A dual-phase homogenization theory for the hysteresis and butterfly-shaped behavior of ferroelectric single crystals

Based on the mechanism of domain switch and a micromechanics-based thermodynamic approach, a dual-phase homogenization theory is developed to calculate the hysteresis loops of ferroelectric single crystals. Unlike the classical phenomenological theories, this dual-phase model is capable of delivering the evolution of new domain concentration, and it further indicates that the driving force for domain switch under a pure electric field increases linearly with the applied E. The theory is then applied to a BaTiO₃ crystal, to examine its hysteresis loop between the electric displacement D vs. the field E, and the butterfly-shaped longitudinal strain ε vs. E. Two types of switching process have been employed for the reversal of ferroelectric domains: a direct 180° switch for the D vs. E, and a two-consecutive 90° switch for the ε vs. E as well as for the D vs. E. The second process enables one to bring about the sharp tails of the butterfly in the ε vs. E curve that otherwise could not be done by the direct 180° switch. Comparisons with the experimental data of a BaTiO₃ crystal also indicate a reasonable agreement.