Immittance Studies of Bi6fe2ti3o18 Ceramics

materials

Article Immittance Studies of Bi6Fe2Ti3O18 Ceramics

Agata Lisi ´nska-Czekaj 1,* , Dionizy Czekaj 1, Barbara Garbarz-Glos 2,3 and Wojciech B ˛ak 2

1 Faculty of Mechanical Engineering, Gda´nskUniversity of Technology, 11/12, Narutowicza St., 80-233 Gda´nsk,Poland; [email protected] 2 Institute of Technology, Pedagogical University of Cracow, 2 Podchor ˛a˙zychStr., 30-084 Kraków, Poland; [email protected] (B.G.-G.); [email protected] (W.B.) 3 Institute of Technology, The Jan Grodek State University in Sanok, 6 Reymonta Str., 38-500 Sanok, Poland * Correspondence: [email protected]

Received: 29 September 2020; Accepted: 20 November 2020; Published: 22 November 2020

Abstract: Results of studies focusing on the electric behavior of Bi6Fe2Ti3O18 (BFTO) ceramics are reported. BFTO ceramics were fabricated by solid state reaction methods. The simple oxides Bi2O3, TiO2, and Fe2O3 were used as starting materials. Immittance spectroscopy was chosen as a method to characterize electric and dielectric properties of polycrystalline ceramics. The experimental data were measured in the frequency range ∆ν = (10 1–107) Hz and the temperature range ∆T = ( 120–200) C. − − ◦ Analysis of immittance data was performed in terms of complex impedance, electric modulus function, and conductivity. The activation energy corresponding to a non-Debye type of relaxation was found to be EA = 0.573 eV, whereas the activation energy of conductivity relaxation frequency was found to be EA = 0.570 eV. An assumption of a hopping conductivity mechanism for BFTO ceramics was studied by ‘universal’ Jonscher’s law. A value of the exponents was found to be within the “Jonscher’s range” (0.54 n 0.72). The dc-conductivity was extracted from the measurements. Activation energy for ≤ ≤ dc-conductivity was calculated to be EDC = 0.78 eV, whereas the dc hopping activation energy was found to be EH = 0.63 eV. The obtained results were discussed in terms of the jump relaxation model.

Keywords: electroceramics; immittance spectroscopy; speciﬁc conductivity; activation energy

1. Introduction Magnetoelectric multiferroics, which are simultaneously ferroelectric and ferromagnetic (or at least show some kind of magnetic ordering), present a somewhat obscure but tempting, from a scientific point-of-view, class of materials. Such materials have all the potential applications of both their parent ferroelectric and ferromagnetic materials. The emerging possibility of controlling the electrical properties by changing the magnetic field and controlling the magnetic properties by changing the electric field opens up new, broad horizons in the device’s design. For instance, memory elements in which data is stored both in the electric and the magnetic polarizations, or novel memory media, might allow the writing of a ferroelectric data bit and the reading of the magnetic field generated by association [1]. Let us consider bismuth ferrite BiFeO3 (BFO) – bismuth titanate (Bi4Ti3O12) compounds. BiFeO3 is known to exhibit a perovskite-type structure and is one of only a few materials in which (anti)ferromagnetism and ferroelectricity coexist at room temperature—which is one of the most desirable features for multiferroics. Therefore, being the only single-phase ABO3-type perovskite compound which possesses multiferroic properties at room temperature, BFO is considered to be the most promising candidate among various multiferroic materials for applications in the next generation memories and spintronics [2]. Bismuth titanate (Bi4Ti3O12) is a bismuth layer-structured ferroelectric material which was discovered by Aurivillius [3]. It is commonly used for various

Materials 2020, 13, 5286; doi:10.3390/ma13225286 www.mdpi.com/journal/materials Materials 2020, 13, 5286 2 of 12 electronic industry applications like capacitors, transducers, non-volatile ferroelectric memory devices, and high temperature piezoelectric sensors. Combining these two materials, namely, bismuth titanate and bismuth ferrite exhibiting different physical properties, one can create novel materials. Thus, achieving rich functionality, Aurivillius phases of the Bi4Ti3O12-BiFeO3 system are known to combine ferroelectric, semiconducting, and ferromagnetic properties and they are potentially attractive for producing high-performance ceramics for information processing and information storage applications [4,5]. The Bi6Fe2Ti3O18 compound has been already synthesized either in a form of bulk polycrystalline ceramics or in a form of thin film. The synthesis of Aurivillius phases has usually been performed by the mixed oxide method, e.g., [6–8], but the molten salt method [9], hydrothermal method [10] and hot pressing method [11] were also utilized. Thin films of Bi6Fe2Ti3O18 were deposited by chemical solution deposition [12,13] or metalorganic decomposition [14] on different substrates (e.g., N-type Si wafers with (111) orientation [14], as well as (111) Pt/TiSiO2/Si and sapphire substrates [12,13] were utilized). The present study was motivated by the high potential applications of Aurivilius phases of the Bi4Ti3O12-BiFeO3 system. Moreover, fundamental physics of multiferroic materials are rich and fascinating. Therefore, the ferromagnetic compound of the Bi6Fe2Ti3O18 (BFTO) composition was chosen as a material of investigation due to its high Curie temperature (T = 805 ◦C) and a second order magnetoelectric effect [10,15]. Although several studies on the impedance spectroscopy study of polycrystalline Bi6Fe2Ti3O18 have already been published, e.g., [6,11–13,16], the results are available for different objects (bulk ceramics or thin films), different methods of fabrication (bulk ceramics), or different conditions of the thin film crystallization. Moreover, the presented data are available for different temperatures or information on the crystal structure and the phase composition is missing, e.g., [6]. Present studies were focused on the dielectric response of polycrystalline BFTO multiferroic ceramics in the frequency range ∆ν = (10 1–107) Hz and the temperature range ∆T = ( 120–200) C. − − ◦ The obtained results were discussed in terms of ac-conductivity as well as complex impedance and complex electric modulus formalisms.

2. Materials and Methods

BFTO ceramics were manufactured by the mixed oxide method (MOM). Bi2O3, TiO2, and Fe2O3 metal oxides (all 99.9% purity, Aldrich Chemical Co., Ltd., Merck KGaA, Darmstadt, Germany) were used for stoichiometric mixture preparation. After the calcination process (Tcalc = 720 ◦C), the pellets were formed and pressed into disks with a diameter of 10 mm and 1 mm thickness. Pressureless sintering was used for final densification of ceramic samples. The sintering temperature was Ts = 980 ◦C, while the dwell time was ts = 2 h. For thermal analysis measurements, the stoichiometric mixture of oxide powders, constituting the Bi6Fe2Ti3O18 composition, was used; whereas for X-ray diffraction analysis, the sintered BFTO ceramic sample was powdered to utilize advantages of the powder diffraction method. For dielectric spectroscopy measurements, the sintered ceramic disks were polished on both sides and covered with silver electrodes to form a disk-shaped parallel-plate capacitor. The Alpha-AN High Performance Frequency Analyzer system combined with a cryogenic temperature control system (Quatro Cryosystem, Novocontrol Technologies GmbH & Co. KG, Montabaur, Germany) was used for broad-band dielectric spectroscopy measurements (BBDS) [17]. A strong point of BBDS is that using an appropriate strategy and various methods of analyzing experimental data, one is possible to separate the impact of individual electrically active areas (e.g., grains and grain boundaries) as well as characterize the dynamics of the electric charge both at the interfacial boundaries and inside the grains of the ceramic material [18,19]. BBDS is used for studying a fine nature of electrical behavior of electroceramic materials including ferroelectrics, solid electrolytes, and materials with mixed conductivity. The measuring temperature range was ∆T = ( 120 ... + 200) ◦C, and the frequency range was 1 7 − ∆ν = (10− ... 10 ) Hz. Before the measurement, the system was cooled down with liquid nitrogen. Materials 2020, 13, 5286 3 of 12 Materials 2020, 13, x FOR PEER REVIEW 3 of 12

