Excitons and Cooper Pairs Two Composite Bosons in Many-Body Physics Monique Combescot Institut des NanoSciences de Paris, Universite Pierre et Marie Curie, CNRS, Paris, France and Shiue-Yuan Shiau Department of Physics and National Centerfor Theoretical Sciences, National Cheng Kung University, Tainan, Taiwan OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Technical aspects 7 1.2 On the possible ways to draw diagrams 9 Part I Excitons 2 The Exciton Concept 15 2.1 The physical picture 16 2.2 Relevant Coulomb processes 19 2.3 Exciton-photon coupling 23 2.4 Many-body effects 24 2.5 Thermal effects 28 2.6 The semiconductor Hamiltonian 30 3 Wannier Excitons 34 3.1 Phenomenological approach 34 3.2 Microscopic derivation 49 3.3 One Wannier exciton 65 3.4 Many-body effects 79 4 Frenkel Excitons 108 4.1 Atomic states and the tight-binding approximation 109 4.2 Second quantization formulation 113 4.3 One Frenkel exciton 131 4.4 Spin and orbital degrees of freedom 139 4.5 Many-body effects 148 5 Elementary Bosons, Wannier Excitons, and Frenkel Excitons 178 5.1 Physical pictures 180 5.2 Commutation relations and Pauli scatterings 181 5.3 Interaction scatterings 183 5.4 Closure relations 187 5.5 Normalization factors 187 5.6 Many-body parameters 188 5.7 Hamiltonian mean values 189 x Contents Part II Cooper Pairs 6 The Cooper Pair Problem 193 6.1 The four main approaches to BCS superconductivity 194 6.2 Effective attraction between two electrons 196 7 The Bardeen-Cooper-Schrieffer Approach 202 7.1 The Cooper problem 204 7.2 The BCS problem 206 7.3 The BCS approach to the BCS problem 207 7.4 Hamiltonian mean value 210 7.5 Mean value minimization 212 7.6 Ground-state energy 214 7.7 Physical meaning of the condensation energy 216 7.8 The energy gap 218 8 The Bogoliubov Approach 222 8.1 The Bogoliubov procedure 223 8.2 Diagonalization of the Bogoliubov Hamiltonian 225 8.3 Eigenstates of the Bogoliubov Hamiltonian 228 8.4 Ground-state energy of the BCS Hamiltonian 231 8.5 Ground-state wave function of the BCS Hamiltonian 234 8.6 Discussion 237 9 The Gorkov Approach 238 9.1 The mean-field Hamiltonian 239 9.2 Gorkov equations for T = 0 241 9.3 The energy gap 244 9.4 Gorkov equations and the energy gap for T ^ 0 245 10 Richardson-Gaudin Exact Solution 247 10.1 Commutator formalism for zero-momentum fermion pairs 250 10.2 One-pair eigenstates (The Cooper problem) 255 10.3 Two-pair eigenstates 257 10.4 Three-pair eigenstates 260 10.5 Richardson-Gaudin equations for N pairs 262 10.6 Analytical solution of the Richardson-Gaudin equations 263 10.7 Hints on the analytical resolution of the Richardson-Gaudin equations 265 10.8 Many-body parameter for Cooper pairs 269 11 Links Between Cooper Pairs and Excitons 270 11.1 Degrees of freedom 272 11.2 Potentials 275 11.3 One composite boson 282 11.4 Two composite bosons 285 Contents xi 11.5 N composite bosons 290 11.6 Many-body parameters 294 11.7 Wave functions 297 11.8 Density regimes 308 Part III Particles Related to Excitons 12 Trions, Biexcitons, and Polaritons 313 12.1 A brief description 313 12.2 Spin and orbital degrees of freedom 315 13 Trions 318 13.1 The X~ trion as an exciton interacting with an electron 320 13.2 Trion creation operator 326 13.3 Trion-photon coupling 330 13.4 More on Sz = 0 trion 337 14 Biexcitons 340 14.1 The biexciton as two interacting excitons 342 14.2 Biexciton creation operator 346 14.3 Biexciton-photon coupling 347 15 Polaritons 351 15.1 Formal description 353 15.2 One polariton 355 15.3 Many-body effects 357 15.4 Microscopic derivation 361 Part IV Bosonic Condensation 16 From Elementary to Composite Boson Condensates 383 16.1 Elementary bosons 385 16.2 Elementary fermions 387 16.3 Composite bosons 388 17 Elementary Bosons 390 17.1 Noninteracting bosons for T = 0 391 17.2 Noninteracting bosons for T ^ 0 391 17.3 Momentum and spin fragmentation of the condensate 397 17.4 Interacting bosons for T = 0 402 18 Elementary Fermions 417 18.1 Free fermions for T = 0 418 18.2 Free fermions for T ^ 0 419 18.3 Interacting electrons for T = 0 420 18.4 Interacting electrons and holes 427 xii Contents 19 Composite Bosons 433 19.1 T = 0 ground state 434 19.2 Momentum, spin, and dark-bright fragmentation 448 Appendix A Some Mathematical Results 465 A. 1 Kronecker symbol and delta function 466 A.2 Fourier transform and series expansion 469 A.3 Coulomb scatterings 471 Appendix B Second Quantization Formalism 473 Appendix C The Hamiltonian for Wannier Excitons 476 C. 1 The semiconductor Hamiltonian in first quantization 477 C.2 Bloch states 478 C. 3 The semiconductor Hamiltonian on the Bloch basis 479 Appendix D Valence Electron Operator Versus Hole Operator 482 D. 1 Valence electron absence 483 D.2 Spin^ 484 D.3 I - 1 orbital momentum 486 Appendix E "The Coboson Bible" 488 Appendix F Direct Coulomb Scatterings for Wannier Excitons 493 E1 Creation potential 494 F.2 Direct Coulomb scatterings 498 F.3 Symmetry properties 501 Appendix G Concerning N Ground-State Wannier Excitons 502 G. 1 Normalization factor 503 G. 2 Hamiltonian mean value 508 Appendix H Photon-Semiconductor Interaction 513 H. 1 Electromagnetic field in vacuum 515 H.2 The electron Hamiltonian in a photon field 516 H.3 Linear coupling 518 H.4 Quadratic coupling 523 H.5 Complex polarization vectors 526 Appendix I Photon-Exciton Interaction 528 1.1 Photon-exciton coupling 529 1.2 The sum rule between photon-exciton couplings 531 References 533 Index 543.