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5 Semiconductor Modeling
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Fig. 5.6. Chipset on Graphics Card

which is a version of the so-called (repulsive) mean-field equation. Clearly,
the drift-diffusion-Poisson (DDP) system has to be supplemented by initial
conditions for the position densities n and p and by boundary conditions for n,
p and V. The boundary of the semiconductor domain D usually splits into two
parts, namely the contact segments (Ohmic, Schottky or Metal-Oxide contacts),
whereDirichletdatafor n,pandV areprescribed,andintoinsulatingorartificial
boundaries, where zero outflow current densities and zero outward electric field
are prescribed resulting in homogeneous Neumann conditions for n, p and V.
Contact voltages determine the Dirichlet condition for the potential V and
Ohmic contacts are assumed to be in thermal equilibrium and to have locally
vanishing space charge density. All in all, mixed Dirichlet–Neumann boundary
conditions for the three unknown functions are prescribed.
Usually, equilibrium data for n and p are given at t = 0(whichrequiresthe
solution of the mean-field equation) and the device is driven out of equilibrium
by applying contact voltages.
The mathematical analysis of the DDP system is in a rather healthy state, we
refer to the references [4] and [5] for classical results and to the author’s publi-
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cation list for more recent work, most of which focuses on various extensions
of the DDP system. Under appropriate assumptions on the data there is exis-
tence and uniqueness of transient solutions, existence of stationary states with
uniqueness for small contact voltages (close to equilibrium) and convergence
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http://homepage.univie.ac.at/peter.markowich/publications.html