vladan mlinar ph.d. defense (2007)
DESCRIPTION
Ph.D. defense at the University of Antwerp, Belgium. The thesis can be downloaded from: http://www.cmt.ua.ac.be/thesis/mlinar.pdfTRANSCRIPT
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Electronic Structure Calculation of Single
and Coupled Self – Assembled
Quantum Dots
Kandidaat:
Vladan Mlinar
Universiteit Antwerpen
Promotor:
Prof. Dr. François Peeters
5. July, [email protected]
Theorie van de Gecondenseerde Materie,
Departement Fysica
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QUANTUM NANOSTRUCTURES:
material 1 material 1material 2
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QUANTUM NANOSTRUCTURES:
material 1 material 1material 2
G6
HH
LH
SO
CB
kz
D
Eg G8
G7
kx
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QUANTUM NANOSTRUCTURES:
material 1 material 1material 2
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QUANTUM NANOSTRUCTURES:
material 1 material 1material 2
• Delta-function atomic
like density of states
• Self-assembly
• strain field
• piezoelectricity
• 3D confinement
• band-mixing
• inter-valley mixing
• Coulomb interaction
• External magnetic field
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APPLICATIONS:
• Quantum dot laser (Kirstaedter et al. (1994))
• (In,Ga)As/GaAs QDs used as an active medium
• Low threshold current density• High characteristic temp.• greater size uniformity• higher QD densities
• Quantum dot infrared photodetector
• high responsivity• high temperature operation• high light coupling to normal
incidence light
• Single polarized photon sources
• Emits one and only one photon in each pulse• Usually InAs/GaAs QDs were investigated for
usage as single photon sources Seguin et al. Appl. Phys. Lett. 95, 263109 (2006)
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OUR TASKS:
• Deeper understanding of the quantum dot electronic structure not only on qualitative but also on quantitative level.
• Modeling of the electronic and optical properties of quantum dots (different theoretical models)
• Comparison with experiment (identification of the results of optical spectroscopy performed on QD systems, exciton complexes etc.)
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OUTLINE:
• Modeling of semiconductor nanostructures
• Electronic and optical properties of QDs• Unstrained QDs in an external magnetic field• Influence of the substrate orientation
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OUTLINE:
• Modeling of semiconductor nanostructures
• Electronic and optical properties of QDs
• What do we know from experiment ?• How do we approach the problem?• k.p model for nanostructures
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WHAT DO WE KNOW FROM EXPERIMENT:
• Growth conditions determine electronic and optical properties of QDs
754 755 756 757 758 759 7600
3000
6000
9000
?
? 5x
3X 4X
2X
X
Wavelength (nm)
PL
In
ten
sit
y (
arb
. u
nit
s)
63593-GaAs -QDs 4 K
?
• Measurements:Group V – sensitive scan Group III – sensitive scan
I. Drouzas, J. Ulloa, D. J. Mowbray
S. Godefroo, et al. JAP 96, 2535 (2004)A. Rastelli et al.
B. Urbaszek, et al., PRL 90, 247403 (2003)
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STEPS IN THE MODELING:
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ENVELOPE FUNCTION APPROACH:
k.p theory?
Envelope function :
HH
LH
SO
CB
kz
D
Eg
G6G8
G7
kx
Hamiltonian:
H = Hk + Hstrain
rFrUr n
n
n
)(
Magnetic field:
H = Hk + Hstrain + HZeeman
(Pikus-Bir
Hamiltonian)
Aekk
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FROM BULK TO NANOSTRUCTURES:
Model is valid at the abrupt interface ?
Conventional approach:
xyyxyx
kxx
NkkNkkkNk
MkkMk
2
1
2
Burt-Foreman approach:
2
' '
x x x
XY x y y x
Mk k Mk
N k N k k N k
material 1 material 1material 2
G. Bastard, PRB 24, 5963 (1981).M. Burt, J. Phys. Condens. Matter, 6651 (1992).
B. A. Foreman, PRB 56, R12748 (1998).
(M, N – effective mass parameters)
Bulk -> nanostructures
j
jxi
k
1
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FROM BULK TO NANOSTRUCTURES:
In the presence of a magnetic field:
]ˆ,ˆ[2
1}ˆ,ˆ{ˆˆ
jijiji xxxxxx kkKkkCkkC
Model is valid at the abrupt interface ?
Operator ordering (nanostructure):
2
' '
x x x
XY x y y x
Mk k Mk
N k N k k N k
Analogy:
The k operators fail to commute with the effective-mass
parameters, whereas in the “bulk” Hamiltonian when a magnetic
field is included, the k operators fail to commute with each other.
