Piezoelectric charge constant

In this system of equations the piezoelectric charge constant d indicates the intensity of the piezo effect is the dielectric constant for constant T and is the elastic compliance for constant E eft is the transpose of matrix d. The mentioned parameters are tensors of the first to fourth order. A simplification is possible by using the symmetry properties of tensors. Usually, the Cartesian coordinate system in Fig. 6.12a is used, with axis 3 pointing in the direction of polarization of the piezo substance (see below) [5,6]. 

The parameters in these equations are the electrical small-signal capacitance C, the small-signal stiffness cp and the effective piezoelectric charge constant dp, compare Fig. 6.130. 

Here the coefficients 7e, 7s = 7a, and 7m correspond to the small-signal capacitance C, the effective piezoelectric charge constant dp and the inverse of the small-signal stiffness cp for a piezoelectric transducer and to the small-signal inductance L, the effective magnetostrictive constant du and the inverse of the small-signal stiffness cm for a magnetostrictive transducer, respectively. 

The piezoelectric voltage constant, also known as the g factor, denotes the electric field generated by materials per unit of mechanical stress applied. Like the piezoelectric charge constant, these values can also be classed in terms of directions (i.e. g y)-... 

Piezoelectric materials generate an electric charge in response to a mechanical stress. This is known as the direct piezoelectric effect, where the charge per unit area or electric displacement D is proportional to the applied stress T. There is also a converse effect, since an applied electric field E produces a proportional strain S in the material, resulting in either expansion or contraction, depending on the field s polarity. For both effects, the proportionality constant is termed the piezoelectric charge constant d ... 

We can estimate the piezoelectric charge and piezoelectric constant in the Bond Orbital Approximation with no additional assumptions or parameters (Harrison, 1974). Return to the geometry of Fig. 8-5 there we found, as shown in Eq. (8-24), a change in every bond length of magnitude dd = — C) d/3. Proceeding just as... 

Piezoelectric Charge and Voltage Coefflcient/-Constant. For a piezoelectric material interactions between the electrical field and mechanical quantities have to be considered. In a good approximation this can be described via the linear context... 

Here D is the vector of the dielectric displacement (size 3x1, unit C/m ), S is the strain (size 6x1, dimension 1), E is a vector of the electric field strength (size 3x1, unit V/m) and T is a vector of the mechanical tension (size 6x1, unit N/m ). As the piezoelectric constants depend on the direction in space they are described as tensors e- is the permittivity constant also called dielectric permittivity at constant mechanical tension T (size 3x3, unit F/m) and 5 , is the elastic compliance matrix (size 6x6, unit m /N). The piezoelectric charge coefficient df " (size 6x3, unit C/N) defines the dielectric displacement per mechanical tension at constant electrical field and (size 3x6, unit m/V) defines the strain per eiectric fieid at constant mechanical tension [84], The first equation describes the direct piezo effect (sensor equation) and the second the inverse piezo effect (actuator equation). 

The piezoelectric charge coefficients are generally expressed by using condensed subscripts, such as djj and c/jj, where the first subscript refers to the electric field direction or direction of polarization and the second subscript refers to the stress or strain direction. The piezoelectric charge coefficients g j. are also expressed in condensed form and are related to the charge coefficients via the dielectric constant K,... 

A nanometer-sized silicon nitride, compacted into bulk sample, showed a strong piezoelectric constant 2613 X 10 12 [C/N]. It is interpreted by the charge accumulation in the interfaces and the surfaces of microvoids (67). 

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. 

Fig. 27 indicates the apparent piezoelectric constant e of roll-drawn PVDF as a function of static bias field E0 (Oshiki and Fukada, 1972). The value of e at E0=0 represents the true piezoelectric constant e. The curve exhibits a hysteresis and the polarity of e changes according to the poling history. If the piezoelectricity in /)-form PVDF originates from the polarization charge due to spontaneous polarization, inversion of polarity of e would mean the inversion of the polarization by the external field and hence /S-form PVDF may be a ferroelectric material, as was first suggested by Nakamura and Wada (1971). 

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],... 

With this background, we have proposed and developed a new purely electrical method for imaging the state of the polarizations in ferroelectric and piezoelectric material and their crystal anisotropy. It involves the measurement of point-to-point variations of the nonlinear dielectric constant of a specimen and is termed scanning nonlinear dielectric microscopy (sndm) [1-7]. This is the first successful purely electrical method for observing the ferroelectric polarization distribution without the influence of the screening effect from free charges. To date, the resolution of this microscope has been improved down to the subnanometer order. 

In Sect. 7.3, Eqs. (18) and (19) describe the Maxwell stress forces acting on a conductive tip when a combined d.c./a.c. voltage is applied. For the PFM set-up we have to complete the total interaction force by the additional effects of piezoelectricity, electrostriction and the spontaneous polarisation. Both electromechanical effects cause an electric field-induced thickness variation and modulate the tip position. The spontaneous polarisation causes surface charges and changes the Maxwell stress force. If the voltage U(t)=U[)c+UAc sin((Ot) is applied, the resulting total force Ftotai(z) consists of three components (see also Eq. 19) Fstatic, F(0 and F2m. Fstatic is the static cantilever deflection which is kept constant by the feedback loop. F2a contains additional information on electrostriction and Maxwell stress and will not be considered in detail here (for details see, e.g. [476]). The relevant component for PFM is F(0 [476, 477] ... 

---