## 3 Results and Discussion
Under an external electric field, the holes in the valence band move due to the hopping transitions of electrons, similar to the behavior in intrinsic semiconductors. A semiconductor doped with an acceptor impurity (nickel atoms, in this case) exhibits only hole conductivity and thus belongs to p-type semiconductors [39]. Thus, NdNa\({}_{2}\)NiMnO\({}_{5}\) exhibits a moderate increase in the bandgap width as temperature rises, which may indicate the stability of its structure within these temperature intervals. On the other hand, NdK\({}_{2}\)NiMnO\({}_{5}\) demonstrates a significantly sharper increase in the bandgap width, which could suggest phase transitions or more pronounced changes in its electronic structure. The differences in the behavior of these compounds might be attributed to the distinct roles of sodium (Na) and potassium (K) ions in the crystal lattice, their influence on electronic interactions, and defect states. An increase in the bandgap width with rising temperature may point to reduced conductivity and a transition to a state with more pronounced insulating properties. The classification of a semiconductor (narrow bandgap or wide bandgap) depends on the bandgap width. For NdNa\({}_{2}\)NiMnO\({}_{5}\), in the range of 293-353 K, it falls into the category of narrow-bandgap semiconductors; in the range of 423-483 K, it transitions closer to the medium bandgap category. For NdK\({}_{2}\)NiMnO\({}_{5}\), in the range of 293-403 K, it also belongs to the narrow-bandgap semiconductor category; at 443-463 K, as the temperature increases, its characteristics shift to those typical of wide-bandgap semiconductors. These materials exhibit a temperature-dependent bandgap width, making them potentially useful for applications that require conductivity variations with temperature changes. It should be emphasized that there is a correlation between the temperature of the second-order phase transition of NdNa\({}_{2}\)NiMnO\({}_{5}\) on the curve C\({}^{0}\)P\({}^{-}\)f(T) at 423 K and the transition temperature from metallic conductivity to semiconductor conductivity on the curve lg\(R\)-f(T) also at 423 K. This, to some extent, indicates the nature of the second-order phase transition. For a comparison of the electrophysical properties of NdNa\({}_{2}\)NiMnO\({}_{5}\) and NdK\({}_{2}\)NiMnO\({}_{5}\) with the reference BaTiO\({}_{3}\) perovskite, we present its electrophysical characteristics measured using the LCR-781 instrument in the temperature range of 293-483 K at a frequency of 1 kHz (Table 9). As shown in Table 9, the dielectric permittivity (\(\varepsilon\)) of NdNa\({}_{2}\)NiMnO\({}_{5}\) at 293 K exceeds that of BaTiO\({}_{3}\) at 293 K by 2.7 times, at 353 K by 106 times, and at 483 K by 193 times. Similarly, the dielectric permittivity of NdK\({}_{2}\)NiMnO\({}_{5}\) at 403 K exceeds that of BaTiO\({}_{3}\) by 870 times. The dielectric permittivity values of NdNa\({}_{2}\)NiMnO\({}_{5}\) at 783 K and NdK\({}_{2}\)NiMnO\({}_{5}\) at 403 K approach levels close to giant dielectric constants (\(\varepsilon\)), making them of interest as memory materials for data storage devices. \begin{table}
\begin{tabular}{c c c c c} \hline
**T, K** & **R, Om** & \(\varepsilon\) & **lg\(\varepsilon\)** & **lgR** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 9: Dependence of electrical capacitance (C), electrical resistance (R), and dielectric capacitivity (\(\varepsilon\)) of BaTiO\({}_{3}\) on temperature and at a frequency of 1 kHz. The reliability of the experimental results presented in this study is substantiated by the following measures. Recalibration of equipment minimizes systematic errors and ensures measurement accuracy throughout the experiment. Conducting a thorough error analysis enhances the interpretation of the results and demonstrates transparency in the experimental approach. Comparing the findings with existing results under similar conditions strengthens the validity of the conclusions by confirming their consistency with established outcomes.