Fabrication and measurement of a flexoelectric micro-pyramid composite

Flexoelectric effect, described as the mechanical strain gradient induced electric polarization and electric field gradient induced mechanical strain,¹ has attracted increasing research interests in the past decade. This effect was triggered by Ma and Cross with their pioneering experimental demonstration of large flexoelectric coefficients in high permittivity ferroelectrics.^2–4 The initial experimental results were well matched with the theoretical prediction, revealing the linear dependence between the flexoelectric coefficients and the dielectric permittivity.^5,6 It is noticed that the size effect inherited from the gradient term enables flexoelectric phenomenon to be much more enhanced in micro/nano scale compared to macro one,^7,8 as interpreted in the constitutional equation of the flexoelectricity

where P_l is the induced polarization, μ_ijkl is the flexoelectric coefficient, a fourth-rank tensor S_ij is the strain and x_k or x_l is the axis; T_ij is the mechanical stress, f_ijkl is the converse flexoelectric coefficient associated with strain and E_k is electric field.

The recent micro/nano scale flexoelectricity study are mainly concentrated on the thin film.⁹ The strain gradient yielded by the lattice misfit between the functional layer and the substrate layer could result in a giant electric polarization, and change the domain configuration of the ferroelectric film accordingly.^10,11 On the other hand, large strain gradient can also be generated in the application of the atomic force microscope (AFM) tip through the mechanical writing force. Typically, up to μN level force can be applied onto the film by the AFM tip in a small circular area of 10 nm in radius. Such a high stress concentrated region could result in a huge out-of plane strain gradient, and thus alter the domain orientation in the film, suggesting a new avenue for non-volatile memory technology.¹² However, the giant flexoelectricity in non-piezoelectric nano structures are yet explicitly developed, due to the fabrication and characterization challenges of ferroelectric nanostructures.

In sub-millimeter level, Fu et al. reported a gradient scaling phenomenon in microsize flexoelectric composites.¹³ They used mechanical dicing method to fabricate truncated pyramid structures with the feature size of 50 μm and 100 μm, respectively, on BST substrates. It is well known that the direct piezoelectric measurement was commonly used for accessing the flexoelectric response of pyramid structure.¹⁴ It can be implemented directly by d₃₃ meter which measures the mechanical force induced electric current of the sample. However, such piezoelectric measurement is difficult to be exerted onto the micro truncated pyramid because of the critical sample clamp condition is always challenged when the sample size scales down. Fortunately, the converse flexoelectric measurement provides a good alternative approach through applying an AC electric voltage across the sample and monitoring the displacement signal that is phase locked at the same driving frequency. Their converse measurement results suggested an approximately double-times relationship between the effective piezoelectric coefficients of two structures, being in good agreement with the scaling effect of flexoelectricity.¹³ 

In order to realize the feasibility of flexoelectric composite as a promising piezoelectric structure, it is necessary to further scale down the structure size of the pyramid units to obtain enhanced effective piezoelectric properties. In the aspect of converse effect, by applying the same voltage, the nanometer size pyramid structure could generate larger displacement compared to the micrometer level counterpart. Nevertheless, the maximum displacement that the nano scale pyramid could achieve is quite small owing to the brittle nature of the ceramic materials, i.e. 0.7 % maximum allowable tensile strain for BST.¹⁵ In this case, in the real application of tens nanometers range displacement, renders that the single layer flexoelectric pyramid structure should have the minimum height of about 10 μm. Pyramid composites at this size level could both exhibit high piezoelectric performance and satisfy the requirement of real application.

For fabricating pyramid of such scale, the conventional dicing saw would not offer a good avenue due to the blade size limit. Conventionally, two measures can be considered for 10 μm range structure fabrication, including top-down and bottom-up methods. Top-down method is based on lithography, electroplating and dry etching processes. This fabrication process was successfully developed by Jiang et al. for fabricating high frequency piezoelectric composites as micro-ultrasound transducers.^16,17 A thick layer of nickel was electroplated through the photoresist pattern, forming the hard mask for the further dry etching step, which involved the deep reactive ion etching with chlorine based gases. However, this method is labor intensive and expensive due to the low etching rate and multiple processing steps. On the other hand, the bottom-up method utilizes sol-gel technology to obtain the thick film.¹⁸ The patterning procedure could rely on either wet etching or dry etching. However, the problem of this method consists of the inferior properties of thick films compared with bulk material.¹⁹ To overcome the drawbacks of these fabrication methods, we developed a new fabrication method combining the conventional precision mechanical dicing and wet etching, in order to generate the 10 μm range pyramid structures based on bulk BST ceramic material.