The measurements were performed during the heating cycle. The impedance spectra were recorded min after reaching the programmed temperature. The step size of the temperature change was T = 5 15 min after reaching the programmed temperature. The step size of the temperature change was °C. The WinDATA Novocontrol software was used for recording, visualization, and processing of ∆T = 5 C. The WinDATA Novocontrol software was used for recording, visualization, and processing experimental◦ data. However, to check the consistency of the experimental data, a computer program of experimental data. However, to check the consistency of the experimental data, a computer program by Boukamp was used [20]. Thus, the Kramers–Kronig test was carried out. by Boukamp was used [20]. Thus, the Kramers–Kronig test was carried out. 3. Results and Discussion 3. Results and Discussion It should be pointed out that bismuth layer-structured multiferroic compounds of the It should be pointed out that bismuth layer-structured multiferroic compounds of the composition composition Bim+1Fem–3Ti3O3m+3 are difficult to produce because these compounds are formed in Bim+1Fem–3Ti3O3m+3 are difficult to produce because these compounds are formed in several stages several stages and their thermal stability is low. Therefore, the process of fabrication of BFTO was and their thermal stability is low. Therefore, the process of fabrication of BFTO was controlled with controlled with both simultaneous thermal analysis (STA) and X-ray diffraction analysis. It was both simultaneous thermal analysis (STA) and X-ray diffraction analysis. It was found by STA that found by STA that the formation of the BFTO phase took place at the temperature range T ≈ the formation of the BFTO phase took place at the temperature range ∆T 650–700 ◦C[7]. For that 650–700 °C [7]. For that reason, the temperature of calcination was chosen≈ as Tc = 720 °C. Results of reason, the temperature of calcination was chosen as Tc = 720 ◦C. Results of the thermal analysis of the thermal analysis of BFTO powder are shown in Figure 1a. BFTO powder are shown in Figure1a. Results of X-ray diffraction studies performed for ceramics sintered at Ts = 850 °C, Ts = 1080 °C Results of X-ray diffraction studies performed for ceramics sintered at Ts = 850 ◦C, Ts = 1080 ◦C[7], [7], and Ts = 980 °C [8,21] proved the essential influence of the sintering temperature on the crystal and Ts = 980 ◦C[8,21] proved the essential influence of the sintering temperature on the crystal structure structure and phase composition of the Bi6Fe2Ti3O18 compound. It was found that the BFTO ceramics and phase composition of the Bi6Fe2Ti3O18 compound. It was found that the BFTO ceramics sintered sintered at Ts = 850 °C, Ts = 1080 °C were multiphase ones and consisted of two phases, namely, the at Ts = 850 C, Ts = 1080 C were multiphase ones and consisted of two phases, namely, the Aurivillius Aurivillius◦ phase with m◦ = 5 layers (i.e., the stoichiometric phase) and m = 4 (i.e., the phase with a phase with m = 5 layers (i.e., the stoichiometric phase) and m = 4 (i.e., the phase with a reduced number reduced number of layers in the slab) [7]. Our earlier studies have shown [8,21] that Bi6Fe2Ti3O18 of layers in the slab) [7]. Our earlier studies have shown [8,21] that Bi6Fe2Ti3O18 ceramics, sintered at ceramics, sintered at Ts = 980 °C, were single-phase and adopted the orthorhombic structure of the Ts = 980 C, were single-phase and adopted the orthorhombic structure of the Aba2 (41) space group. Aba2 (41)◦ space group. Therefore, for the purpose of the present studies, final sintering of BFTO Therefore, for the purpose of the present studies, final sintering of BFTO ceramics was performed at ceramics was performed at Ts = 980 °C. The resulting X-ray diffraction pattern is shown in Figure 1b. Ts = 980 ◦C. The resulting X-ray diffraction pattern is shown in Figure1b.

(a) (b)

FigureFigure 1.1. ((aa)) Thermograms Thermograms of of the the Bi Bi6Fe6Fe2Ti23TiO183O (BFTO18 (BFTO)) powder; powder; (b) X (-bray) X-ray diffraction diﬀraction pattern pattern for BFTO for BFTOcompound. compound.

StructuralStructural analyses analyses based based on the on X-ray the X di-rayffraction diffraction pattern pattern (Figure 1 (Figureb) show 1b)the BFTOshow compoundthe BFTO crystallizedcompound incrystallized an orthorhombic in an orthorhombic structure, Fmm2 structure,space group, Fmm2 withspace the group, following with elementary the following cell parameters:elementary cella = 5.4565(4) parameters: Å, ba == 49.375(3)5.4565(4) Å, Å , andb = c49.375(3)= 5.4816(3) Å , and Å. The c = Rietveld5.4816(3) refinement Å . The Rietveld of the crystalrefinement structure of the was crystal performed structure using wasthe performed ICSD standard using the (ICSD ICSD collection standard code: (ICSD 156257), collection primary code: reference [9]. The average size of crystallites was calculated D = 433.2 Å as well as the average strain 156257), primary reference [9]. The average size of crystallitesh i was calculated D = 433.2 Å as well as ε = 0.043%. It is worth noting that quality parameters of the Rietveld refinement procedure were as hthei average strain ε = 0.043%. It is worth noting that quality parameters of the Rietveld refinement follows:procedure R(expected) were as follows:= 2.81305%; R(expected) R(profile) = = 2.81305%;3.97795%; R(profile) R(weighted = 3.97795%; profile) = 5.52957%; R(weighted goodness profile) of = fit5.52957%; GOF = 3.86392%. goodness of fit GOF = 3.86392 AA SEMSEM micrographmicrograph ofof sinteredsintered BiBi66FeFe22Ti33O18 ceramicsceramics and and EDS EDS spectrum spectrum is is shown shown in in Figure Figure 22.. OneOne can can see see in in Figure Figure2a that 2a that BFTO BFTO grains grains adopt adopt a plate-like a plate shape.-like Measurements shape. Measurements of the average of the grainaverage size grain taken size as taken an average as an average diagonal diagonal of the plate-like of the plate grain-like were grain performed were performed with the with help the of help an ImageJ—Aof an ImageJ public—A public domain domain Java image Java image processing processing program program [22]. It [22]. was It found was found that the that average the average grain sizegrain was size 7.21 wasµm (Figure7.21 μm3a). (Figure Apart from 3a). theApart diagonal, from the thickness diagonal, of the the thickness plates was of measured. the plates It was was measured. It was found that the average thickness of the plate-like grain was 0.44 μm. The resulting distribution of “thicknesses” of the plate-like grains is shown in Figure 3b.

