Piezoelectrics coefficient

The development of active ceramic-polymer composites was undertaken for underwater hydrophones having hydrostatic piezoelectric coefficients larger than those of the commonly used lead zirconate titanate (PZT) ceramics (60—70). It has been demonstrated that certain composite hydrophone materials are two to three orders of magnitude more sensitive than PZT ceramics while satisfying such other requirements as pressure dependency of sensitivity. The idea of composite ferroelectrics has been extended to other appHcations such as ultrasonic transducers for acoustic imaging, thermistors having both negative and positive temperature coefficients of resistance, and active sound absorbers. 

CB04. The spontaneous polarisation was measured by the pulse pyroelectric technique (Ps = 46 nC/cm ). The piezoelectric coefficient evaluated for CB04 was dsi = 1.6 pC/N. The estimation of the efficiency of the second harmonic generation for compound CB04 gives the value three times more than for quartz. 

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Fig. 9.3. Definition of piezoelectric coefficients. A rectangular piece of piezoelectric material, with a voltage V applied across its thickness, causes a strain in the x as well as the z directions. A piezoelectric coefficient is defined as the ratio of a component of the strain with respect to a component of the electrical field intensity.

Because strain is a dimensionless quantity, the piezoelectric coefficients have dimensions of meters/volt in SI units. Their values are extremely small. In the literature, the unit 10 mA is commonly used. For applications in STM, a natural unit is A/V, or 10 m/V. Using primitive means as shown in Fig. 9.2, the Curie brothers (Curie and Curie, 1882) obtained a value of 0.021... 

A/V for the parallel piezoelectric coefficient for quartz, which matches accurately the results of modern measurements (Cady 1946). 

The left-hand side is the forward piezoelectric coefficient, in units of coulombs/newton. The right-hand side is the reverse piezoelectric coefficient, in units of meters/volt. They are equal. The coexistence of forward and reverse piezoelectric effects provides a simple method to test the piezodrive used in STM, which is discussed in Section 9.6. 

Fig. 9.4. Dependence of piezoelectric properties of PbZrOj-PbTiOj on composition. The zirconate-rich phase is rhombohedral, whereas the titanate-rich phase is tetrahedral. The piezoelectric coefficients reach a maximum near the morphotropic phase boundary, approximately 45% PbZrOj and 55% PbTiOj. (After Jaffe et al., 1954.)...

The crystallographic and piezoelectric properties of the ceramics depend dramatically on composition. As shown in Fig. 9.4, the zirconate-rich phase is rhombohedral, and the titanate-rich phase is tetragonal. Near the morphotrophic phase boundary, the piezoelectric coefficient reaches its maximum. Various commercial PZT ceramics are made from a solid solution with a zirconate-titanate ratio near this point, plus a few percent of various additives to fine tune the properties for different applications. 

In addition to the parameters discussed in the previous section, that is, the piezoelectric coefficients dsi, d3i, and the velocity of sound, c, there are several other parameters that are important for applications in STM. 

While designing or using STM at low or high temperatures, the variation of piezoelectric coefficients with temperature has to be considered seriously. The variation differs for different PZT materials. Fig. 9.5 shows measured variations of d-i for several commonly used PZT materials with temperature. 

A typical measuring circuit is shown in Fig. 9.15. A signal generator supplies a sinusoidal signal with 600 fl output impedance. The current is amplified at sensitivity of 10 AfV. The ac voltages are measured by ac digital voltmeters. The experiment is performed with a PZT-4 tube, provided by EBL, Inc., with L = 25.4 mm, D = 12.7 mm, h = 0.50 mm, and Y= 7.5 X 10 N/m. The lowest resonance frequency is 5 kHz. The results of measurements are shown in Fig. 9.16. The current, about 1 xA, can easily be measured with 1% accuracy. The current from the two x quadrants agrees well with that from the two y quadrants. In terms of the units mentioned, the piezoelectric coefficient dn can be obtained from directly measurable quantities as ... 

For small and slow signals, in STM, the piezoelectric coefficients are the only relevant parameters. At relatively high frequencies, the dynamic response of the piezoelectric materials becomes important. The lowest resonance frequencies of the piezodrive are the limiting factor for the scanning speed. 

