High-field electromechanical response of Bi_0.5Na_0.5TiO₃–Bi_0.5K_0.5TiO₃ across its morphotropic phase boundary

The XRD patterns of the BNKTx samples with different values of x are shown in figure 1(a), indicating only perovskite phases for all samples (Miller indices in the figure correspond to the pseudocubic symmetry). The splitting of the (1 1 1) reflection in the 39°–41° 2θ range (figure 1(b)) is due to rhombohedral symmetry (assigned to the (1 1 1)/ $(1\,\bar{1}\,1)$ reflections) like that of BNT, whereas the peak splitting of (0 0 2)/(2 0 0) in the 45°–48° 2θ range (figure 1(c)) is due to tetragonal symmetry like that of BKT. The splitting of both peaks indicates the coexistence of perovskite rhombohedral and tetragonal polymorphs at these MPB compositions in the BNKTx solid solutions as anticipated [6]. By increasing x from 0.18 to 0.24, the potassium-to-sodium ratio is raised, and the diffraction peaks are systematically shifted towards lower 2θ angles as seen in figures 1(b) and (c), indicating the lattice expansion [20]. Not surprisingly, the percentage of tetragonal phase increases with x.

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Figure 2(a) shows a typical microstructure of the BNKT0.20 sample. Since all the samples were prepared using the same processing parameters, particularly, sintering temperature and soaking time, significant differences in grain size distribution or porosity across the MPB region were not found; that is, a dense and homogeneous microstructure with an average grain size of about 1 µm was consistently obtained. EDXS standardless chemical analysis indicated nominal composition (spectrum for BNKT0.20 is given in figure 2(b) as an example). However, this technique is not sensitive enough to detect the small deviations in K/Na ratio among the different samples. The density values of all sintered bodies, relative to the theoretical densities, are above 98%, except for BNKT0.24 that was 95%. These relative densities are presented in table 1.

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Table 1. Dielectric permittivity (ε_r), losses (tan δ), and frequency independent depolarization temperature (T_d) all at 10 kHz, temperature of maximum permittivity (T_m) at 1000 kHz, relative density, electromechanical planar coupling factor (k_p), longitudinal (d₃₃), transverse (d₃₁) and effective $(d_{33}^{\ast})$ piezoelectric coefficients determined for the BNKTx ceramics (x = 0.18–0.24).

Composition	x = 0.18	x = 0.20	x = 0.22	x = 0.24	References
Relative density (%)	98.2	98.5	98.7	95.5	97.6 [44]
εr (at RT)	1344	1743	1518	1209	1030 [6], 1140 [20], 945 [38]
tan δ (at RT)	0.048	0.053	0.060	0.056	0.076 [20], 0.020 [38]
Td (°C)	130	100	71	65	—
Tm (°C)	271	284	313	320	—
d33 (pC N−1)	150	195	160	125	134 [20], 168 [17], 190 [44]
d31 (pC N−1)	−34 + 1.077i	−39.56 + 1.031i	−37.56 + 2.286i	−28 + 1.872i	—
$d_{33}^{\ast} ( {{\rm pm}\,{\rm V}^{{\rm -1}}})$	560	1320	440	360	291 [17]
kp (%)	26.71	27.03	22.47	15.41	35.24 [44]

Figure 3 shows the temperature dependence of dielectric permittivity (ε_r) and losses (tan δ) for all ceramic samples at different frequencies, namely 0.1, 1, 10, 100 and 1000 kHz. In all cases, the maximum of dielectric permittivity (ε_m) is partially masked with a dielectric relaxation, which is better seen in the tan δ curves. This type of dielectric response overlapping with the FE transition anomaly is often observed in perovskite-type oxides, and it is attributed to the presence of defects in the material, such as oxygen vacancies and Maxwell–Wagner type polarization mechanisms due to blocking of mobile ions at interfaces and/or grain boundaries [22].

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Figure 4 shows the results of an Arrhenius-type analysis of the dielectric loss maxima for this dielectric relaxation above T_m, using the following equation:

$$\begin{equation} \omega =\omega_{\rm o} \exp (-E_{\rm a} /k_{\rm B} T) \end{equation} \tag{ 1 }$$

where E_a is the activation energy, k_B the Boltzmann parameter, T is the absolute temperature and ω = 2πf. Results showed activation energies about 2.5 eV with no systematic dependence on the composition. This value is far beyond those usually reported for oxygen vacancy ionization [23], and it could be rather associated with the release of ionic defects trapped at the domain walls when the material undergoes the FE transition [24].

