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Malte Huck
Optical and Electrical Properties of Single Self-Assembled Quantum Dots in Lateral
Electric Fields
ISBN: 978-3-8366-4439-6
Herstellung: Diplomica® Verlag GmbH, Hamburg, 2010
Zugl. Technische Universität München, München, Deutschland, Diplomarbeit, 2009
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© Diplomica Verlag GmbH
http://www.diplomica.de, Hamburg 2010
Das Wetter war stürmisch; der Himmel dicht
bewölkt; die Dunkelheit . . . undurchdringlich
. . . Durch ein solches Laublabyrinth, und in völ-
liger Finsternis, mußten die Falter sich einen
Weg suchen, um ans Ziel ihrer Wallfahrt zu
gelangen.
Kein Käuzchen würde es unter diesen Umstän-
den wagen, die Höhlung seines Ölbaums zu ver-
lassen. Der Schmetterling . . . zieht ohne Zögern
seine Bahn . . . So geschickt lenkt er seinen ver-
schlungenen Flug, dass er, allen Hindernissen
zum Trotz, in einem Zustand vollkommerner
Frische anlangt, die großen Schwingen ganz un-
versehrt . . . Die Dunkelheit ist Licht genug . . .
J. H. Fabre
Contents
1 Motivation: Entangled Photons
1
1.1 Entangled States
. . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2 The Quantum Dot 2X1X Cascade as Entangled Photon Source
4
1.3 Summary and Conclusions
. . . . . . . . . . . . . . . . . . . . .
7
2 Semiconductor Quantum Dots: Artificial Atoms
9
2.1 Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2 Low-Dimensional Semiconductor Nanostructures
. . . . . . . . .
14
2.3 MBE Growth of Self-Assembled InGaAs Quantum Dots
. . . . .
16
2.3.1 Molecular Beam Epitaxy
. . . . . . . . . . . . . . . . . .
16
2.3.2 Stranski-Krastanov Growth Mode
. . . . . . . . . . . . .
18
2.3.3 Sample Design
. . . . . . . . . . . . . . . . . . . . . . .
20
2.4 Energetic Structure of Single Quantum Dots
. . . . . . . . . . .
21
2.5 Optical Properties
. . . . . . . . . . . . . . . . . . . . . . . . .
24
2.5.1 Micro-Photoluminescence Setup
. . . . . . . . . . . . . .
25
2.5.2 Optical Excitation of ElectronHole Pairs
. . . . . . . .
28
2.5.3 From High to Low Quantum Dot Density Material
. . .
32
2.5.4 Photoluminescence of Few Particle States
. . . . . . . . .
35
2.5.5 Line Identification
. . . . . . . . . . . . . . . . . . . . .
38
2.5.6 Initially Charged Quantum Dots
. . . . . . . . . . . . .
40
i
Contents
2.6 Summary and Conclusions
. . . . . . . . . . . . . . . . . . . . .
42
3 Fabrication and Characterization of Lateral Electric Field Devices
43
3.1 Fabrication
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.2 Electric Field Simulation
. . . . . . . . . . . . . . . . . . . . . .
48
3.3 Electrical Characterization and Photocurrent Measurements
. .
50
3.3.1 Electric Field Activated Carrier Loss Mechanisms
. . . .
51
3.3.2 Temperature Dependence
. . . . . . . . . . . . . . . . .
56
3.3.3 Energy Dependence
. . . . . . . . . . . . . . . . . . . . .
61
3.3.4 Power Dependence -- Responsivity
. . . . . . . . . . . .
64
3.3.5 Field Dependent Photoluminescence Quenching
. . . . .
67
3.4 Summary and Conclusions
. . . . . . . . . . . . . . . . . . . . .
70
4 Independent Control of Few Exciton States in Single Quantum
Dots
73
4.1 Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.2 The Quantum Confined Stark Effect
. . . . . . . . . . . . . . .
76
4.3 Electric Field Dependent Micro-Photoluminescence -- Overview
80
4.4 Stark Shifts of Single Lines
. . . . . . . . . . . . . . . . . . . . .
87
4.5 Screening Mechanism
. . . . . . . . . . . . . . . . . . . . . . . .
90
4.6 Determination of the Polarizability
. . . . . . . . . . . . . . . .
92
4.7 Peak Broadening and Lifetime Measurements
. . . . . . . . . .
95
4.8 Summary and Conclusions
. . . . . . . . . . . . . . . . . . . . . 100
5 Outlook
103
ii
List of Figures
1.1 Quantum Dot BiexcitonExciton Energy Cascade
. . . . . . . .
5
2.1 Overview of Different Types of Quantum Dots
. . . . . . . . . .
12
2.2 Low Dimensional Nanostructures -- DOS
. . . . . . . . . . . .
15
2.3 MBE Chamber
. . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.4 Different Heteroepitaxy Growth Modes
. . . . . . . . . . . . . .
19
2.5 TEM and AFM Picture of Self-Assembled Quantum Dots
. . . .
20
2.6 Quantum Dot Sample Design
. . . . . . . . . . . . . . . . . . .
21
2.7 Energetic Band Structure of a Self-Assembled Quantum Dot
. .
