Magnetron sputtering: determining scaling relations towards real power discharges using 3D particle-in-cell Monte Carlo models

Plasma simulation of glow-magnetized discharges with the particle-in-cell Monte Carlo (PICMC) method is constraint to low current densities because of otherwise huge computational requirements. The present work aims to show, how it is nevertheless possible to extrapolate information for higher current densities similar to realistic lab or industrial conditions by applying a scaling strategy on the simulation. This is demonstrated for a DC magnetron sputtering (DCMS) case study involving the following species: Ar, Ar⁺, Ti, Ti⁺ and electrons. The evolution of the electron density is extracted from the simulation and compared with experimental values obtained with a Langmuir probe. A linear relationship between the electron density and the discharge current is highlighted and explained by studying the reaction rates of both ionization and excitation collisions. This allows to scale the reaction rates with the discharge parameters: the Ar-electron impact ionization and excitation rates scale linearly with the discharge current, while the electron impact ionization rate of sputtered species scales quadratically with the discharge current. The simulations also feature propagating plasma instabilities, so-called spokes, but in average, the above-mentioned scaling laws hold. Consequently, the flux of particles at the substrate during a plasma deposition process at realistic power density can be extrapolated from a 3D PICMC simulation at lower power density. Finally, the validity domain of the scaling strategy is discussed in the light of the model constraints.