Table Of ContentQuantum Transport
in Semiconductors
PHYSICS OF SOLIDS AND LIQUIDS
Editorial Board:
Jozef T. Devreese • University of Antwerp, Belgium
Roger P. Evrard • University of Liege, Belgium
Stig Lundqvist • Chalmers University of Technology, Sweden
Gerald D. Mahan • University of Tennessee, USA
Norman H. March • University of Oxford, England
Recent Volumes in the Series:
AMORPHOUS SOLIDS AND THE LIQUID STATE
Edited by Norman H. March, Robert A. Street, and Mario P. Tosi
CHEMICAL BONDS OUTSIDE METAL SURFACES
Norman H. March
CRYSTALLINE SEMICONDUCTING MATERIALS AND DEVICES
Edited by Paul N. Butcher, Norman H. March, and Mario P. Tosi
FRACTALS
Jens Feder
INTERACTION OF ATOMS AND MOLECULES
WITH SOLID SURFACES
Edited by V. Bortolani, N.H. March, and M.P. Tosi
MANY-PARTICLE PHYSICS, Second Edition
Gerald D. Mahan
ORDER AND CHAOS IN NONLINEAR PHYSICAL SYSTEMS
Edited by Stig Lundqvist, Norman H. March, and Mario P. Tosi
POLYMERS, LIQUID CRYSTALS, AND LOW-DIMENSIONAL
SOLIDS
Edited by Norman H. March and Mario P. Tosi
QUANTUM TRANSPORT IN SEMICONDUCTORS
Edited by David K. Ferry and Carlo Jacoboni
THEORY OF THE INHOMOGENEOUS ELECTRON GAS
Edited by Stig Lundqvist and Norman H. March
A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new
volume immediately upon publication. Volumes are billed only upon actual shipment. For further information
please contact the publisher.
Quantum Transport
in Semiconductors
Edited by
David K. Ferry
Arizona State University
Tempe, Arizona
and
Carlo Jacoboni
Universita di Modena
Modena, Italy
Springer Science+ Business Media, LLC
L1brary of Congress catalog1ng-1n-Pub11cat1on Data
Qua~tum transport 10 semiconductors 1 edited by David K. Ferry and
Carlo Jacobon1.
p. cm. -- <Physics of scl1ds and liquidsl
Includes bibllographical references and index.
ISBN 978-0-306-43853-0 ISBN 978-1-4899-2359-2 (eBook)
DOI 10.1007/978-1-4899-2359-2
1. Semlconducto"S. 2. Ouantum theory. I. Ferry. David K.
II. Jacobon1. c. III. Series.
OC611.036 1992
537.6'225--dc20 91-:36620
CIP
This limited facsimile edition has been issued
for the purpose of keeping this title available
to the scientific community.
10 9 8 7 6 5 4 3 2
ISBN 978-0-306-43853-0
© 1992 Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1992
AII rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from.the Publisher
Contributors
N d' Ambrumenil, Department of Physics, University of Warwick, Coventry
CV4 7AL, United Kingdom
David K. Ferry, Center for Solid State Electronics Research, College of
Engineering and Applied Sciences, Arizona State University, Tempe,
Arizona 85287, USA
Gerald J. Iafrate, U.S. Army Research Office, Research Triangle Park, North
Carolina 27709, USA
Carlo Jacoboni, Dipartimento di Fisica, Universita di Modena, 41100
Modena, Italy
Antti-Pekka Jauho, Physics Laboratory, H. C. 0rsted Institute, University
of Copenhagen, DK2100 Copenhagen 0, Denmark. Present address:
Nordita (Nordisk Institut for Teoretisk Fysik), DK2100 Copenhagen 0,
Denmark
M. Jonson, Solid State Division, Oak Ridge National Laboratory, Oak
Ridge, Tennessee 37381-6024, USA. Present address: Institute ofTheoretical
Physics, Chalmers University of Technology, S-41296 Goteborg, Sweden
N C. Kluksdahl, Department of Electrical Engineering, Arizona State
University, Tempe, Arizona 85287, USA. Present address: Ford Aerospace,
Houston, Texas 77058, USA
A M. Kriman, Department of Electrical Engineering, Arizona State Univer
sity, Tempe, Arizona 85287, USA. Present address: Department of Electrical
and Computer Engineering, State University of New York at Bu:ffalo,
Buffalo, New York 14620, USA
v
vi Contributors
Gerald D. Mahan, Department of Physics, University of Tennessee, Knox
ville, Tennessee 37830, USA, and Solid State Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37381-6024, USA
Lino Reggian~ Departimento di Fisica e Centro Interuniversitario di Strut
tura della Materia, Universita di Modena, 41100 Modena, Italy
C. Ringhofer, Department of Mathematics, Arizona State University, Tempe,
Arizona 85287, USA
K. K. Thornber, NEC Research Institute, Princeton, New Jersey 08540, USA
Preface
The majority of the chapters in this volume represent a series of lectures.
