# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 3 Results and Discussion
### Data Validation of Impedance Measurements  
It is commonly known (e.g., see [16; 17; 18; 19]) that impedance spectroscopy is extremely susceptible to random disturbances. Therefore, knowing the quality of the measured impedance data is extremely important to facilitate correct analysis. Kramers-Kronig (K-K) relations are very helpful for data validation [17; 19]. The Kramers-Kronig rule states that the imaginary part of a dispersion is fully determined by the form of the real part of dispersion over the frequency range \(\infty\geq\
u\geq\)0. Similarly, the real part of dispersion is determined by the form of the imaginary part of dispersion [17]. In the present study, an analysis based on the K-K relations was performed with the use of the computer programme created by [PERSON] [17; 19]. Kramers-Kronig transform test results of impedance data measured for 67 BFBT ceramics at room temperature (_RT_) within the frequency range \(\Delta\
u\) = (10\({}^{-1}\)-10\({}^{5}\)) Hz are shown in Figures 4a and 5a. A complex impedance diagram combined with K-K transform test results measured at a signal voltage of \(U\) = 0.1 Vrms is shown in Figure 4b. An Impedance diagram with K-K transform test results measured at a signal voltage of \(U\) = 2.0 Vrms is shown in Figure 5b. One can see in Figures 4a and 5a that the data recorded at RT exhibit a small deviation from K-K behaviour (residuals are less than 0.4%). Upon inspection of the  
Figure 3: Williamson–Hall plot for 67 BFBT ceramics calculated based on X-ray diffraction pattern. Figure 2: X-ray diffraction pattern of 67 BFBT ceramics. Ba\({}_{0.3}\)Bi\({}_{0.7}\)FeO\({}_{2.85}\) was taken as a standard; COD ID:4341652.  
results given in Figures 4b and 5b, it can be determined that there is very good agreement between the measurements (blue circles) and K-K calculations (red crosses). The quality parameter \"_chi-squared_\" reached a value of \(\chi^{2}\) = 3.3 \(\times\) 10\({}^{-7}\)-5.5 \(\times\) 10\({}^{-7}\) for room temperature measurements taken at \(U\) = 0.1 V\({}_{\rm rms}\) (\(E\) = 166 V/m) and \(U\) = 2.0 V\({}_{\rm rms}\) (\(E\) = 3333 V/m), respectively. The above-mentioned results proved the high quality of the measurements and fully justified further analysis of the impedance data. An alternative representation of the impedance data showing the influence increasing electric field intensity on the spectroscopic dependence of the reactance (reactance times pulsation: \(-Z^{\prime\prime}\omega\)) of the piezoelectric ceramic sample 67 BFTO at room temperature is shown in Figure 6a. It can be seen that the spectroscopic plots shift towards higher frequencies as the electric field strength increases. Additionally, the dependence of the real part of complex impedance (\(Z^{\prime}\)) on the imaginary part of complex impedance times pulsation (\(-Z^{\prime\prime}\omega\)) (Figure 6b) shows the substantial dependence on the electric field strength. Figure 4: Results of the Kramers–Kronig validation test for admittance measurements of 67 BFTO ceramics at RT and measuring signal voltage \(U\) = 0.1 V\({}_{\rm rms}\). Panel (**a**) shows the residuals plot; panel (**b**) shows the comparison of measured (blue circles) and calculated values (red crosses) for complex impedance plot (\(-Z^{\prime\prime}\) vs. \(Z^{\prime}\); free scale). Figure 5: Results of the Kramers–Kronig validation test for dielectric measurements of 67 BFTO ceramics at RT and measuring signal voltage \(U\) = 2 V\({}_{\rm rms}\). Panel (**a**) shows the residuals plot; panel (**b**) shows the comparison of measured and calculated values for complex impedance plot (\(-Z^{\prime\prime}\) vs. \(Z^{\prime}\); free scale). As shown in the next paragraph of this paper (Section 3.4), an increase in the electric field strength causes an increase in the circumferential magnetic field strength. In turn, the dependence of electric (and dielectric) parameters on magnetic field strength (and vice versa) is a key feature of multiferroic materials, especially lead-free piezoelectric ceramics such as 67 BFTO.