Fakultät für Naturwissenschaften - Department Physik
COHERENT PROPERTIES OF SINGLE QUANTUM DOT
TRANSITIONS AND SINGLE PHOTON EMISSION
Vom Department Physik der Universität Paderborn zur Erlangung des akademischen
Grades eines Doktors der Naturwissenschaften genehmigte
DISSERTATION
von Dipl. Phys. PATRICK ESTER
Gutachter: Prof. Dr. ARTUR ZRENNER
Zweitgutachter: Prof. Dr. KLAUS LISCHKA
Abgabe der Dissertation: 17. März 2008
Datum der mündlichen Prüfung: 23. April 2008
Contents
1 Preface 3
2 Quantum dot basics 7
2.1 Quantum dot properties ............................. 7
2.2 Growth ....................................... 8
2.3 Energy levels of self-organized quantum dots ................. 9
2.4 Quantum dot occupations and transitions ................... 11
2.5 Quantum dot spectroscopy ........................... 12
3 Theoretical basics 15
3.1 Two level systems ................................. 15
3.1.1 Interaction operator ........................... 15
3.1.2 SCHRÖDINGER-equation ......................... 16
3.1.3 Solution with the rotating-wave-approximation ............. 17
3.1.4 Density matrix .............................. 17
3.1.5 Optical-BLOCH-equation and BLOCH-vector ............. 18
3.1.6 BLOCH-sphere ............................... 19
3.2 Pulsed excitation ................................. 20
3.2.1 RABI oscillations ............................. 20
3.2.2 RABI-solution for the case of exact resonance ............. 21
3.2.3 RABI-solution in real systems ...................... 22
3.2.4 RABI-solution in the case of detuning ................. 22
3.3 Dephasing ..................................... 23
3.4 Double pulse excitation and RAMSEY fringes ................. 25
3.5 Continuous Excitation .............................. 27
3.5.1 Saturation ................................. 27
3.5.2 Power broadening ............................ 29
4 Experimental setup 33
4.1 Excitation ...................................... 35
4.1.1 HeNe-Laser ................................ 35
4.1.2 Ti:Sa-Laser ................................. 35
4.1.3 Optical Setup ............................... 37
4.1.4 Power control ............................... 38
4.1.5 MICHELSON interferometer ....................... 39
4.1.6 Low temperature microscope ...................... 40
4.2 Quantum dot photodiodes ............................ 41
4.2.1 Sample structure ............................. 41
i
Contents
4.2.2 Diode design ............................... 42
4.2.3 Near field shadow masks ........................ 43
4.3 Detection ...................................... 45
4.3.1 Photocurrent technique ......................... 45
4.3.2 Photoluminescence technique ..................... 47
4.3.3 Photoluminescence excitation technique ................ 48
4.3.4 HANBURY-BROWN and TWISS setup .................. 48
5 Characterization of a single quantum dot 51
5.1 Optical properties ................................. 51
5.2 Quantum confined STARK effect ........................ 53
5.3 Photoluminescence excitation spectroscopy .................. 55
6 Single exciton ground state 57
6.1 High resolution absorption spectroscopy ................... 57
6.2 Ground state properties at continuous excitation ............... 59
6.2.1 Fine structure splitting .......................... 59
6.2.2 ZEEMAN splitting ............................. 60
6.2.3 Photocurrent saturation of a single quantum system ......... 62
6.2.4 Power broadening ............................ 64
6.3 Coherent properties ................................ 66
6.3.1 RABI-oscillations ............................. 66
6.3.2 Damping of the RABI-oscillations .................... 69
6.3.3 Detuning .................................. 71
6.4 Quantum interference .............................. 73
6.4.1 Double π/2 pulse excitation ...................... 73
6.4.2 RAMSEY fringes ............................. 77
7 First excited state 81
7.1 Voltage dependent photoluminescence for p-shell excitation ........ 81
7.2 Excitation polarization dependence of the p-shell ............... 83
7.3 p-shell saturation ................................. 85
7.4 p-shell RABI-flopping ............................... 86
7.5 Dephasing of the p-state polarization ...................... 88
8 LO-phonon assisted absorption 91
8.1 Voltage depending photoluminescence ..................... 92
8.2 Polarization analysis of the LO-phonon absorption .............. 93
8.3 Saturation of the LO-phonon assisted transition ............... 93
8.4 Pulsed excitation ................................. 95
8.5 Dephasing of the LO-phonon assisted transition ............... 96
9 Biexciton generation 101
9.1 Photoluminescence excitation of the biexciton ................. 101
9.2 s-shell luminescence for biexciton excitation in the p-shell .......... 102
ii
Contents
10 Single photon emission 105
10.1 First and second order correlation function .................. 105
10.2 Correlation functions of laser, classic light, and quantum-mechanical light 106
10.3 Correlation function of a single photon emitter ................ 107
10.4 Sequential absorption-emission events within one excitation pulse .... 109
10.5 Continuous non-resonant interband excitation ................ 110
10.6 Single photon emission after coherent (p-)state preparation ......... 112
10.7 Single photon emission after GaAs LO-phonon assisted absorption .... 113
10.8 Single photon emission devices ......................... 114
11 Conclusion and outlook 117
Symbols and abbreviations 119
Bibliography 123
List of publications 129
Danksagung 131
iii
Abstract
After the discovery of the self organized generation of quantum dots (QDs), a new field
in the area of semiconductor physics was initiated. Physicists worldwide have been in-
spired by the new possibilities which arise due to this discovery. The creation of self
assembled QDs takes place almost defect free. Thus, these quantum dots exhibit a very
high optical quality. Due to an over-growth with a higher band gap semiconductor, a
confinement potential for excitons can be created. The energy levels of a confined exciton
are quantized, and therefore similar to an atom. However, this quantum system is fixed
in space. Hence, it is fairly easily possible to study the properties of a single quantum
system.
In this work, the properties and the different dephasing mechanisms of single QD tran-
sitions are analyzed. In addition, some applications are presented which arise due to the
properties of the confined exciton. The isolation of a single QD out of the ensemble is
achieved via near field shadow masks, which restricts excitation and QD luminescence
to a single QD. The integration of a QD-layer into a diode structure allows for an analysis
of various dephasing mechanisms of a confined electron hole pair. The single QD is char-
acterized regarding the energy of nearly all possible transitions, e.g. the ground state,
excited states, charged states, multiple occupations, and phonon assisted absorptions. A
very important issue in this content is the voltage dependence of the transition energy
and thereby the ability of tunneling processes of charge carriers in and out of the QD.
The QD-states, which are subject of investigation here, are the single exciton ground
state, the first excited state (p-shell), and the (GaAs-) LO (longitudinal optical) phonon
assisted absorption. By applying a suitable voltage, the resonantly excited ground state
exciton is able to decay by a tunneling process, which reflects the transition energy in
the photocurrent spectra. The p-shell transition decays by a relaxation process into the
ground state, followed by an optical recombination process. The phonon assisted absorp-
tion differs from the p-shell transition. The resonant excitation energy fits to the exciton
ground state energy plus the energy of a GaAs LO phonon. In this case, the single exciton
(ground state) is generated as well as a GaAs LO phonon.
These three states are investigated in different respects, such as different applied volt-
ages, excitation polarizations, excitation intensities, and coherent properties. It is only
possible to treat a QD state as a quantum mechanical two level system if it exhibits spe-
cific coherent and incoherent characteristics, such as occupation saturation (with increas-
ing excitation intensity). In particular, a coherent excitation must be possible. This is
demonstrated for the single exciton ground state by photocurrent experiments and for
the p-shell transition by photoluminescence experiments. The LO-assisted absorption
shows also a saturation behavior, but here coherent state preparation seems not possible
with the available excitation conditions. The comparably long dephasing time of the sin-
gle exciton ground state easily allows for two time separated interactions of laser pulses
1
with the QD. The exciton in the QD is able to interfere with the second laser pulse due
to the storage of the phase information of the first laser pulse. The relative phase of the
QDs exciton can be controlled externally via the bias voltage. This effect is the basis for
the observation of RAMSEY-fringes, which are presented in this work.
The coherent manipulation of the p-shell is the basis for a novel excitation scheme
for single photon emission. In this work it is shown that the first excited state can be
coherently manipulated, similar to the ground state. Due to this defined excitation of
a single exciton in the p-shell, the resulting single photon emission (after the relaxation
and recombination process) appears remarkably clean, as demonstrated by HANBURY-
BROWN and TWISS experiments.
1 Preface
The continuous advancement in the field of semiconductor technology leads to more and
more powerful and smaller devices. The progress in reducing the structure sizes has
been frequently predicted an end. A physical end of the classic design of semiconductor
devices is nowadays predicted due to quantum effects in strongly reduced structure sizes.
The quantum properties of low dimensional semiconductor structures are thereby the
basis for future semiconductor technologies.
The current (computer-) technology is based exclusively on classical physics, where
all currents can be regarded as a continuous charge distribution. The fact that single
charge carriers exhibit wave properties and are furthermore able to perform tunneling
processes through small potential barriers is widely neglected. With further reduction of
the structure sizes, these quantum effects are becoming more and more important. For
the classic design of semiconductor structures, these quantum effects are usually seen
as a handicap [Lun03]. The challenge for future developments is a combination of the
quantum effects and the classic design.
The reduction of the structure sizes in one or more dimensions leads to a quantization
of the exciton states. Every degree of a confinement has its own physics and properties.
A confinement in one dimension is called a quantum well, in two dimensions, quan-
tum wire. A confinement in all three directions of space is called a quantum dot (QD).
The technical realization of these low dimensional systems takes place by structuring-
methods like photo- or electron-beam lithography or by modern epitaxy methods like
molecular beam epitaxy (MBE). The samples used in this work are grown by MBE, which
allow a precise and controlled deposition of semiconductor materials on a substrate with
an atomic resolution in growth direction. A deposition of a semiconductor material on
a semiconductor substrate (different lattice constants) generates strain energy in the co-
herently strained wetting layer. At a critical layer thickness, the strain is removed by the
formation of coherently strained islands. Due to the island overgrowth, the confinement
of the QD arises. The generation of these QDs takes place almost defect free, which is
very important for optical applications. The QDs samples can be annealed in order to
shift their transition energy via diffusion processes.
There are various improvements in device-properties possible, comparing low dimen-
sional systems to a bulk semiconductor device. Due to reduction of the dimensions, the
overlap of electron- and hole-wave function is strongly enhanced, which is a big advan-
tage for optical applications. In the case of one dimensional confinement, the quantum
well structures developed fast to home-used devices. Today, quantum well lasers can be
found in units like CD-players or laser printers. Furthermore low dimensional systems
are very interesting for fundamental research. The quantization in all three directions of
space leads to discrete energy levels, similar to an atom. Hence it is possible to investigate
the properties of an atomic-like single quantum system.
3
1 Preface
These properties are analyzed by means of a coupling or an interaction of the QD
with light, free charge carriers, phonons, electric and magnetic fields, etc. The achieve-
ments of the research in this field are versatile. A QD can be integrated in a diode struc-
ture to analyze tunnel effects, the STARK-effect, and occupation properties. Furthermore
they can be used for optoelectronic devices such as single photon emitters, which is
viable for quantum cryptography. Furthermore, it has been tried to build lasers with
QDs [KLG+94, PY99, LLU+00]. Another interesting feature in the field of QD-devices
is the possibility of a coherent state manipulation. Here, the energy quantization is as
important as the phase of the quantum system. QDs also provide the possibility of an
implementation in quantum gate devices. The utilization of the coherent properties of
such low dimensional systems for quantum information technology is of course subject
of intensive investigation.
A very interesting item in the field of quantum information processing is quantum
cryptography. With this technique, a completely safe transfer of information can be guar-
anteed. The problem of safe communication is reduced to a safe transfer of the encoding
key. In this case, the data communication is done via single photons, where the infor-
mation is stored in the polarization. The main point here is that it is not possible to ex-
actly copy single quantum states, which is expressed in the no-cloning-theorem. An eaves-
dropping attack can thereby be identified by strongly enhanced error rates. Meanwhile
commercial devices for quantum cryptography are available. Usually the single photons
for the information transfer are generated by an attenuated laser. The data transmission
could be improved by using quantum emitters, which are principally not able to emit two
photons at the same time. One proposal for such a single photon emitter is presented in
this work.
Another very interesting issue in this context is quantum computation, which is based
on quantum mechanical effects like entanglement and superposition. For a wide class
of problems, the efficiency in solving problems can be significantly improved compared
to classical information technology. For classic data information processing, a bit can
take two values: zero and one. In the case of a quantum mechanical two level system,
a quantum bit (qubit) can take both states in a coherent superposition. These qubits are
the basic building blocks of a quantum computer. There is an advantage for the quantum
algorithms compared to the classical computation algorithm due to the parallel process-
ing of the coherent state superposition. The theoretical concept is quite promising, but
the technical realization of a quantum computer is extremely complex and challenging.
First success was reported in the year 2001, when the SHOR-algorithm was demonstrated
in a nuclear magnetic resonance based quantum system with seven qubits [VSB+01]. An
expansion of this approximation up to hundred or thousand qubits appears to be hardly
possible. A very good overview about the quantum information processing can be found
in [ABH+01] and [BEZ00].
In this thesis the properties of a single quantum dot are analyzed, regarding the sin-
gle exciton ground state transition, the p-shell transition, and the LO-assisted absorp-
tion. The different quantum dot states exhibit different dephasing mechanisms and time
scales, whereby only the ground state dephasing can be effected by an external parame-
ter. In the case of controllable dephasing, the QD system can be used as a very sensitive
sensor. Here one would like to have a very long dephasing time. The dephasing of the
4
excited state is mainly determined by the relaxation process into the ground state. This
process cannot be controlled by an external parameter. However, for an application such
as a single photon emitter, a fast relaxation process after excitation is preferable.
5
1 Preface
This work is divided into the following chapters:
Chapter two introduces the self organized STRANSKI-KRASTANOW quantum dots. The
basic properties of these quantum dots are discussed, regarding growth methods, ensem-
ble measurements, and single dot isolation.
Chapter three explains the theoretical concept of a quantum mechanical two-level sys-
tem, including the interaction operator, the SCHRÖDINGER equation, the density matrix,
and the optical BLOCH equations. Afterwards, the coherent and incoherent properties,
and the dephasing time constants are discussed. Furthermore, the theoretical basis for
the RAMSEY-fringes is presented.
Chapter four discusses the experimental realization. At the beginning, the excitation
setup is presented. The sample design is a very important item for the photocurrent
spectroscopy. After the sample design, the two different detection techniques for pho-
tocurrent, photoluminescence, and photon correlations are discussed.
Chapter five shows the sample characterization. The spectral position of the ground
state and the excited states are analyzed. The state identification is done by voltage de-
pendent photocurrent and photoluminescence experiments.
Chapter six is dealing with the single exciton ground state. The advantage of the ab-
sorption technique is discussed, followed by the analysis of fine structure splitting of
the ground state. This fine structure splitting is analyzed by polarization dependent ex-
citation and magnetic field measurements. Then the incoherent experiments of power
broadening and saturation are shown. Afterwards, coherent measurements like RABI-
oscillations, dephasing, and RAMSEY-fringes are discussed.
Chapter seven describes the properties of the first excited state. After the resonant ex-
citation process of the p-shell, the exciton relaxes into the ground state and subsequently
decays by optical recombination. The p-shell transition is analyzed regarding the inco-
herent and coherent properties. Finally, the dephasing of the p-shell transition is shown
by double pulse experiments.
Chapter eight is similar to chapter seven, but here the behavior of the LO-phonon as-
sisted absorption state is discussed and compared to the normal p-state transition. With
the excitation of LO-phonon assisted absorption it is also possible to generate the biexci-
ton state. This is shown in a photoluminescence excitation experiment in chapter nine.
Chapter ten shows how the coherent manipulation of the p-state can be used for a very
clean single photon emission. Furthermore the LO-assisted absorptions can also be used
for single photon emission. Both excitation schemes are analyzed in a HANBURY-BROWN
and TWISS experiment.
Chapter eleven gives a summarize of this thesis.
6
2 Quantum dot basics
Not only a single atom can have quantum properties, also a cluster of many more than
one atom (> 104atoms) is able to show quantum effects. Due to the semiconductor band
diagram, it is possible to design a confinement potential for charge carriers with two
different materials. Thereby the complete field of quantum physics in semiconductors
develops. Semiconductor structures, which provide a three dimensional confinement,
are called quantum dots (QDs). The one dimensional confinement is called quantum
well, and a confinement in two dimensions is called quantum wire. In order to define the
kind of confinement, two different (semiconductor-) materials must be combined, which
exhibit different band gaps.
2.1 Quantum dot properties
Every three dimensional confinement potential for charge carriers, which is small enough
to exhibit quantum effects, can be called quantum dot. One kind of QD is a two dimen-
sional electron gas with suitable gate structures (e.g. see reference [HvBV+05]). Another
kind of QD is a fluctuation in a quantum well structure. Here the excitons are located
in regions of the quantum well, which are coincidentally thicker than their environment.
This leads to QDs with a relatively large lateral expansion and weak confinement poten-
tial. The weak confinement potential leads to a large exciton diameter and a low overlap
of the wave functions. Therefore, excitons with a low diameter are more suitable for
optical applications. But the most common system is based on self organized semicon-
ductor QDs. Figure 2.1 c) shows AFM picture of a layer of self organized QDs. These
QDs exhibit a very high optical quality, whereby a spectroscopic analysis becomes pos-
sible. Figure 2.1 a) shows a schematic sketch of the lateral confinement of an exciton in
such a QD. These semiconductors QDs could be easily integrated into a diode structure
and therefore it is possible to build a device. A defined positioning of a single QD would
be a great step for fundamental physics and for all possible applications. Of course this
is subject of intensive research. First successes were achieved by the groups of REUTER
and SCHMIDT (reference [MRM+07] and [KRS05]). They recently processed areas on the
substrate where the QD arrangement is controlled. The next step for the single QD spec-
troscopy is the integration of these QDs into a diode structure where the active region is
reduced to a single QD. The single dot spectroscopy in this work uses samples where QD
isolation is done via shadow masks.
7
2 Quantum dot basics
Figure 2.1: Quantum dot sample: a) schematical illustration of an confined exciton in a
QD b) TEM-picture c) AFM-surface picture with quantum dots
2.2 Growth
For the fabrication of self organized semiconductor QDs, materials with different lattice
constants are needed. The island material must have a larger lattice constant than the
substrate material. For a three dimensional confinement for charge carriers, the island
material should have a smaller band gap than the substrate material. The most common
material system for self organized QD growth is InAs and/or InGaAs.
A key technique for the epitaxial growth of semiconductor QDs is the molecular beam
epitaxy (MBE). The self organized island formation takes place in the STRANSKI-KRA-
STANOW growth mode, which can be characterized in the following way: During the
deposition of (for example) InAs on GaAs, the growth starts in two dimensions (wetting
layer). Thereby the strain energy increases during the deposition of InAs. With this
kind of growth, the epitaxially deposited material has the lateral lattice constant of the
substrate. With MBE, it is possible to add material in growth direction with an atomic
resolution. Further deposition generates more strain energy in the coherently strained
wetting layer. At a critical layer thickness, the transition from layer growth to three-
dimensional island growth takes place. At this layer thickness, the strain is removed by
the formation of coherently strained InAs islands, as shown in figure 2.1 c). An increase
of the surface energy is connected to the island formation. The total energy of the system
remains smaller due to the strongly reduced strain energy in the islands, compared to
a further two-dimensional growth, see reference [SK]. The island formation results in
an almost defect-free island growth and a high optical quality of the QDs (see reference
[EC90]). Due to the island overgrowth, the confinement of the QD arises, as illustrated in
figure 2.1 a). The smaller energy gap of InAs compared to the GaAs substrate generates
a trap for free charge carriers. Due to the (small) size of the QD, the energy levels of a
trapped charge carrier are quantized.
After the growth process, it is possible to tune the QD transition energy by thermal an-
nealing. Due to the thermal energy, Ga atoms from the substrate are able to diffuse into
the QD and replace In atoms. After a while, the QD consists of an enlarged In(1−x)GaxAs
island in a GaAs environment. After this diffusion process, the transition energy of a con-
8
2.3 Energy levels of self-organized quantum dots
Figure 2.2: Comparison of the density of states of a bulk semiconductor and low dimen-
sional systems. Due to the decrease of the dimensions, the density of states
changes to an atomic-like behavior for a three dimensional confinement.
fined exciton is determined by the ratio of In to Ga in the QD and the geometry (shape and
size) of the QD. It is also possible to grow InGaAs QD directly with the MBE-equipment,
but the thermally induced diffusion allows post growth tuning the transition energy to
the desired spectral range.
2.3 Energy levels of self-organized quantum dots
Figure 2.2 shows the change of the density of states from the bulk semiconductor to the
QD. Due to the reduction in the degree of freedom, the density of states results in δ-
functions (similar to an atom) for a zero dimensional system (QD). In the bulk GaAs
semiconductor, the electron hole interaction leads to the formation of the exciton and
to the renormalization energy of about 4.2 meV compared to the band gap energy. The
three-dimensional confinement of electron and hole in a QD leads to a decrease of the ex-
citon diameter as compared to the three dimensional case. The exciton binding energy is
increased compared to the bulk semiconductor. For example, excitons in InGaAs/GaAs
QDs exhibit a characteristic COULOMB energy of approximately 25 meV, as reported by
WOJS et al. [WHFJ96]. For the computation of the energy levels of a QD, which is occu-
pied with several charge carriers, many particle interactions must be considered. The in-
fluence of many particle effects on the energy levels of a QD results in different renormal-
ization energies for different charge configurations. In the examined sample, the renor-
malization energy for a simply negatively charged exciton X−amounts to 5 meV, and the
biexciton binding energy amounts here to 2.7 meV.
9
2 Quantum dot basics
An important fundamental characteristic of the semiconductors GaAs and InAs is that
the conduction band minimum as well as the valence band maximum are located in the
Γ- point of the BRILLOUIN zone. That means that both material systems are direct semi-
conductors. In an atom, the parity of wave functions of the staring state and the ending
state must be different for an optical transition. Here the envelope functions of the con-
duction band and the valence band have the same parity. Due to the BLOCH function,
the optical transition is allowed for the lowest energy levels. The conduction band is s-
shaped, and the valence band behaves p-shaped. Exactly this feature makes the In(Ga)As
QDs very interesting for opto-electronics.
For an accurate quantum-mechanical simulation of the electronic structure of a QD, its
geometry and potential must be known. The microscopic characteristics of an individ-
ual QD can only be determined experimentally. That’s why an accurate calculation of
the energy levels of a specific QD is almost impossible. Experimentally, the QD ground
state energy level can be determined fairly easy by using photoluminescence (PL) spec-
troscopy (see chapter 4.3.2). For the description of the QDs energy levels, one starts with a
separation of electron and hole, and simulate single particle states. The optical selection
rules result in allowed optical transitions (observed in the experiment). The transition
energy will be corrected afterwards, considering the COULOMB-interaction. The single
particle states for electrons and holes in a QD can be simulated with different potentials
and QD-geometries. Many of the models lead to complicated energy spectra. Examples
for models are cylinder, cone [VFSB01], ball [OAM99] or lens shaped [WHFJ96] quantum
boxes of constant potential [DGE+98] up to pyramids with certain crystal faces, consid-
ering the piezo-potentials [SGB99], arising due to the strain.
A fairly simple assumption provides very useful results. The basis of this model is
a two dimensional harmonic oscillator. Detailed computations show that the parabolic
lateral confinement leads to a QD energy scheme similar to a lens-shaped QD. Therefore
this model is a very good approximation [WHFJ96]. In this model, the confinement of
the charge carriers in growth direction can be neglected. The QD can be regarded as a
quasi-two-dimensional system. The strong quantum-mechanical confinement in growth
direction permits the negligence of excited QD levels in this direction. Contrary to the
conduction band, the valence band is degenerated due to its p-character. This results
in heavy and light holes. But only the heavy holes are relevant for the lowest energy
levels of a QD. Due to the restriction to two dimensions, all QD levels consist of a com-
bination of the vertical initial level and the lateral EIGEN-states of the harmonic oscil-
lator. For the description of the lateral confinement, a radial symmetric parabolic po-
tential V=1/2m∗
e/hω2
e/hr2
e/h[JHW98] is used. The effective masses are labeled with
m∗
e/h, and the lateral positions with r2
e/hfor electron (e) and hole (h). The parameter ω2
e/h
describes the strength of the confinement for electron and hole. The energy levels for
electrons and holes correspond to the two-dimensional harmonic oscillator with the two
quantum numbers nand m. The energy EIGEN-values of electron and hole are given by
En,m=¯hωe/h(n+m+1). The two quantum numbers are used for the index of the energy
levels, whereby, in analogy to the assignment of the electron shells in atomic physics, the
assignment l=s,p,d,... for m−n=0,1,... is introduced. The combined main quan-
tum number Nresults from the sum of the quantum numbers (N=n+m) and defines
10
2.4 Quantum dot occupations and transitions
(a) Exciton (X) (b) Biexciton (2X) (c) Exciton in the p-shell
(Xp)(d) Single charged exci-
ton (X−)
Figure 2.3: Schematic illustration of different QD occupations
the degeneracy of the energy levels to 2(N+1). The factor 2 results from the spin de-
generacy. The energy levels En,mare equidistant for electrons and holes, separated by
¯hωeand ¯hωh. Deviations from the cylinder symmetry would lead to different behavior.
With an asymmetric potential, for example, optical transitions between levels with dif-
ferent quantum numbers can occur or the degeneracy of the energy levels can be partly
removed, as reported by TSIPER et. al. in [TWM+97].
2.4 Quantum dot occupations and transitions
In QD optics, the most basic occupancy is an uncharged exciton. This is illustrated in
figure 2.3 a) by an electron in the conduction band and a hole in the valence band (X).
With the confinement of a further charge carrier in the QD, the exciton is singly charged
(X−), as shown in Figure 2.3 d). Due to the interaction of the additional electron with
the existing electron-hole pair, the transition energy shifts. The second electron can take
the second s-state in the QD, with opposite spin. For a further occupation of the QD, the
exciton is either more highly charged (X−2,X−3,...) or developed to a multi-excitonic state
(2X,3X,...). Figure 2.3 b) shows the QD occupation of the QD with two electrons and two
holes. This is called a biexciton 2X.
In(Ga)As-QDs are attributed to the so-called Type IQDs, which represent a potential
minimum for electrons and holes. Therefore electrons and holes can be described in the
single particle picture. The well-known selection rule Δl=±1 for optical transitions is
already fulfilled for se→shh because of the s-character of the BLOCH-waves in the con-
duction band and the p-character of the valence band. Interband transitions are therefore
only allowed between QD levels having the same angular momentum, which results in
the selection rule of Δm=Δn=0. In addition, the spin must also be considered by all
these transitions. Regarding only the lowest level of a QD, the two possible spin settings
are for the electron Sz=±1/2 and for the heavy hole Jz=±3/2. Jdesignates the total
angular momentum, which results from the coupling of spin and angular momentum.
Altogether one receives the four transitions explained in figure 2.4. An electron in the
valence band has a spin of ±3/2. An excitation of the electron into the conduction band
leads to the spin value of ±1/2. The residual hole in the valence band has the opposite
11
2 Quantum dot basics
(a) m=+1;
allowed (b) m=-1;
allowed (c) m=+2;
forbidden (d) m=-2; forbid-
den
Figure 2.4: Spin-adjustments of possible QD excitations of the lowest level of a single
QD. Only a) and b) (green) are optically allowed. In high magnetic fields, the
forbidden states (red) can become allowed.
spin ∓3/2 (of the electron). The combined spin of an electron and a hole ±3/2 →∓1/2
results in a total spin of ±1, which can be carried by a σ±circular polarized photon,
see figure 2.4. The states of ±3/2 and ±1/2 result in a total spin of ±2 and thereby no
optically transitions are allowed. Therefore these levels are also called dark excitons.
