microscopic modelling of the optical properties of quantum ...• refractive index, alfa parameter,...

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