Vladan Mlinar et al., PRB 71, 205305 (2005).
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k.p MODEL FOR NANOSTRUCTURES:
nmhmeV 6,100 nmhmeV 4,150
GaAs/Al0.3Ga0.7As
Vladan Mlinar et al., PRB 71, 205305 (2005).
Hole energy levels
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k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs
meV100
Vladan Mlinar et al., PRB 71, 205305 (2005).
Hole energy levels
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k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs
meV100
B = 40T
Vladan Mlinar et al., PRB 71, 205305 (2005).
Hole energy levels
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k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs systemQuantum dot:E(B) dependence
Quantum well:E(kt) dependence
Vladan Mlinar et al., PRB 71, 205305 (2005).
Hole energy levels
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k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs systemQuantum dot:E(B) dependence
Quantum well:E(kt) dependence
])ˆ,ˆ[2
ˆ,ˆˆˆ(2
3
ˆˆˆˆ
333
33
2
'
jijiij
ijji
xxxxx
j
x
i
xxxx
kkkkkx
ikx
im
kNkkNk
Vladan Mlinar et al., PRB 71, 205305 (2005).
Hole energy levels
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NUMERICAL IMPLEMENTATION:
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SUMMARY (First part):
• Experiment versus theory
• k.p model for nanostructures• “Correct” boundary conditions at the interface
• Existance of non-physical solutions in the conventional k.p
model applied to nanostructures
• 3D model for nanostructures (numerical problems)
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OUTLINE:
• Modeling of semiconductor nanostructures
• Electronic and optical properties of QDs:
• Unstrained QDs in an external magnetic field• Influence of the substrate orientation• Type II QDs
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UNSTRAINED QDs: MOTIVATION
0 10 20 30 40 50
1,620
1,625
1,630
1,635
1,640
1,645
1,650
1,655
en
erg
y (
eV
)
magnetic field
E1
E2
E3
GaAs/AlGaAs QD:
(2) Experimental data (KU Leuven):
(1) XSTM image of GaAs/AlGaAs QD:
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UNSTRAINED GaAs/AlxGa1-xAs QDs:
1600 1620 1640 1660 1680
Energy (meV)
N=0
N=3
In
ten
sity (
arb
. u
nits)
N=5
N=7
N=9
Position of the measured
PL peak
Collaboration with TU Berlin
Vladan Mlinar et al., PRB 75, 205308 (2007).
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UNSTRAINED GaAs/AlxGa1-xAs QDs:
Electron and hole energy level
(with respect to the GaAs
conduction band) as a function of
a magnetic field
The wave function isosurfaces
plotted for 65% probability
density
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COMPARISON WITH EXPERIMENT:
GaAs/AlxGa1-xAs quantum dot:
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SUMMARY (second part):
• Interface roughness was observed to sensitively affect the transition energies, but hardly intrabandenergies.
• For a magnetic field applied in the growth directionand in the direction perpendicular to the growth direction (where B ≤50T), we find good agreementbetween the exciton diamagnetic shift obtained fromour calculations and the experimental data of N. Schildermas et al. (KU Leuven)
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GROWTH ON [11k] MOTIVATION:
M. Schmidbauer et al., PRL 96, 66108 (2006)
P. Caroff et al., APL 87, 243107 (2005)
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INFLUENCE OF SUBSTRATE ORIENTATION:
cossinsinsincos
0cossin
sincossincoscos
U
x
y
z
jiji xUx
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INFLUENCE OF SUBSTRATE ORIENTATION:
cossinsinsincos
0cossin
sincossincoscos
U
4/ tglkh /2,1,1
x
y
z
z´
y´
x´
θ
Φ - For QDs grown on [11k] substrates:
jiji xUx
l
khtg
h
ktg
22
,
- For QDs grown on [hkl] substrates:
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PROBLEM:
• How are the electronic structure and transition energies
influenced by the substrate orientation?
• What is new as compared to [001] grown QDs?
Model QD: lens and truncated pyramidal InAs/GaAs QDs grown on
[11k] substrates, where k=1,2,3.
L1
L2
L3
P1
P2
P3
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[11k] GROWN QDs – strain distribution
• Isotropic strain is increased in [11k] grown flat dots.• The isotropic strain is almost constant in the growth direction
of the larger dots.
