Dielectric Spectroscopy Studies and Modelling of Piezoelectric Properties of Multiferroic Ceramics

: Compounds and solid solutions of bismuth ferrite (BiFeO 3 )—barium titanate (BaTiO 3 ) system are of great scientiﬁc and engineering interest as multiferroic and potential high-temperature lead-free piezoelectric materials. In the present paper, the results of research on the synthesis and characterisation of 0.67Bi 1 . 02 FeO 3 –0.33BaTiO 3 (67BFBT) ceramics in terms of crystal structure and dielectric and piezoelectric properties are reported. It was found that the produced 67BFBT ceramics were characterised by a tetragonal crystal structure described by the P4mm space group, an average crystallite size < D > ≈ 80 nm, and an average strain < ε > = 0.01%. Broad-band dielectric spectroscopy (BBDS) was employed to characterise the dielectric response of polycrystalline ceramics. The frequency range from ν = 10 − 1 Hz to ν = 10 5 Hz was used to characterise the inﬂuence of the electric ﬁeld strength on dielectric response of the ceramic sample at room temperature. The dielectric spectra were checked for consistency with the Kramers–Kronig test, and the high quality of the measurements were conﬁrmed. The electric equivalent circuit method was used to ﬁt the dielectric spectra within the frequency range that corresponded to the occurrence of the resonant spectra of the radial mode for thin disk sample, i.e., from ν = 10 5 Hz to ν = 10 7 Hz and the temperature range from T = − 20 ◦ C to T = 50 ◦ C. The electric equivalent circuit [R s CPE 1 ([L 1 R 1 C 1 ]C 0 )] was used, and good ﬁtting quality was reached. The relevant calculations were performed, and it was found that the piezoelectric charge coefﬁcient exhibited a value of d 31 = 35 pC/N and the planar coupling factor was k p = 31% at room temperature. Analysis of impedance spectra performed in terms of circumferential magnetic ﬁeld made it possible to establish an inﬂuence of magnetic ﬁeld on piezoelectric parameters of 67BFBT multiferroic ceramics. Additionally, the “magnetic” tunability of the modulus of the complex dielectric permittivity makes 67BFBT a sensing material with vast potential.


Introduction
From a scientific point of view, multiferroic materials are a somewhat mysterious yet, at the same time, extremely interesting class of materials. Thanks to the unique properties of these materials, which are characteristic of both ferroelectric and ferromagnetic materials, they have a wide range of potential applications. Multiferroic materials can open new horizons in the design of modern devices, especially those whose operation is based on controlling the electrical properties of materials (and thus influencing the parameters of the electrical circuit) by changing the magnetic field and vice versa, i.e., controlling the magnetic properties of materials by changing electric field strength [1].
Let us consider the terminal elements of bismuth ferrite (BiFeO 3 )-barium titanate (BaTiO 3 ) system. It is well known that BiFeO 3 has an ABO 3 -type perovskite structure and is one of the few materials characterised by the coexistence of ferroelectric and (anti)ferromagnetic properties at room temperature. Because of this essential feature of multiferroics, BiFeO 3 (BFO) is considered the most suitable among all available multiferroic materials for the development of practical spintronic devices and related technologies [2]. BaTiO 3 (BT) belongs to the corner-sharing oxygen octahedral material family (structure), which was discovered during World War II. Since then, it has remained one of the most widely used and systematically studied non-linear dielectric materials and is considered the prototype for ferroelectric ceramics as well as being the first piezoelectric ceramics transducer ever developed [3]. Since the discovery of BT in the early 1940s, it has been continually used in new industrial and commercial applications. Its unique physical characteristics, e.g., values of piezoelectric (d 33~1 90 pC/N) and dielectric (ε r~1 700 and low dielectric losses at room temperature) parameters, enable the use of BT ceramics for capacitors, multilayer capacitors, ultrasonic transducers, pyroelectric infrared sensors, positive temperature coefficient resistors, and energy storage devices [4,5].
However, neither BFO nor BT are actually used in their pure chemical form. To optimise properties for specific applications, they are often combined with various additives that adjust and enhance their basic properties. They are also combined with each other to form either compounds or ceramic-ceramic composites, which have received a lot of scientific interest [6][7][8]. To justify the above-mentioned points, let us point out a bulk ternary BiFeO 3 -BaTiO 3 and PbTiO 3 system [9] that was fabricated and studied as a potential high-temperature piezoelectric ceramics transducer. In case of Tm-doped 0.7Bi 1−x Tm x FeO 3 -0.3BaTiO 3 (x = 0-0.05) ceramics, it was found that Tm-induced structural modifications yielded improvements in the dielectric constant, remnant polarization, and remnant magnetization values [10]. On the other hand, studies of Mn-modified (1 − x)BiFeO 3 -xBaTiO 3 ceramics showed that the addition of Mn improved dielectric losses and increased dc resistivity in bulk BiFeO 3 -BaTiO 3 , greatly enhancing the piezoelectric response; however, the poor stability of the poled domain structure caused rapid ageing and a large difference between low-and high-field piezoelectric charge coefficient d 33 [11].
The main motivation to carry out the present research was the high potential for practical application that the materials of the BiFeO 3 -BaTiO 3 system have. In addition, the issues of the physics and technology of multiferroic materials are extremely rich, interesting, and non-trivial from a scientific point of view. Novel materials of rich functionality can be created by means of combining and properly modifying bismuth ferrite and barium titanate. Therefore, the chemical composition 0.67Bi 1.02 FeO 3 -0.33BaTiO 3 (67BFBT) was chosen as the research material due to its potential use as an alternative to lead-based piezoelectric materials [8,11,12].
The aim of the present research was to synthesise 0.67Bi 1.02 FeO 3 -0.33BaTiO 3 (67BFBT) multiferroic ceramics and characterise the produced ceramics in terms of crystal structure, surface morphology, impedance spectroscopy, and piezoelectric properties, including modelling with the electric equivalent circuit method and the use of a circumferential magnetic field to reveal the effect of a weak magnetic field on the piezoelectric and dielectric properties of 67BFBT ceramics.

