The demands can be roughly categorized.

- automatic mesh generation of complex semiconductor devices within an integrated framework of simulators
- anisotropic geometrical requirements (structure dependent)
- mesh density grading, anisotropic density requirements (application dependent)
- finite element mesh quality and closed control volumes for finite volume methods

Once the mesher is provided with a valid and clean model it has to deal with the topographical as well as topological complexity of the underlying geometry which contains many internal surfaces, edges sharing more than two facets, and thin layers.

**Thin Layer**- Two polygonal surfaces with a large area compared to the maximal local feature size measured inbetween. Less general subtypes are formed when the surfaces are planes. Furthermore, the planes might be parallel or have normals in the direction of a coordinate axis. For the simplest case of two parallel planes their normal distance is small compared to their lateral expansion (Fig. 2.2).

The range of the simulated magnitude (e.g. the concentration during diffusion simulation or in device simulation) covers several exponents and leads to difficulties when discretized with conventional meshes. The variation of the local mesh density over space and over direction (anisotropic mesh density) is the key factor to keep the size of the mesh manageable and at the same time the discrete distribution of the magnitude accurate. A balance between the number, the size, and the quality of the elements must be achieved for such a good mesh grading. The number should be minimized while the size should satisfy local density criteria. Often, the mesh is not optimally fitted and too coarse or too fine elements exist in various areas of the simulation domain. Alternatively, the desired accuracy of the analysis cannot be achieved in all areas, or the subsequent tools and simulators are pushed to limits beyond the scope of an average computer by a generally too fine mesh. An increased flexibility in refinement allowing for rapid changes of the mesh density while keeping the overall element count low would be ideal. This flexibility cannot be increased infinitely. The rapid change of element size is limited under the premise of maintaining a certain element quality.

The finite element method requires that the elements possess a certain geometrical quality. It can have a very bad influence on the convergence of the solution if the elements have extremely obtuse angles. The angle spanned by two planes (dihedral angle) becomes an important measure in three dimensions (Chapter 3). Achieving good bounds on the dihedral angle of the elements becomes a major demand in three-dimensional finite element mesh generation.

The finite volume or control volume method which is often also called
the box integration method is crucial for semiconductor device simulation,
because it can be combined with the Scharfetter-Gummel scheme
[58][119][146] which takes the exponential carrier
concentration into account.
As will be discussed in Chapter 3 ``closed'' control
volumes are required which is usually accomplished by constructing a
*Voronoi* [117] type mesh. The *Voronoi box*
associated with each point from the mesh satisfies the requirements
and possesses some advantages as opposed to e.g. boxes which are
defined by centroids (gravity boxes).
The common approach to obtain such a Voronoi tessellation is to
construct its dual Delaunay Triangulation. (See
Chapter 5 for a description of the Delaunay theory
and its definitions.)
Stable Delaunay meshing of a complex semiconductor structure is a
demanding effort in three dimensions.
The generalization of a two-dimensional to a three-dimensional
Delaunay Triangulation poses not a mere quantitative but rather a
qualitativ challenge (Chapter 5).
Among other reasons is that a bounded dihedral angle between facets of
the input structure becomes a crucial factor for a provably
terminating Delaunay algorithm. With an increasing number of points in three dimensions, keeping the
computational complexity below O() is another crucial
requirement for success.

The tasks can be summarized in relation to their application.

- Meshing of comparatively simple (near dihedral angles
between input facets) but huge structures composed of a large number
of vias and lines.
*Simulation of 3D Interconnects.* - Handling the minimum and possibly faulty information on arbitrary
complex structures provided by topography simulation
with eventually moving structure boundaries [45].
*Semiconductor process simulation.* - Resolving highly non-linear quantities with a directional mesh
density which is not only limited to the three directions of the
cartesian axises (physical anisotropy).
*Semiconductor device simulation.* - Dealing with strongly acute dihedral angles between input facets
and extreme ratios between local edge lengths and local feature size
(geometrical anisotropy).
*Semiconductor process and device simulation.*

2000-01-20