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Optical Engineering for the 21st Century:
Microscopic Simulation of Quantum Cascade Lasers
M.F. Pereira Theory of Semiconductor Materials and OpticsMaterials and Engineering Research Institute
Sheffield Hallam UniversityS1 1WB Sheffield, United Kingdom
Outline
Introduction to Semiconductor Lasers and Interband Optics
Interband vs Intersubband Optics
Fundamentals and Applications
Intersubband Antipolariton - A New Quasiparticle
Introduction to Semiconductor Lasers
From classical oscillators to Keldysh nonequilibrium many body Green’s functions.
Fundamental concepts:Lasing = gain > losses + feedbackWavefunction overlap transition dipole momentsPopulation inversion and gain/absorption calculationsMany body effects
Further applications: pump and probe spectroscopy – nonlinear optics
Laser = Light Amplification by Stimulated Emission of
Radiation
Stimulated emission in a two-level atomic system.
Light Emitting Diodes
pn junction
Light Emitting Diodes
pin junction
Laser Cavity: Mirrors Providing Feedback
Fabry Perot (Edge Emitting) SC Laser
Vertical Cavity SC Laser (VCSEL)
In multi-section Distributed Bragg Reflector (DBR) lasers, the absorption in the unpumped passive sections may prevent lasing.
Simple theories predict that forward biasing leading to carrier injection in the passive sections can reduce the absorption.
Many-Body Effects on DBR Lasers: the feedback is distributed over
several layers
Forward biasing is not a solution!
A. Klehr, G. Erbert, J. Sebastian, H. Wenzel, G. Traenkle, and M.F. Pereira Jr., Appl. Phys. Lett.,76, 2653 (2000).
On the contrary, the absorption increases over a certain range due to Many Particle Effects!!
Many-Body Effects on DBR Lasers
Many-Body Effects on DBR Lasers
Many-Body Effects on DBR Lasers
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr
Semiclassical Optical Response
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr
Fourier Transform
Semiclassical Optical Response
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr 0),(),(
2
2
rDc
rFourier Transform
Semiclassical Optical Response
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr 0),(),(
2
2
rDc
rFourier Transform
),(),(),( rPrrD
Optical Response of a Dielectric
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr 0),(),(
2
2
rDc
rFourier Transform
),(),(),( rPrrD
Displacement field
Optical Response of a Dielectric
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr 0),(),(
2
2
rDc
rFourier Transform
),(),(),( rPrrD
Electric field
Optical Response of a Dielectric
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr 0),(),(
2
2
rDc
rFourier Transform
),(),(),( rPrrD
Polarisation
Optical Response of a Dielectric
),(),())(41(),( rrrD
Optical Response of a Dielectric
),(),())(41(),( rrrD
optical susceptibility
Optical Response of a Dielectric
),(),())(41(),( rrrD
optical dielectric function
Optical Response of a Dielectric
Plane wave propagation:
))()((exp()(),( ikir
Optical Response of a Dielectric
Plane wave propagation:
))()((exp()(),( ikir
wavenumber
cnk /)()( refractive index
Optical Response of a Dielectric
Plane wave propagation:
))()((exp()(),( ikir
)(2)( extinction coefficient
absorption coefficient
Optical Response of a Dielectric
Usually, in semiconductors, the imaginary part of the dielectric function is much smaller then the real part and we can write:
)("4
)(
)(')(
bcn
n
Optical Response of a Dielectric
Microscopic models for the material medium usually yield "
)(")('
dP
)(')("
dP
Kramers-Kronig relations (causality)
Optical Response of a Dielectric
-
+
dE …….
A linearly polarized electric field induces a macroscopic polarization
in the dielectric
Classical Oscillator
dnexn 00
Classical Oscillator
dnexn 00
|| exdipole moment
Classical Oscillator
Electron in an oscillating electric field: Newton’s equation: damped oscillator.
)'()'()(
)'()'(2
)(2
2
2
2
0
2
002
2
0
tettGtx
ttttGtt
m
texmtx
mtx
m
Classical Oscillator
Electron in an oscillating electric field: Newton’s equation: damped oscillator.
