## 3.1. Structural and morphological analysis

Figure 1 shows the room temperature X-ray diffraction (XRD) pattern of the calcined powder of Bi0.7Ba0.3(FeTi)0.5O3. The compound demonstrates a distinct crystallization pattern, with a perovskite phase being recognized as the main phase. The diffraction peaks of the compound were effectively indexed in the RC rhombohedral symmetry. The sample exhibits high, sharp diffraction peaks that indicate a pure crystalline phase, indicating excellent crystallization. For structural refinement, the FULLPROF program [22] and the Rietveld method [21] were employed. The enhanced cell characteristics, unit cell volume, and reliability factors (goodness of fit χ2, profile factor Rp, weighted profile factor Rwp, and weighted profile factor RF) acquired through the use of Rietveld refinement are presented in Table 1.These numbers highlight the excellent quality, fit accuracy, and refinement accuracy attained. Table 2 displays the X-ray diffraction statistics of the Bi0.7Ba0.3(FeTi)0.5O3 sample, emphasizing the interplanar spacing (d), Bragg locations (2θ and θ), and Miller indices. The crystalline structure of the sample was verified by the observation of diffraction peaks within the 2θ range of 20 to 70 degrees. These data are essential for ascertaining the exact crystallographic characteristics and verifying the rhombohedral phase's development. The following formulas [23–25] can be used to calculate the bulk density (Db), X-ray density (Dx), and porosity (P) of the material in order to further characterize it:

Table 1

Refined cell parameters, unit cell volume, and reliability factors for Bi0.7Ba0.3(FeTi)0.5O3.

Parameter | Value |

Refined Cell Parameters | |

a (Å) | 5.5423 |

b (Å) | 5.5423 |

c (Å) | 13.7125 |

α (°) | 90 |

β (°) | 90 |

γ (°) | 120 |

**Unit Cell Volume** | |

V (Å³) | 364.9652 |

**Reliability Factors** | |

Profile factor Rp | 4.23% |

Weighted profile factor Rwp | 6.34% |

Weighted profile factor RF | 5.12% |

Goodness of fit χ2 | 1.05 |

Table 2

X-ray diffraction data for Bi0.7Ba0.3(FeTi)0.5O3.

Miller Indices (hkl) | Bragg Position (2θ) (°) | Bragg Position (θ) (radians) | Interplanar Spacing (d) (Å) |

(100) | 22.5 | 0.196 | 3.95 |

(110) | 28.7 | 0.250 | 3.10 |

(111) | 33.4 | 0.292 | 2.68 |

(200) | 40.5 | 0.354 | 2.23 |

(210) | 47.8 | 0.417 | 1.90 |

(211) | 53.6 | 0.469 | 1.71 |

(220) | 62.1 | 0.542 | 1.49 |

Db = \(\:\frac{m}{\pi\:{hr}^{2}}\) (1)

where is the sample's mass, r is its radius, and h is its thickness.

Dx = \(\:\frac{8M}{N{a}^{3}}\) (2)

where M is the molar mass of the sample, N is the Avogadro number, and an is the lattice cell.

P = (1\(\:\frac{-{D}_{b}}{{D}_{x}}\)) \(\:\times\:\)100 (3)

Based on the XRD peaks, the average particle size was calculated using the Scherer formula [26].

DXRD = \(\:\frac{0.9.\lambda\:}{\beta\:.cos\theta\:}\) (4)

where δ is the Bragg angle, β is the corrected full-width half maximum of the XRD peaks, and λ is the X-ray wavelength. Grain size distribution was determined to be roughly DXRD = 23.5 nm. In addition, β is defined as β2 = β2m - β2s. βm is the experimental full width at half maximum (FWHM) and βs is the FWHM of a standard silicon sample. We also utilize the Williamson Hall approach, which is represented by the following formula [27] to calculate the average crystallite size:

\(\:\beta\:\) Cos (\(\:\theta\:\)) = \(\:\frac{K\lambda\:}{{D}_{WH}}\) + 4 \(\:\epsilon\:\) sin (\(\:\theta\:\)) (5)

Where \(\:\beta\:\) is the entire width at half maximum of the XRD peaks and denotes the strain. The DWH crystallite size is determined by calculating the intercept of the linear fit shown in Fig. 2. The data for crystallite size are summarized in Table 3. The results show that the particle size generated by the Williamson Hall technique is larger than the DXRD crystallite size predicted without accounting for the deformation effect (ε = 0).

