## 3 Results and Discussion
### Electrical Properties
Using Equation (2) and the \(\sigma_{DC}(T)\) conductivity values from Figure 5, we plotted the experimental dependence, \(ln\ \sigma_{DC}\) (\(T^{-1/4}\)), which is shown in Figure 6.  
Slope B of the linear dependence, \(ln\ \sigma_{DC}\) (\(T^{-1/4}\)), was found via the linear fitting of the experimental dependencies from Figure 6. Knowing slope B, the thermal activation energy of electrical conduction, \(E_{A,cond}\), was determined using Equation (3). The temperature dependence of \(E_{A,cond}\) (T) is presented in Figure 7.  
Figure 6: The dependence \(ln\ \sigma_{DC}\)(\(T^{-1/4}\)) of the investigated samples at frequency \(f=1\) kHz. From Figure 7, one can observe a linear increase in \(E_{A,cond}\) with the increase in temperature from 0.177 eV to 0.218 eV for sample 1 (LMO), from 0.175 eV to 0.213 eV for the LMO:Ga sample and from 0.172 eV to 0.210 eV for the LMO:Y sample. As a result, the presence of Ga\({}^{3+}\) or Y\({}^{3+}\) rare earth ions in the LMO perovskite sample leads to a decrease in \(E_{A,cond}\) corresponding to these samples (LMO:Ga and LMO:Y) as compared to the LMO sample. Thus, the increase in conductivity of the doped materials is determined in relation to the electrical conductivity of the undoped sample, as obtained experimentally (see Figure 5). The electrical conduction mechanism of the study samples can be described in terms of the process of charge carriers hopping between the localized states [28; 32]. \(\sigma_{DC}(T)\) may also be expressed as:  
Figure 5.0  
Here, \(T_{0}\) is known as a characteristic temperature coefficient which represents a measure of the degree of disorder [28] and may be expressed as:  
Figure 5.1  
where the constant, \(\lambda\cong 16.6\), is dimensionless [28], \(\alpha\cong 10^{9}\) m\({}^{-1}\) signifies the degree of localization whilst \(N(E_{F})\) stands for the localized states density at the Fermi level \(E_{F}\)[22; 33]. From Equations (2)-(5), after some calculations, the following relation for the \(N(E_{F})\) is expressed:  
Figure 5.2  
Using the values obtained for \(E_{A,cond}\) (\(T\)) from Figure 7 and Equation (6), we computed \(N(E_{F})\) in the temperature range of 28-120 \({}^{\circ}\)C at a low frequency (\(f\) = 1 kHz). For the studied samples, the following values were obtained: \(N(E_{F})\)S1 = 1.1588, 1018 cm\({}^{-3}\) eV-1; \(N(E_{F})\)S2 = 1.2888, 1018 cm\({}^{-3}\) eV-1; and \(N(E_{F})\)S3 = 1.231, 1018 cm\({}^{-3}\) eV-1. These results show that the density of states at the Fermi level \(N(E_{F})\) does not depend on the temperature, remaining constant over the whole investigated domain, as shown recently for other oxide materials (Fe-P or Cu-Mn type) in [34; 35]. Also, the \(N(E_{F})\) for LMO is lower than the values of \(N(E_{F})\) obtained for LMO:Ga and LMO:Y which contain rare earth ions (Ga and Y,  
Figure 5.3 The dependence on temperature of the thermal activation energy of electrical conduction of the investigated samples. respectively) in the investigated temperature range. This result is in agreement with the fact that the thermal activation energy of electrical conduction, \(E_{A,cond}\), of the doped materials is smaller than the \(E_{A,cond}\) of the LMO sample (see Figure 7). As a result, we advance the statement that the reduction in \(E_{A,cond}\) is related to the increase in the density of localized states at Fermi level \(N(E_{F})\), and to support this, we determine two other parameters of the VRH model, i.e., the hopping energy, W, and the hopping distance, R, by means of the following relations [28; 33]:  
Figure 5.4  
Figure 5.5  
Making use of the values of \(N(E_{F})\) in Equation (7), we have computed the hopping distance, \(R\), in every sample and at each measurement temperature within the range of 28-120 \({}^{\circ}\)C. With the hopping distance \(R\) thus computed, from Equation (8), the hopping energy \(W\) was determined for each studied sample.