Materials 2020, 13, x FOR PEER REVIEW 4 of 12 Materials 2020, 13, x FOR PEER REVIEW 4 of 12 Materials 2020, 13, 5286 4 of 12 The stoichiometric composition of the Bi6Fe2Ti3O18 compound expressed in the weight fraction The stoichiometric composition of the Bi6Fe2Ti3O18 compound expressed in the weight fraction of oxides is as follows: Bi2O3—77.7 wt%; Fe2O3—9.0 wt%; and TiO2—13.3 wt% [8]. The EDS foundof oxides that is the as average follows: thickness Bi2O3—77.7 of the wt%; plate-like Fe2O3 grain—9.0 was wt% 0.44; andµ m. TiO The2—13.3 resulting wt% distribution [8]. The EDS of quantitative results, calculated on the base of the spectrum shown in Figure 2b, for BFTO ceramics “thicknesses”quantitative results, of the plate-likecalculated grains on the is base shown of inthe Figure spectrum3b. shown in Figure 2b, for BFTO ceramics were as follows: Bi2O3—78.26 wt%; Fe2O3—8.80 wt%; and TiO2—12.94 wt%. were as follows: Bi2O3—78.26 wt%; Fe2O3—8.80 wt%; and TiO2—12.94 wt%.

(a) (b) (a) (b) Figure 2. (a) Microstructure of BFTO ceramics; (b) the EDS spectrum for BFTO ceramics. Figure 2 2.. ((aa)) Microstructure Microstructure of of BFTO BFTO ceramics; ( b) thethe EDS spectrum for BFTO ceramics.

(a) (b) (a) (b) FigureFigure 3 3.. (a(a) )Analysis Analysis of of the the grain grain size distributiondistribution forfor BFTO BFTO ceramics; ceramics; (b ()b analysis) analysis of of the the thickness thickness size Figure 3. (a) Analysis of the grain size distribution for BFTO ceramics; (b) analysis of the thickness sizedistribution distribution for for plate-like plate-like grains grains of BFTO.of BFTO. size distribution for plate-like grains of BFTO. 3.1. ComplexThe stoichiometric Impedance Analysis composition of the Bi6Fe2Ti3O18 compound expressed in the weight fraction of 3.1. Complex Impedance Analysis oxides is as follows: Bi2O3—77.7 wt%; Fe2O3—9.0 wt%; and TiO2—13.3 wt% [8]. The EDS quantitative Dielectric spectroscopy is extremely susceptible to random disturbances that do not show up in results,Dielectric calculated spe onctroscopy the base is of extremely the spectrum susceptible shown into Figurerandom2b, disturbances for BFTO ceramics that do were not show as follows: up in the impedance spectrum at first glance. Therefore, in order to obtain reliable results as a result of the Bithe2O impedance3—78.26 wt%; spectrum Fe2O3 at—8.80 first wt%;glance. and Therefore, TiO2—12.94 in order wt%. to obtain reliable results as a result of the analysis of impedance data, it is necessary to test the consistency of the recorded measurement data. analysis of impedance data, it is necessary to test the consistency of the recorded measurement data. For3.1. this Complex purpose, Impedance the Kramers Analysis–Kronig (K–K) equations were used to test the quality of measurement For this purpose, the Kramers–Kronig (K–K) equations were used to test the quality of measurement data. Inspection of the results showed high compliance of the measurement with the K–K data.Dielectric Inspection spectroscopy of the results is extremely showed susceptible high compliance to random of disturbances the measurement that do notwith show the up K– inK calculations. The value of the “chi-square” parameter was obtained within the range χ2 = 5 × 10−5–2 × thecalculations. impedance The spectrum value of atthe ﬁrst “chi glance.-square” Therefore, parameter in orderwas obtained to obtain within reliable the results range as χ2 a = result 5 × 10 of−5– the2 × 10−7. Thus, an excellent quality of the measurements was confirmed. Results of the impedance analysis10−7. Thus of, impedancean excellent data, quality it is necessary of the measurements to test the consistency was confirmed. of the recorded Results measurement of the impedance data. measurements are shown in Figure 4. Formeasurements this purpose, are the shown Kramers–Kronig in Figure 4. (K–K) equations were used to test the quality of measurement One can see smooth and monotonic curves of modulus of the complex impedance |Z| that data.One Inspection can see of smooth the results and showed monotonic high compliancecurves of modulus of the measurement of the compl withex impedancethe K–K calculations. |Z| that exhibit weak frequency dispersion in the low frequency range and strong frequency dependence in Theexhibit value weak of thefrequency “chi-square” dispersion parameter in the was low obtained frequency within range the and range strongχ2 =frequency5 10 5–2 dependence10 7. Thus, in the high frequency range (Figure 4a). The changeover point from weak to strong× − dependence× − on antheexcellent high frequency quality ofrange the measurements(Figure 4a). The was changeover conﬁrmed. point Results from of the weak impedance to strong measurements dependence areon frequency, shifts toward higher frequency with an increase in temperature. One can see in Figure 4a shownfrequency in Figure, shifts4 toward. higher frequency with an increase in temperature. One can see in Figure 4a that in the high frequency range |Z| (ν) curves for different temperatures overlap each other. The that inOne the can high see frequency smooth and range monotonic |Z| (ν) curves curves of for modulus different of thetemperatures complex impedance overlap each|Z| that other. exhibit The curves representing a spectroscopic plot of the phase angle (θ) exhibit a sigmoidal shape (in a weakcurves frequency representing dispersion a spectroscopic in the low plotfrequency of the range phase and angle strong (θ) frequency exhibit a dependence sigmoidal shape in the (in high a semi-logarithmic scale). They are also smooth and change monotonically from θ = 0 for low frequencysemi-logarithmic range (Figure scale). 4 Theya). The are changeover also smooth point and from change weak monotonically to strong dependence from θ on= 0 frequency, for low frequency to θ = −90 for high frequency (Figure 4a). They shift toward higher frequency with an shiftsfrequenc towardy to higherθ = −90 frequency for high with frequency an increase (Figure in temperature. 4a). They shift One toward can see inhigher Figure frequency4a that in thewith high an increase in temperature. increase in temperature.

Materials 2020, 13, 5286 5 of 12 frequency range |Z| (ν) curves for diﬀerent temperatures overlap each other. The curves representing a spectroscopic plot of the phase angle (θ) exhibit a sigmoidal shape (in a semi-logarithmic scale). They are also smooth and change monotonically from θ = 0 for low frequency to θ = 90 for high ◦ − ◦ frequencyMaterials 2020 (Figure, 13, x FOR4a). PEER They REVIEW shift toward higher frequency with an increase in temperature. 5 of 12

(a) (b)

