Submitted at Institute of Semiconductor & Solid State Physics Supervisor Univ.Prof. Dr. Armando Rastelli July 2018 JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, ¨Osterreich
Deterministic Fabrication
of Inverted Nanocones
around GaAs Quantum
Dots
Master Thesis
to obtain the academic degree of
Diplom-Ingenieur
in the Master’s Program
Ich erkläre an Eides statt, dass ich die vorliegende Masterarbeit selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Masterarbeit ist mit dem elektronisch übermittelten Textdokument identisch.
To enhance the performance of quantum dots, used as future quantum emitters in quantum comput-ing or quantum cryptography, the creation of nano-structures became indispensable. Furthermore, many quantum dot fabrication techniques, leading to photons with outstanding optical quality, occur without position control. This requires the deterministic positioning of the photonic structure around preselected quantum emitters. In this thesis, an approach to perform quantum dot position mapping and a subsequent creation of inverted nano-cones was worked out. Thereby, a photoluminescence setup, operating with light emitting diodes, was built up, an image processing script was created, and existing nano-structure processing steps were modified in order to enhance processing reproducibility. In the end, the fabricated structures were examined through photoluminescence measurements. As a result, some structures exhibited remarkable intensity enhancement, whereby positioning as well as fabrication techniques were still improvable.
First of all I would like to thank my supervisor Univ. Prof. Dr. Armando Rastelli from the Institute of Semiconductor Physics at JKU in Linz for the possibility to do this extraordinary master thesis. He spent a lot of time for supporting me in all terms of need and did not hesitate to join me in the lab when I could not proceed on my own. Prof. Rastelli was always open for new ideas and steered me through this thesis with his valuable advice.
Furthermore, I want to thank Prof. Rastelli’s great team. Dr. Saimon Filipe Covre da Silva and Huiying Huang MSc. provided the epitaxial grown GaAs quantum dot samples and supported me in the photo luminescence laboratory. Moreover, Huiying and Xueyong Yuan MSc. guided me through the major sample processing steps in the clean room. DI Daniel Huber, DI Marcus Reindl and DI Christian Schimpf had always a sympathetic ear and offered help for urgent questions about optics. Thank goes also to my project partner DI Stefan Gruber, whose task it was to develop the processing of the inverted nano cones. We had many fruitful and productive discussions during our collaboration. Additional thanks are due to the technicians at the Institute of Semiconductor Physics. Alma Halilovic, Ing. Albin Schwarz, Ursula Kainz, Ing. Stephan Bräuer and Ing. Ernst Vorhauer are responsible for the machines and equipment in and outside the cleanroom. They gave worthy in-structions and were always bountiful.
I also want to thank Dr. Nikita Arnold for performing the simulations concerning the inverted nano cones and Mag.a Susanne Schwind for her administrative work.
During my stay at the Institute of Semiconductor Physics, I was lucky to share the office with DI Dorian Ziss, DI Marc Watzinger and DI Stefan Gruber. Besides the work, there was strength giving time for jokes and private conversations. I want to thank you for that.
Most importantly, I want to declare my devout thanks to my parents Gottfried and Elfriede. They did not only give me the opportunity to attend university, but also always trusted in me and my decisions. I can consider myself as lucky to have parents like you, who give me support and inspiration at the same time whenever I need it.
Last but not least, I must express my profound thanks to my wonderful girlfriend and companion Melanie Horner. No matter, how hard the days of studying were or how many setbacks I had, she was always there for me and helped me as best as she can. Thank you for brightening my days.
1 Introduction 1 2 Physical and Technical Background 3
2.1 Basics from Semiconductor Physics . . . 3
2.1.1 Dispersion Relation and Bloch Waves . . . 3
2.1.2 Effective Mass Approximation . . . 4
2.1.3 Bandstructure of GaAs . . . 5
2.2 About Quantum Dots . . . 6
2.2.1 Quantum Dot as Semiconductor Heterostructure . . . 7
2.2.2 Quantum Dot Emission Spectrum . . . 9
2.2.3 Fabrication of Quantum Dots . . . 11
2.3 Performance Enhancement through Nano-structures . . . 12
2.3.1 The Inverted Nanocone . . . 13
2.3.2 A „Prison“ for the Wave Packet . . . 15
3 Quantum Dot Position Mapping 16 3.1 Mapping Principle and Previous Investigations . . . 16
3.2 Sample Preparation . . . 18
3.2.1 Markers by E-beam Lithography . . . 18
3.2.2 Metal Evaporation . . . 20
3.2.3 Lift Off . . . 21
3.3 Position Mapping Setup . . . 21
3.3.1 Dealing with Intensity . . . 23
3.3.2 Imaging Procedure . . . 25
3.3.3 Improvement Possibilities . . . 26
4 Numerical Image Processing 27 4.1 First Image Preparation . . . 27
4.2 Marker Center Detection . . . 28
4.2.1 Image Thresholding for Primary Detection . . . 28
4.2.2 Hough Line Transformation . . . 28
4.3.1 Detecting Local Maxima . . . 32
4.3.2 Dot Selection through Two-Dimensional Gauss Fit . . . 33
4.4 Data Merging and Saving . . . 34
4.5 Determination of Position Uncertainty . . . 36
4.5.1 Accuracy Evaluation Procedure . . . 36
4.5.2 Uncertainty Results and Importance of Image Quality . . . 37
5 Creating Inverted Nanocones 43 5.1 Discs Positioning above Quantum Dots . . . 44
5.2 HBr Wet-etching Procedure . . . 46
5.3 Resulting Nanocone Structure . . . 47
6 Effects of the INCs 50 6.1 Photoluminescence Setup . . . 50
6.2 Photoluminescence Measurement Procedure . . . 51
6.3 Comparison of Setup Modifications . . . 53
6.4 Examining Other Regions . . . 57
6.5 Strain Effects on the INC . . . 58
7 Summary and Outlook 61
List of Figures 63
In 1925 Julius Edgar Lilienfeld created his patent of the first field effect transistor consisting of solid state semiconductor [19]. This invention enabled the integration of electrical circuits, which replaced computing with the former used bipolar junction transistors in the early 1970s. The integration of transistors allowed drastically increasing computation power, which is described by Moore’s law [22]. However, as the dimensions of modern transistors reach down to 14 nm and quantum effects become perceivable, Moore’s law seems to saturate out. To counteract this saturation of processing power, new approaches for future computation techniques, especially quantum computing, are of interest. The main advantage of a quantum computer compared to a classical one lies in the basis of its information unit, the qubit. In contrast to a binary digit (bit), where the information state is either 0 or 1, a qubit is defined by its quantum state, which can be expressed as a superposition of unmixed states. As an example, the quantum state of a photon is determined by its polarization, which can be either horizontally or vertically, circularly left-handed or right-handed. Apart from these binary states, the polarization of a photon can be any linear combination of them. For that reason, future quantum computers shall be able to process certain algorithms with exponentially increasing efficiency. However, current studies on quantum computing are rather counted among fundamental research, since the effective usage of a fully functional quantum computer will probably take several decades of investigation.
Beside others, photons are promising candidates for future qubits. In addition to their quantum state properties mentioned above, photons do not change the information they carry without interaction. This and the fact that photons travel at the speed of light, make them a perfect quantum informa-tion transmitter. Considering quantum cryptography, a linkage between two communicating parties is safe against any eavesdropping, as a distortion of the transmission, e.g. through polarization measurements, would be immediately noticed by the actual recipient. To build up a perfectly secure quantum network over long distances, quantum repeaters become indispensable. These repeaters work by the principle of quantum teleportation, where entangled photon pairs, emitted by isolated ions or high quality quantum dots, are needed [7].
