Mathematical modeling and experimental study on photoresist whirl-coating in convex-surface laser lithography

In this paper, we present a thickness distribution model of photoresist whirl-coating in convex-surface laser lithography based on hydrodynamics and calculus. A multivariable PDE (partial differential equation) of the photoresist-layer thickness distribution has been worked out in constant rate whirl-coating through analyzing and deducing the flow equation on a convex surface. A mathematical model of the photoresist thickness distribution on a convex-surface substrate with respect to various experimental parameters such as the type of photoresist, the viscosity of the photoresist solution, the rotation speed of the spinner, the geometrical dimension of the substrate and the coating time is presented. Furthermore, a homologous uniformity model of the photoresist layer with respect to the rotation speed and the coating time is established during whirl-coating on a convex-surface substrate. By means of a spinner with a high-precision stepping motor, spectroscopy, ellipsometry and profilometry, some experiments were carried out. The results show that the differences between the theoretical thickness distribution model, the uniformity model and their respective measured values are small. According to the first model, it is possible to acquire an arbitrary thickness distribution of the photoresist layer theoretically. The second model could monitor, modify and improve the uniformity of the photoresist-layer distribution on the convex surface so as to facilitate the subsequent lithography procedure and eventually guarantee the line quality in convex-surface laser lithography.