)'()'()(
)'()'(2
)(2
2
2
2
0
2
002
2
0
tettGtx
ttttGtt
m
texmtx
mtx
m
Retarded Green function
Classical Oscillator
iimen
EeGx
et ti
'
0
'
0
'
0
2
0
0
112
)(
)()()()()(
,)(
Classical Oscillator
Even at a very simple classical level:
)()( 2
0 Gen
Classical Oscillator
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
Classical Oscillator
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
22
0
'
0
Classical Oscillator
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
22
0
'
0 renormalized energy dephasing
Classical Oscillator
Even at a very simple classical level:
)()( 2
0 Gen
optical suscpetibility Greens functions
22
0
'
0 renormalized energy dephasing
Current research: Nonequilibrium Keldysh Greens Functions
Selfenergies: energy renormalization & dephasing
Classical Oscillator
The electrons are not in pure states, but in mixed states, described, e.g. by a density matrix
The pure states of electrons in a crystal are eigenstates of
0
nknk nk0
Free Carrier Optical Response in Semiconductors
The electrons are not in pure states, but in mixed states, described, e.g. by a density matrix
The pure states of electrons in a crystal are eigenstates of
0
nknk nk0n band label
k crystal momentum
Free Carrier Optical Response in Semiconductors
k
Free Carrier Optical Response in Semiconductors
The optical polarization is given by
k
nknk nk0
dttrtP )()(
Free Carrier Optical Response in Semiconductors
The optical susceptibility in the Rotating Wave Approximation (RWA) is
k vc
vccv
ikk
kfkfkdL
)()(
)()()(1)(2
3
Free Carrier Optical Response in Semiconductors
sum of oscillator transitions, one for each k-value.
Weighted by the dipole moment
(wavefunction overlap) and by the population inversion:
k
)(kdnl
)()( kfkf vc
Each k-value yields a two-level atom type of transition
Free Carrier Optical Response in Semiconductors
The Keldysh Greens functions are Greens functions for the Dyson equations:
)21()32()13()13(10 GG
Keldysh Greens Functions
The Keldysh Greens functions are Greens functions for the Dyson equations:
)21()32()13()13(10 GG
= +G 0G 0G G
Keldysh Greens Functions
Semiconductor Bloch Equations can be derived from projections of the GF’s
= +G 0G 0G G
)2()1()12( iG
Keldysh Greens Functions
= +G 0G 0G G
)11(),(
)11(),(
)11(),(
eheh
hhh
eee
GitrP
GitrN
GitrN
Keldysh Greens Functions
Start from the equation for the polarization at steady-state
),'()'(2
)(
))()(1(),())()((
'
0
kPkkVkd
kfkfkPikeke
k
scv
hehe
Semiconductor Bloch Equations: Projected Greens Functions
Equations
Start from the equation for the polarization at steady-state
),'()'(2
)(
))()(1(),())()((
'
0
kPkkVkd
kfkfkPikeke
k
scv
hehe
renormalized energies from
Semiconductor Bloch Equations: Projected Greens
Functions Equations
Start from the equation for the polarization at steady-state
),'()'(2
)(
))()(1(),())()((
'
0
kPkkVkd
kfkfkPikeke
k
scv
hehe
dephasing from
Semiconductor Bloch Equations: Projected Greens
Functions Equations
Start from the equation for the polarization at steady-state
),'()'(2
)(
))()(1(),())()((
'
0
kPkkVkd
kfkfkPikeke
k
scv
hehe
Screened potential
Semiconductor Bloch Equations: Projected Greens
Functions Equations
Introduce a susceptibility
2),(),( 0E
kkP
'
0 ),'()'()(
11),(),(
k
s
cv
kkkVkd
kk
Semiconductor Bloch Equations: Projected Greens
Functions Equations
),'(),( 0'
1
', kkk
kk
quasi-free carrier term with bandgap renormalization and dephasing due to scattering mechanims
ikekekfkf
kdkhe
hecv )()(
)()(1)(),(0
Semiconductor Bloch Equations: Projected Greens
Functions Equations
),'(),( 0'
1
', kkk
kk
Coulomb enhancement and nondiagonal dephasing
Sum of oscillator-type responses weighted by dipole moments, population differences and many body effects!
Semiconductor Bloch Equations: Projected Greens
Functions Equations
Pump-Probe Absorption Spectra
Semiconductor Slab
Strong pump laser field generating carriers
Weak probe beam. Susceptibility can be calculated in linear response in the field and arbitrarily nonlinear in the resulting populations due to the pump.
Absorption Spectra of GaAs Quantum Wells
Microscopic Mechanisms for Lasing in II-VI Quantum Wells
Coulomb and nonequilibrium effects are important in semiconductors and can be calculated from first principles with Keldysh Greens functions.
It is possible to understand the resulting optical response as a sum of elementary oscillators weighted by dipole moments, population differences and Coulomb effects.
The resulting macroscopic quantities can be used as starting point for realistic device simulations.
Summary
