My Research Interests
Swiss National Science Foundation Project 2100-57176.99
Collaborators: Dr. Mirela Borici, dipl. ing. Damir Pasalic.
The aim of our work is to develop a device simulator (especially
for the High Electron Mobility Transistor HEMT) which can be
coupled to a full-wave simulation model based on the finite difference
time-domain method (FDTD). The large experience of our institute with
computational electromagnetics will be a big advantage when the
HEMT simulator has to be coupled to the FDTD field solver.
As described in the research plan of our project, our group
had to enter the new field of numerical semiconductor device modelling.
In the first year of our project, we gained experience
with the numerical simulation
of the hydrodynamic model (HDM). This model allows a more accurate
description of electron transport than the conventional drift diffusion
model or energy balance models, which are used in commercial semiconductor
device simulators. In contrast to the drift diffusion model,
the HDM predicts shock waves in the electron gas. This
necessitates the use of shock capturing algorithms (i.e. upwind methods).
You can download here a simple but instructive
which simulates the electron density, electron velocity,
current (and temperature etc etc) inside a ballistic
diode (i.e. an n+ - n -n+ doped piece of silicon) within a full
The units in the program are scaled: E.g., the mass unit is 10^-7 kg,
lenghts are measured in micrometers (see the header of the program).
The thickness of the diode is 0.3 mu and it is split into three equally
thick layers of 0.1 mu. The contact layers are highly doped
(5*10^23 donors/cm^3), the middle layer has a donor concentration of
2*10^21 cm^-3. The applied bias is rising from 0 to 1 volt during
the simulation. Finally, a hydrodynamic shock develops, visible in
the upper curve at grid point 80.
The MATLAB file: hydro.m.
The initialization file: initializer.mat.
Details can be found in this paper.
Furthermore, we developed a Monte Carlo
program which allowed us to obtain the energy-dependent transport parameters
in the HDM for GaAs- or InP-like materials,
following mainly the lines given in the famous textbook of
K. Tomizawa. In this program,
all important scattering mechanisms are taken into account:
Impurity scattering, acoustic phonon scattering, polar and non-polar
optical phonon scattering. Also parameters which describe the
nonparabolicity of the Gamma- and L-bands are taken into
account. As a result, we obtained impulse and energy relaxation
times as a function of the mean electron energy as well as the
energy-dependent electron mass. These results were aproximated by
polynomial fitting functions which made it possible to use them
for hydrodynamic simulations.
Of course, also the dependence between electric field strength
and electron mobility as it is used in drift diffusion models can be
extracted from the Monte Carlo simulations. These Monte Carlo results
are compared to an analytic mobility model used in device simulations
The HDM was applied in one dimension to a silicon
ballistic diode (which models the channel of a MOSFET)
and in two dimensions to a GaAs MESFET structure.
shows a stationary shock wave inside the ballistic diode.
Such shock waves also appear in small MESFET devices and cause
numerical instabilities, if naive finite difference discretization
is used. Using first and second upwind discretization, such
instabilities can be avoided, although the timesteps that
have to be chosen are rather small (of the order of a femtosecond).
But this will play no special role when full wave propagation inside a
device has to be calculated, because the stability of FDTD
simulations makes even smaller timesteps necessary.
The relatively simple structure of the MESFET was well-suited to
gain experience with two-dimensional hydrodynamic simulations,
since the problem of the heterojunction which is present for the HEMT
can be circumvented. Upwind discretization works also in two dimensions
very well. Stability problems were observed near
the Schottky gate, where the electron density can be extremely small
and the electric field strength very large. But we were able to
stabilize this critical region by limiting the electron drift
velocity to physical values without changing the important
characteristics of the device. See also Figure 2.
Our results were in good agreement with the literature,
although there are not too many data available.
For the one-dimensional calculations we used MATLAB as a comfortable
programming language, but for the computationally expensive calculations
for the 2D HDM we used Fortran 90.
The numerical solution of the
Poisson equation was obtained by a multigrid algorithm, which made
it possible to perform the calculations on our local workstations
at the ETH.
The grid used was homogeneous in x- and y-direction, respectively,
and contained three coarser subgrids. An approximate global solution
of the Poisson equation can be calculated by the standard Gauss-Seidel
method on the coarsest grid, which is then tranferred to the finer grids
and improved by several correction cycles.