P1L3
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[11k] GROWN QDs – strain distribution
• Isotropic strain is increased in [11k] grown flat dots.• The isotropic strain is almost constant in the growth direction
of the larger dots.
Biaxial component of the strain is decreased regardless of the dot size!
P1L3
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[11k] GROWN QDs – strain distribution
Simplified picture:
Unstrained
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[11k] GROWN QDs – strain distribution
Simplified picture:
Electron & hole energy levels
of [11k] grown flat dots will
lie energetically higher as
compared to [001] grown QDs
Unstrained
+ isotropic
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[11k] GROWN QDs – strain distribution
Simplified picture:
Electron & hole energy levels
of [11k] grown flat dots will
lie energetically higher as
compared to [001] grown QDs
Unstrained
+ isotropic
+ biaxial
Increased hole band mixing!
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ROLE OF PIEZOELECTRICITY:
•Piezoelectric effect:
• Shear strain leads to piezoelectric polarization PP = eijkεjk
• The polarization induces a fixed charge:
ρP = -divP• Piezoelectric potential VP is obtained from the Poisson equation
ρP = ε0εrΔVP
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ROLE OF PIEZOELECTRICITY:
The asymmetric piezoelectric potential influences the distribution of the electron & hole wavefunction.
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[11k] GROWN QDs – single particle states
P1
L3L1 L2
P2 P3
• Increased hole band mixing!
• The maximum effective-mass
occurs for (111) surfaces (JAP 79, 15
(1996))
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[11k] GROWN QDs:
(i) hydrostatic component of the strain
tensor
(ii) biaxial component of the strain tensor
influencing the degree of the valence band
mixing,
(iii) variation of the hole effective mass with
the substrate orientation, since it can
significantly alter the effects of the size
quantization in QD.
QD size in the growth direction determines the degree of the influence of the
substrate orientation on the electronic and optical properties of [11k] grown
QDs, whereas the influence of the shape is of secondary importance.
Vladan Mlinar and Francois M. Peeters., Appl. Phys. Lett. 89, 261910 (2006);
Vladan Mlinar and Francois M. Peeters, Appl. Phys. Lett 91 (2007).
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COMPARISON WITH EXPERIMENT:
• InAs/GaAs QDs in an external magnetic field
• Experimental data taken from S. Godefroo et al., J. Appl. Phys. 96, 2535 (2004).
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[11k] GROWN QDMs:
Isotropic (hydrostatic) part of strain tensor for [11k] grown QDM:
Piezoelectric
potential of QDM
with isosurfaces
at ±32meV
(blue –32meV,
red +32meV)Model QDM:
-Eight identical lens shaped
InAs/GaAs QDs with
R = 7.91nm, h = 4.52nm
InAs/GaAs QDM
Vladan Mlinar and Francois M. Peeters., J. Mater. Chem (2007).
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[11k] GROWN QDMs:
For [111] grown QDMs, changing the interdot
Distance varies the transition energies up to 50meVV. Mlinar and F.M. Peeters., J. Mater. Chem (2007).
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SUMMARY (third part):
• QDs grown on high index surfaces
• Continuum elastic model for strain calculation
• k·p model for single-particle energy levels
• QD size dependent influence of substrate orientation on
the electronic and optical properties of QDs• the flatter the dot the larger the difference from the reference
[001] case
• Influence of the shape is of secondary importance
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TYPE II QDs: InP/(In,Ga)P QDs
Vladan Mlinar et al., PRB 73, 235336 (2006).
InP/InGaP double
quantum dot molecule:
InP/InGaP
triple QDM
Comparison
with
experiment
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CONCLUSIONS:
Modeling: Substrate orientation: Unstrained QDs:
• Interface roughness was observed
to sensitively affect the transition
energies, but hardly intraband
energies.
• For a magnetic field applied in
the growth direction and in the
direction perpendicular to the
growth direction (where B ≤50T),
we find good agreement between
the exciton diamagnetic shift
obtained from our calculations
and the experimental data of
N. Schildermas et al. (KU Leuven)
• QDs grown on high index
surfaces
-CM model for strain calc
-k.p model for single-particle
energy levels
• QD size dependent influence of
substrate orientation on the
electronic and optical properties
of QDs (the flatter the dot the
larger the difference from the
reference [001] case)
• Experiment versus theory
• k.p model for nanostructures
-“Correct” boundary conditions
at the interface
- Existance of non-physical
solutions in the conventional
k.p model applied to nanostr.
• 3D model for nanostructures
(numerical problems)
Dank u voor uw aandacht!