Technical description

Piezoelectricity is the property of nearly all materials that have a non-centrosymmetric crystal structure.

Some naturally occurring crystalline materials that possess these properties are quartz and tourmaline. Some artificially produced piezoelectric crystals are Rochelle salt, ammonium dihydrogen phosphate and lithium sulphate. Another class of materials possessing these properties is polarized piezoelectric ceramic. In contrast to the naturally occurring piezoelectric crystals, piezoelectric ceramics have a polycrystalline structure.

The most commonly produced piezoelectric ceramics are lead zirconate titanate (PZT), barium titanate and lead titanate.

Ceramic materials have several advantages over single crystals, especially the ease of fabrication into a variety of shapes and sizes. In contrast, single crystals must be cut along certain crystallographic directions, limiting the possible geometric shapes.

PZT (and many other piezoelectric materials) have crystal structures belonging to the perovskite family with the general formula AB03. In the following figure the ideal, cubic perovskite structure (centrosymmetric) and tetragonal (ferroelectric) structure are shown.

A piezoelectric ceramic material consists of small grains (crystallites), each containing domains in which the polar direction of the unit cells are aligned. Before poling, these grains and domains are randomly oriented; hence the net polarization of the material is zero, i.e. the ceramic does not exhibit piezoelectric properties. The application of a sufficiently high DC field (called poling process) will orient the domains in the field direction and lead to a remanent polarization of the material.

The perovskite structure is very tolerant to element substitution (doping) by formation of solid solutions. The possibilities of doping in these materials lead to an unlimited number of possible perovskite-type oxides. Even small amounts of a dopant may cause huge changes in the properties of a material. The coupling of electrical and mechanical energy makes piezoelectric materials useful in a wide range of applications.

Constitutive equations 

The piezoelectric effect depends on directions.
The reference axis, called axis 3, is taken parallel to the direction of poling. Axes 1 and 2 are defined arbitrarily in order to form a direct coordinate system with axis 3. 4, 5 and 6 represent shear movements around axes 1, 2 and 3 respectively.
 

Piezoelectric coefficients

Based on this coordinate system, the piezoelectriceffect can be described in a simplified way by matrix coefficients. The coefficients “d” (piezoelectric change constant 3×6 matrix) and “s^E” (elastic compliance 6×6 matrix) are commonly used.
 

Basic piezoelectric equations 

These coefficients are used to relate the strain “S” (6-components tensor) to the stress “T” (6-components tensor) and electrical field “E” (3-components vector).
 

S = s^E.T + d.E
 

In this equation, the “s^E.T” term describes the mechanical compliance of the component, similarly to any mechanical component. The “d.E” term describes the piezoelectric effect, i.e. strain generated by electrical field.

The above equations are useful for designing a piezoelectric application. However, it must be kept in mind that they represent an approximation.