Effect of Pre-Polarization Process on the Apparent Piezoelectric Response Measured by Point-Ring Method in Ferroelectric Perovskite Oxide Ceramics

Abstract

Both flexoelectricity and piezoelectricity belong to the electromechanical coupling effect. While piezoelectricity only exists in materials whose crystal structure is noncentrosymmetric and a pre-polarization process is necessary for ferroelectric piezoelectric ceramics, flexoelectricity theoretically exists in all dielectric materials and does not require pre-polarization. However, this does not mean that flexoelectricity would not be affected by the pre-polarization process, considering that flexoelectricity is a polarization phenomenon. In this work, we prepared ferroelectric perovskite oxide ceramics Ba_1−xCa_xTiO₃ and revealed a strong effect of the pre-polarization process on the flexoelectric response of the ceramics, characterized by the apparent piezoelectric response measured by the point-ring method. The effective piezoelectric coefficient was separated into the one contributed by the flexoelectric(-like) response and the piezoelectric(-like) response by quasi-static d₃₃ measurement and a two-step point-ring testing method. The effective piezoelectric coefficient contributed by the flexoelectric(-like) response of the ceramics could be largely enhanced to be over 350 pC/N after a 900 V polarization, larger than the standard piezoelectric response. The pre-polarization process was suggested to alter the polarization state and defect distributions, which would further change the overall flexoelectric response (both intrinsic and extrinsic parts) of the samples. Our work indicates a facile method to enhance the apparent piezoelectric response of flexoelectric materials under a bending mode.

1. Introduction

The ability of dielectric materials to produce polarization in response to strain or stress gradients is called the flexoelectric effect (flexoelectricity) [1,2,3,4,5]. The basic equation of the polarization response of a dielectric material to external fields is as follows [6]:

$$P_i={\chi }_{ij}{E}_j+e_{i,kl}{U}_{kl}+{\mu }_{imkl}\frac{\mathrm{\partial }{U}_{kl}}{\mathrm{\partial }{x}_m}$$

(1)

Here, Einstein’s summation convention is adopted for repeated indexes. $P_i$ is the electric polarization intensity, $E_j$ is the external electric field, $U_{kl}$ is the elastic strain tensor and $\frac{\mathrm{\partial }{U}_{kl}}{\mathrm{\partial }{x}_m}$ denotes the strain gradient of the elastic strain; ${\chi }_{ij}$, $e_{i,kl}$, and ${\mu }_{imkl}$ are the dielectric susceptibility tensor, the piezoelectric coefficient tensor, and the flexoelectric coefficient tensor, respectively. The three terms in the RHS of Equation (1) are the dielectric response of materials, piezoelectric response, and flexoelectric response, respectively. Note that the piezoelectric coefficient in Equation (1) is defined under the clamping condition. In practice, it is usual to apply stress to the material. In this case, $P_i=d_{i,kl}{\sigma }_{kl}$, where ${\sigma }_{kl}$ is the stress tensor and $d_{i,kl}$ is the so-called piezoelectric strain coefficient tensor. In the Voigt notation, $d_{i,kl}$ is written as a 3 × 6 matrix. For a thin piezoelectric plate, one can measure the value of the d₃₃ component by normally stressing the plate. Assuming that the area of the plate is A, the total force is F (=σ₃₃A), and the collected charge is Q (=P₃A), one has d₃₃ = Q/F. This is the basic principle of quasi-static d₃₃ measurement. For a thin circular plate bending in the symmetrical flexural mode caused by a force F normally applied to its center, its charge response can be characterized by an effective piezoelectric constant, namely d = Q/F.

