Impedance Spectroscopy of Fe and La-Doped BaTiO₃ Ceramics

Abstract

A wide range of the interesting properties of electroceramics Ba_0.996La_0.004Ti_0.999O₃ (BLT4) undoubtedly deserves differentiation and optimization. For this purpose, the corresponding donor oxide dope Fe₂O₃ was introduced in excess quantities into the base ceramics. In this way, an innovative ceramic material with the general formula of Ba_0.996La_0.004Ti_1−yFe_yO₃ (BLTF), for y = 0.001, 0.002, 0.003, 0.004, has been produced. The crystal structure of BLTF ceramics was investigated using X-ray diffraction. The diffraction peaks in XRD confirm the formation of the tetragonal perovskite phase. The electrical properties of BLTF ceramics have been tested using impedance spectroscopy, in the frequency range of 20 Hz–2 MHz and the temperature range of 20–580 °C. To gain absolute certainty on the consistency of the measured data, the obtained impedance spectra were analyzed using the Kramers–Kronig method. The usage of an equivalent circuit, proposed by the authors, allowed grain and grain boundary resistivity to be obtained. Based on the diagram of the natural logarithm of the mentioned resistivity versus the reciprocal absolute temperature, the activation energies of the conductivity processes have been determined. The values of activation energies indicated that the admixture of iron introduced into the BLT4 ceramics played a crucial role in the conductivity of grains and intergranular borders.

1. Introduction

In the family of ceramic intelligent materials, a special place is occupied by ferroelectrics with a perovskite structure type ABO₃ [1,2,3]. One of the representatives of these materials is barium titanate, an important perovskite ferroelectric material, which has been known since 1940 [4,5,6,7]. This fact is undoubtedly related to the wide range of application possibilities and considering the principles of the so-called green technologies, environmentally friendly. There are many methods for obtaining BaTiO₃ ceramics, e.g., the free sintering method, the sol-gel method, or spark plasma sintering, which has been gaining attention in materials science research [8]. Barium titanate has been the most widely studied classical ferroelectric due to its excellent dielectric, ferroelectric, optical, piezoelectric, and other useful properties [9,10,11]. BaTiO₃ finds practical applications in classic capacitors, thermistors, and varistors [12,13,14,15], piezoelectric sensors, and piezoelectric transducers [16,17,18]. Multilayer ceramic capacitors (MLCCs) with a BaTiO₃ ferroelectric as the core material have a low-temperature coefficient of capacitance over a wide temperature range, and are currently the commercial MLCC with the largest demand in the market [19,20]. Ceramic materials based on BaTiO₃ are also used to build ferroelectric random access memories; the so-called FeRAM [21].

The influence of iron ions on the structure, microstructure, and physical properties of BaTiO₃ ceramics has been widely discussed in recent years [22,23,24]. However, there are no reports in the world literature on the influence of iron admixture on the electrical properties of BLT. In the article, the authors draw attention to changes in electrical phenomena. The literature on the subject shows that barium titanate is a relatively easily modified compound, and one of the most effective additives is lanthanum [25]. The authors [26,27,28] indicate that doping barium titanate with lanthanum ions in an amount of 0.4 mol%. (BLT4) leads to obtaining a ceramic material with a particularly high electrical permittivity value and interesting electrical properties. However, the admixture of iron introduced into BaTiO₃ causes ferromagnetic properties and the appearance of high leakage currents [29]. In light of current research, it seems reasonable to dope BT ceramics simultaneously with lanthanum and iron ions. When considering how to introduce both dopants into the crystal lattice, their ion radii should be taken into account. The ion radius of La³⁺ (r = 1.216 Å) is similar to Ba²⁺ (r = 1.47 Å) in size. The surplus charge of La³⁺ ions is compensated, among others, by creating barium vacancies [30]. This substitution method is confirmed by the increase in electrical permeability noted by both the authors of this paper [31] and other scientists [32,33]. Similarly, Fe³⁺ (0.61 Å) does not enter the barium sites due to its radius but replaces the Ti⁴⁺ ions (0.68 Å). The difference in the charge of the discussed ions is compensated for by creating oxygen vacancies [30]. Moreover, introducing La³⁺ ions into the Ba²⁺ positions reduces the number of covalent bonds, and substituting Fe³⁺ ions in place of Ti⁴⁺ ions leads to a reduction in the movement of oxygen octahedra. This may result in a high possibility of regulating the temperature of the ferroelectric phase transition and a significant increase in the dielectric constant of this compound at room temperature (Tr). The influence of iron ions on the structure, microstructure, and physical properties of BaTiO₃ ceramics has been widely discussed in recent years [34,35,36,37]. Nakayama and others pointed out that the Fe-doped BaTiO₃ ceramics are naturally ferromagnetic (FM) [38]. Another work investigated the simultaneous substitution of iron at the A and B sites in BaTiO₃. It was found that when Fe was substituted at the B-site, it improved the coercive field, while at the A-site substitution, sample saturation magnetization increased [38]. However, there is very little information in the world literature on the effect of iron admixture on the electrical properties of BLT.

