*3.1. Phase Structure of Ceramics*

Figure 1 shows the XRD analysis of the sintered ceramics. As can be seen from Figure 1a, both for the pure BTN ceramic and the BGTN−0.2Cr ceramic, their observed XRD patterns agree with the standard X-ray diffraction powder patterns from JCPDS card No. 39-0233 well. Therefore, both samples were identified as the pure Bi3TiNbO<sup>9</sup> phase with orthorhombic structure and space group of *A21am*. No second phase was found in the XRD pattern of the BGTN−0.2Cr ceramic. The introduced Gd2O<sup>3</sup> and Cr2O<sup>3</sup> formed a complete solid solution with Bi3TiNbO9. The strongest diffraction peak of the BGTN−0.2Cr ceramic was the (1 × 1 × 5) peak, which is consistent with the rule of the strongest diffraction peak of BLSF ceramics (1 × 1 × 2m + 1) [28]. In view of the similar ionic radius, it is believed that Gd3+ (*r* = 0.0938 nm) entered the A-site (Bi3+: *r* = 0.103 nm) of the perovskite unit while Cr3+ (*r* = 0.0615 nm) entered the B-site (Ti4+: *r* = 0.0605 nm). Meanwhile, the stability of the ABO3—type perovskite structure can be described by the tolerance factor (*t*), which can be expressed as [29]:

$$t = \frac{r\_A + r\_o}{\sqrt{2}(r\_B + r\_o)}\tag{1}$$

where *rA*, *rB*, and *r<sup>o</sup>* are the ionic radii of A, B, and the oxygen ion, respectively. The perovskite structure remains stable when *t* is between 0.77 and 1.10. However, the value of the tolerance factor may be further limited in BLSF due to the structural incompatibility between the pseudo perovskite layer and the bismuth-oxygen layer, which is caused by the mismatching of their transverse dimension. Subbarao [1] proposed that when *m* = 2, *t* is limited in the range of 0.81~0.93. We obtained *t* = 0.86 for BTN and *t* = 0.82 for BGTN−0.2Cr. The decrease in the tolerance factor could demonstrate the successful substitution of Gd3+ and Cr3+ for Bi3+ and Ti4+ at the A- and B-sites, respectively, in the perovskite unit of BTN.

The Rietveld refinement was performed for the XRD patterns of the pure BTN ceramic and the BGTN−0.2Cr; the refined factors and cell parameters are shown in Figure 1b,c. When compared with pure BTN, BGTN−0.2Cr showed a contracted unit cell as well as a smaller orthorhombic distortion (a/b). The substitution of Gd3+ for Bi3+ at the A site tended to induce a principle change in the structural distortion of perovskite blocks composed of (Ti, Nb)O<sup>6</sup> octahedrons. However, the substitution of Cr3+ for Ti4+ at the B—site may have decreased the tilting angle of the (Ti, Nb)O<sup>6</sup> octahedron, leading to reduced distortion in the perovskite blocks.

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**Figure 1.** XRD analysis of the sintered ceramics: (**a**) observed XRD patterns of the pure BTN ceramic and the BGTN−0.2Cr ceramic; (**b**) Rietveld XRD refinement of the pure BTN ceramic; (**c**) Rietveld XRD refinement of the BGTN−0.2Cr ceramic. **Figure 1.** XRD analysis of the sintered ceramics: (**a**) observed XRD patterns of the pure BTN ceramic and the BGTN−0.2Cr ceramic; (**b**) Rietveld XRD refinement of the pure BTN ceramic; (**c**) Rietveld XRD refinement of the BGTN−0.2Cr ceramic.