The main advantage of quantum dots over single atoms or ions it that it is much easier to integrate quantum dots (section 2.2) in semiconductor devices. Even if emission properties vary from dot to dot due to slightly different sizes and shapes, much effort was spent in the last decade in creating bright quantum dots which emit highly indistinguishable and entangled photons.
etching at the institute of semiconductor physics at Johannes Kepler University (JKU) have led to dots, emitting indistinguishable photons and featuring very low fine structure (subsection 2.2.2) [12]. By applying strain fields to the quantum dot samples, Prof. Armando Rastelli’s team has succeeded in tuning the quantum dot’s fine structure and create polarization entangled photons on demand [13].
The still remaining issue of photon collection efficiency has been tackled by creating microlenses and later inverted nanocones (INCs) [10]. These nanocones are nano-structures, embodying a photon cavity which shall increase spontaneous emission of the quantum dots and beam the photos in a specific direction. However, since the lateral positions of droplet etched GaAs quantum dots are randomly distributed, the fabrication of any nano-structure needs to be carried out deterministically. To create structures exactly around previously grown quantum emitters, a procedure to map quantum dots in respect to reference markers with light emitting diods (LEDs) has been elaborated [29], [20]. Here, quantum dots and markers are illuminated simultaneously while both, the reflected and emitted light, are imaged. After computational post-processing, one obtains the data, leading to the positions of the quantum dots.
The main objective of this work was to build up an optical quantum dot mapping setup and program the corresponding software for detection. Once, this has been finished, position determination was applied onto GaAs quantum dots to create inverted nanocones deterministically around them. For this, various nano processing techniques including electron beam lithography, wet etching and thin layer depositioning were carried out.
This thesis is composed of seven main chapters, where this introduction is the first one. In chapter 2, further background regarding physical approaches to describe quantum dots in semiconductors and their emission spectra is delineated. Moreover, there is general information about nano-structures and in particular about the inverted nanocone. After this, chapter 3 is about the general procedure of quantum dot position mapping. Furthermore, sample preparation, the optical setup and mapping methods are described. In chapter 4, the used functions for numerical marker and dot detection are presented. Additionally, the merging of created marker and dot data is delineated and statistically generated position uncertainty results are shown. The fabrication steps for the inverted nanocone and resulting structures are exemplified in chapter 5. Hereafter, chapter 6 contains photolumines-cence results, measured on original quantum dots and the processed nano-structures. By comparing the spectra, enhancement effects can be estimated, whereby many INCs exhibit bright emissions. However, there is still space for position mapping and structure creation improvement. Finally, in chapter 7 the thesis is summarized and a brief outlook on possible future investigations is given.
In the following sections, general information about quantum dots, their emission spectra and nanos-trucutres are outlined. The background information from semiconductor physics comes mainly from [9] and [14]. More specific information about the resulting quantum dot emission spectra is taken from [3] and [2]. Additional knowledge is taken from various lectures I participated during the master study.
2.1
Basics from Semiconductor Physics
2.1.1 Dispersion Relation and Bloch Waves
Generally, electrons in a semiconductor material are treated with the adiabatic approximation. There-with the electronic movement takes place in presence of a periodic potential V (r ) = V (r +R) arising from a steady ion core lattice, where R = n1a1+ n2a2+ n3a3 is an arbitrary translation vector of
the Bravais lattice. To gain information about an electron’s wave function Ψ (r ), the appropriate Schrödinger equation −~ 2 2m∇ 2+ V (r ) Ψ(r ) = E Ψ (r ) (2.1)
has to be solved. By choosing an ansatz for the wave function consisting of a linear combination of plane waves, one arrives at a set of wave functions
Ψ(r ) =X G
Ck −Gei(k−G)·r (2.2)
with corresponding eigenvalues Ek. Hereby, the index k represents a certain reciprocal space vector within the first Brillouin zone and G = hb1+ kb2 + lb3 is a reciprocal lattice vector. Due to the
fact that
Ek = En(k) = En(k + Gn), (2.3)
the eigenvalues of the Schrödinger equation can be treated as a function of k. Furthermore, an additional index n, called band index, is introduced to consider all the solutions for a specific k value. The connection between En and k is called dispersion relation. Together with the periodicity in
formula 2.3, it defines the band structure and consequently the density of states (DOS) D(E ) of a semiconductor.
Rewriting the wave function from formula 2.2 as
un,k(r ) =X G
Ck −Ge-iG·r (2.4)
leads to a solution called the Bloch theorem. Since uk(r ) equals a Fourier series over G, the wave function Ψ (r ) is composed of a plane wave ei k·r and a function periodic with the lattice [4]. Written
as Bloch function, this reads
Ψn,k(r ) = un,k(r ) ei k·r with un,k(r ) = un,k(r + R). (2.5)
Summarizing, the wave function solves the Schrödinger equation for a single electron (Bloch electron) in an infinite periodic potential. In contrast to free electrons, the average velocity of a Bloch electron
vn,k = 1
~
∂En(k)
∂k (2.6)
depends on the material’s dispersion relation.
2.1.2 Effective Mass Approximation
To describe electron dynamics within semiconductors in a correct but intuitive way, a semiclassical approach can be used. Thereby, electrons are treated quantum-mechanically though the band struc-ture and external fields are considered classically. Particles can be spatially localized by combining multiple Bloch waves from a certain band index to wave-packages. Searching for a formula similar to F = m · a by using the semiclassical equations of motion ([9]), leads then to the definition of the effective mass tensor
h (m∗ )-1(k)i i j = ~ -2∂2ε(k) ∂ki∂kj . (2.7)
In this equation, ε(k) = En(k) is the band structure of the semiconductor and consequently ∂
2ε(k)
∂ki∂kj corresponds to its curvature. For isotropic band structures, as close to the Γ -point in GaAs, the effective mass tensor reduces to
m∗
= ~
2
d2ε(k)/dk2. (2.8)
With this new definition of the electron’s mass in crystalline material, it is possible to calculate the particle’s wave function in presence of an external perturbation.
Considering the Schrödinger equation 2.1 with a slightly varied potential V (r ) = Vcryst.(r ) + Vext(r ),
where Vcryst.(r ) is the original potential arising from the ion lattice and Vext(r ) is an external
pertur-bation potential, the solving wave function can be expressed as a linear combination of Bloch waves:
Ψ(r ) =X k
an,kΨn,k. (2.9)
If now Vext(r ) is weak enough to neglect interband transitions, meaning that only band n′
approximated by
Ψ(r ) ≈ Ψn′,k0 X
k
an′,kei(k−k0)·r = Ψn′,k0F(r ). (2.10) In words this means that the wave function of an electron in a weakly perturbed crystal potential can be expressed as a Bloch wave, multiplied by an envelope function F (r ). Putting ansatz 2.10 into the Schrödinger equation above and performing some mathematical conversions, results in the Effective Mass Equation (EME):
[En′(-i∇) + Vext(r )] F (r ) = E F (r ). (2.11) Here, the kinetic energy term is replaced by the dispersion relation En′(-i∇) of a certain band
n′
. With this approach, the Schrödinger equation can be solved by considering only the envelope function instead of the whole Bloch wave. Generally, the EME enables solving most of the problems in semiconductor physics, as long as the corresponding dispersion relation is known.