Due to the universality of the flexoelectric effect and its potential in various applications (e.g., energy harvesting devices [7], actuators [8,9], and sensors [10,11]), it has aroused great academic interest. In 1985, Tagantsev theoretically showed that the flexoelectric coefficient $\left({\mu }_{imkl}\right)$ of a dielectric material is positively correlated with its dielectric susceptibility (${\chi }_{ij}$) [1,2]. This prediction has led researchers to search for ferroelectric materials with large flexoelectric coefficients [12,13] as ferroelectric materials generally have large dielectric constants. However, the measured flexoelectric coefficients of ferroelectric ceramics are quite different from the theoretically calculated results. Usually, the flexoelectric coefficients obtained from experiments are several orders of magnitude larger than the theoretical prediction [6,14,15]. This unique phenomenon inspired researchers to explore the reasons for the significant flexoelectric coefficients that are inconsistent with theoretical predictions in experiments [16,17,18,19,20,21], and also aroused researchers’ interest in further exploring methods to enhance the flexoelectric response of materials [22,23,24]. At present, various methods have been reported to improve the material flexoelectric coefficient, such as by vacuum reduction for semiconducting materials [16], by preparing ferroelectric-paraelectric composite materials [18], by introducing local structural heterogeneity [25], and by light absorption [26]. Alternatively, the flexoelectric effect can be introduced by injecting charges to form electrets [27]. By changing the types of electrodes plated on the sample, the flexoelectric response can also be improved [23]. It is well known that piezoelectricity only exists in materials whose crystal structures are noncentrosymmetric, and a pre-polarization process is necessary for ferroelectric piezoelectric ceramics. The polarization process, including polarization voltage, polarization temperature, and polarization time, will have a certain impact on the piezoelectric effect. In contrast, flexoelectricity theoretically exists in all dielectric materials and it can perform without pre-polarization. However, it does not mean that flexoelectricity would not be affected by the pre-polarization process. It is expected that the pre-polarization process would alter the polarization state and defect distributions, which would further change the overall flexoelectric response (both intrinsic and extrinsic parts) of the samples, e.g., by modifying the material dielectric susceptibility and by modifying the piezoelectric mimicries (e.g., surface piezoelectricity and piezoelectricity asymmetry) of flexoelectricity [6,14,20,21]. However, the effect of pre-polarization on the flexoelectric effect remains largely unexplored.

In this work, we prepared ferroelectric perovskite oxide ceramics Ba_1−xCa_xTiO₃ and revealed the effect of the pre-polarization process on the apparent piezoelectric and flexoelectric response of the materials. The effective piezoelectric coefficient was separated into two parts, which are roughly defined as the flexoelectric(-like) part (d^eff,flexo) and the piezoelectric(-like) part (d^eff,piezo), via combining quasi-static d₃₃ measurement and a two-step point-ring testing method [28]. On this basis, the influence of the pre-polarization process on these two effective piezoelectric coefficients was explored. It showed that the pre-polarization process has a significant effect on the flexoelectric(-like) response, and the effect is more complicated than that of the piezoelectric(-like) response. Our result should help understand the influence of the pre-polarization process on the flexoelectric effect of ferroelectric perovskite oxide ceramics and indicates a facile method to enhance the apparent piezoelectric response of ferroelectric materials under a bending mode.

2. Experimental Procedure

In this work, ferroelectric perovskite oxide ceramics Ba_1−xCa_xTiO₃ (BCT) with different proportions x were prepared by the traditional solid-state sintering method. After weighing according to the stoichiometric ratio, BaTiO₃ (purity ≥ 99) and CaTiO₃ (purity ≥ 99) raw materials were mixed uniformly in a ball mill at 450 rpm for 6 h. We then put the powder in a drying oven. The dried powder was kept at 1000 °C for 2 h for pre-sintering, and then ball-milled again for 6 h and dried. Then, 6% PVB (polyvinyl butyral) dissolved in ethanol was added to the powder for granulation. The granulated powder was further poured into a mold and pressed into a disc-shaped body under a cold isostatic press. The ceramic body was kept in a muffle furnace at 500 °C for 2 h and cooled with the furnace for degumming treatment. After that, it was sintered into ceramics at 1320 °C in a muffle furnace for 2 h. The phase analysis and crystal structure characterization of ceramics were carried out by an X-ray powder diffractometer (D8 ADVANCE, Bruker company, Karlsruhe, Germany). The surface morphology and structure of the ceramics were observed by a G500 high-resolution thermal field emission scanning electron microscope (Gemini500, Zeiss/Bruker, Karlsruhe, Germany). We analyzed the surface composition of the samples by an energy-dispersive spectrometer (EDS). The ceramics were polished to 0.8 mm thick. After ultrasonic cleaning and drying, the BY-7270-sintered lead-free conductive silver paste was evenly brushed on both sides of the sample. Then, the samples were placed in the muffle furnace to sinter the electrodes. The ferroelectric properties of the ceramics were tested by a ferroelectric analyzer (Radiant Precision LC II, Albuquerque, NM, USA). The dielectric properties of the ceramics were tested by an LCR meter (Tonghui TH2811D, Shenzhen, China).

