In this section we present some three-dimensional topography simulation results which serve to get insight into the efficiency and time consumption of the simulator.
We begin with a deposition of the material into a rectangular trench shown in Figure 10.1. The transport of the particles is based on the radiosity model. Figure 10.2 shows the simulation result of this material deposition from a plane located above and parallel to the trench leading to the void formation, because of shading effects due to the visibility determination between surface elements and source plane, and between surface elements themselves. Since the angle of visibility of the source is smaller at the bottom of the trench compared to the top of the trench, more material is deposited at the top of the trench. This in turn leads to the effect that the visibility angle of the source plane at the bottom is decreased more and more as the material deposition continues, until the top left and right sides of the trench come together and form a void.
The next example considered is a straightforward simulation of isotropic
etching of the same trench. The simulation is easily performed by setting
the speed function
at each grid point, where
is
negative. As expected, the sides of the trench are etched away cleanly and are
rounded as shown in Figure 10.3.
Finally, Figure 10.4 shows directional etching of the same trench [15]. The total flux at the surface element has been assumed to be a cosine function of the angle between the surface normal and the normal vector of the source plane. Because of considering a mathematical function for the speed function, the visibility and reflection calculation parts of the simulator are turned off. These parts are needed when calculating the flux by using physical models. The trench has been etched less compared to the selective isotropic etching at the sides and tends to be etched more in vertical direction.
Table 10.1 shows a comparison of the simulation times of these different simulation
processes for different grid
resolutions [13]. The most time consuming
simulation is the deposition simulation, because all
expensive steps, e.g., visibility determination, extension of the speed
function, and the iterative solver
are required. For directional etching, extension of the speed function is the only
time consuming part of the simulation. Therefore, the
simulation time is considerably smaller than that of the deposition process. For isotropic etching neither visibility
determination, nor the iterative solver, nor an extension of the speed
function are required. Thus the simulation time
is very small compared to the other simulations. In the third column of Table 10.1 the simulation times for a
grid with twice the resolution of the original resolution are presented. For
instance, the deposition time has increased by about a
factor of 31 when doubling the grid resolution from
3030
30 to 60
60
60. This is in
accordance with the expected minimal factor which we have
estimated in Section 5.13. There we have discussed that
doubling the grid resolution approximately increases the number of
surface elements by a factor of 4 which in turn increases the complexity of the
visibility determination by the radiosity model by a factor of 16. Therefore the simulation time has increased
by a factor
which comes up to the
expectation of a minimum factor of
.
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