Thermo-gravimetric analysis (TGA) as well as differential thermal analysis (DTA) of prepared sample Co_0.5Cd_0.5Fe₂O₄ is accomplished in a temperature range of 0°C–1010°C. The analysis of percentage weight loss against temperature was made with the help of TGA curve, which exhibits that the decomposition process is completed step by step.

A weight loss of about 1.98% is observed in the whole temperature range. Rapid changes in the curve occur at 55°C and 420°C. Below 100°C there is a weight loss of approximately 0.38%, which is due to the evaporation of water.^[24,25] At 358°C the weight loss of about 1.38% occurs which is due to the decomposition of matter, evaporation of water and oxidation of organic matter.^[26,27] Secondly, most of the weight loss is nearly 1.36% occurring in the temperature range of 100°C–420°C. Then in the last step, a 0.17% weight loss is found in the temperature range of 420°C–974°C. From TGA graph it follows that the phase formation starts at 900°C after which there is very small weight loss, beyond 1010°C there exists no weight loss, which confirms the creation of stable phase. The DTA curve shows that the decomposition is carried out step by step due to some exothermic peaks. The first sharp peak is observed at 94°C, the second oneoccurs at 234°C and the third one appears at 405°C, indicating the burning of additional constituents.^[24,28] Figure 1 shows the TGA and DTA curves.

Fig. 1.

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	Fig. 1. TGA and DTA curves.

Bismuth doped cobalt–cadmium ferrites having general composition formula Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are synthesized via micro-emulsion route. To study the crystalline structure in detail and identify the crystalline phase formation, powder XRD is carried out on PANalytical XPert Pro.^[29] It is very basic and important characterization technique to determine certain parameters of crystal structure like, crystalline size, lattice constant, bulk density, x-Ray density, stacking fault, lattice strain, micro-strain and dislocation density, and so on. The crystalline phase of the material is confirmed because the obtained XRD pattern consists of very sharp peaks. At the most intense peaks are observed to have 311 hkl value, this peak is considered to be an ideal for cubic structure.^[30] The peaks corresponding to XRD pattern are studied and are indexed as (220), (311), (222), (400), (422), (511), (440), (531), and (533). These peaks correspond to the FCC spinel structure.^[31,32] For further confirmation, these peaks are verified by JCPDS card number 22-1086. A couple of impure peaks has also been observed: one is at having hkl value 411, the other is at in the sixth sample indicated by (*). The occurrence of these peaks may be because of insolubility of Bi at octahedral site.^[26,33] The XRD patterns of the synthesized ferrite samples are shown in Fig. 2.

Fig. 2.

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	Fig. 2. XRD Analysis of Co0.5Cd0.5BixFe2−xO4 (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25).

The crystalline size is calculated for the hkl value of (311) from the Debye Sherrer’s formula given by 1 where k is the constant having a valueof 0.9, λ is the x-ray beam wavelength and its value is 1.54 Å, β is the full width at half maximum (FWHM) of the (311) peak, and θ is its angle of diffraction.^[4] The crystalline size varies in homogenously with the substitution of Bi. This behavior of inhomogeneity existing in crystalline size may be due to the generation of secondary phase and also relatively small ionic radius of Bi³⁺ as compared with that of the Fe³⁺ which is to be replaced.^[26] Lattice constant is calculated by Nelson Relay Function. This is given as follows: 2

The average lattice constant is calculated to be in a range of 8.46 Å–8.5 Å. The variations in the “a” is described on the basis of Bi³⁺ and Fe³⁺ ion radius. Lattice parameter is found to decrease with the substitution of Bi ion content.