The piezoelectric properties of the ceramics were tested by a quasi-static d₃₃ meter (ZJ-6A, Chinese Academy of Sciences, Beijing, China), as shown in Figure 1a. We used a point-ring method to measure the effective piezoelectric coefficient of the ceramics. The effective piezoelectric coefficient can be separated into two parts, which are roughly defined as the flexoelectric(-like) part (d^eff,flexo), which would not change sign by inverting the sample, and the piezoelectric(-like) part (d^eff,piezo), which would change sign by inverting the sample. In this method, we added a ring-shaped support at the bottom of the sample, as shown in Figure 1b. The top probe of the d₃₃ meter exerts a local stress at the center of the sample surface, and the sample will be bent downward. Compared with the standard d₃₃ measurement, a strain gradient was generated in the thickness direction of the sample to trigger the flexoelectric(-like) response. For samples without residual polarization, the apparent piezoelectric coefficient, as displayed by the quasi-static d₃₃ meter, is totally contributed by the flexoelectric(-like) effect. However, for samples with residual polarization (e.g., after pre-polarization), the apparent piezoelectric coefficient was found to consist of both the flexoelectric(-like) part and the piezoelectric(-like) effect. To separate these two contributions, a two-step point-ring method was used for ceramics after pre-polarization:

(1) After polarization, the residual polarization direction of the sample is put upward. The sample is subjected to the superposition of the flexoelectric(-like) effect and the piezoelectric(-like) effect. The apparent piezoelectric coefficient (d^eff1) is the sum of the effective piezoelectric coefficient contributed by the flexoelectric(-like) effect (d^eff,flexo1) and that contributed by the piezoelectric(-like) effect (d^eff,piezo1). Its mathematical expression is

d^eff1 = d^eff,piezo1 + d^eff,flexo1

(2)

(2) Then, we turn the sample upside down, and the residual polarization direction of the sample is downward. The sample’s apparent piezoelectric response is counteracted by the flexoelectric(-like) effect and piezoelectric(-like) effect. The apparent piezoelectric coefficient (d^eff2) becomes the difference between the effective piezoelectric coefficient contributed by the flexoelectric(-like) effect (d^eff,flexo2) and that contributed by the piezoelectric(-like) effect (d^eff,piezo2). Its mathematical expression is

d^eff2 = −d^eff,piezo2 + d^eff,flexo2

(3)

The effective piezoelectric coefficient contributed by the flexoelectric(-like) effect (d^eff,flexo) and that contributed by the piezoelectric(-like) effect (d^eff,piezo) can then be calculated by Equations (2) and (3):

d^eff,flexo = (d^eff1 + d^eff2)/2

(4)

d^eff,piezo =(d^eff1 − d^eff2)/2

(5)

3. Result and Discussion

According to relevant reports, the solid solution limit of CaTiO₃ in BaTiO₃ at room temperature is 23–33% [29,30]. Figure 2a shows the sintered ceramic samples, BaTiO₃ (BT), Ba₀.₉Ca₀.₁TiO₃ (BCT10), and Ba₀.₆Ca₀.₄TiO₃ (BCT40) from left to right. After polishing, the diameters of the three ceramic samples are all ~1.1 cm. It can be seen from Figure 2b that the three ceramic samples all show an obvious splitting of the (002)/(200) XRD peak around 45°, which is the characteristic peak of tetragonal BaTiO₃, indicating that the main phase of all the ceramics is the tetragonal phase. The XRD patterns of BT and BCT10 are quite similar, indicating that a solid solution structure is formed when the content of CaTiO₃ is less than 10%. With the increase in Ca content, the diffraction peaks are shifted to the high-angle direction obviously, indicating that the lattice constant is decreased. This is because the Ca²⁺ ion radius (1.34 Å) is smaller than the Ba²⁺ ion radius (1.61 Å) [31]. The characteristic XRD peak corresponding to the orthorhombic phase CaTiO₃ is observed in BCT40, as shown in Figure 2b, indicating that when x = 0.4, the ceramic has two phases, the main phase of the tetragonal phase and a small portion of the orthorhombic phase.

Figure 2 depicts the SEM planar images and in situ EDS spectra of BT, BCT10, and BCT40 ceramics. From the SEM images, it can be seen that the prepared ceramics have no obvious cracks and cavities and are in good compactness, indicating the high quality of the sintered ceramics. The difference between BT and BCT10 in Figure 2c,d is not obvious. However, by comparing Figure 2f,g, it can be clearly seen that the Ca element is uniformly distributed in the BCT10 ceramic. Combined with the XRD pattern, we can know that CaTiO₃ is solvable in the BaTiO₃ matrix at this composition, and the phase structure is tetragonal single phase. For BCT40, one can see that there are two phases from the light and dark contrast in Figure 2e. Combined with Figure 2h, we can determine that the bright region in Figure 2e corresponds to the solution with BaTiO₃ dominant, and the dark region corresponds to the solution with CaTiO₃ dominant. There is an obvious two-phase separation at this composition, consistent with previous reports [29,30,32].