22
2.8 Micro-Photoluminescence Setup
. . . . . . . . . . . . . . . . . .
26
2.9 Power-Dependent QD Ensemble PL
. . . . . . . . . . . . . . . .
30
2.10 Excitation Modes
. . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.11 Unrotated Growth -- PL and AFM Wafer Mapping
. . . . . . .
33
2.12 PL Spectra and Sketch of Different Excitonic States in a QD
. .
37
2.13 Power Dependent PL and Poisson Statistics
. . . . . . . . . . .
39
2.14 PL Spectrum of a Charged Quantum Dot
. . . . . . . . . . . . .
41
3.1 Schottky Diode Fabrication Steps
. . . . . . . . . . . . . . . . .
45
3.2 Original Mask File and Microscope Images
. . . . . . . . . . . .
47
3.3 Field Distribution in Cross Section
. . . . . . . . . . . . . . . .
49
3.4 Electric Field in the Quantum Dot Layer
. . . . . . . . . . . . .
50
iii
List of Figures
3.5 Electric Field Activated Carrier Loss Mechanisms
. . . . . . . .
51
3.6 Electron Drift Velocity as a Function of the Electric Field
. . . .
54
3.7 Dark Temperature Dependent FieldCurrent Characteristics
. .
57
3.8 Modified Arrhenius Plots
. . . . . . . . . . . . . . . . . . . . . .
59
3.9 Thermal Activation of Charge Carriers -- Energetic Structure
.
60
3.10 Energy Dependent Field-Current Characteristics
. . . . . . . . .
62
3.11 Excitation Power Dependent Photocurrent -- Responsivity
. . .
65
3.12 Field Dependent Photoluminescence versus Photocurrent
. . . .
68
4.1 Field Dependent PL on Two Lines
. . . . . . . . . . . . . . . . .
75
4.2 Lateral and Vertical Electric Field Applied to a Quantum Dot
.
77
4.3 Quantum Confined Stark Effect
. . . . . . . . . . . . . . . . . .
79
4.4 Field Dependent µ-PL -- Overview
. . . . . . . . . . . . . . . .
81
4.5 Charge Carrier Capture in Initially Charged QDs
. . . . . . . .
84
4.6 Power Dependent PL on a Quantum Dot in Electric Field
. . .
85
4.7 Field Dependent Single QD PL versus Simulation
. . . . . . . .
88
4.8 Screening Mechanism
. . . . . . . . . . . . . . . . . . . . . . . .
91
4.9 Determination of the Polarizability
. . . . . . . . . . . . . . . .
93
4.10 PL Intensity, Peak Width and Lifetime Measurements
. . . . . .
96
4.11 Lorentzian versus Gaussian Fit Function
. . . . . . . . . . . . . 101
5.1 Cross-Correlation Measurement between 2X
0
and 1X
0
. . . . . . 105
iv
When I hear of Schrödinger's cat,
I reach for my gun.
Stephen Hawking
CHAPTER
1
Motivation: Entangled Photons
In this thesis we investigate the optical properties of self-assembled quantum
dots exposed to a lateral electric field. As a result of the electric field the wave
functions of electrons and holes inside the quantum dot are manipulated, which
makes it possible to tune their energy levels and control the optical properties of
the system. The possibility of tuning the emission energy of different few particle
states using this method makes this system very promising for the use of a source
of polarization entangled photons as discussed in the following sections.
In Section
1.1
the concept of entangled states is introduced together with a
brief historical overview. The possibility of using the excitonbiexciton cascade
of a self-assembled quantum dot for the generation of entangled photon pairs is
presented in Section
1.2
.
1
Motivation: Entangled Photons
1.1 Entangled States
The concept of entanglement deemed to be one of the most fascinating and
impressive phenomena in modern quantum physics. Parts of it are well-known
far beyond the borders of the physics world, for example discussions about the
state of health of Schrödinger's cat [
Sch35
] are not only led by physicists.
The history of entanglement began in 1935 when two pioneering papers were
published. First, Einstein, Podolsky and Rosen wrote their famous "EPR" paper
raising the question: "Can Quantum-Mechanical Description of Physical Reality
Be Considered Complete?"
[
Ein35
] The paper discusses the properties of an
entangled two-particle system formed from a radioactive decay. Soon afterward,
Schrödinger published the paper on his famous cat paradox, where the word
entanglement (German: "Verschränkung") was mentioned for the first time to
describe the physical properties, as explained in the following [
Sch35
].
We introduce the concept of entanglement on the basis of a quantum corre-
lated photon pair. As polarization basis we chose horizontal and vertical polar-
ization and designate the polarization states |
and | . These polarizations
can experimentally be determined by using a polarization filter orientated in
the right direction (horizontal or vertical) followed by a single-photon detector.
A basic requirement to achieve entanglement is a photon source that generates
correlated photon pairs
. Such photon pairs have the following properties:
1. The polarization of either photon 1 or photon 2 measured independently
of the other is random. That means by just looking at for example photon
1, the probabilities of measuring the polarizations |
and | are the
same and equal to
1
2
.
2. The polarizations of the photon pair are perfectly correlated. That means
in the case of positive correlation that for both photons the polarization
is either |
or | . In the case of negative correlation, the polarization
of one of the photons is |
while the polarization of the other one has to
be | .
