(1-x)(Na₀.₅Bi₀.₅)TiO₃-xCaTiO₃ ceramics: Investigating structural and microstructural features for enhanced dielectric properties

Abstract

(1-x)(Na₀.₅Bi₀.₅)TiO₃-xCaTiO₃ Lead-free piezoelectric systems, positioned near the morphotropic phase boundary, were synthesized for varying compositions (x = 0.0, 0.05, 0.10, 0.15, and 0.20) using the solid-state reaction route. This study delves into the comprehensive investigation of the compositional effects on phase, structure, and electrical characteristics. Specifically, a morphotropic phase boundary (MPB) involving rhombohedral (R3c) and orthorhombic (Pnma) structures was seen in a (1-x)NBT-xCT crystal structure close to the composition of x = 0.10. Information on the pure phase formation and grain size of the intended composite system has been obtained using Rietveld refinement of the X-ray diffraction (XRD) diagram as well as scanning electron microscopy (SEM). The impact of the CT phase on the NBT lattice was investigated through an analysis of the charge density distribution. Using Williamson-Hall plots from XRD data, the average particle diameter was estimated to be between 131.87 nm and 136.54 nm. The relative permittivity increases with the addition of Ca²⁺, according to dielectric measurements. All ceramics exhibit a diffuse phase transition near (Tm) with a diffusivity range of 1.5–1.8, and a downward shift in depolarization temperature (Td). At the morphotropic phase boundary (MPB), excellent dielectric properties were observed at x = 0.10, which are attributed to the presence of both rhombohedral and orthorhombic structures as well as an appropriate particle size. The conduction process at different temperatures is thermally activated, as determined by the frequency-dependent ac conductivity.

Highlights

• •The present work proposes the calculation of bonding charge density using the GFourier program.
• •The anomalous dielectric effect was studied in (1-x)NBT-xCT.
• •The conductivity of (1-x)NBT-xCT composites is caused by the presence of multiple oxidation states of ions and oxygen vacancies found in the specimen.

1. Introduction

In recent decades, lead-based materials, particularly those derived from the ferroelectric Pb(Zr,Ti)O₃ (PZT) complex, have garnered substantial attention in the realm of electronic micro-devices due to their remarkable properties [1,2]. These materials, known for their exceptional pyroelectric, piezoelectric, ferroelectric, and dielectric characteristics, have found widespread applications in various devices such as actuators, sensors, and pyroelectric transducers [3,4]. However, despite the advantageous properties of lead-based materials like high density, ductility, and low melting point, their use raises significant environmental and health concerns. This has spurred efforts towards eliminating hazardous heavy metals in the development of modern electronic equipment. In light of these environmental challenges, the scientific community has shifted its focus to exploring lead-free, environmentally friendly alternatives. Prominent among these are materials such (K_0.5Na_0.5)NbO₃ (KNN) [5], BaTiO₃ (BT) [6], BiFeO₃ (BF) [7], (K_0.5Bi_0.5)TiO₃(KBT) [8], and (Na_0.5Bi _0.5)TiO₃ (NBT) [9], along with their derivative compositions [[5], [6], [7], [8], [9]]. Among them, Bismuth sodium titanate (NBT)-based lead-free piezoceramics stand out for their adaptable properties, high Curie temperature, and potential as energy storage materials due to their high polarization (exceeding 40 μC/cm²) [10,11]. It represents one of the material systems with great potential to become a replacement for lead-based piezoceramics [12,13]. The NBT exhibits a perovskite structure, and the A site is composed of two cations (Na⁺ and Bi³⁺) and the B site is composed of the titanium (Ti⁴⁺) ion [14]. Moreover, as the temperature rises, the NBT evolves towards a succession of phase transitions [15]. The electrical properties of NBT-based materials are enhanced principally through phase boundary engineering and optimization of the preparation procedure [[15], [16], [17]]. In specific, (1-x)NBT-xBT and (1-x)NBT-xCT systems materials have been regarded as one of the most promising lead-free ferroelectric compounds [18,19]. Subramani et al. investigated the influences of tetragonal CaTiO₃ (CT) addition on the sintering, properties, microstructures, and characteristics of (Na_0.5Bi_0.5)TiO₃ (NBT) materials, and inferred that a little BT is advantageous for enhancing the dielectric performance of NBT materials [20]. According to Maruyama et al. [21] and Chen et al. [22], compositions comprising 0.06–0.07 mol xBT and 0.10–0.15 mol xCT and exhibiting d₃₃ = 125 pC/N and 132 pC/N, respectively. It is due to the extremely large field strain response seen at the morphotropic phase boundary, which divides the ferroelectric rhombohedral (space group: R3c) and ferroelectric orthorhombic (space group: Pnma) phases having xCT of approximately from x(mol.%) = 9 to 15 at ambient temperature. The very elevated piezoelectric characteristic usually results from many phases coexisting, whose free energies are suitable for polarization and enhancement of the piezoelectric performance [23]. However, the influence of the CaTiO₃ (CT) doping content has rarely been reported. Therefore, further investigation of more CT doping will enhance the piezoelectric, electrical and electromechanical properties of NBT-based materials, along with micro-mechanism, i.e. microstructure and domain change, has been of interest to usIn general, a noticeable increase in the piezoelectric constant of modified NBT-based materials is related to a significant decrease in the depolarization temperature (T_d). Indeed, additives in NBT-based compounds would lead to defects such as lattice distortion and lacunae, making easier the domain movement resulting to higher piezoelectric properties yet decreasing (T_d) [24,25]. In this study, we explore the morphotropic phase boundary (MPB) zone within the (1-x)NBT-xCT binary system. The fabrication of this system is achieved through the solid-state method, a robust technique known for its efficacy in producing ceramic materials of high quality. Our investigation focuses on systematically characterizing the influence of CT concentration on various aspects, including crystal structure, dielectric and relaxor responses, and other impedance properties. To ascertain the AC conductivity, we have applied Jonscher's power law universally. These findings enable a meticulous comparison of structural models at the MPB, encompassing considerations such as phase fraction, identified phases, and associated dielectric properties. According to the outcomes of this research, we identify oxygen vacancies and cation carrier hops as the principal factors driving the processes underlying NBT conductivity with CT. Overall, this comprehensive study aims to unravel the complex interplay between the composition, structure, and properties of the (1-x)NBT-xCT system, paving the way for the development of advanced materials for electronic applications.