Ferroelastoelectric Piezoelectric coefficients Electric field and mechanical stress NH4CI... 

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

Composite ferroics. Ingenious experiments have been performed with composites made from a ferroic and another material (Newnham Cross, 1981 Lynn et al, 1981 Rittenmeyer et al, 1982 Safari et al, 1982). For example, in a piezoelectric like PZT, the piezoelectric voltage coefficient g can be defined for a given direction (say Z = 33) thus, 33 = where d and k stand for piezoelectric coefficient... 

Finally, ferroelectricity is manifest in asymmetrical crystals producing domains of spontaneous polarization whose polar axis direction can be reversed in an electric field directed opposite the total dipole moment of the lattice. The two (or more) directions can coexist in a crystal as domain structures comprising millions of unit cells which contain the same electric orientation. The symmetry elements are temperature sensitive in ferroelectric materials [27]. At a particular temperature called the Curie Point the values of the piezoelectric coefficients reach particularly high values. Above the Curie Point the crystal transformation is to a less polar form and the ferroelectric nature disappears. 

There are several types of materials that exhibit the piezoelectric effect. Because it is inexpensive, and because it has a relatively strong piezoelectric coefficient, quartz is the material of choice for most piezoelectric sensor applications. It has a hexagonal crystallographic structure, with no center of symmetry. Both the magnitude of the piezoelectric coefficient and the extent of its temperature dependence are affected by the orientation of the cut of the crystal with respect to the main crystallographic axes. The most popular AT-cut is shown in Fig. 4.2. 

Piezoelectric coefficients need to be measured accurately over a wide range of temperature, drive field amplitude, and frequency, in order to predict device performance appropriately. There are multiple methods available for such characterization in bulk materials and thin films. This paper overviews some of the standard characterization tools, with an emphasis on the methods utilized in the ieee Standard on Piezoelectricity. In addition, several of the evolving methods for making accurate piezoelectric coefficient measurements on thin films are reviewed. Some of the common artifacts in piezoelectric measurements, as well as means of avoiding them, are discussed. 

The piezoelectric effect entails a linear coupling between electrical and mechanical energies. Numerous piezoelectric coefficients are in use, depending on the electrical and mechanical boundary conditions imposed on the part under test. Each of the piezoelectric d, e, g, and h coefficients can be defined in terms of a direct and a converse effect the two sets of coefficients are related by thermodynamics. For example, the piezoelectric charge coefficient, dkjk, can be defined via [1] ... 

The piezoelectric coefficients are third rank tensors, hence the piezoelectric response is anisotropic. A two subscript matrix notation is also widely used. The number of non-zero coefficients is governed by crystal symmetry, as described by Nye [2], In most single crystals, the piezoelectric coefficients are defined in terms of the crystallographic axes in polycrystalline ceramics, by convention the poling axis is referred to as the 3 axis. 

Because the piezoelectric coefficients can each be expressed in two ways, there are in general two different approaches to measuring the piezoelectric response approaches based on measurement of charge (or current), and those based on measurements of displacement (or strain). Choice of which coefficient to measure is often a matter of convenience. 

Piezoelectric coefficients are also temperature dependent quantities. This is true for both the intrinsic and the extrinsic contributions. Typically, the piezoelectric response of a ferroelectric material increases as the transition temperature is approached from below (See Figure 2.3) [3], Where appropriate thermodynamic data are available, the increase in intrinsic dijk coefficients can be calculated on the basis of phenomenology, and reflects the higher polarizability of the lattice near the transition temperature. The extrinsic contributions are also temperature dependent because domain wall motion is a thermally activated process. Thus, extrinsic contributions are lost as the temperature approaches OK [4], As a note, while the temperature dependence of the intrinsic piezoelectric response can be calculated on the basis of phenomenology, there is currently no complete model describing the temperature dependence of the extrinsic contribution to the piezoelectric coefficients. 

In summary, piezoelectric coefficients are complex numbers that depend on the measurement frequency, excitation field, temperature, and time (e. g. time after poling in samples that show finite aging rates). Consequently, in reporting piezoelectric data, it is important to specify how the property was measured. 

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from... 

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