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3.1.3 Benchmark Example

With the three algorithms described above namely the original sphere algorithm ($a$) from [17], the spherical segments algorithm ($b$) from Section 3.1.1, and the line algorithm ($c$) from Section 3.1.2 we have simulated isotropic deposition of aluminium into a 0.5 $\mu\mathrm m$ diameter cylindrical via. The deposition rate was 0.07 $\mu\mathrm m$/s and the deposition time 120 s. In total $160\times 160\times 100$ cells were used. The simulation was performed with two time steps of 60 s. All calculations were done on a 200 MHz DEC Alpha workstation. The initial geometry and the film profiles after the two time steps can be seen in Fig. 7. For more clarity only a quarter of the structure is shown. Table 1 summarizes the times necessary for the different algorithms.


Table 1: CPU time [s] for the deposition routine of different deposition algorithms.
\begin{tabular}{\vert\vert l\vert\vert c\vert c\vert\vert}\hline \hline \multico... ...akebox[2.2cm][r]{23.3} & \makebox[2.2cm][r]{21.6} \hline \hline \end{tabular}


Figure 7: Isotropic deposition of aluminium into a circular via - initial geometry, topography after 60 s and final structure after 120 s deposition time.
\begin{figure}\vspace{-2mm} \psfrag{0.5 \247m}[ct][ct][2]{{0.5~\makebox{$\mu\mat... ...} {\resizebox{!}{8cm}{\includegraphics{circhole2.eps}}} \end{center}\end{figure}

As expected the spherical segment algorithm ($b$) is much faster than the original spherical one ($a$). The drawback of ($b$) is, that the second time step ($2$) takes about twice the time of the first time step ($1$). The reason is that due to the more complex structure after the first time step even the spherical segments algorithm ($b$) introduces some redundancy as described in Section 3.1.1. The figures for the line algorithm ($c$) are misleading, because they include also the time for the orientation calculation required for this algorithm. The advantage of this algorithm is, that the calculation time is quite independent of the complexity of the structure. The difference to ($b$) is, that the interpolation is always performed with quarter circles parallel to the polar angle of the surface orientation of the considered cell and not with eighths of spheres or circles. Therefore the CPU time for the second time step ($2$) of ($c$) is comparable to the first time step ($1$).

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 with high computational costs. Considering overall performance with simulation time, accuracy, and the lack of implicitly given surface orientation the spherical segment algorithm is suited best for the application with the cellular data representation.

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W. Pyka, R. Martins, and S. Selberherr: Optimized Algorithms for Three-Dimensional Cellular Topography Simulation