constant are commonly expressed using the reduced Voigt matrix notation dkm, where k denotes the component of electric displacement D or field E in the Cartesian reference frame (x1, x2, x3), and the index m = 1, …, 6 is used to define the mechanical stress or strain. In this case, m = 1, 2, and 3 correspond to the normal stresses along the x1, x2, and x3 axes, respectively, whereas m = 4, 5, and 6 stand for the shear stresses Τ23, Τ13, and Τ12, respectively. Both d and d* are called the piezoelectric constant or coefficient, but they have different units, which are pC/N and pm/V (here, p stands for 10−9), respectively. It follows from thermodynamic considerations that dkm = dkm*, namely, the coefficients that connect the field and strain are equal to those connecting the stress and the polarization.
In addition to the piezoelectric charge or strain constant, other forms of piezoelectric constants are also used in specialized design cases. Totally, there are four piezoelectric constants including the abovementioned piezoelectric charge or strain coefficient d, which are listed in Table 1.1 with their names and definitions [22]. These piezoelectric constants are defined as partial derivatives evaluated at constant stress (subscript T), constant electrical field (subscript E), constant electrical displacement (subscript D), or constant strain (subscript S). These conditions can be regarded as “mechanically free,” “short circuit,” “open circuit,” and “mechanically clamped,” respectively.
1.3 Ferroelectric Properties and Its Contribution to Piezoelectricity
Since most high‐performance piezoelectric materials are also ferroelectric materials, it is necessary to review ferroelectric properties and their contribution to piezoelectricity [23–28]. Ferroelectricity is a character of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. As illustrated in Figure 1.2, dielectrics are the big family with the core subset being ferroelectrics. Dielectric materials are basically electrical insulators, which become polarized by the peripheral application of electrical field when placed across the plates of a capacitor. Piezoelectric materials belong to the dielectric group, but a stress can create a net separation of positive and negative charges in a piezoelectric crystal that has a non‐centrosymmetric crystal structure. Pyroelectrics are those materials with the ability to generate a temporary voltage when they are heated or cooled, since the polarization magnitude in a pyroelectric crystal can be thermally changed by the temperature change. By comparison, for a piezoelectric crystal, it is the mechanical stimuli resulting in the polarization change and as a consequence, charges build up at its surfaces. Ferroelectrics are an experimental subset of pyroelectric materials. All ferroelectric materials are pyroelectrics, and all pyroelectrics are piezoelectric; however, not all piezoelectric materials are pyroelectric and not all pyroelectrics are ferroelectric. It is known that crystal symmetry governs the aforementioned categorization. All crystalline substances belong to one of the 32 crystallographic point groups. There are 20 piezoelectric point groups and 10 ferroelectric point groups.
Table 1.1 Piezoelectric constants.
| Symbol | Name | Definition |
|---|---|---|
| D | Piezoelectric charge coefficient or piezoelectric strain coefficient |
 |
| G | Piezoelectric voltage coefficient (voltage output constant) |
 |
| E | Piezoelectric stress coefficient |
 |
| H | Piezoelectric stiffness coefficient |
 |
Figure 1.2 The relationship among dielectric, piezoelectric, pyroelectric, and ferroelectric materials.
The direction of electric dipoles in both piezoelectric and pyroelectric (but not ferroelectric) materials cannot be changed, whereas it can be reversed by an electric field for ferroelectric materials. Therefore, the distinguishing feature of ferroelectrics is that the spontaneous polarization can be reversed by a sufficiently high applied electric field along the opposite direction. Furthermore, the polarization is dependent not only on the current electric field but also on its history that the material has experienced, thereby yielding a hysteresis P–E (polarization–electric field) loop, as shown in Figure 1.3. Starting from point A, the polarization initially increases slowly with E‐field, but turns to a sharp rise when the applied field is sufficiently high. Then, after a long and slow stage, the polarization reaches a saturation level (saturation polarization, Ps). The Ps is normally estimated by intersecting the polarization axis with the saturated linear part. The polarization does not go back to the starting point after the removal of E‐field but instead results into non‐zero values, which is defined as the remnant polarization, Pr. In order to reach a zero polarization state, an E‐field applied along the opposite direction is required. This E‐field is named as the coercive field, EC, which stands for the magnitude of the applied electric field to reverse the direction of ferroelectric polarization.
The
