Numerical Issues Involved in Topography Evolution in Etching and Deposition

Within topographic evolutions of etching and deposition, a number of typical problems require attention. These problems include the formation of sharp corners and cusps, the effect of angle-dependent flux functions on propagation rates, topological change, both from splitting and merging, and complexities in three space dimensions. A variety of numerical algorithms are available to advance fronts in etching, deposition and photolithography processes. These methods are not unique to such simulations, and in fact are in use in such areas as dendritic growth and solidification, flame/combustion models, and fluid interfaces. Roughly speaking, they fall into three general categories:

Level set methods, introduced by Osher and Sethian (Osher, S., and Sethian, J.A., Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton--Jacobi Formulations, ournal of Computational Physics, 79, pp. 12--49, 1988), offer a highly robust and accurate method for tracking interfaces moving under complex motions. Their major virtue is that they naturally construct the fundamental weak solution to surface propagation posed by Sethian (see {Sethian, J.A., Level Set Methods, Cambridge University Press, 1996). They work in any number of space dimensions, handle topological merging and breaking naturally, and are easy to program.

These techniques work by approximating the equations of motion for the underlying propagating surface, which resemble Hamilton-Jacobi equations with parabolic right-hand sides. The central mathematical idea is to view the moving front as a particular level set of a higher dimensional function. In this setting, sharp gradients and cusps are easily tracked, and the effects of curvature may be easily incorporated. The key numerical idea is to borrow the technology from the numerical solution of hyperbolic conservation laws and transfer these ideas to the Hamilton-Jacobi setting, which then guarantees that the correct entropy satisfying solution will be obtained.
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