Flexoelectric characterization of BaTiO₃-0.08Bi(Zn_1/2Ti_1/2)O₃

In direct mechanical-electrical coupling, the generation of electric polarization induced by mechanical stress and strain can be defined as

where $Pi$ is the induced polarization; $eijk$ is the piezoelectric constant; $μijkl$ is the flexoelectric coefficient; and $εjk$ is the applied strain. Flexoelectricity refers to the linear coupling between the strain gradient and the electric polarization. Unlike piezoelectricity, which exists only in non-centrosymmetric crystals, flexoelectricity is a universal phenomenon in materials of all symmetry groups.¹ 

The existence of flexoelectricity in solid crystalline materials was originally proposed by Kogan in 1964.² Its expression in a wide selection of materials first gained attention because it allowed centrosymmetric materials to be used in electro-mechanical applications. In addition to its universality, the effect of flexoelectricity does not require a poling process before use and therefore avoids the aging problems associated with poling and de-poling, which usually is a concern for piezoelectricity.^3,4 These advantages are harnessed in certain micro-machined devices,^5,6 where in each device, a cantilever with a single layer of active material replaces the bimorph piezoelectric cantilever. Moreover, the flexoelectric coefficients in solid crystalline materials are unexpectedly large and can be explained by the recent studies on macroscopic centric symmetry breaking mechanisms,⁷ providing further opportunity to study and design flexoelectricity.

There have been experimental studies on the flexoelectricity of materials from a variety of categories, including solid ceramics, amorphous films, liquid crystals, polymers, and biomembranes.^8,9 It has been reported that flexoelectricity is positively correlated with the dielectric constant in solid crystalline materials, especially piezoelectrics, such as barium titanate (BT),¹⁰ barium strontium titanate (BST),^11,12 lead zirconate titanate (PZT),^13,14 and strontium titanate (STO).^5,6 These materials have been widely investigated due to their high dielectric permittivity, particularly at the Curie temperature (⁠$Tc$⁠).

Furthermore, it is expected that flexoelectric materials work in a broader temperature range than piezoelectric materials.^15,16 Nevertheless, the rapid drop in the dielectric permittivity of piezoelectrics (e.g., BST and BT) in their paraelectric cubic phase may lead to a drawback of persistence of flexoelectricity as temperature increases, showing that some of these materials are not suitable for devices in a broader temperature range.

Recent studies on (1-x)BaTiO₃-xBi(Zn_1/2Ti_1/2)O₃ (BT-BZT) perovskite ceramics have shown promising dielectric properties for high-energy and high-temperature (>300 °C) applications.¹⁷ When x = 0.08, BT-8BZT has exhibited a $Tc$ of ∼25 °C, with a frequency-dependent permittivity and good thermal resistance in dielectric properties against temperatures up to 200 °C. Therefore, it would be promising to investigate the flexoelectricity of BT-8BZT in a broad temperature range.

For a cubic crystal, the non-zero components of the flexoelectric coefficients $μijkl$ are $μ1111$⁠, $μ1122$⁠, and $μ1212$ in the tensor notation or $μ11$⁠, $μ12$⁠, and $μ44$ in the reduced (matrix) notation. Each flexoelectric coefficient is associated with a specific mode: $μ11$ for longitudinal, $μ12$ for transverse, and $μ44$ for shear modes, respectively. Separate experiments have been carried out to determine each component for different materials.¹⁸ Among them, the beam bending method is usually adopted to measure the $μ12$ value. The induced polarization is in response to the axial normal strain gradient in the thickness direction.

where $x3$ is the normal axis to the electrode surface and $ε11$ is the in-plane mechanical strain.

In this letter, flexoelectric cantilever beams were fabricated with BT-8BZT and BST ceramics. The dielectric permittivity was examined for BT-8BZT. Then, at room temperature (RT), the flexoelectricity of BT-8BZT was measured by using a bending method, while piezoelectricity was directly measured. Furthermore, at different temperatures, the flexoelectric transverse coefficients of BT-8BZT were determined and compared with BST ceramics.

Bulk polycrystalline samples of BT-8BZT ceramics were prepared using a conventional solid state synthesis technique by Cann's research group at Oregon State University, as described in detail by Triamnak et al.¹⁷ The sintered pellets were cut into ceramic bars with an approximate size of 1 × 5 × 10 mm³ using a diamond wire saw (Well Diamond Wire Saws, Inc., Norcross, GA). The cantilever beams of BT-8BZT were prepared using a dicing process. Parameters of the beam are presented in Table I. Both the bottom and top electrodes consist of 100 nm (Au)/5 nm (Ti) prepared by the e-beam evaporation (Kurt Lesker Electron Beam Evaporation System, Jefferson Hills, PA).

