Grid Generation for 3-D Nonplanar Semiconductor Device Structures

We have developed an automatic grid generator, called fi, for complex nonplanar three-dimensional (3-D) semiconductor device structures. The meshes generated by Q permit an exact geometry modeling of the rather general domain boundaries of modern semiconductor technologies. Avoiding obtuse angles by construction, fi is the ideal preprocessor for a 3-D device simulator. The numerical solution of partial differential equations (PDES) is invaluable in the design and the optimization of semiconductor devices and integrated circuits. The spatial discretization of the structure to be simulated is the key to the accuracy of the computed solution. A reasonable approximation of the geometry to be modeled and of all internal quantities relevant to the solution of the PDEs, such as the doping profile, is extremely important. Additional constraints arise from the discretization schemes used. As classical finite element schemes seem inappropriate, PDEs are usually solved using the control volume or box method [9]. This necessitates that obtuse angles be avoided, another nontrivial condition. In two dimensions(2-D), both rectangles and triangles have been used for the initial coverage of the integration domain. Bank and co-workers have proposed covering the integration domain (a closed polygon) with a carefully chosen triangulation, and to recursively subdivide triangles where higher mesh resolution is needed into four similar triangles, through the addition of four new mid-edge points [2, 3]. Yerry and Shephard proposed to use a modified quadtree data structure: the integration domain is encapsulated in a square, and the square's quadrants are recursively subdivided in quadrants until the mesh density is sufficient to model the domain geometry and internal quantities appropriately [4]. Miiller et al.[13] have recently extended the quadtree idea in the implementation of a fully automatic 2-D mesh generator. In 3-D, the recursive refinement of simplices (i.e., tetrahedra) is much harder than in 2-D [5]. Major problems include the difficulty in generating a well-shaped initial tetrahedra! grid, the impossibility to regularly subdivide tetrahedra in similar sub-elements, and the problems with tetrahedra between dense and coarse mesh regions. Therefore, the modified quadtree approach has been extended successfully to three dimensions [6, 7, 8]. The 3-D domain is enclosed in a cube, whose octants are repeatedly refined until the boundary and corresponding internal quantities arc sufficiently approximated. A slightly different approach for the generation of octree-based Delaunay meshes has been proposed by Schroeder and Shephard [11]. The first version of the grid generator fi 0 ct(^ octree), as presented …