A two-dimensional fluid model for an argon rf discharge

Abstract

A fluid model for an argon rf discharge in a cylindrical discharge chamber is presented. The model contains the particle balances for electrons and ions and the electron energy balance. A nonzero autobias voltage is obtained by imposing the condition that the time-averaged current toward the powered and grounded electrode is zero. Particle densities and ionization profiles peak strongly in front of the smaller, powered electrode. There electric fields are stronger and the electron current density is higher, resulting in more ohmic heating and therefore higher ionization rates. The radial uniformity of the plasma in front of the powered electrode gives a homogeneous ion flux toward this electrode. The asymmetric character of the profiles of the cylindrical geometry is in clear contrast with the essentially one-dimensional infinite parallel-plate geometry, which is fully symmetric with respect to the center of the discharge and has a zero dc autobias voltage. A comparison with results of a one-dimensional model shows that the average ion density, the average ion flux, and the average ionization rate in the cylindrical reactor are comparable to those in a parallel-plate reactor. The numerical treatment of the time evolution of the transport equations and Poisson’s equation needs an implicit method to avoid numerical instabilities. The resulting system of discretized equations is solved by a multigrid technique. The spatial discretization uses the Sharfetter–Gummel scheme.