Study of Sintering Temperature Effects on the Structural and Electrical Properties of Mg_0.6Co_0.4FeCrO₄ Ferrites Synthesized by the Sol–Gel Method

Abstract

In the present study, Mg0.6Co0.4FeCrO4 ferrite samples were synthesized by sol–gel method at 900 °C and 1100 °C. We have studied in detail the effect of the sintering temperature on their structural, infrared, electrical and dielectric properties. The results showed that by increasing the sintering temperature, the lattice constant, the grain size, the intensities of absorption bands, the conductivity and dielectric constants of the samples increase. The X-ray diffraction results show that our samples crystallize in the cubic structure with $\mathrm{Fd}\overline{3}\mathrm{m}$ space group. FTIR spectra showed an absorption band corresponded to metal–oxygen stretching vibration in the tetrahedral sites. A metal–semiconductor transition was observed for the samples at a specific temperature T_MS. When the sintering temperature increases, the value of the electrical conductivity increases, however the transition temperature (T_MS) between metal and semi-conductor behaviors remains unchanged. Dielectric constants rapidly decrease as frequency rises. This phenomenon has been explained using the Maxwell–Wagner theory of interfacial polarization. The electrical modulus and impedance investigation demonstrate that the samples exhibit an electrical relaxation phenomenon that is not of the Debye type. The equivalent electrical circuit with [R_g // CPE_g + R_gb //CPE_gb] configuration is the most appropriate model to modelize the prepared samples. The activation energies calculated from the dc conductivity, impedance, and modulus studies are quite close, indicating that the relaxation and conduction processes are caused by the same defect.

1 Introduction

Ferrites with a spinel structure have a chemical formula of ${(\mathrm{Me}}_{1-\updelta }^{2+} {\mathrm{Fe}}_{\updelta }^{3+})[{(\mathrm{Me}}_{\updelta }^{2+} {\mathrm{Fe}}_{2-\updelta }^{3+}$]${\mathrm{O}}_{4}^{2-}$, where parentheses and square brackets denote the cation sites of tetrahedral (A) and octahedral (B) coordination, respectively [1, 2]. δ represents the degree of inversion expressed as the fraction of the (A) sites occupied by Fe³⁺ ions. Spinel ferrites is classified into three categories (normal spinel structure if δ = 1; inverse spinel structure if δ = 0 and mixed spinel structure if 0 < δ < 1).

Ferrites are the subject of intensive scientific investigation in order to improve their functionality for technological applications [3]. Ferrites have intriguing physical features, such as their structural, electric, dielectric, magnetic, and optical properties, making them valuable materials for technology [4]. Nowadays with the development of technology the ferrites are also promising candidates for various technological applications, such as microwave absorption, ferrofluids, humidity sensor and photo detectors, as well as high density storage devices [5], biosensing, MRI technology with variable magnetic characteristics, electronic devices, and medication delivery [6], biomedical applications like drug delivery system, dye-sensitized solar cells, high frequency transformers, biomedical applications like drug delivery system and so on [7], magnetic refrigeration systems, gas sensing, catalytic application and so on [8]. However, the electrical characteristics of spinel ferrites must be improved in order to enhance their electronic applications. It is crucial to examine their dielectric behavior at various frequencies in this situation.

Spinel ferrites have been prepared using several synthesis methods among them we can cite the electrochemical [9], plasma synthesis [10], co-precipitation [11], hydrothermal [12], sol–gel [13], the reverse micelle [14], and the citrate precursor [15] methods. According to a review of the literature, the preparation technique, concentration, chemical composition, porosity, type of substituted element, sintering temperatures, grain size, microstructure, heat treatment, difference in ionic radii and the distribution of metallic ions among crystallographic crystal lattice sites are the main factors that affect the dielectric properties of ferrites [16, 17].

Cobalt ferrite (CoFe₂O₄), one of the spinel ferrites, has received a lot of attention recently because to its important electrical characteristics, including low eddy current, high dielectric constant, low dielectric losses, and high resistivity [18] and magnetic properties like the highest values of magneto-crystalline anisotropy and magnetostriction [19], high coercivity, moderate saturation magnetization [20]. These characteristics make cobalt ferrite a viable substance for use in the creation of computer memory systems, solar cells, microwave, drug delivery, actuators, Li batteries and super capacitors [21, 22]. Magnesium ferrite (MgFe₂O₄) is a promising candidate for a variety of applications due to its low dielectric losses, low magnetic and high electrical resistivity. These applications include usage in sensors and electronic device, noise filters, fabrication of radio frequency coils [23], catalyst, humidity sensor [24], microwave absorption, semiconductors, magnetic applications including transformers and gas sensor [25].

The studies revealed that the magnetic characteristics of the (MgFe_2-xCr_xO₄) ferrites are significantly altered when iron is replaced with chromium [26]. Mg ferrite with Cr substitution (MgFe_2-xCr_xO₄) is appropriate for the long wave region of the high frequency spectrum because to extraordinarily low dielectric losses [27].

The study of the chromium substituted cobalt ferrites (CoFe_2-xCr_xO₄) show only with increase of Cr³⁺ content the coercivity has decreased [28], magnetostriction coefficient values are increased [29], the magnetization has decreased, and the compounds are being converted into soft magnetic materials [30]. The CoFe_2-xCr_xO₄ ferrites are used in biotechnological application, sensors, high density magnetic information storage, recording tapes, high quality digital printing, switching devices, microwave devices, transducers, actuators [18].

