Quantum Dot Lasers
Andrea Fiore
Ecole Polytechnique Fédérale de Lausanne
Outline:
• Quantum-confined active regions
• Self-assembled quantum dots
• Laser applications
QD
Electronic states in
semiconductors
Schrödinger eq.:
ψ k (r ) = uk (r )χ (r )
 p2


+ Vcrist (r) + Vhet (r ) ψ (r) = Eψ(r)
 2m

uk : Bloch function
χ : envelope function
 p2


+ Vhet (r ) χ (r ) = Eχ (r )
 2m *

E
CB
bulk : Vhet = 0, χ (r ) = eik •r
2 2
k
E=
2m *
QD
k
VB
Andrea Fiore
Energy quantization
QD
k x L = 2m x π
k y L = 2m y π
χ ( r ) = e ik • r
k z L = 2m z π
E
L
(
2
k
4π 2 2
2
2
=
+
+
E=
m
m
m
x
y
z
2m * 2m * L2
2 2
)
4π 2
∆E ≈
2m * L2
2
Wavefunction confinement with heterostructure potential:
1
∆E ∝ 2
L
∆E>>kT at RT ⇒ L<<30 nm
Andrea Fiore
2D nanostructures:
Quantum Wells
kx
z
2> E
2>
ky
1>
1>
kx, ky
AlGaAs GaAs AlGaAs
Vhet = V ( z ), χ (r ) = e
i (k x x + k y y )
E = En +
2
QD
χ (z )
k x2 + k 2y
2m *
(
)
Confinement in one dimension ⇒ Continuum of
energy states
Andrea Fiore
1D nanostructures:
Quantum Wires
2>
ky
1>
AlGaAs GaAs AlGaAs
Vhet = V ( x , z ), χ (r ) = e
E = En +
ik y y
QD
2> E
1>
ky
χ ( x, z )
2 2
ky
2m *
Andrea Fiore
0D nanostructures:
Quantum Dots
2>
QD
E
1>
2>
1>
AlGaAs GaAs AlGaAs
Vhet = V ( x , y , z ), χ (r ) = χ ( x , y , z )
E = En
Only in a 0D nanostructures the energy levels
are completely discrete!
⇒ “Semiconductor atoms”
Andrea Fiore
Electronic transitions
QD
Fermi's golden rule:
w if ∝ f H' i
≈A
2
=A
2
2
∝ uc ( r ) χc ( r ) A ( r ) • p u v ( r ) χ v ( r )
2
≈
uc ( r ) eˆ • p u v ( r ) χ c ( r ) χ v ( r ) + χ c ( r ) eˆ • p χ v ( r ) uc ( r ) u v ( r )
uc ( r ) eˆ • p u v ( r ) χ c ( r ) χ v ( r )
Material
2
2
=
=0
Heterostructure
NB: Dimensionality has little influence on transition
probability (apart from selection rules)
!
Neglecting excitonic effects!
Andrea Fiore
Density of states
QD
Memo:
Born-von-Karman boundary conditions
⇒ k-space density of states g ( k ) = const .
Bulk:
E=
ρ c ( E ) ∝ E − Ec
k
2m *
⇒
g( E )
Density of states
+ dispersion relation:
2 2
E
c
E nerg y
Andrea Fiore
Density of states
ρc ( E ) = const.
QWires
1
ρc ( E ) ∝
E − En
E
2
1
E nerg y
Density of states
Density of states
E
E
QDs
E
1
E nerg y
2
ρ c ( E ) ∝ δ ( E − En )
Density of states
QWs
QD
E
E
2
1
E nerg y
2>
1>
AlGaAs GaAs AlGaAs
More confinement
⇒ Less degrees of freedom
⇒ Density of states more singular
Andrea Fiore
Quantization: What for?
