Lehner, BSc Submitted at Institute of
Semiconductor and Solid State Physics Supervisor Univ.- Prof. Dr. Armando Rastelli September 2019 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, ¨Osterreich www.jku.at DVR 0093696
Magnetophotoluminescence
study of excitons confined
in GaAs quantum dots
Master Thesis
to obtain the academic degree of
Diplom-Ingenieurin
in the Master’s Program
Technische Physik
Ich erkl¨are an Eides statt, dass ich die vorliegende Masterarbeit selbstst¨andig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die w¨ortlich oder sinngem¨aß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Masterarbeit ist mit dem elektronisch ¨ubermittelten Textdoku-ment identisch.
Linz, September 2019
In the field of quantum information technology, single photons are a crucial resource for realizing qubits. An advantage of a photon based qubit compared to other systems is its low interaction with the environment. This allows a decoherence free distribution of quantum states over large distances at the speed of light. Therefore, a lot of effort is put in the research of single photon-sources. Among the different systems under investi-gation, semiconductor quantum dots are promising candidates to act as an on-demand single photon source in advanced quantum technology applications.
While photons are the ideal candidates for quantum information transport, the situation is different when it comes to quantum information storage and processing. Therefore, a transfer from one physical form to another is needed to enable quantum networks and quantum entanglement over long distances. For this purpose, electron spin states confined in quantum dots of a direct bandgap semiconductor are preferred and the con-version of photon qubits into spin qubits is necessary. However, in order to perform a conversion between photonic and spin qubits, elaborate knowledge of the response of excited states in a quantum dot to an external magnetic field is required. This response is specified by the g-factor and the diamagnetic coefficient.
In this thesis magnetic field dependent photoluminescence measurements are performed to study the g-factors of different complexes as well as the diamagnetic shift of optical transitions of an optically excited gallium-arsenide (GaAs) quantum dot. There have al-ready been researches about magneto-optical properties of quantum dots, especially for indium-gallium-arsenide (InGaAs) systems. However, similar works for droplet-etched GaAs quantum dots are still missing. While for strongly confining systems, e.g. InGaAs quantum dots the single-particle Hamiltonian is used to extract the magnetic proper-ties, the question arises if this description is also valid for weak confinement systems like our GaAs quantum dots. This work proves that an evaluation with the single-particle picture is not reasonable. Nevertheless, by investigating the magnetic properties, the various transitions in a QD are studied and assigned to certain charge complexes.
At this point I want to thank the people who supported me during my master thesis at the Institute of Semiconductor and Solid State Physics. First of all I want to thank my supervisor Armando Rastelli, who gave me the opportunity to return after my bachelor thesis for the master and for working on an interesting topic.
Special thanks go to Daniel Huber. He taught and helped me a lot even though he had to finish his PhD. He had always an answer to my questions and was very patient in explaining the things I wanted to know.
Further, I shared the office with Daniel, Marcus, Christian, Simon and Julian. Thank you for the discussions and the help. I had always a lot of fun with you.
I want to thank the rest of team, Saimon, Huiying, Xueyong, Dominic, Tobias and our visitor Michele, for ideas and pleasant group meetings with you.
Thanks go also to our technicians Alma, Ursula, Stephan, Albin and Ernst for keeping things in and around the cleanroom running. I want to thank Susi for the administrative work.
At the end of my master, I want to thank my former colleagues from Primetals Tech-nologies, especially Claudia Hemmelmeir and Nicole Oberschmidleitner, for giving me the opportunity to work during studying. Thank you for the flexible working hours, especially during the master thesis and for the nice counter balance to the university life.
I also want to thank my new colleagues from the open innovation center. I already had some nice coffee breaks with you during writing the thesis.
I want to thank Amadea and of course my fellow students, especially Kathi who gave me extra motivation when walking by her office door. Also Anna and Maxi have become close friends and I could always count on their help.
Finally, I want to thank my family for all the support they gave me. I know I can sometimes be a nuisance, especially when I want to finish things. Last but not least, I want to thank my boyfriend Clemens. You have always a good advice for me, you can calm me down like no one else and I am glad to have you on my side.
1 Introduction 1
2 Concepts and Theoretical Aspects 4
2.1 Band Structure in Solids . . . 4
2.1.1 Band Structure of GaAs . . . 5
2.1.2 Effective Mass Approximation . . . 6
2.2 Quantum Dots . . . 7
2.2.1 A Simple Method to Describe a Quantum Dot . . . 7
2.2.2 Droplet-Etched GaAs Quantum Dots . . . 8
2.3 Multi-particle complexes . . . 9
2.3.1 Fine Structure Splitting . . . 11
2.4 Optical Pumping of a Quantum Dot . . . 12
2.4.1 Resonant Two-Photon Excitation . . . 13
2.5 Response of a QD to an External Magnetic Field . . . 14
2.5.1 The g-factor . . . 14
2.5.2 Diamagnetic Shift . . . 16
2.5.3 Fitting Models . . . 17
2.5.3.1 Exciton States . . . 17
2.5.3.2 Trion States . . . 18
2.5.4 Relative Oscillator Strength . . . 18
3 Photon-To-Spin Conversion 21 4 Measurement Setup 25 4.1 Setup for Photoluminescence Measurements in Magnetic Fields . . . 25
4.2 Vectormagnet . . . 27
4.2.1 Cooldown of the Magnet System . . . 28
4.2.2 Using the magnet . . . 29
4.2.3 Quench . . . 30
4.2.4 Remote operation of the PSU . . . 30
5 Photoluminescence Measurements 31 5.1 Magneto-Photoluminescence Measurements . . . 33
5.1.1 Exciton X . . . 34
5.1.2 Temperature-Dependent Photoluminescence Measurements and Trion States . . . 37
5.1.3 Remaining Spectral Lines . . . 41
5.1.4 Evaluation of Additional Quantum Dots . . . 44
5.1.4.1 Quantum Dot 2 . . . 44
5.1.4.2 Quantum Dot 3 . . . 45
Preface VIII 5.2 Relative Oscillator Strength . . . 46 5.3 Quasi-Resonant Excitation . . . 49
6 Lifetime and Correlation Measurements 50
6.1 Resonant two-photon excitation . . . 50 6.2 Lifetime Measurement of X and XX . . . 51 6.3 Correlation Measurements . . . 53
7 InGaAs Quantum Dots 55
7.1 Sample 1 . . . 56 7.2 Sample 2 . . . 58
8 Possible Improvements and Alternative Methods 61
9 Summary and Outlook 63
Appendix A Evaluated Parameters of GaAs Quantum Dots 65 Appendix B Evaluated Parameters of InGaAs Quantum Dots 68
B.1 Sample 1 . . . 68 B.2 Sample 2 . . . 69
Bibliography 76
List of Figures 80
Day by day, we are used to our smartphones or computers and we are truly depending on that technology for our everyday business. Thanks to enormous progress, computer power and memory have increased tremendously over the past century, but it all started very simple. In 1837 Charles Babbage published the first ideas of a mechanical calcula-tor, the analytical engine. The machine was supposed to work with base-10 fixed-point numbers and he made early drafts of an “internal” memory which should hold 1000 numbers of 40 or 50 digits, each digit represented by a gear wheel. This was larger than the storage capacity of any computer built before 1960, but the steam-driven machine had never been constructed in this way [1]. Decades passed and the next type of mem-ory was used by the Zuse Z3 computer in 1941. It had 1408 relays with a capacity of 64 words with 22 bits each [2]. In the meantime, Julius Edgar Lilienfeld had patented between 1925 and 1927 the first ideas of a field effect transistor. However, back then it was not possible to realize one [3]. In the forties different attempts for realizing a mem-ory were made. Examples are the regenerative capacitor drum memmem-ory or the Williams tube which stored 128 40-bit words.