The ferroelectric properties of the ceramic samples were characterized by a ferroelectric analyzer. Figure 3a–c respectively depict the polarization-electric-field (P-E) hysteresis loops of BT, BCT10, and BCT40 ceramics under different maximum applied voltages (i.e., 1 kV, 2 kV, 3 kV, and 4 kV) at room temperature. It was found that the BT, BCT10, and BCT40 ceramics exhibit standard P-E hysteresis loops, indicating that the ceramics all have good ferroelectric properties. The maximum polarization value that can be achieved by Ba_1−xCa_xTiO₃ ceramics after pre-polarization under the same electric field decreases gradually with the increase in CaTiO₃ content, among which the BT ceramic has the highest maximum polarization value. By comparing Figure 3a,b, it can also be seen that when the content of CaTiO₃ is small, the remanent polarization after pre-polarization under the same electric field does not change significantly; while from Figure 3c, it can be seen that the remanent polarization of the BCT40 ceramic after pre-polarization decreases more. At the same time, the coercive field also increases with the Ca content, which is manifested as a broadening of the loop. One can also see that the polarization of the BCT40 ceramic is less saturated than the other two ceramics after the same maximum electric field application. This is understandable considering that CaTiO₃ is a paraelectric material, which should decrease the stability of the ferroelectric phase. Moreover, with the increase in the content of CaTiO₃, the dielectric strength of the ceramic was found to be significantly improved. This is manifested with the increase in the breakdown strength, and the measurement of the hysteresis loop could be carried out at a higher electric field.

Figure 4 further shows the dielectric constant and dielectric loss (measured at 20 °C and 100 Hz) of Ba_1−xCa_xTiO₃ (x = 0, 0.1, 0.4) ceramic samples after polarization at different electric fields. To perform the measurement, we put the samples in a 120 °C oil bath and polarized them under 100 V–900 V DC voltage (corresponding to electric fields from 0.125 MV/m to 1.25 MV/m) for 20 min. After polarization, we placed the samples for 20 min, and then used the LCR meter to measure the dielectric properties after discharge. It can be seen from Figure 4a,b that the dielectric constant and dielectric loss of the BT ceramic are much larger than those of BCT10 and BCT40. With the increase in polarization voltage, the dielectric constant and dielectric loss of BT ceramic also increase significantly. In contrast, the dielectric constant of BCT10 and BCT40 ceramics remains almost constant at different polarization voltages. Meanwhile, the dielectric loss of BCT10 also maintains low values around 0.016 at different polarization voltages. For BCT40, the dielectric loss after 100 V polarization increases significantly compared with that without polarization (from 0.0125 to 0.021), and then it maintains values around 0.020 at different polarization voltages. It is noteworthy that the BCT10 sample has the lowest dielectric loss after polarization but with a much higher dielectric constant than the BCT40 sample, indicating that it is possible to optimize the dielectric performance of the Ba_1−xCa_xTiO₃ system near the solid solution phase boundary.

Now, we turn to the effect of the pre-polarization process on the electromechanical properties of the Ba_1−xCa_xTiO₃ ceramics. In Figure 5a,b, we depict the apparent piezoelectric coefficients of the samples after polarization under 100 V–900 V DC voltage in a 120 °C oil bath for 20 min, with (a) being the apparent piezoelectric coefficient d^eff1 after the superposition of the flexoelectric(-like) effect and piezoelectric(-like) effect and (b) being the apparent piezoelectric coefficient d^eff2 after the cancellation of the two effects (see the insets in the figures). These two coefficients were directly measured by the two-step point-ring method described in Section 2. Based on the data of d^eff1 and d^eff2, the effective piezoelectric coefficient contributed by the flexoelectric(-like) effect (d^eff,flexo) and that contributed by the piezoelectric(-like) effect (d^eff,piezo) were then calculated according to Equations (4) and (5), and the results are shown in Figure 5c,d, respectively. In Figure 5d, we also plot the standard piezoelectric coefficient (d₃₃) measured by the standard quasi-static d₃₃ method for comparison. It can be seen from Figure 5a that with the increase in polarization voltage, the d^eff1 of both BT and BCT40 increases gradually, while the d^eff1 of BCT10 shows an increase below 300 V and then a decrease above 300 V. From Figure 5b, the change trend of d^eff2 of BCT10 and BCT40 is similar to d^eff1 shown in Figure 5a, while d^eff2 of BT first decreases in the 100 V–500 V stage and increases in the 500 V–900 V stage. Moreover, the d^eff2 of BTO and BCT10 reverse their sign when the polarization voltage increases. This is mainly due to the nonmonotonic dependence on the polarization voltage of the effective piezoelectric coefficient contributed by the flexoelectric(-like) effect d^eff,flexo of these two ceramic samples. As shown in Figure 5c,d, for the two samples, d^eff,flexo and d^eff,piezo have a similar order of magnitude. Hence, the dependence on the polarization voltage of d^eff,flexo of the two samples leads to the nonmonotonic dependence on the polarization voltage and even a sign reversal of the total apparent piezoelectric coefficient measured by the point-ring method.