### The Rietveld refinement was performed for the XRD patterns of the pure BTN *3.2. Grain Morphology and Chemical Composition of Ceramics*

ceramic and the BGTN−0.2Cr; the refined factors and cell parameters are shown in Figure 1b,c. When compared with pure BTN, BGTN−0.2Cr showed a contracted unit cell as well as a smaller orthorhombic distortion (a/b). The substitution of Gd3+ for Bi3+ at the A−site tended to induce a principle change in the structural distortion of perovskite blocks composed of (Ti, Nb)O6 octahedrons. However, the substitution of Cr3+ for Ti4+ at the B−site may have decreased the tilting angle of the (Ti, Nb)O6 octahedron, leading to reduced distortion in the perovskite blocks. *3.2. Grain Morphology and Chemical Composition of Ceramics*  Figure 2 shows the SEM and EDS analysis of the BGTN−0.2Cr ceramic, focused on its thermal−etched surface. As can be seen from the SEM image inserted in the EDS, a dense microstructure with well-defined grain boundaries was formed in the ceramic. All the grains were closely stacked, with random orientation, which agrees with the weaker intensity of the (0 × 0 × *l*) diffracted peaks observed in Figure 1. The microstructure was composed of plate−like grains. Such grain growth possessed a high anisotropy such that the length (*L*) was larger than the thickness (*T*), which can be attributed to the higher grain Figure 2 shows the SEM and EDS analysis of the BGTN−0.2Cr ceramic, focused on its thermal-etched surface. As can be seen from the SEM image inserted in the EDS, a dense microstructure with well-defined grain boundaries was formed in the ceramic. All the grains were closely stacked, with random orientation, which agrees with the weaker intensity of the (0 × 0 × *l*) diffracted peaks observed in Figure 1. The microstructure was composed of plate−like grains. Such grain growth possessed a high anisotropy such that the length (*L*) was larger than the thickness (*T*), which can be attributed to the higher grain growth rate in the direction perpendicular to the *c*-axis of the BLSF grains [30]. It is well known that the crystal grain aspect ratio (*L/T*) has a significant influence on the resistivity of BLSF ceramics. A higher aspect ratio is often related to higher resistivity [31]. The average thickness of these plate-like grains was 1.47 µm, while the length was about 8.93 µm. The high aspect ratio, with a *L/T* value of 6.1, was expected to lend higher resistivity to the BGTN−0.2Cr ceramic. Furthermore, EDS analysis showed that the BGTN−0.2Cr ceramic contained six elements: Bi, Gd, Ti, Nb, Cr, and O. All the elements presented a uniform distribution in the detected zone. Both gadolinium and chromium were successfully incorporated into the BTN.

growth rate in the direction perpendicular to the *c*-axis of the BLSF grains [30]. It is well known that the crystal grain aspect ratio (*L/T*) has a significant influence on the resistivity of BLSF ceramics. A higher aspect ratio is often related to higher resistivity [31]. The average thickness of these plate-like grains was 1.47 μm, while the length was about 8.93 μm. The high aspect ratio, with a *L/T* value of 6.1, was expected to lend higher resistivity to the BGTN−0.2Cr ceramic. Furthermore, EDS analysis showed that the BGTN−0.2Cr ceramic contained six elements: Bi, Gd, Ti, Nb, Cr, and O. All the elements presented a uniform distribution in the detected zone. Both gadolinium and chromium were

successfully incorporated into the BTN.

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**Figure 2.** SEM and EDS analysis of the BGTN−0.2Cr ceramic, focused on its thermal-etched surface (the grain size was determined using the intercept procedure on the basis of the SEM image). **Figure 2.** SEM and EDS analysis of the BGTN−0.2Cr ceramic, focused on its thermal-etched surface (the grain size was determined using the intercept procedure on the basis of the SEM image).