Materials and Methods
Ceramics of 0.67Bi 1.02 FeO 3 -0.33BaTiO 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 • 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 • 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 67BFBT 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 • C with liquid nitrogen. The measurements were performed during the heating cycle up to a maximum temperature of 50 • C. Dielectric spectra were recorded at programmed temperatures with a temperature step of 10 • 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 Boukamp was used [17,19]. The Kramers-Kronig test was carried out for experimental data measured at room temperature within a frequency range from ν = 10 −1 Hz to ν = 10 5 Hz. The amplitude of the test voltage and related electric field strength were as follows: U ac = 0.1 V rms (E ac = 166 V/m), U ac = 0.2 V rms (E ac = 333 V/m), U ac = 1.0 V rms, (E ac = 1666 V/m), and U ac = 2.0 V rms (E 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). Experimental data recorded within a frequency range from ν = 10 5 Hz to ν = 10 7 Hz, which corresponded to the occurrence of radial mode resonance spectra for a sample in the form of a thin disk, were fitted to the electric equivalent circuit [R s CPE 1 ([L 1 R 1 C 1 ]C 0 )]. Complex non-linear least squares method (CNLS) was employed for fitting [17,19].
Piezoelectric parameters were calculated according to resonance (anti-resonance method) [21,22]. For this method, an accurate impedance analyser is a crucial requirement for use in testing (a Novocontrol high-performance frequency analyser was used for the current study's experiments). The resonant frequencies (ν r ) and anti-resonant frequencies (ν a ) were detected based on spectroscopic characteristics measured in the course of BBDS experiments, and the free capacitance C T at 1 kHz was measured [23].

Results and Discussion
3.1. Morphology Studies of 0.67Bi 1.02 FeO 3 -0.33BaTiO 3 Ceramics The morphology of 0.67Bi 1.02 FeO 3 -0.33BaTiO 3 ceramics sintered at 970 • C is shown in Figure 1a, whereas a histogram of the grain size distribution is shown in Figure 1b. Figure 1a, whereas a histogram of the grain size distribution is shown in Figure 1b.
One can see from Figure 1a that several polygonal voids are present in cross-section of the ceramic sample subjected to the SEM investigation preparation p cedure. It is worth noting that all the voids visible in the photo (Figure 1a) exhibit regu mostly pentagonal shapes that resemble the shape of the grains of the ceramics und study. Therefore, it is reasonable to suppose that the voids were "man-made" and crea during the mechanical treatment of the cross-section surface of the ceramic sample (i sample preparation for SEM investigation). It is reasonable to suppose that the grains t initially took their sites were removed from their original positions by mechanical tre ment. The grain size distribution was measured with the help of an image processing a analysis programme (ImageJ [13,14]) on the basis of SEM photos of the ceramic sam (Figure 1a), and the results are given in Figure 1b. The mean and median values of average grain size were approximately 4.7 and 4.4 μm, respectively.

X-ray Diffraction Studies
As an example, an X-ray diffraction pattern of 67BFBT ceramics sintered at 970 °C shown in Figure 2. The search-match process was performed using Match! softw (Crystal Impact, Bonn, Germany) [26]. Phase analysis was performed, and it was fou that the reference pattern of Ba0.3Bi0.7FeO2.85 [27] matches all peaks shown within measuring range 2Θ with Figure  One can see from Figure 1a that several polygonal voids are present in the cross-section of the ceramic sample subjected to the SEM investigation preparation procedure. It is worth noting that all the voids visible in the photo ( Figure 1a) exhibit regular, mostly pentagonal shapes that resemble the shape of the grains of the ceramics under study. Therefore, it is reasonable to suppose that the voids were "man-made" and created during the mechanical treatment of the cross-section surface of the ceramic sample (i.e., sample preparation for SEM investigation). It is reasonable to suppose that the grains that initially took their sites were removed from their original positions by mechanical treatment.
The grain size distribution was measured with the help of an image processing and analysis programme (ImageJ [13,14]) on the basis of SEM photos of the ceramic sample (Figure 1a), and the results are given in Figure 1b. The mean and median values of the average grain size were approximately 4.7 and 4.4 µm, respectively.

