10. About Efficiency

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 $F(t,\mathbf{x})=F$ at each grid point, where $F$ 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 30$\times$30$\times$30 to 60$\times$60$\times$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 $\frac{47.4}{1.54}=30.77$ which comes up to the expectation of a minimum factor of $2^{4}=16$.

Figure 10.1: Initial boundary.

Figure 10.2: Void formation after a deposition process into the initial boundary shown in Figure 10.1.

Figure 10.3: Isotropic etching the initial boundary shown in Figure 10.1.

Figure 10.4: Directional etching the initial boundary shown in Figure 10.1.

Table 10.1: Simulation times for the different simulations presented here.

Grid resolution	30$\times$30$\times$30	60$\times$60$\times$60
Deposition time/step	1.54s	47.4s
Isotropic etching/step	0.53s	2.77s
Directional etching time/step	0.97s	21.1s