Numerical Simulation of Boundary Condition Dependence of Mask Pattern Placement Error by Finite Difference Method in Steady State

摘要

A photomask for semiconductor device production faces serious problems related to the thermal displacement of the mask pattern during exposure. During mask exposure, pattern placement error occurs in the mask. The minimization of thermal in-plane displacement is needed. To estimate thermal displacement, a model based on the three-dimensional steady-state thermal theory and two-dimensional in-plane stress theory was proposed by the finite difference method. Three types of boundary condition, that is, a free edge, a fixed edge, and a combination of a free edge and a fixed edge were investigated to minimize mask pattern placement error. Quantitative relations of the boundary condition and thermal displacement were clarified. It was found that the fixed edge minimizes pattern placement error caused by thermal expansion. Moreover, the combination of the free and fixed edges indicated the directionality of the placement error distribution.