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5 Semiconductor Modeling
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Silicon transistors were invented in the USA, more precisely at Bell Labs
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3
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in the year 1947, by J. Bardeen ,W.AShockley and W. Brattain . The first
transistor had a dimension of about one centimetre, today’s MOS transistors
feature a gate length of less than 100 nanometres! Consequently, the integrated
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circuit was invented in the year 1958, by J. Kilby at Texas Instruments, putting
together a few devices on one chip. Today’s world would be entirely different
without semiconductor devices …
Themathematicalmodelingandsimulationofsemiconductorstructuresacts
on different levels:
1. Modeling of the doping process (= fabrication of the device/chip) by nonlin-
ear diffusion equations. The main goal is to better understand and control
the doping process. We refer to the book [6].
2. Modeling of the electrical functioning of individual semiconductor devices.
Here the doping profile is taken as input function and the current flow
in the device is modeled exploiting insights into the solid state physics of
semiconductors. A wealth of information on this subject can be found in the
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books of the Springer Series on Computational Microelectronics , edited by
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Siegfried Selberherr . In particular we refer to [4] and [5].
3. VLSI circuit modeling. The functioning of a whole chip is modeled by using
the individual device model result as input. Typically very large systems
of ordinary differential equations are obtained which have to be solved
numerically in a fast way. We refer to [2].
In the sequel we shall focus on the second level, that means on mathematical
semiconductor modeling. Clearly, the basis for this is quantum particle physics
and, in a semiclassical framework, solid state physics. There is a semiconduc-
tor modeling hierarchy, based on different scales and accuracy levels of the
description. Roughly speaking, it looks like this:
Quantum Mechanical Modeling relies on the Schrödinger equation and vari-
ousequivalentformulations(Heisenbergformalism,Wignertransportequation,
quantum hydrodynamics). There are (somewhat exotic) semiconductor devices
whose performance is entirely based on quantistic phenomena, e.g. resonant
tunnelling diodes. In many other devices, however, spurious quantistic effects
occur. In both cases, classical mechanics or even a semiclassical framework do
not suffice.
Semiclassical Modeling relies on the Boltzmann equation of solid state
physics. This is a phase-space based integro-differential equation, describing the

2 http://nobelprize.org/physics/laureates/1956/bardeen-bio.html
3 http://nobelprize.org/physics/laureates/1956/shockley-bio.html
4 http://nobelprize.org/physics/laureates/1956/brattain-bio.html
5 http://nobelprize.org/physics/laureates/2000/kilby-autobio.html
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http://www.springer.com/sgw/cda/frontpage/0,11855,1-40109-69-1187595-0,00.html
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http://info.tuwien.ac.at/histu/pers/12152.html