Non-linear Properties of Piezoelectric Ceramics

Invited Paper Non-linear Properties of Piezoelectric Ceramics B.K. Mukherjee *, W. Ren, S-F. Liu, A.J. Masys and G. Yang Department of Physics, Royal Military College of Canada, Kingston, Ontario, Canada, K7K 7B4 ABSTRACT Sensors and actuators based on piezoelectric ceramics are finding an increasingly large variety of applications under a very wide range of environmental conditions and applied signals. Some actuator applications require the piezoelectric materials to support large mechanical loads and produce high strain output. In order to accomplish this requirement of higher strains, large electric fields must be applied. This results in a significant non-linear behaviour and hence affects the performance of the material. It is therefore important to understand the behaviour and properties of these materials over a large range of temperature, frequency and applied electric fields and mechanical stresses. We have measured some of the dielectric, elastic and piezoelectric constants of soft (, EC-76) and hard (EC-64, ) lead zirconate titanate (PZT) piezoelectric ceramics, manufactured by EDO Ceramics, as a function of temperature, frequency, applied field and applied stress. We have also determined the dependence of the piezoelectric constants on an applied DC bias voltage or stress. The time dependence of the piezoelectric response in the piezoelectric ceramics has also been studied. A summary of the results will be presented. Most of these results can be understood on the basis of the extrinsic contributions to the piezoelectric response that arises from the existence of domains in the material. Keywords: Piezoelectric, non-linear, DC bias, AC field, stress, time dependence, temperature dependence, interferometer. 1. INTRODUCTION Sensors and actuators based on piezoelectric ceramics are finding an increasingly wide range of applications. These require the ceramic to be used under a very wide range of environmental conditions and applied signals. In many applications, the material may need to work at a frequency that is different from the resonant condition. In order to achieve large actuation, high electric fields may be applied to the material and some transducer designs require the application of a DC bias on the piezoelectric material. The transducer may be operated at temperatures above or below room temperature and, in high power applications, the temperature of the material may be different from that of the surrounding medium. The design of a transducer or a particular application, such as shape control, may require an actuator to operate under stress. In other applications it may be important to know the time that is required for the full piezoelectric response to appear. The design of piezoelectric smart structures is limited by the accuracy to which the material properties of the sensor/actuator materials are determined. It is therefore useful to know the frequency, electric field, mechanical stress, temperature and time dependence of the properties of the transducer material in order to understand and predict the response of the transducer. Resonance measurements, outlined in the IEEE Standard on Piezoelectricity 1, are commonly used for the characterisation of piezoelectric materials for high frequency applications. The use of complex material constants 2, allows all the losses in the material to be characterised. This is particularly helpful for the proper calculation of the transducer transfer function of lossy piezoelectric materials. Impedance spectra of the radial, thickness, length-thickness, thickness shear, and length extensional modes can be analysed using PRAP software 3 and the 1 independent complex material constants (s E 11, s 12 E, s, s E 33, s 55 ε T 33, ε T 11,, d 31, and d 15 ), which make up the reduced matrix for a C piezoelectric material, can be determined 4. Normally, the resonance measurements are performed at room temperature without DC bias and the analysis is only based on the fundamental resonance. By analysing the fundamental and higher order resonances we have obtained the frequency E 13 E, * Correspondence: E-mail: mukherjee@rmc.ca; Telephone: 613 541 6 x 6348; Fax: 613 541 64. Smart Structures and Materials 1: Active Materials: Behavior and Mechanics, Christopher S. Lynch, Editor, Proceedings of SPIE Vol. 4333 (1) 1 SPIE 277-786X/1/$15. 41

dependence of the material properties 4. The temperature behaviour has been investigated by putting the sample holder into a temperature chamber and controlling the temperature during the test as described in Sherrit et al 5. In addition, a limited amount of DC bias field can be applied to the sample during the resonance characterisation 6 and the field dependence of the piezoelectric properties over the range of bias fields applied can be found using resonance analysis. In order to study the non-linearities in the static and low frequency properties of piezoelectric materials, two experimental systems have been developed to investigate the strain and electric displacement responses of the piezoelectric ceramics under applied electric field and under uniaxial compressive stress, respectively. The stress experiment has been described earlier 7,8 and it allowed us