## 3 Results
Figure 2: Pictures of the microstructures of ceramic Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) samples. (**1**) BiFeO\({}_{3}\); (**2**) \(x\) = 0.10; (**3**) \(x\) = 0.20. The increase in Er concentrations leads to complication of the registered dielectric spectra (their shift, enhancing the dispersion and formation of additional anomalies--ill-defined peaks in the \(\varepsilon/\varepsilon_{0}(T)\) curves at \(T=600\) K (Figure 3c). A further increase in Er concentration leads to a significant decrease in the dielectric constant (Figure 3e). The relaxation analysis, performed using the Cole-Cole relation (Figure 4), made it possible to take into account the effect of reach-through conductivity. From Figure 4b it is clearly seen that at least two relaxation processes take place in the object. When erbium is added, the relaxation spectrum of objects is simplified and described by a single process. The calculations by the Cole-Cole relation allowed us to calculate the activation energy of both relaxation processes and reach-through conductivity (see Table 1). Figure 3: The dependencies of \(\varepsilon^{\prime}/\varepsilon_{0}\) and \(\varepsilon^{\prime\prime}/\varepsilon_{0}\) of Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) ceramic samples on the temperature in the frequency range 25–2 \(\times\) 10\({}^{6}\) Hz. (Arrows show the increase in the frequency, \(f\); (**a**,**b**) BiFeO\({}_{3}\); (**c**,**d**) Er—\(x=0.10\), (**e**,**f**) \(x=0.20\). Figure 5 shows the temperature dependences of thermal diffusivity (\(\chi\)) and specific heat (\(C_{\rm p}\)) of studied ceramic samples. The heat capacity of objects increases and up to 600 K, the experimental data are in satisfactory agreement with those calculated using the Meer-Kelly formula: \(C_{\rm p}(T)=\rm a+bT\)\(-\)\(\rm cT^{-2}\), where \(\rm a\), \(\rm b\), and \(\rm c\) are constant, which are determined from the temperature dependence of the heat capacity at temperatures of 290-320 K. In the temperature range of 640-670 K, covering the region of the antiferromagnetic transition, the behavior of thermophysical characteristics is extreme with a minimum of \(\chi\) and a maximum of \(C_{\rm p}\). The observed effects are in good agreement with the literature data for similar objects [7, 8]. It should be noted that in solid solutions with large amount of Er temperature dependencies of \(C_{\rm p}\) experience at least two maxima, one of which can be referred to antiferromagnetic phase transitions and another one can be associated with the formation of internal stresses due to the ordering of structural defects (Figure 5a). \begin{table}
\begin{tabular}{c c c} \hline \(x\) & \(E_{\rm a}\), \(\bf eV\) (\(\tau\)) & \(E_{\rm a}\), \(\bf eV\) (\(\sigma_{\rm s}\)) \\ \hline
0 & 0.55 & 0.59 \\ \hline
0.10 & 0.73 & 0.76 \\ \hline
0.20 & 0.77 & 0.67 \\ \hline \end{tabular}
\end{table} 
Table 1: Activation energy of relaxation processes (in the case of \(x=0\)—low-frequency process) and reach-through conductivity \(\sigma_{\rm s}\) of Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) ceramic samples with \(x=0\), 0.10, 0.20, respectively, by the Arrhenius law. Figure 4: (**a**,**d**,**g**)—The dependencies of \(\varepsilon^{\prime}/\varepsilon_{0}\) and \(\varepsilon^{\prime\prime}/\varepsilon_{0}\) of Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) ceramic samples with \(x=0\), 0.10, 0.20, respectively, on frequency \(\omega\). (**b**,**e**,**h**)—Cole–Cole plots of Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) ceramic samples with \(x=0\), 0.10, 0.20, respectively. (**c**,**f**,**i**)—Arrhenius plots of relaxation times \(\tau\) and reach-through conductivity \(\sigma_{\rm s}\) of Bi\({}_{1-x}\)Er\({}_{x}\)FeO\({}_{3}\) ceramic samples with \(x=0\), 0.10, 0.20, respectively (Arrows show the increase in the temperature).