La³⁺/Bi³⁺ Co-Doping in BaTiO₃ Ceramics: Structural Evolution and Enhanced Dielectric Properties

Abstract

La³⁺/Bi³⁺ co-doped BaTiO₃ ceramics were synthesized via ball milling followed by heat treatment at 1200 °C according to the Ba_1−3xLa_2xTi_1−3xBi_4xO₃ formula, with dopant levels ranging from x = 0.0 to 0.006. X-ray diffraction and Rietveld refinement confirmed a ferroelectric tetragonal phase for all compositions, with the highest tetragonality (c/a = 1.009) observed for x = 0.001 exceeding that of pure BaTiO₃ (1.0083). High-resolution electron microscopy analysis revealed faceted particles with mean sizes between 362.5 nm and 488.3 nm. Low-doped samples (x = 0.001 and 0.002) exhibited higher permittivity than undoped BaTiO₃, with the maximum dielectric constant (ε_r = 2469.0 at room temperature and 7499.7 at the Curie temperature) recorded for x = 0.001 at 1 kHz. At x = 0.006, minimal permittivity variation indicated a stable dielectric response. A decrease in the Curie temperature was observed with increasing doping levels, indicating a progressive tendency toward the cubic phase. Critical exponent γ values (0.94–1.56) indicated a sharp phase transition for low-doped samples and a diffuse transition for highly doped BaTiO₃. These results showed that La³⁺/Bi³⁺ co-doping effectively tunes the structural and dielectric properties of BaTiO₃ ceramics.

1. Introduction

Tetragonal barium titanate (BaTiO₃) is a ferroelectric perovskite and one of the most significant ceramic materials in electronic applications. At ambient conditions, BaTiO₃ exhibits a tetragonal structure, which transitions to a cubic phase at approximately 120 °C. This transformation has been extensively investigated due to its connection with the change from ferroelectric to paraelectric behavior [1,2,3,4]. The crystal structure of BaTiO₃ plays a critical role in determining its electrical properties. Notably, the substantial dielectric constant exhibited by BaTiO₃ correlates directly with the degree of tetragonal distortion in its crystal structure [5,6].

The ferroelectric properties of BaTiO₃ allow it to sustain a stable electric polarization, which can be reversed by an external electric field. This characteristic makes it essential for capacitors, multilayer ceramic capacitors (MLCCs), and energy storage applications [7]. For MLCC applications, dielectric materials must combine high electrical insulation with a substantial dielectric constant at room temperature, while maintaining stability across operational temperature ranges. However, the temperature-dependent nature of the dielectric properties of BaTiO₃ and the sharp phase transition at its Curie temperature may affect its performance in applications requiring stable dielectric behavior across wide temperature ranges, motivating efforts for property optimization through controlled chemical modification [8,9,10].

Ionic substitution has emerged as the primary strategy for enhancing BaTiO₃ properties, where the partial replacement of Ba²⁺ or Ti⁴⁺ ions with small concentrations of appropriately sized dopants can induce beneficial structural and microstructural modifications [11,12]. While individual doping with various ions has shown improvements, single-dopant systems often involve competing effects between different properties depending on the specific dopant, concentration, and processing conditions [13]. These limitations have motivated research into co-doping strategies that could potentially overcome single-dopant constraints.

The incorporation of rare-earth ions as dopants has been extensively studied to enhance dielectric performance of BaTiO₃ ceramic materials [14,15]. The electrical behavior of BaTiO₃ is highly influenced by the specific lattice sites these ions occupy within its structure [16]. Rare-earth ions with a greater ionic radius, such as Nd³⁺ and La³⁺, tend to replace Ba²⁺, exhibiting donor-like characteristics, whereas smaller ions like Yb³⁺ preferentially substitute Ti⁴⁺, resulting in acceptor-type doping [17,18].

According to the literature, trivalent species like La³⁺ and Bi³⁺ exhibit dual donor-acceptor behavior in BaTiO₃, with their ionic radii governing the ultimate material characteristics [11,13]. La³⁺ and Bi³⁺ exhibit high solubility in BaTiO₃ and are frequently used to improve its dielectric properties [19,20,21,22]. The valence state and ionic radii of La³⁺ (1.032 Å for a 6 coordination number, CN, or 1.36 Å for CN = 12) and Bi³⁺ (1.03 Å for a CN = 6) are intermediate between those of Ba²⁺ (1.61 Å for a CN = 12) and Ti⁴⁺ (0.605 Å for CN = 6) [23], indicating that La³⁺ and Bi³⁺ can substitute either Ba²⁺ or Ti⁴⁺ sites depending on the Ba/Ti molar ratio [24,25].

