Piezoelectric Detection and Response Characteristics of Barium Titanate

This paper studies the detection and response characteristics of barium titanate (BT) unimorph cantilevers under alternating current (AC) electric fields both numerically and experimentally. The unimorph cantilever consists of the BT layer and Cu shim, and the BT layer has sensing, grounding and driving electrodes. The deflection and output voltage of the cantilevers were predicted by three dimensional finite element analysis (FEA). The deflection and output voltage were also measured, and a comparison was made between prediction and experiment. The deflection, stress, output voltage and output power were then discussed. Piezoelectric Detection and Response Characteristics of Barium Titanate Unimorph Cantilevers Under AC Electric Fields Publication History: Received: March 11, 2014 Accepted: April 18, 2015 Published: April 20, 2015


Introduction
Piezoelectric ceramics, polymers and composites are widely used in sensors and actuators in the field of smart materials and structures [1][2][3].With an increasing concern about global warming, the direction of research has changed slightly from just sensors and actuators to an interest in extracting and storing energy from the environment [4,5].Some investigations have also focused on the design of self-powering, self-sensing and self-controlling devices.Recently, Wang and Inman [6] examined the concept and design of a multifunctional composite sandwich structure for simultaneous energy harvesting and vibration control.They proposed the glass fiber reinforced composite materials with lead zirconate titanate (PZT) harvester, PZT sensor and PZT actuator, and studied the configurations, locations and operating modes of PZT materials for optimal power generation.
Lead oxide-based ferroelectrics, especially PZT, exhibit high piezoelectric properties.Due to lead toxicity, however, the introduction of legislation in Europe to limit the usage of lead in automotive and electronic products has led to a worldwide search for lead-free compounds [7].BT (BaTiO 3 ) is one of the most widely investigated lead-free ceramics.Recently, BT-based piezoelectric ceramics have been prepared, and compared with commercial hard PZT materials [8].Although many researchers are developing the multifunctional composites using PZT materials, few of the BT-based multifunctional composites have been investigated.
In this paper, we discuss the detection and response characteristics of BT unimorph cantilevers with sensing, grounding and driving electrodes under AC electric fields.A three-dimensional FEA was performed to predict the deflection and output voltage in the BT unimorph cantilevers.The deflection and output voltage were also measured, and numerical results were compared with measured values.The deflection, stress, output voltage and output power were then examined in detail.

Basic equations
Equations of motion and Gauss' law are given by (1) (2) where σ ij is the stress tensor, u i is the displacement vector, ρ is the mass density, D i is the electric displacement vector, and a comma denotes partial differentiation with respect to the coordinates x i (i =1,2,3)or the time t.We have employed Cartesian tensor notation and the summation convention for repeated tensor indices.Constitutive relations can be written as

International Journal of Metallurgical & Materials Engineering
where ε ij is the strain tensor, E i is the electric field intensity vector, and s ijkl , d kij and ε ik are the elastic compliance, direct piezoelectric coefficient and permittivity at constant stress, which satisfy the following symmetry relations: (5) The relation between the strain tensor and the displacement vector is given by (6) and the electric field intensity vector is (7) where ϕ is the electric potential.The constitutive relations (3) and ( 4) for piezoelectric ceramics poled in the x 3 -direction are given in Appendix A.

Finite element method
A unimorph cantilever constructed of BT layer and Cu shim is shown in Figure 1a.Let the coordinate axes x = x 1 and y = x 2 be chosen such that they coincide with the interface and the z = x 3 axis is perpendicular to this plane.The origin of the coordinate system is BT layer has sensing, grounding and driving electrodes.The sensing and driving electrodes are located at the center and outer parts, respectively.To suppress coupling, there is a grounding electrode between these two electrodes [9].On the other hand, only a whole grounding electrode is located on the opposite side.
We performed three-dimensional finite element calculations.The commercial software ANSYS with 3D eight-node elements was used in the analysis.For simplicity, the electrode layers were not incorporated into the model.This is because the thickness of the electrode layer is much smaller than the thickness of the BT layer.The electric potential on the top driving electrode surface equals the applied voltage ϕ = V 0 exp(iωt), where V 0 is the amplitude of applied voltage and ω is angular frequency.The top and bottom grounding electrode surfaces are connected to the ground, so that ϕ=0.The deflection and internal stress for the cantilever were then calculated.The output voltage of sensing electrode was also predicted.