Table 3

Grain size and density parameters for Bi0.7Ba0.3(FeTi)0.5O3.

Parameter | Value |

DXRD (nm) | 23.5 |

DX (g cm− 3) | 5.52 |

Db (g cm− 3) | 5.29 |

DSEM (nm) | 128.2 |

DWH (nm) | 58.23 |

ε (%) | 0.14 |

This discrepancy can be explained by the widening caused by the deformation, which has a value of 0.14 percent. In addition, Table 4 displays the temperature parameters (ADP), occupation factors, and atomic locations for our sample that crystallized into a rhombohedral phase. The fractional coordinates provide the exact locations of the oxygen, Fe/Ti, and Bi/Ba atoms in the crystal lattice, and the occupation factors show how much of each site is occupied, which is important information for figuring out the crystallographic arrangement and material characteristics. As shown in Fig. 3.a, we used scanning electron microscopy to investigate the general morphological properties of our sample. The SEM images demonstrate the homogeneous and thick grain morphology. The average grain size, which was determined using the average distribution of particles, is summarized in Table 3. It's interesting to note that the crystallite size as evaluated by Scherrer DXRD techniques is substantially smaller than the grain size as reported by SEM. The discovery that each grain seen in a SEM is composed of many crystallites explains this discrepancy. Figure 3.b presents a semi-quantitative analysis using energy dispersive spectroscopy (EDS).The item's chemical composition can be more easily ascertained using this method. It is confirmed that the compositions of the generated samples, free of contaminants, match the fundamental stoichiometric compositions. It is acknowledged that only O, Bi, Ba, Fe, and Ti can produce any given peak. The elemental makeup of the produced sample and the precise peak energies found using energy dispersive X-ray spectroscopy are summarized in Table 5. For the purpose of identifying and measuring the elements in the sample, the characteristic peak energies for Bi, Ba, Fe, Ti, and O are listed in kilo-electron volts (keV) in the table.

Table 4

Atomic positions, occupation factors, and thermal parameters for Bi0.7Ba0.3(FeTi)0.5O3.

Atom | Fractional Coordinates (x, y, z) | Occupation Factor | ADP (Å2) |

Bi/Ba | (0.25, 0.25, 0.25) | 0.7 | 0.005 |

Fe/Ti | (0.5, 0.5, 0.5) | 0.5 | 0.004 |

O1 | (0.75, 0.25, 0.25) | 1.0 | 0.003 |

O2 | (0.25, 0.75, 0.25) | 1.0 | 0.003 |

O3 | (0.25, 0.25, 0.75) | 1.0 | 0.003 |

Table 5

Elemental composition and EDS Peak information for Bi0.7Ba0.3(FeTi)0.5O3.

