Calculation Of Light Scatter From Structures On Silicon Surfaces

Abstract

This paper demonstrates numerical solutions of Maxwell's equations for laser diffraction and scattering from submicron objects on silicon wafers. Discrete numerical solvers using time domain, explicit finite element (FE) and finite difference (FD) methods have been implemented and vectorized for supercomputers. Both 2-D and 3-D FE and FD simulations have been performed for fibers and spheres (.5 to 1.6 microns) in free-space and on bare silicon, illuminated at normal incidence by a He-Ne laser beam. The wave solvers were verified against exact solutions for cylinders and spheres in free-space. Laboratory scattering experiments have been conducted with latex spheres on bare silicon, and the results compared to the corresponding numerical simulations. Simulations of glass and silicon fibers on a layer of photoresist over silicon examined differences in exposure patterns as well as the implementation of bleaching in nonlinear calculations. The results indicate that there is no inherent limitation to rigorously simulating many of the optical problems associated with IC fabrication, inspection, and design. The idea here is not to obtain just "exact" solutions for unsolved problems, but rather to provide the analyst with a broader capability for numerical experiments and simulations. In this way computers and mathematical algorithms can augment ad hoc approximations and laboratory experiments. The ultimate goal of this research is to simulate imaging systems and resist exposure patterns, and incorporate some of these results in process simulation codes like SAMPLE.