In this regard the study of the properties of Cr-substituted MgCoFe₂O₄ with formula (Mg_0.6Co_0.4FeCrO₄) is interesting. In present work, the investigation is concerned with the preparation of the (Mg_0.6Co_0.4FeCrO₄) ferrite and the effect of sintering temperatures (900 and 1100 °C) on the structural characteristics, on the IR bands as well as the dielectric properties of samples. The sol–gel technique is used to create the sample. It is one of the most exploited methods for the preparation of ferrite samples. It is a simple and affordable chemical process which is mainly used to generate ultrafine powder with limited size distribution in a fairly short time and at low temperature. It also improves the product's purity, stoichiometry and homogeneity [31, 32]. At room temperature, the structural characterization was carried out and the impedance spectroscopy method was utilized to ascertain the dielectric properties as a function of frequency and temperature.

2 Experimental Methods

2.1 Preparation

The sol–gel process was used to prepare the Mg_0.6Co_0.4FeCrO₄ ferrites using stoichiometric quantities of Mg(NO₃)₂·6H₂O, Co(NO₃)₂·6H₂O, Fe(NO₃)₃·9H₂O, Cr(NO₃)₃·9H₂O precursors and citric acid of formula C₆H₈O₇H₂O. All these compounds have a purity of 99.9%. The samples whole synthesis procedure is shown in Fig. 1. The samples were synthesized by mixing stoichiometric amounts of metal nitrates in 150 ml distilled water while thermally stirring at 95 °C to create a mixed solution. After the nitrates had completely dissolved in the solution, controlled amounts of citric acid, which is employed as a complexing agent for various metal ions, were added and dissolved with stirring. Nitrates and citric acid were mixed at a 1:1 molar ratio. After that, a small amount of ammonia was added to the solution to bring the pH level down to roughly 7. Following this step, ethylene glycol (which has been employed as an agent for polymerization) was added and the resultant solution was heated for 4 h on a hot plate with vigorous stirring until the gel was produced. The latter was heated in oven for 12 h at 250 °C to obtain a foamy dry which is ground with a mortar. Then, the powders were ground and calcined for 24 h at 800 °C after being annealed at 600 °C for 12 h. The final step was to press the resulting Mg_0.6Co_0.4FeCrO₄ powders into pellets with a diameter of 10 mm and a thickness of roughly 2 mm under 100 MPa. The pellets were then split into two portions and sintered separately for 24 h at 900 °C and 1100 °C, respectively.

Fig. 1

Detailed schematic representation of the synthesis for Mg_0.6Co_0.4FeCrO₄ ferrite by the sol–gel method

2.2 Characterizations

The crystal structure and phase purity of each sample was examined by X-ray diffraction (XRD) using a “PANalytical X'Pert Pro” diffractometer with CuKα radiation (λ = 1.5406 Å) and a nickel filter to block the K_β ray in the 20°–80° range. The FTIR spectra studies were carried out at room temperature at room temperature with a wave number range of 450 to 4000 cm⁻¹ and a resolution of 1 cm⁻¹ using a Shimadzu Fourier Transform Infrared Spectrophotometer (FTIR-8400S). Electrical characterizations have been done by using an instrument called N4L-NumetriQ (model PSM1735) analyzer at different temperatures ranging from 300 to 600 K in the 10²–10⁶ Hz frequency range.

3 Results and Discussions

3.1 Structural Analyses

The X-ray diffraction patterns for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at two different temperatures (900 °C and 1100 °C) are shown in Fig. 2. It can be noticed that all of the Bragg reflections have the following designations: (220), (311), (222), (400), (422), (511), (440), (620), (533), (622) and (444) [33]. Our samples reveal a cubic crystal structure with the $\mathrm{Fd}\overline{3}\mathrm{m}$ space group, and there are no apparent secondary crystallographic phases of impurities. The absence of further impurity-related peaks shows that our samples are of excellent purity. The Table 1 presents all the refined structural parameters of the samples. We suggested the cation distribution ${{({\mathrm{Fe}}_{1}^{3+})}_{\mathrm{A}}{{[\mathrm{Mg}}_{0.6}^{2+}{\mathrm{Co}}_{0.4}^{2+}{\mathrm{Cr}}_{1}^{3+}]}_{\mathrm{B}}\mathrm{O}}_{4}^{2-}$ for the Rietveld refinement technique. The atomic positions for the cations occupying the tetrahedral site (A) are 8a (1/4, 1/4, 1/4), those for the cations occupying the octahedral site [B] are 16d (1/2, 1/ 2, 1/2) and those for O are 32e (x, y, z). We have shown from the XRD patterns that the intense peak (3 1 1) exhibits a little shift towards a lower diffraction angle (2θ) as the sintering temperature rises. This leads us to the conclusion that the lattice parameters increase slightly with the sintering temperature. Moreover, we see that the width of the peak decreases and its intensity increases as the sintering temperature increases. This one is due to the thermally activated enhancement of crystalline structure. The lattice parameter and cell volume in the samples increased with increasing sintering temperature. This shows the relaxation of the lattice structure at higher sintering temperatures [34]. The samples sintered at 900 °C and 1100 °C have lower lattice constants than those of the parent compound Mg_0.6Co_0.4Fe₂O₄ [35]. This is due to the lower ionic radius of Cr³⁺ (${\mathrm{r}}_{{\mathrm{Cr}}^{3+}}$= 0.63 Å) than Fe³⁺ (${\mathrm{r}}_{{\mathrm{Fe}}^{3+}}$= 0.67 Å) [36]. The following equation was used to calculate the X-ray density [37]:

$${\mathrm{d}}_{\mathrm{x}}=\frac{8\mathrm{M}}{{\mathrm{Na}}^{3 }}$$

(1)

where N is the Avogadro’s number, M is the sample’s molecular weight, and 8 is the number of molecules in a spinel lattice unit cell. Using the most intense diffraction peak (311), the average crystallite size was calculated using XRD patterns in accordance with the Debye–Scherrer relation [38, 39]:

$${\mathrm{C}}_{\mathrm{s}}=\frac{0.9\uplambda }{\mathrm{\beta cos\theta }}$$

(2)

where β is the full-width at half-maximum of the (311) peak, θ is the diffraction angle and λ is the X-ray wavelength of the Cu-k_α radiation (λ = 1.5418 Å). Figure 3 shows good agreement between the calculated and observed profiles. The results presented in Table 1 show that when the sintering temperature increases, the grain size increases. During sintering, many crystallites with the same orientation fuse and form a large grain. The increase in grain size with increasing sintering temperature is due to recrystallization of samples and increase in unit cell volume. Indeed, the tendency of crystallites to join together and constitute large grains is evidently realized in the sample sintered at 1100 °C than the one sintered at 900 °C [40].

Fig. 2

The XRD patterns for Mg0.6Co0.4FeCrO4 ferrites sintered at 900 °C and 1100 °C

Table 1 Structural parameters determined after structural refinement according to the Rietveld method for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °

Fig. 3

Typical example for the Rietveld refinement of the XRD pattern for  Mg_0.6Co_0.4FeCrO₄ ferrite sintered at 1100 °C

3.2 FTIR Spectrum Analysis

Infrared spectroscopy is used to locate ions in crystals and identify functional groups in materials by analyzing vibrational modes [41, 42]. In general, the FTIR spectra of ferrites include two distinct absorption peaks, which are related to oxygen bond vibrations with metal cations in tetrahedral and octahedral sites [43, 44]. Generally, the strongest and most apparent peaks are between 350 and 700 cm⁻¹. Their location and intensities depend on the type of cations present and how they are distributed among the sublattices [45]. FTIR spectrum for Mg_0.6Co_0.4FeCrO₄ ferrites measured in the 450 to 4000 cm⁻¹ frequency range as depicted in Fig. 4. The spectra revealed the existence of two vibration bands at about 582.876 cm⁻¹ for the sample sintered at 900 °C and 597.383 cm⁻¹ for the one sintered at 1100 °C, which are related to the stretching vibration caused by the interactions between Metal–oxygen in the tetrahedral site [46]. As the sintering temperature rises, the values of ν_A rise. This rise is explained by a reduction in the R_A distances between the cations at the tetrahedral sites and oxygen as a result of an increase in sintering temperature [47]. When the sintering temperature is high, the value of ν_A shifts to a higher frequency due to a significant improvement in the crystallinity of the samples [48]. However, the FTIR spectra also show the presence of a peak at 883 cm⁻¹, which indicates the presence of a C-H bending vibration [49]. The clear band that appears at 1100 cm⁻¹ is mainly due to the coupling bending vibrations O–H and stretching vibrations C–O [50]. The observed peak at frequency 1635 cm⁻¹ is attributed to both the stretching vibration and the bending vibration of the H–O–H bond [51]. The band between 3570 and 3900 cm⁻¹ is due to the stretching vibrations of the water which is absorbed by the surfaces of the samples [48]. The force constant K_T for the tetrahedral site is determined using the following equation [52]:

$${\mathrm{K}}_{\mathrm{T}}=7.62 {\mathrm{M}}_{\mathrm{A}} {\upnu }_{\mathrm{A }}^{2}{10}^{-3}(\mathrm{dyne}/\mathrm{cm})$$

(3)

where MA represents the average molecular mass of the entity formed by the cations existing in the tetrahedral site. The determined values of the force constant KT are grouped in Table 2. According to the results presented in this table, it can be seen that the KT value increases as the sintering temperature increase. The increase in KT with increasing sintering temperature is also due to the decrease in bond length RA, which is established between the cations and the oxygen of the tetrahedral sites [53].

Fig. 4

FTIR spectra for  Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

Table 2 The values of the force constants K_T  obtained at room temperature for  Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

3.3 Electrical Conductivity

Figure 5 shows the variation of electrical conductivity (σ) with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites. The following equation is used to calculate the conductivity of these samples [54]:

$$\sigma = \left( \frac{e}{s} \right)\left[ {\frac{{Z^{\prime 2}}}{{Z^{\prime 2} + Z^{\prime \prime 2}}}} \right]$$

(4)

where e is the thickness of the pellets and s is its area, (Z′) is the real part and (Z″) the imaginary part of the complex impedance. Moreover, the following formula analyzes the phenomenon of dispersion of the conductivity [54]:

$$\upsigma =\left[\frac{{\upsigma }_{\mathrm{s}}}{{(1+\uptau }^{2}{\upomega }^{2})}\right]+\left[\frac{({\upsigma }_{\infty }{\uptau }^{2}{\upomega }^{2})}{{(1+\uptau }^{2}{\upomega }^{2})}\right] +\mathrm{ A }{\upomega }^{\mathrm{s}}$$