2. Experimental

(1-x)NBT-xCT ferroelectric materials were produced employing a conventional solid-state method implying the following chemical reaction (Equation (1)):

$$\left(1-x\right)\left({\text{Na}}_{0.5}{\text{Bi}}_{0.5}\right)\text{Ti}{\mathrm{O}}_3-x\text{CaTi}{\mathrm{O}}_3+\left[\left(1+3\mathrm{x}\right)/4\right]\mathrm{C}{\mathrm{O}}_2\uparrow$$ (1)

Na₂CO₃, CaCO₃ (Sigma-Aldrich 98.99%), titanium oxide TiO₂ (Sigma-Aldrich 98.99%) and bismuth oxide Bi₂O₃ (Sigma-Aldrich 98.99%) were employed as starting materials. The raw elements were weighed in the appropriate stoichiometric proportion and then carefully mixed to obtain a homogeneous mixture. The resulting mixture was annealed at different temperatures up to 1050 °C for 4 h to prepare the sample. The calcination process was conducted in a Nabertherm furnace, using ambient air as the atmosphere and applying a heating rate of 5 °C/min. All specimens were air cooled to room temperature after calcination. To perform microstructural analysis (SEM pictures) and dielectric measurements, we uniaxially pressed the calcined powders into pellets and sintered them at 1200 °C for 4 h in a programmable furnace. According to various earlier investigations, the intended rhombohedral structure has been effectively essentially formed at the 1050 °C/4h temperature and doesn't change at the sintering temperature. This heat treatment is quite essential for microstructural analysis; dielectric measurements, and should not affect the structural characteristics of our specimens [26,27]. Sintering consists of thermally consolidating a powder product obtained by melting together one or more of its different elements, resulting in a densification of the product according to the temperature and thus making it possible to increase the mechanical resistance and overall to limit the porosity of the ceramic. X-ray powder diffraction was carried out using an XPERT-PRO diffractometer equipped with a Cu anode (Cu-Kα radiation of 1.54186 Å) to determine the structure of the polycrystalline powders. We conducted Rietveld refinement to analyze crystal phases via XRD over a scan range of 2θ ∼ 10–90°, employing the rhombohedral compound NBT as the model structure. The determination of crystallite sizes and lattice constants was accomplished through Rietveld refinement, utilizing the FULLPROF Suite programs [28], in conjunction with our experimental scatter data. The combination of a scanning electron microscope (SEM) (Quanta 200) permitted examination of the phase purity and microstructure of the sintered ceramics. An impedance analyzer (Agilent 4284A) was used to do dielectric measurements over the temperature range (ambient-500 °C) and the frequency interval (100 kHz–400 kHz).

