Finite Element Method for Designing Plasma Reactors

The finite element method has proven to be versatile in the analysis of various electromagnetic phenomena. In this paper, we investigate the utility of the finite element method for guiding the design of plasma reactors. These reactors are used for processing integrated circuit substrates. The design of such reactors is based on the complex interactions between the electromagnetic fields and a plasmafilled region. The reactor geometry, the feed mechanism, and the current state of the plasma determine the electromagnetic fields. This paper investigates the details and difficulties of modeling plasmas within a closed cavity. Introduction The finite element-boundary integral method has been used in the electromagnetics analysis community for more than two decades, especially for electrostatics and magnetostatics analysis. Solution of dynamic problems, such as antenna characterization, radar cross section calculation, and closed cavity analysis, require the introduction of edge-based (also known as vector) finite elements. However, even with the use of vector finite elements, implementations of a hybrid finite element method were scarce and limited in their utility. Hybrid finite element methods utilize a boundary integral to close the finite element mesh by providing the relationship between the tangential electric and magnetic fields on the surface of the mesh. The boundary integral explicitly couples each unknown (or edge) on the surface to all the other unknowns resulting in a fully-populated matrix. Hence, this hybrid finite element-boundary integral (FE-BI) method had limited utility in an era of Intel 386-based computers with up to 32 MB of RAM. Fortunately, for modeling plasma-loaded cavities, the feed coaxial aperture requiring a boundary integral is of limited extent and hence results in a relatively light memory demand. However, the relatively large volume of a plasma reactor results in a large number of unknowns to characterize the electric field within the cavity. This led to a intractable computational demand for average plasma reactor users even five years ago. As computer resources have improved (today, one can purchase a dual processor computer with 1 GB of RAM for less than $10,000) making what was once called a supercomputer available to average users. Significant effort has been spent over the past few years to make user-friendly computer programs that allow significant simulation capability with a minimal user effort. Specifically, computer program designers are utilizing triangular surface meshes and either right prism or tetrahedra meshes to solve rather complex electromagnetic design problems. This paper presents the formulation, problem to be solved, and preliminary results for plasma reactor cavities. Hybrid Finite Element-Boundary Integral Method The FE-BI equations ([1],[2]) for a total electric field formulation may be written as [ ] [ ] ( ) ( ) [ ] ∇ × ⋅ ⋅ ∇ × − ⋅ ⋅ + × ⋅ ×         − ⋅ × × ⋅ ′ = + − ∫ ∫ ∫ ∫ ∫ W W W W W W W W i r j i r j R i R j