Stability, Evolution and Switching of Ferroelectric Domain Structures in Lead-free BaZr$_{0.2}$Ti$_{0.8}$O$_3$-Ba$_{0.7}$Ca$_{0.3}$TiO$_3$ System: Thermodynamic Analysis and Phase-field Simulations

Enhanced room-temperature electromechanical coupling in the lead-free ferroelectric system $(1-x)$BaZr$_{0.2}$Ti$_{0.8}$O$_{3}$ - $x$Ba$_{0.7}$Ca$_{0.3}$TiO$_{3}$ (abbreviated as BZCT) at $x=0.5$ is attributed to the existence of a morphotropic phase region (MPR) containing an intermediate orthorhombic ($O$) phase between terminal rhombohedral ($R$) BZT and tetragonal ($T$) BCT phases. However, there is ambiguity regarding the morphotropic phase transition in BZCT at room temperature - while some experiments suggest a single $O$ phase within the MPR, others indicate coexistence of three polar phases ($T+R+O$). Therefore, to understand the thermodynamic stability of polar phases and its relation to electromechanical switching during morphotropic phase transition in BZCT, we develop a Landau potential based on the theory of polar anisotropy. Since intrinsic electrostrictive anisotropy changes as a function of electromechanical processing, we establish a correlation between the parameters of our potential and the coefficients of electrostriction. We also conducted phase-field simulations based on this potential to demonstrate changes in domain configuration from single-phase $O$ to three-phase $T+R+O$ at the equimolar composition with the increase in electrostrictive anisotropy. Diffusionless phase diagrams and the corresponding piezoelectric coefficients obtained from our model compare well with the experimental findings. Increase in electrostrictive anisotropy increases the degeneracy of the free energy at ambient temperature and pressure leading to decreasing polar anisotropy, although there is an accompanying increase in the electromechanical anisotropy manifested by an increase in the difference between effective longitudinal and transverse piezo-coefficients, $d_{33}$ and $d_{31}$.