X-ray Diffraction Studies
As an example, an X-ray diffraction pattern of 67BFBT ceramics sintered at 970 • C is shown in Figure 2. The search-match process was performed using Match! software (Crystal Impact, Bonn, Germany) [26]. Phase analysis was performed, and it was found that the reference pattern of Ba 0.3 Bi 0.7 FeO 2.85 [27] matches all peaks shown within the measuring range 2Θ with Figure It is worth noting that the reference pattern (source of entry: Crystallography Open Database; COD ID 4341652) exhibited a tetragonal structure (space group P4mm) with the following unit cell parameters: a = 3.9963 Å and c = 4.0032 Å. Based on Archimedes' method and calculations of theoretical density, it was found that 67BFBT ceramics reached a relative density of 94.5%.
Line profile analysis was performed, and the resulting Williamson-Hall plot is shown in Figure 3.
One can see from the Williamson-Hall plot that the average size of the crystallites is <D> = 801 Å. The average strain (<ε>), which is a measure of micro-deformations, is <ε> = 0.01%. It is worth noting that the reference pattern (source of entry: Crystallography O Database; COD ID 4341652) exhibited a tetragonal structure (space group P4mm) with following unit cell parameters: a = 3.9963 Å and c = 4.0032 Å. Based on Archime method and calculations of theoretical density, it was found that 67BFBT cera reached a relative density of 94.5%.
Line profile analysis was performed, and the resulting Williamson-Hall pl shown in Figure 3. One can see from the Williamson-Hall plot that the average size of the crystallit <D> = 801 Å. The average strain (<ε>), which is a measure of micro-deformations, is < 0.01%.

Data Validation of Impedance Measurements
It is commonly known (e.g., see [16][17][18][19]) that impedance spectroscopy is extrem susceptible to random disturbances. Therefore, knowing the quality of the meas impedance data is extremely important to facilitate correct analysis. Kramers-Kr (K-K) relations are very helpful for data validation [17,19]. The Kramers-Kronig states that the imaginary part of a dispersion is fully determined by the form of the  It is worth noting that the reference pattern (source of entry: Crystallography Database; COD ID 4341652) exhibited a tetragonal structure (space group P4mm) w following unit cell parameters: a = 3.9963 Å and c = 4.0032 Å. Based on Archi method and calculations of theoretical density, it was found that 67BFBT ce reached a relative density of 94.5%.
Line profile analysis was performed, and the resulting Williamson-Hall shown in Figure 3. One can see from the Williamson-Hall plot that the average size of the crysta <D> = 801 Å. The average strain (<ε>), which is a measure of micro-deformations, 0.01%.

Data Validation of Impedance Measurements
It is commonly known (e.g., see [16][17][18][19]) that impedance spectroscopy is ext susceptible to random disturbances. Therefore, knowing the quality of the me impedance data is extremely important to facilitate correct analysis. Kramers-(K-K) relations are very helpful for data validation [17,19]. The Kramers-Kron states that the imaginary part of a dispersion is fully determined by the form of t part of dispersion over the frequency range ∞ ≥ ν ≥0. Similarly, the real part of disp is determined by the form of the imaginary part of dispersion [17]. In the present