A QD can be resonantly excited as well as non-resonantly. When the QD is non-
resonantly excited (interband), the generated excitons can be confined in the QD after
relaxation process into the lowest unoccupied level. An empty QD can (resonantly) ab-
sorb a photon of the energy E1Xand produce an exciton as shown in figure 2.3 a). An
optical transitions can then take place by a recombination process with the energy E1X.
If the QD is already occupied with an exciton, the QD can not absorb a further photon
of the energy E1X, however one of the energy E2X−E1Xand thus generate the biexciton
(2X) state, see figure 2.3 b). The biexciton state can decay into a photon and a single exci-
ton, which subsequently decays into a photon with the energy E1X. The electrons and the
holes of the biexciton exhibit opposite spin. A nonlinear two photon absorption process
can also produce a biexciton. The two absorbed photons (complementary circular polar-
ized) each having the energy of (E1X+E2X)/2 (see [SME+06]). With high (interband)
excitation intensity, the generation of a third exciton (3X) is possible. Two excitons oc-
cupy the sstate, and the third exciton is in the p-state. In the case of a resonant excitation,
it is possible to directly excite an exciton in the p-shell. This is illustrated in figure 2.3 c).
This state will relax into the ground state, typically on a ps time scale.
2.5 Quantum dot spectroscopy
A commonly used method for characterizing semiconductors and semiconductor QDs is
photoluminescence (PL) spectroscopy. This technique provides access to the optical qual-
ity and electronic characteristics of the QDs. Due to the discrete energy structure of single
QDs, discrete optical transition energies are expected. By measuring the PL, it turns out
that the sharp spectral characteristics of discrete transitions are only visible in the single
QD limit. Due to self organization of the growth, every QD has an individual size and
12
2.5 Quantum dot spectroscopy
Figure 2.5: Comparing the luminescence of a QD ensemble and a single QD (from
[Beh03]). Due to the statistic fluctuations of the self organized growth the
ensemble luminescence is inhomogeneous broadened. The discrete spectra is
only visible by view on a single QD.
therefore an individual transition energy. The distribution of shape and size follows a
statistic process. On a QD ensemble measurement, every QD contributes with its own
EIGEN-energies. This results in an inhomogeneous broadening of the optical transition
energies. Therefore always relatively broad spectral lines are observed for QD ensembles.
The spectral width of the PL-signal can be used as a measure of the homogeneity of the
manufactured QDs. Typically, the FWHM (full width half maximum) of a InGaAs QD
ensemble is approximately 100 meV, which is relatively large as compared to InAs QDs
with typically 20-50 meV. Since the size of the QD is directly connected to the transition
energy of the exciton, the spectral position of the radiative recombination will vary from
QD to QD. Therefore each sample will be characterized using PL spectroscopy. The emis-
sion signal will not exhibit discrete spectra, due to the statistic fluctuation of the exciton
energy. This is only the case after the isolation of an individual QD. Figure 2.5 shows the
difference in the spectrum between a single QD and an ensemble of QDs. The isolation
of an individual QD is achieved by an aluminum shadow mask with holes having a size
of some 100 nm diameter, see chapter 4.2.1. Ideally only one individual QD should be
located under a single hole. Therefore the QD density must be sufficiently low. It is possi-
ble to investigate an individual QD, when excitation and detection are limited to a single
hole. The luminescence of an excited state of a single QD cannot directly be detected due
to the fast relaxation process into the ground state. For example, the direct luminescence
of the p-shell could only be observed, if the QD is occupied with a 3X. The higher en-
ergy levels of a single exciton can be analyzed using the photoluminescence excitation
spectroscopy, discussed in chapter 4.3.3.
13
2 Quantum dot basics
14
3 Theoretical basics
3.1 Two level systems
Two level systems represent simple but nontrivial quantum mechanical systems. They
are the basis of theoretic and experimental research. The interaction between light and
the two level systems is treated in numerous textbooks, e.g. see reference [AE75]. This
chapter gives an overview of the relevant results of the theoretical description of a two
level system. It is irrelevant which material system is taken into account, e.g. an exciton
in a QD, single atomic transitions, single spins,... It is not important (in a first approx-
imation) weather the system consists in fact of more than two levels. Important is the
limitation of the excitation to a single quantum transition. This is the case if the transition
is resonantly excited and the next transition is clearly separated.
In this work, the two level system is represented by the transition from the empty QD
to (i) a single exciton in ground state, (ii) the first excited state and (iii) the LO-phonon
assisted absorption. It is also possible to describe the transition from the empty QD to
the biexciton state 2Xas a two level system, which is shown in reference [SME+06].
3.1.1 Interaction operator
The basis for the interaction of an incident laser is the existence or the change of an elec-
trical dipole moment. Due to the fixed position of the QD, the dependence of an electrical
field from the space coordinate could be neglected.
E(
r,t)=
E(t)(3.1)
This approximation is called optical dipole approximation. The oscillating electrical field of
the laser radiation at the QD is described as:
E(t)=
E0cos(ωLt)(3.2)
The interaction operator can therefore be written as:
V=−
d·
E(t)(3.3)
with the electrical dipole operator
d=−e
r. The polarization vector corresponds to
and
ωLto the frequency. E0is the electrical field amplitude.
15
3 Theoretical basics
3.1.2 SCHRÖDINGER-equation
The aim of this section is the prediction of the time evolution of a two level system under
the influence of light. The time evolution of a state is described by the time dependent
SCHRÖDINGER-equation:
i¯h∂
∂tψ(r,t)=Hψ(r,t)=(HQD +V)ψ(r,t)(3.4)
The basis for the description of the interaction between light and the QD is the HAMIL-
TON-operator. The HAMILTON-operator is composed of the HAMILTON operator of the
undisturbed state HQD and the interaction part V. The approximation for the wave func-
tion is generally:
ψ(r,t)=c1(t)e−iω1tu1(
r)+c2(t)e−iω2tu2(
r)(3.5)
ψ(r,t)=c1(t)|1+c2(t)|2(3.6)
Here c1(t)and c2(t)correspond to the time dependent amplitudes of the two stationary
EIGEN-functions u1(
r)and u2(
r). If formula 3.6 is used in formula 3.4, this result in:
∂
∂tψ=˙
c1(t)|1+˙
c2(t)|2+c1(t)(−iω1)|1+c2(t)(−iω2)|2(3.7a)
HQDψ=c1E1|1+c2E2|2(3.7b)
Vψ=−
d
εE0cos(ωLt)(c1|1+c2|2)(3.7c)
The three equations 3.7 are added in 3.4 and multiplied scalar with 2|(and afterward
with 1|) from the left.
i¯h˙
c2+c2¯hω2
=E2
=c2E2−E0cos(ωLt)c1<2|
d|1>
ε
with: 1|2=2|1=0 and 1|1=2|2=1. This results in two differential equations
for the time dependent coefficients:
˙
c1=i
¯hdε
21E0e−iω21tcos(ωLt)c2(3.8a)
˙
c2=i
¯hdε
21E0eiω21tcos(ωLt)c1(3.8b)
where dε
21e−iω21t=2|
d|1
is the projection of the dipole matrix element on the polariza-
tion vector. The dipole matrix element indicates the coupling strength of the transition. If
the dipole matrix element vanishes, the transition is forbidden. The differential equations
describe the time evolution in the ideal case.
16
3.1 Two level systems
3.1.3 Solution with the rotating-wave-approximation
In the following, the dipole matrix dε
21 is written as d(d=dε
21), and the RABI-frequency
is introduced.
Ω0=dE0
¯h(3.9)
Equation 3.8 is rewritten with the EULER-formula to:
˙
c1(t)=iΩ0
2(ei(ωL−ω21)t+e−i(ωL+ω21)t
→0
)c2(t)(3.10a)
˙
c2(t)=iΩ0
2(e−i(ωL−ω21)t+ei(ωL+ω21)t
→0
)c1(t)(3.10b)
Within the approximation of a near resonant light-QD coupling the relation ω21 ≈ωL
becomes valid. The terms with ω21 +ωLoscillate very quickly and do not contribute to
the coupling of the two level system. Therefore they are neglected in the following. This
approximation is called rotating-wave-approximation. The detuning of the laser energy
with respect to the transition is defined as:
δ=ωL−ω21 (3.11)
Therefore, equation 3.10 can be written as:
˙
c1(t)=iΩ0
2eiδtc2(t)(3.12a)
˙
c2(t)=iΩ0
2e−iδtc1(t)(3.12b)
The time dependent constants directly correspond to the occupation of the system. The
solution of the time evolution is shown later in this chapter.
3.1.4 Density matrix
Up to now, the effects of finite lifetime on the time evolution of a two level system have
not been included. The decay of the excited state into the ground state under the emission
of a photon is called spontaneous emission. Such spontaneous emission processes cannot
correctly be described by the SCHROEDINGER equation, because this equation can only
characterize pure states. But it is experimentally not guaranteed that only pure states are
prepared, and no emission processes lead to a superposition of two states. Therefore the
density matrix is very important. It is possible to describe coherent as well as incoherent
superposition of states with the density matrix. The density matrix is defined:
ρ=ρ11 ρ12
ρ21 ρ22 =c1c∗
1c1c∗
2
c2c∗
1c2c∗
2(3.13)
17
3 Theoretical basics
The non diagonal elements are called the coherences of the system. They only result for
coherent state superpositions. If the system is, with the statistical probability pk, in the
quantum state ψk, then the density operator is
ˆ
ρ=∑pk|ψkψk|(3.14)
The corresponding density matrix elements ρij are:
ρij =i|ˆ
ρ|j(3.15)
The matrix elements ρ11 and ρ22 describe the occupation of the states. That means they
give directly the probability to find the system in the state |1or |2. The matrix elements
ρ12 and ρ21 correspond to the coherence. If they vanish, the phase between |1and |2is
totally undefined.
3.1.5 Optical-BLOCH-equation and BLOCH-vector
Important is the time evolution of the density matrix. It is obvious that a two-level system
must be in the second state if it is not in the first state. Therefore it is possible to show:
d
dtρ11 =−d
dtρ22 ⇒ρ11 +ρ22 =constant =1 (3.16)
and
ρ12 =ρ∗
21 (3.17)
Therefore the four components of the density matrix are no longer independent. One can
now define the components of the BLOCH-vector as:
u=2Re(ρ12e−iδt)(3.18a)
v=2Im(ρ12e−iδt)(3.18b)
w=ρ22 −ρ11 (3.18c)
uis the dispersive component of the BLOCH-vector, vis the absorptive component of the
BLOCH-vector and wcorresponds to the inversion.wdirectly describes the occupation
of the system because ρ22 −ρ11 is the probability of the energetic higher state minus the
probability of the energetic lower state. The equations 3.18 are called optical-BLOCH-
equations (OBE), which were developed from FELIX BLOCH (Nobel prize laureate) for
nuclear spin resonance effects. The OBE can be written more simply with the help of the
BLOCH-vector (here in the case without damping):
d
dt ⎛
⎝
u
v
w
⎞
⎠=−⎛
⎝
Ω0
0
δ
⎞
⎠×⎛
⎝
u
v
w
⎞
⎠(3.19)
18
3.1 Two level systems
For the case without damping, it can be shown that the BLOCH-vector always moves on
the surface of a sphere with the radius of one:
u2+v2+w2=1. (3.20)
3.1.6 BLOCH-sphere
The states of a single two-level system can be represented as points on the surface of a
sphere. The BLOCH sphere is in some kind analogous to the probability interval [0,1] for
the classical bit. The lowest and the highest point of the sphere (the poles) correspond
to the states where exact states can be measured. Figure 3.1 shows the BLOCH sphere.
The upper pole corresponds to an occupation of 100% and the lower pole to 0%. These
states are called the pure states (in quantum mechanics); for the case of a spin 1/2 particle
e.g. spin down (spin up). This is valid for all two level systems. The points which
are not on the surface, but inside u2+v2+w2<1, can be interpreted too. There is
no complete information about these states. These states are called the mixed states. The
point in the middle of the sphere corresponds to the state without any information. Every
measurement of a state on the BLOCH sphere corresponds to a projection of the state to
the w-axis. Then the measurement correlates directly to the occupation probability of
the state. There is no access to the phase with a simple measurement of the occupation
probability.
The BLOCH-vector formalism can be applied to the optical resonance of two-level sys-
tems. This approach delivers a picture of the coherent processes in the quantum system,
which is described by the motion of a vector in the three dimensional space. Therefore
the BLOCH-vector is defined by the components which contain the dipole moment and
the inversion, as well as the coherent properties of the state. As mentioned before, the
time evolution is given by the time dependent SCHRÖDINGER-equation. With a practical
approximation1, it is possible to create a set of equations of motion, which are known as
the optical BLOCH equations (OBE) 3.21.
˙
u=−δ·v(3.21a)
˙
v=δ·u+μ·E·w(3.21b)
˙
w=−μ·E·v(3.21c)
Figure 3.1 shows the BLOCH-sphere. The states 1 and 0 correspond to the poles of the
sphere. The component wrepresents the occupation probability of the two level system.
uand vrefer to the components of the complex transition matrix element, where ucor-
respond to the dispersive and vto the absorptive component. vis called the absorptive
component because the occupation probability wcan be changed due to absorption of
electromagnetic radiation, see 3.21c. δis the energy detuning (between the transition fre-
quency ω0and the frequency of the incident light ωL). Ecorresponds to the envelope
1Half classical theory of light matter interaction without quantization of the beam field and the use of the
rotating wave approximation
19
3 Theoretical basics
(a) A laser pulse rotates the BLOCH-vector
with an angle of Θ.(b) Excitation of the two level system with
π-pulse followed by a π/2-pulse with
phase shift of 90◦.
Figure 3.1: BLOCH-vector shown on the BLOCH sphere.
of the electrical field amplitude. Figure 3.1 a) shows the BLOCH vector on the BLOCH
sphere. With the rotating frame approximation, the components uand vdefine the phase
of the system. As shown in equations 3.21a and 3.21b, a detuning δdirectly affects the
components uand vand thus the phase of the system.
The electromagnetic field is E(t)=E(t)·[eiωt+c.c.]. The variable μis directly pro-
portional to the dipole transition matrix d≡¯h·μ/2 of the regarded optical transition.
Analytic solutions of the OBE can only be obtained with further approximations. The
incident electromagnetic field is supposed to be perfectly monochromatic and resonant
to the two level system. With the approximation of a constant electrical field E(t)=E0,
the components u,vand wshow an oscillation behavior. This case is known as the RABI
solution of the optical BLOCH equations. It is called RABI oscillation, analogous to the
first evidence of an occupation oscillation in magnetic resonance phenomena.
3.2 Pulsed excitation
3.2.1 RABI oscillations
As mentioned before, an electromagnetic wave of a laser is able to excite a quantum me-
chanical two level system, with two discrete energy levels (|0>and |1>), if the incident
laser frequency is resonant to the transition frequency. If the system is already excited,
further excitation results in stimulated emission. Absorption and stimulated emission
follow up periodically. The oscillation takes place with the RABI-frequency, which is pro-
portional to the laser field strength. Spontaneous emission leads to a damping of the
RABI-oscillation. The spontaneous emission is not taken into account, up to now. The
damping can be neglected if the laser field is so intense that the RABI frequency is much
20
3.2 Pulsed excitation
higher than the decay rate caused by spontaneous emission. These oscillations were ob-
served for the first time by RABI in the year 1936 in his works on spin systems.
3.2.2 RABI-solution for the case of exact resonance
The simplest solution of the OBE is obtained for the case of resonant excitation. The
detuning δ=ωL−ω0is zero in this case.
˙
u=0 (3.22a)
˙
v=μ·E·w(3.22b)
˙
w=−μ·E·v(3.22c)
The dispersive component uof the BLOCH vector is therefore constant in time. The
solution of equations 3.22b and 3.22c results in oscillations for vand w. From:
¨
w=−μE˙
v
we obtain:
¨
w=−μ2E2w.
With the ansatz w=A·eiωtwe further obtain:
ω=±μE=±Ω0.
with the already introduced RABI-frequency Ω0. As solution one gets:
→w=A(eiΩ0t+e−Ω0t)=2Acos(Ω0t).
with the boundary condition w(t=0)=−1 (occupation of the upper level is zero at the
time zero) Ais determined to:
→A=−1
2.
Using the addition theorem sin2ωt
2=1
2(1−cosωt)the solution results in:
w=2sin2Ω0t
2−1
The frequency of this oscillation corresponds to the RABI-frequency. It is directly propor-
tional to the transition probability and to the electric field strength. For the case of an
exciton two level system, the occupation probability of the |1Xstate is given by
P|1X=w+1
2
which results to:
P|1X=sin2Ω0t
2=sin2Φ
2(3.23)
21
3 Theoretical basics
μdescribes the component of the dipole matrix element parallel to E.Ecorresponds to
the envelope peak amplitude of the electromagnetic field.
Φ=Ω0tis called the pulse area. For a time dependent field E(e.g. pulsed excitation),
this results is given by:
Φ(t)=μt
0E(t)dt. (3.24)
For this case, wperforms a RABI-oscillation with varying RABI-frequency. For the de-
scription of the final state of the two level system after an incident laser pulse with the
length tp, the pulse area is nevertheless given by Φ=Ω0tp. Because the field amplitude
is time dependent at a GAUSS-shaped pulse (E(t)=E0exp(−(t−t0)2/2σ2)), Ω0and tp
are taken as adequate averages. For this case, the relevant relations Φ∝Eand Φ∝tp
are also valid in the integral description. If a two level system is resonantly excited, the
BLOCH-vector rotates around the u-axis with the RABI-frequency Ω0. The rotating angle
of the BLOCH-vector after the pulse is given at the time twith Ω0t. Figure 3.1 a) illustrates
the motion of the BLOCH-vector after the interaction of a laser pulse. The oscillating oc-
cupancy probability is a function of the laser field strength (fixed pulse length). With a
defined laser pulse, it is possible to excite the system with the probability of 1. The pulse
area for a single inversion is called π-pulse. This is shown with the green function in
figure 3.2 at Φ=π.
3.2.3 RABI-solution in real systems
In reality, genuine two-level systems do not exist. There is always a certain interaction
with other states and charge carriers. Even if the transition probability into other states
could be neglected, this could lead to a dephasing of the system. This means that the
coherence with the applied periodical excitation is lost. This leads to a damping of the
RABI-oscillations, depending on the incident pulse area. At high excitation intensities,
the same occupation probability arises for both states (see figure 3.2 (black curve)). On
the BLOCH-sphere this corresponds to w=0. If one takes the damping into account, the
relation for the occupation probability (3.23) must be phenomenologically adapted to:
P|1X=e−γΩ0tsin2Ω0t
2−1
2+1
2(3.25)
γcorresponds to the damping parameter. The modification with the two factors 1
2gives
the same occupancy probability for both states at high excitation intensities. Therefore
the occupation probability function P|1X>will approximate 1/2 at high pulse areas.
3.2.4 RABI-solution in the case of detuning
Due to the fast time evolution of the system, the laser energy must not exactly fit to the
energy difference of the two energy levels (δ=0) 2. The exact energy conservation is only
correct here in the limit for long pule width. A detuning of the resonance in relation to the
2Consequence of the uncertain principle
22
3.3 Dephasing
Figure 3.2: green: RABI-oscillations without dephasing and detuning; black: RABI-
oscillations with damping and without detuning; blue: RABI-oscillations with
dephasing and slight detuning
laser frequency leads, in this case, to a modification of the frequency and the amplitude
of the RABI-oscillation.
P|1X=Ω2
0
Ω2sin2Ωt
2with Ω=Ω2
0+δ2(3.26)
where P|1Xis assumed to be P|1X>(t=0)=0. Ω0=μEis still valid in this case. Ωcor-
responds to a generalized RABI frequency where the detuning δis factored in. Figure 3.2
shows the modifications of the RABI-frequency when a detuning is taken into account.
In principle, the formula 3.26 is only correct for a constant RABI frequency Ω0, e.g. for an
excitation with a rectangular laser pulse in time. Due to the factor of Ω2
0/Ω2, the approx-
imation of Ω0as an effective mean frequency is not generally correct. Additionally, the
approximation of a perfect monochromatic pulsed laser is not realistic for a real experi-
ment, where pulsed excitation with a spectral broadening is applied. A simulation of this
situation, taking both effects into account, is non-trivial and not discussed here.
3.3 Dephasing
In reality, the coherent occupation oscillation of a two level system is disturbed more or
less due to additional interactions with other states or charge carriers or due to a decay
process. This leads to a loss of coherence, which results in a damping of the RABI oscilla-
tion. The OBEs (formulas 3.18) describe the two level system under coherent excitation. If
aπ-pulse excites the two level system to a occupation probability of 1, the system would
23
3 Theoretical basics
stay in this state forever if no dephasing process takes place. Therefore the optical recom-
bination time sets an upper time limit for which coherent manipulations are possible. The
analysis of the dephasing processes of a single quantum system is very interesting and
also required in order to build a complete theoretical model. As a first approximation for
the description of the damping in a quantum system, decay terms with empirical time
constants are inducted into the OBE.
˙
u=−δ·v−u
T2(3.27a)
˙
v=δ·u+μ·E·w−v
T2(3.27b)
˙
w=−μ·E·v−δ
T1(3.27c)
The decay time T1corresponds to the occupation component of the BLOCH sphere w(in-
version) which is called longitudinal lifetime. This describes all inelastic interactions with
the quantum system. A second decay time for the components uand vis introduced
which is called the transversal lifetime T2. In addition to the inversion, an elastic interac-
tion can also lead to loss of coherence. For example, a phonon scattering process can lead
to a loss of coherence but the occupation of the system can be retained. In this case, the
oscillation would exhibit strong damping and a reduced transversal lifetime.
The transversal decay time T2implies a loss of the phase of the induced dipole moment.
Here all processes which can lead to a loss of the phase relation between the incident
electromagnetic field and the exciton dipole moment have to be regarded. A damping of
the RABI oscillation can origin from T1and T2processes as well as from pure-dephasing.
Both decay times are linked by:
1
T2=1
2T1+∑
i
1
T∗
i
(3.28)
The time constant T∗
irefers to all processes which will lead to a loss of the phase coher-
ence but not to a change of the occupation. In the case without pure-dephasing:
∑
i
1
T∗
i≈0. (3.29)
Under this condition the relation 1
T2=1
2T1(3.30)
is valid. The in this work analyzed QD-states are able to decay by different processes,
which exhibit different decay times. The ground state exciton is able to decay by a (bias
voltage dependent) tunneling process (tesc) or by an optical recombination (tr). An excited
exciton state can relax into the lowest energetic level (trel) or also decay by an optical
recombination. Thereby, the occupation life time T1is mainly determined by the fastest
decay process. In the case of the coherent manipulation of the ground state, no relaxation
24
3.4 Double pulse excitation and RAMSEY fringes
processes can take place (trel =∞). The occupation life time is determined by:
1
T1=1
τr
+1
τesc
+1
trel
(3.31)
Where τris constant, τesc =τesc(V)is a function of the applied voltage, and 1/trel is zero.
Then the dephasing of the single exciton ground state is given by:
1
T2=1
21
τr+τesc(V)(3.32)
If the tunneling time τexc reaches values shorter than the optical recombination time τr,
then the dephasing time T2is mainly determined by tunneling processes. The dephasing
time of the QD state gives a lower limit for the homogeneous line width Γhom [Wog97]:
Γhom =2¯h
T2(3.33)
For a typical lifetime of T = 300 ps of the exciton in a QD, the linewidth is very small
Γhom =4,4 μeV. Experimentally, the homogeneous linewidth can only be determined if
the detection setup has a higher energy resolution than the transition linewidth. When
the ground state of one QD is resonantly excited, the absorbed and emitted photons have
the same energy. Therefore, it is impossible in experiments with resonant excitation to
separate the luminescence from the scattered light. In this work, the photocurrent tech-
nique is used for high resolution spectroscopy. Here, the QD system is resonantly excited
by a narrow laser; the detection takes place in the photocurrent (tunneling processes).
3.4 Double pulse excitation and RAMSEY fringes
Another possibility for a manipulation of the BLOCH-vector is a double pulse excitation.
Under the condition of the initial state |0, the phase of the system is defined by the first
pulse. If the second pulse is in phase to the first one, the rotation of the BLOCH vector
will continue with the same rotation direction as before. This means a further rotation
around the u-axis, or in the case of a slight detuning, a slight inclined rotation axis. But
if the phase of the second pulse is shifted with the angle of ϕwith respect to the first,
the rotation axis of the second manipulation is also shifted with the angle of ϕin the u-
v-plane. With an excitation of a π-pulse followed by a π/2-pulse with a phase shift of
90◦, one would be observed (a rotation on the BLOCH-sphere from w=−1tow=1)
followed by a rotation from w=1tou=−1, see figure 3.1 b).
Up to now, a detuning of the system has not been considered. The case of a small de-
tuning in a double pulse experiment is discussed in the following. Thereby the detuning
is much less than the spectral width of the excitation pulses: δ/τpulse 1. The detuning
is so small that the pulses can be regarded as almost resonant to the transition, which
means the first manipulation basically takes place around the u-axis. The first pulse de-
fines again the phase of the two level system. The BLOCH-vector rotates in the rotating-
frame-approximation around the waxis with the angular velocity ˙
ϕ. At the arrival of the
25
3 Theoretical basics
Figure 3.3: Origin of the RAMSEY-fringes. Comparing the cases of constructive (a) and
destructive (b) interference.
second pulse (with the time τdelay) the BLOCH-vector is turned in the u-v-layer with the
angle: ϕ=˙
ϕ·τdelay.
It is now considered that both pulses have the same pulse area of π/2 and the same
phase. The first pulse rotates the BLOCH-vector from w=−1tov=−1 (counterclock-
wise around the u-axis). For a detuning condition which results in the same phase at
the arrival of the second pulse as for the first, ˙
ϕ=2πn/τdelay, the rotation will be con-
tinued to the point w=1, see figure 3.3 a). For a detuning of ˙
ϕ=(2n+1)π/τdelay,
the BLOCH-vector is located at v=1 at the arrival of the second pulse. Therefore the
second manipulation will transfer the BLOCH-vector into the initial state. For the cases
˙
ϕ=(n+1/2)π/τdelay, the BLOCH-vector is located at the point u=±1. At this point
on the BLOCH sphere, no further manipulation can take place. The occupation probabil-
ity after the double pulse manipulation oscillates therefore depending on ˙
ϕand the time
26
3.5 Continuous Excitation
delay between the two pulses τdelay:
P|1≈cos2(˙
ϕτdelay/2)(3.34)
These oscillations are known as RAMSEY-interferences [Ram90]. The main advantage
of the double pulse setup is the high sensitivity of the system on a small detuning δin
particular at long time delays. It is thereby irrelevant whether the detuning is done by
changing the excitation energy or by variation of the transition energy of the two level
system. In this work, the possibility to tune the transition energy with the quantum
confined STARK effect (QCSE) is used. As a matter of fact, the time delayed interference
in this double pulse interference becomes possible, because the phase information of the
first pulse is stored within the dephasing time in the quantum system.
