Mathematical methods for thin film deposition simulations

Abstract

We have discussed the theory of moving solid boundaries and numerical algorithms to simulate them. In Sec. II, the problem description (i.e., Hamilton-Jacobi type equation and entropy conditions) is presented. We have emphasized the fact that the Hamilton-Jacobi type equation (and therefore the ray-tracing method or method of characteristics) in itself cannot determine the physically plausible solid surface evolution. It is important to recognize that the additional physical mechanism (i.e., infinitesimal surface diffusion) is needed to derive the entropy conditions, which together with the Hamilton-Jacobi type equation determine the physical solution. A geometric method to construct the physical solution is presented in Sec. II.F, based on the entropy conditions of Eq. (20) or Oleinik construction. The shock-tracking algorithm ( 1 ) presented in Sec. III explicitly uses this geometric construction method to determine the correct moving boundary solution. Earlier work on etching/deposition topography simulation focused primarily on the method of characteristics or ray tracing ( 26-31 ), but paid little attention to the entropy conditions. Without the entropy conditions, however, some other procedures must be invoked to avoid seemingly unphysical solutions ( 32,33 ). In many cases, such procedures are introduced based on the modeler's intuition of what the correct physical solutions should be, rather than additional physical processes such as surface diffusion. In the latter half of this article, we have treated more specific applications, that is dry processes for etching and deposition, and presented some representative examples. The spatial scale treated in these problems is mesoscopic, that is, sufficiently smaller than macroscopic scales such as mean free paths of gaseous atoms, but sufficiently larger than molecular scales. For semiconductor dry processes, the representative scale for microfeatures is typically 1 or less, in which one can still approximate the solid by a continuum, rather than aggregates of molecules. In this mesoscopic continuum model, ion and neutral fluxes impinging on the surface are also treated as continuums. Under some realistic assumptions, such fluxes are easily evaluated, as shown in Sec. IV. For simplicity, we have only discussed the two-energy/single-species model, but extending this model to a multienergy/multispecies model is straightforward. Since the goal of this article is to present the basic theory of moving boundary problems in dry processes, no attempt was made to discuss details of various processes used in industry. For instance, readers interested in more details of the examples shown in Sec. V are referred to Refs. 4, 5, 19, and 20 .