FigureFigure 4. 4(.a )(a Bode) Bode plot plot of impedance of impedance data; (data;b) complex (b) complex impedance impedance plot at 175 plot◦C–200 at 175◦C; °C (c)–200 Cole–Cole °C; (c) plotCole of–Cole impedance plot of dataimpedance at the temperature data at the temperature range from range120 Cfrom to −12055 C; °C (d to) spectroscopic−55 °C; (d) spectroscopic plot of the − ◦ − ◦ realplot part of the of dielectricreal part of permittivity dielectric permittivity at 120 C to at −12055 C.°C to −55 °C. − ◦ − ◦ The dependence of the imaginary part of complex impedance ( Z”) on the real part of complex The dependence of the imaginary part of complex impedance −(−Z″) on the real part of complex impedanceimpedance ((ZZ0’)) isis shownshown in Figure Figure 44b.b. One One can can see see that that despite despite the the isotropic isotropic linear linear scale scale used used,, the theexperimental experimental curves curves (Figure (Figure 4b)4b) are are rarely rarely ideal ideal semicircles. semicircles. In the present case,case, thethe curves curves resemble resemble arcsarcs (deformed, (deformed, ﬂattened flattened semicircles) semicircles) with with the the centers centers below below the the real real axis. axis. The The frequency frequency at which at which the arcthe reaches arc reaches a maximum a maximum corresponds corresponds to the to relaxation the relaxation frequency, frequency e.g.,, [e.g.,16]: [16]:

ωmτmm=m 1,1 , (1)(1)

where ωm, τm—relaxation frequency and relaxation time, respectively. As the temperature rises, the where ωm, τm—relaxation frequency and relaxation time, respectively. As the temperature rises, depression angle increases and the arc radius decreases, which indicates the thermal activation of the depression angle increases and the arc radius decreases, which indicates the thermal activation of the conduction mechanism. The shape of the Nyquist diagram (Figure 4b) shows that the relaxation time of the polarization processes taking place in BFTO ceramics cannot be defined as a single quantity, but as a quantity with a certain distribution around the mean value. Impedance data for the temperature range T = −120 °C–55 °C are shown in Figure 1c and Figure 1d in a form of the Cole–Cole plot (Figure 4c) and the spectroscopic dependence of the real part of dielectric permittivity (Figure 4d). Almost ideal semicircles, shown in Figure 4c, mean that the phenomena responsible for relaxation in the BFTO ceramics can be described by a simple Debye

Materials 2020, 13, 5286 6 of 12 the conduction mechanism. The shape of the Nyquist diagram (Figure4b) shows that the relaxation time of the polarization processes taking place in BFTO ceramics cannot be defined as a single quantity, but as a quantity with a certain distribution around the mean value. MaterialsImpedance 2020, 13, x FOR data PEER for REVIEW the temperature range ∆T = 120 C–55 C are shown in Figure1 6in of a12 − ◦ ◦ form of the Cole–Cole plot (Figure4c) and the spectroscopic dependence of the real part of dielectric modelpermittivity with one (Figure relaxation4d). Almost time. idealThe main semicircles, advantage shown of such in Figure presentation4c, mean is that a possibility the phenomena to read fromresponsible the plot for values relaxation for both in the the BFTO static ceramics (εs) and canhigh be frequency described (ε by) alimits simple of Debyedielectric model permittivity. with one Therelaxation phenomena time. The of main dielectric advantage relaxation, of such however, presentation become is a possibility significantly to read more from complicated the plot values with increasingfor both the temperature. static (εs) and As high the frequency temperature (ε increases,) limits of a dielectric straight permittivity. line appears The on phenomenathe Cole–Cole of ∞ diagradielectricm (for relaxation, low frequencies). however, Figure become 4d significantly demonstrates more the complicatedfrequency dependence with increasing of the temperature.real part (ε’) ofAs the temperature complex permittivity increases, a for straight the BFTO line appears ceramics on the at various Cole–Cole temperatures diagram (for chosen low frequencies). from the temperatureFigure4d demonstrates range T = the −120 frequency °C–55 °C. dependence It is clear offrom the Figure real part 4d ( εthat0) of the the BFTO complex ceramics permittivity exhibit for a lowthe- BFTOfrequency ceramics dispersion. at various However, temperatures what may chosen attract from our the attention temperature is the range step∆-Tlike= decrease120 C–55 in εC.’ − ◦ ◦ thatIt is clearshifts from to higher Figure frequency4d that the with BFTO increasing ceramics the exhibit temperature, a low-frequency indicating dispersion. the thermally However, activated what mechanism.may attract our attention is the step-like decrease in ε0 that shifts to higher frequency with increasing the temperature,Figure 5a shows indicating the dependence the thermally of the activated imaginary mechanism. part of impedance (−Z”) of BFTO ceramics on the frequency.Figure5a shows One thecan dependence see that as of the the temperature imaginary part rises, of impedance the maximum ( Z ”) of ofthe BFTO Z″-curve ceramics shifts on − towardsthe frequency. higher One values can of see frequency. that as the The temperature curves taken rises, at thedifferent maximum temperatures of the Z”-curve are not shifts symm towardsetrical andhigher wide values (FWHM of frequency. ≈ 2 decades The curves of frequency).taken at diff erentThe temperatures dependence ofare ν notmax symmetricalon inverse and absolute wide temperature(FWHM 2 decadesis shown of in frequency). Figure 5b. TheThe dependenceactivation energy of νmax correspondingon inverse absolute to relaxation temperature was found is shown to ≈ bein FigureEA = 0.5735b. TheeV. activation energy corresponding to relaxation was found to be EA = 0.573 eV.

(a) (b)

Figure 5 5.. ((aa)) Dependence Dependence of of im imaginaryaginary part part of of impedance impedance (− (Z″)Z ”)on on frequency frequency for for BFTO BFTO ceramics ceramics at − 140at 140 °C–◦200C–200 °C; ◦(bC;) dependence (b) dependence of frequency of frequency (νmax) ( νcorrespondingmax) corresponding to the maximum to the maximum −Z″(ν) onZ inverse”(ν) on − temperature.inverse temperature.

Presentation of impedance data in the complex Z” Z plane (the so-called Nyquist diagram) or Presentation of impedance data in the complex −−Z″−Z′0 plane (the so-called Nyquist diagram) or inin the form of a spectroscopicspectroscopic dependence Z”(″(ν)) (the so so-called-called Bode diagram) allows to distinguish thethe contribution contribution of of the the areas areas with with the the highest highest resistance resistance to to the the total total impedance impedance response response of of the the tested tested system. They They are are useful useful when when the therelaxation relaxation frequencies frequencies of polarization of polarization processes processes taking takingplace inside place theinside grains the ( grainsνb), at (theνb), grain at the boundaries grain boundaries (νgb) and (ν ingb )the and electrode in the electrode areas (νe areas), differ (νe significantly), differ significantly (e.g., νb >>(e.g., νgbν >>b >> νel).ν gbHowever,>> νel). they However, may not they be maysufficient not be to su definefficient the to contribution define the contribution of the areas ofcomprising the areas thecomprising interior of the the interior grains of (bulk) the grains with (bulk)highly withresistive highly grain resistive boundaries. grain boundaries.They are also They of little are alsouse whenof little the use contributions when the contributions from individual from individual areas of theareas ceramic of the ceramic microstructure microstructure (grains (grains and grain and boundaries)grain boundaries) overlap overlap (see: (see: flattened, flattened, deformed deformed arcs arcs in Figure in Figure 4b).4b). In In such such cases, cases, it it is is advisable to to represent the impedance data data using using an an electric electric modulus modulus function function that that is is sensitive sensitive to to input input from from low low capacitance areas [23]. [23].

3.2. Electric Modulus Analysis In dielectric spectroscopy, a function called the electric modulus is often used. This function is complex and is defined as the reciprocal of the dielectric constant:

1 1 M*   ' j"  M' jM", (2)  * where ε’, ε”—real and imaginary part of dielectric constant, respectively.

