## 3 Results and Discussion
### Impedance Spectroscopic Studies  
As was mentioned in the introduction, impedance studies allow us to understand the contribution of individual microstructural elements to electrical properties. Although the studies were carried out over a wide temperature range (from \(T\) = 473 to \(T\) = 824 K), for the purposes of this publication, only the results obtained at high temperatures are discussed, where the resulting characteristics of the imaginary part of impedance (\(Z^{\prime\prime}\)) as a function of the real part (\(Z^{\prime}\)) exhibit semicircular arcs (rather than only arcs), allowing for the identification of the contributions from grains and grain boundaries. Investigations described in our previous publications have proven that extrapolating the results obtained at high temperatures to room temperature provides accurate information about the resistance of microstructure components. The obtained results were directly confirmed using atomic force microscopy techniques [37]. Figure 5 present the real and imaginary parts of the impedance as a function of frequency on a double logarithmic scale at several temperatures for PBZT 25/70/30 ceramics modified by the minimum and maximum amounts of lanthanum. The characteristics of the remaining discussed ceramic materials are included in Supplements S1 and S2. It was observed that \(Z^{\prime}\) decreases slowly in the first frequency range, and then the rate of decrease increases significantly. The frequency boundary between these two regions shifts to higher values as the temperature increases. Moreover, the increase in temperature also leads to a decrease in \(Z^{\prime}\) values across the entire frequency range studied, which is a consequence of the increase in electrical conductivity [38]. In the frequency dependencies of \(Z^{\prime\prime}\) appears a maximum which decreases in value and moves to a high-frequency region with increasing temperatures. The movement of the peak results from a temperature-dependent relaxation process occurring in the sample [39]. In turn, the asymmetry of the observed peaks suggests the presence of electrical processes in the material with a distribution of relaxation times. The source of this relaxation may likely be electrons or immobile charges at lower temperatures and defects at higher temperatures. The absolute values of the high-frequency slopes are, indeed, very close to unity and appear to be independent of temperature. Similar behavior has also been observed in other ceramic materials, for example, in PLZT 2/95/5 ceramics [40]. The merging of the log\(Z^{\prime\prime}\)(logf) characteristics into a single curve observed in the high-frequency range indicates significant space charge accumulation [41; 42]. Figure 4: EDS spectrum of PBZT-25/70/30 ceramics doped with La\({}^{3+}\) at concentration \(x=1\) at.%. Figure 6 shows the impedance data taken over a wide range of frequencies at several temperatures, presented as a Nyquist diagram (complex impedance spectrum). The diagrams take the shape of highly distorted semicircles with lowered centers, indicating the existence of separate impedance responses from the grains and grain boundaries. The similar effect of lanthanum ions on the shape of implantation characteristics was noted, among others, by the authors of the study [43].The interior of the ceramics cannot be treated as a homogeneous whole. Therefore, the equivalent electrical circuit used to describe the impedance response consists of two branches. For pure PBZT 25/70/30 ceramics, each branch was a classic Voigt element, i.e., a capacitor and a resistor connected in parallel. In the case of lanthanum-modified ceramics, the first branch, describing the dielectric properties of the grains, is a modified Voigt element, which includes not only resistance and capacitance but also a CPE element. The second branch, corresponding to the electrical properties of the grain boundaries, is a classical Voigt element (Figure 7).