Studies of Structural, Dielectric, and Impedance Spectroscopy of KBT-Modified Sodium Bismuth Titanate Lead-Free Ceramics

Abstract

Lead-free ceramic materials produced from bismuth sodium titanate (Na_0.5Bi_0.5TiO₃, NBT)–bismuth potassium titanate (K_0.5Bi_0.5TiO₃, KBT) have been developed through a solid-state reaction technique. The structural, dielectric, and piezoelectric characteristics of the ceramic materials were analyzed. Based on the XRD investigation, the morphotropic phase boundary (MPB) was determined for the composition (x (%) = 16 and 20). The effects of the KBT phase on the NBT lattice were examined using the charge density distribution. Furthermore, the dielectric properties indicated the presence of a negative dielectric constant (ε_r′) as a function of frequency between 1 kHz and 2 MHz. Negative permittivity was observed globally in the (1 – x)NBT–xKBT ceramic which reflects the effect of the dielectric resonance. The grain conduction effect is revealed through the complex impedance spectrum in the form of a semicircular arc within the Nyquist plot. In addition, the samples studied revealed a non-Debye relaxation phenomenon. The relaxation time was determined based on the Vogel-Fulcher law for all samples. DC conductivity was carried out on the ceramics material and revealed that the resistance decreases with increasing temperature indicating a negative temperature coefficient of resistance. The AC conductivity as a function of frequency for different temperatures suggests the presence of a thermally activated conduction mechanism. The activation energy has been determined based on the Arrhenius plot of the DC electrical conductivity as well as the relaxation frequency.

1. Introduction

Lead-based ceramics are the most commonly utilized piezoelectric materials for piezoelectric sensors, actuators, and transducers on account of their excellent piezoelectric characteristics.^1,2 Considering the toxicity and volatility of lead oxides in materials, there is an increased demand for lead-free materials as well as biocompatible and environmentally friendly materials.^3,4 Recently, several lead-free piezoceramics have been produced; within them, perovskite lead-free solid solutions which are based on K_0.5Na_0.5NbO₃ (KNN), BaTiO₃ (BT), (Na_0.5Bi_0.5)TiO₃ (NBT), and (K_0.5 Bi_0.5)TiO₃ (KBT) are very promising eco-friendly piezoceramics.⁵⁻⁸ Nevertheless, these solid solutions need to be analyzed to be utilized in a commercial way. The principal challenge for BT-based piezoceramics is their comparatively lower curie temperature (T_c), making the operating temperature range limited, whereas for KNN- and KBT-based piezoceramics, the difficulty lies in producing densely packed ceramics in a reliable manner.^9,10 Sodium bismuth titanate ((Na_0.5Bi_0.5)TiO₃, abbreviated as NBT)^11,12 is a very well-known antiferroelectric material with a rhombohedral perovskite R3c structure. It is regarded as a great potential candidate as a primary material for lead-free piezoelectric ceramics because it is a highly ferroelectric material with a large Curie temperature (T_c = 320 °C), a highly remanent polarization (P_r = 38 μC/cm²), and a significant coercivity field (E_c = 73kV/cm) at ambient temperature. Nevertheless, because of its significant coercive field and comparatively high conductivity, NBT is not easily polarized, and its piezoelectric properties are not desirable. Hence, NBT-based solid solutions have been investigated to enhance the overall piezoelectric properties, since on one hand, it offers a larger degree of freedom to change the compositions, and on the other hand, it is significantly easier to improve the electrical and electromechanical properties.^13,14 Due to its excellent piezoelectric properties, the solid solution (1 – x)(Na_0.5Bi_0.5)TiO₃–x(K_0.5 Bi_0.5)TiO₃ is one of the most attractive and least explored systems by researchers. Kreisel et al.¹⁵ also defined the MPB region as well as the optimized composition of the binary material (1 – x)NBT–xKBT. Takenaka et al.¹⁶ and Jiang et al.¹⁷ reported that compositions in the near MPB region of 0.05–0.07 mol BT exhibit a piezoelectric constant in the range of d₃₃ = 125pC/N and 0.16–0.20 mol KBT with d₃₃ = 123pC/N, respectively. In particular, the piezoelectric properties can be enhanced when the solid solutions are in the morphotropic phase boundary (MPB) zone. These ceramics have attracted the interest of scientists around the world.¹⁸⁻²⁰ In previous research, the permittivity of ceramic materials was generally positive. In reality, it has been observed that the permittivity could be as well negative.²¹⁻²³ Up to now, negative permittivity generally results from the dielectric resonance of dipoles due to negative phase angle and electromechanical coupling at high frequencies.²⁴⁻²⁶ NBT exhibits a diffuse phase transition nature leading from its nonpolar ferroelectric tetragonal transitions to its ferroelectric rhombohedral transitions.²⁷⁻²⁹ Recently much work involved the enhancement of the electrical properties of (1 – x)NBT–xKBT by substitutions, investigation of impedance data to differentiate between the bulk contribution (grains) and the influence of the grain boundaries, and identifying the elements which are required for the conduction phenomenon in (1 – x)NBT–xKBT ceramics at high temperature.^30,31 Complex impedance spectroscopy (CIS) can be used as a convenient and useful nondestructive experimental method³² which allows us to (i) study the physical processes that reveal the electrical and dielectric properties of compounds, (ii) differentiate the contributions of grains and grain boundaries to the transport properties of compounds, (iii) identify the relaxation frequency/the relaxation time of the charge carriers in the material, and (iv) further investigate conductivity, relative permittivity, and dielectric losses versus frequency.^33,34 Motivated by the previous consideration, we prepared a (1 – x)NBT–xKBT system by solid-state reaction method (x(%) = 0, 12, 16, 20, 30, and 100) with compositions close to MPB. This study aims to examine in detail the influence of the addition of KBT on the generation mechanism of negative permittivity behavior, as well as the impedance and conductivity of NBT over a broad range of temperatures and frequencies. To understand the relaxor mechanism, three different methods, namely Debye, Vogel-Fulcher (V-F), and Arrhenius law, were applied in the dielectric study. This study indicates that the mechanism of NBT conductivity with KBT is mainly caused by oxygen vacancies and cation carrier hops.