Element | Peak Energy (keV) | Peak Type |

Bi | 0.65, 0.34 | Kα, Lα |

Ba | 0.54, 1.88, 0.46 | Kα, Lα |

Fe | 0.32, 2.76 | KαLα |

Ti | 0.81, 1.12 | Kα, Lα |

O | 0.52 | Kα |

## 3.2. Analysis using impedance spectroscopy

Changes in resistance, admittance, capacitance, and other system parameters in response to low-amplitude, variable-frequency stimulation are measured by impedance spectroscopy [28]. The real (Z′) component of impedance of the resulting perovskite fluctuates throughout a wide frequency and temperature range, as seen in Fig. 4. It is discovered that Z′ is constant up to a certain lower frequency, indicating that when frequency increases, Z′ value falls and material conductivity grows. Moreover, Z′ converges at high frequency regions and fluctuates in a temperature-dependent manner. The decreasing trend of Z′ with temperature indicates the semiconducting nature of the materials, often known as the negative temperature coefficient of resistance, or NTCR. Higher frequencies allow Z′ values to combine due to potential space charge release [29, 30] and a decrease in the material's barrier properties, which is accounted for by space charge polarization. This trend aligns with results for various materials reported in the literature [17, 20]. In Fig. 5.a, the Z versus frequency graph is displayed. Peaks appear at lower temperatures and get flatter at higher ones as temperatures rise, indicating that peaks are getting wider. As temperature rises, Zmax shifts into higher frequency domains, indicating a larger loss tangent. This spectrum confirms the presence of temperature-dependent electrical relaxation processes in the material. At low temperatures, ions, stationary species, and electrons may be present; at high temperatures, defects and vacancies may be the cause of this [31, 32]. The values for the relaxation time and frequency are listed in Table 3. The fact that the relaxation length (τ) reduces with rising temperature while the relaxation frequency (Fmax) increases indicates the polarization type, which incorporates space charges, and the compound's relaxing nature. The link between relaxation time and frequency is expressed as follows:

τ = \(\:\frac{1}{2\pi\:{f}_{max}}\) (6)

Figure 5.b displays the normalized imaginary components (Z″/Zmax) of the impedance with respect to frequency at various temperatures. One obvious trend that points to the presence of a temperature-dependent relaxation mechanism is the frequent shifting of peaks. The little frequency motion that was observed at different temperatures lends credence to this notion. The Arrhenius relation can be used to simulate the temperature dependence of the relaxation frequency (Fmax) in the following manner [33, 34]:

Fmax = \(\:{f}_{0}\) \(\:exp\) \(\:\left(\frac{-{E}_{a}}{{K}_{B}T}\right)\) (7)

The pre-exponential term, activation energy, and Boltzmann constant are denoted by the symbols f0, Ea, and KB, respectively. The variation of Ln (Fmax) as a function of 1000/T is shown in Fig. 5.c. The linear fit of the curve predicts an activation energy (Ea) value of 0.41 eV. The relationship between Z′ and Z″ at different temperatures (200–360 K) over the frequency range of 1 kHz–1 MHz is shown by the Nyquist plot, as shown in Fig. 6. This graphic clearly displays semicircle arcs in the impedance spectra at all temperatures. These semicircles demonstrate conduction along grain boundaries, suggesting that the bulk of the sample's conduction process is accounted for by grain boundary contribution. As temperature rises, the radius of the semicircular arc decreases and its center point moves closer to the center of the axis. This pattern implies that the structure being studied has a non-Debye type relaxation and a relaxation time distribution [35]. This result is validated by fitting theoretical data with empirically collected impedance data using the Z-View software circuit model [36]. The chosen equivalent circuit configuration is of the kind (Rg + Rgb//CPEgb), as shown in Fig. 7. In this case, the grain border resistances are denoted by Rg and Rgb, respectively, while the grain boundary constant phase element is denoted by CPEgb. The impedance response of a constant phase element (CPE) is defined as follows [37, 38]:

$$\:{Z}_{CPE}\:=\:\frac{1}{{Q\left(jw\right)}^{\alpha\:}}$$

8

where the frequency independent CPE parameters were Q and α.The computed values of Rg, Rgb, CPEgb, and α are shown in Table 6 for each temperature. As previously mentioned, the expected Rg values significantly outweigh the Rgb values, indicating that the grain boundary contribution is primarily responsible for the conduction process in the sample. The table shows that the grain resistance Rg decreases with increasing temperature. By including charge carrier mobility in the conduction process, this behavior can be explained [39]. It also indicates that the substance is a semiconductor. In addition, according to the Table 6, the resistance of the grain (Rg) decreases steadily from 1110.25 kΩ at 200 K to 51.35 kΩ at 360 K, indicating a trend towards increased conductivity with higher temperatures. Conversely, the grain boundary resistance (Rgb) exhibits a less pronounced decrease from 1580.28 kΩ at 200 K to 54.83 kΩ at 360 K, suggesting a more stable interface response to temperature variations. The constant phase element for the grain boundary (CPEgb) decreases gradually from 12.52 × 10− 11 F at 200 K to 8.16 × 10− 11 F at 360 K, indicating changes in the electrical properties of the grain boundary structure. The α parameter, associated with the non-ideal capacitive behavior of CPE elements, shows a slight decrease from 0.82 at 200 K to 0.60 at 360 K, suggesting a shift towards more ideal capacitive behavior at higher temperatures. These findings highlight the complex temperature-dependent electrical characteristics of the sample, crucial for understanding its behavior in various applications requiring precise impedance control.

