Driven by Moore's law semiconductor manufacturers are permanently investigating
new fabrication technologies and developing next generation semiconductor
devices. In order to deal with these challenges the impact of technology
computer-aided design (TCAD) has been increasing enormously in recent
years. Reliable numerical simulation of semiconductor processes and devices
enables the manufacturers to significantly reduce costs but also
to virtually explore novel ``cutting-edge'' devices at an early pathfinding
stage. Two- and three-dimensional simulations and optimizations allow to make
faster decisions and to reduce the number of expensive fab experiments.
Almost any state-of-the-art computational TCAD method depends on a spatial
discretization of the simulation domain, generally referred to as mesh. It is a
known fact that the quality of the mesh has the most influence on the numerical
solution and is therefore the weak spot in a typical analysis application.
If a dense mesh is given, almost any solver can compute an accurate solution. On
the other hand side, if too few points are used in critical locations, the
calculation of correct results can be problematic, erroneous, or even
impossible. In contrast, too many mesh points slow down the numerical
evaluation and increase the memory consumption. Both are crucial parameters
and must be tracked very meticulously.
In order to strike a balance between accuracy and the consumption of
computational resources meshing should be done judiciously. This means
that only in regions of particular interests a fine mesh is used whereby other
domains are of coarse granularity. To put this leading record into practice
three sophisticated mesh refinement strategies have been developed and
presented in the context of typical TCAD applications.
In addition a novel approach is presented which allows the incorporation of
spatial distortion in the sense of anisotropic mesh elements. Compared to
strict regular meshes with the same mesh density in all spatial directions,
appropriate anisotropic meshes significantly reduce the consumption of
computational resources while increasing or at least preserving the
accuracy.
This thesis consists of two main parts. The first one is a mostly
theoretical presentation of recent developments in the area of refinement
strategies. Various introductory examples are given, illustrated with colorful
interpretations of theoretical concepts. The second part shows
different mesh refinement applications in the field of diffusion,
electromigration, and full band Monte Carlo simulations.
To focus on the improvement of accuracy a heuristic error estimator is
presented for diffusion simulation, which is based on the idea of gradient
recovery estimators. Furthermore, a combination of error estimation and a mesh
perfection by the so-called gradient refinement method is exhibited.
For the simulation of electromigration a combination of mesh refinement and a
retrogressive coarsening (coarsement) has been developed.
The third area of application deals with full band Monte Carlo simulations and the readaptation
of static
-space meshes as pre-processing step. Complex band structures of
semiconductor materials require a general concept of
-space tessellation which is
flexible enough to treat different materials. The idea behind this strategy
is illustrated on the band structure of silicon.