2. Experimental Section

2.1. Material Synthesis and Characterization

(1 – x)NBT–xKBT samples were synthesized by a solid-state ceramic process by blending a stoichiometric amount of dried powders of high-purity (99.9 atom %) of Na₂CO₃, Bi₂O₃, K₂CO₃, Bi₂O₃, and TiO₂, and subsequently heating them for 4 h at 1000 °C in air. The chemical reaction for this synthesis is represented as follows:

1

The resulting powders were milled and blended with polyvinyl alcohol (PVA) binder solution. To perform the dielectric measurement, the powder was processed into pellets of 12 mm diameter and ∼1–2 mm thickness using pressing at a weight of 8 tons on a hydraulic press. The yellow pellets were sintered at 1100 °C for 4 h in air. The density of the pellet was determined by using the Archimedes technique, and it was revealed to be about ∼97.6% of the theoretical density of the material.

2.2. XRD Analysis

The phase formation investigation was performed using an XPERT-PRO X-ray diffractometer (XRD) with Cu-Kα radiation (λ = 1.54 Å) over the range 2θ ∼ 10–80° with a step size of 0.02°. The crystal phases were further refined from the Rietveld refinement process using the FULLPROF suite software³⁵ and the rhombohedral perovskite (Na_0.5Bi_0.5)TiO₃ (space group: R3c) as the model structure.

2.3. Dielectric Properties Measurement

A coating of silver paste was used on both sides, followed by heat treatment of the sample at 80 °C (30 min) for electrical measurements. The relative permittivity (ε_r) and dielectric loss (tan δ) values as a function of temperature were determined at different temperatures ranging from ambient to 600 °C, using an LCR-meter (Agilent 4284A) operating in the frequency range 1kHz–1 MHz.

2.4. Theory

In general, the AC impedance technique is used to characterize the electrical properties of materials. The data are analyzed using one of four complex admittance functions: complex impedance (Z*), complex permittivity (ε*), complex electric modulus (M*), and complex admittance A* (or Y*).³⁶

2

3

4

Also, ω is the angular frequency (2πf), C is the geometric capacitance of the sample and ε₀ indicates the permittivity of free space (8.854 × 10^–12 F/m). Complex impedance and Cole–Cole plot are utilized to distribute out the contribution of grain and grain boundary effects of the ceramics. Moreover, the temperature dependence of grain boundary influences and bulk capacitance can be investigated. As a result, in order to investigate and comprehend the experimental data, a model is required. The equivalent circuit provides a realistic description of the compound’s electrical behavior.

3. Results and Discussion

3.1. X-ray Diffraction (XRD) Analysis

Figure 1 shows the X-ray powder diffraction patterns of (1 – x)(Na_0.5Bi_0.5)TiO₃–x(K_0.5Bi_0.5)TiO₃ (x(%) = 0.0, 12, 16, 20, 30, and 100) calcined at 1000 °C for 4 h, and no secondary phase is detected. Indexation of the patterns was carried out using the software program Dicvol.³⁵ The NBT exhibits good agreement with the rhombohedral structure referred to in JCPDS card no. 36-0340.³⁷ Accounting for a small rhombohedral distortion, all diffraction patterns were indexed according to the pseudocubic unit cell of the perovskite. The absence of potassium-containing secondary peaks in the XRD profiles suggests that K⁺ ions were successfully integrated into the NBT perovskite structure up to x = 30% increase. Besides, the position of the XRD peaks shifts to lower angles as the KBT increases. This phenomenon indicates the dissolution of KBT within the structure and the formation of a binary solid solution system.³⁷ The phase transition process will be explored in more detail in the following pages. Knowing that the radius of K⁺ (r = 1.33 Å) is larger than that of Na⁺ (r = 0.95 Å),^13,38 the substitution of the relatively larger K⁺ for the smaller Na⁺ may induce an expansion of the unit cells.³⁹ The Rietveld refinement approach with a pseudo-Voigt analytical function was employed to refine the structural and microstructural parameters of the rhombohedral phase (space group R3c) by means of the Fullprof software.^35,40

Figure 1.

XRD patterns of (1 – x)NBT–xKBT samples (x(%) = 0.0, 12, 16, 20, 30, and 100) calcined at 1000 °C for 4 h.