The relative permittivity of the BT-8BZT ceramic was measured at 0.1 kHz, 1 kHz, 10 kHz, 100 kHz, and 1000 kHz, respectively, using an impedance analyzer (Agilent Technologies, 4294A, Santa Clara, CA). The temperature dependence of the dielectric permittivity is shown in Fig. 1. The gradual decline in relative permittivity with respect to the temperature was observed, which corresponds to a previous report.¹⁷ A derivation from classic Curie-Weiss behavior was noticed, which is mainly caused by the diffuse nature of the phase transition due to the addition of BZT.^17,19

Meanwhile, according to Cann et al.,¹⁷ a transition in the phase from ferroelectric to paraelectric was noted at around 25 °C. Concerning piezoelectricity in BT-8BZT, when the temperature of the BT-8BZT sample is above the Curie temperature (25 °C), the material should theoretically not exhibit any ferroelectricity. However, when the temperature is close to the Curie temperature, a weak persistence of the macro-ferroelectricity may exist due to local nano-domains.²⁰ A small displacement was observed even at the Curie temperature, 25 °C, which reflected the residual ferroelectricity.^10,21 So, the contribution of piezoelectricity in the $d33$ mode to the total electric output was verified at room temperature. The 378 V AC electrical signal was applied across the thickness direction of the unpoled ceramics at 10 Hz, and the displacement was detected using a laser vibrometer (Polytec, OFV-5000, Irvine, CA) with a resolution of 100 nm/V. The measured $d33$ was about 39 pm/V, which reflects the residual ferroelectricity. (This value will be discussed later, along with the flexoelectric characterization at room temperature.) However, the piezoelectric contribution to the total electric output was further suppressed in the paraelectric phase as the temperature increased (∼200 °C), suggesting that the polarization output from the bending test described below would be dominated by the flexoelectric effect.

The flexoelectric measurement setup was similar to the one used in our previous work.¹⁶ In additional, a thermal chamber and a thermocouple were used to measure the sample temperature in this work, as shown in Fig. 2. The ceramic beam was clamped rigidly at one end and deflected at the other end by a piezoelectric actuator, which was driven by a 10 Hz-excitation from a function generator (Tektronix, Model AFG3101, Beaverton, OR) along with a power amplifier (Brüel & Kjær, type 2706, Nærum, Denmark). A laser vibrometer was also used to measure the vibrating displacement (⁠$δ$⁠). The generated polarization resulted in the current (⁠$I$⁠), which was amplified by a charge amplifier (Brüel & Kjær, type 2635, Nærum, Denmark) and then monitored using an oscilloscope (Agilent Technologies, DSO7104B, Santa Clara, CA) at a condition of 1 GHz, 4 GSa/s.

According to the Euler-Bernoulli beam theory, the strain gradient can be calculated as

The measured current, $I$⁠, can be derived from the induced polarization P₃.

The flexoelectric coefficient is calculated using the following equation:

where $I$ is the current, $f$ is the frequency, $δ$ is the tip displacement, $A$ is the electrode area, $L0$ is the cantilever length, and $x$ is the distance between the electrode and the clamped end.

Figure 3 shows the real time charge output of BT-8BZT under a sinusoidal load with a peak-to-peak value of 1.5 μm at 10 Hz. In this, it can be observed that the charge output is out-of-phase with the applied load. This is because the beam was bent downward, causing the strain gradient direction of the BT-8BZT to be opposite to the electrode connection polarity. The noise was eliminated by a bandpass filter (5–15 Hz) in signal processing. At the same time, compared to piezoelectricity and flexoelectricity, the associated electrostrictive effect is quadratic dependent on the field, which is featured with a higher frequency (⁠$2f$⁠), 20 Hz in this case, than the exciting signal (⁠$f$⁠), which is 10 Hz, so that it would not interfere with the flexoelectric signal.

