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The prefix “piezo” (pronounced pie-ease-o) comes from the Greek word for pressure or mechanical force. Piezoelectricity refers to the linear coupling between mechanical stress and electric polarization (the direct piezoelectric effect) or between mechanical strain and applied electric field (the converse piezoelectric effect). The equivalence between the direct and converse effects was established earlier using thermodynamic arguments (Section 6.2). The principal piezoelectric coefficient, d, relates polarization, P, to stress, X, in the direct effect (P = dX) and strain, x, to electric field E (x = dE). Thus the units of d are [C/N] or [m/V] which are equivalent to one another. Typical sizes for useful piezoelectric materials range from about 1 pC/N for quartz crystals to about 1000 pC/N for PZT (lead zirconate titanate) ceramics. To understand how the piezoelectric effect varies with direction and how it is affected by symmetry, it is necessary to determine how piezoelectric coefficients transform between coordinate systems. Since polarization is a vector and stress a second rank tensor, the physical property relating these two variables must involve three directions: … Pj = djklXkl … . In the new coordinate system … P'i = aijPj = aijdjklXkl … . Transforming the stress to the new coordinate system gives … P'i= aijdjklamkanlX'mn = d'imnX'mn…. Thus piezoelectricity transforms as a polar third rank tensor… . d'imn = aijamkanldjkl … . In general there are 33 = 27 tensor components, but because the stress tensor is symmetric (Xij = Xji), only 18 of the components are independent. Therefore the piezoelectric effect can be described by a 6 × 3 matrix.

The physical properties discussed thus far are linear relationships between two measured quantities. This is only an approximation to the truth, and often not a very good approximation, especially for materials near a phase transformation. A more accurate description can be obtained by introducing higher order coefficients. To illustrate nonlinearity we discuss electrostriction, magnetostriction, and higher order elastic, and dielectric effects. These phenomena are described in terms of fourth and sixth rank tensors. Many of the recent innovations in the field of electroceramics have exploited the nonlinearities of material properties with factors such as electric field, mechanical stress, temperature, or frequency. The nonlinear dielectric behavior of ferroelectric ceramics (Fig. 15.1), for example, has opened up new markets in electronics and communications. In these materials the electric polarization saturates under high fields. Electric displacement Di varies with applied electric field Ej as … Di = εijEj + εijkEjEk + εijklEjEkEl +· · ·, … where εij is the dielectric permittivity and εijk and εijkl are higher order terms. The data in Fig. 15.1 were collected for a relaxor ferroelectric in its paraelectric state above Tc where the symmetry is centrosymmetric. Therefore the third rank tensor εijk is zero, and the shape of the curve is largely controlled by the first and third terms. For cubic crystals, the fourth rank tensor εijkl is similar in form to the elastic constants discussed in Chapter 13. Tunable microwave devices utilize nonlinear dielectrics in which the polarization saturates as in Fig. 15.1. By applying a DC bias the dielectric constant can be adjusted over a wide range.

Twinned crystals are normally classified according to twin-laws and morphology, or according to their mode of origin, or according to a structural basis, but there is another classification that deserves wider acceptance, one that is based on the tensor properties of the orientation states. An advantage of such a classification is the logical relationship between free energy and twin structures, for it becomes immediately apparent which forces and fields will be effective in moving twin walls. The domain patterns in ferroelectric and ferromagnetic materials are strongly affected by external fields, but there are many other types of twinned crystals with movable twin walls and hysteresis. These materials are classified as ferroelastic, ferrobielastic, and various other ferroic species. As explained in the next section, each type of switching arises from a particular term in the free energy function. Ferroic crystals possess two or more orientation states or domains, and under a suitably chosen driving force the domain walls move, switching the crystal from one domain state to another. Switching may be accomplished by mechanical stress (X), electric field (E), magnetic field (H), or some combination of the three. Ferroelectric, ferroelastic, and ferromagnetic materials are well known examples of primary ferroic crystals in which the orientation states differ in spontaneous polarization (P(s)), spontaneous strain (x(s)), and spontaneous magnetization (I(s)), respectively. It is not necessary, however, that the orientation states differ in the primary quantities (strain, polarization, or magnetization) for the appropriate field to develop a driving force for domain walls. If, for example, the twinning rules between domains lead to a different orientation of the elastic compliance tensor, a suitably chosen stress can then produce different strains in the two domains. This same stress may act upon the difference in induced strain to produce wall motion and domain reorientation. Aizu suggested the term ferrobielastic to distinguish this type of response from ferroelasticity, and illustrated the effect with Dauphine twinning in quartz. Other types of secondary ferroic crystals are listed in Table 16.1, along with the difference between domain states, and the driving fields required to switch between states.