3. Results and discussion

3.1. Structural analysis

The crystal lattice of (1-x)NBT-xCT specimens was highlighted through X-ray diffraction which depends on Ca²⁺ concentration (x = 0.0, 0.05, 0.10, 0.15 and 0.20).

Figure 1a displays the XRD plots of (1-x)NBT-xCT ceramics at ambient temperature. With no secondary or impurity phases, the specimens show a pure ABO₃ structure. This gives us reason to believe all the Ca²⁺ ions diffused through the rhombohedral NBT lattice to create a solid solution having a pure perovskite structure. The XRD patterns obtained for all synthesized solid solutions were found to be either to pure CT or pure NBT. Ca²⁺ concentration rises lead XRD peak patterns to move toward a lower angle (2θ), which denotes an increase in cell volume (see Fig. 1b). Moreover, starting at x = 0.20, there is an observed shift of the (111) peak towards higher angles, indicating a reduction in the interplanar spacing [29]. This phenomenon can be attributed to the variance in average ionic radii between the A and B sites in the lattice structures of NBT and CT [30,31]. In the magnified section of Fig. 1b, the splitting observed in the (111) peaks serves as evidence for the presence of an orthorhombic phase. Roukos et al. [32] used a classical solid-state approach to make 0.90NBT-0.10CT samples. They identified a low proportion of the secondary phase in the high-Ca samples (x = 0.15 and x = 0.20), supporting the observed phase transition at higher dopant concentrations Furthermore, at room temperature, the emergence of a pronounced superlattice (111)o peak signifies the development of an orthorhombic phase within the materials, as indicated by a red circle in Fig. 1b, a phenomenon that has been corroborated by numerous studies in this field [32,33].

Fig. 1a.

XRD patterns of the CT-doped NBT ceramics calcined at 1050 °C for 4 h..

Fig. 1b.

Fin scan of X-ray reflections for (1-x)NBT-xCT in the 2θ ranges of 39°–41°.

3.2. Methodology for rietveld refinement analysis

To ascertain structural parameters and crystal configuration, we compared observed Bragg peaks against a theoretical baseline model. This comparison was conducted using the Rietveld refinement approach, with Fullprof software serving as the chosen open-source platform. The refinement commenced with background adjustments using a cosine function. Subsequently, data analysis and the fine-tuning of diffraction peak profiles were facilitated using Fourier series functions and pseudo-Voigt functions, respectively. This comprehensive refinement yielded a variety of parameters, each playing a crucial role in different aspects of our analysis. These aspects included background adjustments, defining profile half-width parameters (u, v, w) for peak profiling, adjusting thermal parameters, and managing preferred orientation, scale factor, zero shifting, and unit cell dimensions (a, b, c). Furthermore, displacement, atomic positional coordinates (x, y, z), and Wyckoff positions were meticulously refined (Table 1). In our rigorous pursuit of precision and quality, we implemented multiple measures to ensure the refinement process's convergence. This included calculating and scrutinizing concordance indices or profile fit parameters (Rp, Rwp, and χ²), where values below 2% signified a high degree of correlation between experimental and calculated data. A pivotal aspect of our analysis was quantifying the phase fraction and pinpointing the morphotropic phase boundary (MPB) through the structural refinement approach. For a comprehensive understanding, detailed outcomes of the Rietveld refinement, encompassing both data and measurements, are meticulously documented in Table 1. Fig. 2 presents a detailed Rietveld refinement analysis of various (1-x)NBT-xCT ceramic compositions. At standard room temperature, the primary crystalline structure of NBT exhibits a rhombohedral configuration, identified by the R3c space group. Notably, as the xCT component is integrated, there is a discernible increase in the lattice parameters of this structure. This observation leads to a broader inquiry into the behavior of a mixed structural system. In this context, we explore whether the coexistence of two distinct structures, rhombohedral (R3c symmetry) and orthorhombic (Pnma symmetry), within the (1-x)NBT-xCT system, behaves typically in a biphasic domain or demonstrates the characteristics of a morphotropic phase boundary (MPB). To unravel this complexity, a comprehensive Rietveld refinement employing a biphasic structural model, encompassing both the rhombohedral and orthorhombic forms, was utilized. This approach facilitated an in-depth examination of the alterations in phase fraction for each structure, as well as the variations in cell volume across the compounds within the concentration range of 0.05 ≤ x ≤ 0.20. Fig. 2 illustrates a thorough Rietveld refinement analysis carried out on diverse compositions of (1-x)NBT-xCT ceramics. At standard room temperature, the crystalline structure of the NBT compound takes on a rhombohedral configuration, characterized by the space group R3c. Significantly, the lattice parameters of this structure exhibit an increasing tendency with the incorporation of the xCT constituent. While the orthorhombic phase fraction increases from 70% to 77% when x(mol.%) = 10 to 15, the rhombohedral phase fraction diminishes steadily (i.e., by 100%–23%) within the compositional interval 0.0 ≤ x ≤ 0.15, and vanishes altogether (0%) at x(mol) ≥ 20%.