to find the piezoelectric coefficient as a function of stress in the poled direction. Our initial investigations showed that the experimental results were dependent on the material type, history of the specimen, stress level, and the time scale of the measurement 7,8. The experimental system has since been improved; not only the dielectric displacement but also the strain caused by a low frequency dynamic stress excitation can be measured at different prestress values and therefore the piezoelectric coefficient and elastic compliance can both be obtained as a function of applied stress. Together with the dielectric permittivity, ε 33, measured at different prestress values, we are now able to determine the elastic compliance, s E 33, the piezoelectric constant,, as well as the coupling coefficient, k 33, all as a function of the prestress. In another experiment, a ZYGO laser Doppler interferometer system with a heterodyne detection technique has been modified to make direct measurements of the strain of piezoelectric ceramics caused by the application of low frequency applied electric fields 9. These measurements have allowed us to determine the piezoelectric d coefficients of the piezoelectric ceramics as a function of electric field and frequency at room temperature. The measurement can also be made with a small AC electric field superimposed on a DC bias field and this has enabled us to determine the DC bias dependence of the low frequency piezoelectric coefficients. This paper is organised as follows: Section 2 gives a general description of the piezoelectric effect as described by the linear constitutive equations as well as a discussion on domain wall dynamics; Section 3 reports on an interferometric measurement of the non-linearity associated with an applied AC field; Section 4 describes our results on the DC bias dependence obtained by using resonance and interferometric techniques; Section 5 discusses the non-linearity associated with an applied stress; Section 6 reports on an investigation of the time dependence; Section 7 summarises results on the temperature dependence of the piezoelectric constants; and, finally, Section 8 gives a summary of all the results within the context of domain dynamics. 2. THEORY When a small stress and a weak electric field are applied to a piezoelectric ceramic specimen along its poling direction, the resulting electric displacement and strain can be described by the constitutive relations: T D3 = ε33e3 + d33t3, (1) S = d E + s T 3 33 T E where E, D, T, and S, are the electric field, dielectric displacement, stress, and strain, respectively. The constants ε 33, s 33, and are the dielectric permittivity, elastic compliance, and the piezoelectric constant, respectively, with the superscript signifying the variable that is held constant, and the subscript 3 indicating the poling direction. There are two experimental conditions that can be imposed to simplify the measurement. If the stress is set to zero (sample free to expand unhindered), the equations in (1) are no longer coupled and the strain and the electric displacement can be determined as a function of the electric field. Similarly, if the field is set to zero (short circuit condition), the strain and the electric displacement can be determined as a function of the stress. The piezoelectric constant,, is the same in both cases 1. When large signals (stress or electric field) are applied, typical ferroelectric materials show a significant hysteresis due to irreversible domain switching in the materials 7. These irreversible changes take a finite time 11 and therefore the hysteresis curve obtained can also depend on the rate at which the applied signal is ramped up or down. The coefficients are no longer constants but are functions of the amplitude and the frequency of the applied signal. 3 E 33 3 T ε 33, E s 33,and During characterisation experiments, a field or a stress is applied at various levels and the resultant strain and dielectric displacement are monitored as the field or stress increases and decreases. For a typical ferroelectric material a measurable hysteresis is noticed in the resultant strain and dielectric displacement which are due to irreversible domain switching in the 42 Proc. SPIE Vol. 4333

D, S E BIAS E, T Under a large low frequency field domain structure changes irreversibly Under a small high frequency field domain walls vibrate reversibly Figure 1: A qualitative look at the effect of domains on the strain and dielectric displacement versus field. Proc. SPIE Vol. 4333 43