Figure 3.3 illustrates the origin of the RAMSEY fringes using the picture of the BLOCH-
sphere. Here the two extreme cases of constructive and destructive interference are dis-
played. In both cases, the first laser pulse excites the QD with a π/2 pulse. This inter-
action rotates the BLOCH vector to the equator. The difference between the two cases is
a slight detuning, which is done in the used material system by an additional applied
voltage ΔV. In the case of constructive interference (a), the BLOCH vector arrives at the
initial point on the equator (v=−1) at the time when the second interaction takes place.
Then a constructive interference is possible, which leads to a fully excited QD state. In the
other case (destructive interference (b)), a slight different voltage (respectively detuning)
is applied to the QD. The phase evolution results at the end of the time delay in the op-
posite state (v=1) as before. The second π/2 interaction transfers this state back to zero.
With a continuous variation of the detuning, the final state oscillates between the states
|0and one |1. As mentioned before, the tunneling process corresponds to a projection
of the BLOCH vector on the w-axis. Thereby a detection of the final state is possible in the
PC experiment, where the state |1corresponds to a current of 12.8 pA.
3.5 Continuous Excitation
3.5.1 Saturation
In the case of continuous excitation, the statistical occupation of a two level can be de-
scribed using rate equations. There are four different transitions possible, where each can
be specified with different coefficients (called EINSTEIN-coefficients). The normal absorp-
tion is written with the EINSTEIN-coefficient B01, the stimulated emission by B10 and the
spontaneous emission with A10. The indices refer to the initial and the final states. The
initial (lower) state is called "0" and the final (upper) state "1". In the case of photocurrent
measurements, a fourth transition can take place. The tunneling can be described by the
tunneling rate proportional to 1/τtunnel. In the case of p-shell excitation, the decay of the
excited state takes place by a relaxation and a recombination process. The decay rate here
is proportional to 1/(τrel +τr). In order to describe both cases, τdecay is introduced, which
represents τtunnel in the first case or τrelax +τrecom in the second case. For the transition
27
3 Theoretical basics
024681012
0.0
0.1
0.2
0.3
0.4
0.5
~
Occupation of N
1
Excitation power P
Figure 3.4: Occupation probability of the upper level depending on the normalized exci-
tation intensity
P. For high excitation intensities the occupation approximates
the value 1/2.
from the upper level into the lower level one obtains the following rate equation:
dN1
dt =−A10N1+B10N1ρ+B01N2ρ+N1
τdecay
(3.35)
where ρrefers to the energy density to the incident field. In the two level system the
following relations also apply: N0+N1=1 and for the EINSTEIN-coefficients B01 =B10.
In the stationary case considered here, also the following relation holds:
dN1
dt =dN2
dt =0. (3.36)
Only the absorption and the stimulated emission depend on the intensity of the incident
laser field. In the limit of high excitation intensities, this results in the same occupation
probabilities for both states. Therefore one gets for the occupation probability of the level
N1, depending on the excitation intensity:
N1=1
2
P
P+1(3.37)
where
Pcorresponds to a normalized (dimensionless) excitation intensity. This parame-
ter is a basic property of the two level system, which can be also derived from the RABI-
frequency Ω0(see next chapter:
P=Ω2
0T1T2).
28
3.5 Continuous Excitation
3.5.2 Power broadening
As a direct result of the saturation of the transition of the two level system is the so called
power-broadening. This effect describes the power induced linewidth broadening of the
resonance. The derivation of the broadening effect takes place as follows (taken from ref-
erence [AE75] and adapted to an exciton):
It is simple to show that the following equation can be derived from the optical BLOCH
equations, see formula 3.27. If T2is very short, then uand vwill quickly reach the quasi-
steady-state values:
u=−μET2δT2
1+(δT2)2w(3.38)
and
v=μET2δT2
1+(δT2)2w. (3.39)
Under these conditions, the inversion obeys the simple equation:
˙
w=−ξ
Pw
T1−δ
T1(3.40)
where ξis the LORENZIAN factor
ξ=1
1+(δT2)2. (3.41)
and
Pis defined as the dimensionless intensity
P=(μE)2T1T2. The steady state values
(taken by u,v, and w) are of particular interest after all of their temporal oscillations have
been damped out. The steady value for the inversion in the long time limit is:
w=weq
1+ξ
P(3.42)
In absence of any incoherent input of weq is equal to -1. For such a two level system the
steady-state inversion has the line shape:
w=−1+(δT2)2
1+(δT2)2+
P, (3.43)
where δcorresponds to the detuning and
Pto the dimensionless intensity. This expres-
sion shows how the energy stored in the system depends on the intensity of the incident
field. For a given detuning, the steady-state inversion becomes increasingly less negative
as
Pincreases. The maximum occupation (wmax) is then given by:
wmax =−1
1+
P, (3.44)
which is always negative for an incoherent field. In order to derive an analytic expression
29
3 Theoretical basics
-4 -2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
Occupation of N1
Detuning (Γ0)
0246810
1.0
1.5
2.0
2.5
3.0
3.5
~
Linewidth (Γ0)
Excitation power P
Figure 3.5: left: occupation probability N1depending on the detuning between the
laser and the resonance, displayed for different excitation intensities
P=
0,2; 0,5; 1; 2; 5; 10. right: linewidth dependence from
P. The broadening
of the linewidth is a direct result of the power dependent saturation. This
effect is known as power-broadening.
for the width of wdepending on the incident intensity, one has to solve formula 3.43. The
half height of wmax is in this case not 1/2 ×wmax because wmin is -1 and not zero. Thus,
w1/2 is:
w1/2 =1
2wmax −1
2. (3.45)
The equation 3.43 must be solved for the case of formula 3.45.
1
2wmax −1
2=−1+(δT2)2
1+(δT2)2+
P, (3.46)
The detuning δ(at w1/2) corresponds to the linewidth (FWHM) of w, which is then called
Γ(w1/2). The conversion of formula 3.46 leads (after some algebra) to the expression:
Γ(w1/2)= 1
T21+
P=1
T21+T1T2(μE0)2, (3.47)
The width at half height Γ(w1/2)of the profile of wis a function of the incident intensity.
With increasing intensity the linewidth of the profile of wincreases, which is called power
broadening.
Finally, it should be mentioned that wis negative for all values of E0. An incoherent
saturating field can never produce a positive inversion. An important implication of
formula 3.43 for the steady state inversion is that a system near to the resonance is no
pure absorber, since it has been stimulated by E0partially out of their ground state.
30
3.5 Continuous Excitation
Phenomenological one can say that at low excitation intensity the occupation proba-
bility of the upper level increases proportional to the excitation power. The shape of the
linewidth only slightly changes in this region. But for higher excitation intensities the
amplitude of the resonance will not increase proportional to the excitation power, be-
cause the system gets close to its saturation. The middle of the resonance reaches the
saturation value (occupation: 1/2) earlier than the sides by increasing excitation inten-
sity. It is possible that the middle of the resonance has nearly reached the maximum, but
the sides of the resonance are still able to increase. This will result in a broadening of the
absorption line. The power broadened absorption line is again LORENZ- shaped, but with
a larger linewidth of Γ.
Figure 3.5 shows the occupation intensity depending on the detuning between the laser
and the transition of the two level system displayed for different values of
P. The satu-
ration value increases the linewidth of the resonance independent from all dephasing
processes. For all analysis of the measurements, one has to take the limit of the linewidth
for vanishing excitation power in order to use the correct basic linewidth.
31
3 Theoretical basics
32
4 Experimental setup
Figure 4.1: Pattern of the experimental setup.
Figure 4.1 shows the scheme of the experimental setup, beginning at the excitation
parameters, followed by sample parameters, and finally the detection options. There are
different possibilities for laser excitation. A HeNe laser can be used for basic characteriza-
tion (voltage dependent PL at interband excitation see figure 5.1) of the QDs. The tunable
Titan:Sapphire (TiSa) laser can be used for high resolution photocurrent spectroscopy of
the QD ground state or for resonant excitations of higher QD states. If the TiSa laser sys-
tem works in the pulsed mode, the QD state occupation can be coherently excited (and
tested for possible coherent state manipulations). By integrating a MICHELSON interfer-
ometer into the excitation beam, a single pulse can be split into two pulses. By adjusting
the time delay between the two pulses, the dephasing of the QD state can be analyzed.
The intensity of the incident laser beam allows for power dependent investigations. Fur-
thermore, the polarization of the laser beam can be adjusted in continuous wave as well
as in the pulsed mode in order to analyze the absorption properties of the QD-state.
At the sample, several parameters can be adjusted. If the sample is integrated in a
diode structure, an external voltage can be applied. This results in an internal electric
field (additional to the build-in voltage) at the sample. This electric field affects the QD
states via the STARK effect. Similar to the electrical field, the sample can be exposed to
an external magnetic field. The magnetic field leads to a ZEEMANN splitting of the QD
states. Apart from electric and magnetic fields, the temperature of the sample can be
adjusted. The change of temperature is controlled by a heating resistor, attached below
the sample carrier. Increasing temperature will lead to a line broadening and energy shift.
For detection, several methods are available. A simple, reliable and frequently used
experimental method is photoluminescence (PL) spectroscopy. With interband excitation
(HeNe laser - above the GaAs band gap), charge carriers are generated in the region of
the QD. These are able to relax into the QD states. There they can recombine and show
characteristic spectra in the luminescence. Another kind of PL method uses a resonant
excitation of higher QD-states. This is called the photoluminescence excitation (PLE)
spectroscopy. The excitation wavelength is thereby tuned continuously. When the laser
energy fits to a higher state, the QD can be occupied with an exciton. After a relaxation
33
4 Experimental setup
process, the recombination can be detected in the ground state PL.
If the QD is embedded in a diode structure, generated charge carriers can tunnel out
of the QD. This is possible with a suitably applied voltage. In this region, all QD states
which decay by tunneling processes can be investigated. Due to the STARK effect (volt-
age), it is possible to shift the QD state with respect to the laser energy. If the laser energy
has an spectral overlap with the QD state, an exciton can be generated. Then the exciton
can tunnel out of the QD and produce a photocurrent. As a consequence of the STARK
effect, the I-V-characteristics reflects the energy spectra. Therefore, this method is called
photocurrent (PC) spectroscopy. It is also possible to perform this kind of spectroscopy
at excited QD states, which is then called photocurrent excitation (PCE).
Apart from the lifetime, a QD system exhibits different timescales for different prop-
erties, e.g. optical recombination time, lifetime of an excited state, tunneling times, de-
phasing times and so on. Furthermore, there are different timescales of the experimental
setup which have to fit to the QD system, e.g. the laser repetition time, pulse length,
pulse width, and time delay at double pulse excitation. Some of these timescales can be
controlled from outside, whereas others are fixed. The longest time period is the laser
pulse repetition time of 12.5 ns. The fastest process is the dephasing of the LO-phonon
assisted absorption, which is in the range of the laser pulse length (1.7 ps). All processes
in this work are taking place within these four orders of magnitude on the timescale. All
experimental (time-) parameters which are important for this spectroscopy are discussed
in the experimental sections.
This chapter is arranged as shown in figure 4.1. In the first section the excitation setup
is discussed. Then the sample design is explained, followed by the discussion of the
different detection methods.
34
4.1 Excitation
4.1 Excitation
For the first characterization of a QD sample, a HeNe laser is used for photoluminescence
measurements. The HeNe laser can be coupled to an optical fiber, or also directed as a
free beam to the sample. The beam intensity is adjusted by an attenuator cascade, which
is installed before the fiber coupler. For resonant excitation, the HeNe laser is replaced
by the TiSa laser. The upper part of figure 4.2 shows the schematic view of the TiSa laser
system. The TiSa laser is pumped by the Verdi V6. A part of the TiSa laser light is coupled
out by a beam splitter and supplied to an auto-correlator for pulse analysis. Another part
is checked in the spectrometer for the correct excitation wavelength.
4.1.1 HeNe-Laser
For the non-resonant (to a QD state) excitation over the band gap of the bulk semiconduc-
tor (GaAs), a HeNe laser with an emission wavelength of 632,8 nm (1,96 eV) is used. This
laser radiation generates electrons and holes all over the semiconductor bulk. The charge
carriers can then recombine from the GaAs, from the wetting layer and from the QD.
Each of those recombination processes has its own characteristic luminescence, which is
always helpful for setup adjustment. An interference filter is mounted behind the laser,
which lets only the actual laser line pass through.
4.1.2 Ti:Sa-Laser
For resonant investigations of an QD state, a commercial titanium sapphire (TiSa) laser
system is used (type: Mira 900 from
coherent
). Since the titanium sapphire crystal pos-
sesses a very large amplification range, the emission wavelength of such a laser system
is tunable in a range from 700 nm to 980 nm, which covers the area of the InGaAs QD
states. The TiSa can work in two different modes, pulsed and continuous wave. The
laser system used here can be (quasi-) continuously tuned by a birefringence filter (LYOT
filter). In order to control the wavelength also computer based, the originally installed
micrometer for filter adjustment was replaced by a motorized unit. The laser system pro-
vides pulses in the range from 1 to 3 ps. The pulse length is not directly tunable, but
varies slightly with adjustment. The resonator length is optimized for a laser pulse fre-
quency of 80 MHz. The maximum average optical output power is approximately 1 W.
The pumping laser is a diode-pumped, frequency-doubled Nd:YV04-ring-laser (Coher-
ent Verdi V6), which provides a high stability of power output and outstanding beam
quality due to single longitudinal mode emission. The maximum optical power output
is 6.5 W at a fixed wavelength of λ=532 nm. The appropriate adjustment is done auto-
matically with the integrated β-lock system, so that the wavelength of the (modelocked)
laser can be varied within a certain range without interruption.
Pulse analysis
By means of a beam splitter, a part of the laser beam is directed into an auto-correlator
(model "mini" of the company APE) for pulse analysis. In the auto-correlator, the laser
beam is divided into two parts in an interferometer arrangement, whereby one arm of the
35
4 Experimental setup
Figure 4.2: Scheme of the experimental setup. All devices which are controlled by the
computer are signed. See text for a detailed description.
36
4.1 Excitation
pulse shape I(t) ΔtP/ΔtAK ΔtPΔν
rectangular I(t)=1; |t|≤tP/2
I(t)=0; |t|>tP/2 1 1
GAUSS-shaped I(t)=exp(−4ln2·t2
Δt2
P
)0,707 0,441
hyperbolic secant I(t)=sech21,76t
ΔtP0,648 0,315
LORENTZ-shaped I(t)= 1
1+4t2/Δt2
P0,5 0,221
Table 4.1: Autocorrelator product for different pulse shapes. ΔtPcorresponds to the full
width at half maximum (FWHM) of the intensity envelope of the pulse, ΔtAK
and Δνthe FWHM of the autocorrelator function respective the pulse spectrum.
interferometer is periodically varied in distance. This length variation can be converted
over the speed of light into a temporal variation of a partial pulse with respect to the
other. Thus, the time measurement in the picoseconds range is transferred to a substan-
tially simpler linear measurement in the 0,1 mm range. The detection of the interferome-
ter signal is done by a nonlinear crystal, in which a frequency-doubled signal is produced
only at overlapping partial pulses. The resulting signal is a result of a convolution of the
laser pulse with itself: I2ω(τ)∝Iω(t)·Iω(t+τ)dt. This is generally broader than the
original signal and must therefore be multiplied by an appropriate factor to receive the
actual half width of the laser pulse, see table 4.1. Figure 4.3 shows the spectrum of a
laser pulse in the picosecond mode (crosses). The appropriate sech2-fitcurve (line) shows
a very good agreement with the experimental data. In theoretical publications, GAUSS
pulses are frequently assumed. These do not differ substantially from sech2-pulses, but
the sech2-pulses exhibit here a slight better agreement. The sech2shape refers to the in-
tensity distribution. The physically important, but directly not accessible amplitude of
the electrical field vector (sech-envelope) has to be calculated. The pulse form in the time
domain can be obtained directly from the FOURIER transformation of the frequency spec-
trum. A comparison of measured time range with the indicated values in table 4.1 shows
that the used laser pulses are basically FOURIER transform limited. Both for the time and
for the spectral distribution of the laser pulses, a hyperbolic secant form is assumed.
4.1.3 Optical Setup
Figure 4.2 shows the general experimental setup. The excitation intensity can be adjusted
with the help of different neutral density filters (coarse) and a computer controlled filter
wheel (fine). The back of the filter wheel is not sufficiently anti-reflection coated, at least
in the close-infrared spectral region. This causes unwanted interferences. Therefore the
filter wheel is placed diagonally to the incident beam. Because of the slow adjustment
speed of the filter wheel, an additional power control unit was developed (see chapter
4.1.4). The power control suppresses strong intensity jumps which arise by tuning the
laser energy. After the power control unit, the beam can be directed into a MICHEL-
SON interferometer in order to generate double pulses, shown in the upper left side of
figure 4.2 (for description see chapter 4.1.5). The free space setup easily makes a polar-
37
4 Experimental setup
926.0 926.5 927.0 927.5 928.0 928.5
0
1000
2000
3000
4000
Laserspectrum
sech
2
-fitcurve
Intensity (a.u.)
Wavelength (nm)
Figure 4.3: Spectrum of the Ti:Sapphire laser in the picosecond mode (crosses) and a
sech2-fitcurve (line).
ization control of the excitation beam possible (linear: λ/2 and circular λ/4). The beam
is steered towards the sample holder with the help of different mirrors. At the top of
the sample holder, the laser beam is split in two parts by a pellicle beam splitter (92 %
transmission) or beam splitter cube (50 % transmission). The transmitted part is directed
onto a photodiode in order to measure the relative intensity of the laser beam. Figure 4.2
shows the experimental setup schematically. In the real setup, the laser beam is directed
onto the sample holder from the side. The reflected part is directed into the dewar to-
wards the low temperature microscope (see 4.1.6). The luminescence from the sample
passes the beam splitter again. The transmitted part is directed towards a CCD camera
to check the laser focus geometry via a flip mirror. Otherwise the luminescence is cou-
pled into the spectrometer via free beam or fiber optics. At the sample holder itself, an
inspection unit is mounted. An IR-LED can illuminate the sample via a flip mirror. The
reflected light can be displayed on a CCD camera in order to position the sample in the
middle of the objective. All measurements are controlled by a single computer.
4.1.4 Power control
As already mentioned, the laser energy does not remain constant by tuning the wave-
length. In PLE measurements, intensity variations of the excitation generate noise or
ghost peaks along the energy axis. In order to compensate intensity variations, a power
control unit was developed. The laser beam (0,8 mm in diameter) is focused by an as-
pheric lens onto a pinhole. The focus size depends on the beam diameter and the focal
length of the lens: D=λf
r. Here, fcorresponds to the focal length, rto the input beam
diameter (1/e2) and λto the wavelength. For the parameters here, a focus size of 10 μm
(1/e decay) results. The pinhole diameter is selected slightly smaller than this focus size,
here: 5 ±1μm. The aspheric lens can be pre-positioned in three directions in space. In
addition, the distance pinhole-lens (Δxin figure 4.4) can be controlled with a closed loop
nano-positioner (80 μmwalk). If the distance is changed from the focal condition, the
38
4.1 Excitation
Figure 4.4: Power control unit and beam expansion. The output intensity is a function of
the distance between the first lens and the pinhole. A reduced distance leads
to reduced laser intensity.
transmitted intensity is inversely-proportional to the square of the distance I∝1/(Δx)2.
The laser beam is collimated by a second lens (achromatic) after the pinhole. On the sam-
ple holder, a photodiode is mounted where the laser intensity can be measured and hence
stabilized. Due to the high intensities, a high power pinhole must be used. In addition,
the laser intensity must still be reduced with a neutral density filter before the power con-
trol, in order to avoid damage to the pinhole. By proper choice of the focal length of the
output lens, the beam expansion can be controlled. The output lens is fixed with respect
to the pinhole. A beam expansion is necessary to guarantee optimum illumination of the
objective at the sample holder.
4.1.5 MICHELSON interferometer
For the measurement of the dephasing time, two equal and successive laser pulses with
precise and adjustable timing are needed. This is realized by an interferometer, which is
positioned behind the filter wheel (see figure 4.2). One interferometer mirror is attached
on a controllable one dimensional positioner. For the dephasing measurements, two dif-
ferent regime of changes of distance must be examined. The desired change of distance
within the μm range is controlled by the closed loop nano-positioner. In order to mea-
sure the decrease of the interference ability depending on the time delay between the
two pulses, larger time delays have to be provided. Therefore the interferometer mirror
is adjusted with the help of a long range positioner, where the position can be controlled
within the range of ≈0.1 m. The MICHELSON interferometer is used for the analysis of the
QD state dephasing. The QD-state is excited with two time delayed π/2-pulses. A small
change of the distance of one interferometer mirror within the range of 0 and 100 μm with
an accuracy of 25 nm affects the phase of the second pulse in relation to the first pulse.
This corresponds to 1/40 wavelength and a time resolution of 0.17 fs. The variation of the
relative phase (piezo positioner) leads to an interferogram. The change of the amplitude
of this interferogram over a longer time scale is detected by a point to point change of the
long range positioner.
39
4 Experimental setup
4.1.6 Low temperature microscope
All measurements presented here are performed in a liquid helium (He) dewar at 4.2 K.
The beam, which is directed into the He dewar, is focused by an objective. In order to
position the sample under the laser focus, a three dimensional piezo positioner is used
(see figure 4.2). The z-dimension controls the focus length, and the xand yaxis, the lateral
position of the sample. The objective and the piezo positioner form the low temperature
microscope, which is inserted in an outer tube. This tube must be evacuated to avoid the
condensation of humidity when cooling down the sample holder. After the evacuation
the tube is filled with a small quantity of gaseous helium. This is done for a better thermal
exchange between the low temperature microscope and the liquid helium in the dewar.
The LT-microscope is mounted on an inner tube. The electrical connections from the
sample to the ADC and the DAC are integrated in the inner tube with two coaxial cables.
A very important item here is the choice of the best compromise for the objective. In order
to get high PL intensities, the objective should have a high numerical aperture (NA). But
on the other hand, objectives with a high NA have a very low working distance. This can
be a problem, if the objective makes contact with the bond wires. Another problem can be
an insufficient color correction of the objective. A bad color correction results in different
focus length for excitation and luminescence, which cannot be optimized at the same
time. One has to make a compromise between good excitation and good luminescence
output. Here a ZEISS objective (100x; 0.75 NA) is used with a comparatively good color
correction and a working distance of about 0.3 mm.
40
4.2 Quantum dot photodiodes
4.2 Quantum dot photodiodes
This chapter gives a general introduction into the structure and the optical characteris-
tics of the investigated sample. The presented measurements refer to the particular QD,
which is examined in the further chapters. The sample was developed in 1999 at the
WALTER SCHOTTKY institute in Garching, in particular for photocurrent spectroscopy.
In the following, the requirements for PC spectroscopy on a single QD are described,
followed by remarks on the band-diagram, the diode design, and the near field shadow
mask.
4.2.1 Sample structure
For PC spectroscopy on single QDs, the samples must fulfill some principle conditions.
These concern both the layer sequence of the samples and the characteristics of the QDs
themselves. Additionally, strict requirements also apply to the crystal quality, the spec-
tral position, and the QD-density. Since electrical tuning is necessary for the PC mea-
surement, the QDs are integrated into a diode structure. The QDs are embedded in a
layer of intrinsic bulk material, see figure 4.5 b). The electric tunability is one of the most
important elements of this work. By means of the STARK effect, the transition energy of
the QD can be controlled extremely precisely. This is the basis for the high resolution
spectroscopy method, which is discussed in chapter 6.1. For the photocurrent measure-
ments, two contacts are necessary, above and below the QD-layer. This can be provided
by doped regions on both sides of the QD layer (p-i-ndiodes or n-i-pdiodes). With
n-ior p-idiodes on the other hand, it is possible to use an doped substrate as a back
contact, and a metal-semiconductor contact (SCHOTTKY contact) as front contact. The
photo diodes should exhibit a high blocking ability, so that the PC signal is not accom-
panied by too high dark-current. Furthermore, the blocking behavior of the diodes must
be sufficient for the field strengths required for the PC measurement. The required field
strengths depend on the tunneling probability of the charge carriers. The electric field for
PC amounts to approximately 30 kV/cm at the examined InGaAs/GaAs QDs sample.
The tunneling probability depends on the barrier height and distance. A short distance
leads to a short lifetime in the QD and therefore to strong broadened resonances in the
photocurrent signal. Thus, a large distance between QD layer and back contact leads to
longer lifetimes. The breakdown field strengths of the used n-i-photodiodes are clearly
higher than 150 kV/cm (see reference [Fin01] and [Beh03]). A detailed description of
the individual processing steps and sample parameters can be found in [Fin01] (sample
120799.2). For PC measurements of the QD-ground states, the sample structure should be
designed in such a way that no charge carriers are permanently in the QD. With the ex-
amined n-i-diodes, a durable charging of the QDs with charge carriers can be prevented
by a suitable reverse voltage.
The layer sequence of the examined n-i-photodiode is shown in figure 4.5 a). The dis-
tance between QD-layer and back contact is 40 nm. Other samples with other distances
were examined in [Fin01] and [Beh03]. The sample consists of the following layers: On
a doped n+-GaAs-substrate a buffer layer of 3000 Å n-doped GaAs is grown, followed
by an AlAs/GaAs super-lattice and further n-doped GaAs layers. Subsequently, intrinsic
41
4 Experimental setup
(a) Layer sequence of the n-i-photodiode. (b) Band structure at applied reverse bias voltage.
Figure 4.5: Sample design
GaAs with a thickness of 400 Å and a 22.6 Å undoped In0.5Ga0.5As were deposited (for-
mation of self organized QDs). After 2700 Å of intrinsic GaAs, a 400 Å thick Al0.3Ga0.7As-
layer and a 100 Å surface layer (i-GaAs) follow. Figure 4.5 b) shows the band-diagram
under reverse bias of the photodiode. The electric field strength within the intrinsic re-
gion of the diode results from the built-in voltage Vbi and the applied voltage VB. For
the connection between electrical field strength Fand the voltage VBapplied between the
SCHOTTKY-contact and the doped back contact, one gets the following relation:
|
F|∼
=(VB+Vbi)·27.78 ·1031
cm.
Here VBcorresponds to the applied bias voltage which is added to the built-in voltage
(Vbi ≈0.8 V). In the following, the properties of the charge carriers are discussed on the
basis of this band diagram. These are, for example, the tunneling of charge carriers from
the QDs of high electric field strengths and the charging of the QDs from the n-doped
back contact.
4.2.2 Diode design
For a good design of the n-i-photodiodes, some parameters must be suitably selected.
Special attention applies to the location of the QDs and the thickness of the intrinsic
region. A change in the total thickness of the i-layer affects, as already mentioned, the
resonance width in the PC spectroscopy and determines the voltages needed for the PC
measurements. The voltage range of a diode follows directly from the electrical field
strengths. With too large i-layer thickness, the dark current can increase and limit the
42
4.2 Quantum dot photodiodes
Figure 4.6: Sample pictures. a) sample on a watch. b) (Au-)bonded sample with a view
from the top. A single field has a size of 300·400 μm2. c): Al-shadow mask
and boundary marks.
field strength range usable for PC measurement. The distance between the n-doping
and the QDs, as well as the electric field strength, controlls the occupation of the QDs
with electrons. The influence of the spacer layer thickness was examined by voltage-
dependent PL-measurements of individual QDs in [Fin01]. There it was shown that for
low field strengths the QDs are occupied on average with additional electrons from the n-
area. With doubling of the spacer layer thickness from 20 nm to 40 nm, this threshold of,
for instance F=57 kV/cm, is reduced to F=30 kV/cm. This reduction can be explained by
the relative energetic level of the electron ground state in the QD and the FERMI-energy
in the n-area. This spacer thickness determines the electron occupation of the QDs. Up
to this point, the QDs are occupied with additional electrons (single electron charging).
