Equivalent modeling technique for predicting the transient thermomechanical response of optical reticles during exposure

To extend optical lithography technology to the sub-65 nm linewidth regime, all mask-related distortions must be eliminated or minimized. Thermal distortion during the scanning exposure process can be a significant contribution to the total pattern placement error budget for advanced photomasks. This paper presents an equivalent finite element modeling technique for use in predicting the transient thermal and structural response of an optical reticle during exposure. The equivalent model significantly reduces the computational time required to calculate the thermomechanical response. It is a fundamental approach that can be used in similar problems. Full three-dimensional finite element heat transfer and structural models are developed to simulate both the “actual” and the “equivalent” scanning processes. The results from the actual models and the equivalent models are compared for a test case and it is found that both models predict virtually the same results. The equivalent models are subsequently used for predicting the transient and periodic steady-state temperature and distortion distributions for typical exposure duty cycles.