A theory of double hysteresis for ferroelectric crystals

Abstract

A ferroelectric crystal is known to exhibit the usual single hysteresis below its Curie point TC, but above this temperature its electric displacement D versus electric field E plot tends to form double loops. We first point out that there is a fundamental difference in the formation of double loops from the single one: the single loop is formed solely by polar reorientation, but in the double loops the right branch of its upper loop is formed by phase transition and only the left branch is formed by polar reorientation (the process is reversed for the lower loop). In this study we take the view that both cubic→tetragonal phase transition and the polar reorientation of ferroelectric domain are thermodynamics-driving process and use this concept to develop a micromechanics-based thermodynamic model to simulate the double hysteresis behavior of the crystal. We first derive the thermodynamic driving force for both spontaneous polarization and domain switch at a given level of temperature, stress, electric field, and new domain concentration c1 and then establish the kinetic equations for domain growth. A dual-phase homogenization theory is then introduced to calculate the overall electric displacement and mechanical strain of the crystal. This approach differs from the classical Landau-Ginzburg-Devonshire theory in at least two significant aspects: (i) it is developed with a micromechanics-based thermodynamics principle, and (ii) it can provide the evolution of new domain concentration. The developed theory is applied to a BaTiO3 crystal. The calculated results show a single loop below its TC and double loops above it but with a diminishing width at higher temperature. Furthermore, the longitudinal strain ε vs E loop is found to exhibit the usual butterfly-shape relation below TC, but above it the loop shows a new, overlapping double-well picture. Good agreement with available test data is also observed.