Piezoelectricity converse piezoelectric coefficient

Ferroelectrics. Among the 32 crystal classes, 11 possess a centre of symmetry and are centrosymmetric and therefore do not possess polar properties. Of the 21 noncentrosymmetric classes, 20 of them exhibit electric polarity when subjected to a stress and are called piezoelectric one of the noncentrosymmetric classes (cubic 432) has other symmetry elements which combine to exclude piezoelectric character. Piezoelectric crystals obey a linear relationship P,- = gijFj between polarization P and force F, where is the piezoelectric coefficient. An inverse piezoelectric effect leads to mechanical deformation or strain under the influence of an electric field. Ten of the 20 piezoelectric classes possess a unique polar axis. In nonconducting crystals, a change in polarization can be observed by a change in temperature, and they are referred to as pyroelectric crystals. If the polarity of a pyroelectric crystal can be reversed by the application on an electric field, we call such a crystal a ferroelectric. A knowledge of the crystal class is therefore sufficient to establish the piezoelectric or the pyroelectric nature of a solid, but reversible polarization is a necessary condition for ferroelectricity. While all ferroelectric materials are also piezoelectric, the converse is not true for example, quartz is piezoelectric, but not ferroelectric. 

The piezoelectric effect entails a linear coupling between electrical and mechanical energies. Numerous piezoelectric coefficients are in use, depending on the electrical and mechanical boundary conditions imposed on the part under test. Each of the piezoelectric d, e, g, and h coefficients can be defined in terms of a direct and a converse effect the two sets of coefficients are related by thermodynamics. For example, the piezoelectric charge coefficient, dkjk, can be defined via [1] ... 

This paper described a number of the means for measuring the piezoelectric coefficients of bulk materials and thin films. In bulk materials, excellent references are available. Numerous means have been used over the years to measure the piezoelectric coefficients, which can be loosely grouped as charge-based and displacement-based. Accurate data can be obtained by many of the techniques, and agreement between measurement types is usually reasonable, provided that comparable excitation levels are utilized. In contrast, for thin films attached to substrates, the mechanical boundary conditions differ in charge and displacement based techniques. As a result, the direct and converse coefficients are not identical. In addition, perhaps because of the relative immaturity of the field, the numerous possible artifacts are not always accounted for, which can lead to erroneous results in thin film measurements. 

The term piezoelectric nonlinearity is used here to describe relationship between mechanical and electrical fields (charge density D vs. stress a, strain x vs. electric field E) in which the proportionality constant d, is dependent on the driving field, Figure 13.1. Thus, for the direct piezoelectric effect one may write D = d(a)a and for the converse effect x = d(E)E. Similar relationships may be defined for other piezoelectric coefficients (g, h, and e) and combination of electro-mechanical variables. The piezoelectric nonlinearity is usually accompanied by the electro-mechanical (D vs. a or x vs. E) hysteresis, as shown in Figure 13.2. By hysteresis we shall simply mean, in the first approximation, that there is a phase lag between the driving field and the response. This phase lag may be accompanied by complex nonlinear processes leading to a more general definition of the hysteresis [2],... 

Figure 13.1 Examples of the field dependence of piezoelectric coefficients (a) direct effect in ferroelectric ceramics, measured with a dynamic press (b) converse effect in rhombohedral 60/40 pzt thin films with different orientations, measured with an optical interferometer [1], correspond to pseudocubic axes.

In the converse piezoelectric effect one usually applies voltage V or electric field E on the sample and measures displacement AZ or strain A///. From relation Al = 0Z33 V for the longitudinal effect, we see that even for materials with exceptionally high piezoelectric coefficient (do3 = 2000pm/V in pzn-pt) the displacement Al is only around 2 nm if 1 V is applied on the sample. For the same voltage the displacement is reduced to 0.2 nm in a typical pzt composition and to only tn 2 pm in quartz. The displacement can be increased by application... 

S]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term Elasticity. Numbers in square brackets show the ranks of the crystal property tensors the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify Ist-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rai tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables. 

The coefficient is called the transverse piezoelectric coefficient. The converse piezoelectric effect, relating strain, e to the applied electric field E is similarly simplified to... 

Direct methods for measuring the strain that results from applying a field or vice versa, applying a strain, and measming the accumulated charge are ahim-dant. Interferometers, dilatometers, fiher-optic sensors, optical levers, linear variable displacement transducers, and optical methods are employed to evaluate the piezoelectric strain (converse effect) (69-72). The out-of-plane or thickness piezoelectric coefficient dss can he ascertained as a function of the driving field and frequency. The coefficient is measmed based on the equation... 