Beyond this threshold, the QDs are no longer occupied with electrons.
4.2.3 Near field shadow masks
In order to realize PC spectroscopy for individual QDs, an optical isolation of a single
QDs is necessary. The sample used in this work exhibits near field apertures to address
individual QDs by selective optical excitation. On the surface of the photodiode, a semi-
transparent top gate is applied (5 nm Ti), followed by 80 nm of Al, see figure 4.5 a).
Subsequently, small holes are opened in the Al-layer, which allow for local optical access
to the semiconductor. The size of the holes is in the sub μm range. For light in the close
infrared spectral region, these holes are nano-apertures with low optical transmission,
since the wavelength in this case exceeds the aperture size. Propagation of the excitation
light can take place only in the near-field optical limit. The 100 to 500 nm big holes work
therefore as near field apertures for the excitation laser light. Therefore, the QD layer
should be located near to the shadow mask. The sample, introduced here, uses QDs
which are 320 nm under the surface. Together with the layers lying between shadow
mask and semiconductor (20 nm), this near field condition is therefore well fulfilled. For
the production of the shadow mask, an 80 nm thick aluminum layer was used. The near
field apertures were defined by means of electron beam lithography and reactive ion
etching.
43
4 Experimental setup
For protection of the semiconductor from etching damages, a 15 nm thick Si3N4stop
layer is deposited under the aluminum layer, see figure 4.5 a). For PC measurements
on individual QDs, further process steps are necessary, apart from the production of the
shadow masks. This concerns the electrical contacts of the photodiode. In order to guar-
antee a homogeneous electrical field at the QD, a surface metalization of the sample is
necessary, around the shadow mask area.
Figure 4.6 b) shows a top view of the sample with four active (bonded) photodiodes
(300·400 μm2). The bond contact is located on the titanium gold metalization. Within the
aluminum metalization, one recognizes the shadow mask field, which contain the near
field apertures. Figure 4.6 c) shows a series of 4 μm big markers, which form a U-shape
around the apertures. During experiments, these markers turned out to be very helpful,
in order to navigate on the sample and perform test measurements of a QD ensemble.
Finally, the photo diodes are separated in 300×400 μm2mesa structures by a further
etching step.
The size of the near field apertures is compared with the average QD-distance in order
to estimate the degree of the optical selection. With a density of the self organized InGaAs
QDs of 100-200 μm−2used, the distance between the QDs amounts, on average, to about
100 nm. Within a 300 nm large aperture, there should be nine QDs. Apart from this
purely geometrical selection of the QDs by the shadow mask, the number of the QDs
measured at the same time is limited furthermore due to the individual transition energy
of each QD. With the help of the spectrally narrow excitation energy, only those QDs are
addressed, which can absorb the selected energy. Due to the inhomogeneous ensemble
broadening, one receives, for the single QD spectroscopy, sufficient energetic separation
of the QD-absorption lines. Contrary to the optically selective excitation of individual
QDs, the electrical access to the QDs is not selective. All QDs within the processed photo
diode (about 12 million QDs in the 300·400-μm2large structure) are electrically contacted
in parallel.
44
4.3 Detection
4.3 Detection
The detection method depends on the exciton decay mechanism. If the QD state is reso-
nantly excited and the exciton is able to tunnel out of the QD, the detection takes place
in the photocurrent. If the decay mechanism is an optical recombination process, the de-
tection takes place in the photoluminescence. In this case, an GaAs-interband excitation
is as possible as an excitation of a higher QD-state. The applied bias voltage controlls
whether the tunneling process or the optical recombination process is dominant.
4.3.1 Photocurrent technique
In this section, the photocurrent (PC) spectroscopy on individual QDs is presented. The
main advantage of this method is the high resolution as well as the direct coupling of
the QD-states to electrical signals. Since detection in the PC spectroscopy takes places
electrically, the ground state can also be investigated by resonant optical excitation.
The photocurrent mechanism is based on the principle of a resonant excitation of an
individual QD. Therefore the isolation of an individual QD with near field a shadow
mask (see chapter 4.2.3) is necessary. The single QD-states must be characterized and
assigned before by voltage-dependent PL in order to receive the ground state energy.
The resonant excitation must be tuned to the energy of the QD state. Then the QD can
be excited and occupied with an exciton. After the excitation process, a tunnel process
for both charge carriers can take place due to the applied voltage. After that, the QD is
unoccupied and can be re-excited again. A measurable current results from this sequence
of exciton generation and tunneling processes. The maximum current depends only on
the repetition frequency of this sequence. A shorter life time and a faster re-occupation
leads to higher currents. An applied voltage changes the tunneling probability of the
charge carriers. An increase of the laser intensity leads to a faster re-occupation of the QD.
The PC is affected by both parameters. If the QD is already excited, the laser can induce
a stimulated emission. In the case of cw excitation, this leads to an equal occupation of
the states for high laser intensities and thus to a saturation of the PC.
In case of pulsed excitation, generation and stimulated emission can follow up within
one pulse. This leads to the RABI oscillations, as discussed in chapter 6.3. With appro-
priate electrical field strength, electron and hole are able to tunnel through the potential
barrier. The electron and the hole are separated in the electrical field. Here the detectable
photocurrent is determined by the repetition frequency of the laser system and the exci-
ton occupation probability after the laser pulse interaction. Thus, the optical generated
excitons are electrically detected with the help of simple DC current measurement.
An AlGaAs barrier is arranged in the intrinsic layer between the QDs and the front con-
tact, see layer sequence in figure 4.5 a). In principle, this barrier could affect the transport
of the tunneled holes to the front contact. An estimation of the ballistic transportation
length of the holes shows, however, that the distance of 270 nm can be easily overcome.
With a hole mobility of μh∝2·104cm2/Vs a ballistic transport length of λbal ∼
=10 μm
results at T = 10 K [TMT+85] for heavy holes in the bulk semiconductor GaAs in the case
of F=30 kV/cm. This is much more than the distance between the QD layer and the
front contact. Therefore a ballistic transport through the AlGaAs barrier should be easily
45
4 Experimental setup
possible. At a temperature of 4.2 K, the thermionic emission from the QD is negligible.
The thermionic influence on the QD state is a very interesting aspect, which is examined
also in the group of Prof. ZRENNER.
The charge carriers generated in the QD are subject of investigations, depending on
the examined states, outside conditions, and different decay processes, e.g. tunneling,
relaxation or recombination. If one regards various absorption events, then the behavior
of the charge carriers can be described statistically, with the help of the time constants
of these processes. The optical recombination of the single exciton ground state takes
place typically on a time scale up to 1 ns [OAO+96, BLH01]. In the case of resonant
occupation of a higher excited QD state, a relaxation of the charge carriers takes place into
the energetically lower lying state on a time scale in the ps range [OAO+96, HTK+02b].
To understand the PC spectra, these times must be compared to the tunneling time of the
charge carriers. Contrary to the recombination and relaxation time, the tunneling time
can be easily controlled by VB. The tunnel barrier depends on the electrical field strength.
The tunneling time can take values from less than 1 ps to more than 1 ns, and in principle
become infinitely long. Please note that the tunnel time of electron and hole is different
at the same electrical field. This is because of different effective masses. For tunneling
times of t» 1 ns, optical recombination takes place before the exciton can tunnel out of
the QD, so that the QD-absorption can no longer be detected in the PC measurement.
Photocurrent detection setup
The measurement of the PC signal is done by a time integrating DC measurement. Fig-
ure 4.7 shows the corresponding circuit. Between front and back contact of the QD-
photodiode, a computer-controlled voltage is applied via a digital to analogue converter
(DAC). The current flowing perpendicularly through the sample is measured at the same
time by a current-voltage-converter (IVC). The PC signals can be in the range of 1 and
100 pA. The diode current flows into am OP-amp with 100 MΩfeedback-resistor, pro-
duces an amplification of 108V/A, see figure 4.7. This signal is fed into an integrating
analogue-digital converter (ADC). The resulting dynamic range extends from -5 to +5 nA
with a resolution of 0.15 pA (16 bit).
•All components, which are in direct electrical contact with the sample (ADC, DAC,
IVC) are implemented in a battery-operated housing with common electric ground.
•Communication with the computer is done via in a digital way fiber optics.
•In the sample holder, coaxial cables are used. All unshielded cables are as short as
possible and if necessary twisted.
•Extremely low-noise units are used, the ADC provides an integration time of 50 ms.
Thus a current measurement with a stable ground potential becomes possible, which
remains uninfluenced by potential fluctuations of normal net operated devices. The con-
trol and data exchange between the measuring computer and the voltage supply and/or
the ADC takes place over optical data communication by means of optical fibers (broken
46
4.3 Detection
Figure 4.7: Photocurrent measurement setup. The resistance of 100 MΩleads to a ampli-
fication of 108V/A.
lines in figure 4.7). Thus a sensitive current measurement becomes possible at the sub-pA
range, in a structure which is well protected from external influences. For the measure-
ment of these very small currents, the IVC contains an operation amplifier, which features
extremely low noise and very high input impedance.
The PC signal contains the dark current of the photodiode. The n-i- diode used has
however a high blocking ability, so that the dark current does not play a large role. With
the typical voltages used for the PC measurement, the diode dark current is in the sub
pA range.
The ADC used has two similar input ports which are integrated at the same time. The
first channel is used, as described, for the measurement of the sample current. The second
channel converts a photodiode signal, which is proportional to the laser intensity at the
sample holder (see figure 4.2). In intensity dependent measurements, this signal is used
for the definition of the x-axis, while the PC of the sample defines the y-value.
4.3.2 Photoluminescence technique
The photoluminescence (PL) spectroscopy is a very sensitive method, in order to examine
optical transitions in semiconductors. By excitation with photons of the energy hν>Egap,
electrons are lifted from the conduction band into the valence band. Pairs of electrons and
holes are produced. The excited electrons and holes can recombine after a certain time.
The resulting energy can thereby be released in different forms, as photons or less likely
as phonons or also as AUGER electrons. In the PL spectroscopy, photons are detected
which are emitted by a radiative recombination mechanism. Most PL-measurements are
performed at low temperatures, in order to prevent thermal ionization and to avoid the
broadening of the PL-lines by lattice vibrations (phonons). The luminescence of the In-
GaAs QDs which can be examined is reached by light excitation.
47
4 Experimental setup
Figure 4.8: Detection setup for luminescence measurements with resonant excitation of
higher QD-states. An extra grating is mounted before the spectrometer in
order to separate laser and QD luminescence.
4.3.3 Photoluminescence excitation technique
Contrary to the normal PL, the photoluminescence excitation (PLE) method is sensitive
to higher shells of the QD. In order to analyze higher QD shell-structure, the excitation
energy is tuned continuously above the exciton ground state. If the laser energy matches
a higher state of the QD, the exciton is directly generated in this state. After a relaxation
process into the ground state, the exciton decays by optical recombination processes. The
experimental setup must separate the (comparatively) weak photoluminescence intensity
and the strong excitation before the detector. A single monochromator is not sufficient.
If the laser gets into the monochromator, too much stray light will be generated. In order
to suppress the stray light, an extra grating is mounted before the monochromator (see
figure 4.8). Detection can be done with a Si-CCD camera or with the HANBURY-BROWN
and TWISS setup at the second exit of the monochromator. The energy range which will
be examined is between the single exciton ground state and the energy gap of the sur-
rounding material. Similar to the PL, the PLE can be performed voltage, intensity and
polarization dependent.
4.3.4 HANBURY-BROWN and TWISS setup
The proof of a single photon emission is done with the HANBURY-BROWN and TWISS
(HBT) setup (figure 4.9). The luminescence of the single photon emitter must be spec-
trally filtered (suppression of the excitation laser) analogous to the PLE measurements,
(see figure 4.8). The QD luminescence is collimated and directed on a 50:50 beam splitter
cube. Both parts of the split luminescence are focused into optical fibers and directed on
single photon counter units, here two similar avalanche photodiodes (APD). If a single
photon is counted, an electrical pulse is generated and send to the TimeHarp 300 (Pi-
coquant). The TimeHarp acts as a time to amplitude converter (TAC) which is able to
display a histogram of correlation events with a maximum time resolution of 39 ps.
The histogram of a single photon emission under continuous and pulsed excitation is
discussed in chapter 10. The proof of single photon emission can be shortly explained
as follows. A single photon can not be transmitted and reflected at the same time in
48
4.3 Detection
(a) Schematic view of the HBT (b) Picture of the experimental realization
Figure 4.9: HANBURY-BROWN and TWISS setup for proof of single photon emission. A
single incident photon cannot be transmitted and reflected at the same time.
Thus, the histogram of the correlation events should show a clear reduction at
zero time delay.
a beam splitter. It can only be detected at one APD. Therefore, the two APDs cannot
count photons at the same time. If the luminescence only consists of single photons, the
histogram of the TAC must show a clear reduction (dip) at zero time delay. In order to
reach a symmetrical histogram, one electrical connection has an integrated time delay.
This will only shift the measuring window to negative times.
One experimental detail should be discussed here. The APD emits a light flash at each
single photon counting event. This effect was analyzed by the group of Prof. WEIN-
FURTER ([KZMW01]). The light flash is called avalanche breakdown flash. The wavelength
of this light flash overlaps with the range of the QD luminescence. It is therefore possible
that the breakdown flash can be sent in backward direction to the sample. The light can
be reflected on the sample surface and then be detected like a normal photon. Due to the
spectral overlap, the photons from the QD and the photons from the APD cannot be dis-
tinguished. This can result in an additional structure in the histogram. The time constant
of this structure corresponds to the optical path length from one APD to the sample and
back. Another effect of the breakdown flash is the generation of background counts in
the histogram. The reflected breakdown photons can initiate a correlation, not only with
the photon which triggered the breakdown, but also with the QD photons.
49
4 Experimental setup
50
5 Characterization of a single quantum dot
5.1 Optical properties
For sample characterization, PL spectroscopy is performed on every hole of the shadow
mask. The PL signal exhibits different information about the sample. By non-resonant
interband excitation (HeNe laser), the spectral distribution of the QD emission can be
measured. A single QD contributes to the luminescence with more than one spectral line
(e.g. charged states). The number of discrete peaks in the PL spectra is proportional to
the number of QDs which contribute to the luminescence. The analysis of a single QD
spectrum will be discussed in the following. In order to identify the different states of the
QD, a voltage dependent PL series is performed.
Figure 5.1 shows a color plot of the voltage dependent PL of the best single QD spec-
trum of the sample. Here mainly the luminescence of a single QD is visible, but a small
contribution of a second QD can be noticed at a slightly higher energy. All other holes
of the shadow mask show the luminescence of more than one QD or no PL signal at all.
One clearly notices three bright states which correspond to the main states of a QD.
Actually, only the lowest energetic state of the single QD should be visible at one time.
But this is clearly not the case. This can be explained by the statistical process of the
non-resonant excitation. The excited charge carriers relax into the QD. Additional to the
luminescence of the neutral exciton, it is possible that only three charge carriers (one
exciton and one further charge carrier) of a non-resonant interband excitation are relaxed
into the QD. Then it is possible to observe the statistical luminescence of a charged QD.
Therefore both emission peaks can be observed at the same voltage in this time integrated
spectrum. Both states will probably not emit photons at the same time. Furthermore, it
is possible that an extra electron can tunnel into the QD from the doped back contact at
an adequate voltage.
This behavior is illustrated in figure 5.2. In case a), the applied voltage is high enough
to lift the ground state over the quasi FERMI-level. Here it is not possible that electrons
from the n-doped back contact can tunnel into the QD. In this region, only the lumines-
cence of the neutral exciton (X) can be observed. For a slightly lower voltage, as shown
in case b), the exciton ground state is equal to the quasi FERMI-energy. Electrons from
the doped back contact are able to tunnel into the QD. The occupation of this state with
an electron from the doped back contact takes place in the thermodynamic equilibrium.
Here the luminescence of Xand X−can be observed. In case c), the applied voltage is
low enough so that the ground state can be occupied with two electrons.
The first indicator for the state identification is the exciton-biexciton power depen-
dence. At non-resonant interband excitation, the biexciton (2X) state can only contribute
to the PL when the QD is occupied by two excitons. This is the case at higher excitation
intensities. After the 2Xrecombination process, the charge carriers of the 1Xstate can
51
5 Characterization of a single quantum dot
Figure 5.1: Voltage depending photoluminescence at non-resonant interband excitation.
Marked are the three brightest states, the single neutral charged exciton X, the
single negative charged exciton X−and the biexciton 2X.
recombine after a characteristic time (radiative recombination time), or a further absorp-
tion process leads again to the occupation of the 2Xstate. For low excitation intensities,
only the 1Xstate can contribute to the luminescence. But the PL of the Xwill saturate
with increasing excitation intensity. Simultaneously the PL intensity of the 2Xstate will
increase.
Thus, the neutral exciton can be identified. The luminescence of the 1Xstate can be ob-
served in the voltage range of -0.35 V to 0.05 V. Below -0.35 V, the luminescence will
vanish due to a higher tunneling probability for the charge carriers. At a voltage of
about -0.2 V, the quasi FERMI-levels of the single exciton ground state and the n+GaAs
back contact line up. At non-resonant excitation, the luminescence of the single negative
charged exciton (X−) can be observed. The slight higher intensity of the X−state (com-
pared to the 1Xstate) can be explained by the high tunneling probability of electrons
from the doped back contact into the QD and the statistical occupation with a charged
exciton (explained above). This state is separated by ≈5 meV (corresponds to 3.5 nm in
wavelength) from the ground state. The strong broadening of the X−state at a voltage of
>0.2 V can be observed in the luminescence. The line broadening and shift in energy is
52
5.2 Quantum confined STARK effect
Figure 5.2: Band structure at different applied voltages. a) The applied voltage is high
enough to lift the ground state over the FERMI-energy. b) The exciton ground
state is in the region of the FERMI-energy. Electrons from the doped back
contact are able to tunnel into the QD. c) The applied voltage is low enough
that the ground state can be occupied with two electrons.
probably an influence of charged wetting layer states to the QD.
The power- and the voltage dependences are the main indicators for the state identi-
fication. The identification of multiexcitonic states is of course more difficult. The weak
state (at V < -0.18 V) has a slightly higher energy as the single exciton ground state.
This state could be tentatively assigned to the last charged s-shell transition, the positive
charged exciton X+. This assignment is supported by the fact that the state vanishes at
the same voltage as electrons from the back contact can tunnel into the QD. Due to the fact
that holes are not able to tunnel into the QD, this state can be generated by a confinement
of an exciton and an extra hole of a non-resonant optical excitation.
But it is also possible that each measured peak of the PL consists of a fine struc-
ture which cannot be resolved with this PL-setup. The fine structure splitting and the
linewidth of the Xstate can be analyzed with PC spectroscopy (in the voltage region of
the PC). The transition energies show a slight red shift with increasing (reverse) bias volt-
age. This results from the quantum confined STARK effect (QCSE). The QCSE affects the
exciton-states in the PL region as well as in the PC region.
5.2 Quantum confined STARK effect
For PC measurements, the excitation setup has to be changed to the tunable TiSa laser
system. The laser energy must be tuned to a slightly lower energy (due to the QCSE) than
the measured PL energy. The energy of the ground state can be tuned via the bias voltage.
If the fixed laser energy fits to the transition energy, an absorption process can take place.
The excited charge carriers are able to tunnel out of the QD and generate a current if the
applied (reverse) voltage is high enough. Therefore the absorption characteristics appear
in the photo- I-V-characteristics of the n-i-SCHOTTKY-diode. The behavior of the QD
with varying different parameters like excitation intensity, excitation polarization and
applied voltage can be investigated. For the basic characterization, the influence of the
applied voltage on the transition energy (and the tunneling time) is important.
In this section, the influence of an electrical field (F) in the z- (growth-) direction is
discussed. The xand ydirections are irrelevant in the first approximation and neglected
here. The description takes places analogous to a quantum well. As mentioned before,
only the lowest quantized energy state in z-direction is examined. In an electric field, the
53
5 Characterization of a single quantum dot
Figure 5.3: Schematic view of the quantum confined STARK effect. left: flat band condi-
tions; right: high applied electric field. (from [BZF+01])
wave functions of electrons and holes in a QD are shifted and slightly deformed. Figure
5.3 shows a schematic sketch of the wave functions with and without an applied elec-
trical field. The deformation results in an induced dipole moment μwhich reduces the
transition energy in an electric field of ΔE=μF. The decrease of the energy is quadratic
with increasing electrical field. For high electrical fields, the shift of the dipole moment is
limited by the size of the QD. Therefore the energy shift is almost linear for higher electric
fields. This is only a coarse approximation, which however correctly reflects the behav-
ior of the quantum confined STARK effect (QCSE). For an exact description, the reduction
of the COULOMB-interaction (due to a reduced electron hole distance) and the change of
the quantization energy have to be taken into account. The shift of the electron and hole
wave functions into different directions reduces the overlap and therefore the transition
matrix element. The QSCE allows for a very accurate tuning of the transition energy. The
tuning range is relatively small (some meV), but the adjustment fidelity only depends on
the stability of the voltage source [HSK+04, SEZB04].
Up to a reverse voltage of about 0.4 V, optical recombination processes are dominating.
Here the QCSE can be observed in the PL, see figure 5.4. At higher voltages, the tunnel-
ing probability increases. The energy of the Xtransition can then be measured in the PC.
Figure 5.4 shows the QSCE, which continuously covers the whole PL and PC region. An
example of a PC resonance is displayed in the inset (upper right) of figure 5.4. The shape
of the resonances and the voltage dependence is discussed in chapter 6.2. The PL mea-
surements are performed at non-resonant interband excitation. The PC measurements
work with resonant excitation and electrical detection. The insets in the figure show the
different processes. The conversion from voltage into energy scales is a result from the fit
curve, see figure 5.4:
E=1.33833 eV −0.00121 eV
V·VB−5.81615 ·10−4eV
V2·V2
B(5.1)
More important than the absolute energy is the relative energy variation ΔEat the tuning
54
5.3 Photoluminescence excitation spectroscopy
Figure 5.4: Quantum confined STARK shift of the single exciton ground state. The en-
ergy shift covers the whole PL (<0.4 V) and PC region (>0.4 V). (published in
[SEZ06])
of the QD energy levels via bias voltage. This variation of energy is given in μeV:
ΔE=−1210 ±17 μeV
V−1163 ±26 μeV
V2·VB·ΔVB(5.2)
The energy of the QD transition decreases with increasing reverse voltage VB.
5.3 Photoluminescence excitation spectroscopy
Since the spectral position of higher excited states of the QD are not visible with the
normal PL technique, a photoluminescence-excitation (PLE) experiment has to be per-
formed. The spectral position of the p-shell has, of course, to be known for all further
investigations, e.g. analysis of the relaxation behavior or for coherent manipulations. For
excitation a tunable Ti:Sa laser is used, which is tuned here in the region of 10 to 50 meV
above the ground state.
Figure 5.5 shows the PLE spectrum of the investigated QD. Of course, the PLE spec-
trum strongly depends on the applied conditions, such as voltage and excitation inten-
sity. The behavior of a QD under an applied voltage is discussed in chapter 7. Very high
excitation intensity could lead to unwanted off resonant excitation of the QD. Therefore,
zero applied voltage and an excitation intensity of approximately 4 μW at the sample are
chosen. Furthermore, one has to be sure that the excitation intensity is constant over the
whole tuning range.
55
5 Characterization of a single quantum dot
15 20 25 30 35 40 45 50
0
500
1000
1500
2000
PL
PD (arb. u.)
Edet= 1337 meV
PL intensity (a.u.)
Differential excitation energy (meV)
|Xp>
|X
LO
>
Figure 5.5: PLE-spectrum of the single exciton ground state. The energy is displayed with
respect to the QD ground state. The strongest peak at 29 meV is assigned to
the p-shell absorption (|Xp) and the resonances around 36.7 meV correspond
to (LO) phonon assisted absorption (|XLO). The PD-signal shows stable ex-
citation intensity in the whole range. Inset left: The laser excites the QD in
the p-shell. After the relaxation process the exciton recombines in the s-shell.
Inset right: LO-phonon assisted absorption. The laser generates the exciton di-
rect in the ground state with a simultaneous generation of GaAs-LO phonons.
(published in [ELdV+07b, ELdV+07a])
The spectrum shows two main resonances. The largest peak at 29 meV above the
ground state (EGS =1.337 eV) is assigned to the p-state (|Xp) and the second largest
peak (42 meV) is found to be the next excited state (probably the d-shell). The small
peaks around 36.7 meV are assigned to GaAs phonon-assisted absorptions (|XLO). The
phonon assisted absorption matches the GaAs LO phonon energy very well. The re-
sults of these PLE measurements are comparable to the results reported in [FZBA00].
The much weaker resonance around 33 meV above the ground state can be tentatively
assigned to an InAs phonon mode, which where also observed in ensemble PLE exper-
iments in reference [HVL+97]. The insets in figure 5.5 illustrate the different excitation
schemes for p-shell excitation and LO-phonon assisted absorption. In the left case, the
excitation energy fits to the p-state. After the absorption process, the exciton relaxes into
the ground state, probably by emitting acoustic phonons. After the relaxation process,
the exciton decays due to an optical recombination. In the left case, the excitation energy
fits to the ground state energy plus the energy of a GaAs-LO phonon. The ground state
is thereby generated with the help of an emitted GaAs-LO phonon. The ground state is
again able to decay by optical recombination.
56
6 Single exciton ground state
6.1 High resolution absorption spectroscopy
The linewidth of a QD ground state transition at low temperatures is very small and
hardly resolvable with optical techniques. If one analyzes the QD ground state optically,
the resolution of the detection setup must be higher than the linewidth of the emission. It
is simpler to analyze the absorption properties instead of the emission properties. There-
fore, one has to use a very narrow linewidth (laser) for excitation. This is quite easy with
the use of a TiSa laser in its continuous wave mode. Due to the integration of the QD into
a diode structure, it is possible to shift the QD energy over the laser energy by means of
the STARK effect. There are two options for the detection of the ground state resonances.
One can analyze the transmitted laser intensity or detect the generated charge carriers
in the photocurrent. When the transmitted laser photons are detected, an AC voltage is
applied at the QD. The detection intensity is then plotted as a function of the applied
voltage in a histogram. The diagram shows a very small dip (only a decrease of 10−5)
when a QD ground state absorbs the incident photon. But the resolution is very high
because only the laser line width determines the resolution. A disadvantage of these
measurements is the comparatively long integration time [SLG+02]. The other option
(photocurrent detection) is used here in order to analyze the QD ground state. The use
of the STARK effect for the energy tuning sets the lower limit for the resolution of the
measurement setup. The use of a 16 Bit DAC results in an increment of 0,3 μeV. There-
fore every linewidth larger than approximately one μeV appears not broadened due to
the experimental setup. Thus, the increment of the used DAC has no influence on the
measured linewidth. The measured linewidth in the photocurrent is a result of a convo-
lution of the incident laser linewidth and the QD ground state. For a very narrow laser
linewidth, the resulting signal is mainly determined by the QD-state.
