optical engineering for the 21st century: microscopic simulation of quantum cascade lasers m.f....
TRANSCRIPT
![Page 1: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/1.jpg)
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
![Page 2: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/2.jpg)
Outline
Introduction to Semiconductor Lasers and Interband Optics
Interband vs Intersubband Optics
Fundamentals and Applications
Intersubband Antipolariton - A New Quasiparticle
![Page 3: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/3.jpg)
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
![Page 4: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/4.jpg)
Laser = Light Amplification by Stimulated Emission of
Radiation
Stimulated emission in a two-level atomic system.
![Page 5: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/5.jpg)
Light Emitting Diodes
pn junction
![Page 6: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/6.jpg)
Light Emitting Diodes
pin junction
![Page 7: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/7.jpg)
Laser Cavity: Mirrors Providing Feedback
![Page 8: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/8.jpg)
Fabry Perot (Edge Emitting) SC Laser
![Page 9: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/9.jpg)
Vertical Cavity SC Laser (VCSEL)
![Page 10: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/10.jpg)
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
![Page 11: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/11.jpg)
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
![Page 12: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/12.jpg)
Many-Body Effects on DBR Lasers
![Page 13: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/13.jpg)
Many-Body Effects on DBR Lasers
![Page 14: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/14.jpg)
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr
Semiclassical Optical Response
![Page 15: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/15.jpg)
A classical transverse optical field propagating in dielectric satisfies the wave equation:
2
2
2 ),(/1),(
dt
trDctr
Fourier Transform
Semiclassical Optical Response
![Page 16: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/16.jpg)
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
![Page 17: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/17.jpg)
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
![Page 18: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/18.jpg)
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
![Page 19: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/19.jpg)
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
![Page 20: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/20.jpg)
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
![Page 21: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/21.jpg)
),(),())(41(),( rrrD
Optical Response of a Dielectric
![Page 22: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/22.jpg)
),(),())(41(),( rrrD
optical susceptibility
Optical Response of a Dielectric
![Page 23: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/23.jpg)
),(),())(41(),( rrrD
optical dielectric function
Optical Response of a Dielectric
![Page 24: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/24.jpg)
Plane wave propagation:
))()((exp()(),( ikir
Optical Response of a Dielectric
![Page 25: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/25.jpg)
Plane wave propagation:
))()((exp()(),( ikir
wavenumber
cnk /)()( refractive index
Optical Response of a Dielectric
![Page 26: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/26.jpg)
Plane wave propagation:
))()((exp()(),( ikir
)(2)( extinction coefficient
absorption coefficient
Optical Response of a Dielectric
![Page 27: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/27.jpg)
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
![Page 28: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/28.jpg)
Microscopic models for the material medium usually yield "
)(")('
dP
)(')("
dP
Kramers-Kronig relations (causality)
Optical Response of a Dielectric
![Page 29: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/29.jpg)
-
+
dE …….
A linearly polarized electric field induces a macroscopic polarization
in the dielectric
Classical Oscillator
![Page 30: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/30.jpg)
dnexn 00
Classical Oscillator
![Page 31: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/31.jpg)
dnexn 00
|| exdipole moment
Classical Oscillator
![Page 32: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/32.jpg)
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
![Page 33: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/33.jpg)
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
![Page 34: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/34.jpg)
iimen
EeGx
et ti
'
0
'
0
'
0
2
0
0
112
)(
)()()()()(
,)(
Classical Oscillator
![Page 35: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/35.jpg)
Even at a very simple classical level:
)()( 2
0 Gen
Classical Oscillator
![Page 36: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/36.jpg)
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
Classical Oscillator
![Page 37: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/37.jpg)
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
22
0
'
0
Classical Oscillator
![Page 38: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/38.jpg)
Even at a very simple classical level:
)()( 2
0 Gen
optical susceptibility Greens functions
22
0
'
0 renormalized energy dephasing
Classical Oscillator
![Page 39: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/39.jpg)
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
![Page 40: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/40.jpg)
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
![Page 41: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/41.jpg)
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
![Page 42: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/42.jpg)
k
Free Carrier Optical Response in Semiconductors
![Page 43: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/43.jpg)
The optical polarization is given by
k
nknk nk0
dttrtP )()(
Free Carrier Optical Response in Semiconductors
![Page 44: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/44.jpg)
The optical susceptibility in the Rotating Wave Approximation (RWA) is
k vc
vccv
ikk
kfkfkdL
)()(
)()()(1)(2
3
Free Carrier Optical Response in Semiconductors
![Page 45: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/45.jpg)
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
![Page 46: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/46.jpg)
The Keldysh Greens functions are Greens functions for the Dyson equations:
)21()32()13()13(10 GG
Keldysh Greens Functions
![Page 47: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/47.jpg)
The Keldysh Greens functions are Greens functions for the Dyson equations:
)21()32()13()13(10 GG
= +G 0G 0G G
Keldysh Greens Functions
![Page 48: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/48.jpg)
Semiconductor Bloch Equations can be derived from projections of the GF’s
= +G 0G 0G G
)2()1()12( iG
Keldysh Greens Functions
![Page 49: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/49.jpg)
= +G 0G 0G G
)11(),(
)11(),(
)11(),(
eheh
hhh
eee
GitrP
GitrN
GitrN
Keldysh Greens Functions
![Page 50: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/50.jpg)
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
![Page 51: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/51.jpg)
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
![Page 52: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/52.jpg)
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
![Page 53: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/53.jpg)
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
![Page 54: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/54.jpg)
Introduce a susceptibility
2),(),( 0E
kkP
'
0 ),'()'()(
11),(),(
k
s
cv
kkkVkd
kk
Semiconductor Bloch Equations: Projected Greens
Functions Equations
![Page 55: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/55.jpg)
),'(),( 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
![Page 56: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/56.jpg)
),'(),( 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
![Page 57: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/57.jpg)
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.
![Page 58: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/58.jpg)
Absorption Spectra of GaAs Quantum Wells
![Page 59: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/59.jpg)
Microscopic Mechanisms for Lasing in II-VI Quantum Wells
![Page 60: Optical Engineering for the 21st Century: Microscopic Simulation of Quantum Cascade Lasers M.F. Pereira Theory of Semiconductor Materials and Optics Materials](https://reader031.vdocument.in/reader031/viewer/2022032703/56649d0c5503460f949dff3d/html5/thumbnails/60.jpg)
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