The determination of the values of the lattice parameters, microstrain, particle size, etc. was performed by fitting the backgrounds of the different XRD profiles through a polynomial function of degree 6. The goodness of fit was enhanced via the Marquardt least-squares process.⁴¹ The minimization of the difference between the experimental (I_e) and calculated (I_c) data was evaluated as a report () called Goodness of Fit (GoF). The two parameters of the reliability index (R_wp and R_exp) are respectively the weighted residual error and the expected residual error. The refinement was carried out until the GoF value was found to be equal to ∼1. In addition, some important parameters are refined: scale factor, including zero correction, displacement, cell parameters and Wyckoff positions, half-width parameters of U, V, and W, and computed and preset atomic site occupancy. The obtained Rietveld-refined output profiles are shown in Figure 2. Compositions with low percentages of potassium (x = 0.0 and x = 12%) show peaks indexed in a rhombohedral lattice of space group R3c, whereas samples with high percentages of potassium (x = 30% and KBT) are indexed in a tetragonal lattice of space group P4mm. Furthermore, we find that the mixing of the phases (Rh and T) is in the region of 16% ≤ x ≤ 20%. To conclude, 0.84NBT–0.16KBT exhibits a morphotropic phase boundary (MPB) at room temperature, in which rhombohedral and tetragonal symmetries coexist.^42,43 Furthermore, the coexistence of tetragonal and rhombohedral phases with the P4mm and R3c space group in the morphotropic phase boundary (MPB) region at room temperature has also been reported by several authors.^28,38,44,45

Figure 2.

Rietveld refined powder XRD profile of (1 – x)NBT–xKBT samples.

In order to calculate, for each of our samples, the respective fractions of the rhombohedral and tetragonal phases, we used the X’pert High Score Plus software.⁴⁶ It should be mentioned that the crystallite size computed using the Williamson-Hall method is slightly larger than that computed by the Debye–Scherrer equation, since the broadening effect caused by the microstrain is entirely ruled out in the latter. The relative fractions of both phases are indicated in Table 1. The crystallite size, D, was calculated by the width of the X-ray diffraction peaks by means of the Williamson-Hall (W-H) method.⁴⁷ The total broadening, β_t, of the peaks is in fact due to the combined effects of crystallite size, β_D, and microstrain, β_ϵ.

5

The crystallite size is calculated on the basis of the Scherrer equation: , whereas the microstrain is obtained from the Stokes and Wilson formula.⁴⁷β_strain = 4ε tan θ. λ = 0.154 nm is the wavelength of the X-ray employed, K = 0.89 is the Scherrer constant, θ is the diffraction angle that allows to reach the position of the peak, in radians, and D is the crystallite size. Associating these latter equations (Sherrer and Stokes and Wilson) with equation 5 and multiplying both sides by cos θ, we obtain the following total broadening:

6

Table 1. Lattice Parameters, Strain, Phase Fraction (%), Crystallite Size, and Cell Size of the Pure and K-Doped NBTs.

Sample xKBT (%)	Structure	Phase fraction (%)	Lattice parameter (nm)	Crystallite size (D) (nm)	Strain (ε) ×10–2	Cell Size (Å3)	Reliability Parameters χ2 (%)
0.0	Rhombohedral	R3c 100%	a = b = 5.4856	131.91	0.0057	352.86	1.45
			c = 13.5402
12	Rhombohedral	R3c 100%	a = b = 5.4963	132.17	0.0061	355.08	1.19
			c = 13.5718
16	Rh+T	R3c 73%	a = b = 5.502	134.02	0.0065	356.090	1.42
			c = 13.5830			59.50
		P4mm 27%	a = b = 3.891
			c = 3.9298
20	Rh+T	R3c 71.13%	a = b = 5.5059	137.52	0.0067	356.74	1.45
			c = 13.5878			59.65
		P4mm 28.8%	a = b = 3.8897
			c = 3.9425
30	T	P4mm 100%	a = b = 3.9005	127.45	0.0073	60.10	1.59
			c = 3.9504
100	Tetragonal	P4mm 100%	a = b = 3.9247	129.13	0.0079	61.37	1.71
			c = 3.9844

Formula 6 allows a straight line to be drawn (y = ax + b). Thus, the average crystallite size, D, is determined from the intercept of the fitting line associated with the following values β_t cos θ versus 4 sin θ (see Figure 3). The resulting crystallite size and strain from this method for (1 – x)(Na_0.5Bi_0.5)TiO₃–x(K_0.5Bi_0.5)TiO₃ are reported in Table 1. The calculated values of strain for x(%) = 0.0, 12, 16, 20, 30, and pure KBT are 0.0057, 0.0061, 0.0065, 0.0067, 0.0073, and 0.0079, respectively. The calculated particle size of the samples as obtained ranges is from 129.13 to 137.52 nm. It is found that the value of D and the microstrain increase with increasing percentage of potassium in the solid solution and lead to the finding that due to the presence of potassium cations, the crystallites expand compared to those of the NBT sample.

Figure 3.

Williamson-Hall plot for (1 – x)NBT–xKBT (x(%) = 0.0, 12, 16, 20, 30, and 100) composites.

This result is due to the small ionic radius of Na⁺ (r = 0.95Å) compared to K⁺ (r = 1.33Å).¹³ Moreover, it might be attributed to the enhanced mechanical contact among the grains of different phases.

3.2. Bonding Charge Density

Figure 4 shows the 3D view of the ((1 – x)NBT–xKBT unit cell (x(%) = 0.0, 16, and KBT) generated by Vesta software taking into account the tilt of the TiO₆ octahedra, where the Ti⁴⁺ ion is in the center of the octahedron composed of six O^2– ions, whereas the Na⁺/Bi³⁺/K⁺ ions are in the space between the TiO₆ octahedra.

Figure 4.

Three-dimensional unit cell of (1 – x)NBT–xKBT (x(%) = 0.0, 16, and 100) with TiO₆ octahedra.