This year (2019) we celebrate the 50-year anniversary of the moon landing. The Apollo Guidance Computer (AGC) used there was equipped with a magnetic core memory as random-access memory (RAM), which had a capacity of 4 KiB [4]. The first patent for magnetic core memories was granted in 1947 and it became the predominant memory type until the seventies. It consists of magnetically hard ferrite cores and each core was capable of storing 1 bit, mostly manually connected by hand through wires. Therefore, the costs started at around one dollar per bit. In the late 1960s a density of around 1130 KiB per 1 m3 was achieved, the costs declined to 0.01 $ per bit. In the 1970s
mag-netic core memory was replaced by semiconductor memory. After all, magmag-netic core memory was still used in military equipment and for space travelling e.g. in the space shuttle, since it was insensitive to radiation [5]. Nevertheless, semiconductor industry was thriving. First, bipolar transistors were used, later on metal-oxide semiconductor field-effect transistors (MOSFETS) were applied for memories (1964) [6]. In 1965 Gor-don Moore already realized the trend of the semiconductor development. He stated that every two years the number of transistors on integrated circuits will double [7]. Nowa-days, transistors have a size of about 14 nm and laptops are available with up to 64 GiB RAM. Sooner or later, this will lead to difficulties, since quantum effects are arising at the nm-scale. Gordon Moore was aware of this fact too and people started to develop new concepts.
In 1980 to 1982 first attempts of developing a quantum computer were made by Richard Feynman and Paul Benioff [8, 9]. Compared to a classical computer, the information unit is a quantum bit (qubit) instead of a bit. In contrast to the bit where the informa-tion state is 0 or 1, the qubit is defined by its quantum state, which can be 0, 1 or any
2
superposition of these states. Therefore, quantum computers could provide solutions for specific problems, since certain algorithms could be executed much faster than with classical computers. However, quantum computing is still in the early stages of develop-ment, no large-scale quantum hardware has evolved yet and fundamental research is a big issue. We have already seen how bits were realized in the past and how long it took to store information properly. The question arises of how should we deal with qubits. A way to realize qubits are photons since they have ideal properties, like the fact that they provide the fastest possible way to transmit information, since photons are travelling at the speed of light. Moreover, the polarization state of a photon can represent the states 0 and 1 as well as any superposition. A Bloch sphere (Fig. 1.0.1) illustrates the possible states, where 0 corresponds to horizontal |Hi and 1 to vertical polarization |V i.
Figure 1.0.1: A qubit can be repre-sented by the Bloch sphere. The states |Hi, |V i and any superposition |ψi= α |Hi + β |V i is possible [10].
Nowadays, quantum computers are an impor-tant research field. However, in order to build up a quantum network also quantum communica-tion and cryptography is needed [11]. The topic of making information transfer absolutely secure is of special interest, since the commonly used way of information encryption is indeed hard to decipher but still not impossible. Here, also pho-tons play a major role. A perfectly secure com-munication of messages between two parties can be achieved by encrypting the information with the polarization of photons. A possible eaves-dropper who wants to spy out the encryption key would reveal himself, since he distorts the trans-mission through his polarization measurements. To realize such an encryption in a commercially useable way, long-distance quantum networks have to be established, which is a task far from being easy and it is essential to use sophisticated devices as the quantum re-peater [12].
A photon source used for quantum computing and quantum cryptography has to meet several requirements. The source has to generate single photons as well as single en-tangled photon-pairs by ”pushing a button” with zero multiphoton emission probability (on-demand generation). Furthermore, the emitted single photons (entangled-photon-pairs) have to be indistinguishable from all the other photons (photon-(entangled-photon-pairs) emitted by the source in all degrees of freedom [13]. Last but not least, a high extraction efficiency of the radiated photons is needed. A promising candidate for a photon source meeting all these requirements are semiconductor-based self-assembled quantum dots (QDs). QDs are nanostructures, where the motion of charge carriers like electrons or holes is confined in all dimensions of space. This is done by embedding a semiconductor struc-ture with a size, smaller than the free exciton Bohr radius in a semiconductor matrix with a larger bandgap. This leads to discrete energy levels like in an atom and the recombination of electrons and holes generates single photons. The benefit to single
atoms is that QDs are easy to integrate in on-chip devices. Within this work, we focus on droplet-etched gallium arsenide (GaAs) QDs (section 2.2) grown by molecu-lar beam epitaxy. These kind of QDs allow to generate highly indistinguishable and strongly polarization-entangled photons on demand [14, 15]. For applications not only the transmission of a qubit is important, moreover, it has to be stored and eventually manipulated. Photon qubits are suitable for transmission as they do not easily interact with the environment [16]. Nevertheless, manipulation and storage of a photonic qubit is difficult. In contrast to that, matter qubits offer benefits regarding manipulation and storage functions. Hence, quantum networks rely on a conversion from photon qubits to matter qubits [17]. Exactly at this point QDs can again provide help. The electrons or holes confined in a QD have a spin, which can be used as an alternative for matter qubits [17]. Other approaches are rubidium and cesium vapor based quantum memories [18, 19]. However, QDs are preferred since established semiconductor techniques can be used and QDs host interband excitations required for photon-to-spin conversion[17]. To put it in a nutshell, QDs are the natural interface between a solid-state and a photonic quantum system [17]. Further, electron spins in a QD provide long coherence times, which is necessary to prevent qubit errors [20]. In order to transform a photonic qubit into a spin qubit, knowledge about the response of excited states in a QD to an external applied magnetic field is required, since electron and hole states have to be defined for a coherently photon-to-spin conversion. Especially the electron and hole g-factors (see chapter 3), which describe this response, have to be investigated.
Photon-to-spin conversion is the main motivation for this thesis, which deals with the study of electron and hole g-factors in weakly confining GaAs/AlGaAs QDs obtained by the droplet etching method. There have already been studies to investigate the magneto-optical properties of QDs, but comparable works for this material system are to our knowledge virtually absent. Furthermore, it is disputable whether approaches and theoretical descriptions from previous studies [21, 22] are applicable to our GaAs QDs. In this work magnetic field dependent photoluminescence (PL) measurements were performed to analyze the g-factors in our QD system. In order to identify different excited states, also polarization-resolved and temperature-dependent PL measurements were made. Finally, this allows a conclusion, whether the applied theory is correctly describing the QD system or not.
This thesis is divided into nine main chapters. Followed by this introduction, chapter 2 gives an overview on semiconductor QDs and the theory on magneto-optical properties. Chapter 3 introduces briefly how a QD can be used to store a qubit, followed by chap-ter 4 where the setup for magneto-optical PL measurements is discussed. The results of these measurements are presented in detail in chapter 5 where several QDs have been investigated. Further experiments on the properties of QDs are presented in chapter 6, where the lifetime of the exciton and biexciton has been studied and correlation mea-surements for studying unknown transition were carried out. After finishing the analysis of GaAs QDs, chapter 7 presents results of magneto-PL measurements of InGaAs QDs for comparison. Finally, possible improvements and alternative methods are shown in chapter 8 followed by a summary (chapter 9). The evaluated data of all studied QDs can be found in Appendix A and B.
2 Concepts and Theoretical Aspects
In this chapter, general information about semiconductor quantum dots and their emis-sion spectra are discussed. Further, theoretical aspects concerning magneto-optical mea-surements are given, which is necessary for evaluating the measurement data (see chap-ter 5).