The authors of the present paper started the investigation on barium titanate modified simultaneously by La³⁺ and Fe³⁺ ions a couple of years ago. The previous paper widely described the influence of simultaneous modifications on electrical and dielectric properties caused by introducing the La³⁺ and Fe³⁺ ions to the crystal lattice [39]. In another paper, the authors described changes in the piezoelectric and elastic properties and increased barium titanate conductivity [40].

In the article, the authors draw attention to changes in electrical phenomena occurring in BLT4 ceramics under the influence of the iron admixture. Iron ions replace Ti sites in the perovskite structure and change the space charge concentration. The dopant modifies the process of accumulation and migration of charged defects and species to the grain boundary [37]. A comprehensive impedance analysis was performed to obtain information about the electrical properties of the tested material. This technique allows for the separation of the individual contribution of grains and grain boundaries in the total impedance spectrum. Also, it allows the examination of the influence of the iron admixture on the dynamics of both bound and free charges inside and in the electrode layers of the ceramic material [41].

2. Materials and Methods

Iron dopant was added to the base material Ba_0.996La0_.004Ti_0.999O₃ (BLT4). The modified ceramic materials with the general formula of Ba_0.996La_0.004Ti_1−yFe_yO₃ (BLTF), for y = 0.1 mol.% (BLTF1), 0.2 mol.% (BLTF2), 0.3 mol.% (BLTF3), and 0.4 mol.% (BLTF4), were obtained using the conventional mixed oxides method. The proper amounts of substrates: BaCO₃, La₂O₃, TiO₂, and Fe₂O₃ (purity ≥ 99%) were weighed in a stoichiometric ratio considering the expected chemical reaction (1):

(1−x)BaCO₃ + (1/2x)La₂O₃ + (1−y)TiO₂ + (1/2y)Fe₂O₃→Ba_1−xLa_xTi_1−yFe_yO₃ + (1−x)CO₂↑

(1)

The stoichiometric formulas of discussed ceramics contain the real amount of ingredients:

Ba_0.996La_0.004Ti_0.999Fe_0.001O₃⇒ BLTF1;

Ba_0.996La_0.004Ti_0.998Fe_0.002O₃⇒ BLTF2;

Ba_0.996La_0.004Ti_0.997Fe_0.003O₃⇒ BLTF3;

Ba_0.996La_0.004Ti_0.996Fe_0.004O₃ ⇒ BLTF4.

The mixture of powders with the addition of ethyl alcohol C₂H₅OH was subjected to grinding in a high-energy ball mill (300 rpm) for 8 h, using zirconium yttrium balls as grinders. The synthesis was conducted at T = 950 °C in a corundum crucible in an air atmosphere for 6 h. The synthesized ceramic materials were then thoroughly ground, pressed and formed into disks with the diameter of d = 10 mm and the thickness of p = 1 mm, under the pressure of p = 30 MPa. BLTF ceramics were sintered at the temperature of T_I = 1250 °C, T_II = 1300 °C, and T_III = 1350 °C for 2 h. All samples were furnace-cooled to room temperature. Basic material parameters, namely real density obtained using the Archimedes method with distilled water and porosity, are summarized in

Table 1. The real density (ρ_r) and open porosity (P) of BLTF ceramics.

