Vacancy modeling in lead titanate and lead zirconate titanate

Precise prediction of lattice constants is critical for various fields including lattice matching for heteroepitaxy. The exact nature of A-site vacancies in lead titanate, lead zirconate titanate and perovskites generally remains somewhat unclear. In this study, it has been shown experimentally via Le Bail refinements of x-ray diffraction data that such vacancies in leadbased perovskites have an effective size which is a function of their concentration. In the case of lead titanate perovskites, the effective size increases with concentration. This effect has been modeled mathematically with great precision (R² = 0.99995); however, A-site vacancies in lead zirconate titanate perovskites have an effective size which decreases as their concentration increases. This effect has also been modeled mathematically (R² = 0.99993). A simultaneous decrease in the effective size of oxygen ions, ostensibly due to the slight decrease in their coordination, was also observed and quantified in both systems. When both effects are considered, cell volume can be predicted from stoichiometry and published ionic radii data alone to within ∼0.2% accuracy for Pb_1-3xLa_2x□_x(Ti_1-yZr_y)O₃ x≤ 0.15, y≈ 0.6 and < 2% all the way to x = 1/3, y = 0. The model may eventually provide a template for substrate engineering and facilitate property tailoring (magnetic, dielectric, and other) based on improved structure predictions.