Investigations of microstructural and impedance spectroscopic properties of Mg0.5Co0.5Fe1.6Al0.4O4 ferrite prepared using sol–gel method

Polycrystalline Mg0.5Co0.5Fe1.6Al0.4O4 ferrite was prepared using sol–gel method. This sample was characterized by powder X-ray diffraction (XRD), Scanning electron microscopy, and impedance spectroscopy. XRD analysis combined with the Rietveld refinement confirmed the cubic-spinel structure (SG: Fd3-m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Fd\stackrel{-}{3}m$$\end{document}) for the prepared sample. Electrical conductivity obeying the Jonscher power law indicates that the prepared material exhibits semiconductor behavior, and the conduction process follows the “non-overlapping small polaron tunnelling, NSPT” model between neighbors’ sites. The behavior of dielectric constants such as permittivity and loss coefficient has been interpreted based on the Maxwell–Wagner’s theory of interfacial polarization. The curves of imaginary parts of impedance (Z″) and modulus (M″) show dielectric-relaxation phenomenon in the sample with activation energy near to that determined from the dc conductivity study. Nyquist plots (− Z″ vs. Z′) show a monotonic decrease in both grain resistance (Rg) and grain boundary resistance (Rgb) with increasing temperature such as Rgb ≫ Rg. This result confirms that the transport mechanism in Mg0.5Co0.5Fe1.6Al0.4O4 compound is governed by the grain boundaries effect.


Introduction
Among the most important classes of materials possessing a diversity in their magnetic, electrical, and optical properties are the spinel ferrites [1]. Presently, magnetic oxides made a great attention on scientists in the sake of their wide-ranging applications from industrial use to fundamental research [2,3]. Spinel ferrites with these properties can be profitable in ferrofluids, gas sensors, transformer cores, biomedical and magnetic refrigeration [4][5][6][7][8]. The spinel ferrites have general formula of MFe 2 O 4 , where M is a divalent metal cation [9]. They represent a face-centered cubic structure with merging crystallographic sublattices between tetrahedral (A) and octahedral [B] sites. The spins in these sites are oriented antiparallel to each other. The nature of spinel structure like normal, inverse, or mixed structure relies on the lattice occupancy of the divalent metallic ions in the crystallographic sites [10]. The ionic occupancy as well as preparation method, annealing temperature, stoichiometry, valency etc., present the critical aspect in determining structural, magnetic, and electrical properties in spinel ferrites [11][12][13].
Among the most interesting spinel systems, Mg 1-x Co x Fe 2 O 4 ferrites are widely used for electronic devices due to their high frequency-permeability, high electrical resistivity, chemical resistance, and cost effectiveness. These interesting properties make these materials good candidates for high-frequency applications and microwave devices [14]. In addition, Mg 1-x Co x Fe 2 O 4 materials are also potential candidates for size-antennas miniaturization [15]. The bibliographic study noted that fewer studies were presented in the literature on these ferrite materials with other substitutions. Particularly, noinvestigation has been cited on the physical properties of Mg-Co ferrites substituted with trivalent metal cation such as Al 3? . Although many metal cations have been studied for their effects on electrical and magnetic properties, Al 3? is a less explored ion. In this report, we presented an investigation on microstructural and impedance spectroscopic properties of Mg 0.5 Co 0.5-Fe 1.6 Al 0.4 O 4 ferrite prepared by sol-gel technique. The selection of Al 3? cation is due to its strong preference to occupy the octahedral B-sites. For different ferrite systems, it was reported that the electrical and dielectric properties such as electrical resistivity, dielectric constant and dielectric loss can be improved by Al substitution [16,17]. These features make the Al-substituted ferrites good choice for high frequency applications and micro-wave devices. The results of the morphological characterization for Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 sample are first presented, to then discuss and interpret the structural characterization by XRD technique and Rietveld method. Electrical measurements by impedance spectroscopy technique as a function of frequency and temperature were also presented. Variations of electrical conductivity, dielectric constants, electrical modulus, and electrical impedance were discussed.

Experiment
The Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 spinel ferrite was synthesized through sol-gel process as displayed in Fig. 1 Then, a few amounts of ammonia were added to adjust the pH-solution to around 7. Ethylene glycol (HO(CH 2 ) 3 OH, 99.9%) was added to the obtained homogeneous solution. The mixtures were maintained under magnetic agitation for 4 h at 90°C to become a gel. To eliminate any residual solvent, the collected gel was dried in a furnace at 250°C for 12 h. The obtained precursor was grained in air for 3 h, and then calcined at 600°C for 24 h, and it was grounded again and calcined at 800°C for 24 h. We can notice that the cooling was very slow. In the end, we ground the final calcined precursor, and the resulting powder was undergone a cycle of (grinding ? pelleting ? sintering) at 1000°C for 24 h.
