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P. Huai, Feb. 18, 2005
Electron
Phonon Photon
Light-ElectronInteraction
Semiclassical: Dipole Interaction + Maxwell EquationQuantum: Electron-Photon Coupling
Quantum Theory of Optical Properties of Semiconductors
Interacting PhotonSemiconductor System
Carrier-CarrierInteraction
Coulomb Interaction (many-body effect)
Carrier-PhononInteraction
Scattering-induced Dephasing (ps)
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Research on Optical Properties of Semiconductorin 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 approximationNonlinear 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, 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)
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Recent Progress in Koch’s group (1)
Entanglement between a Photon and a Quantum WellHoyer et al, PRL93, 067401, (2004)
Free Particle
CoulombInteraction
Carrier-PhotonInteraction
Carrier-PhononInteraction
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Recent Progress in Koch’s group (2)
Exciton-Population Inversion and Terahertz Gain in Semiconductors Excited to ResonanceKira & Koch, PRL93, 076402, (2004)
1s2p
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 byCluster Expansion
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Time-dependent response induced terahertz absorption following non-resonant optical excitationKira 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, 155113 (2003)
systematic study on conditions for a significant amount of excitons generated from an incoherent electron-hole plasma
coupled carrier-phonon-light systemsolved by cluster expansion.
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Electron-Photon Coupled Quantum System
Free Photon
Electron-Electron & Electron-Photon Coupling
gauge transformation
Dipole Interaction in crystal
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Equations of motion for photons and carriers
Hartree-Fock approximation and Random Phase Approximation e.g.
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Semiconductor Luminescence Equations
With the renormalized Rabi energy
Electron-hole pair recombination by emitting a photon
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Example Solution of The Semiconductor Luminescence Equations
Approximation: carrier-occupation functions -> Fermi-Dirac distributionsQuasi-equilibrium condition
M. Kira et al. / Progress in Quantum Electronics 23 (1999) 189
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Semiconductor Bloch Equationsin Classical Light-Field
Pk : Polarization componentne,k (ne,k) : Carrier distribution of electron (hole)
Long-time scale: Quasi-equilibrium ne,k (ne,k) -> thermal distribution
Ultrafast process: Non-equilibrium
*Details given in the following sheets
Mechanism of Dephasing1. carrier-carrier Coulomb scattering (high excitation intensity)2. carrier-phonon scattering (low excitation intensity)3. finite radiative lifetime
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Optical Processes of 2-Band Semiconductor System
Egħ
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, .
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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
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Equations of Motions of Interband Polarization and Carrier Distribution
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Semiconductor Bloch Equations
Treatment of 4-Operator Terms by HF & RPA approximation, e.g.
Generalized Rabi Frequency
Renormalized Single-particle Energies
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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
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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 functionAbsorption Index of refraction
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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
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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 & ExchangeInteraction
VertexCorrection
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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.
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Optical Spectra by Matrix Inversion in 2-D System
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Optical Spectra by Matrix Inversion in 1-D System
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Band-Gap Renormalization in 1-D System
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Optical Spectra by Nonequilibrium Green’s Function Techniquein 1-D System