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Theory ofQuantum Dot Lasers
M. Grundmann
Institut für Experimentelle Physik IIFakultät für Physik und GeowissenschaftenUniversität Leipzig
grundmann@physik.uni-leipzig.dewww.uni-leipzig.de/~hlp/
SemiconductorPhysics Group
Content
SemiconductorPhysics Group
Introduction
Electronic levels, Gain
Carrier distribution function
Laser propertiesstaticdynamic
Conclusion
Scheme QD Laser (Edge Emitter)
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[001]
[110]
n-GaAsn-GaAsn-GaAs
p-GaAs
p-AlGaAs
p-GaAs
n-GaAs
n-AlGaAs
+
−Ni-Ge-Au
Au-Zn-Au-Ti-Pt-Au
Layer Sequence
SemiconductorPhysics Group
GaA
s:S
i buf
fer
with
AlG
aAs/
GaA
s S
PS
0.8
-1.0
µm
AlG
aAs:
Si
clad
ding
laye
r
0.8
- 1.
0 µm
AlG
aAs
: Zn
clad
ding
laye
r
AlG
aAs/
GaA
s S
PS
AlG
aAs/
GaA
s S
PS
70 n
m G
aAs
barr
ier
QD
s
70 n
m G
aAs
barr
ier
300
- 60
0 nm
GaA
s:Z
nco
ntac
t lay
er
700 °C 600/650 °C
505 °C( 640 °C)⇒
640 °C
quantumdot sheets
T :Gr
Simple Picture of Density of States
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bulk QW QD
E
D(E)
E
D(E)
EcE
D(E)
Ec Ec
|1>
EEc
|0>
|000> |010>
|011> |111>
Simplest Theory
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Threshold current density: j thr ~ (1 ... 2)e ×××× nQD/ττττQD
Characteristic temperature: infinite (perfect confi nement)i.e. j thr is T-independent
Scheme
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electronic states
strainconfinement(bi-)excitonsoscillator strength
carrier dynamics
captureinter-sublevel relaxationrecombinationthermal escapedephasing/scattering
QD ensemble effects
inhomogeneous broadeningcarrier distribution functionlateral arrangement
Threshold conditionLaser operation
Single Particles States
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b=13.6 nm
175.2
1359.3
1452.7
1371.4
0
1518
145.1165.1
1273.5
EcGaAs
EvGaAs
1.09
8 eV
1.19
4 eV
5 nm
V2
V3
V1
C2
C3
C1
valence bandconduction band3D strain calculation8-band kp-theory
M. Grundmann et al., PRB 52, 11969 (1995) O. Stier, MG, D. Bimberg, PRB 59, 5688 (1999)
Conventional Rate Equation Model (CRE)
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1
2
τ0
G
τr
τr
τ0 0→ G r< 1 / τ
f G r1 = τ f2 0=
using ensemble averagedstate populations f
incorrect results
Generation rate: GRadiative recombination: ττττrIntersublevel relaxation: ττττ0
Master Equations of Microstates (MEM)
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Mean field Theory Microstates
Differentsituations
are describedby identicalparameters
n =1/4n =1/4
e
h
n =1/4n =1/4
e
h
Precisedescription
of thesituation
N(0,0)=0N(1,0)=1N(0,1)=1N(1,1)=0
N(0,0)=1N(1,0)=0N(0,1)=0N(1,1)=1
Impact on
cw-spectratransients (decay)
gainthreshold
2 QD's:
MEM - Dynamics within a Single Dot
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τr/2
τrτr/2
τr
τr
τ0 τrτr
τ0/2
n=2n=1
n=2n=1
Model: Excitons in ground and excited states n=1,2Radiative recombination: ττττrIntersublevel relaxation: ττττ0
Current - MEM vs. CRE
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State Filling
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Strain induced quantum dotsM. Sopanen, H. Lipsanen, J. AhopeltoAppl. Phys. Lett. 65, 1662 (1995)
State-Filling: MEM vs. CRE
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0.9 1.0 1.1 1.2 1.30
1
2
3
4
5
6
12
6
2
RP, τ0=0
RE, τ0=τ
r/100
Lum
ines
cenc
e In
tens
ity (
arb.
