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Theory ofQuantum Dot Lasers

M. Grundmann

Institut für Experimentelle Physik IIFakultät für Physik und GeowissenschaftenUniversität Leipzig

[email protected]/~hlp/

SemiconductorPhysics Group

Content


Introduction

Electronic levels, Gain

Carrier distribution function

Laser propertiesstaticdynamic

Conclusion

Scheme QD Laser (Edge Emitter)


[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


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


bulk QW QD

E

D(E)

E

D(E)

EcE

D(E)

Ec Ec

|1>

EEc

|0>

|000> |010>

|011> |111>

Simplest Theory


Threshold current density: j thr ~ (1 ... 2)e ×××× nQD/ττττQD

Characteristic temperature: infinite (perfect confi nement)i.e. j thr is T-independent

Scheme


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


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)


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)


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


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


State Filling


Strain induced quantum dotsM. Sopanen, H. Lipsanen, J. AhopeltoAppl. Phys. Lett. 65, 1662 (1995)

State-Filling: MEM vs. CRE


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


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)



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


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


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


CRE only correct in the limit ofsmall excitation

CRE overestimates gain

CRE overestimatesinter-sublevel relaxation time

MEM Summary


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


Electronic structurelevel positionsinhomogeneous broadeninghomogeneous broadeningoscillator strengthbarrier levels

Recombinationexcitonic

Carrier distribution functionpopulation of micro-statesmaster equationsthermal excitationnon-equilibrium distribution

Gain


[ ]

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


L.V. Asryan, M. Grundmann et al.,J. Appl. Phys. 90, 1666 (2001) several excited transitions

Gain


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


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


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


QD's

barrier

τc τe τc τeτc τeE2 E3E1

E2 E3E1

Size dependentcapture time (?)escape time (!)

Size distribution functionGaussian

Carrier Distribution Function


-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



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


-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


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


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


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


L.V. Asryan, M. Grundmann et al.,J. Appl. Phys. 90, 1666 (2001)

Gain on Excited States


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


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


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


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


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


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


8 x j thr

High Power Lasing Spectra


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


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


-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


Lasing on inter-subleveltransition in the MIR

Pumping of upper levelfrom barrier

Depletion of lower levelby interband lasing

Polarization: NIR and FIR


[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


-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


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


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



Homogeneous broadening leads to collective behavior

σ τ=20 meV, =100ps0

M. Grundmann,APL 77, 1428 (2000)



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



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


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


J. Oksanen, J. Tulkki,J. Appl. Phys. 94, 1963 (2003)

QD

QW

LinewidthEnhancement Factor


Smaller for QD than for QW

can be zero for QD laser

temperature effects!

Linewidth Enhancement Factor


J. Oksanen, J. Tulkki,J. Appl. Phys. 94, 1963 (2003)

Impact of Fermi level

Spatio-Temporal Dynamics


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


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


E. Gehrig, O. Hess et al.Appl. Phys. Lett. 84, 1650 (2003)

Smaller M 2 for QD laserfor same stripegeometry

andfor same injectionconditions

Summary


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...


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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