Subsections

• 1.1.1 Numerical Computing
• 1.1.2 Typical Computational Analysis Cycle

1.1 TCAD Tools

The dramatic increase in the use of calculating machines began with 'Third Generation' computers. These relied on Jack St. Claire Kilby's invention of the integrated circuit (or microchip). The first integrated circuit was produced in September 1958, but computers using them did not begin to appear until 1963. Some of their early uses were in embedded systems, notably exploited by the National Aeronautics and Space Administration (NASA) for the Apollo Guidance Computer. It is obvious that in the beginning the computer was part of research itself. This changed over the last two decades and the computer is steadily becoming a general tool for all imaginable uses.

Technology computer-aided design in the context of semiconductor component manufacturing is understood as the exploration, by means of simulation, of fabrication processes and electrical device behavior. Related to this, the development of TCAD tools, or more precisely computer programs, also called simulators, starts with understanding of physical phenomenons and their modeling, mapped onto mathematical representations which can be handled by calculating machines.

TCAD tools are well accepted in semiconductor industry, since simulations can help to improve existing technologies and simultaneously reduce research and development time, which has a direct impact on cost reduction [1]. The simulation of fabrication processes and device behavior as logical loop, enables to prepare, run, optimize, and analyze semiconductor experiments, which helps the development of optimal manufacturing recipes and device targets.

1.1.1 Numerical Computing

An essential part of any TCAD tool is the description of physical phenomenons by means of mathematical modeling. Mathematical models are usually a set of continuous, contingently nonlinear partial differential or integro-differential equations, which are difficult to handle because of their complexity. One very common way to solve or to get at least an approximation of the solution is to use numerical methods, such as finite differences or finite elements, but also statistical methods like the Monte Carlo method are widely established.

To be suitable for numerical computing, continuous models and their mathematical equations have to be transferred into discrete counterparts by means of sampling. This process is usually carried out as a first step toward making physical models accessible to numerical evolution. And so a new discipline comes into play, which deals mostly with the generation and adaptation of spatial discretizations. In general spatial tessellations are referred to as mesh constructed on elementary geometrical objects like points and edges. Over the years different meshing approaches have been developed with different emphases on the underlying numerical computation scheme.

An overview, especially of unstructured mesh generation algorithms, which are utilized today, is given in Figure 1.1. Based on the topology of the mesh element, one can distinguish between two widely used element groups, namely quadrilateral and trilateral elements in two spatial dimensions and hexahedral and tetrahedral elements in three-dimensional space. These element groups yield different meshing approaches which are described in detail, for e.g., in [2].

Figure 1.1: Overview of algorithms for unstructured mesh generation with emphasis on four widely used mesh element types in two and three spatial dimensions (picture partly adapted from [3]).

In this work the main focus is not on mesh generation. Rather, it is all about mesh adaptation, more precisely the modification of an existing mesh by refinement, especially in the field of TCAD simulations for semiconductor structures. The setup of such simulations and how mesh adaptation can improve accuracy and reduce computational time is part of the following.

1.1.2 Typical Computational Analysis Cycle

For TCAD simulations a typical computational analysis cycle can be divided into two phases. The first phase, namely the pre-analysis phase, is used to define the spatial simulation domain in the sense of a mesh with initial data. Such a mesh is produced from a computer-aided design (CAD) model by a mesh generator. One can think about such a CAD model as the definition of a two or three-dimensional geometric shape, defined by basic geometric objects like points and edges or a combination of solids, like spheres, cones, and bricks. Based on these shapes the mesh generator subdivides the spatial domain, defined by the CAD model, into small pieces, so-called mesh elements. To be prepared for a subsequent analysis process, additional data is stored on the mesh, which is referred to as setting initial conditions, as depicted in Figure 1.2.

Figure 1.2: Typical computational analysis cycle which can be divided into the pre-analysis and the analysis phase. The incorporation of error estimation and mesh adaptation improves the quality of the numerical solution.

The second phase, called analysis phase, is the literal computational stage where different scenarios are possible. The simplest case uses a static mesh which is not modified after the analysis process. In this unlooped case the engineer has to validate computed results with the help of post-processing and visualization tools. An accuracy quantification of computed results is difficult to handle and must be done in a separate step. To improve accuracy, a total `hand made' remeshing step must be carried out usually by the engineer, since the mesh density has most influence on reliability of computed results. This drops the progress back to the pre-analysis phase in the case of poor quality solutions.

It is obvious that this procedure is very time consuming and demands from the analyst engineer also high meshing skills. Another critical part is the mesh generator, because this process forces a mesh generator which allows the control of mesh parameters like mesh density or even harder directional dependent mesh densities. These constraints in the meshing process can cause a total malfunction of the mesh generator. Nevertheless, the unlooped case is still in use and daily routine of TCAD engineers.

A more sophisticated scenario can be reached with the incorporation of an error estimation and a mesh adaptation step in the analysis phase. As depicted with a red, bold arrow in Figure 1.2 this yields a loop in the analysis phase and is therefore referred as the looped case. This disburdens the TCAD engineer dramatically, because the solution quality can be tuned more self-acting. The main focus in this scene is on the mesh adaptation routines, which is part of the following.