QD
Photonics:
• Shape of density of states
⇒ Gain spectral width
• Number of states
⇒ Transparency current
• 3D carrier confinement
• Energy tuning
Andrea Fiore
Density of states and
carrier distribution
QD
n( E ) = g ( E ) f ( E )
Density of states
Fermi distribution
Carrier distribution
Bulk:
f(E)
g(E)
Problem with bulk:
Continuous and increasing g(E)
n(E)
⇒ Broad carrier distribution
Energy
Andrea Fiore
g(E)
n(E)
Energy
QWires
f(E)
n(E)
Energy
g(E)
Carrier distribution
f(E)
QWs
Carrier distribution
Carrier distribution
Carrier distribution in
nanostructures
QDs
f(E)
QD
g(E)
n(E)
Energy
Gain calculation:
Proposed by Asada et al.,
JQE 1986
NB:
Idealized
picture !!!
Andrea Fiore
Number of states
QD
In a semiconductor laser: Need to reach transparency
⇒ must fill at least half of the available states
Transparency current: I tr =
eg ( E )∆E
∝ g(E )
2τ
N quantum dots ⇒ 2N states
With low-dimensional heterostructures we can
"engineer" the number of states ⇒ Reduce Itr
NB: Small g(E) ⇒ Small max gain !
Andrea Fiore
Carrier confinement
QD
Lateral carrier confinement in 0D nanostructures
⇒ Suppressed carrier diffusion
Can fabricate smaller devices
Andrea Fiore
Outline
QD
• Quantum-confined active regions
• Self-assembled quantum dots: growth and
physical properties
• Laser applications
Andrea Fiore
Nanostructure fabrication:
Quantum Dots
QD
"Straightforward" approach: Fabrication on the nanoscale
• Need high-resolution lithography
• Etching ⇒ Nonradiative defects
"Self-organized" methods: Making Nature work for us !
• Growth on patterned substrates
substrate
• Strain-driven growth on planar
substrates
substrate
Andrea Fiore
Self-assembled growth
QD
Epitaxial growth: Interplay of surface energy and strain
"Standard" (Frank-van der Merve) 2D epitaxial growth:
GaAs
AlAs
substrate
Stranski-Krastanow 3D growth mode (strain-driven):
InAs
substrate
Andrea Fiore
Self-assembled
quantum dots
QD
MBE growth of InAs on GaAs:
15 nm
200 nm
Atomic Force Microscopy
Transmission Electron Microscopy
☺ Simple growth technique, no
pre-growth patterning required
Random nucleation sites ⇒ no
control of island position
☺ High crystalline quality ⇒
High radiative efficiency at RT
Size dispersion ⇒ Emission energy
dispersion
☺ 1.3 µm operation possible
Andrea Fiore
Self-assembled
quantum dots
QD
Idealized picture:
E
∆E>>kT
Real QDs:
15 nm
200 nm
PL intensity (arb. un.)
Arakawa, Sakaki 1982
Asada et al., 1986
35
ideal
meas.
E
30
25
20
15
10
E
FW H M =
31 m eV
5
0
0.95
1
1.05
1.1
Andrea
Fiore
E nergy (eV
)
Electronic structure
holes
QD
electrons
s-like
p-like
d-like
Atomistic pseudo-potential approach
(Williamson et al, PRB 2000)
Andrea Fiore
QD0
QD1
QD2
Intensity (arb. un.)
Emission spectrum
QD
Low excitation level
7 10
-8
6 10
-8
293 K
5 10
-8
21 A/cm
4 10
-8
3 10
-8
2 10
-8
1 10
-8
QD0
2
QD1
0
0.85
0.95
1.05
1.15
QD filling:
max 2 electrons per state
QD2
QD1
QD0
Intensity (arb. un.)
Energy (eV)
2 10
-7
1.5 10
-7
1 10
-7
QD2
QD1
QW
7 .2 kA/cm
QD0
90 0 A/cm
5 10
2
-8
11 0 A /cm
0
0.9
1
2
GaAs
2
1.1 1.2 1.3 1.4 1.5
Energy (eV)
Andrea Fiore
Relaxation bottleneck?
QD
Prediction of very slow relaxation from excited
states to ground state (Benisty et al., PRB 1991)
If ∆E≠h ωopt ph ⇒ cannot relax from
excited to ground state ⇒ no PL
QD2
QD1
QD0
∆E
∆E
Experiment: Picosecond relaxation times!
Explanation: Auger effect
P av =110 mW, 3.1x10
Intensity (a.u.)
