Simultaneous Potential and Circuit Solution for Two-Dimensional Bounded Plasma Simulation Codes

An algorithm for coupling external circuit elements to be bounded two-dimensional electrostatic plasma simulation codes is developed. In general, the external circuit equations provide a mixture of Dirichlet and Neumann boundary conditions for the Poisson equation, which is solved each time step for the internal plasma potential. We rewrite the coupling between the plasma and the external circuit parameters as an algebraic or ordinary differential equation for the potential on the boundary. This scheme allows decomposition of the field solve into a Laplace solver with boundary conditions (e.g., applied potentials) and a Poisson solver with zero boundary conditions. We present the details of the external circuit coupling to an explicit electrostatic planar two-dimensional particle-in-cell code called PDP2, and discuss briefly how the coupling can be done in an implicit electrostatic code. The decomposition replaces the iterative coupling with a direct coupling and reduces the amount of computational time spent in the field solver. We use PDP2 to simulate a dually excited capacitively coupled RF discharge and show how such a system can be used as a plasma processing tool with separate control over ion flux and ion bombarding energy.