When using radiosity models for simulating the transport of particles
above the wafer in the case where the length of the mean free path is
greater than the size of the feature (cf.
Section 12.3), two operations consume the
most part of the computation time. The first operation is determining
the visibility between all surface elements, which is an
operation, where
denotes the number of surface elements extracted
from the level set grid. The second operation is solving a certain
system of linear equations, which leads to calculating the inverse of
a matrix with
elements, which is an
operation.
Obviously increasing the number of surface elements is not a remedy in cases where high resolution is required. High resolution is needed, e.g., near the trench opening, and the bottom of the trench, and for the simulation of micro-trenching and side wall push back. One approach is to devise a refinement and coarsening strategy for unstructured grids at the level of the level set implementation and the algorithms working on it. This, however, complicates the fast marching algorithm necessary for extending the speed function. Here a different approach was taken by coarsening the surfaces after having been extracted from the level set grid.
The algorithm works by walking down the list of surface elements
extracted as the zero level set and calculating the angle
between two neighboring surface elements. Whenever
is
below a certain threshold value of a few degrees, the neighboring
elements are coalesced into one. After one sweep through the list,
the algorithm can be reapplied for further coarsening. After
coarsening sweeps, at most
surface elements are coalesced into
one. The resulting longer surface elements are used for the radiosity
calculation, after which the fluxes are translated back from the
coarsened elements to the original ones.
Table 12.1 in
Section 12.7 will show the relative speed up that
was achieved in a typical deposition simulation.
Clemens Heitzinger 2003-05-08