that were given at a workshop on quantum transport in ultrasmall electron
devices, held at San Miniato, Italy, in March 1987. These have, of course,
been extended and updated during the period that has elapsed since the
workshop was held, and have been supplemented with additional chapters
devoted to the tunneling process in semiconductor quantum-well structures.
The aim of this work is to review and present the current understanding in
nonequilibrium quantum transport appropriate to semiconductors. Gen
erally, the field of interest can be categorized as that appropriate to
inhomogeneous transport in strong applied fields. These fields are most
likely to be strongly varying in both space and time.
Most of the literature on quantum transport in semiconductors (or in
metallic systems, for that matter) is restricted to the equilibrium approach,
in which spectral densities are maintained as semiclassical energy
conserving delta functions, or perhaps incorporating some form of collision
broadening through a Lorentzian shape, and the distribution functions are
kept in the equilibrium Fermi-Dirac form. The most familiar field of
nonequilibrium transport, at least for the semiconductor world, is that of
hot carriers in semiconductors. Here, the major problem is actually determin
ing the form of the distribution function that arises in the high fields, and
this is generally done by solving the Boltzmann transport equation with the
scattering processes introduced through the Fermi golden rule and an
energy-conserving delta function for the spectral density. When one then
moves to nonequilibrium quantum transport, the situation is complicated
by the fact that now the spectral density deviates from the simple forms
and must also be determined along with the actual distribution function.
The existence of a far-from-equilibrium steady state in the strong applied
fields depends upon achieving a balance between the driving forces and
the dissipative forces (as well as the boundary condition), and this usually
entails a self-consistent approach to determining the spectral densities and
vii
viii Preface
the distribution function. As in the semi-classical regime, many approaches
and approximations have been taken to try to achieve this end. In this
volume we review a number of these different approaches. This is still a
rapidly evolving field though, and no single approach has yet to begin to
provide the required degree of understanding that one would desire. It is
hoped that this volume will stimulate others to take up the field and that,
through much more work, understanding will be achieved.
The editors of this volume would like to thank Dr. Larry Cooper of
the Office of Naval Research for making the original workshop happen and
for providing continuing support for the work leading to this volume. In
addition, many people helped throughout the preparation of the final
chapters. Of particular note, though, was the work of Drs. Rosella Brunetti,
Lino Reggiani, and AI Kriman, and we extend our thanks to them.
David K. Ferry
Carlo Jacoboni
Contents
1. Principles of Quantum Transport
Carlo Jacoboni
1.1. Introduction . . . . . . . . . . . . 1
1.2. The General Problem . . . . . . . . 3
1.3. Quantum Dynamics and Representations 4
1.4. The Density Matrix . 6
1.5. Second Quantization 8
1.6. Green's Functions 10
1. 7. Wigner Functions . . 11
1.8. Kinetic Equations and Irreversibility 14
2. The Kubo Formula and Linear Response
David K Ferry
2.1. Linear Response Theory . . . . . 18
2.2. The Zero-Frequency Form .... 21
2.3. Relaxation and Green's Functions 22
2.4. Some Examples for the Conductivity 23
2.4.1. The Metallic Conductivity . . 23
2.4.2. Localized Conductivity in the Site Approximation 25
2.5. Extension to Two-Time Functions . . . • • . 28
2.5.1. The Quasiequilibrium Statistical Operator 29
2.5.2. The Balance Equations 32
References 35
3. Path Integral Method: Use of Feynman Path
Integrals in Quantum Transport Theory
K K Thornber
3.1. Introduction ..•.... 37
3.2. Formulation of the Problem 38
ix
