Accurate semiempirical analytical formulas for spontaneous polarization by crystallographic parameters of SrTiO₃-BaTiO₃ system by ab initio calculations

Spontaneous polarizations (P_S’s) of BaTiO₃ and SrTiO₃ under various conditions are calculated ab initio using different exchange-correlation functionals. The extensive theoretical sets of P_S vs. ion positions are found to lie on a single curve, despite the chemical differences and the wide variations of P_S and lattice parameters. This uncovers accurate simple analytical formulas of P_S of SrTiO₃-BaTiO₃ system expressed by ion positions; a single formula predicts both macroscopic and atomic-scale P_S of SrTiO₃, BaTiO₃ and SrTiO₃-BaTiO alloys. The accuracy of the formula is demonstrated by the application to experiments, BaTiO₃-SrTiO₃ (-CaTiO₃) alloys, Sr₄Ti₄O₁₂ with P_S // a-axis, a parallel domain, and a headon domain. In addition, the present results verify empirically that oxygen displacement is the primary identifier and the origin of P_S of SrTiO₃ and BaTiO₃ and indicate that BaTiO₃ and SrTiO₃ may transforms into new state by an extremely large strain, e.g., −3%. Furthermore, the earlier prediction of headon domain without aid of defects was confirmed. The present procedures for finding formulas can be applied to other materials.