Doping BaTiO₃ with La³⁺ or Bi³⁺ modifies the ferroelectric-to-paraelectric phase transition, commonly accompanied by a Curie temperature reduction. This transformation can be tailored to obtain desired material properties [19].

Despite extensive research on individual La³⁺ or Bi³⁺ doping of BaTiO₃, systematic investigation of their combined effects, through controlled co-doping, remains limited. Previous studies have primarily focused on single-dopant systems, leaving a significant knowledge gap regarding the potential synergistic effects, optimal co-doping ratios, and underlying mechanisms of simultaneous La³⁺/Bi³⁺ incorporation. Understanding these aspects is crucial for developing next-generation BaTiO₃-based materials with improved properties for advanced electronic applications.

In this study, the effects of controlled La³⁺/Bi³⁺ co-doping on the structural evolution and dielectric properties of BaTiO₃ ceramics are investigated. The formula Ba_1−3xLa_2xTi_1−3xBi_4xO₃ with varied dopant concentrations was employed to explore the potential synergistic effects of combining these two dopant ions. The synthesis was accomplished through solid-state processing using ball milling and controlled heat treatment, providing a scalable route for producing co-doped powders [26]. The characterization included structural analysis through X-ray diffraction and Rietveld refinement, microstructural evaluation via scanning electron microscopy, and dielectric measurements as a function of temperature and frequency.

2. Materials and Methods

BaTiO₃ ceramics co-doped with La³⁺ and Bi³⁺ were synthesized according to the formula Ba_1−3xLa_2xTi_1−3xBi_4xO₃, with doping concentrations of x = 0.0, 0.001, 0.002, 0.004, and 0.006. The synthesis process involved ball milling followed by heat treatment. The precursors—barium carbonate (BaCO₃, 99.0% purity, CAS: 513-77-9, mean particle size d_p = 7.69 µm), titanium dioxide (TiO₂, 99.0% purity, CAS: 13463-67-7, d_p = 0.76 µm), lanthanum oxide (La₂O₃, 99.9% purity, CAS: 1312-81-8, d_p = 5.67 µm), and bismuth oxide (Bi₂O₃, 99.999% purity, CAS: 1304-76-3, d_p = 15.03 µm)—were dried at 200 °C for 24 h to remove moisture. The dried precursors for each composition were placed in a cylindrical polypropylene container (44.8 mm diameter, 127.7 mm height) with zirconia balls of three different diameters (2.95, 5.05, and 6.48 mm) and acetone as a dispersion medium to improve homogeneity. A ball-to-powder weight ratio of 15:1 was used, with 5 g of powder precursors per batch. The Gaudin–Schumann equation was used to calculate the optimal ball charge distribution for the milling process [27].

$$Y=100\times {\left({\displaystyle \frac{d}{d_{max}}}\right)}^{3.8}$$

(1)

where Y is the cumulative contribution of a ball of diameter d relative to the largest ball d_max. The mixture was milled for 6 h at a working speed (W_s) of 140 rpm, determined from:

$$W_s= {\displaystyle \frac{(70\% C_s)}{100}}$$

(2)

$$C_s={\displaystyle \frac{42.3}{\sqrt{D}}}$$

(3)

where C_s is the critical speed (rpm) and D is the container diameter (m).

After milling, powders were allowed to settle for 24 h and dried under ambient conditions. They were then calcined in an Al₂O₃ crucible at 1200 °C for 5 h with heating and cooling rates of 4 °C/min. Crystalline phases were analyzed by X-ray diffraction (XRD, Inel Equinox 2000, Co Kα1, λ = 1.789 Å, 30 kV, 20 mA, 110° curved detector, resolution 0.095° FWHM, Artenay, France). Rietveld refinements were performed using FullProf 4.2 [28] with a pseudo-Voigt profile, and refinement quality was assessed via R_wp, R_exp, and χ² [29].

Powder morphology and particle size were analyzed using high-resolution scanning electron microscopy (HRSEM, JEOL 6701F, JEOL Ltd., Tokyo, Japan) combined with image processing techniques. Feret diameter measurements were performed using ImageJ 1.54p software to determine the mean particle diameter [30].

