Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter,...
Transcript of Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter,...
![Page 1: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/1.jpg)
Microscopic Modelling of the Optical Properties of Quantum-Well Semiconductor Lasers
• OVERVIEW- Outline of Theory- Gain/Absorption- Luminescence - Radiative and Auger Losses- Nonequilibrium Effects
• COLLABORATORS- A. Thränhardt, I. Kutzentsova, S.Becker, Marburg - J. Hader, J. V. Moloney et al., Tucson- W. Chow, Sandia
• SUPPORT- AFOSR F49620-02-1-0380, DFG
Stephan W. Koch Department of Physics
Philipps University, Marburg/Germany
![Page 2: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/2.jpg)
Overview
simple models: easy to use
• often poorly controlled, ad hoc assumptions• rely on major experimental input• (semi-)empirical fits of experimental data• many and adjustable parameters• limited potential for predictions• ….
microscopic theory: numerically demanding
• systematic approach based on fundamental physics• controlled approximations• predicts experiments• no need for detailed experimental input• wide range of applicability• can be used as basis for controllably simplified models• ….
![Page 3: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/3.jpg)
Semiconductor Optics: Semiclassical Theory
MAXWELL’S WAVE EQUATION
macroscopic optical polarization
semiconductors: Bloch basis
general theory: Haug/Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed., World Scientific (2004)
![Page 4: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/4.jpg)
Semiconductor Density Matrix
two-band model, one QW subband
… e-h-pair transition amplitude
… occupation probabilities
![Page 5: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/5.jpg)
H = H0 + HCoul + Hdip
H0 single particle (band structure)
HCoul Coulomb interaction between carriers
Hdip dipole interaction with optical field
Hamiltonian
![Page 6: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/6.jpg)
Calculation of Optical Response
structure of resulting equations:
Heisenberg equation of motion:
infinite hierarchy of many-body correlations
systematic CLUSTER expansion
![Page 7: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/7.jpg)
• two-point level (uncorrelated single particles)
• four-point level (two particle correlations)
Schematics of Cluster Expansion
![Page 8: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/8.jpg)
Semiconductor Bloch Equations
• nonlinearities: phase-space filling, gap reduction, Coulomb enhancement
• correlation contributions: scattering, dephasing, screening
HF field renormalization
HF energy renormalization
all contributions beyond singlets
![Page 9: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/9.jpg)
Second Born-Markov Approximation
• intraband Coulomb scattering
• phonon scattering
Born-Markov limit: Boltzmann equation
Coulomb scattering
phonon scattering
![Page 10: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/10.jpg)
Coulomb Scattering I
scattering rates:
![Page 11: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/11.jpg)
Coulomb Scattering II
Fermi-Dirac distribution:
quasi-equilibrium:detailed balance
![Page 12: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/12.jpg)
Excitation Induced Dephasing
� important for excitation saturation: Jahnke et al., PRL 77, 5257 (1996).and gain calculations: Chow/Koch, Semiconductor-Laser Fundamentals, Springer Verlag (1999).
• and are calculated from all terms quadratic in the screened Coulomb interaction
• strong compensation between diagonal and off-diagonal terms
Re … off-diagonal dephasing
Im … off-diagonal energy shift
)(', tkkΓ)(', tkkΓ
Re … diagonal dephasing � generalized T2 time
Im … diagonal energy shift � generalized band-gap shift
)(tkΓ
)(tkΓ
![Page 13: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/13.jpg)
Density Dependent Absorption
F. Jahnke, M. Kira, and S.W. Koch, Z. Physik B 104, 559 (1997)
Detuning
Abs
orpt
ion
• experiment: InGaAs/GaAs QW• Khitrova, Gibbs, Jahnke, Kira, Koch, Rev. Mod. Phys. 71, 1591 (1999)• EID first observed in 4-wave mixing, Wang et al. PRL 71, 1261 (1993)
![Page 14: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/14.jpg)
Lineshape Problem
0.2
0.0
-0.4
-0.2
-0.6 -20 -10 0 10
Detuning
Abs
orpt
ion
[103 /
cm]
full calculation
dephasing rate approx.