Ceramics	ρr [g/cm3]	P [%]
BLTF1	5.68	0.17
BLTF2	5.65	0.18
BLTF3	5.63	0.18
BLTF4	5.61	0.19

. Previous studies have shown that incorporating a 4 mol.% lanthanum dopant significantly affects the density of BaTiO₃ ceramics, causing an increase from the value of 5.39 g/cm³ (pure BaTiO₃) to 5.63 g/cm³ (ceramic containing 0.4 mol% La). This problem was discussed in [42]. Introducing iron ions at concentrations equal to 0.1 and 0.2 mol% La slightly increased the density of the discussed material (Table 1); however, further increasing the dopant concentration led to a decrease in density to a value of 5.61 g/cm³. Similar behavior was observed in other materials based on iron-modified BaTiO₃ ceramics [39]. The obtained real density is approximately equal to 95% of the theoretical density.

For electric measurements, the surfaces of the sintered pellets were ground and polished, followed by the application of a silver conducting paste on two opposite surfaces of the sample. The crystalline structure of the investigated ceramics was studied using the X-ray diffraction technique (XRD). The measurements were performed at room temperature with the use of the X’Pert Pro diffractometer equipped with CuKα_1and2 radiation (40 kV, 30 mA). Diffraction patterns were measured in the step-scan mode in the angular range from 10° to 140° 2θ, with a measuring step of 0.05° and a counting time of 10 s per step. The measured diffraction patterns were fitted using the Rietveld method implemented in the LHPM computer program (version 4.2. Lucas Heights Research Laboratories ANSTO, Sydney, Australia) [43]. SEM studies of the BLTF ceramic samples, as well as analysis of their chemical composition, were performed by scanning electron microscopy SEM, JEOL JSM-7100 TTL, Tokyo, Japan equipped with an energy dispersive X-ray spectrometer (EDS)). The linear intercept length method was used to determine the grain size. The impedance studies were performed using a computerized measuring system, the integral part of which was an Agilent E4980A LCR meter. This system allowed the performance of complex impedance tests as a function of frequency (in the range of 20 Hz–2 MHz) and temperature (in the range from room temperature to 580 °C). The method using the K-K test V1.01 program confirmed the consistency of experimental data based on the Kramers–Kronig method [41,44]. The method is based on the Kramers–Kronig relationship, which is a very useful tool for data validation [45,46,47]. According to B.A. Boukamp [44], the dataset complied with the Kramers–Kronig transformation rules, which mean it must be:

-

casual, which means that the measured response is a consequence only of the applied signal,

-

a linear response, i.e., the system cannot generate an output frequency higher than the input frequency,

-

stable, which means that the system must be unchanged in the time, not continue to oscillate after the excitation is stopped,

-

finite for all frequencies, including ω→0 and ω→∞.

The Kramers–Kronig rules state that the imaginary part of impedance spectroscopy is completely determined by the form of the real part over the frequency range 0 ≤ ω ≤ ∞ and vice versa; the real part of impedance spectroscopy is determined by the imaginary part in the same frequency range. The sentence is described by Equations (2) and (3):

$$Z_{re}\left(\omega \right)=R_{\infty }+\frac{2}{\pi }{\int }_0^{\infty }\frac{{xZ}_{im}\left(x\right)-{\omega Z}_{im}\left(\omega \right)}{x^2-{\omega }^2}dx$$

(2)

$$Z_{im}\left(\omega \right)=\frac{2\omega }{\pi }\int \frac{Z_{re}\left(x\right)-Z_{re}\left(\omega \right)}{x^2-{\omega }^2}dx$$

(3)

where Z_re (ω)—is the real part of impedance obtained from the imaginary part,

Z_im (ω)—is the imaginary part of impedance obtained from the real part.

