Verification of the flexoelectricity in barium strontium titanate through d₃₃ meter

Flexoelectricity denotes the linear electromechanical coupling between strain gradient with electric polarization, given in the form of¹ 

where P_l is the induced polarization, $μijkl$ is the direct flexoelectric coefficient with respect to strain gradient, a fourth-rank tensor, S_ij is the strain and x_k is the axis of coordinate, fijkl is the direct flexoelectric coefficient with respect to stress gradient, also a fourth-rank tensor, and X_ij is the stress. Unlike piezoelectricity which only exists in non-centrosymmetric materials, flexoelectricity is not limited by crystal symmetry and can appear in all sufficiently insulating materials.² Theoretical models show that the flexoelectric response scales inversely with size, and can be engineered to be as large as the piezoelectric effect in nanoscale structures.³ On the other hand, large local strain gradient induced flexoelectric polarization was invoked to explain nanoscale inversion of polarization in BaTiO₃ films under mechanical pressure.⁴ It has been employed as the physical principle for various sensing applications.^5–7 

Though flexoelectricity in solid materials was found several decades ago, it has not aroused much scientific interest until the early stage of new century, owning to the negligible magnitude (∼10^-10 C/m) in simple solids.⁸ Nevertheless, research in this field has been increasingly growing since the experimental observation of giant flexoelectric coefficients in ferroelectric materials with high dielectric constant by Cross et al.^1,9 Generally, the flexoelectric coefficients reach the highest value near the Curie temperature of the ferroelectrics and diminish as the temperature moves away from this critical temperature. The dielectric permittivity of the ferroelectric materials follows the similar trend, thus their work verified the theoretical prediction of the linear relationship between flexoelectric coefficient and the dielectric permittivity, which was first proposed by Tagantsev in the form of¹⁰ 

where $χij$ is the susceptibility of the dielectric, $γkl$ is a constant material parameter tensor, e is the charge of the electron and a is the atomic dimension of the unit cell of the dielectric.

On the other side, in the cantilever situation, since the flexoelectric property of ferroelectric materials was measured under ferroelectric phase as well as the paraelectric phase, a flip-over experiment was conducted to rule out the possible existence of any spontaneous remnant polarization, in other words, the piezoelectricity when the temperature was below the Curie temperature.¹ This is because that piezoelectricity has an inborn orientation with respect to the materials while it is not the case for flexoelectricity, therefore the piezoelectricity induced current should switch the direction but the flexoelectric component should remain the same when the sample is flipped over, given the identical mechanical stimulus.

Nevertheless, the experimental measurement of the giant flexoelectricity in ferroelectric materials contradicts with the theoretical estimation based on the first principle calculation,^11,12 which suggests values three orders of magnitude smaller. Such discrepancy drove researchers to exploit the origin of the giant measurement results, in addition to the contribution from the intrinsic flexoelectricity based on Kogan-Tagantsev model. In spite of being far away from reaching an agreement, till now, two main mechanisms have been proposed as the explanation. The first mechanism attributes the giant flexoelectricity in paraelectric phases of ferroelectrics to the residual ferroelectricity.¹³ Biancoli et al. reported an observation of the breaking of macroscopic centric symmetry in paraelectric phases of (Ba, Sr)TiO₃.¹⁴ They claimed that high-temperature processing of the materials induces inhomogeneity, leaving piezoelectricity to exist above Curie temperature, which could be erroneously regarded as flexoelectric contribution. The second mechanism that could account for the giant flexoelectricity is related with polar nanodomains.^15,16 Polar nanodomains could persist above the Curie temperature until the onset of anelastic softening of the crystals. Within this temperature range, the polar nanodomains could reorientate themselves under external stress, thus enhancing the bending-induced polarization.

In this paper, we focused on the proceeding controversy and systematically studied the flexoelectric behaviors of the famous flexoelectric material-BST ceramics by the undisputed measurement. Our results would, in a new sight, prove the pure intrinsic flexoelectricity in this kind of materials and hence help elucidate the origin of the giant flexoelectricity in paraelectric phase of ferroelectrics.

Ba_0.67Sr_0.33TiO₃ (BST) powders were prepared by solid state reaction method using high-purity BaCO₃ (99%) and SrCO₃ (99%) as well as rutile TiO₂ (99%) (Analytical reagent, Beijing Sinopharm Chemical Reagent Co. Ltd., Beijing, China) as starting materials. Stoichiometrically weighed powders were wet-milled with alcohol and zirconia ball grinding media for 6 h, dried and calcined at 1150 °C for 3 h and followed by second ball milling. Pellets with 10mm in diameter and 1–2 mm in thickness were pressed using 5 % polyvinylacetate (PVA). After that, the pellets were fired at 600°C for 4 h to remove the organic binder and then sintered at 1375 °C for 2 h.

