WAVE SCATTERING AND MESOSCOPIC PHENOMENA
CONTENTS INTRODUCTION RELEVANT LENGTH SCALES DI?USIVE TRANSPORT COHERENT BACKSCATTERING AND THE APPROACH TO LOCALIZATION SAMPLE SIZE DEPENDENCE LOCALIZATION AND SCALING SPATIAL DIMENSIONALITY IN LOCALIZATION AND DI?USION MESOSCOPIC PHENOMENA LOCALIZATION VS CON?NEMENT TOPICS NOT COVERED QUANTUM AND CLASSICAL WAVES PRELIMINARIES QUANTUM AND CLASSICAL WAVE EQUATIONS GREEN FUNCTIONS FOR WAVES IN A UNIFORM MEDIUM GREEN FUNCTION FOR THE UNIFORM MEDIUM GREEN FUNCTION AND THE DENSITY OF STATES REAL SPACE GREEN FUNCTIONS WAVES ON A DISCRETE LATTICE DISPERSION RELATION RANDOM DISCRETE LATTICES: THE ANDERSON MODEL LATTICE GREEN FUNCTIONS 1D LATTICE GREEN FUNCTION 2D LATTICE GREEN FUNCTION 3D LATTICE GREEN FUNCTION TREATING CONTINUUM PROBLEMS ON A LATTICE PROBLEMS AND SOLUTIONS WAVE SCATTERING AND THE COHERENT POTENTIAL APPROXIMATION AN OVERVIEW OF THE APPROACH INFORMATION FROM THE FIRST MOMENT INFORMATION FROM THE SECOND MOMENT INFORMATION FROM HIGHER MOMENTS WAVE SCATTERING FORMALISM THE OPERATOR NOTATION THE T OPERATOR CON?GURATIONAL AVERAGING THE SELF-ENERGY SINGLE SCATTERER: THE LATTICE CASE IMPURITY BOUND STATE SINGLE SCATTERER: THE CONTINUUM CASE WAVE FUNCTION IN THE SINGLE SCATTERER CASE A DIGRESSION ON NOTATION SCATTERING AMPLITUDE AND THE T OPERATOR SCATTERING AMPLITUDE AND THE OPTICAL THEOREM: 3D CASE SCATTERING AMPLITUDE AND THE OPTICAL THEOREM: 2D CASE SCATTERING AMPLITUDE AND THE OPTICAL THEOREM: 1D CASE IN?NITE NUMBER OF SCATTERERS: THE E?ECTIVE MEDIUM MEANING OF A K-INDEPENDENT SELF ENERGY THE MEAN FREE PATH THE COHERENT POTENTIAL APPROXIMATION (CPA) ACCURACY OF THE CPA COHERENT WAVES AND E?ECTIVE MEDIA COHERENCE AND HOMOGENIZATION CPA: THE ANDERSON MODEL E?CIENT GREEN FUNCTION EVALUATION FROM DOS CPA: THE CLASSICAL WAVES THE ROLE OF MICROSTRUCTURE THE SYMMETRIC MICROSTRUCTURE THE DISPERSION MICROSTRUCTURE E?ECTIVE MEDIUM MODELING OF INHOMOGENEOUS MATERIALS GRANULAR METALS SEDIMENTARY ROCKS THE SPECTRAL FUNCTION APPROACH: COHERENT QUASIMODES FORMULATION COHERENT ACOUSTIC WAVES IN THE INTERMEDIATE WAVELENGTH REGIME NOVEL ACOUSTIC MODES IN COLLOIDAL SUSPENSIONS GROUP VELOCITY IN STRONGLY SCATTERING MEDIA ACCURACY OF THE SPECTRAL FUNCTION APPROACH CONTENTS XI DI?USIVE WAVES BEYOND THE COHERENT REGIME PULSE INTENSITY EVOLUTION IN A RANDOM MEDIUM THE BETHE-SALPETER EQUATION AND ITS SOLUTION BY MOMENTS NOTATIONS IRREDUCIBLE VERTEX FUNCTION AND THE BETHE– SALPETER EQUATION DI?USIVE BEHAVIOR FROM A SIMPLI?ED MODEL THE WARD IDENTITY IN THE SIMPLI?ED MODEL FORMAL DERIVATION OF THE DI?USIVE POLE AND THE DI?USION CONSTANT HIGHER MOMENT OF THE TRANSPORT EQUATION EXPANSION OF ?K SOLUTION FOR THE DI?USIVE POLE AND THE DI?USION CONSTANT CON?GURATIONAL AVERAGING AND DISSIPATION THE VERTEX FUNCTION THE REDUCIBLE VERTEX AND DIAGRAMMATIC REPRESENTATIONS CPA FOR ?: THE LADDER DIAGRAMS AND THE IRREDUCIBLE VERTEX IRREDUCIBLE VERTEX: THE ANDERSON MODEL IRREDUCIBLE VERTEX: CLASSICAL WAVES THE WARD IDENTITY