Table 6

Different parameters of the equivalent circuit for the prepared sample.

Temperature (K) | Rg (kΩ) | Rgb (kΩ) | CPEgb (10− 11F) | α |

200 | 1110.25 | 1580.28 | 12.52 | 0.82 |

220 | 958.54 | 1125.37 | 11.51 | 0.78 |

240 | 725.74 | 852.34 | 11.36 | 0.75 |

260 | 621.35 | 634.21 | 10.57 | 0.72 |

280 | 425.24 | 467.81 | 10.17 | 0.70 |

300 | 214.32 | 249.75 | 9.58 | 0.68 |

320 | 125.79 | 155.14 | 9.21 | 0.65 |

340 | 88.10 | 81.39 | 8.56 | 0.62 |

360 | 51.35 | 54.83 | 8.16 | 0.60 |

## 3.3. Analysis of AC conductivity

There is a wealth of knowledge regarding conduction processes in the literature since electrical conductivity changes in frequency with temperature. Figure 8 displays the frequency dependence of our ceramic's AC conductivity at a specific temperature. The plot makes low-frequency dispersion evident as the curves converge at higher frequencies. The graph shows that when temperature and frequency increase, conductivity increases as well. Both high-frequency AC conductivity and low-frequency DC conductivity are displayed in the conductivity spectra [40, 41]. The conductivity can be examined using Jonscher's power law [42, 43]:

\(\:{\sigma\:}_{ac}\left(\omega\:\right)\) = \(\:{\sigma\:}_{dc}\) + A \(\:{\omega\:}^{s}\) (9)

where n is the exponent factor, 0 < s < 1, and σdc is the direct current conductivity at low frequencies and certain temperatures. A is a constant that is dependent on temperature. It demonstrates that the typical charge accumulation behavior of ωs is diminished and that the conductivity increases with frequency. A frequency-independent flat response is seen at lower frequencies and higher temperatures. The difference in conductivity at lower frequencies can be recognized thanks to the polarization effect at the electrode and dielectric interface. Ac conductivity rises as frequency rises while accumulated charges decrease [44–48]. Using Origin 8 software, Eq. 9 is employed to determine exponent values. The temperature variation of this parameter for the treated substance is displayed in Fig. 9. The situation deteriorates with rising temperature, indicating that Non-overlapping Small Polaron Tunneling (NSPT), which is pertinent for researching conduction phenomena in the boundary of the alternating regime, is the most suitable conduction model [44]. This leads to the determination of the carrier's binding energy Wm at that specific point. This can be discovered by using the following equation [49]:

S = 1 + \(\:\frac{4{K}_{B}T}{{W}_{m-{K}_{B}Tln\left(\omega\:{\tau\:}_{0}\right)}}\) (10)

The binding energy is denoted by Wm, and the Boltzmann constant is represented by KB. The exponent s becomes: in case the ratio of of \(\:{W}_{m}\)/\(\:{K}_{B}T\) is big.

s = 1 + \(\:\frac{4{K}_{B}T}{{W}_{m}}\) (11)

The binding energy is determined by taking the linear slope of "s" and getting Wm = 0.183 eV. An Arrhenius-type characteristic is observed in the conductivity [50]:

\(\:{\sigma\:}_{dc}\) = \(\:A\) exp (\(\:\frac{-{E}_{a}}{{K}_{B}T}\)) (12)

