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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