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 ∞ ≥ ν ≥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 Boukamp [17,19].
Kramers-Kronig transform test results of impedance data measured for 67BFBT ceramics at room temperature (RT) within the frequency range ∆ν = (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 V rms is shown in Figure 4b. An Impedance diagram with K-K transform test results measured at a signal voltage of U = 2.0 V rms 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 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).
in Figures 4a and 5a. A complex impedance diagram combined with K-K transform results measured at a signal voltage of U = 0.1 Vrms is shown in Figure 4b. An Imped diagram with K-K transform test results measured at a signal voltage of U = 2.0 V shown in Figure 5b. One can see in Figures 4a and 5a that the data recorded at RT ex a small deviation from K-K behaviour (residuals are less than 0.4%). Upon inspectio the results given in Figures 4b and 5b, it can be determined that there is very g agreement between the measurements (blue circles) and K-K calculations (red crosse The quality parameter "chi-squared" reached a value of χ 2 = 3.3 × 10 −7 − 5.5 × 10 room temperature measurements taken at U = 0.1 Vrms (E = 166 V/m) and U = 2.0 Vrm 3333 V/m), respectively. The above-mentioned results proved the high quality o measurements and fully justified further analysis of the impedance data.
An alternative representation of the impedance data showing the influence creasing electric field intensity on the spectroscopic dependence of the reactance ( tance times pulsation: −Z"ω) of the piezoelectric ceramic sample 67BFTO at room perature is shown in Figure 6a. It can be seen that the spectroscopic plots shift tow higher frequencies as the electric field strength increases. Additionally, the dependen diagram with K-K transform test results measured at a signal voltage of U = 2.0 V shown in Figure 5b. One can see in Figures 4a and 5a that the data recorded at RT ex a small deviation from K-K behaviour (residuals are less than 0.4%). Upon inspecti the results given in Figures 4b and 5b, it can be determined that there is very agreement between the measurements (blue circles) and K-K calculations (red cross  The quality parameter "chi-squared" reached a value of χ 2 = 3.3 × 10 −7 − 5.5 × 10 room temperature measurements taken at U = 0.1 Vrms (E = 166 V/m) and U = 2.0 Vrm 3333 V/m), respectively. The above-mentioned results proved the high quality o measurements and fully justified further analysis of the impedance data.
An alternative representation of the impedance data showing the influenc creasing electric field intensity on the spectroscopic dependence of the reactance ( tance times pulsation: −Z"ω) of the piezoelectric ceramic sample 67BFTO at room perature is shown in Figure 6a. It can be seen that the spectroscopic plots shift tow higher frequencies as the electric field strength increases. Additionally, the dependen The quality parameter "chi-squared" reached a value of χ 2 = 3.3 × 10 −7 -5.5 × 10 −7 for room temperature measurements taken at U = 0.1 V rms (E = 166 V/m) and U = 2.0 V 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 ω) of the piezoelectric ceramic sample 67BFTO 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 ) on the imaginary part of complex impedance times pulsation (−Z ω) (Figure 6b) shows the substantial dependence on the electric field strength. the real part of complex impedance (Z') on the imaginary part of complex impedance times pulsation (−Z"ω) (Figure 6b) shows the substantial dependence on the electric field strength.
(a) (b) 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 67BFTO.

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 67BFBT ceramics within the following frequency range: ν = 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 Vrms and U = 2.0 Vrms was taken. Taking into account all the impedances in the circuit, the measuring sinusoidal signal leads to electric currents (iac) through the sample of 67BFBT ceramics in the range following range: iac = 1.2 mA (at Uac = 0.1 Vrms, ν = 10 MHz)-11.7 mA (at U = 2.0 Vrms, ν = 10 MHz). The ac field amplitude (rms) generated by these currents in a radial point r on the sample can be calculated as: 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 Vrms and U = 2.0 Vrms, ν = 10 MHz, respectively) [28]. The results of the calculated radial magnetic field intensity Hac are given in Figure 7. It is worth noting that the resonance behaviour of the 67BFBT ceramic sample is also reflected in Figure 7. Positions of the resonances are closely related to the piezoelectric properties of the material. 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 67BFTO.

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 67BFBT ceramics within the following frequency range: ν = 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 rms and U = 2.0 V rms was taken. Taking into account all the impedances in the circuit, the measuring sinusoidal signal leads to electric currents (i ac ) through the sample of 67BFBT ceramics in the range following range: i ac = 1.2 mA (at U ac = 0.1 V rms , ν = 10 MHz)-11.7 mA (at U = 2.0 V rms , ν = 10 MHz). The ac field amplitude (rms) generated by these currents in a radial point r on the sample can be calculated as: 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 rms and U = 2.0 V rms , ν = 10 MHz, respectively) [28].
The results of the calculated radial magnetic field intensity H ac are given in Figure 7. It is worth noting that the resonance behaviour of the 67BFBT 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 0.67Bi 1.02 FeO 3 -0.33BaTiO 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 • C to 50 • C. It should be noted that the frequency peaks visible at about 3-4 × 10 5 Hz (Figure 8) are radial resonances, whereas the frequency peaks visible at about 3-5 × 10 6 Hz (also in Figure 8) are related to the thickness mode resonance.
Appl. Sci. 2023, 13, x FOR PEER REVIEW Figure 7. Dependence of circumferential magnetic field intensity Hac on frequency of the mea field for 67BFBT ceramics studied at temperature 20 °C; measuring voltage 0.1 Vrms and 2.0 V Figure 8 shows the frequency response of a 0.67Bi1.02FeO3-0.33BaTiO3 cerami disk that was 7.52 mm in diameter and 0.6 mm thick. Electrodes were deposited both faces of the disk, and then the disk was poled in the direction perpendicular faces of the disk. The measurements were taken within a temperature range from − to 50 °C. It should be noted that the frequency peaks visible at about 3-4 × 10 5 Hz (F 8) are radial resonances, whereas the frequency peaks visible at about 3-5 × 10 6 Hz in Figure 8) are related to the thickness mode resonance.   Figure 8 shows the frequency response of a 0.67Bi1.02FeO3-0.33BaTiO3 ceramic t disk that was 7.52 mm in diameter and 0.6 mm thick. Electrodes were deposited o both faces of the disk, and then the disk was poled in the direction perpendicular to faces of the disk. The measurements were taken within a temperature range from −20 to 50 °C. It should be noted that the frequency peaks visible at about 3-4 × 10 5 Hz (Fig  8) are radial resonances, whereas the frequency peaks visible at about 3-5 × 10 6 Hz (a in Figure 8) are related to the thickness mode resonance.  Spectroscopic plots of modulus of complex impedance |Z| exhibit the characteristic frequencies, namely ν min when the impedance |Z| is at its minimum (|Z| min ) and ν max when the impedance is at its maximum (|Z| max ) (Figure 8a,b). At the same time, the phase angle (Θ) given in Figure 8c,d tends to have a value of "zero."

