# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 4 Conclusions  
Multiferroic 0.67 Bi\({}_{1.02}\)FeO\({}_{3}\)-0.33 BaTiO\({}_{3}\) (67 BFBT) ceramics were fabricated via a solid-state reaction process and subsequent sintering in a box furnace. They were studied in terms  
Figure 14: (**a**) Dependence of the real part of complex dielectric permittivity (\(\epsilon^{\prime}\)) on frequency. (**b**) Dependence of the imaginary part of complex dielectric permittivity (\(\epsilon^{\prime\prime}\)) on frequency; measurements were performed under electric field strength \(E\) = 1666 V/m (\(U\) = 1.0 V\({}_{\mathrm{rms}}\)) and \(E\) = 3333 V/m (\(U\) = 2.0 V\({}_{\mathrm{rms}}\)) corresponding to radial magnetic field \(H_{\mathrm{ac}}\) = 0.012 A/m and \(H_{\mathrm{ac}}\) = 0.023 A/m (at \(\
u\) = 100 kHz, respectively). Figure 15: Dependence of the difference of circumferential magnetic field strength on frequency of the measuring electric field (**a**). Dependence of modulus of complex dielectric permittivity on circumferential magnetic field strength (**b**). of structure and dielectric and piezoelectric properties. The fabricated 67 BFBT ceramics adopted the tetragonal crystal structure described by the _P4 mm_ space group. The average crystallite size was <\(D\)> \(\approx\) 80 nm, and the average strain was <\(\varepsilon\)> = 0.01%. Broad-band dielectric spectroscopy (BBDS) was used for measurements whereas the electric equivalent circuit method was used to fit the impedance-frequency spectra within the frequency range of the radial resonances. Complex non-linear least squares method (CNLS) was employed to fit the parameters of the electric equivalent circuit [RsCPE\({}_{1}\)([L\({}_{1}\)R\({}_{1}\)C\({}_{1}\)]C\({}_{0}\)]. The high quality of the fitting procedure was confirmed by \"_chi-squared_\" (\(\chi^{2}\)) and weighted sum of squares (_WSS_) parameters. Measurements and subsequent fitting and simulations have shown that the planar coupling factor was \(k_{\rm p}\approx 31\)%, and the mechanical quality was within the range \(Q_{\rm mech}\approx 21\). The piezoelectric charge coefficient for the radial vibration mode of a thin disk (excited through the piezoelectric effect across the thickness of the disk) exhibited a value of \(d_{31}\approx 35\)pC/N at room temperature. The idea of using a measuring signal of different amplitudes (BBDS measurements) was to check the possible influence of the radial magnetic field caused by electric currents flowing through the sample on the piezoelectric properties of multiferroic 67 BFBT ceramics. The rms range of ac measuring fields led to circumferential magnetic fields ranging from 0.05 A/m to 0.49 A/m. It was established that an increase in the intensity of the circumferential magnetic field caused the suppression of the real (\(\epsilon^{\prime}\)) and imaginary (\(\epsilon^{\prime\prime}\)) parts of the complex dielectric permittivity and piezoelectric charge coefficient (\(d_{31}\)) of the multiferroic 67 BFTO ceramics within the range of the measurement parameters used (i.e., frequency, temperature, and radial magnetic field). The \"magnetic\" tunability of the dielectric properties of 67 BFBT ceramics was also found, as evidenced by the possibility of adjusting the modulus of the complex dielectric permittivity (and thus the capacitance value) by means of the magnetic field strength. Changes in dielectric parameters can be used to obtain \"magnetically\" tuneable functional multiferroic devices, e.g., tuneable filters or phase shifters. Taking into account that the strength of the magnetic field used in the current study was about 6 mOe (in CGS units), one can conclude that multiferroic 67 BFBT ceramics are very sensitive to magnetic field influence, which makes 67 BFBT sensing material with vast potential. Conceptualization, D.C. and A.L.-C.; methodology, A.L.-C. and D.C.; software, D.C. and W.B. ; validation, A.L.-C. and J.-H.J. ; formal analysis, A.L.-C., D.C. and J.-H.J. ; investigation, A.L.-C., D.C., T.T.Z., J.-H.J., B.G.-G. and W.B. ; resources, A.L.-C., D.C., B.G.-G. and W.B.