*3.5. Electrical Modulus Spectroscopy of Ceramics*

The impedance spectrum data emphasized only the maximum resistance element of microscopic components. In order to better understand the relaxation behaviors in BGTN−0.2Cr ceramic, electrical modulus analysis and impedance analysis complemented each other. The modulus spectrum handles the minimum capacitive element of the microscopic components and can suppress electrode interface effects [44,45]. Physically, the electrical modulus corresponds to the relaxation of the electric field in the material when the electric displacement remains constant. Therefore, the electrical modulus represents the real dielectric relaxation process, which can be expressed as:

$$M = M' + j\,M'' = j\omega \mathbb{C}\_0 \mathbb{Z} \tag{16}$$

$$M' = \omega \mathbb{C}\_0 \mathbb{Z}''\tag{17}$$

$$M'' = j\omega \mathbb{C}\_0 \mathbb{Z}\prime \tag{18}$$

where *C*<sup>0</sup> is the capacitance of free space given by *C*<sup>0</sup> = *ε*0*A/d* [46].

**Figure 6.** Temperature dependence of *R<sup>g</sup>* and *Rgb* of the BGTN−0.2Cr ceramic (inset shows the Arrhenius fitting of the plots of resistivity vs. temperature). **Figure 6.** Temperature dependence of *R<sup>g</sup>* and *Rgb* of the BGTN−0.2Cr ceramic (inset shows the Arrhenius fitting of the plots of resistivity vs. temperature).

*3.5. Electrical Modulus Spectroscopy of Ceramics* The impedance spectrum data emphasized only the maximum resistance element of microscopic components. In order to better understand the relaxation behaviors in BGTN−0.2Cr ceramic, electrical modulus analysis and impedance analysis complemented each other. The modulus spectrum handles the minimum capacitive element of the microscopic components and can suppress electrode interface effects [44,45]. Physically, the electrical modulus corresponds to the relaxation of the electric field in the material when the electric displacement remains constant. Therefore, the electrical modulus represents the real dielectric relaxation process, which can be expressed as: = ′ + ′′ = 0 (16) ′ = 0′′ (17) In Figure 7, the electrical modulus spectroscopy of the BGTN−0.2Cr ceramic at high temperature is shown as a function of the frequency with temperature changing. It is obvious from Figure 7a that every temperature showed an identical trend—as frequency rose, *M*0 values increased, gradually slowing (*M*0 gradually in a fixed value at higher frequencies). *M*0 showed asymmetry because of the tensile index characteristics of the relaxation time of materials. The monotonic dispersion in the low-frequency area may be due to the short-range jump of the carriers. This result may be related to the lack of recovery power of the charge carrier migration under the control of the polarization electric field [47]. On the other hand, *M*00 increased as frequency rose and reached a peak of relaxation because of the bulk grain and grain boundary behaviors. In physics, the electrical modulus peak can determine the area that the charge carrier can be migrated long distance. The asymmetrical and wide *M*00 peaks imply that the nonexponential behaviors of the grain and grain boundary relaxation deviated from Debye-type relaxation. The behavior indicates that ion migration occurred by jumping accompanying the corresponding timedependent mobility of other nearby charge carriers [48]. These relaxation peaks moved towards higher frequencies as the temperature increased. With the equation *ωτ* = 1, we can obtain the relaxation time *τ*<sup>0</sup> 0 related to the electrical modulus and perform Arrhenius fitting to *τ*<sup>0</sup> 0 . Comparing the activation energies obtained from the impedance and modulus spectra (Figure 8), we found that the two processes had similar activation energies and preexponential factors. This indicates that the two had a common relaxation mechanism—both were dominated by similar carriers. The grain boundaries contributed to low-frequency relaxation, and the grains contributed to high-frequency relaxation.

to high-frequency relaxation.

**Figure 7.** Electrical modulus spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) –*M*′; (**b**) *M*″; (**c**) *M*″/*M*″*max*; (**d**) temperature dependence of *β.*  **Figure 7.** Electrical modulus spectroscopy of the BGTN−0.2Cr ceramic at high temperature: (**a**) −*M*<sup>0</sup> ; (**b**) *M*00; (**c**) *M*00/*M*00 *max*; (**d**) temperature dependence of *β*.

The *M*00 curves at different temperatures were normalized to a master curve with the peak position (*f max*) and the peak height (*M*00 *max*) to research the relaxation process (Figure 7c). The shape of this peak was asymmetrical and wider than Debye-type relaxation. This phenomenon is well described by the Bergman formula [49]:

the two had a common relaxation mechanism—both were dominated by similar carriers. The grain boundaries contributed to low-frequency relaxation, and the grains contributed

$$M''(\omega) = \frac{M''\_{\max}}{1 - \beta + \frac{\beta}{1 + \beta} [\beta(\omega\_{\max}/\omega) + (\omega/\omega\_{\max})]^\beta} \tag{19}$$

where *M*00 *max* is the maximum value of *M*00 , *ωmax* is the angular frequency corresponding to *M*00 *max*, and *β* indicates the extent of deviation from the ideal Debye model—the smaller the value of *β*, the greater the deviation from Debye−type relaxation (*β* = 1, see [48]). *β* increased as temperature rose (Figure 7d), which proves that the relaxation behavior of the BGTN−0.2Cr ceramic grew closer to Debye−type relaxation at higher temperature.