The octahedral tilt allows also the presence of enhanced ferroelectric properties in the compound. The appearance of a mixed phase in the specimens near the morphotropic phase boundary (x(%) = 16) suggests a complex bonding between different ions, leading ultimately to a transition in structural properties. As a result, the bond lengths and angles become significantly influenced and the crystal structure changes away from the structure of pristine NBT.^48,49

Another important result obtained by the Rietveld refinement process is the mapping of the electron density by means of the GFourier program, which allows the characterization and visualization of the electron density in the unit cell. The electronic charge density reveals the nature of the chemical bonds among the ions forming the (1 – x)NBT–xKBT powders and identifies the atomic positions of the component elements in the unit cell for crystals, i.e., the heavier electron density contours indicate the position of a larger element among the principal elements in the unit cell. The diffusion density ρ(x,y,z) is usually obtained according to the equation:

7

where ρ(x,y,z) represents the electron density at a point (x,y,z) within a unit cell of volume V, α_hkl represents the phase angle of each Bragg reflection, and F(hkl) represents the amplitude of the structure factor. Figure 5 displays the obtained 2-D contour maps of the charge density distributions for all structure of (1 – x)NBT–xKBT samples in the x–y plane. Two-dimensional maps are conventionally drawn using contours (and possibly color) to indicate various density layers, whereas three-dimensional maps use a “chicken-wire” mesh representing a single level. Although the regions colored from red to violet-brown indicate increasing levels of electron density about the Na/Bi cation, the zero level density contour is displayed in black. In the (1 – x)NBT–xKBT system under examination, the Ti and Na/Bi/K cations are located at (0, 0, 0.5) and (0.031, 0.28, 0.998), respectively. The electronic charge density map on all powders shows a strong charge localization along the radius of the circles about Na and Bi atoms interacting with the adjacent O atoms, which confirms the prevalence of the ionic character of the single bonds Na/K-O and Bi-O.^29,50,51 This ionic bond comes from a great difference in the Pauling electronegativity of O and Ti with the Na/K and Bi elements.^29,52,53 We also find that the bonding interaction of both atoms O and Bi is higher compared to that of Na/K. The maximum electron density was found for K (Z = 19) and Bi (Z = 83) which is ∼93 e/ų along the xy direction about K/Bi atoms, as illustrated in Figure 5. This results from the fact that the K/Bi atoms have a high atomic number compared to the other elements present in the compound.^50,54

Figure 5.

2-D electron density map of individual atoms on the xy plane in the unit cell of (1 – x)NBT–xKBT; the electron density is measured in electrons per cubic angstrom, e/ų.

The density map changes significantly after 30% potassium doping. The substitution of Na⁺ by K⁺ ion, that exhibits a partially filled orbital (4S¹), produced an improved K–O bond length and increased the charge density for this composition. This could be possibly associated with the new diffusion center brought by higher doping concentration. Figure 6 shows 3D Fourier maps of single atoms in the unit cell for all specimens. On the 3D charge density map, the site of Na/Bi/k, Ti, and O atoms and the electron clouds surrounding them are clearly shown. The high positive peaks associated with the 2b sites for Na/Bi cations (A-site) are present in all samples. An intermediate electronic density is present between the two peaks associated with the 2a sites for the Ti⁴⁺ cation (B-site).

Figure 6.

Three-dimensional electron density distribution in the unit cell of (1 – x)NBT–xKBT (x(%) = 0.0, 16, and 100).

On the other hand, electron density contour maps provide insight into the relationship between conductivity and structure.⁵⁵ Generally, in ABO₃-type ferroelectric perovskites, the anion conduction pathway is supposed to occur via a hopping mechanism by which evolving anions go via the bottleneck or the beginning of the critical radius of a triangle via socket points which are found between the A-site and B-site ions.⁵⁶ According to Singh et al,⁵⁵ the oxide ion migrates around the edges of TiO₆ octahedra via −O₁–O₁– routes, as well as through interstitials and saddle points within the crystal’s unit cell. The oxide ion migration mechanism favors the formation of a large number of defects between dopant and oxygen vacancies, which affects the conductivity of the material. This mechanism is in agreement with the conductivity behavior of our samples which will be examined in more detail later.

3.3. Study of Dielectric Properties

Figure 7 shows the real permittivity spectra of (1 – x)NBT–xKBT ceramics with different content of K⁺ ((x(%) = 0, 12, 16, 20, 30, and 100) as a function of frequency in the range from 1 kHz to 2 MHz. As indicated above, the dielectric constant of all specimens stays positive up to the resonant frequency (f_r) and negative above this frequency.^57,58 In the case of pure NBT, as the frequency increases, the dielectric constant increases steadily, then, in the vicinity of the anomaly noted A, it increases and reaches a maximum, at this frequency, which corresponds to the so-called resonance frequency (f_r), and then decreases rapidly for a frequency in the vicinity of the anomaly B, known as antiresonance (f_a). This particular type of peaking behavior is associated with resonance mechanisms.⁵⁹⁻⁶¹

Figure 7.

Relative permittivity at different temperatures for different xKBT contents in (1 – x)NBT–xKBT ceramics.

In addition, the resonance frequency obtained allows us to specify the frequency at which the dipole moment vector changes direction. This evolution reveals the presence of charged species (oxygen vacancies, grain boundary effect, etc.) unable to follow the variation of the alternating electric field for high frequencies.⁶² As the temperature increases, the amplitude of the dielectric constant also increases, since an increase in temperature causes the orientation of the corresponding dipoles. Moreover, the value of the anomaly A shifts to a lower frequency until a temperature of T = 350 °C, at which there is an inverse shift of the resonance frequency (f_r) to a higher frequency. As the xKBT content increases, there is a slight decrease in the dielectric permittivity. However, over the morphotropic limit (MPB), the maximum dielectric permittivity increases for the xKBT rich region (x = 30% and KBT), as shown in Table 2. Furthermore, The increase in the charge density values along the Na/Bi/K-O₁ and k-O₁ bonds (verified previously) is good evidence for the increase of charge carriers (Na⁺/B³⁺ ions and K+ ions) in the potassium-substituted system, which can be attributed to the increase of dielectric properties over the morphotropic limit (MPB), the maximum dielectric permittivity increases for the xKBT rich region (x = 30% and KBT), and the alternative conductivity. Consequently, we observe that the addition of xKBT reduces the dielectric constant in the morphotropic zone (MPB) and promotes it beyond this zone. On the other hand, it is necessary to identify the origin of this behavior by complex impedance spectroscopy, i.e., analysis of the imaginary and real part of the complex impedance, a study of the AC conductivity, and examination of the dielectric modulus.