2.1 Band Structure in Solids
First, we will discuss the band structure in solids. Therefore, it is necessary to look at the dispersion relation of an electron in a crystal, where a large number of atoms is arranged in a periodic order in all dimensions of space. Hence, one hast to solve the time-independent Schr¨odinger equation for a single electron
"
−~2
2m 52+V (r)
#
Ψ(r) = EΨ(r), (2.1.1)
where ~ denotes the Planck constant, m the mass and V a potential. Ψ(r) is the wave function as a function of the position r and E represents the eigenenergy of the system. Further a potential
V(r) = V (r + R), (2.1.2)
which is periodic with the lattice, is considered, where the translational vector R is given by
R= n1a1+ n2a2 + n3a3, (2.1.3)
where a1, a2, a3 are the primitive translation vectors and n1, n2 and n3 are integer
numbers. The general ansatz for the wave function is a linear combination of plane waves
Ψ(r) =X
G
Ck−Get(k−G)·r, (2.1.4)
with the vector k in the reciprocal space within the first Brillouin zone and the reciprocal lattice vector G. The corresponding eigenvalues are then
Ek = En(k) = En(k + Gn) (2.1.5)
where n is the band index to numerate all solutions for a specific k. The connection between E and k is called the dispersion relation. With Eq. 2.1.5 the band structure is defined. The different bands are separated by forbidden regions, so-called bandgaps.
By filling up the states at zero temperature with electrons, some bands are filled com-pletely and others stay empty. Between the highest filled (valence) and the lowest empty (conduction) band an energy gap for all k-vectors exits. The Fermi energy marks the highest occupied energy. For insulators and semiconductors it lies between valence and conduction band.
At a temperature T 6= 0 an electron can be excited to the conduction band and leaves a vacancy called hole in the valence band. Further, the movement of all remaining elec-trons in the valance band due to the influence of an electric field, can also be described equally by the movement of a hole, i.e. as a particle with positive charge. In conclusion electrons and holes are responsible for the optical and electric properties.
2.1.1 Band Structure of GaAs
The quantum dots used in this thesis are mostly GaAs quantum dots. Therefore, we want to have a look at the band structure of GaAs in this section. Since the band structure is quite complex, we will first discuss the parabolic approximation of a direct semiconductor. Hence, we just consider the highest valence band and the lowest conduc-tion band at the origin of the Brillouin zone (Γ-point). The valence and the conducconduc-tion band can be well approximated by the following parabolas, whereas the the semicon-ductor is supposed to be isotropic and the dispersion relation is equal in all directions of the k-space: Ec(k) = Ec+ ~ 2k2 2m∗ e (2.1.6) Ev(k) = Ev− ~ 2k2 2m∗ h . (2.1.7)
Here, Ec− Ev = Eg denotes the energy gap and m∗e and m
∗
h are the effective masses of an electron and a hole respectively, which leads to different curvatures of the bands. The effective mass for electrons and holes moving inside the bands is given by
m∗e = ~ 2 ∂2E c ∂k2 , (2.1.8) m∗h = ~ 2 ∂2Ev ∂k2 . (2.1.9)
Further, some semiconductors e.g. GaAs, show degenerated bands as depicted in Fig. 2.1.1. The different valance bands can be described by different effective masses in the disper-sion relation in Eq. 2.1.7. For the heavy hole band, the effective mass is m∗
HH, for the light hole band m∗
6 2.1. Band Structure in Solids
(a) (b)
Figure 2.1.1: Band Structure of (a) GaAs, whereas (b) shows only the bands of GaAs at the Γ-Point. One can see that GaAs has degenerated valence bands with different curvatures. Heavy hole (HH), light hole (LH) and split-off band (SO) are visible. Adapted from [23, 24].
For creating semiconductor heterostructures such as quantum dots, one needs materials with different bandgaps.
2.1.2 Effective Mass Approximation
Mathematically the behaviour of electronic motion can be described by the effective mass approximation. It is a semiclassical model, where electrons are described quantum-mechanically and the external fields classically. Further, the wave functions are linear combinations of Bloch waves describing now wave-packages. Instead of using the scalar mass m, the semiclassical equations of motion leads to the definition of an effective mass tensor. Additionally, a slowly varying potential Vext(r) is assumed, so that interband transitions can be neglected and only band n is contributing. The wave function which is localized in the k-space is approximated by
Ψ(r) ≈ Ψn,k0
X k
an,kei(k−k0)·r = Ψn,k0F(r). (2.1.10)
Now, this equation describes the wave function of an electron in a weakly perturbed crystal potential in form of Bloch waves, multiplied by an envelope function F (r). We can plug this into the Schr¨odinger equation, and get the Effective Mass Equation (EME)
[En(−i5) + Vext(r)] F (r) = E F (r). (2.1.11)
The kinetic term of the Schr¨odinger equation is now replaced by the dispersion relation
En(−i5) of a specific band n. This brings the benefit that not the actual wave function has to be treated. Instead, the envelope function F (r) can be used. A detailed derivation can be found in Ref. [25].
Quantum dots, which will be discussed in the next section can be described with the EME.
2.2 Quantum Dots
In principle a quantum dot is a system where the motion of charge carriers is confined in every dimension of space. These systems are zero-dimensional semiconductor het-erostructures. Since quantum confinement effects lead to discrete energy states, which is shown in section 2.2.1, QDs are also called artificial atoms. The main difference to atoms or molecules is that the energy levels of a quantum dot can be tuned due to the size dependency of the confinement effect, whereas the energy levels of atoms or molecules are mainly depending on the elements.
2.2.1 A Simple Method to Describe a Quantum Dot
The mathematical description of a QD is challenging, since fully analytical solutions do not exist. The system of a QD can be described by using the EME from section 2.1.2. Nevertheless, there are simplified models [26]. The easiest way of describing the fun-damental principles of a QD is the particle in a box problem, with an infinitely high potential and the length L in x,y and z-direction. Therefore, the Schr¨odinger equation Eq. 2.1.1 has to be solved for a potential
V(r) =
(0 for |x|, |y|, |z| < L/2
∞ else. (2.2.1)
The full derivation of the solution can be found in [27, 28]. For the wave function we get the following solution:
Ψ(y, x, z) = s 8 L3 sin n xπx L sinnyπy L sinnzπz L , (2.2.2)
where nx, ny, nz are quantum numbers according to the three spatial dimensions. The eigenenergies are given by
E(nx, ny, nz) =
h
2m∗L(n 2
x+ n2y + n2z). (2.2.3)
This energy spectrum is similar to that of an atom, therefore, QDs are also called arti-ficial atoms. Fig. 2.2.1 shows the energy levels, conduction and valence band. Now, the EME from section 2.1.2 allows us to draw the shown band structure over the directions
x, y, z instead of using the wave vector k.
The energy levels are denoted as s-, p-, d-shell and so forth, whereas the s-shell refers to the first valence and conduction band level. Note, that this labeling does not correspond to orbital symmetries. Further, we can introduce the exciton X Bohr radius, which is given by aXB = r me µ ! aB, (2.2.4)
where ris the relative permittivity, methe free electron mass, aBthe Bohr radius and µ the reduced mass of electron and hole, which is given by the respective effective masses
µ= m ∗ em∗hh m∗ e+ m∗hh . (2.2.5)
8 2.2. Quantum Dots By using r = 12.9, m∗e = 0.063me and m∗hh = 0.51me for GaAs [29], an exciton Bohr radius of 12 nm is obtained. Due to the size of the GaAs QDs used in this thesis, the QD electronic states have a lateral extension much larger than the exciton Bohr radius for GaAs [30]. This results in laterally weak confined X states. Therefore, quantum confinement effects are overwhelmed by Coulomb interaction [31].
Figure 2.2.1: Schematics of conduction (CB) and valence band (VB) edges in an optically active quantum dot formed by semiconductors A and B with different bandgaps. Adapted from [16].