2
1.1 Entangled States
The two particles are said to be in an entangled state if their wave function
cannot be factorized into a product of the two individual particles. For the cor-
related photon pair, the wave function in the case of perfect positive correlation
has to be written in the form:
|
±
=
1
2
(|
1
,
2
± |
1
,
2
)
(1.1)
and in the case of perfect negative correlation it has to be written in the form:
|
±
=
1
2
(|
1
,
2
± |
1
,
2
) .
(1.2)
The subscripts 1 and 2 refer to the individual photons.
Consequently, entanglement leads to the fact that the quantum states (in this
case the polarization) of the two photons no longer can be regarded as being
independent from each other. Their quantum states are said to be entangled.
The measurement of the polarization of photon 1 simultaneously determines the
polarization of photon 2, even if the system is spatially separated.
This behavior appears surprising because at first glance, it seems to violate
the rule that no information can be transmitted faster than the speed of light.
In fact, this no communication theorem [
Per04
] is not violated since no informa-
tion can be transmitted with a measurement like this. As defined by the first
condition for correlated photon pairs, the measurement of either |
or | can
not be enforced and has to be random, which makes it impossible to generate
information and process it with such an EPR pair.
The "EPR paradox" as this phenomenon is often referred to, immediately
raises another question: Does the polarization measurement of the first photon
really determine the polarization of the second one, or is there a hidden variable
that determines the polarizations of the particles from the beginning on? To
answer that question, John Bell set up an equation in 1946, which is referred
to as Bell's inequality [
Bel64
]. The violation of this inequality would prove
the absence of hidden variables and support the quantum mechanical point of
view as explained above. Actually, three experiments could prove the violation
of Bell's inequality between 1981 and 1982 [
Asp81
,
Asp82b
,
Asp82a
]. These
experiments were a fundamental step toward modern quantum theory, since they
3
Motivation: Entangled Photons
are part of the historical intersection between classical and quantum physics.
Besides the fascinating properties of entangled photon pairs explained above,
they have also possible uses in applications like quantum teleportation. Here, the
quantum state of one particle is transfered to another one which is physically
separated from it. A possible use for quantum teleportation is the transfer of
quantum states between spatially separated quantum computers [
Kni01
]. A pos-
sible scheme for photon quantum teleportation was proposed in 1993 by Bennett
et al
. [
Ben93
]. Four years later, quantum teleportation could be experimentally
demonstrated by Bouwmeester et al. [
Bou97
] and shortly afterwards by Boschi
et al
. [
Bos98
]. A detailed introduction to quantum teleportation can be found
in Reference [
Fox06
].
1.2 The Quantum Dot BiexcitonExciton Cascade
as Entangled Photon Source
As explained in the previous section, the creation of entangled photon pairs
requires a source of correlated photons. In the following we describe how the
biexcitonexciton cascade of a quantum dot can be used for this purpose. Details
about the growth and optical properties of self-assembled quantum dots can be
found in Chapter
2
. The influence of an electric field on the quantum dots is
introduced in Chapter
3
. Finally, the tuning of different few particle states is
presented in Chapter
4
.
Quantum dots are very promising for the generation of single photons due to
their atom-like energetic structure. This led to the proposal of a single photon
turnstile device in 1994 by Imamo¯glu et al. [
Ima94
]. The experimental realiza-
tion followed in 2000 by Michler et al. [
Mic00
] and one year later by Santori et al.
[
San01
]. Single photons are the basic requirement for the realization of quan-
tum information processes such as quantum cryptography [
Gis02
] and quantum
computation [
Kni01
].
Besides the emission of single photons, quantum dots can emit sequences
of photons in a radiative cascade [
Bac99
,
Dek00
]. This two-photon cascade
is depicted schematically in Figure
1.1
(a) on the facing page. A biexciton
singlet state |2X (two electrons and two holes) decays radiatively to one of
4
1.2 The Quantum Dot 2X1X Cascade as Entangled Photon Source
|2Xñ
|1Xñ
|cgsñ
FSS
ideal QD
real QD
2X & 1X in
resonance
lateral
E-field
+
s
+
s
-
s
-
s
H
H
V
V
V
H
V
H
(a)
(b)
(c)
E
E
1
E
2
E
a
E
b
E
a
E
b
Figure 1.1: Quantum Dot BiexcitonExciton energy cascade. A biexciton state |2X
decays to one of two exciton states |1X and subsequently to the crystal
ground state |cgs by the emission of one photon in each decay step.
(a) In an ideal quantum dot, both |1X states are degenerate and the two
photons of the energy cascade are polarization entangled. (b) Asymme-
try of the quantum dot gives rise to a fine structure splitting (FSS) of the
|1X states and, thus, destroys the entanglement. Path information can
be gained by measuring the energy of a photon since all four energies
are different. (c) By applying a lateral electric field, the |1X and |2X
states can be modified in such a way that the |1X state is in the middle
of the cascade and again two energies are equal. For example, the biex-
citon energy in the vertical polarized decay path (E
a
) is the same as the
exciton energy in the horizontal polarized decay path. Entanglement is
possible through time reordering.
two optically active single exciton states |1X (one electron and one hole) by
emitting a photon. In a second recombination process, the |1X state decays
to the crystal ground state |cgs by emitting another photon. The energies
of the emitted photons depend strongly on the used material system. For self-
assembled InGaAs quantum dots, they are typically in the range of 1.0 1.4 eV.