The final powder was characterized by X-ray diffractometer (Panalytical X'Pert Pro System) using a monochromatic CuKa radiation (k = 1.5406 Å ) in a range of 15 B 2h B 80°with a step of 0.02°. The XRD diffractogram was refined adopting the FULLPROF program [18] founded on the Rietveld analysis [19]. Scanning electron microscopy [SEM, Philips XL30 microscope] was used to probe into the surface morphology of the sample at 20 kV in secondary electrons mode. For the electrical characterizations, silver electrodes were sputtered on both sides of the sample (in disk form with diameter of 10 mm and thickness of approximately 2 mm). The electrical measurements were obtained over a wide temperature range using an Agilent 4294 analyzer under vacuum, in dark and in 100 Hz-5 MHz frequency range.
3 Results and discussion 3.1 Microstructure analysis Figure 2a shows the SEM image for Mg 0.5 Co 0.5 Fe 1.6-Al 0.4 O 4 ferrite. This morphological analysis shows the existence of homogeneous distribution of the spherical grains. We can also observe minor voids associated to the porosity of the sample. Generally, sintering temperature affects the size distribution and the shape of grains along with agglomeration in the synthesized materials. In our case, the sample was prepared at sufficiently higher sintering-temperature (1000°C). This indicates that this sintering temperature is favorable for good crystallinity and microstructural homogenization for the Mg 0.5 Co 0.5-Fe 1.6 Al 0.4 O 4 sample. The statistical particle size distribution using ImageJ software is shown in Fig. 2b. The histogram fitting using the Gaussian function gives an average particle size around 0.66 lm. The phase purity and structure of Mg 0.5 Co 0.5 Fe 1.6-Al 0.4 O 4 sample were approved by analysis of the Xray diffraction pattern as shown in Fig. 3a. All peaks can be indexed to the face-centered cubic phase with Fd À 3 m space group. We also note that the sample was described as a pure single phase without other secondary crystallographic phases related to impurities. The peaks are indexed as (110), (220), (311), (222), (400), (422), (511), (440), (620), (533), and (622) which are characteristics of the cubic spinel structure. The average grains size (crystallites) of the studied system is determined from the most intense (311) diffractionpeak (inset of Fig. 3a), in conformity with the Debye-Scherrer formula given by Eq. (1) [20]: where, k is a constant such that k = 0.9, k is the X-ray wavelength (1.54059 Å ), b 1/2 is the full width at half maximum (FWHM) and h is the Bragg angle of the peak. The measured average grains size is approximately 37 nm. Hence, the change of growth grain size behavior according to morphological analysis in comparison with that given by the structural analysis can be associated with the difference in particle nature. This latter is secondary particle for morphological analysis whereas for structural one, it is primary particle. To investigate the structural transformation in more details, we realized a simulation on the measured XRD pattern based on the Rietveld refinement technique. The data analysis using the Rietveld method is presented in Fig. 3b. Generally, for AB 2 O 4 The inset shows the profile of (3 1 1) peak. b Observed and calculated patterns obtained from Rietveld analysis spinel materials, certain experimental techniques such as Mö ssbauer spectroscopy have been carried out to establish the cations distribution between both tetrahedral (A) and octahedral (B) sites. The characterization by Mössbauer technique of Mg 1-x Co x Fe 2 O 4 ferrites [21,22], and certain Al-substituted ferrites [23] showed that Co 2? , Mg 2? and Al 3? ions are preferred to occupy the octahedral (B) sites, while Fe 3? ions are distributed between both (A) and (B) sites. The distribution of all these cations agrees with those presented in previous studies [24,25]. Accordingly, the ions distribution in Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 compound can be written as: Otherwise, the cations distribution, and the atomic compositions among the tetrahedral (A) and octahedral (B) sites for spinel ferrites can be obtained from an analysis of the XRD patterns. Several methods have been developed to estimate the cations distribution and the atomic compositions in AB 2 O 4 materials from the XRD results. Among the presented methods we can cite: (i) the Rietveld method based on the refinement of the atomic occupancy values [26], and (ii) the Bertaut method based on the comparison of the experimental and calculated intensity ratios for reflections whose intensities are nearly independent of the oxygen parameter, vary with the cation distribution in opposite ways, and do not significantly differ [27]. In our case we used the Rietveld method, based on the arguments mentioned above, to confirm the cations distribution and the atomic compositions for Mg 0.5-Co 0.5 Fe 1.6 Al 0.4 O 4 sample. The refined values of structural parameters, including occupancy factors are grouped in Table 1. The results show that the refined occupancy factors for (Fe) in the tetrahedral A-sites and the octahedral B-sites, and the refined occupancy factors for (Mg, Co, Al) in the B-sites are close to the nominal ones which confirms well the distribution. From this cation's distribution, we can deduce that the inversion degree 1 in the Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 sample is equal to one so our sample presents a normal spinel structure. On the other hand, the obtained lattice constants are lower than those presented in the literature for the parent compound Mg 0.5 Co 0.5 Fe 2 O 4 [24,29]. This is due to the smaller ionic radius of Al 3? (r 3þ Al ¼ 0; 535Å) by comparing with that of Fe 3? (r 3þ Figure 4a presents the conductivity spectra according to the frequency and temperature for Mg 0.5 Co 0.5-Fe 1.6 Al 0.4 O 4 ferrite. According to this figure, the conductivity spectra can be categorically divided into two regions. Almost constant conductivity values were observed at low-frequency region extending up to a frequency, known as the hopping frequency (m H ).

