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Spontaneous polarization in Ferroelectrics(In ferroelectrics)
editFerroelectrics are insulating materials characterized by switchable polarization. At equilibrium, a FE material is a non-centrosymmetric structure ie. it lacks an inversion center. This broken symmetry leads to a polarization P at equilibrium also known as spontaneous polarization. However, experimentalists do not have access to the value of spontaneous polarization directly but rely on the macroscopic switching of polarization to determine its value. Most Ferroelectrics have different structures for which the polarizations are equal in modulus but point along the corresponding enantiomorphous(wiki)mirror images of one another but not the same) symmetry direction. In a typical experiment, application of an electric field switches the polarization from P to -P .ie the polarization reversal is due to a change from one enantiomorphous structure to another. By symmetry this difference is equal to twice the spontaneous polarization. A typical ferroelectric is the perovskite oxide PbTiO3 whose equilibrium structure at zero temperature is tetragonal. There are six enantiomorphous structures of which two of them could be involved in polarization reversal. Figure[add] shows a typical hysteresis loop characteristic of experiment. When the system is driven from A to B, the difference (PB−PA)/2 defines the spontaneous polarization in the vertical direction. Theoretically, we can define the same as the polarization difference between the broken symmetry structure at B and the centrosymmetric structure which has no polarization since nuclear displacements are zero. Since the difference of polarization is being measured, we can formulate the following definition of spontaneous polarization as
where λ = 0 corresponds to the centrosymmetric structure and λ = 1 corresponds to the broken structure. One important condition for Eq.() to hold is that the system remain insulating at all times when we vary λ. Otherwise the current being a transient phenomenon will not be well defined. The perturbation corresponds to turning on the sublattice displacements which leads to an adiabatic electronic and nuclear current which flows through the crystal.