Table 1.

Refined structural parameters of sodium bismuth titanate (NBT).

NBT
Atom	Site	x	y	z	B(Å)	Occupancy
Na/Bi	6a	0	0	0.252	0.79	0.5/0.5
Ti	6a	0	0	0.007	0.47	1
O	18b	0.127	0.251	0.071	1.91	1

Fig. 2.

Rietveld refinement of powder XRD profile for ceramics containing (1-x)NBT-xCT: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT

The evidence shows that a heterogeneous combination of two solid solutions exhibits typical behavior in the biphasic domain in terms of phase ratios. It really is known that within a biphasic domain separated by two monophasic domains, the percentage of the two phases fluctuates with x, but the cell volume of each phase stays constant [34]. Nevertheless, in present instance, the cell volume of the R3c and Pnma phases varies gradually as Ca²⁺ content. In the specific two-phase region exhibited in the (1-x)NBT-xCT ceramics (0.10 ≤ x ≤ 0.15), the solution solid was revealed to be composed of a mixture of two structures (Rhombohedra + orthorhombic), and the fraction of the two phases changes along as xCT (Table 2).This suggests that based on their distribution, there may be a considerable interaction between these two phases. This finding suggests that the distinct biphasic region between x(mol.%) = 10 and x(mol.%) = 15, which was observed in the (1-x)NBT-xCT, is only a morphotropic phase boundary (MPB). In addition, studies conducted through neutron diffraction on (1-x)NBT-xCT [33], with a primary focus on the superstructure and the calculation of the phase tilt angle of the oxygen octahedra, have recently uncovered the coexistence of rhombohedral and orthorhombic (R3c + Pnma) phases when x equals 0.15 in the system. In this system, the structure is rhombohedral (R3c) at x ≤ 0.05, and transitions to an orthorhombic (Pnma) phase at x ≥ 0.15. Additionally, the research by Du et al. [35] pinpointed the morphotropic phase boundary (MPB) within the range of 0.08 ≤ x ≤ 0.14, a region where rhombohedral and orthorhombic symmetries are concurrently present. In the specific biphasic range observed in the (1-x)NBT-xCT system (0.09 ≤ x < 0.15), it was demonstrated that the solid is constituted by a blend of two phases (R3c + Pnma), where he proportion of these two phases fluctuates with x. Considering this, a significant interaction might take place between these two phases, depending on their distribution. Well-established is the fact that solid solutions with compositions proximate to the morphotropic phase boundary (MPB) showcase intriguing physical characteristics. As per Glazer's findings, there is no evident group-subgroup relationship between the rhombohedral (R3c) and orthorhombic (Pnma) phases [36,37]. Nonetheless, a substantial correlation between these phases within the morphotropic phase boundary (MPB) region has been suggested. This investigation provides empirical support for the idea that the orthorhombic phase (Pnma) contributes to the rhombohedral phase (R3c), as indicated by the observed dielectric response. This fresh insight contributes to a more thorough understanding of the intricate relationship between the structural configuration and dielectric properties of lead-free (1-x)NBT-xCT polycrystalline ceramics.

Table 2.

Refined structural parameters, average crystallite size, and cell volume of the (1-x)NBT-xCT specimens.

x	Structure	Phase fraction (%)	Lattice parameter (Å)	Average crystallite size (nm)	Strain (ɛ) x 10−2	Cell Volume (Å3)	GOF
0.0	Rhombohedral	R3c 100 %	a = b = 5.4634 c = 13.4271	131.87	0.0056	350.31	1.33
0.05	Rhombohedral	R3c 100%	a = b = 5.653 c = 13.7915	131.81	0.0058	352.73	1.27
0.10	Rh + O	R3c 29.85% Pnma 70.15%	a = b = 5.4829 c = 13.5616 a = b = 3.750 c = 3.8172	134.12	0.0064	353.25 57.43	1.29
0.15	Rh + O	R3c 22.82% Pnma 77.18%	a = b = 5.4874 c = 13.4784 a = b = 3.7951 c = 3.8423	135.79	0.0065	355.21 58.54	1.24
0.20	Orthorhombic	Pnma 100%	a = b = 3.8124 c = 3.8563	136.54	0.0072	59.32	1.17