S.M. Ghazaly used a SOR method, but on a massively parallel
machine (MasPar MP-2) with an array of 64 times 128 processing elements.
It is known that the multigrid method is the fastest method for solving
the Poisson equation; we also observed that it is not necessary
to solve the Poisson equation at each time step, which made it possible
to speed up the simulation times by a factor of about ten.
It is well possible that multigrid methods will also provide a means
to simulate HEMT devices, where the heterojunction region has to be described
on a much smaller scale than the surrounding layers.
Thus we were able to follow the lines given by our project schedule
for phase 1 of our project. Since the presence of heterojunctions e.g. in a
HEMT poses additional problems from the numerical and physical point
of view, we also concentrated on this subject.
In this case we deviated slightly
from the initial research plan, where we intended to consider heterostructures
in HEMTs in the second year of our project. We performed also one-dimensional
simulations of quantum well structures, based on relatively simple
(thermoionic) physical models for the heterojunction region and the drift
diffusion model for the homogeneous semiconductor material.
This way we gained some experience which
will become important when we turn our attention to
numerical HEMT modelling.
During the first half of this century, achievements in Europe dominated
progress in the physics, from the discovery of the electron to the atomic
nucleus and its constituents, from special relativity to quantum mechanics.
Sadly, the conflicts of the 1930s and 40s interrupted this as many scientists
had to leave for calmer shores. The return of peace heralded some decisive
changes. By the early 50s, the Americans had understood that further progress
needed more sophisticated instruments, and that investment in basic science
could drive economic and technological development. While scientists in Europe
still relied on simple equipment based on radioactivity and cosmic rays,
powerful accelerators were being built in the US. Table-top experiments were
being overtaken by projects involving large teams of scientists and engineers.
A few far-sighted physicists, such as Rabi, Amaldi, Auger and de Rougemont,
perceived that co-operation was the only way forward for front-line research in
Europe. Despite fine intellectual traditions and prestigious universities, no
European country could cope alone. The creation of a European Laboratory was
recommended at a UNESCO meeting in Florence in 1950, and less than three years
later a Convention was signed by 12 countries of the Conseil Européen
pour la Recherche Nucléaire. CERN was born, the prototype of a chain of
European institutions in space, astronomy and molecular biology, and Europe was
poised to regain its illustrious place on the scientific map.
CERN exists primarily to provide European physicists with accelerators that meet
research demands at the limits of human knowledge. In the quest for higher
interaction energies, the Laboratory has played a leading role in developing
colliding beam machines. Notable "firsts" were the Intersecting Storage Rings
(ISR) proton-proton collider commissioned in 1971, and the proton-antiproton
collider at the Super Proton Synchrotron (SPS), which came on the air in 1981
and produced the massive W and Z particles two years later, confirming the
unified theory of electromagnetic and weak forces.
The main impetus at present if from the Large Electron-Positron Collider (LEP),
where measurements unsurpassed in quantity and quality are testing our best
description of sub-atomic Nature, the Standard Model, to a fraction of 1% soon
to reach one part in a thousand. By 1996, the LEP energy was doubled to 90 GeV
per beam in LEPII, opening up an important new discovery domain. More high
precision results are expected in abundance throughout the rest of the decade,
which should substantially improve our present understanding. The LEP/LEPII
missions will by then be largely completed.
LEP data are so accurate that they are sensitive to phenomena that occur at
energies beyond those of the machine itself; rather like delicate measurement
of earthquake tremors far from an epicentre. This gives us a "preview" of exciting
discoveries that may be made at higher energies, and allow us to calculate the
parameters of a machine that can make these discoveries. All evidence indicates
that new physics, and answers to some of the most profound questions of our time,
lie at energies around 1 TeV (1 TeV = 1,000 GeV).
To look for this new physics, the next research instrument in Europe's particle
physics armoury is the LHC. In keeping CERN's cost-effective strategy of
building on previous investments, it is designed to share the 27-kilometre LEP
tunnel, and be fed by existing particle sources and pre-accelerators. A
challenging machine, the LHC will use the most advanced superconducting magnet
and accelerator technologies ever employed. LHC experiments are, of course,
being designed to look for theoretically predicted phenomena. However, they
must also be prepared, as far as is possible, for surprises. This will require
great ingenuity on the part of the physicists and engineers.