Due to the fact that the measured spectra are not influenced (broadened) by the exper-
imental setup and the detection efficiency in the photocurrent is almost perfect (100%),
all measured criteria can be referred back to basic effects of the quantum system. In a
closed circuit, no charge carriers (electrons) could be lost. Every charge carrier which
is tunneled out of the QD contributes to the photocurrent. Only excitons which recom-
bine optically (after the optical life time) cannot be detected. The detection limit for the
current is determined by the PC pre-amplifier and the used ADC, see chapter 4.3.1. The
occupation of the upper state of this two level system can be described with a rate equa-
tion (see chapter 3.5.1). For tunneling times longer than the radiative lifetime, the decay
of the ground state is mainly determined by the optical recombination process. Thus the
lower limit for PC measurements is determined by the optical recombination time (ap-
proximately 500 ps). The PC pre-amplifier must be selected suitable to detect these low
currents. A tunneling time of one ns results in a PC of 1 pA, which can be detected with
57
6 Single exciton ground state
0.4 0.6 0.8 1.0 1.2
0
5
10
15
20
25
laser
928.03 nm
927.62 nm
= 927.35 nm
λ
928.45 nm
927.1 nm
Photocurrent (pA)
(reverse) Bias voltage (V)
Figure 6.1: Photocurrent resonances of the single exciton ground state transition at differ-
ent excitation wavelengths. At low voltages, one can observe the fine struc-
ture splitting (resulting from the QD asymmetry). At higher voltages, the line
width is broadened due to a reduced tunneling times and thus covers up the
fine structure slitting. (published in [SEZB04, ESdV+06a, ZSE+06, KSS+05])
the used setup.
Figure 6.1 shows photocurrent resonances of the single exciton ground state transition.
Each spectrum is measured at fixed laser energy and a varying bias voltage. If the laser
energy is tuned to a slightly lower energy, the resonance is shifted to a higher voltage.
•The resonances become broadened at higher voltages. A higher voltage induces a
higher tunneling probability and a shorter life time.
•At low voltages, a double resonance can be clearly observed. This is an effect of the
asymmetry of the QD, which is called fine structure splitting.
•The integral intensity of the absorption line is constant for voltages higher than 0,4
V.
•The absorption resonance of the ground state cannot be shifted into the region of
the photoluminescence. The resonance at 0,4 V is located in the transition region
between the photoluminescence and the photocurrent. Therefore the PC amplitude
is reduced.
58
6.2 Ground state properties at continuous excitation
Figure 6.2: Fine structure splitting of the single exciton ground state analyzed by different
linear excitation polarization. The fine structure splitting arises due to the
asymmetry of the QD. Each line can be clearly suppressed with respect to the
other.
6.2 Ground state properties at continuous excitation
6.2.1 Fine structure splitting
In figures 6.1 and 6.2, one can clearly see the fine structure splitting of the single exciton
ground state (double absorption line). As mentioned before, the STARK effect allows for
a conversion of the voltage into an energy scale. Thereby the fine structure splitting cor-
responds to an energy splitting of 30 μeV. This is caused by a slight asymmetry of the
QD concerning different crystal directions. One can see in figure 6.2 that each absorption
line can be clearly suppressed with respect to the other by an excitation with linear polar-
ized laser ([110]or [1¯
10]direction). A complete suppression is possible by using a slight
elliptical polarized excitation.
The polarization dependence of the resonances also remains by changing the excitation
to the pulsed mode. This is shown in double pulse experiments, where the fine structure
splitting leads to a quantum beating (see chapter 6.4.1). At low bias voltages, it is possible
to address only one single resonance, but for higher voltages the two resonances cannot
be fully distinguished. At higher voltages (0.8 V), one always gets a mixing of both states,
which always leads to a quantum beat behavior.
59
6 Single exciton ground state
Figure 6.3: ZEEMANN splitting of the single exciton ground state. The fine structure split-
ting (30 μeV at B= 0 T) increases with increasing magnetic field to a Zeeman
splitting of 0.85 meV at B=10T).
6.2.2 ZEEMAN splitting
In the previous section, the influence of an applied electric field was discussed. Not
only an electrical field, but also an applied magnetic field affects the states of the single
QD. Figure 6.3 shows 21 different characteristic curves of the QD measured at different
magnetic fields from zero to ten Tesla (0.5 T step width). All measurements are performed
at 4.2 K. The laser wavelength was fixed during this measurement. One can clearly see
that the splitting of the ground state increases with higher magnetic fields. The tunneling
processes of the charge carriers are in first order not affected by the magnetic field. The
normal fine structure splitting was determined to 30 μeV without magnetic field. The
splitting increases to 0.85 meV at a field of 10 T as a result of the ZEEMAN interaction.
The linewidth of the two resonances is normally broadened by the applied voltages and
not further affected by the magnetic field. Please note that the variation of peak heights
(at different magnetic fields) has no physical origin. The magnetic field slightly moves
the sample holder and the laser focus with respect to the QD position. The focus of the
objective must be re-adjusted at each magnetic-field setting. Thereby a slightly different
adjustment could lead to a different maximum peak height. The spectral position of the
two resonances is not affected by the displacement of the sample holder. Due to the high
resolution of the photocurrent technique, the energy of the two transitions can be clearly
determined with an increasing magnetic field. The magnetic splitting of the ground state
measured in the PL was shown before [BKF+99].
Figure 6.4 shows the spectral position of the two resonances depending on the applied
magnetic field. One can observe a clear quadratic dependence of the state energy with
60
6.2 Ground state properties at continuous excitation
(a) Energy of the fine structure levels with increas-
ing magnetic field. One can see an anticrossing of
the levels at 0 T.
(b) Energy splitting with increasing magnetic field.
For low fields the normal fine structure splitting is
dominant, but for higher magnetic fields the split-
ting increases linear.
Figure 6.4: ZEEMANN splitting with increasing magnetic field.
increasing magnetic field. The magnetic field dependence of the transition energy results
from the diamagnetic shift of the exciton plus the ZEEMAN energy. The interaction of the
exciton spin with B-filed depends on the spin orientation. In strongly confined QDs, the
exciton diamagnetic shift is determined by the single particle wave functions and is inde-
pendent of the exciton spin orientation. The different Bdependences of the photocurrent
resonances therefore result from their ZEEMAN splitting. Therefore, the behavior can be
well described by a square dependence on the magnetic field.
The (heavy hole) exciton is fourfold degenerate at zero magnetic field. The exciton
states are characterized by the total angular momentum malong the zdirection, m=
Se,z±Jh,zwith the electron spin Se,z=±1/2 and the hole spin Jh,z=±3/2. The states
with the total angular momentum of m=±1 are called bright excitons (m=±2are
called dark excitons). In this measurement, one can only observe (even at high magnetic
fields) the bright excitons. The luminescence of dark excitons in high magnetic field mea-
surements in a QD ensemble is reported by M. BAYER in [BKF+99]. Here, the dark exciton
could be observed at higher excitation intensities and other magnetic field geometries.
The breaking of the quantum dot symmetry by a magnetic field can be tested by tilt-
ing the magnetic field out of the [001] direction. The magneto-PL experiments (in the
so-called VOIGT configuration) is also discussed by M. BAYER in [BOS+02] for varying
magnetic-field strengths. At B= 0 T, a single emission line corresponding to emission
from the |M|=±1 excitons was observed. On its low-energy side an additional spec-
tral line appears at B= 2 T which originates from the |M|=±2 excitons (dark-exciton)
at zero field. For higher fields each of the two emission lines splits into a doublet. The
energy splitting between them increases with increasing B.
The field dependence of the splitting between the two states is given by
ΔE=(ge+gh)2μ2
BB2+E2
as
61
6 Single exciton ground state
Figure 6.5: Comparison of measured line width (open circles) and lorenzian fit curve.
where geand ghrespectively correspond to the electron and hole gfactor (formula taken
from [BKF+99]). Eas refers to the asymmetry (fine structure) splitting. At low magnetic
fields, the splitting increases quadratically, and then transforms into a linear dependence
(in magnetic field). The crossover occurs at rather small magnetic fields of about 1 T, be-
cause the ZEEMAN energy then becomes considerably larger than the asymmetry split-
ting. This behavior can be clearly seen in figure 6.4 b), where the energy splitting at low
magnetic field is only determined by the asymmetry splitting. By analyzing the reso-
nances even at negative magnetic fields, one can see that the two states perform an anti-
crossing with a minimum energy splitting which corresponds to the asymmetry energy.
A fit of the splitting as a function of the magnetic field (in the linear regime) leads to a
ZEEMAN splitting of 99.47 μeV/T. Therefore the absolute value of the sum of the electron
and hole g-factor is about 1.72 ±0.01. Here it is not possible to determine the underlying
electron and hole g-factors separately. The dashed line in figure 6.4 a) corresponds to the
mean value (diamagnetic shift) of both states.
6.2.3 Photocurrent saturation of a single quantum system
In this section, the behavior of the QD for increasing laser excitation is discussed. Fig-
ure 6.5 shows a measured fine structure splitting and a LORENZ-fit curve. As it can be
seen, the fit curve matches the measured fine structure line shapes very well. For further
analysis the linewidth of the fit curve, the peak height and the peak position are taken
from those fit curves. Figure 6.6 shows different spectra of the fine structure splitting
with increasing excitation intensity. One can clearly see that the resonances are broad-
ened up to higher excitation intensity. For low intensities, both resonances are clearly
separated (30 μeV). But for higher intensities, the two peaks have a clear overlap. For the
analysis of the saturation, the peak heights of the fit curves are displayed over the excita-
tion intensity. The occupation of the single exciton ground state shows a clear saturation
62
6.2 Ground state properties at continuous excitation
Figure 6.6: Fine structure resonances with increasing excitation intensity. A clear broad-
ening of the resonances can be observed. (published in [SEZB04, ESdV+06a])
behavior.
I=N1e
τtunnel
where eis the elementary charge, N1is th upper state occupation probability, and τtunnel
is the tunneling time of the state.
The saturation behavior for continuous excitation can be written as:
I=ISat
P
P+1
with
ISat =e
2τtunnel
.
Figure 6.7 shows the comparison between PC peak height and the fit curve. One can
observe a very good agreement between measurements and fit curve. Please note that
the applied voltage of 0.4 V is in the region between the photoluminescence and the
photocurrent. Therefore the tunneling time should be in the region of the optical recom-
bination time. The analysis of the fit parameters results in a tunneling time of 5,3 ns. The
normalized excitation intensity
P=1 corresponds to an intensity of only 35 nW at the
sample.
The tunneling time of 5.3 ns is not in contrast to the optical recombination time of
63
6 Single exciton ground state
Figure 6.7: Saturation of the single exciton ground state at a bias voltage of 0.4 V. The
peak heights of the fit curves (square points) are displayed over normalized
excitation intensity. The fit curve is taken from formula 6.2.3. (published in
[SEZB04, ESdV+06a, ZSE+06, KSS+05])
about 500 ps. The exciton consists of two charge carriers, the electron and the hole. It
is plausible to think that both charge carriers have a different effective masses, different
confinement potentials and therefore also different tunneling times. The saturation value
is only determined by the tunneling time of the slower charge carrier, which is probably
the hole. The initial state (empty dot) is reached when both charge carriers are tunneled
out of the dot. Therefore the slower tunneling time determines the saturation value. The
energy level of the QD in the presence of an extra hole is separated well enough from the
laser energy, so that no further absorption can take place. Thus, further absorption can
only take place when the QD is again in the initial state.
The power dependence of the peaks heights corresponds well to the theoretical model.
Even at low bias voltages, a saturation can be observed for very low excitation intensities
(<100 nW). Other measurements with higher bias voltages lead to the same qualitative
behavior. A higher voltage leads to shorter tunneling time for both charge carriers and
therefore to higher saturation value (37 pA at 0.5 V and 75 pA at 0.58 V). A good overview
of the saturation behavior which includes the theoretical description of the rate equations
can be found in [BZF+01].
6.2.4 Power broadening
The effect of the excitation power induced broadening of the absorption linewidth has
been known for a long time from atomic optics. The observation of the power broadening
in a single QD requires a very high resolution of the experimental setup. The technique
of the photocurrent spectroscopy allows for a very detailed observation of the power
induced line broadening.
As theoretically discussed in chapter 3.5.2, one expects a broadening of the absorption
64
6.2 Ground state properties at continuous excitation
Figure 6.8: Broadening of the absorption linewidth with increasing excitation intensity.
The only free parameter for the fit curve here is the y-axis segment. Due to the
very good agreement from the measurement and the fit curve it is guaranteed
that no other line broadening effects are relevant here. (published in [SEZB04,
ESdV+06a, ZSE+06, KSS+05])
linewidth as a direct consequence from the saturation with: Γ=Γ01+
P(Formula
3.37). Figure 6.8 shows measurements of the absorption line of the same state for increas-
ing excitation amplitudes. The data which is analyzed here, corresponds to the same set
of data as used for the saturation analysis (see figure 6.7). The full line is the theoretical
fitcurve. The very good agreement between the measurement and the theoretical fitcurve
is obvious. The scaling of the x-axis is taken from the figure 6.7. The only free parameter
for the fitcurve here is the y-axis-value (at
P=0). This value corresponds to the linewidth
for vanishing excitation power Γ0. The increase of the fitcurve is determined by formula
3.37, because the conversion of the laser power into the normalized excitation power
Pis
identical to the analysis of the saturation.
Due to the fact that the measured linewidth matches extremely well the fitcurve, it can
be concluded that the line broadening is only determined by the saturation. Thereby
all other possible line broadening mechanisms can be neglected. One can see in fig-
ure 6.8 that the power induced saturation broadening can cause a significant increase
in the linewidth. Compared to the linewidth for vanishing excitation power Γ0, a power
of about 100 nW results in a doubling of the linewidth. Thus, it is possible that details
of a resonance (e.g. a fine structure splitting) can be washed out by applying too high
excitation intensity. The theory, which is described in chapter 3.5.2, only includes the line
broadening effect of the saturation, a dephasing effect is not included. In order to de-
termine the dephasing time T2from the linewidth, one has to take the saturation broad-
ening into account. For the correct determination of this value, one has to extrapolate
the linewidth to vanishing excitation power. This method is used in figure 6.16 for the
determination of the T2-times of the quantum system resulting from the linewidth.
65
6 Single exciton ground state
Figure 6.9: Relevant timescales for coherent control experiments in the photocurrent. The
tunneling time of both charge carriers strongly depend on the applied voltage.
The tunneling time of the electron is longer than the pulse length (adjusted by
the voltage). The hole tunneling time is comparatively long, but shorter than
the laser repetition time.
6.3 Coherent properties
In the case of continuous wave excitation, it is only possible to occupy the QD with a
statistical probability of 0.5. The use of pulsed excitation allows for a defined state prepa-
ration of the two level system within a single interaction. The fundamental effect in the
coherent regime is the RABI-oscillation. The occupation of the two-level system is thereby
measured depending on the incident pulse area 1. With increasing excitation intensity, an
oscillation behavior of the occupancy can be observed. The physical content of this effect
is fairly easy. At low excitation intensities, it is possible to excite the two level system.
The excitation probability is proportional to the square of the pulse area in the low in-
tensity regime. If the excitation intensity increases, it is possible that stimulated emission
follows the excitation within one pulse. Therefore the measured signal will oscillate (with
increasing excitation intensity) between the two extremes 0 (no exciton generated) and 1
(every pulse generates an exciton). In recent years, a lot of works have been published
concerning coherent experiments on semiconductor QDs [SLS+01a, KGT+01, HTK+02a,
ZBS+02, BLS+02, BBM03, MWB+04, WMB+05, UML+05, PLW05].
6.3.1 RABI-oscillations
The theoretical description of the RABI-oscillations is discussed in chapter 3, here the
experimental results are presented. The pulse area (or the rotation angle on the BLOCH
sphere) is adjusted by the attenuation of the laser field amplitude. The RABI-oscillations
are affected by the amplitude of the exciting laser field, which corresponds to a quadratic
dependence of the RABI-oscillations from the laser power.
For coherent control experiments, one has to take some important time scales into ac-
count. It must be guaranteed that each laser pulse interacts with the QD in its initial state.
1The pulse area is the result of the integration of the laser field amplitude during a single pulse.
66
6.3 Coherent properties
Figure 6.10: RABI-oscillations of a single exciton ground state. The photocurrent mea-
sured is directly proportional to the occupancy of the two level system. The
signal shows a very periodical oscillation with a very slight damping of the
signal (background corrected). (published in [SEZB05, ZSE+06, KSS+05])
That means the repetition time period of the laser must be longer than any fundamental
process in the QD. In the case of coherent control of the single exciton ground state, the
laser repetition time has to be longer than the radiative recombination time. On the other
side, a low repetition frequency leads to a low current, see formula 6.1. Additionally,
the performance of the coherent control is only good when the coherence of the QD is
maintained during the interaction. The fastest dephasing process is, in the case of the
coherent ground state control, the tunneling process of the electron. Both tunneling pro-
cesses (electron and hole) are strongly dependent on the applied voltage and range from
10 ps to 300 ps (electron), which is longer than the used laser pulse width (2 to 4 ps).
The slowest tunneling process is in any case faster than the repetition period of the laser.
Therefore each laser pulse interacts with the QD in its initial state. In the case of pulsed
excitation, any generated exciton is able to tunnel out of the QD (in the PC regime). Every
tunnel process contributes to the PC with one elementary charge. The maximum current
is thereby defined by the laser repetition frequency and the elementary charge [ZBS+02].
I=P1X·fLaser ·e(6.1)
fLaser corresponds to the laser repetition frequency, and P1Xto the occupation probabil-
ity. This maximum value (12.8 ps for flaser = 80 MHz) can only be reached if no dephasing
process takes place during the interaction, for example optical recombination in the tran-
sition regime of photocurrent and photoluminescence.
Figure 6.10 displays the RABI-oscillation of the single exciton ground state (photocur-
rent versus excitation pulse area). This measurement is performed at a voltage of about
67
6 Single exciton ground state
Figure 6.11: RABI-oscillation without background correction. Probably due to stray light
absorptions close to the QD a background signal is generated, which is linear
to the laser power.
0.6 V and circular polarized excitation. The peak width of the auto correlation signal
amounts to 3,45 ps, which corresponds to spectral width of 0.6 meV. With the assump-
tion of a sech2pulse-shape one gets a pulse length of 2.25 ps. The pulse area of one π
corresponds to a time averaged power of 2 μW at the sample. With increasing excitation
intensity, a background current increases continuously. This is displayed in figure 6.11.
All further PC-measurements of RABI-oscillations shown in this work are corrected for
this quadratic background signal. The background signal increases linear to the excita-
tion power and is probably caused by incoherent absorption processes in a wider area
around the nano aperture. At low excitation intensities, the incoherent signal is very
low (6% at π-pulse excitation). But for higher excitation intensities, the incoherent signal
strongly exceeds the coherent signal. In order to improve this, the sample design must be
corrected. The active region of the SCHOTTKY-diode should be reduced as much as pos-
sible. Due to multiple laser reflections between back contact and shadow mask, a lot of
unwanted off-resonance absorptions are possible. All absorptions in the mesa structure
(300 ·400 μm2) contribute to the background. A reduced active region should improve
the background signal notably.
The measured RABI-oscillations are slight damped. Comparing the results to other
measured RABI-oscillations [ZBS+02], the damping is very low. One can clearly see that
coherent manipulations of up to 9 πare possible within one pulse. The improvements in
comparison to previous results maybe caused by one of the following points:
•The self-organized grown InGaAs/GaAs QDs own a much higher confinement
potential than, for example, GaAs/AlGaAs interface fluctuations [SLS+01a]. The
stronger damping of the RABI-oscillation is likely occurred due to the presence
68
6.3 Coherent properties
of nearby states that are not well confined. QDs with stronger confinement are
expected to be less sensitive to these effects. Thereby the dephasing of the In-
GaAs/GaAs QDs is much lower and this results in an enhanced coherent control.
•Single QD measurements are not affected by inhomogeneous broadening effects, as
in ensemble measurements [BLS+02].
•The ground state exhibits a much longer dephasing time than any excited state,
see chapter 7.4 (p-shell RABI-flopping). [KGT+01, HTK+02a, BBM03, MWB+04,
WMB+05].
•Comparing these results to (former) measurements on similar structures, reported
in [ZBS+02], one has to notice that the electrical and optical setup was strongly im-
proved; especially the laser system for the excitation provides a much more stable
signal.
6.3.2 Damping of the RABI-oscillations
The pulse area (needed for RABI-oscillations) is simply increased by a change of the atten-
uation of the excitation amplitude. Thereby the damping of the RABI-oscillation is only
affected by the excitation amplitude. A constant low damping of the RABI-oscillations
is observed in the range between 0.4 V and 0.75 V. This is remarkable, because the de-
phasing time in this region changes from 320 ps to 50 ps (see figure 6.16). With further
increase of the applied voltage, and thereby reduced tunneling times, the increase of the
photocurrent up to the first maximum is similar as compared to low voltages. But the
decrease to the minimum is reduced, shown in figure 6.12. Similar to the measurements
presented before, the real PC value (at high pulse areas) of the signal could be slightly dif-
ferent due to the subtraction of the incoherent background. However, the results shown
here are the most probable, because the background is assumed to be monotonous. It is
remarkable that the photocurrent does not approach the value 1/2 for occupation. This
can be explained by the fact that the tunneling process is the main dephasing process in
this voltage region. If a tunneling process takes place during the interaction, the QD is
then again in the initial state. The residual pulse area for this interaction is then reduced.
This statistically leads to an increased damping and to a higher incoherent signal. The
average occupation for high pulse areas is then higher than 1/2 for a single interaction.
Therefore, there is a difference in the damping behavior of RABI-oscillations between
short and long dephasing times, see figure 6.10 and 6.12. The strong damping, which
arises from the dephasing, can only be observed if the dephasing time reaches the mag-
nitude of the pulse length. Then the two effects of power induced damping and damping
due to short dephasing times are overlaid. The influence of the fast dephasing processes
during or shortly after the excitation is also subject of investigations of other groups,
see for example [BLS+01, VAK03]. There are different theoretical approaches in order to
identify the damping mechanisms. Due to the oscillation up to high pulse areas (9 π),
these results are helpful for theoretical simulations.
One approach for the explanation is the electron-phonon interaction. J. FÖRSTNER et.
al. calculated a density matrix theory of the electron-phonon interaction up to the second
69
6 Single exciton ground state
Figure 6.12: RABI-oscillations at 1,2 V applied voltage. The oscillation exhibits a stronger
damping due to the short tunneling time of only 10 ps. The photocurrent
does not approach the occupation value of 1/2 for high excitation intensity.
order of a correlation expansion [FWDK03]. The presented phonon-induced pure dephas-
ing exhibits a reduced damping of RABI oscillations in QDs. For pulse durations on the
time scale below the phonon-induced dephasing, a pure single exponential approxima-
tion can be applied. But for longer pulses, the full quantum kinetic dynamics must be
included.
In reference [MJ04] P. MACHNIKOWSKI and L. JACAK studied the carrier-lattice dynam-
ics for optically induced RABI oscillations of exciton occupation in a QD. They showed
that the lattice response is resonantly driven by a combination of the linear pulse spec-
trum and the RABI-frequency. The RABI-oscillations are almost perfect for very short
pulses (1 ps), then lose their quality for longer pulse durations (10 ps). It is surprising
that in this theory the quality of oscillations is again nearly perfect for longer pulses (50
ps).
J. M. VILLAS-BOAS et. al. studied the influence of multiexciton and wetting layer
states [VBUG05]. They described that for short pulses the two-level model fails and
higher levels should be taken into account. Thus, the damping observed cannot be
explained using constant rates with fixed pulse duration. The damping of the RABI-
oscillations could be induced by an off-resonant excitation to or from the continuum of
wetting layer states.
70
6.3 Coherent properties
Figure 6.13: RABI-oscillations for different detuning (concerning laser - resonance at
VB=0,7 V). The central curve at 927.6 nm corresponds to resonant excita-
tion. With increasing detuning, the amplitude of the oscillation decreases.
6.3.3 Detuning
In chapter 3, the solution of the optical BLOCH equations with and without detuning is
shown. For the case of a slight detuned excitation this leads to [MS91]:
P|1X=Ω0
Ω02+δ2sin2(Ω02+δ2t
2). (6.2)
If the detuning δis higher than the RABI-frequency Ω0for resonant excitation, this re-
sults in a notably reduced occupation probability N1for the upper level. In addition, the
RABI-frequency Ω02+δ2is enhanced. Experimentally, the RABI-oscillations are mea-
sured as a function of the excitation amplitude, which is proportional to Ω0. For a strong
excitation (or Ω0>δ), the occupation of the upper level converges fast to the values
known from resonant excitation. The maximum occupation probability almost reaches
100 %. This prediction cannot be verified in the experiment. While the simplified and
analytically solvable theoretical model supplies a good description in almost all regions,
the spectral broadening of the short pulses influences the measurements.
Figure 6.13 shows the RABI-oscillations for different detunings. The central curve cor-
71
6 Single exciton ground state
responds to the case of resonant excitation. The detuning changes in steps of 0.1 nm of
laser wavelength (0.144 meV). The peak width at FWHM was (in this experiment) 0.4 nm.
Therefore the curves displayed cover the whole width of spectral pulse range. With in-
creasing detuning, a continuous damping of the RABI-oscillations can be observed. Con-
trary to the expectation of the simple theoretical model, the amplitude only slightly ap-
proaches the resonant case. In the case of a small detuning, a significant change of the
RABI-frequency is neither measured nor theoretically expected. Even in the strongest
detuning (δ=0,23 Ω0), one only expects a shift of maximum 3 %.
72
6.4 Quantum interference
6.4 Quantum interference
6.4.1 Double π/2 pulse excitation
In this section, the coherent behavior of the single exciton ground state is used in two
time delayed interactions. While a measurement of RABI-oscillations represents the oc-
cupancy of a two-level system, one has to perform quantum interference experiments to
also gain access to the phase of coherent excitations. If the QD is excited in a coherent
process, the generated exciton will oscillate with the excitation frequency. Then the phase
of the excitation pulse is stored in the QD. The phase of the excitation will be lost after
the dephasing time. The main dephasing mechanism is the tunneling process, which
is strongly depending on the applied voltage. This leads to a loss of the phase relation
between the exciton and the laser pulse. In order to determine the dephasing time it is
necessary to test the interference ability of the exciton with a second laser pulse. For
long time delays between the excitation pulse and the delayed test pulse, the detectable
interference decreases.
The basis for measurements of the dephasing time is the coherent state preparation.
The first manipulation excites the two level system with a π/2 pulse and thereby defines
a phase. The excitation pulse of two times π/2 corresponds to a pulse area of 1π(sin-
gle pulse) at complete overlapping pulses and constructive interference. The π/2-pulse
corresponds to a rotation of the BLOCH-vector from the zero state to the equator. Thus,
the first pulse creates a coherent superposition of the "0" and "1" states. For the test of the
interference ability, the second pulse follows up with a variable delay in the range of 0 to
1000 ps. The relative phase of the second pulse can be controlled via an additional fine
delay with sub femto-second resolution (25 nm delay of one interferometer arm). At time
separated excitation pulses, the QDs superposition state is expected to be transferred into
the pure "1" or "0" states, depending on whether the excitons phase and the laser phase
interfere constructively or destructively. By varying the phase continuously (via the fine
delay), an oscillation of the final state can be observed. After the interaction of the second
pulse, the exciton is able to tunnel out of the QD and the final state is displayed in the PC.
For long delay times, both interactions lead to independent tunneling processes (with a
current of two times π/2) and no interference can be observed. The observed oscillation
in the PC has the same period as the excitation wavelength.
Figure 6.14 shows a measurement of the quantum interference of the QDs exciton and
the laser pulse. The time delay here is 133 ps at an applied voltage of VB=−0.59 V.