The direct effect coefficients are defined by the derivatives (5D/SX) = d (piezoelectric strain coefficient), (5D/5x) = e, -(5E/5X) = g (piezoelectric voltage constant) and -(5E/5x) = h. The converse-effect coefficients are defined by the derivatives (8x/5E) = d, (5x/5D) = g, -(5X/5E) = e, and -(5X/5D) = h. As the piezoelectric coefficients are higher-rank tensors, their mathematical treatment is rather tedious. Fortunately, in higher symmetric crystals the number of tensorial components will be drastically reduced due to symmetry constraints. An example is shown below. 

The thermodynamical derivation of piezoelectricity includes two steps (1) The relevant mechanical or electrical quantities are calculated as partial derivatives of the Gibbs free energy with respect either to one of the two mechanical or to one of the two electrical observables, respectively. (2) The second partial derivative of the Gibbs free energy with respect to the other domain (electrical or mechanical, respectively) yields one of the piezoelectric coefficients. Because there is one intensive (force-like or voltage-like) observable, namely, mechanical stress and electrical field, and one extensive (displacement-like) observable, namely, mechanical strain and electrical displacement, in each of the two domains, we have four possible combinations of one mechanical and one electrical observable in total. Thus, we obtain four different piezoelectric coefficients that are usually abbreviated as d, e, g, and h. As the sequence of the two partial derivations can be reversed, we arrive at two different expressions for each coefficient one for direct piezoelectricity (mechanical stimulus leads to an electrical response) and one for inverse or converse piezoelectricity (electrical stimulus leads to a mechanical response). For example, the piezoelectric d coefficient is given by the two alternative terms ... 

We see from Figure 8.13 that a number of liquid crystal phases, SmC of chiral rod-shape and the tilted columnar phase of chiral disc shape molecules, as well as the SmCP of achiral bent-core, and the tilted bowl-shape molecules all have C2 symmetry with eight independent piezoelectric coefficients. The direct and converse i- piezoelectric effects have been mostly studied in the fluid SmC liquid crystals. 

The piezoelectric coefficients discussed so far are the direct piezoelectric ooefficieats. When an electric field is applied it can produce a strain, and the ooefficieots which describe the strain are refen to as converse piezoelectric coefficients. 

A more complete description of these definitions is provided in a book by Nye (8). With these definitions it is now possible to describe the converse piezoelectric coefficient which relates the strain induced by an electric field lo the magnitude of the electric field E at constant stress ... 

Also, it is importsnl to point out that the direct and convene piezoekctric coefficients are not equal [101], as is usually assumed. The difference tentk to be small. - 10%, in hard and brittle fenoekcirics, but as will be demonstrated later in this chapter, the correction term can dominate in soft, pliable polymcn. The relatioa between tlie direct and converse piezoelectric coefficient is typically determined by deriving a Maxwell relation from a thermodynamic potential. A thorough description of Maxwell relations and thermodynamic potentials is presented in a book by Cailen (90. ... 

Thking the second partial derivatives with respect to the opposite variables In Eq. (44) and making use of the fact that the order of differentiation makes no difference, the relationship of the direct and converse piezoelectric coefficients is found to be... 

The results of measurements by Kepler and Anderson [81] using the static technique are presented in Thble 1. All the data presented in thb and other tables in thb chapter are the experimentally determined values, uncorrccted for changes in electrode area, and therefore arc not the true piezoelectric coeffidents. As we showed earlier in thb chapter, it b the experimentally determined value of the direct piezoelectric coefficient that should be equal to the converse piezoelectric coefficient and. in general, it appears that it b the experimentally determined values of other coeffidenb that should be used for comparison and calculations. 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

All piezoelectric crystals should have a good temperature coefficient, that is. should show as little change in resonant frequency as possible under large variations in temperature. Ideally. Ihe piezoelectric constant of proportionality between the mechanical and electrical variables must be the same for both direct (pressure-to-electricily) and converse effects... 

If the film cannot be freely deformed in its plane, the piezoelectric current is called t/33 or dj. If the variation in the electric field is measured per unit of stress, g coefficients are obtained that are connected by the correlation of g = d/e where e is the dielectric constant depending on the film thickness. Constants g and tf are most widely used in the design of electromechanical transducers. The yield from the conversion of mechanical energy into electrical energy is represented by the electromechanical coupling coefficient ATjby Eq. (3.3). 

The piezoelectric generator coefficient d indicates the mechanical strains that are induced by electric potentials and the piezoelectric motorcoefficient e is a measure of the dipole moments that are created by strains. These two parameters are similar in nature and proportional to one another. The electromechanical coupling coefficient k is proportional to both d and e, its square giving the efficiency of conversion between mechanical and electrical energy that can be achieved with the ferroelectric... 

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