Piezoelectric nonlinear

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],... 

Investigation of the piezoelectric nonlinearity in PZT thin films using optical interferometry 255... 

Figure 134 Illustration of piezoelectric nonlinearity and hysteresis in a PZT thin film [3].

Barium sodium niobium oxide [12323-03-4], Ba2NaNb5015, finds application for its dielectric, piezoelectric, nonlinear crystal and electro-optic properties (35,36). It has been used in conjunction with lasers for second harmonic generation and frequency doubling. The crystalline material can be grown at high temperature, mp ca 1450°C (37). 

Shvartsman, V.V., Kholkin, A.L, and Pertsev, N.A. (2002) Piezoelectric nonlinearity of Pb(Zr, TiJOs thin films probed by scanning force microscopy. 

It is far beyond the scope of this chapter to review the electronic structures and properties of all metal oxides, or even all of the important metal oxide stmc-ture types. Instead, this section covers some featnres of one stmctural family, perovskite, in some detail. In doing so, it is hoped that the important concepts will be illnstrated in snch a way that they can be widely appUed. Of course, the choice of the perovskite stmctnre as an illnstrative example is not a random choice. The perovskite family of componnds is very extensive, encompassing most of the periodic table. Fnrthermore, perovskites exhibit nearly every type of interesting electronic or magnetic behavior seen in oxides (ferromagnetism, ferroelectricity, piezoelectricity, nonlinear optical behavior, metaUic condnctivity, snpercondnct-ivity, colossal magnetoresistance, ionic conductivity, photoluminescence, etc.). One important property that is not readily found among perovskites, transparent conductivity, is the focus of Section 6.7. 

In this chapter nonlinear piezoelectric and dielectric behavior shock-induced electrical conductance semiconductors elastic physical properties. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

Chen et al. [76C02] and Lawrence and Davison [77L01] have placed the fully coupled nonlinear theory of uniaxial piezoelectric response in a form that is convenient for numerical solution of problems and have simulated a number of experiments in terms of this theory. An example of the results obtained is given below. 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

Several structural theories of piezoelectricity [72M01, 72M02, 72A05, 74H03] have been proposed but apparently none have been found entirely satisfactory, and nonlinear piezoelectricity is not explicitly treated. With such limited second-order theories, physical interpretations of higher-order piezoelectric constants are speculative, but such speculations may help to place some constraints on an acceptable piezoelectric theory. 

Nonlinear properties of normal dielectrics can be studied in the elastic regime by the method of shock compression in much the same way nonlinear piezoelectric properties have been studied. In the earlier analysis it was shown that the shape of the current pulse delivered to a short circuit by a shock-compressed piezoelectric disk was influenced by strain-induced changes in permittivity. When a normal dielectric disk is biased by an electric field and is subjected to shock compression, a current pulse is also delivered into an external circuit. In the short-circuit approximation, the amplitude of this current pulse provides a direct measure of the shock-induced change in permittivity of the dielectric. 

The compressibility of polymers is strongly nonlinear at pressures of a few GPa. In order to consider the nonlinearity of the piezoelectric effect at shock pressure, it is of interest to consider the piezoelectric polarization in terms of the volume compression as shown in Fig. 5.9. The pressure-versus-volume relation for PVDF is not accurately known, but the available data certainly provide a relative measure of changes in compressibility. When considered versus volume, the piezoelectric polarization is found to to be remarkably linear. Thus, large volume compression does not appear to introduce large nonlinearities. Such a behavior will need to be considered when the theory of piezoelectricity for the heterogeneous piezoelectric polymer is developed. 

Fig. 5.9. The piezoelectric polarization of Fig. 5.7 is found to increase with volume compression. Hence, the large decrease in change of charge with stress is a manifestation of the highly nonlinear stress-volume relation of PVDF, not nonlinear piezoelectricity.

The piezoelectric polymer investigations give new physical insight into the nature of the physical process in this class of ferroelectric polymers. The strong nonlinearities in polarization with stress are apparently more a representation of nonlinear compressibility than nonlinear electrical effects. Piezoelectric polarization appears to be linear with stress to volume compressions of tens of percent. The combination of past work on PVDF and future work on copolymers, that have quite different physical features promises to provide an unusually detailed study of such polymers under very large compression. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

The semiconducting properties of the compounds of the SbSI type (see Table XXVIII) were predicted by Mooser and Pearson in 1958 228). They were first confirmed for SbSI, for which photoconductivity was found in 1960 243). The breakthrough was the observation of fer-roelectricity in this material 117) and other SbSI type compounds 244 see Table XXIX), in addition to phase transitions 184), nonlinear optical behavior 156), piezoelectric behavior 44), and electromechanical 183) and other properties. These photoconductors exhibit abnormally large temperature-coefficients for their band gaps they are strongly piezoelectric. Some are ferroelectric (see Table XXIX). They have anomalous electrooptic and optomechanical properties, namely, elongation or contraction under illumination. As already mentioned, these fields cannot be treated in any detail in this review for those interested in ferroelectricity, review articles 224, 352) are mentioned. The heat capacity of SbSI has been measured from - 180 to -l- 40°C and, from these data, the excess entropy of the ferro-paraelectric transition... 

Theoretical estimations and experimental investigations tirmly established (J ) that large electron delocalization is a perequisite for large values of the nonlinear optical coefficients and this can be met with the ir-electrons in conjugated molecules and polymers where also charge asymmetry can be adequately introduced in order to obtain non-centrosymmetric structures. Since the electronic density distribution of these systems seems to be easily modified by their interaction with the molecular vibrations we anticipate that these materials may possess large piezoelectric, pyroelectric and photoacoustic coefficients. 

The above conclusions introduce intrinsic limitations to the use of the ID conjugated systems in nonlinear optical devices. Although these may benefit (38) from the high nonlinearities,their response speed will be limited by the motion of such defects. These may also be formed by other means than light and this will clearly have implications on photoelastic, pyroelectric and piezoelectric effects as well. We point out that materials like polydiacetylenes may show appreciable quadrupolar pyroelectric effect (39). 

The nonlinear optical and dielectric properties of polymers find increasing use in devices, such as cladding and coatings for optical fibres, piezoelectric and optical fibre sensors, frequency doublers, and thin films for integrated optics applications. It is therefore important to understand the dielectric, optical and mechanical response of polymeric materials to optimize their usage. The parameters that are important to evaluate these properties of polymers are their dipole moment polarizability a, hyperpolarizabilities 0... 

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