One can see a clear and symmetric (around the mean value) oscillation with varying fine
delay. The oscillation frequency corresponds to the excitation wavelength. The optical
path length is two times the coarse delay (back and forth). For the analysis of the de-
phasing time of the system, the decay of the oscillation amplitude must be investigated
as a function of the time delay. The amplitude of these oscillations versus delay time is
displayed in figure 6.15. At low time delays, the interference amplitude corresponds to
a full πpulse, but for longer time delays, the amplitude decreases due to the dephasing
process. Due to the strong voltage dependence of the tunneling time, this measurement
must be performed at different voltages.
Figure 6.15 shows the result of the analysis of the oscillation amplitude depending on
73
6 Single exciton ground state
01234
6
7
8
9
10
x = 20 mm
Photocurrent (pA)
Fine delay (µm)
Figure 6.14: Interference pattern in the photocurrent at double pulse excitation (applied
voltage:VB=−0.59 V and time delay: 133 ps).
the time delay. The data (represented by squares) correspond to an excitation polarization
where the excitation is limited to only one of the asymmetry split levels. A fit to these
data points reveals a purely exponential decay, corresponding to a dephasing time of
T2= 322 ±5 ps at an applied (reverse) bias voltage of 0.4 V. The analysis of the signal in
the timescale at overlapping pulses is complicated and neglected here. It is still possible
to determine the dephasing time without using the data by overlapping pulses. The
circles correspond to an excitation polarization at which both states of the asymmetry
split doublet are excited. Here an additional beating can be observed with a period of
T = 133 ps. Due to the fact that the experimental setup is insensitive to a phase shift of
180◦, the data has to be fitted to an exponential decay modulated by the absolute of a
cosine function cos(πt/T). The period Tcan be converted into an energy difference by
using ΔE=h/T=31μeV, which is in good agreement with the value observed in a
direct measurement as displayed in figure 6.2.
Another feature of the double pulse technique is the possibility to compare the de-
phasing times measured by quantum interference experiments with those derived by
linewidth analysis. Especially at low applied bias voltages (corresponding to long tun-
neling times), one has to consider that the linewidth is mainly determined by the power
induced line broadening. One has to perform a full power broadening analysis with an
extrapolation to zero excitation for all measurements up to 0.6 V. At higher bias voltages,
the PC saturation value is high enough so that the linewidth of a low-power spectrum
already exhibits the correct results. The linewidth can be converted into a dephasing time
T2via T2=2¯h/Γ. The corresponding data are plotted versus bias voltage in figure 6.16
(full circles).
74
6.4 Quantum interference
Figure 6.15: Dephasing of the QD ground state at 0.4 V. The interference amplitude de-
creases with increasing time delay due to the dephasing processes. The ad-
ditional (quantum) beating can be observed if both fine structure states are
excited. The beating frequency matches the fine structure energy splitting
very well. (published in [SEZB05, ZSE+06])
The strong voltage dependence indicates that the dephasing time is mainly determined
by the tunneling time. Optical recombination processes are only of significance at very
low voltage levels, in the transition region from photocurrent to photoluminescence. In
the investigated regime, it is therefore possible to control the main dephasing mechanism
of the quantum system via the bias voltage. T2times derived from the quantum inter-
ference experiments are shown in the same figure, indicated by triangles. Both sets of
data show excellent agreement up to a bias voltage of 0.7 V. This indicates that dephasing
times for excitation with two π/2 pulses are exactly the same as for vanishing excitation
power. Both sets of data agree in a range from 80 to 320 ps. At higher voltages, one
observes quantum beats independent of the choice of polarization.
A possible explanation for this behavior is based on the strongly reduced tunneling
times at high applied voltages. At a (reverse) bias voltage of 0.8 V, the dephasing time of
linewidth analysis and quantum interference differ. Here the dephasing time amounts to
approximately 60 ps, see figure 6.16. But this is also the first minimum of the quantum
beating, see figure 6.152. At (reverse) voltages higher than 0.8 V the linear polarization
2In this figure the beating is shown at a (reverse) bias voltage of 0.4 V. The beating frequency is not influ-
enced by the applied voltage.
75
6 Single exciton ground state
Figure 6.16: Comparison of the dephasing times derived from line width analysis and
double pulse interference. Up to 0.8 V the dephasing time of the line width
analysis matches the results of the double pulse experiments very well. (pub-
lished in [SEZB05, ZSE+06])
of the fine structure levels may no longer be developed due to the short tunneling times.
Even in cw-resonances, one can see that the fine structure splitting vanishes for voltages
higher than 0.8 V, due to the tunneling induced line broadening. Here it is no longer
possible to analyze the polarization dependence of both states. If the linear polarization
of the fine structure states is no longer maintained, a single state is not addressable by
the orientation of the linear excitation polarization. Thereby the pulsed excitation would
always excite both fine structure states at high (reverse) bias voltages and a quantum
beating could not be suppressed.
The linewidth of the ground-state resonance corresponds very well to the decay time of
quantum interference. The asymmetry-induced splitting of energy levels is obtained in
the period of quantum beats. The polarizations at which these effects can be suppressed
are the same in both types of experiments at voltages lower then 0.8 V. From saturation
and power broadening experiments, one derives a characteristic dimensionless power
level ˜
P=Ω2T1T2.AsT1and T2times are also obtained from these experiments, it is
possible to compare the resulting RABI-frequency Ωto a direct measurement of RABI-
oscillations. In cw measurements, one gets a typical value of Ω≈0.2 GHz at a laser power
of P = 100 nW. A comparison of different measurements delivers a more general ratio of
Ω/√P=0.19 ±0.4THz/mW1/2. In a measurement of RABI-oscillations as shown in
figure 6.10, the laser power for an inversion (π-pulse) corresponds to 2 μW. After a con-
version into continuous excitation this value results in Ω/√P=0.25 ±0.3THz/mW1/2.
Both values of the RABI-frequency agree remarkably well.
76
6.4 Quantum interference
6.4.2 RAMSEY fringes
In this section, the coherent control of a QD two level system in a double-pulse RAMSEY
experiment is presented. The experimental setup for the RAMSEY experiment is nearly
the same as the double pulse experiments. The only difference here is the fixed time delay
between the two laser pulses and the variation of the applied voltage. A change of the
voltage directly affects the phase evolution of the exciton.
History
The original experiment, performed by NORMAN F. RAMSEY, consists of the interaction
of two time delayed electromagnetic pulses with an incomplete inversion (π/2-pulses)
of quantum systems. For the quantum systems, an atomic beam was used, which was
excited by two microwave pulses. The time delay was reached due to spatially separated
interaction zones where the atoms need time to pass through. The required detuning
between the quantum systems and the excitation was achieved by the detuning of the
microwave field compared to the transition energy. The first state manipulation takes
place in the first interaction zone with a pulse area of π/2. The second manipulation
takes place in the second interaction zone. Now it depends on the phase of the quantum
system whether the interaction takes place constructively or destructively. The final state
(respectively the phase before the second interaction) thereby depends on the quantum
state detuning. The transition energy of the quantum system is fixed and is not influenced
by an external parameter. Thereby it was possible to stabilize the microwave frequency
with the optically measured final state. This concept is the basis for extremely accurate
atomic clocks, where the time is triggered by the stabilized microwave frequency. This
effect was first described by NORMAN F. RAMSEY [Ram90] and it was honored in 1989
with the NOBEL prize.
RAMSEY effect in an electric field tunable device
With double pulse excitation in an electric field tunable device, it is possible to control
the occupancy and the phase of the quantum mechanical two level system entirely via
the bias voltage. Comparing the original RAMSEY experiment with the setup used here,
the laser frequency (quantum state excitation) is fixed, and the detuning is done via the
QCSE, see chapter 5.2. The QD transition energy decreases with increasing (reverse) bias
voltage. A detailed measurement of the STARK effect gives a conversion of voltage into
energy scales. Therefore it is possible to tune the QD energy precisely with respect to a
fixed laser wavelength. In the present case, the final quantum state is detected in the PC.
Thus, the detuning is switched from the excitation to the quantum system.
Figure 6.17 (a) shows the difference in the PC response between pulsed and continuous
excitation of the exciton ground state in the PC. In the case of continuous excitation, the
observed spectral response corresponds to the homogeneous linewidth of the system.
For pulsed excitation, the PC signal directly displays the power spectrum of the laser.
The FWHM of 0.47 meV corresponds to the transform limit of a 2.6 ps laser pulse. The
QD can therefore act as a spectrum analyzer in the PC regime.
77
6 Single exciton ground state
Figure 6.17: (a) Photocurrent spectra of the single exciton ground state comparing con-
tinuous and pulsed (π/2) excitation. (b) Schematic picture of the phase
relation between the two laser pulses (central) and the two cases of de-
tuned (upper curve) and resonant (lower curve) QD excitation. (published
in [SEZ06, ESdV+06b, ZSE+06])
The exciton ground state of the QD represents a two level system. The lower state
|0is defined by an empty dot, and the upper state |1by a single exciton ground state
occupancy. A πpulse switches the occupation from |0to |1or vice versa. The time sep-
aration of the two π/2 pulses is achieved by the MICHELSON interferometer, described in
section 4.1.5. Figure 6.17 (b) schematically displays the phase evolution of the quantum
state for two laser pulses. The central curve corresponds to the two laser pulses with the
frequency ωlaser separated by a delay time tdelay. In the lower case, the QD oscillates in ex-
act resonance (ωQD =ωlaser) with the laser. After the first laser pulse, the QD is therefore
in a coherent superposition between |0and |1. The second laser pulse is then in phase
and the system will be transferred into the pure state |1. The upper curve corresponds
to the same situation with a slightly detuned QD. The QD phase is defined again by the
first laser pulse. But the phase evolution will evolve with a slightly different frequency
(ω
QD =ωlaser −Δωwith Δωω). When the second interaction occurs, the QD can
have a different phase when compared to the previous case (opposite phase in this exam-
ple), resulting in the final state |0. Within the pulse duration, the phase relation between
the QD and the laser can be regarded as fixed. The final state occupancy will oscillate as
a function of the detuning Δωwith a period of 2π/tdelay. The frequency of the spectral
fringes increases directly proportional to the delay time between the two pulses.
Figure 6.18 displays RAMSEY fringes with different time delays in a range from 33 to
167 ps. The oscillation is caused by a voltage dependent detuning of the QD between the
two interactions. Obviously the oscillation frequency increases directly proportionally to
the delay time between the two pulses. There is also a slight increase in the frequency
of the RAMSEY-fringes observable towards higher bias voltages. This is a result of the
nonlinear (quadratic) STARK effect. The envelope of the PC spectra corresponds to the
78
6.4 Quantum interference
Figure 6.18: RAMSEY fringes measured in the photocurrent with double pulse excitations
for different delay times. (published in [SEZ06, ESdV+06b, ZSE+06])
79
6 Single exciton ground state
spectrum of a single pulse with a pulse area of two times π/2. At a fixed bias voltage,
the interference contrast decreases due to the finite dephasing time of the system toward
longer delay time. Also at a fixed time delay, the interference contrast decreases with
increasing bias voltage. This is caused by the strong voltage dependence of the dephasing
time. At long time delays and high (reverse) voltages, both pulses generate a PC signal of
π/2 (as shown in figure 6.17 a). In this regime, the tunneling time is shorter than the delay
time between the two laser pulses. The interference will be lost if the exciton tunnels
out of the QD before an interaction with the second pulse becomes possible. RAMSEY
fringes are visible at delay times even longer than the dephasing time of the QD system.
This is shown in the inset of figure 6.18. The half period of the RAMSEY fringes at a
voltage of 0.43 V and a delay time of 670 ps is only about 3 μeV, which is smaller than
the homogeneous linewidth of the system (5 μeV). The spectral resolution of the double
pulse experiment hence exceeds any possible single pulse experiment.
The RAMSEY method can also be applied to a quantum mechanical two qubit gate
operation. As shown in figure 6.18, very small energy shifts are already sufficient for a
full 1πvariation of the final state rotation angle. This energy shift can be achieved by
e.g. a dipole-dipole interaction between neighboring dots ([UML+05]). Thus a relatively
weak coupling of states could be used for a conditional qubit rotation. It is possible to fit
the gate operation time to any coupling strength simply by varying the delay time. But
any change in the environment of the QD, which has an effect on the exciton transition
energy, will lead to a detuning of the system. This will result in an observable variation
of the final state occupancy at a constant bias voltage. The QD can therefore act as a
highly sensitive quantum sensor for interactions occurring in the time interval between
the two laser pulses. The RAMSEY setup offers a unique switch between constructive
and destructive interference simply by a small variation of the bias voltage. For delay
times longer than the dephasing time, the spectral resolution of the RAMSEY setup is
increased beyond the continuous excitation limit (i.e. the homogeneous linewidth). Due
to the RAMSEY method, here a new link between voltage based control and the coherent
control of single quantum system is demonstrated.
80
7 First excited state
7.1 Voltage dependent photoluminescence for p-shell excitation
In chapter 5.3, the spectral position of the excited states of the single QD was analyzed.
Thus, it is possible to investigate the p-shell transition for different external parameters,
e.g. voltage, polarization, excitation intensity, etc. Due to the integration of the QD into
an-i-SCHOTTKY diode, it is possible to perform the PLE measurements at different volt-
ages. The results of this measurement series is shown in figure 7.1 for voltages from 0 V
down to -0.5 V. The spectral position of the p-state shifts as expected from 0.0 V to -0.1 V
due to the QSCE. From -0.1 V to -0.2 V, however, the p-state signal shows a strong shift
of about 2 meV into the red. Going further to higher negative voltages, the p-state peak
position again shifts continuously with the applied bias voltage. The jump of the p-shell
peak position from -0.1 V to -0.2 V is an effect of a change of charge carrier in a QD.
Figure 5.1 shows that the X−-state vanishes between -0.1 V to -0.2 V. Here the electron
ground state is shifted above the quasi-FERMI-energy of the back contact, and electrons
from the doped back contact are no longer able to tunnel into the QD. An extra elec-
tron leads to energy renormalization also for the excited state. The PLE peak intensity
decreases as expected towards higher applied voltages. The probability of a tunneling
process for the excited exciton dominates the optical recombination process at higher
applied voltages.
The LO-phonon absorption peak (1.376 eV) decreases and shifts with the applied volt-
age, as expected. The PL signal of the peak at 1.38 eV (probably the d-shell) decreases
very quickly. This can be explained by the higher tunneling probability (much lower tun-
neling barrier) for the charge carriers at higher excited QD states. Other measurements
performed on this sample (by MARC.C.HÜBNER), concerning the photocurrent signal
of excited QD states, verify the fact that the tunneling probability is much higher for ex-
cited states. Two further resonances appear between -0.2 V and -0.3 V, which couldn’t be
assigned up to now.
Another interesting feature of the p-shell excitation is shown in figure 7.2, which shows
the voltage dependent ground state luminescence at p-shell excitation. For normal volt-
age dependent PL (at non-resonant interband excitation, figure 5.1), the luminescence
of the exciton vanishes at -0.35 V. In the case of resonant p-shell excitation, one can ob-
serve luminescence of the ground state (and the positive charged exciton) up to a voltage
of -0.8 V. At a voltage of -0.35 V, the intensity of exciton is strongly reduced. Here the
tunneling time decreases to values shorter than the radiative lifetime. It is clear that the
generated charge carriers are able to tunnel out of the QD direct from the p-shell (strongly
reduced barrier) or from the ground state. Comparing the non-resonant interband and
the p-shell excitation, the non-resonantly generated interband excitons are able to tunnel
out from even higher levels. Therefore the ground state PL at p-shell excitation can be
81
7 First excited state
1.35 1.36 1.37 1.38 1.39
0
2000
4000
6000
8000
10000
-0,1 V
-0,2 V
-0,3 V
-0,4 V
PL intensity (a.u.)
Excitation energy (eV)
-0,5 V
0 V
Pexc = 80 µW
Figure 7.1: Voltage dependence of the PLE signal at ground state detection (neutral ex-
citon) and resonant p-shell excitation. Up to higher voltages the normal lu-
minescence vanishes due to change to tunneling processes. In the step from
-0.1 V to -0.2 V the p-shell transition shifts 2 meV into the blue. In this region
the QD cannot be charged from electrons from the back contact.
82
7.2 Excitation polarization dependence of the p-shell
Figure 7.2: Voltage dependence of the ground state by resonant excitation of the p-shell.
The luminescence of th X+can be observed probably due to a biexciton gen-
eration and a tunneled exciton.
seen even at higher applied (reverse) voltages as compared to non-resonant interband
excitation.
In figure 7.2 one can also see the luminescence from the recombination of the positive
charged exciton. This is not possible if only a single exciton is generated. In the voltage
region of the positive charged exciton, only electrons are able to tunnel out of the QD.
It is not possible that a hole tunnels into the QD in order to create a positively charged
exciton. Therefore it is probable that the X+is generated by a biexciton excitation and an
electron tunneling process. The measurements of the photocurrent saturation and power
broadening (chapter 6.2.3 and 6.2.4) have shown that the tunneling times are not equal
for electrons and holes, which intensities this excitation scheme. The PL-intensity of the
Xand X+PL is nearly equal and decreases slowly up to higher negative voltages. One
can see that the STARK effect is not equally strong for Xand X+. The excitation intensity
here is rather high (1.2 mW before pellicle, which corresponds to an intensity at the QD of
approximately 50 μW). The generation scheme for the biexciton is discussed in chapter 9.
In order to analyze the excitation of the biexciton, a PLE measurement is performed with
the detection on the biexciton luminescence which is presented later (see figure 9.1).
7.2 Excitation polarization dependence of the p-shell
In the photocurrent regime, the possibility of fine tuning of the QD state (via QCSE)
allows for high resolution spectroscopy of the QD states. Thereby it is possible to re-
83
7 First excited state
1.365 1.370 1.375
0
500
1000
1500
2000
2500
3000
λ
/4 = 190°
λ
/4 = 235°
PL intensity (a.u.)
Excitation energy (eV)
PLE XP
(a) Dependence of circular polarization
1.365 1.370 1.375
0
1000
2000
3000
4000
5000
6000
7000
λ
/2 = 220°
λ
/2 = 265°
PLE XP
PL intensity (a.u.)
Excitation energy (eV)
(b) Dependence of linear polarization
Figure 7.3: Properties of the ground state luminescence at different excitation
polarization.
solve the fine structure splitting of the QD ground state. In the PLE measurement, the
resolution of the excitation is limited by the used laser system (step width for spectral
tuning). Furthermore, the short lifetime of the excited state leads to a spectral broadened
linewidth. In the ground state, one state of the fine structure could be suppressed by a
linear polarized excitation, see figure 6.2. In order to analyze a possible fine structure
splitting of the p-shell, a polarization dependent PLE measurement (linear and circular)
was performed. One tuning step of the laser energy corresponds to an energy of 50 μeV.
The results of the polarization dependent PLE analysis are shown in figure 7.3. In case
a), the excitation of the linear polarized laser is turned into circular polarization by using
aλ/4 plate in the excitation path. By tilting the fast axis 22.5◦from the laser axis, one
generates circular left (and respectively circular right). One can see that the PL intensity
shows a slight shift of the peak position between circular left and right polarization. The
PL intensity can be reduced to nearly half intensity. Figure 7.3 b) compares the two linear
polarizations (vertical and horizontal). The change of the linear polarization only slightly
affects the peak height.
A clear fine structure splitting is not observed here, as it is possible in the ground state.
A possible p-shell fine structure splitting is here washed out due to the line broadening
(fast relaxation into the ground state). The p-shell linewidth here is 0.5 meV, which is
also the upper limit for an p-shell fine structure splitting. The ground state shows a
strong dependence on linear polarized excitation, whereby the p-state is affected more
by circular polarized excitation.
84
7.3 p-shell saturation
Figure 7.4: Power dependence of the PL intensity from the QD ground state for cw p-shell
excitation. The PL signal shows a clear saturation behavior, as expected for a
two level system. (published in [ELdV+07b])
7.3 p-shell saturation
The saturation behavior of the s-shell emission with increasing p-shell excitation intensity
is an important indicator for the quantum mechanical two level character of the p-shell
transition. Figure 7.4 shows an analysis of the PL peak intensity versus excitation power.
The p-shell transition is resonantly excited by the TiSa laser in the continuous wave mode.
A clearly nonlinear power dependence can be observed, which is well described by the
fitcurve
I=Isat
P
P+1.
The formula is a result of the saturation of a two-level system, which is discussed in the
theoretical chapter 3.5.1. Here Irefers to the PL peak amplitude (PL intensity), Isat to
its saturation value, and Pcorresponds to a normalized excitation power. At low excita-
tion intensities, the s-shell luminescence increases linear to the excitation power. But for
higher excitation intensities, the s-shell luminescence approaches a saturation value (see
chapter 6.2.3 for photocurrent saturation of the ground state). The physical reason for PL
saturation can be derived fairly easily: If the QD is already excited by one exciton, no
further absorption can take place due to a renormalization of the energy levels caused by
few particle interactions. This also holds true if the exciton relaxes to the ground state.
Further absorption in the p-shell can only take place after a relaxation and recombina-
tion process. With the detection efficiency of the experimental setup (discussed later)
and the saturation value of ISat =10400 (a.u.), one can conclude that the total time for
85
7 First excited state
0π 1π 2π 3π 4π 5π
0
100
200
300
400
PL intensity (s-shell) (a.u.)
Pulse area
E
exc
= 1.368 eV
E
det
= 1.337 eV
V
b
= 0.018 V
Figure 7.5: p-shell RABI-oscillations detected in the ground state PL. The RABI-
oscillations have a clear developed π-pulse maximum, followed by a substan-
tial damping at higher pulse areas. (published in [ELdV+07b])
relaxation and recombination takes around 500 ps. This value matches the upper limit of
the recombination time of 700 ps very well, which is inferred from the analysis of anti-
bunching data using cw excitation, see chapter 10.5. Therefore the relaxation process is,
as expected, much faster than the recombination process. The observed cw-saturation be-
havior underlines the two level character of the p-shell transition; a second requirement
is the ability of the coherent control of the two level system.
7.4 p-shell RABI-flopping
The fundamental experiment in the coherent regime is aiming for the observation of
RABI-oscillations, analogous to the ground state. RABI-oscillations from excitons con-
fined to single QDs with optical detection have been reported by STIEVATER et. al.
[SLS+01b]. Here ps-laser pulses (τp≈2ps) have been used for excitation, where the
spectral pulse width is sufficiently low to avoid possible off resonance excitations. The
main problem for the coherent control of the excited state is a dephasing process faster
than the excitation. If the QD is resonantly excited in the p-shell, the dephasing is mainly
caused by relaxation to the s-shell. The projected occupancy of a two level system under
coherent resonant excitation corresponds to the measured PL intensity and is given by
I=sin2(Ωt/2). The RABI-frequency Ωis again proportional to the square root of the
laser intensity (pulse area) and tcorresponds to the pulse length. The pulse area (the
rotation angle Θ=Ωt) is defined here by adjusting the excitation amplitude.
86
7.4 p-shell RABI-flopping
Figure 7.6: Comparison between p-(a) and s-(b) shell RABI-oscillations. The p-shell RABI
oscillation is detected in the ground state PL, and the s-shell RABI-oscillation
in the PC. Due to a fast dephasing (relaxation), the RABI oscillation of the p-
shell is substantially damped when compared to the s-shell. (published in
[ELdV+07b])
Figure 7.5 shows the RABI-oscillations of the p-state measured in the s-shell PL. The PL
intensity shows a clear nonlinear behavior, whereby the oscillations imply that the excita-
tion of the QD can be controlled by the strength of the incident laser field. The measure-
ments exhibit a strong damping of the oscillation, but the first maximum is clearly visible.
The data obtained here is comparable to the results reported in [SLS+01b], which have
been achieved by pump-probe techniques. The damping indicates enhanced dephasing
rates compared to ground state RABI-oscillations 6.10.
The kind of damping is similar to the damping of the ground state RABI-oscillations
at high applied bias voltages, see figure 6.12. The high voltage leads to a fast tunneling
process and thus a short dephasing time. In this case, the RABI-oscillations does not ap-
proach the value 1/2for occupation. The similar kind of damping of the p-shell implies
also a very short dephasing of the p-shell transition, which is verified by double pulse ex-
periments, see chapter 7.5. Thus, it cannot be guaranteed that the occupation approaches
the value 1/2for high excitation amplitudes. In the PL, it is not possible to determine an
absolute value due to the detection efficiency. This is only possible in the PC.
Aπ-pulse typically corresponds to an average laser power on the sample of about 2
μW at a pulse length of 2.3 ps and a laser repetition frequency of fLaser= 80 MHz. For this
condition, nearly every excitation pulse should lead to the emission of one photon from
the s-shell. Therefore the maximum PL intensity can be used for an estimation of the
87
7 First excited state
0.0 0.5 1.0 1.5 2.0 2.5
200
250
300
350
400
450
500
550
600
PL intensity (s-shell) (a.u.)
Fine delay (
μ
m)
p-shell inference at complete overlapping pulses
(a) Ground state PL at overlapping p-shell-
excitation
0.0 0.5 1.0 1.5 2.0 2.5
300
350
400
450
500 p-shell inference at 5.46 ps delay
PL intensity (s-shell)(a.u.)
Fine delay (µm)
(b) Ground state PL with 5.46 ps time delay
between the pulses
Figure 7.7: Interference of the p-shell detected in the ground state PL
collection efficiency of the experimental setup. Assuming that every π-pulse generates
one photon, the collection efficiency of this system approaches 10−5(collection through
shadow mask, semitransparent gate and a NA = 0.75 objective). The measurements re-
ported in this section demonstrate coherent control of the first excited state.
In this section the coherent behavior of the s- and p-shell of the single QD is compared.
Due to the use of a tunable sample, it is possible to choose the dominant dephasing pro-
cess simply by the applied voltage and excitation conditions (p-ors- shell). In the first
case, the QD is resonantly excited in the p-shell (low voltage). The main dephasing pro-
cess here is relaxation into the ground state. After the recombination process, the pro-
jected state can be observed in the ground state PL, which is shown in figure 7.6 a). In the
second case, a (reverse) bias voltage of about 0.6 V is applied, which switches the main
dephasing process to tunneling. Thereby one can perform the coherent manipulation in
the ground state and detect the projected state in the PC (see figure 7.6 b)). The PC signal
shows more than eight full inversions (only five inversions are plotted here) at high ex-
citation intensities within each laser pulse. The PL intensity (p-shell excitation) shows an
enhanced damping compared to the PC signal (s-shell excitation). Comparing the p-shell
π-pulse to the ground state π-pulse, it turns out that the RABI-flop needs about the same
excitation intensity (about 2 μW at the sample). This means that the associated s- and
p-matrix elements are approximately equal in this special QD.
7.5 Dephasing of the p-state polarization
With double pulse excitation, it is possible to get access to the phase of the QDs exciton.
Similar to the measurement of the dephasing time of the ground state (see chapter 6.4.1),
it is possible to determine the dephasing time of the p-state of the QD, which is probably
much faster. A first laser pulse excites the QD in the p-shell and defines the phase of the
exciton. With the second laser pulse, the ability of interference between the QD and the
laser pulse is tested by changing the phase in relation to the first pulse. Depending on
the time delay between the two pulses, the inference will decrease due to the dephasing.
88
7.5 Dephasing of the p-state polarization
0
50
100
150
200
250
300
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
024681012
Coarse delay (mm)
measurement
fitcurve (exponential decay)
Oscillation amplitude (arb. u.)
τ
= 6.02 ps
Time delay (ps)
Figure 7.8: Dephasing analysis of the p-state. The interference amplitude at double pulse
excitation is displayed versus the time delay. The PL signal at overlapping
pulses is not considered.