Based on the values calculated from the presented equation as well as experimentally measured, the residuals could be defined in the following relationship (4) and (5):

$${∆}_{re,i}=\frac{Z_{re,i}-Z_{re}\left({\omega }_i{,a}_k\right)}{\left|Z\left({\omega }_i,a_k\right)\right|}$$

(4)

$${∆}_{im,j}=\frac{Z_{im,J}-Z_{im}\left({\omega }_i{,a}_k\right)}{\left|Z\left({\omega }_i,a_k\right)\right|}$$

(5)

For a good match between the data and the model, the residuals should be randomly distributed around the log(ω) axis, and their value should not exceed 5%.

A detailed analysis of the experimental data was performed using the Zview software (version 3.5, ScribnerAssociates Inc., Southern Pines, NC, USA).

3. Results

3.1. X-ray Diffraction

X-ray diffraction patterns of the investigated materials are presented in Figure 1. First, phase identification was performed using the ICDD (International Centre for Diffraction Data) PDF-4 database. For all samples, XRD diffraction patterns show the presence of a single perovskite phase without any secondary phase peaks. The layout of diffraction lines is characteristic of a tetragonal perovskite crystal structure with space group P4mm (ICDD card number 01-005-0626).

To explain the influence of Fe content on the crystallographic structure, the obtained results were subjected to refinement using the Rietveld method. Crystallographic data from ICDD PDF-4 file no. 01-005-0626 was used to build a theoretical model of the unit cell. By the assumptions of the chemical composition, the Ba positions were taken by La with an appropriate share modeled by the “site occupation” parameter. A similar approach was performed in the case of Ti, which Fe replaced. Lattice parameters calculated from refinement are collected in

Table 2. The values of lattice parameters in BLTF composite.

Composite	a0 = b0 [nm]	c0 [nm]	δT [nm]	V∙10−30 [m3]
BLTF1	0.3993	0.4032	1.0098	64.3
BLTF2	0.3992	0.4031	1.0098	64.2
BLTF3	0.3993	0.4031	1.0095	64.2
BLTF4	0.3993	0.4030	1.0093	64.2

. The parameters of the unit cell practically do not change with the increase of iron dopant. The cell volume of the BLTF1 is VV = 64.3 × 10⁻³⁰ m³ and unit cell volume in BLTF4 is V = 64.2 × 10⁻³⁰ m³. Obtained results show that the introduced dopants (La, Fe) led to the tetragonalization of the crystal lattice, as indicated by the parameter (δ_T). The increase in iron concentration causes a slight decrease in the δ_T parameter, which is caused by a decrease in the c₀ parameter.

3.2. Microstructure and EDS Analysis

The study of Ba_0.996La_0.004Ti_1−yFe_yO₃ ceramics morphology was carried out to determine the effect of Fe³⁺ admixture on the microstructure of the based ceramics. Figure 2 presents SEM images of BLTF ceramic fractures, for the various concentrations of added iron admixture.

The presented images indicated that BLTF ceramic materials are characterized by a densely packed, fine-grained microstructure with well-formed angular grains and a small number of pores. The results agree with the values of obtained density and porosity. The size of the grains increases with the increasing iron dopant concentration from d = 0.5 μm (for BLTF1) to d = 2 μm (for BLTF4). Therefore, the grain size of all discussed ceramics stays significantly lower than those observed in unmodified Ba_0.996La_0.004Ti_0.999O₃. The BLT4 ceramics are characterized by well-shaped large angular grains (d = 5 µm), with a tendency to spiral hexagonal growth [33]. The appearance of the microstructure indicates that, during the fractures, some cracks occurred through the grains (transcrystalline cracks). These kind of fractures were observed in the other materials [48,49,50]. This behavior indicates high mechanical strength in the grain boundaries at the expense of the interior of the grains [51].

The qualitative and quantitative chemical composition of all the discussed BLTF ceramics was tested with the energy-dispersive X-ray spectroscopy method. The obtained spectra are graphically presented in Figure 3, while for the pure compound in this work, see [14].