The LHC is a remarkably versatile accelerator. It can collide proton beams with
energies around 7-on-7 TeV and beam crossing points of unsurpassed brightness,
providing the experiments with high interaction rates. It can also collide
beams of heavy ions such as lead with a total collision energy in excess of
1,250 TeV, about thirty times higher than at the Relativistic Heavy Ion
Collider (RHIC) under construction at the Brookhaven Laboratory in the US.
Joint LHC/LEP operation can supply proton-electron collisions with 1.5 TeV
energy, some five times higher than presently available at HERA in the DESY
laboratory, Germany. The research, technical and educational potential of the
LHC and its experiments is enormous.
Anyhow, at these "incredible" particle energies that are achieved in the
LHC, where the protons have an energy which is 7000 times their rest mass
energy, the Coulomb field of the particles in the accelerator has the shape
of a flat disk for the external observer and it contains photons of high
energy. If the fields of two heavy ions collide, important processes can
take place. It is e.g. possible that a photon-photon fusion creates an
electron-positron pair such that the negatively charged electron is
captured by the positively charged stripped nucleus. This nucleus is
lost for the collider beam, since its charge is now reduced by one
elemental charge and the accelerator fields can no longer hold the particle
on its foreseen trajectory. Pair production with capture therefore restricts
the maximum beam luminosity of the LHC; the described process has been
Phys. Rev. A 50, 3980 (1994),
and plays also a role for the production of
antihydrogen. Antihydrogen atoms
are still the most complex form of antimatter which has been produced
artificially up to now.
A further interesting process is the fusion of two photons into
a Higgs particle, the last particle in the Stardard model which has
not been detected yet experimentally (besides the graviton and many other
hypothetic particles, but this is another story).
Causal perturbation theory is one more attempt to put the theoretical
setup of (perturbative)
Quantum Field Theory on a sound mathematical basis.
One of the main problems that is solved in the (finite) causal perturbation
theory (fCPT) framework are the ultraviolet and infrared divergences
which show up in the calculation of so-called Feynman diagrams.
Using only well-defined mathematical objects, it is in fact possible to
avoid the problem of divergent quantities in fCPT from the start (if the
theory under consideration is meaningful). Therefore, subsequent
renormalization of divergent expressions is not necessary. It is a very
difficult question to what extent the different renormalization methods
and fCPT are in correspondence. But there are fundamental and conceptual
differences between the methods.
Applying fCPT to gauge theories leads to an important observation. It is
possible to derive the classical properties of QFT's from purely
quantum mechanical input as e.g. Lorentz covariance and unitarity
(Int. Journal Mod. Phys. A, vol. 14, no. 21, 3421 (1999)) .
This is a more satisfactory strategy as it is usually given in
literature, since classical physics should be derived from the
more fundamental quantum principles.
Also the notion of spontaneous symmetry breaking becomes obsolete in fCPT:
The Higgs particle is merely a consequence of the particle masses, and not
(Ann. Phys. 8, No. 5, 389 (Leipzig,1999)) .
This does of course not mean that there is no interconnection
between the Higgs particle the the mass generation mechanism.
Some of the people who worked or are still working on causal
perturbation theory and related research areas are:
Henry Epstein, Vladimir Jurko Glaser, Raymond Stora,
Günter Scharf, Dan Radu Grigore, Michael Dütsch, Andreas Aste,
Tobias Hurth, Frank Krahe, Nicola Grillo, Bert Schroer,
Jeferson De Lima Tomazelli, Jose Tadeu Teles Lunardi,
Bruto Max Pimentel, Marc Wellmann, Kostas Skenderis, Urs Walther,
Michel Dubois-Violette, Claude Viallet, L. A. Manzoni,
Florin Constantinescu, Dirk Prange, Markus Gut, Ivo Schorn,
Adrian Müller, Franz-Marc Boas, Gudrun Pinter, Walter F. Wreszinski,
José M. Gracia-Bondia, Serge Lazzarini, Klaus Bresser, Sergio Doplicher,
Klaus Fredenhagen, D. Bahns, Romeo Brunetti, Kostas Skenderis.
Created August 2000 by
Last updated January, 2003.
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