### *3.3. Electrical Conduction Behaviors of Ceramics 3.3. Electrical Conduction Behaviors of Ceramics*

In order to further understand the conduction mechanism at high temperature, the results of electrical conduction spectroscopy of the BGTN−0.2Cr ceramic at high temperature are shown in Figure 3. It can be seen from Figure 3a that the conductivity of the BGTN−0.2Cr ceramic did not change with frequency in the low−frequency section at various temperatures, while in the high-frequency section, the conductivity increased with the increase in frequency. This is consistent with the jump relaxation model proposed by Funke [32]. In the low-frequency section, the migration of charge carriers was mainly implemented through long-distance jump, which led to direct current conductivity. As the frequency increased, the mobility of charge carriers was gradually limited, and the conductivity became positively related to frequency. The functional relationship between conductivity and frequency is consistent with the general Jonscher's theory [33]: In order to further understand the conduction mechanism at high temperature, the results of electrical conduction spectroscopy of the BGTN−0.2Cr ceramic at high temperature are shown in Figure 3. It can be seen from Figure 3a that the conductivity of the BGTN−0.2Cr ceramic did not change with frequency in the low−frequency section at various temperatures, while in the high-frequency section, the conductivity increased with the increase in frequency. This is consistent with the jump relaxation model proposed by Funke [32]. In the low-frequency section, the migration of charge carriers was mainly implemented through long-distance jump, which led to direct current conductivity. As the frequency increased, the mobility of charge carriers was gradually limited, and the conductivity became positively related to frequency. The functional relationship between conductivity and frequency is consistent with the general Jonscher's theory [33]:

(2) (1~0 = ݊) ߱ܣ + (ܶ)ௗߪ = (߱)ߪ

$$
\sigma(\omega) = \sigma\_{dc}(T) + A\omega^{\text{\textquotedblleft}}\left(\mathfrak{n} = \mathbf{0} \sim \mathbf{1}\right) \tag{2}
$$

where *σ*(*ω*) is the total conductivity, *σdc*(*T*) is the dc conductivity, *A* is a constant with temperature dependence, ω is the angular frequency, *n* is the frequency index factor, and *Aω<sup>n</sup>* represents the ac conductivity. *σdc*(*T*), *n*, and *A* can be fitted by Equation (2). Based on Jonscher's theory, the frequency dependence of ac conductivity originated from the relaxation of the ionic atmosphere after the movement of charge carriers, as shown in Figure 3a. where ߪ)߱ (is the total conductivity, ߪௗ(ܶ) is the dc conductivity, *A* is a constant with temperature dependence, ω is the angular frequency, *n* is the frequency index factor, and ܣ߱ represents the ac conductivity. ߪௗ(ܶ), *n*, and *A* can be fitted by Equation (2). Based on Jonscher's theory, the frequency dependence of ac conductivity originated from the relaxation of the ionic atmosphere after the movement of charge carriers, as shown in Figure 3a.

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**Figure 3.** Electrical conduction spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) frequency dependence of the total conductivity; (**b**) relationship between ac conductivity and angular frequency; (**c**) values of the frequency index factor *n* calculated by the general Jonscher' s theory; (**d**) Arrhenius fitting of the plots of lnσdc and lnωp vs. 1000/*T*. **Figure 3.** Electrical conduction spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) frequency dependence of the total conductivity; (**b**) relationship between ac conductivity and angular frequency; (**c**) values of the frequency index factor *n* calculated by the general Jonscher's theory; (**d**) Arrhenius fitting of the plots of lnσdc and lnω*<sup>p</sup>* vs. 1000/*T*.

The changes in the temperature-related frequency index factor *n* provide information for the origins of conductance. The value of *n* in the correlated barrier−hopping (CBH) model decreases with temperature rising [34]. The results obtained for the BGTN−0.2Cr ceramic are shown in Figure 3b,c and are well in line with the CBH model. The first-order approximation of the frequency index factor *n* of the CBH model is given as Equation (3) The changes in the temperature-related frequency index factor *n* provide information for the origins of conductance. The value of *n* in the correlated barrier−hopping (CBH) model decreases with temperature rising [34]. The results obtained for the BGTN−0.2Cr ceramic are shown in Figure 3b,c and are well in line with the CBH model. The first-order approximation of the frequency index factor *n* of the CBH model is given as Equation (3) [35]:

where *WM* is the maximum barrier height. The calculated *WM* (~0.63 eV) was slightly less than the oxygen vacancy's activation energy (~0.6–1.2 eV), indicating that oxygen vacancies were the main charge carriers between local states. This may be related to the