Table 2. Resonant Frequency and Permittivity of (1 – x)NBT–xKBT Samples at Different Temperatures.

T (°C)	NBT		12%		16%		20%		30%		KBT
	fr (MHz)	εr	fr (MHz)	εr	fr (MHz)	εr	fr (MHz)	εr	fr (MHz)	εr	fr (MHz)	εr
280	1.618	3283	1.937	1928	1.998	3053	1.817	4427	1.990	3549	1.381	5559
300	1.582	3678	1.916	2747	1.977	3705	1.657	4535	1.974	4058	1.234	5380
320	1.341	4391	1.892	3137	1.875	3875	1.535	4625	1.941	4334	1.183	5299
350	1.118	5143	1.823	3598	1.830	3596	1.520	4371	1.964	3975	1.125	5189
400	1.142	3384	1.906	3422	1.881	3160	1.567	3829	1.978	2638	1.135	5110
450	1.283	3203	1.947	2511	1.945	2196	1.740	3431	1.988	1796	1.199	3138

The temperature dependence of the dielectric constant (ε_r) and dielectric loss (tan δ) at various frequencies for (1 – x)NBT–xKBT ceramics sintered at a temperature of 1100 °C is depicted in Figure 8. This figure displays two significant dielectric peaks of the (1 – x)NBT–xKBT ceramic materials. The first transition temperature is associated with the depolarization temperature (T_d), and the second transition temperature is related to T_m and is near the maximum value of (ε_rmax). Jo et al. suggested that the two detected local dielectric maxima (T_d and T_m) in NBT-based ceramics are not related to phase transitions, though they can be attributed to the thermal behavior of the local ferroelectric polar nanometer-size regions (PNRs) of rhombohedral R3c and tetragonal P4bm symmetries.⁶³T_d is the depolarization temperature that corresponds to the transition from a ferroelectric state to an antiferroelectric state and is related to the thermal behavior of R3c and P4bm of polar nanometer-size (PNRs), which is associated to with the transformation from classic ferroelectric to relaxor ferroelectric.⁶⁴ The depolarization temperature (T_d), having been proposed as a measure of the stability of ferroelectric domains, has an important role in the practical application of NBT-based ceramics.⁶⁵ The temperature of dielectric maximum (T_m) is the temperature at which the transition from an antiferroelectric state to a paraelectric state occurs. At T_m, the dielectric curve exhibits broad dielectric maxima and moves toward the highest temperature along with the dielectric constant decreasing with increasing frequency. It is also the temperature at which the thermal evolution of the relaxation time distribution or correlation length distribution of polar nanometer-size regions occurs (PNRs).^63,64,65,66 At these temperatures, the dielectric properties become frequency dependent, indicating that the (1 – x)NBT–xKBT materials are ferroelectric relaxors. Furthermore, upon increasing frequency, the dielectric peaks displace slightly further along the high-temperature direction, a behavior characteristic of dielectric relaxation, i.e., frequency dispersion. Similar findings have also been reported by various research groups, it has been claimed in the NBT system in the vicinity of the MPB composition that the phase region displaying P4bm symmetry with integrated nanodomains in the R3c matrix participates in the relaxor nature.⁶⁷⁻⁶⁹ Dielectric losses are low between ambient temperature and 300 °C. At high temperatures, there is a sharp rise in dielectric loss that could be connected to an increase in electrical conductivity. Therefore, it is proposed that (1 – x)NBT–xKBT ceramics may be a candidate for high-temperature capacitors. This results from the inhomogeneity of the cation distribution which generates a random local field that destabilizes the long-range ferroelectric order, and the system is fragmented into polar nanoregions (PNRs).^70,71

Figure 8.

Temperature dependences of dielectric properties for the (1 – x)NBT–xKBT ceramics over −40 °C–600 °C at various frequencies.

The relaxation nature of a system can, after all, also be verified by various methods such as the formation of polar nanometer-size regions (PNRs),⁶⁷ PE loops,⁷² by comparing the variation of ε″, M″, and Z″ with frequency,⁷³ Vogel-Fulcher relation,⁷⁴ and from diffuse phase transition.⁷⁵

In addition, many ABO₃-type perovskites and ceramics based on NBT have the diffuse phase transition reported to occur when the A or B sites are occupied by at least two cations.⁶⁵ The dielectric behavior of complex ferroelectrics with diffuse phase transition may be understood using the modified Curie–Weiss law.⁷⁶