material. A qualitative look at the effect of domains on the strain and dielectric displacement versus field curves is shown in Fig. 1 12. When the applied signal is small, only reversible effects occur so that the piezoelectric response is small and is reversible. When a large field (electric or stress) is applied, the domain structure will change to maintain a minimum in the domain energy. Contributions from the electrostatic and domain wall energies counteract each other and the underlying domain structure (size, shape, and density) changes. In the process, some of the domains engulf other domains or change shape irreversibly which contributes to the net strain and polarisation. As a result there is a measurable hysteresis in the strain and electric displacement of piezoelectric materials when the applied field or stress is large. When large signals are applied, the piezoelectric coefficient,, can be defined in different ways depending on the desired application 7. An average may be defined as the average slope of the hysteresis curve up to a particular, maximum value of applied signal. The slope of the hysteresis curve would define a differential at a particular value of the applied signal. Both the average and differential values of include reversible and irreversible contributions to the piezoelectric response. In some applications, a substantial DC bias signal is applied and a relatively small AC signal is superimposed on the bias. The piezoelectric response to the AC signal is then mainly determined by reversible effects and it can be used to define a dynamic. In this paper we present our experimental results on the AC field, DC bias field, stress, temperature and time dependences of the piezoelectric response of both hard and soft lead zirconate titanate (PZT) ceramics. 3. AC FIELD DEPENDENCE The use of laser Doppler interferometry to measure piezoelectric strains has been reported earlier 13. Our measurement system uses a Zygo ZMI heterodyne laser Doppler interferometer with a resolution of.62 nm. The experimental arrangement of the system is schematically drawn in Fig. 2. The system is controlled by a computer using NI-VME interfaces. A sine-wave voltage generated by a HP3321A function generator is amplified by a Trek 2/2A amplifier and then applied to the ceramic samples. The polarised laser beam from a He-Ne laser with wavelength of 633 nm has two frequency components f 1 and f 2 which are orthogonally linearly polarised with a frequency difference of 2 MHz. The beam is split into two beams by a beam-splitter (BS). One beam goes through the Interferometer 1 and only the component f 1 incidents on the sample top surface to measure longitudinal displacement. The second beam goes through the Interferometer 2 and the beam component f 1 incidents on the side surface of the sample to measure transverse displacement. A frequency shift f 1 in the reflection beam from a moving surface at velocity ν is given by the equation: f 1 = 2ν / λ, Laser f 1, f 2 BS Mirror Ref.optic fiber Meas.optic fiber 1 Meas.optic fiber 2 f 2 - f 1 f 2 (f 1 ± f 1 ) f 2 (f 1 ± f 1 ) Phase Detector Computer Interferometer 1 Interferometer 2 Power Supply Sample Stage Mirror 2 Figure 2: Schematic of the ZYGO interferometer. Figure 2: Schematic of the Zygo interferometer. 44 Proc. SPIE Vol. 4333

where λ is the laser wavelength. The reflection beam (f 1 ± f 1 ) from the sample is recombined with the component f 2 in the interferometer. The output beam [f 2 -(f 1 ± f 1 )] is sent to a phase detector with a reference beam (f 2 - f 1 ) through optical fibres. The decoded frequency shift f 1 is converted to a voltage that is proportional to the velocity of the sample surface. The displacement and strain of the sample can then be calculated. The system is very stable and allows measurements to made fairly easily. The Zygo interferometer system was used to determine the longitudinal, transverse and shear strains induced in piezoelectric materials as a function of applied electric fields under stress free conditions. The various piezoelectric d coefficients of the ceramic were then calculated from the measured variation of strain as a function of the applied field. Figure 3 shows the variation of the, d 31 and d 15 for the two types of PZT as a function of the electric field. It can be seen that, in general, the d coefficients increase with electric field with the effect being more marked in the case of the soft PZT. In our results for the hard PZT,, the small increase in d 15 is perhaps due to some de-pinning of domain walls (de-ageing) that increases the extrinsic contribution, but in general even the high fields do not cause much de-pinning. Soft PZT is characterised by a high piezoelectric constant and mobile domain boundaries 14, and consequently extrinsic contributions are inherently more important in soft PZT 15. The observed increase in the d coefficients in soft PZT is likely due to the larger extrinsic contribution resulting from increased domain switching under the influence of larger fields.,-d 31,d 15 5 PZT @ Hz d 15 -d 31 1-2 1-1 1 Field (MV/m) d33, -d31, d15, 18 16 1 1 8 6 PZT @ Hz d 15 -d 31 1-2 1-1 1 Field (MV/m) Figure 3: Variation of the piezoelectric constants as a function of applied field Figure 4 shows the variation of the, d 31 and d 15 for the two types of PZT as a function of the frequency. The d coefficients show a very small decrease as the frequency increases from.1 Hz to Hz. As discussed in Sherrit et al 11, the non-18 domain changes take a finite time to occur. Only the domain wall motion that can keep pace with the changing field strength will contribute to the measured displacement at any frequency. Therefore as the frequency is increased, less domains have time to re-align and the piezoelectric response is smaller. Over the small frequency range of our measurements, the decrease in the d coefficients is quite small. 4. MEASUREMENTS UNDER DC BIAS 4.1 Resonance Measurements The effects of DC bias fields on the material properties have been studied by carrying out radial mode resonance measurements on disk specimens while a DC bias field was applied to the specimen. The impedance resonance measurements were made using an HP 4194A impedance analyser and, with the protection circuit used, the DC bias voltages had to be limited to the range between +3 kv and - 3 kv, in order to protect the analyser. The disk specimens used for the radial resonance measurements conformed with the geometric aspect ratios recommended by the IEEE standard 1. The fundamental resonance mode impedance spectra were analysed using PRAP software 3 to obtain the following material Proc. SPIE Vol. 4333 45

