A Criterion for Case II Diffusion in Resist Development by Dissolution

The development of a polymeric resist by dissolution may be described as a two-step process (1). First, the smaller solvent molecules must diffuse into the polymer matrix. The resulting increase in internal pressure causes the rigid, interpenetrating polymer molecules to loosen and swell. The increased mobility of the polymer mole-cules typically causes an i crease in the mobility of the sol-vent molecules, in which case the diffusive penetration of the solvent will autoaccelerate. Second, the polymer mole-cules become sufficiently separated from each other to dif-fuse into the solvent. Since the mobility of the solvent molecules depends upon both the concentration of the solvent and the length of time the polymer molecules have been loosening, the diffusion step generally cannot be described by Fick's law unless the diffusion coefficient has both a concentra-tion and history dependence. There a other effects which add to the difficulty of developing a widely applicable, theoretical model of the development process, such as sur-face induction (2) and adhesion (3). Because of the compli-cation arising from these effects, the development process is usually described by an empirical model in which a bulk development rate is modified to include the necessary depth dependence (3-4). One type of development behavior that appears to be very non-Fickian is Case II or relaxation-balanced diffu-sion, in which the solvent polymer interface travels into the polymer at a constant velocity with little diffusion of solvent beyond the interface (Fig. 1). The dissolution of PMMA by MIBK, a resist-solvent combination frequently used in electron-beam lithography, is probably the most important example of this behavior. Model and Solution For Case II diffusion, a one-dimensional model of the solvent-polymer system will be considered (Fig. 2). The solvent concentration c in the polymer is described b