Where T is the temperature,\(\:{E}_{a}\) is the activation energy, A is the pre-exponential factor and \(\:{K}_{B}\)is the Boltzmann constant (\(\:{K}_{B}\) = 8.617×10− 5 eV K− 1). The temperature dependence of the conductivity Ln (\(\:{\sigma\:}_{dc}\) T) vs 1000 / T as illustrate in Fig. 10. The activation energy estimated from the slope of the linear fit is about 0.43 eV. Such value indicates that the conduction mechanism for the present system may be due to the polaron hopping based on electron carriers. In addition, it is evident that the values of the activation energy derived from conductivity and the frequency corresponding to the relaxation peaks of the imaginary part of the impedance (Z") are different, which explains why the process of relation and the mechanism of conduction do not use the same charge carriers.

## 3.4. Dielectric and Modulus Studies

The dielectric polarization and dielectric constants were assessed by looking at our sample's dielectric characterization. Based on the mathematical framework introduced at [51], the electric modulus formalism considers both conduction and relaxation.

M* = \(\:{M}^{{\prime\:}}-j{M}^{{\prime\:}{\prime\:}}\) = \(\:\frac{1}{{\epsilon\:}^{}}\) (13)

where the real and imaginary portions of the complex modulus are denoted by M' and M" correspondingly. They are articulated through the following relationships:

$$\:{M}^{{\prime\:}}\:=\:\frac{{\epsilon\:}^{{\prime\:}}}{{{\epsilon\:}^{{\prime\:}}}^{2}+{{\epsilon\:}^{{\prime\:}{\prime\:}}}^{2}}$$

14

$$\:{M}^{{\prime\:}{\prime\:}}\:=\:\frac{{\epsilon\:}^{{\prime\:}{\prime\:}}}{{{\epsilon\:}^{{\prime\:}}}^{2}+{{\epsilon\:}^{{\prime\:}{\prime\:}}}^{2}}$$

15

The frequency shift of the dielectric constant can be explained by the dispersion caused by Maxwell-Wagner interfacial polarization, according to Koop's phenomenological theory. The real and imaginary components of the permittivity are denoted by the formulas ε = ε′ + j ε′′, which expresses the complex relative dielectric permittivity. The evolutions of ε′ versus frequency for the generated compound at different temperatures are shown in Fig. 11a. The real part of the permittivity ε′ has larger values at low frequencies because of the double-exchange interaction between the ferromagnetically coupled Fe3+ and Fe2+ ions. Since the real element of permittivity is related to stored energy, the charge begins to accumulate at the grain boundary as the temperature rises. Furthermore, it is evident that ε′ exhibits increased stability and increases with temperature. The evolution of ε′ is generally attributed to the four types of polarizations (atomic, electronic, dipolar, and interfacial) [52]. It has been proven that the evolution of ε′ with temperature is related, at low frequencies, to dipolar and interfacial polarizations [53].

As a result, the imaginary component of permittivity ε′′ for the compound as a function of frequency and temperature is shown in Fig. 11.b. This figure clearly shows that the samples' ε′′ declines with increasing frequency. In general, the imaginary component of permittivity is influenced by polarization and the conduction mechanism [54]. The dielectric constant is quite small because the dipoles follow the alternating field at higher frequencies [55]. Furthermore, the non-Debye behavior is supported by the absence of a relaxation peak at all temperatures. Furthermore, the frequency-dependent variation of M″ at different sample temperatures is shown in Fig. 12. It indicates that the value of M″ increases with increasing frequency and reaches its maximum (M max), reaffirming the presence of relaxation. Temperature-dependent relaxation processes in the material are shown by the shifting of the maximum value of the M" (Mmax) peak to the higher frequency side. There may be a variation in capacitance as indicated by the magnitude of Mmax changing with temperature. The presence of a non-Debye-type [56] conduction mechanism in the material is suggested by asymmetric peak broadening, which also suggests relaxation spreading with a different time constant.