8

T_m is the temperature at which r reaches its highest value, ε_r,max is the permittivity’s maximum value, and γ (1 < γ < 2) is a parameter that indicates the degree of phase transition diffusion. Additionally, when the diffusivity value is equal to one, we discover that samples displaying this sort of phase transition are classical ferroelectrics; relaxor ferroelectrics when the diffusivity parameter is 1< γ < 2; and ideal relaxor ferroelectrics when the diffusivity parameter is equal to two. Graphs of ε_r,max/ε_r versus (T – T_m)^γ, at 100 kHz, for (1 – x)NBT–xKBT systems sintered at 1100 °C for 4 hours are plotted in Figure 9. Equation 8 was fitted in the region of T_m in accordance with findings obtained from the experimental data for the dielectric constant. The fact that the several testing sites all neatly line up on a straight line indicates that these materials do in fact satisfy this modified law. With an increase in xKBT, the diffusivity parameter increases from 1.6 for pure NBT to 1.8 for NBT–0.20KBT, as can be seen in the findings (see Figure 9). Table 3 displays the calculated values of γ and δ. It is most probable that a disorder at the A-site (Na⁺/K⁺ and Bi³⁺) is what is causing the high values around the morphotropic phase boundary (MPB) (NBT–0.20KBT). In light of this, the dispersive behavior may be brought on by the cationic disorder in the A-site, which might result in relaxation. These findings back up previous dielectric measurements published in the literature.^77,78

Figure 9.

Dielectric data fitted with Uchino’s law for (1 – x – y)NBT–xKBT ceramics.

Table 3. Values of γ and δ as a Function of Potassium Content in (1 – x – y)NBT–xKBT Systems.

Samples	NBT	KBT	NBT-0.12KBT	NBT-0.16KBT	NBT-0.20KBT	NBT-0.30KBT
γ	1.6	1.5	1.6	1.7	1.8	1.6
δ	96	110	124	127	135	125

3.4. Impedance

Cole–Cole plots for (1 – x)NBT–xKBT samples (x(%) = 0, 12, 16, 20, 30, and KBT) as a function of temperature in the range of 280 to 450 °C are presented in Figure 10. In low temperature, the Z values are extremely high, suggesting a high material resistance. The Z″ vs Z′ patterns for low temperature line up linearly with the Z″ axis (which is not shown in the Figure 10). These curves exhibit the bending of the graphs around the Z′ axis as the temperature rises, and these graphs change to almost semicircular arcs as the as a function of temperature. The area surrounded within the semicircle decreases as the temperature increases, suggesting a reduction in the resistivity of the material. The centers of the semicircular arcs found under the real axis indicate a non-Debye type of relaxation in the material.⁷⁹⁻⁸¹

Figure 10.

Complex impedance diagram at different temperatures for different xKBT contents in (1 – x)NBT–xKBT samples.

Table 4 displays the fitted parameters (R_g, R_gb, C_g, and C_gb) for all samples at 400 °C using an equivalent circuit consisting of a resistor and a capacitor connected in parallel. With increasing potassium doping rate, the (1 – x)NBT–xKBT system’s R_g decreases while its C_g increases. It is also discovered that as the (K⁺) doping rate increases, the grain boundary resistance increases. This can be explained by the introduction of potassium, which causes a decrease in grain size, resulting in an increase in the number of grain boundaries. Christie et al.⁸² and Badwal et al.⁸³ observed the same behavior in rare-earth doped perovskites. It may be also observed that the value of R_g decreases with rising temperature, which confirms the negative temperature coefficient of resistance (NTCR) behavior or semiconductor behavior.^30,84,85 According to M. Benyoussef et al.,⁸⁴ these results confirms the materials NTCR (Negative Temperature Coefficient of Resistivity) behavior and reflects the semiconductive nature of ceramics. The reason for this is the increase of mobility of space charges with rising temperature.^30,86 On the other hand, the asymmetric shrinkage of the semicircles as a function of temperature suggests an inhomogeneous distribution of the relaxation time (τ = 1/ω).^18,87,88

Table 4. Values for Grain Resistance (R_g), Grain Joint Resistance (R_gb), Grain Capacity (C_g), and Grain Joint Capacity (C_gb) of the (1 – x)NBT–xKBT System.

Sample (%)	Rg (KΩ)	Rgb (KΩ)	Cg (pF)	Cgb (pF)
NBT	10.45	1.23	0.78	0.21
x = 12	9.78	1.89	0.87	0.11
x = 16	7.59.	2.74	0.92	0.48
x = 20	7.90	3.24	1.16	0.52
x = 30	7.47	5.38	1.78	0.67
KBT	6.13	2.11	1.98	0.81

Figure 11 presents the variation of the imaginary part of the impedance (Z″) with frequency at different temperatures for (1 – x)NBT–xKBT (x(%) = 0, 12, 16, 20, 30, and 100 compounds). The single curve represents the overall response of the samples. The grain boundary effects become negligible in the presence of large grains. These plots present broad curves at temperatures above 300 °C as a function of xKBT concentration, and the width of the peak was shown to increase with temperature. Furthermore, the value of the width at half-maximum (fwhm) revealed from the normalized spectrum is superior to 1.144 decades (ideal Debye relaxation). This confirms the non-Debye behavior in the (1 – x)NBT–xKBT material.⁸⁹ The broad peak is associated with the cation disorder problem resulting from the random distribution of A and B site cations over the equivalent lattice positions.⁹⁰ In these graphs, we can observe the effect of increasing temperature on the value of Z″. Z″ increases with frequency and reaches a maximum value at a particular frequency, which is designated as the relaxation frequency. Its associated relaxation time is determined by the following equation: ω_maxτ = 1, where ω_max is the angular frequency which corresponds to the relaxation frequency and τ is the relaxation time necessary for the emergence and disappearance of the polarization. The relaxation peak shifts toward a lower frequency with rising temperature owing to the increase in relaxation time, suggesting the presence of a temperature-dependent electrical relaxation phenomenon in the sample.^91,92 The peaks broaden asymmetrically, indicating the presence of particular electrical processes within the material with a dispersed relaxation time, which will be detailed in the following section. The reason for this may be the appearance of immobile species at low temperatures and defects at high temperatures.^92,93 Since these investigations are carried out at higher temperatures, it is possible that some relaxation species, such as defects, are the cause of electrical conduction in the product through the jumping of electrons/ oxygen ion vacancy/defects between the located sites present. The relaxation times are calculated from the plot of Z″ against the frequency. Moreover, the variation of τ as a function of the inverse of the absolute temperature is presented in the following section.

