Properties of piezoceramic materials in high electric field actuator applications

We obtained the piezoceramic samples from five different manufacturers: American Piezo, Johnson Matthey, Ekulit GmbH, PI Ceramics and TRS Technologies. The materials are listed in table 1 along with the values of the material parameters from the datasheet. Ekulit-PZT and AP-PZT856 have no material datasheet available. The material name as given in table 1 will be used throughout this paper to denote the materials. The piezoceramic from Ekulit GmbH comes in the form of a piezo buzzer where we separated the piezo layer from its metal base by dissolving the glue in Dichloromethane. All other materials are supplied in monolayer sheets with electrode layers on both sides. We added a PMN-PT material from TRS Technologies to the study to compare its performance to PZT ceramics. According to the mechanical quality factor (Q_m) which we measured for the materials whose quality factor was not provided by the manufacturer, all materials except JM-M1334 were soft PZT materials since their quality factor was below 150 and JM-M1334 has a quality factor of 220.

Table 1.  The materials used in the study, their manufacturers and the material parameters as given in the datasheet.

Material	Manufacturer	Density	Tc	r	Ec	d31	Thickness
		(Kg m–3)	(°C)		(kV mm–1)	(pm V–1)	($\mu {\rm{m}}$)
AP-PZT850	American piezo	7600	360	1900	—	175	100
AP-PZT856	American piezo	—	—	—	—	—	300
EKULIT-PZT	Ekulit GmbH	—	—	—	—	—	120
JM-M1100	Johnson Matthey	8100	177	4750	0.57	315	120
JM-M1334	Johnson Matthey	7900	200	3500	0.615	230	150
TRS-207HD	TRS technologies	7950	370	—	1.6	202	150
TRS-610HD	TRS technologies	7950	210	—	0.78	380	150
TRS-PMNPT	TRS technologies	—	—	—	0.45	1200 (d33)	205
PI-PIC252	PI ceramics	7800	350	1650	—	180	110

We chose a configuration with a unimorph beam that uses an active layer of piezoelectric material and 100 μm thick borosilicate glass (D263T-eco Thin Glass from Schott AG) as a passive layer. We evaporated a thin metal layer of 10 nm chromium and 300 nm aluminum on to the glass layer to provide a high-quality optically reflective surface to reliably measure the surface with an optical profilometer (figure 1 (a)) as described in more detail in sections 2.3 and 2.4. The glass and PZT were structured by UV laser and glued using epoxy resin (HTG-240 from Resoltech) . The resin was cured for 8 h at 40 °C and then at 80 °C for 2 h, which resulted in a glass transition temperature of the glue layer (T_g) of about 150 °C. Even though it is possible to achieve higher T_g by increasing the curing temperature of the glue, we chose a low curing temperature as the materials in this study have the Curie temperature in the range of 140 °C–400 °C. We glued the finished unimorph beams to a silicon mount as shown in figure 1 (b) using an alignment structure; both also structured using the UV laser. The beams had a free effective length of 12–15 mm and width of 2–3 mm depending on the size of the sample available.

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One of the main material parameter affecting our COMSOL simulation is the relative permittivity of the material, which we measured by structuring the piezoceramic samples to a defined electrode surface area using a UV laser and measuring the capacitance across the electrodes. This gives us the ε_r at constant mechanical stress which is shown in table 2.

To evaluate the value for ${\varepsilon }_{r}$ at different electric fields, we applied a sinusoidal signal with different amplitudes in the direction of polarization to a sample single layer of the piezo material of JM-M1334 of known area. The frequency of the signal was kept at 100 mHz in order to measure the voltage and current in a well-defined static regime of the material. We then obtained the high-field relative permittivity at each voltage cycle from the integrated V–I relation. The results are discussed in section 2.4.

The Young's modulus of a material can be measured using static measurement techniques like tensile, bending or torsion tests, but also doing dynamic measurement methods by measuring the resonance frequency of the material. As the forces or strains in static tests may be non-trivial to measure, we adopted the latter by measuring the resonance frequency of the unimorph beams. We excited the beams with low electric fields of about 0.05 kV mm⁻¹ over a frequency sweep and recorded the amplitude of the deflection using a high-speed triangulation sensor (Keyence LK-H022K) to obtain the resonance frequency. To obtain the Young's modulus from the resonance frequency, we further needed the density of the material. Even though the datasheet of the materials provides us with a value for the density, we also measured it using the traditional water displacement method.