uni
ts)
Energy (eV)
State-Filling of Self-Assembled QD's
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0.9 1.0 1.1 1.2 1.3 1.4 1.5
101
102
103
104 Quantum Dots Wettinglayer GaAs
0.5
5
50
500
I (W/cm2
300 KP
L-I
nte
nsi
ty (
arb
. un
its)
0.9 0.81.4 1.3 1.2 1.1 1.0
Wavelength (µm)
Energy (eV)
State-Filling of Self-Assembled QD's
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101
102
103
104
0.55 50 500 W/cm2
tav
T=8K
InAs/GaAs=1.0nm
PL
-In
tens
it y(a
rb.u
n it s
)
0.9 1.0 1.1 1.2 1.3 1.4 1.5Energy (eV)
125
State-Filling
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10-3 10-2 10-1 100 10110-5
10-4
10-3
10-2
10-1
100
101
I0 I1 I3MEM, τ0=35 ps
PL
Inte
nsity
Excitation (X/ (QD / τ))
Photoluminescence of mesa (d=30 µm)homogeneous excitation density
MEM vs. CRE
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0.0 0.2 0.4 0.6 0.80.01
0.1
1
Exp. |001>MEM, τ0=30psCRE, τ0=30ps
PL
Inte
nsity
(arb
. uni
ts)
Time (ns)
Time-resolved photoluminescence
Ground State Gain - MEM vs. CRE
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CRE only correct in the limit ofsmall excitation
CRE overestimates gain
CRE overestimatesinter-sublevel relaxation time
MEM Summary
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Master equations for the transitionsbetween micro-states are theconceptually correct model todescribe the dynamics in quantum dots
Modeling of the finite inter-levelscattering time with conventionalrate equations for the averagelevel population can lead to wrongresults, especially for t0<<tr.
Experiments on quantum dots withfast and slow inter-level relaxationhave been fitted.
Gain
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Electronic structurelevel positionsinhomogeneous broadeninghomogeneous broadeningoscillator strengthbarrier levels
Recombinationexcitonic
Carrier distribution functionpopulation of micro-statesmaster equationsthermal excitationnon-equilibrium distribution
Gain
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[ ]
g
e
m c nM
VE E f f d
rg i c v
i n
i ni
( )
( ) ( ) ( )/
( ),
h
h
ωπε ω
δ ε ε ε πω ε
ε
=
− − −− +∑∫
2
02
0
2
00 2 2
2 ΓΓ
pre-factor
homogeneousbroadening
inhomogeneousbroadening
carrierdistribution
function
DOS
oscillatorstrength
Saturated Gain
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L.V. Asryan, M. Grundmann et al.,J. Appl. Phys. 90, 1666 (2001) several excited transitions
Gain
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0 1 2-1
0
1
N/ND
0 1 2 3
-1
0
1
Gai
n (
g )
max
Injection current ( e N / )D rτ
e+h(eh)
Effect of correlated capture Only ground state is considered
Gain - p-doped
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Effect of static hole population Only ground state is considered
O.B. Shchekin, D.G. DeppeAppl. Phys. Lett. 80, 2758 (2002)
Threshold Current vs. Coverage
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Effect of correlated capture Only ground state is considered
0.01 0.1 1
101
102
103
e+h(eh)
Coverage ζζmin
Th r
esho
l dcu
r ren
t(A
/cm
)2
α=10cm-1
MEM - Dynamics in an Ensemble
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QD's
barrier
τc τe τc τeτc τeE2 E3E1
E2 E3E1
Size dependentcapture time (?)escape time (!)