ܹெ

$$m = 1 - \frac{6K\_B T}{W\_M} \tag{3}$$

[35]:

where *W<sup>M</sup>* is the maximum barrier height. The calculated *W<sup>M</sup>* (~0.63 eV) was slightly less than the oxygen vacancy's activation energy (~0.6–1.2 eV), indicating that oxygen vacancies were the main charge carriers between local states. This may be related to the segmental ionization of the first- and second-order oxygen vacancies, which can be expressed by Equations (4) and (5), respectively:

$$V\_O \leftrightarrow V\_{\dot{O}} + e'\tag{4}$$

$$V\_{\dot{O}} \leftrightarrow V\_{\ddot{O}} + e'\tag{5}$$

where *V* . *O* and *V* .. *O* are the first- and second-stage ionized oxygen vacancies, respectively. In many BLSF materials, the activation energies of the first- and second-stage ionized oxygen vacancies (E<sup>I</sup> and EII) have been reported as ~0.5 eV and ~1.2 eV, respectively [36]. For instance, E<sup>I</sup> and EII in SBTW0.04 are 0.57 eV and 0.74 eV [37], and EII in Bi2.8Nd0.2NbTiO<sup>9</sup> is 0.9 eV [38]. The conductive and relaxation behaviors related to these BLSF materials can be attributed to long-range/local migration of two-stage ionized oxygen vacancies.

The conductivity of ion conductive materials is related not only to movable ion concentrations but also to ion jump frequencies [39]. The hopping angular frequency ω*<sup>p</sup>* of ac conductivity can be fitted by Equation (6), and the activation energy of dc conduction and hopping conduction can be determined and calculated by Arrhenius fitting shown in Equations (7) and (8):

$$
\omega\_p = \left(\frac{\sigma\_{\rm ac}}{A}\right)^{1/n} \tag{6}
$$

$$
\omega\_p = \omega\_0 \exp\left(-E\_p/kT\right) \tag{7}
$$

$$
\sigma\_{dc} = \sigma\_0 \exp(-E\_{dc}/kT) \tag{8}
$$

where *ω*<sup>0</sup> and *σ*<sup>0</sup> are pre−exponential factors, *k* is the Boltzmann constant, *E<sup>p</sup>* is the activation energy of hopping conduction, and *Edc* is the dc conduction activation energy. The calculated *Edc* (~1.09 eV) in Figure 3d was slightly smaller than the activation energy (~1.2 eV) of the second ionized oxygen vacancies, which indicates that *V* . *O* and *V* .. *<sup>O</sup>* were involved in conduction during the dc conductance process. Furthermore, the reduction of *E<sup>p</sup>* (0.73 eV) indicates that when the oxygen vacancies migrated from long- to short-range jumps, the activation energy decreased, which may have increased the carrier mobility during the ac conduction process. *E<sup>p</sup>* was slightly greater than *WM*, indicating that carriers were over the barrier height and then had a short-distance jump participation in the ac conduction process. The higher *E<sup>p</sup>* may be also caused by relaxation.

When poled ferroelectric ceramics are used as piezoelectric elements, over time, the polarization change caused by the applied stress is offset by charge movement caused by internal conduction inside the material. At high frequencies, the charge compensation caused by conductivity can be ignored, because the change rate of charge caused by the applied stress is much faster than the time constant (*RC*). However, at low frequencies, signals from sensors or generators may be significantly attenuated. The minimum useful frequency or lower limit frequency (*f LL*) is inversely proportional to the time constant, which can be calculated by Equation (9):