The fabrication process started from a Ba_0.67Sr_0.33TiO₃ plate with the dimension size of 5 mm × 5 mm × 600 μm. The Curie temperature of the raw material is 21 °C.^2,4 All measurements in this study were conducted at room temperature of 23 °C to ensure the paraelectric phase of the materials. The top and bottom surfaces were both lapped and polished to obtain a good surface finishing. A mechanical dicing saw (Disco, DAD320, Santa Clara, CA) was employed to cut the top surface into line arrays. The line post width and kerf width are both 30 μm, corresponding to a pitch of 60 μm. The line post depth is 40 μm. Then the sample went through a wet etching process with a recipe developed following the work reported on BST thin film etching using buffered oxide etchant (BOE) with strong acids, e.g. HNO₃, HCl, H₂ SO₄, as the catalysts.^20–22 A solution with the composition of BOE (10:1): HCl = 80%: 20% was prepared for wet etching of BST in our work. However, a residue layer was observed on the surface after etching for a short period of time. This residue layer, acted as a protection layer and hindered the further etching, has yet been reported before. The crystalline structure of residue material was examined by the X-ray diffraction (XRD) (Rigaku SmartLab). As shown in Fig. 1, cubic structure of the BST substrate accompanied with the pattern of Ba_4.67Cl_1.33F₈ were observed on the residual material. This etching residue is not soluble in water. To remove this residue layer, a two-step recipe was developed by dipping the sample in a pure HCl bath for 10 seconds after every 5 minutes etching in BOE based etchant. It was found that the residue material can be dissolved instantly in HCl solution, similar to the PZT wet etching process.²³ According to this recipe, the etching rate was measured to be 100 nm/min and a clean surface finish can be achieved. Two samples were prepared with one etched for 20 minutes and the other for 30 minutes. A third sample was used as a reference sample without dicing or etching treatment. The cross sections of two samples were captured by scanning electron microscope (SEM) (JEOL 2000FX) as shown in Fig. 2. Wet etching is known to be isotropic for ceramic materials. The convex shape corner can be etched faster due to the large exposed area to the etchant, while the concave corner exhibits a slower etching rate because of the small exposed area. Also, the etchant in the groove area has less agitation compared to the top surfaces. All these factors enable wet etching to generate pyramid shapes out of the diced straight line post structures. Note that the observed cracks, mainly caused by the handling impact, only existed near the edge of the samples. The main parts of the structure were intact with smooth pyramid shape. The dimension of the two pyramid structures were listed in Table I.

In order to generate an electric field gradient along the pyramid structure, the sidewalls of the pyramid should be protected from the electrode coverage. Epoxy (Epo-Tek 301, Epoxy Technology, Billerica, MA) was casted onto the pyramids to fill the kerfs. The residual epoxy on the top surface was removed by lapping. Top and bottom surfaces of two samples were coated with Ti/Au (10 nm/100 nm) as electrodes through electron beam evaporation.

In principle, the effective piezoelectric constant of the pyramid layer can be calculated as

where a₁ and a₂ are the top and bottom width, t is the height and c₁₁ is the elastic constant of BST with the value of 1.66 × 10¹¹ N/m². The schematic view of the 1D pyramid composite is displayed in Fig. 3 with the critical feature sizes. The μ₁₁ of Ba_0.67Sr_0.33TiO₃ has been measured by Cross et al. to be 120 μC/m.⁴ Based on the dimension and the material properties, the effective d₃₃ of two samples were calculated to be 31 pm/V and 87 pm/V, respectively.

It is worth mentioning that owing to the existence of the substrate layer, only a portion of the electric voltage would fall upon the pyramid layer. To attain the exact amount of voltage percentages on the pyramid, COMSOL Multiphysics was used to simulate the electric field distribution in the composite structure. As shown in Fig. 4, the relative dielectric constants of BST and the polymer are set as 12000 and 5, respectively. 1 V voltage was applied across the whole sample (geometry adopted here corresponds to BST I), and electric field gradient can be apparently observed in Fig. 4(a). The percentage of the voltage falling on the pyramid is about 15%. For the BST II, the part falling on the pyramid is 13% percent.