Size distribution functionGaussian
Carrier Distribution Function
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-150 -100 -50 00.0
0.2
0.4
0.6
0.8
1.0
0.08
0.4 nA/QD
0.24
0.32
0.16
300 K
77 K Fermi
Pro
babi
lity
Energy (meV)excitedstate
groundstate
Low temperatures:Strong deviation fromFermi-function
Room temperature:Small deviation from
Fermi-functionShift of E F with
increasing injection
State-Filling of Self-Assembled QD's
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101
102
103
104
0.55 50 500 W/cm2
tav
T=8K
InAs/GaAs=1.0nm
PL
-I n
tens
ity(a
rb.u
nits
)
0.9 1.0 1.1 1.2 1.3 1.4 1.5Energy (eV)
125
Non-thermal carrier distribution!
Gain Spectrum
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-150 -100 -50
0.0
0.5
1.0
300 K77 K
Gai
n
Energy (meV)
0.16
0.08 nA/QD
0.24
0.4 nA/QD
0.32Low temperature:
No Fermi distributionSmall shift of gain
maximumLarger gain maximum
Room temperature:Shift of wavelength
and gain maximum
Gain - Extremes: NTC vs. TC
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Gai
n
0.0
0.5
1.0
gmax
fn+fp-1=0.86 fn+fp-1=0.69 fn+fp-1=0.39
-4 -3 -2 -1 0 1 2 3 4(E-E0) / σE
-0.5
0.0
0.5
1.0
kBT=σE
µ=E0+3σE
µ=E0+2σE
µ=E0+σE
µ=E0
0 1 2 3 41.0
1.5
2.0
N/ND
µ-E0/σE
NTC
TC
Gai
n
NTC: non-thermaldistribution
all QD's have the samepopulation regardless of theground state energy
TC: thermal distribution
QD population is given byFermi function
Experimental Gain at Low Temperature
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12151215 12201220 12251225 12301230 12351235 12401240 12451245 12501250 12551255 12601260 12651265 12701270 12751275-15-15
-10-10
-5-5
00
55
1010
15
2020
2525
3030
3535
4040
Energy (meV)
PL (arb. units) 100 Acm 100 Acm-2
90 Acm 90 Acm-2
80 Acm 80 Acm-2
70 Acm 70 Acm-2
60 Acm 60 Acm-2
Gai
n (c
m)
-1
T=77K
NON-thermalcarrierdistributionfunction
Gain ~Gaussian × j
Gain at High Temperature
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Thermalcarrierdistributionfunction
Gain=Gaussian Fermi
×
1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220-50
-40
-30
-20
-10
0
10
20
EF
Gai
n (c
m)
-1
Energy (meV)
350 Acm-2
300 Acm-2
200 Acm-2
150 Acm-2
100 Acm-2
60 Acm-2
250 Acm-2
T=300K
QD laser emission
Gain of 2nd Excited State
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L.V. Asryan, M. Grundmann et al.,J. Appl. Phys. 90, 1666 (2001)
Gain on Excited States
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1
10
100 NTC
(a)
kBT=σE
ξ=15%ξ=10%ξ=5%
j th(A
/cm2
)
1
10
100
RT
NTC
(b)
ξ=10%
ξ=10%
kB T=2σEkB T=σEkB T=σE /2
j th(A
/cm2
)
0.01 0.1 1-2
-1
0
1
2
3kB T=2σEkB T=σEkB T=σE /2
NTC
(c)
(Em
ax-E
0)
/ E 0
Area coverageζ
For increasing losses ordecreasing gain
Shift of laser emission toexcited stateshigher energies
continouslydiscontinuously
Gain Saturation
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0.0
0.4
0.8
1.2
1.6
j th(k
A/c
m2) T=77K 1 Layer
6 Layers
2 1 0.5 0.25
QD
WL
Cavity Length (mm)
History of Diode Laser Threshold
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1960 1970 1980 1990 2000 2010
101
102
103
104
SCH-QW
Theory
Year
strainedQW
Thr
esho
ld c
urre
nt d
ensi
ty (
A/c
m)2 Quantum
DotsDH
293 K
Temperature Dependence of Gain
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0 50 100 150 200 250 300 350 4000.5
0.6
0.7
0.8
0.9
1.0
Current e/QD/τ 2 4
Temperature (K)
Gai
n (
scal
ed u
nits
)
-80K: negative T80-150K: very high T>150K: positive T 0
0
0
Temperature Dependence of Threshold
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0 50 100 150 200 250 300 350 4001.5
2.0
2.5
3.0
Master equationsfor micro-statesincl. thermal emission
T0 =500KT0 =-500K
T0~∞∞∞∞Thresholdcurrent(e/QD/) T
hres
hold
curr
ent(
e/Q
D/τ
)
g= 0.7 gmax
No T-dependentcarrier loss in the barrier!ττττbarr=ττττQD
Small T 0 values at RTLeakage current!
