Semiquantitative Model of Piezoelectricity

The brothers Jacques and Pierre Curie are credited with the discovery of piezoelectricity in a number of hemiedric crystals (Curie and Curie, 1880). Today, piezoelectrics are utiUzed in acousto-electronic devices and sensors based on bulk and surface acoustic waves, piezomechanical sensors to monitor pressure, power, and acceleration, as actuators for micropositioning devices, band pass filters with low insertion losses, as electro-optic devices for optical memories, displays for high-definition televisions, and possibly as transparent piezoelectric speaker membranes as well as miniaturized piezoelectric transformers and motors. As the classic piezoelectric material is a-quartz, the basic relationships are detailed below using it as a model structure. Further details on the piezoelectric properties of quartz, and of its history, discovery and utilization, are available elsewhere (Ballato, 2009). 

As shown in Table 8.1, the piezoelectric effect causes the creation of charges in a dielectric and ferroic materiaL respectively, in response to an applied stress field. The opposite effect-that is, the induction of strain (deformation) by applying an outside electric field-is called the inverse piezoelectric effect. Piezoelectricity requires that no symmetry center exists in the crystal structure. The piezoelectric properties of ceramic materials are described by four parameters (i) the dielectric displacement D (ii) the electric field strength E (iii) the applied stress X and (iv) the strain (deformation) x. These are related by two equations that apply to the (direct) piezoelectric effect D = e x and E = h x, and two equations that apply to the inverse piezoelectric effect x = g D and x = d E. The four coefficients e, h, g, and d are termed the piezoelectric coefficients. 

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. 

Materials-dependent figure of merits in piezoelectric ceramics are  

Barium titanate (point group 4mm below the Curie temperature of 130 °C) has the (reduced) piezoelectric strain constant (d) matrix 

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