This page intentionally left blank 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