To be able to deduce quantum effects in nanostructures, some information about the corresponding bandstructure is needed. Since the used samples consist of GaAs quantum dots in an AlxGa1-xAs
matrix, especially these materials will be examined.
2.1.3 Bandstructure of GaAs
Gallium arsenide is a crystalline, intrinsic semiconductor with zincblende structure. Its corresponding band diagram is shown in figure 2.1. One sees the bandgap Eg between the conduction band
minimum (Γ6C) and the valence band maximum (Γ8V). These points are called band edges and lie
at the same k-value (Γ -point), which makes GaAs a direct semiconductor. Near the Γ -point, the
conduction and valence band are well approximated by the parabolas EC(k) = ~2k2 2m∗ e + Eg and EV(k) = -~2k2 2m∗ h . (2.12) Here, m∗
e is the effective mass of an electron in the conduction band and m ∗
his the effective mass of a
hole in the valence band. This primitive dispersion relation simplifies further calculations. However, some semiconductors show degenerated bands. GaAs for example exhibits two valence bands with different curvature, the light hole (LH) and the heavy hole (HH) band (figure 2.2). They are
Figure 2.2: Bandstructure of GaAs around the Γ -point with degenerated maximum in the conduction band. [8]
considered in the dispersion relation 2.12 by different effective masses m∗
LH and m ∗
HH. Both bands
contribute to the materials density of states.
To create semiconductor heterostructures, materials with different energy bandgaps are needed. In case of GaAs this is made possible by replacing gallium atoms by aluminium. AlxGa1-xAs is, similar to
GaAs, an intrinsic semiconductor with zincblende structure. Up to an aluminium content of x = 45% its bandgap is direct and amounts at 300 K
Eg(x ) = 1.424 + 1.155x + 0.37x2. (2.13)
For higher aluminium contents, the conduction band minimum lies in the X -valley which makes AlxGa1-xAs an indirect semiconductor [8].
2.2
About Quantum Dots
Quantum dots are zero-dimensional semiconductor heterostructures, where a material dependant bandgap difference leads to quantum confinement effects and discrete energy states. Electron-hole pairs, which are generated electrically or through higher energetic photons, get trapped in these states and can recombine. This way, a photon of specific wavelength is emitted through spontaneous
emission. For that reason, quantum dots are single photon emitters and of interest for quantum optics and future quantum applications.
In the following the appearance of quantum confinement and subsequently discrete states in quantum dots is derived. This is carried out by applying the EME to an artificially created quantum well.
2.2.1 Quantum Dot as Semiconductor Heterostructure
Quantum confinement effects are obtained by trapping electrons and holes and constraining their motion. This is achieved by creating a quantum well out of semiconductor material with dimensions comparable to the quantum carrier’s de Broglie wavelength. The kinetic energy of a fermion rises then due to the spatial constraint. The energies of the corresponding confined states Econf. are
discrete and can be derived by solving the particle in a box problem.
Figure 2.3: Sketch of GaAs and AlxGa1-xAs heterostructure with band edges and resulting discrete
states. Electrons and holes in conduction band (CB) and valence band (VB) fall into the potential well. Once they reach their confinement state, they recombine and a photon of specific wavelength is emitted.
In figure 2.3, the conduction and valence band edges of a GaAs-AlGaAs-heterojunction in one dimen-sion are shown. The different bandgaps Eg for GaAs and Eg′ for AlxGa1-xAs act as a potential well of
length Lz for electrons and holes. In quantum dots, this well occurs in all three spatial dimensions.
Interpreting the band edge mismatch as an external potential Vext(r ), one can describe the quantum
dot system by the EME. To simplify the problem, the well is assumed to be infinite with
Vext(r ) = 0 for 0 < x < Lx, 0 < y < Ly, 0 < z < Lz, ∞ else. (2.14)
is analogous. Hereby, formula 2.11 with the corresponding band structure (2.12) results in -~2 2m∗ e ∇2− e Vext(r ) F(r ) = Econf.F(r ) (2.15)
with Econf. = E − Eg. This eigenvalue equation is solvable by applying a separation ansatz. Thereby
the envelope function is split up into three single functions F (r ) = fx(x ) · fy(y ) · fz(z), where each of
them describes one spatial dimension x , y and z. Moreover, every dimension contributes separately to the final eigenenergy Econf. = Ex + Ey + Ez. With this approach, equation 2.15 can be split up
and treated as one dimensional problem: -~2 2m∗ e ∂2 ∂x2 − e Vext(x ) fx(x ) = Exfx(x ). (2.16)
Considering the boundary condition fx(0) = fx(Lx) = 0, originating from the infinite potential, the
solutions for fx(x ) need to be sine functions following
fx(x ) = r 2 Lx sin nxπ Lx x with nx ∈ N. (2.17)
In other words, an arbitrary number of half of the wavefunction’s wavelength needs to fit into the quantum well, which is only possible for discrete wavenumbers kx(nx) = nLxxπ. Consequently, also
the corresponding eigenenergies are discrete with
Ex = ~2kx2 2m∗ e = ~ 2π2 2m∗ e n2x L2 x . (2.18)
The smaller the quantum well, the higher are the confinement energies.
Coming back to the three dimensional solution, the general envelope function and the total electron energy read F(r ) = s 8 LxLyLz sin nxπ Lx x sin nyπ Ly y sin nzπ Lz z (2.19) and E(nx, ny, nz) = ~2π2 2m∗ e " nx Lx 2 + ny Ly 2 + nz Lz 2# + Eg. (2.20)
One sees that the quantum dot energy states are discrete with the corresponding quantum numbers
nx, ny and nz, similar as in atoms. For this reason, quantum dots are also called „artificial atom“.
However, the potential depth of real quantum wells is finite, which varies the actual wavefunctions and energies. Summing up the confinement effects of electrons and holes, one comes to the quantum dots emission wavelength. In figure 2.3, a transition between an electron and a hole in the quantum well ground state (nx = ny = nz= 1) is depicted. The energy of the emitted photon ~ω is equal to
the total energy difference of the fermions in the heterostructure, which results in ~ω = Econf.,e−− Econf.,h++ Eg = ~2π2 2 1 m∗ e + 1 m∗ h 1 L2 x + 1 L2 y + 1 L2 z + Eg. (2.21)
The above treatment neglects the finite depth of the confinement potential, the complexity of the valence band and the electron-hole interactions, which modifies the transition energy. Information about quantum dot emission spectra is examined in the following subsection.
2.2.2 Quantum Dot Emission Spectrum
To examine the behaviour of fermions in a quantum dot in more detail, it is helpful to look at its photoluminesce (PL) spectrum. This rather complex topic is kept assessable by assuming only the ground states populated. To meet this requirement, the sample is excited with reasonable laser power and cooled down below 10 K. Due to the fermion nature of electrons and holes, the ground state can accommodate two of the particles with different spin respectively. This results in four
Figure 2.4: Possible types of excitons in the ground states of a quantum dot. From left to right: neutral exciton, positive trion, negative trion and biexciton
possible emission combinations (figure 2.4). If only one electron and one hole are in the ground states, the resulting complex is called neutral exciton. Here, both particles are mutually attracted through the Coulomb interaction. The exciton can recombine, if the spins of electron and hole are inverse. Thereby a photon with wavelength λX is emitted.
