This work focused on some issues for accurate numerical simulations of electronic devices with the main attention on three-dimensional concepts. Due to the rapidly increasing miniaturization and complexity of the devices, the use of three-dimensional simulations is often inevitable. Starting with the diffusion simulation of dopants, three-dimensional effects, which are often parasitic, cannot be neglected any longer. With the developed tool for diffusion simulation the fast computation of simplified diffusion processes was enabled. On the basis of those, statements about device characteristics can be chosen. Additionally, the three-dimensional diffusion simulation is a fundamental basis for three-dimensional electrical simulations of semiconductors. The second part of this work deals with the simulation meshes which are required for device simulation. The developed method allows to tune the grid density in wide ranges. Due to the almost orthogonal potential distributions, which are the basis for the placement of the grid points, the density of the grids can be independently controlled in three directions of space. Since the equipotential surfaces are conforming to the geometry surfaces, next to the surfaces of the geometry, grid elements with certain preferred directions can be obtained. And as a result of nearly cuboidal grid elements, the Delaunay criterion is almost fulfilled and the following tetrahedrization does not cause difficulties. With this method, the automatic generation of specially suited grids for three-dimensional semiconductor device simulation was enabled.

There still remain unresolved issues in the field of fully automatic grid generation. Especially for three-dimensional grid generation a wide range of questions is left open and needs a lot of further work. Usually a general approach of meshing cannot be given and the development of the grid generators follows different ways.

Several anisotropic grid generation methods are under development, with the main goal to deliver grids with as few as possible grid points, but not neglecting the necessity of an accurate solution on the based model. That means that different kinds of grids are necessary for process simulation, diffusion simulation, and device simulation.

In summary, further development steps are:

- Surface coarsement algorithms: Often between two kinds of simulation, the surface representation must be adapted to the new demands, for example between a topological and a diffusion simulation. As the physically based addition and deletion of material influences the surface directly and the grid in the inner of the regions is of minor interest, the demands for a diffusion or electrical simulation shift to the interior grid. The surface grid should be coarsened as much as possible, but structural edges must be detected and preserved automatically.
- Implementation of grid refinement strategies: The grids should be adapted to the different discretization methods and different simulation models. Especially edge splitting algorithms which produce anisotropic grid densities must be implemented.
- Development of coarsement algorithms of the volume mesh: On the one hand in combination with refinement strategies, the generation of adaptive meshes for transient simulations must be improved. On the other hand, the matrix sizes can be reduced by coarsening the grid in regions of minor interest.
- Finally the handling of thin material layers must be improved. Such thin segment structures are found frequently within electronic semiconductor device structures, for example the field oxide layer under the polysilicon segment of the gate contact of MOS transistors. For the generation of a Delaunay grid, point insertion on the surface is often necessary. With standard point insertion algorithms a lot of additional grid points are generated on the surface of thin layers. To prevent this effect, a projection of grid points from one to the other thin layer surface gives better results.

For further extensions of the potential method described in this work, investigations on point reduction algorithms are required. As described before, an additional equipotential surface causes additional grid points at all intersections with the other existing potential ticks. Therefore an algorithm, which works similarly to a terminating lines algorithm based on otho-grids is inevitable. These intersectional points in regions of small interest must be prevented. Additionally, grid points at the surface, which arise from structural edges or are eventually inserted by the grid generator to satisfy the Delaunay criterion, affect the grid quality near the boundaries. To prevent this usually adverse influence on the grid quality, a continuation of these surface points as potential ticks inside the segment region keeps the orthogonality of the grid lines. In addition to this continuation, the number of grid points inside the structures caused by these potential ticks must be limited by terminating lines algorithms which maintain the Delaunay criterion.

J. Cervenka: Three-Dimensional Mesh Generation for Device and Process Simulation