Lead-Free Piezoelectric Materials. Jing-Feng Li

Figure 1.1 (a) The direct piezoelectric effect provides an electric charge upon application of a mechanical stress, whereas (b) the converse piezoelectric effect describes the situation where strain develops under an applied electric field.

      Source: Reproduced with permission from Roedel and Li [17]. Copyright 2018, Cambridge University Press.

      In a narrow sense, piezoelectricity can be understood as a linear electromechanical interaction between the mechanical and the electrical states. The constant for such a linearly proportional relation is defined as the piezoelectric coefficient d, which is a third‐rank tensor coupling the first‐rank tensor or vector (electric displacement or field) and the second‐rank tensor (stress or strain). Hence, the piezoelectric equations may be written in the following form (i, j, k = 1, 2, 3) [22]

      (1.1)

      (1.2)

      where D_i is electric displacement (C/m²), E_i is electric field component (V/m), S_ij is strain component, Τ_ij is stress component (N/m²), and d_kij or d_kij^* is component of the piezoelectric charge or strain constant. It should be noted that the subscripts of piezoelectric constant are commonly expressed using the reduced Voigt matrix notation d_km, 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 d_km = d_km^*, 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