Temperature (K)
Temperature Dependence of Threshold
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Reduction of T 0 due toT-dependent quantumefficiency ηηηηbarr in the barrier
ττττQD=1 ns
0 50 100 150 200 250 300 35010
100
RT:τbarr=τ
QD
ηbarr=37%
ηbarr=5%
Thr
esho
l dc u
rre n
tde n
s ity
(A/c
m)
2
Temperature (K)
T ~500K0
T=5
4K0
T =114K
0
High Power Laser Performance
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8 x j thr
High Power Lasing Spectra
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Spe
ctra
l pow
er d
ensi
ty (
W/n
m)
1070 1080 1090 1100 1110 1120
10-5
10-4
10-3
10-2
10-1
18.210.5 4.7
1.3
1.0
0.8
0.5
QD laser3×InAs/GaAs
Wavelength (nm)
Increasing width of modespectrum with power dueto inhomogeneous broadening
Saturation value:12.5 nW per QD
refill time < 14ps
High Power Simulation
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Inhomogeneousbroadeningdominates
"Hat"-like spectral shape> finally all QDs participate for which
the gain is larger than the losses
Saturation at high injection current> dependent on relaxation bottleneck
-240 -230 -220 -210 -200 -190 -180 -170 -16010
10
10
10
10
10
10
10
10
10
10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1.2×Ithr
0.8×Ithr
0.4×Ithr
Lase
r in
tens
ity (
arb.
uni
ts)
Energy (meV)
σ=20 meVΓ=0
High Power Simulation
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-240 -220 -200 -180 -16010-10
10 -9
10 -8
10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
0.96 I× thr
Lase
r in
tens
ity (
arb.
uni
ts)
1.04 I× thr
Energy (eV)
σ=20 meVΓ=20 meV
Inhomogeneousbroadening
Homogeneousbroadening ofsimilar size
Sharp spectral shape in the center of thegain spectrum> off-resonant QDs participate in lasing> collective action of the QD ensemble
Principle of bipolar MIR QD Laser
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Lasing on inter-subleveltransition in the MIR
Pumping of upper levelfrom barrier
Depletion of lower levelby interband lasing
Polarization: NIR and FIR
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[001]
C1-V1
Band edge(1.12 eV, 1100nm)
C1-C2 C1-C3×5
FIR Inter-sublevel transitions(90 meV, 13.8 µm)
Simulation of MIR-Laser - Population
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-1.0
-0.5
0.0
0.5
1.00.0
0.2
0.4
0.6
0.8
1.0
f1f2
E1E2MIR
Gai
n( g
)m
axP
opul
a tio
n
αα12=8cm, =44 cm-1-1αα12=14cm, =14 cm-1-1
0 5 10 15 20 25 0 5 10 15 20 25Injection current (e/ )τrInjection current (e/ )τr
Simulation of MIR-Laser - Emission
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0
5
10
15
0 5 10 15 20 250
5
10
15
20
0 5 10 15 20 25
0 5 100.0
0.5
Current
MI R
10×6
0 5 100.0
0.5
Current
MI R
10×6
Spo
ntan
eous
MIR
Lase
ro u
tput
( N/e
)ph
Injection current (e/ )τ rInjection current (e/ )τr
NIR
MIR
NIRNIR
α α1 2=8cm , =44 cm-1 -1α α1 2=14cm , =14 cm-1 -1
Relaxation Oscillations
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0 1 2 31
2
3
4
5
6
T=293 KL=265µm
√Power (√mW)0 1000 2000
0
5
10
15
20
25
P (mW)2.902.411.981.491.01