5 PZT PZT 9,-d 31,d 15 d 15 -d 31 1-2 1-1 1 1 1 1 2 1 3 Frequency (Hz),-d 31,d 15 8 7 6 5 d 15 -d 31 1-2 1-1 1 1 1 1 2 1 3 Frequency (Hz) Figure 4: Variation of the piezoelectric constants as a function of frequency E E T constants: s 11, s 12, ε 33, d 31, and k p. The results on three types of PZT samples manufactured by EDO Ceramics, namely (hard), (soft), and EC-76 (very soft), are plotted in Fig. 5. In all cases, the piezoelectric d 31 constant shows a slight decrease when a positive DC bias is applied on the specimen. When a negative DC bias is applied, d 31 increases at first and then begins to drop when the negative bias field is sufficiently high to cause depoling of the material. Within the range of the DC bias fields that we were able to apply, depoling of the hard PZT is not evident. For the relatively soft ceramic, we found significant depoling at a bias field of -.8 MV/m. As for the soft PZT, EC-76, due to the very soft nature of the material, depoling occurs at very low values of bias. As a result only positive DC bias results are shown for this material. 4.2 DC Biased Low Frequency Measurement The ZYGO laser interferometer system described in section 3 has been used to determine the dynamic piezoelectric d coefficients of PZT ceramics as a function of bias electric field at different frequencies. Figure 6 shows the DC field dependence of for the hard PZT,, and the soft PZT,, at various frequencies. The AC field amplitude and frequencies used for the measurements are shown in the figure. When a positive DC bias (along the poling direction) is applied to the well poled hard PZT,, a very small decrease in is observed. However, when a negative DC bias is applied to this hard ceramic, there was a marked increase in the as the applied negative bias was increased up to 2 MV/m, without any depoling being observed. For soft PZT, the also showed a slight increase as a function of a negative bias field until the sample depoled at around 1 MV/m. A further increase in the negative bias causes a repoling of the specimen as shown in Fig. 6. Our results suggest that for a well-poled ceramic, an additional positive bias field does not appreciably increase the total polarisation but it probably contributes to the electrical clamping of domain walls that causes a small reduction of the piezoelectric response. On the other hand, a negative DC bias, when it does not cause depoling, can produce field induced de-ageing 17 (unclamping of domain walls) that results in an increase in the piezoelectric coefficient, the dielectric constant and the elastic compliance. Measurements were also made of the AC field dependence for various DC bias fields and Fig. 7 shows the results for the hard PZT,. Of particular note is the non-linear increase in with an increasing AC field. As the DC bias is increased the increases up to a point where depoling causes a significant decrease in. Figure 8 shows similar results for the soft PZT,. The increase in as a function of the AC field is even more non-linear in this case. These results correlate with the AC field dependence in the absence of a DC bias that has been discussed above. 46 Proc. SPIE Vol. 4333

18-3 17 16-4 s E 11 (1-12 m 2 /N) 15 14 13 12 11 EC-76-2. -1.5-1. -.5..5 1. 1.5 2. DC Bias Field (MV/m) s E 12 (1-12 m 2 /N) -5-6 -7 EC-76-8 -2. -1.5-1. -.5..5 1. 1.5 2. DC Bias Field (MV/m) (a) (b) 25 EC-76 3 25 EC-76 d 31 (pc/n) 15 5-2. -1.5-1. -.5..5 1. 1.5 2. DC Bias Field (MV/m) ε T 33 (nf/m) 2 15 1-2. -1.5-1. -.5..5 1. 1.5 2. DC Bias Field (MV/m) (c) (d).7.65.6.55 k p.5.45.4 EC-76-2. -1.5-1. -.5..5 1. 1.5 2. DC Bias Field (MV/m) (e) Figure 5 (a-e): DC Bias field dependence of the material constants obtained with the radial mode resonance measurement. Proc. SPIE Vol. 4333 47

38 36 34 32 28 26 PZT EC 69 @.26 MV/m AC Hz 1 Hz 1 Hz.1 Hz 6 5 PZT @.2 MV/m AC Hz 1 Hz 1 Hz 24 22-2 -1 1 2-3 -2-1 1 2 3 DC Bias (MV/m) DC Bias (MV/m) Figure 6: DC bias field dependence of for and PZTs at various frequencies. 5 45 35 25 15 5 PZT AC Dependence @ Hz MV/m -.5 MV/m -1. MV/m -1.5 MV/m -2.1 MV/m.1 1 AC Field (MV/m) 1 1 1 9 8 7 6 5 PZT AC Dependence @ Hz MV/m -.13 MV/m -.26 MV/m -.79 MV/m -1.3 MV/m -3. MV/m.1 1 AC Field (MV/m) Figure 7: AC field dependence of for PZT under DC bias fields. Figure 8: AC field dependence of for PZT under DC bias fields. It is clear from our results that the field dependent non-linearities play an important role in applications where large driving fields are applied to piezoelectric materials. Soft PZT materials provide the higher piezoelectric constants but are susceptible to depoling; in such materials, the application of a positive bias can allow the application of larger AC driving fields without causing depoling. The positive bias also helps to stabilise the polarisation alignment and prevents depoling due to stress or temperature variations. In hard PZT materials, depoling is not a big issue but their piezoelectric coefficients are smaller than those of softer materials; in this case, the piezoelectric response can be increased somewhat by the application of a negative bias leading to the possibility of "tuning" the piezoelectric response within limits. 48 Proc. SPIE Vol. 4333