We then modeled the unimorph beams in COMSOL and carried out an eigenfrequency simulation with a parameter sweep over the Young's modulus by scaling the compliance matrix of the material. By taking the inverse of the (1, 1)th value of the compliance matrix corresponding to the measured resonance frequency, we got the Young's modulus of the material, which is given in table 2. The uncertainty in the values of the Young's modulus originates from the variations in the fabrication process of the beams that affect the effective length of the beam, the width, and the thickness and also the uncertainty of the density value. Since PZT is a highly nonlinear material, we furthermore measured the Young's modulus for different amplitudes of the displacement by increasing the electric field. To best quantify the result, we use the volume average strain of the PZT at the amplitude of the resonance in the simulations to represent the strain at which the Young's modulus was measured. In figure 2 we see that the measured Young's modulus decreases with increasing strain.

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The piezoelectric charge coefficient or piezo modulus (d_ij) is defined as the ratio of the charge displacement in a certain direction (i) to an applied force (j), or the strain (j) to the applied electric field (i). The three independent charge coefficients, d₃₃, d₃₁, and d₁₅ then describe the longitudinal, transverse and shear deformation of polarized ceramics respectively. The measurement of d₃₃ and d₁₅ is not covered in this study as most bending actuator applications utilize the transverse strain, i.e. the charge coefficient, d₃₁.

To evaluate this charge coefficients at different electric fields, we excited the beams with a low-frequency sinusoidal signal of 1 Hz and different amplitudes in the direction of polarization. We then measured the bending profile of the beams dynamically using an optical profilometer equipped with a chromatic confocal distance sensor with a vertical and lateral resolution of 75 nm and 4 μm, respectively. We used a 100 μm raster and averaged the voltage-dependent displacement over five voltage cycles to minimize the error and extracted the bending profile at the maximum of the corresponding voltage cycle and fitted a second-order polynomial function

$$\begin{eqnarray}&&f(x)=a+{bx}+{c}_{{meas}}{x}^{2}\end{eqnarray} \tag{ 3 }$$

on segments of 2 mm along the length of the beam to obtain the local curvature coefficient. To avoid edge effects, we did not consider the outer 200 μm of the beam.

To determine the charge coefficient, we modeled the beam in a COMSOL multiphysics FEM simulation. As we could only measure one component of the compliance matrix and the permittivity matrix, we scaled the matrices of the built-in PZT-5H material with our measured values. The coupling matrix was kept at the default value with a charge coefficient of −274 pm V⁻¹ (d₃₁^sim). We simulated the bending profile (figures 1(c) and (d)) for electric fields as in the physical measurement and fitted the simulated curvature with a second-order polynomial function, extracting the simulated curvature coefficient (c_sim) in the same method as in the measurement. Since the ratio of measured curvature (c_meas) to the simulated curvature (c_sim) at the same electric field E, is equal to the ratio of charge coefficient d₃₁ to the charge coefficient used in the simulation (d₃₁^sim), d₃₁ becomes:

$$\begin{eqnarray}&&{d}_{31}=\displaystyle \frac{{c}_{{meas}}}{{c}_{{sim}}}{d}_{31}^{{sim}}\cdot \end{eqnarray} \tag{ 4 }$$

This relation holds up as the charge coefficient and the resulting curvature have a linear relationship.

The obtained charge coefficient for different applied electric fields can be seen in figure 3. The maximum deflection of the beams that we observed in the measurements is ∼400 μm for Ekulit-PZT, which corresponds to the strain of 1.8 × 10⁻⁴. The corresponding variation in the Young's modulus is ∼2 GPa (figure 2) which causes a change in our measurement of the transverse charge coefficient of approximately 1 pm V⁻¹. Hence, we neglect this change in the measurements as this is smaller than the other error sources of that measurement. The piezo-material AP-PZT856 from American piezo has a thickness of 300 μm, and we could reliably measure its deflection only at high electric fields. We limited the maximum electric field for each material to the point where an increase in field strength resulted in electrostatic breakdown of the beam or through the air at the side of the beam. Ekulit-PZT ceramics shows the highest d₃₁ of −487 pm V⁻¹ when actuated with an electric field of 1 kV mm⁻¹ followed by JM-M1334 and JM-M1100. We will discuss the performance more in detail in section 4, where we also take into account the coercive field strengths.

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It turns out that the linear scaling of equation (4) applies in simulations for large scaling factors only if we simultaneously scale the relative permittivity proportional to ${d}_{31}^{2}$. I.e. if we want to use the d₃₁ measured above to simulate the piezoelectric material, we need to scale the relative permittivity as ${\varepsilon }_{{rnew}}={\varepsilon }_{r}{\left(\tfrac{{d}_{31}}{{d}_{31}^{{\rm{PZT}}5H}}\right)}^{2}$.

