Understanding the voltage-pressure relationship in DC magnetron sputtering via 2D-RZ PIC simulation and a 1D fluid model

We show that 2D-RZ particle-in-cell (PIC) simulation can accurately reproduce the voltage-pressure (V-P) dependence in DC magnetron sputtering (DCMS). Our PIC simulations capture the monotonic voltage decrease with increasing pressure observed consistently in experiments with constant current. By systematically varying the electron reflection coefficient, we also demonstrate that the previously-accepted explanation based on electron recapture cannot account for the V-P dependence. To explain the V-P dependence, we develop a steady-state 1D-axial fluid model that treats electron transport via Pedersen conductivity. The model reveals that the V-P dependence arises from particle balance in the presheath: as pressure decreases, the presheath plasma density also decreases, so the presheath voltage must increase to energize the electrons, which increases the ionization cross-section, thus maintaining the global ionization rate, which is fixed by the discharge current. The fluid model agrees with both the PIC results and experimental data within 15%, and provides instantaneous results and physical intuition. This work establishes validated tools for DCMS optimization, which may be extended to other ExB discharges, including Hall thrusters and high-power impulse magnetron sputtering (HiPIMS).