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5 Semiconductor Modeling
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that the doping profile C, which determines the electrical characteristics of the
device under consideration, only enters in the Poisson equation for the potential.
We remark that this Poisson equation models the repulsive electrical inter-
action between equally charged particles. A corresponding attractive (gravita-
tional) model is obtained by reversing the sign of ΔV in the Poisson equation,
as used, for example, in the modeling of biological cell motion by chemotaxis in
Chapter 4.
Equilibrium states (i.e. stationary states with vanishing current densities) are
Maxwell distributed:
2
2
n e = δ exp(V e ), p e = δ exp(−V e ),
where δ is a positive device-dependent parameter. Note that, by basic solid state
physics, the recombination-generation rate R vanishes in equilibrium where
4
n e p e = δ holds. The equilibrium Poisson equation then becomes semilinear:
2
2
2
λ ΔV e = δ exp(V e )− δ exp(−V e )− C(x), x ∈ D

Fig. 5.5. RAM (Random Access Memory) Array