This biexcitonexciton cascade of an ideal quantum dot was proposed as a
source for polarization entangled photons by Benson et al. in 2000 [
Ben00
]. Two
possible biexcitonexciton decay paths exist in a quantum dot. In an ideal quan-
tum dot as shown in Figure
1.1
(a), the emitted photons are circularly polarized.
For each decay path, the polarizations of the first and the second photon are
anticorrelated. If the first photon is
+
polarized, the second one has
-
polar-
ization and vice versa. This behavior is exactly condition number two for the
5
Motivation: Entangled Photons
generation of a correlated photon pair, as discussed in Section
1.1
. Condition
number one is also fulfilled, since the cascade either decays over path one or
path two with the same probability. The two decay paths are indistinguishable
since for both paths the emission energy of the biexciton and the exciton de-
cay are the same (E
1
and E
2
). Thus, the system can be used as a source for
polarization entangled photon pairs.
In practice, an asymmetric dot shape, strain and piezoelectric effects [
Sti99
]
destroy the anticorrelated behavior explained above. The |1X state is split
through the electronhole interaction giving rise to an anisotropic exchange
splitting
[
Bay99
,
Bes03
] destroying the degeneracy of the two states [
Kul99
] (c. f.
Figure
1.1
(b)). These split states couple to photons having orthogonal linear po-
larizations (H and V) [
San02
]. The splittig, often also referred to as fine structure
splitting
(FSS), is typically in the order of 10 100 µeV [
Bay02
,
Ger07
,
Vog07
].
Since the homogeneous linewidth of self-assembled quantum dots is typically
1 µeV [
H¨04
], the fine structure splitting is energetically resolvable and entan-
glement is destroyed, since the two paths become energetically distinguishable.
Several successful attempts were made to tune the fine structure splitting in
a way that it is smaller than the homogeneous linewidth of the emission peaks
and again the situation from Figure
1.1
(a) is valid. An in-plane magnetic field
[
Ste06
] and an in situ uniaxial stress [
Sei06
] were applied to reduce the fine
structure splitting of quantum dots. The fine structure splitting is also reduced
in smaller quantum dots that can be obtained via growth [
Seg05
] or subsequent
annealing [
Tar04
]. Furthermore, the application of a lateral electric field can
lead to a reduction and even to the vanishing of the fine structure splitting
[
Ger07
,
Kow05
,
Kow07
,
Vog07
].
In this thesis we follow another approach involving a lateral electric field. In
2008 Avron et al. proposed a possibility of using the biexcitonexciton cascade
for the generation of entangled photon pairs [
Avr08
] even if a finite fine structure
splitting is present. The proposed level scheme is shown in Figure
1.1
(c). With
a lateral electric field the exciton and biexciton energy states can be tuned in
such a way that the fine structure splitting is in the middle of the cascade
and again two energies are equal (labeled E
a
and E
b
in Figure
1.1
(c)). The
indistinguishability of the two decay paths is restored. However, this makes a
time reordering of the photons necessary, since otherwise path information can
6
1.3 Summary and Conclusions
be gained by measuring which photon is emitted first. For detailed discussion
on this time reordering, we refer the reader to Reference [
Avr08
].
In the following thesis we will show that we are able to tune the biexciton
and exciton emission energies into resonance. Thus, the use of quantum dots
in lateral electric fields is a very promising system for future devices creating
entangled photon pairs on demand.
1.3 Summary and Conclusions
In this chapter we have introduced the concept of entanglement and we gave
a brief historical overview on the subject. We have presented two different
approaches to use the biexcitonexciton cascade of a self-assembled quantum
dot for the generation of entangled photon pairs. A concept was introduced
that proposes the generation of polarization entangled photon pairs by using
a lateral electric field to tune the biexciton and exciton emission energy into
resonance.
7
Director :
Up and atom!
Rainier Wolfcastle: Up and at them.
Director :
Up and ATOM!
Rainier Wolfcastle: Up and atdem!
Director :
UP AND ATOM!
Rainier Wolfcastle: UP AND ATEM!
Director :
Better.
The Simpsons -- Radioactive Man
CHAPTER
2
Semiconductor Quantum Dots: Artificial Atoms
In this chapter we introduce the concept of quantum dots and demonstrate
their optical emission properties. In Section
2.1
the quantum dot is introduced
as a three-dimensional charge carrier trap. Several types of quantum dots are
presented in an overview.
In Section
2.2
we discuss the physical effects that occur on the way from bulk
semiconductor material to the three-dimensional charge carrier confinement in
the case of quantum dots. The growth of self-assembled quantum dot samples
is the topic of Section
2.3
, where the technique of molecular beam epitaxy is
introduced (Section
2.3.1
). This technique is used to grow the semiconductor
quantum dots via heteroepitaxy in the Stranski-Krastanov growth mode (Sec-
tion
2.3.2
).