Materials 2020, 13, 5286 7 of 12

3.2. Electric Modulus Analysis In dielectric spectroscopy, a function called the electric modulus is often used. This function is complex and is deﬁned as the reciprocal of the dielectric constant:

Materials 2020, 13, x FOR PEER REVIEW 1 1 7 of 12 M = = (ε0 jε00 )− = M0 + jM00 , (2) ∗ ε − The electric modulus function is neither∗ a directly measurable quantity nor directly related to where , ”—real and imaginary part of dielectric constant, respectively. any microscopicε0 ε physical processes. However, it completes the set of four interrelated to each other The electric modulus function is neither a directly measurable quantity nor directly related electrical relaxation functions, which are complex permittivity, complex conductivity, complex to any microscopic physical processes. However, it completes the set of four interrelated to each resistivity, and complex electric modulus [24]. The advantages of an analysis based on the use of other electrical relaxation functions, which are complex permittivity, complex conductivity, complex electrical modulus functions are apparent when there is a need to distinguish the electrode resistivity,polarization and phenomena complex electricor assess modulus the relaxation [24]. The time advantages of electrical of conductivity an analysis [25]. based Presentation on the use of electrical modulus functions are apparent when there is a need to distinguish the electrode polarization the impedance data in the form of a spectroscopic dependence of the electric modulus often allows phenomenato emphasize or theassess contribution the relaxation of polarization time of electrical phenomena conductivity occurring [25]. Presentation inside the grains of the impedance (bulk) and datacharacterized in the form by ofthe a lowest spectroscopic capacitance. dependence A contribution of the electric of other modulus polarization often allowsprocesses to emphasizeexhibiting thesmall contribution differences of in polarization capacitance phenomenamay also be revealed occurring [23]. inside the grains (bulk) and characterized by the lowestFigure capacitance. 6 shows the A dependence contribution of other the real polarization part of electric processes modulus exhibiting M′ (Figure small di 6a)ﬀerences and the in capacitanceimaginary part may of also electric be revealed modulus [23 M]. ″ (Figure 6b) on frequency at diverse temperatures for BFTO M ceramics.Figure 6 shows the dependence of the real part of electric modulus 0 (Figure6a) and the imaginary part of electric modulus M” (Figure6b) on frequency at diverse temperatures for BFTO ceramics.

(a) (b)

Figure 6.6. ((aa)) DependenceDependence of of the the real real part part of of electric electric modulus modulus (M (0M) on′) on frequency; frequency; (b) ( dependenceb) dependence of the of imaginarythe imaginar party of part electric of electric modulus modulus (M”) on ( frequencyM″) on frequency at various at temperatures various temperatures for BFTO ceramics. for BFTO ceramics. One can see from Figure6a that in the low frequency range M0 reaches very low values. A drive of M0Onetowards can see zero from points Figure out that6a that the in force the restoringlow frequency the flow range of charge M′ reaches decays very as thelow frequency values. A ofdrive the measuringof M′ towards electric zero field points decreases. out that Atthe the force same restoring time, it the indicates flow of that charge the electrodedecays as and the/ orfrequency interfacial of processes,the measuring which electric usually field manifest decreases. themselves At the within same the time, low it frequency indicates range, that the do electrode not significantly and/or contributeinterfacial to processes, the total dielectricwhich usually response manifest of BFTO themselves ceramics [26 within]. In the the high low frequency frequency range, range,M do0 tends not tosignificantly a constant contribute value the same to the for tot allal measurementsdielectric response carried of BFTO out at ceramics different [26]. temperatures. In the high This frequency in turn representsrange, M′ thetends reciprocal to a constant value of value the frequency the same independent for all measurements dielectric constant carried characterizing out at different the boundtemperatures. charge response.This in turn The represents changeover the point reciprocal from a strongvalue of to the weak frequency dependence independent of M0 on frequencydielectric shiftsconstant toward characterizing higher frequency the bound with charge an increase response. in temperature. The changeover point from a strong to weak dependenceThe dependence of M′ on offrequency the imaginary shifts parttoward of the higher electric frequency modulus with on frequencyan increase presented in temperature. in Figure 6b allowsThe to dependence highlight the of relaxation the imaginary processes part takingof the electric place inside modulus the grains on frequency (bulk). It presented can be seen in thatFigure as the6b allows temperature to highlight increases, the therelaxation maxima processes on M” curves taking shift place towards inside higher the grains frequencies (bulk). andIt can the be height seen ofthat the as maximum the temperature decreases increases, slightly. the It ismaxima worth notingon M″ curves that the shift dependence towards ofhigherM” on frequencies frequency and should the beheight of the of Lorentzthe maximum type if decreases the relaxation slightly. process It is isworth the exponential noting that one.the dependence Nevertheless, of aM deviation″ on frequency from theshould Debye be behavior of the Lorentz of relaxation type if phenomena the relaxation in BFTO process ceramics is the has exponential been noted. one. The Nevertheless, Lorentz fit of a thedeviationM” spectroscopic from the Debye dependence behavior showed of relaxation that the phenomena maxima are in wideBFTO (FWHM ceramics was has aboutbeen noted. 2 decades) The Lorentz fit of the M″ spectroscopic dependence showed that the maxima are wide (FWHM was about 2 decades) and asymmetric. Two possible reasons for such behavior observed in BFTO ceramics can be assumed, namely, the presence of a distribution of relaxation times or stretching of the relaxation times [25]. The spectroscopic M″ data were normalized along the ordinate axis and abscissa axis (Figure 7a). Normalization along the ordinate axis consisted in dividing the current values of M″ by the maximum value of the imaginary component of the electric modulus (M″/Mmax). Normalization along the abscissa axis consisted in dividing the current value of the measuring frequency by the

Materials 2020, 13, 5286 8 of 12 and asymmetric. Two possible reasons for such behavior observed in BFTO ceramics can be assumed, namely, the presence of a distribution of relaxation times or stretching of the relaxation times [25]. The spectroscopic M” data were normalized along the ordinate axis and abscissa axis (Figure7a). Normalization along the ordinate axis consisted in dividing the current values of M” by the maximum valueMaterials of the 2020 imaginary, 13, x FOR PEER component REVIEWof the electric modulus (M”/Mmax). Normalization along the abscissa8 of 12 axis consisted in dividing the current value of the measuring frequency by the frequency corresponding tofrequency the maximum corresponding value of theto the imaginary maximum component value of the of imaginary the electric component modulus ( νof/ν themax electric) known modulus as the conduction(ν/νmax) known relaxation as the frequency conduction (νmax relaxation). As a result frequency of such ( dataνmax). treatment, As a result it wasof such found data that treatment the proﬁle, it ofwas the found spectroscopic that the curveprofile of of the the electric spectroscopic modulus curve (M”) of does the notelectric change modulus with temperature.(M″) does not All change the Mwith”-curves temperature. were superimposed All the M″-curves on one were so-called superimposed primary curve. on one so-called primary curve.

(a) (b)

FigureFigure 7. 7. ((aa)) Dependence Dependence of normalized modulus modulus ( (MM″”/M/M″”maxmax) ) on on so so-called-called frequency frequency (ν (/νmax/νmax); ();b) (bd)ependence dependence of of frequency frequency (ν (maxνmax) corresponding) corresponding to tothe the maximum maximum M″(Mν”() onν) oninverse inverse temperature. temperature.