Electrical conductivity
The frequency independence of conductivity in lowfrequency range is related to the higher activity of grain boundaries which weaken the electrons hopping. Since the conductivity in this range is virtually constant, although the frequency varies, it is often noted, r dc , as ''direct current'' conductivity. According to Fig. 4a, r dc grows with temperature rise which indicates that the conduction process is thermally activated, and the prepared material exhibits semiconductor behavior over the entire temperature range. Above m H , the conductivity increases exponentially with frequency due to the activity of the conductive grains. This promotes the conduction process in the sample due to the increase of electrons hopping. The conductivity at frequencies higher than m H corresponds to the conductivity in alternating current (r ac ). The phenomenon of the conductivity dispersion observed for the sample is analyzed by using the universal law of Jonscher [31] given by Eq. (2): where, r dc (T) represents the dc conductivity. A(T) and ''s'' are the pre-exponential and the frequency-exponent parameters, respectively. The electrical conductivity results are well fitted as depicted by continuous red lines in Fig. 4a values plotted in Fig. 4b are all superior than 1. So, we can conclude from this figure that the electrons hopping involves translational movement between neighboring sites. Such result was found in other works [33,34]. In addition, an increasing tendency in s values was observed with temperature rise. This is a criterion for a typical thermally activated conduction process. This indicates also that the ''non-overlapping small polaron tunnelling'' (NSPT model) [35] is valid in the case of Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 sample. According to this model, the s exponent can be formulated in Eq. (3) as [35]: where k B is the Boltzmann constant, T is the temperature, and W m represents the polaron binding energy (i.e. the binding energy of the carrier in its localized site). The obtained W m value is around 0.134 eV. The obtained fitting parameters using Jonscher law were used to describe the hopping frequency (m H ). This frequency, defined as the frequency at which the conductivity relaxation begins to appear, can be formulated in Eq. (4) as follows: where, r dc is the dc conductivity, A(T) is the pre-exponential factor, and s is the frequency-exponent. Basically, dc and ac conductivities were correlated by comparing both r dc (T) and m H (T) curves which are 3.82 The log(r dc ) vs. log(m H ) curve presents a linear variation with slope close to unity. This implies that dc and ac conductivities are well correlated [36]. In the other hand, Fig. 5a shows the same temperature dependence of both r dc (T) and m H (T) curves. In fact, the dc conductivity, and the hopping frequency increase with increasing temperature. This confirms the utility of the Arrhenius' law to explain the temperature variation of r dc and m H . Then, the Arrhenius law was used to analyze the r dc (T) and m H (T) behaviors according to Eqs. (5) and (6) presented as follows [37]:  where r 0 and m 0 are pre-exponential factors, E a is the activation energy required in the sample, k B is the Boltzmann constant, and T is the temperature. Equations (5) and (6) are used to calculate the E a values by plotting log(r dc 9 T) and log(m H 9 T) vs.
(1000/T) in Fig. 5b. The activation energies values derived from the linear fit of both curves are shown in the same figure. It can be clearly seen that the E a value deduced from dc conductivity is very close to that estimated from frequency hopping. This shows that charge carriers must overcome the same energy barriers while conducting and relaxing [38]. On the other hand, the obtained E a value for Mg 0.5 Co 0.5-Fe 1.6 Al 0.4 O 4 sample (see Table 2) is smaller than the reported ones for various compositions of Mg 1-x-Co x Fe 2 O 4 system [39][40][41]. Obviously, the addition of Al 3? decreases the activation energy in Mg 1-x Co x-Fe 2 O 4 system. Then, we can conclude that the substitution of Al in Mg 1-x Co x Fe 2 O 4 ferrites can improve their electrical-conduction process and thereby their dielectric properties. This is in good agreement with the results found for other Al doped ferrites [16,17].
