Theory of
Quantum Dot Lasers
M. Grundmann
Institut für Experimentelle Physik II
Fakultät für Physik und Geowissenschaften
Universität Leipzig
grundmann@physik.uni-leipzig.de
www.uni-leipzig.de/~hlp/
Semiconductor
Physics Group
Content
Introduction
Electronic levels, Gain
Carrier distribution function
Laser properties
static
dynamic
Conclusion
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Physics Group
Scheme QD Laser (Edge Emitter)
+
Au-Zn-AuTi-Pt-Au
p-GaAs
p-AlGaAs
p-GaAs
n-GaAs
n-AlGaAs
[001]
n-GaAs
[110]
Ni-Ge-Au
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Physics Group
−
TGr:
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Physics Group
700 °C
600/
650 °C
505 °C
(⇒ 640 °C)
640 °C
300 - 600 nm GaAs:Zn
contact layer
0.8 - 1.0 µm AlGaAs : Zn
cladding layer
AlGaAs/GaAs SPS
70 nm GaAs barrier
QDs
70 nm GaAs barrier
AlGaAs/GaAs SPS
0.8 -1.0 µm AlGaAs:Si
cladding layer
GaAs:Si buffer
with AlGaAs/GaAs SPS
Layer Sequence
quantum
dot sheets
Simple Picture of Density of States
bulk
QD
QW
E
|1>
Ec
|000>
|010>
|011>
|111>
|0>
D(E)
D(E)
Ec
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Physics Group
E
D(E)
Ec
E
Ec
E
Simplest Theory
Threshold current density:
jthr ~ (1 ... 2)e × nQD/ττQD
Characteristic temperature: infinite (perfect confinement)
i.e. jthr is T-independent
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Scheme
electronic states
carrier dynamics
strain
confinement
(bi-)excitons
oscillator strength
capture
inter-sublevel relaxation
recombination
thermal escape
dephasing/scattering
Threshold condition
Laser operation
QD ensemble effects
inhomogeneous broadening
carrier distribution function
lateral arrangement
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Single Particles States
EcGaAs
1452.7
1518
1371.4
1.194 eV
1.098 eV
1359.3
1273.5
175.2
145.1
165.1
Ev
0
C1
V1
C2
V2
C3
V3
GaAs
b=13.6 nm
5 nm
3D strain calculation
8-band kp-theory
conduction band
valence band
M. Grundmann et al., PRB 52, 11969 (1995)
O. Stier, MG, D. Bimberg, PRB 59, 5688 (1999)
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Physics Group
Conventional Rate Equation Model (CRE)
G
τr
2
using ensemble averaged
state populations f
τ0
1
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Physics Group
Generation rate:
G
Radiative recombination:ττr
Intersublevel relaxation: τ0
τr
incorrect results
τ0 → 0
G < 1 / τr
f 1 = Gτ r
f2 = 0
Master Equations of Microstates (MEM)
Mean field Theory
2 QD's:
Microstates
ne=1/4
nh=1/4
N(0,0)=0
N(1,0)=1
N(0,1)=1
N(1,1)=0
ne=1/4
nh=1/4
N(0,0)=1
N(1,0)=0
N(0,1)=0
N(1,1)=1
Different
situations
are described
by identical
parameters
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Physics Group
Impact on
cw-spectra
transients (decay)
gain
threshold
Precise
description
of the
situation
MEM - Dynamics within a Single Dot
Model: Excitons in ground and excited states n=1,2
Radiative recombination:
τr
Intersublevel relaxation:
τ0
τr/2
n=2
n=1
τr
n=2
n=1
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Physics Group
τ0
τr/2
τr
τr
τ0/2
τr
τr
Current - MEM vs. CRE
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State Filling
Strain induced quantum dots
M. Sopanen, H. Lipsanen, J. Ahopelto
Appl. Phys. Lett. 65, 1662 (1995)
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Physics Group
Luminescence Intensity (arb. units)
State-Filling: MEM vs. CRE
6
5
4
12
3
6
2
2