2.2.2 Droplet-Etched GaAs Quantum Dots
The GaAs quantum dots used in this thesis are fabricated by the droplet etching method as shown in Fig. 2.2.2 and embedded in an AlGaAs matrix. First Al is evaporated, which forms a droplet on the AlGaAs surface. Due to an arsenic gradient, nanoholes are etched
AlGaAs AlGaAs AlGaAs Nanohole Al Droplet Inverted GaAs GaAs QD (a) (b)
Figure 2.2.2: (a) Fabrication of droplet-etched GaAs quantum dots with molecular beam epitaxy (MBE). (b) Atomic force microscopy image (250 nm x 250 nm) of a nanohole. The color corresponds to the height. Adapted from [16].
into the surface. These nanoholes are then filled with GaAs which forms the QD. In the end, the layer is capped with a AlGaAs layer, which forms the top barrier [32, 33]. We obtain lens-shaped GaAs QDs. To increase the photon extraction efficiency, which is limited by total internal reflection on the AlGaAs/Air interface, we use a planar cavity structure with like shown in Fig. 2.2.3. The cavity consists of top and bottom distributed Bragg reflector (DBR) mirrors. The DBR consists of Al0.95Ga0.05As/Al0.2Ga0.8As pairs.
Two pairs (top) and nine pairs (bottom) were used. We define the growth direction [001] as z (α = 0◦). The in-plane directions are x and y. The angle α between x and z
is important for the orientation of the applied magnetic fields in this thesis.
Figure 2.2.3: Sketch of the used sample structure. The GaAs QDs are embedded in a Al0.4Ga0.6As matrix. The distributed Bragg reflectors are enhancing the
photon extraction efficiency. We label the growth direction [001] as z. The in-plane directions [110], [1-10] as x and y, respectively.
2.3 Multi-particle complexes
The fundamental optical transitions involves carriers in the first energy level in the CB and the VB, also called s-shell. Due to Pauli’s exclusion principle, one energy level can be occupied by two electron e− or two holes h+ with opposite spin. Commonly, the first
h+ level is a heavy hole (HH) like state as a result of quantum confinement effects [26].
Fig. 2.3.1 shows multi-particle complexes that are formed by Coulomb interaction. A single e− and h+ are forming the fundamental complex, a neutral exciton X. The X is
decaying via the emission of a single photon. An e− has a total angular momentum
projection of
se,z = ±1/2, (2.3.1)
whereas the heavy-holes have a total spin of
jhh,z = ±3/2. (2.3.2)
Therefore, e−and h+ needs opposite spin in order to preserve the spin at recombination,
10 2.3. Multi-particle complexes
Figure 2.3.1: Multi-particle complexes in a QD. Filled circles correspond to electrons, empty circles correspond to holes. Not all possible spin configurations are shown here. Especially the different spin configurations of the excited X+ (e− - h+ pair with an extra h+ in the p-shell), which lead to the
configurations X1+∗, X3+∗ and X4+∗ are not displayed here.
optically inactive and called dark states. Further complexes are the biexciton XX, which is formed by two e− and two h+, the negative trion X– (two e− and one h+)
and the positive trion X+ (one e− and two h+). We are especially interested in the
excited states of the X+, the so called hot trions, which are formed by an e− - h+
pair in the s-shell and an additional e− or h+ in the p-shell. These states provide a
dominant optical transition in the spectrum of the GaAs QDs (chapter 5) and are of special interest during our g-factor studies, since the hole in the p-shell has already 20% light hole character due to heavy hole-light hole mixing [34, 35]. As we will discuss later, a higher g-factor is expected for these states. In total there are23 combinations
of forming an excited X+ state. Due to the exchange interaction (discussed later) we
get four degenerate doublets [36]. The energetic order is difficult to identify. Therefore, the order is taken from Ref. [36]. The states are labeled as X1+∗, X2+∗, X3+∗ and X4+∗,
where the subscript 1 is denoting the state lowest and 4 the state highest in energy. |X+∗1 i= ↑s⇑s⇑p jz = (+7/2) ↓s⇓s⇓p jz = (−7/2) |X+∗3 i= ↑s(⇑s⇓p + ⇓s⇑p) jz = (+1/2) ↓s(⇑s⇓p + ⇓s⇑p) jz = (−1/2) |X+∗2 i= ↓s⇑s⇑p jz = (+5/2) ↑s⇓s⇓p jz = (−5/2) |X+∗4 i= ↑s(⇑s⇓p − ⇓s⇑p) jz = (+1/2) ↓s(⇑s⇓p − ⇓s⇑p) jz = (−1/2) (2.3.3) The spin configuration of an e− is denoted by ↑ or ↓ and that of a h+ by ⇑ and ⇓.
The respective shell is noted as index s or p. The X1+∗ state does not fulfill the dipole
selection rules and is therefore forbidden. Also negative hot trions are existing, which are similar to the positive ones. The different states will be treated in chapter 5, when the effect of a magnetic field is discussed.
2.3.1 Fine Structure Splitting
There are four different spin configurations of an e− - h+ pair (exciton state) formed
by a heavy hole (HH) with the spin projection jhh,z = ±3/2 and by an electron with a spin projection of se,z = ±1/2 in the s-shell. The four different exciton states are having different total spin projections jz of
|+1i = |↓⇑i , |−1i = |↑⇓i ,
|+2i = |↑⇑i , |−2i = |↓⇓i , (2.3.4)
where again ↑ or ↓ denotes the spin configuration of an e− and ⇑ and ⇓ the spin
configuration of a h+.
The states with total angular momentum |±2i are optically not active (dark states), whereas the states |±1i are optically active (bright states). Depending on the symmetry of the QD confinement potential, the exchange interaction leads to a coupling between |+1i and |−1i and we obtain the two eigenstates √1
2(|+1i ± |−1i) which are split by
an energy called fine structure splitting (FSS) [15, 26]. Since only the bright states can be observed in the absence of a magnetic field, two lines with different polarization will be visible in a photoluminescence (PL) spectrum. Independently from the symmetry of the confinement potential, a splitting between the dark states is always existing [26]. Fig. 2.3.2 shows the possible radiative decay paths with and without FSS.
(a) (b)
Figure 2.3.2: Scheme of the radiative decay of a biexciton XX state in a quantum dot. (a) In general, the X state of a QD is lifted by the FSS. (b) In a QD without FSS the paths are indistinguishable. This leads to an entangled state and the emitted photon pairs are cross-circularly polarized. Adapted with changes from Ref. [37].
The exchange interaction between e− and h+ can be described by the exchange
12 2.4. Optical Pumping of a Quantum Dot and can be written according to Ref. [38] as
Hexchange= 1 2 +δ0 +δ1 0 0 +δ1 +δ0 0 0 0 0 −δ0 +δ2 0 0 +δ2 −δ0 , (2.3.5)
where δ0 is the splitting between bright and dark states, δ1 the FSS of a bright exciton
and δ2the equivalent splitting for a dark exciton. For other complexes, like the biexciton
or the trions no exchange interaction is present [21].
2.4 Optical Pumping of a Quantum Dot
The generation of excited states within a QD can be accomplished in different ways. If we restrict the discussion to optical excitation, we mainly have three distinct oppor-tunities namely non-resonant, quasi-resonant and resonant excitation. An example for the first one is above-bandgap excitation is shown in Fig. 2.4.1(a) and was used for most of the measurements in this thesis. Pumping with a laser excites carriers in the barrier material, which are captured by the QD. They relax to the s-shell via phonons [26]. In contrast to above-bandgap excitation, with quasi-resonant excitation shown in Fig. 2.4.1(a), e− - h+ pairs are created in a higher shell of the QD, for example the
p-shell. Usually, the carriers are experiencing a fast relaxation process to the s-shell. The third way to excite carries in the dot is the resonant excitation (Fig. 2.4.1(b)), where e−- h+ pairs are directly generated in the s-shell [16]. For a detailed discussion of
(a) (b) (c)
Figure 2.4.1: Pumping of a quantum dot under (a) above-bandgap, (b) quasi-resonant and (c) resonant excitation with the laser energy EL.
the advantages/disadvantages of the different excitation schemes the interested reader is referred to Ref. [39].