To determine the average crystallite size, the Williamson-Hall (W–H) method was utilized, which involves analyzing the width of X-ray diffraction peaks [[38], [39], [40]]. Moreover, this method reveals that the overall XRD peak width arises from both crystal size broadening and microstrain, which can be represented by the following equation. The measured crystallite size is marginally increased compared to that predicted by the Debye-Scherrer equation because the latter does not account for the microstrain-induced broadening effect. The overall broadening of the peaks is a result of the combined effects of microstrain, ${\beta }_{strain}$, and crystallite size, ${\beta }_{size}$ (Equations (2), (3))):

$${\beta }_{total}={\beta }_{size}+{\beta }_{strain}$$ (2)

The Scherrer equation is used to compute the crystallite size:

$${\beta }_{size}=\frac{K\times \lambda }{D\times \mathit{COS}\theta }$$ (3)

The crystallite size was obtained based on the fit line intercept related to the resulting values: ${\beta }_t\mathit{cos}\theta$ versus 4 $\mathit{sin}\theta$ (see Fig. 3). The results obtained from this method for (1-x)NBT-xCT are presented in Table 2, including the crystallite size and strain. Microstrain values were calculated for NBT, NBT-0.05CT, NBT-0.10CT, NBT-0.15CT, and NBT-0.20CT, resulting in 0.0057, 0.0059, 0.0067, 0.0068, and 0.0074, respectively. Furthermore, the average crystallite size increased from 132 nm for NBT to 138 nm for NBT-0.20CT. This increase could be attributed to a change in the structural phase (O to Rh + O) and an increase in residual stresses with the concentration of xCT.

Fig. 3.

Williamson-Hall plot for (1-x)NBT-xCT ceramics: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT

3.3. Charge density studies by means of structure factors

The VESTA software was utilized to create a 3D model of the crystal structure and perform electronic state calculations. Basic structural parameters for NBT, CT, and NBT-0.10CT were used to generate the 3D structures in VESTA. Fig. 4 displays the 3D structure of NBT-0.10CT, while the other composites have a similar structure. Additionally, the Rietveld refinement produced an electron density map used with the GFourier program to characterize and visualize the electron density in the crystal lattice. The electron charge density is used to identify the atomic locations of the elements in the crystal unit cells and to learn more about the type of chemical bonds holding the ions of (1-x)NBT-xCT powders together. Heavy electron density contours reveal the location of larger elements within the unit cell's principal elements. Fig. 5 displays a 3D Fourier map of each atom in the unit cell for pure NBT, pure CT, and the composite, NBT-0.10CT. The contour colors indicate the ascending levels of electron density for the Na/Bi cation, with black denoting zero density. The NBT-0.10CT system under investigation shows Ti and Na/Bi/Ca cations located at (0, 0, 0.5) and (0.034, 0.30, 0.879), respectively. Fig. 5 also highlights the highest electron density for Ca (Z = 20) and Bi (Z = 83), which is 74 e/ų in the xy direction. Furthermore, the density map undergoes substantial changes following a 10% calcium (Ca²⁺) doping. The substitution of Na⁺/Bi³⁺ with Ca²⁺ ions, which have a closed-shell [Ar] electron configuration, results in an enhanced Ca–O bond length and an increased charge density for this composition.

Fig. 4.

Three-dimensional crystal diagram: (a) rhombohedral (R3c) for pure NBT, (b) Orthorhombic (Pnma) for CT, and (c) rhombohedral (R3c) as well as Orthorhombic (Pnma) for composite NBT-0.10CT

Fig. 5.

Electron density distribution in the unit cell of (1-x)NBT-xCT in 3 dimensions: (a) Pure NBT, (b) CaTiO₃, and (c) NBT-0.10CT

This phenomenon may be attributed to the introduction of new diffusion centers associated with the higher doping concentration [41,42]. Furthermore, the spatial distribution of electron density, particularly in the context of calcium doping, plays a crucial role in understanding the variation of the dielectric constant. Regions with high electron density, more pronounced in materials doped with Ca²⁺, represent polar areas where the polarization is elevated and typically exhibit higher local dielectric responses. This correlation between elevated electron density and increased polarization is pivotal in analyzing the changes in dielectric properties as a function of Ca²⁺ doping levels. The variation of doping levels and their impact on these properties will be further examined in the subsequent sections of this article to verify this observation.