1
0
0.9
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Physics Group
RP, τ0=0
RE, τ0=τr/100
1.0
1.1
Energy (eV)
1.2
1.3
State-Filling of Self-Assembled QD's
1.4 1.3
PL-Intensity (arb. units)
10
Wavelength (µm)
1.1
1.0
1.2
0.8
300 K
Wetting
layer GaAs
Quantum Dots
4
0.9
I (W/cm2
10 3
500
50
10
2
5
10 1
0.5
0.9
Semiconductor
Physics Group
1.0
1.1
1.2
1.3
Energy (eV)
1.4
1.5
PL - Intensity (arb. units)
State-Filling of Self-Assembled QD's
10
InAs/GaAs
tav=1.0nm
4
T=8K
103
2
5 50 125 500 W/cm
0.5
102
101
0.9
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Physics Group
1.0
1.1
1.2
1.3
Energy (eV)
1.4
1.5
State-Filling
1
10
I0
I1
I3
MEM,τ0 =35 ps
0
PL Intensity
10
-1
10
-2
10
-3
10
-4
10
-5
10
-3
10
-2
10
-1
10
0
10
Excitation (X/ (QD / τ))
Photoluminescence of mesa (d=30 µm)
homogeneous excitation density
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Physics Group
1
10
MEM vs. CRE
PL Intensity (arb. units)
1
Exp. |001>
MEM, τ0=30ps
CRE, τ0 =30ps
0.1
0.01
0.0
0.2
0.4
0.6
Time (ns)
0.8
Time-resolved photoluminescence
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Ground State Gain - MEM vs. CRE
CRE only correct in the limit of
small excitation
CRE overestimates gain
CRE overestimates
inter-sublevel relaxation time
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MEM Summary
Master equations for the transitions
between micro-states are the
conceptually correct model to
describe the dynamics in quantum dots
Modeling of the finite inter-level
scattering time with conventional
rate equations for the average
level population can lead to wrong
results, especially for t0<<tr.
Experiments on quantum dots with
fast and slow inter-level relaxation
have been fitted.
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Physics Group
Gain
Electronic structure
level positions
inhomogeneous broadening
homogeneous broadening
oscillator strength
barrier levels
Recombination
excitonic
Carrier distribution function
population of micro-states
master equations
thermal excitation
non-equilibrium distribution
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Physics Group
Gain
pre-factor
πe2
m 02 ε 0 c n r ω
∫ ∑i
g (hω ) =
M
2
2
δ (ε − E
V0
DOS
inhomogeneous oscillator
broadening
strength
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Physics Group
g
− E
0 ,i
)[ f c ( ε ) − f v ( ε ) ]
Γ in / π
dε
( h ω − ε ) 2 + Γ i 2n
carrier
homogeneous
distribution
broadening
function
Saturated Gain
L.V. Asryan, M. Grundmann et al.,
J. Appl. Phys. 90, 1666 (2001)
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several excited transitions
Gain
Gain ( gmax )
1
e+h
(eh)
0
1
0
-1
-1
0
1 N/N D 2
2
Injection current ( e ND / τr )
Effect of correlated capture
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Physics Group
1
0
3
Only ground state is considered
Gain - p-doped
O.B. Shchekin, D.G. Deppe
Appl. Phys. Lett. 80, 2758 (2002)
Effect of static hole population
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Physics Group
Only ground state is considered
Threshold Current vs. Coverage
Threshold current
2
(A/cm )
3
10
e+h
(eh)
2
10
α=10cm-1
1
10
0.01
ζmin
Effect of correlated capture
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Physics Group
0.1
Coverage
ζ
1
Only ground state is considered
MEM - Dynamics in an Ensemble
E1
τc
E2
τeE
1
E3
τc τEe
2
barrier
Size dependent
capture time (?)
escape time (!)
