Towards scalable sources of entangled photon pairs relying on GaAs quantum dots embedded in circular Bragg resonators / submitted by Tobias Maria Krieger, BSc

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(1)Submitted by Tobias Maria Krieger, BSc. Towards scalable sources of entangled photon pairs relying on GaAs quantum dots embedded in circular Bragg resonators. Submitted at Institute of Semiconductor and Solid State Physics Supervisor Univ.-Prof. Dr. Armando Rastelli November 2020. Master Thesis to obtain the academic degree of. Diplom-Ingenieur in the Master’s Program. Technische Physik. JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Österreich www.jku.at DVR 0093696.

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(3) iii. Eidesstattliche Erklärung 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. Tobias Maria Krieger.

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(5) v. Abstract Tobias Maria Krieger Towards Scalable Sources of Entangled Photon Pairs Relying on GaAs Quantum Dots Embedded in Circular Bragg Resonators In the recent decade of research, semiconductor quantum dots have proven to be promising candidates for the use as node points in quantum information networks, as tunable on-demand sources of pairs of polarization-entangled photons featuring a near unity degree of entanglement, quasi-perfect single photon purity and a high indistinguishability. However, as-grown quantum dots lack a sufficient photon yield and suffer from total internal reflection within their embedding matrix. To enhance the extraction efficiency, the implementation of photonic cavities seems to be inevitable. The circular Bragg resonator photonic cavity has been reported to show outstanding results in extraction efficiency while also enhancing the spontaneous radiative emission rate, taking advantage of the Purcell effect. The cavity can be fabricated deterministically on quantum dots with pre-recorded positions through a reactive-ion dryetching process. In this thesis, an already existing quantum dot position mapping system was optimized, achieving a state-of-the-art detection precision and patterning accuracy. Furthermore, processing techniques were improved for an enhanced membrane fabrication success rate near 100 % and a recipe for dry-etching electron beam lithography patterned structures on GaAs membranes was established. The resulting circular Bragg resonators show broadband cavity modes in µ-reflectivity measurements, carried out polarization-resolved and in a temperature series..

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(7) vii. Acknowledgements I want to take this chance to thank all the people who contributed in their many different styles and ways to this master thesis, may it be technically or personally, for it would not have been possible to finish this work, as it is, without them. First of all, I want to thank my supervisor Univ.-Prof. Dr. Armando Rastelli, who gave me the opportunity to complete the master program by incorporating me into the Nanoscale Semiconductor group and offering me this position in research on this incredibly intriguing topic. It really makes me feel like working on something promising and meaningful. I also want to thank Assoc.-Prof. Dr. Rinaldo Trotta, who I met for my bachelor thesis for the first time on the very same institute, still being a significant partner with this group in Linz, although he is teaching in Rome now. Within this collaboration, M.Sc. Michele Rota joined the group in Linz and without hesitation, accepted me as a mentee and taught me "how to science", both technically and personally. A big thanks for that, I really appreciate it. Furthermore, I want to thank Prof. Rastelli’s great team in the cleanroom, Dr. Saimon Covre da Silva, Dr. Xueyong Yuan and M.Sc. Huiying Huang for their assistance with the machines, processing techniques and for funny chats in a clean environment. Thanks are also due to Dr. Santanu Manna for providing helpful simulations, adapted to our exact device. Additionally, I would like to thank Dr. Marcus Reindl, Dr. Daniel Huber and DI Christian Schimpf for fruitful discussions and advices regarding the optical systems. Thanks also to DI Christoph Kohlberger for his precedent work on the optical setup and the software, which for sure saved me from many sleepless nights. I am especially grateful to Mag.a Susanne Schwind who treated any administrative problem of mine as one of hers, working hard and being an incredibly valuable part of the institute. A special thanks goes also to the technicians of the clean room Alma Halilovic, Ing. Albin Schwarz, Ing. Stefan Bräuer, Ursula Kainz and Ing. Ernst Vorhauer for teaching about and maintaining the machines and also for sharing their great experience. Thanks is also due to Mag.a Esther Wöckinger from the International Office of JKU, doing her best in assisting me to do an Erasmus+ Traineeship with Prof. Trotta’s group at Sapienza University in Rome. Taking part in this group was an amazing opportunity for me that I enjoyed so much. I want to thank Dr. Francesco Basso Basset, Dr. Davide Tedeschi, Dr. Emanuele Roccia, M.Sc. Matteo Savaresi, DIin Julia Neuwirth, Francesco Salusti, B.Sc and Giuseppe Ronco, B.Sc. for inviting me to discover Rome and the Italian culture. Also Dr. Giorgio Pettinari of CNR-IFN Rome was very welcoming and contributed significantly with his great knowledge in the cleanroom..

(8) viii Since this master thesis was written during the challenging times of the outbreak of the COVID-19 pandemic, so many things changed in this world and, therefore, I want to express my deep gratitude to the medics, nurses, cashiers, pharmacists, truck drivers and all the other essential workers, keeping the society running while exposing themselves to a great risk without hesitation. You are performing incredible work, although lacking fair remuneration. On this point, thanks are also due to the rectorate of the JKU for making final examinations still possible within these uncertain times without creating a significant delay of my studies. Last but not least, I want to thank my beloved parents Sabine and Werner, for giving me the opportunity and support of attending university, although I know that I should not take this for granted. Yet, you made it possible which makes me so proud and grateful. I could not wish for better parents..

(9) ix. Contents Contents. ix. 1 Introduction and Motivation. 1. 2 Theoretical Background 2.1 The Physics of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Crystal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Bloch Waves and Bandstructure . . . . . . . . . . . . . . . . . . . 2.1.3 Effective Mass Approximation . . . . . . . . . . . . . . . . . . . . . 2.1.4 Gallium Arsenide . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Quantum Dot Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Growth of Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Electronic Structure of a Quantum Dot . . . . . . . . . . . . . . . 2.2.3 Few-Particle States . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Entangled Photon Pairs via the Biexciton-Exciton Cascade . . . . . 2.3 Photonic Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Quality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Light-Matter Interaction . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Purcell Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Circular Bragg Resonator and Highly-Efficient Broadband Reflector. . . . . . . . . . . . . . . .. 5 5 5 7 10 11 13 13 14 17 17 21 21 21 22 23. 3 Fabrication of the Circular Bragg Resonator 3.1 Fabricating Membranes . . . . . . . . . . . . 3.1.1 Atomic Layer Deposition of Aluminum 3.1.2 Physical Vapor Deposition of Gold . . 3.1.3 Bonding . . . . . . . . . . . . . . . . 3.1.4 Back-Etching . . . . . . . . . . . . . 3.1.5 Quantum Dot Membranes . . . . . . 3.2 Processing of the Circular Bragg Resonator . 3.2.1 Electron-Beam Lithography . . . . . . 3.2.2 Reactive Ion Etching . . . . . . . . .. . . . . . . . . .. 25 25 27 28 29 30 34 35 35 38. . . . . Oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . ..

(10) x. CONTENTS. 4 Analysis of the Reactive-Ion Etched Circular Bragg Resonators 4.1 µ-Reflectivity Measurement Setup . . . . . . . . . . . . . . . 4.2 Cavity Resonance Analysis . . . . . . . . . . . . . . . . . . . 4.2.1 Cavity Modes . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Polarization-Resolved Measurements . . . . . . . . . . 4.2.3 Temperature-Resolved Measurements . . . . . . . . .. . . . . .. 41 41 43 44 47 49. 5 Finding Quantum Dot Positions Using Automated Image Processing 5.1 Principles of Marker-based Automatic Recognition of Point-like Light Emitters 5.2 Markers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Light and Dark Markers . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Sample Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Fabrication of Markers . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Mapping of QD Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 High-Resolution Imaging Setup . . . . . . . . . . . . . . . . . . . . . 5.3.2 Marple Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Deterministic patterning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Considerations of Improvement . . . . . . . . . . . . . . . . . . . . . . . . .. 51 51 52 52 52 54 56 56 60 65 66. 6 Conclusion and Outlook. 67. A Fabrication of HBR-backed QD membranes A.1 Fabrication of Membranes with a HBR . . . . . . A.1.1 Sample cutting . . . . . . . . . . . . . . A.1.2 Sample cleaning . . . . . . . . . . . . . . A.1.3 Substrate choice and preparation . . . . . A.1.4 Oxide deposition . . . . . . . . . . . . . . A.1.5 Metal evaporation . . . . . . . . . . . . . A.1.6 Flip-Chip Bonding . . . . . . . . . . . . . A.1.7 Back-Etching . . . . . . . . . . . . . . . A.2 Transfer to a Six-Legged Piezoelectrical Actuator List of Figures Bibliography. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . .. . . . . . . . . .. . . . . . . . . .. I I I II II II IV IV VI VIII XI XIII.

