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MESOSCOPICPHYSICSOFELECTRONSANDPHOTONS
Quantum mesoscopic physics covers a whole class of interference effects related to the
propagation of waves in complex and random media. These effects are ubiquitous in
physics, from the behavior of electrons in metals and semiconductors to the propagation
ofelectromagneticwavesinsuspensionssuchascolloids,andquantumsystemslikecold
atomicgases.Thisbookisamodernaccountoftheproblemofcoherentwavepropagation
inrandommedia.
As a solid introduction to quantum mesoscopic physics, this book provides a unified
overviewofthebasictheoreticaltoolsandmethods.Ithighlightsthecommonaspectsof
thevariousopticalandelectronicphenomenainvolved.Withover200figures,andexercises
throughout,thebookisidealforgraduatestudentsinphysics,electricalengineering,optics,
acousticsandastrophysics.Itpresentsalargenumberofexperimentalresultsthatcovera
widerangeofphenomenafromsemiconductorstooptics,acoustics,andatomicphysics.It
willalsobeanimportantreferenceforresearchersinthisrapidlyevolvingfield.
EricAkkermansisProfessorofPhysicsintheDepartmentofPhysicsattheTechnion,
IsraelInstituteofTechnology,Israel.GillesMontambauxisDirecteurdeRechercheatthe
CNRS,LaboratoiredePhysiquedesSolides,UniversitéParis-Sud,France.Theirresearch
interestsincludethetheoryofcondensedmatterphysics,mesoscopicquantumphysics,and
coherenteffectsinthepropagationofwavesinrandommedia.
Mesoscopic Physics
of Electrons and Photons
EricAkkermans
Technion,IsraelInstituteofTechnology
GillesMontambaux
CNRS,UniversitéParis-Sud
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521855129
© E. Akkermans and G. Montambaux 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-29016-9 eBook (NetLibrary)
ISBN-10 0-511-29016-0 eBook (NetLibrary)
ISBN-13 978-0-521-85512-9 hardback
ISBN-10 0-521-85512-8 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface pagexiii
Howtousethisbook xv
1 Introduction:mesoscopicphysics 1
1.1 Interferenceanddisorder 1
1.2 TheAharonov–Bohmeffect 4
1.3 Phasecoherenceandtheeffectofdisorder 7
1.4 Averagecoherenceandmultiplescattering 9
1.5 Phasecoherenceandself-averaging:universalfluctuations 12
1.6 Spectralcorrelations 14
1.7 Classicalprobabilityandquantumcrossings 15
1.7.1 Quantumcrossings 17
1.8 Objectives 18
2 Waveequationsinrandommedia 31
2.1 Waveequations 31
2.1.1 Electronsinadisorderedmetal 31
2.1.2 Electromagneticwaveequation–Helmholtzequation 32
2.1.3 Otherexamplesofwaveequations 33
2.2 Modelsofdisorder 36
2.2.1 TheGaussianmodel 37
2.2.2 Localizedimpurities:theEdwardsmodel 39
2.2.3 TheAndersonmodel 41
AppendixA2.1:Theoryofelasticcollisionsandsinglescattering 43
A2.1.1 Asymptoticformofthesolutions 44
A2.1.2 Scatteringcrosssectionandscatteredflux 46
A2.1.3 Opticaltheorem 47
A2.1.4 Bornapproximation 51
AppendixA2.2:Reciprocitytheorem 54
AppendixA2.3:Lightscattering 56
A2.3.1 ClassicalRayleighscattering 56
v
vi Contents
A2.3.2 Miescattering 60
A2.3.3 Atom–photonscatteringinthedipoleapproximation 61
3 Perturbationtheory 70
3.1 Green’sfunctions 71
3.1.1 Green’sfunctionfortheSchrödingerequation 71
3.1.2 Green’sfunctionfortheHelmholtzequation 77
3.2 Multiplescatteringexpansion 79
3.2.1 Dysonequation 79
3.2.2 Self-energy 82
3.3 AverageGreen’sfunctionandaveragedensityofstates 86
AppendixA3.1: Shortrangecorrelations 88
4 Probabilityofquantumdiffusion 92
4.1 Definition 92
4.2 Freepropagation 95
4.3 Drude–Boltzmannapproximation 96
4.4 Diffusonorladderapproximation 97
4.5 TheDiffusonatthediffusionapproximation 102
4.6 Coherentpropagation:theCooperon 104
4.7 Radiativetransfer 110
AppendixA4.1:DiffusonandCooperoninreciprocalspace 113
A4.1.1 CollisionlessprobabilityP (q,ω) 114
0
A4.1.2 TheDiffuson 115
A4.1.3 TheCooperon 117
AppendixA4.2:HikamiboxesandDiffusoncrossings 120
A4.2.1 Hikamiboxes 120
