P. Huai, Feb. 18, 2005

1. P. Huai, Feb. 18, 2005 Quantum Theory of Optical Properties of Semiconductors Electron Interacting Photon Semiconductor System Phonon Photon Carrier-Carrier Interaction Coulomb Interaction (many-body effect) Scattering-induced Dephasing (ps) Carrier-Phonon Interaction Semiclassical: Dipole Interaction + Maxwell Equation Quantum: Electron-Photon Coupling Light-Electron Interaction

2. Research on Optical Properties of Semiconductor in S. W. Koch’s group • Semiclassical Approach: Semiconductor Bloch Equation • Hartree-Fock & Random Phase Approximation. • Coulombic effect : bandgap & field renormalization • Treatment of Correlation effect • Dynamics-controlled truncation (DCT) • Four-wave-mixing signal, Lindberg et al. PRB50, 18060 (1994) • Nonequalibrium Green’s function with second Born approximation • Nonlinear saturation of the excitonic normal-mode coupling, Jahnke et al. PRL77, 5257 (1996) • Cluster Expansion • Influence of Coulomb and phonon interaction on the exciton formation dynamicsin semiconductorheterostructures, Hoyer et al. PRB67, 155113 (2003) • Fully Quantum Mechanical Approach: • Coupled Semiconductor Bloch and Luminescence Equation • PL & Absorption, e.g. Kira et al. PRL81, 3263 (1998) • Exciton correlations, formation rates, distribution functions, e.g. Kira et al. PRL87, 176401 (2001) • *Review paper: Kira et al. Prog. Quan. Elec. 23, 189 (1999)

3. Recent Progress in Koch’s group (1) Entanglement between a Photon and a QuantumWell Hoyer et al, PRL93, 067401, (2004) Free Particle Coulomb Interaction Carrier-Photon Interaction Carrier-Phonon Interaction

4. 1s 2p Recent Progress in Koch’s group (2) Exciton-Population Inversion and Terahertz Gain in Semiconductors Excited to Resonance Kira & Koch, PRL93, 076402, (2004) Carrier + Phonon: Quantum Light-Field : Classical Equation of motion decoupled by Cluster Expansion Formation of excitons in 2p statesfor excitation aroundthe 2s resonance. exciton-population inversion between the 2pand 1s states

5. Recent Progress in Koch’s group (3) Time-dependentresponse induced terahertz absorptionfollowing non-resonant optical excitation Kira et al. Solid State Commun. 129, 733 (2004) Influence of Coulomb and phonon interaction on the exciton formation dynamicsin semiconductorheterostructures Hoyer et al. PRB67, 155113 (2003) systematic study on conditions for asignificant amount of excitons generated from an incoherentelectron-hole plasma coupled carrier-phonon-light system solved by cluster expansion.

6. Electron-Photon Coupled Quantum System Free Photon Electron-Electron & Electron-Photon Coupling gauge transformation in crystal Dipole Interaction

7. Equations of motion for photons and carriers Hartree-Fock approximation and Random Phase Approximation e.g.

8. Semiconductor Luminescence Equations Electron-hole pair recombination by emitting a photon With the renormalized Rabi energy

9. Example Solution of The Semiconductor Luminescence Equations Approximation: carrier-occupation functions -> Fermi-Dirac distributions Quasi-equilibrium condition M. Kira et al. / Progress in Quantum Electronics 23 (1999) 189

10. Semiconductor Bloch Equations in Classical Light-Field *Details given in the following sheets Pk : Polarization component ne,k (ne,k): Carrier distribution of electron (hole) Long-time scale: Quasi-equilibrium ne,k (ne,k)-> thermal distribution Ultrafast process: Non-equilibrium Mechanism of Dephasing 1. carrier-carrierCoulomb scattering (high excitation intensity) 2.carrier-phonon scattering (low excitation intensity) 3. finite radiative lifetime

11. Eg ħw ħw Optical Processes of 2-Band Semiconductor System Conduction Band Valence Band ------ Coupling with classical light field See chapters 8,10, 12, 15 of “Quantum Theory of the Optical andElectronic Properties of Semiconductors”, 4th ed. World Scientific,Singapore, 2004 by H. Haug and S. W. Koch, .

12. Equations of Motions of 2-band System Bloch functions Here 2 bands l=c,v are taken into account Diagonal and off-diagonal elements of reduced single-particle density matrix Equation of motion

13. Equations of Motions of Interband Polarization and Carrier Distribution

14. Semiconductor Bloch Equations Treatment of 4-Operator Terms by HF & RPA approximation, e.g. Generalized Rabi Frequency Renormalized Single-particle Energies

15. Optical Properties of Quasi-Equilibrium System Electron (hole) reach thermal distributions Quasi-static screening taking into account screening effect due to Coulomb interaction phenomenologically Polarization equation in quasi-equilibrium

16. Solution of Polarization by Numerical Matrix Inversion Define : Angle-averaged potential susceptibility free-carrier susceptibility Improvement: finite damping rate without the detailed mechanism Vertex integral equation complex susceptibility Absorption Index of refraction Dielectric function

17. Correlation Effect of Coulomb Interaction Omit the correlation -> Lack of screening and carrier-carrier scattering Solution: – Nonequilibrium (Keldysh) Green’s function – Dynamics-controlled truncation – Cluster Expansion Exciton formation, Ultrafast Femtosecond build-up of screening

18. Nonequilibrium Green’s function Quantum kinetic collision integral generalized Kadanoff-Baym ansatz • Second Born Approximation • Off-diagonal spectral function decayed in long-time limit • Quasi-stationary conditions • Markov approximation Direct & Exchange Interaction Vertex Correction

19. Optical Spectra by Matrix Inversion in 3-D System Beakdown of thermalized carrier distribution, which is only valid in weak recombination, i.e., no lasing takes place.

20. Optical Spectra by Matrix Inversion in 2-D System

21. Optical Spectra by Matrix Inversion in 1-D System

22. Band-Gap Renormalization in 1-D System

23. Optical Spectra by Nonequilibrium Green’s Function Technique in 1-D System