(11) 1. Chapter 1. Introduction and Motivation When looking at the current scientific and technological progress, it is often said that the role which the electron played for the technical developments of the twentieth century, will be inherited by the photon in the coming one [1]. The concept of a photon, i.e., the quantum of light, follows from the quantization of the electromagnetic field. Some processes, especially in the condensed matter, require the treatment of light-matter interaction in a fully quantum mechanical picture, where the classical approach is not sufficient. A quantum optics scientist investigates in such phenomena theoretically and experimentally. While the completeness of quantum mechanics was heavily doubted by the famous Einstein-Podolsky-Rosen (EPR) paper in 1935 [2], Bell counterposed in 1964 [3] with the proposition of Bell’s inequality, which would prove the existence of local hidden-variable theories, if the inequality was satisfied. However, it was first shown by Freedman-Clauser [4] experimentally that Bell’s inequality is violated, thus proofing Bell’s theorem that quantum mechanics is in fact a non-local theory and moreover, disclosing the concept of quantum entanglement. Entanglement can be imagined as an instantaneous, non-local link between two or more quantum particles sharing a common state which is not separable, i.e., it can not be written as a product state in a multi-dimensional Hilbert space. This fundamental concept enables the possibility to transfer and manipulate information exploiting advantages of quantum mechanical properties. The most important feature is the no-cloning theorem. Tempering with a quantum state, like measuring or perturbing it in any sense, causes the wave function of the state to collapse into an eigenstate. This important property can be taken as an advantage for quantum networks in general and quantum cryptography in specific. Following from there and with a rational distribution of a key for en- and decryption, communication based on quantum systems can be made absolutely secure against eavesdropping. The first method using such a concept was proposed by Bennett and Brassard [5], the BB84 quantum key distribution scheme, while the first protocol using quantum entanglement for quantum key distribution (QKD) was reported by Ekert in 1991 [6], commonly referred to as E91. The idea to use states in a quantum system to hold information gives rise to the concept of the quantum bit, abbreviated as qubit. To implement any method of sharing such qubits, the unperturbed transport of a quantum state is necessary. To carry such, photons can be used, as they are much more robust than other quantum systems, like atomic spins, when it.

(12) 2. Chapter 1. Introduction and Motivation. comes to preserve coherence over space and time and, furthermore, they are capable of being distributed over the existing world-wide fiber network, making them an ideal candidate to be used as a transmittive "flying" qubit. As in classical communication, some factors like signal and coherence loss have to be compensated by a repeater when going over large distances. However, problems arise due to the no-cloning theorem, which implies that information held in a qubit is lost when measured, making amplification unfeasible. Therefore, a quantum repeater [7] is needed. Such a novel device uses the concept of quantum teleportation [8] and entanglement swapping [9] together with a quantum memory [10]. In a quantum teleportation no matter is teleported but rather the quantum state |Ψ ⟩ in which a photon is prepared in. To teleport this state, that photon interacts with one partner of an entangled photon pair in a interference experiment, making this scheme a three-particle process. Such a two-photon interference is called a Bell state measurement (BSM). The result of this BSM indicates which unitary transformation needs to be applied to the remaining photon, i.e., the second partner of the entangled pair, and recovers the initially prepared quantum state of the first photon. As this last bit of information can only be transferred classically, the speed of a teleportation of quantum information can not exceed the speed of light and, therefore, does not violate causality and is in accordance to Einstein’s theory of special relativity. To overcome even greater distances, entanglement can be shared between particles that never interacted before through the swapping of entanglement. For this process, two entangled photon pairs are involved in which one of each pair are brought to interference, swapping the entanglement to the remaining photons. Together with using quantum memories maintaining the coherence of those photons for some time, quantum repeaters can be used to overcome great distances on a worldwide scale. The working principle of a quantum teleporter using a repeater, is sketched in Figure 1.1. The list of requirements towards a scalable EPR source, i.e., a source of entangled pairs, to be used within a quantum network includes the on-demand generation of entangled photon pairs with high purity, high indistinguishability, high efficiency and a high degree of entanglement [11], favorably embedded in a semiconductor environment to take advantage of existing optoelectronic technologies. This leads to the promising concept of semiconductor quantum . . Ψ. . . . . . . . Ψ. . Figure 1.1 – Sketch of a quantum teleportation of the qubit |Ψ ⟩ using a quantum repeater for entanglement swapping. A coincidence measurement (BSM) of the inner photons swaps the entanglement to the outer ones, being stored in a quantum memory (Q-MEM). EPR denotes an entangled particle source like a quantum dot..

(13) Chapter 1. Introduction and Motivation. 3. dots (QDs) that emit highly polarization-entangled photon pairs on demand [12] and feature a tunable emissions wavelength [13]. In addition to the small size of QDs, a feasible implementation in everyday-devices is within reach in the near future and seems much more likely to be implemented than other bulky systems. Scientific progress has been increasing tremendously in this field in the last years, including successful implementations of quantum teleportation [14] and entanglement swapping [15], using droplet-etched GaAs QDs [16]. To cope with errors made in the transmission between Alice and Bob, quantum error correction methods can be applied [17][18], however, many entangled photon pairs are needed for those. Therefore, one must assure to maintain a high collection efficiency from the QD. As photons are emitted isotropically, lenses and photonic cavities should be used to direct the beam towards a collecting objective or a fiber coupler. Keeping a high photon count rate within the experimental setup is also utterly important, to keep runtimes of experiments within a reasonable duration. In the Nanoscale Semiconductors group of Prof. A. Rastelli, the QD emission is currently collected with the use of distributed Bragg reflectors (DBR) underneath and above the QD layer, forming a planar cavity, and a solid immersion lens (SIL) on top of the surface. This prevents, to some extent, total internal reflection of the generated photons within the highrefractive index semiconductor matrix and raises the extraction efficiency to approximately 10 %, compared to < 1 % when not using any structure at all. In recent years, many possible candidates emerged in order to enhance the source’s brightness by using microcavities like microlenses [19][20], micropillars [21][22], microscopic mesa structures [23] and inverted nanocones [24], with much work also done at the JKU in Linz [25][26][27]. The aim of this master thesis is to implement a planar photonic cavity, the circular Bragg resonator (CBR) also called the "bullseye" cavity, like sketched in Figure 1.2. The CBR features a resonance with a broad spectral bandwidth, a high extraction efficiency and a radiative rate enhancement. The first realization of a suspended bullseye cavity was reported by Davanço et al. [29] in 2011. Following the presentation of a revised design of the bullseye cavity by Yao et al. [30] in 2018, Liu et al. [28] and Wang et al. [31] reported outstanding values for the photon pair extraction efficiency (65 % [28]), entanglement fidelity (90 % [31]),. Figure 1.2 – A sketch of a CBR cavity with a quantum dot in the middle. An oxide and gold layer on the bottom further boosts extraction efficiency. Image taken from Ref. [28]..