It is also clear from Figure 5c that the polarization process significantly affects the flexoelectric(-like) response of the samples. For BT, d^eff,flexo after 900 V polarization is 355 pC/N, which is 5.6 times that of that without polarization (63 pC/N) and 2.2 times that of the standard piezoelectric coefficient d^eff,piezo after 900 V polarization (161 pC/N). However, the d^eff,flexo of BCT10 increases first and then decreases with the polarization voltage, with the maximum value being 203 pC/N and the minimum value being −44 pC/N. This indicates that there exists a competitive contribution with that of the intrinsic flexoelectric effect to the overall flexoelectric response. A possible competitive contribution is that caused by the field-dependent piezoelectric mimicries (e.g., surface piezoelectricity and piezoelectricity asymmetry) of flexoelectricity in the sample during the polarization process, as both the polarization near the surfaces and the defect distribution are expected to be modified by the electric field. During bending, the polarity of this flexoelectric-like response will interact with the polarity generated by the intrinsic flexoelectric effect, so as to enhance or weaken the total flexoelectric response of the material. In addition, according to Figure 4a, it can be seen that the dielectric constant of the ceramics is affected by a certain extent after polarization at different voltages. Combined with Figure 5c, it is found that the variation trend of the equivalent piezoelectric coefficient d^eff,flexo of Ba_1−xCa_xTiO₃ ceramics is very close to that of dielectric constant, indicating that the dielectric property of Ba_1−xCa_xTiO₃ ceramics also has a certain impact on the flexoelectric property.

In contrast to the complicated dependence of the flexoelectric response on the polarization voltage, the effect of pre-polarization on the standard piezoelectric response is much more normal. As shown in Figure 5d, with the increase in polarization voltage, the standard piezoelectric coefficients d₃₃ of BT, BCT10, and BCT40 have the same monotonic increasing trend of change with the effective piezoelectric coefficients d^eff,piezo obtained by the two-step point-ring method. The difference between d₃₃ and d^eff,piezo is small, verifying the plausibility of the two-step point-ring method.

In order to verify whether the effect of the pre-polarization process on the effective piezoelectric coefficient contributed by the flexoelectric(-like) effect and that by the piezoelectric(-like) effect is stable, we performed fatigue experiments after the samples were polarized at 400 V and 800 V. As shown in Figure 6a, the time dependence curves of d^eff,flexo and d^eff,piezo of BCT10 and BCT40 after polarization at 400 V maintain basically a straight line within 48 h, while those of the BT sample show an oscillation and a slight decrease with time. After the 800 V polarization, the time dependence curves of d^eff,flexo and d^eff,piezo of BCT10 and BCT40 also maintain basically a straight line, as shown in Figure 6b. The d^eff,flexo of the BT sample after the 800 V polarization decreases significantly within the first few hours and then tends to remain at a value of ~100 pC/N; meanwhile, the time stability of its piezoelectric(-like) part (d^eff,piezo) is quite good. Combined with Figure 6a,b, it is suggested that adding CaTiO₃ into BaTiO₃ can help improve the stability of the flexoelectric response after polarization, which is beneficial for the application of BCT ceramics in practical devices. From Figure 6a–d, one can also see that the time dependence of the standard piezoelectric coefficients d₃₃ is quite similar to that of the d^eff,piezo.

4. Conclusions

In summary, via using quasi-static d₃₃ measurement and a two-step point-ring method, we investigated the effect of the pre-polarization process on the apparent piezoelectric response measured by the point-ring method of Ba_1−xCa_xTiO₃ ceramics. The results show that the effective piezoelectric coefficient contributed by the flexoelectric(-like) response of the ceramics can be largely enhanced by over 350 pC/N after 900 V polarization, larger than that of the standard piezoelectric response. Meanwhile, the effective piezoelectric coefficient contributed by the flexoelectric(-like) response shows a nonmonotonic dependence on polarization voltage and even shows a sign reverse. This indicates that there exists a competitive contribution with that of the intrinsic flexoelectric effect to the overall flexoelectric response. In addition, samples with doping CaTiO₃ are found to show a better temporal stability of the flexoelectric response after polarization. Our work thus reveals that the pre-polarization process is an effective way to enhance the apparent piezoelectric response of ferroelectric perovskite oxide ceramics under a bending mode.