To verify that this scaling also applies in nature, we show in figure 4 the relative permittivity depending on the corresponding charge coefficient of JM-M1334, in addition to the best square law fit, ${\varepsilon }_{r}=a\ {d}_{31}^{2}$ and a quadratic scaling of the standard values in COMSOL. We see that the measured values are, within their uncertainties, in good agreement with a square law, and that this fit also agrees within 3σ of the scaled built-in values, validating the approach to scale the permittivity quadratically when scaling the d₃₁ in a simulation. The measured permittivity showed similar scaling with a mean deviation of 8% from the data found in the literature which is measured within a lower electric field of 0.15–0.45 kV mm⁻¹ [23].

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Above the Curie temperature, ferroelectric ceramics lose their spontaneous polarization. Near the Curie temperature, the dielectric constant ε_r diverges, and at $T\gtrsim {T}_{c}$, it is described by the Curie–Weiss law, in terms of the Curie constant K_Curie, temperature T and Curie temperature T_c:

$$\begin{eqnarray}&&{\epsilon }_{r}=\displaystyle \frac{{K}_{{\rm{Curie}}}}{T-{T}_{c}}+1\cdot \end{eqnarray} \tag{ 5 }$$

From an engineering point of view, however, it is more interesting to measure the Curie temperature when approached from below. So we again measured the capacitance across the electrodes of the piezoceramic while ramping the temperature to 300 °C. For continuity in the complex plane, we expect the divergence also to be of 1st order when approached from below, so we fit a generic first order divergence

$$\begin{eqnarray}&&C=a+b\ T+\displaystyle \frac{c}{{T}_{c}-T}\end{eqnarray} \tag{ 6 }$$

as shown in figure 5(a). The fit was done over the region with largest variation in capacitance. The choice of that region causes a deviation of 1.0 °C. We also fitted the usual Curie–Weiss law

$$\begin{eqnarray}&&C=a+\displaystyle \frac{c}{T-{T}_{c}}\end{eqnarray} \tag{ 7 }$$

at T > T_c, for materials where we had sufficient data. The variation in the fitting region causes a deviation of 6.3 °C when fitted above T_c, and we found good agreement between both methods with an average (rms) deviation of just 8.7 °C. The resulting values for the Curie temperature are given in table 2.

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When the temperature increases, the material becomes softer and at the same time, the Weiss domains may become more flexible, affecting also the charge coefficient d₃₁. We evaluated this change in Young's modulus depending on the operating temperature by measuring the resonance frequency as mentioned in section 2.3 at different temperatures. As an example, we kept an Ekulit-PZT beam in a mount milled out of aluminum with an attached resistive heater. A temperature controller (UR484802 from Wachendorff ) was connected to the resistive heater. A temperature sensor (Pt100) was placed near the beam as the feedback to the controller and the mount was closed using a glass cover. The mount was designed to be filled with oil to achieve a uniform temperature, but this will affect the resonance frequency and also the measurement using the optical profilometer as the refractive index of the oil changes with temperature. Hence, even though the resistive heater was able to achieve up to 90 °C, the temperature values were stable only up to 60 °C.

Similarly, we measured the deflection of the tip of the cantilever in the quasistatic limit at 1 Hz to measure the temperature scaling of the charge coefficient, taking into account the change in the Young's modulus. For both measurements, we assumed the Young's modulus of the glass to be approximately constant as the PZT is assumed to have stronger variations due to the proximity to the Curie temperature. In figure 6 we see an almost linear dependence of the Young's modulus of the material on the temperature with a coefficient α_Y ∼ −0.0014 K⁻¹. The charge coefficient changes linearly with ${\alpha }_{{d}_{31}}\sim 0.002\,{{\rm{K}}}^{-1}$, agreeing well with the values measured by Wang et al (∼0.002 K⁻¹) in the same temperature range [24].

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The electric current flowing through piezoelectric ceramics when applying an AC signal is a result of aligning Weiss domains inside the ceramic thereby polarizing the material in the direction of the applied electric field; the coercive field is the electric field required to bring the average polarization in the piezoceramic back to zero. To measure the coercive field, we applied a sinusoidal voltage signal with a frequency of 100 mHz and an amplitude that is high enough so that the domain switching can be reliably observed in both in-and-against the direction of original polarization. We kept the sample in the same mount described in section 3.2 this time filled with oil to reduce the risk of electrostatic breakdown due to high electric fields and to optimize the uniformity of the temperature that we varied in steps of 10 °C.

