Important piezoelectric constants

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]  

The piezoelectric coefficients are third rank tensors, hence the piezoelectric response is anisotropic. A two subscript matrix notation is also widely used. The number of non-zero coefficients is governed by crystal symmetry, as described by Nye [2], In most single crystals, the piezoelectric coefficients are defined in terms of the crystallographic axes in polycrystalline ceramics, by convention the poling axis is referred to as the 3 axis. 

Because the piezoelectric coefficients can each be expressed in two ways, there are in general two different approaches to measuring the piezoelectric response approaches based on measurement of charge (or current), and those based on measurements of displacement (or strain). Choice of which coefficient to measure is often a matter of convenience. 

In many ferroelectric materials, the net piezoelectric effect is a result of both intrinsic and extrinsic responses. Here, intrinsic refers to the response that would result from an appropriately oriented single crystal (or ensemble thereof, in a polycrystalline sample). The extrinsic response is typically the result of motion of non-180° domain walls. The principle of these 

It should be noted that in practice, the piezoelectric response will typically not continue to rise all the way to the transition temperature, as elevated temperatures induce depoling of the ferroelectric, unless appropriate care is taken to insure that the material remains polarized (e. g. by application of a bias electric field). Depoling of this type is often important at temperatures of 1/2 of the Curie temperature, making high transition temperature materials interesting both for the decreased temperature dependence in the response, and the wider use range that can be achieved. 

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In the usual experiment where E0 = 0, the term kSE in Eq. (9) does not make any contribution as far as the electrical response with the same frequency as the mechanical excitation is concerned. However, as will be described in 2.2 and 2.4, the piezoelectric constant of a polymer film is sometimes a function of the electrostriction constant which plays an important role in the anisotropy and relaxational behavior of the piezoelectric effect. 

The vibration induced by E allows the frequency of an electric resonant circuit (used, for example, in a quartz watch) to be stabilized. Quartz is also used in the manufacture of acoustic generators. The theory which accounts for this dynamic effect comprises the calculation of the elastic vibrations of the crystal (Section 4.4.3) induced by an alternating electric field E cos cot parallel to in the absence of external stresses. The importance of quartz is due to the fact that it is chemically inert and very stable, and, even if the elastic and piezoelectric constants vary with temperature, it is possible to cut crystal plates with orientations such that their natural frequencies are constant over a large temperature... 

In this section we discuss the properties of FLCEs that are important for applications. Two options are possible either a mechanical deformation generates a change of size or orientation of the polarizatimi of the FLCE (an electric signal), or an applied electric field generates a deformation. In the case that an electric signal is detected, the piezoelectric constant is obtained. If the sample is deformed, the generated strain is of interest. Some characteristic values are collected in Table 1. 

An important feature of the strain characteristics of BaTiOj is that as the temperature is raised above its Curie temperature the piezoelectric constants disappear because the symmetry becomes cubic. The DJx curve thus becomes truly parabolic (oc P ), while at lower temperatures the strain is related to even powers of the polarization by a relation of the type... 

At the same time that Nakamura was making his major breakthroughs in GaN-based materials and devices, Vanderbilt was developing the quantum theory of polarization [13,14]. In this landmark work, Vanderbilt showed that polarization in a solid is a bulk property and can be determined quantum mechanically with knowledge of the phase of the valence electron wavefxmctions. Subsequently, Bernardini, Fiorintini, and Vanderbilt calculated the spontaneous polarization and piezoelectric constants for GaN, AlN, and InN [15] - this work is the standard used today for these important physical properties. 

Because of the crystalline symmetry of cBN (F 3m), only three elastic constants, one piezoelectric constant, and one dielectric constant in the equation are of importance or have... 

Alkaline-Earth Titanates. Some physical properties of representative alkaline-earth titanates ate Hsted in Table 15. The most important apphcations of these titanates are in the manufacture of electronic components (109). The most important member of the class is barium titanate, BaTi03, which owes its significance to its exceptionally high dielectric constant and its piezoelectric and ferroelectric properties. Further, because barium titanate easily forms solid solutions with strontium titanate, lead titanate, zirconium oxide, and tin oxide, the electrical properties can be modified within wide limits. Barium titanate may be made by, eg, cocalcination of barium carbonate and titanium dioxide at ca 1200°C. With the exception of Ba2Ti04, barium orthotitanate, titanates do not contain discrete TiO ions but ate mixed oxides. Ba2Ti04 has the P-K SO stmcture in which distorted tetrahedral TiO ions occur. 

An important class of materials that originates from the precursor core-shell particles is hollow capsules. Hollow capsules (or shells ) can be routinely produced upon removal of the core material using chemical and physical methods. Much of the research conducted in the production of uniform-size hollow capsules arises from their scientific and technological interest. Hollow capsules are widely utilized for the encapsulation and controlled release of various substances (e.g., drugs, cosmetics, dyes, and inks), in catalysis and acoustic insulation, in the development of piezoelectric transducers and low-dielectric-constant materials, and for the manufacture of advanced materials [14],... 

Barium titanate has many important commercial apphcations. It has both ferroelectric and piezoelectric properties. Also, it has a very high dielectric constant (about 1,000 times that of water). The compound has five crystalline modifications, each of which is stable over a particular temperature range. Ceramic bodies of barium titanate find wide applications in dielectric amplifiers, magnetic amplifiers, and capacitors. These storage devices are used in digital calculators, radio and television sets, ultrasonic apparatus, crystal microphone and telephone, sonar equipment, and many other electronic devices. 

As will be shown in the theory, the electrostriction effect plays an important role in the piezoelectric effect of polymer films. Moreover, a knowledge of the complex electrostriction constant as a function of frequency reveals a new aspect of the relaxational behavior of polymers. In this review a new method for measuring complex electrostriction constant with varying frequency will be presented with some results for poly(vinylidene fluoride). 

When the film is short-circuited and heated to high temperatures at which the molecules attain a sufficiently high mobility, a current is observed in the external circuit. This phenomenon is called pyroelectric effect, thermally stimulated current, or, when the film has been polarized by a static field prior to measurement, depolarization current. The conventional definition of pyroelectricity is the temperature dependence of spontaneous polarization Ps, and the pyroelectric constant is defined as dPJdd (6 = temperature). In this review, however, the term will be used in a broader definition than usual. The pyroelectric current results from the motion of true charge and/or polarization charge in the film. Since the piezoelectricity of a polymer film is in some cases caused by these charges, the relation between piezoelectricity and pyroelectricity is an important clue to the origin of piezoelectricity. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

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