Free energy of BaTiO3 single crystal

The thermodynamic description of a ferroelectric material is generally done using the approach of Landau. This latter is based on the development of the free energy around the para/ferro-electric transition temperature. Following the formulation of Devonshire (1949), many authors have tried to improve the description level. In this context, the free energy due to the elastic behavior of the material has been included in models. Despite this progress, the development of the free energy in successive power of the polarization still depends on parameters that must be set in an ad hoc manner. We propose here a formulation of the free energy for displacive ferroelectrics which takes direct account of the polarization but also of the coupling with temperature and of the electrostrictive coupling. We apply this formalism to the case of BaTiO₃ single crystal. A comparison with experiment and with previous works is performed. The proposed approach allows for example to properly account for the evolution of the spontaneous polarization and lattice parameters as a function of temperature in all polar phases.