(14) 4. Chapter 1. Introduction and Motivation. Purcell enhancement (11.3 [31]) and the photon indistinguishability (90 % [28][31]) for selfassembled InAs [31] and GaAs [28] QDs. However, these QDs are not site-controlled grown, therefore, a quantum emitter position mapping system is needed, to fabricate microcavities deterministically on specific positions [32][33]. This thesis is structured into six main chapters and an appendix. After the introductory chapter 1, some necessary background information and fundamental physical theory is covered in chapter 2. The following chapter 3 shows results of optimizing suitable processing recipes for fabricating membranes and the dry-etching of the cavity. The analysis of etched CBR cavities placed in arrays is carried out by µ-reflectivity measurements and results are presented in chapter 4. The method and setup for finding QD positions, both from hardware and software perspectives is presented in chapter 5. As the cavities were not placed on recorded QD positions when finishing my laboratory experience, an outlook on this project and also a brief summary of the thesis are presented in the final chapter 6. Since there is still work in progress, there will be a chapter in the appendix which has a rather instructional character to have important numbers and parameters handy when processing in a cleanroom environment. Therefore, information about the fabrication and processing of membranes is provided in Appendix A..

(15) 5. Chapter 2. Theoretical Background This chapter provides some relevant preliminary information in order to understand the principles underlying this master thesis. In the first section 2.1, the focus will be on solid state and semiconductor physics in general, with most of the information taken from the book of Gross and Marx [34], Ashcroft and Mermin [35] and the lectures in solid state physics and semiconductor physics held by Univ.-Prof. Dr. Armando Rastelli. In the second section 2.2 about quantum dots, I will mainly rely on the eponymous lecture by Assoc.-Prof. Dr. Rinaldo Trotta and on the book of Michler et al. [36]. The third section 2.3 is based on the book Microcavities from Kavokin et al. [37].. 2.1. The Physics of Semiconductors. In order to understand what semiconductors are and to learn about their properties, we need to take a step back and look at well-ordered matter in the solid state and describe their electronic structure.. 2.1.1. Crystal Lattice. A crystal is a type of condensed matter which can be constructed by stacking identical blocks following a certain rule such that we can obtain a 3-dimensional, periodic array of atoms. A piece of such a crystal is usually found polycrystalline in nature, meaning that several periodic domains form one block of matter, however, it can also be in a monocrystalline state. Since single crystals can be produced and are widely used in industry and research, we want to consider only those from now on. The Bravais lattice is a model to fully describe the position of all lattice points in a crystalline material. Any lattice point R in 3D can be found by a set of integer multiples n1 , n2 , n3 of primitive lattice vectors a1 , a2 , a3 , which define the primitive unit cell, i.e., the smallest possible volume sufficient to recover the whole crystal when stacked along those vectors. Any lattice point R is a linear combination, obeying R = n1 a1 + n2 a2 + n3 a3 .. (2.1).

(16) 6. Chapter 2. Theoretical Background. The absolute values |ai | = ai are called lattice constants. The absolute value of the triple product (a1 × a2 ) · a3 is the volume of the primitive unit cell. In general, a crystal structure is described by a Bravais lattice plus a basis. The basis is either mono- or poly-atomic. The material of relevance in this thesis is GaAs (AlGaAs), which crystallizes in the zincblende structure, consisting of two shifted face centered cubic (fcc) lattices, where the side of each cube is called the lattice parameter a. One fcc lattice consists of Ga (Ga and Al) atoms and is situated at (0,0,0), while another fcc lattice, describing the As atoms, is shifted to a/4 (1, 1, 1). Reciprocal Lattice While the crystal lattice describes the position of lattice points in the direct space, the reciprocal lattice is an entity of the reciprocal space, i.e., the k-space. Its introduction is useful for diffraction experiments but also for the Fourier-analysis of periodic functions. The definition of the reciprocal lattice is given by eiG·R = 1. ⇔. G · R = 2πn. with integer n. (2.2). where R is any of the Bravais lattice vectors and G a reciprocal lattice vector. The definition induces that any function that is periodic with the Bravais lattice, i.e., f (r) = f (r + R), can be expanded as a Fourier series containing only reciprocal lattice vectors G as wavevectors. A reciprocal lattice point G can be written in the basis b1 , b2 , b3 with the so-called Miller indices h, k, l, as in G = hb1 + kb2 + lb3 . (2.3) The reciprocal basis vectors can be obtained from Bravais basis vectors, as described in textbooks [35] [34]. Looking at a plane wave with wavevector k (not to be confused with the miller index k) propagating through a lattice in direct space, there is a maximum wavelength λmax for which it still holds that the plane wave has the same value at all lattice points Ri on the same lattice plane. There is a family of those equidistant lattice planes separated by the distance d and perpendicular to k, which must contain all lattice points of the whole crystal. Since R is a Bravais lattice point, the wavevector of such a wave is a reciprocal lattice vector G with an absolute value |Gmin | = 2π/d. Planes and directions in a crystal There are families of equidistant lattice planes which contain all lattice points of the crystal and are perpendicular to a reciprocal lattice vector G. They can be addressed to with the Miller indices written in round brackets (hkl). If there are equivalent planes due to the crystal symmetry, e.g., all surfaces of cubic lattices, they are denoted with curly brackets {hkl}. Also crystal directions can be expressed with the parameters of the Bravais lattice, giving the direction of R. As introduced in 2.1, one writes square brackets [n1 n2 n3 ] and is able to include all equivalent directions, again due to crystal symmetry, with the notation using angle brackets ⟨n1 n2 n3 ⟩. Negative values used in crystal planes and directions are denoted with a line on top, e. g. (1¯11), for better readability..

(17) 2.1. The Physics of Semiconductors. 7. Brillouin zone The Brillouin zone (BZ) of first order is a volume in the reciprocal space defined around a reciprocal lattice point where it holds that this region is closer to that point than to any other point in the lattice. An equivalent volume can be formulated in direct space and is called the Wigner-Seitz cell (WS cell). The surfaces of the BZ are called Bragg planes, since the Bragg condition is valid there, enabling constructive interference in diffraction experiments. The WS cell of a fcc lattice is a a rhombic dodecahedron, centered in r = 0. The corresponding first BZ is a truncated octahedron centered in k = 0. Since the reciprocal lattice of fcc is bcc and vice versa, the first BZ and WS cell are the exact opposite for bcc.. 2.1.2. Bloch Waves and Bandstructure. In the previous subsection, the Bravais lattice was introduced to map atomic positions of a whole crystal through a discrete and periodic translation of the unit cell. We want to see how electrons can move around the lattice sites, which are positively charged atomic cores. Starting with the many-particle Hamiltonian . ˆ = H. . X pˆ2 X pˆ2 X X X  ZI ZJ ZI e2  1 i I   , (2.4) + + + −  2MI 2m0 4πϵ0 I,J |RI − RJ | |ri − rj | |RI − ri |  i i,j I I,i I<J. i<j. with M and R (m0 and r) being the mass and position of atomic cores (electrons), ϵ0 the electric field constant, Z the atomic number, pˆ the momentum operator and using uppercase summation letters for atomic cores and lowercase summation letters for electrons, it is evident that there is no chance in solving the Schrödinger equation without approximations when applied to a massive wavefunction ψ = ψ(...RI , RJ ...ri , rj ...). Beginning with the adiabatic Born-Oppenheimer approximation, where the movement of the much heavier cores is decoupled from the movement of electrons, i.e., the potential caused by the atomic cores seems constant for electrons in every moment, the Schrödinger equation separates. Since all electrons should behave similarly, we push the electron-electron interaction together with the atomic potential into V (r) and solve the one-particle Schrödinger equation using an Ansatz of a linear combination of plane waves with wavevectors k, fulfilling the boundary conditions of the crystal with length Li along the lattice unit vector ai , such that ki = 2πni /Li , with ni ∈ N and i = 1, 2, 3, and write ψ(r) =. X k. Ck eik·r. and V (r) =. X. VG eiG·r ,. (2.5). G. since we saw previously that lattice periodic functions, as V (r) = V (r + R) is such, can be Fourier expanded with reciprocal lattice vectors G. The Schrödinger equation in Fourier space yields an infinite number of solutions for each value of k which makes it necessary to label them with the integer index n which from now on we call band index. Although k is a discrete quantum number in principle, the spacing for macroscopic crystals is so dense that.

