Quantum Kinetic Theory and Applications - Electrons, Photons, Phonons

790 Pages·2005·4.713 MB·English

by  Vasko F.T., Raichev O.E.

Quantum Kinetic Theory
and Applications
Electrons, Photons, Phonons
Quantum Kinetic Theory
and Applications
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 ofCongress Control Number:2005926337
ISBN-10:0-387-26028-5 e-ISBN:0-387-28041-3
ISBN-13:978-0387-26028-0
Printed on acid-free paper.
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Preface
Physicalkineticsisthefinalsectionofthecourseoftheoreticalphysics
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, arecurrently under investigation. Thequantumtheory 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-
nomena.
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-
v
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
ofkineticphenomenaunderstrongexternalfieldsandinnanostructures.
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
physics.
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,theresearcherswhowanttounderstandtheback-
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
thequasiparticlesunderconsideration, anddiagramtechnique. Thepre-
sentation of the key approaches is always accompanied by solutions of
concrete problems, to illustrate applications of the theory. The remain-
ingchaptersaredevotedtovariousmanifestationsofquantumtransport
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
chaptersarechosentohelpthereadertostudythematerialofthemono-
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.
Theseappliedaspectsofphysicalkineticsarenotcoveredindetaileither.
However, the methods presented in this monograph provide the theoret-
ical background both for analysis of experimental results and for device
PREFACE vii
simulation. In the recent years, these theoretical methods were applied
fortheabove-mentionedpurposessoextensivelythatanycomprehensive
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
reviews.
Fedir T. Vasko
Oleg E. Raichev
Kiev, December 2004
Monographs:
1. J. M. Ziman, Electrons and Phonons, the Theory of Transport Phenomena in
Solids, Oxford University Press, 1960.
2. L.P.KadanoffandG.Baym,QuantumStatisticalMechanics,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,
1981.
7. H.BottgerandV.V.Bryksin,Hopping Conduction in Solids,VCHPublishers,
Akademie-Verlag Berlin, 1985.
8. V.L.Gurevich,TransportinPhononSystems(ModernProblemsinCondensed
Matter Sciences, Vol. 18), Elsevier Science Ltd., 1988.
9. V.F.GantmakherandY.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,Gordonand
Breach, 1992.
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(FrontiersinPhysics,Vol. 99),West-
view Press, 1998.
21. T. Dittrich, P. H¨anggi, G.-L. Ingold, B. Kramer, G. Sch¨on, and W. Zverger,
Quantum Transport and Dissipation, Wiley-VCH, Weinheim, 1998.
22. B.K.Ridley,QuantumProcessesinSemiconductors,OxfordUniversityPress,
1999.
23. D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Infor-
mation, Springer, Berlin, Heidelberg, New York, 2000.
Reviews:
1. D. N. Zubarev, Double-Time Green’s Functions, Sov. Phys. - Uspekhi 3, 320
(1960).
2. R. N. Gurzhi and A. P. Kopeliovich, Low-Temperature Electrical Conductivity
of Pure Metals, Sov. Phys. - Uspekhi 133, 33 (1981).
3. T.Ando,A.B.Fowler,andF.Stern,ElectronicPropertiesofTwo-Dimensional
Systems, Rev. Mod. Phys. 54, 437 (1982).
4. J. Rammer and H. Smith, Quantum Field-Theoretical Methods in Transport
Theory of Metals,Rev. Mod. Phys. 58,323(1986);J.Rammer,Quantum Transport
Theory of Electrons in Solids: A Single-Particle Approach,Rev. Mod. Phys. 63,781
(1991).
5. G. D. Mahan, Quantum Transport Equation for Electric and Magnetic Fields,
