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2.3 State of the Art

Most of the research in mesh generation is done in the fields of computational fluid dynamics and stress mechanics. Three-dimensional mesh generation for applications in TCAD and for semiconductor process and device simulation is a recent topic and leaves room for much innovation. The following is a summary of modern approaches respectively in the order of cartesian-based, octree-based, unstructured tetrahedral, advancing front, hybrid, and Delaunay methods.

Aftosmis et al. [2] developed cartesian meshes for applications in computational fluid dynamics. They incorporate complex boundaries with a rigorous method to intersect the cartesian cells with the boundary representation. A limited flexibility in refinement is experienced, due to the cartesian nature of the mesh and the isotropic refinement technique. Each further refinement level yields about 2-4 times more elements. With intial 1.6 million elements a further increase in accuracy by adding one more refinement level would result in 3.3-6.6 million elements according to one of the examples in [2]. Still, in some areas the achieved accuracy is not optimal and refinement would be appropriate. Such a high point propagation due to refinement and the rigorous intersection method which greatly increases the number of surface elements is less likely to be afforded on a typical computer available to a user for everyday simulation purposes.

The cartesian-based mesh generator OMEGA well known for semiconductor simulation applications is described in [53]. This intersection-based method originates from previous works using bisection-based and octree decomposition techniques [66,65,35]. Improvements include the ability to tessellate the cartesian cells into Delaunay tetrahedra and a more anisotropic refinement approach. Subdivisions for higher refinement levels are performed by splitting the nodes at intersections or geometry-specific vertices instead of splitting the nodes always at bisections. However, if vertices of the geometry lie closely together without conveying important geometrical information as for instance in the case of staircase-like approximations of slopes, too thin elements are created and unnecessarily propagated throughout the mesh. The key issue with cartesian methods is to merge the coordinate system aligned cells with the unstructured surface of the geometry. It seems that without rigorously generating a cell-consistent surface triangulation as in [2], difficulties are experienced dealing with the boundaries and interfaces of a semiconductor device. To attain optimal flexibility regarding both the mesh density distribution as well as the fitting of the boundary it will be desireable to achieve anisotropy in arbitrary directions (not just along the three coordinate axises).

Johnson et al. [76] use an unstructured tetrahedral method for three-dimensional flow problems. The mesh generator is capable to efficiently deal with moving objects in fluids. It is linked to an input modeler for non-uniform rational B-spline (NURBS) surface patches which however makes it less suitable for TCAD applications.

Advancing front methods [19,94] are mostly used because of their good point placement properties which result in better boundary-fitted meshes as opposed to mere boundary consistent meshes. Elements are not intersected with the boundary, instead they are grown from the boundary and are therefore well aligned with it. An interesting technique to utilize the efficient and well defined Delaunay Triangulation as a background mesh2.1 for advancing front style mesh generation has been developed by P.L. George et al. at INRIA, France [50]. Generally, it can be of interest to construct protection layers around physically crucial boundaries or interfaces to provide the required orthogonal mesh resolution for a correct representation of a physical magnitude of the solution. It remains an open problem how to effectively apply such methods to the complexity of a three-dimensional semiconductor device. In two dimensions approaches to enforce a certain anisotropy along inversion layers or otherwise highly non-linear interfaces have been investigated by [4,112,177,82,133]. Most methods have come to a stage where they more or less rely on a triangulation engine, e.g. TRIANGLE [160], to generate a valid tessellation of a set of supporting grid lines and/or mesh points supplied together with the boundary. Conformal mapping techniques to ensure orthogonal boundary-fitted meshes by means of mathematical transformations of parametrized non-planar boundaries pose computation difficulties in three dimensions. The related partial differential equation (PDE) gridding method2.2 seems better extendable for three-dimensional TCAD applications. So far the two-dimensional implementation CGG [26] of the PDE method has proven powerful for semiconductor device simulation.

Pure octree-based solutions combined with tetrahedral templates are hardly used for todays TCAD applications. Considering the variety of existing approaches in three dimensions one can observe that recently most methods have evolved to become hybrid methods and require some sort of general tetrahedralization [90,77,178,81]. The software package LAGRIT formerly known as X3D provides a universal mesh generation toolkit [102]. Another hybrid mesh generator is MESHise [54] which is a further development of OMEGA.

For moving boundary situations level set methods have gained great importance [90,1,27,127]. They require a mesh to define the magnitude which describes the moving boundary. At some stage a boundary consistent mesh must be derived from the level set representation by means of intersection and eventually tetrahedralization.

The history of methodologies in two-dimensional process and device simulation leads to the observation that an unrestricted and stable triangulation engine is one of the most important tools for mesh generation purposes. The theory of Delaunay provides a mathematical foundation for provably terminating triangulation algorithms [15]. While the two-dimensional Delaunay Triangulation poses less difficulties and a vast amount of literature exists [41,70,145,175,86], the integration of boundaries into a three-dimensional Delaunay Triangulation remains a very active research area. An extremely thin oxide layer on top of a comparatively big silicon block calls for very sophisticated algorithms to automatically generate an unstructured Delaunay mesh consistent with the boundaries and interfaces. This is one of the reasons why most of the software for a three-dimensional Delaunay Triangulation available to the public domain is less suitable for semiconductor simulation purposes. A Delaunay method using local transformations is included in GEOMPACK by B. Joe [74,72]. QHULL is a code which constructs a Delaunay Triangulation by computing the convex hull in higher dimensions [12]. Excellent work related to Delaunay refinement mesh generation can be found in [164]. The QMG software package for quality mesh generation based on octrees is described in [109]. A detailed and much more complete survey of the worldwide research activities in mesh generation is maintained by R. Schneiders [147]. Regarding future aspects of TCAD and automation issues one may refer to [34,83].

next up previous contents
Next: 3. Mesh Generation Up: 2. Challenge and Demands Previous: 2.2 Semiconductor Process and
Peter Fleischmann