5. STRESS DEPENDENCE The experimental arrangement for the stress measurements been described elsewhere 11,16. It has since been improved to facilitate the measurement of the stress dependence of strain and dielectric permittivity, ε 33, together with the electric displacement. The strain is measured by using a strain gauge attached to the specimen and the dielectric permittivity is measured at a frequency of 1 khz by using a Hewlett Packard precision LCR meter model 4284A. Thus we are now able to not only determine the coefficient but also the elastic compliance s E 33 and the permittivity ε 33 as a function of the applied stress; the coupling coefficient k 33 can then be calculated and its variation as a function of the stress can be found. Our measurements of the average and the differential of piezoelectric materials have been reported earlier 7. Here, we report on our investigation of the dynamic of one soft and one hard PZT ceramic as a function of an applied bias stress. We also discuss the importance of time dependent effects in the measurement. Details of the stress dependence of ε 33 and s E 33 are particularly helpful in our obtaining a better understanding of our results on the stress dependence of the piezoelectric constant. Figure 9 shows the experimental results for the set of parameters as a function of the prestress up to 16 MPa in steps of 1 MPa for (hard) and (soft) PZT ceramics. The dynamic values are measured with an AC stress of amplitude 1.5 MPa at a frequency of.2 Hz. The polarisation, and the dynamic, are measured about 5 minutes after changing the value of the stress. This wait is important as the domain changes that result from the application of stress take time to occur 11. A longer wait can produce ageing effects. These time-dependent effects will be discussed in the next section. Both materials show a non-linear behaviour with an initial increase in as the stress is increased followed by a significant decrease. After a stress cycle, the stress-free of the soft ceramic dropped significantly suggesting that significant depoling had taken place during the stress cycle resulting in the observed decrease of under high stress. On the other hand, a net increase of was found in the hard ceramic after a stress cycle. This indicates the possibility of domain switching and the generation of non-18 domain walls and de-ageing as a result of the applied stress 7,8. The observed decrease in under high stress is perhaps mainly due to the effects of stress clamping on domain mobility. The stress dependence of ε 33 and s E 33 of the two materials were also very different. A significant drop of s E 33 has been found in after a stress cycle, which was another proof of the depoling that occurred during the cycle. On the other hand, EC- 69 shows a strong increase in its ε 33 but much less change in its s E 33. The increase of ε 33 is normally an indication of the deageing in the material 17. It is now well known that the piezoelectric properties in PZT ceramics are caused by both intrinsic and extrinsic contributions 15,18. The piezoelectric properties of a single domain crystal are defined as the intrinsic properties under the appropriate conditions, whereas the piezoelectric responses that originate from sources other than the intrinsic contribution are lumped as the extrinsic properties. It is believed that the majority of the extrinsic contributions are related to the domain wall motion in the ceramic 15. As the 18 domain walls are purely ferroelectric walls and the non-18 domain walls are both ferroelectric and ferroelastic, only the non-18 domain wall motion contributes to piezoelectric properties. Investigation 15,18 has also shown that the extrinsic contributions are more significant in soft PZT than in hard PZT. For a fresh sample at zero stress, the of a hard PZT is mainly due to intrinsic contributions while the of a soft PZT includes significant extrinsic contributions 15. If a ferroelectric material is exposed to a mechanical stress or an electric field, the domain structure (size, shape and density) will change to maintain a minimum in the domain energy. In the process some of the domains engulf other domains or change shape irreversibly. Under a uniaxial stress, the domain structure in PZT ceramics may undergo domain switching 19,2,21,22, domain wall mobility clamping 2, deageing 1,17,2, and depoling 1,17, 2. As a compressive stress along the poling direction is applied to a sample, some degree of polarisation will move away from the poling direction. As a consequence, there will be changes in the domain structures. If the poling level of the sample is