Quantum Kinetic Theory
Electrons, Photons, Phonons
Quantum Kinetic Theory
Electrons, Photons, Phonons
FEDIR T. VASKO
OLEG E. RAICHEV
Institute of Semiconductor Physics
NAS of Ukraine, Kiev
Fedir T. Vasko Oleg E. Raichev
Institute of Semiconductor Physics, NAS Institute of Semiconductor Physics, NAS
45 Prospekt Nauki 45 Prospekt Nauki
Kiev 03028 Ukraine Kiev 03028 Ukraine
Library of Congress Control Number: 2005926337
ISBN-10: 0-387-26028-5 e-ISBN: 0-387-28041-3
Printed on acid-free paper.
©2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street,
New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly
analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,
computer software, or by similar or dissimilar methodology now known or hereafter developed
The use in this publication of trade names, trademarks, service marks and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they
are subject to proprietary rights.
Printed in the United States of America. (HAM)
9 8 7 6 5 4 3 2 1
Physical kinetics is the final section of the course of theoretical physics
in its standard presentation. It stays at the boundary between gen-
eral theories and their applications (solid state theory, theory of gases,
plasma, and so on), because the treatment of kinetic phenomena always
depends on specific structural features of materials. On the other hand,
the physical kinetics as a part of the quantum theory of macroscopic
systems is far from being complete. A number of its fundamental is-
sues, such as the problem of irreversibility and mechanisms of chaotic
responses, are now attracting considerable attention. Other important
sections, for example, kinetic phenomena in disordered and/or strongly
non-equilibrium systems and, in particular, phase transitions in these
systems, are currently under investigation. The quantum theory of mea-
surements and quantum information processing actively developing in
the last decade are based on the quantum kinetic theory.
Because a deductive theoretical exposition of the subject is not con-
venient, the authors restrict themselves to a lecture-style presentation.
Now the physical kinetics seems to be at the stage of development when,
according to Newton, studying examples is more instructive than learn-
ing rules. In view of these circumstances, the methods of the kinetic
theory are presented here not in a general form but as applications for
description of specific systems and treatment of particular kinetic phe-
The quantum features of kinetic phenomena can arise for several rea-
sons. One naturally meets them in strongly correlated systems, when it
is impossible to introduce weakly interacting quasiparticles (for exam-
ple, in a non-ideal plasma), or in more complicated conditions, such as
in the vicinity of the phase transitions. Next, owing to complexity of
the systems like superconductors, ferromagnets, and so on, the manifes-
tations of kinetic phenomena change qualitatively. The theoretical con-
vi QUANTUM KINETIC THEORY
sideration of these cases can be found in the literature. Another reason
for studying quantum features of transport and optical phenomena has
emerged in the past decades, in connection with extensive investigation
of kinetic phenomena under strong external fields and in nanostructures.
The quantum features of these phenomena follow from non-classical dy-
namics of quasiparticles, and these are the cases the present monograph
takes care of, apart from consideration of standard problems of quan-
tum transport theory. Owing to intensive development of the physics of
nanostructures and wide application of strong external (both stationary
and time-dependent) fields for studying various properties of solids, the
theoretical methods presented herein are of current importance for anal-
ysis and interpretation of the experimental results of modern solid state
This monograph is addressed to several categories of readers. First,
it will be useful for graduate students studying theory. Second, the top-
ics we cover should be interesting for postgraduate students of various
specializations. Third, the researchers who want to understand the back-
ground of modern theoretical issues in more detail can find a number
of useful results here. The phenomena we consider involve kinetics of
electron, phonon, and photon systems in solids. The dynamical prop-
erties and interactions of electrons, phonons, and photons are briefly
described in Chapter 1. Further, in Chapters 2−8, we present main the-
oretical methods: linear response theory, various kinetic equations for
the quasiparticles under consideration, and diagram technique. The pre-
sentation of the key approaches is always accompanied by solutions of
concrete problems, to illustrate applications of the theory. The remain-
ing chapters are devoted to various manifestations of quantum transport
in solids. The choice of particular topics (their list can be found in the
Contents) is determined by their scientific importance and methodolog-
ical value. The 268 supplementary problems presented at the end of the
chapters are chosen to help the reader to study the material of the mono-
graph. Focusing our attention on the methodical aspects and discussing
a great diversity of kinetic phenomena in line with the guiding principle
“a method is more important than a result,” we had to minimize both
detailed discussion of physical mechanisms of the phenomena considered
and comparison of theoretical results to experimental data.