In Fig. 5c we present the Arrhenius plots for r ac at two different frequencies (1 MHz and 5 MHz). It's clear that the r ac values increase with frequency. The deduced activation energies (E ac ) values which are presented in Fig. 5c are lower than the one determined from the r dc curve. In fact, due to the higher activity of grain boundaries in the dc conductivity regime the electrons hopping requires more energy; however the conductive grains are more active in the ac conductivity regime and consequently low activation energy is required for the electrons hopping [42,43].

Dielectric study
The imaginary part of permittivity (e 00 ) is presented in Fig. 6a versus temperature and frequency for Mg 0.5-Co 0.5 Fe 1.6 Al 0.4 O 4 ferrite. The e 00 values decrease with increasing frequency. This behavior can be interpreted considering the Maxwell-Wagner interfacial polarization [44,45] and Koop's phenomenological theory [46] of dielectrics. Indeed, the dielectric structure of ferrites is assumed to be composed of good conductive grains, separated by grain boundaries slowing the movement of charge carriers, which drops the inter-grains conductivity. During this exchange mechanism, electrons must pass through the grains and grain boundaries of the sample. Due to the high resistance grain boundaries, electrons accumulate at the grain boundaries and produce space charge polarization. Figure 6a also shows that e'' values decrease rapidly at low frequencies and becomes frequency independent at higher frequencies. This behavior is since the effect of grain boundaries is greater in the low frequency region while that of grains is more preponderant in the high frequency region [46]. The non-frequency dependence of e'' for higher frequencies, can be attributed to the fact that beyond a certain frequency, the electrons in the grain can no longer follow the alternating field and consequently, the medium ceases to polarize, hence the constancy of e''. In addition, Fig. 6a shows an increase of e'' values with increasing temperature because the hopping of charge carriers is thermally activated. On the other hand, the frequency and temperature dependence of e 00 can be analyzed according to Giuntini model [47] which is given in Eq. (7) as: where e 0 and e ? are the static and optical dielectric constants, respectively; N is the concentration of localized sites, n is the number of electrons that hop, e is the elementary charge, k B is the Boltzmann constant, T is the temperature, s 0 is the pre-exponential factor, m is the frequency-exponent, and W m is the maximum barrier height (the energy required to move the electron from a site to the infinity). Equation (7) can be simplified in Eq. (8) as: where B is a pre-exponential factor. The exponent m was calculated from the negative slopes of the obtained lines in Fig. 6a and it was presented in Fig. 6b versus temperature. It emerges that m values increase linearly with temperature. According to Giuntini model, the behaviors of m(T) (dielectric properties) and the exponents s(T) (ac measurements) show the same tendency by varying temperature [48].
In addition, the exponent m(T) can be expressed according to Eq. (9) as follows: The calculated value of W m is around 0.138 eV. This value agrees well with that obtained from Eq. (3). From Fig. 6c, one can notice that the loss factor (tand) decreases rapidly at low frequencies and it becomes almost frequency independent in the higher frequencies' values. As mentioned above, the grain boundaries are active in the low frequency region which requires more energy for the electrons exchange, and therefore, the loss is higher. However, in higher frequencies region which corresponds to the conductive grains, low energy is required for the electrons transfer. In addition, the tand values increase with temperature due to the thermally activated conduction process. Figure 7a shows the behavior of the real part of the modulus (M 0 ) versus frequency and temperature for Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 ferrite. In the low frequency region, the M 0 values are very negligible. This confirms that the electrode polarization makes a negligible contribution in the material [49]. With increasing frequency, the M 0 values increase reaching a saturation at high frequencies especially at low temperatures. This behavior may be due to the shortrange mobility of charge carriers [50]. Figure 7b shows the variation of the imaginary component of modulus (M 00 ). The M 00 (f) spectra are characterized by peaks at characteristic frequencies (f max M 00 ) for all temperatures, and the positions of these peaks shift to higher frequencies with increasing temperature. This means that the relaxation time for this process increases with temperature. The M 00 (f) curves were modeled using the Kohlrausch, Williams and Watts (KWW) function which is presented in Eq. (10) as below [51]:

Electrical modulus
where M 00 max is the peak maxima, and f max its corresponding maximum frequency. b is the stretching factor (0 \ b \ 1) which describes the dielectric-relaxation nature (Debye or non-Debye) [52]. The fitting parameters of M''(f), given in Table 3, shows that b values are less than unity, showing so a non-Debye nature of the relaxation phenomenon for the sample.

