2.2 Semiconductor Process and Device Simulation

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

Combined full scale three-dimensional process and device simulation has suffered from the lack of available TCAD tools for the various established data formats in the field. A framework of such a TCAD environment providing the necessary tools is itself still a subject to research and development [79,172,182]. Aside from preliminary means to couple and control some of the simulators [173,121,185], powerful geometry processors are missing. Considerable person power is consumed to detect inconsistencies in a tedious manner, and to debug structures that the mesher should, but does not mesh. Such structures typical for semiconductor simulation purposes often exhibit extreme ratios between the size of the smallest and the largest features, where no structural simplifications can be afforded. Only quantitative data reduction may be performed to decrease the amount of redundant data.

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 $\mathcal{L}_{\max}$ 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 $d = \alpha \mathcal{L}, \alpha \in [1..2]$ is small compared to their lateral expansion (Fig. 2.2).

It is relevant especially in three dimensions to detect such precisely defined thin layers in order to avoid an unnecessary high number of mesh elements. While standard isotropic elements are still manageable in two dimensions, the number of isotropic elements in three dimensions roughly increases quadratically with the ratio of lateral edge length to $\mathcal{L}_{\max}$. It will be necessary independent of the physical application or solution quantity and due to a pure geometrical feasibility that the mesh generator enforces anisotropic elements at automatically detected regions which convey anisotropic geometrical information. The most general and sophisticated algorithms have to be applied in order for the mesh generator to deal with automatically generated structures. Computer generated topographies not only evolve from etching and depositon modules, but also through three-dimensional extraction of geometrical data from two-dimensional simulations and through solid modelers deriving their information from layout data. An automatic geometry preprocessor and mesh generator can complete the integrated framework system and enable the fast and efficient optimization of various design and manufacturing parameters.

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 $n$ of points in three dimensions, keeping the computational complexity below O($n^{2}$) is another crucial requirement for success.

The tasks can be summarized in relation to their application.

• Meshing of comparatively simple (near $90^{\circ}$ 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.