3.4. Microstructural study

The SEM images of (1-x)NBT-xCT ceramics are presented in Fig. 6, revealing a fully dense microstructure with rectangular or squared-shaped grains and grain boundaries, indicating their polycrystalline nature. The Image J software was utilized to analyze micrographs obtained from scanning electron microscopy (SEM) in order to determine the average grain size of the ceramic specimens.

Fig. 6.

Surface Morphologies of sintered (1-x)NBT-xCT ceramics: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT

The relative density (ρ_r) of sintered (1-x)NBT-xCT ceramics, measured by Archimedes' principle, ranges between 96% and 98%. The grain sizes for NBT, NBT-0.05CT, NBT-0.10CT, NBT-0.15CT, and NBT-0.20CT are 745 nm, 653 nm, 812 nm, 894 nm, and 625 nm, respectively. The grain size exhibits a zigzag trend as a function of varying Ca²⁺ dopant levels.

Among all compositions, the maximum grain size and ρ_r are observed in NBT-0.15CT. The coexistence of a non-polar phase (Pnma) alongside polar phases (R3c) within the solid solution effectively lowers the energy barrier, facilitating enhanced polarization switching through an extended polarization mechanism [43]. This unique structural arrangement, coupled with the nanometric grain size, significantly boosts the material's performance, making it a standout candidate for efficient energy storage [44,45]. Generally, the electrical properties of polycrystalline ferroelectric compounds are strongly influenced by their grain size and density [23].

3.5. Dielectric analysis

Fig. 7 displays the variations in the dielectric constant (ε_r) with temperature for (1-x)NBT-xCT ceramics, each containing different _xCT proportions. These properties were measured at several frequencies, specifically at 100 kHz, 200 kHz, 300 kHz, and 400 kHz. The resulting curves show two significant anomalous dielectric peaks. Notably, the first of these peaks is broad and shows a frequency-independent nature, occurring at around (Tm = 363 °C). Tm represents the temperature at which there is a transition from an antiferroelectric state to a paraelectric state [46,47]. The second notable anomaly was observed around (Td∼226 °C) and is associated with the depolarization temperature, which corresponds to the transition from ferroelectric to antiferroelectric states [48,49]. In this context, the antiferroelectric behavior of NBT in the vicinity of 226 °C can be explained by the reorientation of the polar vector due to the emergence of a new orthorhombic phase [50,51]. The detected decrease in Td and rise in Tm suggest an enlargement of the anti-ferroelectric region due to the integration of Ca²⁺ into the NBT-CT matrix, further promoting the potential reorientation of polar domains, a key factor in enhancing piezoelectric performance [52]. The Td value holds significant relevance in the practical applications of a range of materials. Jo, W et al. proposed that the identified local dielectric maxima (Td and Tm) in NBT-based ceramics are not associated with phase transitions. Instead, they suggest that these maxima can be ascribed to the thermal characteristics of local ferroelectric polar nanometer-size regions (PNRs) with rhombohedral R3c and tetragonal P4bm symmetries [53]. In our current investigation, we established the depolarization temperature (Td) for all ceramic samples based on the peak dielectric constant (Ɛ_rmax), and these findings are in line with those reported in the existing literature [52,61]. A summary of this information is presented in Table 3.

Fig. 7.

Dielectric properties as a function of temperature at different frequencies (1-x)NBT-xCT ceramics: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT

Table 3.

Physical properties (1-x)NBT-xCT systems Measured at 100 kHz.

Sample	Ɛr at room temperature	Ɛrmax at Tm	Td (°C)	Tm (°C)
NBT	987	3429	226	363
NBT-0.05CT	2748	4261	223	368
NBT-0.10CT	3117	5274	218	381
NBT-0.15CT	2741	4426	213	387
NBT-0.20CT	1234	3175	206	384

The dielectric constant value for pure and doped (1-x)NBT-xCT at x = 0.10 has demonstrated improvement, increasing from 3429 to 5274 at the temperature corresponding to Tm. As the value of x increases, the Tm (temperature corresponding to the peak in dielectric constant) of (1-x)NBT-xCT ceramics shifts towards higher temperatures. Since the composition of (1-x)NBT-xCT is characterized by a heterogeneous mixture, including a relaxor ferroelectric phase (NBT) and a paraelectric phase (CT) within the morphotropic phase boundary (MPB) region. However, the permittivity increases as the CT content (x) rises, especially in compositions near the morphotropic phase boundary (MPB). These novel findings may be attributed to the interactions between the rhombohedral (R3c) and orthorhombic (Pnma) phases, resulting in a nonlinear response dependent on the variable x. We propose that the incorporation of CT in NBT alters nano polar domains into micro ones, thereby amplifying the polarization within the sample in the vicinity of the morphotropic phase boundary (MPB). Moreover, it is experimentally demonstrated that the dielectric breakdown strength of bulk ceramics is influenced by various microstructural aspects, such as grain size, grain boundary conditions, and porosity, among which the role of grain size is especially pronounced [54]. Hence, relaxor ferroelectrics are considered promising candidates for applications in the realm of energy storage [55,56].

