| Preface |
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xiii | |
| How to use this book |
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xv | |
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Introduction: mesoscopic physics |
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1 | (30) |
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Interference and disorder |
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1 | (3) |
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The Aharonov--Bohm effect |
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4 | (3) |
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Phase coherence and the effect of disorder |
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7 | (2) |
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Average coherence and multiple scattering |
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9 | (3) |
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Phase coherence and self-averaging: universal fluctuations |
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12 | (2) |
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14 | (1) |
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Classical probability and quantum crossings |
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15 | (3) |
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17 | (1) |
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18 | (13) |
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Wave equations in random media |
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31 | (39) |
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31 | (5) |
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Electrons in a disordered metal |
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31 | (1) |
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Electromagnetic wave equation -- Helmoltz equation |
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32 | (1) |
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Other examples of wave equations |
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33 | (3) |
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36 | (7) |
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37 | (2) |
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Localized impurities: the Edwards model |
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39 | (2) |
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41 | (2) |
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Appendix A2.1: Theory of elastic collisions and single scattering |
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43 | (11) |
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Asymptotic form of the solutions |
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44 | (2) |
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Scattering cross section and scattered flux |
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46 | (1) |
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47 | (4) |
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51 | (3) |
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Appendix A2.2: Reciprocity theorem |
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54 | (2) |
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Appendix A2.3: Light scattering |
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56 | (14) |
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Classical Rayleigh scattering |
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56 | (4) |
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60 | (1) |
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Atom-photon scattering in the dipole approximation |
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61 | (9) |
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70 | (22) |
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71 | (8) |
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Green's function for the Schrodinger equation |
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71 | (6) |
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Green's function for the Helmholtz equation |
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77 | (2) |
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Multiple scattering expansion |
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79 | (7) |
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79 | (3) |
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82 | (4) |
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Average Green's function and average density of states |
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86 | (2) |
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Appendix A3.1: Short range correlations |
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88 | (4) |
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Probability of quantum diffusion |
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92 | (56) |
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92 | (3) |
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95 | (1) |
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Drude-Boltzmann approximation |
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96 | (1) |
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Diffusion or ladder approximation |
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97 | (5) |
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The Diffusion at the diffusion approximation |
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102 | (2) |
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Coherent propagation: the Cooperon |
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104 | (6) |
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110 | (3) |
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Appendix A4.1: Diffusion and Cooperon in reciprocal space |
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113 | (7) |
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Collisionless probability P0(q, ω) |
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114 | (1) |
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115 | (2) |
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117 | (3) |
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Appendix A4.2: Hikami boxes and Diffusion crossings |
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120 | (12) |
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120 | (5) |
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Normalization of the probability and renormalization of the diffusion coefficient |
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125 | (3) |
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Crossing of two Diffusions |
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128 | (4) |
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Appendix A4.3: Anisotropic collisions and transport mean free path |
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132 | (6) |
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Appendix A4.4: Correlation of diagonal Green's functions |
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138 | (4) |
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Appendix A4.5: Other correlation functions |
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142 | (6) |
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Correlations of Green's functions |
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142 | (3) |
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145 | (1) |
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Correlations of wave functions |
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145 | (3) |
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Properties of the diffusion equation |
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148 | (47) |
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148 | (1) |
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Heat kernel and recurrence time |
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149 | (3) |
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Heat kernel -- probability of return to the origin |
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149 | (2) |
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151 | (1) |
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152 | (3) |
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Diffusion in a periodic box |
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155 | (1) |
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Diffusion in finite systems |
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156 | (3) |
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Diffusion time and Thouless energy |
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156 | (1) |
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Boundary conditions for the diffusion equation |
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156 | (1) |
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Finite volume and ``zero mode'' |
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157 | (1) |
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Diffusion in an anisotropic domain |
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158 | (1) |
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One-dimensional diffusion |
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159 | (6) |
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The ring: periodic boundary conditions |
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160 | (1) |
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Absorbing boundaries: connected wire |
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161 | (1) |
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Reflecting boundaries: isolated wire |
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162 | (2) |
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164 | (1) |