$$f\_{LL} = \frac{1}{2\pi \text{RC}} = \frac{\sigma}{2\pi \text{c}'} \tag{9}$$

where *σ* is the dc conductivity, which can be deduced from AC fields below 100 Hz, and *ε* 0 is the real part of the complex dielectric permittivity. It is well known that the *RC* time constant of BLSF ceramics tends to become very low at high temperature because of the sharp decline of resistivity [40]. In line with Equation (9), the values of *f LL* in the temperature range of 450–650 ◦C were calculated for the BGTN−0.2Cr ceramic and are shown in Figure 4. The values of *f LL* showed a sharp rise when the temperature increased from 450 to 500 ◦C, and then the increase in *f LL* with temperature began to slow down when the temperature exceeded 500 ◦C. In addition to the significant decrease in resistance with temperature, the capacitance of the ferroelectric material is closely related to the temperature. Therefore, the temperature correlation of *f LL* combines the effects of capacitance and resistance, which can be considered as a useful quality factor for evaluating the service performance of ferroelectric materials. capacitance and resistance, which can be considered as a useful quality factor for evaluating the service performance of ferroelectric materials.

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**Figure 4.** Lower limit frequency (*fLL*) of the BGTN−0.2Cr ceramic as a function of temperature. **Figure 4.** Lower limit frequency (*f LL*) of the BGTN−0.2Cr ceramic as a function of temperature.

## *3.4. Electrical Impedance Spectroscopy of Ceramics 3.4. Electrical Impedance Spectroscopy of Ceramics*

In order to study electrical behavior and distinguish the contribution of grains and grain boundaries to the conductivity of BGTN−0.2Cr ceramic, we analyzed the complex impedance data of BGTN−0.2Cr ceramic, considering both the real (*Z*′) and imaginary (*Z*″) parts. The correlation function relationship can be expressed as follows: In order to study electrical behavior and distinguish the contribution of grains and grain boundaries to the conductivity of BGTN−0.2Cr ceramic, we analyzed the complex impedance data of BGTN−0.2Cr ceramic, considering both the real (*Z* 0 ) and imaginary (*Z* 00) parts. The correlation function relationship can be expressed as follows:

$$\mathbf{Z} = \mathbf{Z}' - \mathbf{j}\mathbf{Z}'' \tag{10}$$

$$Z' = \frac{R\left(1 + (\omega \tau)^{1-\alpha} \cos\left[(1-\alpha)\frac{\pi}{2}\right]\right)}{1 + (\omega \tau)^{2(1-\alpha)} + 2(\omega \tau)^{1-\alpha} \cos\left[(1-\alpha)\frac{\pi}{2}\right]} \tag{11}$$

$$1 + (\omega \tau)^{-1} \tau^{-1} + 2(\omega \tau)^{-1} \cos \left[ (1 - \alpha) \frac{\pi}{2} \right]$$

$$Z^{\prime \prime} = \frac{R(\omega \tau)^{1 - \alpha} \sin \left[ (1 - \alpha) \frac{\pi}{2} \right]}{1 + (\omega \tau)^{2(1 - \alpha)} + 2(\omega \tau)^{1 - \alpha} \cos \left[ (1 - \alpha) \frac{\pi}{2} \right]} \tag{12}$$

1 + (߱߬)ଶ(ଵିఈ) + 2(߱߬)ଵିఈ cos ቂ(1−ߙ (<sup>ߨ</sup> 2ቃ where *τ* = *RC* and *α =* 0~1 is the proportion of the relaxation time distribution.

