Effect of Co²⁺ substitution on the structural, terahertz and magnetic characterization of NiZn ferrites

We have used analytical grade reagents; Ni nitrate [Ni(NO₃)₂·6H₂O] (purity: ⩾97.0%, Sigma Aldrich), Co nitrate [Co(NO₃)₂·6H₂O] (purity: ⩾98%, Sigma Aldrich), Zn nitrate [Zn(NO₃)₂·6H₂O] (purity: 98%, Sigma Aldrich), ferric nitrate [Fe(NO₃)₃·9H₂O] (purity: ⩾98%, Sigma Aldrich) as oxidizing agents, and glycine [NH₂CH₂COOH] (purity: ⩾99%, Sigma Aldrich) as a reducing agent and deionized water.

The compositions of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) are synthesized using the chemical sol–gel combustion route as shown in figure 1. In this method, the high purity metals; Ni, Co, Zn and Fe nitrates are used. Glycine is selected as a reducing agent to drive the combustion reaction due to its more cost-effectiveness and greater combustion rate (−3.24 kcal g⁻¹) compared to urea (−2.98 kcal g⁻¹) and citric acid (−2.76 kcal g⁻¹) [16, 17]. Stoichiometric ratio of fuel to nitrate (1:1.25) for sol–gel combustion is calculated based on the total reducing and oxidizing valencies of the fuel and oxidizer. Initially, a homogenous solution is prepared by dissolving the metal nitrates and glycine in a minute volume of deionized water. The resultant solution is poured into a beaker and heated at 100 °C for 4 h under constant stirring to form a viscous gel. Afterwards, the beaker is kept in a muffle furnace at 350 °C for auto-ignition. The whole combustion process is completed within 30 min whereas the actual time of ignition is small (nearly ⩽5 s). During the combustion, a huge amount of gases; N₂, CO₂, and O₂, and H₂O vapors are emitted and a loose-brownish powder is formed as a product. These powders are crushed and ground manually and calcined at 800 °C for 15 min in a microwave furnace. Subsequently, the powders are mixed with 2 wt% polyvinyl alcohol and pressed them into disc-shaped pellets (10 mm dia) by using hydraulic press which is used to apply a uniform pressure of 6 tons. Finally, the pellets are sintered at 1250 °C for 2 h under an ambient air atmosphere in a high-temperature muffle furnace. The furnace temperature is ramped at a rate of 5 °C min⁻¹.

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Based on propellant chemistry to prepare stoichiometric redox mixtures, the ratio of net reducing valency of the fuel to the net oxidizing valency of metal nitrates should be one [18, 19]. Normally, in the case of ferrites, the divalent and trivalent metal nitrates have the oxidizing valencies as 10− and 15−, respectively and the glycine has the reducing valency as 9+. The net oxidizing valency of the individual nitrate should be balanced by the net reducing valency of glycine. In our experiment, the redox reactions for Ni_0.5Co_x Zn_0.5−x Fe₂O₄ at x = 0.1, 0.2 and 0.3 can be expressed as follows:

$$\begin{align} 0.5{\text{ }}&\left[ {{\text{Ni(N}}{{\text{O}}_3}{)_2} \cdot 6{{\text{H}}_{\text{2}}}{\text{O}}} \right]{\text{ }}({\text{c}}) + x\;\left[ {{\text{Co(N}}{{\text{O}}_3}{)_2} \cdot 6{{\text{H}}_{\text{2}}}{\text{O}}} \right]{\text{ (c)}}\nonumber\\ & + (0.5 - x){\text{ }}\left[ {{\text{Zn(N}}{{\text{O}}_3}{)_2} \cdot 6{{\text{H}}_{\text{2}}}{\text{O}}} \right]{\text{ (c)}} \nonumber\\ & + 2{\text{ }}\left[ {{\text{Fe}}{{({\text{N}}{{\text{O}}_3})}_3} \cdot 9{{\text{H}}_{\text{2}}}{\text{O}}} \right]{\text{ (c)}} + 1.25{\text{ }}\left[ {{\text{N}}{{\text{H}}_{\text{2}}}{\text{C}}{{\text{H}}_{\text{2}}}{\text{COOH}}} \right]{\text{ (c)}}\nonumber\\ & \to \left[ {{\text{N}}{{\text{i}}_{0.5}}{\text{C}}{{\text{o}}_x}{\text{Z}}{{\text{n}}_{0.5 - x}}{\text{F}}{{\text{e}}_{\text{2}}}{{\text{O}}_4}} \right]{\text{ (c) }} \nonumber\\ & + 4.625{\text{ }}\left[ {{{\text{N}}_2}} \right]{\text{ (g)}} + 2.5{\text{ }}\left[ {{\text{C}}{{\text{O}}_2}} \right]{\text{ (g)}} + 27.125{\text{ }}\left[ {{{\text{H}}_{\text{2}}}{\text{O}}} \right]{\text{ (g)}}\nonumber\\ & + 7.1875{\text{ }}\left[ {{{\text{O}}_2}} \right]{\text{ (g),}} \end{align} \tag{ 1 }$$

where (c) = crystalline, (g) = gas.

The ferrite powders are calcined in a microwave muffle furnace (CEM Phoenix). X-ray diffractograms of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) are recorded on Rigaku Smart Lab x-ray diffractometer with CuKα (λ = 1.5406 Å) radiation in 2θ range from 10° to 80° with a step size of 0.01° at room temperature. The elemental compositions of the samples are obtained using a field emission‐scanning electron microscope (FEG-SEM: JEOL, JSM-7600F) equipped with an Oxford Instruments EDS Microanalysis X-Max-80 mm² system.

THz-TDS measurement is performed using a homemade THz time-domain spectroscopy (TDS) setup in a frequency range from 0.2 to 2.5 THz. The setup has a mode locked Ti:sapphire laser (Femtosource Synergy) having ultrashort pulses of duration 10 fs, wavelength centered at 800 nm and a repetition rate of 78 MHz. These pulses are focused on low temperature-GaAs (LT-GaAs) based THz source (BATOP GmbH) to generate THz pulses. The THz spectra are experimentally obtained by transmitting THz through the fabricated NiCoZn ferrite samples at room temperature. The transmitted THz radiation is obtained using ZnTe based electro-optic technique. In our THz measurement, we use NiCoZn ferrite disc-shaped pellets with a thickness of 500 μm of each. A THz signal passing through the polycrystalline sample with two flat and parallel sides at a normal angle of incidence, assuming that there are no reflections, can be considered as a sample signal, that is given below [20, 21],

$$\begin{equation}{E_{\text{s}}}(\omega ) = \eta \frac{{4{{\hat n}_{\text{s}}}(\omega )}}{{{{\left[ {{{\hat n}_{\text{s}}}(\omega ) + 1} \right]}^2}}} \cdot \exp { }\left[ { - {\text{i}}{{\hat n}_{\text{s}}}(\omega )\frac{{\omega { }t}}{c}} \right] \cdot { }{E_0}(\omega ),\end{equation} \tag{ 2 }$$

where η is the transmission factor of the free space surrounding the sample, the complex refractive index, $\hat{n}_{\text{s}}$= n_s + iκ_s, where n_s is the index of refraction, κ_s is the extinction coefficient, ω is the angular frequency, t is the sample thickness, c is the speed of light in vacuum and E₀(ω) is the incident wave. The spectral component of the electric field of the THz signal passes in the absence of the sample can be considered as a reference signal,

$$\begin{equation}{E_{{\text{ref}}}}(\omega ) = \eta \cdot \exp { }\left[ { - {\text{ i}}\frac{{\omega { }t}}{c}} \right] \cdot { }{E_0}(\omega ){\text{. }}\end{equation} \tag{ 3 }$$

The magnetic measurement is carried out using superconducting quantum interference device (SQUID) magnetometer (Quantum Design, USA) at a different temperature range from 300 to 50 K.

