Multigroup Equations for the Description of the Particle Transport in Semiconductors (Series on Advances in Mathematics for Applied Sciences, V. 70) (Series ... in Mathematics for Applied Sciences)

247 Pages·2005·8.85 MB·English

by  Martin Galler

MULTIGROUP EQUATIONS
FOR THE DESCRIPTION OF
THE PARTICLE TRANSPORT
IN SEMICONDUCTORS
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Series on Advances in Mathematics for Applied Sciences - Vol. 70
MULTIGROUP EQUATIONS
FOR THE DESCRIPTION OF
THE PARTICLE TRANSPORT
IN SEMICONDUCTORS
Martin Galler
Graz University of Technology, Austria
r p
World Scientific
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NEW JERSEY LONOON SINGAPORE * BElJlNG SHANGHAI HONG KONG TAIPEI CHENNAI
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Library of Congress Cataloging-in-PublicationD ata
Galler, Martin, 1977-
Multigroup equations for the description of the particle transport in
semiconductors/ Martin Galler.
p. cm. -- (Series on advances in mathematics for applied sciences ; v. 70)
ISBN 981-256-355-5 (alk.p aper)
1. Transport theory--Mathematics. 2. Semiconductors--Mathematics. I.
Title. 11. Series.
QC793.3.V G35 2005
530.13’8-dc22
200504943 I
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Preface
Accurate semiconductor device simulation is mainly based on Monte Carlo
methods. However, there are essential advantages gained by directly solving
the Bloch-Boltzmann-Peierls equations, which govern the dynamics of car-
riers and phonons in semiconductors. In this book, an attempt is made to
introduce such deterministic solution techniques, called multigroup model
equations, especially for describing the particle transport in 111-V com-
pound semiconductors.
First, we present a multigroup model to the Boltzmann equations gov-
erning the transient transport regime in polar semiconductors. Special
effort is invested in an accurate description of the coupled hot-electron hot-
phonon system. The related conservation laws for the electron density and
the total energy density of the multigroup model equations are deduced.
This physically motivated, discrete model is used for studying the transport
properties of indium phosphide and gallium arsenide in response to a time-
depending external electric field. The results are compared to experimental
and theoretical data.
Second, a multigroup transport model for describing degenerated car-
rier gases is deduced. These model equations are based on a general carrier
dispersion law and contain the full quantum statistics of both, the carriers
and the phonons. We prove the boundedness of the solution according to
the Pauli principle and study the conservational properties of the multi-
group equations. Moreover, the existence of a Lyapounov functional to the
proposed model equations is proved and expressions for the equilibrium
solution are given.
Furthermore, the two-dimensional electron transport at an AlGaN/GaN
heterojunction in the presence of strain polarization fields is simulated with
the help of a multigroup model. The envelope wave functions for the con-
vii
viii Multigroup Equations for Particle Transport in Semiconductors
fined electrons are calculated using a self-consistent Poisson-Schrodinger
solver. The electron gas degeneracy and hot phonons are included in these
transport equations.
Finally, a multigroup-WEN0 solver for the non-stationary Boltzmann-
Poisson system for semiconductor device simulation is constructed. The
proposed numerical technique is applied for investigating the carrier trans-
port in bulk silicon, in a silicon n+ - n - n+ diode, in a silicon MESFET and
in a silicon MOSFET as well as in bulk GaAs, in a GaAs n+ - n - n+ diode
and in a GaAs MESFET. Additionally, the obtained results are compared
to those of a full WEN0 solver and Monte Carlo simulations.
This book is based in the doctoral thesis, which I wrote at the Insti-
tute of Theoretical and Computational Physics of the Graz University of
Technology. First of all I would like to thank my supervisors, Prof. Dr. Fer-
dinand Schiirrer and Prof. Dr. Armando Majorana. It was Prof. Schiirrer
who encouraged me to write both the diploma thesis and the doctoral the-
sis in the field of transport theory. His special way of motivating persons,
his critical reading of my work and the various discussions, which often
exceeded his personal frame of time, proves to be invaluable for me. Thank
you.
During my doctoral studies I got the possibility to participate in the
IHP-project "Hyperbolic and Kinetic Equations: Asymptotics, Numerics,
Analysis (HYKE)" of the European Community. In the course of this
project, I had the pleasure to enjoy the Italian hospitality of Prof. Majo-
rana and his colleagues at the Dipartimento di Matematica e Informatica
dell'Universit8 di Catania. My very exciting and interesting three months
stay in Catania submontane the Etna greatly enriched both my mathemat-
ical experience and the knowledge of the Italian culture. Thank you very
much.
Moreover, I would like to thank my family and my friends who supported
me in a way only they can do.
Finally, I acknowledge the financial support of my doctoral thesis by
the Fond zur Forderung der wissenschaftlichen Forschung, Vienna, contract
numbers P14669-TPH and P17438-N08, and by the European community
program IHP, under the contract number HPRN-CT-2002-00282 on behalf
of the CNR.
M. Galler
Contents
Preface vii
1. Introduction 1
2 . The Bloch-Boltzmann-Peierls Equations 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Electrons in Semiconductors . . . . . . . . . . . . . . . . . . 5
2.3 Phonons in Semiconductors . . . . . . . . . . . . . . . . . . 9
2.4 Scattering Mechanisms . . . . . . . . . . . . . . . . . . . . . 11
2.4.1 General Theory of Scattering . . . . . . . . . . . . . 12
2.4.2 Phonon Scattering . . . . . . . . . . . . . . . . . . . 14
2.4.2.1 Non-polar Phonon Scattering . . . . . . . . 15
2.4.2.2 Polar Phonon Scattering . . . . . . . . . . . 19
2.4.3 Ionized Impurity Scattering . . . . . . . . . . . . . . 22
2.5 Semiclassical Dynamics of Electrons . . . . . . . . . . . . . 24
2.6 The Bloch-Boltzmann-Peierls Equations . . . . . . . . . . . 26
2.7 Mathematical Properties of the BBP Equations . . . . . . . 32
3 . Multigroup Model Equations for Polar Semiconductors 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Multigroup Equations to the Bloch-Boltzmann-Peierls Equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 The Electron Boltzmann Equation . . . . . . . . . . 40
3.2.2 The LO Phonon Boltzmann Equation . . . . . . . . 45
3.2.3 The Coupling POP Interaction Term . . . . . . . . . 47
3.2.4 The Evaluation of the Collision Coefficients . . . . . 51
ix