, Comilla University, Cumilla; Atomic Energy Centre, Dhaka (AECD), and Ministry of Science and Technology, Government of the People’s Republic of Bangladesh.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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