(18) 8. Chapter 2. Theoretical Background. it can be treated as a continuous variable. Because of the periodicity with the length of a reciprocal lattice vector, we find the same solution for any multiple of G Enk = En (k) = En (k − Gn ).. (2.6). The wavefunction therefore reads ψnk (r) = unk (r) eik·r. with unk (r) =. X. Ck−G e−iG·r ,. (2.7). G. with the lattice periodic Bloch wave function unk (r). The wavefunction itself however is not lattice periodic, since it is modulated with a plane wave. Band structure and dispersion relation To construct the band structure, it turns out convenient to examine the effect of the periodicity of the problem without an active potential first. The dispersion relation of a free electron as a function of k reads En (k) =. ~2 (kx2 + ky2 + kz2 ) 2m0. (2.8). and is parabolic in every direction. Following from 2.6, we find one parabola after the other with increasing n. Consider the graphs in Figure 2.1 in one dimension for better clarity. (a) shows a parabola of a free electron for n = 0 while in (b), one can additionally find a parabola centered in G where n = 1. In fact, there are parabolas for every solution n which stack. Figure 2.1 – Construction of the band structure. Adapted from Ref. [35]..

(19) 2.1. The Physics of Semiconductors. 9. up fast on top of the parabola centered in the first BZ ∈ [− 12 G, 12 G]. One can see that at the edges of the BZ, the Bragg planes, degeneracies pop up. As it is plotted in (c), the degeneracy when two bands intersect is lifted at the Bragg planes. Qualitatively, this can be explained by a Bragg reflection of two plane waves with wavevectors ± 12 G that are interfering and developing a symmetric (s-like) and anti-symmetric (p-like) standing wave. Due to their coupling energy UG (Fourier coefficient of the potential corresponding to a wavevector G) the degeneracy lifts. In (d), it is clearly visible how the lifted degeneracy alters the n = 0 band. Continuing systematically on further Bragg planes in (e), one finds the so-called extendedzone scheme. The plot including the contributions of higher bands but in the first BZ only is called reduced-zone scheme as in (f) and within many BZs repeated-zone scheme in (g). In (e), (f) and (g), one can see the appearance of band gaps, which is an important property to classify materials. For intrinsic semiconductors, i.e., materials that do not feature any doping and, therefore, have an equal number of free positive and negative charge carriers, the Fermi energy EF lies within this gap. Following Fermi-Dirac statistics, this means that all (most) negative charge carriers reside below EF at 0 K (finite temperatures). Bands below the gap are called valence bands (VBs) and above conduction bands (CBs). When the gap can be overcome by charge carriers and the material be made conductive, it is a semiconductor. If such surmounting of the band gap is not possible, the material is considered an insulator and if there are bands that are not full with electrons, i.e., EF does not lie within the gap, it is a conductive metal. An electron crossing the band gap from the VB to the CB leaves a vacancy which is called a hole and can also contribute to conduction. A hole is treated as a quasi-particle with opposite charge and spin compared to the removed electron, with a negative mass and a negative energy. As soon as the band structure of a material is given, one can try to model regions around minima and maxima for one specific band of En (k) in a harmonic approximation. Considering a Taylor expansion around an extremum at k0

(20). 1 X ∂ 2 E

(21)

(22) E(k) = E(k0 ) +

(23) ∆ki ∆kj 2 i,j ∂ki ∂kj

(24) k. with i, j = x, y, z,. (2.9). 0. we can define the effective mass tensor in terms of the second derivative of the energy m∗ij = ~2.

(25) !−1. ∂ 2 E

(26)

(27)

(28) ∂ki ∂kj

(29) k. (2.10) 0. and also see the relation to the curvature of the band, which is proportional to the inverse of the effective mass. The tensor is symmetric and in some cases, as for at the bottom of the lowest lying conduction band of GaAs, isotropic, i.e., can be treated as a number. Due to the orientation of the band, the effective mass can also be negative, yielding a hole-like.

(30) 10. Chapter 2. Theoretical Background. behavior. In the case of an electron close to an isotropic local minimum at k0 , the energy dispersion takes the form ~2 (∆k)2 E(k) = E(k0 ) + , (2.11) 2m∗ with ∆k being the absolute value of the distance between k and k0 . The effective mass and the energy of a particle in a minimum are positive, in a maximum negative.. 2.1.3. Effective Mass Approximation. The effective mass approximation is an important concept throughout semiconductor physics to treat problems deviating from a perfect crystal, taking Bloch electrons [38] into account. Trying to solve the Schrödinger equation for such a perfect crystal with a small perturbation caused by an impurity and acting as a potential Vîmp . . ˆ per + Vîmp ψ(r) = E(r)ψ(r), H. (2.12). one wants to reduce complexity. Consider the one-dimensional formulation of the problem. Suppose we have found the solution of the Hamiltonian of a perfect crystal ˆ per ϕnk (z) = εn (k)ϕnk (z) H. (2.13). with the wave vector k and the energy level n as quantum numbers. The solutions ϕnk (z) of this eigenvalue problem form a complete set and can be expanded, as ψ(z) =. ∞ Z X. π a. −π a. n=0. χ(k)ϕ ˜ nk (z). dk , 2π. (2.14). with the Fourier coefficient χ. ˜ This is yet no simplification to the initial problem 2.12, which is why we want to consider (1) only one band, i.e., drop the sum over n. Therefore, this model does not allow interband transitions and assumes only weak perturbations and no degeneracy. Furthermore, we want to assume (2) that k lies at the bottom of the CB, close to k = 0 and only a small range of k values contribute to the unknown wavefunction ψ. Assuming that the Bloch function stays constant for a small region of k and the only k-dependent part comes from the plane wave, ψ(z) takes the form of an inverse Fourier transform, as ψ(z). (1)+(2). ≈. Z. ϕn0 (z). π a. −π a. χ(k) ˜ eikz. dk = ϕn0 (z)χ(z). 2π. (2.15). Equation 2.15 shows the first results of these approximations: The solution of problem 2.12 can be separated into the Bloch wave part ϕn0 (z) and an envelope function χ(z). Note that the χ(z) varies only slowly in real space compared to the Bloch wave, since only a small range of wave numbers are considered. For hetero-nanostructures with sizes ζ ≈ 10 nm ≫ a ≈ 0.5 nm, i.e., much larger than the lattice constant a, this assumption seems justified..