Figure 2.5: Four typical quantum dot PL spectra with light intensity over emission energy. Each spectrum has four well defined peaks, which stand for the different exciton types. [34]
Excitons with an additional electron or hole in the quantum wells are called trions. Depending on the overall charge, it is referred to as either positive or negative. In trions, the charge of the third particle affects the emission wavelength usually to λX− < λX and λX+ > λX.
If the ground states of the quantum dot are fully occupied, i.e. with two electrons and holes, the formed complex is named biexciton. Since here two electron hole pairs with different spins are available, also two photons can be emitted. After the first emission, where all four charge carriers affect the resulting wavelength λXX, the second pair recombines as a neutral exciton.
Characteristic emission spectra of four InGaAs/GaAs quantum dots are depicted in figure 2.5. Here one sees the intensity peaks of the exciton, the trions and the biexciton. Examining the neutral exciton and the biexciton, their energy order varies randomly between different quantum dots. This is not explicable by considering only the Coulomb interaction. In fact, electrons, holes and especially
Figure 2.6: Summary of interaction effects on exciton binding energies in quantum dots. On the abscissa, interactions are gradually cut in. The ordinbate shows the corresponding binding energy levels. States with larger energy than the bright exciton are called antibinding. Lower energy states are stated binding. [31]
their spins interact through the Fermi and Coulomb correlation [31]. Thereby it is possible, that the biexciton emission peak is energetically lower than the exciton (figure 2.6).
Treating the exciton state further, it is split up by the exchange interaction of indistinguishable particles. This can be understood by considering the electron and the hole, which is actually nothing but a missing electron, as a perturbation of the remaining electron system [31]. Formulating the exchange Hamiltonian and applying the spin states on it results in four distinguishable exciton energy states, where two of them have a total momentum of |±1i (bright) and two exhibit |±2i (dark). Only the bright exciton states can couple to photons due to their spin of |±1i. Subsequently, there are two distinct polarized emission lines for the neutral exciton. Their energy difference ∆FSS is
Figure 2.7: Decay path of a biexciton in a quantum dot. a) If the intermediate exciton state exhibits FSS, the biexciton decays to one of the two polarization states by emitting a horizontally or vertically polarized photon. The photon emitted by the remaining neutral exciton has the same polarization as the first one. b) Without FSS, the two emitted photons are mutually circularly polarized. [39]
In comparison to the neutral exciton, trions and the biexciton do not show up Fine-structure. Never-theless, also the biexciton emission is polarized because of its decay process (figure 2.7). Depending on which path the biexciton decays, its emission energy changes with that of the exciton (figure 2.6). If the biexciton’s decay paths are totally indistinguishable, as in quantum dots without FSS, the emitted photons are entangled. To reach the condition ∆FSS = 0, quantum dot needs to fulfil C3v
-symmetry [15]. Since this -symmetry is never met perfectly in as-grown dots, techniques to lower the FSS were developed at JKU by Prof. Rastelli and his coworkers [35]. In this work, quantum dots are exposed to a three directional strain field by piezoelectric actuators which enables control over optical properties as FSS and the dot’s emission wavelengths.
2.2.3 Fabrication of Quantum Dots
In the past three decades, various methods for creating semiconductor quantum dots were developed. Thereby, heterostructures with different results in emission peak width, corresponding to the optical quality of a quantum dot, or confinement energy were created. Depending on the approach of choice the quantum dot positions can be engineered or not. Growth methods, where positioning is possible are e.g. quantum well deep etching, local laser annealing, modulated H-diffusion in GaAsN/GaAs or quantum dots in inverted pyramids. Nevertheless, the quantum dots resulting from these procedures mostly emit photons of poor optical quality or their confinement energy reaches only 100 meV. Methods, where the quantum dots emerge at random positions like thickness fluctuations in quantum wells, quantum dots by the Stranski-Krastanow growth mode, phase separation or droplet etching usually exhibit better optical properties [3].
The GaAs quantum dots, treated in this thesis, are created through local aluminium droplet etching by molecular beam epitaxy (MBE). The resulting, self assembled heterostructures feature confinement potentials deeper than 200 meV and emit photons with good optical quality. Local aluminium droplet etching works according to the principle of diffusion. Thereby, tiny aluminium droplets are deposited
Figure 2.8: Schematic sketch of aluminium droplet etching procedure. a) Aluminium droplets are deposited onto AlGaAs. b) Material from the AlGaAs layer diffuses into the dropled during annealing. c) After dissipating the whole droplet, a hole in the AlGaAs remains. d, e) Atomic force microscopy images of the resulting holes. [16]
onto a previously grown AlGaAs layer (figure 2.8). By heating up the sample (annealing) arsenic from the AlGaAs layer diffuses into the aluminium droplet and drops out as AlAs around the droplet. At the same time the aluminium spreads on the layer beneath and combines with provided arsenic. A hole surrounded by AlAs remains. To create the quantum dots, GaAs is deposited onto the sample and the etched hole is filled. After depositing another layer of AlGaAs, a heterostructure with quantum confinement in all three spatial dimensions is formed. [16]
At the institute of semiconductor physics at JKU, these quantum dots are grown by Dr. Saimon Covre da Silva. During his cooperation with Prof. Rastelli he managed to create highly symmetric dots with a very low FSS down to the resolution limit [12].
2.3
Performance Enhancement through Nano-structures
Even if quantum dots emit with unity quantum efficiency, photons still need to escape the semicon-ductor host material to be utilized. Without any wastages, a typical exciton lifetime of about 200 ps results in 5 · 109 photon counts per second. However, in real measurements quantum dot sources are never that bright. Various material and setup dependent losses reduce the intensity of the single photon emitter to only a few thousand counts and exacerbate quantum optic experiments.
Generally, the components and light paths of scientific setups are optimized in terms of photon collection. Nevertheless, most of the emitted photons are scattered and thereby lost within the semiconductor. A high refractive index of e.g. aluminium gallium arsenide for wavelengths of 800 nm and an aluminium contant of 40% (nAlGaAs = 3.4) leads to a total reflection angle of αtot, AlGaAs =
17.1◦
. According to this, photons escape the material only within a solid angle fraction of 2.35%. Furthermore, the collection efficiency is limited by the used objectives. Since the quantum dot samples are usually located in a cryostat, the usage of immersion objectives or objectives with short
working distance is not possible. To still collect a maximum number of photons, it is important to use objectives with comparable high numerical aperture. Anyhow, the finally resulting effective collection efficiency is far below 1% [10].
An encouraging approach to solve this problem is to create photonic nano-structures with quantum dots inside. As a result, a local increase of the electromagnetic mode density shall raise the exciton recombination rate and enhance its optical properties on the one hand. On the other hand, the emitted light shall be guided by the photonic structure and beamed in a certain direction to increase collection efficiency. Therefore, different structure shapes, generated by various nano technologies were elaborated in the past decades. In some of these methods, the quantum dots are created
(a) (b) (c)
Figure 2.9: Different nano-structures for quantum dot emission enhancement. (a) Nanowire with implanted qunatum dot [37]; (b) Engineered qunatum dot in the middle of a photonic crystal [30]; (c) Circular grating created deterministically around quantum dot [29]
directly during structure processing as for e.g. quantum dots in nanowires (figure 2.9a) [37] or samples containing distributed Bragg reflectors (DBRs). If the dots are positionable, it is possible to engineer the emitters into previously prepared nano-structures. An example for this would be the integration of silicon/germanium quantum dots into photonic crystal cavities (figure 2.9b) [30]. However, in case of dots which originate at random positions, nano-structures need to be created deterministically. This can be performed, by creating structures in-situ, as in [11], or after mapping the as-grown quantum dot sample (figure 2.9c) [29], [20].