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165 | (1) |
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Appendix A5.1: Validity of the diffusion approximation in an infinite medium |
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166 | (2) |
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Appendix A5.2: Radiative transfer equation |
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168 | (9) |
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168 | (2) |
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170 | (2) |
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172 | (3) |
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Slab illuminated by an extended source |
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175 | (1) |
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Semi-infinite medium illuminated by a collimated beam |
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176 | (1) |
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Appendix A5.3: Multiple scattering in a finite medium |
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177 | (5) |
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Multiple scattering in a half-space: the Milne problem |
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177 | (3) |
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Diffusion in a finite medium |
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180 | (2) |
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Appendix A5.4: Spectral determinant |
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182 | (2) |
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Appendix A5.5: Diffusion in a domain of arbitrary shape -- Weyl expansion |
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184 | (3) |
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Appendix A5.6: Diffusion on graphs |
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187 | (8) |
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Spectral determinant on a graph |
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187 | (4) |
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191 | (2) |
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Thermodynamics, transport and spectral determinant |
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193 | (2) |
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195 | (75) |
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Dephasing and multiple scattering |
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195 | (4) |
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195 | (1) |
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Mechanisms for dephasing: introduction |
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196 | (3) |
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199 | (1) |
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Magnetic field and the Cooperon |
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199 | (4) |
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Probability of return to the origin in a uniform magnetic field |
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203 | (2) |
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Probability of return to the origin for an Aharonov--Bohm flux |
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205 | (5) |
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206 | (2) |
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208 | (2) |
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Spin-orbit coupling and magnetic impurities |
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210 | (16) |
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Transition amplitude and effective interaction potential |
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210 | (2) |
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212 | (2) |
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214 | (5) |
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219 | (2) |
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221 | (2) |
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The diffusion probability |
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223 | (1) |
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224 | (2) |
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Polarization of electromagnetic waves |
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226 | (8) |
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227 | (1) |
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228 | (3) |
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231 | (2) |
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233 | (1) |
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Dephasing and motion of scatterers |
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234 | (4) |
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General expression for the phase shift |
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234 | (3) |
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Dephasing associated with the Brownian motion of the scatterers |
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237 | (1) |
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Dephasing or decoherence? |
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238 | (2) |
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Appendix A6.1: Aharonov-Bohm effect in an infinite plane |
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240 | (2) |
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Appendix A6.2: Functional representation of the diffusion equation |
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242 | (5) |
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Functional representation |
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242 | (2) |
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Brownian motion and magnetic field |
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244 | (3) |
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Appendix A6.3: The Cooperon in a time-dependent field |
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247 | (4) |
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Appendix A6.4: Spin-orbit coupling and magnetic impurities, a heuristic point of view |
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251 | (5) |
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251 | (3) |
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254 | (2) |
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Appendix A6.5: Decoherence in multiple scattering of light by cold atoms |
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256 | (14) |
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Scattering amplitude and atomic collision time |
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256 | (1) |
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257 | (5) |
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262 | (8) |
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270 | (50) |
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270 | (3) |
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Incoherent contribution to conductivity |
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273 | (6) |
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Drude-Boltzmann approximation |
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273 | (3) |
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The multiple scattering regime: the Diffusion |
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276 | (2) |
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Transport time and vertex renormalization |
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278 | (1) |
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279 | (2) |
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The weak localization regime |
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281 | (5) |
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282 | (2) |
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284 | (1) |
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285 | (1) |
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Weak localization in a magnetic field |
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286 | (6) |
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Negative magnetoresistance |
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286 | (4) |
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Spin-orbit coupling and magnetic impurities |
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290 | (2) |
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Magnetoresistance and Aharonov--Bohm flux |
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292 | (4) |
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293 | (1) |
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Long cylinder: the Sharvin--Sharvin effect |
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294 | (1) |
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Remark on the Webb and Sharvin-Sharvin experiments: φ0 versus φ0/2 |
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295 | (1) |
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The Aharonov--Bohm effect in an infinite plane |
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296 | (1) |
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Appendix A7.1: Kubo formulae |
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296 | (6) |
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Conductivity and dissipation |
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296 | (5) |
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Density-density response function |
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301 | (1) |
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Appendix A7.2: Conductance and transmission |
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302 | (13) |