The spectra confirm the presence of La, Ti, Ba, Fe, and O in all the tested samples. No other element was detected. The graph shows the additional maximum, derived from the graphite layer sprayed on the sample. This maximum was not taken into consideration during the quantitative analysis. The chemical composition of the ceramic materials, determined on the basis of the results of the EDS analysis, remains in good agreement with the assumed stoichiometry. The percentage of individual components in the tested ceramics showed very slight deviations from the theoretically assumed quantities of oxides, and amounted to ±1 wt.%.

3.3. Impedance Analysis

Impedance analysis is a powerful technique that which provides information about the contribution of the microstructure components (such as the grain, grain boundary, and electrode interface) contribution to the electrical conductivity process. The technique was also used to determine the influence of the iron admixture on the electrical properties of the grains and grain boundaries of the BLT4 ceramics. The impedance spectra (SI) were determined at a wide range of temperatures for all of the discussed ceramics. Figure 4 presents frequency dependencies of real and imaginary impedance components at selected temperatures. The obtained characteristics have a smooth course, which indicates properly conducted measurements. In order to be absolutely certain about the consistency of the measured data, the obtained impedance spectra were analyzed using the Kramers–Kronig method. The sample residual spectra calculated for the temperature of 350 °C are shown in Figure 5. The test quality parameter was the coefficient χ² [52].

The residual values of the results analyzed did not exceed 1%, whereas the distribution relative to the frequency axis was accidental, indicating the consistency of the measurement data. Further analysis of the impedance spectra is therefore most sensible.

The frequency characteristics of complex impedance for BLTF1 ceramics are significantly different from others presented. Namely, only one broadened maximum exists on the frequency dependencies of the imaginary part of impedance, and a single plateau is visible on the frequency dependencies of the real part of impedance. Moreover, the increase in the value of the real and imaginary part of the impedance of the discussed samples should be noted, in comparison to the values recorded for the basic ceramics BLT4. The shape of both discussed characteristics changes diametrically for the further considered concentrations of the admixture of iron. In the case of the BLT4 ceramics containing 0.2 mol.% of iron, the outline of the second maximum appears on the logZ_im(f) dependencies, in the range of high frequencies. The maximum becomes more visible for ceramics containing 0.3 and 0.4 mol.% Fe. Moreover, for ceramics with the admixture of iron exceeding 0.1 mol.%, a significant influence of temperature on the shape of the real and imaginary parts of the impedance spectra is visible.

To fully describe the electrically active areas of BLT ceramics doped with iron, the obtained results are presented in the form of the so-called Nyquist graphs (Figure 6), which significantly facilitates the selection of a suitable equivalent circuit.

The dependence of Z_im(Z_re) for ceramics containing 0.1 mol.% Fe obtained at different temperatures has the shape of deformed, symmetrical circles, the centers below the axis representing the real part of the impedance. The shape of the Nyquist chart, as well as the Z_re(f) and Z_im(f) dependencies pointing at the circuit consisting of single serially connected R CPE elements, is the best equivalent circuit to describe the electrical behavior of BLTF1 (Figure 7).

The CPE element has an impedance response that displays non-ideal frequency dependent properties and a constant phase over the entire range of frequencies. The impedance (Z) of CPE is defined by the CPE constant T (CPE-T) and CPE index p (CPE-P), which are parameters given by the following relation (6):

$$Z=\frac{1}{\left[T{\left(j\omega \right)}^P\right]}$$

(6)

The CPE is identical to a capacitance component when the exponent p equals 1, and to a resistance component when p = 0 [53,54].

The complex impedance in the proposed electrical equivalent circuit is expressed by Equation (7):

$$Z=\frac{R}{1+TR{\left(j\omega \right)}^P}$$

(7)

The result of the fitting procedure for several selected temperatures is shown in Figure 8. All fitted parameters for BLTF1 at selected temperatures are presented in

Table 3. Parameters of the components of the applied electrical circuit for the impedance response of the ceramic BLTF1.