This is due to the replacement of large radius of Fe³⁺ ion (0.76 Å) by smaller radius of Bi³⁺ ion (0.74 Å) at the octahedral site.^[34–36] Lattice constant first decreases rapidly with the substitution and then increases, in the end it again decreases, which s may be due to the segregation of Bi³⁺ ions on grain boundaries.^[1] The x-ray density of synthesized ferrite is found from the following formula: 3 where M represents the molecular weight, N the Avogadro number, and is a unit cell volume. The relation between x-ray density and concentration is almost linear. With the increase of Bi³⁺ concentration, the x-ray density also increases: this is attributed to the greater molar weight of Bi³⁺ (208.98) than that of Fe³⁺ (55.845). Bulk density of the prepared pallets is calculated from the formula 4 where m represents the mass of pallet and ‘v’ its volume. Lattice strain (ε) of the prepared nanoparticles is calculated from the Stokes–Wilson equation 5 where β is the FWHM of the most intense XRD peak. The micro-strain of prepared nanoparticle is found from the following equation: 6 Dislocation density of the synthesized ferrite is calculated from the equation 7 where D represents the crystalline size. Stacking fault is calculated for the prepared ferrite samples with the help of the following formula: 8 Stacking fault fluctuates repeatedly: first it decreases then it increases to a certain value and in the end it again decreases and then increases. This inhomogeneous behavior can be due to annealing temperature.^[37] Table 1 shows all the structural parameters dependent on the doping concentration.

Table 1.

Table 1.

Table 1. Structural parameters of Co0.5Cd0.5BixFe2−xO4 (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25). . Parameter x = 0.0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25 Crystalline size/nm 15.4157 14.1147 14.8737 11.7253 12.8087 13.4281 Lattice constant a/Å 8.50931 8.4783 8.46595 8.49676 8.49952 8.4948 Cell volume ( ) 616.145 609.433 606.774 613.422 614.020 612.998 x-ray density/ 5.63509 5.86405 6.05738 6.15756 6.31720 6.49367 Bulk density/ 1.17301 1.26855 1.35518 1.24530 1.24331 1.27261 Lattice strain/ 7.49341 8.15242 7.72146 9.84644 9.006118 8.593068 Microstrain/ (lines−2/m−4) 2.24769 2.45487 2.32960 2.95514 2.70519 2.580409 Dislocation density/1015 (lines/m2) 4.20794 5.01940 4.52018 7.27363 6.09522 5.54588 Stacking fault 0.45118 0.45022 0.44974 0.45104 0.45083 0.45090

Table 1. Structural parameters of Co0.5Cd0.5BixFe2−xO4 (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25). .

Fourier transform infrared spectrum (FTIR) is recorded the on iS50 FT-IR for further detailed analysis of crystal structure. The spinel structures of the prepared ferrites having compositional formula Co_0.5Cd_0.5Bi_xFe O₄ (x =0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are investigated. The two frequency bands are recorded: one is high frequency bands (at around 500 cm⁻¹) because of tetrahedral site of intrinsic stretching vibrations, and the other is low frequency band which is (at around 400 cm⁻¹) because of octahedral stretching bands. These frequency bands represent the characteristic feature of spinel structure.^[26,38] The changes in are observed in a span of 530 cm⁻¹–538 cm⁻¹ whereas remains static. The high frequency band and low frequency band are formed because of the Fe³⁺–O²⁻ stretching vibrations at tetrahedral and octahedral site. The shifting of frequency band slightly to the high frequency is due to the variation generated in lattice parameter, and these changes affect the stretching vibrations of Fe³⁺–O²⁻ due to which the band position may be changed.^[24,26] Figure 3 shows the FTIR spectra observed in 400 cm⁻¹–1000 cm⁻¹ range.

Fig. 3.

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	Fig. 3. FTIR analysis of Co0.5Cd0.5BixFe2−xO4 (x= 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25).