It should be emphasized that the kinetic properties are the impor-
tant source of information about the structure of materials, and many
peculiarities of the kinetic phenomena are used for device applications.
These applied aspects of physical kinetics are not covered in detail either.
However, the methods presented in this monograph provide the theoret-
ical background both for analysis of experimental results and for device
simulation. In the recent years, these theoretical methods were applied
for the above-mentioned purposes so extensively that any comprehensive
review of the literature seems to be impossible in this book. For this
reason, we list below only a limited number of relevant monographs and
Fedir T. Vasko
Oleg E. Raichev
Kiev, December 2004
1. J. M. Ziman, Electrons and Phonons, the Theory of Transport Phenomena in
Solids, Oxford University Press, 1960.
2. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, W. A. Benjamin,
Inc., New York, 1962.
3. A. A. Abrikosov, L. P. Gor’kov and I. E. Dzialoszynski, Methods of Quantum
Field Theory in Statistical Physics, Prentice-Hall, 1963.
4. S. Fujita, Introduction to Non-Equilibrium Quantum Statistical Mechanics,
Saunders, PA, USA, 1966.
5. D. N. Zubarev, Nonequilibrium Statistical Thermodynamics, Consultants Bu-
reau, New York, 1974.
6. E. M. Lifshitz and L. P. Pitaevski, Physical Kinetics, Pergamon Press, Oxford,
7. H. Bottger and V. V. Bryksin, Hopping Conduction in Solids, VCH Publishers,
Akademie-Verlag Berlin, 1985.
8. V. L. Gurevich, Transport in Phonon Systems (Modern Problems in Condensed
Matter Sciences, Vol. 18), Elsevier Science Ltd., 1988.
9. V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Metals and Semi-
conductors (Modern Problems in Condensed Matter Sciences, Vol. 19), Elsevier Sci-
ence Ltd., 1987.
10. A. A. Abrikosov, Fundamentals of the Theory of Metals, North-Holland, 1988.
11. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic
Properties of Semiconductors, World Scientific, Singapore, 1990.
12. N. N. Bogolubov, Introduction to Quantum Statistical Mechanics, Gordon and
13. G. D. Mahan, Many Particle Physics, Plenum, New York, 1993.
14. H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of
Semiconductors, Springer, Berlin, 1997.
15. Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 1997.
16. D. K. Ferry and S. M. Goodnick, Transport in Nanostructures, Cambridge
University Press, New York, 1997.
17. R. P. Feynmann, Statistical Mechanics, Addison-Wesley, 1998.
18. A. M. Zagoskin, Quantum Theory of Many-Body Systems: Techniques and
Applications, Springer-Verlag, New York, 1998.
19. F. T. Vasko and A. V. Kuznetsov, Electron States and Optical Transitions in
Semiconductor Heterostructures, Springer, New York, 1998.
viii QUANTUM KINETIC THEORY
20. J. Rammer, Quantum Transport Theory (Frontiers in Physics, Vol. 99), West-
view Press, 1998.
21. T. Dittrich, P. Hänggi, G.-L. Ingold, B. Kramer, G. Schön, and W. Zverger,
Quantum Transport and Dissipation, Wiley-VCH, Weinheim, 1998.
22. B. K. Ridley, Quantum Processes in Semiconductors, Oxford University Press,
23. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-
mation, Springer, Berlin, Heidelberg, New York, 2000.
1. D. N. Zubarev, Double-Time Green’s Functions, Sov. Phys. - Uspekhi 3, 320
2. R. N. Gurzhi and A. P. Kopeliovich, Low-Temperature Electrical Conductivity
of Pure Meta