Two-dimensional analytical description of the plasma potential in a magnetron discharge

Simple analytical formulas are proposed to describe the plasma potential in a steady-state magnetron discharge, based on the results of various experiments and numerical simulations reported in the literature. The description is two-dimensional (2D), covering two main regions, the cathode sheath and the ionization region, both contributing to electron energization. A parabolic potential in the axial direction governs the cathode sheath. The thickness of the cathode sheath is obtained from the 1D collisionless Child-Langmuir law. A parabolic or linear potential in the axial direction characterizes the ionization region. The local ion current density to the cathode, estimated from the target erosion profile, sets the radial dependence of the potential. The proposed formulas use a set of input parameters that can be experimentally obtained. The analytical description captures all characteristics of the highly inhomogeneous plasma potential of a steady-state magnetron discharge operated in a reduced magnetic field B_RT/p lower than 0.1 T/Pa, as revealed by the comparison to self-consistent 2D numerical simulations.