From the frequency band data the force constants are also calculated, which are K_t and K_o for tetrahedral and octahedral sites by the following formulas: 9 10 where M denotes the molecular weight, and are frequency bands. The force constants are analyzed to enhance the doping concentration, which indicates the possible strengthening of interionic bonding.^[39] The tetrahedral radius and the octahedral radius are also obtained from the following formulas: 11 12 where R_tetra is the tetrahedral radii, R_octa is the octahedral radii, u is the oxygen position parameter, and a represents the lattice parameter. The value of oxygen position parameter is 0.375 for FCC structure.^[40]

These tetrahedral and octahedral radii are seen to decrease inhomogenously, which proves the decrease in bond length with the concentration of Bi³⁺ increasing.^[24] The values of these parameters are given in Table 2.

Table 2.

Table 2.

Table 2. Different parameters of FTIR analysis. . Parameters x=0.0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25 Molecular weight/(gm/mol) 261 269 276 284 291 299 /cm−1 530 537 533 533 538 534 /cm−2 402 402 402 402 402 402 /(dyne/cm 1.35644 1.36066 1.36467 1.36850 1.37213 1.37561 /(dyne/cm 2.52909 2.57048 2.55886 2.56602 2.59698 2.58419 0.80733 0.79958 0.79649 0.80419 0.80488 0.8037 0.52232 0.51561 0.51293 0.5196 0.5202 0.51918

Table 2. Different parameters of FTIR analysis. .

The dielectric properties of prepared ferrite samples having general formula of composition Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x =0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are calculated with the help of impedance analyzer in a span of frequency from 1 MHz to 3 GHz at room temperature (as shown in Table 3). The dielectric properties are of great interest because of their suitability in high frequency appliances. These properties strictly depend on the synthesis method, composition of material, and the cation positioning.^[26] In order to obtain the information about conduction mechanism and polarization, the AC conductivity and the real and imaginary part of dielectric constant must be studied in a high frequency range.^[41]

Table 3.

Table 3.

Table 3. Dielectric parameters for Co0.5Cd0.5BixFe2−xO4 (x=0.0, 0.05, 0.1, 0.15, 0.2 and 0.25). . Parameters Frequency x=0.0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25 Dielectric constant 1 MHz 4.74 3.69 2.86 5.07 2.53 2.49 1 GHz 4.02 2.63 2.20 3.45 1.83 2.18 3 GHz 4.18 2.49 2.17 3.55 1.73 2.18 Dielectric loss 1 MHz 0.163 0.113 0.110 1.18 0.419 0.315 1 GHz 0.149 0.0250 0.0291 0.0702 0.00977 0.0213 3 GHz 0.359 0.00308 0.0623 0.230 0.00203 0.0793 Tan loss 1 MHz 0.0344 0.0307 0.0385 0.232 0.165 0.126 1 GHz 0.0372 0.00951 0.0132 0.0204 0.00534 0.0097 3 GHz 0.0858 0.00124 0.0287 0.0646 0.00117 0.0364

Table 3. Dielectric parameters for Co0.5Cd0.5BixFe2−xO4 (x=0.0, 0.05, 0.1, 0.15, 0.2 and 0.25). .

3.4.1. Dielectric constant, dielectric loss and tangent loss

Dielectric constant, dielectric loss, and tangent loss of Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are calculated with respect to the frequency of applied field.

Figure 4 and 5 show these parameters as a function of frequency. These graphs exhibit the dispersion behaviors at high frequencies, such a kind of behavior is because of Maxwell Wagner interfacial polarization and also Koop’s phenomenological theory. These models explain that the grain boundaries are more active at low frequencies whereas grains are more operational at high frequencies. While annealing the samples, the irregular spreading of oxygen ion at grains and grain boundaries lead to the ionic and electronic polarization at high frequency and the interfacial polarization at low frequency. As the frequency rises, the polarization goes on decreasing in strength and hence the dielectric constant continues to decrease. As is well known, the main charge carriers in ferrites are electrons, the electronic exchange occurs between Fe³⁺ and Fe²⁺ which reside at octahedral sites. This process of electronic exchange is known as electron hopping. This behavior of dielectric constant decreasing is accomplished due to the electron hopping because it does not obey the pattern of applied alternating field at very high frequency.^[24,42–44]

Fig. 4.