(31) 2.1. The Physics of Semiconductors. 11. To find an equation for χ(z), we want to insert the n-th term of the expansion 2.14 into ˆ per on ϕnk and expand the solution εn in a Taylor the Schrödinger equation 2.12, apply H series around k = 0. Using approximation (2) yields ˆ per ψ(z) ≈ ϕn0 H. Z. π a. −π a. χ(k) ˜. ∞ X. am k m eikz. m=0. dk . 2π. (2.16). It is evident that the Fourier transform of (−i) times the derivative of a function with respect to its spatial variables is exactly k. If we now substitute back into the power series and abbreviate it with εn in a second step, we find ˆ per ψ(z) ≈ ϕn0 H. ∞ X m=0. d −i dz. m. . χ(z) = ϕn0 εn. . d χ(z). −i dz. (2.17). Substituting back into equation 2.12, canceling the common factor ϕn0 and expanding the problem to three dimensions, we find the effective mass equation (EME): h. i. εn (−i∇) + Vîmp (r) χ(r) = Eχ(r). (2.18). The kinetic term takes the form of the dispersion relation of the electron within the crystal for one certain band, as long as the band structure is known. We hereby found a "Schrödinger equation" only for the envelope function. Often, the band structure of the semiconductor lies all in the effective mass, hence the name "effective mass approximation". With the EME, one can find a fast and easy approximation of energies for donor or acceptor charge carriers, bulk excitons and hetero-nanostructures like QDs.. 2.1.4. Gallium Arsenide. GaAs and the alloy AlGaAs are the materials used for QDs studied within this thesis. Therefore in this subsection, selected important properties of GaAs, and to some extent of AlGaAs, are listed. Additionally to the main sources of the whole chapter, the numbers will originate from the website of Ioffe NMS [39]. In Figure 2.2 the bandstructure of GaAs is plotted next to a sketch of the first BZ with points of high symmetry, e.g., the Γ -point in the origin, where k = 0. The material’s chemical bond is mostly covalent. Valence electrons are the group of outermost charge carriers of atoms which contribute to bonding. There are 8 valence electrons per unit cell (containing one Ga and one As atom), summing up to 8N electrons in a crystal with N cells but that is also exactly the number of available k values per band. Due to Pauli’s exclusion principle, only two electrons with opposite spin can occupy the same quantum state. Therefore, the valence electrons fill the 4 lowermost bands which are the VBs and their uppermost point is defined to be the origin of the energy scale. The infinite bands higher in energy are the CBs, which contribute to a current if negative charge carriers overcome the band gap energy which separates VB and CB..

(32) 12. Chapter 2. Theoretical Background. Looking more closely at the upper VBs, they have p-orbital character due to the involved valence electrons and an angular momentum L = 1 (in units of ~) which mixes with the electron spin S = 1/2 to the total angular momentum J = L ± S. The spin-orbit interaction is a relativistic effect, shifting the split-off (SO) band (J = 1/2, mJ = ±1/2) down from the heavy hole (HH) band (J = 3/2, mJ = ±3/2) and light hole (LH) band (J = 3/2, mJ = ±1/2), that are degenerate at the Γ -point, with an energy difference ∆ = 0.341 eV. The effective mass for charge carriers in the Γ -valley in GaAs, as defined in 2.10, is for electrons m∗e = 0.063 m0 , heavy holes m∗hh = 0.51 m0 , light holes m∗lh = 0.082 m0 and split-off holes m∗soh = 0.15 m0 , where m0 is the mass of a free electron. GaAs is a direct band gap semiconductor. Direct band gap means that the minimum of the CB is on top of the maximum of the VB, greatly enhancing the probability for optical transitions. The band gap energy is temperature dependent. The phenomenological Varshni equation 2.19 with parameters a, b shows a quadratic (linear) behavior for low (high) temperature. aT 2 Eg (T ) = Eg (0) − (2.19) T +b For GaAs, a = 5.405 × 10−4 eV K−1 and b = 204 K. At T = 0 we find Eg = 1.519 eV and for room temperature Eg = 1.424 eV. The alloy of GaAs and AlAs is Alx Ga1−x As. AlAs has nearly the same lattice constant a as GaAs however, it is an indirect semiconductor. The alloy is considered direct still for x < 0.45. At room temperature it holds Eg (x) = 1.424 + 1.247x eV. for x < 0.45,. a(x) = 5.65325 + 0.0078x Å.. (2.20) (2.21). Figure 2.2 – The lowest bands of the calculated band structure of GaAs. Points of high symmetry in the BZ are plotted besides. Bandstructure taken from Ref. [40]..

(33) 2.2. Quantum Dot Nanostructures. 2.2. 13. Quantum Dot Nanostructures. A quantum dot (QD) is a semiconductor hetero-nanostructure with a size of a few up to a few tens of nanometer which leads to specific electronic and occasionally optical properties. A confinement of the motion of charge carriers in all spatial directions results in the necessity of a quantum mechanical treatment, whereas the dominating property is the quantization of energy states. Electrons can occupy these discrete energy levels, can be excited with various techniques, e.g., with a laser or with positioning in the active region of a light emitting diode (LED), and can also show radiative transitions, similar to an atom. This is the reason why QDs are also called "artificial atoms". QDs come in different shapes and sizes but are usually a crystalline semiconductor cluster of a few thousand atoms, embedded in another semiconductor material, called matrix, fixing the position of the QD in space, providing a huge advantage compared to the demanding control of an ion trap. The difference of the band gap energies of the QD and matrix semiconductor material generates a potential well in energy for the three spatial directions. The resulting energy levels give room to electrons to be excited into discrete states in the CB which leave holes in the VB. The electron-hole pair can be bound via Coulomb interaction and is then called exciton (X).. 2.2.1. Growth of Quantum Dots. A much studied material system is the InAs(GaAs) QD fabricated with Stranski-Krastanow (SK) growth, featuring good optical properties. Huo, Rastelli and Schmidt [16] however, have reported that local droplet etching (LDE) can yield ultra-symmetric nanoholes and strain-free quantum dots which are desirable properties for the generation of highly-entangled photon pairs. The samples are grown by molecular beam epitaxy (MBE) on GaAs (001) substrates. A sketch of the process is given in Figure 2.3. The matrix is made of Al0.4 Ga0.6 As where some excess aluminum is deposited under the suppression of the As flux, forming selfassembled droplets by Volmer-Weber (VW) growth at high temperature. During annealing and while ensuring a constant As pressure for keeping the As concentration inside the sample at equilibrium, the As gradient at the interface between AlGaAs and Al causes the diffusion of As atoms into the droplet, etching a very shallow but highly symmetric nanohole into the AlGaAs matrix and forming an optically inactive AlAs crater around it [41]. The nanohole is filled with GaAs, then annealed, and, eventually, the stack is capped with a top barrier of AlGaAs. The GaAs QDs maintain their shapes due to the weak intermixing of GaAs and AlGaAs. An atomic force microscope image of such nanohole is shown in Figure 2.4.

(34) . . . . . . . . . Figure 2.3 – Cross-sectional view of the fabrication of a GaAs QD on an AlGaAs film..

(35) 14. Chapter 2. Theoretical Background. Figure 2.4 – An atomic force microscope (AFM) image of a highly symmetric nanohole made by aluminum LDE (before filling up the hole) and the corresponding line-scan both for the [110] and [1-10] crystal direction. Image and graph taken from Ref. [16].. 2.2.2. Electronic Structure of a Quantum Dot. In the former section, the QD was introduced qualitatively as a three-dimensional potential well formed by the difference in band gap energies of QD material and surrounding matrix. To justify this simple picture, electronic energy states of a QD are derived starting from the effective mass equation. Quantization of Energy States in a Quantum Dot Trying to understand the quantization of energy better, we want to consider a flat QD, which means that there is a main ζ quantization axis, which is parallel to the growth direction z. A cross-section of a QD is sketched in Figure 2.5. The QD’s lateral dimensions wx , wy satisfy wx , wy ≫ ζ, with ζ Figure 2.5 – Crossbeing the peak height of the QD, while its shape is given section of a quantum dot. by the function L(x, y). Starting with the EME (2.18), we strive to find the energy levels that an electron in a GaAs crystal occupies, when confined in a potential Vˆ (x, y, z). We define the energy scale to be 0 eV at the maximum of the VB of the QD material and assume an infinitely high potential barrier in the matrix, although, in the case of GaAs and Al0.4 Ga0.6 As, the difference in Eg is only ≈ 0.5 eV. If one is attempting to solve the EME for heterostructures, m∗ (z) and boundary conditions need to be considered, making the equation non-Hermitian. One would need to transform the equation into a Ben Daniel-Duke partial differential equation, which is typically solved numerically to obtain exact results. Therefore, we want to use this simplification and treat the potential and effective mass as finite constants inside the well: ". #. −~2 2 ˆ ∇ + V (x, y, z) Ψ (x, y, z) = EΨ (x, y, z) 2m∗ (. Vˆ (x, y, z) =. Eg ∞. 0 < z < Lz (x, y) otherwise. (2.22).