not changed significantly, this change in the domain structure may generate more non-18 domain walls, which will increase the piezoelectric and dielectric responses of the ceramic. Domain wall mobility clamping under a compressive stress is a reversible process in which the domain wall motion is mechanically clamped by a stress. The clamping of domain wall motion reduces the extrinsic contribution and therefore reduces both dielectric and piezoelectric responses. After poling, PZT ceramics go through an ageing process in which the domain walls are pinned by impurities and structural discreteness 23. Stress applied on ageing samples has a deageing effect in which the stress causes redistribution of impurities and changes of structure. As a result, a domain wall motion which was pinned during ageing can be activated. Thus deageing of the samples increases both dielectric and piezoelectric responses. Mechanical depoling involves all stress induced processes which lead to a total or partial loss of the preferable orientation of the polarisation in the material which was attained during the poling of the ceramic. The contribution of each mechanism in PZT ceramics depends on the type of dopants, the stress level, and the time scale of the experiment used. Proc. SPIE Vol. 4333 49

s E 33 (normalized) 1.2 1..8.6.4.2 dielectric constant (normalized) 2. 1.8 1.6 1.4 1.2 1.. 2 4 6 8 12 14 16 2 4 6 8 1 12 14 16 stress (MPa) stress (MPa) 1.4 (normalized) 1.2 1..8.6.4.2. 2 4 6 8 12 14 16 stress (MPa) k 33 (normallized) 1.4 1.2 1..8.6.4.2-2 2 4 6 8 12 14 16 18 stress (MPa) Figure 9: Stress dependence of the electromechanical properties of two PZT ceramics. When a compressive stress is applied to a hard PZT sample, such as, along the polarisation direction, domain switching processes may produce new non-18 domain walls, which increase the extrinsic contribution. The deageing of the sample due to the stress may also have a contribution. Both effects lead to an initial increase in as seen in Fig. 9. The increase of the extrinsic contribution also causes the dielectric constant and the loss to increase, as has been observed earlier 17,2. As the applied stress increases, the domain wall motion becomes partially and progressively clamped by stress. Consequently the shows a decrease with increasing stress. Upon the release of the stress, the recovers due to a reduction of the clamping effect. A net increase of was found in hard PZT specimens after a stress cycle. This indicates the possibility of the generation of new non-18 domain walls and deageing as a result of the applied stress. Up to the maximum stress applied to the hard PZT specimens, the observed decrease in under high stress is perhaps mainly due to the effects of stress clamping on domain wall mobility and not due to the depoling. 5 Proc. SPIE Vol. 4333

In the case of soft PZT ceramics, such as, the initial increase of seen in Fig. 9 can be interpreted in the same way as for hard PZT, i.e. it is caused by domain switching and deageing. This increase of was not significant compared with that of the hard PZT. The decrease of under increasing stress is accounted for by the depoling process. The depoling of soft PZT starts at a relatively low stress level and it is a dominant effect at high stress levels. The slight recovery of during the stress release is probably due a reduction of the clamping effect and deageing. But because of the severe depoling of soft PZT samples under a high stress level, the is dramatically reduced after a stress cycle. 6. TIME DEPENDENCE OF THE PIEZOELECTRIC RESPONSE In earlier work carried out in our laboratory, Sherrit et al 11 have measured the time dependence of the polarisation caused by the application of a uniaxial step stress to PZT (Navy I and III) specimens along the poling direction. The polarisation was found to include two parts: one part reflected the response of 18 domain changes and occurred very quickly, while the other part reflected the non-18 domain changes and had a logarithmic time dependence. By measuring this time dependence as a function of temperature, Sherrit et al were able to show the effects of thermal activation and they could then determine the average activation energies that governed the domain changes in the different types of PZT ceramic investigated. It should be noted that the time dependence of the extrinsic response found by Sherrit et al explains the frequency dependence of the piezoelectric coefficients noted towards the end of section 3. 6.1 Time Dependence of the Stress-induced Enhancement of In our investigations of the effects of stress on the piezoelectric response that have been described in the previous section, the measurements of the polarisation were made dynamically after waiting for a period of 5 minutes after the application of the appropriate stress so that the full effects of the stress appeared before the measurements were made. We have observed that a longer wait causes ageing effects to occur and these effects have also been investigated. 