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	Fig. 4. Variations of dielectric constant versus frequency.

Fig. 5.

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	Fig. 5. Variations of dielectric loss versus frequency.

Figure 6 shows the variations in tangent loss with frequency. This variation in tangent loss is analyzed to be like that in dielectric constant and dielectric loss. In a feeble frequency range the electron hopping frequency follows the applied field frequency, that is why the loss is calculated to be comparatively high, but as soon as frequency of applied field is enhanced further the electron hopping cannot follow the applied field frequency. Therefore after certain critical frequency, the tangent loss decreases significantly.^[24] These dielectric parameters are listed in Fig. 3.

Fig. 6.

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	Fig. 6. Tangent loss versus frequency.

3.4.2. AC Conductivity

The AC conductivity is one of the important characteristics of dielectric material. The AC conductivity of the prepared ferrite samples Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are calculated in a frequency range from 1 MHz to 3 GHz. ssThe formula to be used for the calculation of AC conductivity is where t represents the thickness of pallet, A is its area, and represent the real and imaginary impedance parts, respectively. The plot of AC conductivity versus frequency for each of prepared samples is shown in Fig. 7.

Fig. 7.

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	Fig. 7. AC Conductivity versus frequency.

From the figure it can be observed that for most of samples the AC conductivity has an increasing trend in the low frequency range, whereas in the high frequency range it presents the dispersive behavior . Maxwell–Wagner model and Koop’s phenomenological theory both explain that the ferrites materials have high conducting grains which are separated by grain boundaries having resistive nature.^[45] At low frequency the behaviors of all samples seem to be planar and the same due to high resistance of grain boundaries. At high frequencies, as a result of the effects of grains and the hopping phenomena in Fe²⁺ and Fe³⁺ at octahedral sites, the conductivity is enhanced. The increasing behavior of conductivity may also be due to the reduction of porosity. At low frequencies it is seen that the conductivity has a grain boundary effect whereas in the high frequency range the conducting effects of grains are observed, therefore the dispersion trend can be seen.^[1,24,26]

3.4.3. Real and imaginary impedance

Impedance analysis is a powerful tool to relate dielectric properties of synthesized material with its microstructural composition.^[46,47] The real and imaginary impedance part are calculated for each of the ferrite Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25). These impedance parts dependent on the applied field frequency are calculated as depicted in Figs. 8 and 9.

Fig. 8.

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	Fig. 8. Real part of impedance versus log of frequency.

Fig. 9.

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	Fig. 9. Imaginary part of impedance versus log of frequency.

Impedance spectrum shows that the increase of applied frequency reduces the real and imaginary impedance parts. With the further increase of frequency the impedance curves of all samples merge with each other and at higher frequency impedance show constant behavior, which is attributed to the release of space charges.^[48] These space charges are produced because of the disparity of concentration and also due to the inhomogeneity of applied field which tends to accumulate these charges on the grain boundaries. The decreasing real and imaginary impedance parts show the improvement of conductivity with the increase of field frequency.^[47,48]

3.4.4. The real, imaginary modulus, and Cole–Cole (Nyquist) diagram interpretation

Real and imaginary modulus of the prepared ferrite having compositional formula Co_0.5Cd_0.5Bi_xFe_2−xO₄ (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) are calculated as a function of applied frequency. The relations used for the calculation of real and imaginary modulus are 13 14

Figure 10 and 11 demonstrate the plots of real and imaginary electric modulus versus applied field frequency in a range from 1 MHz to 3 GHz. The ferroelectric materials have an electrical response which can be analyzed by using electric modulus, and this electrical response is based on the phenomenon of electric polarization. In some of homogenous materials the bulk (grain) and grain boundary effects can be described on the basis of complex electric modulus formalism. Electric modulus is used to analyze the behavior of interfacial polarization depending on the applied field frequency in Co_0.5Cd_0.5Bi_xFe_2−xO₄.

Fig. 10.

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	Fig. 10. Real electric modulus versus frequency graph.

