The behavior of dopants during diffusion can be described by a set of coupled nonlinear PDEs. Since there is no analytical solution scheme possible for diffusion problems, except for some problems with special boundary conditions [Yos74], numerical methods are used to solve the diffusion equations.
Before we classify the procedures necessary for the numerical solution of PDEs we give some basic definitions of terms used in the context of simulation tools and needed for further specifications. These definitions are based on the Profile Interchange Format (PIF) which was proposed by S. Duvall as a hierarchical concept for simulation data management [Duv88]. This original concept was adapted and implemented at the Institute for Microelectronics and is being used as basic data format within the Viennese Integrated System for TCAD Applications (VISTA) framework [Fas91] [Hal93] [Fas94b].
After definition of some basics, we proceed with the main steps to obtain the numerical solution of the diffusion problem. They can be classified as follows:
The simulation geometry can be given by the previous simulation tools, it can be specified by a graphical editor [Rie95], or must be defined textually in the worst case. The physical properties of the different involved semiconductor materials is stored in segment attributes. The doping information is needed to built up the initial values for all solution quantities. Also boundary condition information must be given to ensure the complete description of the diffusion problem. Boundary conditions are defined on interfaces and borders of the simulation domain, i.e. on geometry lines.
Once the geometry and dopant attributes have been obtained, the simulation domain must be tessellated, i.e. divided into small elements or cells which form a grid. The distribution of these grid points is related to both geometrical and doping information, see for instance the next section.
The diffusion equation can be implemented numerically on the mesh by using either a finite-difference method (FD), a box-integration method (BM) or a finite-element method (FEM). The choice of the appropriate discretization method depends on the geometry, the grid type available, and on the type of PDEs to solve (see Section 4.1.2). After discretization of the physical equations we have a large system of nonlinear algebraic equations to solve.
Because of the nonlinearity of the algebraic equation system we are forced to use iterative methods to solve them. For a strongly coupled equation system the Newton iteration scheme can be applied. If the system is weakly coupled, the PDEs can be solved independently from each other with a block iterative method, like the Gauß-Seidel iteration method. The advantage of Newton's method is that convergence is generally much faster than with the decoupled method. On the other hand, the initial guess for the solution quantities must be reasonably good for Newton's method, where the decoupled scheme usually converges with a wide variety of starting values. Due to generation/recombination terms, the transient diffusion problem is a strongly coupled system, hence, the Newton iteration method is obligatory.
The correct interpretation of the final result requires the graphical representation. Therefore, it is necessary to have a flexible analysis tool which can handle scalar or vector-valued attributes. This task is captured in a strong manner by the functionality of TCAD framework tools [Hal95].