X-ray diffractograms of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3), samples sintered at 1250 °C are shown in figure 2(a). The x-ray diffraction (XRD) data of these samples are matched with the standard patterns (ICDD: 01-078-6781, ICDD: 01-074-6403, and ICDD: 01-071-4920) of NiFe₂O₄, CoFe₂O₄ and ZnFe₂O₄, respectively. The main reflections are observed at hkl values of (111), (220), (311), (222), (400), (422), (511), (440), (620), (533), (622) and (444) and labeled them. The XRD results of the samples have confirmed the formation of FCC single-phase spinel structure at room temperature and revealed that the samples are polycrystalline. In figure 2(b), it is showed that there is a continuous shift in 2θ towards the higher angle side with the increase in Co²⁺ content in NiZn ferrite. Therefore, this angle shift towards the higher values is ascribed to decrease in lattice constant. The lattice constants of the samples are obtained from the highest intensity-indexed reflection (311) to minimize the error in lattice constant-calculation using equation (4). The crystallite sizes of the samples are determined from the peak-broadening, detected in the XRD patterns using the Debye–Scherrer formula (equation (5)) [22],

$$\begin{equation}a = {d_{hkl}}{\text{ }}\sqrt {{h^2} + {k^2} + {l^2}} {\text{,}}\end{equation} \tag{ 4 }$$

$$\begin{equation}D = \frac{{0.9\lambda }}{{\beta \cos \theta }}{\text{ ,}}\end{equation} \tag{ 5 }$$

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where λ is the wavelength of x-rays (1.5406 Å), β is the full-width at half-maxima in radian and θ is the angle of diffraction. The x-ray density (ρ_x) of the pellets is calculated using equation (6) given by Smith and Wijn [23],

$$\begin{equation}{\rho _{\text{x}}} = \frac{{nM}}{{{N_{\text{A}}}{ }{a^3}}}\end{equation} \tag{ 6 }$$

where M is the molecular weight of the sample, n is the number of formula units present in a unit cell (for NiCoZn ferrite, n = 8) and N_A is the Avogadro's number.

In Archimedes' method, the experimental density (ρ_exp) of the samples is measured by using the formula, ${\rho _{\exp }} = ({w_{{\text{s,a}}}}/{w_{\text{l}}}){\rho _{\text{l}}},$ where w_s,a is the weight of sample in air, w_l is the weight of liquid displaced by the sample and ρ_l is the density of liquid (distilled water). The weight of displaced liquid can be extracted by measuring the weight of sample in air (w_s,a) and in liquid (w_s,l) and considering the difference between them, as described below [24],

$$\begin{equation}{w_{\text{l}}} = ({w_{{\text{s,a}}}} - {w_{{\text{s,l}}}}),\end{equation} \tag{ 7 }$$

$$\begin{equation}{\rho _{\exp }} = \left( {\frac{{{w_{{\text{s,a}}}}}}{{{w_{{\text{s,a}}}} - {w_{{\text{s,l}}}}}}} \right){ }{\rho _{\text{l}}}{\text{. }}\end{equation} \tag{ 8 }$$

Thus, the obtained densities of the samples were around 90%–93% of that estimated by x-ray method. This 7%–10% loss of densities could be due to the presence of binder, possible porosity in the pellets and other anomalies arising from the material synthesis process. The porosity in percentage of the samples is then determined by using the experimental density and x-ray density. That is given by [25],

$$\begin{equation}P = 100{ }\left[ {1 - \frac{{{\rho _{\text{l}}}}}{{{\rho _{\text{x}}}}}\left( {\frac{{{w_{{\text{s,a}}}}}}{{{w_{{\text{s,a}}}} - {w_{{\text{s,l}}}}}}} \right)} \right]{ }{\text{. }}\end{equation} \tag{ 9 }$$

The lattice constants, lattice volumes, x-ray densities, experimental densities, percentage of porosities and the average crystallite sizes of the samples as a function of x are tabulated in table 1.