Green pellets were prepared by placing the powders directly in an 8 mm die. A drop of acetone was added to improve compaction, and the pellets were then uniaxially pressed at 35 kN, producing pellets with a thickness of 3 mm. The green pellets were sintered in air atmosphere at 1250 °C for 6 h with heating and cooling rates of 4 °C/min. Surfaces were polished with SiC papers up to 2000 grit and ultrasonically cleaned in acetone. A commercial silver-platinum (Ag-Pt) paste was applied to both opposite surfaces as electrodes and cured at 200 °C for 30 min; subsequently, a thin platinum wire was attached to each surface for electrical contact. Capacitance measurements were performed using an LCR meter (East Tester ET4410) (Hangzhou, China) at 1 kHz and 100 kHz from 25 °C to 200 °C. The Curie temperature (T_c) was determined from the maximum of the dielectric constant vs. temperature curves, while the Curie–Weiss temperature (T₀), the Curie constant (C), the critical exponent of nonlinearity (γ), and the Curie-like constant (C’) were obtained from the Curie–Weiss law and its modified form.

3. Results and Discussion

3.1. X-Ray Diffraction Analysis

The crystal structures of the ceramic samples with compositions x = 0.0, 0.001, 0.002, 0.004, and 0.006 were analyzed by X-ray diffraction (XRD). Figure 1 presents the diffraction patterns obtained for BaTiO₃ powders. In all compositions, the XRD patterns revealed a crystalline phase consistent with that of pure BaTiO₃ in its tetragonal phase at room temperature (JCPDS 96-152-5438), with characteristic diffraction peaks at 2θ ≈ 26.12°, 37.06°, 45.80°, 53.49°, 60.30°, 66.67°, 78.56°, 84.42°, 89.56°, and 90.44°. No secondary phases were detected, indicating the formation of a single-phase tetragonal structure.

Figure 2 shows expanded views of the XRD patterns in the 2θ ranges of 52–54.5° and 88.5–92°. The crystallinity of BaTiO₃ was identified by evaluating peak intensity and the characteristic splitting of the (200) reflection from cubic into tetragonal (200) and (002) components. As shown in Figure 2a, all samples exhibited the (200)/(002) peak splitting, confirming the presence of a tetragonal ferroelectric phase in the BaTiO₃ crystal structure. However, for the sample with x = 0.006, a reduction in the intensity of the (002) peak was observed, indicating a progressive tendency toward the cubic phase with increasing dopant content.

Figure 2b reveals a decrease in the intensity of the peak corresponding to the (103) plane with increasing dopant content. These results align with literature reports indicating that the tetragonal-to-cubic phase transition, associated with the ferroelectric–paraelectric transformation, occurs at room temperature in BaTiO₃ ceramics co-doped with high concentrations of La³⁺ or Bi³⁺ [31,32].

Changes in unit cell volume and lattice parameters (a and c) were evaluated to investigate the structural evolution of Ba_1−3xLa_2xTi_1−3xBi_4xO₃ solid solutions.

Table 1. Structural parameters from Rietveld refinement of La³⁺/Bi³⁺ co-doped BaTiO₃ powders (0.0 ≤ x ≤ 0.006).

x	Lattice Parameters		Tetragonality c/a	Parameters Characterizing the Refinement Quality		Unit Cell Volume (Å3)
	a = b (Å)	c (Å)		χ2	Rp (%)
0.0	3.9947	4.0279	1.0083	1.80	2.88	64.2757
0.001	3.9863	4.0220	1.0090	1.49	2.87	63.9119
0.002	3.9807	4.0153	1.0087	1.69	2.80	63.6263
0.004	3.9798	4.0096	1.0074	1.58	3.16	63.5073
0.006	3.9772	4.0050	1.0070	1.56	3.89	63.3516

