*3.3. Electrical Conduction Behaviors*

The growth of the ferroelectric phase and the movement of charge carriers are affected by conductivity to a certain extent. The study of conductivity not only helps to clarify the influence of conductivity on domain structure and its motion, but also helps to clarify the properties of carriers. The AC conductivity (*σ*ac) of CBT-*x*Gd/0.2Mn ceramics was studied to better understand the relaxation-conduction behaviors of the system. The AC conductivity *σ*ac of dielectrics could be calculated using the following relation

$$
\sigma\_{ac} = \omega \varepsilon\_0 \varepsilon\_r \bullet \tan \delta \tag{1}
$$

where *ω* is the frequency of the applied electric field, *ε*<sup>0</sup> is the permittivity of free space, *ε*<sup>r</sup> is the dielectric constant, and tan *δ* is the dissipation factor. Frequency dependence of *σ*ac at various temperatures is shown in Figure 4.

**Figure 4.** Frequency dependence of AC conductivity of the CBT−*x*Gd/0.2Mn ceramics measured at different temperatures.

It can be seen from Figure 4a–f that the conductivity spectra of all samples exhibited the following characteristics: (i) The conductivity curved at lower temperatures and higher frequencies were frequency dependent, whereas at higher temperatures and lower frequencies these plots showed the frequency independence. (ii) The characteristic frequency (*f* <sup>h</sup> as marked by the arrow), where the conductivity became dependent on frequency and independent on frequency, moved to a higher frequency with the temperature increasing. (iii) In the high-frequency region, the dispersion of conductivity was less and all the curves tended to merge with a single slope. The peak observed at the low frequency was due to the application of low frequency AC electric field on the high concentration doped samples, which caused the de-coupling of a large number of internal defect dipoles, resulting in a relaxed dielectric loss peak. For the perovskite-type ferroelectrics, the increase of conductivity with increasing of frequency and temperature was usually attributed to the hopping of charge carriers through the barrier or the moving of ionic defects as space charges [29,30].

The frequency dependence of conductivity has long been found to obey the following Jonscher's power law [31]:

$$
\sigma\_{ac} = \sigma\_0 + A(T)\omega^{s(T)} \tag{2}
$$

where *σac* is the AC conductivity, *σ*<sup>0</sup> is the frequency independent (i. e. DC) conductivity, which can be obtained by extrapolating these plots in the low-frequency region, *ω* (=2*πf*) is the angular frequency of the AC electric field in the high-frequency region, *A* is a characteristic parameter assigning the polarization strength, and *s* is a dimensionless exponent to evaluate the degree of interaction between mobile charge carriers and surrounding lattice. Both *A* and the exponent *s* are the temperature and material intrinsic property dependent constants, which can be obtained from the fitting of the frequency dependence of conductivity according to Equation (2). The frequency and temperature dependance of ac conductivity of CBT-*x*Gd/Mn ceramics had been carried out by Jonscher's theory, as shown in Figure 4a–f.

Ion concentrations and ion jump frequency have an influence on the conductivity of ion conductivity. The ac conductivity can be obtained by Equation (3), and the *ω*<sup>p</sup> and dc conductivity are calculated by Arrhenius Equations (4) and (5), respectively [32]:

$$
\omega\_p = \left(\frac{\sigma\_{ac}}{A}\right)^{1/s} \tag{3}
$$

$$
\omega\_p = \omega\_0 \exp\left(-E\_\hbar \mathcal{l} k\_B T\right) \tag{4}
$$

$$
\sigma\_{d\varepsilon} = \sigma\_0 \exp\left(-E\_{d\varepsilon}/k\_B T\right) \tag{5}
$$

