A hybrid finite element‐finite difference method for thermal analysis in X‐ray lithography

Abstract

X‐ray lithography is an important technique in micro fabrication used to obtain structures and devices with a high aspect ratio. The X‐ray exposure takes place in a system 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 three‐dimensional numerical method for obtaining the temperature profile in an X‐ray irradiation process by using a hybrid finite element‐finite difference scheme for solving three‐dimensional parabolic equations on thin layers. A domain decomposition algorithm is then obtained based on a parallel Gaussian elimination for solving block tridiagonal linear systems. The method is illustrated by a numerical method.