BLTF1
T [°C]		402	452	502	552
R	Value [Ω]	1.626 × 106	504,400	171,000	58,967
	Relative error [Ω]	29,415	2158.4	235.94	98.275
	Absolute error [%]	1.809	0.42791	0.13798	0.18911
CPE-T	Value [F]	4.314 × 10−9	6.427 × 10−9	8.676 × 10−9	1.0661 × 10−8
	Relative error [F]	5.7289 × 10−11	4.0278 × 10−11	2.9931 × 10−11	8.5592 × 10−11
	Absolute error [%]	1.3281	0.62668	0.34497	0.80285
CPE-P	Value [a.u.]	0.79131	0.76821	0.75357	0.74612
	Relative error [a.u.]	0.0011	0.0005	0.0003	0.0006
	Absolute error [%]	0.14	0.07	0.09	0.08
χ2		9.8 × 10−4	4.7 × 10−4	1.2 × 10−4	4.7 × 10−4

, together with the errors of the fits for each parameter.

The obtained bulk resistance has been presented as lnR(1/T) dependence. The dependence is linear, which points the activation character of the conduction process (Figure 9). Based on the Arrhenius formula, the activation energies (E_a) of the process in question were determined. The value is equal to 1.03 eV.

In the case of the BLT4 ceramics modified with 0.2 mol.% of iron, the outline of the second semi-circle appears on Z_im(Z_re) dependencies in the high frequency range, which is connected with the electrical response of grains strongly suppressed by the electrical response of the grain boundaries With the further increase of iron concentration, the discussed semicircle, associated with the electrical response of grains, becomes more visible on the characteristics of Z_im(Z_re). Nevertheless it remains much smaller than the semi-circle representing the electrical response of the grain boundaries. The shapes Z_im(Z_re) dependencies pointing at the circuit consisting of two serially connected RC elements is the best equivalent circuit to describe the electrical behavior of BLT4 ceramics modified by iron ions. However, taking into consideration the value of the χ² coefficient, the capacitance of both RC elements is replaced by CPE constant phase element (Figure 10).

The complex impedance in the proposed electrical equivalent circuit (Figure 10) is expressed by the Equation (8):

$$Z\left(\omega \right)=Z_G+Z_{GB}$$

(8)

where (9), (10):

$$Z_G=\frac{R_G}{1+T_G{R}_G{\left(j\omega \right)}^{P_G}}$$

(9)

$$Z_{GB}=\frac{R_{GB}}{1+T_{GB}{R}_{GB}{\left(j\omega \right)}^{P_{GB}}}$$

(10)

The modification has significantly reduced the value of the χ² coefficient, which means an improvement in the fitting quality.

The use of the above-described system allowed us to distinguish the contribution of grains and grain boundaries to electrical conductivity. The example result of the fitting procedure for BLTF4 ceramics at several selected temperatures is shown in Figure 11. All fitted parameters for BLTF2, BLTF3, and BLTF4 at selected temperatures are presented in

Table 4. Parameters of the components of the applied electrical circuit for the impedance response of the ceramic BLTF2, BLTF3, and BLTF4.