Modelling of Impedance-Frequency Characteristics of the Piezoelectric Equivalent Circuit
Let us first explain the notation used. In the adopted notation, square or box brackets [ ] denote that elements are in series-connected, whereas round brackets or parentheses ( ) denote the parallel connection of electric elements. According to the adopted notation, (RC) is a parallel circuit, while [RC] is a series connection of the elements R and C.
To accurately approximate the behaviour of the piezoelectric specimen close to its fundamental resonance, it can be represented by the electric equivalent circuit ([L 1 R 1 C 1 ]C 0 ) consisting of a "mechanical arm" (L, C, and R connected in series) and C 0 (which corresponds to the electrical capacitance of the specimen) connected in parallel [24]. In this connection, it is worth remembering that the impedance of the parallel circuit can be represented by the equivalent series circuit consisting of equivalent in series-connected resistance and reactance values.
In the case of our simulation and fitting, the modified electric equivalent circuit [RsCPE 1 ([L 1 R 1 C 1 ]C 0 )] including resistance Rs and constant phase element CPE 1 connected in series with the "piezoelectric" equivalent circuit was used. Figure 9 shows the electric equivalent circuit used for the simulation and fitting of the impedance response of the ceramic specimen vibrating close to its fundamental resonance.
Spectroscopic plots of modulus of complex impedance |Z| exhibit the ch frequencies, namely νmin when the impedance |Z| is at its minimum (|Z|m when the impedance is at its maximum (|Z|max) (Figure 8a,b). At the same time angle (Θ) given in Figure 8c,d tends to have a value of "zero."