presents the refined lattice parameters along with the reliability factors, including the goodness of fit (χ²) and profile R-factor (R_p), used to evaluate refinement accuracy. Rietveld refinement confirmed that all compositions within the range 0.0 ≤ x ≤ 0.006 crystallized in a pure tetragonal phase. A slight reduction in lattice parameters was observed in the ferroelectric tetragonal BaTiO₃ samples co-doped with La³⁺ and Bi³⁺ at concentrations between 0.001 ≤ x ≤ 0.006, in comparison with the undoped BaTiO₃ (x = 0.0). This reduction was associated with partial La³⁺ and Bi³⁺ incorporation into the BaTiO₃ lattice. La³⁺ and Bi³⁺ ionic radii are considerably larger than that of Ti⁴⁺ but smaller than Ba²⁺, which significantly influenced the structural modifications. At equilibrium, La³⁺ and Bi³⁺ are expected to primarily substitute Ba²⁺ sites. Given their difference in valence, this substitution is predominantly compensated by cation vacancy formation [25,33,34]. Since the dopant ionic radii are smaller than Ba²⁺, lattice parameter contraction was expected. Additionally, the introduction of dopants and associated vacancies increases structural disorder in the perovskite lattice. This disorder progressively destabilizes the tetragonal phase, explaining the observed reduction in tetragonality at higher doping levels.

Table 1 (fourth column) presents the tetragonality ratio (c/a) evolution, a key parameter related to the ferroelectric behavior of the ceramic. A value of 1 signifies cubic symmetry, indicating the absence of ferroelectricity characteristic of the tetragonal phase. An initial tetragonality enhancement was observed for x = 0.001, where the ratio rose from 1.0083 in the undoped sample to 1.009. This value surpasses the minimum tetragonality threshold (1.008) required for multilayer ceramic capacitors (MLCCs). However, as the dopant concentration increased (x = 0.004 and x = 0.006), the tetragonality ratio declined. This trend is consistent with the XRD data in Figure 2, which suggests the onset of a phase transition toward the cubic structure. Specifically, this transition is evidenced by a progressive reduction in the intensity of a characteristic BaTiO₃ peak at 2θ ≈ 53.49°, typically associated with the tetragonal phase.

The unit cell volume (V) evolution for each sample is reported in Table 1. A reduction in unit cell volume was observed for all doped compositions relative to the undoped BaTiO₃ (x = 0.0), which can be attributed to the decrease in lattice parameters, as previously discussed. This trend agrees with literature data on BaTiO₃ doped with La³⁺ or Bi³⁺, where a gradual shift in diffraction peaks toward higher 2θ angles has been correlated with a reduction in unit cell volume [35,36].

3.2. Scanning Electron Microscopy

Figure 3 presents the HRSEM micrographs of BaTiO₃ ceramics co-doped with La³⁺ and Bi³⁺, calcined at 1200 °C for 5 h. The micrographs were obtained at magnifications of 10,000× and 20,000×. The microstructure of the BaTiO₃ powders with doping levels of x = 0.0, 0.001, 0.002, 0.004 and 0.006 revealed agglomerates of particles with varying sizes and morphologies. The results indicated that the incorporation of La³⁺ and Bi³⁺ at these concentrations did not produce significant alterations in the microstructure. In general, the particles exhibited a faceted morphology, a characteristic commonly observed in both doped and undoped BaTiO₃ powders [13].

The characterization of individual particles was performed using microscopy techniques, enabling visualization and dimensional analysis. ImageJ 1.54p software was employed to measure mean particle size via the Feret diameter method, offering precise dimensional data for individual particles [30]. The Feret diameter represents the linear distance between two parallel lines tangential to opposite sides of a particle, measured along a specific orientation. Since this value can vary depending on the direction, measurements were taken in two perpendicular directions to obtain a representative average. A total of 400 particles were analyzed from HRSEM micrographs corresponding to samples within the compositional range 0.0 ≤ x ≤ 0.006. The results, including mean particle diameter and standard deviation for each composition, are reported in

Table 2. Particle size distribution parameters of La³⁺/Bi³⁺ co-doped BaTiO₃ powders (0.0 ≤ x ≤ 0.006).

x	Mean (nm)	Standard Deviation (nm)	Aspect Ratio (AR)
0.0	362.5	143.8	0.7725
0.001	450.8	145.2	0.7628
0.002	457.0	132.7	0.8128
0.004	470.5	162.0	0.7895
0.006	488.3	171.2	0.7836

.

The increase in mean particle size observed across the samples suggests that particle growth was promoted as the dopant concentration increased. Although previous studies have reported a reduction in particle size with increasing La³⁺ content in BaTiO₃ [37], other reports indicate that Bi³⁺ doping at low concentrations results in grain growth, consistent with the trend observed in this work [20]. Thus, the particle growth observed in these samples appears to be primarily associated with the incorporation of Bi³⁺.