A4.2.2 Normalizationoftheprobabilityandrenormalizationofthe
diffusioncoefficient 125
A4.2.3 CrossingoftwoDiffusons 128
AppendixA4.3:Anisotropiccollisionsandtransportmeanfreepath 132
AppendixA4.4:CorrelationofdiagonalGreen’sfunctions 138
AppendixA4.5:Othercorrelationfunctions 142
A4.5.1 CorrelationsofGreen’sfunctions 142
A4.5.2 AWardidentity 145
A4.5.3 Correlationsofwavefunctions 145
5 Propertiesofthediffusionequation 148
5.1 Introduction 148
5.2 Heatkernelandrecurrencetime 149
5.2.1 Heatkernel–probabilityofreturntotheorigin 149
5.2.2 Recurrencetime 151
5.3 Freediffusion 152
5.4 Diffusioninaperiodicbox 155
Contents vii
5.5 Diffusioninfinitesystems 156
5.5.1 DiffusiontimeandThoulessenergy 156
5.5.2 Boundaryconditionsforthediffusionequation 156
5.5.3 Finitevolumeand“zeromode” 157
5.5.4 Diffusioninananisotropicdomain 158
5.6 One-dimensionaldiffusion 159
5.6.1 Thering:periodicboundaryconditions 160
5.6.2 Absorbingboundaries:connectedwire 161
5.6.3 Reflectingboundaries:isolatedwire 162
5.6.4 Semi-infinitewire 164
5.7 Theimagemethod 165
AppendixA5.1:Validityofthediffusionapproximationinaninfinite
medium 166
AppendixA5.2:Radiativetransferequation 168
A5.2.1 Totalintensity 168
A5.2.2 Diffuseintensity 170
A5.2.3 Boundaryconditions 172
A5.2.4 Slabilluminatedbyanextendedsource 175
A5.2.5 Semi-infinitemediumilluminatedbyacollimatedbeam 176
AppendixA5.3:Multiplescatteringinafinitemedium 177
A5.3.1 Multiplescatteringinahalf-space:theMilneproblem 177
A5.3.2 Diffusioninafinitemedium 180
AppendixA5.4:Spectraldeterminant 182
AppendixA5.5:Diffusioninadomainofarbitraryshape–Weylexpansion 184
AppendixA5.6:Diffusionongraphs 187
A5.6.1 Spectraldeterminantonagraph 187
A5.6.2 Examples 191
A5.6.3 Thermodynamics,transportandspectraldeterminant 193
6 Dephasing 195
6.1 Dephasingandmultiplescattering 195
6.1.1 Generalities 195
6.1.2 Mechanismsfordephasing:introduction 196
6.1.3 TheGoldstonemode 199
6.2 MagneticfieldandtheCooperon 199
6.3 Probabilityofreturntotheorigininauniformmagneticfield 203
6.4 ProbabilityofreturntotheoriginforanAharonov–Bohmflux 205
6.4.1 Thering 206
6.4.2 Thecylinder 208
6.5 Spin-orbitcouplingandmagneticimpurities 210
6.5.1 Transitionamplitudeandeffectiveinteractionpotential 210
6.5.2 Totalscatteringtime 212
6.5.3 Structurefactor 214
viii Contents
6.5.4 TheDiffuson 219
6.5.5 TheCooperon 221
6.5.6 Thediffusionprobability 223
6.5.7 TheCooperonX 224
c
6.6 Polarizationofelectromagneticwaves 226
6.6.1 Elasticmeanfreepath 227
6.6.2 Structurefactor 228
6.6.3 Classicalintensity 231
6.6.4 Coherentbackscattering 233
6.7 Dephasingandmotionofscatterers 234
6.7.1 Generalexpressionforthephaseshift 234
6.7.2 DephasingassociatedwiththeBrownianmotionofthe
scatterers 237
6.8 Dephasingordecoherence? 238
AppendixA6.1:Aharonov–Bohmeffectinaninfiniteplane 240
AppendixA6.2:Functionalrepresentationofthediffusionequation 242
A6.2.1 Functionalrepresentation 242
A6.2.2 Brownianmotionandmagneticfield 244
AppendixA6.3:TheCooperoninatime-dependentfield 247
AppendixA6.4:Spin-orbitcouplingandmagneticimpurities,
aheuristicpointofview 251
A6.4.1 Spin-orbitcoupling 251
A6.4.2 Magneticimpurities 254
AppendixA6.5:Decoherenceinmultiplescatteringoflightbycoldatoms 256
A6.5.1 Scatteringamplitudeandatomiccollisiontime 256
A6.5.2 Elementaryatomicvertex 257
A6.5.3 Structurefactor 262
7 Electronictransport 270
7.1 Introduction 270
7.2 Incoherentcontributiontoconductivity 273
7.2.1 Drude–Boltzmannapproximation 273
7.2.2 Themultiplescatteringregime:theDiffuson 276
7.2.3 Transporttimeandvertexrenormalization 278
7.3 Cooperoncontribution 279
7.4 Theweaklocalizationregime 281
7.4.1 Dimensionalityeffect 282
7.4.2 Finitesizeconductors 284
7.4.3 Temperaturedependence 285
7.5 Weaklocalizationinamagneticfield 286
7.5.1 Negativemagnetoresistance 286
7.5.2 Spin-orbitcouplingandmagneticimpurities 290
