Discretization of Semiconductor Device Problems (I)

This chapter discusses discretization of semiconductor device problems (I). It is possible to view classical or semiclassical modeling of transport in semiconductors as a hierarchical structure, sweeping from the Boltzmann Transport Equation (BTE) down to the Drift-Diffusion (DD) model, passing through all the systems derived from the moments of the BTE with respect to increasing powers of the carrier group velocity. The Hydrodynamic (HD) model and its inertial limit, that is a type of Energy-Transport (ET) model, are examples, among others, of these intermediate steps. From a mathematical viewpoint, when appropriate scalings are employed for the quantities appearing in the DD system, one finds that the conservation part of the system is convection dominated. In particular, there is a scaling, which makes the order of magnitude of the electron and hole concentrations equal to one. Looking at the structure of the Gummel map, one can recognize a nonlinear block Gauss–Seidel iteration that can be subdivided into two main loops: (1) a nonlinear inner iteration for solving the semilinear Poisson equation, and (2) two iterations for solving the continuity equations.