The Discretization of the Basic Semiconductor Equations

Abstract

The system of partial differential equations which forms the basic semiconductor equations together with appropriate boundary conditions has been investigated and characterized analytically in the previous chapter. This system cannot be solved explicitly in general. Therefore, the solution must be calculated by means of numerical approaches. We shall consider in this chapter such Solution procedures for the sealed equations which read:

$$\lambda ^2 \cdot \text{div}\,\text{grad}\,\psi - (n - p - C) = 0$$

(6-1)

$$\text{div}(D_n \cdot \text{grad}\,n - \mu _\text{n} \cdot n \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial n}} {{\partial t}}$$

(6-2)

$$\text{div}(D_p \cdot \text{grad}\,p - \mu _\text{p} \cdot p \cdot \text{grad}\,\psi ) - R(\psi ,n,p) = \frac{{\partial p}} {{\partial t}}$$

(6-3)

Any numerical approach for the Solution of such a system consists essentially of three tasks. First, the domain, i.e. the Simulation geometry of the device, has to be partitioned into a finite number of subdomains, in which the Solution can be approximated easily with a desired accuracy. Secondly, the differential equations have to be approximated in each of the subdomains by algebraic equations which involve only values of the continuous dependent variables at discrete points in the domain and knowledge of the structure of the chosen funetions which approximate the dependent variables within each of the subdomains. In that way one obtains a fairly large system of, in general nonlinear, algebraic equations with unknowns comprised of approximations of the continuous dependent variables at discrete points. The Solution of this system is the final third task to be carried out. As this problem can be viewed rather independently it will be treated separately in Chapter 7.