TheThe analysis analysis of of the the dependence dependence of of the the conduction conduction relaxation relaxation frequency frequency of of BFTO BFTO ceramics ceramics on on the the inverseinverse absolute absolute temperature temperature (Figure(Figure7 7b)b) allowed allowed toto calculatecalculate the activation energy energy of of the the process process EA EA= 0.570= 0.570 eV. eV.

3.3.3.3. Electric Electric Conductivity Conductivity Analysis Analysis FigureFigure8a 8a shows shows the the dependence dependence of ofac ac-conductivity-conductivity ( σ ()σ) on on angular angular frequency frequency ( ω (ω)) at at various various temperatures for Bi Fe Ti O ceramics. One can see in Figure8a that the behavior of BFTO ceramics temperatures for Bi6 6Fe2 2Ti3 3O1818 ceramics. One can see in Figure 8a that the behavior of BFTO ceramics isis typical typical for for ionic ionic materials. materials. It can It be can seen be that seen in the that low-frequency in the low region,-frequency the electrical region, conductivity the electrical hardlyconductivity depends hardly on the frequencydepends on (plateau the frequency is present) (plateau and takes is apresent value equal) and totakes thevolume a value conductivity equal to the ofvolume the sample conductivityσ(0). With of an the increase sample in theσ(0) measuring. With an frequency, increase in a strongthe measuring high-frequency frequency, dependence a strong ofhigh the- electricfrequency conductivity dependence of theof the sample electric appears. conductivity On the of other the hand,sample the appears.dc-plateau On isthe temperature other hand, dependent—thethe dc-plateau is raise temperature of conductivity dependent is observed—the raise while of temperature conductivity increases. is observed The while plotin temperature Figure8b showsincreases. the dependence The plot in ofFigure conductivity 8b showsσ(0) theT (assumingdependence a hoppingof conductivity conduction σ(0) mechanism)T (assuming ona hopping 1000/T (Tconductionis an absolute mechanism) temperature). on 1000/T (T is an absolute temperature). The expression “hopping” refers to the sudden movement of an electric charge carrier from one location to another in its vicinity. The process of electric current conduction based on the hopping mechanism does not take place through the conduction band. It occurs through the use of localized energy states within the band gap and it is characteristic for dielectrics, like, e.g., BFTO ceramics. It can be seen in Figure8a that the conductivity spectroscopic curve bent strongly starting from a characteristic point. At that bent point, not only the marked dependence of the conductivity on frequency begins, but also the relaxation of the conductivity starts. Geometrically, it can be found as the intersection of two tangent lines to the conductivity spectroscopic curve, namely, one tangent plotted in the low frequency region and the other tangent plotted in the high frequency region. The frequency that corresponds to that characteristic point is referred as a hopping rate (νH), e.g., [27]. Values of the hopping rate for BFTO ceramics were read out from Figure8a and plotted against inverse absolute temperature (the insertion in Figure8a). (a) (b)

Figure 8. (a) Dependence of ac-conductivity on frequency. The insertion shows dependence of the hopping frequency on inverse absolute temperature; (b) dependence of dc-conductivity σ(0)T on inverse temperature.

The expression “hopping” refers to the sudden movement of an electric charge carrier from one location to another in its vicinity. The process of electric current conduction based on the hopping

Materials 2020, 13, x FOR PEER REVIEW 8 of 12

frequency corresponding to the maximum value of the imaginary component of the electric modulus (ν/νmax) known as the conduction relaxation frequency (νmax). As a result of such data treatment, it was found that the profile of the spectroscopic curve of the electric modulus (M″) does not change with temperature. All the M″-curves were superimposed on one so-called primary curve.

(a) (b)

Figure 7. (a) Dependence of normalized modulus (M″/M″max) on so-called frequency (ν/νmax); (b) dependence of frequency (νmax) corresponding to the maximum M″(ν) on inverse temperature.

The analysis of the dependence of the conduction relaxation frequency of BFTO ceramics on the inverse absolute temperature (Figure 7b) allowed to calculate the activation energy of the process EA = 0.570 eV.

3.3. Electric Conductivity Analysis Figure 8a shows the dependence of ac-conductivity (σ) on angular frequency (ω) at various temperatures for Bi6Fe2Ti3O18 ceramics. One can see in Figure 8a that the behavior of BFTO ceramics is typical for ionic materials. It can be seen that in the low-frequency region, the electrical conductivity hardly depends on the frequency (plateau is present) and takes a value equal to the volume conductivity of the sample σ(0). With an increase in the measuring frequency, a strong high-frequency dependence of the electric conductivity of the sample appears. On the other hand, the dc-plateau is temperature dependent—the raise of conductivity is observed while temperature increases. The plot in Figure 8b shows the dependence of conductivity σ(0)T (assuming a hopping Materialsconduction2020, 13 mechanism), 5286 on 1000/T (T is an absolute temperature). 9 of 12

(a) (b)

FigureFigure 8. 8(.a )(a Dependence) Dependence of ofac -conductivityac-conductivity on on frequency. frequency. The The insertion insertion shows shows dependence dependence of of the the hoppinghopping frequency frequency on on inverse inverse absolute absolute temperature; temperature; (b ()b dependence) dependence of ofdc -conductivitydc-conductivityσ (0)σ(0)T Ton on inverseinverse temperature. temperature.

TheThedc expression-conductivity “hopping” activation refers energy to the was sudden found movement to be EDC of= an0.78 electric eV, whereascharge carrier the hopping from one activationlocation to energy another was in found its vicini to bety.E HThe= 0.63process eV. Thisof electric result current suggests conduction that the charge based carriers on the havehopping to overcome the diﬀerent energy barriers while conducting or relaxing [28]. It can be explained in terms of a jump relaxation model [29] which allows for two competing relaxation processes. The ﬁrst is the forward-backward hop of the ion. The second is the relaxation of the crystal lattice in the immediate vicinity of the ion and the creation of a new energetically convenient vacant location for the ion to hop. For most ceramic materials, the dependence of ionic conductivity on frequency can be described by Jonscher’s law of dielectric response [30] as

σ(ω) = σ(0) + Aωn, (3) where σ(0) is the dc-conductivity, A is a constant, and the exponent n is within 0 < n < 1. Experimental data were subjected to the fitting procedure according to Equation (3). One can see from Table1 that fitting quality parameters, namely, χ2 (chi-squared) and R2 (r-squared), state that the fitting quality improves with increasing temperature.

Table 1. Results of ﬁtting the experimental data to the Jonscher equation.

