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