The general rule for nano-structures containing single photon emitters is that the structure should exhibit similar dimensions as the photon emission wavelength. This requirement makes processing difficult, since typical quantum dot emission wavelengths are within the range of visible to near infrared light. It is often impossible to create nano-structures by optical lithography due to the diffraction limit. In these cases, electron beam lithography is the method of choice.
2.3.1 The Inverted Nanocone
The structure treated in this work and in the thesis of Stefan Gruber [10] is called the inverted nanocone (INC) consisting of Al0.4Ga0.6As surrounded by a gold mirror with a quantum dot in the
Figure 2.10: Cross-section sketch of the inverted nanocone. A covering dielectric oxide (HfO) and gold layer results in the desired cavity.
To optimize the structure of the INC, Dr. Nikita Arnold performed finite element simulations with different structure dimensions and layer thicknesses for a dot emission wavelength of 785 nm and a material refractive index of n = 3.3. As a result, the ideal size for the cone bottom radius is
r = 179 nm. The final cone height and the vertical dot position, measured from the cone bottom, are calculated to h = 307 nm and hd= 155 nm. A 10 nm thick oxide layer between semiconductor
(a) (b) (c)
Figure 2.11: Results on INC simulations provided by Dr. Nikita Arnold. (a,b) Cross-section and topview E-field distribution within the INC. Red areas stand for strong fields, blue ones for weak. (c) Solid angle dependant emission power distribution.
and gold suppresses plasmon polariton coupling effects without disturbing photon emission. In figure 2.11, the calculated distribution of the electric field within the cone and the solid angle dependant emission power is shown. In the latter on sees that the optimized INC structure provides a well directed emission beam. Using an objective with numerical aperture of 0.65, one can then reach a collection efficiency of 35.8% [10]. Additionally a Purcell factor (subsection 2.3.2) of FP ≈ 13.57
further increases the all-over photon gain.
In the ideal case, the INC promises outstanding results compared to the creation effort. However, when dealing with randomly grown GaAs quantum dots, one needs to make sure, that the structure midpoint is located exactly over a quantum dot (∆r ≈ 20 nm). Even if the spatial dot density is high and if multiple thousand structures are created, it is hardly possible to hit a dot accurately without positioning. Therefore, quantum dots position mapping turned to be indispensable for INC creation.
2.3.2 A „Prison“ for the Wave Packet
The background of the high improvement potential in nano-structures with sizes of a few hundred nanometers lies within Fermi’s golden rule [25]. The corresponding relation
Γi →f =
2π
~ ̺0(Ef) |hf |V |ii|
2 (2.22)
originates from time dependant perturbation theory and states that the transition rate Γi →f, from an
initial |ii to a final state |f i, depends linearly on the final density of states ̺0(Ef) and the squared
transition matrix element hf |V |ii. This means that a change in the electromagnetic DOS of an exciton in a quantum dot directly affects its emission rate.
From quantum electrodynamics, one can deduce that the electromagnetic DOS within a resonant cavity increases for wavelengths fitting into the cavity and decreases for others. This circumstance can be utilized and was discovered by E. M. Purcell in 1946 [28]. The resulting spontaneous emission rate of an electron hole pair in a cavity is raised by the Purcell factor
FP= 3 4π2 λc n 3 Q V . (2.23) In this formula, λc
n is the wavelength in the cavity material and V is the cavity’s mode volume. The
quality factor Q depends on cavity properties as the reflectance of the used mirrors or the shape of it.
To create inverted nanocones precisely around randomly grown gallium arsenide quantum dots, it is necessary to map the dot positions. For this, quantum dots were excited by blue light to luminesce and infrared light is reflected on gold markers on the sample. The originating light was then collected by the objective and focussed into a highly sensitive CMOS camera. During the imaging process, camera settings and light intensities required specific attention. In the following sections, details about the principle of quantum dot position mapping, gold marker deposition, the finally built up photoluminescence setup and the imaging procedure are presented.
3.1
Mapping Principle and Previous Investigations
A method to detect self-assembled quantum dots is described in the references [29] and [20]. Here, two light emitting diodes (LEDs) are used to excite quantum dots and illuminate gold markers
simul-Figure 3.1: Sketch of position mapping setup with LEDs, objective over the sample and camera combined by two beamsplitters.
taneously. Thereby, the imaging setup does not need to be modified during the measurement, which results in reliable maps of the quantum emitters within the semiconductor bulk. The experimental setup of [20] can be seen as a modification of the one in [29]. The improved measurement construc-tion in reference [20] consists of an objective implemented within the cryostat, whereas objective and cryostat are separated in the first generation setup. A major advantage of a joint system is that the cryostat window can be omitted, which results in sharper images, higher magnification and a larger numerical aperture.
Basically, the used setup consisted of two LEDs, an objective, a cryostat, a camera and two beamsplit-ters, which combined the illumination with the imaging path and the light of the LEDs (figure 3.1). One LED emitted at shorter wavelength to excite the quantum dots, the other LED at a wavelength similar to the dot emission. The resulting light illuminated the sample within the cryostat, cooled down below 10 K. An objective collected light coming from the sample, which was further reflected by another beamsplitter. In the imaging path, a bandpass filter blocked out disturbing emissions and let through the wanted infrared light, which was focussed to the camera’s chip. Resulting pictures were saved on a computer for further processing.
The main advantage of using LEDs for sample illumination instead of lasers was that light emitting diodes enable a homogeneous light distribution across the whole image area. To use laser light for large area acquisitions, it needed to be defocussed by lenses. Therewith, unwanted inhomogeneities (speckles) occurred due to laser light interference mechanisms. Even if LED light provided constant illumination brightness, it was tricky to gather enough power to excite the quantum dots. Acceptable exposure times could only be reached, if the LED light intensity impinging on the sample was maximized. This was one of the major issues for building up the LED photoluminescence setup.
Figure 3.2: Layer structure of sample AS271. Gallium arsenide quantum dots are embedded in an aluminium gallium arsenide matrix.
3.2
Sample Preparation
For INC creation, samples with a special structure were created. The layers of sample AS271, grown by Dr. Saimon Covre da Silva, are depicted in figure 3.2. Here, the distance between surface and sacrificial layer (100 nm) corresponds to the height of the inverted nanocones (310 nm). The GaAs quantum dots emitted at about 795 nm and lay between two 30 nm deposits of Al40Ga60As which
were further located in an Al20Ga80As matrix. To meet the depth of the quantum dot’s barycenter
within the final structure as calculated in the simulations (subsection 2.3.1), the distance between GaAs filling layer and sacrificial layer amounted to 140 nm. On top, a 5 nm thick GaAs capping layer protected the sample from oxidization.
Obviously, every mapping procedure presupposes references on which the measured objects can be correlated. For this, gold reference markers were deposited onto the samples by electron beam
lithog-Figure 3.3: Main e-beam lithography steps for the definition of marker structures on the sample surface. From left to right: illumination of the e-beam resist on the sample, development of the resist, gold layer deposition and lift-off.
raphy and subsequent metal evaporation. In figure 3.3, the principle of electron beam lithography with positive resist is depicted. Here one can see the sample illumination, development, thin layer deposition and the lift off. In the end, the gold should remain where the electron beam impinged. The needed processing steps are described in more detail in the following subsection.