The Curie-Weiss formula (Equation (4)) was employed to model the relaxation coefficient, providing insights into the nature of diffusion in the (1-x)NBT-xCT ceramics' relaxation properties.

$$\frac{\mathbf{1}}{{\mathit{\epsilon }}_{\mathit{r}}}-\frac{\mathbf{1}}{{\mathit{\epsilon }}_{\mathit{r}\mathit{max}}}=\left[\frac{{\left(\mathit{T}-{\mathit{T}}_{\mathit{m}}\right)}^{\mathit{\gamma }}}{\mathbf{C}}\right]$$ (4)

In this formula, 'C' and 'γ' stand for the Curie-Weiss constants and the diffusion degree of the phase transition (where 1 ≤ γ ≤ 2), respectively. Fig. 8 illustrates the values of γ for the (1-x)NBT-xCT ceramics. The results reveal an increase in the diffusion coefficient from 1.5 in pure NBT to 1.8 in the 0.90NBT-0.10CT composition, as depicted in Fig. 8. Table 4 presents the determined values of γ and δ. This enhancement in diffusion levels, especially near the morphotropic phase boundary (MPB), is likely attributed to disturbances at the A-site, involving Na⁺/Bi³⁺ and Ca²⁺ ions [18,57].

Fig. 8.

Illustratethe Curie-Weiss law deviation of (1-x)NBT-xCT ceramics: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT

Table 4.

Values of γ and δ as a function of potassium content in (1-x)NBT-xCT systems.

x	NBT	0.95NBT-0.05CT	0.90NBT-0.10CT	0.85NBT-0.15CT	0.80NBT-0.20CT
Γ	1.5	1.6	1.8	1.7	1.6
Δ	102	96	112	119	121

3.6. Electrical conductivity

To gain a deeper understanding of the electrical charge carrier behavior in Ca²⁺ doped (1-x)NBT-xCT samples, the relationship between AC conductivity and frequency was examined. Fig. 9 displays how AC conductivity changes with frequency at various temperatures for composites like NBT, NBT-0.05CT, NBT-0.10CT, NBT-0.15CT, and NBT-0.20CT. A trend is observed where conductivity increases with increasing frequency. This escalating trend in conductivity seems to be in accordance with Jonscher's law (Equation (5)) [[58], [59], [60]].

$${\mathrm{\sigma }}_{\text{ac}=}{\mathrm{\sigma }}_{\text{dc}+\mathrm{A}{\mathrm{\omega }}^{\mathrm{n}}}$$ (5)

Fig. 9.

AC conductivity versus frequency for different temperatures of (1-x)NBT-xCT cearmics: (a) Pure NBT, (b) 0.95NBT-0.05CT, (c) 0.90NBT-0.10CT, (d) 0.85NBT-0.15CT, and (e) 0.80NBT-0.20CT.

Where the term “σ_ac” stands for frequency-dependent ac conductivity and “σ_dc” stands for dc conductivity. The polarization force is denoted by the term “A,” while the temperature-dependent coefficient is denoted by the term “n”. To examine the conduction mechanism, the temperature-dependent development of the parameter “n” is used. At any temperature taken into account, there are two separate areas of the AC conductivity. We may see a first zone at low frequencies where the conductivity does not change with frequency.