where *ω<sup>p</sup>* is the hopping angular frequency, *k*<sup>B</sup> is the Boltzmann constant, and *T* is the absolute temperature (K). Both *ω*<sup>0</sup> and *σ*<sup>0</sup> are the pre-exponential factor. *E*<sup>h</sup> and *E*dc are the activation energy of hopping conduction and dc conduction activation energy, respectively. Figure 5a shows the fitting of the *σac*-*f* curves for the composition with *x* = 0.06 measured at different temperatures (500~700 ◦C). It can be seen that the values of s decreased with temperature increasing, indicating that the electrical conduction was a thermally activated process, which agreed with the correlated barrier hopping (CBH) model [33]. The result that s < 1 (s = back hop rare/site relaxation rate, which was defined by the jump relaxation model [34]) indicated that the time of the charge carriers returning to initial position was longer than its relaxation times. Oxygen vacancy and bismuth vacancy may cause the decrease of s value at high temperature, and the free movement of these charge carriers reduces the probability of back hoping rate.

**Figure 5.** Fitting for the temperature and frequency dependence of conduction parameters of the composition with *x* = 0.06.

According to the CBH model, the hopping of electrons between the charged defects was limited in finite clusters, where they were bound to various defects different from the free carriers. The conduction could be attributed to the short-range hopping of localized charge carriers over trap sites separated by energy barriers of different heights. The maximum barrier height *W*M, defined as the energy required to remove the electrons completely from one site to another [35], could be evaluated by using the following equation:

$$s = 1 - \frac{6k\_B T}{W\_M} \tag{6}$$

Figure 5b shows the fitting of the *s*-*T* curve for the composition of *x* = 0.06, according to the equation above. Before 600 ◦C, the obtained value of *W*<sup>M</sup> (~0.61 eV) agreed well with the activation energy (*E*<sup>a</sup> = 0.3~0.5 eV [36]) of single-ionized oxygen vacancies (V<sup>O</sup> • ), which confirmed the single-polaron hopping of electrons from the localized oxygen vacancies to the double-ionized oxygen vacancies (V<sup>O</sup> •→V<sup>O</sup> ••+*e* 0 ) in this material. At low frequencies, electrons underwent successive and successful hopping motions for long time periods, but the ratio between successful and unsuccessful hopping, along with the relaxation of the surrounding charged carriers caused the dispersion of conductivity at high frequencies [37]. In the high-frequency region, the conductivity increase d with the increase of frequency, which may have been due to the hopping of charge carriers in finite clusters. The frequency at which the change in a slope occurs is known as the hopping frequency *ω*<sup>h</sup> (=2*πf* <sup>h</sup>), which obeyed the Arrhenius relation. The plots of *lnf* <sup>p</sup> vs. 1000/*T* was depicted for the composition of *x* = 0.06 in Figure 5c and the value of *E*<sup>h</sup> was calculated from the slope of the fitting line according to Equation (4). The *E*<sup>h</sup> value is calculated to be 1.64 eV for the sample.

At low frequencies and high temperatures, the long-range migration of charge carriers contributed to the DC conductivity (*σ*dc). With the increase of temperature, an increase in charge carrier due to thermal ionization resulted in an increased *σ*dc. Therefore, the temperature dependence of DC conductivity could be described by the Arrhenius relation as Equation (5). Figure 5d shows the plots of *lnσ*dc vs. 1000/*T* and the value of *E*dc was estimated from the fitting of the *σ*dc~*T* curve based on the equation above. The fitting result estimated the value of *E*dc to be 1.87 eV for the composition. Here, a small difference between the values of *E*dc and *E*<sup>h</sup> in the same temperature region indicated the similar type of localized charge carriers responsible for the DC and AC conduction. However, because the activation energy for the conduction process was the sum of diffusion activation (*E*dc) and the formation energy (*E*h) of charge carriers. *E*<sup>h</sup> < *E*dc, indicated that the hopping distance of charge carriers (usually limited in a unit-cell) was always shorter than their diffusing distance (including bulk/intragranular diffusion and grain boundary diffusion).