CANCELLATION OF THE IMPURITY SCATTERING TERMS REDUCTION TO THE SIMPLI?ED MODEL MEANING OF THE WARD IDENTITY DI?USION CONSTANT MODI?CATION FOR CLASSICAL WAVES EVALUATION OF THE WAVE DI?USION CONSTANT QUANTUM CASE: THE ANDERSON MODEL CLASSICAL WAVE DI?USION CONSTANT ANISOTROPIC SCATTERING AND THE TRANSPORT MEAN FREE PATH APPLICATION: DI?USIVE WAVE SPECTROSCOPY THE SIEGERT RELATION AVERAGING OVER THE DI?USIVE MOTION OF SCATTERERS AVERAGING OVER THE SCATTERING ANGLES DI?USIVE TRANSPORT OF WAVES BOUNDARY CONDITIONS FOR DI?USIVE LIGHT FLUX SOLUTIONS FOR THE SLAB GEOMETRY THE COHERENT BACKSCATTERING E?ECT WAVE DI?USION VERSUS CLASSICAL DI?USION COHERENCE IN THE BACKSCATTERING DIRECTION TIME REVERSAL INVARIANCE ANGULAR PRO?LE OF THE COHERENT BACKSCATTERING SINGLE CON?GURATION AND CON?GURATIONAL AVERAGING COUNTING THE SCATTERING PATHS BACKSCATTERING FROM A RANDOM HALF SPACE FEATURES OF THE COHERENT BACKSCATTERING PEAK BACKSCATTERING FROM A FINITE THICKNESS SLAB SAMPLE SIZE (PATH LENGTH) DEPENDENCE E?ECT OF ABSORPTION RENORMALIZED DI?USION COHERENT BACKSCATTERING E?ECT IN THE DIAGRAMMATIC REPRESENTATION EVALUATION OF THE MAXIMALLY CROSSED DIAGRAMS MOMENTA TRANSFORMATIONS SUMMATION OF MAXIMALLY CROSSED DIAGRAMS RENORMALIZED DI?USION CONSTANT SAMPLE SIZE AND SPATIAL DIMENSIONALITY DEPENDENCIES OF WAVE DI?USION CONDUCTIVITY CORRECTION DI?USION CONSTANT CORRECTION THE EMERGENCE OF LOCALIZATION LENGTH LOCALIZATION IN 3D : THE IO?E – REGEL CRITERION LOCALIZATION IN ONE DIMENSION: THE HERBERT–JONES–THOULESS FORMULA LOCALIZATION AND INTENSITY TRANSMISSION FLUCTUATIONS IN INTENSITY TRANSMISSION RANDOMNESS IS NOT ALWAYS A SU?CIENT CONDITION FOR 1D LOCALIZATION LOCALIZATION AND EIGENFUNCTIONS IN 1D: THE CHARACTERISTIC FUNCTION RELATION BETWEEN THE 1D LOCALIZATION LENGTH AND DENSITY OF STATES HJT FORMULA FOR THE 1D ANDERSON MODEL HJT FORMULA FOR 1D CLASSICAL SCALAR WAVES HIGH FREQUENCY AND LOW FREQUENCY LIMITING BEHAVIORS THE SCALING THEORY OF LOCALIZATION DISTINGUISHING A LOCALIZED STATE FROM AN EXTENDED STATE THE SCALING ARGUMENT AND SPATIAL DIMENSIONALITY DIMENSIONLESS CONDUCTANCE THE SCALING HYPOTHESIS AND ITS CONSEQUENCES DE?NING THE SCALING FUNCTION SCALING AND THE E?ECT OF RANDOMNESS BEHAVIOR OF THE SCALING FUNCTION WHEN SYSTEM SIZE IS LESS THAN THE LOCALIZATION LENGTH WAVE TRANSPORT NEAR THE 3D MOBILITY EDGE ANOMALOUS DI?USION AND THE MINIMUM METALLIC CONDUCTANCE IN 3D SCALING BEHAVIOR IN THE 3D LOCALIZED REGIME NUMERICAL EVALUATION OF THE SCALING FUNCTION THE FINITE-SIZE SCALING FUNCTION LIMITING BEHAVIORS RELATION TO THE SCALING FUNCTIONS IN 1D, 2D, AND 3D UNIVERSALITY AND LIMITATIONS OF THE SCALING THEORY RESULTS LOCALIZATION IN 3D : LATTICE VS CONTINUUM IMPLICIT ASSUMPTIONS OF THE SCALING THEORY LOCALIZED STATES AND THE APPROACH TO LOCALIZATION THE SELF-CONSISTENT THEORY OF LOCALIZATION TRANSITION TO THE STRONG SCATTERING REGIME RESPONSE OF THE LOCALIZED STATE AT FINITE MODULATION FREQUENCY QUALITATIVE EVALUATION OF THE CORRECTION TERM