Pre-requisite Capacitive behavior Polarization Dielectric Loss Insulating behavior

1. Dielectric Materials • Pre-requisite • Capacitive behavior • Polarization • Dielectric Loss • Insulating behavior • Dielectric Breakdown • Refractive Index • Piezoelectricity & Ferroelectricity

2. Piezoelectricity • Piezoelectricity (or pressure electricity): •  Unusual phenomena in which polarization is induced and an electric field is established across a sample when it is mechanically stressed. [Fig. (a) & (b)] •  Similarly, the same crystal also exhibits mechanical strain when it experiences an electric field. [Fig. (c) & (d)] •  The direction of mechanical deformation (extension or compression) depends on the direction of the applied field, or the polarity of the voltage. (a) (b) (c) (d)

3. Criteria of Piezoelectricity: Center of symmetry Dielectric with center of symmetry has no piezoelectricity • Example: the cubic unit cell • When unstressed [Fig. (a)], center of mass (c.m. of the –ve charges at the corner of unit cell coincides with +ve charge at center  therefore, no net polarization occurs and P=0. • Under stress, unit cell becomes strained [Fig. (b)]. However, c.m. of the –ve charges still coincides with +ve cahrge and net polarization is still 0. P=0 for stained crystal. This is generally true for crystals with center of symmetry. If we draw a vector from O (position of an arbitrary point charge) to any charge, then the reverse vector will point to the same type of charge  we call O, or any other point charge, a center of symmetry

4. Piezoelectricity: Dielectric with noncentrosymmetric If we draw a vector from O to any charge, then the reverse vector will point to an opposite charge. The unit cell is said to be noncentrosymmetric. • - When unstressed [Fig. (a)], c.m. of –ve charges coincides with c.m. of the +ve charges, both at O  therefore, no net polarization occurs and P=0. • Under stress [Fig. (b)], the +ve charge at A and –ve charge at B both become displaced inwards to A’ and B’ respectively. The two c.m.’s become shifted and there is now a net polarization P for the strained crystal.

5. Piezoelectricity: Dielectric with noncentrosymmetric • Direction of induced polarization: depends on direction of applied stress. In the above case, P appears in the same direction as applied stress along y. • If the stress is applied along x, A and B are displaced outwards to A’’ and B’’ respectively, resulting in shift of c.m.’s away from each other in y direction  therefore P appears along y direction. • Generally, an applied stress in one direction can give rise to induced polarization in other crystal directions and reversing the stress reverses the polarization  crystals with no center of symmetry exhibit piezoelectricity

6. Piezoelectric Coefficients • If Tj is the applied mechanical stress along some j direction and Pi is the induced polarization along some i direction, we relate them via • Pi = dij Tj • where dij are called piezoelectric coefficients and T can represent either tensile or shear stresses. • An equivalent relation between the strain Sj along the j direction and the electric field Ei along the i direction is given by • Sj = dij Ei

7. Piezoelectricity: Applications • Piezoelectric crystals are essentially electromechanical transducers as they convert electrical signals to mechanical signals, strain or vice versa. • Typical engineering applications: ultrasonic transducers, microphones, sonar detectors, accelerometers, frequency control of oscillators and filters, monitoring of thin film deposition. • Example: In phonographic pick-ups; stylus traverses grooves of record  pressure variation imposed on a piezoelectric material located in cartridge  transformed into electrical signal  amplified and broadcasted through speaker. • - Efficiency of conversion between electrical and mechanical energy is given by the electromechanical conversion factor K defined in terms of K2 by • Common piezoelectric materials include quartz and PZT (lead zirconate titanate) ceramic which is a solid solution of lead zirconate and lead titanate.

8. Properties of some comment piezoelectric materials Music making with Piezos! Quartz watch

9. Ferroelectricity Certain crystals are permanently polarized even without an applied field. The crystal possess a finite polarization vector due to the separation of +ve and –ve charges in the crystal and is said to be ferroelectric. • Above 130oC, crystal structure is cubic unit cell, c.m.’s of –ve charges (O2-) and +ve charges (Ba2+ and Ti4+) coincide at Ti4+ ion  therefore no net polarization and P=0. BaTiO3 is not ferroelectric and exhibits no permanent polarization above 130oC. • Below 130oC, structure of BaTiO3 is tetragonal (cube that has been elongated slightly in one direction). The Ti4+ ion is not located at the c.m. of the –ve charges  crystal has a net polarization P  BaTiO3 is ferroelectric.

10. Ferroelectricity: Curie temperature and Mechanism • Curie temperature: a critical temperature above which the ferroelectric property is lost is termed the Curie temperature, Tc. Below Tc, crystal becomes spontaneously polarized. • Mechanism • Below the Curie temperature, • the Ti4+ ion displaces spontaneously (due to long range interactions between ions outside the simple unit cell) • the slight displacement of the Ti4+ ion along the c direction in each unit cell results in the lowering the energy of the crystal • this elongates the cubic structure which becomes tetragonal • a dipole moment is generated in each unit cell and the interaction energy of all these dipoles when all aligned in the same direction lowers the energy of the whole tetragonal crystal

11. Ferroelectricity: polarization change • Poling • If we apply a temporary field E and let the crystal cool to below 130oC, we can induce the spontaneous polarization P to develop along the field direction, i.e. able to define c axis by imposing a temporary external field. This process is called poling. The c axis is the polar axis along which P develops and is also called the ferroelectric axis. • Polarization: since there is already a permanent polarization (below Tc), • cannot be used. Instead, we have • where ΔP is the change in polarization and ΔE is the change in applied field.

12. Ferroelectricity: some comments • Experiments have shown that an applied field along a axis can displace the Ti4+ ion more easily than that along c axis. r ~4100 along a axis and r~160 along c axis. Due to their high dielectric constants, ferroelectric ceramics are often used as high-K dielectrics in capacitors, thus having significantly smaller sizes compared to capacitors made from other materials. • Other materials that display ferroelectricity include potassium dihydrogen phosphate (KH2PO4), Rochelle salt (NaKC4H4O6.4H2O), lead zirconate titanate (Pb[ZrO3TiO3]), and potassium niobate (KNbO3). • All ferroelectric crystals are also piezoelectric, but the reverse is not true: NOT all piezoelectric crystals are ferroelectric.

13. Ferroelectricity: Ferroelectric crystals are also piezoelectric • When a stress along y is applied to the BaTiO3 crystal, the crystal is stretched along y and the Ti4+ ion becomes shifted. However, there is no shift in the c.m. of the –ve charges. Therefore there is a change ΔP in the polarization vector along y. • When a stress is applied along x, then the change in the polarization is along y. In both cases, ΔP is proportional to stress