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

1 September 1986

journal article
Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Electron Devices

Vol. 33 (9) , 1263-1269
https://doi.org/10.1109/t-ed.1986.22656

Abstract

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 thicknessHas a function of solvent concentration c for transient timetis generally expressed asH(t, r) = K(\Omega t, v(c)^{- 1/2} \Omega^{1/2}r) v(c)^{1/2}\Omega^{-1/2}, Kbeing a dimensionless function. Herer; v; \ΩandRare 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.

Keywords

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