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References
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model PDE systems, the so called doping profile, is obtained from process mod-
eling (nonlinear diffusion), and device model results are then assembled into
models for integrated circuits (large ODE systems). As discussed in detail the
drift-diffusion-Poissonsystemconsistsoftwoparabolicdifferentialequationsof
Fokker–Planck type for the position densities of (negatively charged) electrons
and (positively charged) holes, coupled nonlinearly to a Poisson equation for the
electrical potential. An important extension of the drift-diffusion equations is
given by energy transport systems, which take into account convection driven by
particle temperatures. The temperatures satisfy nonlinear transport equations,
too, which are coupled to the drift-diffusion equations for the particle densities.
We refer to [1] for detail. Further modeling detail is provided by hydrodynamic
semiconductor systems [3], which are – for each carrier species – compress-
ible Euler equations (with position density, particle velocity and temperature as
unknowns)withconvectiveforcing provided by theelectricfieldand with mo-
mentum and energy relaxation due to the dominant inelastic collisions with the
semiconductor crystal lattice (phonons). All the above mentioned charge carrier
transport models are based and can be derived from the semiclassical semicon-
ductor Boltzmann equation. This is a phase space model, posed in position-wave
vector space, balancing charge carrier transport in semiconductor crystals (tak-
ing into account the crystal’s energy band structure) with dominantly inelastic
collisions (typically elastic particle-particle collisions are a second order effect
in semiconductors). Fully quantistic models like the quantum-drift-diffusion
system [3], the quantum hydrodynamics system and quantum Boltzmann equa-
tions are able to model spurious quantistic effects in current device technology
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as well as quantum semiconductor devices like tunnelling diodes .Due to their
high complexity these models were not and are still not (yet in 2006) used for
general design-oriented simulations of semiconductor devices as they occur in
the chipsets depicted in the Images 5.1–5.6.

References

[1] F.Brezzi,L.D.Marini,S.Micheletti,P.Pietra,R.Sacco,andS.Wang,Finiteel-
ement and finite volume discretizations of Drift-Diffusion type fluid models
for semiconductors, in Handbook of Numerical Analysis, Volume XIII: Spe-
cial Volume: Numerical Methods in Electromagnetics, W.H.A. Schilders,
E.J.W. ter Maten, Guest ed., P.G. Ciarlet ed., Elsevier, Amsterdam, 317–441,
2005
[2] G.F. Carey, B. Mulvaney, W.B. Richardson and C.S.Reed, CIRCUIT, DE-
VICE, AND PROCESS SIMULATION: Mathematical and Numerical Aspects,
John Wiley & Sons, 1996

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