9
Semiconductor Quantum Dots: Artificial Atoms
Quantum dots are commonly referred to as artificial atoms due to their atom-
like emission features. The origin for this expression is explained in Section
2.4
on the basis of the energetic structure of self-assembled quantum dots.
The optical properties of quantum dots are discussed in Section
2.5
, begin-
ning with an introduction to the experimental setup that has been used to in-
vestigate the quantum dots during this thesis (Section
2.5.1
). Different optical
excitation modes are presented in Section
2.5.2
and in Section
2.5.3
we discuss,
how to achieve a low enough quantum dot density on the analyzed samples.
Section
2.5.4
deals with the photoluminescence of different exciton states and in
Section
2.5.5
we present how these lines can be identified via power dependent
measurements. Finally, the concept of initial charges in self-assembled quantum
dots is presented in Section
2.5.6
.
10
2.1 Motivation
2.1 Motivation
Quantum dots are nanometer sized potential traps which provide full three-
dimensional confinement to charge carriers (electrons and/or holes). When the
size of this multidimensional quantum well [
Ara82
] is so small that it is com-
parable to the de Broglie wavelength of an electron, the quantum confinement
leads to an atom-like discrete energy structure that shows the same shell filling
behavior as an atom does [
Wie02
]. Thus, quantum dots are often referred to
as artificial atoms and experiments are similarly possible with real atoms and
quantum dots.
The concept of the quantum dot was first proposed in 1982 by Arakawa and
Sakaki [
Ara82
], where it was called a three-dimensional quantum well. One
year later, the name quantum box was proposed, which developed to the word
quantum dot
-- the most common expression for these structures since the early
1990s.
Quantum dots are ideal single photon sources for quantum information pro-
cessing, due to their discrete energy structure. A turnstile device for heralded
single photons
was first proposed in 1994 by Imamo¯glu et al [
Ima94
]. Experi-
mentally, this single photon emission from an individual quantum dot was first
demonstrated by Michler et al. in 2000 [
Mic00
] and one year later by Santori
et al
. [
San01
]. Quantum dots are important for device applications, such as
quantum dot lasers, since they show a low lasing threshold [
Ara82
]. They also
build the basis for several applications in quantum information processing. Pos-
sible applications are for example quantum cryptography [
Gis02
] and quantum
computation [
Kni01
]. The ability to control the quantum dot energy levels after
the fabrication is the core of this thesis.
Different Types of Quantum Dots
Figure
2.1 on the next page
shows five possible ways of realizing quantum dots
in semiconductor material. In most of them the potential minimum is achieved
by embedding a small amount of quantum dot material (e. g. InGaAs) into a
surrounding material with larger band gap (e. g. GaAs). The five concepts are
in detail:
11
Semiconductor Quantum Dots: Artificial Atoms
[Elz04]
[Sch97]
[Har99]
30 nm
x
y
(a)
(b)
(e)
(c)
(d)
[Evi]
Figure 2.1: Overview of different types of quantum dots: (a) Self-assembled quan-
tum dot, (b) electrostatically defined quantum dot, (c) colloidal quantum
dot, (d) quantum dot formed by cleaved edge overgrowth, and (e) litho-
graphically defined quantum dot.
12
2.1 Motivation
(a) Self-assembled quantum dots [
Sta04
], for example grown in the Stranski-
Krastanov
growth mode (c. f. Section
2.3
) [
Str39
]. Figure
2.1
(a) shows an
atomic microscope picture of an uncapped InGaAs self-assembled quantum
dot. Those self-assembled InGaAs quantum dots are used in this thesis
and shall be discussed in more detail later in this chapter.
(b) Electrostatically defined quantum dots as shown in Figure
2.1
(b) from
Reference [
Elz04
]. The basis of this kind of quantum dots is a two-
dimensional electron or hole gas in the semiconductor, which is electro-
statically manipulated with nanometer-size contacts, so that the required
potential minimum forms. Since those quantum dots can only confine ei-
ther electrons or holes, they cannot readily be used for optical recombina-
tion measurements. They are favored in quantum transport experiments
due to the possibility of "switching the dot on and off" and the ability to
control the coupling between two of them just by changing a voltage.
(c) Colloidal quantum dots as shown in Figure
2.1
(c) are nanocrystals that are
chemically synthesized from precursor compounds dissolved in solutions
[
Evi
]. The advantage of these dots is, that they are by far the cheapest
and since they can be bought in solution, they can be placed easily on
any material system that shall be studied or even incorporated into living
cells [
Mic05
].
(d) Cleaved edge overgrowth quantum dots are formed in the intersection of
three quantum wells, as shown in Figure
2.1
(d) [
Sch97
]. This situation is
achieved by cleaving the sample in situ in an ultra high vacuum molecular
beam epitaxy system and growing the next quantum well layer on top of
the cleaved edge. This process, executed two times, leads to a precisely
positioned quantum dot with a well defined geometry at the intersection
of the quantum wells.
13
Semiconductor Quantum Dots: Artificial Atoms
(e) Pyramidal quantum dots defined by lithography are presented in Fig-
ure
2.1
(e) [
Har99
]. In this case the surface is pre-patterned via electron
beam lithography and directs the positions where the quantum dots shall
form during the growth process. It is possible to define the spatial po-
sition of the quantum dot with high accuracy, but as with cleaved edge
overgrowth, the resulting quantum dots possess defects and are not very
clean.