τc τEe
3
QD's
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Physics Group
Size distribution function
Gaussian
Carrier Distribution Function
1.0
0.4 nA/QD
0.32
Probability
0.8
0.6
0.4
300 K
Fermi
77 K
Low temperatures:
Strong deviation from
Fermi-function
0.24
0.16
Room temperature:
Small deviation from
Fermi-function
Shift of EF with
increasing injection
0.2 0.08
0.0
-150
-100
-50
Energy (meV)
ground
state
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Physics Group
excited
state
0
PL - Intensity (arb. units)
State-Filling of Self-Assembled QD's
10
InAs/GaAs
tav=1.0nm
4
T=8K
103
2
5 50 125 500 W/cm
0.5
102
101
0.9
1.0
1.1
1.2
1.3
Energy (eV)
1.4
Non-thermal carrier distribution!
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Physics Group
1.5
Gain Spectrum
1.0
0.4 nA/QD
300 K
77 K
Low temperature:
No Fermi distribution
Small shift of gain
maximum
Larger gain maximum
0.32
Gain
0.24
0.5
0.16
Room temperature:
Shift of wavelength
and gain maximum
0.08 nA/QD
0.0
-150
-100
Energy (meV)
Semiconductor
Physics Group
-50
Gain - Extremes: NTC vs. TC
fn+fp-1=0.86
fn+fp-1=0.69
fn+fp-1=0.39
Gain
1.0
NTC:
NTC
gmax
0.5
all QD's have the same
population regardless of the
ground state energy
0.0
1.0
Gain
non-thermal
distribution
µ=E0+3σE
µ=E0+2σE
µ=E0+σE
µ=E0
2.0
TC:
N/ND
1.5
1.0
0 1 2 3 4
µ-E0/σE
0.5
0.0
kBT=σE
-0.5
-4
-3
-2
-1 0
1
(E-E0) / σE
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Physics Group
TC
2
3
4
thermal distribution
QD population is given by
Fermi function
Experimental Gain at Low Temperature
40
35
30
20
-1
Gain (cm )
25
T=77K
-2
100 Acm
-2
90 Acm
80 Acm -2
70 Acm -2
60 Acm -2
PL
(arb. units)
NON-thermal
carrier
distribution
function
15
10
Gain ~
Gaussian × j
5
0
-5
-10
-15
1215 1220 1225 1230 1235 1240 1245 1250 1255 1260 1265 1270 1275
Energy (meV)
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Physics Group
Gain at High Temperature
QD laser emission
20
T=300K
10
Thermal
carrier
distribution
function
Gain (cm-1)
0
-10
-2
-20
-30
-40
350 Acm
-2
300 Acm
-2
250 Acm
-2
200 Acm
-2
150 Acm
-2
100 Acm
-2
60 Acm
Gain=
Gaussian ×
Fermi
EF
-50
1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220
Energy (meV)
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Gain of 2nd Excited State
L.V. Asryan, M. Grundmann et al.,
J. Appl. Phys. 90, 1666 (2001)
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Physics Group
2
jth (A/cm)
ξ=15%
ξ=10%
ξ=5%
100
Gain on Excited States
(a)
NTC
10
kB T=σE
kB T=2σE
kB T=σE
kB T=σE /2
100
2
jth (A/cm)
1
For increasing losses or
decreasing gain
(b)
NTC
Shift of laser emission to
excited states
higher energies
RT
10
ξ=10%
1
(Emax-E0 ) / E0
3
kB T=2σE
kB T=σE
kB T=σE /2
2
continously
discontinuously
(c)
1
0
-1
-2
NTC
ξ=10%
0.01
0.1
Area coverageζ
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Physics Group
1
Gain Saturation
2
jth (kA/cm)
1.6
T=77K
1.2
1 Layer
6 Layers
0.8
QD
0.4
0.0
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Physics Group
WL
2
1
0.5
Cavity Length (mm)
0.25
History of Diode Laser Threshold
Threshold current density (A/cm2)
10
4
Quantum
Dots
DH
10
3
SCH-QW
102
strained
QW
10
Theory
1
293 K
1960
Semiconductor
Physics Group
1970
1980
1990
Year
2000
2010
Temperature Dependence of Gain
Gain (scaled units)
1.0
0.9
-80K:
negative T0
80-150K: very high T0
>150K: positive T0
0.8
0.7
0.6
0.5
0
Current e/QD/τ
2
4
50
100
150
200
250
Temperature (K)
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Physics Group
300
350
400
Temperature Dependence of Threshold
Threshold current (e/QD/τ)
3.0
Master equations
for micro-states
incl. thermal emission
No T-dependent
carrier loss in the barrier!