(5)

where ${\upsigma }_{\mathrm{s}}$ and ${\upsigma }_{\infty }$ denote respectively the values of the conductivity at low and high frequencies,$\uptau$ is the relaxation time, A is a pre-factor which is influenced by both composition and temperature, ‘s’ represents the frequency exponent and $\upomega$= 2 $\uppi$ f represents the angular frequency. The change of exponents with the evolution of the temperature gives an idea of the possible conduction model. There is a physical significance to the value of ‘s’. When s ≤ 1, the electron is abruptly hopping while also traveling in translation. While s > 1 shows that the motion includes limited hopping between close locations. The exponent ‘s’ depends on both the temperature and the nature of the material, but it is independent of the frequency [55]. The frequency-dependent change of conductivity at various temperatures shows that below the temperature T_MS (440 K for the two investigated samples) the conductivity decreases with the frequency. This one demonstrates that the behavior of our samples is metallic [56]. Above the T_MS temperature, the conductivity increases with increasing frequency. In this case, our samples exhibit semiconductor behavior [57]. T_MS is therefore the transition temperature from metallic behavior to semiconductor behavior. The increase in conductivity with increasing frequencies is explained by the pumping force which allows charge carriers to be moved between different localized states while also allowing the trapped charges to be released from different trapping centers. These charge carriers, together with the electrons generated by the valence exchange of various metal ions, engage in conduction [58]. The experimental results of electrical conductivity shown in Fig. 5 are fitted according to Eq. (5). Symbols in this figure are the experimental data and red lines are the best fits. It can be seen in this figure that there is a very good agreement between the experimental measurements and their adjustments using Eq. (5) over the entire frequency range. By studying the evolution of the exponent ‘s’ as a function of temperature, it is therefore necessary to examine the different results seen in the zone where the conductivity depends on ω^s. Figure 6a shows that the values of ‘s’ decrease until they reach a minimum and then increase. In this case, the most suitable model for describing the electrical conduction process in these materials is the Overlapping-Large Polaron Tunneling (OLPT) model. Beyond T_MS, the values of ‘s’ decrease as the temperature increases. We then deduce that the appropriate model to explain the transport properties in these materials is the Correlated Barrier Hopping (CBH) [59]. For the two samples sintered 900 °C and 1100 °C, the values of ‘s’ are greater than 1. This suggests that charge carrier hopping happens between two nearby sites [60]. Figure 6b illustrates the variation of the dc conductivity with temperature. We notice from this figure that for temperatures lower than T_MS, the values of σ_s decrease during the increase in temperature, then they increase when the temperature increases for temperatures above T_MS. These results clearly illustrate that our materials exhibit a metal–semiconductor transition at the T_MS temperature. It is also noticed that when the sintering temperature increases, the values of σ_s are increased in the whole explored temperature range and the T_MS does not change. This non-change of the transition temperature is in good agreement with other work [61]. The experimental results of the dc conductivity σ_s are fitted using the Arrhenius law by the following relation [62]:

$${\upsigma }_{\mathrm{s}}= {\upsigma }_{0}\mathrm{exp}\left(-\frac{{\mathrm{E}}_{\mathrm{dc}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)$$

(6)

where σ₀ is a pre-exponential factor, E_a represents the activation energy and k_B is the Boltzmann constant and T is the absolute temperature. Figure 6c displays the plots of Ln(σ_s × T) versus (1000/T). For the two samples sintered at 900 °C and 1100 °C, the values of E_dc derived from the slope of the linear fitting plot are equal to 0.794 eV and 0.524 eV, respectively. The high E_dc value for the two samples investigated is owing to the high degree of disorder in these two samples induced by the large number of doping elements (Mg, Co, Cr) in these samples [63]. We can observe that the values of the activation energies decrease when the sintering temperature is high. This decrease in the activation energy means that the electrons during their jump, they require less energy. This one is due to the increase in the temperature of sintering. These different values of activation energies of our samples are lower than those obtained by other studies on ferrites compounds [63]. Therefore, these samples are classified as good conductors.

Fig. 5

Variation of electrical conductivity (σ) with frequency at various temperatures for  Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C. Fit to experimental results is represented by red lines (Color figure online)

Fig. 6

a Variation of dc conductivity (σs) with temperature for  Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C. b The temperature dependence of the exponent (s) for the two samples. c Ln(σdc × T) as a function of (1000/T) at a frequency of 100 Hz. Red lines represent the linear fit of conductivity data (Color figure online)

3.4 Dielectric Properties

The dielectric properties of ferrite nanoparticles are influenced by several factors including the synthesis technique, chemical composition, phase purity, crystal structure, the ratio of Fe²⁺/Fe³⁺, morphology, oxygen anion vacancies in lattices, cations distribution, frequency of the applied electric field, grain size and sintering temperature [64, 65]. The adaptation of these parameters makes it possible to obtain a material with the desired dielectric behavior. In this case, the dielectric response of the ferrite nanoparticles is considered as a complex quantity which is the complex permittivity, its expression is translated by the following formula ε* = ε′-jε″, where (ε′) and (ε″) respectively designate the real part and the imaginary part of the dielectric constant, with (ε′) represents the stored energy and (ε″) designates the energy dissipated in the dielectric medium. The equation shown below is used to calculate the complex permittivity [66, 67]:

$${\upvarepsilon }^{*}=\frac{1}{\mathrm{j\omega }{\mathrm{C}}_{0}{\mathrm{Z}}^{*}}$$

