Modeling of the chemical x-parameter in pseudobinary alloys from the shift of the x-ray bragg angle: Application to cubic Ba_xSr_1-xTiO₃ films

Pseudobinary alloys of the type A_xB_1-xC are expected to obey the Vegard’s law that states a linear relationship between the chemical parameter x and the lattice parameter. However, in many cases, deviations from the Vegard’s are found, and the calculation of x requires compositional characterization, often complicated in the presence of secondary phases of the same material. In the present work, Ba_xSr_(1−x)TiO₃ (BSTO) films were deposited by RF co-sputtering from BaTiO₃ and SrTiO₃ targets to obtain differently Ba⁄Sr ratios considering the different RF-power applied to each target. We describe a mathematical model based on the Boltzmann equation to calculate the x parameter of the cubic phase of the BSTO films in the 0 ≤ x ≤ 1 range. That was made by following the 2Θ diffraction angular shift of the plane (111) upon the increase of substitutional Ba²⁺ in the pseudobinary alloy, as a function of the RF applied power and fitting the interplanar distances and lattice parameter in the alloys were calculated onto the fitted sigmoid trajectory. The model showed the region where the lattice parameter a vs x obeys the Vegard’s law and correctly fitted “a” in the entire 0<x<1 range. The model can be applicable to other pseudobinary systems with deviations from the Vegard’s law, from a simple X-ray diffraction analysis, providing fast chemical information, complementary with further compositional characterization.