3.2.1 Markers by E-beam Lithography
The electron beam lithography (EBL) procedure can be divided up into three major parts. In the fist step, the sample needed to be cleaned and spin coated with the e-beam resist. After this, the sample could be loaded in the nano-lithography system „e-LINE Plus“ from Raith. In here, the resist was illuminated by electrons with nanometer precision. The last step of electron beam processing was the resist development. In the following, this whole procedure is described in more detail.
First of all, the GaAs sample needed to be cut from the 2 inch wafer AS271. Thereby a quadratic piece with dimensions of about 4.5 × 4.5 mm2 was broken along its lattice planes within a flowbox. After putting the sample into acetone and into an ultrasonic cleaner for two minutes, it was rinsed with isopropanol and dried. In the spin coater, about 10 µl of the positive e-beam resist AR-P 6200.09 (Allresist CSAR 62) were put on top of the sample. Subsequently, the specimen was spun for 40 seconds at 4000 rpm, which resulted in a resist thickness of about 200 nm [26]. Before the
sample could be put into the load-lock of the e-LINE Plus, it needed to be prebaked on a hotplate for 5 minutes at 150◦
C. After this, the machine’s loading procedure was initiated.
Starting the e-beam lithography system, the electron gun was ramped up to 20 kV and the aperture diameter was set to 10 µm. Before the resist could be illuminated, multiple system components needed to be aligned. Thereby, the electron beam was focussed on the sample surface and stigmator and aperture position were adjusted properly. Up next, the coordinate origin was set to the bottom left edge of the sample and the abscissa was aligned along the bottom sample edge. To ensure precise lithography within and over multiple write fields, write field and beam tracking alignment procedures were performed iteratively. Once, all these adjustments were done, the beam current was measured. With this and the charge dose for the CSAR 62 resist (E0 = 27.5 cmµC2), the EBL system calculated the exposure times for the wanted structures. These were created within a GDSII file in the e-LINE
(a) Whole marker structure with alignment marks in the corners
(b) E-LINE writefield consisting of nine marker fields with corresponding numbering
Figure 3.4: Images of GDSII marker structures written by electron beam lithography. After gold marker deposition and lift off, these structures remain on the sample.
software or could be imported from various file types. The required reference markers consisted of crosses, which were separated by 50 microns in x - and y -direction (figure 3.4a). In the top left corner, as well as in the bottom left and the bottom right, alignment markers were created. These should help to align the sample properly, when creating the INCs after dot position mapping. To distinguish between the rows and columns, there were letters and numbers inside of four neighbouring crosses (figure 3.4b). Onwards, these areas are called marker fields. In the end, the designed structures were added to a position list and subsequently scanned. As soon as lithography was finished and the electron gun was shut down, the sample could be unloaded.
The illuminated specimen was developed by Allresist AR 600-549. After one minute within the alka-line developer, the sample was put into isopropanol for at least 30 seconds to stop the development reaction. On the dried sample one saw how the resist vanished on places, where the electron beam
Figure 3.5: Image of marker field after development. At areas, where the electron beam impinged, the e-beam resist vanishes. At not illuminated parts, it remains.
impinged (figure 3.5). The structure quality highly depended on the chosen electron dose. For too high doses, inner structure edges (e.g. in the middle of the crosses) got rounded due to the proxim-ity effect [18]. If the charge dose was too low, the resist did not fully vanish at illuminated areas. Therefore, multiple dose tests were performed to gain the optimum beam settings.
3.2.2 Metal Evaporation
The illuminated structures needed to be coated with metal to see them during quantum dot positions mapping. For infrared wavelengths around 800 nm, a layer of 50 nm gold results in a reflectivity of R ≈ 95 %. To improve the layer’s sticking properties, a small amount of chromium (3 nm to 5 nm) was deposited onto the GaAs cap before the gold. The metal was applied by physical vapour deposition within the „PLS 570“ from „Pfeiffer - Balzers“.
Generally, physical vapour deposition takes place in the evaporation chamber, which is pumped down to high vacuum (< 2.5 · 10−6
mbar). In here, metal is evaporated and deposited on the sample which is mounted up side down on a sample holder on top of the chamber. The high vacuum enables the vapour to reach the sample simply by its kinetic energy. It is generated by different kinds of evaporators, as e.g. thermal or electron beam evaporators in the used machine.
Thermal evaporators consist of a conductive vessel („boat“) containing the desired metal. It is heated electrically until the metal evaporates. If the boat is not used, a shutter protects the material from contamination by other metals. In contrast to this, the material in electron beam evaporators is melted and vaporized by a high energy electron beam. On top of the vaporizer, different metals are stored within a turret. Depending on which metal is needed, the turret rotates to the right position, before the electrons impinge on the material. Also the electron beam evaporator is covered by a shutter.
During evaporation, the deposit thickness is controlled in-situ by the quartz micro balance technique. Hereby, the resonance frequency of a quartz crystal changes continuously with increasing oscillation mass. By measuring the change of this frequency, one can deduce to the deposited layer thickness.
3.2.3 Lift Off
After coating the sample with 3 nm chromium and 50 nm gold, a lift off needed to be performed. Thereby, the remaining resist was removed by acetone together with the gold on it. At resist free areas, the deposited metal stuck on the GaAs. First lift off experiments had shown that CSAR 62 dissolved poorly in acetone at room temperature. To expedite this, the solvent was heated up to 60◦
C within a water bath. After leaving the sample in the heated acetone for about 30 minutes, the
Figure 3.6: Photographs of the sample surface after gold deposition and successful lift off. The numbers and letters stick well on the sample and are clean of remaining gold flakes.
Left: Single marker field, Right: Multiple marker fields and alignment marker;
samples were put into an ultrasonic cleaner to remove the first gold flakes. Another 30 minutes in the solvent and about one minute in the ultrasonic bath resulted in well defined gold structures with clear edges (figure 3.6).
With the resulting reference markers on the sample, quantum dot position mapping could be started. During the time between lift off and loading to the cryostat, it was important to store the sample in vacuum or in a nitrogen box to prevent the contained aluminium from oxidation.
3.3
Position Mapping Setup
Since the Institute of Semiconductor Physics in Linz did not posses an objective-in-cryostat system as in [20], the built up setup in the photoluminescence laboratory was comparable to the one in [29]. Furthermore, the stage within the cryostat moved only in x- and y-direction. Therefore, the imaging focus was adjusted by the objective’s z-position. In contrast to the samples in the references (λ[29] ≈ 940 nm, λ[20]≈ 930 nm), the QDs of sample AS271 emitted at around 795 nm. This lead
to generally different optical components as LEDs, lenses and beamsplitters. Another critical issue was the maximization of light intensity, why the positions of lenses, mirrors and the objective needed to be adjustable.
Due to the properties of the used sample, the emission wavelengths of the LEDs were chosen to 470 nm for the blue excitation LED (1) and 810 nm for the IR illumination LED (2) (figure 3.7). Thereby the mounted LED systems M470L3 and M810L3 and other components from Thorlabs were
Figure 3.7: Sketch of the finally realized PL mapping setup. The used components (green numbers) are described in the main text. Major differences to the setups in the references are the dichroic mirror, the LEDs and the objective.
acquired. The maximum power of the emitted blue light was 650 mW at 1000 mA, whereas the IR emission power was at maximum 325 mW with 500 mA. The light of both LEDs was parallelized by collimator lenses (ACL2520U) with a focal length of about 21 mm and joined by a cut microscope slide acting as a 5:95 beamsplitter (3). Here, 5% of the IR light were reflected and 95% of the blue light transmitted. This resulted in a maximization of the 470 nm dot excitation intensity. All these components were connected by the cage cube C4W to keep the LED illumination part handy. The light coming out of this cube went through another lens (f = 120 mm), a dichroic mirror (4) and was focussed to the objectives exit pupil. A dichrioc mirror reflects or transmits, depending on the wavelength of the impinging light. In case of the short pass mirror DMSP650, the average transmission rate for light wavelengths of 400 nm to 633 nm is over 90%. Light wavelengths from 685 nm to 1600 nm are reflected on average with more than 95% [33]. Here, a large portion of IR light intensity was lost, since on average only 5% of the photons were transmitted.