24000
12
cm
-2
el-hole pairs per pulse
resolution:
3.5 ps
20000
5K
16000
12000
8000
1.064 eV
1.158 eV
1.247 eV
laser
4000
0
20
40
60
Time (ps)
80
100
Morris et al, APL 1999
Andrea Fiore
Radiative properties of
optimized QDs
110 0
10 W /cm
5K
100K
150K
200K
250K
300K
2
20
15
10
5
0
1100
Decay time (ps)
Intensity (arb. un.)
25
QD
100 0
900
800
700
400
600
200
300
100
500
0
0
400
1200
1300
1400
300
100K
0
50
1000
2000
T im e [ps]
100
150
200
250
300
Tem perature (K)
W avelength (nm )
Markus et al., APL 80, 911 (2002)
Material properties:
• High radiative efficiency ≈20% at RT
• Long carrier lifetime ≈ 1 ns
• 1300 nm emission at RT
• Density ≈ 109 - 1010 cm-2
Andrea Fiore
QD excitons:
Homogeneous linewidth
Single-QD photoluminescence:
2
2
Γ=
=
+ Γ phon (T )
T2 τ life
Radiative dephasing Phonon scattering
QD
(Bayer et al., PRB 2002)
T<2 K: lifetime-limited
2 K<T<60 K: acoustic phonons
T>60 K: optical phonons
Andrea Fiore
Outline
QD
• Quantum-confined active regions
• Self-assembled quantum dots: growth and
physical properties
• Laser applications
Andrea Fiore
QDs: Laser applications
QD
Potential advantages:
• Narrow gain spectrum?
⇒ No, inhomogeneous broadening
• Wavelength tuning: Reaching 1.3-1.55 µm on GaAs
InAs
substrate
Local In content can be
high with low average
strain
• Low density of states ⇒ Low transparency current
Andrea Fiore
QDs vs QWs: Gain
QD
Maximum gain per pass in a 2D system:
g max
QWs:
QWs:
πe2 x 2cv ω
=
ρ(E)
e 0nc
E
ρ(E) :
density of states
per unit energy
per unit area
QDs:
InAs
QDs
GaAs
InGaAs
GaAs
GaAs
GaAs
kt
m*
ρ QW (E ) = 2
π
gS
ρ QD (E ) ≈ 2
∆E inh
g S : areal dot density
∆E inh : in hom ogeneous broadening
Andrea Fiore
Max gain
g QD
g QW
≈
ρ QD
ρQW
QD
π 2 2g S
≈ *
≈ 0.1
m ∆E inh
( g S = 3 x1010 cm −2 , ∆Einh = 20meV )
gQW(1 pass) ≈ 1%
gQD (1 pass) ≈ 0.1%
(1 quantum well)
(1 QD layer, 3x1010 cm-2)
(measured: gQD=0.03% with 3x1010 cm-2, ∆ E=35 meV (Birkedal et al, 2000))
QDs have lower gain than QWs! (at 1300 nm)
Laser applications:
• Minimize losses
• Stack many QD layers
Andrea Fiore
QD stacking for max gain
10 nm spacing: QDs aligned
50nm
QD
25 nm spacing: QDs not aligned
50nm
5
100 K
Intensity (a.u.)
4
3 QDs
d=25 nm
1 QD
3
2
3 QDs
d=10 nm
1
0
1100
1200
Strain interaction between dot
layers produces vertical
alignment and PL broadening
1300
Wavelength (nm)
Andrea Fiore
QD lasers
EPFL QD lasers:
QD
Laser performance:
• Jth<100 A/cm2
• ηext=30-40%
• CW up to 80 C
7 10
-4
6 10
-4
5 10
-4
4 10
-4
3 10
-4
2 10
-4
1 10
-4
0
600 µm, HR coated 95%:
2
J (A/cm )
2 m m , 3 Q D layers, RT
147 A/cm2
136 A/cm2
0 10
110 0 1150 120 0 1250 130 0 1350 140 0
W avelength (nm )
35
Pow er (m W)
Intensity (a.u.)