•
• gain of two-band bulk material
• nondiagonal scattering contributions → lineshape modification,no absorption below the gap
Hughes, Knorr, Koch, Binder, Indik, and MoloneySolid State Comm. 100, 555 (1996)
![Page 15: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/15.jpg)
abso
rptio
n (x
103 /
cm)
abso
rptio
n/ga
in [1
/cm
]
abso
rptio
n/ga
in
Photon Energy (eV)
Current 0-20mA… Density 0.6-3.0x1012cm-2
exp: C. Ellmers et al., theory: A. Girndt et al. MarburgCourtesy of W.W. Chow, P.M. Smowton, P. Blood, A. Grindt, F. Jahnke, and S.W. Koch
8nm InGaAs/AlGaAs 6.8 nm In0.4Ga0.5P/(Al0.5Ga0.5)In0.51P0.49
10nm (9.2nm) InGaAs/AlGaAs
N=1.6, 2.2, 2.5, 3.0*1012/cm2
exp: D. Bossert et al., theory: A. Girndt et al., Marburg
Detuning
Detuning
Optical Gain: Theory and Experiment
![Page 16: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/16.jpg)
Wide Bandgap Materials: InGaN/GaN
K300TGaAsAsGaInnm5 8020 =,/..K300TGaNNGaInnm2 8020 =,/..
×20 .
×60 . ×1 ×2×3
212 cm104 −×
3
1.1 1.2 1.3 1.4 1.5 1.62.9 3.0 3.1 3.2
Photon Energy (eV) Photon Energy (eV)
-1
0
1
2
unexcited qw bandgap
212 cm10 −
212 cm108 −×
unexcited qw bandgap
-5
0
5
10
15
20TE
abso
rptio
n (
10 c
m
)4
-1
• strong Coulomb effects: gapshift and broadening
• excitonic effects even at elevated densities and temperatures
![Page 17: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/17.jpg)
Semiconductor Luminescence
where is proportional to dipole matrix element and mode strength at the QW position
![Page 18: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/18.jpg)
Semiconductor Luminescence Equations I
• photon operator dynamics
• polarization operator dynamics
• Coulomb hierarchy problem (as in semiclassical theory)
• photon-carrier coupling hierarchy
• Cluster expansion to treat hierarchies on equal footing
![Page 19: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/19.jpg)
Semiconductor Luminescence Equations II
photon assisted interband polarization
incoherent source
coherent source
feedback (cavity)
excitonic signatures
![Page 20: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/20.jpg)
• excitonic resonances independent of source term• relative peak height depends on populations
with •
• „true“ excitons
•
PRL 92, 067402 (2004)
Quasi Stationary: Luminescence Elliott Equation
![Page 21: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/21.jpg)
Gain/Absorption and Luminescence
5nm Ga0.8In0.2As/GaAs pin-MQWgain: theory: 1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875 PL: theory: 0.17, 0.30, 0.40, 0.53, 0.68, 0.86 [1012 / cm2]
experiment: 6.0, 6.5, 7.0, 7.5, 8.3, 9.0 [mA] experiment: 12, 16, 18, 21, 24 [mW]
![Page 22: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/22.jpg)
Luminescence Analysis
Input:
• nominal structural parameters• experimental luminescence spectra at two moderate intensities
Analysis:
calculate ideal, homogeneously broadened luminescence spectra for given structure
compare to experimental luminescence
determine from deviations the inhomogeneous broadening, actual structural parameters, quality of device
use optimized parameters to calculate gain/absorption spectra, refractive index, differential gain, linewidth enhancement factor, …
Ref.: Hader et al., IEEE Phot. Techn. Lett. 14, 762 (2002)
![Page 23: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/23.jpg)
Luminescence Analysis II
• sample with three quantum wells
• nominal composition: 5nm Ga(.8)In(.2)As/GaAs
• sandwiched between n and p doped buffer layersexperiment: 12, 16, 18, 21, 24 mWtheory: 0.1, 0.17, 0.23, 0.3, 0.4, 0.53, 0.68, 0.86 1012cm2
• needed: slight adjustment of composition
• inhomogeneous broadening
no broadening, no shifting
![Page 24: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/24.jpg)
Luminescence Analysis
Input:
• nominal structural parameters• experimental luminescence spectra at two moderate intensities
Analysis:
calculate ideal, homogeneously broadened luminescence spectra for given structure
compare to experimental luminescence
determine from deviations the inhomogeneous broadening, actual structural parameters, quality of device
![Page 25: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/25.jpg)
Luminescence Analysis III
• sample with three quantum wells
• nominal composition: 5nm Ga(.8)In(.2)As/GaAs
• sandwiched between n and p doped buffer layersexperiment: 12, 16, 18, 21, 24 mWtheory: 0.1, 0.17, 0.23, 0.3, 0.4, 0.53, 0.68, 0.86 1012cm2
• good fit to experiment yields actual composition
• less In (0.19) or thinner (- one monolayer)
![Page 26: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/26.jpg)
Luminescence Analysis
Input:
• nominal structural parameters• experimental luminescence spectra at two moderate intensities
Analysis:
calculate ideal, homogeneously broadened luminescence spectra for given structure
compare to experimental luminescence
determine from deviations the inhomogeneous broadening, actual structural parameters, quality of device
use optimized parameters to calculate gain/absorption spectra
![Page 27: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/27.jpg)
Absorption/Gain
• using optimized material parameters
• experimental inhomogeneous broadening
• other results:
• refractive index, alfa parameter, etc.
theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2]
1000
900
800
700
600
500
400
300
200
100
0
-100
-200850 870 890 910 930 950 970 990 1010
wavelength [nm]
mod
al a
bsor
ptio
n [1
/cm
]
![Page 28: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/28.jpg)
Test of Predicted Gain Spectra
• independently measured gain spectra
• NO additional fitting!
• very good theory/experiment agreement
• verification of microscopic theory
• predictive capabilities
experiment: 5.0, 6.0, 6.5, 7.0, 7.5, 8.0, 8.6, 10.0 [mV]theory: 1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875 [1012cm2]
![Page 29: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/29.jpg)
k
E
k
E
hωωωω
k
E
Classical Parametrization of Loss Current Jloss:
defect-recombination spontaneous emission Auger recombination non-capture, escape
usually negligible in high
quality crystal growth
absent in optically
pumped devices
Problems with A, B, C - Parametrization:
usually dominant
� parameters only very roughly known and only for special cases;
depend on well- and barrier-materials, layer widths, temperatures, densities...
� simple density-dependence far from reality
Losses in Semiconductor Lasers
![Page 30: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/30.jpg)
Spontaneous Recombination Current: Examples
increases only linear with N at high densities
![Page 31: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/31.jpg)
Auger Recombination I
Inclusion of particle number non-conserving terms:Impact
Ionization
Auger Recombination
k
E
![Page 32: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/32.jpg)
Auger Recombination II
Quantum-Boltzmann scatteringin 2. Born-Markov approximation:
Impact Ionization
Auger Recombination
![Page 33: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/33.jpg)
Auger Recombination: Examples
increases far less than cubic with N sometimes even less than quadratic
![Page 34: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/34.jpg)
Theoretical Procedure:
� calculate gain for various densities� search for density that overcomes intrinsic losses (mirror losses) = threshold density� calculate spontaneous emission and Auger recombination for this density
� comparison with experimental data without adjustment
Theory-Experiment Comparison I
(experimental data from
R.Fehse, et al., IEEE J. Sel.
Topics Quantum Electron.
8, 801 (2002))
![Page 35: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/35.jpg)
Theory-Experiment Comparison II
experimental data:
A.F. Phillips et al.,
IEEE J. Sel. Topics
Quantum Electron. 5,
401 (1999))
![Page 36: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/36.jpg)
Laser Modelling: VECSEL
Numerical simulation issues:
Vertical External Cavity Surface Emitting Laser
• optical pumping = input of carriers and kinetic energy • nonequilibrium carrier distributions, nonequilibrium gain • dynamic response …..