In energy dispersive x-ray spectroscopy (EDX) analysis, a bulk amount of these samples are mounted on conductive paint-coated aluminum stubs and then coated with 10 nm thick iridium layer. EDX spectra of the samples are displayed at 10 keV in figures 3(a)–(c). The compositional results (in wt% and at%) for all the elements present in the samples at x = 0.1, 0.2 and 0.3 are given in the tables insets of figures 3(a)–(c), respectively. The spectra of the samples specified the presence of C, O, Fe, Co, Ni, Zn and Ir elements as the major elements elucidated with the non-appearance of other impurities. The presence of C and Ir are due to the conductive paint (carbon tape) and iridium-coating, respectively, which are applied before the measurement. However, the x-ray emission from the conductive layer and conducting paint as well as from the stub itself obstructs an accurate detection of Ir, C and Al [26, 27].

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3.3.1. Optical parameter extraction: complex transmission coefficient, refractive index and optical dielectric constant.

The optical properties of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) compositions are characterized by the frequency dependent complex transmission coefficient (T), refractive index ($\hat{n}_{\text{s}}$) and optical dielectric constant (${\hat\varepsilon}_\text{s}$). The parameter extraction process of the material necessitates two spectra, E_s(ω) and E_ref(ω) which are the Fourier transforms of time-domain measurements. The complex transmission coefficient of the samples in frequency domain is obtained by dividing the sample signal E_s(ω) to the reference signal E_ref(ω) which is expressed as,

$$\begin{align}T(\omega ) = \frac{{{E_{\text{s}}}(\omega )}}{{{E_{{\text{ref}}}}(\omega )}} & = \frac{{4{{\hat n}_{\text{s}}}(\omega )}}{{{{\left[ {{{\hat n}_{\text{s}}}(\omega ) + 1} \right]}^2}}} \cdot \exp \left[ { - { }{\kappa _{\text{s}}}(\omega )\frac{{\omega { }t}}{c}} \right]{ }\nonumber\\ & \quad \cdot \exp \left[ { - {\text{i}}\left\{ {{n_{\text{s}}}(\omega ) - {n_{{\text{air}}}}(\omega )} \right\}\frac{{\omega { }t}}{c}} \right],\end{align} \tag{ 10 }$$

where n_air is the refractive index of air. Considering a weakly absorbing medium in which $\hat{n}_{\text{s}}$(ω) ≫ κ_s(ω), the complex transmission coefficient without Fabry–Perot term is given by replacing $\hat{n}_{\text{s}}$(ω) with n_s(ω). In equation (10), the real and imaginary parts of the complex transmission coefficient are dispersive and are given by n_s(ω) and κ_s(ω). The transmission amplitude of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) samples as a function of frequency is shown in figure 4. It is found that at longer frequencies, the transmission decreases in the polycrystalline samples. The maximum transmission amplitude is observed for the minimum substitution of Co²⁺ content at x = 0.1 in NiZn ferrite. Accordingly, as the Co²⁺ content is increased, the transmittance decreases. Now it is a well-known fact that the conduction in ferrites occurs due to the electron hopping between Fe³⁺ and Fe²⁺ at octahedral sites [28]. In the case of spinel ferrites, beyond a certain frequency of electric field, an electronic exchange, Fe³⁺ ↔ Fe²⁺ cannot follow the alternating field of THz radiation, decreasing the response to the incident field, due to which the resulting polarization decreases with increasing frequency when there is no cobalt present. Based on Rezlescu model, the conduction mechanism can be attributed to the presence of combined contribution of p-type (holes) and n-type (electrons) charge carriers [29]. The substitution of cobalt in NiZn ferrite results in hopping of holes between Co²⁺ and Co³⁺ and hopping of electrons between Fe²⁺ and Fe³⁺ that can be expressed as, Co²⁺ + Fe³⁺ ↔ Co³⁺ + Fe²⁺. Thus the increasing amount of Co²⁺ content would cause an increase in the number of charge carriers that lead to the increase of conduction and the decrease of transmission of THz radiation thereby causing increase in polarizability and optical dielectric constant [30].