Timet (ps)
3dB cutoff: 8.2 GHz
L=265 µm laserI =40 A/cm2
= 91%tr
iηMOCVD3 InAs/GaAs×
Coupling ofcarrier densityphoton density
Relaxation Oscillations
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Homogeneous broadening leads to collective behavior
σ τ=20 meV, =100ps0
M. Grundmann,APL 77, 1428 (2000)
Relaxation Oscillations
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Homogeneous broadening leads to collective behavior
-30 0 300.5
1.0
1.5
2.0
Tim
e (n
s)
Γ=30 meVΓ=5 meVΓ=0.5 meV
-30 0 30-30 0 30
Energy (meV) Energy (meV)
0.6
0.7
0.8
0.9
1.0
0.5
Energy (meV)
Gro
und
stat
e fil
ling
σ τ=20 meV, =100ps0
Relaxation Oscillations
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30 40 50 60 70 802
3
4
5
6
7
σ=20 meVI=1nA/QD
0.5 meV 30 meV
RO
Fre
quen
cy (
GH
z)
Gain (cm-1)
Impact of time constants Impact of gain
M. Grundmann,Electr. Lett. 36, 1851 (2000)
Chirp - Simple Picture
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0
10
20
1.20 1.25 1.30-1
0
1
2
symmetric QD Ensemble
asymmetric QD Ensemble
α
dn (
%)
r
α=0
-0.2
0.0
0.2
Energy (eV)
gain
(cm
)-1
Ig
I
LNn
Nn
neti
r
∆∆∆∆⋅
⋅−≈
∂∂∂∂
≡/
/2
/
/ λδλ
πα
αααα is also calledlinewidthenhancement factor
Absorption - QD vs. QW
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J. Oksanen, J. Tulkki,J. Appl. Phys. 94, 1963 (2003)
QD
QW
LinewidthEnhancement Factor
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Smaller for QD than for QW
can be zero for QD laser
temperature effects!
Linewidth Enhancement Factor
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J. Oksanen, J. Tulkki,J. Appl. Phys. 94, 1963 (2003)
Impact of Fermi level
Spatio-Temporal Dynamics
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E. Gehrig, O. Hess,Phys. Rev. A 65, 033804 (2002)
Mesoscopic theory
QD fluctuations
spatially inhomogeneouslight propagation
dynamic scattering
Maxwell + QD-Blochequations
Spatio-Temporal Dynamics
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E. Gehrig, O. Hess et al.Appl. Phys. Lett. 84, 1650 (2003)
Th.
Exp.
Near fieldcharacteristicsshow lessfilamentationfor QD laser
due to smallamplitude-phasecoupling
60mW
Beam Quality M 2
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E. Gehrig, O. Hess et al.Appl. Phys. Lett. 84, 1650 (2003)
Smaller M 2 for QD laserfor same stripegeometry
andfor same injectionconditions
Summary
SemiconductorPhysics Group
Single QD properties and dynamics
QD fluctuations, ensemble average
at room-T: Fermi is a good approximationotherwise: non-thermal carriers
Spatially dependent light field
Realistic description of QD laser propertiesand agreement with experimental results
Thanks to...
SemiconductorPhysics Group
the many colleagues I enjoy(ed) working withon quantum dot lasers over the last 10 years,in particular:(in alphabetical order)
M.-H. MaoCh. RibbatA. SchliwaO. StierV. UstinovA. Weber
Zh.I. AlferovL.V. AsryanD. BimbergF. HeinrichsdorffR. HeitzN. KirstaedterN.N. Ledentsov
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