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5 Semiconductor Modeling
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of close-to-equilibrium transient solutions to the unique thermal equilibrium
state (determined by the mean field equation) in the large-time limit. (if, say,
contact voltages are turned off after some time). These qualitative results are
complemented by a series of quantitative results of singular perturbation type,
more specifically in the limit λ tending to zero. Note that, typically, at inter-
faces between positively and negatively doped device subdomains (so called
pn-junctions), the doping profile has a very large gradient and, in fact, is very
well approximated by a discontinuous function. When λ is set to zero in the
Poisson equation, then the global charge-neutrality equation

0 = n − p − C(x), x ∈ D

results, which implies that at least one of the limiting densities n or p has to
be discontinuous (it turns out that both are!). Elliptic and parabolic regularity
theory implies, however, that – for λ small but nonzero – the functions n, p
and V are continuous in the interior of D! This is a well-known phenomenon in
singular perturbation theory, which indicates that there is a very thin layer (of
width O(λ) roughly speaking) around pn-junctions, within which the densities n
and p have a very large gradient in orthogonal direction to the junction. Clearly,
this interior layer structure of the solutions has to be well taken into account
when numerical discretisation schemes are devised and it renders the design of
discretisation schemes and grids highly nontrivial. We refer to the webpage of
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Paola Pietra for references on this subject.
Recent mathematical/numerical efforts have gone into inverse and optimisa-
tion problems concerned with the DDP system. In particular the identification
of the doping profile from current-voltage or from capacitance measurements
for the sake of quality control in device manufacture and the design of doping
profiles according to certain optimality criteria is of great practical importance
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(see the author’s publication list and the webpage of Martin Burger for more
information).
Comments on the Images 5.1–5.6 The Images 5.1–5.6 show chipsets of modern
computers. Each of them consists of a huge number (many millions) of semicon-
ductor devices, typically MOS-transistors. The continuous and rapid advance
of computer technology relies on an interplay of numerical simulations and
engineering insights used for the design of prototypes of new semiconductor
technology, which then becomes absorbed into new mainstream chipsets. At the
very basis of this is the modeling of individual semiconductor devices (typically
MOS technology) using the drift-diffusion-Poisson system, energy transport
and hydrodynamical models. Simulation runs of the semiconductor Boltzmann
equation are often used to provide benchmarks for the macroscopic param-
eters like diffusivities and charge carrier mobilities. The input for the device

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http://www.imati.cnr.it/∼pietra
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http://www.indmath.uni-linz.ac.at/people/burger