Modelling of Impedance-Frequency Characteristics of the Piezoelectric Equivale
Let us first explain the notation used. In the adopted notation, square or ets [ ] denote that elements are in series-connected, whereas round bracket theses ( ) denote the parallel connection of electric elements. According to t notation, (RC) is a parallel circuit, while [RC] is a series connection of the elem C.
To accurately approximate the behaviour of the piezoelectric specimen fundamental resonance, it can be represented by the electric equivalent circuit consisting of a "mechanical arm" (L, C, and R connected in series) and C0 (w sponds to the electrical capacitance of the specimen) connected in parallel [ connection, it is worth remembering that the impedance of the parallel cir represented by the equivalent series circuit consisting of equivalent in series resistance and reactance values.
In the case of our simulation and fitting, the modified electric equiva [RsCPE1([L1R1C1]C0)] including resistance Rs and constant phase element CPE in series with the "piezoelectric" equivalent circuit was used. Figure 9 shows equivalent circuit used for the simulation and fitting of the impedance resp ceramic specimen vibrating close to its fundamental resonance. It should be noted that the resonance frequency νr and antiresonance fr correspond to the "zero" value of reactance for the electric equivalent circuit The reactance of the "mechanical arm" is zero at the series resonant frequency where ω is the angular frequency (ω = 2 πν).
The reactance of the parallel circuit is zero at the parallel resonant freq The parallel resonance νp occurs when the currents flowing in the two arms phase, which is when: In this connection, it must be pointed out that the relation be above-mentioned characteristic frequencies of the equivalent circuit is as follo < νr. However, the difference between them is very small (νmin~νs~νr) [24]. Simi exists between antiresonance, parallel resonance, and |Z|max frequencies: νa (νa~νp~νmax). What is important is that values of νmin an νmax can be readily meas an impedance analyser (Alpha-AN High Performance Frequency Analyzer).
An example of the fitting results obtained for impedance characteristics under the influence of a weak circumferential magnetic field is shown in Fig  ure 10a shows the spectra measured at Hac = 0.05 A/m, (E = 166 V/m; U = 0.1 Vr It should be noted that the resonance frequency ν r and antiresonance frequency ν a correspond to the "zero" value of reactance for the electric equivalent circuit (Figure 9). The reactance of the "mechanical arm" is zero at the series resonant frequency ν s when: where ω is the angular frequency (ω = 2 πν). The reactance of the parallel circuit is zero at the parallel resonant frequency (ν p ). The parallel resonance ν p occurs when the currents flowing in the two arms are in antiphase, which is when: In this connection, it must be pointed out that the relation between the above-mentioned characteristic frequencies of the equivalent circuit is as follows: ν min < ν s < ν r . However, the difference between them is very small (ν min~νs~νr ) [24]. Similar relation exists between antiresonance, parallel resonance, and |Z| max frequencies: ν a < ν p < ν max (ν a~νp~νmax ). What is important is that values of ν min an ν max can be readily measured using an impedance analyser (Alpha-AN High Performance Frequency Analyzer).
An example of the fitting results obtained for impedance characteristics measured under the influence of a weak circumferential magnetic field is shown in Figure 10. Experimental data were fitted to the electric equivalent circuit using the Z programme (Scribner Associates, Inc. Southern Pines, North Carolina, USA). Com non-linear least squares method (CNLS) was employed for the analysis of the im ance/frequency data of the electroceramics [17]. The fitting procedure was limited t frequency range of the radial resonances ∆ν = (2-6) × 10 5 Hz. The quality of the f procedure was estimated according to the following parameters: "chi-squared" (χ 2 weighted sum of squares (WSS) [29]. In the case of the fitting results shown in Figu the parameters were as follows: "chi-squared" was χ 2 = 2.78 × 10 −3 and χ 2 = 3.23 × 10 −3 = 0.1 Vrms and U = 2.0 Vrms, respectively. The weighted sum of squares was WSS = 0.31 WSS = 0.177 for U = 0.1 Vrms and U = 2.0 Vrms, respectively. Figure 11 shows the dependence of the fitting quality parameters, na "chi-squared" (χ 2 ) and weighted sum of squares (WSS), on the temperatures at whic impedance/frequency characteristics of 67BFBT ceramics were recorded.
(a) (b) Figure 11. Fitting quality parameters χ 2 ("chi-squared") (a) and weighted sum of squares (WS obtained as a result of fitting the experimental resonance spectra and response of the e equivalent circuit given in Figure 8; amplitude of the measuring signal: U = 0.1 Vrms (blue squ and U = 2.0 Vrms (red circles).
One can see in Figure 11 that both χ 2 ("chi-squared") fitting quality parameter (F 11a) and WSS parameter (Figure 11b) change within one order of magnitude. Upon ual inspection, it can be seen that the linear approximations used for the χ 2 and WSS approximation show that a higher electric field strength of the measuring signal (an Experimental data were fitted to the electric equivalent circuit using the ZView programme (Scribner Associates, Inc. Southern Pines, NC, USA). Complex non-linear least squares method (CNLS) was employed for the analysis of the impedance/frequency data of the electroceramics [17]. The fitting procedure was limited to the frequency range of the radial resonances ∆ν = (2-6) × 10 5 Hz. The quality of the fitting procedure was estimated according to the following parameters: "chi-squared" (χ 2 ) and weighted sum of squares (WSS) [29]. In the case of the fitting results shown in Figure 10, the parameters were as follows: "chi-squared" was χ 2 = 2.78 × 10 −3 and χ 2 = 3.23 × 10 −3 for U = 0.1 V rms and U = 2.0 V rms , respectively. The weighted sum of squares was WSS = 0.314 and WSS = 0.177 for U = 0.1 V rms and U = 2.0 V rms , respectively. Figure 11 shows the dependence of the fitting quality parameters, namely "chisquared" (χ 2 ) and weighted sum of squares (WSS), on the temperatures at which the impedance/frequency characteristics of 67BFBT ceramics were recorded. Experimental data were fitted to the electric equivalent circuit using the programme (Scribner Associates, Inc. Southern Pines, North Carolina, USA). Co non-linear least squares method (CNLS) was employed for the analysis of the i ance/frequency data of the electroceramics [17]. The fitting procedure was limited frequency range of the radial resonances ∆ν = (2-6) × 10 5 Hz. The quality of the procedure was estimated according to the following parameters: "chi-squared" (χ weighted sum of squares (WSS) [29]. In the case of the fitting results shown in  Figure 11 shows the dependence of the fitting quality parameters, n "chi-squared" (χ 2 ) and weighted sum of squares (WSS), on the temperatures at wh impedance/frequency characteristics of 67BFBT ceramics were recorded.
(a) (b) Figure 11. Fitting quality parameters χ 2 ("chi-squared") (a) and weighted sum of squares (W obtained as a result of fitting the experimental resonance spectra and response of the equivalent circuit given in Figure 8; amplitude of the measuring signal: U = 0.1 Vrms (blue s and U = 2.0 Vrms (red circles).
One can see in Figure 11 that both χ 2 ("chi-squared") fitting quality parameter ( 11a) and WSS parameter (Figure 11b) change within one order of magnitude. Up ual inspection, it can be seen that the linear approximations used for the χ 2 and WS approximation show that a higher electric field strength of the measuring signal ( the same time, a higher value of the circumferential magnetic field strength) impro Figure 11. Fitting quality parameters χ 2 ("chi-squared") (a) and weighted sum of squares (WSS) (b) obtained as a result of fitting the experimental resonance spectra and response of the electric equivalent circuit given in Figure 8; amplitude of the measuring signal: U = 0.1 V rms (blue squares) and U = 2.0 V rms (red circles).
One can see in Figure 11 that both χ 2 ("chi-squared") fitting quality parameter (Figure 11a) and WSS parameter (Figure 11b) change within one order of magnitude. Upon visual inspection, it can be seen that the linear approximations used for the χ 2 and WSS data approximation show that a higher electric field strength of the measuring signal (and, at the same time, a higher value of the circumferential magnetic field strength) improves the quality of the fitting procedure. This also means that the data is less susceptible to external interference. A comparison of the statistical characteristics of the obtained "chi-squared" fitting quality parameters showed that the standard deviation was SD = 0.00192 and SD = 0.00212 for the low value of the electric field strength E = 166 V/m (U = 0.1 V rms ) and high electric field strength E = 3333 V/m (U = 2.0 V rms ), respectively. The WSS parameter showed that the standard deviation was SD = 0.21743 and SD = 0.11077 for E = 166 V/m and E = 3333 V/m, respectively. An increase in the intensity of the measuring field led to a substantial improvement in the quality of further data simulation.