The infinity corrected plan apochromat objective „M Plan Apo NIR HR 50x“ from Mitutoyo (5) with 50× magnification, a numerical aperture (N.A.) of 0.65 and a working distance of 10 mm collected a large amount of light despite the wide distance to the observed sample [21]. As the model name indicates, it is a high resolution objective, corrected for the near infrared field. With a standard tube length of 200 mm, the objective’s focal length equals fobj.= 4 mm. An homogeneous illumination of
If the LED’s image appeared at this back focal plane, the light equally spread on the front focal plane (ffp), matching the sample’s surface. The focus could be adjusted by a micrometer screw and a piezoelectric actuator.
At the cooled sample (6), the impinging light simultaneously excited the GaAs quantum dots and illuminated the gold markers on the sample. The dot emission and the reflected light were then collected by the same objective, which was used for illumination. The cryostat (T < 10 K) including the prepared specimen, was located on an x y -stage. This enabled scanning multiple samples within the cryostat with sub-micrometer accuracy.
Since the wavelengths of quantum dot and quantum well emission of AS271 and the reflected light of the gold markers lay between 750 nm and 820 nm, nearly all of the light was reflected at the dichroic mirror. This IR signal was either passed to the spectrometer to observe the spectral emission properties of the dots or it was reflected by a flip mirror (7) into the imaging path. Here, the light was filtered by a bandpass filter (8) with a center frequency of 800 nm (FB800-10) and focussed into the highly sensitive, monochromatic CMOS camera „ASI 290MM COOLED“ from ZWO (9). This camera contains a 1/2.8” CMOS IMX290LLR sensor with 1936 × 1096 pixels and can be cooled down -40◦
C below ambient. Furthermore, the implemented sensor’s quantum efficiency at 800 nm is comparably high for CMOS chips [1].
The total focal length of the used imaging lens system was chosen to fi = 220 mm, which resulted in
an overall magnification of β = 55. This was realized by two lenses f1 = 250 mm and f2= 500 mm.
To calculate the distance between these, the formula for the effective focal length 1 fi = 1 f1 + 1 f2 − e f1f2
was considered [32]. Thereby, both lenses were assumed to be thin. Placing the corresponding focal lengths into this equation, the lenses needed to be separated about e = 180 mm to reach the required total focal length fi. Since the pixel size of the camera sensor was 2.9 µm × 2.9 µm, the field of
view on the sample resulted in approximately 102 µm × 58 µm. This was sufficient for the created gold markers and the following image processing.
3.3.1 Dealing with Intensity
Observing the paper of Sapienza et al. ([29]), one can perform a rough estimate of the needed 630 nm LED light intensity. On page two, one can find that the maximum power is about 40 mW, which corresponds to an intensity of 130 W/cm2 = 1.3 µW/µm2. This approximately conforms to the assumed intensity value for the dot illumination of sample AS271 (Imin ≈ 1 µW/µm2). If now
about 75% of the light was collimated and all the LED light could be collected by the objective, the maximum LED power had to be around 660 mW. Comparing this value with the available power of the blue 470 nm LED in the setup above (650 mW), it seemed promising to reach the desired intensity Imin. However, there were multiple parameters to adjust, to obtain the maximum intensity
Figure 3.8: Sketch of light collection path with criti-cal component and beam diameters for a proper in-tensity estimation.
First of all it was again assumed that 75% of the LED power was collected and paralleled by the collimator lens. This could be argued by the collection angle and the angular dependent light emission of the LED. Since data for the LEDs in usage were not available, a collection efficiency of 75% was estimated as a minimum value. In the first setup, the two LED sources were combined by a 50/50 beamsplitter. In front of the objective, a 90/10 beamsplitter was used. With a pupil diameter of
Dobj = 2 N.A. fobj= 5.2 mm
and an image diameter of the LED emission diode of about 8 mm at the pupil, approximately 42% of the light went through the objective [27]. If now these 42% of the blue reached the sample surface, the total impinging power (without considering the losses at the cryostat window) calculated to less than P = 10.3 mW. With a minimum illumination spot diameter of d = 300 µm (empirical value), the reachable intensity counted I ≈ 0.15 µW/µm2. This could render
imaging impossible or would at least result in very long acquisition times. For this reason the setup needed to be modified, whereby the most enhancement potential lay in changing the beamsplitters.
As described in the paragraph above and depicted in figure 3.8, the beamsplitter merging the LEDs was replaced by a microscope slide (T = 0.95). Combining this with a dichroic mirror (T<650 =
0.9) which replaces the former 90/10 beamsplitter, the intensity impinging on the sample could be enhanced to I = 2.5 µW/µm2. However, this was just an estimation of the resulting power outcome.
In reality, it was very difficult to maximize the light focussed into the objectives pupil. On the one hand, placing the LED further away from the objective led to more parallel light, which resulted in smaller illumination spots at the sample. On the other hand, this also implied a larger image of the emitting diode at the back pupil, which made it hard to collect light with the objective. In the end, one needed to find proper distances and lenses for sufficient short acquisition times.
Using a dichroic mirror instead of a beamsplitter at the objectives back brought the advantage that firstly nearly all of the excitation light could pass to the objective and secondly a comparable percentage infrared light from the sample was reflected to the imaging path. Nevertheless, this replacement also had its downsides. Considering the IR LED light, used for sample illumination, nearly all the intensity got lost due to reflection. This could create severe imaging problems by using low power LEDs, since most of the IR light was already wasted by LED light merging at the microscope slide. Furthermore, the dichroic coating created unwanted polarizing effects, which could distort polarization-resolved spectroscopy measurements. Another trivial disadvantage was the loss of multi-functionality. Since the dichrioc mirror acted as short pass filter, the setup could only be used for a comparatively small wavelength range.
3.3.2 Imaging Procedure
Once all light paths were aligned and the distances between components were adjusted in respect to the focus lengths, imaging could start. Thereby, the objective’s focus was checked on every edge of the marker structure on the sample. Slight tilts of the specimen resulted in blurred marker crosses afterwards. To obtain an optimum mapping picture, the focus was adjusted for every marker field using the piezoelectric actuator at the objective. To avoid inhomogeneous light distribution, reflections at the dichroic mirror mount were suppressed with black adhesive tape. For imaging, the blue LED was turned up to full and the IR LED to half the maximum power. Hereby it was important to find the right light intensity for the infrared, to not overexpose the markers.
During the measurement of a single field, at least eight frames were stacked together to one final image. Thereby the exposure time for each frame was two seconds. Splitting up the total exposure time of 16 seconds brings the advantage of not saturating the pixels. Nevertheless, it was critical to keep the optical table and the camera free from vibrations. The gain value was automatically set to
Figure 3.9: Resulting mapping image of a marker field, combined from multiple stacked frames.
