Ph.D. defense at the University of Antwerp, Belgium. The thesis can be downloaded from: http://www.cmt.ua.ac.be/thesis/mlinar.pdf

1. Universiteit Antwerpen
Electronic Structure Calculation of Single
and Coupled Self – Assembled
Quantum Dots
Kandidaat: Promotor:
Vladan Mlinar Prof. Dr. François Peeters
Theorie van de Gecondenseerde Materie,
Departement Fysica
vladan.mlinar@ua.ac.be 5. July, 2007

2. QUANTUM NANOSTRUCTURES:
material 1 material 2 material 1

3. QUANTUM NANOSTRUCTURES:
material 1 material 2 material 1
CB
G6
kx Eg G8 kz
HH
D
G7
LH
SO

4. QUANTUM NANOSTRUCTURES:
material 1 material 2 material 1

5. QUANTUM NANOSTRUCTURES:
material 1 material 2 material 1
• Delta-function atomic
like density of states
• Self-assembly
• strain field
• piezoelectricity
• 3D confinement
• band-mixing
• inter-valley mixing
• Coulomb interaction
• External magnetic field

6. 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)

7. 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.)

8. OUTLINE:
• Modeling of semiconductor nanostructures
• Electronic and optical properties of QDs
• Unstrained QDs in an external magnetic field
• Influence of the substrate orientation

9. OUTLINE:
• Modeling of semiconductor nanostructures
• What do we know from experiment ?
• How do we approach the problem?
• k.p model for nanostructures
• Electronic and optical properties of QDs

10. WHAT DO WE KNOW FROM EXPERIMENT:
• Growth conditions determine electronic and optical properties
of QDs
I. Drouzas, J. Ulloa, D. J. Mowbray
Group V – sensitive scan Group III – sensitive scan
• Measurements:
PL Intensity (arb. units)
9000 63593-GaAs -QDs 4K
X
6000
?
2X
3000
4X ?
3X
5x
?
0
B. Urbaszek, et al., PRL 90, 247403 (2003) S. Godefroo, et al. JAP 96, 2535 (2004) 754 755 756 757 758 759 760
Wavelength (nm)
A. Rastelli et al.

11. STEPS IN THE MODELING:

12. ENVELOPE FUNCTION APPROACH:
k.p theory?
  
Envelope function : (r )  U n r Fn r 
n
Hamiltonian: CB
H = Hk + Hstrain (Pikus-Bir
Hamiltonian)
G6
kx Eg G8 kz
Magnetic field:
HH
H = Hk + Hstrain + HZeeman D
G7

  eA LH
k k 
 SO

13. FROM BULK TO NANOSTRUCTURES:
Model is valid at the abrupt interface ?
1 
Bulk -> nanostructures kj 
i x j
Conventional approach: Burt-Foreman approach:
Mk x2  k x Mkk Mk x2  k x Mk x
Nk x k y 
1
k x Nk y  k y Nk x  N XY  k x N  k y  k y N  k x
' '
2
M. Burt, J. Phys. Condens. Matter, 6651 (1992).
G. Bastard, PRB 24, 5963 (1981). B. A. Foreman, PRB 56, R12748 (1998).
material 1 material 2 material 1
(M, N – effective mass parameters)

14. FROM BULK TO NANOSTRUCTURES:
Model is valid at the abrupt interface ?
Operator ordering (nanostructure):
Mk x2  k x Mk x
N XY  k x N  k y  k y N  k x
' '
In the presence of a magnetic field:
ˆ k  C{k , k }  1 K [k , k ]
Ck xi ˆx j ˆ ˆ ˆ ˆ
xi xj xi xj
2
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).

15. k.p MODEL FOR NANOSTRUCTURES:
GaAs/Al0.3Ga0.7As
0  10meV , h  6nm 0  15meV , h  4nm
Hole energy
Vladan Mlinar et al., PRB 71, 205305 (2005).
levels

16. k.p MODEL FOR NANOSTRUCTURES:
0  10meV
Hole energy
levels
InAs/GaAs
Vladan Mlinar et al., PRB 71, 205305 (2005).

17. k.p MODEL FOR NANOSTRUCTURES:
B = 40T
0  10meV
Hole energy
levels
InAs/GaAs
Vladan Mlinar et al., PRB 71, 205305 (2005).

18. k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs system
Quantum dot: Quantum well:
E(B) dependence E(kt) dependence
Hole energy
levels
Vladan Mlinar et al., PRB 71, 205305 (2005).

19. k.p MODEL FOR NANOSTRUCTURES:
InAs/GaAs system
Quantum dot: Quantum well:
E(B) dependence E(kt) dependence
ˆ 'ˆ ˆ ˆ
k xi N  k x j  k x j N  k xi 
 3 ˆ  3 ˆ
 
 3   3 ˆ ˆ

2 ˆ ˆ
3 (i kx j  i k xi   3 k xi , k x j  [k xi , k x j ])
2m xi x j 2
Hole energy
levels
Vladan Mlinar et al., PRB 71, 205305 (2005).