Figure 11.

Variation of Z″ with frequency for (1 – x)NBT–xKBT samples.

3.5. Model Fitting

To better understand the dispersive dielectric relaxation exhibited for (1 – x)NBT–xKBT, the distribution of relaxation time being temperature-dependent or not was examined. The relaxation time of the materials are determined by the following relationship:

9

where τ₀, K_B, and E_a are pre-exponential values, Boltzmann constant, and the activation energy, respectively. The relaxation time τ is determined as follows:

10

In which the frequency, f_max, is equal to Z″ maxima. The plot between the log τ vs 1000/T is presented in Figure 12, and the activation energies obtained from the linear fit of the graphs are reported in Table 5. Moreover, the activation energy value for relaxation decreases with increasing xKBT content in these materials. The activation energy values of the materials are in the same range as those of NBT and KBT. In the present study, disorder was gradually introduced by breaking up the long-range symmetry of the ferroelectric KBT lattice in the NBT lattice and thus observing the relaxor behavior in the (1 – x)NBT–xKBT samples. However, many researchers have proven that dipoles are not free and that there can be an interaction between dipoles.^40,86,94 This interaction results in the freezing of the dipoles at a particular temperature denoted T_vF (freezing temperature). This disagreement of the Debye model is accounted for in the Vogel-Fulcher (V-F) model which will be discussed in detail in the next section. The Vogel-Fulcher law was used in this work to determine the average relaxation time. (The fitting curves of ln f(Hz) versus T_m are given in Figure 13).

Figure 12.

Variation of relaxation time with absolute temperature (1 – x)NBT–xKBT samples.

Table 5. Variation of Activation Energies for NBT, KBT, and (1 – x)NBT–xKBT Samples.

Sample	DC Activation energies (eV)	AC Activation energies (eV) for conduction	AC Activation energies (eV) for relaxation (τR)	Activation energies (eV) for relaxation (τM)
NBT	0.56	0.97	0.90	2.11
12%	0.60	0.70	0.63	0.57
16%	0.83	0.74	0.69	0.78
20%	0.91	0.76	0.94	0.98
30%	1.02	0.67	1.08	1.21
KBT	1.05	0.68	1.19	1.27

Figure 13.

Variation of ln (f) with temperature (T_m) and fitted curve of Vogel-Fulcher relation for (1 – x)NBT–xKBT. (x(%) = 0, 0.12, 0.16, 0.20, 0.30, and 100 ceramics).

The relaxation behavior of the system may also be examined by means of the Vogel-Fulcher (V-F) law³² being fitted to the samples, which is illustrated in Figure 13. The nonlinear VF fit provides diverse parameters for the quantitative measurement of the relaxant characteristics of all materials. It takes into consideration the interaction between the dipoles. These interactions cause the dipoles to freeze at a specific temperature called the Vogel-Fulcher temperature. This model is due to thermally activated polarization fluctuations related to polar nanometer-size regions (PNRs).^66,71

The relaxation times of the materials follow the equation

11

In which τ₀ is the pre-exponential value, k_B is the Boltzmann constant, T is the temperature, E_a is the activation energy, and T_vF (freezing temperature) is the temperature from which dynamic reorientation of the dipolar cluster polarization becomes thermally activated. Figure 13 displays the temperature dependence of the average relaxation time for all studied ceramics. Based on an analysis of the V-F fit, we have found the activation energy and some other parameters; among them E_a = 0.09 eV, τ₀ = 1.5910^–7 and T = 593 k with R² = 0.998 indicate that NBT presents a relaxing behavior. The fitted parameters for the samples are listed in Table 6. The distribution of relaxation times and the activation energy obtained from the Vogel–Fulcher law increase when the potassium content in the ceramics increases while the freezing temperature decreases. At T – T_vF temperatures, glassy relaxations may actually be detected.⁹⁵ It is noted that the T_vF decreases by adding KBT to the ceramics. Furthermore, the difference in τ₀ and E_a is coherent within a thermally activated polarization fluctuation that might originate from the large grain size difference such as an effect of the glass.⁹⁶

Table 6. Vogel Fulcher Fitting Parameters for (1 – x)NBT–xKBT Samples.

Sample	Freezing temperature (TvF) (K)	Activation energy (Ea) (eV)	Pre exponential Factor (10–8 s)	Relaxation time (τ)
NBT	593	0.09	0.275	5.07 × 10–4 s
12%	589	0.12	0.389	9.98 × 10–4 s
16%	571	0.15	0.412	12.37 × 10–4 s
20%	563	0.19	0.512	4.09 × 10–4 s
30%	552	0.27	0.433	16.22 × 10–4 s
KBT	547	0.32	0.312	12.32 × 10–4 s

3.6. Electrical Conductivity

Electrical conductivity refers to the extent of charge movement within the material. The properties (insulator, semiconductor, conductor) of the product can be determined according to the movement of the charge.³² Based on the dielectric data measured, the AC conductivity (σac) was evaluated according to the following relationship:

12

The different symbols have a common meaning. The variation of AC conductivity as a function of frequency for (1 – x)NBT–xKBT ceramics is shown in Figure 14. From this plot it can be seen that for all ceramics, the AC conductivity increases with frequency. The AC conductivity at every temperature considered has two distinct regions. For low frequencies, we can observe a first region where the conductivity remains unchanged with the frequency. In the second region (>10⁶ Hz), the conductivity increases as the frequency increases. In addition, it was found that the conductivity increases with the potassium content. This behavior can be attributed to the effect of grain boundaries having a large number of defects and oxygen vacancies.⁹⁷ These defects/gaps result in the addition of free charge carriers which can increase the conductivity at higher frequencies under the effect of the applied field. Such AC conductivity is explained according to the Jonscher’s power law.^32,80,98

13

Figure 14.