LOCALIZATION BEHAVIOR OF THE ANDERSON MODEL EVALUATION OF THE MAXIMALLY CROSSED DIAGRAMS’ CONTRIBUTION MOBILITY EDGE IN 3D ANDERSON MODEL LOCALIZATION LENGTH AND CORRELATION LENGTH IN THE 3D ANDERSON MODEL 2D ANDERSON MODEL LOCALIZATION LENGTH IN THE 2D ANDERSON MODEL 1D ANDERSON MODEL LOCALIZATION LENGTH IN THE 1D ANDERSON MODEL INPUT PARAMETERS TO LOCALIZATION CALCULATIONS CLASSICAL SCALAR WAVE LOCALIZATION ONE PARAMETER CRITERION FOR THE CLASSICAL WAVE 3D MOBILITY EDGE INPUT PARAMETER VALUES AND WAVE SCATTERING REGIMES SCATTERING AND LOCALIZATION OF CLASSICAL WAVES IN 3D PERIODICITY IN SCATTERING STRUCTURES AND 3D LOCALIZATION LOCALIZATION OF 2D CLASSICAL SCALAR WAVE LOCALIZATION OF 1D CLASSICAL SCALAR WAVE TRANSPORT VELOCITY OF CLASSICAL SCALAR WAVES TRANSPORT VELOCITY AND SCATTERING RESONANCES E?ECTIVE MEDIUM BASED ON ENERGY HOMOGENIZATION THE SCALING FUNCTION EVALUATION FINITE SIZE MODI?CATION OF THE DI?USION CONSTANT FINITE SIZE E?ECT FOR DENSITY OF STATES SCALING FUNCTION AND ITS LIMITING BEHAVIORS LOCALIZATION PHENOMENA IN ELECTRONIC SYSTEMS 101 FINITE TEMPERATURES AND THE E?ECT OF INELASTIC SCATTERING 1011 INELASTIC SCATTERING IN THE CLASSICAL WAVE CASE 1012 INELASTIC SCATTERING IN THE ELECTRONIC CASE 102 TEMPERATURE DEPENDENCE OF THE RESISTANCE IN 2D DISORDERED FILMS 1021 MANIFESTATION OF LOCALIZATION CORRECTION IN 2D CONDUCTIVITY 103 MAGNETORESISTANCE OF DISORDERED METALLIC FILMS 1031 E?ECT OF A MAGNETIC FIELD 1032 MAGNETIC FIELD CORRECTION TO THE COHERENT BACKSCATTERING E?ECT 1033 THICK FILM CASE 1034 THIN FILM CASE 1035 DI?USION AND INELASTIC RELAXATION 1036 MAGNETORESISTANCE AND THE DISTANCE TO LOCALIZATION 1037 SPIN-ORBIT COUPLING 104 TRANSPORT OF LOCALIZED STATES AT FINITE TEMPERATURES: HOPPING CONDUCTION 1041 CONSTRUCTING THE CRITICAL PERCOLATING PATH OF LEAST RESISTANCE 1042 TEMPERATURE DEPENDENCE OF THE CRITICAL PATH CONDUCTANCE 1043 VARIABLE RANGE HOPPING 111 WHAT IS “MESOSCOPIC”? 1111 DISSIPATION AND MESOSCOPIC CONDUCTANCE 112 INTENSITY DISTRIBUTION OF THE SPECKLE PATTERN 1121 STATISTICAL INDEPENDENCE AND GAUSSIAN DISTRIBUTION 113 CORRELATIONS IN THE DI?USIVE INTENSITY 1131 FREQUENCY CORRELATION OF THE DI?USIVE INTENSITY 1132 SPATIAL CORRELATION IN THE DI?USIVE INTENSITY 1133 INTENSITY FLUCTUATIONS FOR CLASSICAL WAVES IN 3D 114 LONG-RANGE CORRELATION IN INTENSITY FLUCTUATIONS 1141 REDUCTION TO A DI?USION PROBLEM WITH RANDOM SOURCES 1142 SOLUTION IN THE SLAB GEOMETRY 1143 SLOW DECAY OF TRANSMITTED INTENSITY FLUCTUATIONS 115 LANDAUER FORMULA AND QUANTIZED CONDUCTANCE CONTENTS XV 1151 QUANTIZED CONDUCTANCE 1152 TWO-TERMINAL VERSUS FOUR-TERMINAL 1153 THE SINGLE CHANNEL (1D) CASE 1154 LANDAUER–BUTTIKER FORMULA FOR THE MULTICHANNEL CASE 116 CHARACTERISTICS OF MESOSCOPIC CONDUCTANCE 1161 DISORDERED 2D METALLIC FILM 1162 DISTRIBUTION OF THE DIMENSIONLESS CONDUCTANCES 1163 LOCALIZATION LENGTH, CONDUCTANCE FLUCTUATIONS AND TRANSPORT REGIMES 1164 HEURISTIC EXPLANATION OF UNIVERSAL CONDUCTANCE FLUCTUATIONS REFERENCES