For the magneto-PL measurements we will make use of above-band excitation, since electron and hole mobility in the AlGaAs barrier influences the formation probability of different charge complexes, in which we are interested. Nevertheless, we will need resonant two-photon excitation (TPE) in chapter 6 to populate the biexciton XX state. Hence, in the next section we want to discuss TPE in more detail.
2.4.1 Resonant Two-Photon Excitation
By driving a QD under above-band excitation, the XX in our QDs is hardly visible, since the probability of having two e− and two h+ in the s-shell is almost zero because
of short lifetime of the optical transition. A way to directly create a XX state is the two-photon excitation. Due to dipole selection rules a two-photon absorption process is needed to populate the XX state, where the photons (with energy E1 and E2) have to
fulfill the condition [16, 40]
E1+ E2 = EX+ EXX, (2.4.1)
where EX and EXX are the emission energies of X and XX according to Fig. 2.4.2. In
Figure 2.4.2: Scheme of two-photon excitation of the biexciton XX. The laser energy is tuned in order to fulfill EL = (EXX + EX)/2. Adapted with changes from [41].
an experimental setup this is realized by a laser which is tuned to the energy EL where
EL = E1 = E2 =
EXX+ EX
2 , (2.4.2)
is valid. Fig. 2.4.2 shows the energy scheme of the process. The emission energy of the XX differs by the relative binding energy EB from the emission energy of the single X and can be calculated via
EB = EX− EXX. (2.4.3)
The XX is called antibinding for EB <0 and binding for EB >0 which is depending on the strength of the Coulomb interaction [16]. In chapter 6 we will need TPE in order to resonantly drive the XX-X cascade, locate the XX and calculate the binding energy EB.
14 2.5. Response of a QD to an External Magnetic Field
2.5 Response of a QD to an External Magnetic Field
In this section we will discuss the change of the emission properties of a QD due to an applied magnetic field. First, the g-factor and the diamagnetic shift are introduced. Then fitting models will be derived for evaluating experimental data.2.5.1 The g-factor
The g-factor combines the spin magnetic moment and the angular momentum of a particle. For example, the electron spin g-factor ge is defined by
µS = ge
µB
~ S, (2.5.1)
where µS is the magnetic moment from the spin of an electron, S the spin angular
mo-mentum and µB the Bohr magneton. For a free e−the value of ge ≈2 is well known [42]. In a bulk semiconductor a strong spin-orbit interaction, leads to huge negative electron g-factors e.g. in InAs −14.7, GaAs −0.44 or in InSb 51.3 [43, 44].
In general, the spin-orbit interaction can be derived from the non-relativistic limit of the dirac equation [45] and influences the magnetic moment µ of an e− by a contribution
from the orbital motion additionally to the spin angular momentum [46]. The coupling between µ with an external magnetic field B is given by [47]
−µB = 1
2µBσgB, (2.5.2)
where σ is the Pauli vector and g a tensor. Its components are called g-factors. The
g describes the spin and orbital contributions to the magnetic moment [46]. Therefore,
knowledge of the g-factors gives an understanding of spin-orbit interaction as well as an insight to the effective orbital motion related with a quantum state [48].
In Fig. 2.5.1 the change of the g-factor due to spin-orbit coupling in different systems is shown. The free e− g-factor from Eq. 2.5.1 is shown left in Fig. 2.5.1 whereas for an
atom the g tensor reduces to the Land´e g-factor, since the magnetic moment is now given by adding spin and orbital angular momentum. On the contrary, the lattice pe-riodicity in solids leads to extended states, described by the Bloch theorem. Therefore, the magnetic moment has also contributions of the envelope wave functions and the g-factors are changing substantially in bulk crystals [46]. They can be calculated by Roth’s formula [49]. However, we will not further focus on the bulk crystal. In QDs, the g-factor is altered due to the three-dimensional (3D) confinement, which changes the envelope orbital momentum of the envelope wave function. The envelope wave function is quenched in contrast to the bulk semiconductor [46]. The in-plane and out-of-plane anisotropy of lens-shaped QDs leads to an anisotropic g-tensor.
As mentioned in the introduction, we perform magneto-PL measurements to determine the g-factors of the QD system. The emission properties of a QD are modified by the applied magnetic field and the most dominant processes are the Zeeman interaction and the diamagnetic shift. In the following, we will discuss the description of the system
Figure 2.5.1: The g-factor of an electron is changing depending on the system due to spin-orbit interaction. Adapted from Ref. [46].
with the single-particle Zeeman Hamiltonian.
If an external magnetic field Bz is applied in Faraday configuration, i.e. normal to the sample surface (z-direction), which corresponds to the growth direction [001] of our QDs, electron and hole spin couple via the Zeeman Hamiltonian to the magnetic field
Bz. The four exciton states (Eq. 2.3.4) are used as basis and we get for the Zeeman Hamiltonian [38] HBz = µBBz 2 gX,+1 0 0 0 0 gX,−1 0 0 0 0 gX,+2 0 0 0 0 gX,−2 , (2.5.3)
where gX,±1 = ±(ge,z+gh,z) and gX,±2 = ±(ge,z−gh,z) are the expressions for the exciton g-factors for bright and dark excitons in z-direction.
Applying a magnetic field in Voigt configuration, i.e. parallel to the sample surface in [110] direction (x-direction) is described by the Hamiltonian
HBx = µBBx 2 0 0 ge,x gh,x 0 0 gh,x ge,x ge,x gh,x 0 0 gh,x ge,x 0 0 . (2.5.4)
The corresponding Hamilton for the y-direction is [38]
HBy = iµBBy 2 0 0 ge,y −gh,y 0 0 gh,y −ge,y −ge,y gh,y 0 0 −gh,y ge,y 0 0 . (2.5.5)
Due to the in-plane symmetry of the QDs, we consider only Hx B.
For describing the X state, we have to combine the exchange Hamilton Eq. 2.3.5 with
Hz
B and/or HBx depending on the direction of the applied magnetic field. The total Hamilton is given by
HtotX(α) = Hexchange+ HBx ·sin(α) + H z
16 2.5. Response of a QD to an External Magnetic Field where α is the angle of the magnetic field. For a magnetic field in z-direction α = 0◦,
for the x-direction α = 90◦.
For trion states Eq. 2.5.6 changes to
HtotT(α) = HBx ·sin(α) + H z
B·cos(α), (2.5.7)
since no exchange interaction is present [21].
For a magnetic field along z-direction the spin degeneracy is lifted, whereas a field along x is also breaking the symmetry of the system and a coupling between different states (e.g. coupling between bright and dark exciton states) is emerging in addition to the spin splitting. The off-diagonal terms in Eq. 2.5.4 are responsible for the coupling among the bright and dark states [21]. Under magnetic field, the coupled dark states will be visible. For simplicity, we will denote the coupled dark states just as dark states. In the absence of magnetic field we expect a linear horizontal/vertical polarization of the X transistion. At high magnetic fields along z (e.g. 7 T), the eigenstates couple and we expect left and right circularly polarized light for the X, which results in constant intensity [50]. As shown later, we will not observe this behaviour in our measurements with a field up to ∼ 3 T.
2.5.2 Diamagnetic Shift
An electron or a hole is not only affected by the Zeeman interaction but also by the diamagnetic shift. The change in energy is given in first approximation by
∆E = γB2, (2.5.8)
where γ is the diamagnetic coefficient [51]. This shift in energy originates from circu-lating currents in the ground states induced by the magnetic field. This results in a magnetic moment opposed to the field, which couples to the magnetic field itself and generates the quadratic dependence in B [46]. The shift is anisotropic, since it reflects the spatial extent of the excitonic wave function, which depends on the spatial con-finement of the QD and the interaction between the particles [38, 52, 53]. By using
k · p - theory and first order perturbation theory [54, 55], the diamagnetic coefficient of
a carrier in a semiconductor is [56]
γ ∝ hr
2i
m∗ , (2.5.9)
where m∗ is the carrier’s effective mass and hr2i the average lateral extension of the
wave function perpendicular to the applied magnetic field. Therefore, γx describes the spatial extent of the wave function in z, and γz the extent in x.