Particle shape can be quantitatively described through dimensionless parameters known as shape factors, which include metrics such as axial lengths, areas, perimeters, and geometric moments. One commonly used shape factor is the aspect ratio, defined as the ratio between the longest and shortest diameters of a particle, and is calculated according to the following expression:

$${\mathrm{A}}_{\mathrm{R}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{L}}}{{\mathrm{d}}_{\mathrm{S}}}}$$

(4)

where A_R denotes the aspect ratio, d_L corresponds to the largest diameter, and d_S to the smallest diameter. The aspect ratio values are listed in Table 2, demonstrating that doped ceramics maintain relatively stable aspect ratios with increasing dopant content.

Furthermore, Figure 4 shows the elemental distribution maps corresponding to the composition with x = 0.006. These results provide additional evidence of uniform dopant incorporation within the BaTiO₃ lattice.

To confirm the successful incorporation of La³⁺ and Bi³⁺ dopants and verify the stoichiometry of the synthesized ceramics, semi-quantitative EDS analysis was performed for all compositions.

Table 3. Semi-quantitative EDS analysis of La³⁺/Bi³⁺ co-doped BaTiO₃ powders (0.0 ≤ x ≤ 0.006).

x	Ba (at.%) Exp./Theo.	Ti (at.%) Exp./Theo.	O (at.%) Exp./Theo.	La (at.%) Exp./Theo.	Bi (at.%) Exp./Theo.
0.0	20.26/20.00	19.37/20.00	60.37/60.00	0.00/0.00	0.00/0.00
0.001	20.13/19.88	21.51/19.88	58.18/59.76	0.07/0.04	0.11/0.08
0.002	18.76/19.76	21.67/19.76	59.26/59.52	0.10/0.08	0.21/0.16
0.004	17.84/19.52	20.18/19.52	61.46/59.05	0.14/0.16	0.38/0.32
0.006	19.01/19.28	21.48/19.28	58.92/58.58	0.18/0.24	0.41/0.47

presents the elemental composition results, comparing experimental atomic percentages with theoretical values calculated from the formula Ba_1−3xLa_2xTi_1−3xBi_4xO₃. The experimental values are in good agreement with the theoretical stoichiometry, confirming successful dopant incorporation into the BaTiO₃ structure.

3.3. Electrical Properties

The dielectric properties were evaluated using capacitance measurements at fixed frequencies of 1 kHz and 100 kHz. The dielectric constant was calculated using the following formula [38]:

$${\epsilon }_r={\displaystyle \frac{Cd}{{\epsilon }_{0}A}}$$

(5)

where d represents the pellet thickness, C is the capacitance, ε₀ is the permittivity of free space, and A is the pellet area. The opposite faces of the sintered pellets were painted with silver-platinum paint, and a thin wire of platinum was placed on each side to act as electrodes to determine the electrical properties (Figure 5).

The variation in the dielectric constant as a function of temperature was evaluated in the range of 25 °C to 200 °C at 1 kHz, and the results are presented in Figure 6a and

Table 4. Dielectric parameters for La³⁺/Bi³⁺ co-doped BaTiO₃ ceramics at 1 kHz and 100 kHz.

x	1 kHz			100 kHz
	εr (T25 °C)	εr (Tc)	Tc (°C)	εr (T25 °C)	εr (Tc)	Tc (°C)
0.0	827.8	2919.2	105	663.0	2785.6	114
0.001	2469.0	7499.7	103	1822.4	4434.6	114
0.002	1740.3	4338.8	99	1457.4	3718.7	114
0.004	634.0	1267.6	94	435.6	783.4	101
0.006	334.8	442.4	95	234.1	367.1	99

. The dielectric permittivity as a function of temperature clearly illustrates significant differences in behavior between samples with low and high doping concentrations. The pure BaTiO₃ ceramic sample (x = 0.0) and low-doped samples (x = 0.001, 0.002) displayed sharp peaks, indicating a well-defined ferroelectric-to-paraelectric phase transition at the Curie temperature. In contrast, highly doped samples (x = 0.004, 0.006) exhibited broadened peaks and more stable electrical behavior. These results are consistent with literature reports for La³⁺-doped BaTiO₃, where permittivity initially increases with dopant concentration up to an optimal value (x = 0.001 in this study), beyond which it decreases, accompanied by progressively less pronounced peaks at the Curie temperature [10].