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Introduction: Landauer formula |
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302 | (3) |
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305 | (2) |
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Average conductance and transmission |
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307 | (4) |
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Boundary conditions and impedance matching |
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311 | (2) |
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Weak localization correction in the Landauer formalism |
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313 | (1) |
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Landauer formalism for waves |
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314 | (1) |
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Appendix A7.3: Real space description of conductivity |
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315 | (2) |
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Appendix A7.4: Weak localization correction and anisotropic collisions |
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317 | (3) |
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Coherent backscattering of light |
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320 | (34) |
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320 | (1) |
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The geometry of the albedo |
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321 | (3) |
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321 | (1) |
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Albedo of a diffusive medium |
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322 | (2) |
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324 | (6) |
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Incoherent albedo: contribution of the Diffusion |
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324 | (2) |
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The coherent albedo: contribution of the Cooperon |
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326 | (4) |
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Time dependence of the albedo and study of the triangular cusp |
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330 | (3) |
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333 | (1) |
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Coherent albedo and anisotropic collisions |
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334 | (2) |
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The effect of polarization |
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336 | (3) |
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Depolarization coefficients |
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336 | (1) |
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Coherent albedo of a polarized wave |
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337 | (2) |
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339 | (8) |
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340 | (1) |
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Decrease of the height of the cone |
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341 | (2) |
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343 | (4) |
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Coherent backscattering at large |
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347 | (7) |
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Coherent backscattering and the ``glory'' effect |
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347 | (1) |
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Coherent backscattering and opposition effect in astrophysics |
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348 | (1) |
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Coherent backscattering by cold atomic gases |
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349 | (3) |
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Coherent backscattering effect in acoustics |
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352 | (2) |
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Diffusing wave spectroscopy |
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354 | (16) |
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354 | (1) |
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Dynamic correlations of intensity |
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355 | (2) |
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Single scattering: quasi-elastic light scattering |
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357 | (1) |
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Multiple scattering: diffusing wave spectroscopy |
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358 | (1) |
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Influence of the geometry on the time correlation function |
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359 | (8) |
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Reflection by a semi-infinite medium |
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359 | (2) |
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Comparison between Gr1(T) and αc(θ) |
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361 | (2) |
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Reflection from a finite slab |
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363 | (1) |
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364 | (3) |
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Appendix A9.1: Collective motion of scatterers |
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367 | (3) |
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Spectral properties of disordered metals |
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370 | (26) |
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370 | (4) |
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Level repulsion and integrability |
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371 | (2) |
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Energy spectrum of a disordered metal |
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373 | (1) |
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Characteristics of spectral correlations |
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374 | (2) |
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376 | (1) |
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Universality of spectral correlations: random matrix theory |
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377 | (8) |
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Level repulsion in 2 x 2 matrices |
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377 | (3) |
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Distribution of eigenvalues for N x N matrices |
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380 | (2) |
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Spectral properties of random matrices |
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382 | (3) |
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Spectral correlations in the diffusive regime |
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385 | (9) |
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Two-point correlation function |
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386 | (4) |
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390 | (1) |
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391 | (3) |
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Appendix A10.1: The GOE-GUE transition |
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394 | (2) |
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Universal conductance fluctuations |
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396 | (31) |
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396 | (3) |
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Conductivity fluctuations |
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399 | (7) |
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Fluctuations of the density of states |
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402 | (3) |
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Fluctuations of the diffusion coefficient |
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405 | (1) |
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Universal conductance fluctuations |
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406 | (3) |
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Effect of external parameters |
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409 | (13) |
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409 | (1) |
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409 | (2) |
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Phase coherence and the mesoscopic regime |
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411 | (4) |
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Magnetic field dependence |
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415 | (3) |
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418 | (1) |
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Spin-orbit coupling and magnetic impurities |
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419 | (3) |
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Appendix A11.1: Universal conductance fluctuations and anisotropic collisions |
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422 | (2) |
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Appendix A11.2: Conductance fluctuations in the Landauer formalism |
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424 | (3) |
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Correlations of speckle patterns |
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427 | (38) |