![Page 37: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/37.jpg)
Nonequilibrium Gain Simulations
• Semiconductor Bloch Equations
•Reduced Wave Equation
APPROACH• Calculate rates for scattering/relaxation/dephasing• Solve coupled equations • Interesting situations: pulsed and/or CW excitation
Full numerical solution way too time consuming
![Page 38: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/38.jpg)
Relaxation Rate Model for Scattering
Parameters
• Relaxation rates are obtained by fits to microscopic calculations
• is calculated considering energy and particle conservation
• is calculated considering particle conservation
• Carrier-carrier scattering is an order of magnitude faster than carrier-phonon scattering
• Band structure is taken from microscopic calculations
![Page 39: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/39.jpg)
Carrier Dynamics
10nm Ga0.764In0.236As QW between 20.3nm GaN0.007As0.993,Excitation by a 50fs pulse at 1.35eV at t=0
optical excitation
0 50 100 150 200 250 3000.0
0.2
0.4
0.6
0.8
1.0
Ele
ctro
n D
ensi
ty
Energy (meV)
-500fs0fs100fs250fs500fs
![Page 40: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/40.jpg)
Effective Carrier-Carrier Scattering Times
-100 0 100 200 300 400 500 600 7000.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
Microscopic Calculation Exponential Fit, τ = 349fs
f k,P
eak-f Fe
rmi
Time (fs)
• simple evaluation
• exponential fits work well
![Page 41: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/41.jpg)
Electron-Hole Scattering Times
• Hole scattering faster because of larger effective mass
• Variation of scattering times with excitation energy (band structure vs availability of scattering partners)
• Increased scattering time for lower densities (e.g. for 1.5 * 1012 cm-2 �e = 354 fs, �h = 94 fs, instead of �e = 349 fs, �h = 87 fs at 2.0 * 1012 cm-2)
(Initial Density 2.0*1012/cm-2)
0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.760
50
100
150
200
250
300
350
400
450
Hole scattering times
Electron scattering times
τ (f
s)
Excitation Energy (eV)
![Page 42: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/42.jpg)
Switch-On of Optically Pumped SCL
• dynamics of electron and hole distributions
kinetic hole
opticalexcitation
• distinct kinetic hole in electron distribution
• small kinetic hole in hole distribution due to faster scattering time
0.00 0.05 0.10 0.15 0.20 0.25 0.300.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Ele
ctro
n D
istr
ibut
ion
Energy (eV)
2.5ps 5ps 25ps 175ps
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Hol
e D
istri
butio
n
Energy (eV)
2.5ps 5ps 25ps 175ps
![Page 43: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/43.jpg)
Gain Dynamics
• initial absorption turns into gain
• transient gain overshoot
cw gain
1.0 1.1 1.2 1.3-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
αL
Energy (eV)
2.5ps 5ps 25ps 175ps
![Page 44: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/44.jpg)
Carrier Temperature and Laser Intensity
0 100 200 3000
2
4
6
8
10
12
14
16
Intr
acav
ity In
tens
ity (M
W/c
m2 )
Time (ps)
0 50 100 150 200 250 300340
360
380
400
372K
383K379K
Tem
pera
ture
(K)
Time (ps)
Plasma Temperature Electron Temperature Hole Temperature
![Page 45: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/45.jpg)
Excitation Induced Temperature Change
0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74-10
-505
10152025303540455055
∆T (
K)
Excitation Energy (eV)
2.*1012cm-2
1.5*1012cm-2
0.60 0.65 0.70500
1000
1500
2000
T =
(Eex
c-Ega
p)/k
Eexc
• “hot”carriers (relative to the bandgap) can still cool down carrier distribution• excitation energy = 0.61 eV => Excitation at energy chemical potential => �T=0• hole chemical potential always in the bandgap
![Page 46: Microscopic Modelling of the Optical Properties of Quantum ...• refractive index, alfa parameter, etc. theory: 0.5, 0.75, 1.0, 1.25, …, 4.25, 4.5 [1012/cm2] 1000 900 800 700 600](https://reader036.fdocuments.us/reader036/viewer/2022081407/604e37a8d3ddc42b1f5f8330/html5/thumbnails/46.jpg)
Summary
• predictive microscopic theory• application to semiconductor laser gain media• luminescence, Auger loss rates• nonequilibrium gain, laser dynamics
• non-classical properties (quantum optics, quantum information science, …)
• modified photonic environment (phot. x-tals, …)• microscopic modelling of disorder
MANY CHALLENGES:
References and more informations:
http://www.physik.uni-marburg.de (look for: semiconductor theory)http://www.nlcstr.com