*3.4. Electrical Impedance Analysis*

To further study the dielectric relaxation behavior of the CBT-*x*Gd/0.2Mn ceramics, Figure 6 shows the electrical modulus of *x* = 0.06 at a different temperature. The complex electrical modulus (*M\**) was calculated from the electrical modulus measured based on the following equations:

$$M^\* = M' + jM'' = j\omega \mathbb{C}\_0 \mathbb{Z}^\* \tag{7}$$

$$\mathbf{M}' = j\omega \mathbf{C}\_0 \mathbf{Z}'' \tag{8}$$

$$\mathbf{M}'' = j\omega \mathbb{C}\_0 \mathbf{Z}'\tag{9}$$

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

where *C*<sup>0</sup> is the capacitance of free apace given by *C*<sup>0</sup> = *ε0A/d* [41], *Z* \* is the complex electrical impedance, and *Z* 0 and *Z*" is its real part and imaginary part, respectively. It is shown that the *M*0 values increased quickly with frequency rising at low temperature, and slowly increased gradually with frequency increasing. Moreover, the reason for the relative dispersion in low frequency region may be related to short range hopping of charge carriers and lack of recovery energy [42]. Besides, only one single peak can be seen from Figure 6b, which results from only the grain response was observed [19] at the temperature and frequencies. *M*00 increased sharply and reached the top may be related to both grain size and grain boundary relaxation. However, the peak value of *M*00 declined with the increase of temperature, which demonstrated the relaxation deviated from Debye-type relaxation. The results showed that the ions move in a hopping manner along with other related carriers [43].

**Figure 6.** Electrical modulus spectroscopy of the composition with *x* = 0.06 measured at different temperatures: (**a**) −*M*<sup>0</sup> ; (**b**) *M*00; (**c**) *M*00*/M*00max; (**d**) temperature dependence of *β*.

To further clarify the dielectric relaxation mechanism, the electric modules *M*00 were normalized to research the relaxation process (Figure 6c). The shape of the curves were asymmetrical and higher than Debye-type relaxation. The Bergman formula [44] can explain the phenomenon:

$$M''(\omega) = \frac{M\_{\text{max}}^{\prime\prime}}{1 - \beta + \frac{\beta}{1 + \beta} [\beta(\omega\_{\text{max}}/\omega) + (\omega/\omega\_{\text{max}})]^\beta} \tag{11}$$

where *M*00 max is the maximum value of *M*00 , *ω*max is the angular frequency corresponding to *M*00 max, and *β* indicates the ideal Debye model—the closer the *β* value is to 1, the more it consistent with Debye-type relaxation [41]. *β* tended to increase to 1 as the temperature rose from 500 ◦C to 600 ◦C (Figure 6d), which showed that the relaxation type of the sample was closer to the Debye-type relaxation. A higher value of *β* indicated a weaker interaction between charge carriers. However, *β* began to decrease with increasing temperature at 600 ◦C, showing the dielectric relaxation behavior began to deviate from the Debye-type relaxation, which may be owing to the increased leakage current at the temperature above 600 ◦C. This phenomenon was consistent with that the peak value of *M*00 was found to decrease faster when the temperature exceeded 600 ◦C.