Due to the well understood growth technique and the good emission prop-
erties, self-assembled quantum dots as shown in Figure
2.1
(a) were chosen for
this thesis. In the following we concentrate on this type of quantum dots and
introduce their properties in some more detail. However, these properties are
very similar for all the mentioned types of quantum dots.
2.2 Low-Dimensional Semiconductor
Nanostructures
It is possible to achieve spatial confinement of charge carriers in one or more
dimensions via semiconductor band gap engineering [
Kro01
]. The fundamental
requirement for confinement is a localized potential well, which is deep enough
to prevent the confined charge carriers from tunneling out of the well. The most
common method of realizing such a quantum well is to embed a lower band gap
semiconductor material inside a surrounding one of higher band gap. If one
assumes an ideal quantum well situation, it is well known from basic quantum
mechanics that quantized energy levels will form inside the well [
Sch07
]. This
change in the electron density of states is the most important property of such a
nanostructure because it determines the allowed states for an electron in k-space.
Confinement can be defined in either one, two or all three dimensions, as
depicted in Figure
2.2 on the facing page
. The lower part shows the electronic
density of states for confinement in different dimensions, while in the upper part
the corresponding geometry is sketched.
Figure
2.2
(a) shows the situation for no confinement. The corresponding
density of states has the normal square root behavior and electrons are free to
14
2.2 Low-Dimensional Semiconductor Nanostructures
Quantum
Well
Quantum
Wire
Quantum
Dot
t
g (E)
0D
E
E
c
E
0
E
1
g (E)
2D
E
g (E)
1D
E
E
00
E
nm
"E "
nlm
2D
1D
"0D"
Bulk
g (E)
3D
E
3D
(a)
(b)
(c)
(d)
Figure 2.2: Shape and electronic density of states of (a) bulk semiconductor ma-
terial, (b) a two-dimensional quantum well, (c) a nanowire, and (d) a
quantum dot. For comparison, the bulk density of states is indicated on
all plots as a black dotted line.
move in all three directions. In the case of a two-dimensional quantum well (b)
electrons are only free to move in a two-dimensional plane (indicated in blue)
and the density of states shows stepwise behavior. If the states are populated
by modulation doping, a two dimensional electron gas (2DEG) can be formed.
Such systems are used to measure quantum transport effects, for example the
quantum Hall effect
was discovered in such a two-dimensional system [
Kli80
].
An electrostatically defined 2DEG is responsible for the electron transport in
the metaloxidesemiconductor field-effect transistor (MOSFET), which is one
of the most common semiconductor devices. Limiting the electron motion in
one more dimension results in a quantum wire, as presented in Figure
2.2
(c).
Quantum wires can be grown via molecular beam epitaxy [
Spi08
] and placed on
various substrates to study their electrical and optical properties. The density
of states in this case has van Hove singularities with energy tails.
Finally, the quantum dot is shown in Figure
2.2
(d). In this case the electron
(and/or) hole wave function is confined to a single point. That means that elec-
trons have "zero dimensions" of free movement. The density of states consists of
delta functions at certain energies and, therefore, only discrete energy levels can
15
Semiconductor Quantum Dots: Artificial Atoms
be populated in a quantum dot. This fact makes quantum dots very promising
for spectroscopy experiments and, thus, they are the basis of this work.
The formulas for the electronic density of states g(E) of the situations ex-
plained above can be derived by taking into account the possible states in k-space
inside the three-, two-, one- and zero-dimensional Fermi-"sphere" [
Dav97
]. For
parabolic quasi free electron bands, the results can be written as:
g
3D
(E) =
1
2
2
2m
2
3/2
E - E
C
(2.1a)
g
2D
(E) =
m
2
(2.1b)
g
1D
(E) =
1
m
E - E
C
(2.1c)
g
0D
(E) = 2(E - E
C
)
(2.1d)
for the bulk material, the two-dimensional quantum well, the quantum wire and
the quantum dot, respectively. Here, m
denotes the effective mass of an electron
(or hole) and is the Planck constant. E
C
is the energy of the conduction band
edge and depends on the used material. denotes the Dirac delta function.
2.3 MBE Growth of Self-Assembled InGaAs
Quantum Dots
2.3.1 Molecular Beam Epitaxy
In order to realize charge carrier confinement in all three dimensions, a material
with smaller band gap is embedded into a surrounding matrix of a higher band
gap material, which is in our case GaAs (E
GaAs
=
1.51 eV at T = 15 K). As
quantum dot material we chose In
0.5
Ga
0.5
As (E
In
0.5
Ga
0.5
As
=
0.85 eV at T = 2 K
[
Goe83
]). In the following, the heteroepitaxial growth of this material system is
discussed.
The self-assembled InGaAs quantum dots used for this thesis were grown in
a molecular beam epitaxy (MBE) machine. The formation of InGaAs quantum
dots using MBE was first demonstrated in 1985 by Goldstein et al. [
Gol85
].
