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5 Semiconductor Modeling
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proportional (Einstein relations), such that they may be taken equal after scal-
ing. Then, inserting the current relations into the continuity equations gives the
system of three nonlinearly coupled partial differential equations for the three
unknown functions n, p and V:

n t = div D n (grad n − n grad V) + R(n, p)

p t = div D p (grad p + p grad V) + R(n, p)
2
λ ΔV = n − p − C(x).
The equations for the densities n and p are parabolic (assuming for a moment
that V is known) and the equation for the potential is elliptic. Note that the
potential V depends linearly but in a nonlocal way on the densities n and p and

Fig. 5.4. Intel Celeron Processor (upside)