Electronic Structure Calculation of Single

and Coupled Self – Assembled

Quantum Dots

Kandidaat:

Vladan Mlinar

Universiteit Antwerpen

Promotor:

Prof. Dr. François Peeters

5. July, 2007vladan.mlinar@ua.ac.be

Theorie van de Gecondenseerde Materie,

Departement Fysica

QUANTUM NANOSTRUCTURES:

material 1 material 1material 2

QUANTUM NANOSTRUCTURES:

material 1 material 1material 2

G6

HH

LH

SO

CB

kz

D

Eg G8

G7

kx

QUANTUM NANOSTRUCTURES:

material 1 material 1material 2

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

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)

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.)

OUTLINE:

• Modeling of semiconductor nanostructures

• Electronic and optical properties of QDs• Unstrained QDs in an external magnetic field• Influence of the substrate orientation

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

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)

STEPS IN THE MODELING:

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

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

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).

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

k.p MODEL FOR NANOSTRUCTURES:

InAs/GaAs

meV100

Vladan Mlinar et al., PRB 71, 205305 (2005).

Hole energy levels

k.p MODEL FOR NANOSTRUCTURES:

InAs/GaAs

meV100

B = 40T

Vladan Mlinar et al., PRB 71, 205305 (2005).

Hole energy levels

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

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

NUMERICAL IMPLEMENTATION:

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)

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

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:

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).

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

COMPARISON WITH EXPERIMENT:

GaAs/AlxGa1-xAs quantum dot:

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)

GROWTH ON [11k] MOTIVATION:

M. Schmidbauer et al., PRL 96, 66108 (2006)

P. Caroff et al., APL 87, 243107 (2005)

INFLUENCE OF SUBSTRATE ORIENTATION:

cossinsinsincos

0cossin

sincossincoscos

U

x

y

z

jiji xUx

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:

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

[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

[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

[11k] GROWN QDs – strain distribution

Simplified picture:

Unstrained

[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

[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!

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

ROLE OF PIEZOELECTRICITY:

The asymmetric piezoelectric potential influences the distribution of the electron & hole wavefunction.

[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))

[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).

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).

[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).

[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).

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

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

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!