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

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, (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 dynamics in semiconductor heterostructures, Hoyer et al. PRB67, (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, (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 Quantum Well Hoyer et al, PRL93, , (2004) Free Particle Coulomb Interaction Carrier-Photon Interaction Carrier-Phonon Interaction

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

5 Time-dependent response induced terahertz absorption following non-resonant optical excitation Kira et al. Solid State Commun. 129, 733 (2004) Recent Progress in Koch’s group (3) Influence of Coulomb and phonon interaction on the exciton formation dynamics in semiconductor heterostructures Hoyer et al. PRB67, (2003) systematic study on conditions for a significant amount of excitons generated from an incoherent electron-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 Dipole Interaction in crystal

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

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

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 P k : Polarization component n e,k (n e,k ) : Carrier distribution of electron (hole) Long-time scale: Quasi-equilibrium n e,k (n e,k ) -> thermal distribution Ultrafast process: Non-equilibrium *Details given in the following sheets Mechanism of Dephasing 1. carrier-carrier Coulomb scattering (high excitation intensity) 2. carrier-phonon scattering (low excitation intensity) 3. finite radiative lifetime

11 Optical Processes of 2-Band Semiconductor System EgEg ħħ ħħ Coupling with classical light field Valence Band Conduction Band See chapters 8,10, 12, 15 of “Quantum Theory of the Optical and Electronic 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 =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 Dielectric function Absorption Index of refraction

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 Second Born Approximation Off-diagonal spectral function decayed in long-time limit Quasi-stationary conditions Markov approximation Quantum kinetic collision integral generalized Kadanoff-Baym ansatz 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