microscopic modelling of the optical properties of quantum ...• refractive index, alfa parameter,...
TRANSCRIPT
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
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• ….
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)
Semiconductor Density Matrix
two-band model, one QW subband
… e-h-pair transition amplitude
… occupation probabilities
H = H0 + HCoul + Hdip
H0 single particle (band structure)
HCoul Coulomb interaction between carriers
Hdip dipole interaction with optical field
Hamiltonian
Calculation of Optical Response
structure of resulting equations:
Heisenberg equation of motion:
infinite hierarchy of many-body correlations
systematic CLUSTER expansion
• two-point level (uncorrelated single particles)
• four-point level (two particle correlations)
Schematics of Cluster Expansion
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
Second Born-Markov Approximation
• intraband Coulomb scattering
• phonon scattering
Born-Markov limit: Boltzmann equation
Coulomb scattering
phonon scattering
Coulomb Scattering I
scattering rates:
Coulomb Scattering II
Fermi-Dirac distribution:
quasi-equilibrium:detailed balance
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
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)
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)
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
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
Semiconductor Luminescence
where is proportional to dipole matrix element and mode strength at the QW position
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
Semiconductor Luminescence Equations II
photon assisted interband polarization
incoherent source
coherent source
feedback (cavity)
excitonic signatures
• excitonic resonances independent of source term• relative peak height depends on populations
with •
• „true“ excitons
•
PRL 92, 067402 (2004)
Quasi Stationary: Luminescence Elliott Equation
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]
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)
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
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
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)
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
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
]
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]
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
Spontaneous Recombination Current: Examples
increases only linear with N at high densities
Auger Recombination I
Inclusion of particle number non-conserving terms:Impact
Ionization
Auger Recombination
k
E
Auger Recombination II
Quantum-Boltzmann scatteringin 2. Born-Markov approximation:
Impact Ionization
Auger Recombination
Auger Recombination: Examples
increases far less than cubic with N sometimes even less than quadratic
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))
Theory-Experiment Comparison II
experimental data:
A.F. Phillips et al.,
IEEE J. Sel. Topics
Quantum Electron. 5,
401 (1999))
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 …..
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
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
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
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
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)
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
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
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
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
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