# 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:
• Marker/String Methods: In these methods, a discrete parametrized version of the interface boundary is used. In two dimensions, marker particles are used; in three dimensions, a nodal triangularization of the interface is often developed. The positions of the nodes are then updated by determining front information about the normals and curvature from the marker representation. Such representations can be quite accurate, however, limitations exist for complex motions. To begin, if corners and cusps develop in the evolving front, markers usually form ``swallowtail'' solutions which must be removed through delooping techniques which attempt to enforce an entropy condition inherent in such motion (see Sethian, J.A., Curvature and the Evolution of Fronts, Comm. in Math. Phys., 101, pp. 487--499, 1985). Second, topological changes are difficult to handle; when regions merge, some markers must be removed. Third, significant instabilities in the front can result, since the underlying marker particle motions represent a weakly ill-posed initial value problem. Finally, extensions of such methods to three dimensions require additional work.

• Cell-Based Methods: In these methods, the computational domain is divided into a set of cells which contain ``volume fractions'' These volume fractions are numbers between 0 and 1, and represent the fraction of each cell containing the physical material. At any time, the front can be reconstructed from these volume fractions. Advantages of such techniques include the ability to easily handle topological changes, adaptive mesh methods, and extensions to three dimensions. However, determination of geometric quantities such as normals and curvature can be inaccurate.

• Characteristic Methods: In these methods, ``ray-trace''-like techniques are used. The characteristic equations for the propagating interface are used, and the entropy condition at forming corners is formally enforced by constructing the envelope of the evolving characteristics. Such methods handle the looping problems more naturally, but may be complex in three-dimensions and require adaptive adding and removing rays, which can cause instabilities and/or oversmoothing.

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.