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Nevertheless, and back to the transition, the temperature of maximum permittivity (T_m) can be clearly distinguished in the curves at 1000 kHz, a frequency at which the relaxation has shifted well above T_m. On the other hand, the permittivity curves in all cases, aside from the peak of maximum permittivity, exhibit a lower temperature shoulder which is the least pronounced for the BNKT0.18 sample. It is observed that an increase in the potassium-to-sodium ratio results in higher T_m values, which are also tabulated in table 1. Note that BNT and BKT show the FE transition at 300–320 °C and 380 °C, respectively, so an increase across the solid solution is expected.

The broad maxima observed in the permittivity curves might indicate the existence of a diffuse phase transition or even a high temperature relaxor state in the BNKTx system, as previously reported [25]. In fact, a modified Curie–Weiss law can be applied as

$$\begin{equation} \frac{1}{\varepsilon_{\rm r}}-\frac{1}{\varepsilon_{\rm m}}=C\left( {T-T_{\rm m}} \right)^{\gamma} \end{equation} \tag{ 2 }$$

where C and γ are the Curie constant and the diffusion exponent, respectively [26]. γ ranges between 1 and 2 for FE materials. When γ = 1, the above relation reduces to the simple Curie–Weiss law and the material is a classic perovskite FE and when it approaches 2, a relaxor-type FE behaviour is obtained. Intermediate values are associated with diffuse phase transitions. The ln((1/ε_r) − (1/ε_m)) versus ln(T − T_m) is plotted in figure 5 for BNKTx at 1000 kHz. The slopes of the lines (γ values) for different compositions lie in the 1.83–1.98 interval which is very close to the value for relaxor ferroelectrics.

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In relation to the lower temperature anomaly, it has been previously associated with the depolarization event, and a number of different methods have been suggested to define the characteristic temperature (T_d) for a ceramic sample [27, 28]. Unfortunately, a consensus has not been reached but the work of Hiruma et al [29] provides a good basis for estimating the values of T_d from the first peak of tan δ curves in fully poled samples. Figure 6 illustrates the plots of the dielectric permittivity versus temperature for both the poled and unpoled BNKTx samples at the frequency of 10 kHz. There is a discontinuity in tan δ versus temperature which manifests itself as a spike in poled samples at lower temperatures. These spikes, unlike the first maxima in tan δ curves of the unpoled samples are frequency independent and better indicators of the depolarization temperatures (T_d) according to these authors [17]. It is observed that an increase in the potassium-to-sodium ratio results in lower T_d values, which are also tabulated in table 1.

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In table 1, the values ε_r and tan δ for all the compositions at RT have been compared with similar reports. The reported values are for the samples in the unpoled state. Note that permittivity values are significantly higher than those in the literature, while comparable losses are observed, which suggest the high quality of the samples. Note that dispersion is not caused by electrical conduction for losses increase with frequency—a characteristic typical of relaxor systems. This suggests the existence of different scale domains within the FE phase, a phenomenon widely observed in relaxor-based ferroelectrics such as Pb(Mg_1/3Nb_2/3)O₃–PbTiO₃ [30]. A maximum of permittivity occurs at x = 0.20, where the MPB is thought to be placed.

The P–E hysteresis loops of BNKTx samples for different values of x are plotted in figure 7(a). The measurements were carried out on the unpoled samples. Conventional loops close to saturation were obtained for all samples except for BNKT0.24, for which a pinched loop was plotted. The coercive field (E_c) is observed to systematically decrease with x, from 3.2 kV mm⁻¹ for x = 0.18 down to 1.3 kV mm⁻¹ for x = 0.24, with an abrupt reduction for x = 0.22 (see figure 7(b)), from which pinched loops start to develop. Remanent polarizations between 28 and 37.5 µC cm⁻² are obtained, with a maximum for x = 0.20. Like permittivity figures, these values are significantly larger than those previously reported for BNKT system [6]. It is observed that increasing the potassium-to-sodium ratio, in general, results in thinner hysteresis loops. In several reports, the presence of AFE order in BNT-based ceramics has been presumed to be the cause of the pinched loops [31, 32]. However, similar loops have also been reported in relaxor systems such as PLZT and PMN-0.1PT, at the onset of FE long-range order [33, 34].