Physics Reports 145, 251 (1987).
6. W. R. Frensley, Boundary Conditions for Open Quantum Systems Driven Far
from Equilibrium, Rev. Mod. Phys. 62, 745 (1990).
7. B. Kramer and A. MacKinnon, Localization: Theory and Experiment, Rep.
Prog. Phys. 56, 1469 (1993).
8. C. H. Henry and R. F. Kazarinov, Quantum Noise in Photonics, Rev. Mod.
Phys. 68, 801 (1996).
9. C.W.J.Beenakker,Random-MatrixTheoryofQuantumTransport,Rev. Mod.
Phys. 69, 731 (1997).
10. Ya. M. Blanter and M. Buttiker, Shot Noise in Mesoscopic Conductors,
Physics Reports 336, 1 (2000).
11. P. Lipavsky, K. Morawetz, and V. Spicka, Kinetic Equation for Strongly In-
teracting Dense Fermi Systems, Annales de Physique 26, 1 (2001).
Contents
Preface v
1. ELEMENTS OF QUANTUM DYNAMICS 1
1. Dynamical Equations 1
2. S-Operator and Probability of Transitions 6
3. Photons in Medium 11
4. Many-Electron System 17
5. Electrons under External Fields 25
6. Long-Wavelength Phonons 35
Problems 46
2. ELECTRON-IMPURITY SYSTEM 51
7. Kinetic Equation for Weak Scattering 51
8. Relaxation Rates and Conductivity 57
9. Quasi-Classical Kinetic Equation 65
10. Multi-Photon Processes 72
11. Balance Equations 79
12. Conductance of Microcontacts 87
Problems 94
3. LINEAR RESPONSE THEORY 99
13. Kubo Formula 99
14. Diagram Technique 107
15. Bethe-Salpeter Equation 115
16. Green’s Function as a Path Integral 123
17. Dispersion of Dielectric Permittivity 130
ix
x QUANTUM KINETIC THEORY
18. Interband Absorption under External Fields 139
Problems 147
4. BOSONS INTERACTING
WITH ELECTRONS 155
19. Kinetic Equation for Boson Modes 155
20. Spontaneous and Stimulated Radiation 162
21. Phonon Instabilities 169
22. Boson Emission by 2D Electrons 176
Problems 184
5. INTERACTING PHONON SYSTEMS 189
23. Phonon-Phonon Collisions 189
24. Thermal Conductivity of Insulators 197
25. Balance Equations for Phonons 201
26. Relaxation of Long-Wavelength Phonons 207
27. Polaritons and Dielectric Function of Ionic Crystals 215
Problems 224
6. EFFECTS OF ELECTRON-ELECTRON
INTERACTION 229
28. Hartree-Fock Approximation 229
29. Shift of Intersubband Resonance 235
30. Exciton Absorption 243
31. Electron-Electron Collision Integral 250
32. Coulomb Drag Between 2D Electrons 256
33. Dynamical Screening 262
Problems 273
7. NON-EQUILIBRIUM ELECTRONS 281
34. Electron-Boson Collision Integral 281
35. Quasi-Isotropic and Streaming Distributions 289
36. Diffusion, Drift, and Energy Balance 299
37. Heating under High-Frequency Field 309
38. Relaxation of Population 321
Problems 331
8. NON-EQUILIBRIUM DIAGRAM
TECHNIQUE 341
Contents xi
39. Matrix Green’s Function 341
40. Generalized Kinetic Equation 347
41. General Formulation of NDT 355
42. NDT Formalism for Electron-Boson System 364
43. Weak Localization under External Fields 374
Problems 381
9. KINETICS OF BOUNDED SYSTEMS 391
44. Boundary Conditions at Non-Ideal Surface 391
45. Size-Dependent Conductivity 398
46. Thermal Conductivity of Bounded Insulators 405
47. Electron Relaxation by Near-Surface Phonons 411
Problems 420
10. QUANTUM MAGNETOTRANSPORT 425
48. Method of Iterations 426
49. Green’s Function Approach 434
50. Quasi-Classical Conductivity 443
51. Quantum Hall Effect 452
52. Magnetooptics 465
Problems 476
11. PHOTOEXCITATION 483
53. Photogeneration Rate 483
54. Response to Ultrafast Excitation 490
55. Partially Inverted Electron Distribution 497
56. Photoinduced Interband Hybridization 508
57. Excitation of Coherent Phonons 520
Problems 528
12. BALLISTIC AND HOPPING TRANSPORT 537
58. Quantized Conductance 538
59. One-Dimensional Conductors 549
60. Tunneling Current 562
61. Coulomb Blockade 574
62. Polaronic Transport 585
Problems 596