If the grain boundary region occupies a greater volume, then the graph between and can provide good information about the semicircle.^[49] This kind of attitude is confirmed by obtaining the relation between grain boundary and the formation of peak.^[1] The appearance of each peak vs. frequency in imaginary electric modulus is depicted in Fig. 11. The values of AC conductivity, impedance and modulus are represented in Table 4.

Fig. 11.

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	Fig. 11. Imaginary electric modulus versus frequency graph.

Table 4.

Table 4.

Table 4. AC conductivity, impedance, and modulus values for Co0.5Cd0.5BixFe2−xO4 (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) in the frequency range from 1 MHz to 3 GHz. . Parameters Frequency x=0.0 x=0.05 x=0.1 x=0.15 x=0.2 x=0.25 AC Conductivity 1 MHZ 5.79866 × 10−6 7.87803 × 10−7 7.2124 × 10−7 9.54509 × 10−6 1.91567 × 10−6 5.26607 × 10−6 1 GHZ 1.883699 × 10−3 3.96262 × 10−4 4.596 × 10−4 9.48 × 10−4 1.6695 × 10−4 3.30598 × 10−4 3 GHZ 1.2092363 × 10−2 4.14805 × 10−4 2.84216 × 10−3 8.553363 × 10−3 3.5165 × 10−6 3.395429 × 10−3 1 MHZ 9.41 × 103 2.23 × 103 1.98 × 103 1.08 × 104 7.61 × 103 1.76 × 104 1 GHZ 3.31 1.27 1.90 2.08 8.88 × 10−1 1.38 3 GHZ 2.45 1.64 × 10−1 1.34 2.02 2.26 1.59 1 MHZ 8.50 × 104 1.13 × 105 1.11 × 105 7.04 × 104 1.33 × 105 1.21 × 105 1 GHZ 8.89 × 101 1.20 × 102 1.36 × 102 9.93 × 101 1.55 × 102 1.37 × 102 3 GHZ 3.01 × 101 4.21 × 101 4.60 × 101 3.25 × 101 5.38 × 101 4.59 × 101 M’ 1 MHZ 1.05039 1.39519 1.37548 0.870514 1.650339 1.501691 1 GHZ 1.10524 1.493224 1.694277 1.234792 1.923289 1.705278 3 GHZ 1.11549 1.562217 1.704708 1.205494 1.994711 1.703632 M” 1 MHZ 0.11640 0.027573 0.024533 0.13304 0.094080 0.217934 1 GHZ 0.04118 0.015793 0.023585 0.025844 0.011037 0.017183 3 GHZ 9.074 × 10−2 6.065 × 10−3 4.9524 × 10−2 7.4754 × 10−2 8.3825 × 10−5 5.9111 × 10−2

Table 4. AC conductivity, impedance, and modulus values for Co0.5Cd0.5BixFe2−xO4 (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25) in the frequency range from 1 MHz to 3 GHz. .

The Cole–Cole diagram is shown in Fig. 12, which provides the information about grain and grain boundaries. This graph can be plotted on any of these parameters such as electric modulus, permittivity and impedance. These parameters are interrelated to each other in terms of characteristics. The Cole–Cole plot shows only single semicircle, which confirms that the conduction behavior is due to grain boundaries only. Furthermore, the complex modulus spectra show that is inversely related to the capacitance due to the fact that the bulk (grain) effects are also seen. The variations in the semicircle radius may be due to the substitution of Bi³⁺. The obtained spectra can be explained by jump relaxation model (JRM) as well. According to this model, the conduction process at low frequency is due to successful hopping of charge carriers between A site and B site but this hopping mechanism is greatly restricted at high frequency, so the dispersion behavior is observed.^[1,50]

Fig. 12.

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	Fig. 12. Cole–Cole plot variations for Co0.5Cd0.5BixFe2−xO4 ferrite (x = 0.0, 0.05, 0.1, 0.15, 0.2, and 0.25).