Heat Conduction to Photoresist on Top of Wafer during Post Exposure Bake Process: I. Numerical Approach

Abstract

The post exposure bake (PEB) of a chemically amplified resist is one of the key processes for fabricating very small features of semiconductor devices. The use of photogenerated acid enables the de-protection of protected polymers, and this de-protection highly depends on PEB temperature and time. The diffusion length of acid is also strongly dependent on PEB temperature and time. As the line width of a device decreases, a smaller diffusion length is required to reduce the roughness of the line edge and width, and an acid diffusion length less than 20 nm is needed. One of the key factors for determining de-protection and acid diffusion is the initial temperature rise of the resist. The unpredictable temperature rise to the preset temperature mainly causes a variation in line width. In addition, in order to accurately predict the PEB temperature and time dependencies of line width, heat transfer from the hot plate to the resist on the top of a silicon wafer has to be calculated since reaction and diffusion occur inside the resist, not on the top of the bare silicon wafer. Heat transfer includes multiscale conductivity and thickness, so that we need an accurate and reliable approach. For this purpose, a novel numerical approach incorporated with analytic method is proposed to solve the heat conduction problem. Since this approach is incorporated with an analytic method, the number of unknowns can be markedly reduced. Indeed, only the interface temperatures are unknowns in this method and we can derive a system of Volterra-type integral equations for the same number of unknowns. Accordingly, this method has many advantages over other methods. Since it is not a difference method but an integral method, it is stable and robust in time step. The unknowns for temperature are located only at the interfaces between layers, so that this approach is fast and effective. The discretization in time variable is flexible enough to readily achieve the accuracy of the numerical solutions over time, i.e., in both ranges of short and long times. In this paper, we calculated the time consumed for the resist to attain the prescribed PEB temperature. Additionally, the effects of the different layer stacks and thicknesses are also investigated through our numerical approach.