Mathematical model of Boltzmann’s sigmoidal equation applicable to the set-up of the RF-magnetron co-sputtering in thin films deposition of $\hbox {Ba}_{{x}}\hbox {Sr}_{1-{x}} \hbox {TiO}_{3}$

Abstract

In this work, we present the stoichiometric behaviour of $\hbox {Ba}^{2+}$ and $\hbox {Sr}^{2+}$ when they are deposited to make a solid solution of barium strontium titanate. $\hbox {Ba}_{{x}}\hbox {Sr}_{1-{x}} \hbox {TiO}_{3}$ (BST) thin films of nanometric order on a quartz substrate were obtained by means of in-situ RF-magnetron co-sputtering at 495${^{\circ }}$C temperature, applying a total power of 120 W divided into intervals of 15 W that was distributed between two magnetron sputtering cathodes containing targets of $\hbox {BaTiO}_{3}$ and $\hbox {SrTiO}_{3}$, as follows: 0–120, 15–105, 30–90, 45–75, 60–60, 75–45, 90–30, 105–15 and 120–0 W. Boltzmann’s sigmoidal modified equation (Boltzmann’s profile) is proposed to explain the behaviour and the deposition ratio Ba/Sr of the BST as a function of the RF-magnetron power. The Boltzmann’s profile proposal shows concordance with experimental data of deposits of BST on substrates of nichrome under the same experimental conditions, showing differences in the ratio Ba/Sr of the BST due to the influence of the substrate.