300. Since not the whole camera chip was needed for a marker field, only a region of interest of 1448 × 1096 pixels was captured. This saved computation time in numerical post-processing. To enhance the camera’s sensitivity, the integrated cooling system was set to -5◦
C. In figure 3.9, one can see an example of the final images, which were saved as .fits files. Here, the markers and orientation letters were prominent, whereas the dot luminescence seemed rather weak. Generally one can say that the dot density of sample AS271 was too high for proper position mapping. Nevertheless, the optical quality of the dots was good why the deterministic creation of INCs continued on this sample. After imaging 45 fields per sample, the numerical position mapping of the quantum dots could be started. Before this is delineated in the next chapter, a few setup improvement possibilities are outlined.
3.3.3 Improvement Possibilities
Observing the imaging results in figure 3.9, one sees that the marker edges did not look sharply defined. The microscope images of the e-beam photo resist or the gold markers (figure 3.5 and 3.6) looked much clearer, than the depiction taken during mapping. A main reason for this was the blurring effect of the cryostat window. In [20], Jin Liu et al. shows differences in uncertainty results for mapping with and without an objective in cryostat system. With this and a more sophisticated mapping software, they reach dot position uncertainties down to 4.5 nm, which would be sufficient to create the INCs within the needed tolerances. Compared to this, the predecessor setup in [29] reaches uncertainties of 28 nm. Knowing this, brings out that an objective in cryostat system is favourable to precisely position nanostructures around quantum dots.
Nevertheless, there are still other, maybe cheaper things that can be done to enhance the performance of the home-built system. First of all, the objective, used in this work, was not glass-corrected. This means that its use in connection with a cryostat window resulted in aberration. Replacing the ob-jective with a glass-corrected one with higher numerical aperture should lead to sharper and brighter images. To further reduce the imaging exposure time, more powerful LEDs can be purchased, which could result in a better signal to noise ratio. Finally, the imaging procedure could be automated. Thereby the goal should be to load the preprocessed sample and let the system measure automati-cally.
The next step to generate position data of the randomly grown quantum dots is to post process the acquired pictures with an image processing software. This was created within the open source coding language „Python“, where the most important package was the „scikit-image“-package [36]. Its numerous methods allowed scientific image processing based an well understandable, mathematical functions.
In general, data generation could be split up into four major parts. At first there was the image loading and preparation step, followed by the detection of the reference markers and their center points. The third part consisted of the quantum dot position detection. In the end, the marker data and dot position data were combined to real positions and saved in different file formats. The following sections describe these steps in more detail.
4.1
First Image Preparation
Figure 4.1: Original image of a marker field (left) and contrast improved image after intensity rescale (right).
Before image processing was started, all taken pictures were stored in a common folder as .fits-files with a resolution of 1448 × 1096. The images were named by the wafer number and the field position for automatic identification by the program. Every file within the folder was loaded as a float pixel array, where the array values stood for the brightness of the pixel. These brightness values lay between 0, for a black pixel, and 1, for a white one. In most photographically taken images, these
values were not fully distributed over the whole brightness range. To enhance the images contrast, an intensity rescale was performed (figure 4.1). With this prepared image, the data generation steps could be started.
4.2
Marker Center Detection
For determining the marker centers, two lines of every marker cross were detected by a so called „Hough line transformation“. After joining the results of all four crosses, linear Gauss fits were performed perpendicular to the detected lines. The maxima of these fits, were then processed to retrieve to the final marker lines and the corresponding field corners.
4.2.1 Image Thresholding for Primary Detection
(a) Original image (b) Thresholded image and detected lines
Figure 4.2: Cross images prepared for hough line detection. This is performed for all four crosses of the field.
To identify the lines of the crosses, every marker field image was split up into four equally sized cross images. One of them is shown in figure 4.2a. For these pictures a multilevel thresholding procedure was performed based on the method of Yen J.C. [38]. Pixel values above the calculated threshold value were changed into white ones, pixels with a value below were turned to black (figure 4.2b). Now the Hough line transformation could be performed.
4.2.2 Hough Line Transformation
The scikit-image-package in Python provided multiple methods around Hough’s transformation pro-cedure. For performing a straight line Hough transformation, the method „hough_line“ was chosen. Its mathematical background lies within the mapping of lines in an x -y -plane to points in the pa-rameter space (Hough-space). Thereby a line in the image plane is expressed through its normal distance to the images origin ρ and the angle between its perpendicular and the abscissa θ (figure 4.3a). Observing an image point P, all lines intersecting in P correspond to a sinusodial curve in
(a) Image space (b) Hough space
Figure 4.3: Hough transform exemplified by two different plots of the same algebraic objects. A line in the image plane is represented as a point in Hough space, x -y -points are transformed to sinusoidal curves. Intersections in the parameter space are collinear points in the image.
Hough-space, where ρ = xPcos(θ) + yPsin(θ) [6]. Multiple curves have a common intersection in ρ-θ-space, if their corresponding points are collinear in the image (figure 4.3b). The parameters of this intersection point describe the line in the image plane on which the congruent points lie. Practically, the program transformed the input image into Hough space and looked for parameter points with many intersections. The scikit-image-method „hough_line_peaks“ gave out those in-tersection parameters. They were interpreted as the lines of interest. Certainly, the marker lines of photographically taken images do not only consist of collinear points. Thin lines get blurred due to the diffraction limit and inhomogeneous light intensities. Thus, „hough_line_peaks“ detected more than two lines per cross image (figure 4.2b). To determine those which belong to the marker cross, the two lines, most perpendicular to each other, were selected. Thereby, the reciprocals of the horizontal line slopes were compared with the slope values of the verticals. The line pair with the minimum difference in the compared quantities were then used for further processing.
The next step in locating the marker lines, was to join the four single crosses. For this, the slopes and y-intercepts of related lines were averaged. Hereby it was important to rotate the vertical lines by 90◦
before the means were evaluated. Slight differences in the small slopes could otherwise cause large shifts of the y-intercepts. By rotating the vertical lines, the y-intercept average was calculated properly. After a back rotation of the verticals, the results of all four detected lines were saved and plotted. Examining this outcome (figure 4.4), one sees that the located lines and cross centers did not perfectly coincide with those of the image. To cancel out this inaccuracy, further treatment was needed.
4.2.3 Accuracy Improvement through Linear Gauss Fits
(a) Marker field image and Gauss means of suc-cessful fits (blue crosses). The red lines are gen-erated by a linear regression method.
(b) Scaled intensity versus position and corre-sponding gaussian fit.
Figure 4.5: Principles of marker line fitting procedure.
In the following, linear Gauss fits were performed perpendicular to the markers according to the non-linear least squares method. The procedure is based on the assumption that the pixel values of a marker line’s cross-section corresponds to a Gaussian function. This assumption holds for most of the marker images, as long as the infrared intensity is homogeneous and not overexposing while position mapping (subsection 3.3.2). For better results, the fitting procedure was iterated im = 3
times. Thereby the result of the former fit was used as input for the next one.
Before the Gauss fits started, data sets of 120 pixels (≈ 6 µm) in x - or y -direction were generated. They consisted of arrays including the pixel’s brightness and their positions. The distance between two data sets along a marker line amounted 20 pixels (≈ 1 µm). Data in the cross center and outside the markers was not considered (figure 4.5a). A one-dimensional Gauss curve with offset was chosen for the fitting equation. The corresponding formula read
B = A e-12( x −µ
σ ) 2
+ B0,