(7)

where $\upomega$ is the angular frequency, ${\mathrm{Z}}^{*}$ is the complex impedance spectroscopy of the sample and ${\mathrm{C}}_{0}={\upvarepsilon }_{0 }\mathrm{S}/\mathrm{e}$ represents its geometric capacity, with ${\upvarepsilon }_{0}$ represents the permittivity for free space. However, the real part (ε′) and the imaginary part (ε″) of the dielectric constant as well as the dielectric loss tangent (tan δ) are determined using the equations shown below [68].

$${\upvarepsilon }^{\mathrm{^{\prime}}}=\frac{{\mathrm{Z}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}}{\upomega {\mathrm{C}}_{0}({\mathrm{Z}}^{\mathrm{^{\prime}} 2}+{\mathrm{Z}}^{\mathrm{^{\prime}}\mathrm{^{\prime}} 2})}$$

(8)

$${\upvarepsilon }^{\mathrm{^{\prime}}\mathrm{^{\prime}} }=\frac{{\mathrm{Z}}^{\mathrm{^{\prime}}}}{\upomega {\mathrm{C}}_{0}({\mathrm{Z}}^{\mathrm{^{\prime}} 2}+{\mathrm{Z}}^{\mathrm{^{\prime}}\mathrm{^{\prime}} 2})}$$

(9)

$$\mathrm{tan \delta }=\frac{{\upvarepsilon }^{\mathrm{^{\prime}}\mathrm{^{\prime}}}}{{\upvarepsilon }^{\mathrm{^{\prime}}}}$$

(10)

where ${\mathrm{Z}}^{\mathrm{^{\prime}}}=|\mathrm{Z}|\mathrm{cos}\Phi$ and ${\mathrm{Z}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}=|\mathrm{Z}|\mathrm{ sin}\Phi$ are respectively the real part and the imaginary part of the complex impedance Z, and $\Phi$ is the argument defined as $\Phi =\frac{\pi }{2}-\delta$ where δ is the phase shift. Figures 7 and 8 illustrate, respectively, the variations of the real and imaginary parts of the dielectric constant as a function of frequency at different temperature for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C. It is clear from the two exposed examples that the values of the real part (ε′) and the imaginary part (ε″) of the dielectric constant are high at low frequency, and then they decrease as frequency increases, becoming extremely low in the high frequency zone. In this region the values of (ε') and (ε″) approach the same values and become independent of the frequency. Similar findings have been seen in other studies [53, 69]. The dependence of the dielectric permittivity observed during the increase of frequency reflects that there is dielectric dispersion [70]. Dielectric dispersion with frequency in ferrite can be used to explain on the basis of Koop’s theory [71] and Maxwell–Wagner interfacial type of polarization [72]. From to these theories, the ferrite material's dielectric structure may be thought of as a heterogeneous structure made of highly conductive grains and weakly conducting grain boundaries. Grain boundaries with low conductivity are considered useful at low frequencies, while grains with high conductivity are considered useful at high frequencies [73]. Under the effect of an applied alternating electric field, electric charges move in the grains and grain boundaries of the ferrite materials. Therefore, by application of the electric field, the movement of charges through grains and grain boundaries is interrupted at the grain boundaries and they exhibit accumulation of charges at the grain boundaries. Since the grain boundaries are very resistive, this accumulation of charges leads to an interfacial polarization [74]. Therefore, at low frequency, the dielectric constant has a high value. Moreover, when the frequency of the applied field increases, the exchange of electrons between the Fe²⁺ and Fe³⁺ ions does not follow the evolution of the external field applied at a certain frequency in the high frequency region. Consequently, the probability of reaching the grain boundary by the electrons decreases and this leads to a decrease in the interfacial polarization. Indeed, at high frequency, the dielectric constant becomes minimal and at the same time it does not depend on the frequency. As a result of the low dielectric constant values in the high frequency region, these samples are especially well suited for applications in this region [75]. With increasing sintering temperature, the diffuse and clumped atoms of ferrite compounds receive energy during sintering. This energy increases with increasing sintering temperature. As the energy increases, then more atoms clump together and therefore the size of the crystals increases from 48 to 56 nm [76]. Furthermore, the larger crystallites size, the greater the contact between neighboring grains, then the movement of electrons will be easier, and they can flow easily to grain boundaries which leads to an increase in the polarization. Moreover, increasing the sintering temperature causes an increase in the generation of Fe²⁺ ions and hence the jumping process is stimulated. Similarly, the increase in temperature also releases more of the localized dipoles, which are therefore aligned by the applied electric field in its direction [77]. Consequently, raising the sintering temperature increases the dielectric polarization and therefore the permittivity increases [78]. Figure 9 displays the evolution of the dielectric loss tangent (tanδ) for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C as a function of frequency at various temperatures. According to Fig. 9, the two samples exhibit relaxations in the metallic regime (T < T_MS), with a loss peak at specific frequencies appearing at lower frequencies and disappearing at higher frequencies. The loss peak is observed when the electron hopping frequency between Fe²⁺ and Fe³⁺ and the frequency of the established electric field are equal. The condition for having a resonance peak is given by the following relation [79]:

$$\omega \tau \, = \,{1}$$

(11)

