PIEZOELECTRIC MATERIALS
4.2 PHYSICAL BASIS FOR ELECTROMECHANICAL COUPLING IN PIEZOELECTRIC MATERIALS
4.2.1 Manufacturing of Piezoelectric Materials
Piezoelectricity is a phenomenon that is present in a number of natural materials.
As discussed in Chapter 1, the phenomenon of piezoelectricity was first discovered
+ –
– + – + – +
+
–
– +
+
– +
+ – –
ferroelectric
material +
–
electrical dipole due to molecular
structure negative
charge
positive charge
Figure 4.5 Electric dipoles that lead to electromechanical coupling in piezoelectric materials.
in a natural crystal called Rochelle salt in the late nineteenth century. For a number of years the only piezoelectric materials that were studied were natural crystals that exhibited only weak piezoelectricity. It was not until the mid-twentieth century that synthetic piezoelectric materials with increased coupling properties enabled practical applications.
The manufacture of synthetic piezoelectric materials typically begins with the constituent materials in powder form. A typical mixture of materials that exhibit piezoelectric properties are lead (with the chemical symbol Pb), zirconium (Zr), and titanium (Ti). These materials produce the common piezoelectric material lead–
zirconium–titinate, typically referred to as PZT. Other types of piezoelectric materials are barium titinate and sodium–potassium niobates.
The processing of a piezoelectric ceramic typically begins by heating the powders to temperatures in the range 1200 to 1500◦C. The heated materials are then formed and dimensioned with conventional methods such as grinding or abrasive media.
The result of this process is generally a wafer of dimensions on the order of a few centimeters on two sides and thicknesses in the range 100 to 300µm. Electrodes are placed on the wafers by painting a thin silver paint onto the surface. The resulting wafer can be cut with a diamond saw or joined with other layers to produce a multilayer device.
As discussed in Section 4.1, the piezoelectric effect is strongly coupled to the existence of electric dipoles in the crystal structure of the ceramic. Generally, after processing the raw material does not exhibit strong piezoelectric properties, due to the fact that the electric dipoles in the material are pointing in random directions. Thus, the net dipole properties of the material are very small at the conclusion of the fabrication process. The orientation of the individual electric dipoles in a piezoelectric material must be aligned for the material to exhibit strong electromechanical coupling.
The dipoles are oriented with respect to one another through a process called poling. Poling requires that the piezoelectric material be heated up above itsCurie temperatureand then placed in a strong electric field (typically, 2000 V/mm). The combination of heating and electric field produces motion of the electronic dipoles.
Heating the material allows the dipoles to rotate freely, since the material is softer at higher temperatures. The electric field produces an alignment of the dipoles along the direction of the electric field as shown in Figure 4.6. Quickly reducing the temperature and removing the electric field produces a material whose electric dipoles are oriented in the same direction. This direction is referred to as the poling direction of the material.
Orienting the dipoles has the effect of enhancing the piezoelectric effect in the material. Now an applied electric field will produce similar rotations throughout the material. This results in a summation of strain due to the applied field. Conversely, we see that strain induced in a particular direction will produce a summation of apparent charge flow in the material, resulting in an increase in the charge output of the material.
The basic properties of a piezoelectric material are expressed mathematically as a relationship between two mechanical variables, stress and strain, and two electrical variables, electric field and electric displacement. The direct and converse piezoelec- tric effects are written as the set of linear equations in equations (4.1) to (4.4). The
+ –
+ –
+ –
+ –
+ –
+ –
+ –
+ –
poling direction
+ –
+ –
+ –
+ –
+ –
+ –
+ –
+ – +++++
- - - - -
Applied field (2000 V/mm) material
expansion and contraction
material is heated above its Curie
temperature
Figure 4.6 Poling process associated with piezoelectric materials.
expressions for the direct and converse piezoelectric effect can be combined into one matrix expression by writing the relationship between strain and electric displacement as a function of applied stress and applied field:
S D
= s d
d ε T E
. (4.5)
The top partition of equation (4.5) represents an equation for the converse piezoelectric effect, whereas the bottom partition represents an expression of the direct effect.
Writing the expressions as a matrix highlights some fundamental concepts of piezo- electric materials. Examining the matrix expression, we see that the on-diagonal terms represent the constitutive relationships of a mechanical and an electrical material, re- spectively. For example, the (1,1) term in the matrix,s, represents the mechanical constitutive relationship between stress and strain, whereas the (2,2) termε repre- sents the electrical constitutive equation. These constitutive relationships would exist in a material that was either purely elastic or purely dielectric.
The electromechanical coupling in the material is represented by the off-diagonal terms of equation (4.5). A larger off-diagonal term will result in a material that produces more strain for an applied electric field and more electric displacment for an applied mechanical stress. For these reasons, the piezoelectric strain coefficient is an important parameter for comparing the relative strength of different types of piezoelectric materials. In the limit asdapproaches zero, we are left with a material that exhibits very little electromechanical coupling. Examining equation (4.5) we see that the coefficient matrix is symmetric. The symmetry is not simply a coincidence, we will see that symmetry in the coefficient matrix represents reciprocity between the electromechanical transductions mechanisms in the material. This will naturally arise when we discuss the energy formulation of the piezoelectric consitutive equations in Chapter 5.
There is no reason why equation (4.5) has to be expressed with stress and electric field as the independent variables and strain and electric displacement as the dependent variables. Equation (4.5) can be inverted to write the expressions with stress and field as the dependent variables and strain and electric displacement as the independent variables. Taking the inverse of the 2×2 matrix produces the expression
T E
= 1 sε−d2
ε −d
−d s S D
. (4.6)
The determinant can be incorporated into the matrix to produce T
E
=
1 s
1
1−d2/sε − d/sε 1−d2/sε
− d/sε 1−d2/sε
1 ε
1 1−d2/sε
S
D
. (4.7)
The term d2/sε appears quite often in an analysis of piezoelectric materials. The square root of this term is called thepiezoelectric coupling coefficientand is denoted
k= d
√sε. (4.8)
An important property of the piezoelectric coupling coefficient is that it is always positive and bounded between 0 and 1. The bounds on the coupling coefficient are related to the energy conversion properties in the piezoelectric material, and the bounds of 0 and 1 represent the fact that only a fraction of the energy is converted between mechanical and electrical domains. The piezoelectric coupling coefficient quantifies the electromechanical energy conversion. The rationale for these bounds will become clearer when we derive the constitutive equations from energy principles in Chapter 5.
Substituting the definition of the piezoelectric coupling coefficient into equa- tion (4.7) yields
T E
=
1 s
1 1−k2 −1
d k2 1−k2
−1 d
k2 1−k2
1 ε
1 1−k2
S
D
. (4.9)
Simplifying the expression yields T
E
= 1 1−k2
s−1 −d−1k2
−d−1k2 ε−1 S D
. (4.10)
The fact that 0<k2<1 implies that the term 1/(1−k2) must be greater than 1.
Example 4.2 A coupling coefficient ofk=0.6 has been measured for the piezo- electric material considered in Example 4.1. Compute the dielectric permittivity of the sample.
Solution Solving equation (4.8) for dielectric permittivity yields ε= d2
sk2.
Substituting the values for the piezoelectric strain coefficient and mechanical com- pliance into the expression yields
ε= (550×10−12m/V)2
(20×10−12m2/N)(0.62)=42.0×10−9F/m.
The dielectric permittivity is often quoted in reference to the permittivity of a vacuum, εo=8.85×10−12F/m. The relative permittivity is defined as
εr = ε εo
= 42.0×10−9F/m
8.85×10−12F/m =4747 and is a nondimensional quantity.