# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 3 Results and Discussion
### Piezoelectric Ceramics Characterisation with the Resonant Method  
It is commonly known (e.g., see [21; 24]) that values of the piezoelectric properties of a material can be derived from the resonance behaviour of suitably shaped specimens subjected to a sinusoidally varying electric field. Therefore, the impedance measurements were performed for 67 BFBET ceramics within the following frequency range: \(\
u\) = 100 kHz-10 MHz, which corresponded to the frequency ranges of resonant spectra of the radial mode and the thickness extension mode for thin disk sample [21; 22]. To reveal the influence of a weak circumferential magnetic field on the resonance behaviour of multiferroic lead-free material, the measuring voltage \(U\) = 0.1 V\({}_{\rm rms}\) and \(U\) = 2.0 V\({}_{\rm rms}\) was taken. Taking into account all the impedances in the circuit, the measuring sinusoidal signal leads to electric currents (\(i_{\rm ac}\)) through the sample of 67 BFBET ceramics in the range following range: \(i_{\rm ac}\) = 1.2 mA (at \(U_{\rm ac}\) = 0.1 V\({}_{\rm rms}\), \(\
u\) = 10 MHz)-11.7 mA (at \(U\) = 2.0 V\({}_{\rm rms}\), \(\
u\) = 10 MHz). The ac field amplitude (rms) generated by these currents in a radial point \(r\) on the sample can be calculated as:  
\[H_{ac} = \
icefrac{{i_{\rm ac}r}}{{2\pi a^{2}}} \tag{1}\]  
where \(r\) is the radial point considered on the sample cross-section, and \(a\) is its total radius. The rms range of ac-measuring fields leads to circumferential magnetic fields (i.e., on the sample edges--the lateral surface of the disk-shaped sample--is where it is highest) between 0.05 and 0.49 A/m (at \(U\) = 0.1 V\({}_{\rm rms}\) and \(U\) = 2.0 V\({}_{\rm rms}\), \(\
u\) = 10 MHz, respectively) [28]. The results of the calculated radial magnetic field intensity \(H_{\rm ac}\) are given in Figure 7. It is worth noting that the resonance behaviour of the 67 BFBET ceramic sample is also reflected in Figure 7. Positions of the resonances are closely related to the piezoelectric properties of the material. Figure 8 shows the frequency response of a 0.67 Bi\({}_{1.02}\)FeO\({}_{3}\)-0.33 BaTiO\({}_{3}\) ceramic thin disk that was 7.52 mm in diameter and 0.6 mm thick. Electrodes were deposited onto both faces of the disk, and then the disk was poled in the direction perpendicular to the faces of the disk. The measurements were taken within a temperature range from \(-\)20 \({}^{\circ}\)C to 50 \({}^{\circ}\)C. It should be noted that the frequency peaks visible at about 3-4 \(\times\) 10\({}^{5}\) Hz (Figure 8)  
Figure 6: Alternative representation of the impedance data collected at room temperature and signal amplitude \(U\) = 0.1 V\({}_{\rm rms}\), \(U\) = 0.2 V\({}_{\rm rms}\), \(U\) = 1.0 V\({}_{\rm rms}\), and \(U\) = 2.0 V\({}_{\rm rms}\). Panel (**a**) shows the dependence of reactance times pulsation (\(-\)\(Z^{\prime\prime}\omega\)) on frequency (\(\
u\)); panel (**b**) shows the dependence of resistance (\(Z^{\prime}\)) on reactance times pulsation (\(-\)\(Z^{\prime\prime}\omega\)). are radial resonances, whereas the frequency peaks visible at about 3-5 \(\times\) 10\({}^{6}\) Hz (also in Figure 8) are related to the thickness mode resonance. Figure 8: Dependence of impedance modulus \(|Z|\) as well as the phase angle (\(\Theta\)) on frequency for 67 BFBT ceramics; measuring voltages: \(U=0.1\) V\({}_{\rm rms}\) (**a**,**c**) and \(U=2.0\) V\({}_{\rm rms}\) (**b**,**d**). Figure 7: Dependence of circumferential magnetic field intensity \(H_{\rm ac}\) on frequency of the measuring field for 67 BFBT ceramics studied at temperature 20 \({}^{\circ}\)C; measuring voltage 0.1 V\({}_{\rm rms}\) and 2.0 V\({}_{\rm rms}\). Spectroscopic plots of modulus of complex impedance \(|\,Z\,|\,\) exhibit the characteristic frequencies, namely \(\
u_{\min}\) when the impedance \(|\,Z\,|\,\) is at its minimum (\(|\,Z\,|\,_{\min}\)) and \(\
u_{\max}\) when the impedance is at its maximum (\(|\,Z\,|\,_{\max}\)) (Figure 8a,b).