Mathematical model of Boltzmann's sigmoidal equation applicable to the set-up of the RF-magnetron co-sputtering in thin films deposition of Ba_xSr_1-xTiO₃

In this work, we present the stoichiometric behaviour of Ba²⁺ and Sr²⁺ when they are deposited to make a solid solution of barium strontium titanate. Ba_xSr_1-xTiO₃ (BST) thin films of nanometric order on a quartz substrate were obtained by means of in-situ RF-magnetron co-sputtering at 495°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 BaTiO₃ and SrTiO₃, 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.