Table 2 listed the electrical conduction parameters of the CBT-*x*Gd/0.2Mn ceramics calculated according to the method above. As for the pure CBT ceramic, the estimated value of DC activation energy (*E*dc = 1.28 eV) was less than half the band gap value of CBT (*E*<sup>g</sup> = 3.36 eV), indicating an extrinsic conduction process existing in the ceramic [38]. The activation energy was generally associated with the acceptor or donor levels. For the CBT-*x*Gd/Mn ceramics, the values of *W*M, *E*<sup>p</sup> and *E*dc presented a mostly consistent varying trend with the doping content of Gd. The estimated value of *E*dc was found to increase from 1.31 eV to 1.87 eV with an increase in *x* from 0 to 0.06 and then showed a decrease to 1.59 eV till *x* = 0.11. The composition of *x* = 0.06, *E*dc reached to the maximum value of 1.87 eV (associated with a relatively high *E*<sup>h</sup> value of 1.64 eV), so that the composition with *x* = 0.06 could obtain the lowest *<sup>σ</sup>*dc value of 1.8 <sup>×</sup> <sup>10</sup>−<sup>5</sup> S/m among the CBT-*x*Gd/Mn ceramics.

**Table 2.** Electrical conduction parameters of the CBT-*x*Gd/0.2Mn ceramics.


In References. [7,15], Z-Y. Shen et al. prepared a kind of Nd/Mn co-doped CBT ceramics, where the activation energy (*E*<sup>a</sup> = 1.2–1.3 eV) in the temperature range 300~600 ◦C were suggested to be closed to the high temperature dc conductivity activation energy (*E*dc) reported for other BLSF ceramics, which was predominated by the conduction mechanism of intrinsic charge carriers. As compared with his works, the Gd/Mn co-doped CBT ceramics prepared in this work presented a higher activation energy (*E*dc = 1.31–1.87 eV). At high temperatures, when the intrinsic conduction predominates the material, the nominal activation was the sum of diffusion activation (*E*d) and the formation energy of charge carrier (*E*<sup>f</sup> ). Therefore, a higher activation energy observed in our material may be owing to the different doping effect between Gd and Nd in the CBT lattice. Considering the substitution of Gd3+ and Nd3+ for Ca2+ at A-site, a stronger Gd–O bonds compared to Nd–O bonds might induce an increase in the formation energy of oxygen vacancies.

It was well known that the primitive BLSFs were usually not stoichiometric, since that contained amounts of inherent defects, such as oxygen vacancies and bismuth vacancies, et al. This was owing to that the unavoidable volatilization of Bi2O<sup>3</sup> during the high-temperature sintering of ceramics would produce the complexes of bismuth and oxygen vacancy in the (Bi2O2) 2+ layers. Therefore, the doubly positively charged defects, oxygen vacancy V<sup>O</sup> ••, was considered to be the most mobile intrinsic ionic defect in the perovskite-type ferroelectrics. Their long-range migration in the octahedra of any perovskite structure, which was evidenced through greatly enhanced conductivity and activation energy of ~1 eV [39], contributed to the intrinsic ionic conduction in the temperature region of ~300 ◦C to ~700 ◦C [39].

Alternatively, according to the experimental data presented in Table 2, the variation of conductivity of the CBT-*x*Gd/0.2Mn ceramics with the doping content of Gd (*x*) did not seem to be regular. The lowest conductivity value at a high temperature (600 ◦C) was observed at *x* = 0.06. For a single phase material with a homogenous microstructure, the electrical conductivity, σ, depended on both the concentration (*n*) and mobility (*µ*) of charge carriers and obeyed the following simplified equation *σ* = *nqµ*. Here, *q* was the number of charges per charge carrier. With increasing *x*, although the oxygen vacancy concentration tended to be decreased by the donor substitution of Gd3+ for Ca2+, the change in the mobility of oxygen vacancies could not be determined. As a result, the conductivity value at a high temperature was not expected to present a regular trend with change in *x* for the CBT-*x*Gd/0.2Mn ceramics. Similar to the investigation on the doping effect in layer structured SrBi2Nb2O<sup>9</sup> ferroelectrics [40], the experimental results in this work suggested that the doping effects of A-site (Ca/Gd) on the dc conduction were complex, and further analysis is required to achieve a better understanding.