As-cleaved, RT:
0
400
800
1200
1600
p ulsed
30
25
20
15
2 0C
3 0C
4 0C
5 0C
6 0C
7 0C
8 0C
10
5
0
0
0.2
0.4
0.6
0.8
1
Current (A)
HR coatings and measurements:
O. Gauthier-Lafaye, Opto+ Andrea Fiore
QD
8.4 µm x 4 mm
2 QD layers
Voltage (V)
Single facet power (mW)
Record threshold
current density
J
Room-temperature:
Jth=33 A/cm2
Jtr = 9 A/cm2
The lowest Jtr of any
semiconductor laser?
Current (mA)
Huang et al., Electron. Lett. 2000
Theoretical estimate for Jtr:
J tr = N QD
(NQD=2, gS=3x1010 cm-2, τ=800 ps)
eg S
τ
= 12 A/cm 2
Andrea Fiore
Gain problem in QD lasers
QD
Intensity (a.u.)
Ground state lasing only for low loss (L>1.5 mm): N QD gth = α + 1 ln 1
L R
7 10
-4
6 10
-4
5 10
-4
4 10
-4
3 10
-4
2 10
-4
1 10
-4
3 Q D layers
2 mm
High T ⇒ Higher optical loss
⇒ Wavelength switch
QD2
600 µ m
0
L= 600µm, HR/ HR coa ted, P= 3 mW.
Device width= 3µm
QD1
QD0
0
0 10
1100
-1 0
45°C
1200
1300
-2 0
Po we r (dBm )
W avelen gth (nm )
Wavelength (nm )
25°C
1350
Ground state
1300
1250
65°C
-3 0
-4 0
-5 0
1200
-6 0
Excited state
1150
1100
0
2
4
Device length (m m )
-7 0
1150
1200
1250
1300
1350
Wa ve le ng th (nm )
Andrea Fiore
Max modal gain in QD lasers
QD
gmod ≈ 3 cm-1 per QD layer
Klopf et al., PTL 2001
1 1
N QD g = α + ln
L R
Lmin =
ln(1 / R )
N QD g − α
≈ 600 µm
⇒ Need a factor of two in gain for applications
Andrea Fiore
Effect of inhomogeneous
broadening
3 Q D layers, 2 mm, pulsed
0.01
Intensity (arb. un.)
QD
4 00 m A
0.001
293 K
2 00 m A
1 00 m A
0.0001
40 m A
10
-5
10
-6
in creasing
current
10
-7
10
-8
10
-9
1150
-
--
-
--
hν2 hν3 hν1 hν2 hν3 hν1
1250
W avelength (nm )
1350
Inhomogeneous broadening +
absence of thermal equilibrium
⇒ No population clamping at threshold
⇒ Broad laser line = many independent lasers!
Andrea Fiore
Dual-wavelength lasing
QD
Markus et al., APL 2003
Violates population
clamping theory ???
light
population
bias
Andrea Fiore
Interpretation of
dual-λ lasing
QD
"Slow" intraband relaxation
(Benisty et al., PRB 1991)
τ0
τ0 ≈10 ps in our QDs (PL rise time)
Photoluminescence:
1
τ0
τspont
τ0
>>
Laser:
1 − fGS
1
τ spont
⇒ no relax.
bottleneck
τ0
τstim
Behaviour predicted by Grundmann et al., APL 2000
τ0
≈
1
τ stim
⇒ ES
population
Andrea Fiore
Modeling 2-λ lasing
Rate equation model:
fES
ES
τ0
Model
τ0 ≈ 8 ps
fGS
GS
Photon num ber
0.35
fWL
WL
QD
total
2 mm
0.3
0.25
0.2
GS
0.15
0.1
0.05
0
ES
0
8
16
24
32
40
C arrier injection rate (e/ τ )
2
Population
2 mm
1.5
1
GS
ES threshold
0.5
0
Exper.