20. NUMERICAL IMPLEMENTATION:

21. 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)

22. 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

23. UNSTRAINED QDs: MOTIVATION
GaAs/AlGaAs QD: (1) XSTM image of GaAs/AlGaAs QD:
(2) Experimental data (KU Leuven):
E1
E2
1,655 E3
1,650
1,645
energy (eV)
1,640
1,635
1,630
1,625
1,620
0 10 20 30 40 50
magnetic field

24. UNSTRAINED GaAs/AlxGa1-xAs QDs:
Collaboration with TU Berlin
N=9
Intensity (arb. units)
N=7
N=5
N=3
N=0
1600 1620 1640 1660 1680
Energy (meV)
Position of the measured
PL peak
Vladan Mlinar et al., PRB 75, 205308 (2007).

25. UNSTRAINED GaAs/AlxGa1-xAs QDs:
The wave function isosurfaces
Electron and hole energy level
plotted for 65% probability
(with respect to the GaAs
density
conduction band) as a function of
a magnetic field

26. COMPARISON WITH EXPERIMENT:
GaAs/AlxGa1-xAs quantum dot:

27. SUMMARY (second part):
• 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)

28. GROWTH ON [11k] MOTIVATION:
P. Caroff et al., APL 87, 243107 (2005)
M. Schmidbauer et al., PRL 96, 66108 (2006)

29. INFLUENCE OF SUBSTRATE ORIENTATION:
 cos  cos  sin  cos   sin  
 
xi  U ij x j U    sin 
 cos  sin 
cos  0 
 sin  sin  cos   
z
y
x

30. INFLUENCE OF SUBSTRATE ORIENTATION:
 cos  cos  sin  cos   sin  
 
xi  U ij x j U    sin 
 cos  sin 
cos  0 
 sin  sin  cos   
z - For QDs grown on [hkl] substrates:
z´
θ
k h2  k 2
tg  , tg 
y
h l
x´
x Φ - For QDs grown on [11k] substrates:
y´
   / 4 h  1, k  1, l  2 / tg

31. 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 P1
L2 P2
L3 P3

32. [11k] GROWN QDs – strain distribution
L3 P1
• Isotropic strain is increased in [11k] grown flat dots.
• The isotropic strain is almost constant in the growth direction
of the larger dots.

33. [11k] GROWN QDs – strain distribution
L3 P1
• 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!

34. [11k] GROWN QDs – strain distribution
Simplified picture:
Unstrained

35. [11k] GROWN QDs – strain distribution
Simplified picture:
Unstrained Electron & hole energy levels
+ isotropic of [11k] grown flat dots will
lie energetically higher as
compared to [001] grown QDs

36. [11k] GROWN QDs – strain distribution
Simplified picture:
Unstrained Electron & hole energy levels
+ isotropic of [11k] grown flat dots will
+ biaxial lie energetically higher as
compared to [001] grown QDs
Increased hole band mixing!

37. ROLE OF PIEZOELECTRICITY:
•Piezoelectric effect:
• Shear strain leads to piezoelectric polarization P
P = eijkεjk
• The polarization induces a fixed charge:
ρP = -divP
• Piezoelectric potential VP is obtained from the Poisson equation
ρP = ε0εrΔVP

38. ROLE OF PIEZOELECTRICITY:
The asymmetric piezoelectric potential influences the
distribution of the electron & hole wavefunction.

39. [11k] GROWN QDs – single particle states
L1 L2 L3
• Increased hole band mixing!
• The maximum effective-mass
occurs for (111) surfaces (JAP 79, 15
(1996))
P1 P2 P3

40. [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).

41. 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).

42. [11k] GROWN QDMs:
Isotropic (hydrostatic) part of strain tensor for [11k] grown QDM:
InAs/GaAs QDM
Piezoelectric
potential of QDM
with isosurfaces
at ±32meV
(blue –32meV,
Model QDM: red +32meV)
-Eight identical lens shaped
InAs/GaAs QDs with
R = 7.91nm, h = 4.52nm
Vladan Mlinar and Francois M. Peeters., J. Mater. Chem (2007).

43. [11k] GROWN QDMs:
For [111] grown QDMs, changing the interdot
Distance varies the transition energies up to 50meV
V. Mlinar and F.M. Peeters., J. Mater. Chem (2007).

44. 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

45. TYPE II QDs: InP/(In,Ga)P QDs
InP/InGaP double
quantum dot molecule:
InP/InGaP
triple QDM
Comparison
with
experiment
Vladan Mlinar et al., PRB 73, 235336 (2006).

46. CONCLUSIONS:
Modeling: Substrate orientation: Unstrained QDs:
• Experiment versus theory • QDs grown on high index • Interface roughness was observed
surfaces to sensitively affect the transition
• k.p model for nanostructures -CM model for strain calc energies, but hardly intraband
-“Correct” boundary conditions -k.p model for single-particle energies.
at the interface energy levels
- Existance of non-physical • For a magnetic field applied in
solutions in the conventional • QD size dependent influence of the growth direction and in the
k.p model applied to nanostr. substrate orientation on the direction perpendicular to the
electronic and optical properties growth direction (where B ≤50T),
• 3D model for nanostructures of QDs (the flatter the dot the we find good agreement between
(numerical problems) larger the difference from the the exciton diamagnetic shift
reference [001] case) obtained from our calculations
and the experimental data of
N. Schildermas et al. (KU Leuven)
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