In Figure 8, the frequency dependence of *Z* 00/*Z* 00 *max* and *M*00/*M*00 *max* at 575 ◦C is shown. It was discovered that the impedance and electrical modulus each showed only one peak of relaxation behaviors. Because of the different focuses of the impedance and modulus, the impedance spectrum with the more resistive grain-boundary component displayed only a single peak, while the single peak of *M*00/*M*00 *max* may be due to the contribution of the grains. The *Z* 00/*Z* 00 *max* peak appeared at a lower frequency, which suggests that the grain boundary gathered a large amount of oxygen vacancies and space charges. The *M*00/*M*00 *max* peak appeared at a higher frequency, which suggests that the grain has a carrier

concentration or migration speed below the grain boundary. *E<sup>a</sup>* dominated by *Z* was slightly larger than *Ea*' dominated by *M*, which once again proves that the activation energy of the grain boundary was slightly larger than that of the grain, as shown in Figure 6. *Materials* **2021**, *14*, x FOR PEER REVIEW 13 of 16

**Figure 8.** Comparison between *Z*″/*Z*″*max* peak and *M*″/*M*″*max* peak for the BGTN−0.2Cr ceramic (inset shows the Arrhenius fitting of the plots of relaxation time vs. temperature). **Figure 8.** Comparison between *Z* 00/*Z* 00 *max* peak and *M*00/*M*00 *max* peak for the BGTN−0.2Cr ceramic (inset shows the Arrhenius fitting of the plots of relaxation time vs. temperature).

### The *M*′′ curves at different temperatures were normalized to a master curve with the *3.6. Electromechanical Resonance Spectroscopy of Ceramics*

peak position (*fmax*) and the peak height (*M*′′*max*) to research the relaxation process (Figure 7c). The shape of this peak was asymmetrical and wider than Debye-type relaxation. This phenomenon is well described by the Bergman formula [49]: ௫′′ܯ = (߱)ᇱᇱܯ ߚ +ߚ1− 1+ߚ] ߚ)߱௫/߱) + (߱/߱௫)]ఉ (19) where *M*′′*max* is the maximum value of *M*′′, *ωmax* is the angular frequency corresponding to *M*′′*max*, and *β* indicates the extent of deviation from the ideal Debye model—the smaller the value of *β*, the greater the deviation from Debye−type relaxation (*β* = 1, see [48]). *β* increased as temperature rose (Figure 7d), which proves that the relaxation behavior of the BGTN−0.2Cr ceramic grew closer to Debye−type relaxation at higher temperature. Figure 9 shows the electromechanical resonance spectroscopy results for the BTGN−0.2Cr ceramic at room temperature. This measured the frequency dependence of impedance |*Z*| and the phase angle *θ* of the piezoceramic as a resonator. The paired peak of resonanceantiresonance around 285.5 kHz showed that the sample was in the radial-extensional vibration mode. In an ideal poling state, piezoelectric materials should exhibit an impedance phase angle *θ* approaching 90◦ in the frequency range between the resonance (*fr*) and antiresonance frequencies (*f <sup>a</sup>*). As can be seen, the maximum phase angle *θmax* was only 36.4◦ , indicating the insufficient poling of the sample. Since BTN has a high coercive electric field, as well as a low resistivity, a complete domain switching is difficultly achieved by the poling electric field. Things that give satisfaction is that the *d*<sup>33</sup> value of the BTGN−0.2Cr ceramic has reached 18 pC/N, which provides it with a large competitiveness used as sensing materials for high−temperature piezoelectric sensors.

In Figure 8, the frequency dependence of *Z*″/*Z*″*max* and *M*″/*M*″*max* at 575 °C is shown. It was discovered that the impedance and electrical modulus each showed only one peak of relaxation behaviors. Because of the different focuses of the impedance and modulus, the impedance spectrum with the more resistive grain-boundary component displayed only a single peak, while the single peak of *M*″/*M*″*max* may be due to the contribution of the grains. The *Z*″/*Z*″*max* peak appeared at a lower frequency, which suggests that the grain boundary gathered a large amount of oxygen vacancies and space charges. The *M*″/*M*″*max* peak appeared at a higher frequency, which suggests that the grain has a carrier concentration or migration speed below the grain boundary. *Ea* dominated by *Z* was slightly larger than *Ea*' dominated by *M*, which once again proves that the activation energy of the grain boundary was slightly larger than that of the grain, as shown in Figure

Figure 9 shows the electromechanical resonance spectroscopy results for the BTGN−0.2Cr ceramic at room temperature. This measured the frequency dependence of

*3.6. Electromechanical Resonance Spectroscopy of Ceramics* 

6.

extensional vibration mode. In an ideal poling state, piezoelectric materials should exhibit an impedance phase angle *θ* approaching 90° in the frequency range between the resonance (*fr*) and antiresonance frequencies (*fa*). As can be seen, the maximum phase angle *θmax* was only 36.4°, indicating the insufficient poling of the sample. Since BTN has a high coercive electric field, as well as a low resistivity, a complete domain switching is difficultly achieved by the poling electric field. Things that give satisfaction is that the *d*<sup>33</sup> value of the BTGN−0.2Cr ceramic has reached 18 pC/N, which provides it with a large competitiveness used as sensing materials for high−temperature piezoelectric sensors.

**Figure 9.** Electromechanical resonance spectroscopy of the BGTN−0.2Cr ceramic at room **Figure 9.** Electromechanical resonance spectroscopy of the BGTN−0.2Cr ceramic at room temperature.