Experimental Procedure
The unimorph cantilever was fabricated using BT layer (NEC/Tokin Co. Ltd., Japan) and Cu substrate.Table 1 lists the material properties of BT layer.Elastic compliances s 11 , s 33 , direct piezoelectric coefficients d 31, d 33 , dielectric permittivity є 33 and mass density ρ can be found in the published data, while the remaining properties are assumed to be the same as those of BT reported by Jaffe and Berlincourt [10].The elastic compliance (inverse of Young's modulus), Poisson's ratio and mass density of Cu shim are 7.69×10 -12 m 2 /N, 0.34 and 8920 kg/m 3 , respectively.Electrodes were coated on both sides of BT layer, and the BT layer was bonded to the upper surface of Cu shim by conductive bonding, as shown in Figure 1.A resistive load was then connected to the cantilever.
An AC voltage was applied to the driving electrode surface of the BT layer, while the grounding electrode surfaces were earthed.The amplitude of tip deflection w tip of the cantilever was measured with a laser displacement sensor (LK-G10, KEYENCE, Co. Ltd., Japan).Output peak voltage V out of the sensing electrode was also measured for the cantilever using an oscilloscope (GDS-1062A, Good Will Instrument Co., Ltd., Japan).Then, the output power P out from the cantilever can be calculated using the output peak voltage V out and load resistance R.  2, and the coercive electric field is about 2 MV/m.Note that the tip deflection for PZT C-203 unimorph cantilever was predicted considering domain wall motion [11].A nonlinear relationship between the tip deflection and electric field is observed for PZT C-203 cantilever.On the other hand, the tip   T T deflection for BT unimorph cantilever increases linearly as the electric field increases, and the results of the FEA are in excellent agreement with measured data.It is therefore expected that the domain wall motion may not occur in the BT layer under AC electric fields.The formation of defect complexes during manufacturing of the BT layer are thought to reduce the mobility of ferroelectric domain wall.Figure 3 shows the predicted and measured output peak voltage V out of the sensing electrode as a function of E 0 for BT unimorph cantilever without load resistance at f = 50 Hz.Also shown is the predicted data for PZT C-203 unimorph cantilever.The output peak voltage for BT unimorph cantilever increases linearly as the electric field increases, similar to the tip deflection.It can be seen that agreement between FEA and test is fair.The results show that the output voltage is capable of self-powering or self-sensing.

Results and Discussion
The variation of normal stress amplitude |σ xx | along the length direction is calculated for BT unimorph cantilever without load resistance at a chosen point (y = 0 mm and z = 1 mm here) and the result is shown in Figure 4.All calculations were done for E 0 = 0.1 MV/m at f = 50 Hz.The output peak voltage of the cantilever under E 0 = 0.1 MV/m at f = 50 Hz is about V out = 2.0 V.A high normal stress occurs near the center of the BT unimorph cantilever.Figure 5 shows the variation of normal stress amplitude |σ xx | along the width direction for BT unimorph cantilever without load resistance at x = 25 mm and z = 1 mm.All calculation was done for E 0 = 0.1 MV/m at f = 50 Hz.There is a substantial difference in the stress near the edge of driving electrode.It is noted that the normal stress for the sensing part is much higher than that for the driving part.The results for the evaluation of stresses may help multielectroded BT cantilever designers to estimate the fracture risk and to optimize in-service loading conditions.The variation of the measured output peak voltage V out of the sensing electrode with changing load resistance R is shown in Figure 6 for BT unimorph cantilever under E 0 = 0.1 MV/m at f = 50 Hz.The output voltage increases with load resistance.Figure 7 shows the output power P out for BT unimorph cantilever under E 0 = 0.1 MV/m at f = 50 Hz.The output power tends to increase with load resistance reaching a peak and then to decrease in magnitude.It can be seen that the optimal load resistance for BT unimorph cantilever is about 2 MΩ.

Conclusion
This paper presents the results of numerical and experimental study in the BT unimorph cantilevers with sensing, grounding and driving electrodes under AC electric fields.On the basis of the study conducted, the following conclusions seem to be justified.
1.The tip deflection of the cantilever and the output peak voltage of the sensing electrode increase linearly without domain wall motion as the AC electric field amplitude increases.
2. The stress in the sensing part of BT unimorph cantilever is significantly higher than that in the driving part for the same AC electric fields.
3. The output power in the sensing electrode of BT unimorph cantilever under 0.1 MV/m can achieve about 0.4 μW at 50 Hz.
By knowing how the sensing and driving electrodes interact, we can design self-powering, self-sensing and self-controlling devices to achieve particular overall properties.Our present study offers a method for aiding the design of new BT-based multifunctional composites, and provides a basis for refining the real device design in order to reduce internal stresses and increase output voltage and output power.
center of the bottom left side of BT layer, and the edge at x = 0 is clamped.The geometry and dimension are shown in Figure1b.BT layer of length 50 mm, width 25 mm and thickness 1 mm is added to the upper surface of Cu shim of length 50 mm, width 25 mm and thickness 0.5 mm.The total thickness is 1.5 mm.

Figure 2
Figure 2 shows the amplitude of the tip deflection amplitude |w tip | as a function of AC electric field amplitude E 0 for BT unimorph cantilever without load resistance at frequency f = ω/2 = 50Hz.The solid line represents the value of the tip deflection predicted by the FEA, and the solid circle denotes the measured data.For comparison, the predicted deflection for PZT C-203 unimorph cantilever (Fuji Ceramics Co. Ltd., Japan) is also shown as dashed line.The material properties of C-203 are listed in Table2, and the coercive electric field is about 2 MV/m.Note that the tip deflection for PZT C-203 unimorph cantilever was predicted considering domain wall motion[11].A nonlinear relationship between the tip deflection and electric field is observed for PZT C-203 cantilever.On the other hand, the tip

Citation:Page 2 of 4 Figure 1 :
Figure 1: Schematic drawing of (a) BT unimorph cantilever and (b) geometry and dimensions.

Figure 3 :
Figure 3: Output peak voltage versus electric field amplitude.

Figure 4 :
Figure 4: Distribution of normal stress amplitude along the length direction.

Figure 5 :
Figure 5: Distribution of normal stress amplitude along the width direction.

Table 1 :
Material properties of BT.