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Many-Body Correlations and Excitonic Effects in Semiconductor Spectroscopy

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ARTICLE IN PRESS Progress in Quantum Electronics 30 (2006) 155–296 www.elsevier.com/locate/pquantelec Review Many-body correlations and excitonic effects in semiconductor spectroscopy M. KiraÃ, S.W. Koch Department of Physics and Material Sciences Center, Philipps-University Marburg, Renthof 5, D-35032 Marburg, Germany Abstract The optically excited system of electronic excitations in semiconductor nanostructures is analyzed theoretically. A many-body theory based on an equation-of-motion approach for the interacting electron, hole, photon, and phonon system is reviewed. The infinite hierarchy of coupled equations for the relevant correlation functions is systematically truncated using a cluster-expansion scheme. The resulting system of equations describes the optical generation of semiconductor quasi-particle configurations with classical or quantum mechanical light sources, as well as their photon-assisted spontaneous recombination. The theory is evaluated numerically to study semiclassical and quantum excitation under different resonant and non-resonant conditions for a wide range of intensities. The generation of a correlated electron–hole plasma and exciton populations is investigated. It is shown how these states can be identified using direct quasi-particle spectroscopy with sources in the terahertz range of the electromagnetic spectrum. The concept of quantum–optical spectroscopy is introduced and it is predicted that semiconductor excitation with suitable incoherent light directly generates quantum-degenerate exciton states. The phase space for this exciton condensate is identified and its experimental signatures are discussed. r 2007 Elsevier Ltd. All rights reserved. PACS: 42.50.Àp; 78.55.Àm; 03.75.Kk; 42.25.Kb; 71.35.Ày; 78.47.+p Keywords: Semiconductor quantum optics; Many-body correlations; Excitons; Electron–hole plasma; Exciton condensate ÃCorresponding author. Tel.: +49 6421 282422; fax: +49 6421 2827076. E-mail address: [email protected] (M. Kira). 0079-6727/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.pquantelec.2006.12.002 ARTICLE IN PRESS 156 M. Kira, S.W. Koch / Progress in Quantum Electronics 30 (2006) 155–296 Contents 1. Introduction . 157 2. Quantum-electrodynamic theory . 159 2.1. Quantization of carriers, photons, and phonons. 160 2.2. Many-body Hamiltonian . 163 3. Equations of motion hierarchy . 165 3.1. General operator dynamics . 165 3.2. Cluster expansion . 167 4. Singlet–doublet correlations . 170 4.1. Singlet–doublet quantities for homogeneous excitation . 170 4.2. Singlet dynamics . 172 4.3. Doublet dynamics . 174 4.3.1. Pure photon and phonon correlations. 174 4.3.2. Mixed carrier–photon–phonon correlations. 175 4.3.3. Pure carrier correlations . 177 5. Excitonic effects in semiconductor optics . 183 5.1. Maxwell-semiconductor Bloch equations. 184 5.2. Excitonic states. 186 5.3. Coherent excitonic polarization . 187 5.4. Linear optical response . 189 5.5. Elliott formula . 190 5.6. Self-consistent transmission and reflection . 191 5.7. Radiative polarization decay . 195 5.8. Coherent and incoherent carrier correlations . 197 5.9. Linear optical polarization. 198 5.10. Excitation induced dephasing . 201 6. From polarization to incoherent populations . 204 6.1. Energy transfer in the carrier system. 204 6.2. From coherent to incoherent excitations . 205 6.3. Dynamics of exciton correlations . 210 6.4. Incoherent excitons . 211 6.5. Correlated electron–hole plasma. 213 6.6. Terahertz spectroscopy and excitons . 218 7. Optical semiconductor excitations . 224 7.1. Resonant 1s-excitation. 226 7.1.1. Weak excitation . 227 7.1.2. Strong excitation . 228 7.2. Resonant 2s-excitation. 230 7.2.1. Terahertz gain . 232 7.2.2. Formation of 2p-excitons . 234 7.3. Nonresonant excitation . 237 8. Exciton formation . 240 8.1. Numerical studies . 241 8.2. Microscopic analysis . 244 8.3. Phase space . 245 8.4. Thermodynamic limit . 246 9. Quantum-optical semiconductor excitations . 249 9.1. Semiconductor luminescence equations . 249 9.2. Radiative recombination of carriers and exciton populations . 253 9.3. Concept of quantum-optical spectroscopy . 254 ARTICLE IN PRESS M. Kira, S.W. Koch / Progress in Quantum Electronics 30 (2006) 155–296 157 9.4. Quantum description of exciting light fields. 255 9.5. Matching classical and quantum light sources . 256 9.6. Classical vs. quantum excitation. 258 9.7. Emission from the condensate . 261 9.8. Exciton condensate under non-ideal conditions . 263 9.9. General principle of quantum-optical spectroscopy . 266 10. Summary and outlook . 266 Acknowledgements . 268 Appendix A. System Hamiltonian . 268 Appendix B. Implicit-notation formalism . 270 Appendix C. Relevant singlet–doublet factorizations . 272 C.1. Two-particle factorizations. 273 C.2. Three-particle factorizations. 274 C.3. Constraints in homogeneously excited systems. 275 Appendix D. Singlet–doublet dynamics for carriers . 275 D.1. Dynamics of two-particle correlations. 276 D.2. Formal aspects of the three-particle scattering term . 279 Appendix E. Classification of correlations . 280 E.1. Spin-related symmetries . 282 E.2. Coherent vs. incoherent contributions. 282 E.3. Fermionic exchange symmetry . 283 Appendix F. Exciton-correlation dynamics . 283 F.1. Single-particle source. 284 F.2. Two-particle correlations . 284 F.3. Energy transfer . 286 Appendix G. Excitation induced dephasing. 287 References . 290 1. Introduction Semiconductors and their nanostructures efficiently interact with the light field if the optical frequency is in the spectral vicinity of the direct material band gap. This light–matter coupling is mediated by transitions of electrons between the valence and the conduction band. Since the missing valence-band electrons are treated as ‘‘holes’’, we speak of optically assisted electron–hole-pair transitions. The properties of the valence-band holes are those of the missing electrons, i.e. they are Fermions and have a spin, charge, effective mass and so on, which are opposite to those of the valence-band electrons. Since the electron charge is.
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