where ω = 2πf represents the angular frequency, f is the frequency of the applied field and τ represents the relaxation time. According to Iwauchi [80], the presence of the peak in the dielectric loss tangent (tanδ) as a function of frequency can be explained by the close link between the hopping conduction process and the dielectric behavior of ferrites. The loss peaks showed a shift to the high frequencies for both samples with high sintering temperatures. As the temperature drops, these peaks shift to a higher frequency at the same time. Moreover, it has been shown that increase in temperature is accompanied by a decrease in peak height. In the semiconductor regime (T > T_MS), the dielectric loss tangent (tanδ) of the two samples decreases at lower frequencies, then it becomes completely independent of frequency at higher frequencies. The loss is maximum when the frequency of the electric field applied is clearly lower than the frequency of the jump of electrons between the Fe²⁺ and Fe³⁺ ions on the neighboring octahedral sites and in this case the electrons obey the evolution of this electric field. It is minimal when the frequency of the electric field applied is greater than the jump frequency of electron exchange between the Fe²⁺ and Fe³⁺ ions and in this case the electrons do not obey the evolution of this electric field [81]. At low frequencies, the grain boundaries have low conductivity, so the charge carrier jump process requires more electrical energy, hence the loss is high. At high frequencies, the grain boundaries have high conductivity, so the charge carrier jumping process requires less electrical energy, hence the loss is low [82]. In this study, it can be deduced that the value of (tanδ) strongly depends on the sintering temperature.

Fig. 7

Variations of ε’ with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

Fig. 8

Variations of ε” with frequency at various temperatures for  Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

Fig. 9

Variations of the dielectric loss tangent (tanδ) with frequency at various temperatures for  Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

3.5 Electric Modulus Analysis

The study of the complex dielectric modulus gives information on the nature of the polycrystalline samples, whether they are homogeneous or heterogeneous and also to have global ideas on the electrical relaxation process of a conducting material [83]. The adoption of a complex electrical modulus formalism aims to eliminate the electrode effect [84]. The following formula is employed to determine the complex modulus M* [85]:

$${\mathrm{M}}^{*}=\frac{1}{{\upvarepsilon }^{*}} =\frac{{\upvarepsilon }^{\mathrm{^{\prime}}}}{{\upvarepsilon }^{\mathrm{^{\prime}}2}+{\upvarepsilon }^{\prime\prime 2}}+\mathrm{ j }\frac{{\upvarepsilon }^{\prime \prime}}{{\upvarepsilon }^{\mathrm{^{\prime}}2}+{\upvarepsilon }^{\prime\prime 2}}= {\mathrm{M}}^{\mathrm{^{\prime}}}+\mathrm{j }{\mathrm{M}}^{\prime\prime}$$

(12)

where M' and M" represent respectively the real and imaginary parts of the complex electrical module. Figure 10 depicts the variation of the real component M′ of the modulus with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites. It has been shown that the values of M' in the low frequency domain are extremely low, demonstrating that the polarization of the electrodes has little effect on the material [86]. It was found that the values of M′ increased exponentially at intermediate frequencies. For high frequencies, we observe that the variation of the values of M' is independent of the frequency and that they tend towards a single value. This leads to the phenomenon of conduction due to the short-range mobility of charge carriers under the effect of the applied electric field [87, 88]. This leads to the conclusion that under the effect of the applied electric field, the conduction phenomenon is due to the short-range mobility of the charge carriers.

Fig. 10

Variations of M′ with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

Figure 11 shows the imaginary component of the complex electrical modulus M″ for Mg_0.6Co_0.4FeCrO₄ ferrites. We can see from this figure that the profile of the M′′ curves at different temperatures is the same. The values of M" initially increase with frequency until they reach their maximum value at (f_max). Then they begin to decrease as frequency increases. At different temperatures, the frequency-dependent spectra of M″ contain peaks that appear at specific frequencies (f_max). When the temperature falls below T_MS, the peaks shift to the high frequency band. However, once the temperature increases above T_MS, the peaks shift to the high frequency band. This suggests that these samples' dielectric relaxation is thermally activated. These peaks reveal an increasing frequency shift in the mobility of charge carriers from long-range to short-range. The frequency band below the peak maximum is the low frequency region. Charge carriers at this location can travel great distances. While the area of frequencies above the peak maxima is the high frequency area. In this zone, the charge carriers are spatially confined to their potential sinks, and they move over short distances [89]. The broadening asymmetry exhibited by the peaks at different temperatures shows that the propagation of relaxation times is with different time constants hence there is in these samples a non-Debye type of relaxation [90]. The following function is used to fit the curves of variation of M′′ as a function of frequency at various temperatures [91]:

$${\mathrm{M}}^{^{\prime\prime} }={\mathrm{M}}_{\mathrm{max}}^{^{\prime\prime} } / \left[1-\upbeta +\left(\frac{\upbeta }{1+\upbeta }\right)\left(\upbeta \left(\frac{{\mathrm{f}}_{\mathrm{max}}}{\mathrm{f}}\right)+{\left(\frac{\mathrm{f}}{{\mathrm{f}}_{\mathrm{max}}}\right)}^{\upbeta }\right)\right]$$

(13)

where M″_max is the maximum of the peak, f_max is its frequency and β is the stretching factor (varying from 0 to 1). This factor identifies the type of relaxation Debye or non-Debye that occurs in a particular material [92]. The parameters for adjusting the data of M″ as a function of frequency using Eq. (13) are given in Table 3. The values of β are less than one, as shown in this table, indicating that these samples exhibit non-Debye dielectric behavior [93]. The position of the peaks also makes it possible to obtain the relaxation time (τ) using this relation [94]:

Fig. 11

Variations of M″ with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C. Fit to experimental results is represented by red lines

Table 3 Fitting parameters derived from the experimental data of imaginary part of modulus versus frequency at different temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

$${\uptau }_{\mathrm{M}^{\prime\prime}}=\frac{1}{{2\mathrm{\pi f}}_{\mathrm{max }} }$$