The dielectric constant reached maximum values for x = 0.001, with ε_r = 2469.0 at room temperature and ε_r = 7499.7 at the Curie temperature. The sample with x = 0.002 also exhibited enhanced permittivity (ε_r = 1740.3 and 4338.8, respectively). Both compositions surpassed undoped BaTiO₃ across the entire temperature range. Beyond the optimal concentration (x = 0.001), the permittivity decreased progressively, reaching minimum values for x = 0.006 (ε_r = 334.8 and 442.4 at room temperature and Curie temperature, respectively).

A well-documented relationship exists between the tetragonality of BaTiO₃ and its dielectric constant. The results obtained in this study reinforced this correlation, as the permittivity reached its maximum at x = 0.001—the same composition exhibiting the highest tetragonality. Beyond this critical concentration, both tetragonality and dielectric constant declined, with the effect being particularly evident at x = 0.006, where the permittivity was notably lower compared to the other compositions.

For samples with higher dopant concentrations (x = 0.006), dielectric permittivity exhibited minimal variation from room temperature up to the Curie temperature, maintaining an almost flat profile. Across the examined temperature range (25–200 °C), the total permittivity variation was below 33%, indicating nearly temperature-independent dielectric behavior. This characteristic is particularly relevant for applications requiring thermal stability in dielectric properties. The trend is quantified by the ε_max/ε_min ratio, which compares the maximum permittivity at the Curie temperature (ε_max) to the permittivity at room temperature (ε_min). This ratio decreased from 3.5 in undoped BaTiO₃ (x = 0.0) to 1.3 for x = 0.006, confirming the suppression of permittivity variation at higher dopant levels.

The Curie temperature at 1 kHz, corresponding to the maximum dielectric constant observed in the permittivity-temperature curve, varied between 94 and 105 °C, decreasing with increasing La³⁺ and Bi³⁺ doping levels. This reduction in T_c is attributed to the smaller ionic radii of La³⁺ and Bi³⁺ compared to Ba²⁺. The partial substitution of these dopants for Ba²⁺ reduced the mean ionic radius, leading to paraelectric cubic phase stabilization at lower temperatures. Similar behavior has been documented by other researchers [19].

Polarization effects explain the dielectric behavior of the BaTiO₃ system. Below the Curie temperature, the material becomes spontaneously polarized due to enhanced dipole interactions that lead to an internal field aligning the dipoles. At the Curie temperature, a phase transition from the ferroelectric to the paraelectric state occurs. Above T_c, the material enters the paraelectric phase, where elementary dipoles are randomly oriented [39]. As shown in Figure 6a, this transition becomes increasingly diffuse for highly doped samples (x = 0.006), reflecting the progressive disorder introduced into the perovskite structure by dopant incorporation.

The results for the dielectric constant as a function of temperature over the range of 25 °C to 200 °C at 100 kHz are presented in Figure 6b and Table 4. These results indicated a reduction in permittivity at both room temperature and the Curie temperature for all concentrations compared to the measurements at 1 kHz. In general, ferroelectric materials exhibit lower dielectric constants at higher frequencies because dipoles cannot follow the applied AC electric field. Conversely, at lower frequencies, interfacial and dipolar polarization contribute to higher dielectric constant values. Additionally, an increase in the Curie temperature (T_c) was observed for all concentrations compared to those measured at 1 kHz. This frequency-dependent shift occurs because ferroelectric domains require more thermal energy to respond as the frequency increases, resulting in an apparent displacement of the Curie temperature to higher values.

Dielectric parameters such as Curie constant (C) and Curie–Weiss temperature (T₀) were calculated using the Curie–Weiss law

$${\epsilon }_r={\displaystyle \frac{C}{T-T_0}}$$

(6)

where C is the Curie constant, and T₀ is the Curie–Weiss temperature, which is close to the Curie temperature.

T₀ was obtained from the x-axis intercept of the linear extrapolation of 1/ε_r vs. T above T_c (Figure 7), while C was determined from the slope of this linear region. T₀ decreased for all doped samples (

Table 5. Dielectric parameters obtained from the Curie–Weiss law and its modified form for La³⁺/Bi³⁺ co-doped BaTiO₃ ceramics at 1 kHz.

x	T0 (°C)	C × 104 (K)	C’ × 104 (K)	γ
0.0	86.9	5.23	5.84	1.04
0.001	50.2	39.6	35.4	0.94
0.002	74.9	11.6	47.8	1.38
0.004	55.3	4.85	16.6	1.40
0.006	−210	13.9	168	1.56

), while the highest C value (39.6 × 10⁴ K) was measured for x = 0.001. The Curie constant is directly related to the strength of ferroelectric behavior and the degree of dipolar interactions within the material, with higher values indicating stronger ferroelectric coupling and more pronounced phase transitions. The significant increase in C for x = 0.001 suggests enhanced ferroelectric properties, consistent with this sample exhibiting the maximum dielectric constant. Notably, the concentration x = 0.006 does not follow the Curie–Weiss law, as evidenced by the negative T₀ value in Table 5; therefore, the C value at this concentration is not physically meaningful.