16
2.3 MBE Growth of Self-Assembled InGaAs Quantum Dots
motor
window
transfer system
RHEED screen
shutter
effusion cells
cryopanel (liquid N )
2
RHEED electron gun
main shutter
ion gauge
valve
substrate holder
Figure 2.3: Schematic of an MBE chamber. Different materials can be grown on the
sample which is mounted on a rotatable sample holder. Atomic mono-
layer accuracy is achieved by monitoring the growth via reflection high-
energy electron diffraction (RHEED).
Nowadays, it is a well-established method for a large variety of purposes and
has become the most important technique for the growth of low-dimensional
nanostructures like quantum wells, quantum wires and quantum dots with IIIV
semiconductor materials.
Therefore, the growth of self-assembled quantum dots in an MBE machine is
discussed in some more detail in the following. Figure
2.3
shows a sketch of the
vacuum chamber of such an MBE machine.
In order to achieve high purity single layer growth, it is necessary to pump
this chamber down to ultra high vacuum (< 10
-10
mbar). A mono-crystalline
substrate is mounted on a rotatable sample holder and heated up to 590
C, so
that atoms can diffuse on the growth surface to arrange in monolayers. The layer
thickness is controlled via reflection high-energy electron diffraction (RHEED),
where a focused electron beam is sent to the growth surface and the diffracted
beam is monitored in a fluorescent window. Here, a light detector is mounted,
which transfers the intensity signal to a computer screen. Sinusodial intensity
variations occur because of the different diffraction on the surface at half and
fully filled monolayers. By counting these sinus periods it is possible to deter-
17
Semiconductor Quantum Dots: Artificial Atoms
mine a growth rate and to adjust the growth thickness in the range to single
atom layer precision. Shutters enable rapid switching between different growth
materials, which are arranged in effusion cells around the growth chamber. For
the growth of the structures used in this work, cells for gallium, indium and
aluminum were needed.
Self-assembled growth of quantum dots occurs due to strain effects caused
by the lattice mismatch of the two materials, which are in our case GaAs and
In
x
Ga
(1-x)
As, an alloy which has an indium content x and a gallium content (1-
x
). The lattice constants for the two basic materials GaAs and InAs are at room
temperature (300 K) a
GaAs
=
5.653 Å and a
InAs
=
6.058 Å, respectively [
Sin93
].
Depending on the indium content x, the lattice constant of the In
x
Ga
(1-x)
As
alloy obeys Vegard's law [
Sin93
]:
a
In
x
Ga
(1-x)
As
= x · a
InAs
+ (1 - x) a
GaAs
(2.2)
For the used indium content of x = 0.5, this leads to a lattice constant of
a
In
0.5
Ga
0.5
As
=
5.856 Å which corresponds to a GaAs/InGaAs lattice mismatch
of a/a = 3.5 %.
Besides the deposition rate and the temperature, the lattice mismatch value
is the crucial factor for the type of arrangement of the second layer on top of
the first one. Three different modes of heteroepitaxy can be distinguished, as
shown in Figure
2.4
(a) on the facing page: Frank-van-der-Merwe (the second
material grows layer by layer on top of the first material without the formation
of islands), Stranski-Krastanov (after a two-dimensional wetting layer, quantum
dot growth occurs) and Vollmer-Weber (islands form directly on top of the first
material).
2.3.2 Stranski-Krastanov Growth Mode
The growth mode occurring in our case (In
0.5
Ga
0.5
As/GaAs quantum dots with
a chosen rate and temperature), is the Stranski-Krastanov mode as shown
schematically in Figure
2.4
(b) [
Str39
]. First a few monolayers of In
0.5
Ga
0.5
As
grow on top of the GaAs, forming the two-dimensional wetting layer (c. f. Fig-
ure
2.4
(ii)). This layer will later be important when characterizing the optical
18
2.3 MBE Growth of Self-Assembled InGaAs Quantum Dots
(i)
(ii)
(iii)
(iv)
GaAs
InGaAs
InGaAs
quantum dot
GaAs
capping layer
GaAs
substrate
2D InGaAs
wetting layer
Frank-van-der-Merwe
Stranski-Krastanov
(a)
(b)
z
V o l l m e r - W e b e r
Figure 2.4: (a) Different growth modes of heteroepitaxy. (b) Growth steps of the
Stranski-Krastanov mode showing the formation of a self-assembled
quantum dot due to strain effects (after Reference [
Kan09a
]).
properties, since electronhole pairs can recombine within and it can also be
used for quasi-resonant excitation of the quantum dots. After a critical thick-
ness d
crit
, the strain energy E
strain
reaches a critical value. Passed this energy,
the system releases some stress by the formation of islands (quantum dots) on
top of the wetting layer (c. f. Figure
2.4
(iii)). For complete three-dimensional
confinement, the dot is capped with GaAs(c. f. Figure
2.4
(iv)).
Figure
2.5 on the next page
shows in panel (a) a transmission electron micro-
scope
(TEM) picture of a single self-assembled InGaAs quantum dot embedded
in a GaAs matrix [
Kre06
]. The dot is clearly lens-shaped with a lateral diameter
of 30 nm and a height of 8 nm. In Figure
2.5
(b) we show an atomic force
microscope
(AFM) image of a 1 µm × 1 µm piece of a low dot density sample
( 10 QDs/µm
2
).
