리소그래피과정 중 PEB단계의 다중스케일문제에 대한 준 해석적 방법

Abstract

In the post exposure bake (PEB) process of semiconductor production, when heat transfer occurs from one layer to another, there is some heat leakage into the surrounding air. We propose a new mathematical heat conduction model in which the effect of heat loss is added during the PEB process for getting the suitable baking temperature and time and compute it by using a numerical method. Computation of the heat conduction needs high accuracy since the multi-scale thermal diffusivity and thickness of stacked sub-layers of the wafer are involved. The proposed method markedly reduces the number of unknowns. Therefore, it brings cost effective benefits. In addition, systems of Volterra-type integral equations for the unknowns are driven and the integral scheme is stable and robust. Moreover, we reduce the computational cost by using the technique of extracting singularities. These are points of advantage of our method. There are previous works on this topic (Kim et al., 2008 and Oh et al., 2008). The previous papers stated that the PEB wafer stack was insulated from the surrounding air temperature. There was no consideration about heat loss. Finally, we obtain the accurate and efficient results. We demonstrate the reliability of our work with comparison tests between numerical and theoretical solutions. Reliable results are shown for heat conduction to photoresist on top of wafer during the PEB process.