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The deposition rate was 0.07/s and the deposition time 120s. In total
cells were used for the simulation domain of
. The simulation was performed with
two time-steps of 60s. All calculations were done on a 200MHz DEC Alpha
workstation.
Table 2.1 summarizes the calculation times obtained with the different algorithms. Two major observations are obtainable from the figures: Firstly, both spherical segments (b) as well as line elements (c) lead to an acceleration in the range of two orders of magnitude. Thereby the spherical segments algorithm (b) is the faster one. The second information which can be derived is that the line element algorithm is rather independent on the complexity of the geometry, which differs between the two time-steps. The required simulation time for the two time-steps is comparable for the line algorithm (c), whereas it doubles for the second time-step of the spherical segments simulation (b).
Let's consider the number of operations needed for the different methods. The
three-dimensional case is investigated, since this is the challenging
part. The considerations for the two-dimensional case are similar. Roughly
speaking each exponent in the following equations can be diminished by one
(dimension) for the two-dimensional analogon. We assume a cubic simulation
domain with unit length. All three methods have in common, that they finally
have to scan the whole material array and to switch the index of the marked
cells. The number of required operations is proportional to the cell resolution
used for the simulation. If is the number of cells per unit length,
switching operations are necessary for our assumptions. Fortunately this
has to be done only once per time-step. Suppose a grid of
cells. Switching all
cells does not really take a long
time on state-of-the-art processors running at clock frequencies of 500MHz
(
instructions per second). This switching of the indices is not the
limiting factor. Moreover it is common to all three methods and therefore
omitted in the following considerations.
Before switching the marked cells it is necessary to decide which
ones have to be marked. This is done with the structuring element applied at
each surface cell. Basically, there are about surface cells needing
scanning operations in their vicinity. Using the original implementation
it has to be tested for
cells in the vicinity of a surface cell whether
they are less than
away from
the considered surface cell.
is the local etch or deposition rate which
determines the extension of the structuring element and
is the diameter
of the structuring element given as
in number of cells. This makes in sum
operations. For the assumed 200 cells per unit length and a
rate
resolved by
cells this makes already
operations. These are almost 20 times more operations than
necessary for the index-switching.
For the spherical segment method the number of scanning operations is reduced
from to simply
for line elements at surface cells in
a plane parallel to the planes of the coordinate system, to
for two-dimensional quarter-circle elements at edge cells,
and to
for arbitrary located cells. Here enters the
complexity of the geometry resulting in a combined overall exponent
depending on the ratio between plane, edge, and corner cells contained in the
actual structure. The overall equation for the number of operations derives to
, with
. The prefactor of
is kept, since
has the biggest influence on the overall
acceleration. Again, for the worst case the acceleration factor is 8 (eights of
spheres instead of spheres) with
. With
, which is a very
cautious assumption, the number of operations is reduced to
,
which is less than the number of index switching operations and approximately
120 times less than the
operations needed for scanning the
complete spheres. With a rough estimation the benchmark example rather suggests
an exponent of
for the first time-step on the initial structure
which consists only of vertical and horizontal planes with sharp corners and
of
for the second, already more complex geometry including curved
regions. (2b) in Table 2.1 is about 125 times faster than (1b) for
. This is in very good agreement with the factor of 120 derived above
for
assumed as 2. It explains that the algorithm gives an acceleration in
the range of two orders of magnitude and that the second time-step takes twice
as long as the first one, which is due to the more complex geometry for the
second time-step, resulting in a higher exponent
.
Now let's consider the situation for the line algorithm. Again there are
surface cells, but now only
operations are needed for the
linear structuring elements applied at each surface cell. Still, two terms are
missing for this case. Firstly, this method needs interpolation at convex
corners. For each surface cell located at a convex corner about
interpolation operations are
necessary in the implementation used for this comparison. This includes the
operations from above.
is the angle between the
surface normals of two planes forming a convex corner and
is the
angular step width of the interpolation necessary to maintain a smooth
surface. The second factor is the calculation of the surface normals, which
requires
operations for summing up the vectors of the cube face normals
in the vicinity of the surface cells. This makes in sum about
operations if
is assumed to be the ratio between convex surface cells and the total
number of surface cells. For the numerical assumptions of
,
,
resolved by
cells,
, and
the time needed for the simulations taken from Table 2.1,
can be
estimated as 0.15 for both time-steps by a simple division assuming a constant
time per operation throughout all simulations.
Taking a look at Fig. 2.6 clearly reveals, that cannot be similar
in the two time-steps as derived in the above estimation. For the first time
step only the upper rim of the cylindrical via has convex surface cells with two
exposed cube faces, whereas for the second time-step, all surface cells forming
the curvature around the initial rim have to be considered as convex corners in
terms of surface cells. In the latter case the bottom and sidewall areas of the
feature become smaller, which saves linear operations without interpolation and
screens the proper calculation of the ratio
. The reason therefore is
the simplified assumption of
surface cells, which does not consider
vertical sidewalls and curved regions. Additionally, this simplified assumption
explains, why this algorithm shows less dependence on the complexity of the
structure, since the increasing number of convex corner cells is partially
compensated by the shrinking size of the feature. However, the derivation of
the required number of operations is applicable to the estimation of the
acceleration which can be gained with this method.
A further remark is necessary for the line element algorithm. It turned out, that the linear structuring elements imply two contradicting requirements. For minimizing the discretization error, the lines are always applied at the center of an exposed cube surface, whereas for representing the correct growth direction they have to be applied in the center of the surface cell. Since only one origin of the structuring element is possible, this leads to the consequence, that the final geometry obtained with this method slightly differs from the results from the previous methods. The discrepancy prevents the method from being applied to isotropic or anisotropic models. Nevertheless, the line algorithm is used in our cellular topography simulator, namely, for cases where the rate vector is not given by the surface normal but by the directionality inherent to the process to be modeled. Line elements are therefore implemented for unidirectional etching and deposition as well as for sputter deposition where the rate is given as vector integral of the particle distribution function over the visible solid angle. A detailed explanation of this implementation will be given in Chapter 6.
With regard to CPU time, the surface orientation dependent linear element
algorithm could be the method of choice if additional surface information is
given, e.g., in a triangular format, and has not to be calculated internally
with high computational costs. In this way the dependence could be
discarded from the estimation of the number of required operations. Considering
overall performance taking into account simulation time, accuracy, and the lack
of information about surface orientation, the spherical segment algorithm is
suited best for the application to isotropic etching and deposition steps
within the cellular data representation. Nevertheless, linear structuring
elements find application where the direction of the lines is not determined by
the surface normal but by the directionality of the etching or deposition
process, which is the case for unidirectional etching and deposition
(cf. Section 3.2.3) and for sputter deposition (cf. Section 6.1.5).
These considerations conclude the introduction of the cellular structuring element approach and its recent developments. How the improved algorithms are applied to different models for etching and deposition processes will be demonstrated later in this thesis. Basic models will be found in Section 3.2, the description of more complex applications starts in Chapter 6. However, before starting etching and deposition simulations we need some input geometries on which to apply the processes, i.e., we need some silicon and a method to apply patterns onto the wafers. This is absolutely necessary in order to obtain selective etching and deposition forming three-dimensional structures on the wafer and needed for the simulation in the same way as for manufacturing of the desired devices. So let's first address the generation of three-dimensional geometries.
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