## 3 Results and Discussion
### Electrical Properties
\[Z^{\prime}=\frac{R_{g}}{1+\left(R_{g}C_{g}\right)^{2}}+\frac{R_{gb}}{1+\left(R_ {gb}C_{gb}\right)^{2}}+\frac{R_{el}}{1+\left(R_{el}C_{el}\right)^{2}} \tag{1}\]  
\[Z^{\prime\prime}=\frac{\omega R_{b}^{2}C_{b}}{\left[1+\left(\omega R_{g}C_{g} \right)^{2}\right]}+\frac{\omega R_{gb}^{2}C_{gb}}{\left[1+\left(\omega R_{gb }C_{gb}\right)^{2}\right]}+\frac{\omega R_{el}^{2}C_{el}}{\left[1+\left(\omega R _{el}C_{el}\right)^{2}\right]} \tag{2}\]  
where \(\omega\) is the angular frequency. The electrical behavior of the semiconducting material can be explained by the semicircles and their points of intercept on the real (\(Z^{\prime}\)) axis. These intercepts give the values of the bulk (\(R_{g}\)), grain boundaries (\(R_{gb}\)), and material-electrode interface (\(R_{el}\)). Figure 3: XPS spectra of Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) for Fe. Figure 5 helps to illustrate the frequency-dependent Z\({}^{\prime}\) and Z\({}^{\prime\prime}\) fits, which correspond to the proposed equivalent circuit. These spectra assist in validating good agreement between the theoretical and experimental data and in elucidating the electrical properties of the sample through the proposed electrical circuit [32]. Figure 4: Nyquist plots of Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) at different temperatures. Inset shows the equivalent circuit used to fit the data, where R1, R2, and R3 are contributions due to grain (\(R_{g}\)), grain boundary (\(R_{gb}\)), and interface (\(R_{d}\)), respectively. Similarly, CPE1, CPE2, and CPE3 represent the capacitance due to grain (\(C_{g}\)), boundary (\(C_{gb}\)), and interface (\(C_{d}\)), respectively. (Black, red, blue, and green represent the measurements at 25, 100, 150, and 200 \({}^{\circ}\)C, respectively. The data measured at 250 and 300 \({}^{\circ}\)C are not seen due to overlap near the origin with lower temperature data.) Figure 5: Variation of Z\({}^{\prime}\) and Z\({}^{\prime\prime}\) with frequency at 25 \({}^{\circ}\)C. Inset is the cole-cole plot at the same temperature for the compound Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\). The intercept of the semicircular curve on the low-frequency region in Figure 4 gives the resistance of the material under study. This resistance (R) provides the conductivity with \(\sigma\) of the bulk material (circular pellet) of the thickness (L) and surface area (A) according to Equation (3) [33]:  
\[\sigma=\mathrm{L}/(\mathrm{RA}) \tag{3}\]  
The total resistance \(\mathrm{R_{total}}\) is calculated using the intercept of the semicircle on the low-frequency region of the real axis (\(Z^{\prime}\)), which is equivalent to the total of the individual contributions such that  
\[R_{g}+R_{gb}+R_{el}=\mathrm{R_{total}} \tag{4}\]  
Table 2 displays the values of \(R_{g}\), \(R_{gb}\), and \(R_{el}\) for the material Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) at different temperatures. Figure 6 clearly shows how the grain, grain boundary, and interface contributions were affected by the temperature variation. As seen in Figure 6, the grain and electrode-material interface showed similar patterns of temperature effects, where the contributions first increased from 25 \({}^{\circ}\)C to 100 \({}^{\circ}\)C and then decreased until 300 \({}^{\circ}\)C. However, the grain boundary had a different temperature effect where the contribution first reduced from 25 \({}^{\circ}\)C to 100 \({}^{\circ}\)C and then very slowly increased until 300 \({}^{\circ}\)C. It seems that the contributions were almost similar at 300 \({}^{\circ}\)C for \(R_{g}\), \(R_{gb}\), and \(R_{el}\). The values were almost identical at 300 \({}^{\circ}\)C, as seen in Table 2.  
The frequency and temperature-dependent imaginary part of the impedance (\(Z^{\prime\prime}\)) for Ca\({}_{2}\)Fe\({}_{2}\)O\({}_{5}\) are shown in Figure 2:0 Each curve demonstrates a peak (at the Z\({}^{\prime\prime}\) max value) at a frequency, and this peak position is conventionally known as the relaxation frequency (\(\mathrm{f_{R}}\)). Hence, our material showed the existence of relaxation phenomena due to the presence of the peaks.