## 3 Results and Discussion
### Electrical Properties  
With the measured values of \(Z^{\prime}\) and \(Z^{\prime\prime}\), the electrical conductivity \(\sigma\), can be computed for each temperature, \(T\), using the following equation:  
\[\sigma\ =\ \frac{Z^{\prime}}{|Z^{*}|^{2}}\cdot\frac{d}{A} \tag{1}\]  
Here, \(A\) and \(d\) stand for the transversal sectional area and the length of the sample, respectively, and \(|Z^{*}|\ =\ \sqrt{Z^{\prime 2}+Z^{\prime\prime}2}\) represents the modulus of the complex impedance of the sample. It is known that [21] in the low-frequency range, (\(f\leq 1\) kHz), the value of \(\sigma\), determined by relation (1), represents the DC electrical conductivity, \(\sigma_{DC}(T)\), which does not depend on frequency but on temperature. For each sample, the temperature dependence of static conductivity, \(\sigma_{DC}(T)\), was plotted by using the conductivity values obtained at the frequency of \(f=1\) kHz at temperatures ranging from 28 \({}^{\circ}\)C to 120 \({}^{\circ}\)C. This plot is depicted in Figure 5.  
Figure 5 shows that the static conductivity, \(\sigma_{DC}(T)\), rises with temperature across all samples, demonstrating a thermally activated conduction process within the measurement temperature range of 28-120 \({}^{\circ}\)C. The electrical conductivities of LMO:Ga and LMO:Y, respectively, are higher than that of the LMO sample, which shows that the presence of rare earth ions of the Ga\({}^{3+}\) or Y\({}^{3+}\) types in the LaMnO\({}_{3}\) perovskite sample leads to the electrical conductivity of these samples increasing compared to the pristine sample (LMO). On the other hand, the substituted Ga\({}^{3+}\) ions in the LaMnO\({}_{3}\) perovskite crystal lattice behave as acceptor dopants replacing the Mn\({}^{3+}\) ions in the octahedral B-site [22], which leads to the compensation of the charge imbalance following their oxidation. As a result, gallium doping slightly reduces the octahedral distortion and diminishes the Jahn-Teller effect by enhancing electronic correlations [23; 24], resulting in a larger increase in the electrical conductivity [25] of the LMO system doped with Ga ions, as can be observed in  
Figure 5: Dependence on temperature of the static conductivity, \(\sigma_{DC}(T)\), of the investigated samples. Figure 5. The yttrium ions replace the lanthanum ions in the tetrahedral A-site of the LMO perovskite crystal lattice [26], and as a result, the yttrium substitution leads to a lowering of the mean ionic size at the La-site, which leads to distortion of the MnO\({}_{6}\) octahedra. The Mn\({}^{3+}\)-O-Mn\({}^{4+}\) bond angle thus changes from 180\({}^{\circ}\), which leads to a decrease in the probability of electron hopping between Mn\({}^{3+}\) and Mn\({}^{4+}\) ions [27], and will result in a lesser rise in the electrical conductivity of the LMO system doped with Y ions, as was determined experimentally (see Figure 5). Also, the rise in \(\sigma_{\rm DC}\) with temperature stems from the growth of the drift mobility of charge carriers in the sample, in agreement with [PERSON]'s model of variable-range hopping (VRH) [28]. This type of behavior with the temperature of the conductivity \(\sigma_{\rm DC}(T)\) has also been observed for other perovskite materials, such as NaTaO\({}_{3}\)[29] doped or not doped with metallic ions (Ag, Fe, Cu) [11; 30; 31], which shows that the most suitable electrical conduction mechanism in these materials is the VRH mechanism. According to the VRH model, the \(\sigma_{DC}(T)\) conductivity is given by:  
\[\sigma_{DC}\ =\ \sigma_{0}\exp\left[-\frac{B}{\left(T\right)^{1/4}}\right] \tag{2}\]  
where \(\sigma_{0}\) is the pre-exponential factor and  
\[B\ =\ \frac{4E_{A,cond}}{kT^{3/4}} \tag{3}\]  
In Equation (3), \(k\) stands for the Boltzmann's constant, whilst \(E_{A,cond}\) represents the thermal activation energy of electrical conduction [28].