(36) 2.2. Quantum Dot Nanostructures. 15. Since we concentrate on flat QDs, the kinetic energy is dominated by the motion along z which leads to the separation of vertical and lateral motion with the adiabatic approximation Ψ (x, y, z) = ψ(x, y)ϕ(z) and we find −~2 1 ∗ 2m ψ(x, y). ∂ 2 ψ(x, y) ∂ 2 ψ(x, y) + ∂x2 ∂y 2. !. +. −~2 1 ∂ 2 ϕ(z) = Ex,y + Ez − Eg , 2m∗ ϕ(z) ∂z 2. (2.23). within the QD. The eigenvalues of ϕ(z) are found by restricting the free particle’s infinite amount of possible kz numbers to as many which fit into the potential well satisfying ϕ = 0 on the boundaries, i.e., kz = nz π/Lz with nz ∈ N. Paying attention to the CB offset, the energy reads ~2 π 2 n2z Ez = Eg + . (2.24) 2m∗ L2z The vertical confinement is x, y-dependent due to Lz (x, y) and acts as a potential for the lateral part of the Schrödinger equation 2.23. We now rename Ez → V (x, y) and Taylor expand to the second order (harmonic approximation) around x = y = 0 and find

(37).

(38). ~2 π 2 n2z ~2 π 2 n2z ∂ 2 Lz

(39)

(40) ~2 π 2 n2z ∂ 2 Lz

(41)

(42) 2 V (x, y) ≈ Eg + − x −

(43)

(44) y2, 2m∗ ζ 2 2m∗ ζ 3 ∂x2

(45) 0,0 2m∗ ζ 3 ∂y 2

(46) 0,0. (2.25). where Lz (0, 0) = ζ. A potential’s quadratic dependence on the position resembles the quantum harmonic oscillator and its solutions are well known from literature, so we can write . . .

(47). . −~2 π 2 n2z ∂ 2 Lz

(48)

(49) with =

(50) . m∗2 ζ 3 ∂l2

(51) 0,0 (2.26) Since ωx and ωy have no explicit form yet, we consider a lens-shaped QD, parameterized by 1 1 ~2 π 2 n2z En = Eg + + nx ~ωx + + ny ~ωy + 2 2 2m∗ ζ 2. 4x2 4y 2 Lz (x, y) = ζ 1 − 2 − 2 wx wy. ωl2 l=x,y. !. .. (2.27). Assuming an electron in the ground state inside a symmetric QD, i.e., nx = ny = 0, nz = 1 and wx = wy = w, the ground state energy is given by √ 2 2~2 π ~2 π 2 n2z + E0 = Eg + . (2.28) 2m∗ ζ 2 m∗ ζw From equation 2.28, it is evident that the dominating part of the electronic ground state energy is Eg . However, there is still an energy contribution raising the state’s energy, mainly due to the potential well which scales with ζ −2 and partly the harmonic oscillator. Considering a GaAs QD with ζ = 7 nm and w = 50 nm as an example, the energy spacing for the harmonic oscillator is 31 meV, while the spacing due to the quantum well is quadratic and.

(52) 16. Chapter 2. Theoretical Background. yields 365 meV between nz = 1 and nz = 2. Since the QD is an "artificial atom" its energy levels are denoted with the shell numbers s,p,d,f... like an atom. Excitonic Transition Energy To get a rough idea of what to expect from the emission of a QD, we want to look at the optical active electron-hole complex X. For that, the hole is included and the problem is treated with the two particle EME, as in !. ~2 ∇2rh −~2 ∇2re e2 − − χ(re , rh ) = (E − E0,e − E0,h ) χ(re , rh ), (2.29) 2m∗e 2m∗h 4πϵ0 ϵr |re − rh | where the differential operators act on the electron and hole terms, respectively. The electronhole pair is bound via Coulomb interaction, shielded by the semiconductor relative permittivity ϵr . The energy offset is the sum of the energy of the electronic state E0,e (including Eg , as in 2.28) and E0,h , the energy of the hole. The latter can be obtained by inserting m∗h into 2.28 as well but neglecting Eg this time. Introducing a center-of-mass frame of reference as in classical physics, the two-particle problem decouples and can be solved independently. M ∗ = m∗e + m∗h. µ∗ =. m∗e m∗h m∗e + m∗h. R=. re m∗e + rh m∗h M∗. r = re − rh. (2.30). The differential operators transform accordingly. The center of mass problem is equivalent to the problem of a free particle with effective mass given by M ∗ and the relative motions corresponds to the Hydrogen atom problem but with different constants. The total energy of equation 2.29 reads Enexc. ~2 K 2 R∗ − 2 = E0,e + E0,h + 2M ∗ n. with K = ke + kh. µ∗ and R = m0 ∗. . 1 ϵr. 2. R, (2.31). with the Rydberg energy R ≈ 13.6 eV. Excitons in GaAs QDs typically recombine when the electron is in the Γ -valley and the hole is on the top of the heavy hole band [11]. Therefore, we can assume K = 0 and with the corresponding numbers for GaAs, taken from Ioffe NSM [39] and assuming ζ = 7 nm, w = 50 nm, the ground state energy of a bound exciton reads √ ~2 π 2 (4 2ζ + w) exc E = Eg + − R∗ ≈ 1.59 eV → λexc = 780 nm. (2.32) 2µ∗ ζ 2 w The result of these last pages is astoundingly close to actually observed values of the exciton emission. However, the exciton confinement is not as strong as it would be in an infinitely high potential barrier as assumed in equation 2.22, making the envelope of the wave function larger. Also the temperature dependence of Eg was neglected, although spectra of QDs are actually measured at low temperature. These deviations are therefore similar but opposite in.

(53) 2.2. Quantum Dot Nanostructures. 17. their contribution. Finally, it is known that the binding energy of excitons in nanostructures is increased compared to the bulk value because of the improved electron-hole overlap.. 2.2.3. Few-Particle States. Starting with the total angular momentum operator J = L ± S, an electron in the CB occupies a band constructed from s-like (L = 0) atomic orbitals, with a spin of Se = 1/2, so it follows Je = 1/2, jz,e = ±1/2. As seen in subsection 2.1.4, the VBs separate by the spin-orbit coupling, and yield the split-off band (Jso = 1/2, jz,so = ±1/2), separated from the VB edge by ∆ = 0.341 eV, i.e., the HH band (Jhh = 3/2, jz,hh = ±3/2) and the LH band (Jlh = 3/2, jz,lh = ±1/2). In a bulk semiconductor heavy and light hole bands are degenerate for a wave-vector k = 0 but in QDs they are separated because of different hole effective masses in confined states, see equation 2.28. Not only is the HH closer to the band gap, it has also a 3 times higher transition probability, according to optical selection rules in the dipole approximation. Thus, the LH is neglected further on. A right (left) circular polarized photon, denoted as σ + or R (σ − or L), carries an angular momentum of +1 (−1) and no spin S = 0. Therefore a photon can couple, if ∆jz = ±1 and ∆s = 0. The bright (dark) X (D) states, i.e., excitons that can (not) be addressed by or emit photons, when represented in the basis of the total angular momentum |jz ⟩ and where jz is the eigenvalue of J = Je + Jhh , are:

(54)

(55)

(56)

(57)

(58) 1

(59) 3

(60) 1

(61) 3

(62)

(63)

(64)

(65) |Xσ+ ⟩ =

(66) − ⊗ + = |+1⟩ and |Xσ− ⟩ =

(67) + ⊗ − = |−1⟩ 2 e

(68) 2 hh 2 e

(69) 2 hh

(70)

(71)

(72)

(73)

(74) 1

(75) 3

(76) 1

(77) 3 |Dσ2+ ⟩ =

(78)

(79) + ⊗

(80)

(81) + = |+2⟩ and |Dσ2− ⟩ =

(82)

(83) − ⊗

(84)

(85) − = |−2⟩ 2 e 2 hh 2 e 2 hh. (2.33) (2.34). Because of Pauli’s exclusion principle, two fermionic particles can only occupy the same state if their spin is opposite. Therefore, other combinations, like the biexciton (XX) or the charged trion state (X+ or X− ) can form in the QD s-shell, as one can see in Figure 2.6 The charge carriers are bound via Coulomb interaction and therefore, all transitions show different energies. In order to minimize this interaction however, charge carriers can move spatially [42], re-arranging the energetic order of radiative recombinations in the observed spectrum. The emission spectra are captured through photoluminescence (PL) spectroscopy. In the simplest non-resonant configuration, the sample is illuminated with light with an energy higher than the energy band gap of the barrier material, to generate excitons in the matrix, relaxing to the QD s-shell via phonon decay. Other methods of QD s-shell population are the resonant excitation of neutral or charged excitons [43], the resonant two-photon excitation (TPE) on the XX level or phonon-assisted excitation [44].. 2.2.4. Entangled Photon Pairs via the Biexciton-Exciton Cascade. To understand how entanglement arises, suppose that the system is prepared in the XX state. When one electron recombines with a hole, a photon is emitted which has a right or left-hand.

