Analytical solutions for the deformation of a photoresist film

Modern photoresists shrink during exposure and post-exposure bake processes. The shrinkage is of an elastic nature, occurring below the glass transition temperatures when material does not flow. This elasticity problem is usually modeled numerically using finite element method. Here this problem is solved analytically for a one-, two-, and three- dimensional distribution of displacements. Exact analytical solutions in the form of Fourier series by the lateral film coordinates were found. The film shape and lateral displacements after shrinking are found to be convolutions on the distribution of voids that are formed after evaporation of volatile chemicals. Convolution is a fast numerical algorithm easily incorporated into compact process models for optical proximity correction and applied to the full chip processing. Photoresist shrinking effects are analyzed using this apparatus for modern bright and dark tone processes.