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The real and imaginary parts of refractive index of the samples are extracted by taking the argument and logarithm of the complex transmission coefficient in thick-sample approximation, that is given by,

$$\begin{equation}\angle T(\omega ) = - { }\left[ {{n_{\text{s}}}(\omega ) - 1} \right]\frac{{\omega { }t}}{c},\end{equation} \tag{ 11 }$$

$$\begin{equation}\ln \left| {T(\omega )} \right| = \ln \left[ {\frac{{4{n_{\text{s}}}(\omega )}}{{{{\left\{ {{n_{\text{s}}}(\omega ) + 1} \right\}}^2}}}} \right] - {\kappa _{\text{s}}}(\omega )\frac{{\omega { }t}}{c},\end{equation} \tag{ 12 }$$

$$\begin{equation}{n_{\text{s}}} = 1 - \frac{c}{{\omega { }t}}\angle T(\omega ),{ }\end{equation} \tag{ 13 }$$

$$\begin{equation}{\kappa _{\text{s}}} = \frac{c}{{\omega { }t}}\left\{ {\ln \left[ {\frac{{4{n_{\text{s}}}(\omega )}}{{{{\left\{ {{n_{\text{s}}}(\omega ) + 1} \right\}}^2}}}} \right] - \ln \left| {T(\omega )} \right|} \right\}{\text{.}}\end{equation} \tag{ 14 }$$

Figure 5 shows the refractive index of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) samples in the frequency range 0.2–2.5 THz. It is observed that the index of refraction (n_s) and the extinction coefficient (κ_s) are slightly varied between about 3.1–3.4 and 0.01–0.1, respectively, for the increasing amount of Co²⁺ content. After extracting the refractive index, we obtain the optical dielectric constants of the samples by using the relation, ${\hat \varepsilon _{\text{s}}}(\omega ) = {\varepsilon ^{\prime}_{\text{s}}}(\omega ) + {\text{i}}{\varepsilon^{\prime\prime}_{\text{s}}}(\omega ) = {\hat n_{\text{s}}}^2(\omega ).$ These ferrites reveal almost constant complex optical dielectric constants ∼9.6 + i0.07, ∼10.9 + i 0.2 and ∼11.7 + i0.5 within 5% error for the substitution of Co²⁺ content at x = 0.1, 0.2 and 0.3, respectively. Based on Lorentz model, the frequency dependent complex optical dielectric constant is expressed as [31–33],

$$\begin{equation}{\hat \varepsilon _s}(\omega ) = {\varepsilon _\infty } + \sum\limits_{j = 1}^m {\frac{{{g_j}}}{{\left( {\omega _j^2 - {\omega ^2}} \right) + i\omega {\gamma _j}}}} ,\end{equation} \tag{ 15 }$$

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where _∞ is the high frequency dielectric constant, g_j (=Δ_jω_j ² ) represents the oscillator strength for the jth resonance, Δ_j is the dielectric contribution, ω (=2π f) is the angular frequency, ω_j is the resonance frequency and γ_j is the damping factor

$$\begin{align}{\varepsilon ^{\prime}_{\text{s}}}(\omega ) = {\varepsilon _\infty } + &\left[ \frac{{{g_1}\left( {\omega _1^2 - {\omega ^2}} \right)}}{{{\omega ^4} - 2{ }{\omega ^2}\omega _1^2 + {\omega ^2}\gamma _1^2 + \omega _1^4}}\right. \nonumber\\ & \left.\quad + \frac{{{g_2}\left( {\omega _2^2 - {\omega ^2}} \right)}}{{{\omega ^4} - 2{ }{\omega ^2}\omega _2^2 + {\omega ^2}\gamma _2^2 + \omega _2^4}} \right],\end{align} \tag{ 16 }$$

$$\begin{equation}{\varepsilon^{\prime\prime}_{\text{s}}}(\omega ) = \left[ {\frac{{{g_1}\omega {\gamma _1}}}{{{{\left( {\omega _1^2 - {\omega ^2}} \right)}^2} + {\omega ^2}\gamma _1^2}} + \frac{{{g_2}\omega {\gamma _2}}}{{{{\left( {\omega _2^2 - {\omega ^2}} \right)}^2} + {\omega ^2}\gamma _2^2}}} \right]{ }{\text{.}}\end{equation} \tag{ 17 }$$