To quantify the diffuseness of the phase transition, the equation proposed by Uchino and Nomura [40] was used:

$${\displaystyle \frac{1}{{\epsilon }_r}}-{\displaystyle \frac{1}{{\epsilon }_{max}}}={\displaystyle \frac{{(T-T_{max})}^{\gamma }}{C\prime }}$$

(7)

where ε_r is the dielectric constant, ε_max is the maximum dielectric constant, T_max is the temperature at which ε_max occurs, γ represents the critical exponent of nonlinearity, and C’ the Curie-like constant. This equation constitutes a modified form of the Curie–Weiss law.

The calculated parameters are presented in Table 5. The critical exponent γ was obtained from the slope of the linear fit to the ln(1/ε_r − 1/ε_max) vs. ln(T − T_max) plot (Figure 8). The value of γ ranges from 1, indicating a sharp phase transition, to 2, characteristic of a diffuse phase transformation.

For low dopant concentration (x = 0.001), γ reached its minimum value of 0.94, consistent with the sharp ferroelectric-to-paraelectric phase transition observed in the ε_r vs. T plots. The critical exponent increased with dopant concentration, reaching its maximum value (γ = 1.56) for x = 0.006, indicating a diffuse phase transformation in heavily doped samples. As shown in Table 5, the Curie-like constant C’ generally increases with La³⁺ and Bi³⁺ content, with the highest value observed for the x = 0.006 sample.

The observed trends in the critical exponent γ are consistent with reports on La-doped BaTiO₃ systems. Similar behavior has been documented for La-BaTiO₃ [41], where increasing dopant concentration leads to a progressive transition from sharp to diffuse phase transformations, with γ values increasing as structural disorder intensifies.

4. Conclusions

XRD analysis confirmed the successful synthesis of single-phase tetragonal BaTiO₃ for all compositions (0.0 ≤ x ≤ 0.006), with no secondary phases detected. The incorporation of La³⁺ and Bi³⁺ dopants significantly influenced the lattice parameters, causing a reduction in unit cell volume compared to undoped BaTiO₃. The sample with x = 0.001 exhibited the highest tetragonality ratio (1.009), exceeding the minimum threshold required for multilayer ceramic capacitors. At higher doping levels, a gradual reduction in tetragonality was observed, indicating a progressive tendency toward cubic symmetry. HRSEM analysis revealed an increase in mean particle size from 362.5 nm (x = 0.0) to 488.3 nm (x = 0.006), with all samples exhibiting faceted morphology independent of dopant concentration.

Dielectric measurements demonstrated strong concentration-dependent behavior. The sample with x = 0.001 exhibited optimal performance with the highest dielectric constant at room temperature (ε_r = 2469.0) and Curie temperature (ε_r = 7499.7) at 1 kHz, confirming that controlled La³⁺/Bi³⁺ co-doping effectively enhances dielectric properties. At higher concentrations (x = 0.006), minimal permittivity variation (below 33%) from room temperature to the Curie temperature indicated nearly temperature-independent behavior suitable for applications requiring thermal stability. The Curie temperature decreased progressively with increasing dopant content at 1 kHz.

Analysis using the Curie–Weiss law showed that T₀ decreased for all doped samples compared with undoped BaTiO₃. The highest Curie constant (C = 39.6 × 10⁴ K) was measured for x = 0.001, consistent with enhanced ferroelectric properties. The critical exponent γ varied from 0.94 for x = 0.001, indicating a sharp ferroelectric-to-paraelectric phase transition, to 1.56 for x = 0.006, characteristic of a diffuse phase transition. These results confirm that La³⁺/Bi³⁺ co-doping offers a viable strategy for tailoring BaTiO₃ properties to specific applications, enabling either enhanced dielectric performance at optimal doping levels or temperature-stable behavior at higher concentrations.