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

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.