In the second frequency range above 10⁶ Hz, we observed an increase in conductivity, which further escalates with the addition of calcium. This trend is attributed to the prevalence of various defects and oxygen vacancies at the grain boundaries. Perovskite materials, in general, demonstrate enhanced conductivity primarily due to these oxygen vacancies [19,42]. Commonly, the sintering process facilitates the volatilization of A-site cations, leading to the formation of oxygen vacancies [9]. For the (1-x)NBT-xCT samples, it is either the oxygen ion defects or the vacancies of Bi³⁺ ions that predominantly contribute to conductivity. O⁻² ions are located in the octahedral sites of the material and move at high temperatures, while Bi³⁺ ions occupy the centers of oxygen dodecahedra. Due to their larger size, Bi³⁺ ions need substantial thermal energy to migrate. To investigate the conduction mechanism, we analyzed the variation of the coefficient “n” as a function of temperature. This exponent (n) ranges from 0 to 1. Funke et al. [61] have discussed the significance of the numeric value of “n”. Our findings indicate that n is less than 1, suggesting that the conduction character is not of the Debye type (see Table 5). This leads us to conclude that the correlated barrier hopping (CBH) model aptly describes the charge transport process in these materials. Fig. 10 shows the variation of ln(σDC) with the inverse of temperature (1000/T), consistent with Arrhenius' law (Equation (6)), enabling us to determine the activation energy for electrical conduction in the (1-x)NBT-xCT samples. In the equation:

$${\mathrm{\sigma }}_{\text{DC}}={\mathrm{\sigma }}_0{\mathit{exp}}^{-\frac{E_a}{K_{B}T}}$$ (6)

Table 5.

Parameters obtained from Joncher's plot for (1-x)NBT-xCT ceramics.

x	Temperature (°C)	n
NBT	320	0.40571
	340	0.35525
	360	0.31675
	380	0.30381
NBT-0.10CT	320	0.41093
	340	0.35570
	360	0.38143
	380	0.34718
NBT-0.20CT	320	0.58750
	340	0.47411
	360	0.44138
	380	0.43791

Fig. 10.

Conductivity ln(σ_DC) as a function of 1000/T (a) for (1-x)NBT-xCT ceramics and (b) Pure NBT.

where T represents the temperature in Kelvin, E_a is the activation energy, k_B is the Boltzmann constant, and σ₀ is the pre-exponential factor. The graphical representation illustrates an increase in conductivity with rising temperature, indicating conductive behavior in the samples. Notably, for (1-x)NBT-xCT ceramics, two distinct activation energies were identified: one associated with grain boundaries and the other with the bulk grains. Furthermore, the activation energy within the grains is notably higher compared to that at the grain boundaries. This is evident from the substantial variation in the samples' activation energies, ranging from 0.25 to 0.41 eV, highlighting the predominant role of internal oxygen vacancies in defect formation within each sample. The lower activation energy values imply that the incorporation of Ca²⁺ acts as an inhibitor, reducing the formation of oxygen vacancies in the (1-x)NBT-xCT specimens, thereby influencing their electrical properties.

4. Conclusion

In the current research, sintered (1-x)NBT-xCT specimens were examined for different compositions (NBT, NBT-0.05CT, NBT-0.10CT, NBT-0.15CT, and NBT-0.20CT). The structural examination of the (1-x)NBT-xCT solid solution was carried out to identify the influence of calcium substitution on the structure's cation distribution, microstructure, dielectric behavior, and electrical conductivity. XRD results indicate the incorporation of Ca²⁺ ions into the NBT structure. The average particle diameter of different perovskite compounds, determined using the refinement technique, ranged between 131.87 and 136.54 nm according to the Williamson-Hall plot. Additionally, the charge density distribution and the type of bonds were studied qualitatively and quantitatively using the structure factors that were generated. The MPB related to the coexistence of the rhombohedral (space group: R3c) and orthorhombic (space group: Pnma) structures were identified at x(mol.%) = 10–15, using Rietveld refinement via the Full-Prof approach. VESTA was used to create the crystal structure in three dimensions. A homogeneous microstructure with grains that were almost the same size, strong densification, and little porosity was shown in the SEM pictures. The dielectric constant shows an increase in the ferroelectric phase and exhibits an improvement over the two constituent phases in the NBT-0.10CT system, indicating its MPB (morphotropic phase boundary) nature. The NBT-0.10CT system's dielectric constant has very high values, making it suitable for high-temperature ceramic capacitors. The dispersive behavior of the diffusive phase transition is indicated by the diffuse value and modeled using the modified Curie-Weiss law relation. This behavior seems to be associated with the distribution of cations within the structure. Conductivity and impedance analyses are used to confirm the non-Debye relaxation process, and Arrhenius' law has been used to estimate the activation energy for relaxation. The Joncher power-law parameter suggests that the charge carriers undergo a similar barrier hopping (CBH) process. Two separate activation energy zones were identified by fitting the Arrhenius formula at high temperatures.

Funding

No funding was received from any public, commercial or non-profit organization for this research.

CRediT authorship contribution statement

M. Mesrar: Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Majdoubi H: Software, Methodology, Data curation. Yan Le: Writing – review & editing, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.