# Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics
## 2 Materials and Methods  
Ceramics of 0.67 Bi\({}_{1.02}\)FeO\({}_{3}\)-0.33 BaTiO\({}_{3}\) composition were synthesised by a conventional solid-state reaction process. Chemically pure Bi\({}_{2}\)O\({}_{3}\) (Sigma Aldrich, Burlington, MA, USA, 99.9%), Fe\({}_{2}\)O\({}_{3}\) (Sigma Aldrich, USA, 99%), TiO\({}_{2}\) (Sigma Aldrich, USA, 99.8%) metal oxides, and BaCO\({}_{3}\) (Sigma Aldrich, USA, 99%) were used as raw materials. To obtain a homogeneous mixture, stoichiometric amounts of oxide and carbonate precursors were first carefully weighted. Then, they were mixed thoroughly by ball-milling in a polyethene bottle with yttria-stabilised ZrO\({}_{2}\) (YSZ) balls for 24 h in ethanol dispersion reagent. After that, they were calcined at 900 \({}^{\circ}\)C for 6 h in a box furnace. Later on, the calcined powder was ball-milled again for 24 h, and then the dried powder was slightly pressed into compacts in a stainless steel die into disks with a diameter of 10 mm using a uniaxial press, followed by cold isostatic pressing (CIP) at 200 MPa for 5 min. The green pellets were sintered at 970 \({}^{\circ}\)C for 3 h in a box furnace. The relative density, crystal structure, and microstructure of the sintered sample were examined by using the Archimedes method, an X-ray diffractometer (XRD, D/Max-2500, Rigaku, Japan), and a scanning electron microscope (SEM, JEOL JSM-7900F), respectively. Two-dimensional grain sizes of the sintered 67 BFBT ceramics were calculated by measuring more than 100 grains area in the SEM image using ImageJ--a public domain Java image processing and analysis programme [13,14]. The obtained values were then converted to three-dimensional grain sizes by multiplying by a factor of 1.68 [15]. Broad-band dielectric spectroscopy measurements (BBDS) [16] were carried out using a system consisting of a high performance frequency analyser (Alpha-AN) and a temperature control system (Quatro Cryosystem) via Novocontrol. It is well known (see [17,18]) that, by applying the suitable strategy and alternative methods of analysing the experimental immittance data obtained as a result of the BBDS measurement, it is possible to characterise the electrical and dielectric properties of the ceramic material. Spectroscopic measurements were carried out on disk-shaped pieces that were 7.52 mm in diameter and 0.6 mm thick. To ensure a good electric contact, the sample ends were cleaned with a soft acid solution and silver electrodes (silver paste) were deposited. Before the measurement, the system was cooled down to \(-20\)\({}^{\circ}\)C with liquid nitrogen. The measurements were performed during the heating cycle up to a maximum temperature of 50 \({}^{\circ}\)C. Dielectric spectra were recorded at programmed temperatures with a temperature step of 10 \({}^{\circ}\)C 15 min after the temperature had been stabilised. The measurement software allowed a frequency run of 40 points per decade in the measuring frequency range. WinDATA Novocontrol software was used for the recording, visualization, and processing of experimental data. However, to check the consistency of the experimental data, a computer programme created by [PERSON] was used [17,19]. The Kramers-Kronig test was carried out for experimental data measured at room temperature within a frequency range from \(\
u=10^{-1}\) Hz to \(\
u=10^{5}\) Hz. The amplitude of the test voltage and related electric field strength were as follows: \(U_{\text{ac}}=0.1\) V\({}_{\text{rms}}\) (\(E_{\text{ac}}=166\) V/m), \(U_{\text{ac}}=0.2\) V\({}_{\text{rms}}\) (\(E_{\text{ac}}=333\) V/m), \(U_{\text{ac}}=1.0\) V\({}_{\text{rms}}\) (\(E_{\text{ac}}=1666\) V/m), and \(U_{\text{ac}}=2.0\) V\({}_{\text{rms}}\) (\(E_{\text{ac}}=3333\) V/m) (root mean square--rms). Practical application of the impedance data validation method in the study of electroceramics was described by us in detail elsewhere (in [20] and the references cited therein).