Temperature, σ(0) 108, Exponent n, A 1011, χ2 1015, R2, × 1 ×1 n × [◦C] [Scm− ] [1] [Scm− rad− ] [1] [1] 140 1.3523 0.69896 4.0590 17.007 0.98101 160 2.9900 0.72308 2.9343 7.9279 0.99482 180 6.5479 0.53998 53.609 4.9587 0.99605 200 19.928 0.58696 40.520 1.3978 0.99917

It is worth noting that conductivity is a fundamental physical quantity. Therefore, presentation and analysis of the dielectric data of ionic conductors using the concept of ac-conductivity is preferable. For that reason, an attempt was made to scale the experimental data of ac-conductivity according to the approach given in the literature [31]. The results are shown in Figure9a. Materials 2020, 13, x FOR PEER REVIEW 9 of 12

mechanism does not take place through the conduction band. It occurs through the use of localized energy states within the band gap and it is characteristic for dielectrics, like, e.g., BFTO ceramics. It can be seen in Figure 8a that the conductivity spectroscopic curve bent strongly starting from a characteristic point. At that bent point, not only the marked dependence of the conductivity on frequency begins, but also the relaxation of the conductivity starts. Geometrically, it can be found as the intersection of two tangent lines to the conductivity spectroscopic curve, namely, one tangent plotted in the low frequency region and the other tangent plotted in the high frequency region. The frequency that corresponds to that characteristic point is referred as a hopping rate (νH), e.g., [27]. Values of the hopping rate for BFTO ceramics were read out from Figure 8a and plotted against inverse absolute temperature (the insertion in Figure 8a). The dc-conductivity activation energy was found to be EDC = 0.78 eV, whereas the hopping activation energy was found to be EH = 0.63 eV. This result suggests that the charge carriers have to overcome the different energy barriers while conducting or relaxing [28]. It can be explained in terms of a jump relaxation model [29] which allows for two competing relaxation processes. The first is the forward-backward hop of the ion. The second is the relaxation of the crystal lattice in the immediate vicinity of the ion and the creation of a new energetically convenient vacant location for the ion to hop. For most ceramic materials, the dependence of ionic conductivity on frequency can be described by Jonscher’s law of dielectric response [30] as n  ()   (0)  A , (3) where σ(0) is the dc-conductivity, A is a constant, and the exponent n is within 0 < n < 1. Experimental data were subjected to the fitting procedure according to Equation (3). One can see from Table 1 that fitting quality parameters, namely, χ2 (chi-squared) and R2 (r-squared), state that the fitting quality improves with increasing temperature.

Table 1. Results of fitting the experimental data to the Jonscher equation.

σ(0) × 108, Exponent n, A × 1011, χ2 × 1015, R2, Temperature, [°C] [Scm−1] [1] [Scm−1rad−n] [1] [1] 140 1.3523 0.69896 4.0590 17.007 0.98101 160 2.9900 0.72308 2.9343 7.9279 0.99482 180 6.5479 0.53998 53.609 4.9587 0.99605 200 19.928 0.58696 40.520 1.3978 0.99917

It is worth noting that conductivity is a fundamental physical quantity. Therefore, presentation and analysis of the dielectric data of ionic conductors using the concept of ac-conductivity is preferable. For that reason, an attempt was made to scale the experimental data of ac-conductivity Materialsaccording2020, 13 to, 5286 the approach given in the literature [31]. The results are shown in Figure 9a. 10 of 12

(a) (b)

FigureFigure 9. ( a9). Conductivity(a) Conductivity master master curve curve of BFTO of BFTO ceramics; ceramics; (b) dependence(b) dependence of dc of-conductivity dc-conductivity on theon the reciprocalreciprocal of the of fourththe fourth root root of the of absolutethe absolute temperature temperature for BFTOfor BFTO ceramics. ceramics.

It can be seen in Figure9a that the curves measured at di fferent temperatures and showing the dependence of the normalized specific ac-conductivity (σ/σDC) on the circular frequency divided by σ(0)T (i.e., ionic conductivity for hopping mechanism) lie on the same primary curve. The fact that the ac-conductivity curves recorded at different temperatures are superimposed and their profile remains unchanged, allows us to conclude that the conductivity relaxation mechanism is independent on temperature [25]. An experimental confirmation for conduction taking place according to the hopping mechanism is the linear dependence of conductivity, or electric resistivity in relation to the absolute temperature raised to the power ( 1/4). One can see in Figure9b that the linear dependence is present − but for temperature higher than T 17 C. Below that temperature, the dependence can hardly be ≈ − ◦ fitted with a linear function exhibiting the same slope. Considering the existence of structural defects such as oxygen deficiency, bismuth volatilization, and imperfect arrangement of Ti and Fe ions in the crystal lattice of BFTO ceramics valence transfer may occur between Fe3+ and Fe2+ in order to maintain the electrical neutrality of the system—as reported in literature [12,32]. Therefore, one may suppose that the relaxation was due to the hopping electron between Fe2+ and Fe3+, executing short-range movement. However, taking into consideration the atoms’ positions in the orthorhombic structure, Fmm2 space group [9], one can see that when Fe ions are located at Ti4+ sites, the next neighbor oxygen can be vacant forming a neutral center which relaxes the hopping conduction ions.

4. Conclusions Almost the same value of activation energies were obtained from the analyses of Z” and M” − spectroscopic data. The activation energy corresponding to non-Debye type of relaxation calculated on the base of the complex impedance analysis was found to be EA = 0.573 eV. On the base of complex electric modulus formalisms, the activation energy of conductivity relaxation frequency was found to be EA = 0.570 eV. The ac-conductivity data of BFTO multiferroic ceramics were found to obey Jonscher’s universal power law of dielectric response. A value of the exponents was 0.54 n 0.72. ≤ ≤ The hopping frequency was found to be temperature dependent and the hopping activation energy was found to be EH = 0.63 eV. On the base of conductivity approach, the dc-conductivity activation energy was extracted from the frequency independent plateau region. It was found to be EDC = 0.78 eV. The dc-hopping mechanism was conﬁrmed with the linear dependence of conductivity on the inverse fourth root of temperature (in Kelvin). A diﬀerence between the hopping activation energy (EH) and the dc-conductivity activation energy (EDC) can be ascribed to two competing mechanisms of relaxation appropriate for the jump relaxation model. Materials 2020, 13, 5286 11 of 12

Author Contributions: Conceptualization, D.C. and A.L.-C.; methodology, A.L.-C., D.C., B.G.-G., and W.B.; software, D.C. and W.B.; validation, A.L.-C., B.G.-G., and W.B.; formal analysis, A.L.-C., D.C., B.G.-G., and W.B.; investigation, A.L.-C., D.C., B.G.-G., and W.B.; resources, A.L.-C., D.C., B.G.-G., and W.B.; data curation, A.L.-C., B.G.-G., and W.B.; writing—original draft preparation, A.L.-C., D.C., B.G.-G., and W.B.; writing—review and editing, A.L.-C., B.G.-G., and W.B.; visualization, A.L.-C., B.G.-G., and W.B.; supervision, D.C.; project administration, A.L.-C.; funding acquisition, A.L.-C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the POLISH NATIONAL SCIENCE CENTRE (NCN), grant number N N507 446934. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Hill, N.A. Why are there so few magnetic ferroelectrics? J. Phys. Chem. B 2000, 104, 6694–6709. [CrossRef] 2. Wu, J.; Fan, Z.; Xiao, D.; Zhu, J.; Wang, J. Multiferroic bismuth ferrite-based materials for multifunctional applications: Ceramic bulks, thin ﬁlms and nanostructures. Prog. Mater. Sci. 2016, 84, 335–402. [CrossRef]