τbarr=ττQD
2.5
T0 =-500K
T0 =500K
2.0
T0~∞
Q
)D
/h
(e
lrsu
o
td
n
cT
g= 0.7 gmax
1.5
0
50
100
150
200
250
Temperature (K)
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Physics Group
300
350
400
Small T0 values at RT
Leakage current!
RT:
100
τbarr=τQD
ηbarr=37%
ηbarr=5%
T0 =
54
K
2
Threshold current density (A/cm )
Temperature Dependence of Threshold
14
1
T 0=
Reduction of T0 due to
T-dependent quantum
efficiency ηbarr in the barrier
K
0K
0
5
T0 ~
10
0
50
100 150 200 250 300 350
Temperature (K)
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Physics Group
τQD=1 ns
High Power Laser Performance
8 x jthr
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Physics Group
High Power Lasing Spectra
QD laser
3×InAs/GaAs
Spectral power density (W/nm)
10-1
Increasing width of mode
spectrum with power due
to inhomogeneous broadening
-2
10
18.2
10.5 4.7
-3
10
1.3
1.0
Saturation value:
12.5 nW per QD
refill time < 14ps
10-4
0.8
0.5
-5
10
1070 1080 1090 1100 1110
Wavelength (nm)
Semiconductor
Physics Group
1120
Laser intensity (arb. units)
High Power Simulation
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
10
-8
10
-9
10
-10
-240
σ=20 meV
Γ=0
Inhomogeneous
broadening
dominates
1.2×Ithr
0.8×Ithr
0.4×Ithr
-230
-220
-210
-200
-190
-180
-170
-160
Energy (meV)
"Hat"-like spectral shape
Saturation at high injection current
> finally all QDs participate for which > dependent on relaxation bottleneck
the gain is larger than the losses
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Laser intensity (arb. units)
High Power Simulation
10
0
10
-1
10 -2
σ=20 meV
Γ=20 meV
Inhomogeneous
broadening
Homogeneous
broadening of
similar size
10 -3
10
-4
1.04×Ithr
10
-5
0.96×Ithr
10 -6
10 -7
10 -8
10
-9
-10
10
-240
-220
-200
-180
-160
Energy (eV)
Sharp spectral shape in the center of the
gain spectrum
> off-resonant QDs participate in lasing
> collective action of the QD ensemble
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Principle of bipolar MIR QD Laser
Lasing on inter-sublevel
transition in the MIR
Pumping of upper level
from barrier
Depletion of lower level
by interband lasing
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Polarization: NIR and FIR
Band edge
(1.12 eV, 1100nm)
FIR Inter-sublevel transitions
(90 meV, 13.8 µm)
[001]
×5
C1-V1
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Physics Group
C1-C2
C1-C3
Simulation of MIR-Laser - Population
-1-1
αα
=14cm,
=14 cm
12
-1-1
αα
=8cm,
=44 cm
12
Population
1.0
0.8
0.6
f1
f2
0.4
0.2
0.5
max
Gain (g )
0.0
1.0
0.0
E1
E2
MIR
-0.5
-1.0
0
5 10 15 20 25
Injection current (e/τr)
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Physics Group
0
5 10 15 20 25
Injection current (e/τr)
Simulation of MIR-Laser - Emission
Spontaneous MIR
0.5
6
NIR
10
0.0
0
5
5
Current
-1
-1
0.5
MIR
0.0
0
10
5
Current
10
NIR
NIR
0
20
15
10
5
0
0
5
10 15 20 25
Injection current (e/ τr )
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Physics Group
α1=8cm , α2=44 cm
6
15
-1
MIR ×10
Laser output (Nph /e)
-1
MIR ×10
