P. Huai, Feb. 18, 2005
Quantum Theory of Optical Properties of Semiconductors
Electron
Interacting Photon
Semiconductor System
Phonon
Carrier-Carrier
Interaction
Photon
Coulomb Interaction
(many-body effect)
Scattering-induced Dephasing (ps)
Carrier-Phonon
Interaction
Light-Electron
Interaction
Semiclassical: Dipole Interaction + Maxwell Equation
Quantum: Electron-Photon Coupling
1
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 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)
2
*Review paper: Kira et al. Prog. Quan. Elec. 23, 189 (1999)
Recent Progress in Koch’s group (1)
Entanglement between a Photon and a Quantum Well
Hoyer et al, PRL93, 067401, (2004)
Free
Particle
Coulomb Carrier-Photon Carrier-Phonon
Interaction
Interaction
Interaction
3
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
1s
2p
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
4
Recent Progress in Koch’s 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.
5
Electron-Photon Coupled Quantum System
Free Photon
Electron-Electron &
Electron-Photon Coupling
gauge transformation
Dipole Interaction
in crystal
6
Equations of motion for photons and carriers
Hartree-Fock approximation and
Random Phase Approximation e.g.
7
Semiconductor Luminescence Equations
Electron-hole pair
recombination by
emitting a photon
With the renormalized Rabi energy
8
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
9
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-carrier Coulomb scattering (high excitation intensity)
2. carrier-phonon scattering (low excitation intensity)
3. finite radiative lifetime
10
Optical Processes of 2-Band Semiconductor System
Conduction Band
ħw
Eg
ħw
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, .
11
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
12
Equations of Motions of
Interband Polarization and Carrier Distribution
13
Semiconductor Bloch Equations
Treatment of 4-Operator Terms by HF & RPA approximation, e.g.
Generalized Rabi Frequency
Renormalized Single-particle Energies
14
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
15
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
16
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
17
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
18
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.
19
Optical Spectra by Matrix Inversion in 2-D System
20
Optical Spectra by Matrix Inversion in 1-D System
21
Band-Gap Renormalization in 1-D System
22
Optical Spectra by Nonequilibrium Green’s Function Technique
in 1-D System
23