BLTF2
T [°C]		552	502	452	402
RG	Value [Ω]	987	2086	5609	18,602
	Relative error [Ω]	8.07	23.39	103.08	587.2
	Absolute error [%]	0.82	1.12	1.83	3.15
CPEG-T	Value [F]	1.81 × 10−9	8.51 × 10−10	6.26 × 10−10	5.19 × 10−10
	Relative error [F]	1.11 × 10−10	4.99 × 10−11	4.47 × 10−11	5.11 × 10−11
	Absolute error [%]	6.13	5.86	7.10	9.84
CPEG-P	Value [a.u.]	0.844	0.896	0.923	0.942
	Relative error [a.u.]	0.004	0.004	0.005	0.008
	Absolute error [%]	0.46	0.44	0.58	0.88
RGB	Value [Ω]	8299	28,812	109,150	381,300
	Relative error [Ω]	34.30	189.61	1614.4	15,156
	Absolute error [%]	0.41	0.66	1.48	3.97
CPEGB-T	Value [F]	5.11 × 10−8	5.45 × 10−8	4.57 × 10−8	2.52 × 10−8
	Relative error [F]	1.27 × 10−9	1.65 × 10−9	2.01 × 10−9	2.00 × 10−9
	Absolute error [%]	2.50	3.04	4.39	7.93
CPEGB-P	Value [a.u.]	0.749	0.723	0.711	0.745
	Relative error [a.u.]	0.002	0.003	0.005	0.009
	Absolute error [%]	0.31	0.43	0.69	1.33
χ2		0.00009	0.00041	0.0011	0.0041
BLTF3
RG	Value [Ω]	459	798	2420	9409
	Relative error [Ω]	0.70	1.29	3.56	16.19
	Absolute error [%]	0.15	0.16	0.15	0.17
CPEG-T	Value [F]	2.14 × 10−10	1.87 × 10−10	1.59 × 10−10	1.54 × 10−10
	Relative error [F]	1.31 × 10−12	7.71 × 10−12	3.01 × 10−12	2.00 × 10−12
	Absolute error [%]	6.08	4.12	1.89	1.30
CPEG-P	Value [a.u.]	0.945	0.957	0.973	0.985
	Relative error [a.u.]	0.004	0.003	0.001	0.001
	Absolute error [%]	0.40	0.27	0.13	0.09
RGB	Value [Ω]	3285	6436	20,643	90,007
	Relative error [Ω]	5.96	14.16	54.61	360.5
	Absolute error [%]	0.18	0.22	0.26	0.40
CPEGB-T	Value [F]	1.28 × 10−7	1.21 × 10−7	1.03 × 10−7	7.58 × 10−8
	Relative error [F]	1.29 × 10−9	1.31 × 10−9	1.06 × 10−9	8.48 × 10−9
	Absolute error [%]	1.003	1.08	1.03	1.12
CPEGB-P	Value [a.u.]	0.802	0.798	0.794	0.799
	Relative error [a.u.]	0.001	0.001	0.001	0.001
	Absolute error [%]	0.12	0.13	0.14	0.18
χ2		8.33 × 10−5	0.00012	0.00014	0.00025
BLTF4
RG	Value [Ω]	523	1142	2996	9559
	Relative error [Ω]	0.87	1.87	3.79	21.46
	Absolute error [%]	0.17	0.16	0.13	0.22
CPEG-T	Value [F]	1.02 × 10−10	1.22 × 10−10	1.38 × 10−10	1.33 × 10−10
	Relative error [F]	6.7 × 10−12	4.25 × 10−12	2.31 × 10−12	2.12 × 10−12
	Absolute error [%]	6.5	3.47	1.72	1.59
CPEG-P	Value [a.u.]	0.971	0.965	0.964	0.974
	Relative error [a.u.]	0.004	0.002	0.001	0.001
	Absolute error [%]	0.42	0.26	0.12	0.11
RGB	Value [Ω]	5884	16,566	50,118	214,680
	Relative error [Ω]	5.72	21.21	75.61	1290
	Absolute error [%]	0.097	0.13	0.15	0.80
CPEGB-T	Value [F]	4.14 × 10−8	3.5 × 10−8	2.82 × 10−8	2.19 × 10−8
	Relative error [F]	2.75 × 10−10	2.49 × 10−10	1.64 × 10−10	2.45 × 10−10
	Absolute error [%]	0.67	0.70	0.58	1.21
CPEGB-P	Value [a.u.]	0.797	0.796	0.812	0.819
	Relative error [a.u.]	0.0005	0.001	0.0006	0.001
	Absolute error [%]	0.007	0.13	0.073	0.157
χ2		4.11 × 10−5	7.43 × 10−5	5.99 × 10−5	0.0002

, together with the errors of the fits for each individual parameter.

The obtained grain and grain boundary resistance in individual temperatures allowed us to determine the dependence of lnR_G(1/T) and lnR_GB(1/T) as shown in Figure 12.