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What is a speckle pattern? |
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427 | (1) |
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How to analyze a speckle pattern |
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428 | (5) |
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Average transmission coefficient |
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433 | (2) |
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Angular correlations of the transmitted light |
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435 | (11) |
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Short range C(1) correlations |
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435 | (4) |
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Long range correlations C(2) |
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439 | (2) |
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Two-crossing contribution and C(3) correlation |
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441 | (4) |
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Relation with universal conductance fluctuations |
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445 | (1) |
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Speckle correlations in the time domain |
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446 | (6) |
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Time dependent correlations C(1)(t) and C(2)(t) |
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447 | (3) |
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Time dependent correlation C(3)(t) |
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450 | (2) |
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Spectral correlations of speckle patterns |
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452 | (1) |
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Distribution function of the transmission coefficients |
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453 | (5) |
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Rayleigh distribution law |
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454 | (1) |
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Gaussian distribution of the transmission coefficient Ta |
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455 | (1) |
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Gaussian distribution of the electrical conductance |
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456 | (2) |
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Appendix A12.1: Spatial correlations of light intensity |
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458 | (7) |
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459 | (2) |
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461 | (4) |
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Interactions and diffusion |
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465 | (51) |
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465 | (1) |
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Screened Coulomb interaction |
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466 | (2) |
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Hartree--Fock approximation |
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468 | (2) |
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Density of states anomaly |
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470 | (13) |
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470 | (5) |
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Tunnel conductance and density of states anomaly |
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475 | (3) |
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Dynamically screened interaction |
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478 | (3) |
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481 | (2) |
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Correction to the conductivity |
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483 | (4) |
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Lifetime of a quasiparticle |
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487 | (11) |
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Introduction: Landau theory and disorder |
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487 | (1) |
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Lifetime at zero temperature |
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487 | (7) |
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Quasiparticle lifetime at finite temperature |
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494 | (1) |
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Quasiparticle lifetime at the Fermi level |
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495 | (3) |
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498 | (18) |
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498 | (1) |
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Phase coherence in a fluctuating electric field |
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499 | (3) |
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Phase coherence time in dimension d = 1 |
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502 | (4) |
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Phase coherence and quasiparticle relaxation |
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506 | (3) |
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Phase coherence time in dimensions d = 2 and d = 3 |
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509 | (1) |
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Measurements of the phase coherence time τeeφ |
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510 | (2) |
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Appendix A13.1: Screened Coulomb potential in confined geometry |
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512 | (2) |
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Appendix A13.2: Lifetime in the absence of disorder |
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514 | (2) |
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Orbital magnetism and persistent currents |
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516 | (31) |
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516 | (2) |
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Free electron gas in a uniform field |
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518 | (6) |
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A reminder: the case of no disorder |
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518 | (3) |
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521 | (1) |
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Fluctuations of the magnetization |
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522 | (2) |
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Effect of Coulomb interaction |
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524 | (4) |
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Hartree--Fock approximation |
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525 | (1) |
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526 | (2) |
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528 | (1) |
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Persistent current in a ring |
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528 | (8) |
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Clean one-dimensional ring: periodicity and parity effects |
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529 | (5) |
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534 | (2) |
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Diffusive limit and persistent current |
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536 | (9) |
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Typical current of a disordered ring |
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537 | (2) |
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Effect of the Coulomb interaction on the average current |
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539 | (3) |
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Persistent current and spin-orbit coupling |
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542 | (1) |
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A brief overview of experiments |
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543 | (2) |
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Appendix A14.1: Average persistent current in the canonical ensemble |
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545 | (2) |
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547 | (14) |
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Density of states and conductance |
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547 | (1) |
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Fourier transforms: definitions |
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548 | (1) |
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Collisionless probability P0(r, r' , t) |
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548 | (1) |
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548 | (3) |
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Wigner-Eckart theorem and 3j-symbols |
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551 | (1) |
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552 | (6) |
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558 | (1) |
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558 | (1) |
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Characteristic times introduced in this book |
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559 | (2) |
| References |
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561 | (21) |
| Index |
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582 | |