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In these perovskite ferroelectrics, the pinched loops are not attributed to the presence of AFE phases but rather to the presence of a 'non-polar' or 'weakly polar' phase converting to a polar phase upon the application of the electric field [33]. With regard to the significant values of P_r, even for the BNKT0.24 (figure 7), the constriction may be due to the presence of a non-polar or weakly polar phase in the BNKT system. Indeed, there has been no solid confirmation of a FE–AFE transition in BNT-based ceramics above T_d. It is possible that the constriction in the hysteresis loop of BNKT0.24 is due to electromechanical interaction between the polar and non-polar regions [35, 36]. As a matter of fact, the latest works on BNT-based system relate the depolarization event to a destruction of FE long-range order at a transition to a non-identified short-range polar state [37]. Whatever the mechanism, pinched loops correlate with the continuous decrease of the depolarization temperature.

The longitudinal and transverse piezoelectric coefficients (d₃₃, d₃₁) and the electromechanical planar coupling factor (k_p) of all compositions have been compared with those of previous reports in table 1. Figures favourably compare with previously reported values for the system, as permittivity and polarization ones did [17, 38]. To estimate the suitability of BNKTx for actuator applications the strain responses of the samples under unipolar electric field were measured. Figure 8(a) shows the unipolar electric field–induced strain curves of ceramics for different values of x under 1 kV mm⁻¹. Induced-field strain shows a maximum for x = 0.20, like the d₃₃ piezoelectric coefficient, confirming this composition to be the MPB one. A strain as high as S = 0.13%, was obtained for x = 0.20 under this field. The corresponding effective piezoelectric coefficients $(d_{33}^{\ast})$ as a function of increasing field are plotted in figure 8(b). Note the remarkable value of 1000 pm V⁻¹ under 0.2 kV mm⁻¹ that increases up to 1320 pm V⁻¹ under 1 kV mm⁻¹. This value is significantly higher than those of soft PZT and approaches those of textured PMN-PT [39].

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The x = 0.20 composition was further tested under higher electric fields (figure 9(a)) and as a function of temperature (figure 9(b)). An outstanding strain level of 0.38% was obtained under 2.4 kV mm⁻¹ at RT $(d_{33}^{\ast}$ of 1650 pm V⁻¹). Furthermore, a similar strain can be developed at 100 °C under only 1 kV mm⁻¹. This is the temperature at which strain is maximum, and it is also the depolarization temperature for this composition. Giant strains as high as 0.45%, have been reported for lead-free compositions such as BNT–BT–KNN under an electric field of 8 kV mm⁻¹ [31], BNT–KN under 9 kV mm⁻¹ (0.4%) [17], and BNT–SrTiO₃ under 6 kV mm⁻¹ (0.29%) [40].

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This giant strains in BNT-based systems are thought to occur for compositions with depolarization temperature close to RT, associated with an electric field induced phase change between the short-range polar state and long-range FE order [41]. Similar strain levels have been obtained here for BNKT0.20. The S–E curves (figure 9(b)) show the typical features of a phase-change electromechanical response; an initial increase of slope with temperature at a fixed electric field up to T_d showing a large hysteresis which then decreases with increasing temperature [42]. Therefore, one can assume that the same mechanism is active in the BNKT system, as already suggested [43]. The development of such high strain at a much lower electric field for this system is a major novelty, which is a clear advantage for actuation.

It is not clear at this point why BNKT is able to deliver the phase-change strain at a much lower electric field than other materials. Indeed, previous reports for BNKT0.20 [43] or BNKT0.22 [45] indicate much larger threshold fields. This might be related to an effect of microstructure or even defect chemistry that has not been addressed up to now. The low-temperature FE state has clearly an important role because larger strains might be expected for x = 0.24 than for 0.20 because of its lower depolarization temperature. The RT phases of BNKTx across the MPB have been recently shown to present large complexity with a hierarchical structure at different scales [46]. Effects of grain size on phases have been described in lead-containing MPB systems [47]. Therefore, the electromechanical response of a specific BNT-based system seems to result from a delicate interplay between composition, defect chemistry and microstructure.