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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, P_s). The P_s 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, P_r. 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, E_C, which stands for the magnitude of the applied electric field to reverse the direction of ferroelectric polarization.

The appearance of such a P–E loop is an important criterion to distinguish whether a material is ferroelectric or not. Ferroelectric materials display such a hysteretic behavior as a result of the response of electric domains to electric field, analogous to that of magnetic domains of a ferromagnetic material against a magnetic field. It should be emphasized that a polar material may be piezo‐/pyro‐electric but not ferroelectric if the direction of its dipoles is not switchable even under exceedingly high external electrical fields. For example, single crystalline quartz is a conventional piezoelectric material, but has no ferroelectric properties. Similarly, ZnO is a piezoelectric but non‐ferroelectric material in general.

Figure 1.3 Polarization vs. electric field hysteresis loop in ferroelectric materials.

Once a ferroelectric crystal is cooled across the Curie temperature, a polarization develops. The ferroelectric phase transition is a structural phase transition, during which the displacements of ions produce lattice distortions and change the symmetry of the crystal. The magnitude of the ion displacements along certain crystallographic directions in the materials is specific to a given crystal structure and composition. If the polarization develops uniformly throughout the whole crystal, a depolarizing electric field will be produced. To minimize the electrostatic energy associated with this field, the crystal often splits into regions, called domains; a region in which the polarization is uniform is called a domain. The regions between two adjacent domains are called domain walls. Their thickness is typically of the order of 10–100 Å. Domain represents a region within a ferroelectric material in which the direction of polarization is uniform. The saturation polarization, P_s, corresponds to the total polarization at an extreme state where (almost) all domains are aligned along the direction of applied electric field. Some of these domains stay at the same direction even after the removal of electric field, resulting in the remnant polarization. It can be readily envisaged that a ferroelectric material at a state with remnant polarization can be used a piezoelectric material, since it can generate electric charges when subjected to mechanical stress. In other words, if a ferroelectric material, at least polycrystalline bulk materials should show no piezoelectric response if it has not been subjected to an electric field. This is because the charges will be canceled collectively if the domains are randomly distributed along different directions, resulting in zero change when the whole material receives mechanical deformation. As such, piezoelectricity can be regarded as one of the functionalities of ferroelectric materials, and in general, ferroelectric materials need to be poled before they can be used as piezoelectric materials. Therefore, electrical poling is an indispensable process for ferroelectric piezoelectric materials. During poling, a strong electric field is applied across ferroelectric materials and consequently, a majority of the domains switch their pristine polarization and become aligned along the electric field direction. Figure 1.4 schematically shows the poling process. The virgin materials are subjected to an electric field, which should be sufficiently higher than the coercive field (E_C), so that the domains can be re‐orientated almost along the same direction. As shown in Figure 1.4b, the poling process is accompanied with an expansion of the poled materials or tensile strains, which is basically consistent with the converse piezoelectric phenomenon. As shown in Figure 1.4c, although most domains are kept along the poling direction, part of them revert back or change their orientations after the removal of the poling electrical field in order to reduce the mechanical strains. After the poling treatment, the material possesses a macroscopic polarization, which is equal to the remnant polarization (P_r) in the P–E loop shown earlier. Therefore, the poling process is very important for piezoelectric materials. Even for the same materials, if not completely poled, the resultant piezoelectric properties, especially piezoelectric charge coefficient (d₃₃), will be very low. Also, it is clear that the poling process is not applicable to non‐ferroelectric materials. That is why high‐performance piezoelectric materials must be ferroelectric in the first place.

Figure 1.4 The schematic illustrations showing the alignment of ferroelectric domain and macroscopic strains when a ferroelectric material is subjected to a poling treatment under an electric field. (a) Virgin state. (b) Saturation state. (c) Remnant state.