All presented dependencies possess linear character, which has allowed us to determine the activation energy. The activation energy connected with the electrical conductivity of the grain boundary stays practically unchanged with the increase of iron concentration. The activation energy of grains is slightly less than the activation energy associated with the conductivity process in grain boundaries. Noteworthy is its sharp decline for the highest concentration of iron admixture. When interpreting the obtained impedance results, we must consider the direct impact of iron ions on the conductivity processes and the indirect impact, i.e., changes in the microstructure of the modified materials. The admixture of iron caused an increase in the grain size. In the undoped BaTiO₃ material, increasing the grain size causes a significant decrease in both the grain resistance and the activation energy value [55]. In the case of the materials described in this work, the changes in activation energy are much more complicated. Namely, the admixture of iron causes an increase in the activation energy from the value of 0.87 eV (for the base Ba_0.996La_0.004Ti_0.999O₃ ceramics [33]) to the value of 1.03 eV (reported for ceramics containing 0.2 mol.% of iron). A further increase in iron content reduces the activation energy, but its value does not reach the initial value. However, the grain resistance shows an increasing tendency. The phenomena of electric charge transport are undoubtedly also influenced by the degree of porosity of the ceramic material. Based on the research presented by the authors of the work [56], significant changes in the values of electrical permittivity and loss tangent are visible only when the pore content exceeds 11%. These values are related to the commonly known relationships with complex impedance. Therefore, considering slight differences in the porosity of the materials presented in this paper, it can be assumed that the influence of porosity is negligible.

Considering the valence of iron ions and their ionic radius, it was assumed that they substitute in place of Ti⁴⁺ ions. The results of temperature tests of the tangent dependence of the loss angle, widely discussed in the paper [39], clearly confirmed the correctness of these assumptions. Namely, the tangent of the loss angle decreased with the increased concentration of iron ions. This change can be attributed to the decreasing number of unstable Ti⁴⁺ ions because Fe³⁺ occupying the position of Ti⁴⁺ ions reduces the possibility of changing the oxidation state of Ti⁴⁺ ions [57]. The reduction is responsible for the change in the concentration of space charge. The main reservoir of space charge is the grain boundaries. Therefore, changing the concentration of space charge should significantly change their resistance. Such changes occur in the case of discussed ceramic materials. The research results presented in this paper show that the resistance of grain boundaries increases rapidly. For example, grain boundary resistance increases at a temperature of 402 K from 26,620 Ω for unmodified ceramics [33] to 214,680 Ω for ceramics with 0.4 mol% Fe³⁺ ions. The grain resistance increase is another confirmation of the proposed explanation’s correctness. At a temperature of 402 K, the grain resistance changed from 2701 Ω for pure ceramics [33] to 9559 Ω for BLTF 4 ceramics.

Considering the previously published results of dielectric and electrical measurements [39] and the slight PTCR effect observed, the most promising material is BLT1 ceramics.

The research widely discussed in the present paper revealed that these ceramics are homogeneous from the point of view of conductivity—there is no distinction between grain conductivity and grain boundaries. This fact may be an additional advantage in the previously discussed applications.

4. Conclusions

The analysis of the microstructure made it possible to reveal a fine microstructure with well-formed, angular, and homogeneous crystalline grains containing dissolved donor admixtures, which confirmed that the sintering conditions used promoted the synthesis of the material. The tendency of the grains’ growth was observed with the increasing concentrations of the Fe³⁺ admixture. The elements of the microstructure, namely grains, and intergranular borders contribute to electrical conductivity. To determine the effect of iron on the electrical properties of BLT4 ceramics, impedance spectroscopy was used as a function of temperature, which confirmed the thesis of the complexity the mechanism of electric charge transport in the tested ceramic materials. On the basis of Nyquist’s graphs, the mutual overlap of two semi-circles connected with two microstructure components, namely grains and grain boundaries, was observed in the case of BLT4 ceramics containing 0.2, 0.3, and 0.4 mol.% of iron admixture. The dominant semi-circle is associated with the electrical response of the grain boundaries. The energy of activation of the conductivity of grains is smaller than the values of the activation energy associated with conduction in the grain boundary.