8 The dynamic of the hard PZT,, has been measured as a function of time after a stress of 9 MPa was applied and the results are shown in Fig. 1. The stress of 9 MPa corresponds approximately to the maximum in the stress-induced enhancement of the piezoelectric response in this material as may be seen in Fig. 9. Over the period of the observations, the time dependence of can be approximately described by the relation: where d 33 is the stress free value of, Fig. 1 using (2) gives k = 1.52, and d t () t k d d ln( t ) =, (2) 33 33 33 t is a time coefficient, and k is a proportionality factor. The fitting of the data in t = 8.45pC/N (for, the stress-free value of is d 33 = 22 pc/n). The decrease of with time can be attributed to the phenomenon of stress induced ageing and its logarithmic dependence parallels the decrease in the extrinsic contribution to the polarisation of the material determined from measurements of the short circuit current generated in the specimen after the application of a step stress. 11 6.2 Time Dependence of the Effects of DC Bias In section 4 we have described our determination of the dynamic piezoelectric d coefficients as a function of an applied DC bias field. We have also investigated the influence of time on these measurements by determining the value of as a function of time elapsed after the application of the DC bias field. For the hard PZT,, no time dependence has been found, for either positive or negative bias, for DC bias levels up to 2 MV/m, as shown in Fig. 11. In the case of the soft PZT,, Fig. 12 shows that a clear time dependence is observed for the DC bias field strengths of ±.79 MV/m and ±.92 MV/m and it will be clear from Fig. 6 that these values correspond respectively to the depoling and repoling of the ceramic. Some time dependence can also be observed for DC bias fields that are close to the values mentioned. Not surprisingly, in the light of our earlier results, the domain changes that cause poling and depoling require a finite time to occur. Proc. SPIE Vol. 4333 51

1.3 @ 9MPa (reduced units) 1.2 1.1 1. Times (s) Figure 1: of PZT as a function of time following the application of an 9 MPa stress. d33 38 36 34 32 28 26 24 22 18 16 PZT Time Dependence @.26 MV/m AC MV/m 2. MV/m -2. MV/m 5 6 Time (s) 55 5 45 35 25 15 5 PZT Time Dependence @ Hz 5 6 Time (s) MV/m.4 MV/m.52 MV/m.66 MV/m.79 MV/m.92 MV/m 1. MV/m 1.3 MV/m 3. MV/m Figure 11: Time dependence of for under various applied DC bias fields. Figure 12: Time dependence of for under various applied DC bias fields. TEMPERATURE DEPENDENCE OF THE PIEZOELECTRIC RESPONSE Because of the wide variety of environmental conditions under which piezoelectric transducers are being used, it is useful to understand the behaviour of the piezoelectric response over a wide range of temperatures. In work reported earlier 5, a PZT- 4D type lead zirconate titanate piezoelectric ceramic manufactured by Morgan Matroc Ltd, was characterised over the temperature range from C to C, which was a suitable range for undersea applications. Some applications, such as those in space, involve a much wider variation in temperature. We have now modified our resonance impedance measurement system so as to be able to mount the specimen inside a Thermotron Temperature Controlled Chamber which allows the temperature to be controlled in the range from -72 C to 17 C. The full resonance characterisations of a hard PZT,, and a soft PZT, EC-76, manufactured by EDO Corporation, have been carried out over this temperature range. 52 Proc. SPIE Vol. 4333

Figure 13 shows the variations of the different piezoelectric constants of the two materials as a function of temperature. It can be seen from the figure that the piezoelectric coefficients increase as the temperature is increased for both materials. The increase is steeper in the case of the softer ceramic, EC-76, and this is due to the fact that extrinsic contributions have a greater role in the piezoelectric response of softer materials. Thermal hysteresis has been observed in the piezoelectric response. Full details of the hysteresis and results showing the variations of all the dielectric, elastic and piezoelectric coefficients as a function of the temperature will be published elsewhere 24. di j (pc/n) 45 EDO ij=15 (TS) 35 25 ij=33 (LE) 15 ij=31 (RAD) - -5 5 15 Temperature ( C) di j (pc/n) 1 9 8 7 6 5 ij=15(ts) ij=33(le) ij=31(rad) EDO EC-76 - -5 5 Temperature ( C) Figure 13: Temperature dependence of the piezoelectric constants for hard PZT and soft PZT EC-76. 8. SUMMARY The non-linear behaviour of soft and hard piezoelectric ceramics has been investigated. The variation of the piezoelectric response as a function applied AC field amplitude, applied DC bias field strength, stress and temperature have been experimentally determined for typical hard and soft PZT ceramics. The time dependence of the piezoelectric response has been investigated and it is clear that this dependence plays a critical role in determining how the piezoelectric response can be measured. Most of the non-linear properties are related to the extrinsic contribution to the piezoelectric response and our understanding of the non-linear properties is closely tied to our understanding of domain behaviour in these materials. 9. ACKNOWLEDGEMENTS The authors thankfully acknowledge funding support from the US Office of Naval Research. 1. REFERENCES 1. IEEE Standard on Piezoelectrics, ANSI/IEEE Std. 176-1987. 2. S. Sherrit and B.K. Mukherjee, "The Use of Complex Material Constants to Model the Dynamic Response of Piezoelectric Materials", Proceedings of the 1998 IEEE Ultrasonics Symposium, pp. 63-64, Sendai, Japan, IEEE, Piscataway, NJ, USA, 1998. 3. Piezoelectric Resonance Analysis Program (Version 2.11), available from TASI Technical Software (www. tasitechnical. com). 4. S. Sherrit, H.D. Wiederick and B.K. Mukherjee, "A complete characterization of the piezoelectric, dielectric and elastic properties of Motorola PZT323HD ceramic including losses and dispersion", Ultrasonic Transducer Engineering: SPIE Proceedings Volume 337, pp. 158-169, SPIE, Bellingham, WA 98227, USA, 1997. Proc. SPIE Vol. 4333 53

