Analytical Investigations About the Basic Semiconductor Equations

Abstract

In this chapter we review some of the existing analytical results which characterize the basic semiconductor equations. Of particular concern will be the questions of existence, uniqueness and structure and smoothness of solutions. These are of importance in both the theoretical context and the practical context, since the knowledge of the structure and smoothness properties of solutions is indeed essential for the selection of appropriate numerical Solution procedures. The basic semiconductor equations as given in Chapter2 are:

$$\text{div}\,\text{grad}\,\psi = \frac{q} {\varepsilon } \cdot (n - p - c)$$

(5-1)

$$\text{div}\,\vec J_n - q \cdot \frac{{\partial n}} {{\partial t}} = q \cdot R(\psi ,n,p)$$

(5-2)

$$\text{div}\,\vec J_p + q \cdot \frac{{\partial n}} {{\partial t}} = - q \cdot R(\psi ,n,p)$$

(5-3)

$$\vec J_n = - q \cdot (\mu _n \cdot n \cdot \text{grad}\,\psi - D_n \cdot \text{grad}\,n)$$

(5-4)

$$\vec J_n = - q \cdot (\mu _p \cdot p \cdot \text{grad}\,\psi - D_p \cdot \text{grad}\,p)$$

(5-5)

We have omitted in the current relations (5-4) and (5-5) terms which account for current components caused by bandgap narrowing and temperature gradients. All these effects are considered to be only small perturbations which just make the essential analytical results about the basic semiconductor equations less transparent. One should also bear in mind that the current relations will become potentially incorrect if one of the above cited effects would change the equations in a dominating manner (cf. Section 2.3). We shall also ignore the impact of a non- homogeneous temperature distribution on the basic semiconductor equations for the following analytical investigations.