(86) 18. Chapter 2. Theoretical Background. . . . . . Figure 2.6 – (a) Electron hole pair configurations with different strengths of the Coulomb interaction (light gray), (b) resulting in different emission lines of the radiative decay with energies ~ω. Correlation effects can lead to reordering of the expected spectral distribution, while the exchange interaction introduces energy splittings for recombinations that include the X state. (c) Four QD PL-spectra showing a different ordering of the few-particle states on the very same sample. Image (b) taken from Ref. [42], (c) from Ref. [45].. circular polarization, leaving the system in the X state. Then it can relax to the ground state by emitting another photon with the other polarization. Since there is no preference, if the spin-up electron (|L⟩XX decay) or the spin-down electron (|R⟩XX decay) recombines first, we find a polarization-entangled XX-X photon pair, as a consequence of the underlying "which-path" principle of the biexciton cascade. Figure 2.7 (a) illustrates this phenomenon. The states can not be written separately in a product state and can be represented using the Bell states (2.35)

(87) + as a basis in the two-dimensional Hilbert space. GaAs QDs used for this thesis emit

(88) ϕ states.

(89) E 1

(90) + = √ (|H⟩XX |H⟩X + |V⟩XX |V⟩X )

(91) ϕ 2

(92) E 1

(93) + = √ (|H⟩XX |V⟩X + |V⟩XX |H⟩X )

(94) ψ. 2.

(95) −

(96) ϕ = √1 (|H⟩ |H⟩ − |V⟩ |V⟩ ) XX X XX X. 2.

(97) − 1

(98) ψ = √ (|H⟩XX |V⟩X − |V⟩XX |H⟩X ). (2.35). 2. Fine Structure Splitting As soon as the exchange interaction of (quasi-)particles with non-zero spin is not neglected anymore, the fine structure splitting emerges [46]. The exchange Hamiltonian Hexc with the exchange interaction coupling constants ai , bi reads . Hexc. . +δ +δ1 0 0   0   X +δ +δ 0 0 1 1 0   3 =− ai Jh,i Se,i + bi Jh,i Se,i =  ,  2 0 0 −δ0 +δ2  i=x,y,z  0 0 +δ2 −δ0. (2.36).

(99) 2.2. Quantum Dot Nanostructures. . . φ+. . . . . . . . . . . . . . . . . . . . . . . . . . φ+. . . . 19. . .

(100) .

(101)

(102) . Figure 2.7 – a) The biexciton-exciton cascade yields polarization-entangled photon pairs due to the indistinguishability of the two possible decay paths. (b) Anisotropic exchange interaction lifts the degeneracy of the X level and introduces the X+ and X− states, strongly suppressing the entanglement. (c) Recovery of the entanglement in ultra-symmetric QDs or external perturbations of the FSS.. when written as a matrix in the basis of bright and dark exciton states, as introduced in equations 2.33 and 2.34, and only including short-range interactions, with abbreviations δ0 = 1.5(az + 2.25bz ), δ1 = 0.75(bx − by ) and δ2 = 0.75(bx + by ). Following from 2.36, the bright and dark X states separate by δ0 . Because of the block matrices one can see that bright and dark exciton states do not mix. When Hexc is not neglected, the dark X states are always separated by δ2 , however, there is no further treatment of dark excitons in this thesis. Bright X states are degenerate for bx = by , i.e., structural symmetry of the dot satisfying the D2d group, and are eigenstates of the matrix operator for that case. However, if there is lower symmetry (bx ̸= by ), then the following bright eigenstates emerge 1 |X+ ⟩ = √ (|+1⟩ + |−1⟩) 2 1 |X− ⟩ = √ (|+1⟩ − |−1⟩) 2. 1 with ∆E = (δ0 + δ1 ), 2 1 with ∆E = (δ0 − δ1 ), 2. (2.37) (2.38). where ∆E is the energy difference to the former exciton |±1⟩ state and δ1 = EFSS is the fine structure splitting (FSS) energy gap between the excitonic states. Due to Kramers theorem, which states that in a system with half-integer value of

(103) the sum of all spins, all states have to be at least two-fold degenerate, the trion states

(104) X + and |X − ⟩ (± in superscript) underlie no FSS. Also the biexciton shows no fine structure since the total sum of spins is 0. Since it decays into an exciton state being affected by the exchange interaction, the XX recombination is characterized by two lines split by the FSS..

(105) 20. Chapter 2. Theoretical Background. Such biexcitons and excitons show emission of linearly polarized photons with distinguishable energies, decreasing the entanglement significantly. Figure 2.7 (b) provides a sketch of the energy levels underlying such phenomenon. The degeneracy of the bright exciton state can be recovered if the FSS is smaller than the radiative linewidth, such that the two decay paths are indistinguishable as illustrated

(106) again,. in Figure 2.7 (c). If so, a polarization-entangled photon pair in the

(107) ϕ+ state [47] can be expected. Tuning of the FSS of QDs with low structural symmetry can be achieved through external perturbations [12]. For this purpose a sample can be placed on a piezo-electric actuator and, through strong electric fields, QDs initially lacking symmetry can show a very small FSS. Similarly, a six legged piezo-electric actuator can induce strain on the sample in three directions and is capable of simultaneously tuning the FSS and the emission wavelength of a QD [13]. Fortunately, droplet etched GaAs QDs are initially highly symmetric and can show an ultra-small excitonic fine structure splitting, even without external perturbations [16]. Another approach is to implement Fourier broadening of the spectral linewidth due to a shortening of the radiative lifetime. This can be achieved by the Purcell effect in a photonic cavity, boosting the emission rate and the degree of entanglement. The circular Bragg resonator is such a cavity [28]. Effects of Polarized Light Emission on the Entanglement Fidelity Quantum dots embedded into photonic cavities can experience a coupling to polarized cavity modes [48]. Here, we want to take a closer look on how the entanglement fidelity, i.e., the similarity to a Bell state, changes when a certain polarization is preferred

(108) in the emission process. Considering an ideal source of entangled photon pairs, emitting

(109) ϕ+ states in the H,V polarization basis, the density operator ρˆ takes the form . 1 

(110) ED

(111)  1 0

(112)

(113) ρˆ =

(114) ϕ+ ϕ+

(115) =  2 0 1. . 0 0 0 0. 0 0 0 0. 1  0   0  1. with.

(116) E 1

(117) + = √ (|HH⟩ + |VV⟩) .

(118) ϕ. 2. (2.39). Assuming a polarization dependent emission intensity I according

(119) +′ to IV = (1 − p)IH with a horizontal polarization parameter p, we form the pure state

(120) ϕ = CH |HH⟩ + CV |VV⟩ with the density operator   |CH |2 0 0 CH CV∗    0  0 0 0   (2.40) ρˆ′ =  .  0 0 0 0    CH∗ CV 0 0 |CV |2.

