# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 3 Results and Discussion
### Calculation of Piezoelectric Parameters of 67 Bfto Ceramics
One can see from Figure 13a that the piezoelectric charge coefficient \(d_{31}\) increases from \(d_{31}\approx 30\) pC/N to \(d_{31}\approx 45\) pC/N at the following temperature range: \(\Delta T=(-20\)\(\rightarrow\)\(40)\({}^{\circ}\)C. The courses of the \(d_{31}\) curves calculated for data measured at radial magnetic field \(H_{\rm ac}=0.05\) A/m (blue squares; solid line; \(U=0.1\) V\({}_{\rm rms}\)) and \(H_{\rm ac}=0.49\) A/m (\(U=2.0\) V\({}_{\rm rms}\); red stars; dashed line) are almost identical. Additionally, the linear regression fits for the experimental data cannot easily be discerned in Figure 13a. Linear regression (\(Y=A+B\times X\)) for the data measured at \(H_{\rm ac}=0.05\) A/m (\(U=0.1\) V\({}_{\rm rms}\)) exhibited the following parameters: \(A=3.44221\times 10^{-11}\); \(B=2.57759\times 10^{-13}\); and \(R=0.99097\). Alternatively, linear regression for the data measured at \(H_{\rm ac}=0.49\) A/m (\(U=2.0\) V\({}_{\rm rms}\)) showed the following values: \(A=3.4412\times 10^{-11}\); \(B=2.57385\times 10^{-13}\); and \(R=0.99102\). Therefore, to show the possible influence of the circumferential magnetic field on the piezoelectric charge coefficient \(d_{31}\), the relative change in the \(d_{31}\) piezoelectric modulus was introduced, and the results are plotted in Figure 13b. One can see from Figure 13b that the plot of the relative change in the \(d_{31}\) piezoelectric modulus exhibits non-monotonic behaviour with increasing temperature. Within the measuring temperature range, one local maximum and one local minimum are clearly visible in the plot. One can see from Figure 13b that, according to linear fit, the relative change in the piezoelectric modulus has a positive sign within the whole measuring temperature range. This means that an increase in the intensity of the circumferential magnetic field generated by electric currents through the sample causes the suppression of the piezoelectric response of the multiferroic ceramic sample under study. The above-mentioned influence of circumferential magnetic field on the piezoelectric charge coefficient \(d_{31}\) can explain the dielectric properties of multiferroic ceramics. The point is that the magnetic field influences both real (\(\epsilon^{\prime}\)) and imaginary (\(\epsilon^{\prime\prime}\)) parts of the complex dielectric permittivity. It was found that the higher the radial magnetic field, the lower the dielectric permittivity (at a given temperature). As an example, the dependence of the real and imaginary parts of the complex dielectric permittivity on frequency (below resonance) for radial magnetic field \(H_{\rm ac}=0.023\) A/m (\(U=2.0\) V\({}_{\rm rms}\); \(\
u=10^{5}\) Hz) and \(H_{\rm ac}=0.012\) A/m (\(U=1.0\) V\({}_{\rm rms}\); \(\
u=10^{5}\) Hz) at 20 \({}^{\circ}\)C are shown in Figure 14.  
The influence of the circumferential magnetic field on the dielectric properties of 67 BFBT ceramics was also revealed when the difference of the modulus of dielectric permittivity was plotted against the difference of circumferential magnetic field caused by the electric currents (\(i_{\mathrm{ac}}\)) flowing through the ceramic sample (Figure 15). One can see in Figure 15a that the difference in circumferential magnetic fields generated by the electric currents flowing through the sample linearly depends (in log-log scale) on the frequency of the measuring sinusoidal signal. On the other hand, Figure 15b shows that the change in magnetic field strength causes a change in the modulus of the complex dielectric permittivity. Thus, the possibility of adjusting the dielectric permittivity (and therefore capacitance value) via changes in magnetic field intensity was obtained for 67 BFBT ceramics. It is worth noting that changing the capacitance of a capacitor in an electric circuit has a predictable effect on the complex impedance and phase angle in the circuit. These parameter changes can be exploited to yield tuneable impedance-matching networks, tuneable filters, phase shifters, and other functional multiferroic devices.