(14)

Figure 12 shows the Ln (${\uptau }_{\mathrm{M}^{\prime\prime}}$) vs. (1000/T) curves which have also been modelized using the following Arrhenius relation [82, 95]:

$${\uptau }_{\mathrm{M}^{\prime\prime}} = {\uptau }_{0}\mathrm{exp}\left(\frac{{\mathrm{E}}_{\mathrm{M}^{\prime\prime}} }{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)$$

(15)

with Ea being the activation energy, T being the temperature, kB being the Boltzmann constant, and τ0 being a pre-exponential factor. According to the linear adjustments of the curves, the activation energies EM" are 0.761 eV and 0.493 eV for the samples sintered at 900 °C and 1100 °C, respectively. The values of this energy are fairly close to those established by the investigation of dc conductivity ${\upsigma }_{\mathrm{s}}$. Figure 13 depicts Cole–Cole plots for Mg_0.6Co_0.4FeCrO₄ ferrites at various temperatures. We can notice from Cole–Cole plots that the patterns are characterized by the presence of little asymmetric. This suggests that these samples exhibit electrical relaxation phenomena [96].

Fig. 12

Variation of the Ln(${\uptau }_{\mathrm{M}^{\prime\prime}}$) as a function (1000/T) for Mg_0.6Co_0.4FeCrO₄  ferrite sintered at 900 °C and 1100 °C. Fit to experimental results is represented by red lines (Color figure online)

Fig. 13

Complex modulus spectra at different temperatures for Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

3.6 Complex Impedance Analysis

The evolution of the real component of the impedance Z′ with frequency at various temperatures is shown in Fig. 14. It can be seen from this figure that for temperatures above T_MS, the values of Z′ are high in the low frequency band, then they increase when the frequency increases. Below T_MS, they decrease with decreasing temperature. This suggests that the ac conductivity increased with increasing temperature and frequency [95]. This phenomenon could be explained by an increase in the mobility of charge carriers and a decrease in the density of confined charges [73]. At high frequencies, the Z′ values at all temperatures begin to converge. This phenomenon can be explained by the existence of space charge polarization [60]. It's important to also mention that Z′ decreased as the sintering temperature increases. Other studies on ferrite compounds have shown a comparable variation in Z′ [47, 97]. The decrease in Z′ values indicates that the resistive properties of samples have decreased. This phenomenon is caused by the existence of space charge polarization in the samples [98].

Fig. 14

Variations of Z′ with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

The variation of the imaginary component Z′′ of impedance with frequency at various temperatures is depicted in Fig. 15. It can be shown that, for temperatures over T_MS, the spectra of Z′′ are characterized by the presence of sharp peaks that, as temperature rises, travel closer to the high-frequency zone. This indicates that a relaxation phenomenon exists in these samples [55] and that these relaxations are thermally activated [99]. But for temperatures below T_MS, these peaks shift to the high frequency region as the temperature decreases. This behavior can be also attributed to the fact that the polarization establishment time (i.e., the relaxation time) in the metallic region increases with increasing temperature; however, in the semiconductor region the relaxation time decreases. In addition, at higher frequencies, all the curves Z″ coincide; this phenomenon is caused by the accumulation of space charge in these compounds [100, 101]. It is also noted that for these two samples sintered at 900 °C and at 1100 °C, that at different temperatures, the values of Z″ increases when increasing the frequency, but for a certain frequency, these values gradually decrease which is due to the decrease in the polarization which does not follow the frequency of the alternating electric field applied [48]. In this instance, we also observed that the relaxation peaks migrated into the high frequency region as the sintering temperature increases. This behavior demonstrates that when the sintering temperature rises, the relaxation time shortens and the mobility of the hopping charge carriers rises, which results in a loss of material strength [101].

Fig. 15

Variations of Z″ with frequency at various temperatures for Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

Figure 16 shows the Nyquist plots at various temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites. Two arcs of circles that are not centered on the real axis can be seen in the impedance spectra, which may indicate non-Debye relaxation [65]. The first arc, which is a complete semicircle, is seen at high frequencies, but the second arc, which is an incomplete semicircle with a larger radius, is seen at low frequencies. This second arc corresponds to the contribution of the grain boundary resistance (R_gb), whereas the first arc denotes the contribution of the grain [102]. The two arcs seen in the impedance spectra suggest the presence of two types of relaxation [103]. From the spectra, it is observed that for temperatures above T_MS, half-arc radii decrease as temperature increases. The reduction in half-arc radii as temperature increases proves that electrical conductivity increases as temperature increases [104].

Fig. 16

Nyquist curves at various temperatures of Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C

Zview software is used to fit impedance data for all temperatures [69]. Red lines in Fig. 16 represent the appropriate fit, is made using an equivalent circuit composed of a serial association of two contributions which is represented in the Fig. 17 [69]. The interpretation of the impedance spectra has been based on the first contribution, which is related to the grains (R_g-CPE_g), and the second contribution, which is related to the grain boundaries (R_gb-CPE_gb), where R_g and R_gb denote grain and grain boundary resistance respectively, and CPE_g and CPE_gb denote grain and grain boundary constant phase elements respectively.

