A three‐dimensional numerical method for thermal analysis in X‐ray lithography

Abstract

X‐ray irradiation of photoresists, such as polymethylmethacrylate (PMMA), on a silicon substrate is an important technique in micro fabrication used to obtain structures and devices with a high aspect ratio. The process is composed of a mask and a photoresist deposited on a substrate (with a gap between mask and resist). Predictions of the temperature distribution in three dimensions in the different layers (mask, gap, photoresist and substrate) and of the potential temperature rise are essential for determining the effect of high flux X‐ray exposure on distortions in the photoresist due to thermal expansion. In this study, we develop a numerical method for obtaining the steady state temperature profile in an X‐ray irradiation process by using a preconditioned Richardson method for the Poisson equation in the micro‐scale. A domain decomposition algorithm is then obtained based on the parallel “divide and conquer” procedure for tridiagonal linear systems. Numerical results show that such a method is efficient.