Electron cumulative distribution function in a magnetron sputtering discharge

This study investigates electron kinetics in a direct current magnetron sputtering discharge, advocating for the use of the cumulative distribution function (CDF) as a superior alternative to conventional probability distribution functions (PDFs), such as the electron energy distribution function. A three-dimensional simulation model is employed to track electron dynamics under balanced and two unbalanced magnetic field configurations. The analysis contrasts two critical phases: a prebreakdown stage with a vacuum electric field (Plasma Off) and a sustained discharge stage incorporating a strong sheath potential (Plasma On-). Results from the PDF analysis confirm that plasma formation profoundly modifies the electron distributions, leading to a substantial increase in the low-energy electron population and a general rise in the effective electron temperature due to sheath acceleration. Among the tested topologies, the unbalanced typeI (UB1) configuration consistently generates the most energetic electrons. However, the CDF provides a more intuitive and quantitatively robust analysis, free from binning artifacts. The rightward shift of the electron energy CDF (EECDF) for the UB1 configuration unequivocally confirms its higher median and mean energies. This approach reveals nuanced dynamics obscured by PDFs, such as a decrease in mean energy for the UB1 case upon plasma formation, while it increased for the other configurations. Furthermore, the CDF allows for precise quantification of distribution skewness; for instance, the fraction of electrons with energy below the mean in the UB1 configuration increased from approximately 60% in the prebreakdown phase to 80% in the sustained discharge. This work establishes the CDF as a powerful tool for characterizing non-Maxwellian plasmas, offering unambiguous insights into electron properties critical for optimizing magnetic confinement and process control in plasma-based technologies.