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 angledependent 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. 487499, 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 illposed initial
value problem.
Finally, extensions of such methods to three dimensions require
additional work.

CellBased 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, ``raytrace''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
threedimensions 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 CurvatureDependent Speed: Algorithms Based on
HamiltonJacobi Formulations, ournal of Computational Physics, 79,
pp. 1249, 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 HamiltonJacobi equations with parabolic righthand 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 HamiltonJacobi setting, which then guarantees that the
correct entropy satisfying solution will be obtained.
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