The coercive field is the null point on the polarization curve, which in turn is given by the integrated current. As the pre-polarized piezo films, however, have asymmetric polarization curves, we instead use the voltage at which the piezoceramic material draws the maximum current. The peak current indicates that the domain switching is taking place and this domain switching current is very large compared to the leakage current at the maximum electric field in the ferroelectric materials [28]. Strictly speaking, this gives the field strength of the fastest re-poling but in practice, it will be very close to the coercive field strength. In figure 7 we see that all materials show an approximately linear decrease in the coercive field against the direction of the pre-polarization with increasing temperature.

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The main limitation on the operation of piezoceramics resulting from the coercive field, is the maximum electric field that we can apply against the polarization direction of the piezo materials without risking depolarization. A common assumption has been that the maximum electric field against the direction of polarization should be kept smaller than 33% of the coercive field strength to prevent depolarization [29]. To investigate this limit further, we excited the beam with a 1 Hz signal of different amplitudes and measured the deflection averaged over 5 cycles in two configurations: negative cycles, where the applied electric field is against the direction of polarization and symmetric cycles, where the piezo beams were excited with a symmetric electric field in-and-against the direction of polarization. All the measurements were performed on piezoelectric beams fabricated using PIC252 from PI Ceramics.

Figure 8 shows the mean strain in the piezo obtained from the displacement at the tip of the beam for different voltage cycles. For the negative cycles (figure 8(a)), we see that the piezo unimorph beams were able to operate down to 61% of the coercive field without any significant loss in polarization. When going above 61%, we observe a reduction in the slope of the hysteresis curve and finally the reversal of the polarization direction when the electric field increases above the coercive field. For the symmetric cycles in figure 8(b), we see that the operating region can be extended to approximately 95% of the coercive field against the direction of polarization. When applying field strengths above 95% against the direction of polarization, the material starts to show first depolarizing effects. Even though the symmetric driving at 95% of E_c introduces larger hysteresis in the material, this kind of actuation can be useful for systems with closed loop control or for systems with periodic operation as in the case of a micropump [5, 30].

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Given this observation, we introduced a quick re-poling method to improve the working range against the polarization direction, where a voltage pulse is given in the direction of polarization at the end of a negative voltage cycle. The duration of the pulse was kept about 20 times the resonance frequency of the beam with an amplitude of 150% of the coercive field. Figure 9(a) shows the quick re-poling cycle with the pulse at the end. The short duration of the pulse is critical in order to repolarize the domains without giving the material enough time to react mechanically to the pulse. In figure 9(b), we see that it was possible to increase the operating range against the direction of polarization to approximately 75% of the coercive field, above which the slope of the hysteresis curve decreases until the reversal of the polarization direction at 115% of the coercive field. A short resonance can still be observed as shown at one example in figure 9(c) due to the polarizing pulse, but we assume that this will be suppressed in a system with higher damping or by decreasing the pulse width, which was in our case limited by the function generator.

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To evaluate the long-term effects of these operation cycles, we actuated the piezoelectric beam in different configurations for 1 million cycles. First, we polarized the piezo-beam by applying 150% of the coercive field (E_c) at 1 Hz in the direction of polarization for five minutes. Afterwards, we subjected the piezoelectric beams to negative cycles of amplitude 33% of E_c at 10 Hz and measured the deflection at the tip of the beam every five minutes. After operating for one million cycles, we polarized the beam again by applying 150% of E_c and repeated the process for 50%–75% of E_c in negative cycles, then for 66%–95% of E_c applied symmetrically in-and-against the direction of polarization, and finally, 66%–85% of E_c with the quick re-poling method. To evaluate the repeatability of the process, we subjected another beam made from the same material (PI-PIC252) to 66% of E_c in symmetric cycles.

Figure 10(a) shows the amplitude of the deflection over the 1 million cycles observed every 3000 cycles. We see that, if we apply a maximum of 33% of E_c against the direction of polarization, as suggested by Ruschmeyer [29], the maximum deflection amplitude, and hence the charge coefficient, decreases by 6% after operating 1 million cycles. In comparison, if we apply 66% of E_c in negative cycles, we observe a decrease in the deflection of 26% with no indication of saturation. Symmetric cycles measured up to 95% E_c, however, show a decrease in the amplitude of less than 3%. Moreover, applying 95% of E_c was more stable over time than driving with 85% of E_c or less. With the quick re-poling pulse, the beams can be driven with 75% of E_c with a decrease of the amplitude similar to the negative cycles of 33% E_c.

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