where *τ* = *RC* and *α =* 0~1 is the proportion of the relaxation time distribution. It can be seen from Figure 5a that the impedance value had a monotonous reduction as temperature and frequency rose in the low frequency range (≤10 kHz). The reduction of *Z*′ as the temperature rose in the low−frequency part suggests that conductivity increased with temperature. It was also seen that while *Z*′ decreased as frequency increased, after reaching a fixed frequency (≥200 kHz), the value of *Z*′ became higher as the temperature increased and merged when the frequency increased further. This sudden change represents a possibility that the conductivity increased as frequency and temperature rose because of the fixed carriers at low temperatures and the defects at high temperatures. The combination of *Z*′ at all temperatures at high frequencies may be due to the release of space charges, which would have led to a decrease in the resistance of the material. Figure 5b shows the change of *Z″* with frequency at different temperatures. Obviously, as the frequency increased, *Z″* reached a maximum value, which points to the relaxation process in the system. As the temperature increased, the maximum value shifted to higher frequencies. This shows that the relaxation was related to both temperature and frequency. The appropriate temperature activated the particles to cause It can be seen from Figure 5a that the impedance value had a monotonous reduction as temperature and frequency rose in the low frequency range (≤10 kHz). The reduction of *Z* <sup>0</sup> as the temperature rose in the low−frequency part suggests that conductivity increased with temperature. It was also seen that while *Z* 0 decreased as frequency increased, after reaching a fixed frequency (≥200 kHz), the value of *Z* 0 became higher as the temperature increased and merged when the frequency increased further. This sudden change represents a possibility that the conductivity increased as frequency and temperature rose because of the fixed carriers at low temperatures and the defects at high temperatures. The combination of *Z* 0 at all temperatures at high frequencies may be due to the release of space charges, which would have led to a decrease in the resistance of the material. Figure 5b shows the change of *Z* 00 with frequency at different temperatures. Obviously, as the frequency increased, *Z* 00 reached a maximum value, which points to the relaxation process in the system. As the temperature increased, the maximum value shifted to higher frequencies. This shows that the relaxation was related to both temperature and frequency. The appropriate temperature activated the particles to cause large motions, and the appropriate frequency caused resonance. When the temperature matched the frequency, the relaxation phenomenon induced was the most obvious (*Z* 00 maximum). It is well known that in

large motions, and the appropriate frequency caused resonance. When the temperature

perovskite−type compounds, the short-range motion of oxygen vacancies is a common phenomenon that contributes to high-temperature relaxation [41]. Because of the dispersion of bulk grains, *Z* 00 merged at higher frequencies, which signified that the space charge was released. At the same time, the peak value decreased as the temperature increased and tended to become wider. The widening of the peak at higher temperatures indicates the existence of temperature-dependent relaxation. The substances causing relaxation of the material at high temperature may be vacancies or defects [42]. of oxygen vacancies is a common phenomenon that contributes to high-temperature relaxation [41]. Because of the dispersion of bulk grains, *Z*″ merged at higher frequencies, which signified that the space charge was released. At the same time, the peak value decreased as the temperature increased and tended to become wider. The widening of the peak at higher temperatures indicates the existence of temperature-dependent relaxation. The substances causing relaxation of the material at high temperature may be vacancies or defects [42].

matched the frequency, the relaxation phenomenon induced was the most obvious (*Z″* maximum). It is well known that in perovskite−type compounds, the short-range motion

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**Figure 5.** Electrical impedance spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) *Z*'; (**b**) −*Z″*; (**c**) −*Z″* vs. *Z*′ (inset shows the impedance curve measured below 550 °C); (**d**) fitting for the Nyquist plot at 600 °C. **Figure 5.** Electrical impedance spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) *Z* 0 ; (**b**) −*Z* <sup>00</sup>; (**c**) −*Z* 00 vs. *Z* 0 (inset shows the impedance curve measured below 550 ◦C); (**d**) fitting for the Nyquist plot at 600 ◦C.