Thus it is possible to infer a dephasing time. The main dephasing process here is the
relaxation into the ground state. The relaxation process will probably not depend on the
applied voltage. Thereby it is not possible to influence the dephasing time by an external
parameter, like it is possible in the ground state. In this case, the optical delay time will
be much shorter compared to ground state analysis.
The double pulses are generated by the MICHELSON interferometer, shown in chap-
ter 4.1.5. The excitation amplitude used here corresponds to a single π-pulse at complete
overlap of the two pulses and constructive interference (two π/2-pulses). Figure 7.7
shows the interference of the p-shell by varying the fine delay between the two pulses.
The final state of this variation is detected in the ground state PL after the relaxation (and
recombination processes). At zero time delay (Figure 7.7 a)), one can observe a very clear
oscillation in the ground state PL which has a range from zero (200 a.u. correspond to
zero (Si:CCD-offset)) to the full π-pulse intensity. The oscillation pattern is sinus shaped.
This pattern-shape of the interference is not surprising, but the same analysis of the de-
phasing time of the LO-phonon assisted absorption leads to a different behavior, which
will be discussed in chapter 8.5. At a delay time of 5.46 ps, the interference oscillation
amplitude is lower than before, but the oscillation stays sinus shaped.
Figure 7.8 shows the decrease of the oscillation amplitude (the dephasing) of the sys-
tem. One can clearly see a decrease of the PL oscillation amplitude within a range of a
few mm delay of the second pulse. A mirror shift of about 1.4 mm corresponds to a time
delay of ten ps. The fit of an exponential decay results in a dephasing time of the p-state
of 6.02 ps, which is about 3 times as large as the applied pulse width (1.7 ps). One can
89
7 First excited state
see that the data follow a simple exponential decay law with a decay time of 6.02 ps. This
dephasing time is much too short to resolve quantum beating effects in the p-shell. The
dephasing of the p-shell is much faster than the ground state dephasing and cannot be
affect by external parameters. The possibility of the coherent control allows for a defined
excitation of an excited state, which is used for a novel excitation scheme of single photon
emission in chapter 10.
90
8 LO-phonon assisted absorption
In the previous chapter, it was shown that the p-shell transition could also be treated as a
quantum mechanical two level system, like the exciton ground state. The main difference
between the s- and the p-shell is the strongly reduced dephasing time due to the different
dephasing processes. The p-shell dephasing is, however, determined by the relaxation
process, which is much faster than the dephasing due to tunneling processes. The LO-
phonon absorption resonance is different to both, s- and p-shell, dephasing processes.
Here LO-phonons are generated during the excitation process of the exciton. The excita-
tion scheme is shown in figure 8.1. The excitation energy fits to the ground state energy
plus the energy of a GaAs-LO phonon. The ground state is thereby generated with the
help of an emitted GaAs-LO phonon. The ground state is then able to decay by optical
recombination. It is possible that the GaAs-LO phonon propagates in the bulk GaAs crys-
tal. There are some interesting questions about the properties of the LO-phonon state.
•How long does the phonon assisted transition maintain coherent?
•Is it possible to perform a coherent manipulation with the LO-phonon assisted ab-
sorption?
•Can the LO-phonon assisted transition be treated as a two-level system?
•Is the phonon assisted absorption polarization sensitive?
Figure 8.1: Scheme of LO-phonon assisted excitation of the ground state.
91
8 LO-phonon assisted absorption
Figure 8.2: Voltage depending photoluminescence by LO-pumping. The luminescence of
the X+-state is again visible due to the biexciton generation and a tunneled
electron.
8.1 Voltage depending photoluminescence
In order to compare the p-shell transition with the LO-phonon assisted transition, a volt-
age dependent photoluminescence measurement is performed. The region investigated
here is again between -0.9 V and 0.1 V. The excitation power is 1.2 mW (before pellicle) at
an energy of 1.375 eV. Figure 8.2 shows that the luminescence of the ground state is visi-
ble in the same region (from 0.05 V to -0.8 V) as for the p-shell transition (see figure 7.2).
With this experimental setup, the positively charged exciton X+(which is close to the lu-
minescence of the neutral exciton) is also detected. The luminescence of the X+vanishes
when electrons from the doped back contact are able to tunnel into the QD (similar to the
voltage dependence at non-resonant interband excitation). Both lines show a (different)
red shift, arising from the QCSE (see chapter 5.2). The luminescence of both lines can be
observed at even higher electrical fields than at non-resonant interband excitation. The
exciton ground state is, in this case, directly generated with the help of LO-phonon as-
sisted absorption. At p-shell excitation, the exciton is able to tunnel out of the QD from
the p-shell and the s-shell. In the case of LO-phonon assisted absorption, the excitons are
only able to tunnel out of the QD from the ground state. Therefore the luminescence of
the ground state can be observed up to high electrical fields (-0.8 V). The PL intensity of
the positive charged exciton X+is higher in the case of LO assisted absorption than at p-
shell excitation. This could be explained by a higher probability of biexciton generation,
which is discussed in chapter 9.
92
8.2 Polarization analysis of the LO-phonon absorption
(a) Dependence of circular polarization (b) Dependence of linear polarization
Figure 8.3: Properties of the ground state PL and polarized excitation of the LO assisted
absorption.
8.2 Polarization analysis of the LO-phonon absorption
The LO-phonon resonance is quite broad (0.8 meV) compared to the p-shell resonance.
Using formula 3.33, this cw linewidth results in a very short dephasing time of only
1.7 ps. If the LO-phonon resonance could be affected by a specific polarization of the
excitation, one should be able to observe a change in the PLE spectrum (e.g. a shift, or
partly suppression). Due to the combined exciton-phonon excitation and a the very short
dephasing time a polarization dependence of the LO assisted excitation is not expected.
Figure 8.3 shows the results of the polarization dependent excitation. A change of the lin-
ear polarized excitation has no effect on the PLE intensity of the ground state figure 8.3
b). But in figure 8.3 a) one can see a slight energy shift of the resonance by changing the
circular polarization. The origin of this effect is unknown so far. However, the combined
generation of exciton and phonon is possible with any excitation polarization. Apart
from the identification of the GaAs LO-phonon resonance by the energy match, this ab-
sorption resonance is different as compared to a normal shell transition. This confirms
the identification of the LO-phonon assisted transition.
8.3 Saturation of the LO-phonon assisted transition
Since the p- and s-shell occupation show a saturation behavior with increasing (cw-) exci-
tation intensity, one would like to know if the LO-phonon assisted transition also shows
saturation behavior. This is an important indicator for the two level character of the LO-
phonon assisted absorption. For the case that the LO-phonon resonance behaves as a
93
8 LO-phonon assisted absorption
Figure 8.4: Saturation of the LO-absorption at continuous excitation. The signal shows a
normal saturation behavior, but a reduced saturation intensity of only the half
of the p-saturation value.
quantum mechanical two level system, its power saturation should follow the formula
I=Isat P
(P+1).
Figure 8.4 shows an analysis of the PL peak intensity versus excitation power. Here
Irefers to the PL peak amplitude (PL intensity), Isat to its saturation value, and Pcor-
responds to a normalized excitation power. At low excitation intensities, the s-shell lu-
minescence increases linear to the excitation power. But for higher excitation intensities,
a clearly nonlinear power dependence can be observed. Up to an excitation power of
0.1 a.u., the PL intensity of the ground state increases with the excitation intensity. In this
regime, the behavior could be well described by the fit-formula. For excitation intensities
larger than 0.1 a.u. (see figure 8.4), the PL intensity shows slight variations from the fit
curve. Comparing the LO-phonon PL signal to the p-shell transition, one can see that
the data here exhibit more noise than the p-shell transition (especially at higher excita-
tion amplitudes). Nevertheless, the PL-signal shows saturation behavior. The saturation
value Isat here is about 5066 a.u., which is only half of the saturation value of the p-
shell (same adjustment). The excitation intensity for the saturation is much larger for the
LO-phonon assisted transition than for the p-shell. At an excitation power of 0.07 a.u.,
the p-shell pumped PL is close to the saturation value (90%), whereas the LO-phonon
pumped luminescence is still in the linear regime (40% of the saturation value).
In the case of the p-shell excitation, it was shown that the maximum intensity of the
PL is determined by the total time of recombination and relaxation. In the high excita-
tion power limit, the re-excitation takes place immediately after the recombination. For
the case of LO-phonon assisted absorption, the maximum PL intensity is only half of
the intensity of the p-shell pumped excitation (same experimental setup). This indicates
94
8.4 Pulsed excitation
Figure 8.5: Pulsed excitation of the LO assisted absorption. No RABI-oscillations could
be observed like in the p-shell. The signal seems to be over damped. The
intensities values of I50, I75, I100, and I200 are shown, which are relevant in
the following.
that the average time between two ground state emissions is twice as long in the LO-
phonon case. But it is unlikely that the optical recombination time depends on the kind
of excitation. More probable is the excitation of the biexciton in this case. If the QD is
occupied with a single exciton, further excitation can take place in the case of LO assisted
absorption. If the biexciton is generated too, the average time between two ground state
recombinations is extended due to the (not detected) life time of the biexciton state. The
biexciton excitation can be proven with a PLE measurement where the detection applies
to the biexciton state, see chapter 9.
8.4 Pulsed excitation
The possibility of a coherent manipulation of a phonon assisted transition is a very in-
teresting question, especially for a theoretical description. Is it possible to perform a
stimulation of the combined exciton-phonon state after the excitation? In this section, a
coherent manipulation of the LO-phonon assisted absorption is analyzed (similar to the
p-shell, see figure 7.5).
Figure 8.5 shows the PL intensity of the ground state with increasing pulse area (√I),
where Icorresponds to the excitation intensity. For low excitation intensities, the PL in-
tensity increases (quadratic, from 0 to 0.05 a.u.) and then switches to a linear increase
with incident pulse area (up to 0.15 a.u.). At an intensity of more than 0.15 a.u., the PL
intensity increases to a maximum followed by a slight decrease. The origin of this satu-
95
8 LO-phonon assisted absorption
Figure 8.6: Power dependent dephasing analysis. The decrease of the interference am-
plitude with varying delay times is shown for different excitation intensities.
The dephasing at an intensity of I200 shows a stronger decrease than the other
excitation intensities.
ration behavior is not known up to now. A possible explanation of the missing coherence
is for example the biexciton generation. Furthermore, the generated GaAs LO-phonons
are able to propagate in the bulk GaAs crystal. Thus, the phase relation of the excitation
and the combined exciton-phonon state is lost very fast. However, the PL signal shows
only a strong damping and no coherent properties. But this is no sufficient evidence that
a coherent manipulation is generally not possible with a LO-phonon assisted transition.
The pulse length for excitation used here is about 1.7 ps. Shorter excitation pulses may
be useful for a possible coherent control, but also unfavorable for this quantum system.
The probability for the generation of the biexciton is strong enhanced for high excitation
powers. But maybe much shorter excitation pulses could be helpful to observe a coherent
behavior before the excitation intensity is high enough for the generation of the biexciton.
8.5 Dephasing of the LO-phonon assisted transition
In the previous sections, the dephasing of different QD states (s- and p-shell) was dis-
cussed which originates from tunneling, relaxation, and recombination processes. In the
case of the LO-phonon assisted absorption, the dephasing mechanism is different. GaAs
LO-phonons with an energy of 36 meV are generated in order to excite the exciton ground
state. Since it is not possible to perform a coherent manipulation with ps laser pulses, one
expects a very fast loss of the phase relation between the laser pulse and the LO-phonon
assisted transition. In order to analyze this dephasing behavior, double pulse experi-
96
8.5 Dephasing of the LO-phonon assisted transition
0.4 0.6 0.8 1.0 1.2
20
40
60
80
100
120
140
160
Oscillation amplitude
Fit (exponential decay)
Oscilation amplitude (a.u.)
Coarse delay (mm)
Dephasing after excitation
τ = 3.5 ps
Figure 8.7: Dephasing analysis of the LO assisted absorption at an intensity of I100. The
decrease of the interference amplitude is fitted by a normal exponential func-
tion. The PL signal at overlapping pulses is not considered.
ments have been performed. Due to the fact that no RABI-oscillations could be observed
on this transition, it is not possible to define a π-pulse. Therefore the double pulse exper-
iment is performed with a couple of selected excitation intensities. Here the excitation
power for maximum PL intensity (0.2 a.u.)(see figure 8.5) is defined as intensity = 100 %
(I100). The double pulse experiments are performed at excitation intensities of 50 %, 75
%, 100 %, and 200 %.
Figure 8.6 shows the oscillation amplitudes of the interference depending on the pulse
delay for different excitation powers. One can see a fast decrease of the interference am-
plitude for all excitation intensities. The maximum intensity (zero time delay) increases
with increasing excitation intensity as expected (up to 100% intensity). The decay of the
oscillation amplitude is nearly the same for all excitation intensities ≤100 %. The exci-
tation pulse length of 1.7 ps corresponds here to 0.25 mm pulse delay. For the case of
an excitation amplitude of 200 %, one can observe a very fast dephasing of the system.
In this case, the system is saturated already and therefore a faster dephasing for high
excitation intensities is not surprising.
In order to determine the dephasing time of the system, the decay of the oscillation
amplitude at an intensity of 100 % is analyzed in figure 8.7. The experimental data are
fitted with a simple exponential decay function. For complete overlapping pulses, the
two laser pulses will interfere with each other. For a correct analysis of the dephasing
time, here only the data for delay times longer than the pulse length are considered. One
can see a very fast dephasing of the system which is only 3.5 ps. This is only half of the
dephasing time of the p-shell (6.02 ps).
97
8 LO-phonon assisted absorption
Figure 8.8: Interference pattern at double pulse excitation on resonance to the LO-assisted
absorption at different excitation intensity. For higher excitation intensities,
the interference pattern shows a variation from a symmetric oscillation. A
plateau at high PL intensity with very small dips origins.
Figure 8.9: Comparison of the oscillation pattern of the PL at overlapping and time sep-
arated laser pulses. The oscillation amplitude is reduced, but the plateau is
still visible. This effect is probable a result of excitation intensity which is also
visible at time separated pulses.
98
8.5 Dephasing of the LO-phonon assisted transition
If one takes a closer look at the shape of the oscillation pattern of the interference,
one can observe some changes compared to the p-shell excitation. Figure 8.8 shows the
oscillation of the ground state PL for different excitation amplitudes, 50 %, 100 %, and
200 %. In the case of 50 % excitation amplitude, the QD is still in the linear regime (see
figure 8.5). Here, the PL intensity increases directly with the excitation intensity. The
interference pattern is very symmetrical around its mean value (350 a.u.) (figure 8.8 blue
line). For increasing excitation amplitudes (100 %), the oscillation amplitude increases
as expected (Figure 8.8 red line). But here, the form of the interference oscillation has
changed and is no longer symmetric around the mean value. At constructive interference,
the PL builds up some kind of plateau. The PL shows "destructive interference" only in
a smaller region of the fine delay. This effect develops even more at higher excitation
amplitudes (see figure 8.8 black line for 200 % excitation amplitude). The plateau of
the "high PL state" has developed even more, and the "dip" to the "low state" is even
smaller. Thus, this change in the oscillation pattern becomes significant if the excitation
intensity reaches saturation (see figure 8.5). The origin for this is effect is unknown up
to now. The explanation of this effect may be possible with theoretical description and
simulation. The group of J. FÖRSTNER has started to simulate the interaction of the LO-
phonon and the exciton for this kind of systems. These results have to show whether pure
LO-phonon interaction is responsible for this effect or if one has to take the biexciton state
into account.
Figure 8.9 compares the oscillation pattern of the PL for overlapping and time sepa-
rated (time delay = pulse length) pulses. Here the excitation amplitude is 200 %. One
can see that the oscillation amplitude decreases due to the dephasing. But the oscillation
pattern remains unsymmetric around the mean value with a plateau in the high state.
Thus the change in the oscillation pattern is not an effect of overlapping pulses.
In summary, the LO-phonon assisted absorption has different properties compared to
the p-shell:
•It is possible to saturate the LO-phonon assisted transition.
–The excitation power needed for the saturation is much higher than the p-shell
saturation intensity.
–The saturation value is approximately half of the p-shell saturation. This indi-
cates the excitation of the biexciton state.
•A coherent manipulation of the LO-phonon assisted absorption state is not possible
with the used excitation conditions.
•The dephasing time of the LO-phonon state (3.5 ps) is faster than the dephasing
due to relaxation (6.02 ps). High excitation intensities lead to enhanced dephasing
rates.
•The interference pattern for double pulse excitation is different in the LO-phonon
resonance as compared to the p-shell resonance.
99
8 LO-phonon assisted absorption
100
9 Biexciton generation
In chapters 7 and 8, the coherent excitation of the p-shell and the LO-assisted absorption
was discussed. The p-shell shows a clearly developed maximum of the RABI-oscillations,
whereby a coherent manipulation of the LO-assisted absorption was not possible with
the avaiable excitation condition. An additionally generated biexciton could lead to en-
hanced dephasing rates. Thus, it is possible that the LO phonon assisted absorption
cannot be coherently manipulated due to the generation of the biexciton. This chapter
discusses the possibilities of the biexciton generation with resonant excitation of higher
QD-states.
9.1 Photoluminescence excitation of the biexciton
Figure 9.1 shows the PLE spectra for the biexciton state. The diagram shows the PL inten-
sity of the biexciton state (2X) depending on the excitation energy. In order to compare
this data with the PLE spectra from the ground state (see figure 5.5), the energy is directly
plotted as energy difference with respect to the ground (1X). One can see that there are
various possibilities to excite the biexciton. It has to be noticed that this experiment is not
sensitive to the kind of excitation (resonant, half resonant, non-resonant/background).
Every excitation which leads to the biexciton ground state is detected. Here one kind
of excitation is a resonant two photon process in a higher state followed by a relaxation
process into the biexciton ground state. It is also possible that the QD is only occupied
with one exciton (e.g. after the biexciton decay process), and the second exciton is gen-
erated considering the renormalization energy. Please note that the excitation power is
much higher as compared to figure 5.5 (700 μW before pellicle/30 μW at the sample). An
identification of all these states is not easy.
The large peak 32 meV above the ground state could be the biexciton p-shell. Here
it may be possible that both excitons are created in the p-shell by a two photon absorp-
tion process. Due to the renormalized energy, the resonance is shifted to a higher energy
(E2Xp−EXp=2.4 meV). In this case the renormalization energy is positive. The phonon
absorption peak here is again 36 meV above the ground state. One can clearly see that the
biexciton can be generated by pumping the GaAs LO-phonon state. At the same excita-
tion power, the biexciton PL intensity (at ground state Xp-state) is much lower (only 1/6).
Therefore it is sure that the biexciton can be generated by pumping the LO phonon state
(which is also demonstrated in [FZBA00]). If the QD is already excited by one exciton,
further absorption can take place. This could explain the fact that the PL saturation value
of the LO phonon state is much lower than the p-shell.
A model for sequential phonon assisted generation of a biexciton in the QD and the
sequential decay is discussed in the following: At the resonant laser energy (EXLO ), an
101
9 Biexciton generation
20 25 30 35 40 45 50
200
400
600
800
1000
1200
1400
1600
|X
LO
>
PL intensity (arb. u.)
E
exc
- E
1X
(meV)
|X
P
>
Detection on the XX line
P
exc
= 700 µW
Figure 9.1: Photoluminescence intensity of the biexciton transition with varying excita-
tion energy. One can see that the biexciton can be created with many different
laser energies. The highest peak here is probably the biexcition p-shell, which
is excited by two-photon absorption process. The biexciton generation is more
probable at LO-phonon excitation than at p-shell excitation.
exciton in the QD and a phonon with energy ELO are generated. As long as the dot is
occupied with one exciton, this transition is renormalized by the amount of the biexci-
ton binding energy and hence blocked for further absorption at the excitation energy of
EXLO . However, for unchanged excitation energy, a second exciton can be generated by
absorption under participation of a different phonon with energy EXLO+ΔE2X. This second
absorption process is made possible by the weak background absorption. Each absorp-
tion process leading from the single to the two exciton state can contribute here and does
not have to be generally phonon assisted. Otherwise it is also possible that the second
absorption takes place assisted by two phonons (optical and acoustical), even if the prob-
ability is low.
9.2 s-shell luminescence for biexciton excitation in the p-shell
Figure 9.2 shows the s-shell luminescence of the single QD for biexciton p-shell excitation.
Please note that the detection setup is similar to the PLE-measurements. This means
that the luminescence intensities for all detected states are not comparable due to the
additionally installed grating before the spectrometer. The energy detection window is
limited by the spectrometer entrance slit and the intensity function of the blazed grating.
Thus, it is not ensured that all s-shell states can be detected. Only the states near the the
ground state ±3 meV are detected at the same time. A luminescence of more s-shell states
102
9.2 s-shell luminescence for biexciton excitation in the p-shell
Figure 9.2: s-shell luminescence at a strong excitation of biexciton p-shell. At least two
charge carriers (one electron and one hole) must relax into the s-shell for the
recombination. There are different possibilities for the other two charge carri-
ers. All configurations lead to different renormalization energies. Each state
shows a different STARK shift.
outside this window is excluded by random measurements with different adjustments of
the blazed grating. One can observe some additional states, if the PL is compared to
the normal non-resonant interband excitation PL, see figure 5.1. The excitation of the
biexciton state always needs a higher excitation intensity, therefore the excitation power
here is about 3 mW (120 μW at the sample).
There are different recombination schemes which are all separated by different renor-
malization energies. The excitation probably takes place by a two photon absorption pro-
cess in the p-shell. Thus, the generation scheme starts with two excitons in the p-state.
The detection takes place in the s-shell luminescence. The recombination luminescence
is detected from the electrons and holes which have already relaxed into the s-shell. All
direct p-shell transitions are separated in energy and cannot be detected here. There are
multiple possibilities for recombination configurations, e.g. only three (of four) charge
carriers have relaxed into the ground state. An assignment of all these transitions in
figure 9.2 is hardly possible.
103
9 Biexciton generation
104
10 Single photon emission
Since the experimental setup allows for electrical and optical detection, it is possible to
operate the single QD as a single photon emitter. Single quantum systems are often the
basis for good single photon emitters. In this dissertation, the advantage of coherent
state preparation as compared to the commonly used non-resonant interband excitation
is presented. The main advantage of using a coherent manipulation of a higher QD state
for excitation is the defined occupation of the QD with a single exciton.
In the present work, two possibilities for the occupation of the QD with a single ex-
citon in the ground state are presented: The first option uses an excited state excitation,
followed by a relaxation process into the ground state. The second option uses phonon
assisted absorption to create the desired ground state occupation. In both cases, excita-
tion and recombination are separated in energy. Finally, the recombination process from
the ground state will generate the desired single photon.
10.1 First and second order correlation function
The correlation of light is usually described by means of coherence. This corresponds
to the amplitude and the phase characteristics of an electromagnetic field. An example
for this is the well known experiment of the YOUNG double-slit. The interference and
the contrast results from the first order correlation function which is defined on the basis
of the E-field. The first order correlation function is represented as spatial or as temporal
correlation. The spatial correlation function in the first oder g(1)(
r1,
r2)is defined as:
g(1)(
r1,
r2)=U1(
r1,t)U∗
2(
r2,t)
(I(
r1)I(
r2))1/2 .
The temporal first order correlation function g(1)(τ)is defined as:
g(1)(τ)= U(t)U∗(t+τ)
U(t)U∗(t+τ)
where Uand U∗are correlated to the E-field of the electromagnetic field. For example, a
light beam is split into two parts, and one part has to go a longer path length. Then the
coherence time can be analyzed in the loss of interference.
The photon statistics reveals whether a given light field originates from a classical or a
quantum source, such as a single quantum system. These sources can be distinguished
by their intensity correlation function, g(2). The second order correlation function gives
the probability that a photon is detected at a time tand another photon at the time (t+τ).
105
10 Single photon emission
Figure 10.1: Second order correlation function of a) laser light, b) classic light, and c) a
single quantum system
The second order correlation function g(2)(τ)is given by the normalized convolution
g(2)(τ)= I(t)I(t+τ)
I(t)I(t+τ)(10.1)
where I(t)corresponds to the emission intensity [ZBJ+01]. The second order correla-
tion function is measured in the HANBURY-BROWN and TWISS setup (presented in chap-
ter 4.3.4) in a photon counting experiment. The luminescence, which has to be tested for
correlation, is split up into two parts. Each part is detected by a single photon counter
device (avalanche photo diode (APD)). A single photon counting event of the first de-
vice starts a time measurement (done by time to amplitude converter), a single photon
detection at the second device stops the time measurement. The time between the two
photon counting events is the time interval τ. The measurement of many of these cor-
relation events results in a histogram which corresponds to the second order correlation
function. A single photon can only be detected at one counter. For the case of a single
photon emitter, the two APDs cannot count photon events at the same time.
10.2 Correlation functions of laser, classic light, and
quantum-mechanical light
Each given light source has its own characteristic second order correlation function. For
example, laser radiation exhibits constant g(2)over time. The probability for photon de-
tection is constant in time. Thus, the correlation event rate depends not on the time delay
τ. The normalized second order correlation function is equal to one, see figure 10.1 a).
The second example, a light bulb, has a different correlation function. The photon emis-
sion of such a thermal light source shows strong fluctuations of the photon emission
rate (classic light). The probability for two detection events is not constant in time, it
is higher for short time intervals. For this case, the equation g(2)(0)≥g(2)(τ)is valid.
Figure 10.1 b) shows the two cases of LORENTZIAN (red) and GAUSSIAN (green) classic
light. The classic light is said to be bunched. Classical light fulfills the CAUCHY-SCHWARZ
106
10.3 Correlation function of a single photon emitter
inequality, g(2)(0)≥g(2)(τ). On the other hand, light that violates this inequality is
characteristic for a single quantum emitter. If a local minimum is found at t=0, i.e.
g(2)(0)<g(2)(τ), this light is said to be antibunched, which results in a dip in the corre-
lation function, see figure 10.1 c). In this case, the probability for two photon detection
events close in the time domain is low. For a single photon emission device, g(2)(0)is
ideally zero.
10.3 Correlation function of a single photon emitter
If the luminescence consists only of single photons, the histogram must show a clear
reduction at zero time delay. The dip (cw-case) in the histogram is an effect of the an-
tibunching. This indicates a luminescent source as a single photon emitter. In order to
reach a symmetrical histogram, one electrical connection from the APD to the time to
amplitude converter contains is subjected to a time delay. This only shifts the detection
window to negative times.
Under continuous excitation, a single quantum emitter would generate a dip in the
histogram at zero delay time (τ=0). This means that the quantum system, that gener-
ates the luminescence, must be re-excited after emission before a second photon can be
emitted. For long time delays, the histogram is constant, just as in the case of a constant
photon emission rate (see figure 10.1 c)). This is illustrated in figure 10.2 a). Ideally, the
exponential decay constant τcorresponds to the systems life time of the quantum system.
g(2)(t)=1−1−g(2)(0)·exp−|t|
τ(10.2)
Contrary to the antibunching effect, there can be also bunching effects (g(2)(0)≥g(2)(τ)).
For example, a single laser pulse consists of many photons within one pulse. It is there-
fore possible to count photon events on both APDs within one pulse. Furthermore, corre-
lations can be measured between two different pulses. This results in a periodic structure
of the measured histogram with a time constant of the delay time between two pulses.