Complex impedance analysis
To distinguish between the grain and the grain boundary contributions to electrical properties for Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 ferrite, Nyquist plots have been analyzed as depicted in Fig. 8. The Nyquist plots are represented as semicircular arcs not centered on the real axis. This decentralization indicates a relaxation process of non-Debye type. The diameters of the semicircles, limited by the total resistance of the material, decrease with temperature, in relation to the pronounced increase in r dc . The appearance of a single semicircle for each temperature means that the electrical process obeys a single dipole relaxation [53]. Indeed, the analysis of the dielectric relaxation linked to the grain boundaries in the temperature range 300-500 K has clearly shown that the conduction is controlled by the grain boundaries which explains the appearance of only one semicircle in the Nyquist diagram. These semicircles inform us about the electrical processes occurring in the sample and its correlation with the microstructure of the sample, when we modeled the sample by an equivalent electrical circuit [54,55]. The experimental data of these semicircles were well fitted by ZView software   [56] using an equivalent circuit formed by a grain resistance (R g ) in series with a parallel combination of a grain boundary resistance (R gb ) and constant phase element impedance (Z CPE ). Such equivalent configuration can be represented as (R g ? R gb //CPE gb ) [57], as indicated in the insert of Fig. 8. All adjusted parameters are summarized in Table 4. We can notice in this table that R g values are very low, therefore the total resistance R T of the sample which is the sum of those of R g and R gb [57] is approximately equal to that of the grains-boundaries. Also, the R gb values decrease with increasing temperature, which indicates the semiconductor behavior for the prepared sample. Such behavior has been observed in other materials [37,58]. It appears that with the increase in temperature, the grain boundary effect facilitates the passage of charge carriers by lowering the barrier, thus paving the way for increased electrical transport. The normalized representation of the imaginary part of impedance (Z 00 /Z 00 max vs. frequency) is shown in Fig. 9a. The curves are characterized by well resolved peaks moving to the higher frequencies with temperature rise. Such behavior indicates the existence of a relaxation phenomenon in our material. The peaks maximum allows us the calculation of the relaxation frequency (f max ), and consequently the estimation of the relaxation time (s) by using Eq. (11): The Arrhenius variation of log(s) versus (1000/T) is given in Fig. 9b according to Eq. (12) as follows [34]: where s 0 is the pre-exponential factor, E Z 00 is the activation energy, k B is the Boltzmann constant, and T is the temperature. The linear fit of the curve gives a value of E Z 00 of the order of 0.206 eV which is close to that found from the dc conductivity analysis (E a-= 0.218 eV). The slight difference between the two activation energies may be since the relaxation process involves only the hopping energy of charge carriers between localized states, however, the conduction mechanism involves the hopping energy, the disorder as well as the binding energy of polarons [59]. This convergence in values confirms that both relaxation and conduction processes may be attributed to the same type of charge carriers. Figure 9c shows the plot Z 00 /Z 00 max vs. log(f/f max ). As shown in this figure, all normalized curves collapse into one curve which indicates that the distribution of the relaxation time is temperature independent [60]. For comparison, the variations of Z 00 (f) and M 00 (f) are depicted in Fig. 10 for two different temperatures (300 K and 400 K). At room temperature (T = 300 K), the curves show single peak which appears approximately at the same frequency value. With increasing temperature (T = 400 K), the gap between both Z 00 and M 00 peaks increases confirming the non-Debyetype relaxation behavior for the prepared sample [61].

Conclusion
In the present paper, Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 spinel ferrite was synthesized using sol-gel method, and its microstructural and electrical properties were successively investigated. SEM analysis reveals that sample presents homogeneous surface with grains of regular shape. Rietveld analysis of XRD pattern reveals the cubic spinel structure of the sample with Fd3m space group. According to the Jonscher-law analysis of electrical conductivity, the conduction process for sample having semiconductor behavior occurred between neighboring sites and follows the NSPT model. Dielectric constants decrease with frequency and increase with temperature. Such behavior was interpreted according to the Maxwell-Wagner's theory of interfacial polarization. Results of modeling of Nyquist plots show that the conduction process in the prepared sample is governed by the grain boundaries effect. In conclusion, the synthesized material presents several advantages such as its high resistivity, good electrical properties, low cost and above all its easy synthesis. These features make the Mg 0.5 Co 0.5 Fe 1.6 Al 0.4 O 4 ferrite, by comparing with the properties of Mg 1-x Co x Fe 2 O 4 ferrites, good candidate in electronic devices such as for high frequency applications and microwave devices.