ES
G S threshold
0
8
16
24
32
40
C arrier injection rate (e/ τ )
r
Integr. int. (arb. un.)
r
0.14
2 mm
0.12
total
293 K
0.1
0.08
0.06
GS
0.04
0.02
0
ES
0
200
400
600
800
C urrent (m A)
1000
Andrea Fiore
3D carrier confinement
in QDs
QD
Carrier diffusion limits device scaling:
QW
Al2O3
QDs
300 nm current aperture
oxid. edge
oxid. edge
Au
GaAs
AlGaAs
QDs
Fiore et al., Appl. Phys. Lett. Andrea
(2002) Fiore
Ultrasmall LEDs: Scaling
4
10
2
10
0
10
Area = aperture area
2
10
QWs
293 K
-2
10
-4
10
-6
10
-8
9.0 µm
2.0 µm
940 nm
720 nm
530 nm
0
0.2 0.4 0.6 0.8
1
1.2 1.4
Voltage (V)
2
Current density (A/cm )
Quantum dots:
Current density (A/cm )
2
Current density (A/cm )
InGaAs quantum well:
QD
Ldiff
10
4
10
2
10
0
10
-2
10
-4
10
-6
10
-8
Area = aperture area
Q Ds
293 K
30 µm
9.1 µm
1.8 µm
1.0 µm
830 nm
600 nm
0
0.2
0.4
0.6
0.8
1
1.2
Voltage (V )
Area = aperture area + diffusion
10
2
10
0
10
-2
10
-4
10
-6
10
-8
Q W s, L =2.7 µ m
diff
293 K
9.0 µm
2.0 µm
940 n m
720 n m
530 n m
0
0.2
0.4
0.6
0.8
1
Voltage (V)
1.2
1.4
Diffusion length:
≈ 2.7 µm in QWs
< 100 nm in QDs !
QDs for nanoscale devices
Andrea Fiore
Real applications for
QD lasers
QD
Disadvantage: Low gain ⇒ Difficult to get VCSELs
Advantages:
• Low threshold currents
• Low linewidth enhancement factor ?
Low chirp, insensitivity to optical feedback
• Broad gain spectrum & high saturation power
SOAs, tunable lasers, superluminescent LEDs
Andrea Fiore
Laser chirp
QD
Direct laser modulation:
01011...
01011...
Problem: Spectral broadening
laser mode
λm =
gain
∆I
λ
∆λ ∆n
2nLcav
⇒
=
λ
m
n
⇒
∆N ⇒
∆ g

∆ n
KramersKronig
I = I (t ) ⇒ n = n (t) ⇒ λ = λ (t)
Laser "chirp":
dn / dN
α
∆ν =
Γv g g1∆N with : α ≡ −
4π
dn i / dN
linewidth-enhancement
factor
Andrea Fiore
Gain
Ideally:
4π dneff / dN
α =
=0
λ dg / dN
QD
Refractive index
Linewidth enhancement
factor in QD lasers
Energy
At low bias:
GS population
only ⇒ low α
PTL 1999
Andrea Fiore
Linewidth enhancement
factor in QD lasers
Experiment:
12
25 C
10
G =1 0
α -factor
Gain (arb. un.)
A more complete picture...
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0.9
G =5
G =2
G =0
0.95
QD
8
6
4
2
1
1.05
1.1
1.15
Energ y (eV)
Model
0
0
10
20
30
40
50
C urrent (m A )
Markus et al., JSTQE 2003
12
α -factor
10
8
Need to operate well
below GS saturation
to have low α
6
4
2
0
0
5
10
G (el./dot/ τ
rad
15
)
Andrea Fiore
Key features for SOA
applications
Inhomogeneous
broadening from size
dispersion:
QD
Broad gain
spectrum
WL
Large reservoir of
carriers for
replenishing the QDs:
ES2
ES1
GS
Large saturation
power
Polarisation sensitivity ? Preliminary evidence of
2004)
polarisation control by shape engineering (Jayavel, APLAndrea
Fiore
QD Semiconductor Optical
Amplifiers
QD
Fujitsu, presented at OFC 2004:
Large gain bandwidth:
Large saturation power:
Andrea Fiore
QD lasers: Real
applications coming up?
QD
• QDs do not compete with QWs for high-gain devices
(e.g. VCSELs)
• Threshold current lower in QDs, but no so useful
• Practical advantages over QWs for:
• Broad gain devices (amplifiers, tunable lasers)
• Long-wavelength on GaAs (1.3 µm OK - 1.55 µm?)
• Temperature-insensitive operation ?
Andrea Fiore