P. Huai, Feb. 18, 2005

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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
Light-Electron Interaction
Semiclassical Dipole Interaction Maxwell
Equation Quantum Electron-Photon Coupling
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Research on Optical Properties of
Semiconductor in S. W. Kochs 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 Greens 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, 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)

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Recent Progress in Kochs group (1)
Entanglement between a Photon and a Quantum
Well Hoyer et al, PRL93, 067401, (2004)
Free Particle
Coulomb Interaction
Carrier-Photon Interaction
Carrier-Phonon Interaction
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Recent Progress in Kochs 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 states for excitation
around the 2s resonance. exciton-population
inversion between the 2p and 1s states
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Recent Progress in Kochs group (3)
Time-dependent response induced terahertz
absorption following non-resonant optical
excitation Kira et al. Solid State Commun. 129,
733 (2004)
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 system solved by
cluster expansion.
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Electron-Photon Coupled Quantum System
Free Photon
Electron-Electron Electron-Photon Coupling
gauge transformation
in crystal
Dipole Interaction
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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
Electron-hole pair recombination by emitting a
photon
With the renormalized Rabi energy
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Example Solution of The Semiconductor
Luminescence Equations
Approximation carrier-occupation functions -gt
Fermi-Dirac distributions Quasi-equilibrium
condition
M. Kira et al. / Progress in Quantum Electronics
23 (1999) 189
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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) -gt thermal
distribution Ultrafast process
Non-equilibrium
Mechanism of Dephasing 1. 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
Conduction Band
Valence Band
------ Coupling with classical light field
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 lc,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
Absorption Index of refraction
Dielectric function
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Correlation Effect of Coulomb Interaction
Omit the correlation -gt Lack of screening and
carrier-carrier scattering
Solution Nonequilibrium (Keldysh) Greens
function Dynamics-controlled
truncation Cluster Expansion
Exciton formation, Ultrafast Femtosecond build-up
of screening
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Nonequilibrium Greens 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
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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 Greens
Function Technique in 1-D System