Note, that Eq. 2.5.8 is only valid for small magnetic fields, where the magnetic length
lm =
s
~
eB, (2.5.10)
is larger than the spatial extent of the exciton wave function lwf [57]. With the GaAs QDs used in this thesis it is possible that the magnetic length lm ≈ 15 nm is smaller than lwf at an applied magnetic field of 3 T. In this case, a deviation of the diamagnetic shift from the parabolic dependence would be the consequence.
2.5.3 Fitting Models
For evaluating the measurement data, we will need the eigenvalues of Eq. 2.5.6 and 2.5.7, since they represent the fitting models needed in chapter 5.
Note, that with these models it is not possible to determine the sign of the g-factors. By performing measurements in z-direction only the squared sum of (ge,z + gh,z)2 can be determined. Moreover, a separation of ge,z and gh,z is only possible by applying a magnetic field 0◦ < α > 90◦ e.g. 45◦, since at a measurement with a field along z the
dark states stay dark. By combining x and z magnetic field, a fitting with all four eigenvalues is possible since the dark states become visible.
The sign can be determined, by analyzing the polarization (Stokes measurement [58]). For measurements with a magnetic field in x-direction the situation is more complicated and usually, numerical models are used for obtaining the sign of the g-factors [56]. In this thesis we will deal with absolute values of the g-factors. More information on determining the sign of the g-factors can be found in Refs. [56, 59, 60].
2.5.3.1 Exciton States
For the exciton state and a magnetic field in z-direction (α = 0◦), the eigenvalues of
Eq. 2.5.6 for the bright states are
EV ± = ± 1 2
q
δ12+ (ge,z+ gh,z)2µ2BB2. (2.5.11) By adding the energy E0 at absence of magnetic field and the diamagnetic shift, we gain
our fitting formula
E±= E0+ γzB2± 1 2
q
δ12+ (ge,z + gh,z)2µ2BB2, (2.5.12) where δ1 is the fine structure splitting and γz the diamagnetic shift for the z magnetic field. As shown in Eq. 2.5.4, a magnetic field along x-direction (α = 90◦) couples
the bright states with the dark states and all four eigenvalues are necessary for the evaluation. To obtain the fitting formula for the bright states we again add the energy offset E0 and the diamagnetic shift γxB2 to the eigenenergies and get
Eb± = E0+ γxB2± 1 4(δ1+ δ2) + 1 4 q 4µ2 BB2(ge,x± gh,x)2 + (2δ20± δ1∓ δ2)2 . (2.5.13) Similarly we get the equations for the dark states, which are given by
Ed± = E0+ γxB2± 1 4(δ1+ δ2) − 1 4 q 4µ2 BB2(ge,x± gh,x)2+ (2δ0± δ1 ∓ δ2)2 , (2.5.14) Fig. 2.5.2 shows the eigenvalues of Eq. 2.5.13 and 2.5.14. For a vivid description δ0 =
105 µeV, δ1 = 15 µeV and δ2 = 10 µeV and γx = 6 µeV/T2 were used. The energy offset is E0 = 1.5 eV. We see that the dark states are lower in energy. Note, that by applying
a magnetic field along z, only the the sum |ge,z+gh,z|can be determined with Eq. 2.5.12. In order to separate these g-factors, a measurement with a field at α = 45◦ is necessary.
18 2.5. Response of a QD to an External Magnetic Field 0 1 2 3 1 . 4 9 9 9 3 1 . 5 0 0 0 0 1 . 5 0 0 0 8 E n e rg y ( e V ) B x ( T ) X b 1 X b 2 X d 1 X d 2 δ0 δ1 δ2
Figure 2.5.2: Eigenenergies vs. magnetic field along x-direction. For a vivid description large splittings of δ0 = 105 µeV, δ1 = 15 µeV and δ2 = 10 µeV and γx = 6 µeV/T2 were used. The energy offset is E
0 = 1.5 eV.
Therefore, the eigenvalues of Eq. 2.5.6 are needed for fitting the 45◦ measurement. They
are not given here, due to their complexity. However, all four g-factors ge,x, gh,x, ge,z and
gh,z are appearing in these eigenvalues. By using ge,x and gh,x obtained from a x-field measurement (Eq. 2.5.13 and 2.5.14) the separation of ge,z and gh,z is possible.
2.5.3.2 Trion States
In contrast to the exciton transition, the positive (X+) and negative (X-) trion states
are not affected by the exchange interaction. Therefore, equations 2.5.12-2.5.14 simplify to [61]
E1,2 = ±1
2µBB(ge,i+ gh,i), (2.5.15)
E3,4 = ±1
2µBB(ge,i− gh,i), (2.5.16)
where i = x, z denoting the direction of the magnetic field. With the high E1 and low
E2 energy component of the split trion states, we can obtain the g-factor gi by
gi =
E1− E2
µBB
, (2.5.17)
where gi = ge,i+ gh,i. To extract the diamagnetic shift we use
E = E0± 1
2µBBgi+ γB2. (2.5.18)
2.5.4 Relative Oscillator Strength
As already mentioned, a magnetic field in x-direction results in the coupling of bright and dark exciton X states. A parameter which is describing the degree of mixing between bright and dark states is the relative oscillator strength R. In this section we will derive an expression for R. Therefore, we take Eq. 2.5.6 for α = 90◦, which gives us the
Hamiltonian Hx
totX for describing the X at a magnetic field in x-direction. Furthermore, we change from the basis shown in Eq. 2.3.4 to the following
|Xi= |−1i − |+1i√ 2 , |iY i= − |−1i + |+1i √ 2 , (2.5.19) |X2− Y2i= |+2i + |−2i√ 2 , |2iXY i = |+2i − |−2i √ 2 , (2.5.20)
where |Xi and |iY i corresponds to the bright states and |X2− Y2i and |2iXY i to the
dark states [22, 31]. We transform the Hamiltonian Hx
totX by the operator P , which is given by P = −√1 2 −√12 0 0 1 √ 2 −√12 0 0 0 0 √1 2 √12 0 0 √1 2 −√12 , (2.5.21) so that HtotXx0 = P−1HtotXx P, (2.5.22)
is valid. Next, we calculate the eigenstates of Hx0
totX. The vectors |b1i, |b2ibelong to the bright states and |d1i and |d2i to the dark states and are given by
|b1i= 0 η1 1 0 , |b2i= η2 0 0 1 , (2.5.23) |d1i= ϑ1 0 0 1 , |d2i= 0 ϑ2 1 0 . (2.5.24)
The states with index 1 are higher in energy than the states with index 2. The variables
α in the bright states are given by η1 = −2δ0+ δ1− δ2+ q (2δ0+ δ1 − δ2)2+ 4µ2BB2(ge,x+ gh,x)2 2µBB(ge,x+ gh,x) , (2.5.25) η2 = −2δ0− δ1+ δ2+ q (2δ0− δ1+ δ2)2+ 4µ2BB2(ge,x− gh,x)2 2µBB(ge,x− gh,x) . (2.5.26)
For the dark states, ϑ1 and ϑ2 are
ϑ1 = −2δ0+ δ1− δ2+ q (2δ0− δ1+ δ2)2+ 4µ2BB2(ge,x− gh,x)2 2µBB(ge,x− gh,x) , (2.5.27) ϑ2 = −2δ0− δ1+ δ2+ q (2δ0+ δ1 − δ2)2+ 4µ2BB2(ge,x+ gh,x)2 2µBB(ge,x+ gh,x) . (2.5.28)
20 2.5. Response of a QD to an External Magnetic Field The relative oscillator strength is then obtained by
Rd1 = | hX|d1i |2, (2.5.29)
Rd2 = | hiY |d2i |2, (2.5.30)
and we get the following expressions for the relative oscillator strengths
Rd1 = −2δ0+ δ1− δ2+q(2δ0− δ1+ δ2)2+ 4µ2BB2(ge,x− gh,x)2 2µBB(ge,x− gh,x) )Nd1 2 , (2.5.31) Rd2 = −2δ0− δ1 + δ2+q(2δ0+ δ1 − δ2)2+ 4µ2BB2(ge,x+ gh,x)2 2µBB(ge,x+ gh,x) Nd2 2 , (2.5.32)
where N is a normalization factor given by
Nd1 = q 1 1 + |ϑ1|2 (2.5.33) Nd2 = 1 q 1 + |ϑ2|2 (2.5.34) After calculating the g-factors, Eq. 2.5.31 and 2.5.32 allow us to determine the strength of the coupling between bright and dark states. We will need these equations in chap-ter 5 again, where the measurements of this thesis are discussed in detail.