Calculation of Piezoelectric Parameters of 67BFTO Ceramics
The entry parameters used to calculate the piezoelectric parameters of BFBT ceramics, namely resonant frequencies ν r , anti-resonant frequencies ν a , impedance, and free capacitance C T on 1 kHz were measured by using the aforementioned impedance analyser. As shown in Figure 8, a thorough analysis of the experimental data (i.e., measured impedance spectra) was also performed. The procedure for the calculation of single coefficients is described in detail in classical textbooks (e.g., see [21,24,25]), scientific papers (e.g., see [22,23]), or standards (e.g., see [30]). The results of the calculations are given in Figures 12 and 13. quality of the fitting procedure. This also means that the data is less susceptible to ternal interference. A comparison of the statistical characteristics of the obta "chi-squared" fitting quality parameters showed that the standard deviation was S 0.00192 and SD = 0.00212 for the low value of the electric field strength E = 166 V/m 0.1 Vrms) and high electric field strength E = 3333 V/m (U = 2.0 Vrms), respectively. The W parameter showed that the standard deviation was SD = 0.21743 and SD = 0.11077 fo 166 V/m and E = 3333 V/m, respectively. An increase in the intensity of the measu field led to a substantial improvement in the quality of further data simulation.

Calculation of Piezoelectric Parameters of 67BFTO Ceramics
The entry parameters used to calculate the piezoelectric parameters of BFBT cer ics, namely resonant frequencies νr, anti-resonant frequencies νa, impedance, and capacitance C T on 1 kHz were measured by using the aforementioned impedance lyser. As shown in Figure 8, a thorough analysis of the experimental data (i.e., measu impedance spectra) was also performed. The procedure for the calculation of single efficients is described in detail in classical textbooks (e.g., see [21,24,25]), scientific pa (e.g., see [22,23]), or standards (e.g., see [30]). The results of the calculations are give Figures 12 and 13.  Figure 13b shows the formula for calcula of the relative change in piezoelectric modulus d31. Figure 12a shows dependence of the planar coupling factor kp for the vibration a the radial direction in a circle-shaped disk of 67BFTO ceramics on temperature. The quality of the fitting procedure. This also means that the data is less susceptible to ternal interference. A comparison of the statistical characteristics of the obtai "chi-squared" fitting quality parameters showed that the standard deviation was S 0.00192 and SD = 0.00212 for the low value of the electric field strength E = 166 V/m 0.1 Vrms) and high electric field strength E = 3333 V/m (U = 2.0 Vrms), respectively. The W parameter showed that the standard deviation was SD = 0.21743 and SD = 0.11077 for 166 V/m and E = 3333 V/m, respectively. An increase in the intensity of the measu field led to a substantial improvement in the quality of further data simulation.