From TEM and AFM measurements we know that quantum dots with lat-
eral diameters of 30 40 nm and heights of 4 8 nm can be expected in
our samples. Between these values the size of the quantum dots is randomly
distributed, which is visible in AFM pictures like Figure
2.5
(b).
19
Semiconductor Quantum Dots: Artificial Atoms
10 nm
z
(a)
(b)
y
x
500n
m
Figure 2.5: (a) Transmission electron microscope image of a single self-assembled
In
0.5
Ga
0.5
As quantum dot grown in the Stranski-Krastanov mode
[
Kre06
]. (b) Atomic force microscope image of a low dot density sam-
ple with uncapped quantum dots on the surface.
2.3.3 Sample Design
The above explained Stranski-Krastanov growth of self-assembled quantum dots,
is part of a bigger MBE fabrication process, involving other important growth
steps. An overview of these layers is given in Figure
2.6 on the facing page
.
An intrinsic (100) monocrystalline GaAs wafer forms the basis for all further
growth layers.
First, a 300 nm thick GaAs buffer is grown on top of this substrate, guarantee-
ing a smooth surface with as few defects as possible. The next grown layer is an
AlAs/GaAs superlattice, consisting of 25 periods of 2.5 nm thick layers of AlAs
and GaAs each. This superlattice avoids carrier diffusion into the substrate
and thus increases the optical output of the sample. Finally, the 197.5 nm thick
active GaAs region is deposited. In its middle (100 nm below the surface) a
single layer of self-assembled In
0.5
Ga
0.5
As quantum dots is grown, as discussed
in the previous section. For the growth of the quantum dot layer unrotated
growth
is used during the epitaxy of the InGaAs quantum dot material (c. f.
Section
2.5.3
).
On samples that are destined for AFM measurements an additional uncapped
quantum dot layer is grown on the surface. Due to charging effects, these
samples can not be used for lateral electric field measurements.
20
2.4 Energetic Structure of Single Quantum Dots
i-GaAs substrate
GaAs buffer
300 nm
25 x
2.5nm/2.5nm
97.5 nm
100 nm
GaAs with single
layer of In Ga As QDs
0.5
0.5
AlAs/GaAs superlattice
z
Figure 2.6: Quantum dot sample design as it was used for the lateral field samples
in this thesis.
2.4 Energetic Structure of Single Quantum Dots
As mentioned in Section
2.2
, quantum dots provide the possibility of confining
charge carriers in all three dimensions resulting in the formation of discrete
energy levels. In this section we discuss this energy spectrum and its origin in
more detail.
Since the self-assembled quantum dots used in this thesis form a lens-shaped
geometry (c. f. Figure
2.5 on the preceding page
), different potential approxi-
mations have to be used for the in-plane (x, y) and out-of-plane (z) directions:
The in-plane energetic structure can be approximated by a weak parabolic po-
tential, while for the out-of-plane approximation a much stronger quantum-
well-like confinement is assumed [
Hoh00
,
Haw99
]. These potentials with their
resulting energy levels are shown schematically in Figure
2.7 on the following
page
[
Bim99
,
Sab04
].
Figure
2.7
(a) shows the strong carrier confinement in the z-direction, includ-
ing the two-dimensional wetting layer resulting from the Stranski-Krastanov
MBE growth (c. f. Section
2.3
). The wetting layer has typically an energy sep-
aration of E
W L
1.40 1.46 eV, depending on its width. The GaAs band gap
energy at 4.2 K is E
GaAs
=
1.515 eV.
21
Semiconductor Quantum Dots: Artificial Atoms
~40-60 meV
~20-40 meV
~1000-1300 meV
E (GaAs)
1515 meV
»
E (GaAs) »
1515 meV
E (WL)
1400-
1460 meV
»
E (QD)
1000-
1380 meV
»
z
x
y
(a)
(b)
s
s
p
p
WL QD
QD
Figure 2.7: (a) Out-of-plane and (b) in-plane energetic band structure of a self-
assembled InGaAs quantum dot.
For the quantum dot potential in the out-of-plane direction, the situation of
a one-dimensional infinite quantum well can be assumed [
Sch07
]:
E
z,l
=
2 2
l
2
2m
L
2
,
l N,
(2.3)
with m
being the effective mass of the electron in the conduction band (the hole
in the valence band, respectively) and L the quantum dot height. Increasing
the integer number l leads to discrete energy levels. In the case of lens-shaped
quantum dots, this height L is small compared to the lateral dimensions. Thus,
the resulting confinement in the out-of-plane direction is much stronger than in
the in-plane direction. Therefore, the energy splitting between the quantized
states in the out-of-plane direction is so big that it is reasonable to just consider
the first state to be populated (l = 1). This leads to a constant energy offset
E
z,1
, which depends strongly on the quantum dot height L. Thus, the quantized
states of smaller quantum dots are at higher energy than the ones of larger
quantum dots.
22
Ende der Leseprobe aus 133 Seiten
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Details
- Arbeit zitieren
- , 2009, Optical and Electrical Properties of Single Self-Assembled Quantum Dots in Lateral Electric Fields, Hamburg, Bedey Media GmbH, https://www.diplom.de/document/227686
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