In Fig. 2.5.3 an example of the relative oscillator strengths Rd1 and Rd2 depending on the g-factors is shown. Rd1,2 increases with rising magnetic field. For B → ∞, Eq. 2.5.31 and 2.5.32 will reach 0.5.
0 2 4 6 8 1 0 0 . 0 0 . 2 0 . 4 0 . 6 R e l. o s c ill a to r s tr e n g th B x ( T ) R d 1 R d 2 (a) 0 2 4 6 8 1 0 0 . 0 0 . 2 0 . 4 0 . 6 R e l. o s c ill a to r s tr e n g th B x ( T ) R d 1 R d 2 (b)
Figure 2.5.3: Simulation of the relative oscillator strengths Rd1 and Rd2 for (a) ge,x= 1 and gh,x = 0.5 and (b) ge,x = 0.5 and gh,x = 0.05. The curvature changes with the g-factors. Rd1,2 increases with rising magnetic field. For B → ∞, the relative oscillator strengths reaches 0.5. δ0 = 105 µeV, δ1 = 4.2 µeV,
In the field of quantum computing and quantum communication it will be necessary to convert quantum information from one physical form to another. For example a photon in a coherent superposition of polarization states is suitable for the quantum information transport, while electron or hole spins could be favored for the storage and processing of quantum information [20]. A way to convert a photon qubit into a spin qubit is the photon-to-spin conversion, which we will discuss in detail within this chapter.
As stated above, spins of electrons and holes confined in QDs of a direct bandgap semi-conductor are attractive, since interband excitations are necessary for photon-to-spin conversion [17]. Further, long spin coherence times of about 200 µs have been reported for electron spin qubits in GaAs [17].
In general, there are two documented possibilities of coherent photon-to-spin conver-sion which include engineering of the g-factors in semiconductor QD structures, namely the polarization scheme and the time-bin scheme, whereas we will focus on the former [20, 62]. The following information is mainly taken from [17] and [20].
A scheme of the photon-to-spin state conversion is shown in Fig. 3.0.1(a). The photon is depicted by the yellow arrow and we assume that the qubit state is encoded using circular polarization which is given by
|phi= α |σ+i+ β |σ−i , (3.0.1)
where |σ+i and |σ−i are the circular right and left polarization states and α and β
complex numbers to fulfill the normalization |α|2+ |β|2 = 1. The goal is now to obtain
a spin qubit in a QD (red sphere in Fig. 3.0.1(a)) with the spin superposition
|φi = α |↑i + β |↓i , (3.0.2)
where |↑i and |↓i denotes the spin up and spin down states. Therefore, the photon has to be absorbed by the semiconductor material in the first place. This creates an electron-hole pair, which leads us to optical selection rules. Fig. 3.0.1(b) shows the single-particle spin states with their projections of the total angular momentum
jz = lz+ sz, (3.0.3)
where lz is the projection of the angular momentum and sz the projection of the spin. For electrons lz = 0, therefore |sz = +1/2i and |sz = −1/2i is written. Under magnetic field B the spin states are separated due to the Zeeman splitting E(e)
z = geµBB, with the electron g-factor ge. Likewise, a similar splitting is assumed for heavy holes and light holes. The optical selection rules are obtained by comparing the angular momentum of the incoming photon (+1 for σ+ and −1 for σ−) to that of the generated electron-hole
pair. For simplicity, electron-hole interaction and exciton fine structure are neglected
22
(a) (b)
Figure 3.0.1: Photon-to-spin conversion. (a) The state of a photonic qubit (yellow ar-row) is a superposition of circular polarization states. On the contrary, the qubit spin state (red sphere) is a superposition of spin up and spin down states. The particle is confined in a QD (purple disk). A magnetic field B is applied in Faraday configuration (parallel to direction of light propaga-tion). (b) Spin states belonging to the electron (|sei), heavy hole (|jhhi) and light hole (|jlhi) subbands. In light blue, the semiconductor bandgap is shown. The electron (hole) spin doublets are split by the Zeeman energy. Adapted from Ref. [17]
in this chapter. Fig. 3.0.2 depicts the optical selection rules for an electron-hole pair generated by circularly polarized light. Now, the total angular momentum projection
jz(eh) = +1 of the electron-hole pair created by absorbing a σ+photon can be obtained by
two different ways, either by |se= −1/2i |jhh = +3/2i or by |se= +1/2i |jlh = +1/2i. Likewise for j(eh)
z = −1. All possibilities are shown in Fig. 3.0.2(a) by the black arrows. As a result, each of the allowed transitions will have different energies (indicated by different arrow lengths), due to non-zero g-factors. This is very undesired, since σ+ and
σ− of the photon state would need to have different energies. For a photon, generated
by a narrow band source as desired for quantum information technology, this would lead to decoherence and information would be lost.
A possible solution is shown in Fig. 3.0.2(b), which requires a magnetic-field in Voigt configuration (x-direction), therefore, the states labeled with jz are no longer good eigenstates. Nevertheless, we can use the electron states |szi if we tune the electron g-factor ge to approximately zero. The states are then degenerated. We can connect the electron states optically to the light hole states, which are now given by linear combinations
|±ilh = √1
2(|jlh = −1/2i ± |jlh = +1/2i) . (3.0.4)
and polarization of the light is not changed. In this configuration each of the light hole states can support the photogeneration process with both polarizations. With the configuration shown in Fig. 3.0.2(b), where the energy of the photon is chosen to be resonant with the transitions depicted as black arrows, we obtain the following form of the absorption process:
|phi= α |σ+i+ β |σ−i −→ |φi= (α |se= +1/2i + β |se= −1/2i) |+ilh. (3.0.5) The electron-hole pair is not entangled, since the light hole state can be factorized out. Further, coherence is preserved and we successfully transformed the photonic |σ+i into
an electronic spin up |se = +1/2i and the |σ−istate into a spin down |se = −1/2i term. Now we see the necessity in studying the g-factor. Converting a photon qubit into a spin qubit requires a tuning of the g-factor (ge ≈0). This can be accomplished by controlling size and shape of the nanostructures [63, 64]. A more flexible solution is to use external perturbations such as electric fields and strain [59, 61, 65–68].
(a) (b)
Figure 3.0.2: (a) Optical selection rules in the Faraday configuration. The black arrows denote electron-hole pairs with total angular momentum projection of −1 or +1, created through the absorption of a σ− or σ+ photon, respectively.
Non-zero g-factors are assumed. (b) Photon-to-spin conversion with elec-tron and light hole states with a magnetic field in x-direction. In order to convert the photonic qubit state into an electron spin state a zero electron g-factor is used. Adapted from Ref. [17].
