An Orthogonal Collocation Technique for Rapid Thermal Processing System Discretization

Abstract

A model of a multiple heating zone Rapid Thermal Processing (RTP) system is developed to study wafer thermal dynamics during a processing cycle. The system is discretized with trial functions generated from the linearized wafer energy balance equation eigenfunctions, and careful analysis of the solution residual reveals a slow, but predictable, convergence rate. A modified set of trial functions is derived from a subset of the original eigenfunctions combined with the dominant modes identified by the Karhunen-Loeve expansion of the wafer temperature variance component that contributes most to the slow convergence. Since the wafer temperature variance is computed explicitly from an eigenfunction expansion solution of the linearized system with specified processing statistics, the collocation procedure effectively links RTP model reduction and simulation in one discretization procedure. The convergence rate of the modified collocation method is shown to be superior to collocation methods based on the original eigenfunction and polynomial sequences.