The measurement setup was laid down on a floating optical table (Newport, ATS, Irvine, CA) to eliminate the vibrational noise. The AC voltage was generated by a power amplifier (Trek, 2220, Lockport, NY) under the excitation from a function generator (Tectronix, AFG3101, Lake Mary, FL). To measure the small flexoelectric displacement, the bottom surface of the sample was clamped on the table. The axial deformation was measured using a high resolution (<10 pm) laser vibrometer (Polytec, OFV-5000, Irvine, CA) and a lock-in amplifier (Stanford Research System, SR830, Sunnyvale, CA). The applied voltage was ranged from 20 V to 60 V, in order to avoid the saturation of the relative dielectric permittivity of BST at high electric field.¹⁴ The excitation frequency was swept from 1 Hz to 10 Hz. Higher frequency was not studied due to the concern of affecting the clamping condition of the sample by vibration. In principle, when the BST is above the Curie temperature, the material in this experiment should not exhibit any piezoelectric effect. Nevertheless, when the temperature is very close to the Curie temperature, there may still exist a weak persistence of macroferroeletric regions due to local nano domains.²⁴ Moreover, the electrostrictive effect is coexisted and would inevitably contributes to the total displacement. In this case, the application of the electric field would results in the displacement sum of three superposition.

where f₁₁₁₁ is the axial converse flexoelectric coefficient and should equals to μ₁₁/c₁₁, E₁ is the electric field, ω is the frequency, and M₁₁₁₁ is the electrostrictive coefficient. Hereafter, the piezoelectric, flexoelectric, and electrostriction contribution is labeled as P, F, and M, respectively.²⁵ 

As interpreted in Eq. (3), the electrostrictive displacement is a 2ω-signal with a DC bias. This can be ruled out by choosing the reference frequency of the lock-in amplifier to be ω, leaving alone the flexoelectric and piezoelectric displacements. Based on this setting, the measured displacements from the BST I, BST II and the reference sample are plotted as a function of the applied voltage, as shown in Fig. 5(a). The displacement response at low frequencies exhibit a good frequency-independence and the data present in this figure were obtained at 10 Hz. The reference frequency of the lock-in amplifier was the same as the driving frequency from the function generator. A linear trend was observed in the displacement data of all three samples. Linear fittings of the data exhibit negligible intercepts, meaning the trends all go through the origin. To extract the flexoelectric part from the mixed signal, the displacement of the reference sample was subtracted from the total displacement of the BST I and BST II. This calculation is based upon an assumption of identical dielectric and electrical properties in all three samples. The same mother sample source and preparation process ensures the homogeneity of the samples. The remaining displacement signal should be the pure flexoelectric response from the pyramid structures. The effective d₃₃ values can be calculated with the help of the apportioned electric potential in the pyramids discussed above, as shown in Fig. 5(b). The calculated effective d₃₃ values of the BST I and BST II are 39.8 pm/V and 85.8 pm/V, respectively.

Intriguingly, the measured effective piezoelectric coefficients of BST I are higher than the previous estimation, while the measured values are close to the expectation for BST II. There could be several reasons for the discrepancy in the measured and estimated values of BST I. First, a straight sidewall was considered for estimating the electric field gradient and the flexoelectric response in Eq. (2). However, the pyramid sidewalls created by the wet etching process are in curved and smooth shapes. This could assist to engender a higher electric field gradient inside, and thus to improve the effective piezoelectric output. Second, the etching profile may not be uniform everywhere, and compared to the captured cross section geometry in Fig. 2, larger aspect ratio of the pyramid could exist at other places. This could also enlarge the overall response as measured. Taking all these factors into account, the theoretical calculation provides a moderate estimation of the electromechanical response of flexoelectric pyramid composite. On the other side, in order to further improve the effective piezoelectric properties, 2D pyramids could be employed due to the steeper area variation along depth direction over 1D pyramid structures. Instead of generating line arrays, 2D post arrays could be fabricated by adding a dicing trace, which will become the 2D pyramids during the subsequent wet etching process.⁴ For actuating application, the apportion effect of the substrate is possible to be diminished by decreasing the substrate thickness. In addition, multilayer configuration of the composite structure can be employed to take better advantage of the applied voltage.²⁶ 

In summary, a fabrication method by combining mechanical dicing and wet etching was developed to make micro pyramid structure based on bulk BST ceramic materials. We investigated the effective piezoelectric properties of flexoelectric pyramid structures in ten-micrometer-scale. A scaling effect of the flexoelectric response was demonstrated as the size shrinks down. The results do suggest the great potential of flexoelectric micro pyramid as a good alternative to lead-free piezoelectric material with properties comparable to the widely used PZT compositions. Future work could focus on the 2D pyramid arrays and multilayer composite structures, which could further promote the electromechanical output from flexoelectricity.