AC conductivity as a function of frequency at different temperatures for (1 – x)NBT–xKBT ceramics.

The σ_dc specifies the DC conductivity, and the second term is the frequency dependent AC conductivity. “A” refers to the polarization force, and “n” is the temperature-dependent parameter. The evolution of the parameter n as a function of temperature is employed to evaluate the conduction mechanism. The physical significance of the parameter n numerical values has been explained by Funke et al.⁹⁹ For n > 1, conduction through the lattice is due to the Maxwell–Wagner (M-W) mechanism, while at n ≤ 1, it is attributed to the correlated barrier jump (CBH) mechanism. The exponent value (n) is between 0 and 1. Since we have calculated n < 1, the conduction mechanism is of the non-Debye type (Table 7). These results suggest that correlated barrier hopping (CBH) is the most adequate model to explain the mechanism of charge transport in these materials.

Table 7. Parameters Obtained from Joncher’s Plot for (1 – x)NBT–xKBT Ceramics.

Potassium content (%)	Temp (°C)	n
NBT	300	0.50782
	350	0.46636
	400	0.42786
	450	0.41492
16	300	0.52114
	350	0.47761
	400	0.44274
	450	0.40821
KBT	300	0.62171
	350	0.58727
	400	0.55241
	450	0.54473

Figure 15 displays the variation of ln(σ_DC) against the inverse of temperature (1000/T), which provides the activation energy value for electrical conduction of (1 – x)NBT–xKBT ceramics at 100 kHz, determined from Arrhenius’ law.

14

where σ₀ is the pre-exponential factor, k_B is the Boltzmann constant, T is the temperature (K), and E_a is the activation energy. The plots show that an increase in conductivity increases with temperature, indicating that the samples have conductive behavior. Two different activation energies were identified for (1 – x)NBT–xKBT ceramics, namely, at grains and at grain boundaries. Moreover, the activation energy of the grain is significantly higher than that of the grains’ boundaries. Here again, this indicates that the grains exhibit a higher resistance than the grains’ boundaries, as obtained earlier from the Nyquist plots (see Table 4). The value of E_a for these samples is significantly in the range ∼1.12 to ∼1.37 eV, revealing that the creation of internal defects for all samples is primarily associated with internal oxygen vacancies. Consequently, for NBT ceramics, the conductivity is essentially attributable to oxygen ion vacancies or Bi ion vacancies. Indeed, O^2– ions are located in the octahedral sites of these perovskite ceramics and transport at high temperatures. Meanwhile, Bi ions are at the center of these oxygen dodecahedra, and the high size needs a high thermal energy to be activated to mobilize them.

Figure 15.

Conductivity ln(σ_DC) as a function of 1000/T for (1 – x)NBT–xKBT ceramics (a) for grains and (b) for grain boundaries at 100 kHz.

4. Conclusions

In this study, (1 – x)NBT–xKBT ceramics near the morphotropic phase boundary were synthesized by a solid-state reaction method and sintered. The structural analysis was carried out on the X-ray diffraction data employing the Rietveld refinement process. The structure factors yielded by the refinement technique were employed both to obtain the average particle diameter of various perovskite compounds, and revealed a range between 129.13 and 137.52 nm from the Williamson-Hall plot, and qualitatively and quantitatively study the charge density distribution and the nature of the bonds. Dielectric investigations reveal two anomalies marked A and B depending on the frequency for all specimens, which correspond to the frequency of resonance (f_r) in A, and for a frequency near the anomaly B, the frequency is referred to as antiresonance (f_a). As a function of increasing xKBT content, we find that an increase in the dielectric permittivity value is found. Nevertheless, the dielectric permittivity decreases above the morphotropic phase boundary (MPB) in the xKBT rich region (x = 30% and KBT). The imaginary part of the impedance spectra (Z″) shows that the distribution of relaxation time is temperature independent. The relaxation frequency (f_max = 1/τ) varies accordingly with the temperature as well as the composition of the xKBT. Complex impedance spectroscopy was carried out across a large frequency interval (500–2 MHz) at different temperatures to improve comprehension of the electrical properties and their relationship to the grains and grain boundaries of the ceramics. In addition, the experimental data of the Nyquist plot have been explained correctly by means of theoretical simulation, and both grains and grain boundaries have been identified as contributing to the conduction process. The V-F and power-law models were employed to investigate the activation energy anomaly. Using these models, the freezing temperature and the degree of relaxation were also obtained. The non-Debye relaxation process is confirmed by employing conductivity and impedance analyses, and the activation energy for the relaxation has been derived according to Arrhenius’ law. The Joncher power-law parameter indicates a related barrier hopping (CBH) mechanism for the charge carriers. Two different activation energy regions were identified using a fit of the Arrhenius formula at high temperatures. The activation energies that are calculated from Arrhenius’ law are usually larger than the VF activation energies.

The authors declare no competing financial interest.