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6. Conclusion and Outlook

In this work a general solver is presented that accounts for most necessary features in developing powerful physical models to accurately simulate semiconductor process steps, and, of course, models of other physical disciplines. The basic development for an analytical description of partial differential equations that also provides the independence of the kind of discretization have been implemented. The usual trade off between flexibility of a system and performance can be kept within acceptable bounds due to the implemented optimizing routines. For further improvement of performance a C-code extraction has been implemented that minimizes the speed loss, so that nearly no difference between `hand coded' specialized simulation tools written for a specific problem and AMIGOS appears. An already existing solver library supporting Gaussian elimination and iterative stabilized biconjugate gradient method in conjunction with the equation system analyzer set up a flexible possibility to solve successfully a developed analytical model. Finally, the implemented adaptive grid algorithm supporting both, refinement and coarsement, is a powerful means to calculate complex three-dimensional coupled differential equations, since the density of discretization points can be adapted to the calculated distribution.

To demonstrate the abilities of AMIGOS several examples have been presented including various disciplines in semiconductor device simulation (e.g. pair-diffusion, in-segregation, temperature distribution in a wired frame, adaptive grid relaxation, oxidation). In the last chapter of the thesis a closer inspection of the oxidation is given and a new diffusion coupled oxidation model is introduced that exploits all the features of AMIGOS. Thus it has been possible to simulate successfully three-dimensional oxidation steps for non trivial geometries.

Although the basics are considered and several examples are treated many topics are open to be developed further. As a direction to future work the most important ones are listed below.

TCAD Framework

Till now AMIGOS is a stand alone simulation tool that is not integrated in any TCAD framework and therefore only isolated problems can be solved. For efficient usability and general acceptance an integration of AMIGOS into VISTA, a framework for technology purposes [Pic93][Str97], is necessary where complete devices are simulated starting from a pure silicon block resulting in precalculated device characteristics. Because of AMIGOS generality it might be possible to replace several `hand-coded' simulation tools integrated into the TCAD framework, thus reducing the overall complexity of the system. But to extract process dependent conditions concerning a given physical problem a preprocessor must be developed that automatically sets up the conditions for AMIGOS to support full integration into the TCAD framework.

Of great interest is the optimization of several process and device steps by parameterizing various models. Fitting these parameters manually is a rather ungrateful and time consuming process. The available optimization package [Kha95a][Pla97] within the VISTA framework is another task that has to be linked to AMIGOS in the near future to increase the rigorousity of the complete system.

Model Library

Increasingly complex physical formulations are required to account for effects that were not important in simulating previous generations of technology. To accurately simulate modern semiconductor process steps, a simulation tool must include a variety of physical models that can be joined together without high effort or changing any line of source code.

The C-code extraction is already the first step to develop a physical based model library for simulation purposes. In combination with the simultaneously developed model management library [Mle97], which supports inheritance and several mechanisms to combine models to a new one also without necessity to have access to the source code, new possibilities for rapid model development are offered to keep the pace of advance. Therefore, it will be a future task to develop a model library that supports the most striking models.

Mesh Generation

Although AMIGOS is equipped with a hierarchical mesh adaptation algorithm an integration of a mesh generator is advisable. Just look at the example adaptive mesh relaxation (Chapter 4.8.3) where the importance to be able to change the connectivity of the elements is obvious. Especially in moving grid problems, such as mechanical deformations e.g. during oxidation, retriangulation is often necessary to preserve a tolerable element quality. Unfortunately, there is no meshing tool available that supports all three spatial dimensions and, therefore, a rigorous integration is rather complex. Whereas in two dimensions it is acceptable and often executed to communicate with a meshing tool via file access, it seems to be inconceivable to go the same way in three dimensions because of the enormous size of the files. That is why an integration of a three-dimensional meshing tool [Fle96] seems to be more important.


Solving complex partial differential equations especially in three dimensions is still a non trivial task, and memory consumption and simulation time is extremely high. Even if Moore's Law is right and memory and system performance continues increasing during the next years the complexity of models increases even faster. Nowadays it seems to be the only perspective for simulation tools to dramatically increase the overall performance by parallelization. Due to AMIGOS object oriented approach a parallelization is promising and not only the solver library but also the auto assembling library (evaluation of the analytical model per element) is predestinated for parallelization. Tests have shown that about 97% of CPU consumption is used in these two parts of the algorithm - this equals in an invitation for parallelization in the future.

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Mustafa Radi