Fig. 17

The equivalent circuit used for the analysis of the impedance data

The following equation determines the impedance of a CPE [45]:

$${\mathrm{Z}}_{\mathrm{CPE}}=\frac{1}{\mathrm{Q}{(\mathrm{j\omega })}^{\mathrm{\alpha }}}$$

(16)

where ω is the angular frequency, Q denotes the capacitance value of the CPE element and α are frequency independent parameters. The real and imaginary parts of the impedance are determined using the following formulas [105]:

$${\mathrm{Z}}^{\mathrm{^{\prime}}}=\frac{{\mathrm{R}}_{\mathrm{g}}+{{\mathrm{R}}_{\mathrm{g}}}^{2}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2})}{[1+{\mathrm{R}}_{\mathrm{g}}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2}){]}^{2}+[{\mathrm{R}}_{\mathrm{g}}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2}){]}^{2}}+\frac{{\mathrm{R}}_{\mathrm{gb}}+{{\mathrm{R}}_{\mathrm{gb}}}^{2}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2})}{[1+{\mathrm{R}}_{\mathrm{gb}}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2}){]}^{2}+[{\mathrm{R}}_{\mathrm{gb}}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2}){]}^{2}}$$

(17)

$$-{\mathrm{Z}}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}=\frac{{{\mathrm{R}}_{\mathrm{g}}}^{2}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2})}{[1+{\mathrm{R}}_{\mathrm{g}}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2}){]}^{2}+[{\mathrm{R}}_{\mathrm{g}}{\mathrm{Q}}_{\mathrm{g}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{g}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{g}}\frac{\uppi }{2}){]}^{2}}+\frac{{{\mathrm{R}}_{\mathrm{gb}}}^{2}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2})}{[1+{\mathrm{R}}_{\mathrm{gb}}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{cos}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2}){]}^{2}+[{\mathrm{R}}_{\mathrm{gb}}{\mathrm{Q}}_{\mathrm{gb}}{\upomega }^{{\mathrm{\alpha }}_{\mathrm{gb}}}\mathrm{sin}({\mathrm{\alpha }}_{\mathrm{gb}}\frac{\uppi }{2}){]}^{2}}$$

(18)

Table 4 displays the values of all the fitted parameters. This table shows that at temperatures lower than T_MS, the grain boundary resistance R_gb, the grain boundary R_g and the total resistance (R = R_g + R_gb) increase with temperature, which confirms the metallic behavior of these compounds, but at temperatures above T_MS these resistances decrease with temperature; this confirms the semiconductor behavior of this compound. The observed decrease can be attributed to the grain boundary effect, which reduced the barrier to charge carrier hopping [94]. It has been observed that at different temperatures the resistances of Rgb grains have higher values than that of Rg grains. It is because of the atomic arrangement, which is disordered near the grain boundary, causing a marked increase in electron scattering [55, 106]. As a consequence, we may conclude that the grain boundary contribution is mostly responsible for the conduction mechanism in the samples. It should be observed that the R_g and R_gb values are higher for the sample sintered at 900 °C than for that sintered at 1100 °C. Figure 18 shows the logarithmic evolution of the total resistance (R) for the two samples as a function of the inverse of the temperature. The activation energy is determined from the following Arrhenius’ law [99]:

$$\mathrm{R }= {\mathrm{R}}_{0}\mathrm{exp}\left(\frac{{\mathrm{E}}_{\mathrm{R}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)$$

(19)

where k_B represents the Boltzmann constant, R₀ is a pre-exponential term and E_a is the activation energy. For T > T_MS, the curves in Fig. 18 are fitted by red lines using Eq. (19); from which the values of the activation energies of these samples will be determined. For the sample sintered at 900 °C, the activation energy is E_R = 0.462 eV and that for the sample sintered at 1100 °C is E_R = 0.462 eV. These results are similar to those obtained via the study of dc conductivity ${\upsigma }_{\mathrm{s}}$.

Table 4 Electrical parameters obtained from Nyquist plots at different temperatures for Mg_0.6Co_0.4FeCrO₄ ferrites sintered at 900 °C and 1100 °C

Fig. 18

Variation of Ln(R) versus (1000/T) for Mg_0.6Co_0.4FeCrO₄  ferrites sintered at 900 °C and 1100 °C. Fit to experimental results is represented by red lines (Color figure online)

4 Conclusion

In conclusion, we investigated the structural, infrared, and electrical properties of Mg_0.6Co_0.4FeCrO₄ ferrites synthesized by the sol gel process and sintered at 900 °C and 1100 °C. According to X-ray diffraction measurements, our samples crystallize in the cubic structure with the space group $\mathrm{Fd}\overline{3}\mathrm{m}$. In FTIR spectra, only peaks associated with tetrahedral stretching are observable. The increase in sintering temperature was accompanied by an increase in cell parameters, cristallites size, absorption bands, conductivity, dielectric constants, and dielectric loss tangent. A semiconductor–metal transition is observed in the tow samples at a specific temperature T_MS. We observed that the T_MS is not affected by the sintering temperature. The conduction process for the two samples follows both the NSPT and CBH models. The values of dielectric constants decrease gradually as frequency increases. The decrease in dielectric constants as frequency increases is interpreted by Koop's theory and the Maxwell–Wagner type of interfacial polarization. The variation of the imaginary components of modulus and impedance suggests that both samples exhibit non-Debye dielectric relaxation. The Nyquist curves are adjusted using an equivalent circuit composed of a series association of two circuits [(R_gb //CPEgb) and (R_g //CPE_g)]. The estimated activation energies derived from the electrical modulus spectrum, impedance, and dc conductivity analyses are found to be quite similar, suggesting that the phenomenon of relaxation process and electrical conduction are both related to the same defect.