Calculation of Piezoelectric Parameters of 67BFTO Ceramics
The entry parameters used to calculate the piezoelectric parameters of BFBT cer ics, namely resonant frequencies νr, anti-resonant frequencies νa, impedance, and capacitance C T on 1 kHz were measured by using the aforementioned impedance lyser. As shown in Figure 8, a thorough analysis of the experimental data (i.e., measu impedance spectra) was also performed. The procedure for the calculation of single efficients is described in detail in classical textbooks (e.g., see [21,24,25]), scientific pa (e.g., see [22,23]), or standards (e.g., see [30]). The results of the calculations are give Figures 12 and 13.  Figure 13b shows the formula for calcula of the relative change in piezoelectric modulus d31. Figure 12a shows dependence of the planar coupling factor kp for the vibration al the radial direction in a circle-shaped disk of 67BFTO ceramics on temperature. The sults of the calculations of the mechanical quality factor Qm are given in Figure 12b.  Figure 13b shows the formula for calculation of the relative change in piezoelectric modulus d 31 . Figure 12a shows dependence of the planar coupling factor k p for the vibration along the radial direction in a circle-shaped disk of 67BFTO ceramics on temperature. The results of the calculations of the mechanical quality factor Q m are given in Figure 12b.
One can see from Figure 12a that the coupling factor for both magnetic circumferential fields has a value of about k p = 31%. Local extremes visible on the plots at −10 • C and −30 • C differ from the average value by about 1%. Therefore, it is reasonable to use linear regression fit for data analysis. The difference between the linear fit plots is rather small but can easily be discerned. Moreover, the linear regression fit for a higher magnetic field (H ac = 0.49 A/m) shows better stability with temperature within the measured temperature range.
One can see from Figure 12b that the mechanical quality factor Q m exhibits nonmonotonic behaviour with increasing temperature for both of the used values of magnetic circumferential field intensity. The local extremes are clearly visible. Within the temperature range ∆T = (−30-+20) • C, the courses of the Q m curves are almost identical. The influence of magnetic field intensity on Q m becomes noticeable at about room temperature (20 • C). The difference between the linear fit plots can easily be discerned in Figure 12b. One can see that the linear regression fit for higher magnetic field intensity (H ac = 0.49 A/m; red line in Figure 12b) shows a negative slope with temperature within the measured temperature range, whereas the linear regression fit for Q m behaviour at a smaller magnetic circumferential field exhibits a positive one (H ac = 0.05 A/m; blue squares; blue line in Figure 12b). Figure 13a shows the dependence of piezoelectric charge coefficient d 31 on the radial vibration mode of a thin disk (excited through the piezoelectric effect across the thickness of the disk). One can see from Figure 13a  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 (ε ) and imaginary (ε ) 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 ac = 0.023 A/m (U = 2.0 V rms ; ν = 10 5 Hz) and H ac = 0.012 A/m (U = 1.0 V rms ; ν = 10 5 Hz) at 20 • C are shown in Figure 14. The influence of the circumferential magnetic field on the dielectric properties o 67BFBT ceramics was also revealed when the difference of the modulus of dielectri permittivity was plotted against the difference of circumferential magnetic field caused by the electric currents (iac) flowing through the ceramic sample ( Figure 15). One can see in Figure 15a that the difference in circumferential magnetic fields gen erated by the electric currents flowing through the sample linearly depends (in log-lo scale) on the frequency of the measuring sinusoidal signal. On the other hand, Figure 15 shows that the change in magnetic field strength causes a change in the modulus of th complex dielectric permittivity. Thus, the possibility of adjusting the dielectric permit tivity (and therefore capacitance value) via changes in magnetic field intensity was ob tained for 67BFBT ceramics. It is worth noting that changing the capacitance of a capaci tor in an electric circuit has a predictable effect on the complex impedance and phas angle in the circuit. These parameter changes can be exploited to yield tuneable imped ance-matching networks, tuneable filters, phase shifters, and other functional multifer roic devices. The influence of the circumferential magnetic field on the dielectric properties of 67BFBT 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 ac ) flowing through the ceramic sample ( Figure 15). The influence of the circumferential magnetic field on the dielectric properties of 67BFBT 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 (iac) 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 67BFBT 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.

Conclusions
Multiferroic 0.67Bi1.02FeO3-0.33BaTiO3 (67BFBT) ceramics were fabricated via a solid-state reaction process and subsequent sintering in a box furnace. They were studied in terms of structure and dielectric and piezoelectric properties. The fabricated 67BFBT ce- 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 67BFBT 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.

Conclusions
Multiferroic 0.67Bi 1.02 FeO 3 -0.33BaTiO 3 (67BFBT) ceramics were fabricated via a solidstate reaction process and subsequent sintering in a box furnace. They were studied in terms of structure and dielectric and piezoelectric properties. The fabricated 67BFBT ceramics adopted the tetragonal crystal structure described by the P4mm space group. The average crystallite size was <D> ≈ 80 nm, and the average strain was <ε> = 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" (χ 2 ) and weighted sum of squares (WSS) parameters. Measurements and subsequent fitting and simulations have shown that the planar coupling factor was k p ≈ 31%, and the mechanical quality was within the range Q mech ≈ 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 ≈ 35pC/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 67BFBT 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 (ε ) and imaginary (ε ) parts of the complex dielectric permittivity and piezoelectric charge coefficient (d 31 ) of the multiferroic 67BFTO 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 67BFBT 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 67BFBT ceramics are very sensitive to magnetic field influence, which makes 67BFBT sensing material with vast potential.

Data Availability Statement:
The data presented in this paper are available on request from the corresponding author.