Transient behavior of spin-coated resist film thickness based on invariance of viscous fluid under similarity transformation

The spin-coating method that has been used as a standard procedure in the semiconductor industry is investigated theoretically. A phenomenological model is derived to describe transient behaviors of the resist film thickness on a rotating circular wafer. The model is constructed by leaving invariant the Navier-Stokes and continuity equations that describe the viscous resist fluid under the similarity transformation of variables. The film thickness as a function of solvent concentration c for transient time is generally expressed as being a dimensionless function. Here and are the two-dimensional space vector in the wafer, viscosity, rotational speed, and wafer radius, respectively. This relationship is used to yield the transient behaviors of the film thickness that is described by a simple partial differential equation similar to a Fokker-Planck equation by assuming azimuthal symmetry in the wafer. By applying the singular perturbation technique, we obtain an analytic approximate solution that exhibits various important features such as a time development of the film thickness. The calculated film thickness is qualitatively consistent with experiments.