The Nyquist plot (*Z″* vs. *Z*′) from 200 to 650 °C is shown in Figure 5c. The impedance curve at low temperature (~200 °C) was close to the *y*-axis (imaginary part), which shows the high insulation properties of the ceramic at low temperatures. As the temperature increased, the impedance curve deviated from the *y*-axis and gradually curved toward the *x*-axis (real part) to form a deformed, asymmetric, semicircular arc. The asymmetry from the ideal semicircular arc suggests the possibility of multiple relaxation behaviors in BGTN−0.2Cr ceramic. The temperature continued to rise, and the radius of the deformed semicircular arc gradually decreased, indicating that the resistance of the ceramic gradually decreased as the temperature increased. The Nyquist plot (*Z* 00 vs. *Z* 0 ) from 200 to 650 ◦C is shown in Figure 5c. The impedance curve at low temperature (~200 ◦C) was close to the *y*-axis (imaginary part), which shows the high insulation properties of the ceramic at low temperatures. As the temperature increased, the impedance curve deviated from the *y*-axis and gradually curved toward the *x*-axis (real part) to form a deformed, asymmetric, semicircular arc. The asymmetry from the ideal semicircular arc suggests the possibility of multiple relaxation behaviors in BGTN−0.2Cr ceramic. The temperature continued to rise, and the radius of the deformed semicircular arc gradually decreased, indicating that the resistance of the ceramic gradually decreased as the temperature increased.

In order to judge whether multiple relaxation processes existed in BGTN−0.2Cr ceramic, the complex impedance spectrum was analyzed by z-view simulation software. The results are shown in Figure 5d. In the illustration, a resistor *R<sup>g</sup>* and a constant phase element *Q<sup>g</sup>* (CPE) connected in parallel represent the contribution of the crystal grain, and

the other parallel element (*R*gb and *Qgb*) represents the contribution of the grain boundary. The total resistance of this equivalent circuit can be expressed as:

$$Z^\* = \frac{R\_{\mathcal{S}}}{\left[1 + R\_{\mathcal{S}}T\_1(i\omega)^{n\_1}\right]} + \frac{R\_{\mathcal{S}b}}{\left[1 + R\_{\mathcal{S}b}T\_2(i\omega)^{n\_2}\right]}\tag{13}$$

where *Rg*, *Rgb*, *T*1, and *T*<sup>2</sup> are the resistance and variables related to the relaxation time distribution from the grain and grain boundaries, respectively; *n* (0~1) is the distribution of relaxation time; and *n* = 1 is the ideal Debye relaxation response. The fitting curve simulated by the Z-View software matched our experimental data well, which confirmed that two kinds of relaxation mechanisms were involved in the BGTN−0.2Cr ceramic, i.e., grain boundaries contributed to the relaxation at low frequency, while grains contributed at high frequency [43]. The results of the simulation analysis showed that the grain boundary resistance (*Rgb*) was far greater than the bulk grain resistance (*R*g) as shown in Figure 5d. This means that at high temperatures, the concentration of oxygen vacancies and captured electronics in the grain boundary was lower. *n*<sup>2</sup> was greater than *n*1, which indicates that the impedance behavior at the grain boundaries was closer to the ideal Debye relaxation model.

The temperature dependence of *R<sup>g</sup>* and *Rgb* of the BGTN−0.2Cr ceramic is displayed in Figure 6. The rate at which the grain's resistance decreased as the temperature rose was slower than that of the grain boundary. The calculation of resistivity and its temperature dependence can be described by the following formulas:

$$
\rho = \frac{RS}{L} \tag{14}
$$

$$
\rho = \rho\_0 \exp(E\_{\rm con}/kT) \tag{15}
$$

where *Econ* is the activation energy for conduction, *ρ*<sup>0</sup> is the pre-exponential factor, and *k* is the Boltzmann constant. As shown in Figure 6, as the temperature rose, *E*<sup>g</sup> increased from 0.49 to 1.15 eV, while *Egb* increases from 0.64 to 1.34 eV. Such a significant increase in the activation energy confirmed that the charge carriers responsible for the electrical conduction process at high temperature changed from the primary−ionized oxygen vacancies to the secondary-ionized ones. No matter what kind of charge carriers dominated the electrical conduction, *E<sup>g</sup>* was always lower than *Egb*, which implies that the carrier concentration or migration speed of the grains was higher than that of the grain boundaries.