As per the Lorentzian expressions, the real and imaginary components of the optical dielectric constant for the ferrites sintered at 1250 °C for j= 1 and 2, are used to obtain the optical dielectric parameters using equations (16) and (17). The damping factor obtained from the fit to the complex optical dielectric constant describes the scattering time (τ_j or dielectric relaxation constant) of the modes present. The fitting parameters for Ni_0.5Co_x Zn_0.5−xFe₂O₄ (x = 0.1, 0.2 and 0.3) samples are tabulated in table 2. The theoretical curves of the complex optical dielectric constant are used to fit the real and imaginary parts of experimentally observed optical dielectric constant.

Table 2. Optical dielectric parameters of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) extracted from fitting the experimental data of '_s (ω) and ''_s (ω).

Sample	x = 0.1	x = 0.2	x = 0.3
∞	9.6 ± 0.5	10.7 ± 0.6	11.1 ± 0.6
Δ1	1.4 ± 0.1	2 ± 0.1	2.6 ± 0.1
Δ2	0.01 ± 0.001	0.2 ± 0.01	0.6 ± 0.03
ω1/2π (THz)	0.3 ± 0.01	0.4 ± 0.02	0.9 ± 0.04
ω2/2π (THz)	1.9 ± 0.1	1.7 ± 0.1	1.7 ± 0.1
$\tau_{\text{1}}$ (ps)	1.2 ± 0.1	0.1 ± 0.01	0.06 ± 0.003
$\tau_{\text{2}}$ (ps)	1 ± 0.1	0.7 ± 0.03	0.5 ± 0.02

Figure 6 shows that the experimental data is fitted well for the substitution of Co²⁺ content at x = 0.1 and 0.2, however, we find some deviation of the fit curve at x = 0.3 of the real component (left; y-axis) of the optical dielectric constant. This is due to the low signal to noise ratio as we go towards high THz frequency limit. Excellent fits are obtained for the substitution of Co²⁺ content at x = 0.1, 0.2 and 0.3 to the imaginary component (right; y-axis). The complex optical dielectric constants are expressed in terms of high frequency dielectric constant (_∞), two distinct values of dielectric contribution (Δ₁) and (Δ₂), resonance frequencies (f₁) and (f₂), and the dielectric relaxation constants (τ₁) and (τ₂) are listed in table 2.

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We experimentally find that the substitution of Co²⁺ content in NiZn ferrite enhances the conduction in the samples due to the additional charge carriers. This reduces the transmission of THz through the samples. From the parameters obtained in table 2, we find that the substitution of Co²⁺ content leads to the decrease in the higher mode frequency and increase in the lower mode frequency in the dielectric response of the samples. The strength of these modes also becomes higher with the increase in the substitutional Co²⁺ content due to availability of more carriers. This is consistent with the more carriers, decreasing the tight binding in the lattice. It is also evident from the lower mode gaining more strength and blue shifting while higher mode is red shifting as seen in the table.

Magnetic hysteresis loops of Ni_0.5Co_x Zn_0.5−x Fe₂O₄ (x = 0.1, 0.2 and 0.3) samples are presented in figures 7(a)–(c). These are obtained at a temperature range from 300 to 50 K under an applied magnetic field of ±5 tesla. It is found from the figures that M-H loops are S-shaped and ferromagnetic. Low values of coercivity (or narrow-shaped curves) confirm the soft magnetic nature of the ferrites which are depicted in the insets of M-H plots. The experimental values of magnetic parameters; saturation magnetization (M_s), remanent magnetization (M_r) and coercivity (H_c) of the samples are summarized in tables 3(a)–(c). The saturation magnetization as a function of the applied field has been found to decrease significantly with increasing Co²⁺ content at room temperature as revealed in figure 8.