α1=14cm , α2=14 cm
0
5
10 15 20 25
Injection current (e/ τ r )
Relaxation Oscillations
25
6
20
5
P (mW)
2.90
2.41
1.98
1.49
1.01
15
10
5
0
Coupling of
carrier density
photon density
0
1000
2000
Timet (ps)
MOCVD
3×InAs/GaAs
Semiconductor
Physics Group
4
3
2
1
T=293 K
L=265µm
0
1
2
3
√ Power (√mW)
3dB cutoff: 8.2 GHz
L=265 µm laser
Itr=40 A/cm2
ηi = 91%
Relaxation Oscillations
σ=20 meV, τ0=100ps
Homogeneous broadening leads to collective behavior
Semiconductor
Physics Group
M. Grundmann,
APL 77, 1428 (2000)
Relaxation Oscillations
Γ=0.5 meV
Γ=5 meV
Γ=30 meV
2.0
Time (ns)
0.9
1.5
0.8
0.7
1.0
0.6
0.5
-30
0
30 -30
0
30 -30
0
30
Energy (meV)
Energy (meV)
Energy (meV)
Homogeneous broadening leads to collective behavior
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Physics Group
Ground state filling
1.0
0.5
σ=20 meV, τ0=100ps
Relaxation Oscillations
RO Frequency (GHz)
7
0.5 meV
30 meV
6
5
4
σ=20 meV
3
I=1nA/QD
2
30
40
50
60
70
80
-1
Gain (cm )
M. Grundmann,
Electr. Lett. 36, 1851 (2000)
Impact of time constants
Semiconductor
Physics Group
Impact of gain
Chirp - Simple Picture
0.2
10
0.0
dnr (%)
gain (cm-1)
20
α≡
∂nr / ∂N
2π
∆λ / ∆I
≈−
⋅
∂ni / ∂N
δλ ⋅ L ∆g net / ∆I
-0.2
0
2
asymmetric
QD Ensemble
α
1
0
symmetric
QD Ensemble
α=0
-1
1.20
1.25
1.30
Energy (eV)
Semiconductor
Physics Group
α is also called
linewidth
enhancement factor
Absorption - QD vs. QW
QD
QW
J. Oksanen, J. Tulkki,
J. Appl. Phys. 94, 1963 (2003)
Semiconductor
Physics Group
Linewidth
Enhancement Factor
Smaller for QD than for QW
can be zero for QD laser
temperature effects!
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Physics Group
Linewidth Enhancement Factor
Impact of Fermi level
J. Oksanen, J. Tulkki,
J. Appl. Phys. 94, 1963 (2003)
Semiconductor
Physics Group
Spatio-Temporal Dynamics
Mesoscopic theory
QD fluctuations
spatially inhomogeneous
light propagation
dynamic scattering
Maxwell + QD-Bloch
equations
E. Gehrig, O. Hess,
Phys. Rev. A 65, 033804 (2002)
Semiconductor
Physics Group
Spatio-Temporal Dynamics
60mW
Th.
Near field
characteristics
show less
filamentation
for QD laser
due to small
amplitude-phase
coupling
Exp.
E. Gehrig, O. Hess et al.
Appl. Phys. Lett. 84, 1650 (2003)
Semiconductor
Physics Group
Beam Quality M2
Smaller M2 for QD laser
for same stripe
geometry
and
for same injection
conditions
E. Gehrig, O. Hess et al.
Appl. Phys. Lett. 84, 1650 (2003)
Semiconductor
Physics Group
Summary
Single QD properties and dynamics
QD fluctuations, ensemble average
at room-T: Fermi is a good approximation
otherwise: non-thermal carriers
Spatially dependent light field
Realistic description of QD laser properties
and agreement with experimental results
Semiconductor
Physics Group
Thanks to...
the many colleagues I enjoy(ed) working with
on quantum dot lasers over the last 10 years,
in particular:
(in alphabetical order)
Zh.I. Alferov
L.V. Asryan
D. Bimberg
F. Heinrichsdorff
R. Heitz
N. Kirstaedter
N.N. Ledentsov
Semiconductor
Physics Group
M.-H. Mao
Ch. Ribbat
A. Schliwa
O. Stier
V. Ustinov
A. Weber