Figure 4 shows that the polarization induced by the flexoelectric effect is linearly proportional to the strain gradient calculated from Eq. (3), which indicates the average $μ12$ (slope). The effective $μ12$ value of BT-8BZT is calculated to be ∼25 μC/m at room temperature. The effective $d33$ value is calculated¹² to be ∼67 pm/V. Because the Curie temperature of BT-8BZT is close to room temperature, there is still the remnant piezoelectric contribution. Compared to the $d33$ value (39 pm/V) from the direct measurement mentioned earlier, the flexoelectric contribution to the effective mechanical-electrical coupling $d33$ is comparable to the part from remnant piezoelectricity at room temperature.

In addition to the room temperature set-up, an extra thermal chamber built with a ceramic fiber blanket was included for flexoelectric measurements at elevated temperatures, as shown in Fig. 2. Both the thermal couple (TC) and the infrared thermal meter (TM) were used to monitor the temperature inside the chamber. Hot air generated by a heat gun (Proheat, PH-1300, LaGrange, KY) was blown into the chamber for temperature control.

Both BST and BT-8BZT cantilever beams were tested at temperatures that ranged from room temperature to 200 °C, where both the materials were in the paraelectric phase. The calculated $μ12$ values of BST and BT-8BZT are plotted in Fig. 5. Both BST and BT-8BZT have high flexoelectric coefficients, 25 μC/m and 22 μC/m at 25 °C, respectively. In BST, an increase in temperature brings the flexoelectricity down rapidly, while the flexoelectric coefficient of BT-8BZT changes more gradually. As the result, the flexoelectric coefficient $μ12$ of BT-8BZT remains at around 12 μC/m at 200 °C, while $μ12$ of BST is less than 3 μC/m at 200 °C. Since the temperature range is above the Curie temperature, weak macro-ferroelectric regions may exist due to the presence of local nano-domains which gradually disappear as the temperature is increased. To verify the disappearance of this residual ferroelectricity, the direct piezoelectric response of BT-8BZT was measured over the same temperature range, and the data are shown in Fig. 6. As indicated in the figure, the piezoelectric coefficient $d33$ decayed rapidly, which indicates that the piezoelectric effect eventually disappears in the paraelectric phase. Thus, the flexoelectric contribution at high temperatures dominates the mechanical-electrical coupling.

Also, it is worth mentioning that compared to the $μ12$ value of lead contained perovskites, such as PZT, PMN, BT, and SrTiO₃ at room temperature (RT), as shown in Table II, the flexoelectric coefficient of BT-8BZT is attractive at both room temperature and elevated temperatures (e.g., 200 °C).

Our experimental results showed there is a relatively less variation in $μ12$ in BT-8BZT compared to that in BST, which is known with the highest $μ12$ value among all the reported ferroelectric materials. This finding is believed to correspond to the broad dielectric maximum associated with the diffuse phase transition exhibited by BT-8BZT.¹⁷ Hence, the thermal dependence of flexoelectricity in BT-8BZT is much more stable than in BST. The previous work by Cross's group¹⁰ showed that the flexoelectric coefficient exactly follows the temperature dependent dielectric properties. Our work with BT-8BZT again proved that its flexoelectric ability and its dielectric permittivity are correlated in such materials. The broaden ferroelectric-paraelectric peaks of BT-8BZT have enhanced the flexoelectric response with good temperature stability.²³ Furthermore, the thermal stability of the enhanced flexoelectric response in BT-8BZT would make it a potential candidate for sensing and detecting at high temperatures.

In conclusion, the gradual decline of flexoelectric coefficient $μ12$ with increasing temperature in BT-8BZT was reported in this letter, confirming the advantage of a high dielectric permittivity of BT-8BZT in the paraelectric phase. The measured flexoelectric coefficient $μ12$ of BT-8BZT is 25 μC/m at 25 °C and can remain above 12 μC/m at temperatures up to 200 °C, which suggests potential applications for high temperature micro/nano-sensing. Other compositions in the BiMeO₃-BaTiO₃ system exhibit temperature stable permittivity over a much wider range.²⁴ These findings suggest that these materials may be more effective as flexoelectric-based sensors over a broader temperature range.

This work was supported by the U.S. Army Research Laboratory and the U.S. Army Research Office under Contract/Grant No. W911NF-11-1-0516. This work was performed in part at the NCSU Nanofabrication Facility (NNF), a member of the North Carolina Research Triangle Nanotechnology Network (RTNN), which was supported by the National Science Foundation (Grant No. ECCS-1542015) as part of the National Nanotechnology Coordinated Infrastructure (NNCI).