5. S. Sherrit, G. Yang, H.D. Wiederick, and B.K. Mukherjee, Temperature Dependence of the Dielectric, Elastic and Piezoelectric Material Constants of Lead Zirconate Titanate Ceramics, Proceedings of the International Conference on Smart Materials, Structures and Systems, pp. 121-126, Allied Publishers, Mumbai 38, India, 1999. 6. G. Yang, W. Ren, S-F. Liu, A.J. Masys and B.K. Mukherjee, to be published in the Proceedings of the IEEE Ultrasonics Symposium, San Juan, Puerto Rico, October (in printing). 7. G. Yang, S.-F. Liu, W. Ren, and B.K. Mukherjee, Uniaxial stress dependence of the piezoelectric properties of lead zirconate titanate ceramics, Active Materials: Behavior and Mechanics, SPIE Proceedings Volume 3992, pp.13-113, SPIE, Bellingham, WA 98227, USA,. 8 G. Yang, S.-F. Liu, W. Ren, and B.K. Mukherjee, Uniaxial stress dependence of the piezoelectric properties of lead zirconate titanate ceramics, to be published in the Proceedings of the 12th IEEE International Symposium on the Applications of Ferroelectrics, (ISAF'), Honolulu, HI, USA, July-August (in printing). 9. A.J. Masys, W. Ren, G. Yang, and B.K. Mukherjee, The variation of piezoelectric and electrostrictive strain as a function of frequency and applied electric field using an interferometric technique, Proceedings of the 3 rd CanSmart Workshop on Smart Materials and Structures, pp.21-3, St. Hubert, Quebec,. 1. Q.M. Zhang, J. Zhao, K. Uchino, and J. Zheng, Change of the weak-field properties of Pb(ZrTi)O 3 piezoceramics with compressive uniaxial stress and its links to the effect of dopants on the stability of the polarizations in the materials, J. Mater. Res., 12, pp. 225-234, 1997. 11. S. Sherrit, D.B. Van Nice, J.T. Graham, B.K. Mukherjee, and H.D. Wiederick, Domain wall motion in piezoelectric materials under high stress, in the Proceedings of the 8th International Symposium on the Applications of Ferroelectrics, (ISAF 92), pp. 167-17, IEEE, Piscataway, N. J., USA, 1992. 12. S. Sherrit, H. D. Wiederick, B.K. Mukherjee, and M. Sayer, Field dependence of the complex piezoelectric, dielectric, and elastic constants of Motorola PZT 323 HD ceramic, Smart Materials Technologies, SPIE Proceedings Volume 34, pp. 99-19, SPIE, Bellingham, WA 98227, USA (1997). 13. K.M. Rittenmyer, and P.S. Dubbelday, "Direct Measurements of the temperature-dependent piezoelectric coefficients of composite materials by laser doppler vibrometry", J. Acoust. Soc. Am., 91(4), pp. 2254-226, 1992. 14. Q.M. Zhang, W.Y. Pan, L.E. Cross, Laser interferometer for the study of piezoelectric and electrostrictive strains, J. Appl. Phys., 63, pp. 2492-2496, 1988. 15. Q.M. Zhang, H. Wang, N. Kim, and L.E. Cross, "Direct evaluation of domain-wall and intrinsic contributions to the dielectric and piezoelectric response and their temperature dependence on lead zirconate titanate ceramics", J. Appl. Phys., 75, pp. 454-459, 1994. 16. S. Sherrit, R. B. Stimpson, H. D. Wiederick and B. K. Mukherjee, Stress and temperature dependence of the direct piezoelectric charge coefficient in lead zirconate titanate ceramics, Smart Materials, Structures and MEMS, SPIE Proceedings Volume 3321, pp. 74-81, SPIE, Bellingham, WA 98227, USA, 1996. 17. Q.M. Zhang and J. Zhao, Electromechanical properties of lead zirconate titanate piezoceramics under the Influence of mechanical stresses, IEEE Trans. on Ultrasonics, Ferroelectrics and Frequency Control, 46, pp. 1518-1526, 1999. 18. Q. M. Zhang, W. Y. Pan, S. J. Jang, and L. E. Cross, Domain wall excitations and their contributions to the weak-signal response of doped lead zirconate titanate ceramics, J. Appl. Phys., 64, pp. 6445-6451, 1988. 19. R. Y. Nishi, Effects of one-dimensional pressure on the properties of several transducer ceramics", J. Acoust. Soc. Am., 4, pp. 486-495, 1966. 2. H. H. A. Krueger, Stress sensitivity of piezoelectric ceramics: part 1: sensitivity to compressive stress parallel to the polar axis, J. Acoust. Soc. Am., 42, pp. 636-645, 1967. 21. H. H. A. Krueger, Stress sensitivity of piezoelectric ceramics: part 2: heat treatment, J. Acoust. Soc. Am., 43, pp. 576-582, 1968. 22. H. H. A. Krueger, Stress sensitivity of piezoelectric ceramics: part 3: sensitivity to compressive stress perpendicular to the polar axis, J. Acoust. Soc. Am., 43, pp. 583-591, 1968. 23. B. Jaffe, W. R. Cook, Jr., and H. Jaffe, Piezoelectric Ceramics, Academic Press, London, 1971. 24. G. Yang, S.-F. Liu, W. Ren, and B.K. Mukherjee, Temperature dependence of the piezoelectric, dielectric, and elastic properties of lead zirconate titanate ceramics ceramics, to be published. 54 Proc. SPIE Vol. 4333