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Generally, the M_s depends upon several factors; superexchange (A-B exchange) interaction, moment of atoms, anisotropy constant of magnetic crystals, size of grains, temperatures and a technique for material synthesis [34, 35]. The superexchange interaction between the metal ions at A- and B-sites in the spinel structure plays an important role for magnetic order [1]. In NiCoZn ferrite, Zn²⁺ often occupies tetrahedral site (A-site) due to its electronic configuration (orbital orientations) that favors tetrahedral bonding with the oxygen ions and Ni²⁺ often occupies octahedral site (B-site) because of higher crystal field stabilization energy [36, 37]. Fe³⁺ and Co²⁺ contents can occupy both the sides; either the A- or the B-sites. Based on lattice constant variation, it is concluded that Co²⁺ content prefers octahedral site [7]. The cationic distribution depends upon the ionic radius of the cations and the method for material preparation [38, 39]. Neel had proposed a two sub-lattice model of collinear spin arrangement. According to this model, the net magnetic moment (M) in ferrites is given by [1, 6],

$$\begin{equation}M = \left| {{M_{\text{B}}} - {M_{\text{A}}}} \right|{\text{,}}\end{equation} \tag{ 18 }$$

where M_A and M_B are the magnetic moments on A- and B-sites in Bohr magneton (μ_B), respectively. In the experimental design, Co nitrate decomposes into Co₃O₄ (Co³⁺) during the pre-sintering process. Further decomposition of Co₃O₄; Co₃O₄ $\to$ 3CoO + $\frac{1}{2}$ O₂, leads to the formation of CoO (Co²⁺) at the sintering temperature of 1250 °C. As a result, Co³⁺ transforms to Co²⁺ because of the reducibility of Fe²⁺ as indicated by the reaction, Co³⁺ + Fe²⁺ $\to$ Co²⁺ + Fe³⁺. Therefore, the number of Fe³⁺ content increases due to the replacement of Fe²⁺ content by the small amount of Co³⁺ content at B-site. When the reduction reaction is completed, the more number of Co²⁺ content are added meanwhile the resultant magnetic moment decreases. The decreasing trend of magnetic moment is due to the magnetic moment at A-site is constant while the magnetic moment at B-site decreases at the same time. This decrease of magnetic moment is led by the magnetic moment of Co²⁺ content that replaced Fe³⁺ content at B-site [40–42]. According to equation (18), the theoretical values of magnetic moment per formula unit in Bohr magneton of the samples at room temperature are calculated 5.3 μ_B, 4.6 μ_B and 3.9 μ_B for x = 0.1, 0.2 and 0.3, respectively, using the cation distribution and their estimated room temperature magnetic moments of Ni²⁺ (2 μ_B), Co²⁺ (3 μ_B), Zn²⁺ (0 μ_B) and Fe³⁺ (5 μ_B). On the other hand, the measured values of the magnetic moment per formula unit in the Bohr magneton are estimated around 3.68 μ_B, 3.53 μ_B and 3.50 μ_B using equation (19) for x = 0.1, 0.2 and 0.3, respectively [23],

$$\begin{equation}n_{\text{B}}^{\text{m}} = \frac{{{M_{\text{w}}}{M_{\text{s}}}}}{{5585}},\end{equation} \tag{ 19 }$$

where M is the molecular weight of the sample, and ${M_{\text{s}}} \propto (1/{a^3})$. Thus according to equations (18) and (19), the variation in lattice constant (a) with increasing Co²⁺ content results in decrease of the saturation magnetization. So this decreasing behavior of M_s depends upon various factors such as; nonmagnetic ions, composition, crystal structure and defects [43]. The magnetic parameters for nickel–cobalt–zinc ferrites prepared by dissimilar methods and sintered at different temperatures are reported in table 4.