(121) 2.3. Photonic Cavities. 21. Following the properties of a density operator, we find Tr(ˆ ρ′ ) = |CH |2 + |CV |2 = 1 and 2 relate the equation to the polarization parameter p by |CV | = (1 − p)|CH |2 to express the 1 polarization state probabilities as |CH |2 = 2−p and |CV |2 = 1−p ˆ′ as a function 2−p . Expressing ρ

(122). of p, we want to evaluate the ϕ+ fidelity f ϕ+ of the

(123) ϕ+′ state by taking the trace of the matrix product of both density operators f ϕ+ = Tr (ˆ ρ′ ρˆ) and find √ (1 + 1 − p)2 ϕ+ , (2.41) f = 2(2 − p) which yields f ϕ+ = 1 for p = 0, i.e., a maximally entangled state, and f ϕ+ = 0.5 for p = 1, reaching the classical limit, where all entanglement correlation vanishes. Considering p = 0.2 as example, the fidelity yields f ϕ+ = 0.9969 and f ϕ+ = 0.9714 for p = 0.5.. 2.3. Photonic Cavities. A photonic microcavity is an optical resonator with dimensions that are similar to the wavelength of the light they couple to. A cavity introduces cavity modes (CMs) which are resonances of distinct linewidth and spacing. Confinement of light modes in microcavities can come from a high reflectivity on the cavity borders, which is achievable with mirrors, like the Fabry-Pérot resonator, or a large contrast of the refractive index of the cavity material compared to the ambient, like for the CBR. Also a periodic lattice, modulating the refractive index, like in photonic crystal cavities, can restrict the propagation of light and, therefore, form a cavity. [49] If a cavity photon is absorbed faster than decaying due to strong coupling, it forms new modes called polaritons that are half-light and half-matter. Weak coupling cavities, however, feature a preferential emission of photons, making light collection out of solid-state sources more efficient and consequently, is of higher interest within this thesis.. 2.3.1. Quality Factor. The quality factor is a useful quantity, describing the amount of optical energy dissipation from the cavity, and is dependent on the linewidth at the full width at half maximum (FWHM) δωc of a CM at ωc : ωc , (2.42) Q= δωc. 2.3.2. Light-Matter Interaction. In order to understand optical resonators better, it is necessary to briefly discuss basic principles and equations of light-matter interaction. In a semi-classical approach, light couples to quantized matter within three processes, spontaneous emission with the probability density p(t) of decay p(t) = exp(−t/τ ) and the emitter’s characteristic mean lifetime τ , stimulated emission, where the presence of an external photon promotes the emission of a clone and absorption which is the inverse, yet symmetric process of stimulated emission..

(124) 22. Chapter 2. Theoretical Background. Fermi’s Golden Rule In a full quantum-mechanical interaction picture, one can make use of Fermi’s golden rule which is the result of a weak time-dependent perturbation H ′ on a system with a known spectrum of the unperturbed Hamiltonian and reads

(125) 2 2π

(126) 1 = 2

(127) ⟨ f | H ′ | i ⟩

(128) ρ f , τ ~. (2.43). with the transition matrix element of initial to final state and the density of final states ρ f . In the case of an excitation in a single atom in vacuum, the atom’s dipole d couples to the (empty) photon states of the vacuum field, E kn with wavevectors kn , and emits with a probability related to the transition dipole matrix element, i.e., the off-diagonal elements of the perturbation matrix Mkn ∝ |d · E kn |. Using the electric dipole approximation, i.e., the field wavelength is much larger than the size of the atom, one can find the decay rate τ0−1 of an excited state of an atom in vacuum with the transition frequency ω0 and the dipole magnitude d, representing the coupling strength: 1 ω03 = d2 , τ0 3π~c3 ϵ0. 2.3.3. (2.44). Purcell Effect. When studying the physics of weak coupling cavities, the Purcell effect is of great interest, since it is a figure of merit of a photonic cavity, stating that a change of the photon density of states that couple to the emitter’s dipole can enhance or inhibit its emission rate. A weak coupling justifies the use of a perturbation theory and, therefore, of Fermi’s golden rule. Changing equation 2.44 from vacuum to a medium with refractive index n and effective volume Veff confining the electric field, including the Lorentzian mode density within the cavity and taking the ratio of τ0 and the lifetime τc inside the cavity, one finds τ0 δλ2c |E(r)|2 = FP 2 τc δλc + 4(λc − λe )2 |Emax |2. . d·E dE. 2. ,. with FP =. 3Λ3 Q , 4π 2 Veff. (2.45). where Λ = λnc . This effect was reported by Purcell in 1946 [50] and its factor FP , called Purcell factor, only features cavity properties, making it a handy number for designing optical resonators. Equation 2.45 shows also limiting factors of a potential lifetime reduction, namely the spectral matching of emitter λe and cavity mode λc within the Lorentzian, as well as the spatial matching of the emitter to regions with high electric field density. The last term shows that the Purcell enhancement also appears polarization-selective, as seen in Wang et al. [48]. Furthermore, the lifetime shortening effect can also be limited by non-radiative recombination channels and leaking modes..

(129) 2.3. Photonic Cavities. 2.3.4. 23. Circular Bragg Resonator and Highly-Efficient Broadband Reflector. The circular Bragg resonator (CBR) is a broadband microcavity, showing an optimized extraction efficiency of embedded light emitters, while simultaneously enhancing the radiative rate through the Purcell effect [29][51]. The nanostructure operates by including three mechanisms. A circular second-order dielectric Bragg grating with a period Λ = λ/neff enables, contrary to a first-order grating, not only lateral reflection of emitted light back to the center but also surface-normal radiation [52], greatly boosting the signal yield [53]. The center of the rings is basically a grating defect that acts as a cavity confining the electromagnetic field in a disc with diameter 2Λ and effective volume Veff to exploit Purcell enhancement. Underneath the structure, a highly-efficient broadband reflector (HBR), consisting of an oxide layer and an Ag or Au film acting as a mirror, makes light emitted towards the substrate also available for collection by an objective or an optical fiber [30]. In Figure 2.8 (a) and (b), one can see a schematic and a micrograph of the microcavity nanostructure, while (c) provides characteristic cavity properties. To design the cavity [28], a simulation of the 1D radial section of the grating on an oxide layer is carried out, altering the grating period such that perpendicular reflection of a lateral applied electric field yields a broadband surface-normal radiation, as in Figure 2.9 (a). Then, the radius of the central cylinder is tuned by a 3D finite difference time domain (FDTD) simulation to obtain a Purcell factor matching the typical emitter wavelength, cf. Figure 2.9 (b). Finally an Au reflector is added underneath and, through optimization of the oxide spacer, leaking modes are reflected back to the cavity, being made accessible for light collection. Figure 2.9 (c) shows the cross-sectional electric field distribution and a highly directed beam. The inset shows the Gaussian far-field profile of the cavity emission. To gain flexibility with the material system, Dr. Santanu Manna carried out further simulations to change the material of the HBR dielectric from SiO2 to Al2 O3 , using FDTD within the software Lumerical. The results are plotted in Figure 2.10, showing that a change of the oxide material is possible with similar results. The parameters used in the experiment for the structures for both material systems are given in Table 2.1. Simulations also show that. Λ. . Λ. 1 µm. 1.0 0.9.

(130) . 25. Collection efficiency Purcell factor. 20 15. 0.8. 10. 0.7. Y. 5. 0.6 740. Purcell factor. Collection efficiency. . 760. 780. 800. Wavelength (nm). 0 820. Figure 2.8 – (a) Cross-sectional and (b) SEM top view (taken from Ref. [31]) of the CBRHBR cavity. (c) Simulation results (taken from Ref. [28]) of the collection efficiency using a numerical aperture (NA) objective with NA = 0.65, where NA = sin θ (half angle) in air, and the Purcell factor of a CBR-HBR cavity designed for 785 nm..