We have seen in chapter 2 that the exciton is described by an electron and a heavy-hole. In order to perform photon-to-spin conversion we need a light-hole exciton. GaAs QDs are suitable for this application, since excitons with a light-hole as a ground state can be realized [34]. Actually, this thesis deals with heavy-hole excitons, since g-factors for droplet-etched GaAs QDs are unknown in general and only a couple of works [34, 69, 70] have already dealt with magneto-optical properties of GaAs QDs. Further, there is an
24
approach to separate electron and hole g-factors [21], but it is doubtful if this method can be applied to weakly confining systems.
In addition, we will investigate the excited positive trions X+∗
2 - X+∗4 , since the hole in
the p-shell has already 20% light-hole character due to valence band mixing [34, 35]. We expect, that the light-hole g-factor in x-direction is higher than the heavy-hole g-factor in x, since the heavy hole Bloch state has only a projection in z-direction [59]. Therefore, also a small exciton g-factor in x-direction is anticipated.
The main part of this work are µ-photoluminescence (PL) measurements. Hence, the setup will be discussed in the following sections.
4.1 Setup for Photoluminescence Measurements in
Magnetic Fields
The setup for photoluminescence measurements is shown in Fig. 4.1.1. The sample with GaAs QDs is mounted on a piezo stage and inserted with a stick into a helium bath cryostat which is equipped with a superconducting vector magnet (see section 4.2 for a detailed explanation of the magnet system). The piezo stage consists of three linear nanopositioners that are mounted in the insertion stick. Each nanopositioner has a positioner body out of titanium since the stage has to be nonmagnetic and suitable for low temperature applications [71]. The piezo actuators are lead zirconium titanate (PZT) ceramics. Further, the positioners have integrated resistive encoders, a position resolution of 200 nm and a repeatability of 1 µm at 4 K. A closed-loop control of the stage can be accomplished via USB and the ANC350 Piezo Motion Controller [71]. The sample is mounted on the stage and has to be positioned in the field plane of the aspheric focusing lens (NA = 0.55, focal length 2.75 mm, anti-reflective coating: 600 - 1050 nm) [72], which is also mounted in the stick. Afterwards, the focus of the lens is adjusted. The stick is encapsulated and evacuated. For cooling down, the stick is then filled with 20 mbar gaseous He. Subsequently, the stick is inserted in the He bath cryostat. The sample temperature can be adjusted by a heater.
Next, the setup has to be aligned for the measurement. This is done by a titanium-sapphire (TiSa) laser that has a wavelength range of 700 to 800 nm, which is pumped by a 5 W 532 nm cw pump laser [73, 74]. The wavelength of the TiSa is adjusted to around 785 nm, since this corresponds to the exciton emission wavelength of our GaAs QDs and we have an achromatic objective. The laser is coupled into the stick via mirrors and a beam splitter (BS). Now, the focusing of the laser onto the sample surface is the main part of the aligning procedure, since different samples have different thicknesses and the rough focusing we performed earlier is not sufficient to excite QDs. The laser light reflected by the sample surface (and later also the emitted light from the QD) is collected by the same setup. In contrast to the setup sketched in Fig. 4.1.1, the built up setup in the lab (see Fig. 4.1.2) is also equipped with a camera to see the focus spot of the laser on the sample. The focusing is realized by moving the piezo stage vertically (z-direction) and watching the laser spot with the camera until it is focused. After aligning, the TiSa laser is substituted with a 532 nm continuous wave (cw) diode laser.
26 4.1. Setup for Photoluminescence Measurements in Magnetic Fields LHe 4.2 K Lens Superconducting Magnet Piezostack with sample on top LN2 HWP Polarizer 70:30 BS Laser 532 nm Insert Magnet Chamber M M Magnet PSU Ethernet Connection Trigger Signal USB CCD Spectrometer FC FC BS on flip stage M Camera
Figure 4.1.1: Sketch of the setup for magnetic field dependent PL measurements. The components and abbreviations are explained in the main text. QDs are excited with a 532 nm laser. The QD emission is guided into a spectrom-eter.
For the magneto-PL measurement we excite an arbitrarily chosen QD by above-band excitation. The QDs are then emitting light at around 785 nm, which is collected by the same objective and goes over a cubic 70:30 beamsplitter (BS) to separate laser and QD emission. Further, the emission light goes over mirrors (M), a halfwaveplate (HWP) and a linear polarizer (P) to the fibre coupler (FC). In earlier versions of this setup the light was going via a free path to the spectrometer. During this thesis the free path has been replaced by a polarization-maintaining (PM) fibre for reducing the detection area (fibre core diameter 5 µm), since many dots are excited simultaneously by the laser. The HWP and the polarizer are needed to perform polarization measurements. The light, which is focused in the spectrometer is dispersed by a turnable grating and then reflected upon a CCD that is cooled with liquid nitrogen. The used grating for magneto-PL measurements is a blazed grating with 1800 grooves/mm and has a blaze wavelength of 500 nm [75]. A resolution of ∼ 30 µeV is achieved.
The computer is controlling the power supply units (PSUs) of the magnet, the rotational stage of the HWP and is triggering the spectrometer.
Fig. 4.1.2 shows the setup how it is built up in the lab. The magnet is about 1.5 m beneath the floor. The setup for exciting the QDs and for collecting the photons emitted by the QD is arranged on the table above the magnet (see Fig. 4.1.2).
(a) (b)
Figure 4.1.2: Setup for magnetophotoluminescence measurements built up in the lab. (a) Shows the top of the magnet dewar, the insertion stick and the setup table. The spectrometer is positioned on a second optical table. In (b) the magneto-setup is shown in more detail. The laser light (green arrow) is guided down in the stick. The light emitted by the QDs is collected by the same setup and is later coupled into a fibre (red arrow).
4.2 Vectormagnet
The vector magnet which is used for performing magneto-optical measurements is a superconducting vectormagnet from Oxford Instruments. It consists of a solenoid mag-net for applying a magmag-netic field up to 10 T in z-direction and a split-pair magmag-net for applying a field up to 3 T in x-direction. [76]
To achieve superconductivity of the magnet coils, it must be cooled down to 4.2 K with liquid helium (LHe). Therefore, the magnet is inside a low loss dewar, where the LHe is filled into the inner chamber, which is shielded by a liquid nitrogen reservoir. Further, the dewar is vacuum insulated. The magnet dewar is connected to the helium recovery line, since He is volatile, expensive and always a certain amount of LHe will evaporize. Of course, also the liquid nitrogen chamber has an exhaust port, but it does not have to be recovered and is mainly for pressure compensation. From the top, a stick with the sample can be inserted into the magnet dewar. A simplified illustration of the magnet
28 4.2. Vectormagnet system is shown in Fig. 4.2.1. In the next section we discuss the cooldown procedure, before using the magnet system.
Figure 4.2.1: Sketch of the magnet dewar system. Left the cooling down with liquid helium is shown, when the inner chamber has a temperature higher than 4.2 K. Right the refilling with LHe is shown, when the system is already at operation temperature [77].
4.2.1 Cooldown of the Magnet System
For using the magnet, the system has to be cooled down to 4.2 K. We assume a fully warmed up system, therefore, the magnet has to be precooled at first. First, the outer vacuum chamber is evacuated to around 5·10−5mbar. Then, the inner magnet chamber
is pumped to remove air and moisture through the He exhaust port. Afterwards, the LHe reservoir is filled with He or N2 gas. Subsequently, one can start to fill the main
chamber with LN2 for precooling the system. After filling the main chamber with LN2,
the LN2 has to be blown out again. This is done by filling in gaseous nitrogen over the
exhaust port due to a slight overpressure of N2. The blown out LN2 can be transferred
into the outer nitrogen shield. If all of the LN2is blown out, one has to ensure that there
is no remaining N2. This is done by pumping and flushing the helium reservoir with