This is illustrated in figure 10.2 b). The correlation function of pulsed laser consists of a
series of peaks, separated by the repetition time between two laser pulses (Trep). Without
any time jitter of the APDs, the width of these peaks are correlated to the pulse width of
the laser (2-3 ps). The theoretical correlation function of the pulsed laser is a convolution
(see 10.1) of the auto correlation function
Ipulsed =∑
n
I0
τpulse√2πexp(−(t−nTrep)2
2τ2
pulse
). (10.3)
This again results in a repetitive structure of the correlation function.
If the single quantum system is excited by a pulsed laser, the luminescence only con-
sists of single photons separated in time by Trep and the variation due to the statistical
recombination time. The histogram will again show different peaks with a time delay of
the inverse repetition frequency (12.5 ns). The peak at zero delay time will vanish due to
the single photon emission. This is shown in figure 10.2 c). It is very complex to derive
107
10 Single photon emission
0
1
0
1
-40 -20 0 20 40
0
1
)exp())0(1(1)(
)2()2(
t
gtg −⋅−−=
c
)
a
)
Correlation function
E
)
τ
Trep
Time (ns)
Figure 10.2: Theoretic histogram of the time to amplitude converter (corresponds to the
second order correlation function). a) Antibunching at continuous excitation
of the single photon emitter. The dip in the histogram is an exponential func-
tion in time and is correlated to the life time of the quantum system. b) Corre-
lation of a pulsed laser. The peaks are separated in time due to the repetition
frequency of the excitation laser. c) Perfect clean single photon emission. The
central peak of the repetitive structure is missing completely.
an analytical function of g(2)of the single photon emission for pulsed excitation. Ideally,
there is only a single photon in each pulse. Therefore the time dependent function for
the single photon intensity can only give a probability of the emission. In principle, this
function depends on the excitation pulse function, the generation probability and on the
exponential decay. In the HBT setup, the luminescence is split into two parts. But a single
photon cannot be split up. That’s why a convolution of the time dependent single photon
emission probability would not lead to the correct results. Therefore it is not possible to
give an analytical function for the convolution like it is possible in the case b) (correlation
of pulsed laser light).
108
10.4 Sequential absorption-emission events within one excitation pulse
-1.0 -0.5 0.0 0.5 1.0
0.0
0.1
0.2
0.3
0.4
Time (ns)
τ
decay
= τ
pulse
= 250 ps
Figure 10.3: Illustration for the histogram for single photon emission for same time con-
stants for excitation and decay. See text for a detailed description.
10.4 Sequential absorption-emission events within one
excitation pulse
Up to now, only the ideal case of the single photon emission is discussed, where the
decay time of the quantum state is much longer than the pulse duration for excitation
and much shorter than the repetition time of the excitation laser. A problem can arise
here if the pulse width of the excitation laser has the same order of magnitude as the
decay time of the quantum system. Then it is possible that more than one absorption and
decay processes take place within one excitation pulse. Thus, the resulting correlation
function should show two symmetrical peaks as illustrated in figure 10.3. Here, the case
of equal time constants for decay and excitation is assumed. At zero time delay, there
is a still a dip in the correlation due to the single photon emission. But after the decay
process, a following (second) single photon emission can take place.
In the case of non-resonant interband excitation, one needs strong excitation intensity
in order to avoid the case missing excitation. Due to the POISSON-distribution, there is
a non-negligible probability for the excitation of more than one exciton. An excitation of
two excitons, before the first recombination takes place, leads to the sequential decay of
the biexciton and exciton. The luminescence of the biexciton is separated in energy. It
is also possible that a second exciton is generated after the first recombination process
(within a single excitation pulse). This would lead to a second single photon within one
pulse.
An analytical derivation of this content is very complex. The time dependent prob-
abilities of of the first single photon emission and the sequential (second) single photon
emission have be convoluted. But the time dependent single photon emission probability
is furthermore a convolution of the laser excitation and the decay function. In principle,
one has to calculate the time dependent probability of the first and the second photon
after the laser pulse.
In order to illustrate the effect of equal time constants for excitation and decay, the
correlation function of the single photon emission 10.2 (which is seen as the response of
109
10 Single photon emission
the system) is multiplied with the central pulse (n=0) of the excitation function 10.3.
The multiplication of excitation and system response formula only gives an illustration
for behavior of equal time constants for excitation and emission. In this case, the central
peak is strongly reduced but is not completely vanished. The height and the width of the
central peak is then only determined by the two parameters τdecay and τpulse. This effect
exceeds for short decay times. The second order correlation function will show two peaks
which are symmetric around zero time delay. If one regards the time jitter of the APDs,
the dip in the middle of the structure in figure 10.3 could be washed out. Thus, the single
photon emission appears to be unclean.
This problem can generally be avoided by a coherent excitation of an excited state of
the single quantum system. By using a coherent manipulation of the first excited state of
the single quantum system, one can avoid a second excitation-emission process. Thus,
the effect of an unclean histogram of the single photon emission at short decay times
should be of no significance here. The excitation scheme for the single photon emission
is discussed in chapter 10.
10.5 Continuous non-resonant interband excitation
Single photon emission from semiconductor QDs was first reported by MICHLER et al.
[MKB+00], followed by many other groups. These groups employed optical as well as
electrical pumping. The general concept of the so far employed excitation scheme is
based on strong incoherent, non-resonant excitation and a sequential decay cascade with
final single photon emission. Using this excitation scheme, an interaction between an
exciton and other charge carriers cannot be excluded. The different renormalization en-
ergies of the various multi-exciton states allow for spectral isolation of the last photon
emitted from the previous decay process of the multiexcitonic state. The last exciton is
located in the ground state of the QD. It cannot be guaranteed that the luminescence of
the last exciton is clearly separated in energy from a multiexcitonic state. It is possible that
higher multiexciton states have an spectral overlap with the ground state luminescence.
Therefore, the requirements for the energy separation are quite big. A strong excitation
is necessary to avoid the case of missing population (within the statistical fluctuation of
the created exciton number for this incoherent process).
For the analysis of the single photon emission of the ground state, a photon correlation
measurement is performed under the condition of continuous non-resonant interband
excitation. The time correlation measurement is performed by a time to amplitude con-
verter with a resolution of 300 ps. For a clean single photon emission, one would only
expect a dip in the histogram. A perfect clean single photon emission would lead to van-
ishing correlation counts at zero delay time. The result of this measurement is shown in
figure 10.4. For low excitation intensity, the counting rate and the correlation event rate
are very low. Here, even an integration time of 10 h would not lead to usable results. For
extremely high excitation intensities (non-resonant and interband), the linewidth of the
luminescence is broadened due to the interaction of the QDs exciton with other excited
charge carriers. With increasing excitation intensity, one gets a saturation of the single
ground state exciton and an increase of the biexciton luminescence. The excitation inten-
110
10.5 Continuous non-resonant interband excitation
Figure 10.4: Photon correlation measurement after interband excitation at ground state
detection. The correlations are fitted with the formula for an exponential
ground state decay, see text.
sity in this case is adjusted near to the saturation level of the ground state luminescence.
The correlation counts are normalized to the value 100. The second order correlation
function g(2)in the middle of the histogram has a value of g(2)(t0)=0.61.
The dip in the middle of the histogram is fitted by the formula 10.2 g(2)(t)=100 −
1−g(2)(t0)·exp(−|t−t0|
tc)where g(2)(t0)corresponds to the minimum correlation value
and tcto the decay time of the dip. The formula is the result of the exponential decay of
the QD state in time. Unfortunately, it is very difficult to determine a possible back-
ground. An background could arise from the avalanche breakdown flash (discussed in
chapter 4.3.4) or from the time jitter of the APDs. The exponential decay is not influenced
by a constant background. But the effect of the time jitter of the APD is different. For
example, a correlation measurement of the pulsed laser leads to repetitive peak structure
in the histogram (without antibunching). Ideally, the peak width would be correlated to
the laser pulse length (2-3 ps). But the resulting peaks are broadened due to the time jitter
of the APD (approximately 300 ps). The measurements of a single photon emission (cw)
are similarly affected by the APD time jitter. The resulting dip of the correlation measure-
ment is then less pronounced. Therefore, the results of the fitcurve of the histogram are
only an upper limit for the ground state decay time.
Furthermore, the decay time of this exponential function is the time between two re-
combination processes. The time for a re-occupation of the ground state after the re-
combination process is assumed to be negligibly short. As a result of the fitcurve, the
upper limit for the ground state decay time τis approximately 700 ps. Here the second
111
10 Single photon emission
Figure 10.5: Scheme for single photon emission. The excitation takes place in the p-shell
with a π-pulse. After the relaxation process into the ground state a single
photon can be emitted. (published in [ELdV+07b, ELdV+07a])
order correlation function for continuous excitation is g(2)(0)=0.61. Even if the two
background effects are considered, this doesn’t accord to a clean single photon emission.
10.6 Single photon emission after coherent (p-)state preparation
In contrast to the commonly used non-resonant interband excitation, the coherent state
preparation allows for the generation of a defined optical polarization, which, in case of
π-pulse excitation, results, after the projection, in a single exciton occupancy. The coher-
ent manipulation of the single exciton ground state works very well, see chapter 6.3.1,
but cannot be applied here due to the spectral overlap of ps-excitation and single pho-
ton emission. In order to spectrally separate the emission from the ps-excitation, RABI-
flopping is performed in the p-shell with subsequent relaxation to the ground state, fol-
lowed by single photon emission. The possibility of a coherent state preparation of the
first excited state is shown in chapter 7.4.
The basic sequence of the concept for a defined single photon emission is sketched
in figure 10.5. After the coherent excitation of the p-shell (|Xp) via a π-pulse, the gener-
ated exciton relaxes into the ground state (|Xs) and subsequently decays by spontaneous
emission. The main advantage of this concept is the defined excitation of a single exci-
ton in a coherent process. There is no interaction with other excited charge carriers. The
luminescence must only be filtered from the excitation laser and not from the lumines-
cence of any multiexcitonic state. The pulsed excitation, combined with the coherent state
manipulation, minimizes the possibility of two following excitation and recombination
processes, which is discussed in before 10.1.
The coherent state preparation uses laser pulses which lead to a recurring pattern of
peaks in the histogram. Each photon detected can be correlated with a photon generated
from one of the next laser pulses. Due to the coherent excitation, there is at most one
photon for each laser pulse. Therefore, there cannot be a correlation at zero delay time.
Single photon emission will lead to a missing peak at zero delay time; g(2)(τ=0)should
vanish. Figure 10.6 shows the result of the correlation measurements with an integration
time of 10 h and a time resolution of 300 ps. The correlation peaks have a time interval of
12.5 ns (80 MHz laser repetition frequency). The numbers printed above the peaks cor-
112
10.7 Single photon emission after GaAs LO-phonon assisted absorption
Figure 10.6: Photon correlation measurement under the condition of π-pulse excitation
in the p-shell (10 h integration time). The numbers printed above the peaks
give the value of the normalized correlation function. The central peak of the
periodic pattern (τ=0) is strongly suppressed, which proves clean single
photon emission. (published in [ELdV+07b, ELdV+07a])
respond to the normalized second order correlation function g(2)(τ), whereby g(2)=1
corresponds to the average peak height. The peak at zero delay time is basically miss-
ing, which proves that this excitation scheme results in fact a very defined single photon
emission. Also quantitatively, the single photon emission of a QD under coherent p-shell
excitation appears remarkably clean (g(2)(0)≤0.02).
10.7 Single photon emission after GaAs LO-phonon assisted
absorption
Another possibility for a defined single photon emission is sketched in figure 10.7. Here
the excitation is performed using the GaAs LO-phonon assisted absorption. The LO-
phonon assisted resonance is separated by 7 meV from the p-shell absorption, see fig-
ure 5.5. Due to the non-coherent properties of the LO-phonon absorption (see figure 8.5),
it is not possible to define a π-pulse in this excitation scheme. Thus it is not possible to
declare the occupation probability for the excited exciton. The excitation amplitude for
the photon correlation measurement used here is I100, see figure 8.5. It has to be noticed
that the generation of a second exciton is possible with the LO-phonon state (see chap-
ter 9). But the luminescence of the biexciton state is spectrally filtered. The result of the
113
10 Single photon emission
Figure 10.7: Scheme for single photon emission after photon assisted absorption. The
excitation energy fits to the ground state energy plus the energy of a GaAs
LO-phonon. The ground state is thereby generated with the help of an emit-
ted GaAs-LO phonon. The ground state is again able to decay by optical
recombination and thus the desired single photon is generated.
photon correlation measurement using LO-phonon absorption is shown in figure 10.8.
The pulsed excitation leads again to a recurring pattern of peaks. The correlation peaks
also have a time interval of 12.5 ns (80 MHz laser repetition frequency). The peak at zero
delay time is basically missing again, which proves that this kind of excitation leads also
to a clean single photon emission. The single photon emission of a QD ground state
appears surprisingly clean (g(2)(0)≤0.04). Comparing the two excitation schemes of p-
shell and LO-assisted absorption, one can say that both kinds lead to a very clean single
photon emission.
In conclusion, the coherent p-state excitation scheme has the following advantages
(compared to LO-phonon and non-resonant interband excitation):
•Defined excitation of a single exciton in a coherent process; optimum probability
for the excitation of a single exciton.
•No multiple absorption/emission events within a single pulse; clean single photon
emission in the time domain.
•In principle, no spectral overlap of the single photon emission with a multiexciton
decay; clean single photon emission in the energy domain.
•No interaction with other excited charge carriers.
10.8 Single photon emission devices
There are different possibilities for improvements of the single photon emission, when
one wants to build a single photon emitter device. In this work, the proof of the sys-
tematic excitation scheme for a very clean single photon emission is demonstrated. The
single photon emission is very clean in the time and energy domains.
114
10.8 Single photon emission devices
-40 -20 0 20 40
0
20
40
60
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Correlation counts
Time (ns)
T = 4.2 K
LO phonon assisted state preparation
0.87
1.16
0.98
0.036
0.95
1.20
Second order correlation function
0.77
Figure 10.8: Photon correlation measurement under the condition of excitation in LO-
phonon assisted absorption resonance (10 h integration time). The central
peak of the periodic pattern (τ=0) is strongly suppressed, which proves
clean single photon emission.
The next step for the development of a single photon emitter device is the integration
of a single QD into an optical resonator structure, where this excitation scheme can be
applied. This is possible, for example, with a distributed BRAGG reflector (DBR) in a pil-
lar microcavity. If the QDs are integrated in a planar dielectric microcavity, the emission
properties are modified. Spontaneous emission in the normal direction is enhanced if the
cavity is resonant to the emission. The QDs must be located at an antinode position of the
cavity field. The cavity thickness must correspond to the wavelength at the cavity reso-
nance. If there is no coupling to other modes, the spontaneous emission is inhibited (the
cavity is off-resonant). Due to the enhanced emission coupling into a single cavity mode,
the spontaneous emission will have a good directionality, which makes an out-coupling
more easily. The isolation of a single QD can be done by electron-beam lithography and
reactive ion etching. The so produced mircoposts exhibit sinlge QDs. The fabrication of
InAs quantum dots in AlAs/GaAs DBR pillar microcavities is shown by B. ZHANG in
the group of Y. YAMAMOTO in [ZSP+05].
Without using a microcavity structure the output count rate can be enhanced by using
an objective with a higher numerical aperture and/or with an additional lens on the
surface. The semitransparent top (SCHOTTKY-) gate can be omitted as well as the near
field shadow mask (if the QD density is low enough). The relaxation and recombination
process requires a time of less than 1 ns. Thus, the excitation frequency could be enhanced
by a factor of ten (now 80 MHz).
115
10 Single photon emission
116
11 Conclusion and outlook
Single quantum dots are very interesting systems for experimental as well as theoreti-
cal research. The field of single quantum dot spectroscopy is a very fast growing and
dynamic field, where fundamental and applied research are combined with material sci-
ence.
In this work, the dephasing properties of different transitions of a single quantum dot
have been analyzed. The isolation of a single quantum dot allowed for an analysis of
various dephasing mechanisms of a confined electron hole pair. The dephasing of the
p-shell transition is mainly determined by a relaxation process into the ground state, fol-
lowed by an optical recombination process. The excitation of a quantum dot using GaAs
LO-phonon assisted absorption also results in a ground state exciton. The dephasing of
this transition is much faster than the p-shell transition. Due to the integration of a quan-
tum dot-layer into a diode structure it became possible to apply an internal electric field
perpendicular to the quantum dot. The resonantly excited ground state exciton is then
also able to decay by a tunneling process. In this case, the dephasing of the ground state
transition could be controlled via the applied bias voltage. These dephasing mechanisms
exhibit specific time scales which are important for different applications quantum dots.
The possibility of coherent manipulations of single quantum dot-states is thereby the
basis for versatile applications. Coherent investigations of semiconductor quantum dots
(as well as implementations in the field of quantum information processing) are still at
the very beginning of their development. In view of quantum computing, the challenges
are extremely high, so the question of an actual future feasibility can still not be answered.
The great interest in this new type of material system is certainly motivated by the fact
that any research here is based on fundamental quantum effects and the progress in this
field leads to a completely new technology.
One application based on a coherent property of a single quantum system is single
photon emission. The possibility of a coherent manipulation of the p-shell is thereby
the basis for a novel excitation scheme for single photon emission. It has been shown
that the first excited state can be coherently manipulated analogous to the ground state.
Due to the defined excitation of a single exciton in the p-shell, the resulting single pho-
ton emission (after the relaxation and recombination process) appears remarkably clean,
which was demonstrated in a very clean way in this work. The generation of exactly one
exciton in a quantum dot using a single laser pulse is well controllable.
The main challenges for future applications are now an adapted sample design and the
manufacturing of appropriate samples and components. For example, the integration of
quantum dots into a micro-resonator allows for an efficient and highly directive emission.
Furthermore it is possible to influence properties like the optical recombination time with
a micro-resonator.
117
11 Conclusion and outlook
Other material systems are also very interesting for single photon emission. CdSe/ZnSe
quantum dots exhibit a higher band gap and deeper confinement potential which results
in a higher shell energy separation. With suitable excitation, the CdSe/ZnSe quantum
dots can be used for single photon emission in the visible. Further research on single
photon emitters will concentrate on material systems which allow for an operation at
room temperature.
For an application in the field of quantum computing (as well as for fundamental re-
search), the coupling of two or more quantum dots is necessary but also challenging. For
reasons of scalability, a coupling of several quantum dots in a defined lateral arrange-
ment is very promising. For this proposes, a defined and localized quantum dot growth
is necessary. First results have been reported by the groups of REUTER and SCHMIDT
(reference [MRM+07] and [KRS05]). They recently processed areas on a substrate where
the quantum dot position is controlled. Possible coupling-mechanisms are, for exam-
ple, COULOMB interactions or interactions through an optically induced dynamic dipole
moment (FÖRSTER coupling).
The approach of an electrically tunable quantum system, as shown in this thesis, is in
this context a very promising opportunity. Independent from the experimental realiza-
tion, there is always the problem that in particular systems with very weak coupling to
the environment show long coherence times. Conversely, due to the small interaction
strength, these systems can only be very slowly initialized, manipulated, and read out
(for example nuclear spins). Electrically contacted quantum dots offer the interesting
property that the tunneling time can be varied simply by a voltage change. By using a
time dependent bias voltage, the initialization and the read-out process can be decoupled
from normal dephasing.
In summary, there are many theoretical concepts for quantum dot applications today.
Their experimental implementation is scientifically very interesting, but from a techno-
logical perspective also often extremely challenging. The combination of a single quan-
tum system with electric access, as presented in this work, opens up many new concepts
and applications for semiconductor based quantum optics and coherent opto-electronics.
118
Symbols and abbreviations
δdetuning between laser energy and QD-resonance
(δ=ωLaser −ωQD)
ϕphase angle (angle in the u-v-plane of the BLOCH-sphere)
Γlinewidth
Γ0natural linewidth
λwavelength
μdipole moment
θpulse area
σ±circular polarization
τtime scale, e.g. tunnel time, pulse length, decay time, etc.
ωangular frequency of the laser radiation or the optical transition
Ω,Ω0RABI-frequency (on resonance)
ψwave function
1Xsingle exciton
1X−negative charged exciton
1X+positive charged exciton
2Xbiexciton
A10 EINSTEIN-coefficient for spontaneous emission
AFM atomic force microscope
B01 EINSTEIN-coefficient resonant absorption
119
Symbols and abbreviations
B10 EINSTEIN-coefficient stimulated emission
cw continuous excitation (continuous wave)
eelementary charge (1,6022 ·10−19 C)
e−electron
E,ΔEenergy (difference)
Felectric field
fLaser repetition frequency of the laser (80 MHz)
FWHM full width half maximum
GVD group velocity dispersion
hPLANCK’s constant (4,1357 ·10−15 eVs)
¯hh/2π(6,5821 ·10−16 eVs)
hh heavy hole
Icurrent
Isat saturation current
J,Jztotal angular momentum (z-component)
MBE molecular beam epitaxy
langular momentum (orbit)
N1occupation probability of the 1X-state
NA numerical aperture
P(optical) excitation intensity
Pstandardized excitation intensity
PL photoluminescence (spectroscopy)
PLE photoluminescence-excitation (spectroscopy)
QCSE quantum confined STARK effect
120
QD quantum dot
S,Szspin (z-component)
ttime
Ttime period
T1life time
T2dephasing time
T2∗time scale of pure dephasing (no life time induced dephasing)
TEM transmission electron microscope
udispersive component of the BLOCH-vector
vabsorptive component of the BLOCH-vector
VBbias voltage of the photo diode
woccupation component of the BLOCH-vector
WL wetting layer
Xneutral uncharged ground state exciton
2Xbiexciton (neutral, uncharged, ground state)
X−single charged exciton
Xpneutral exciton in the first excited state (p-shell)
XLO ground state exciton, generated by means of an GaAs-LO-phonon
121
Symbols and abbreviations
122
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List of publications
•S. Stufler, P. Ester, A. Zrenner, and M. Bichler
Power broadening of the exciton linewidth in a single InGaAs/GaAs quantum dot
Appl. Phys. Lett. 85, 4202-4204 (2004)
•S. Stufler, P. Ester, A. Zrenner, and M. Bichler
Quantum optical properties of a single InxGa1−xAs−GaAs quantum dot two-level system
Phys. Rev. B 72, 121301(R) (2005)
•A. Zrenner, S. Stufler, P. Ester, and M. Bichler
Manipulations of a qubit in a semiconductor quantum dot
Advances in Solid State Physics 45, 73-184 (2005)
•H. J. Krenner, S. Stufler, M. Sabathil, E. C. Clark, P. Ester, M. Bichler, G. Abstreiter, J.
J. Finley, and A. Zrenner
Recent advances in exciton-based quantum information processing in quantum dot nanos-
tructures
New Journal of Physics 7, 184 (2005)
•S. Stufler, P. Ester, A. Zrenner, and M. Bichler
Ramsey fringes in an electric-field-tunable quantum dot system
Phys. Rev. Lett. 96, 037402 (2006)
•S. Stufler, P. Machnikowski, P. Ester, M. Bichler, V. M. Axt, T. Kuhn, and A. Zrenner
Two-photon Rabi oscillations in a single InGaAs/GaAs quantum dot
Phys. Rev. B 73, 125304 (2006)
•P. Ester, S. Stufler, S. M. de Vasconcellos, M. Bichler, and A. Zrenner
Ramsey fringes in a single InGaAs/GaAs quantum dot
physica status solidi (b) Vol. 243, Issue 10, p. 2229
•A. Zrenner, S. Stufler, P. Ester, S. Michaelis de Vasconcellos, M. Hübner, and M.Bich-
ler
Recent developments in single dot coherent devices
physica status solidi (b) Vol. 243, Issue 14, p. 3696
129
List of publications
•S. Michaelis de Vasconcellos, S. Stufler, S.-A. Wegner, P. Ester, A. Zrenner, and M.
Bichler
Quantum interferences of a single quantum dot in the case of detuning
Phys. Rev. B 74, 081304 (2006)
•A. Zrenner, S. Stufler, P. Ester, and M. Bichler
Coherent properties of quantum dot two-level systems
ISRAEL JOURNAL OF CHEMISTRY, Volume 46, Issue: 4, p. 349-356 (2006)
•P. Ester, S. Stufler, S. Michaelis de Vasconcellos, M. Bichler, and A. Zrenner
High resolution photocurrent-spectroscopy of a single quantum dot
physica status solidi (c) Vol 3, Issue 11, p. 3722 (2006)
•S. Michaelis de Vasconcellos, S. Stufler, S.-A. Wegner, P. Ester, M. Bichler and A.
Zrenner
Quantum interferences of a single quantum dot in the case of detuning
physica status solidi (c) Vol. 3, Issue 11, p. 3730 (2006)
•P. Ester, L. Lackmann, S. Michaelis de Vasconcellos, M. C. Hübner, A. Zrenner, and
M. Bichler
Single photon emission based on coherent state preparation
Appl. Phys. Lett. 91, 111110 (2007)
•P. Ester, L. Lackmann, M.C. Hübner, S. Michaelis de Vasconcellos, A. Zrenner and
M. Bichler
p-shell Rabi-flopping and single photon emission in an InGaAs/GaAs quantum dot
Physica E: Low-dimensional Systems and Nanostructures, Volume 40, Issue 6, April
2008, Pages 2004-2006
130
Danksagung
Abschließend möchte ich mich bei allen bedanken, die zum Gelingen dieser Doktorarbeit
beigetragen haben. Mein besonderer Dank gilt:
•ARTUR ZRENNER, der mir die Möglichkeit gegeben hat, an diesem interessanten
und innovativem Thema meine Dissertation durchzuführen. Ohne diese fachliche
und technologische Hilfestellung wäre diese Arbeit nicht möglich gewesen.
•Der gesamten Arbeitsgruppe nanostructure optoelectronics. Allen voran den beiden
Quanten-Punkt-Doktoranden MARC C. HÜBNER und STEFFEN J. MICHAELIS DE
VASCONCELLOS für die super Zusammenarbeit, CHRISTIAN PASSLICK,GERHARD
BERTH,HEIKE DEGLER,LYDIA LACKMAN,REINER SCHNEIDER,WOLF-RÜDIGER
SCHULTE,THOMAS HANGLEITER und allen ehemaligen Mitgliedern der Arbeits-
gruppe für eine wirklich angenehme Atmosphäre.
•Der Prüfungskommission: Prof. Dr. TORSTEN MEIER für die Übernahme des
Komissionsvorsitz, Prof. Dr. KLAUS LISCHKA und Prof. Dr. ARTUR ZRENNER
für die Übernahme der Gutachtertätigkeiten, und Dr. CHRISTOF HOENTZSCH als
Vertreter des Mittelbaus.
•Dem gesamten Department Physik, insbesondere den Arbeitsgruppen LISCHKA,
MEIER und FÖRSTNER für den wissenschaftlichen Austausch, sowie der mechanis-
chen Werkstatt für die schnelle Umsetzung meiner Projekte.
•JOHANNES PAULI, unserem schnellen Tieftemperatur-Dealer der immer Zeit hatte
für ein kaltes Kännchen.
•Ganz besonderer Dank gilt meiner Freundin JOHANNA, die mich und meine Arbeit
fantastisch unterstützt hat. Du bist super!
•Meinen Eltern, die mich in meinen Entscheidungen immer gefördert haben.
•Meinen Freunden: ANDI,ANSGAR,BASTIAN,BERND,BURKHARD,CAROLINE,
CHRISTIAN,DAVID,DORITH,FRAUKE,GUNNAR,JENS,JÖRG,JULIA,MARC,NICKI,
SARAH,SABINE,SEBASTIAN,STEFAN,THOMAS, ...
•und allen die in dieser Liste ausversehen vergessen wurden. Danke!
131