3. Aurivillius, B. Mixed bismuth oxides with layer lattices. II Structure of Bi4Ti3O12. Arkiv Kemi 1949, 1, 499–512. 4. Krzhizhanovskaya, M.; Filatov, S.; Gusarov, V.; Pauﬂer, P.; Bubnova, R.; Morozov, M.; Meyer, D. Aurivillius

phases in Bi4Ti3O12/BiFeO3 system: Thermal behaviour and crystal structure. Anorg. Allg. Chem. 2005, 631, 1603–1608. [CrossRef] 5. Lomanova, N.A.; Morozov, M.I.; Ugolkov, V.L.; Gusarov, V.V. Properties of Aurivillius phases in the

Bi4Ti3O12-BiFeO3 system. Inorg. Mater. 2006, 42, 189–195. [CrossRef] 6. Srinivas, K.; Sarah, P.; Suryanarayana, S.V. Impedance spectroscopy study of polycrystalline Bi6Fe2Ti3O18. Bull. Mater. Sci. 2003, 26, 247–253. [CrossRef] 7. Lisi´nska–Czekaj,A.; Lubina, M.; Czekaj, D.; Rerak, M.; Garbarz-Glos, B.; B ˛ak,W. Inﬂuence of processing

conditions on crystal structure of Bi6Fe2Ti3O18 ceramics. Arch. Metall. Mater. 2016, 61, 881–886. [CrossRef] 8. Lisi´nska–Czekaj,A. Fabrication of Bi6Fe2Ti3O18 ceramics by mixed oxide method. Mater. Sci. Forum 2013, 730–732, 100–104. 9. Garcıa-Guaderrama, M.; Fuentes-Montero, L.; Rodriguez, A.; Fuentes, L. Structural Characterization of

Bi6Ti3Fe2O18 Obtained by Molten Salt Synthesis. Integr. Ferroelectr. 2006, 83, 41–47. [CrossRef] 10. Bu´cko,M.; Polnar, J.; Przewo´znik,J.; Zukrowski,˙ J.; Kapusta, C. Magnetic properties of the Bi6Fe2Ti3O18 Aurivillius phase prepared by hydrothermal method. Adv. Sci. Technol. 2010, 67, 170–175. [CrossRef]

11. Lisi´nska–Czekaj,A.; Czekaj, D. Characterization of Bi6Fe2Ti3O18 Ceramics with impedance Spectroscopy. Mater. Sci. Forum 2013, 730–732, 76–81. 12. Bai, W.; Chen, G.; Zhu, J.Y.; Yang, J.; Lin, T.; Meng, X.J.; Chu, J.H. Dielectric responses and scaling behaviors

in Aurivillius Bi6Ti3Fe2O18 multiferroic thin ﬁlms. Appl. Phys. Lett. 2012, 100, 082902. [CrossRef] 13. Bai, W.; Xu, W.F.; Wu, J.; Zhu, J.Y.; Chen, G.; Yang, J.; Chu, J.H. Investigations on electrical, magnetic and

optical behaviors of ﬁve-layered Aurivillius Bi6Ti3Fe2O18 polycrystalline ﬁlms. Thin Solid Films 2012, 525, 195–199. [CrossRef]

14. Lu, J.; Qiao, L.J.; Ma, X.Q.; Chu, W.Y. Magnetodielectric effect of Bi6Fe2Ti3O18 film under an ultra-low magnetic field. J. Phys. Condens. Matter 2006, 18, 4801–4807. [CrossRef] 15. Jartych, E.; Mazurek, M.; Lisińska-Czekaj,A.; Czekaj, D. Hyperfine interactions in some Aurivillius

Bim+1Ti3Fem-3O3m+3 compounds. J. Magn. Magn. Mater. 2010, 322, 51–55. [CrossRef] 16. Lisi´nska–Czekaj,A.; Rerak, M.; Czekaj, D.; Lubina, M.; Garbarz-Glos, B.; B ˛ak,W. Low temperature broad

band dielectric spectroscopy of multiferroic Bi6Fe2Ti3O18 ceramics. Arch. Metall. Mater. 2016, 61, 1101–1106. [CrossRef] 17. Barsoukov, E.; Macdonald, J.R. (Eds.) Impedance Spectroscopy, Theory, Experiment, and Applications, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2005. 18. Boukamp, B.A. Electrochemical impedance spectroscopy in solid state ionics: Recent advances. Solid State Ion. 2004, 169, 65–73. [CrossRef] 19. Abram, E.J.; Sinclair, D.C.; West, A.R. A Strategy for Analysis and Modelling of Impedance Spectroscopy Data of Electroceramics: Doped Lanthanum Gallate. J. Electroceram. 2003, 10, 165–177. [CrossRef] 20. Boukamp, B.A. A linear Kronig-Kramers transform test for immitance data validation. J. Electrochem. Soc. 1995, 142, 1885–1894. [CrossRef] Materials 2020, 13, 5286 12 of 12

21. Lisi´nska-Czekaj,A. Wielofunkcyjne Materiały Ceramiczne na Osnowie Tytanianu Bizmutu; Wydawnictwo Gnome, Uniwersytet Sl´ ˛aski:Katowice, Poland, 2012. (In Polish) 22. ImageJ—A Public Domain Java Image Processing Program. Available online: https://imagej.nih.gov/ij/index. html (accessed on 14 November 2020). 23. Abrantes, J.C.C.; Labrincha, J.A.; Frade, J.R. Representations of impedance spectra of ceramics Part I. Simulated study cases. Mater. Res. Bull. 2000, 35, 955–964. [CrossRef] 24. Hodge, I.M.; Ngai, K.L.; Moynihan, C.T. Comments on the electric modulus function. J. Non-Cryst. Solids 2005, 351, 104–115. [CrossRef]

25. Anantha, P.S.; Hariharan, K. ac Conductivity analysis and dielectric relaxation behaviour of NaNO3–Al2O3 composites. Mater. Sci. Eng. B 2005, 121, 12–19. [CrossRef] 26. Howell, F.S.; Bose, R.A.; Macedo, P.B.; Moynihan, C.T. Electrical Relaxation in a Glass-Forming Molten Salt. J. Phys. Chem. 1974, 78, 639–648. [CrossRef] 27. Almond, D.P.; West, A.R. Mobile ion concentrations in solid electrolytes from an analysis of a.c. conductivity. Solid State Ion. 1983, 9, 277–282. [CrossRef] 28. Reau, J.M.; Jun, X.Y.; Senegas, J.; Le Deit, C.; Poulain, M. Influence of network modifiers on conductivity and relaxation parameters in some series of fluoride glasses containing LiF. Solid State Ion. 1997, 95, 191–199. [CrossRef] 29. Funke, K. Jump relaxation in solid electrolytes. Prog. Solid State Chem. 1993, 22, 111–195. [CrossRef] 30. Jonscher, A.K. The “universal” dielectric response. Nature 1977, 267, 673–679. [CrossRef] 31. Roling, B.; Happe, A.; Funke, K.; Ingram, M.D. Carrier Concentrations and Relaxation Spectroscopy: New Information from Scaling Properties of Conductivity Spectra in Ionically Conducting Glasses. Phys. Rev. Lett. 1997, 78, 2160–2163. [CrossRef] 32. Hunpratub, S.; Thongbai, P.; Yamwong, T.; Yimnirun, R.; Maensiri, S. Dielectric relaxations and dielectric

response in multiferroic BiFeO3 ceramics. Appl. Phys. Lett. 2009, 94, 062904. [CrossRef]

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