The temperature and frequency dependence of dielectric properties of the BST ceramics were measured using a broadband dielectric spectrometer (Concept 40, Novocontrol, Germany) and an impedance analyzer (E4294A, Agilent, Palo Alto, CA), respectively. As shown in Fig. 1, the dielectric constant peaks at almost 302 K with a sharp bandgap. On the other hand, no frequency shift were observed in this kind of materials, revealing its typical ferroelectric nature. In this case, according to the classical ferroelectrics theory, the macro piezoelectricity would disappear because the material is in paraelectric phase in room temperature. However, recent studies suggested such statement is possible to be invalid especially when the temperature is very close to the Curie temperature. Next we further verified the possible influence of BST ceramics by hysteresis loop, irrespective with the conclusion made by typical ferroelectric theory that no loop would be observed in the paraelectric phase of perovskites. The polarization–electric field (P-E) hysteresis loops were measured using a ferroelectric tester (TF Analyzer 2000, aixACCT, Aachen, Germany) at room temperature under 10 Hz. However, even in the paraelectric phase, the measured BST ceramics still exhibited a little loop behavior as shown in Fig. 2. The observed loop is gradually expanded when the applied electric filed is increased. Particularly, when the applied electric field equals to as low as 2 kV, the P-E response become quite narrow, but not a perfect linear line. This experimental result is possible to suggest the existence of a little ferroelectricity in this kind of material when the temperature is slightly higher than Curie temperature.

In this case, taking account of the conventional cantilever system for the flexoelectric measurement is possible to bring some ferroelectric error. Consequently, in this paper, we used the equivalent piezoelectric method for flexoelectric measurement by d₃₃ meter (ZJ-3B/4B, H. C. Materials, Bolingbrook, IL). It is noticed that the resolution of this piezoelectric measurement can be as high as 0.1 pC/N. In order to extract the pure flexoelectricity, the prepared BST blocks were firstly mechanically diced into two groups (trapezoid group and rectangle group) by a dicing saw (Disco DAD320 Mesa, AZ), where the rectangle group is only a reference. With the help of the 0.3 mm thick blade, we precisely diced the materials according to our design. After dicing, the samples were annealed at 500 °C so that the residual stress of the materials could be removed. The geometry details of these two groups were schematically shown in Fig. 3. It is noticed that the design of these two groups aimed at distinguishing the pure flexoelectricity with other factors, because we only changed the geometry of the materials which could generate a strain gradient. Subsequently, we measured the equivalent piezoelectric constants by clamping all these samples between the piezoelectric tip and holder in z direction, while other directions are set free. According to the size effect of the flexoelectricity, the smaller trapezoid sample should display a larger d₃₃ value. It is also noticed that all these samples were then measured by the same d₃₃ meter under the same condition. The experimental result was shown in Fig. 4. For the first clamping method which is named as up-down, the samples exhibited different piezoelectric response. For the non-flexoelectric rectangle groups, the d₃₃ constant of all the samples were measured to be zero. However, for the rectangle samples, the d₃₃ value were measured to be 0.3 pC/N.

Next we flipped the samples and named it as bottom up. The rectangle groups still hold the same value. However, the d₃₃ constant of the smallest rectangle sample occur a sign variation. Other rectangle samples still has non-zero value, but has no sign variation. This experiment would partially demonstrated the existence of flexoelectricity in this kind of materials. On the other hand, in our experiment, the size effect of flexoelectricity is not obvious. No significant value difference was observed in the 2mm rectangle BST sample and 4mm one, suggesting the flexoelectric coefficients is possible to be varied under different strain gradient, or in other words, also a function of strain gradient. According to this experiment, we proposed the flexoelectric behavior was actually an average charge effect which highly relies with the size and the procedure process of materials. However, it is noticed that the verification result presented in this paper still contains some uncertainty. The investigated material is possible to exhibit a piezoelectric response until poled by stress or electric field, which could happen during processing or electrical measurements, and hence bring some interference for our flexoelectric verification.

In conclusion, we experimentally observed the existence of ferroelectricity in the paraelectric phase of BST materials. Such ferroelectricity would be coupled into the cantilever flexoelectric measurement but separated by the d₃₃ meter. Our comparison experiment by d₃₃ meter partially proved the existence of flexoelectricity in BST materials. The measured flexoelectricity suggested BST could be a promising flexoelectric material.

This work is supported by the National Natural Science Foundation of China under grant number 51605052, 11604135 and 11574126, and in part by the Natural Science Foundation of Jiangxi Province (20161BAB216110).